TIPS for Grades 7, 8, and 9 Applied Math · TIPS for Grades 7, 8, and 9 Applied Math is designed to...

TIPS for Grades 7, 8, and 9 Applied Math is designed to be useful to teachers in both Public and Catholic schools, and is intended to support beginning teachers, provide new insights for experienced teachers, and help principals and professional development providers as they work to improve mathematics education. Permission is given to reproduce these materials for educational purposes. Teachers are also encouraged to adapt this material to meet their students’ needs. Any references in this document to particular commercial resources, learning materials, equipment, or technology reflect only the opinions of the writers of this resource, and do not reflect any official endorsement by the Ministry of Education or by the Partnership of School Boards that supported the production of the document. © Queen’s Printer for Ontario, 2003 Acknowledgements

Irene McEvoy (Project Coordinator) Peel DSB (lead board) Georgia Chatzis Mary Kielo Lindy Smith

Jan Crofoot Eric Ma Tania Sterling

Cathy Dunne Reet Sehr

Greater Essex County DSB Paul Cornies Honi Huyck Debbie Price London DCSB Mary Howe (Steering Committee Member) Ottawa Carleton CDSB Marie Lopez Peter Maher Ottawa Carleton DSB Lynn Pacarynuk

Judy Dussiaume (Steering Committee Member) Rainbow DSB Heather Boychuk Linda Goodale

Simcoe DSB Patricia Steele Simcoe Muskoka CDSB Greg Clarke Toronto CDSB Anthony Azzopardi Anna D’Armento Dennis Caron Toronto DSB Trevor Brown

Kevin Maguire Sandy DiLena Silvana Simone

Maria Kowal

Shelley Yearley (Steering Committee Member) Trillium Lakelands DSB Barry Hicks Pat Lightfoot

Upper Grand DSB Anne Yeager Rod Yeager York Region DSB Shirley Dalrymple Susan McCombes Ministry of Education Myrna Ingalls (Steering Committee Member) Faculty of Education, University of Ottawa

Christine Suurtamm

Faculty of Education, University of Western Ontario

Dan Jarvis Barry Onslow

Faculty of Education, Queens University

Lynda Colgan

EQAO Elizabeth Pattison Kaye Appleby (Project Manager) Retired Educators Carol Danbrook Ron Sauer

Contents Section 1: Developing Mathematical Literacy Introduction Theoretical Framework for Program Planning Research Section 2: Continuum and Connections

Mathematical Processes Patterning to Algebraic Modelling Solving Equations and Using Variables as Placeholders Developing Perimeter and Area Formulas

Section 3: Grade-Level Planning Supports

Grade 7 Grade 8 Grade 9

Section 4: TIPS for Teachers Section 5: Posters

Classroom Teachers

TIPS: Section 1 – Developing Mathematical Literacy © Queen’s Printer for Ontario, 2003 Page 1

“In this changing world, those who understand and can do mathematics have significant opportunities and options for shaping their future.”

NCTM 2000 p. 5

A sound understanding of mathematics is one that sees the connections within mathematics and between mathematics and the world.

Developing Mathematical Literacy

A. Introduction Targeted Implementation and Planning Supports (TIPS) offers ways of thinking about mathematics education, resources, and teacher education for those working with students in Grades 7 to 9. “In this changing world, those who understand and can do mathematics have significant opportunities and options for shaping their future.” NCTM 2000 p. 5. Students in this age group are at a critical, transitional stage where their perceptions of mathematics will help to shape their success in secondary mathematics and their career decisions.

Mathematics competence and confidence open doors to productive futures. Implementation of this resource will help to enhance the mathematics knowledge and understanding of students in Grades 7 to 9, and help them develop the critical skills identified in the Conference Board of Canada’s Employability Skills Profile: communicate, think, continue to learn throughout their lives, demonstrate positive attitudes and behaviours, responsibility, and adaptivity, and work with others. One way to think of a person’s understanding of mathematics is that it exists along a continuum. At one end is a rich set of connections. At the other end of the continuum, ideas are isolated or viewed as disconnected bits of information. (Skemp) A sound understanding of mathematics is one that sees the connections within mathematics and between mathematics and the world. However, to lead students to rich connections, “teachers must be agents of change that they did not experience as students” (Anderson, D. S. & Piazza, J. A., 1996.)

The intent is to help both students and teachers develop the “big picture” of mathematics that includes: competence in mathematical skills, substantive understanding of mathematical concepts, and the application of these skills and understandings in problem-solving situations. This resource package helps teachers to see how, in their program and daily lesson planning, they can address the critical connections between:

− instruction and assessment; − one mathematics topic and the next; − one strand and the other strands; − the mathematics done in class and students’ sense-making processes; − learning mathematics and doing mathematics; − the instructional strategy selected for a specific learning goal and research into how students learn

mathematics; − students’ prior learning and new knowledge and understanding; − existing resources and the envisioned program; − the mathematics classroom and home; − mathematics topics and topics in other disciplines.

This resource is intended to: − support beginning teachers; − provide new insights for experienced teachers; − help principals and professional development providers as they work to improve mathematics

education for young people.


This package is a ‘work in progress.’ It will be revised, refined and added to, as teams of teachers come together to engage in collaborative planning, then share their work. It is available electronically at www.curriculum.org

Key Messages There are important messages in TIPS for the many stakeholders involved in the mathematical education of the young adolescent:

TIPS for Teachers • Focus on important mathematical concepts or “big ideas” that cluster expectations. • Include a variety of instructional strategies and assessment strategies. • Value the abilities and needs of the adolescent learner. • Provide a positive environment for learning mathematics through problem solving. • Allow opportunities for students to explore, investigate, and communicate mathematically as well as

opportunities to practise skills. • Encourage a variety of solutions that incorporate different representations, models, and tools. • Incorporate relevant Ontario Catholic School Graduate Expectations.

TIPS for Principals • An effective mathematics program should have a variety of instructional and assessment strategies. • Effective mathematics classrooms are active places where you can hear, see, and touch mathematics. • Principals are encouraged to provide teachers with necessary support for an effective mathematics

program. This can be done through: − student and teacher resources; − providing time for teachers to collaborate and share ideas; − providing in-service opportunities and information about mathematics education for teachers; − celebrating mathematics learning for all students, not just the achievements of a select few; − showing a positive attitude towards mathematics; − demonstrating the importance of life-long learning in mathematics.

TIPS for Coordinators • Mathematics teachers need support. • All teachers require information and in-service about mathematics education. • Teachers need to know how and why these activities promote student learning of mathematics.

TIPS for Researchers Use of this package may bring to light a number of research questions focused on the topics that were the subjects of literature reviews. Synopses of research on a wide variety of themes are included in the Research section (p. 17). Any of these could be expanded into a more complete literature review with accompanying research questions and follow-up research. It is hoped that Ontario-based research that addresses any of the elements featured in Targeted Implementation and Planning Supports (TIPS) will be submitted to the project for inclusion and reference. Please contact the Ontario Curriculum Centre at [email protected], if you have research that you would like to contribute.


Program Planning Supports This package provides grade-specific and cross-grade program planning supports. These are structured to provide opportunities for in-depth, sustained interaction with key mathematical concepts, or “big ideas” that cluster and focus curriculum expectations. The intent is to help students see mathematics as an integrated whole, leading them to make connections and develop conceptual and relational understanding of important mathematics.

Content and Reporting Targets by Grade The package for each grade begins with a one-page overview of the mathematics program that includes a suggested sequencing of curriculum expectation clusters, rationale for the suggested sequence, and in Grades 7 and 8, suggested strands for reporting in each term. Detailed alignment of all curriculum expectations with the identified clusters is included in the Appendix for each grade level.

Continuum and Connections across Grades Continuum and Connections, Section 2 is based on significant themes in mathematics and includes Mathematical Processes and five content-based packages – Patterning to Algebraic Modelling, Solving Equations and Using Variables as Placeholders, Developing Perimeter and Area Formulas, Integers, and Fractions.

In Mathematical Processes – Section 2 • The first page situates the mathematical processes in a larger context and asks key questions about

development of the processes – through the grades, for different types of learners, and using different strategies.

• To answer these key questions, there is a description of student and teacher roles and assessment suggestions for each of the four mathematical processes. These processes and the research that lead to their identification are outlined in Theoretical Framework for Program Planning, p.10 and Research, p. 17.

For each content-based package – Section 2 • The first page situates the mathematical theme in a larger context, shows connections to other

subjects, careers, and authentic tasks, and identifies manipulatives, technology, and web-based resources useful in addressing the theme.

• Connections Across Grades outlines scope and sequence, using before Grade 7, Grade 7, Grade 8, Grade 9, and Grade 10 as organizers.

• Instruction Connections suggests instructional strategies, with examples, for each of Grade 7, Grade 8, and Grade 9 Applied, and includes advice on how to help students develop understanding.

• Connections Across Strands provides a sampling of connections that can be made across the strands, using the theme as an organizer.

• Developing Proficiency presents sample tests and Developing Mathematical Processes, a set of four short-answer questions – one per mathematical process – based on the theme for each of Grade 7, Grade 8, and Grade 9 Applied are included.

• Questions posed in Extend Your Thinking and Is This Always True? help students develop depth of understanding. Answers show a variety of representations and strategies that students may use and that teachers should validate.

Summative Tasks Summative tasks are provided for each of Grade 7, Grade 8, and Grade 9 Applied mathematics. These tasks focus on some of the major concepts of the mathematics program and require facility in several strands that the students have studied throughout the program.


Teachers can use these summative tasks to design other tasks in the program and to recognize some of the important ideas in the program. Teachers can use the summative tasks in planning a program by determining what important mathematical ideas students should understand by the end of the program and then designing the instructional approaches that support those mathematical ideas.

Lesson Planning Supports • The pace and structure of lessons will vary depending on the goals of the lesson. Using a single

sequence of elements, such as taking up homework, teacher demonstration, student practice can be uninspiring or monotonous.

• Mathematics classes should: − provide opportunities for students to become proficient in basic skills; − focus on introducing new skills and concepts through problem solving; − incorporate a balance and variety of teaching strategies, assessment strategies, student groupings,

and types of activities. • Each lesson should appeal to auditory, kinesthetic, and visual learners. Appealing to all these

preferences deepens the understanding of all students. • Lessons should introduce algorithms only after students have constructed meaning for the procedure,

and introduce formulas only after students develop a conceptual understanding of the relationships among measurements.

• Lessons are developed in a template to outline a desired vision for classes.

Lesson Planning Template The following organizers in the template provide for concise communication while allowing the teacher to vary the pace and types of groupings and activities for classes:

Minds on… suggests how to get students mentally engaged in the first minutes of the class and establishes a positive classroom climate, making every minute of the math class count for every student.

Action! suggests how to group students and what instructional strategy to use. This section maps out what students will do and how the teacher can facilitate and pose thought-provoking questions. Suggestions for time management, scaffolding, and extension are included, as appropriate.

Suggested groupings and strategies include:

Groupings:

Carousels, Expert Groups, Groups of 4, Individual, Jigsaw, Pairs, Pair/Share, Small Groups, Think/Pair/Share, Whole Class.

Strategies:

Acting, Brainstorm, Concept Map, Conferencing, Connecting, Demonstration, Discussion, Experiment, Field Trip, Game, Guest Speaker, Guided Exploration, Independent Study, Interview, Investigation, Kinesthetic Activity, Model Making, Note Making, Portfolio, Practice, Presentation, Problem Solving, Reflection, Research, Response Journal, Retelling, Role Playing, Simulation, Survey, Tactile Activity, Visual Activity, Worksheet.


The time clock/circle graphic (Grades 7 and 8) and the time bar graph (Grade 9) suggest the proportion of class time to spend on various parts of the mathematics class.

Consolidate/Debrief suggests ways to ‘pull out the math,’ check for conceptual understanding, and prepare students for the follow-up activity or tomorrow’s lesson. Often this involves whole class discussion and sharing. Students listen to and contribute to reflections on alternate approaches, different solutions, extensions, and connections. Students should be well prepared to do mathematics individually after the three-part lesson.

Home Activity or Further Classroom Consolidation suggests meaningful and appropriate follow-up. These activities provide opportunities: − to consolidate understanding; − to build confidence in doing mathematics independently; − for parents to see the types of math activities students engage in during class and to see

connections between the mathematics being taught and life beyond the classroom; − for giving students some choice through differentiated activities.

• The MATCH (Minds on, Action, Timing, Consolidate, Home Activity or Further Classroom Consolidation) acronym reminds teachers that all of these elements must be considered in lesson planning. It suggests that connection/matches should be made in each lesson to: − program goals; − research and effective instruction; − characteristics of adolescent learners.

H

Interpreting the Lesson Outline Template Download the Lesson Outline Template at www.curriculum.org/occ/tips/downloads.shtml


A brief descriptive lesson title

Lesson Outline: Days 5 – 9 Grade 7

BIG PICTURE Students will: • explore and generalize patterns; • develop an understanding of variables; • investigate and compare different representations of patterns.

Day Lesson Title Description Expectations5 Toothpick Patterns • Review patterning concepts

• Build a growing pattern • Explore multiple representations

7m70, 7m72 CGE 3c, 4f

6 Patterns with Tiles • Build a pattern • Introduce the nth term

7m66, 7m71 CGE 4b

7 Pattern Practice • Continued development of patterning skills 7m67, 7m71, 7m75 CGE 2c, 5e

8 Pattern Exchange • Class sharing of work from previous day. 7m69, 7m75 CGE 2c, 5e

9 Performance Task • Performance Task - individual 7m66, 7m67, 7m73, 7m75 CGE 5g

NOTES a) While planning lessons, teachers must judge whether or not pre-made activities support development of

big ideas and provide opportunities for students to understand and communicate connections to the “Big Picture.”

b) Ontario Catholic School Graduation Expectations (CGEs) are included for use by teachers in Catholic Schools.

c) Consider auditory, kinesthetic, and visual learners in the planning process and create lessons that allow students to learn in different ways.

d) The number of lessons in a group will vary.

e) Schools vary in the amount of time allocated to the mathematics program. The time clock/circle on completed Grade 7 and 8 lessons suggests the fractions of the class to spend on the Minds On, Action!, and Consolidate/Debrief portions of the class. Grade 9 Applied lessons are based on 75-minute classes.

f) Although some assessment is suggested during each lesson, the assessment is often meant to inform adjustments the teacher will make to later parts of the lesson or to future lessons. A variety of more formal assessments of student achievement could be added.

Sequence of Lessons Addressing a Theme

Grade Level

Lessons are planned to help students develop and demonstrate the skills and knowledge detailed in the curriculum expectations.

• To help students value and remember the mathematics they learn, each lesson is connected to and focussed on important mathematics as described in the Big Picture.

• Since students need to be active to develop understanding of these larger ideas, each point begins with a verb. • Sample starter verbs: represent, relate, investigate, generate, explore, develop, design, create, connect, apply

List curriculum expectations (and CGEs)

• Two or three points to describe the content of this lesson. • Points begin with a verb. • Individual lesson plans elaborate on how objectives are met.

http://www.curriculum.org/occ/tips/downloads.shtml

Interpreting the Lesson Planning Template Download the Lesson Planning Template at wwwwww..ccuurrrriiccuulluumm..oorrgg//oocccc//ttiippss//ddoowwnnllooaaddss..sshhttmmll


Day 1: Encouraging Others Grade 8

Description • Practise the social skill of encouraging others. • Identify strategies involving estimation problems. • Set the stage for using estimation as a problem-solving strategy.

Materials • BLM 1.1 • birdseed

Assessment Opportunities

Minds On ...

Whole Group Brainstorm Explain why it is important to encourage others. Explicitly teach the social skill, “Encouraging Others,” through a group brainstorm. Create an anchor chart using the criteria: What does it look like? What does it sound like?

Action! Think/Pair/Share Gather Data Use an overhead of the Think/Pair/Share process (TIP 2.1) and student copies of BLM 1.1. Students gather data. Learning Skill/Observation/Mental Note: Circulate, observing social skills and listening to students. Share with students some of the positive words and actions observed during the activity and invite students to make additions to the anchor chart on Encouraging Others. Whole Class Sharing Based on ‘teachable topics’ during the Think/Pair/Share Activity, e.g., a particularly effective phrase/statement expressed by a student, clarification of the cooperative learning strategy, an interesting result on BLM 1.1, ask representatives of groups to share their results or report on their process.

Consolidate Debrief

Whole Class Discussion Use the posters Inquiry Model Flow Chart, Problem-Solving Strategies, and Understanding the Problem. Discuss how these posters will be of assistance over the next few days as well as during the whole math program. Point out that when students encourage others, it makes it safe for them to try new things and contribute to group activities.

“Learning is socially constructed; we seldom learn isolated from others.” - Bennett & Rolheiser Consider using stickers as a recognition for examples of the social skill being applied by a group. Solving Fermi problems is a way to collect diagnostic assessment data about social skills, academic understandings, and attitudes towards mathematics (see TIP 1.2).

Social Skill Practice Reflection

Home Activity or Further Classroom Consolidation Interview one or more adults about estimation using the following guiding questions and record your responses in a math journal. Summarize what you notice about the responses. You may be asked to share this math journal entry with the class.

Day #: Lesson Title

Grade lleevveell

Tips for the TeacherThese include: - instructional hints - explanations - background - references to

resources - sample responses to

questions/tasks

Materials used in the lesson

Same two or three objectives listed in the lesson outline

Time colour-coded to the three parts of the day’s lesson

Meaningful and appropriate follow-up to the lesson.Focus for the follow-up activity.

• Mentally engages students at start of class • Makes connections between different math strands, previous lessons or

groups of lessons, students’ interests, jobs, etc. • Introduces a problem or a motivating activity • Orients students to an activity or materials.

• Students do mathematics: reflecting, discussing, observing, investigating, exploring, creating, listening, reasoning, making connections, demonstrating understanding, discovering, hypothesizing

• Teachers listen, observe, respond and prompt

“Pulls out´ the math of the activities and investigations

Prepares students for Home/Further Classroom Consolidation

Suggested student grouping teaching/learning strategy for the activity.

Indicates an assessment opportunity - what is assessed/strategy/scoring tool

Indicates suggestedassessment

http://www.curriculum.org/occ/tips/downloads.shtml


It is important to set high standards for all students and to provide the supports and extensions that help all students learn.

Research Base for Planning Templates Each of the organizers in Continuum and Connections (Section 2) and Lesson-Planning Templates, p.6-7 is supported by a one-page research synopsis. It is through a thoughtful combination of decisions that teachers proactively establish Classroom Management (p. 27) and a community of learners and demonstrate their commitment to equity. It is important to set high standards for all students and to provide the supports and extensions that help all students learn. By making thoughtful choices at each decision point in the templates, teachers demonstrate their understanding of Adolescent Mathematics Learners (p.26) as it relates to the students in their class.

Time Although the time allotment for mathematics classes may vary within and between the elementary and secondary panels and between schools, teachers must plan appropriate time for the various components of each lesson. To assist teachers in this stage of planning, research regarding Japanese Lesson Structure (p. 33) is provided as an example of an internationally respected model of student inquiry and related time allotment.

Materials The contemporary mathematics classroom features a combination of traditional materials and more recently developed tools. Concrete Materials (p. 29) provides a description and analysis of effective mathematical manipulatives. Technology (p. 41), teachers can read about the various types and applications of technological tools, and important considerations relating to their implementation.

Type of Learner Research regarding the complex amalgam of students’ abilities and competencies, e.g., Gardner’s Multiple Intelligence research, has had an immense impact on modern educational thinking. Students not only demonstrate many different kinds of skills and creative talents, but possess different learning styles including auditory, visual, and kinaesthetic. The lesson template encourages teachers to mindfully interpret each lesson in terms of these considerations.

Grouping Research has shown that students benefit from a balance of different learning structures. Flexible Grouping (p. 31) informs teachers of the many options available for logistical planning within the classroom. These options include heterogeneous pairs and small groups, homogeneous pairs and small groups, co-operative learning activities, independent research or class work, together with whole-class and small group discussions.

Instructional Strategies A variety of effective instructional strategies are explored within this document. Differentiated Instruction (p.30) describes how teachers can orchestrate different activities for different students simultaneously, for one or more instructional purposes, and to the benefit of all involved. Mental Mathematics and Alternative Algorithms (p. 34) provides insight into the research surrounding estimation and algorithmic exploration. Further valuable strategies such as Scaffolding (p. 38), Student-Centred Investigations (p. 39), and Teacher-Directed Instruction (p. 40) are also detailed.

Communication Communication contributes to development of mathematical understanding in the classroom. Metacognition (p. 35) describes ways in which teachers can encourage students to analyze and alter their own thinking and learning processes. Questioning (p. 37) discusses the complexity involved in, skills required for, and examples of higher-order questioning. In Communication (p. 28), a model is presented for a positive classroom environment in which students are encouraged to question, comment, theorize, discuss, and defend ideas.


Students need opportunities to learn, practise, select their own preferred problem-solving methods and strategies, and demonstrate learning in a variety of ways.

Assessment The purpose of assessment is to improve student learning by Providing Feedback (p. 36) and using assessment data to inform and guide instruction. Principles of Quality Assessment (p. 21) and Linking of Assessment and Instruction (p. 22), describe assessment that models and is embedded in instructional planning connects to student needs and prior knowledge, and promotes achievement. Good assessment practices provide opportunities for students to demonstrate what they “know and can do” and promote their success. Using a variety of assessment strategies and tools assists in providing Balanced Assessment (p. 23) that recognizes that students demonstrate their understanding in many different ways and that provides a complete picture of a student’s mathematical understanding. Students who have difficulty in mathematics may need adjustments or Assessment Accommodations (p. 24) opportunities to demonstrate achievement. Evaluation (p. 25) involves the judging and interpreting of assessment data and the assigning of a grade. Evaluation should be a measure of a student’s current achievement in the important mathematical concepts in the program.

Developing Understanding The facilitation of conceptual understanding as it relates to geometry, is elaborated in a discussion of the van Hiele Model (p. 42). The importance of conceptual understanding is also highlighted in each section of Continuum and Connections.

Home Activity or Further Classroom Consolidation Within the Providing Feedback (p. 36) section, teachers are reminded of the importance of consistent and detailed feedback regarding student progress to both students and parents throughout the school year. The lesson template encourages teachers to plan a variety of activities for follow-up just as they plan a variety of lesson types, depending on the learning goal. Students need opportunities to learn, practise, select their own preferred problem-solving methods and strategies, and demonstrate learning in a variety of ways.


B. Theoretical Framework for Program Planning

Vision

What is mathematical literacy and why it is important? Mathematical literacy, as defined by the Programme for International Student Assessment (PISA, www.pisa.oecd.org/knowledge/summary/b.html) is “measured in terms of students’ capacity to: • recognise and interpret mathematical problems encountered in everyday life; • translate these problems into a mathematical context; • use mathematical knowledge and procedures to solve problems; • interpret the results in terms of the original problem; • reflect on the methods applied; and • formulate and communicate the outcomes.”

Another source suggests that: “Mathematics literacy is an individual’s capacity to identify and understand the role that mathematics plays in the world, to make well-founded mathematical judgments and to engage in mathematics, in ways that meet the needs of that individual’s current and future life as a constructive, concerned and reflective citizen.” (Measuring Up, OECD PISA Study, 2001, p. 10)

Mathematical literacy is valued for many different reasons. Mathematics provides powerful numeric, spatial, temporal, symbolic, and communicative tools. Mathematics is needed for “everyday life” to assist with decision making. As John Allen Paulos pointed out in his book, Innumeracy (2001), “ … numeracy is the ability to deal with fundamental notions of number and chance in order to make sense of mathematical information presented in everyday contexts.” Mathematics and problem solving are needed in the workplace for many professions such as health science workers or graphic artists, as well as statisticians and engineers. Mathematics is also, ultimately, a cultural and intellectual achievement of humankind and should be understood in its aesthetic sense (NCTM, 2000). All people have a right of access to the domain of mathematics.

The need for mathematical literacy for all students requires us to describe what an effective mathematics program looks like – one that fosters mathematical literacy in all students.

What does effective mathematics teaching and learning look like? There are several components to sound mathematics teaching and learning: a) a classroom climate that is conducive to learning b) mathematical activities that engage students in important mathematical concepts c) a focus on problem solvingd) a program that provides a balance of instructional and assessment strategies that sustains

mathematical understanding.

a) A Classroom Climate for Mathematics Learning Learning does not occur by passive absorption of information. Students approach a new task with prior knowledge, assimilate new information, and construct new understandings through the task (Romberg, 1995). The classroom climate must be conducive to such learning. The following factors need to be considered:

• Prior Knowledge: Students come to class with significant but varying prior knowledge. Such prior knowledge should be valued and used during instruction.

• Equity: All students are able to learn mathematics. Excellence in mathematics requires high expectations and strong support for all students.

• Technology: Technology influences the mathematics that we teach and how we teach it. To this end, teachers need to keep up-to-date with technology.


The intent of Targeted Implementation and Planning Supports (TIPS) is to help teachers activate all of the student’s learning systems through the delivery of a balanced mathematics program.

• Social, emotional, and physical considerations: Brain research on learning (Given, 2002) suggests that there are five “theatres of the mind” — emotional, social, physical, cognitive, and reflective. The emotional, social, and physical learning systems tend to be the most powerful in terms of their demands. The level of their functioning determines how effectively the cognitive and reflective systems operate.

• Attitude and confidence: Fostering a positive attitude and building student confidence helps students to develop values that do not limit what they can do or can assimilate from a learning experience. Students’ emotions, beliefs, and attitudes towards mathematics and learning interact with cognition and are instrumental in empowering learners to take control of their own learning and express confidence in their mathematical decisions.

A classroom that is conducive to learning takes all of these areas into consideration so that students have a safe and positive environment for learning. Traditionally, mathematics programs have attended to the cognitive learning system with less attention being paid to the other four learning systems described above. However, mathematics teachers are becoming more aware of the importance of the affective domain – emotional, social, and reflective learning systems. For instance, curriculum policy requires that students learn how to communicate effectively in mathematical contexts. This communication provides perspectives into students’ feelings and values as well as their thinking. The intent of Targeted Implementation and Planning Supports (TIPS) is to help teachers activate all the student’s learning systems through the delivery of a balanced mathematics program.

An appropriate classroom environment in Grade 7, Grade 8, and Grade 9 Applied also considers the characteristics of the adolescent learner. These characteristics include intellectual development such as the ability to form abstract thought and make judgments. Adolescents are going through many physical changes, and may be experiencing emotional uncertainty. Recognition by peers and social status are extremely important to them. In working with adolescents, it is important to recognize signs of disengaged or at-risk learners and to provide support for them. A more extensive discussion of the adolescent learner is included in Adolescent Mathematics Learners, p. 26.

Summary A classroom environment that supports student learning has the following qualities: • Students are supported to take risks, explore different problem-solving

strategies, and communicate their understanding. • Mathematics is seen, heard, and felt. • The teacher models and promotes a spirit of inquiry. • Students’ prior knowledge is valued and built upon. • Students actively explore, test ideas, make conjectures, and offer explanations. • Social skills are developed to promote effective teamwork.


“…conceptual understanding without skills is inefficient.”

(Mathematics Program Advisory, June 1996)

“Skills without conceptual understanding are meaningless.”

(Mathematics Program Advisory, June 1996)

b) Mathematical Activity Mathematical activity should develop mathematical thinking. The successful teaching of mathematics can be characterized as helping students develop mathematics proficiency. In Adding It Up: Helping Children Learn Mathematics (Kilpatrick, Swafford, & Findell, 2001), the concept of mathematical proficiency is described in terms of five intertwining components:

• conceptual understanding – comprehension of mathematical concepts, operations and relations • procedural fluency – skill in carrying out procedures flexibly, accurately, efficiently, and

appropriately • strategic competence – ability to formulate, represent, and solve mathematical problems • adaptive reasoning – capacity for logical thought, reflection, explanation, and justification • productive disposition – habitual inclination to see mathematics as sensible, useful, and worthwhile,

coupled with a belief in diligence and one’s own efficacy

These five competencies capture what it means for someone to learn mathematics successfully. The integrated and balanced development of all five should guide the teaching and learning of school mathematics. Instruction should not be based on extreme positions that students learn, on one hand, solely by internalizing what a teacher or book says, or, on the other hand, solely by inventing mathematics on their own. (Kilpatrick et al., 2001, p. 11)

Mathematical activity should be balanced with time to investigate and explore as well as time to practise skills. As such, mathematical activity can be organized around five different processes as outlined below:

• Knowing Facts and Procedures: “…conceptual understanding without skills is inefficient.” (Mathematics Program Advisory, June 1996) There are basic facts and proficiencies required at each grade level to develop mental mathematics skills including recognizing the reasonableness of a result and the use of symbolic manipulation software and calculators. Some procedures involve manipulation or the performance of calculations and the successful use of algorithms and procedures.

• Reasoning and Proving: Reasoning is essential to mathematics. Developing ideas, making conjectures, exploring phenomena, justifying results, and using mathematical conjectures help students see that mathematics makes sense. As teachers help students learn the norms for mathematical justification and proof, the repertoire of types of reasoning available to students – algebraic and geometric reasoning, proportional reasoning, probabilistic reasoning, statistical reasoning, and so forth – expand (NCTM, 2000). Using manipulatives and technology efficiently and effectively in investigating mathematical ideas and in finding solutions to mathematical problems provides students with problem-solving tools and a way to create a visual representation.

• Communicating: Communication plays an important role in supporting learners by clarifying, refining, and consolidating their thinking. Mathematically literate learners should be able to communicate their mathematical ideas orally and in writing while defending and offering justification for such ideas.

• Making Connections: “Skills without conceptual understanding are meaningless.”(Mathematics Program Advisory, June 1996) Students demonstrate their understanding through application of knowledge and skills and through problem solving. Component skills of application and problem solving are representing, selecting, and sequencing procedures. Representing involves the learner in constructing and alternating between various mathematical models such as equations, matrices, graphs, and other symbolic and graphical forms.

• Valuing Mathematics: Values include learners’ emotions, beliefs, and attitudes towards mathematics and learning. Such affective processes interact with cognition and are instrumental in empowering the learner to express confidence in their mathematical decisions and to take control of their own learning.


It is important to provide students with opportunities to practise these activities individually, and to provide rich and challenging problems that require students to combine these activities in various ways.

Summary Mathematical activity in a mathematics classroom is characterized by: • Computing • Recalling facts • Manipulating • Using manipulatives and technology • Exploring • Hypothesizing • Inferring/concluding • Revising/revisiting/reviewing/reflecting • Making convincing arguments, explanations, and justifications • Using mathematical language, symbols, forms, and conventions • Explaining • Integrating narrative and mathematical forms • Interpreting mathematical instructions, charts, drawings, graphs • Representing a situation mathematically • Selecting and sequencing procedures

c) A Focus on Problem Solving Problem solving is fundamental to mathematics. “Without problem solving, skills and conceptual understanding have no utility.”(Mathematics Program Advisory, June 1996) The mathematical processes can be aligned with various problem-solving models, with the mathematical competencies as defined by Kilpatrick, Swafford, and Findell (2001) and with the categories of the Achievement Chart in the Ontario Ministry of Education Mathematics Curriculum. This illustrates that the many different types of organizing structures can be aligned so that teachers see how they fit together and address similar processes of mathematical thinking. The categories in the elementary Achievement Chart – Problem Solving, Understanding of Concepts, and Application of Procedures – and the categories in the secondary Achievement Chart – Knowledge/Understanding, Application, and Thinking, Inquiry, and Problem Solving – should not be viewed as isolated but rather as component processes of problem solving. Problem solving involves designing a plan and applying a variety of skills to solve the problem. The problem-solving process also requires mathematical communication. Thus, problem solving is seen as the most comprehensive framework in the learning of mathematics. Mathematical activity and learning should be centred on problem solving. NCTM’s Principles and Standards document (2000) points out the importance of problem solving:

“Solving problems is not only a goal of learning mathematics but also a major means of doing so. Problem solving is an integral part of all mathematics learning, and so it should not be an isolated part of the mathematics program. Problem solving in mathematics should involve all the five content areas ... Good problems will integrate multiple topics and will involve significant mathematics.” (p. 52).

“Without problem solving, skills and conceptual understanding have no utility.”

(Mathematics Program Advisory, June 1996).

“Solving problems is not only a goal of learning mathematics but also a major means of doing so.”

NCTM’s Principles and Standards (2000)


Students need regular opportunities to represent, manipulate, and reason in the context of solving problems and conducting mathematical inquiries. It is expected that students use a variety of tools, including technology to solve problems and develop their understanding. Students should regularly communicate their thinking to peers and to the teacher and reflect on their learning. Problem solving is the most effective way to engage students in all of the mathematical processes. In Elementary and Middle School Mathematics, John A. Van de Walle says:

“Most, if not all, important mathematics concepts and procedures can best be taught through problem solving. That is, tasks or problems can and should be posed that engage students in thinking about and developing the important mathematics they need to learn.” (p. 40)

d) A Balanced Mathematics Program In a balanced mathematics program, students become proficient with basic skills, develop conceptual understanding, and become adept at problem solving. All three areas are important and need to be included. Students need to develop both skills and conceptual understanding: problem solving is the medium for students to use to make connections between skills and conceptual understanding (Mathematics Program Advisory, 1996).

This resource gives teachers examples of lessons and assessments that will allow them and their students to experience a balanced mathematics program. Once experienced, a balanced mathematics program empowers both teachers and students as life-long learners of mathematics.

Summary A balanced mathematics program has: • students working in groups, pairs, and individually • a variety of activities for students to engage in all of the mathematical processes • a variety of diagnostic, formative, and summative assessment data to improve

student learning and adjust program • a focus on developing key mathematical concepts or “Big Ideas”

Making It Happen This section highlights some of the student and teacher roles in a mathematics class, and describes the components of an effective lesson.

As problem solving places the focus on the student’s attention to ideas and making sense, student roles are very active. Students are encouraged to:

− make conjectures; − gather data; − explore different strategies; − share their ideas; − challenge and defend ideas and solutions; − consolidate and summarize their understanding; − practise skills that relate to the mathematical concepts being explored.

Teachers’ roles are much more than telling and explaining. Teachers need to: − create a supportive mathematical environment; − pose worthwhile mathematical tasks; − recognize when students need more practice with skills and accommodate those needs; − encourage mathematical discussion and writing; − promote the justification of student answers;


− pose important questions that help students “pull out the math;” − listen actively to student questions and responses and react accordingly; − consolidate and summarize important mathematical concepts and help students make

connections.

The structure of a lesson in mathematics can take a variety of forms. In a lesson that focuses on problem solving, the following components incorporate the mathematical processes that have been discussed: • Teacher poses a complex problem. • Students work on the problem. • Students share solutions. • Teacher consolidates understanding and summarizes discussion. • Students work on follow-up activities.

In a lesson focusing on skill development, the following sequence should motivate the hard work that is sometimes needed to develop proficiency: • Teacher poses an interesting problem that will extend a student’s current skill development. • The problem exposes the need for a particular skill. • Building on students’ prior knowledge, the teacher instructs, guides, or demonstrates the new skill. • The teacher guides students through the solution of the original problem by applying the new skill. • Students practise the skill and apply the skill to similar problems.


References

Anderson, D. S. & Piazza, J. A. (1996). Changing Beliefs: teaching and learning mathematics in Constructive pre-service classrooms (pp. 51-62).

Given, B.K. (2002). Teaching to the Brain’s Natural Learning Systems. Association for Supervision and Curriculum Development (ASCD) ISBN 0-87120-569-6

Hiebert, J.C., & Carpenter, T.P. (1992). “Learning and teaching with understanding.” In D.A. Grouws (ed.), Handbook of research on mathematics teaching and learning (pp. 65–97). New York: Macmillan.

Hiebert, J.C., Carpenter, T.P., Fennema, E., Fuson, K.C., Human, P.G., Murray, H.G., Olivier, A.I., & Wearne, D. (1997). Making sense: Teaching and learning mathematics with understanding. Portsmouth, NH: Heinemann.

Kilpatrick, J., Swafford, J., & Findell, B. (2001). Adding it up: Helping children learn mathematics. Washington, DC: National Academy Press.

Mathematics Program Advisory, June, 1996. www.lucimath.org/files/balanced_program.pdf

National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: NCTM.

National Research Council. (1998). How people learn: Brain, mind, experience and school. Washington, DC: National Academy Press.

Ontario Ministry of Education and Training. (1997). The Ontario Curriculum Grades 1-8: Mathematics. Toronto: Queen’s Printer.

Paulos, J.A. (2001). Innumeracy. New York, NY: Hill and Wang.

Romberg. T.A. & Wilson, L.D. (1995). Issues Related to the Development of an Authentic Assessment System for School Mathematics. In T.A. Romberg (ed.) Reform in School Mathematics and Authentic Assessment. Albany, NY: SUNY Press.

Stenmark, J. (ed.). (1991). Mathematics assessment: Myths, models, good questions, and practical suggestions. Reston, VA: National Council of Teachers of Mathematics.

Van de Walle, J.A. (2001). Elementary and middle school mathematics: Teaching developmentally. (4th ed.). New York, NY: Addison Wesley Longman.


C. Research – Topic Summaries Mathematics and Reform ……………………………………………………………………..…. p. 20

Assessment and Evaluation

a) Principles of Quality Assessment ………………………………………………………… p. 21

• Assessment and instruction should reflect the four categories of the Achievement Chart or actions of mathematics, as well as mathematical competences.

• One way to plan purposeful assessment is to include it when planning a lesson, unit, and course.

b) Linking Assessment and Instruction ……………………………………………………...p. 22

• An assessment plan that is well-aligned with the mathematics curriculum and instruction is built around tasks that are similar to the tasks that form instruction.

• As teachers deepen their understanding of assessment, they will find that almost any setting can be an opportunity for assessment.

c) Balanced Assessment …………………………………………………………………..….p. 23

• Teachers should provide assessment tasks that include mathematical content and mathematical processes such as reasoning, problem solving, and communication.

• Assessment tasks should vary in terms of task type, openness, length, modes of presentation, modes of working (groupings), and modes of response.

d) Assessment Accommodations …………………………………………………………… p. 24

• Teachers may need to adapt their assessment plan to better suit the characteristics and circumstances of students in the class.

• A variety of adaptations, including accommodations, can be used to support at-risk students.

e) Evaluation …………………………………………………………………………………... p. 25

• Evaluation should represent a current and accurate picture of a student’s achievement of the curriculum expectations.

• Evaluation should represent a balance of strands and Achievement Chart categories, and focus on the important mathematical concepts in the course.

Adolescent Mathematics Learners …………………………………………………………….. p. 26

• Adolescent students learn mathematics best in environments which allow for physical activity, social interaction, technological investigations, choice, variety, and meaningful input.

• Patience and awareness are key factors in providing adolescent learners with the kind of emotional and pedagogical support that they require during this time of personal change.

Classroom Management ………………………………………………………………………….p. 27

• Good classroom management results from well planned lessons and well established routines. • A stimulating classroom that engages students through a balanced variety of teaching, learning,

and assessment strategies prevents and reduces behaviour problems.

Communication …………………………………………………………………………………… p. 28

• Communication in mathematics requires the use and interpretation of numbers, symbols, pictures, graphs, and dense text.

• When explaining their solutions and justifying their reasoning, students clarify their own thinking and provide teachers with windows into students’ depth of understanding.

Concrete Materials ………………………………………………………………………………... p. 29

• Effective use of manipulatives helps students move from concrete and visual representations to more abstract cognitive levels.

• Teachers can diagnose gaps in a student’s conceptual understanding through observing the use of manipulatives and listening to the accompanying narrative explanation.


Differentiated Instruction ………………………………………………………………………... p. 30

• Differentiated instruction recognizes and values the wide range of students' interests, learning styles, and abilities and features a variety of strategies based on individual needs.

• It is sometimes appropriate to have students in the same class working on different learning tasks.

Flexible Grouping …………………………………………………………………………………. p. 31

• Flexible groups should be used to maximize learning for all students in the class. • Consider prior knowledge, achievement on particular tasks, social skills, learning skills, gender,

and exceptionalities when determining a grouping for a particular task.

Graphic Organizers ………………………………………………………………………………..p. 32

• Graphic organizers are powerful, visual tools that are effective in teaching technical vocabulary, helping students organize what they are learning, and improving recall.

• Graphic organizers rely heavily on background information and can be particularly helpful for students with learning disabilities.

Japanese Lesson Structure …………………………………………………………………….. p. 33

• Japanese lesson structure often features the posing of a complex problem, small-group generation of possible solutions, a classroom discussion and consolidation of ideas, followed by extended practice.

• In this model, teachers plan lessons collaboratively, anticipate student responses, and watch each other teach.

Mental Mathematics and Alternative Algorithms …………………………………………… p. 34

• Traditional algorithms are an essential part of mathematics learning and should be taught, but only after students have developed understanding of the concept and shared their own approaches to the problem.

• Teachers and students should share their strategies for mental computation.

Metacognition …………………………………………………………………………………….. p. 35

• Students learn to monitor their own understanding when they are constantly challenged to make sense of the mathematics they are learning and to explain their thinking.

• Students who are aware of how they think and learn are better able to apply different strategies to solve problems.

Providing Feedback ………………………………………………………………………………. p. 36

• Students and parents need regular formative feedback on the students’ cognitive development and achievement of mathematical expectations, as well as on their learning skills.

• Formative feedback should be meaningful and encouraging while allowing students to grapple with problems.

Questioning …………………………………………………………………………….………….. p. 37

• Higher level (process) and lower level (product) questioning have their place in the mathematics classroom, with teachers employing both types as the need arises.

• Successful questioning often involves a teacher asking open-ended questions, encouraging the participation of all students, and patiently allowing appropriate time for student responses.

Scaffolding …………………………………………………………………………………………. p. 38

• Scaffolding works best when students are at a loss as to what they should do, but can accomplish a task that is just outside their level of competency, with assistance.

• Successful use of scaffolding requires educators to determine the background knowledge of students, and develop a comfortable, working rapport with each student.


Student-Centred Investigations …………………………………………………….………….. p. 39

• Student-centred investigations are learning contexts which require students to use their prior knowledge to explore mathematical ideas through an extended inquiry, discovery, and research process.

• Student-centred investigations provide a meaningful context in which students must learn and use these skills to solve more complex problems.

Teacher-Directed Instruction …………………………………………………………………… p. 40

• Teacher-directed instruction is a powerful strategy for teaching mathematical vocabulary, facts, and procedures.

• Teachers should plan lessons where students will grapple with problems individually and in small groups, identifying for themselves the need for new skills, before these teacher-directed skills are taught.

Technology ………………………………………………………………………………………… p. 41

• Technology such as graphing calculators, computer software packages (e.g., Geometer’s Sketchpad, Fathom, spreadsheets), motion detectors, and the Internet are powerful tools for learning mathematics.

• Technology, when used properly, enhances and extends mathematical thinking by allowing students time and flexibility to focus on decision making, reflection, reasoning, and problem solving.

van Hiele Model …………………………………………………………………………………….p. 42

• The van Hiele model of geometry learning is comprised of the following sequential conceptual levels: visualization, analysis, informal deduction, formal deduction, and rigor.

• To maximize learning and successfully move students through the levels from concrete to visual to abstract, teachers must be aware of students’ prior knowledge and ongoing understandings.


Mathematics Reform Mathematics education is undergoing significant reform in many countries, including Canada and the United States. Principles and Standards for School Mathematics (NCTM, 2000) highlighted six key principles—Equity, Curriculum, Teaching, Learning, Assessment, and Technology—that must guide meaningful curricular reform efforts. The vision statement which prefaced the document elaborated on the need for the continued improvement of mathematics education:

Evidence from a variety of sources makes it clear that many students are not learning the mathematics they need or are expected to learn (Kenney and Silver, 1997; Mullis et al., 1997, 1998; Beaton et al., 1996). The reasons for this deficiency are many. In some instances, students have not had the opportunity to learn important mathematics. In other instances, the curriculum offered to students does not engage them. Sometimes students lack a commitment to learning. The quality of mathematics teaching is highly variable. There is no question that the effectiveness of mathematics education in the United States and Canada can be improved substantially. (2000, p. 5)

Similarly, The Ontario Curriculum: Mathematics (1997, 1999) documents “have been developed to provide a rigorous and challenging curriculum for students” which includes a “broader range of knowledge and skills” (OMET, 1997, p. 3), and which “integrates appropriate technologies into the learning and doing of mathematics, while recognizing the continuing importance of students mastering essential arithmetic and algebraic skills” (OMET, 1999, p. 3). The Grades 7-9 Mathematics Targeted Implementation and Planning Supports (TIPS) document has been developed to further assist teachers, coordinators, and administrators as they continue to implement these reforms. This section will specifically provide educators with a variety of teaching strategies, that are supported by contemporary research in mathematics education.

References National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author. Ontario Ministry of Education and Training. (1997). The Ontario Curriculum Grades 1-8: Mathematics. Toronto, ON: Queen’s

Printer for Ontario. Ontario Ministry of Education and Training. (1999). The Ontario Curriculum Grades 9 and 10: Mathematics. Toronto, ON:

Queen’s Printer for Ontario.


Assessment and Evaluation: Principles of Quality Assessment What is Quality Assessment? The National Council of Teachers of Mathematics (NCTM) Assessment Standards for School Mathematics (NCTM, 1995) suggest that good assessment should: enhance mathematics learning; promote equity; be an open process; and be a coherent process.

Quality assessment should be continual and promote growth in mathematics over time. Quality instruction and assessment should reflect the actions or competencies of mathematics and value process as well as product.

Important aspects of quality assessment The Ontario Association of Mathematics Educators (OAME) document Linking Assessment and Instruction in the Middle Years (Onslow & Sauer, 2001) suggests that in order to be effective, assessment practices need to: • be ongoing and an integral part of the learning-teaching process, giving students regular opportunities

to demonstrate their learning; • emphasize communication between teacher and student, parent and teacher, student and student; • encourage students to reflect on their own growth and learning; • enable the teacher to describe student growth in the cognitive, physical, social, and emotional

domains, and • provide opportunities for students to connect new learning to previous knowledge. (p. 1)

Quality assessment includes a variety of tools and strategies that assess both the processes and products of mathematics learning and serves a variety of purposes: diagnostic, formative, and summative.

Considerations regarding quality assessment Teachers are continually assessing their students, both informally (through observation and listening) and formally. Teachers need to be sure that structured assessments are appropriate to the instructional tasks and are developmentally appropriate for the students. Teachers also need strategies for making assessment manageable so that it appears to be a seamless component of instruction.

References NCTM. (1995). Assessment Standards for School Mathematics. Reston, VA: NCTM. Onslow, B. & Sauer, B. (2001). Linking Assessment and Instruction in Mathematics. Toronto, ON: OAME.


Assessment and Evaluation: Linking Assessment and Instruction What is linking assessment and instruction? Assessment that is aligned with curriculum and instruction is the most purposeful type of assessment. In linking assessment and instruction, assessment should be well connected to what students do in classrooms every day. Important aspects of linking assessment and instruction Research has shown that linking assessment with instruction increases students’ knowledge. A meta-analysis by Black and William (1998), in which they reviewed approximately 250 studies, indicates that when teachers link assessment with instruction on a daily basis, students’ learning generally improves. Gathering assessment regularly, using a variety of methods, provides teachers with wide-ranging indications of students’ performance, and limits the bias and distortion that can occur through the use of a limited number of assessment strategies at the end of instruction. Multiple strategies, e.g., observations, portfolios, journals, rubrics, tests, projects, self- and peer-assessments, indicate to students that the teacher appreciates their daily contributions, values their reasoning and attitude toward mathematics, and does not base evaluations solely on their ability to be successful on tests.

Stenmark (1991) suggests that: As the forms of mathematics teaching become more diverse – including open-ended investigations, cooperative group activity, and emphasis on thinking and communication – so too must the form of assessment change. (p. 3).

Moving from instructional activities to assessment activities should be seamless: one should look like the other. Linking assessment with instruction allows teachers to involve students as responsible partners in their own learning. Understanding one’s strengths and limitations is a key factor for growth and becoming an independent learner. When students are part of the decision making concerning how their work should be evaluated, e.g., development of a rubric, selection of work to be placed in their portfolio, they are more likely to understand the characteristics that constitute sound mathematical thinking, as well as what is expected of them. Evidence from a variety of sources, in a variety of contexts, assists the teacher in accurate diagnosis and provides the information necessary for helping all students advance their understanding. Formative assessment throughout each day guides teaching, informing us of those who understand and those needing further time and practice. Assessment should guide and improve learning of mathematics and a teacher’s teaching of mathematics on an ongoing basis. An over reliance on any one method, in limited contextual settings, tends to provide an imprecise depiction of a student’s true achievement and disposition towards mathematics. Considerations regarding linking assessment and instruction Teachers must understand which assessment methods are appropriate and compatible with the teaching and learning approaches being used. All students deserve opportunities to demonstrate their mathematical abilities and attitudes. Some factors that can mislead interpretations of a student’s mathematical understanding are culture, developmental level of the student, and language. Methods used to collect assessment information should not be overly time-consuming.

If students have a variety of tools available for instruction, these tools should be available for assessment. If students engage in instructional tasks in a variety of forms and groupings then these should also be part of the forms and groupings for assessment.

References Black, P., & William, D. (1998). Inside the black box: Raising standards through classroom assessment. Phi Delta Kappan, 80

(2), 139-148. National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Onslow, B.

& Sauer, B. (2001). Linking Assessment and Instruction in Mathematics. Toronto, ON: OAME. Stenmark, J. K. (1991). Mathematics Assessment: Mythes, Models, Good Questions, and Practical Suggestions. Reston, VA:

NCTM.


Assessment and Evaluation: Balanced Assessment What is balanced assessment? Balanced assessment is assessment that allows students multiple opportunities to show what they know and can do. Balanced assessment considers different learning styles, different groupings of students, different task types, and focuses on a variety of mathematical concepts and processes.

Important aspects of balanced assessment A program with balanced assessment should include a balance of: • Categories of the Achievement Chart or mathematical processes – Assessment should focus not only

on knowledge and understanding but should also ask students to demonstrate their ability to problem-solve and their application, and communication of mathematics.

• Task types – Balanced assessment is more than tests and quizzes and should include other assessments such as presentations, investigations, performance tasks, projects, portfolios, journals.

• Groupings – Most assessments are administered to students individually. Some tasks such as a performance task can be worked on in small groups. Students submit individual solutions to the task to demonstrate their understanding of the work.

• Purposes – Not all assessment should be summative. Diagnostic and formative assessment helps teachers make instructional decisions and provides feedback to students during the learning process.

Considerations regarding balanced assessment Judicious selection of assessment tasks, strategies, and tools should help to provide a balance of assessment and still maintain its manageability.

References Schoenfeld, A., Burkhardt, H., Daro, P., Ridgway, J. Schwartz, J., & Wilcox, S. (1999). Balanced Assessment for the

Mathematics Curriculum: High School Assessment. White Plains, NY: Dale Seymour Publications. ENC 016281 Suurtamm, C. (2001). Assessment Corner: What Really Counts: Building a Balanced Assessment Plan. Ontario Mathematics

Gazette, 40(1), 21 – 23.


Assessment and Evaluation: Assessment Accommodations What are assessment accommodations? Assessment accommodations provide suitable opportunities for specific students to demonstrate their mathematical understanding and help them to develop a productive disposition. Assessment accommodations recognize that individual students may need changes to regular classroom assessment practices to allow them to show what they know and can do. Such accommodations are not limited to but could include: • extended time on assessment tasks; • assessment done orally rather than in writing, e.g., description, report, presentation, video; • reducing the number of tasks used to assess the same concept or expectation; • observation of achievement during instruction or seatwork as the focus rather than paper-and-pencil

task; • support during assessment tasks such as prompts to improve task completion and to activate prior

knowledge; • assessment tasks performed outside the normal classroom time under teacher supervision; • progress submissions or observations as part of assessment to encourage success for students who are

not submitting written work.

Important aspects of assessment accommodations Teachers need to know the characteristics of the learners in the class to provide opportunities that match students’ needs. For specific students, assessment accommodations could be used for a variety of reasons: • A student may be struggling due to some temporary circumstances, such as a broken arm, or math

anxiety. • A student’s stage of processing a particular mathematical concept may be at a concrete level as

opposed to the abstract level reached by many classmates. • Students may require regular accommodations as part of their IEPs. • ESL students may require accommodations in recognition of their stages of language acquisition. • Some adolescent students are reluctant to hand in completed assignments and the teacher may need to

gather and include observation data based on drafts done in class. • A particular summative task may prove to be too challenging at the time it is given, resulting in the

teacher deciding to provide formative feedback, further instruction and practice, then another summative assessment opportunity.

Considerations regarding assessment accommodations Teachers need to be sure that assessment accommodations for individual students are integrated into the entire assessment process. Assessment planning should include ways to assimilate students without IEPs into the regular types of assessment strategies over time through explicit teaching of strategies and skills. In all cases, assessment accommodations are not a “watering down” of the subject but an opportunity for students to show what they know and can do in a way other than what was originally planned.

References Andrews, J. (1996). Teaching Students with Diverse Needs. Toronto: Nelson Canada. Suurtamm, C. (2003). Now you see it … then they forget to hand it in! OAME Gazette, 41(4).


Assessment and Evaluation: Evaluation What is evaluation? “Evaluation involves the judging and interpreting of the assessment data and, if required, the assigning of a grade.” (Dawson & Suurtamm, 2003, p. 38) Evaluation should occur after sufficient time has been given to learn the relevant concepts and skills.

Important aspects of evaluation “Evaluations of students’ achievement at particular times have several characteristics. They are summative in nature, are usually designed to communicate to audiences beyond the classroom, and are often used to make important educational decisions for the students.” (NCTM, 1995, p. 56).

To evaluate students’ achievement, teachers summarize evidence from multiple sources to form a description or judgment of students’ mathematical understanding as seen through the level of achievement of curriculum expectations. Evaluation requires the use of a teacher’s professional judgment as well as assessment data.

Considerations regarding evaluation There are several ways that teachers track and use student assessment data. In most cases, evaluation policies at the school board level help schools and teachers to maintain consistency. Consistency is further enhanced through teacher discussion and sharing. As well, professional development in assessment and evaluation helps teachers develop confidence in evaluation so that they can support their evaluation in dialogues with students, parents, and administrators.

References Dawson, R., Suurtamm, C. Early Math Strategy: the Report of the Expert Panel on Early Math in Ontario. Toronto: Queen’s

Printer, 2003. National Council of Teachers of Mathematics (NCTM). (1995). Assessment Standards for School Mathematics. Reston, VA:

NCTM.


Adolescent Mathematics Learners What are the characteristics of adolescent mathematics learners? According to Reys et al (2003), students vary greatly in their development and readiness for learning and “teachers play a critical role in judging the developmental stage” of each student and in establishing “rich environments for students to explore mathematics at an appropriate developmental level.” (p. 25) The adolescent mathematics learner (Grades 7-9) is experiencing great changes and challenges in several domains simultaneously. Intellectually, adolescents are refining their ability to form abstract thought, think symbolically, render objective judgments, hypothesize, and combine multiple reactions to a problem to achieve resolution. Wolfe (cited in Franklin, 2003), noted:

By adolescence, students lose about 3 percent of the gray matter in their frontal lobe – this is a natural process where the brain ‘prunes’ away excess materials to make itself more refined and more efficient. Such changes could indicate why adolescents sometimes have difficulty prioritizing tasks or multitasking… . Some neuroscientists feel that the cerebellum may be responsible for coordination or cognitive activity in addition to muscle and balance coordination. If that’s true, then it’s possible that physical activity could increase the effectiveness of the brain and learning. (p. 4)

Physically, adolescents tend to mature at varying rates, e.g., girls developing physically earlier than boys; adolescents are often concerned about their physical appearance; and may experience fluctuations in metabolism causing extreme restlessness and/or lethargy. Emotionally, many adolescents are sensitive to criticism; exhibit erratic emotions and behaviour; feel self-conscious; often lack self-esteem; search for adult identity and acceptance; and strive for a sense of individual uniqueness. Socially, adolescents may be eager to challenge authority figures and test limits; can be confused and frightened by new school settings that are large and impersonal; are fiercely loyal to peer group values; and are sometimes cruel and insensitive to those outside the peer group. Capitalizing on adolescent characteristics Erlauer (2003) has suggested what she refers to as the 20-2-20 Rule for Reflection and Application at the Middle/High School Levels. Twenty (20) minutes into the lesson, when students’ attention is waning, the teacher has the students re-explain what they have just learned (e.g., brief class discussion, sharing with partner, or entry in student journals) with some form of feedback (e.g., from teacher or from classmates) to check for understanding. Within two (2) days of initial learning, the teacher requires students to review and apply the new information, e.g., mind-map, piece of writing, developing a related problem for classmate to solve. And within twenty (20) days, usually at the end of a unit, the teacher has the students reflect on what they have learned and apply the concepts/skills they have learned to a more involved project which is then shared with the whole class or a small group of students. (pp. 84-85) Considerations regarding adolescent characteristics Teachers of Grades 7-9 adolescent mathematics learners, should consider the following questions: • Do students have a role in determining classroom rules and procedures? • Do students feel safe to take risks and participate during mathematics learning? • Do students have opportunities to move around and engage in situations kinesthetically? • Are a variety of groupings used, for particular purposes? • Do students have opportunities to discuss and investigate different ways of thinking about and doing

mathematics? • Do tasks have multiple entry points to accommodate a range of thinkers in the concrete-abstract

continuum?

References Erlauer, L. (2003). The brain-compatible classroom: Using what we know about learning to improve teaching. Alexandria, VA:

Association for Supervision and Curriculum Development. Franklin, J. (2003). Understanding the adolescent mind. Education Update, 45(4), p. 4. Reys, R., Lindquist, M., Lambdin, D., Smith, N., & Suyduam, M. (2003). Helping children learn mathematics (7th ed.).

Hoboken, NJ: John Wiley & Sons.


Classroom Management What is classroom management? Classroom management can best be viewed as both an active and proactive phenomenon. Teachers have to make decisions on a daily basis concerning their teaching, learning and assessment processes, as well as the mathematical content they will stress. These and other important teacher choices directly determine student behaviour.

Important aspects of classroom management Successful teachers establish a community of learners within a positive learning environment by: gaining the respect and trust of students, establishing clear and consistent routines, developing meaningful relationships with each student, understanding and accommodating the variety of student needs, and by selecting a full range of materials and strategies to teach mathematics.

Physical settings often convey to students whether their interactions are welcomed or if the teacher is the authority in transmitting the knowledge students are expected to learn. It is the teacher’s attitude towards learning, more than the physical setting, which is likely to foster a community of learners. Teachers who want to empower students encourage them to take risks, and appreciate the value of their students not knowing, using these occasions as opportunities for growth rather than anxiety. Students are expected to make conjectures, justify their thinking, and respond to alternative perspectives respectfully, rather than memorize arbitrary rules that might not make sense to them.

Teachers present worthwhile tasks that challenge students and hold their interest. These tasks can often be solved in more than one way, and contain flexible entry and exit points, making them suitable for a wide range of students. Teachers who know their students’ abilities and attitudes towards mathematics are likely to challenge them and pique their interest. They know what questions to ask and the appropriate time to ask them; when to remain silent, and when to encourage students to meet the challenge without becoming frustrated.

Considerations regarding classroom management Kaplan, Gheen and Midgley (2002) conducted research with Grade 9 students and found that the emphasis on mastery and performance goals in the classroom is significantly related to students’ patterns of learning and behaviour. Their conclusions are as follows:

This study joins many others (see Urdan, 1997) that have pointed to the benefits of constructing learning environments in which school is thought of as a place where learning, understanding, improvement, and personal and social development are valued and in which social comparison of students’ ability is deemphasized. (p. 206)

A balanced and active approach to teaching and learning allows students to become engaged in mathematics and to learn co-operative and self-management skills (Ares & Gorrell, 2002; Boyer, 2002; Brand, Dunn & Greb, 2002). Each lesson in TIPS includes a variety of student groupings and instructional strategies, depending on the learning task. Lesson headers (Minds On, Action!, Consolidate/Debrief) are reminders to plan active lessons that appeal to auditory, kinesthetic and visual learners, thereby, having a positive impact on classroom management.

References Ares, N., & Gorrell, J. (2002). Middle school students’ understanding of meaningful learning and engaging classroom activities.

Journal of Research in Childhood Education, 16(2), 263-277. Boyer, K. (2002). Using active learning strategies to motivate students. Mathematics Teaching, 8(1), 48-51. Brand, S., Dunn, R., & Greb, F. (2002). Learning styles of students with attention deficit hyperactivity disorder: Who are they

and how can we teach them? The Clearing House, 75(5), 268-273. Kaplan, A., Gheen, M., & Midgley, C. (2002). Classroom goal structure and student disruptive behaviour. The British Journal of

Educational Psychology, 72(2), 191-211.


Communication What is communication in a mathematics classroom? Communication is the process of expressing ideas and mathematical understanding using numbers, pictures, and words, within a variety of audiences including the teacher, a peer, a group, or the class.

Important aspects of communication The NCTM Principles and Standards (2000) highlights the importance of communication as an “essential part of mathematics and mathematical education” (p. 60). It is through communication that “ideas become objects of reflection, refinement, discussion and amendment” and it is this process that “helps build meaning and permanence for ideas and makes them public” (p. 60). The Ontario Curriculum (OMET, 1999) also emphasizes the significance of communication in mathematics, describing it as a priority of both the elementary school and the secondary school programs. It further states, “This curriculum assumes a classroom environment in which students are called upon to explain their reasoning in writing, or orally to the teacher, to the class, or to other students in a group” (p. 4).

Craven (2000) advocates for a strong emphasis on mathematical communication, providing ideas for the recording and sharing of students’ learning (e.g., journal entry/learning log, report, poster, letter, story, three-dimensional model, sketch/drawing with explanation, oral presentation). He concluded by stating:

Children must feel free to explore, talk, create, and write about mathematics in a classroom environment that honours the beauty and importance of the subject. Children must be empowered to take risks and encouraged to explain their thinking. Teachers must construct tasks that will generate discussion and provide an opportunity for students to explain their understanding of mathematical concepts through pictures, words, and numbers. (p. 27)

Whitin and Whitin (2002) conducted research by examining a fourth-grade class under the instruction of a progressive classroom teacher. The authors described how talking helped students explore, express their observations, describe patterns, work through difficult concepts, and propose theories; and how drawing and writing assisted students in recording and clarifying their own thinking. They concluded by noting that communication was “…enhanced, and the children’s understanding developed, when mathematical ideas were represented in different ways, such as through a story, with manipulatives and charts, and through personal metaphors…. In these ways, children can become proficient and articulate in communicating mathematical ideas.” (p. 211)

Considerations regarding communication Franks and Jarvis (2001) maintained that new forms of communication, although potentially liberating and motivating, can initially be difficult and uncomfortable for teachers and students to explore. However, they noted that both of these groups, given sustained support, had rewarding experiences when asked to become playful yet thoughtful risk-takers. (p. 66)

References Craven, S. (2000). Encouraging mathematics communication in children. Ontario Mathematics Gazette, 38(4), 25-27. Franks, D., & Jarvis, D. (2001). Communication in the secondary mathematics classroom: Exploring new ideas. In A. Rogerson

(ed.), The Mathematical Education into the 21st Century Project: International Conference on New Ideas in Mathematics Education (pp. 62-66). Palm Cove, Australia: Autograph.

NCTM. (2000). Principles and standards for school mathematics. Reston, VA: Author. Ontario Ministry of Education and Training. (1999). The Ontario curriculum grades 9 and 10: Mathematics. Toronto, ON:

Queen’s Printer for Ontario. Whitin, P., & Whitin, D. J. (2002). Promoting communication in the mathematics classroom. Teaching Children Mathematics,

9(4), 205-211.


Concrete Materials What are concrete materials? Concrete materials (manipulatives) are physical, three-dimensional objects that can be manipulated by students to increase the likelihood of their understanding mathematical concepts.

Strengths underlying the use of concrete materials Concrete materials have been widely used in mathematics education, particularly in the K-6 elementary panel, but also in the middle school years (see Toliver video series). The results of TIMSS (1996) showed that Grade 8 students from Ontario did less well in the areas of algebra and measurement than other areas. Do Grade 8 students make connections between abstract algebraic concepts and their more concrete, previously learned, arithmetic experiences? Chappell and Strutchens (2001) noted that:

Too many adolescents encounter serious challenges as they delve into fundamental ideas that make up this essential mathematical subject [algebra]. Instead of viewing algebra as a natural extension of their arithmetic experiences, significant numbers of adolescents do not connect algebraic concepts with previously learned ideas. (p. 21)

In response to this situation, they recommended the use of concrete models, such as algebra tiles, to assist students in making connections and to facilitate mathematical understanding. Ross and Kurtz (1993) provided the following recommendations for teachers planning to implement a lesson involving manipulatives: (i) choose manipulatives that support the lesson’s objectives; (ii) ensure significant plans have been made to orient students to the manipulatives and corresponding classroom procedures; (iii) facilitate the active participation of each student; and (iv) include procedures for evaluation that reflect an emphasis on the development of reasoning and processing skills, (e.g., listening to students talking about mathematics, reading their writings about mathematics, and observing them at work on mathematics. (pp. 256-257)

Considerations regarding concrete materials Thompson (1994) maintained that professional development should model selective and reflective use of concrete materials, helping teachers to focus on “what students will come to understand” as a result of using manipulatives, rather than just on “what students will learn to do” (p. 557). Similarly, Moyer (2001) noted that manipulatives can be used in a rote manner with little or no learning of the mathematics, and that the effective use of concrete materials relies heavily on a teacher’s background knowledge and understanding of mathematical representations. Other researchers have warned that manipulatives should not be over-used (Ambrose, 2002); should not be treated as a fun reward activity or trivialized through teachers’ comments (Moyer, 2001); but, should be used effectively and selectively as a means of facilitating the ongoing transitions from the concrete (physical and visual) information involved in student learning to the more abstract knowledge of mental relationships and deep understandings. (Kamii, Lewis, & Kirkland, 2001)

References Chappell, M. F., & Strutchens, M. E. (2001). Creating connections: Promoting algebraic thinking with concrete models.

Mathematics Teaching in the Middle School, 7(1), 20-25. Kamii, C., Lewis, B. A., & Kirkland, L. (2001). Manipulatives: When are they useful? Journal of Mathematical Behaviour,

20(1), 21-31. Moyer, P. S. (2001). Are we having fun yet? How teachers use manipulatives to teach mathematics. Educational Studies in

Mathematics, 47(2), 175-197. Ross, R., & Kurtz, R. (1993). Making manipulatives work: A strategy for success. Arithmetic Teacher, 40(5), 254-257. Thompson, P. W. (1994). Concrete materials and teaching for understanding. The Arithmetic Teacher, 41(9), 556-559. Tolliver, K. (1995). The Kay Toliver files (Parts 1-4) [Motion Picture]. (Available from Foundation for Advancement in Science

and Education, Los Angeles, CA)


Differentiated Instruction What is differentiated instruction? Differentiated instruction is based on the idea that because students differ significantly in their interests, learning styles, abilities, and experiences, teaching strategies and pace should vary accordingly.

Strengths underlying differentiated instruction The September 2000 issue of Educational Leadership focused almost exclusively on issues pertaining to differentiated instruction. Pettig (2000) noted that “differentiated instruction represents a proactive approach to improving classroom learning for all students,” and that it “requires from us [teachers] a persistent honing of our teaching skills plus the courage to significantly change our classroom practices.” (pp. 14, 18) Heuser (2000) presented math and science workshops, both teacher- and student-directed, as a viable strategy for facilitating differentiated instruction. He stated that the philosophy behind these workshops is founded on research and theory that support diverse learners’ understanding of math and science, and can be summarized as follows: (i) children learn best when they are actively involved in math and science and physically interact with their environment; (ii) children develop a deeper understanding of math and science when they are encouraged to construct their own knowledge; (iii) children benefit from choice, both as a motivator and as a mechanism to ensure that students are working at an optimal level of understanding and development, (iv) children need time and encouragement to reflect on and communicate their understanding, and (v) children need considerable and varying amounts of time and experiences to construct scientific and mathematical knowledge. (p. 35) Notwithstanding the fact that “teachers feel torn between an external impetus to cover the standards [curriculum] and a desire to address the diverse academic needs,” Tomlinson (2000) maintained:

There is no contradiction between effective standards-based instruction and differentiation. Curriculum tells us what to teach: Differentiation tells us how. Thus, if we elect to teach a standards-based curriculum, differentiation simply suggests ways in which we can make that curriculum work best for varied learners. In other words, differentiation can show us how to teach the same standard to a range of learners by employing a variety of teaching and learning modes. (pp. 8, 9)

Considerations regarding differentiated instruction Within differentiated instruction, the successful inclusion of all types of learners is facilitated. (Winebrenner, 2000, p. 2000) Karp and Voltz (2000) summarized differentiated instruction in the following way:

As teachers learn and practice various teaching strategies, they expand the possibilities for weaving rich, authentic lessons that are responsive to all students’ needs… the adherence to a single approach will create an instructional situation that will leave some students unravelled and on the fringe. (p. 212)

References Heuser, D. (2000). Reworking the workshop for math and science. Educational Leadership, 58(1), 34-37. Karp, K. S., & Voltz, D. L. (2000). Weaving mathematical instructional strategies into inclusive settings. Intervention in School

and Clinic, 35(4), 206-215. Page, S. W. (2000). When changes for the gifted spur differentiation for all. Educational Leadership, 58(1), 62-64. Pettig, K. L. (2000). On the road to differentiated practice. Educational Leadership, 58(1), 14-18. Tomlinson, C. A. (2000). Reconcilable differences? Standards-based teaching and differentiation. Educational Leadership, 58(1),

6-11. Winebrenner, S. (2000). Gifted students need an education, too. Educational Leadership, 58(1), 52-56.


Flexible Grouping What is flexible grouping? Flexible grouping refers to the practice of varying grouping strategies based on short-term learning goals that are shared with the students, then regrouping once goals are met. Groupings often include individual, partners, student- and teacher-led small groups, and whole-class configurations.

Strengths underlying flexible grouping Linchevski and Kutscher (1998) suggest that during whole class discussions, the teacher could: develop conceptions about what mathematics is; create an appropriate learning environment and foster essential norms of classroom behaviour; legitimize errors as part of the learning process; and allow expressions of ambiguity. “These discussions also allow weaker students to participate, albeit many times passively via “legitimate peripheral participation” (Lave & Wenger, 1991) and “cognitive apprenticeship” (Brown, Collins, & Duguid, 1989), in a challenging intellectual atmosphere.”

The Early Math Strategy (2003) suggests that a reason for independent mathematics, as well as shared and guided mathematics, is that “children demonstrate their understanding, practise a skill, or consolidate learning in a developmentally appropriate manner through independent work…Students need time to consolidate ideas for and by themselves.” (p. 37)

Students need opportunities to learn from each other in small groups and pairs, to try ideas, practise new vocabulary, and construct their own mathematical understanding with others. Large homogeneous groups can engage in differentiated tasks including enrichment topics, remediation, and filling gaps for groups of students who were absent when a concept was taught.

Considerations regarding flexible grouping In reporting the results of three studies that focused on the effects of teaching mathematics in a mixed-ability setting on students’ achievements and teachers’ attitudes, researchers Linchevski and Kutscher (1998) concluded that “it is possible for students of all ability levels to learn mathematics effectively in a heterogeneous class, to the satisfaction of the teacher.” (p. 59) To examine high-achieving students’ interactions and performances on complex mathematics tasks as a function of homogeneous versus heterogeneous pairings, Fuchs, Fuchs, et al. (1998) videotaped ten high achievers working with a high-achieving and a low-achieving classmate on performance assessments. Based on their findings, they recommended that “high achievers, when working on complex material, should have ample opportunity to work with fellow high achievers so that collaborative thinking as well as cognitive conflict and resolution can occur.” (p. 251) However, they also noted that heterogeneous groupings can prove valuable when “used appropriately with less complex tasks, by providing maximal opportunities for high achievers to construct and low achievers to profit from well-reasoned explanations.” (p. 251) Flexible grouping has much to offer, yet also demands careful planning.

References Fuchs, L. S., Fuchs, D., Hamlett, C. L., & Karns, K. (1998). High-achieving students’ interactions and performance on complex

mathematical tasks as a function of homogeneous and heterogeneous pairings. American Educational Research Journal, 35(2), 227-267.

Hoffman, J. (2002). Flexible grouping strategies in the multiage classroom. Theory into practice, 41(1), 47-52. Linchevski, L., & Kutscher, B. (1998). Tell me with whom you’re learning, and I’ll tell you how much you’ve learned: Mixed-

ability versus same-ability grouping in mathematics. Journal for Research in Mathematics Education, 29(5), 533-554. Early Math Strategy: The Report of the Expert Panel on Early Math in Ontario, Ontario Ministy of Educaton, 2003.

ISBN 0-7794-4089-7


Graphic Organizers What are graphic organizers? Graphic organizers are visual representations, models, or illustrations that depict relationships among the key concepts involved in a lesson, unit, or learning task.

Strengths underlying the use of graphic organizers According to Braselton and Decker (1994), mathematics is the most difficult content area material to read because “there are more concepts per word, per sentence, and per paragraph than in any other subject” and because “the mixture of words, numerals, letters, symbols, and graphics require the reader to shift from one type of vocabulary to another.” (p. 276) They concluded by stating, “One strategy that is effective in improving content area reading comprehension is the use of graphic organizers. (Clarke, 1991; Flood, Lapp, & Farnan, 1986; Piccolo, 1987)” (p. 276) Similarly, DiCecco and Gleason (2002) noted that many students with learning disabilities struggle to learn in content area classes, particularly when reading expository text. Since, in their opinion, content textbooks often do not make important connections/relationships adequately explicit for these students, the authors recommended the use of graphic organizers to fill this perceived gap.

Graphic organizers are one method that might achieve what textbooks fail to do. …They include labels that link concepts in order to highlight relationships (Novak & Gowin, 1984). Once these relationships are understood by a learner, that understanding can be referred to as relational knowledge. …Logically, if the source of relational knowledge is structured and organized, it will be more accessible to the learner (Ausubel, 1968). (p. 306)

Monroe and Orme (2002) presented two general methods for teaching mathematical vocabulary: meaningful context and direct teaching. Described as a powerful example of the latter method, the use of graphic organizers was further highlighted as being “closely aligned with current theory about how the brain organizes information” and as one of the more promising approaches regarding the recall of background knowledge of mathematical concepts. (p. 141)

Considerations regarding the use of graphic organizers Monroe and Orme note that the effectiveness of a graphic organizer is limited by its dependence on background knowledge of a concept (Dunston, 1992). As an example, they explain that “if students have not encountered the concept of rhombus, a graphic organizer for the word will not help them to develop meaning.” (p. 141) Also noteworthy on this topic is the fact that many software programs and websites are now being developed which provide students with digital versions of graphic organizers and virtual manipulatives, e.g., interactive animations of 3-dimensional, mathematical learning objects, an increasing number of which are becoming available free of charge for educators and parents.

References Braselton, S., & Decker, B. C. (1994). Using graphic organizers to improve the reading of mathematics. The Reading Teacher,

48(3), 276-282. DiCecco, V. M., & Gleason, M. (2002). Using graphic organizers to attain relational knowledge from expository text. Journal of

Learning Disabilities, 35(4), 306-320. Monroe, E. E., & Orme, M. P. (2002). Developing mathematical vocabulary. Preventing School Failure, 46(3), 139-142.

TIPS References Grade 7: Venn diagrams, Days 12 & 13 Grade 8: Ranking Ladder, Day 3; Yes/No examples, Day 11; Tree Diagram, Day 18 Grade 9 Applied: Placemat, Unit 2, Day 3


Japanese Lesson Structure What are the typical elements of a Japanese mathematics lesson? A typical Japanese mathematics lesson features the following components: (i) the posing of a complex, thought-provoking problem to the class; (ii) individual and/or small group generation of possible approaches for solving the problem; (iii) the communication of strategies and methods by various students to the class; (iv) classroom discussion and collaborative development of the mathematical concepts/understandings; (v) summary and clarification of the findings by the teacher; (vi) consolidation of understanding through the practice of similar and/or more complex problems.

Strengths underlying the Japanese lesson structure Since the release of the Third International Math and Science Study (TIMSS) results in 1996, nations have been devoting a great deal of time, resources, and research to the process of unravelling the diverse findings, and exploring various methods for national mathematical and scientific reform. Since Japanese mathematics students outperformed their global peers, attention has focused on the methods of instruction and classroom management that have been adopted by Japanese educators. Lessons in Japanese classrooms were found to be remarkably different from those in Germany and the U.S., promoting students’ understanding, while U.S. and German teachers seemed to focus more exclusively on the development of skills. (Martinez, 2001; Roulet, 2000; Stigler & Hiebert, 1997)

The considerable time spent in Japanese classrooms on inventing new solutions, engaging in conceptual thinking about mathematics, and communicating ideas has apparently paid rich dividends in terms of students’ understanding and achievement. It is somewhat ironic to note that in light of the fact that TIMSS has been criticized as being, overly skill-based as opposed to featuring more problem-solving content, Japanese students, who have experienced the types of reforms promoted by groups such as the National Council of Teachers of Mathematics, also outperformed the world on more traditional mathematics questions. It appears that the deeper understandings cultivated through these methodologies are not at the expense of technical prowess; rather this form of instruction seems to strengthen both procedural and conceptual knowledge.

Considerations regarding Japanese lesson planning Japanese teachers regularly take part in lesson study and inquiry groups, producing gradual but continual improvement in teaching. (Stigler & Hiebert, 1999) Watanabe (2002) noted that “a wide range of activities characterizes this kind of professional development, offering teachers opportunities to examine all aspects of their teaching – curriculum, lesson plans, instructional materials, and content.” (p. 36) He further recommended that in order to learn from Japanese lesson study, North American teachers should: (i) develop a culture through regular and collective participation; (ii) develop the habit of writing an instruction plan for others; (iii) develop a unit perspective; (iv) anticipate students’ thinking; and (v) learn to observe lessons well.

References Martinez, J. (2001). Exploring, inventing, and discovering mathematics: A pedagogical response to the TIMSS. Mathematics

Teaching in the Middle School, 7(2), 114-119. Roulet, G. (2000). TIMSS: What can we learn about Ontario mathematics? Ontario Mathematics Gazette, 38(3), 15-23. Stigler, J., & Hiebert, J. (1997). Videotape classroom study. Washington, DC: United States Department of Education, National

Center for Educational Statistics. (Available at: http://nces.ed.gov/timss/video.asp) Stigler, J., & Hiebert, J. (1999). The teaching gap: Best ideas from the world’s teachers for improving education in the

classroom. New York, NY: The Free Press. Watanabe, T. (2002). Learning from Japanese lesson study. Educational Leadership, 59(6), 36-39.


Mental Mathematics and Alternative Algorithms What is meant by mental mathematics and alternative algorithms? People who are mathematically literate can generally compose (put together) and decompose (take apart) numbers in a variety of ways. Having flexible thinking skills empowers students and frees them to invent strategies that make sense to them.

Example:

Students who use the traditional subtraction algorithm for 6000 – 1, often get the answer wrong, and generally don’t have a good sense of number or a positive attitude towards mathematics.

When asked to multiply 24 by 25 many people who are comfortable playing with numbers use strategies such as divide 24 by 4 (recognizing 4 quarters make one dollar and so 24 quarters equals $6 or 600 cents); or add 240+240+120 (decomposing: 24 x 10 + 24 x 10 + 24 x 5); or 625 – 25 (they know 25 x 25 then subtract 25).

Strengths underlying mental mathematics and alternative algorithms Research projects have shown that young people who invent their own alternative algorithms to solve computation questions generally have a firm sense of number and place value (Kamii & Dominick, 1997; Van de Walle, 2001). When students invent their own strategies for solving computation questions, the strategies tend to be developed from knowledge and understanding rather than the rote memorization of the teacher’s method. As Kamii and Dominick (1997) explain, knowledge is developed from within and young people can then trust their own powers of reasoning. Kamii and others have shown that students who invent their own algorithms tend to do as well as other students on standardized computation tests, and also have a far better understanding of what is happening mathematically.

Many computational errors and misconceptions are based on students’ misunderstandings of traditional algorithms. When subtracting 48 from 72 they often write 36, subtracting the smaller digits from the larger ones. When asked why zeros are included when using the traditional algorithm to multiply 25 by 33, few students can explain that they multiplying 25 by 3 and 25 by 30. They have an even more difficult time explaining why these two sums are added together. Misconceptions are often the result of following procedures without understanding. Alternative algorithms encourage students to make sense of what they are doing rather than accepting rules based on faith.

Considerations regarding alternative algorithms Many mathematics educators believe that students should be taught traditional algorithms after or along with the opportunity to invent their own algorithms. Traditional algorithms are an essential part of mathematics learning and should be taught, but only after students have developed understanding of the concept and shared their own approaches to the problem.

References Kamii, C. K., & Dominick, A. (1997). To teach or not to teach algorithms. Journal of Mathematical Behavior, 16(2), 51-61. Van de Walle, J. A. (2001). Elementary and middle school mathematics: Teaching developmentally (Fourth Edition). New York,

NY: Longman.


Metacognition What is metacognition? Metacognition is the awareness and understanding of one’s own thought processes; in terms of mathematics education, the ability to apply, evaluate, justify, and modify one’s thinking strategies.

Important aspects of metacognition Metacognition and its implications for mathematics education are described in the National Council of Teachers of Mathematics Principles and Standards (2000) document:

Good problem solvers become aware of what they are doing and frequently monitor, or self-assess, their progress or adjust their strategies as they encounter and solve problems (Bransford et al. 1999). Such reflective skills (called metacognition) are much more likely to develop in a classroom environment that supports them. Teachers play an important role in helping to enable the development of these reflective habits of mind by asking questions such as "Before we go on, are we sure we understand this?" "What are our options?" "Do we have a plan?" "Are we making progress or should we reconsider what we are doing?" "Why do we think this is true?" Such questions help students get in the habit of checking their understanding as they go along. (p. 54)

Kramarski, Mevarech and Lieberman (2001) noted that “For more than a decade, metacognition researchers have sought instructional methods that use metacognitive processes to enhance mathematical reasoning.” (p. 292) For example, Pugalee (2001) described how metacognition has been shown to complement problem solving, enhancing decisions and strategies such as predicting, planning, revising, selecting, checking, guessing, and classifying. (p. 237)

Considerations regarding metacognition Although many students are able to discover thinking and problem-solving strategies independently or detect unannounced strategies that they see others use, some students who have “metacognitive deficits may not even be aware that others are using strategies to successfully complete the task at hand.” (Allsopp et al., 2003, p. 310) Moreover, these deficits “become more evident as students are expected to apply strategies they have learned to new situations, concepts, or skills.” (p. 310) In light of this reality, and the fact that young people typically like to talk about their thinking in the classroom, researchers Stright and Supplee (2002) recommend that small-group instructional contexts be used often to encourage math talk and to facilitate the sharing of varied models of thinking among students. (p. 237) Part of consolidation and debriefing should provide opportunities for students to reflect on the lesson, through writing in journals and/or talking in small groups, or whole-class discussions.

References Allsopp, D., Lovin, L., Green, G., & Savage-Davis, E. (2003). Why students with special needs have difficulty learning

mathematics and what teachers can do to help. Mathematics Teaching in the Middle School, 8(6), 308-314. Kramarski, B., Mevarech, Z. R., & Lieberman, A. (2001). Effects of multilevel versus unilevel metacognitive training on

mathematical reasoning. The Journal of Educational Research, 94(5), 292-300. NCTM. (2000). Principles and standards for school mathematics. Reston, VA: Author. Pugalee, D. K. (2001). Writing, mathematics, and metacognition: Looking for connections through students’ work in

mathematical problem solving. School Science and Mathematics, 101(5), 236-245. Stright, A. D., & Supplee, L. H. (2002). Children’s self-regulatory behaviors during teacher-directed, seat-work, and small-group

instructional contexts. The Journal of Educational Research, 95(4), 235-244.


Providing Feedback What is meant by providing feedback? Providing feedback involves the comprehensive and consistent communication of information regarding a student’s progress in mathematics learning, to both the student and the parent.

Important aspects of providing feedback Mathematics educators are responsible for not only preparing and presenting quality instruction, but also for developing and implementing effective methods of assessment and reporting. Providing regular formative and summative feedback to both students and parents is of importance to help students adjust to the demands of the mathematics program. There is perhaps nothing worse than arriving at the end of a term or semester only to find that one’s marks, or one’s child’s marks, in mathematics are dismally below average, yet without any prior feedback being received. One difficult area for teachers is to know how much assistance to offer students on a particular mathematics problem or task, without giving away too much information. Chatterley and Peck (1995), in their article entitled, “We’re Crippling our Kids with Kindness!”, noted the following:

The key is experience. We cannot, even though with the most kindly of intentions, exclude students from those experiences that come from struggling with a problem. …We cannot cripple our students mentally by taking away from them the struggles that must come before understanding is brought to fruition. If we understand the process necessary to provide the referents within the minds of our students, we will cease to mentally cripple them by being overly kind and sympathetic and by helping too much and often far too soon. (pp. 435-436)

Kewley (1998) emphasized the importance of this notion of cognitive dissonance, or psychological discomfort, by referring to the work of several key educational theorists:

Both Piaget (Ginsburg & Opper, 1979) and Vygotsky (Wertsch & Stone, 1985) believed that disequilibrium was a process necessary to learning, because if everything goes according to plan, nothing rises to the level of consciousness. Disequilibrium, or cognitive imbalance, is a state that occurs when the learner is unable to assimilate an experience or achieve a goal. It motivates the student’s search for better knowledge and a valid solution. (pp. 30-31)

Assistance during the lesson or task must be meaningful and encouraging, but not overstated; teachers must carefully provide students with enough feedback to help them move forward by themselves (see Scaffolding).

Considerations regarding providing feedback Both the NCTM Principles and Standards (2000) and The Ontario Curriculum (1997, 1999) highlight the importance of parental involvement in mathematics education. Because teachers, students, and parents/guardians are considered partners in schooling, the provision of consistent feedback regarding student progress becomes a vital link connecting all members of this team. As Onslow (1992) noted:

If we want parents to understand mathematics as more than the procedural drills of arithmetic then they must be provided with opportunities to explore their children’s mathematics programme. In this way, parents will be in a position to understand not only what we are teaching but why we are teaching the way we do. (p. 25)

References Chatterly, L. J., & Peck, D. M. (1995). We’re crippling our kids with kindness! Journal of mathematical behavior, 14(4),

429-436. Kewley, L. (1998). Peer collaboration versus teacher-directed instruction: How two methodologies engage students in the

learning process. Journal of Research in Childhood Education, 13(1), 27-32. Onslow, B. (1992). Improving the attitude of students and parents through family involvement in mathematics. Mathematics

Education Research Journal, 4(3), 24-31.


Questioning What is questioning? Teacher questioning and teacher listening are closely linked skills that are employed daily in the mathematics classroom to guide both teaching and learning, facilitate participation, and to stimulate higher-order thought. Important aspects of questioning Nicol (1999) conducted research regarding questioning with pre-service teachers. She found that prospective teachers seemed to “struggle with not only how they might ask students questions but also what they might ask and for what purpose.” (p. 53) She concluded that “As prospective teachers… began to consider students’ thinking and create spaces for inquiry through the kinds of questions posed, they began to see and hear possibilities for mathematical exploration that evolved as their relationship with mathematics and with students changed.” (p. 62) Defining a higher order question as a query that asks students to respond at a higher level than factual knowledge, Wimer et al. (2001) conducted research surrounding the higher order questioning of boys and girls in elementary mathematics classrooms. They noted in summary that “higher level and lower level questions have their place in the classroom; vigilant teachers employ both types when the need arises.” (p. 91)

After observing the low levels of achievement of many of his middle school students, Reinhart (2000) decided to implement changes in his teaching methods. He noted, “It was not enough to teach better mathematics; I also had to teach mathematics better. Making changes in instruction proved difficult because I had to learn to teach in ways that I had never observed or experienced, challenging many of the old teaching paradigms.” (p. 478) Understanding his students proved helpful in this process:

Getting middle school students to explain their thinking and become actively involved in classroom discussions can be a challenge. By nature, these students are self-conscious and insecure. This insecurity and the effects of negative peer pressure tend to discourage involvement. To get beyond these and other roadblocks, I have learned to ask the best possible questions and to apply strategies that require all students to participate. (pp. 478-479)

Reinhart offered five suggestions for implementing these positive, yet difficult changes: (i) never say anything a kid can say, (ii) ask good questions, i.e., that require more than recalling a fact or reproducing a skill; the best questions are open-ended, (iii) use more process questions (i.e., that require the student to reflect, analyze, and explain his/her thinking and reasoning) than product questions (i.e., that require short answers, yes/no responses, or rely almost completely on memory), (iv) replace lectures with sets of questions, and (v) be patient, i.e., wait time is very important; increasing it to five seconds or longer results in better responses. (p. 480) Considerations regarding questioning Although the Socratic method of questioning and answering has existed as a longstanding and effective mathematics teaching strategy, it is best used as one method among several. Open-ended questioning, some of it creating disequilibrium, and informal classroom discussions regarding mathematical thinking and processes are valuable. References Nicol, C. (1999). Learning to teach mathematics: Questioning, listening, and responding. Educational Studies in Mathematics,

37(1), 45-66. Reinhart, S. C. (2000). Never say anything a kid can say! Mathematics teaching in the middle school, 5(8), 478-483. Wimer, J. W., Ridenour, C. S., Thomas, K., & Place, A. W. (2001). Higher order teacher questioning of boys and girls in

elementary mathematics classrooms. The Journal of Educational Research, 95(2), 84-92.

TIPS References TIPS for Teachers: “Questioning” includes: Clarifying Questions, Prompting Questions, Reflective Questions, Strategic

Questions, and Inquiry/ Big Questions.


Scaffolding What is scaffolding? Scaffolding is a metaphor that is used as a framework to describe how teachers can guide students through a learning task. Scaffolds may be tools, such as cue cards, analogies, and models; or techniques, such as teacher modeling, prompting, or thinking aloud.

Strengths underlying scaffolding Research has shown that scaffolding is most useful in situations where students are at a loss as to what they should do, but can proceed, and accomplish a task that is just outside their level of competency, with assistance. The successful use of scaffolding requires the educator to determine the background knowledge of each student, and to develop a comfortable, working rapport with each student. One research study in which a mathematics teacher used a “thinking aloud” strategy as a form of scaffolding is described in the following vignette:

In a mathematics study by Schoenfeld (1985), the teacher thought aloud as he went through the steps in solving procedures he was using (for example, making diagrams, breaking the problem into parts). Thus, as Schoenfeld points out, thinking aloud may also provide labels that students can use to call up the same processes in their own thinking. …Through modeling and thinking aloud, he applied problem-solving procedures and revealed his reasoning about the problems he encountered. Students saw the flexibility of the strategies as they were applied to a range of problems and observed that the use of a strategy did not guarantee success. …As individual students accepted more responsibility in the completion of a task, they often modeled and thought aloud for their less capable classmates. Not only did student modeling and think-alouds involve the students actively in the process, but it allowed the teacher to better assess student progress in the use of the strategy. Thinking aloud by the teacher and more capable students provided novice learners with a way to observe “expert thinking” usually hidden from the student. (Rosenshine & Meister, 1992, p. 28)

Although scaffolding was developed as an educational theory particularly in response to the specific needs of students with learning disabilities (Stone, 1998), it has more recently seen application in both general and adult education, as an affective strategy for all learners (Graves et al., 1996). Furthermore, this strategy has also been found effective in teaching higher-order cognitive skills. Rosenshine and Meister (1992) described six components that they believe comprise the successful teaching of higher-order cognitive skills via scaffolding: (i) presenting a new cognitive strategy, (ii) regulating difficulty during guided practice, (iii) varying the context for practice, (iv) providing feedback, (v) increasing student responsibility, and (vi) providing student responsibility. (pp. 26-32)

Considerations regarding scaffolding Stone (1998) documented some cautionary notes. Teachers should be aware that scaffolding is meant to be a temporary, as opposed to long-term strategy; and that scaffolding requires adaptation for individual learners. (pp. 349-350) Scaffolding should not focus exclusively on teacher-directed methods that provide more information than is necessary.

References Graves, M. F., Graves, B. B., & Braaten, S. (1996). Scaffolded reading experiences for inclusive classes. Educational

Leadership, 53(5), 14-16. Rosenshine, B., & Meister, C. (1992). The use of scaffolds for teaching high cognitive strategies. Educational Leadership, 49(7),

26-33. Stone, C. A. (1998). The metaphor of scaffolding: Its utility for the field of learning disabilities. Journal of Learning Disabilities,

31(4), 344-364.


Student-Centred Investigations What are student-centred investigations? Student-centred investigations, referred to as rich investigations or rich tasks, are learning contexts which require students to explore mathematics through inquiry, discovery, and research.

Strengths underlying student-centred investigations Chapin (1998) defined a mathematical investigation as a “multidimensional exploration of a meaningful topic, the goal of which is to discover new ways of thinking about the mathematics inherent in the situation rather than to discover particular answers.” (p. 333) She elaborated on their significance:

Investigations afford students an opportunity for sustained, in-depth study of a topic; students must make sense of their observations and synthesize and analyze their conclusions. …In addition, connections among different areas of mathematics (e.g., algebra, geometry, statistics) can be made, since students tend to bump into related ideas and concepts while pursuing the investigative questions. The “bumping” phenomenon enables a teacher to pursue related topics in parallel with the mathematical investigation – presenting a context for connecting ideas, for clarifying concepts, or for teaching new material. Finally, mathematical investigations can assist in establishing a classroom environment that supports inquiry. Students are expected to explore questions and engage in relevant discourse. By working slowly through many levels of questions and responses, students begin to experience the importance of careful reasoning and disciplined understanding. (pp. 333-334)

Flewelling and Higginson (2000) have developed similar ideas in their work regarding rich tasks. Creativity is presented as a salient feature of these explorations, in contrast to many traditional tasks which “ask students to follow given recipes to expected end-points, giving students little opportunity to consider alternatives and be creative.” (p. 18) Rich learning tasks are designed in such a way that “different students are able to demonstrate (very) different kinds and levels of performance.” (p. 18)Researchers Ares and Gorrell (2002) interviewed teachers and students from five middle schools to gain insight into perceptions surrounding meaningful learning experiences. They reported that: “The overriding message from students is that active learning, rather than passive listening, reading, and note-taking, draws them into subjects and deepens their understanding and appreciation of what they are learning.” (p. 270) Based on their study, they concluded that, “…the assumption underlying management techniques should be that students and teachers value the same things: productive interactions centred on substantive learning.” (p. 275)

Considerations regarding student-centred investigations Some educators advocate the use of student-centred investigations throughout the entire curriculum, introducing traditional mathematics skills, when appropriate, to support the rich task explorations. Others view investigations as one successful strategy among many, to be used several times throughout the term, semester, or course. Teachers new to the idea are encouraged to plan and try one investigation of interest, reflect on and discuss the task with colleagues, and then make changes to the investigation before using it again, based on this professional dialogue.

References Ares, N., & Gorrell, J. (2002). Middle school students’ understanding of meaningful learning and engaging classroom activities.

Journal of Research in Childhood Education, 16(2), 263-277. Chapin, S. (1998). Mathematical investigations: Powerful learning situations. Mathematics Teaching in the Middle School, 3(5),

332-338. Flewelling, G., & Higginson, W. (2000). A handbook on rich learning tasks. Kingston, ON: Queen’s University.


Teacher-Directed Instruction What is teacher-directed instruction? Teacher-directed instruction involves teaching rules, concepts, principles, and problem-solving strategies in an explicit fashion, providing a wide range of examples with extensive review and practice.

Strengths underlying teacher-directed instruction According to Baker, Gersten, and Lee (2002), who synthesized empirical research on teaching mathematics to low-achieving students, “research suggests that principles of direct or explicit instruction can be useful in teaching mathematical concepts and procedures.” (p. 68) Monroe and Orme (2002), focusing on the development of mathematical vocabulary, stated the following:

Direct teaching of selected vocabulary has been advocated for many years (Gray & Holmes, 1938, cited in Chall, 1987; Moore, Readence, & Rickelman, 1989) and is supported by Vacca and Vacca (1996) and Klein (1988). Vacca and Vacca (1996) assert that the most important vocabulary words “need to be taught directly and taught well.” (p. 136) Klein expresses the idea that direct teaching of vocabulary will guide students to assign deeper meaning to words. (p. 140)

The authors also noted the existence of poor and ineffectual methods of teaching vocabulary via direct-instruction, such as the “definition-only” approach. Therefore, they advocated a balanced approach for teaching mathematical vocabulary “that combines meaningful context and direct teaching through the use of a graphic organizer.” (p. 141) Wilson, Majsterek, and Simmons (1996) compared the effects of computer-assisted versus teacher-directed instruction on the multiplication performance of elementary students with learning disabilities. Although their findings suggested that for these students teacher-directed procedures were the more efficient and effective method of achieving basic fact mastery, they also recommended a balanced and context-sensitive approach, involving both teacher-directed and computer-assisted instruction. (p. 389) Whereas the results of Stright and Supplee’s (2002) research suggested that students are more self-regulated learners in small group and seat work settings, the authors also concluded by stating, “In order for children to truly become self-regulated learners, the classroom should include all three contexts [teacher-directed, individual seat work, and group work] to provide direct instruction, independent practice, and the opportunity to practice metacognitive skills in a social context (Slavin, 1987).” (p. 242)

Considerations regarding teacher-directed instruction Kewley (1998) pointed out that, although research indicates that teacher-directed instruction promotes learning, the method also has inherent problems. For example, the “superior adult mentality tends to dominate the proceedings, suppressing the reciprocity of ideas and their coordination. The children are then less stimulated to clarify their own ideas, an essential element in gaining understanding and eventually taking ownership of a concept.” (p. 31) The author asserted that “by varying instructional methods there is a greater possibility that teachers will meet the needs of all students, since some may learn better through one set of mechanisms coming into play than another.” (p. 31)

References Baker, S., Gersten, R., & Lee, D. (2002). A synthesis of empirical research on teaching mathematics to low-achieving students.

The Elementary School Journal, 103(1), 51-73. Kewley, L. (1998). Peer collaboration versus teacher-directed instruction: How two methodologies engage students in the

learning process. Journal of Research in Childhood Education, 13(1), 27-32. Monroe, E. E., & Orme, M. P. (2002). Developing mathematical vocabulary. Preventing School Failure, 46(3), 139-142. Stright, A. D., & Supplee, L. H. (2002). Children’s self-regulatory behaviors during teacher-directed, seat-work, and small-group

instructional contexts. The Journal of Educational Research, 95(4), 235-244. Wilson, R., et al. (1996). The effects of computer-assisted versus teacher-directed instruction on the multiplication performance

of elementary students with learning disabilities. Journal of Learning Disabilities, 29(4), 382-390.


Technology What is technology? Technology, a term derived from the Greek teknologia meaning “systematic treatment,” encompasses both a wide range of products, e.g., calculators, software, hardware, and related systematic processes.

Strengths underlying the use of technology Gilliland (2002) noted that although machines do not think, calculators can “take the drudgery out of computation by performing low-level tasks in mathematics.” (p. 50) In referring to related research, she further explained:

Students who learn paper-and-pencil techniques in conjunction with the use of four-function calculators or other technology, and are tested without calculators, perform as well as, or better than, those who do not use technology in class. Appropriate use of calculators does not result in the atrophy of computational skills; instead, it provides an impetus and opportunity for students to focus on conceptual learning (Heid, 1997). (p. 50)

Despite this type of research, only 18% of Grade 3 students and fewer than 10% of Grade 6 students use calculators on a regular basis in Ontario schools (EQAO, Report of Provincial Results, 2002, pp. 30, 39).

Being aware of the increased emphasis on mathematics problem solving and the poor performance of students with learning disabilities, Babbitt and Miller (1996) documented and explored appropriate methods for teaching these skills with greater effectiveness and efficiency. (p. 392) The researchers recommended the use of computers, and specifically hypermedia, i.e., digital environments that go beyond text and traditional computer-assisted instruction to incorporate sound, animation, photographic images, and video clips in sophisticated ways, to teach mathematical problem solving to students with learning disabilities. (pp. 393-395)

Considerations regarding the use of technology The advent of increased technology in the mathematics classroom does not guarantee improved teaching or increased learning. Although complex in nature, and possessing expansive possibilities, technological products such as calculators and software packages are only as effective as the teacher using them. Among the many considerations regarding the implementation of technology that face administrators and teachers in the 21st Century, the two that are of perhaps greatest import are resources and training. Extended and innovative teacher training programs (Woolley, 1998) are most desirable. Dedicated teachers should be encouraged to continue exploring these strategies and tools, keeping in mind the reality that often the students themselves are a rich resource when it comes to technology in the classroom.

References Babbitt, B. C., & Miller, S. P. (1996). Using hypermedia to improve the mathematics problem-solving skills of students with

learning disabilities. Journal of Learning Disabilities, 29(4), 391-401. Gilliland, K. (2002). Calculators in the classroom. Mathematics teaching in the middle school, 8(3), 150-151. Peck, K. L., & Dorricott, D. (1994). Why use technology? Educational Leadership, 51(7), 11-14. Woolley, G. (1998). In Thailand: Connecting technology and learning. Educational Leadership, 55(5), 62-65.


van Hiele Model What is the van Hiele model? The van Hiele level approach is based on the 1950’s work of a Dutch couple, Dina van Hiele-Geldof and Pierre van Hiele, who developed a model that described five distinctive geometry learning levels: visualization, analysis, informal deduction, formal deduction, and rigor. Being neither age-dependent nor content-based, the van Hiele levels describe how students think about geometry and how this thinking changes over time as students become increasingly competent.

Strengths underlying the use of the van Hiele model Van de Walle (2001) described the five levels in the following way:

Visualization: Students recognize and name figures based on the global, visual characteristics of the figure and are able to make measurements and talk about simple properties of shapes.

Analysis: Students are able to consider all shapes within a class rather than a single shape, but are unable to understand the intricacies of categorical definitions.

Informal Deduction: As students begin to be able to think about properties of geometric objects without the constraints of a particular object, they are able to develop relationships between and among these properties.

Formal Deduction: Students are able to examine more than just the properties of a shape, recognizing the significance of an axiomatic system and able to construct geometric proofs.

Rigor: Students become increasingly capable of understanding a complex system complete with axioms, definitions, theorems, corollaries, and postulates. (pp. 309-310)

Because the van Hiele approach assumes that students progress through the five levels sequentially, and that lessons must be delivered at a level that matches this progress, the necessity for teachers to develop a clear picture of prior knowledge and ongoing student understanding is underscored. According to Malloy (1999): “By the time that students enter the middle grades, most of them are between the concrete [first and second] and informal deduction [third] levels defined by the van Hieles. (p. 87) However, teachers often talk about geometry using third or fourth level language that students cannot understand, leading to a mismatch in teaching and learning. An awareness of this disjuncture allows teachers to modify their presentation of geometric concepts in order to appropriately engage students.

Considerations regarding the use of the van Hiele model To illustrate how educators can help students progress from one van Hiele level to another, Malloy (1999) provided the following recommendations:

The van Hiele model suggests using five phases of instruction to help students in this progression. Students first gather information by working with examples (e.g., finding the perimeter of shapes), then they complete tasks that are related to the information, such as adding tiles to the figure to increase perimeter. The students become aware of relationships and are able to explain them. Finally, students are challenged to move to more complex tasks and to summarize and reflect on what they have learned. The language used by teachers and students is important for students’ progression through the levels from concrete to visual to abstract (Fuys, Geddes, and Tischler, 1988. p. 89)

References Malloy, C. (1999). Perimeter and area through the van Hiele model. Mathematics Teaching in the Middle School, 5(2), 87-90. van de Walle, J. A. (2001). Elementary and middle school mathematics: Teaching developmentally (4th ed.). New York, NY:

Lonman.


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TIPS: Section 2 – Mathematical Processes © Queen’s Printer for Ontario, 2003 Page 1

Mathematical Processes Knowing Facts and Procedures

Reasoning and Proving Communicating

Making Connections

Context – Why is the development of mathematical processes important? “Even if you’ve stopped growing physically, you certainly haven’t stopped growing mentally and emotionally. Nor do you stop learning after you finish school, not as long as there are opportunities for learning and growth all around us. Learning also comes in many — and often surprising — forms. But no matter how it appears, learning is forever and learning is for the future.”

(Ontario Prospects 2002, Ontario’s Guide to Career Planning)

“Half of the jobs that will exist in the year 2005 we don’t even have names for now.” (Ontario Prospects 2002, Ontario’s Guide to Career Planning)

Context Connections

Primary/Junior Intermediate/Senior Next Steps

How do mathematical processes develop through different grade levels? What do they look like, feel like, sound like?

Visual Auditory Kinesthetic

How are mathematical processes taught and assessed in ways to address the different needs of different types of learners?

Guided Shared Independent

How can a variety of groupings and instructional strategies help students develop mathematical processes?

Manipulatives and Technology Research suggests that manipulatives themselves do not magically carry mathematical understanding. Rather, they provide concrete ways for students to give meaning to new knowledge.

(Stein and Bovalino, 2001)

Manipulatives, [including technology] also help the student to describe the mathematics and make the dialogue between students and between teacher and student more accessible. Children’s mathematical thinking becomes more transparent when teachers observe children using manipulatives and listen to children’s conversations.

(Early Math Report, 2003)


Knowing Facts and Procedures

Role of Students Role of Teacher Compute • Mentally compute estimates by substituting

rounded values into formulas. • Mentally add many large numbers, keeping

rough track of rounding error to adjust the total, e.g., 750 + 325 + 910 + 645 + 325 can be thought of as 700 + 300 + 900 + 600 + 300 + approximately another 150 to get 2950.

• Develop and use a personal set of referents, e.g., 1 cm is approximately the width of a baby finger.

• Estimate a square root from known square numbers.

• Facilitate development of mental mathematics and estimation skills by providing regular opportunities for students to apply them.

• Insist that students use mental skills regularly. • Model for students the different mental

strategies that can be used. • Facilitate development of standard referents

by providing opportunities for students to share their own referents when explaining their reasoning.

• Insist on mental as well as pencil-and-paper computational competence rather than allowing constant and inappropriate use of technology.

Recall facts • Memorize essential facts, paying attention to

grade expectations.

• Provide frequent opportunities for students to use the facts that are essential for a mathematically-literate person, e.g., playing games.

• Provide students with a needed but forgotten fact at assessment time so that a student may be able to demonstrate skills using this fact.

Manipulate mathematical models • Use different models to best display

mathematical information. • Practise essential manipulations on their own

until they are mastered.

• Facilitate making connections between particular manipulations and the types of problems that require these manipulations by introducing skills in context, e.g., ‘Factor” can be interpreted as finding the dimensions of a rectangle whose area has been given. 2x2 + x looks like

• Allow students to form their own methods at

the introductory stage, monitoring these for correctness.

• Have students practise skills when they see a need for those skills.

• Insist on development of essential skills by providing many opportunities for students to demonstrate their attainment of those skills. Consider introducing proficiency tests. (See samples in each content-based package.)

• Help students develop procedural fluency by providing differentiated practice/homework, as needed.



Role of Students Role of Teacher Use manipulatives and technology • Learn how to use a calculator, and choose to use

it only when appropriate. • Understand when mental arithmetic or pencil-

and- paper calculations or estimation are more appropriate than technology.

• Use a scientific calculator when an exact answer is needed and computation involves several numbers or numbers with more than one digit, and when the numbers are not easily calculated mentally

• Perform repeated calculations using a spreadsheet.

• Use graphing calculators, spreadsheets, The Geometer’s Sketchpad® to explore, gather, display, manipulate, and present data in a variety of ways.

• Use manipulatives and technology to discover/develop understanding of new concepts.

• Enter computations into a non-scientific calculator using correct order of operations commands. (Note: scientific calculators have built-in order of operations.)

• Use the feedback from technological tools to refine hypotheses.

• Encourage students to use a calculator to solve number questions that are beyond the proficiency expectations for operations using pencil and paper.

• Foster understanding and intuition using technology (not as a replacement for proficiency with basic skills). Insist on appropriate grade-specific proficiencies.

• Plan instructional and assessment tasks in which students do not have access to calculators when the goal is learning how to carry out pencil and paper computations or mental estimation.

• Have technological tools and manipulatives available at assessment time so that students have a choice of tools for problem solving.

• Model the use of technology to explore and discover mathematical concepts and properties and reduce the time needed for mechanical operations.

• Model appropriate use of manipulatives and technologies.

• Use time gained through use of technology to focus on conjecturing, decision making, and problem solving.

• Select a wider range of instructional tasks, allowing for efficiencies gained through use of technology.

• Coach students to extend from specific geometric cases to general cases through the use of Geometer’s Sketchpad, e.g., the Pythagorean relationship can be investigated and confirmed.



Connections to Assessment – (Rubric not recommended)

Criteria Attribute Strategies Tools Type of Data Messages

Computing [using pencil and paper, using mental mathematics and estimation]

Correctness Quizzes

Test questions

Exam questions

Analytic marking scheme

Error circling

Proficient or still developing

A mark out of _____

Recalling facts Correctness Homework questions

Manipulating a mathematical form or carrying out a procedure

Correctness Peer editing

Using manipulatives and technology

Correctness

Knowing facts and procedures is an important aspect of mathematical literacy. This mathematical process is assessed for correctness.

Helpful feedback to students comes in the form of pointing directly to and diagnosing their errors.

Not all students come to a grade with all of the proficiencies of earlier grades. The goal should be to help all students develop necessary proficiencies.

Monitor, diagnose, remediate, and provide multiple opportunities and strategies for students to practise, e.g., games, centres, peer-coaching, and demonstrate their developing proficiencies. (See sample proficiency tests in each content-based packages.)


Reasoning and Proving

Role of Students Role of Teacher Explore • Try things. • Make sure they understand the problem. • Generate some examples. • Start to gather data. • Ask questions. • Organize work with a sequential flow.

• Support and encourage risk-taking, applaud creative approaches, expect that different groups will choose different approaches.

• Encourage independence and interdependence within groups.

• Consider collaborating with students, asking questions or thinking aloud when a student or a group of students is making no progress.

• Differentiate suggestions and scaffold based on knowledge of individual students.

• Act as a strategist when walking around the classroom and observing the different ways individuals and groups are working.

Hypothesize • Use intuition combined with given information

to make a reasoned guess when prompted. • Refine hypothesis as evidence is gathered. • Make a reasoned guess as to: − the answer − the strategy that will most likely lead to a

solution − where and why an attempted solution failed.

• Facilitate sharing of hypotheses. • Guide and nurture improvement in hypothesis

formation. • Maintain the desired standard. • Accept all student suggestions, whether you

know them to be true or false. It is as important for students to see how to refute a hypothesis, as it is to see how to confirm a hypothesis.

• Help students decide what sort of evidence would be needed to confirm or refute their hypotheses.

• Model for students how to adjust a hypothesis that has been refuted by evidence.

• Use hypothesis formation as an instructional strategy once students become comfortable with the idea that they can take the risk of making a hypothesis that turns out to be false, and not fail. Students will take the risk to think out loud. For example: “Now that we have tested all of these numerical values in the inequation 2x + 1>3, does anyone have a hypothesis for the solution to the inequation? What evidence would convince us of the truth of this hypothesis?” “Now that we have worked through all of these examples, what hypothesis can we make about how to determine whether the answer will be positive or negative? How could we confirm or refute this hypothesis?”



Role of Students Role of Teacher Make inferences and conclusions • Use their models and logic to infer/conclude • Adjust their models as needed.

• Listen to what students say and study, and look at what students write to identify misunderstandings and misconceptions.

• Use differentiated instruction to address misconceptions that different groups of students exhibit.

• Provide many opportunities for students to hear the reasoning of other students and to provide cognitive apprenticeship and to develop adaptive reasoning.

Reflect on new skills, concepts, and questions to see how they connect to prior knowledge/ review/revise • Demonstrate understanding of concepts rather

than just memorizing and using them. • Extend knowledge to new situations. • Examine questions and demonstrate flexibility

in choice of procedure based on the nature of the question.

• Facilitate the sharing of student findings. • Give students opportunities to gather data,

detect patterns, make and justify conjectures. • Require students to reflect on their learning and

to clarify/summarize/record their thoughts. These support development of conceptual understanding.

• Generate cognitive dissonance in students who have misunderstood a concept as this may be necessary before the student will change their understanding.

• Use Venn diagrams and concept maps to review and summarize.

• Use instruction that emphasizes the interrelatedness of mathematical ideas. Students not only learn mathematics, they also learn about the utility of mathematics. They come to value mathematics.

• Encourage and require “what-if” questions. • Model for students alternative mental strategies

that can be used. • Model for students alternative procedures using

manipulatives and technology based on the nature of the question.



Role of Students Role of Teacher Make convincing arguments, explanations and justifications • Analyse and evaluate the mathematical thinking

and strategies of others. • Incorporate different strategies based on

appreciation of the strengths of others’ work. • Write for a variety of audiences.

• Include formal and informal discussion of a variety of approaches to a problem through an electronic bulletin board, on the board, or in project presentation format.

• Validate different approaches to the same problem.

• Discuss that there is not always one right answer or strategy.

• Provide opportunities for students to write for a variety of audiences.

• Provide many opportunities for students to hear the reasoning of other students and to provide cognitive apprenticeship and to develop adaptive reasoning.


Reasoning and Proving Connections to Assessment – Generic Rubric Criteria Attributes Below L1 L1 L2 L3 L4

Identification of variable quantities

Inappropriate differentiation between variables and constants

Limited appropriateness of differentiation between variables and constants

Somewhat appropriately differentiates between variables and constants

Appropriately differentiates between variables and constants

Appropriately differentiates between variables and constants, articulating assumptions

Gathering of data connected to the problem or inquiry

Gathers inappropriate data

Gathers limited appropriate data

Gathers some appropriate data

Gathers appropriate data

Gathers a wide range of appropriate data

Exploring

Quality of questions posed

Poses questions unrelated to the problem

Superficial questions

Poses questions of moderate depth

Poses questions of substantive depth

Poses insightful questions

Hypothesizing Reasonable-ness of connection to the problem or situation

Unconnected to the problem or situation

Limited connections to the problem or situation

Moderately connected to the problem or situation

Well connected to the problem or situation

Insightful connections to the problem or situation

Inferring/ concluding

Reasonable-ness of connection to the problem-solving process and representa-tion used

Unconnected to problem-solving process and representation used

Limited connections to problem- solving process and representation used

Moderately connected to the problem- solving process and representation used

Well connected to the problem- solving process and representation used

Insightful connections to the problem-solving process and representation used

Revising/ reviewing/ reflecting

Evidence of self-monitoring

No evidence Limited evidence

Some evidence Evidence Evidence of attention to fine detail

Logical sequence

No evidence of logic

Limited logic evident

Somewhat logical

Logical Highly logical

Completeness No conclusion reached

Major omissions in arriving at a conclusion

Some omissions in arriving at a conclusion

Thorough Complete and extended

Making convincing arguments, explanations and justifications

Constructs meaning for intended audiences effectively and with clarity (e.g., self, peers, teacher, community members, publication)

Sends a message that demonstrates a limited understanding and sense of the audience

Sends a message that demonstrates that he/she is beginning to develop an understanding and sense of the audience

Sends a message that demonstrates a moderate understanding and sense of the audience

Sends a message that demonstrates a developed understanding and sense of the audience

Sends a message that demonstrates a well-developed understanding and sense of the audience


Communicating

Role of Students Role of Teacher Use mathematical language, symbols, forms, and conventions • Use = down the middle when solving equations. • Use = down the left side when simplifying

expressions. • Use ≅ for congruence. • Use a2 + b2 = c2 or a2 + b2 ≠ c2 when testing for

Pythagorean relationships.

• Use for square roots

• Use < and > when making mathematical arguments.

• Use ∠ for naming angles and ° symbol for degree for angle measures.

• Use <, >, ≤ , ≥ , and ≠ when solving inequations.

• Use ⏐⏐ and ⊥ when referring to parallel and perpendicular lines.

• Use P(A) for the probability of an event A. • Use π for pi. • Use brackets (parentheses) to indicate

multiplication. • Use it to indicate answer has been

approximated or rounded.

• Model proper use of symbols, vocabulary, and notations orally, in board notes, and on handouts.

• Expect correct use of mathematical symbols and conventions in student work.

• Coach students on proper use of conventions as needed.

• Post a word wall with new vocabulary as it is introduced.

• Ensure that new mathematical vocabulary is practised in group and whole-class discussions.

• Coach students in proper usage of terminology.


Communicating

Role of Students Role of Teacher Explain • Learn and use the vocabulary that is used to

instruct, so that this same language can be used in explanations. For example, − If given a graphical model: interpolate,

extrapolate, draw a line of best fit, interpret, draw a conclusion.

− If given an algebraic expression: evaluate, factor, expand, simplify.

− If given an equation: solve, rearrange, isolate a particular variable.

− If given a dynamic geometric model: drag a vertex, transform by reflection, rotation, dilatation, measure, construct.

• Include enough detail and clarity that the reader can follow the author’s thinking.

• Be aware of the audience and write formally or informally to suit the audience.

• Create separate sentences for separate ideas. Avoid run-on sentences.

• Avoid shorthand symbols other than mathematical ones.

• Look for patterns and for contrasts in charts and graphs.

• Read and re-read prose to ensure understanding of all of the given information and instructions.

• Decipher key information from prose for problem-solving purposes.

• Refer to scales and labels on charts and graphs in explanations.

• Give over-viewing descriptions and as much detail as possible when interpreting charts and graphs.

• Recognize, model, and develop a mathematical style of dialogue and argument in the classroom.

• Provide informal feedback on an individual student basis, as indicated, even when Communication is not being formally assessed.

• Provide opportunities for students to read, hear, question and discuss other students’ explanations.

• Use cross-curricular applications in Science, History, and Geography or in the media as contexts for problem solving to demonstrate the utility of mathematics and help students develop a productive disposition.


Communicating

Role of Students Role of Teacher Integrate narrative and mathematical forms • Include words with diagrams, e.g., highlight the

part of a chart where a pattern is noticed. • Include a chart or graph with verbal

descriptions.

• Display samples of student work that exhibit the desired integration of forms.

• Provide informal feedback on an individual student basis.

Communicating

Connect to Assessment – Generic Rubric

Criteria Attribute Below L1 L1 L2 L3 L4 Using mathematical language, symbols, forms, and conventions across a larger body of work, e.g., an entire test

Uses conventions accurately, effectively and fluently

Demonstrates an undeveloped use of conventions

Demonstrates minimal skill in the use of conventions

Demonstrates moderate skill in the use of conventions

Demonstrates considerable skill in the use of conventions

Demonstrates a high degree of skill in the use of conventions

Explaining Clarity/focus Unclear/ Confusing

Limited clarity

Some clarity Clear Precise

Conciseness No mathematical explanation offered

Rambling Somewhat concise

Concise High degree of conciseness

Integrating narrative and mathematical forms

Degree of integration

Message demonstrates limited or no integration

Message demonstrates beginning integration

Message demonstrates moderate integration

Message demonstrates developed integration

Message demonstrates well-developed integration


Making Connections

Role of Students Role of Teacher Represent a situation mathematically • Understand when each of the following types of

models is the most appropriate choice and defend your choice: − physical/concrete/manipulative − electronic/pencil and paper − mental − numerical − graphical − dynamic geometry model − scale drawing − not-to-scale diagram − graphical organizers, e.g., Venn diagram,

T-chart, concept map − table of values − equation/algebraic expression/formula − algorithm/logic model

• Understand that various types of models can be used to appropriately represent the same situation and use multiple representations as indicated.

• Understand that different variations of one model can be used to represent a situation, e.g., algebraic expressions may be equivalent yet appear different.

• Understand the role of constants, e.g., pi, and variables, e.g., radius, in formulas and patterning rules.

• Make connections between new and prior knowledge to make sense of what they are learning.

• Ask questions to resolve areas of confusion. • Apply mathematics to contexts outside of

mathematics.

• Value various ways to demonstrate understanding, e.g., talking, writing, explaining, questioning, drawing, acting out.

• Value and use grade-appropriate types of mathematical models.

• Model appropriate use of manipulatives and technology as tools in developing conceptual understanding.

• Require students to demonstrate their understanding in more than one way.

• Introduce new concepts using concrete materials, as students develop concepts in stages: concrete visual symbolic

• Be flexible in allowing students more or less time to work with concrete materials. Meet student needs through differentiating tasks based on regular observation and assessment of concept attainment.

• Facilitate the sharing of student solutions in order that students see the use of multiple representations.

• Require students to use multiple representations, when appropriate.

• Be open and receptive to connections identified by students.

• Be a cheer-leader when a student makes a significant mathematical connection.

• Recognize the importance of students’ seeing mathematics as sensible, useful, and worthwhile, in the development of a productive disposition.

• Coach students who need help.

Select and apply a problem-solving strategy • In earlier grades, follow stages in problem

solving that include: explore, ask questions, predict possibilities, plan and reflect, decide, communicate, evaluate, plan, carry out the plan, and look back to check the results.

• Provide many opportunities for students to work in groups when solving problems in order to provide ‘cognitive apprenticeship’ experiences for less able students, and to facilitate sharing of strategies among students who approach problems in different ways.


Making Connections

Role of Students Role of Teacher Select and apply a problem-solving strategy • In earlier grades, practise and apply, as the need

arises, specific strategies such as: − draw a diagram − make a simpler but similar problem − act it out − create a mathematical model − work backwards − use a formula − look for a pattern − make a scale drawing

• In Grades 7-9, use whichever of these strategies they find most useful, remembering to: − reread and restate the problem; − identify the information given and needed; − compare the problem to previous experience; − explore and hypothesize; − consider possible strategies and select or

blend strategies; − select a strategy or a blend of strategies; − monitor progress and revise as necessary. Is

the answer reasonable? − consider extensions and variations.

• Guide and coach students to complete solutions using the student-selected strategy if that strategy could lead to a correct solution, even if another strategy might be quicker or more elegant.

• Facilitate the strategic sharing of different problem-solving strategies for the same problem.

• Require students to use multiple strategies to solve the same problem, when appropriate.

• Recognize, encourage, and applaud perseverance.

• Require students to work on tasks that demand sustained effort over time, e.g., problem of the week


Making Connections

Connect to Assessment – Generic Rubric

Criteria Attribute Below L1 L1 L2 L3 L4 Quality of fit to the situation

Little or no evidence

Narrow fit Moderate fit Broad fit Very extensive fit

Depth of understanding


Superficial depth Moderate depth Substantial Insightful

Represent a situation mathematically

Complexity of connections made


Simple Moderately complex

Complex Very complex

Appropriateness of strategy selected

Inappropriate-ate for the situation

Limited appropriateness

Moderate appropriateness

Appropriate Highly appropriate

Select and apply a problem-solving strategy

Completeness of a suitable process

Little or no evidence of a conclusion to the process

With major omissions

With some omissions

Complete Complete and includes evidence of reflection on the strategy

TIPS: Section 2 – Patterning to Algebraic Modelling © Queen’s Printer for Ontario, 2003 Page 1

Patterning to Algebraic Modelling Context • Patterning is an integral part of all strands in the elementary curriculum. It is the foundation for the

study and application of relations, the cornerstone of higher-level mathematics. • The mystery of algebra and the fear of “letters” are reduced when patterning is used to develop an

understanding of variables. • A math trail of patterns could include flowers, house numbers, and architectural designs – physical,

numerical, and geometrical patterns. • Curriculum expectations require students to identify, extend, create, analyse, discuss, and explain

patterns to develop understanding that leads to algebraic modelling.

Context Connections

1 1 1

1 2 1 1 3 3 1

1 4 6 4 1 1 5 10 10 5 1

Architecture Calendars Geometric Designs Pascal’s Triangle

Staircases Schedules Music Fibonacci Sequence

Cab Fares Postage Rates Puzzles and Games Other Connections

Manipulatives • pattern blocks • colour tiles • toothpicks • cubes

Technology • spreadsheets • The Geometer’s Sketchpad® • graphing calculators • calculators

Other Resources http://standards.nctm.org/document/chapter6/alg.htm http://matti.usu.edu/nlvm/nav/frames_asid_169_g_1_t_2.html http://standards.nctm.org/document/eexamples/chap4/4.1/ http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibnat.html#Rabbits


Connections Across Grades Selected results of word search using the Ontario Curriculum Unit Planner Search Words: pattern, variable, algebraic, model, differences, rule

Grade 6 Grade 7 Grade 8 Grade 9 Grade 10 • recognize relationships and use them to

summarize and generalize patterns, e.g., in the number pattern 1, 2, 4, 8, 16, . . . , recognize and report that each term is double the term before it

• identify, extend, and create patterns that identify changes in terms of two variables, e.g.,1, 3, 7, 15, 31, . . . double the previous term and add one

• describe patterns encountered in any context, e.g., elevation maps, newspapers, make models of the patterns, and create charts to display the patterns

• identify and extend patterns to solve problems in meaningful contexts, e.g., notes in music, patterns on graphs

• use a calculator and computer applications to explore patterns

• pose and solve problems by recognizing a pattern, e.g., comparing the perimeters of rectangles with equal area

• analyse number patterns and state the rule for any relationships

• discuss and defend the choice of a pattern rule • given a rule expressed in mathematical

language, extend a pattern • state a rule for the relationship between terms in

a given data table of values and (graph the relationship in the first quadrant)

• describe patterns in a variety of sequences using the appropriate language and supporting materials

• extend a pattern, complete a table, and write words to explain the pattern

• recognize patterns and use them to make predictions

• present solutions to patterning problems and explain the thinking behind the solution process

• describe and justify a rule in a pattern

• write an algebraic expression for the nth term of a numeric sequence

• find patterns and describe them using words and algebraic expressions


• evaluate simple algebraic expressions, with up to three terms, by substituting fractions and decimals for the variables

• Academic and Applied

• identify, by calculating finite differences in its table of values, whether a relation is linear or non-linear

• use algebraic modelling as one of several problem-solving strategies in various topics of the course, e.g., relations, measurement, direct and partial variation, the Pythagorean theorem, percent

• compare algebraic modelling with other strategies used for solving the same problem

See notes below for an overview.

Summary of Prior Learning and Next Steps In earlier years, students: build understanding that patterns occur in a variety of contexts, e.g., music, nature, newspapers, games, graphs, etc. use manipulatives and pictures to recognize, create, describe and extend patterns create tables to display patterns transfer patterns from one medium to another, e.g., physical model to table to word rule begin an informal exploration of the concept of function – where the change in one variable, (e.g., term number) results in changes in another

variable, e.g., term use words to describe a pattern rule, e.g., double the previous number and add 3 – often this rule is based on the previous term focus on

developing area and perimeter formulas for triangles and some quadrilaterals, i.e., rectangle, square, trapezoid, parallelogram; In Grade 7, students: continue to identify, create, describe, and extend patterns use words to describe a pattern rule based on the term number (as opposed to the previous term) defend a pattern rule by confirming that the rule generates the given terms and that the rule can be used to extend the pattern explore multiple representations of the same pattern to build understanding that a pattern rule might be expressed in more than one way

Example: 3 + 2(n – 1) + 1 is the same rule as 2n + 2 In Grade 8, students: continue to identify, create, describe, and extend patterns – continue to explore multiple representations of the same pattern develop an algebraic model by using a variable (e.g., n) to represent the term number use substitution skills to generate terms and to verify different but equivalent algebraic models for the same pattern

In Grade 9 Applied, students: express algebraic models in their simplest forms - example: 3 + 2(n – 1) + 1 will be expressed as 2n + 2 build on their understanding of algebraic models to make connections with linear relationships by examining finite differences in tables of

values connect the numerical coefficient in the linear algebraic model to the constant finite differences to the constant rate of change to the

slope connect the constant in the linear algebraic model to the y-intercept (which is not necessarily the first term in the pattern used to generate it) differentiate between direct and partial variation and develop understanding of slope and y-intercept

In Grade 10 students: continue to connect linear and non-linear relations, e.g., algebraic, graphical, numerical, scale, physical create algebraic models for quadratic functions using a variety of representations (e.g., y = a(x – h)2 + k, y = ax2 + bx + c)


Instruction Connections

Suggested Instructional Strategies Helping to Develop Understanding

Grade 7 • Provide a variety of experiences with concrete materials for students as they work individually and in

groups. • Most patterns will be arithmetic (add a constant to get successive terms) to lead to the study of linear

relations in Grade 9: For a Patterning Example, see Short-Answer Questions, p._ Reasoning and Proving. • Use a T-chart or table to organize information and develop understanding that connects the stage to the

perimeter, e.g., The term number to the actual term. See Short-Answer Questions, p. 5 Reasoning and Proving.

• The Understanding column of the chart for the question cited above is a necessary step in building understanding of the connections between the physical and algebraic models.

Example: Stage Perimeter Understanding 1 6 6 2 8 6 – 1 + 3 3 10 6 – 1 + 2 + 3 4 12 6 – 1 + 2 + 2 + 3

Explanation: To determine the perimeter of stage 4 start with 6 (hexagon sides) then subtract 1 since one side is not included once a square is added. Add 2 for each middle square then add 3 for the final square.

• Most patterns can be interpreted in more than one way (see Grade 8 note below) – the understanding part of the chart could vary – encourage different representations.

• Have students create pattern rule(s) in word statements. • Grade 7 sample response: The rule for this pattern is: 6 – 1 added to two times the number of middle

squares added to 3. The number of middle squares is two less than the stage number. (The last sentence must be included for a complete description.)

• Expect students to explain their rules to each other and to justify their rules by showing how their rule(s) work for given stages and later stages/terms.

Grade 8 • Use word descriptions and algebraic expressions to describe patterns. • Have students express the word rule stated above algebraically as 6 – 1 + 2(n – 2) + 3 where n is the stage

number. • The simplified expression is (4 + 2n) students are not expected to show this until Grade 9, however

students in Grade 8 should be able to verify through substitution that different correct algebraic models will yield the same result for any value of n.

• Further develop the concept of variable by encouraging students to discover, justify, and explain alternative algebraic representations based on different interpretations of the pattern.

• Integrate words and math symbols. Example: Count the number of sides for all shapes then subtract the interior sides

(hexagon sides) + 4 × (number of squares) – 2 × (number of squares) For the 4th stage this would be: 6 + 4 (3) – 2(3) which equals 12 For the nth stage this would be: 6 + 4(n – 1) – 2(n – 1)

If n = 10 then: 6 – 1 + 2(n – 2) + 3 = 6 – 1 + 2(10 – 2) + 3 = 6 – 1 + 16 + 3 = 24

6 + 2(n -1) = 6 + 2(10 –1) = 6 + 18 = 24

6 + 4(n – 1) – 2(n – 1) = 6 + 4(10 –1) – 2(10 – 1) = 6 + 36 – 18 = 24

2n + 4 = 2(10) + 4 = 24

• Justify an algebraic pattern rule. Example: Jenna correctly determined that the nth term of the sequence {3, 5, 7, …} is 3 + 2(n – 1). Justify Jenna’s answer. Note that it would be insufficient to show that the expression “works” for a few cases. Students explain that the sequence starts with 3 and then 2 is added (n – 1) times. So the nth term is 3 + (2 + 2 + 2 + … + 2) where the number of 2s is (n – 1).

Grade 9 Applied • Use simplification to show that two algebraic models are equivalent. Example: Each expression above

simplifies to become (2n + 4). • Recall familiar patterns from Grade 8 – calculate first differences – look for patterns in the algebraic

models of linear relations to make connections between tables of values, equations, and graphs. • Use contextual situations to apply algebraic modelling to solve problems, e.g., given information about

rental companies, students find which one offers the best deal.

• Help students understand the

benefits of using a pattern rule based on the term number, e.g., using a functional model, by asking for the one-millionth term.

• Students are usually comfortable with explaining a pattern rule based on the previous term, e.g., using recursion – this method works well when using the “fill down” feature of spreadsheets.

• By the end of Grade 7, students should also be giving verbal descriptions of the relationship between the term number and the value of the term. Encourage students to analyse charts by looking across a row as well as down a column.

• Help students connect new patterns to prior experiences – look for common elements in building the patterns, e.g., using the first term, using repeated terms to create a multiplication, using constants.

• If a student is uncomfortable using a functional model (based on the term number), they can use models, e.g., table, to see the connections between a model with which they are comfortable and a new one they are striving to understand.

• Some students may need the concrete model to establish a pattern.

• Encourage students to add several rows to a chart to see a pattern unfold.

• Students need to develop proficiency at converting simple word statements into algebraic expressions when translating word rules into algebraic expressions, e.g., two less than the term number translates into n – 2, where n is the term number. Many examples can be practised quickly by having peers and small groups check oral translations and provide supportive feedback.


Connections Across Strands Grade 7

Number Sense and Numeration Measurement Geometry and Spatial

Sense Patterning and Algebra Data Management and Probability

• use patterning to develop understanding that repeated multiplication can be represented as exponents (e.g., in the context of area and volume)

• demonstrate an understanding of the order of operations with brackets and apply the order of operations in evaluating expressions that involve whole numbers and decimals

• identify relationships between and among measurement concepts (linear, square, cubic, temporal, monetary)

• use patterning to develop volume formula for rectangular prism

• recognize patterns to help classify geometric figures

• construct and analyse tiling patterns with congruent tiles

• create and analyse designs that include translated, rotated, and reflected two-dimensional images using concrete materials and drawings, and using appropriate computer applications

• develop understanding of the concept of variable

• substitute into and evaluate simple algebraic expressions

• identify, create, and solve simple algebraic equations

• translate simple statements into algebraic expressions or equations (and vice-versa)

• apply and discuss patterning strategies in problem-solving situations

• See Connections Across the Grades p. 2

• collect, organize, analyse, and interpret data

• use computer applications to examine and interpret data in a variety of ways

• use conventional symbols, titles, and labels when displaying data

• use data to solve problems

Grade 8

Number Sense and Numeration Measurement Geometry and Spatial

Sense Patterning and Algebra Data Management and Probability

• use patterning to discover the rules for the multiplication and division of integers

• represent whole numbers in expanded form using powers and scientific notation

• mentally divide numbers by 0.1, 0.01, and 0.001

• express repeated multiplication as powers

• develop the formula for finding the circumference and the formula for finding the area of a circle

• develop the formula for finding the volume of a triangular prism (area of base × height)

• use patterning to classify geometric figures

• identify geometric properties (e.g., pairs of angles within parallel lines and transversals)

• use patterning to investigate the Pythagorean relationship

• See Connections Across the Grades p. 2


• use the concept of variable to write equations and algebraic expressions

• write statements to interpret simple equations

• collect, organize, analyse, and interpret data

• use computer applications to examine and interpret data in a variety of ways

• know that a pattern on a graph may indicate a trend

• use data to solve problems

Grade 9 Applied

Number Sense and Algebra Measurement and Geometry Analytic Geometry Relationships

• use patterning to determine the meaning of negative exponents and of zero as an exponent and to develop exponent rules

• substitute into and evaluate algebraic expressions

• solve first-degree equations

• use algebraic models to describe relationships involving interior and exterior angles and parallel lines

• use formulas

• identify the geometric significance of m and b in the algebraic model y = mx + b

• use the slope and y-intercept, to determine an algebraic model for a linear relationship

• use formulas

• construct tables of values • construct formulas to represent

linear relations • use formulas to interpolate or

extrapolate • identify, by calculating finite

differences in its table of values, whether a relation is linear or non-linear

Summary or synthesis of curriculum expectations is in plain font. Verbatim curriculum expectations are in italics.


Developing Mathematical Processes Grade 7 Short-answer Questions Addressing the Different Mathematical Processes Patterning and Algebra – 7m71

Extend a pattern, complete a table, and write words to explain the pattern.

Name: Date:

Knowing Facts and Procedures Helena created a table to look for a pattern in the given figures. Complete Helena’s table.

Figure number

Number of sides

1 3 2 5 3 7 4 5

500 The figure number in the last row is 500.

Reasoning and Proving The picture shows 4 stages in the construction of a walkway. The walkway starts with a hexagon and continues with squares. Determine the perimeter of the walkway when it has one hexagon and 352 squares. Show your work.

Communicating The picture shows 4 stages in the construction of a walkway. The walkway starts with a hexagon and continues with squares. A table is used to determine the perimeter of the pathway.

Stage Perimeter Ryan’s Pattern 1 6 6 2 8 6 – 1 + 3 3 10 6 – 1 + 2 + 3 4 12 6 – 1 + 2 + 2 + 3 5 14 6 – 1 + 2 + 2 + 2 + 3

Explain why Ryan started each of the shaded entries with 6 – 1.

Making Connections The picture shows 4 stages in the construction of a walkway. The walkway starts with a hexagon and continues with squares. Ryan created a table:

Stage Perimeter Ryan’s Pattern 1 6 6 2 8 6 – 1 + 3 3 10 6 – 1 + 2 + 3 4 12 6 – 1 + 2 + 2 + 3

Ryan correctly found that the perimeter of the 20th stage is 44 units. Explain how he determined this result. Show your work.

1 2 3 4

1 2 3 4 1 2 3 4

Figure 1 Figure 2 Figure 3


Developing Mathematical Processes Grade 8 Short-answer Questions Addressing the Different Mathematical Processes Patterning and Algebra – 8m80

Write an algebraic expression for the nth term of a numeric sequence.

Name: Date:

Knowing Facts and Procedures Helena created a table to look for a pattern in the given figures. Complete Helena’s table by finding an algebraic expression for the number of triangles in the nth figure.

Figure number

Number of small triangles

1 1 2 4 3 9 n


Communicating The picture shows 4 stages in the construction of a walkway. The walkway starts with a hexagon and continues with squares. A table is used to determine the perimeter of the pathway.

Stage Perimeter Dave’s Pattern

1 2 3 4

Figure 1 Figure 2 Figure 3

1 6 6 2 8 6 – 1 + 3 3 10 6 – 1 + 2 + 3 4 12 6 – 1 + 2 + 2 + 3 5 14 6 – 1 + 2 + 2 + 2 + 3

Dave determined an expression for the perimeter of the nth stage. Explain how Dave used the chart to get the expression: 6 – 1 + 2(n – 2) + 3

Making Connections The picture shows 4 stages in the construction of a walkway. The walkway starts with a hexagon and continues with squares. Ryan created a table:

Stage Number Perimeter Ryan’s Pattern

1 6 6 2 8 6 – 1 + 3 3 10 6 – 1 + 2 + 3 4 12 6 – 1 + 2 + 2 + 3

Determine an algebraic expression for the perimeter of the nth stage. Give reasons for your answer.

1 2 3 4 1 2 3 4


Developing Mathematical Processes Grade 9 Applied Short-answer Questions Addressing the Different Mathematical Processes

Name: Date:

Knowing Facts and Procedures Calculate finite differences to determine if the relationship between the width and the perimeter is linear or non-linear. Give reasons for your answer.

Width (cm)

Perimeter (cm)

11 42 13 46 15 50 17 54


Communicating Reyna made an error when she copied a table from the board. Terry copied the table correctly. Reyna’s Table Terry’s Table

x y x y 2 6 2 6 4 9 4 9 6 12 6 12 9 15 8 15

Reyna says the relationship between x and y is non-linear. Terry says the relationship between x and y is linear. Explain why both answers show good reasoning.

Making Connections The picture shows 4 stages in the construction of a pathway. The table shows the relationship between the construction stage and the perimeter of the pathway.

Stage number

Perimeter (sides)

1 6 2 8 3 10 4 12

Is the relationship between the stage number and the perimeter, linear or non-linear? Give reasons for your answer.

1 2 3 4

Relationships – RE3.03 Identify, by calculating finite differences in its table of values, whether a relation is linear or non-linear.

1 2 3 4


Developing Proficiency Grade 7

Expectation: Extend a pattern, complete a table and write words to explain the pattern Name:

Date: Proficiency Target Met [Practise occasionally to maintain or improve your proficiency level.] □ Still Developing [Search out extra help and practise until you are ready for another opportunity to demonstrate proficiency.]

□

Each table shows the first three terms in a pattern. Use the pattern to complete the table. Some term numbers have been skipped in the last row. a)

1 7 2 9 3 11 4 5

252

b) 1 3 2 6 3 9 4 5

357

c) 1 1 2 4 3 9 4 5 40

d) 1 0.1 2 0.2 3 0.3 4 5

222

e) 1 99 2 98 3 97 4 5 87

Show your work. Use the back of this sheet if you need more space.



Expectation: Find patterns and describe them using words and algebraic expressions

Name: Date:

Proficiency Target Met [Practise occasionally to maintain or improve your proficiency level.] □ Still Developing [Search out extra help and practise until you are ready for another opportunity to demonstrate proficiency.]

□

Each table shows the first three terms in a pattern. Use the pattern to complete the table.

a) 1 7 2 9 3 11 4 5 n

b) 1 3 2 6 3 9 4 5 n

c) 1 1 2 4 3 9 4 5 n

d) 1 0.1 2 0.2 3 0.3 4 5 n

e) 1 99 2 98 3 97 4 5 n



Developing Proficiency Grade 9 Applied Expectation: Identify, by

calculating finite differences in its table of values, whether a relation is linear or non-linear

Name: Date:


□

Use first differences to determine if the relation is linear or non-linear. Pay attention to the increments in both variables. a)

number Length (cm)

1 6 2 8 3 10 4 12 5 14

This relation is ______________________ [linear OR non-linear]

b) Time (min.)

Distance (km)

2 22 4 19 6 16 8 12

10 8 This relation is ______________________ [linear OR non-linear]

c) Time (h)

Cost ($)

5 275 10 325 15 375 30 425 60 475


d) Time (h)

Temperature (oC)

1 6 2 1 3 -4 4 -9 5 -14


e) x y -2 4 -1 1 0 0 1 1 2 4




Extend Your Thinking Grade 7 Expectation: Present solution to

patterning problems and explain the thinking behind the solution process

Name: Date:

Find 4 different ways to determine how many toothpicks there would be in the 50th term of the pattern. Give reasons for your answer. 1

2

3

4


Extend Your Thinking Grade 7 (Answers)

No. Visualization Description Understanding

1 Start with 4 toothpicks for the first term, then add 3 toothpicks for each subsequent term.

4 4 + 3 4 + 3 + 3 …The 50th term would be:

1513494 =×+

2 Two horizontal toothpicks and 1 vertical toothpick for each term number plus 1 more vertical toothpick than the number of the term.

2 + 1 + 1 2x2 + 2x1 + 1 2x3 + 3x1 + 1 …The 50th term would be:

1511150502

=+×+×

3 Four toothpicks times the number of the term, take away the extra vertical toothpicks in the interior.

4 4 + 4 – 1 4 + 4 + 4 – 1 – 1 …The 50th term would be:

151149450

=×−×

4 Think of there being 1 vertical toothpick there before I start to build the squares, and I add 3 toothpicks for each term

1 + 3 1 + 3 + 3 1 + 3 + 3 + 3 …The 50th term would be:

1513501 =×+


Extend Your Thinking Grade 8

Name: Date:

Find 4 different ways to determine how many toothpicks there would be in the nth term of the pattern. Give reasons for your answer.

Expectation: Present solution to patterning problems and explain the thinking behind the solution process.

1

2

3

4


Extend Your Thinking Grade 8 (Answers)

No. Visualization Description Understanding Representation of nth Term



1513494 =×+

• Four plus 3 multiplied by one less than the term number • 4 + 3(n – 1)



1511150502

=+×+×

• Two times the number of squares (for the horizontals) plus the one more than the number of squares (for the verticals) • 2n + n + 1

3 Four toothpicks times the number of the term, take away the extra vertical toothpicks in the interior.


151149450

=×−×

• Four multiplied by the term number subtract one less than the term number • 4n – (n – 1)

4

Think of there being 1 vertical toothpick there before I start to build the squares, and I add 3 toothpicks for each term


1513501 =×+

• One plus 3 multiplied by the term number • 1 + 3n


Extend Your Thinking Grade 9 Applied Expectation: Present solution to

patterning problems and explain the thinking behind the solution process.

Name: Date:

Find 4 different ways to determine whether the relationship between the term number and the number of toothpicks is linear or non-linear. Give reasons for your answer. 1

2

3

4


Extend Your Thinking Grade 9 Applied (Answers)

No. Visualization Description Understanding Nth term Representation



1513494 =×+

• 4 plus 3 multiplied by one less than the term number • 4 + 3(n – 1)



1511150502

=+×+×

• Two times the number of squares (for the horizontals) plus the 1 more than the number of squares (for the verticals) • 2n + n + 1

3 4 toothpicks times the number of the term, take away the extra vertical toothpicks in the interior.


151149450

=×−×

• Four multiplied by the term number subtract one less than the term number • 4n – (n – 1)

4

Think of there being 1 vertical toothpick there before I start to build the squares, and I add 3 toothpicks for each term


1513501 =×+

• 1 plus 3 multiplied by the term number • 1 + 3n

• Multiple representations of a pattern can motivate the study of algebraic equivalence. All of the expressions in the right-most column are equivalent since they all represent the same pattern.

• First differences can be used to demonstrate that the relationship between the term number and the number of toothpick needed to build the pattern is linear.

• Data values (1, 4), (2, 7), (3, 10), (4, 13) can be plotted to demonstrate that the relationship is linear.


Is This Always True? Grade 7, 8, 9

Name: Date:

Judy says that as you add a chord to a circle, you always double the maximum possible number of regions. e.g., has 1 region

has 2 regions

has 4 regions

(Answer) No. One counter example will suffice to disprove Judy’s hypothesis. e.g., Seven, not 8, is the maximum number of regions using 3 chords.

TIPS: Section 2 – Solving Equations and Using Variables as Placeholders © Queen’s Printer for Ontario, 2003 Page 1

Solving Equations and Using Variables as Placeholders Context • What is a variable? What does an equal sign mean? What is an expression? What is an equation?

Clear understanding of the answers to these questions will help students develop the algebraic thinking that leads to the study of functions and relations in secondary school.

• Variables are used for placeholding, making rules, or for generalizing. Particular attention must be paid to the development of these concepts to help students develop algebraic fluency.

• Students are introduced in early grades to the concept of a “placeholder” variable in questions using the form □ + 3 = 8 and 2 × □ = 16. This leads to equations using the form a + 3 = 8 and 2n = 16.

• In Grades 7, 8, and 9, students create and solve a variety of equations that have one specific solution, thereby adding the creation and solving of equations to their problem-solving strategies.

• Students also use variables to create equations that define rules for relationships (e.g., a = 2b + 6 or C = 2πr) and equations that state generalizations (e.g., a + b = b + a, where there is no relationship between the variables).

• Students extend their thinking about the equal sign from “get the answer” to being an essential part of a mathematical statement that shows equality between two expressions.

• When known values are substituted into a formula to find an unknown value, the connection is made between the rulemaking equation and the equation to be solved for the unknown placeholder in a specific case.

Context Connections

Ratios Calendars Crop Yields Electricity Formulas

$

Speed/distance/time Exchange Rates Measurement Formulas Time Zones

Wages Angle Relationships Spreadsheets Other Connections

Manipulatives Technology • tiles • balance • cubes

• spreadsheets • The Geometer’s Sketchpad® • graphing calculators • calculators

Other Resources http://www.standards.nctm.org/document/chapter3/alg.htm#bp2 http://www.purplemath.com/modules/variable.htm http://math.rice.edu/~lanius/Lessons/calen.html http://www.uni-klu.ac.at/~gossimit/pap/guest/misconvar.html http://www.uwinnipeg.ca/~jameis/New%20Pages/MYR39.html


Connections Across Grades Selected results of word search using the Ontario Curriculum Unit Planner Search Words: equation, formula, variable, algebraic

Grade 6 Grade 7 Grade 8 Grade 9 Grade 10 • estimate and

calculate the area of a parallelogram and the area of a triangle, using a formula

• determine the value of a missing term or factor in simple formulas using guess-and-test methods, with and without the use of calculators

• identify the relationships between whole numbers and variables


• interpret a variable as a symbol that may be replaced by a given set of numbers

• write statements to interpret simple formulas

• evaluate simple algebraic expressions by substituting natural numbers for the variables

• translate simple statements into algebraic expressions or equations

• solve equations of the form ax = c and ax + b = c by inspection and systematic trial, using whole numbers, with and without the use of a calculator

• solve problems giving rise to first-degree equations with one variable by inspection or by systematic trial

• establish that a solution to an equation makes the equation true (limit to equations with one variable)

• identify the relationships between whole numbers and variables


• use the concept of variable to write equations and algebraic expressions



• translate complex statements into algebraic expressions or equations

• solve and verify first-degree equations with one variable, using various techniques involving whole numbers and decimals

• create problems giving rise to first-degree equations with one variable and solve them by inspection or by systematic trial

• interpret the solution of a given equation as a specific number value that makes the equation true

Academic • rearrange formulas involving

variables in the first degree, with and without substitution, as they arise in topics throughout the course

• solve first-degree equations, including equations with fractional coefficients, using an algebraic method

Applied • substitute into measurement

formulas and solve for one variable, with and without the help of technology

• solve first-degree equations, excluding equations with fractional coefficients, using an algebraic method

Academic and Applied • manipulate first-degree

polynomial expressions to solve first-degree equations

• solve problems, using the strategy of algebraic modelling

• expand and simplify polynomial expressions involving one variable

Applied • solve first-

degree equations in one variable, including those with fractional coefficients, using an algebraic method

• isolate a variable in formulas involving first-degree terms

• solve quadratic equations by factoring

Academic and Applied • solve systems

of two linear equations in two variables by the algebraic methods of substitution and elimination

Summary of Prior Learning and Next Steps In earlier years, students: determine missing values using guess-and-test (e.g., □ + 3 = 8 or 4 × □ = 12) substitute into formulas (equations) to determine one unknown value

In Grade 7, students: develop proficiency in translating simple statements into algebraic expressions with single operations (e.g., 2n represents a number doubled) develop understanding that a first-degree equation has only one unique solution whereas an algebraic expression takes on a set of values that

depend on the set of numbers represented by the variable(s) continue to use formulas but have not developed understanding that formulas define relationships create, solve, and verify equations of the forms ax = c and ax + b = c by inspection and systematic trial, using whole numbers

In Grade 8, students: develop proficiency in translating complex statements into algebraic expressions (e.g., 2n + 3 represents three more than double a number) develop proficiency at substituting fractions and decimals to evaluate simple algebraic expressions (up to 3 terms) create, solve and verify equations of the forms ax = c and ax + b = c, using whole numbers and decimals also solve equations by using concrete representations (e.g., balance, tiles) to develop understanding of the equal sign and to lead to solving

equations algebraically; students translate actions into an algebraic solution develop proficiency in solving equations of the forms ax = c and ax + b = c using an algebraic method

In Grade 9 Applied, students: develop proficiency in creating, solving, and verifying first-degree equations solve more complex first degree equations using an algebraic method solve equations with integer coefficients (do not solve equations with fractional coefficients in Grade 9 Applied) apply equation solving skills with measurement formulas (e.g., given the radius and volume of a cylinder, determine its height)

In Grade 10, students: extend their equation solving skills to quadratic equations and solving systems of linear equations solve first-degree equations with fractional coefficients (Grade 10 Applied)




Grade 7 • Have students explain in words the meaning of simple expressions in context, e.g., 25q cents

represents the value of q quarters, or 10n dollars represents the payment for n hours of work at $10/h. • Discuss the range of values that a variable could represent, e.g., Could it be zero? 1000? 3.5? • Develop understanding that the variable in a first-degree expression is a symbol that may be replaced

by a given set of numbers. • Generate a table of values using an expression and use one possible value to form an equation.

n 1 2 3 4 5 6 7 8 21n 21 42 63 84 105 126 147 168

An equation created from the table of values is 21n = 105. The variable n in the equation is now a placeholder variable that has one unique value (n = 5).

• Connect to prior learning - students compare 21 × � = 105 to 21n = 105; students make the communication transition from “finding the number that goes in the box” to “solving the equation.”

• Solve simple first degree equations of the forms ax = c and ax + b = c with whole number solutions only by inspection and systematic trial – students give reasons for the values they choose to try.

• Guide students to use their knowledge from an incorrect guess to make a more educated subsequent guess, e.g., Is there a larger or smaller difference between the left and right sides of the equation for the current trial compared to the previous trial? Should you use a higher or lower value for the variable in your next trial?

• Discuss multiple representations that yield the same solution, e.g., “The length is 10 m which is 6 m longer than the width” and can be represented as 6 + w = 10 or 10 – w = 6 or 10 – 6 = w or w + 6 = 10 or 6 = 10 – w.

• Ask students to explain in words the meaning of simple formulas, e.g., A = bh, P = 2l + 2w. Use area and perimeter formulas, e.g., If the area of a rectangle is 240 m2 and its base is 60 m long, what equation could you use to determine the height of the rectangle? Answer: Since A = bh is the formula for the area of a rectangle, then the equation is 240 = 60h or 240 = 60 × h.

• Develop understanding that a solution to an equation is a value that makes the left side equivalent to the right side, e.g., the two sides can be equivalent masses, equivalent dollar values, equivalent perimeters, etc. Students explain how they know that their solution makes the equation true.

By the end of Grade 7 students should be proficient at translating from simple word descriptions with one operation to algebraic expressions (and vice versa) and at solving equations of the forms ax = c and ax + b = c by inspection and systematic trial. Grade 8 • Create and solve simple equations that have decimal coefficients and solutions – solving by inspection

or systematic trial can be tedious when looking for a decimal solution like 2.83 – Students need this experience to see the value of learning more formal algebraic methods for equation solving.

• Begin activities that will lead to the use of formal algebraic solutions. • Use a balance and equivalent masses to establish that to preserve balance

in an equation you must add the same quantity to both sides, or take the same quantity away from both sides, or divide both sides by the same number, etc.

• Develop understanding that a first degree equation has a maximum of one solution. Students work an equation that comes from a context and a set of “test” numbers – include the solution as well as numbers above and below the solution – and explain why they think the solution is unique.

By the end of Grade 8 students should be proficient at translating from simple word descriptions with two operations to algebraic expressions (and vice versa) – example: Three more than twice a number 3 + 2n, and at solving equations of the forms ax = c and ax + b = c algebraically. Grade 9 Applied • Create and solve more complex equations – equations have integer coefficients, not fractional

coefficients – solving equations may require students to simplify algebraic expressions. • Use measurement formulas to provide a context for solving equations – students are required to

substitute into formulas and solve for one variable that is not the subject of the formula. By the end of Grade 9 Applied, students should be proficient at translating from word statements with two operations to algebraic equations (and vice versa) and at solving linear equations with whole number coefficients.

• Explain the shorthand use of 3n for 3 × n. Compare this to n + 3; what is the same? what is different?

• Use different variables to help students realize that the equations 21n = 105 and 21t = 105 have the same solution – compare 21n = 105 with 105 = 21n.

• Distinguish between expressions and equations, e.g., 2x is an expression, 2x + 3 = 9 is an equation – formulas are equations.

• Provide frequent feedback on their form as students evaluate expressions and solve equations (= sign down the left as the expression becomes simpler and simpler versus = sign down the middle between expressions as one side of an equation simplifies to just the variable).

• A statement that identifies a variable needs to be very clear – “Let n represent the marbles” does not clearly state which attribute is represented – Is it the number of marbles? The radius of one marble? The total volume of all the marbles? – include units when appropriate.

• Contrast the instructions when students are to evaluate an expression, simplify an expression, and solve an equation.

• When students have a deeper understanding of using balance to solve equations they will naturally eliminate some lines in an algebraic solution. Teaching “shortcuts” detracts from understanding. Sample solution:

2.5x4

1044x

104x37334x

734x

=

=

=+=+−

=−

• Check the solution to a linear equation and emphasize it is the only value that will make the left side equivalent to the right side.


Connections Across Strands Grade 7

Number Sense and Numeration Measurement Geometry and

Spatial Sense Patterning and Algebra Data Management and Probability

• justify the choice of method for calculations: estimation, mental computation, concrete materials, pencil and paper, algorithms (rules for calculations), or calculators, e.g., when solving equations by inspection and systematic trial

• use order of operations in solving and verifying equations

• develop skills with equations through measurement formulas

see Connections Across Grades, p. 2

• express the mean measure of central tendency as a formula (equation)

• use cell names as placeholder variables when working with spreadsheets to organize and analyse data

Grade 8



• solve and explain multi-step problems involving fractions, decimals, integers, percents, and rational numbers

• use estimation to justify or assess the reasonableness of calculations

• use computations skills involving order of operations, fractions, decimals and integers, in solving equations

• create formulas (equations) and solve problems for circumference and area of a circle, surface area and volume of a triangular prism

• construct and solve problems involving lines and angles

• create and solve angle measurement problems for triangles

• solve angle measurement problems involving properties of intersecting line segments, parallel lines, and transversals

See Connections Across Grades, p. 2 • investigate inequalities

and test whether they are true or false by substituting whole number values for the variables 9 (e.g., in 4x ≥ 18, find the whole number values for x)

• manipulate and present data using spreadsheets, and use the quantitative data to solve problems

Grade 9 Applied

Number Sense and Algebra Measurement and Geometry Analytic Geometry Relationships See Connections Across Grades, p. 2 • consolidate numerical skills

(e.g., mental mathematics, estimation, operations with integers, rationals, percents)

• manipulate polynomial expressions (e.g., expand, collect like terms)

• demonstrate facility with integers

• solve problems involving composite plane figures

• solve problems involving measurement formulas

• construct a variety of rectangles, square-based prisms and cylinders (use formulas to determine dimensions)

• illustrate and explain various geometric relationships (using equations)

• communicate the findings of investigations, using appropriate language and mathematical forms (e.g., formulas)

• use the equation of a linear relation to determine intercepts and points on the line

• determine the equation of a line, given the slope and y-intercept, the slope and a point on the line, and two points on the line

• use equations to create tables of values

• communicate the findings of an experiment clearly and concisely, using appropriate mathematical forms (e.g., written explanations, formulas, charts, tables, graphs), and justify the conclusions reached

• determine values of a linear relation by using the formula

• use equations to create tables of values

Summary or synthesis of curriculum expectations is in plain font

Verbatim curriculum expectations are in italics


Developing Mathematical Processes Grade 7 Short-answer Questions Addressing the Different Mathematical Processes

Name: Date:

Knowing Facts and Procedures There are 70 m of fencing around a rectangular field. The length of the field is 20 m. Determine the width of the field by solving this equation: 2(20) + 2(w) = 70

Show your work.

Reasoning and Proving Add any four numbers in a “box” on a calendar and I can tell you the location of the box.

1 2 3 4 5 6 7 8 9 10 11 12 13

14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Example: Tell me the sum is 60, and I know the box begins at 11. I used an equation to get the answer. What equation did I use: (a) 11n + 60 = 181 (b) 4n +16 = 60 (c) 11n = 60 Give reasons for your answer.

Communicating A local restaurant charges a $10 reservation fee and $25 per person for group parties.

Jeremiah used the equation below to determine the number of people at a graduation party that cost a total of $610. 25n + 10 = 610, where n represents the number of people First he tried n = 20. The value of the left side of the equation was 510. Next he tried n = 10.

Did Jeremiah demonstrate good reasoning in his second try? Explain your answer.

Making Connections Bargain Bs is having a $5 sale day. Buy two of the same item and get $5 taken off the total price. Spreadsheet:

Item Reg. Price ($)

Sale Price for 2 items($)

#001 $15 $25 #002 $10 $15 #003 $50 $95

A box is hiding one number. Determine an equation that could be used to find the missing number. Give reasons for your answer.

Patterning and Algebra – 7m79 Solve problems giving rise to first-degree equations with one variable by inspection or by systematic trial.



Name: Date:

Knowing Facts and Procedures There are 70.8 m of fencing around a rectangular field. The length of the field is 20.2 m. Determine the width of the field by solving this equation algebraically 2(20.2) + 2w = 70.8

Show your work.


1 2 3 4 5 6 7 8 9 10 11 12 13

14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Example: Tell me the sum is 60, and I know the box begins at 11. I used an equation to get the answer. What equation did I use: (a) 11 + 4n = 60 (b) 4n +16 = 60 Give reasons for your answer.

Communicating Farrell says that without solving the two equations given below he knows they have the same solution.

3n – 2 = 7 7 = 3a – 2

Do you agree or disagree? List reasons for your answer.

Making Connections Bargain Bs is having a sale. The regular price of every item is reduced by 20%. Spreadsheet:

Item Reg. Price ($) Sale Price for 1 item ($)

#001 $20.00 $16.00 #002 $10.00 $8.00 #003 $50 $40.60

A box is hiding one number. Determine an equation that could be used to find the missing number. Give reasons for your answer.

Patterning and Algebra – 8m88 Solve and verify first-degree equations with one variable, using various techniques involving whole numbers and decimals.



Name: Date:

Knowing Facts and Procedures Solve the given equation. −10m + 8 = −5m − 8 Show your work.


1 2 3 4 5 6 7 8 9 10 11 12 13

14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Example: Tell me the sum is 60, and I know the box begins at 11. I use an equation to get the answer. Develop the equation by letting n represent the number in the first box. Give reasons for your answer.

Communicating -3x is an algebraic expression. 2x + 7 = 12 is an algebraic equation. Make a chart to compare expressions and equations.

Making Connections Bargain Bs is having a sale. The regular price of every item is reduced by 20%. Ali bought a T-shirt on sale for $12.76 Determine the regular price of the T-shirt by using an equation. Show your work.

Number Sense and Algebra – NA3.03 Solve first-degree equations, excluding equations with fractional coefficients, using an algebraic method.

What is the same? What is different?



Name: Date:


□

State an algebraic expression for each phrase. Underline the word(s) that indicate what the variable represents. 1. Two times the number of squares _________________________________

2. One less than the number of fence posts _________________________________

3. Four more than the width of the rectangle _________________________________

4. A number increased by six _________________________________

5. Half of the distance around the figure _________________________________

6. A number increased by itself _________________________________

7. A number multiplied by itself _________________________________

8. Three times her age _________________________________

9. The number of sides for n squares _________________________________

10. The same as the term number of the pattern _________________________________

Expectation: Translate simple statements into algebraic expressions



Name: Date:


□

State an algebraic expression for each phrase. Underline the word(s) that indicate what the variable represents. 1. Two more than three times the number of squares __________________________

2. One less than the twice the number of fence posts __________________________

3. Four more than the half the width __________________________

4. The number of hours increased by six __________________________

5. Three less than half of the distance around the rectangle ____________________

6. A number increased by itself __________________________

7. Double the number of coins, increased by 3 __________________________

8. Three times her age two years ago __________________________

9. The number of sides for n squares and m triangles __________________________

10. The square root of the term number of the pattern __________________________

Expectation: Translate complex statements into algebraic expressions.


Developing Proficiency Grade 9 Applied

Name: Date:


□

State an equation that could be used to solve for the unknown. Underline the word(s) that indicate what the variable represents. 1. Three times a number is equal to 40. ______________________

2. Two less than a number is 8.5. ______________________

3. The product of 2 and a number increased by 7 is 16. ______________________

4. The area of a rectangle is 112 m2 and the length is 22 m. What is the width? ______________________

5. The circumference of a circle is 52 cm. What is the radius? ______________________

6. The perimeter of the rectangle is 40. The length of the rectangle is double the width. What is the width of the rectangle? ______________________

7. This year John had 21 days for holidays which is 3 more than double the number of days he had last year. ______________________

8. Three centimetres less than half of the distance around the rectangle is 36. What is the perimeter of the rectangle? ______________________

9. Three less than double a number is the same as the number increased by 5. ______________________

10. The perimeter of a regular hexagon is 66 cm. What is the length of one side of the hexagon? ______________________

Expectation: Solve problems, using the strategy of algebraic modelling



Name: Date:


□

Solve. Show your work. 1. 3x = 9 2. 2a – 4 = 12

4. 100 + 2p = 112 5. 15w – 25 = 50

6. 7h = 364 7. 21y + 5 = 236

Expectation: Solve equations of the form ax = c and ax + b = c by inspection and systematic trial, using whole numbers, with and without the use of a calculator.



Name: Date:


□

Solve algebraically. Show your work. 1. 25x = 15 2. 5n + 20 = 65

3. 7 + 4y = 31 4. 3c – 7 = 15.5

5. 3.2m – 1.7 = 23.9 6. 12.2a = 79.3

Expectation: Solve and verify first-degree equations with one variable, using various techniques involving whole numbers and decimals.



Name: Date:


□

Solve algebraically. Show your work. 1. 5x – 3 = 7 2. 4 – 2a = – 6

3. 10 = 3a + 2 4. 4a + 8 = 2a – 12

5. –3m + 2 = 6m – 9 6. –3(n + 2) + n = –2

Expectation: Solve first degree equations, excluding equations with fractional coefficients, using an algebraic method.



Name: Date:

Find 2 different ways to determine the solution to 35x + 52 = 647. Show your work.

1 2

Expectation: Solve equations of the form ax = c and ax + b = c by inspection and systematic trial, using whole numbers, with and without the use of a calculator.


Extend Your Thinking Grade 7 (Answers) Possible ways students could respond: 1 Establishing a lower bound first, then building up

Trial value for x Value of 35x + 52 Thinking

More than 10 by inspection

Mentally, 350 + 52 = 402 I need about ½ as much again to increase 402 to 647, and the 52 is not relatively that large, so try 15 for x.

15 35 × 15 + 52 = 577 15 is a bit too small, and 647 is more than 35 more than 577, so try 17 for x.

17 35 × 17 + 52 = 647 The solution to 35x + 52 = 647 is x = 17.

2 Establishing an upper bound first, then moving down Trial value

for x Value of 35x + 52 Thinking

Less than 20 by inspection

Mentally, 35 × 20 = 700

I need less than 20 for x, but not too much less since each time I reduce x by 1, the value of 35x + 52 will decrease by 35.

18 35 × 18 + 52 = 682 18 gives a value that is a bit too big, but only about 35 too big, so decrease x by 1.

17 35 × 17 + 52 = 647 The solution to 35x + 52 = 647 is x = 17.

3 Systematically testing the middle value between upper and lower bounds Trial value

for x Value of 35x + 52 Thinking

10 Mentally, 35 × 10 = 350 which will be too low, even after 52 more is added

I need more than 10 for x.

20 Mentally, 35 × 20 = 700 is already too big, even before adding 52

I need less than 20 for x, so x is between 10 and 20.


17 35 × 17 + 52 = 647 The solution to 35x + 52 = 647 is x = 17

4 Analysing numbers and their relationships deductively • If I know that the solution to the equation is a whole number, then I need a multiple of 35 that is 52

less than 647, or 647 – 52 = 595. • Since 595 has 5 as its units digit, the value for x must be odd (i.e., I need an odd multiple of 35 to

give 595).

• I can see that 35 divides into 595 somewhere between 10 and 20 times. ?1

59535

• I can complete the long division or try, 35 × 13, 35 × 15, 35 × 17, and 35 × 19 until I get 595. • Since 35 × 17 = 595 and 595 + 52 = 647, therefore, x = 17 is the solution to 35x + 52 = 647



Name: Date:

Find 2 different ways to determine the solution to 35x + 52 = 649. Show your work. 1 2

Expectation: Solve and verify first-degree equations with one variable, using various techniques involving whole numbers and decimals.


Extend Your Thinking Grade 8 (Answers) Possible ways students could respond:

Trial value for x Value of 35x + 52 Thinking

More than 10 by inspection

Mentally, 350 + 52 = 402

I need about ½ as much again to increase 402 to 647, and the 52 is not relatively that large, so try 15 for x.


17 35 × 17 + 52 = 647 To get 649 instead of 647, I need to add a small fraction of 35.

17.1 35 × 17.1 + 52 = 650.5 650.5 is a bit too big, so I need to reduce x by a bit. 17.05 35 × 17.05 + 52 = 648.75 648.75 is a bit too small, so x must be between 17.05

and 17.1. It depends on the context how accurate a value is needed for x. The accuracy possible for x depends on how accurate the 35, 52, and 649 are in the given equation. If more accuracy is needed and warranted, I could keep going.

17.06 35 × 17.06 + 52 = 649.1 Since 649 is between 648.75 and 649.1, x must be between 17.05 and 17.06.

17.055 35 × 17.055 + 52 = 648.925 Since 649 is between 648.925 and 649.1, x must be between 17.055 and 17.06. However, chances are this is close enough.

Undo the addition of 52. Simplify.

Undo the multiplication by 35

057.1735

59735

3559735

526495252356495235

=

=

=−=−+

=+

x

xx

xx

Simplify. (Just x is left after undoing the addition of 52 and the multiplication by 35.)

Undo the multiplication by 35. Simplify.

Undo the addition of 3552 .

057.1735

5973552

35649

3552

3552

35649

3552

35649

3552

3535

6495235

=

=

−=−+

=+

=+

=+

x

x

x

x

xx

Simplify. Convert to decimal.

1

2

3


Extend Your Thinking Grade 9 Applied

Name: Date:

Find 2 different ways to determine the solution to 35x + 52 = 649. Show your work. 1 2

Expectation: Solve first-degree equations, excluding equations with fractional coefficients, using an algebraic method.


Extend Your Thinking Grade 9 Applied (Answers) Possible ways students could respond:

Undo the addition of 52, showing the balancing of sides of the equation. Simplify.

Undo the multiplication by 35, showing the balancing of sides.

057.1735

59735

3559735

526495252356495235

=

=

=−=−+

=+

x

xx

xx

Simplify. (Just x is left after undoing the addition of 52 and the multiplication by 35.)

Undo the multiplication of 35, showing the balancing of sides of the equation.

Simplify.

Undo the addition of ,3552 showing the balancing of sides.

Simplify.

057.1735

5973552

35649

3552

3552

35649

3552

35649

3552

3535

6495235

=

=

−=−+

=+

=+

=+

x

x

x

x

xx

Convert to decimal.

Undo the addition of 52 by transposing.

Simplify.

057.1735

59759735

52649356495235

=

=

=−=

=+

x

x

xx

x

Undo the multiplication by dividing. Simplify.

1

2

3


Is This Always True? Grade 7

Name: Date:

1. Trevor says that if you multiply all the numbers in an equation by 2, the solution to the

equation will remain the same. For example: 5x – 12 = 3 has x = 3 as its solution and 2 × 5x – 2 × 12 = 2 × 3 or 10x – 24 = 6 also has x = 3 as its solution. Is this true for all equations? Explain.

2. Irene says that if you subtract fractions of the formn

nn

n 11

−−

+, you always get a numerator

of 1. Is this true for all whole numbers?

Answers 1. Yes. Multiplying each number by 2 doubles each term of the equation and preserves the balance. 2. Yes. The fractions look like:

201

201516

43

54

121

1289

32

43

61

634

21

32

=−

=−

=−

=−

=−

=−

I can see that the pattern for the final numerator is always the square of a number minus the product of the whole numbers 1 above and 1 below it. e.g., 52 – 4 × 6= 1, 62 – 5 × 7 = 1, 72 – 6 × 8 = 1. These are always 1 since we are comparing a square to the rectangle 1 unit shorter and 1 unit wider. For example,



Name: Date:

1. Trevor says that if you multiply all the numbers in an equation by 2, then the solution to the



nn

n 11

−−


of 1. Is this true for all whole numbers? 3. Kaye says that every time you move a number from one side of an equation to the other

side, you change its sign. Is she correct? Explain.


201

201516

43

54

121

1289

32

43

61

634

21

32

=−

=−

=−

=−

=−

=−

3. No. When solving 5x = 35, we get 5

35=x or x = 7. The sign on 5 did not change as we undid the

multiplication by 5 with a division by 5. It is only when undoing addition and subtraction, that the sign appears to change.


Is This Always True? Grade 9 Applied

Name: Date:

1. Trevor says that if you multiply all the numbers in an equation by 2, then the solution to the



nn

n 11

−−


of 1. Is this true for all whole numbers?

3. Kaye says that every time you move a number from one side of an equation to the other

side, you change its sign. Is she correct? Explain.

4. Myrna says that 3x = 2x + x is true for x = 1, and x = 2, and x = 3.

Is 3x = 2x + x always true?


201

201516

43

54

121

1289

32

43

61

634

21

32

=−

=−

=−

=−

=−

=−

3. No. When solving 5x = 35, we get 5

35=x or x = 7. The sign on 5 did not change as we undid the

multiplication by 5 with a division by 5. It is only when undoing addition and subtraction, that the sign appears to change.

4. Yes. Since the expressions on the left side and right side of the equation are equivalent, the values of

the left and right sides will be equivalent for any value of x. This is an example of an identity.

TIPS: Section 2 – Developing Perimeter and Area Formulas © Queen’s Printer for Ontario, 2003 Page 1

Developing Perimeter and Area Formulas

Context • Stand in any room, in any building, or on any street and you can see the context for a measurement

problem. • Students learn through posing their own measurement problems, drawing from their prior knowledge,

and making connections. • When students learn how to develop measurement formulas for the basic shapes, they are building

understanding that can be extended when they encounter a new irregular shape. • Optimization problems, such as maximizing the area enclosed by a length of fence, require the use

and development of formulas.

Context Connections

Signs/Logos/Symbols Designing Irregular Surfaces Sports Field

Packaging Frames Buildings Tracks

Fencing Sewing Fractions Other Connections

Manipulatives • tiles • cubes • 3-D models • nets • geoboards

Technology • The Geometer’s Sketchpad® • Fathom • calculators/graphing calculators • spreadsheet software

Other Resources http://standards.nctm.org/document/chapter6/meas.htm http://mmmproject.org/dp/mainframe.htm http://www.shodor.org/interactivate/activities/perm/index.html


Connections Across Grades Selected results of word search using the Ontario Curriculum Unit Planner Search Words: develop, understand, comparison, relate, investigation, pose, “surface area”, composite, irregular, formula

Grade 6 Grade 7 Grade 8 Grade 9 Grade 10 • solve problems related to the

calculation and comparison of the perimeter and the area of regular polygons

• relate dimensions of rectangles and area to factors and products, e.g., in a rectangle 2 cm by 3 cm the side lengths are factors and the area, 6 cm2, is the product of the factors

• understand the relationship between the area of a parallelogram and the area of a rectangle, between the area of a triangle and the area of a rectangle, and between the area of a triangle and the area of a parallelogram

• understand the relationship between area and lengths of sides and between perimeter and lengths of sides for squares, rectangles, triangles, and parallelograms

• develop rules for calculating the volume of rectangular prisms, generalize rules, and develop formulas, e.g., Volume = surface area of the base × height

• pose problems by recognizing a pattern, e.g., comparing the perimeters of rectangles with equal area

• solve problems related to the calculation and comparison of the perimeter and the area of irregular two-dimensional shapes

• understand that irregular two-dimensional shapes can be decomposed into simple two-dimensional shapes to find the area and perimeter

• develop the formula for finding the area of a trapezoid

• develop the formulas for finding the area of a parallelogram and the area of a triangle

• develop the formula for finding the surface area of a rectangular prism using nets

• estimate and calculate the perimeter and area of an irregular two-dimensional shape, e.g., trapezoid, hexagon

• ask questions to clarify and extend their knowledge of linear measurement, area, volume, capacity, and mass, using appropriate measurement vocabulary

• develop the formula for finding the circumference and the formula for finding the area of a circle

• develop the formula for finding the surface area of a triangular prism using nets

• estimate and calculate the radius, diameter, circumference, and area of a circle, using a formula in a problem-solving context

Academic • identify, through investigation, the

effect of varying the dimensions of a rectangular prism or cylinder on the volume or surface area of the object

• identify, through investigation, the relationships between the volume and surface area of a given rectangular prism or cylinder

• pose a problem involving the relationship between the perimeter and the area of a figure when one of the measures is fixed

• solve simple problems, using the formulas for the surface area and the volume of prisms, pyramids, cylinders, cones, and spheres (Note: Students should develop these concepts.)

Applied • solve problems involving the area

of composite plane figures, e.g., combinations of rectangles, triangles, parallelograms, trapezoids, and circles

• solve problems involving perimeter, area, surface area, volume, and capacity in applications (Note: Students should develop these concepts.)

Academic and Applied • solve simple problems, using the

formulas for the surface area of prisms and cylinders and for the volume of prisms, cylinders, cones, and spheres

Academic • determine the

properties of similar triangles, e.g., the correspondence and equality of angles, the ratio of corresponding sides, the ratio of areas, through investigation, using dynamic geometry software

Applied • determine some

properties of similar triangles through investigation, using dynamic geometry software

• construct tables of values, sketch graphs, and write equations of the form y = ax2 + b to represent quadratic functions derived from descriptions of realistic situations, e.g., vary the side length of a cube and observe the effect on the surface area of the cube

Summary of Prior Learning and Next Steps In earlier years, students: • build understanding and familiarity with measurement attributes (length, height, width, perimeter, area, etc.); • become familiar with both standard (metric) and non-standard units of measure (e.g., area of a page is four pencil cases); • estimate regular and irregular areas using grid paper (e.g., surface area of a puddle); • explain the difference between perimeter and area and understand when each measure should be used; • develop rules and formulas for perimeter and area of rectangles, squares (Grade 5). Note: Students use area formulas for triangles and parallelograms in Grade 6 before the formulas are developed in Grade 7. In Grade 7, students: • focus on developing area and perimeter formulas for triangles and some quadrilaterals (rectangle, square, trapezoid, parallelogram); • determine perimeter and area of complex shapes by decomposing into simple 2-D shapes which leads to determining the surface area of a

rectangular prism by constructing the net; • after demonstrating understanding of the grade-level formulas, should become proficient in the use of the formulas. In Grade 8, students: focus on developing formulas for the area and circumference of a circle; after demonstrating understanding of the circle, should become proficient in the use of formulas.

In Grade 9 Applied, students: • are introduced to cones, cylinders, pyramids, and spheres – surface area and volume formulas are developed and used; • use formulas developed in earlier grades as a good foundation for grade-level investigations; • after demonstrating understanding, should become proficient in the use of the formulas for surface area of prisms, pyramids, cylinders, cones,

and spheres. In Grade 10, students: proficiency is assumed – concepts become tools for investigations and building new knowledge.



Suggested Instructional Strategies Helping to Develop Understanding Grade 7 • Use non-standard units for measuring length, area, and volume – students

must think conceptually about the required measurement to measure a desk. Ask students which type of measurement they are going to make and then choose an appropriate non-standard unit.

Attribute Unit of Measurement Length length of a ruler, length of a piece of paper Area area of one face of a book, area of a sticky

note • Cut a piece of paper into several pieces – reassemble and compare the

perimeter and areas to help students understand that two different shapes can have the same area but different perimeters.

• Use transparent grid paper to estimate areas of irregular shapes, e.g., picture of a puddle – experiment with different types of grid paper to develop understanding of units of measure.

• Discuss appropriate units of measure, reasons for standard measures, e.g., cm, and selection of appropriate units depending on precision requirements, e.g., would you measure the length of a pencil and the distance to the next city with the same standard measure? Explain.

• Develop the formula for the area of a rectangle by using geoboards – collect data in charts – start with whole number dimensions then use grid paper to move to non-whole number dimensions.

• Make connections between the two common formulas for the area of a rectangle. (Area = l × w and Area = b × h) – the formula Area = b × h (base × height) connects to the formulas for the area of a triangle, a trapezoid, and a parallelogram.

• Develop the formula for the area of a parallelogram by using paper models – students can use their knowledge about rectangles to determine the area of a parallelogram, e.g., any parallelogram can “become” a rectangle therefore the area formula is the same (A = b × h).

• Use The Geometer’s Sketchpad® sketches to facilitate the development of area formulas (the pictures above can become dynamic with GSP).

• Develop the formula for the area of a triangle using the formula for the area of a parallelogram – two identical paper triangles can be arranged to form a parallelogram – since 2 triangles are used the area of one triangle is one-half of the area of a parallelogram – A = ½ (base × height).

Grade 8 • Ask questions to clarify understanding about measurement, attributes, units

of measure, and unit choices. • Develop the formulas for the circumference and area of a circle through

investigation using paper circles, string/rope, and The Geometer’s Sketchpad® – use the formulas only after students have developed them.

• Connect students’ knowledge about rectangles and circles to surface area of cylinders. Although the formula for surface area of cylinders is not part of the Grade 8 expectations, all necessary concepts are components of the Grade 8 curriculum.

• Construct nets for triangular prisms, use prior knowledge about composite 2-D shapes to determine the net area then develop a formula.

Grade 9 Applied • Develop the formulas for surface area of prisms, pyramids, cylinders, cones,

and spheres before using the formulas. • Use concrete materials – students pose problems. • Investigate relationships between perimeter and area by graphing data with

graphing calculators – students estimate and hypothesize. • Pose “what if” questions. • Support students in developing proficiency in using formulas.

• When grid paper is used to estimate areas, the area is expressed in terms of grid squares. It may be necessary to apply a scale factor to approximate the actual area represented by the drawing or diagram.

• To establish a conceptual basis for understanding perimeter and area, students can use physical models to measure the perimeter (units needed to go around) and area (units needed to cover).

• Height, length, slant height, side, and base are confusing to students, e.g., any side of a parallelogram can be called a base. Demonstrate that if you cut out the shape and placed any base on the floor, the height would be the perpendicular distance from the floor.

• Shapes can be decomposed (separated) into simple 2-D shapes to find area or perimeter. However, when finding the perimeter of the illustrated shape, students may need to be reminded not to add the interior side of the rectangle and/or the interior diameter of the semi-circle.

• For composite shapes, have students write word statements then replace the words with appropriate variables or formulas, e.g., in the previous diagram if the circle had a diameter of 4 m and the base of the rectangle was 10 m then:Perimeter = (perimeter of a semi-circle) + 2 sides + 1 side

= πd ÷ 2 + 2b + d = π(4) ÷ 2 + 2(10) + (4) …etc.

• When students can’t remember a formula encourage them to recall how they developed the formula.

• Use manipulatives, e.g., interlocking cubes, to help students understand that the volume of a rectangular prism is the area of its base times its height – students need to understand that this basic formula is true for any prism.

• For a 2-D polygon the base can be any side; however, for a 3-D prism the base is the face that ‘stacks’ to create the prism. This face determines the name of the prism. Discuss.

• Give students opportunities to progress through different representations (concrete→diagrams→symbolic) – use formulas only after students have personally developed them.


Connections Across Strands

Grade 7 Number Sense

and Numeration Measurement Geometry and Spatial Sense Patterning and Algebra Data Management and Probability

• understand fractions by connecting them to area

• use order of operations in measurement formulas

• represent perfect squares and their square roots in a variety of ways, e.g., by using blocks, grids

See Connections Across Grades, p. 2

• identify, describe, compare, and classify geometric figures

• identify, draw, and construct three-dimensional geometric figures from nets

• recognize and sketch three-dimensional figures

• build three-dimensional figures and objects from nets

• explain why two shapes are congruent, e.g., bases of prisms

• use patterns • interpret a variable as a

symbol that may be replaced by a given set of numbers

• write statements to interpret simple formulas

• translate simple statements into formulas

• systematically collect, organize, and analyse data

• use conventional symbols, titles, and labels when displaying data

• connect area to bar and circle graphs


and Numeration Measurement Geometry and Spatial Sense Patterning and Algebra Data Management and Probability

• use order of operations in measurement formulas

• use exponents in measurement formulas

• develop understanding of squares and square roots through formula for area of circle

• develop estimation skills in working with π


• identify, describe, compare, and classify geometric figures

• identify, draw, and construct nets for 3-D figures

• investigate the Pythagorean relationship using area models and diagrams

• apply the Pythagorean relationship to area problems

• construct heights (line segments) of parallelograms, trapezoids and triangles using a variety of methods including paper folding

• use patterns in algebraic terms

• identify, create, and solve simple algebraic equations [formulas]

• use the concept of variable to write equations [formulas] and algebraic expressions

• write statements to interpret simple equations [formulas]

• evaluate formulas • translate complex

statements into formulas

• systematically collect, organize, and analyse primary data

• connect area to bar and circle graphs

Grade 9 Applied Number Sense and Algebra Measurement and Geometry Analytic Geometry Relationships

• manipulate first-degree polynomial expressions to solve first-degree equations

• demonstrate facility in operations with percent, ratio and rate, and rational numbers, as necessary to support other topics of the course, e.g., analytic geometry, measurement

• use a scientific calculator to evaluate formulas

• use the Pythagorean theorem • substitute into measurement

formulas and solve for one variable, with and without the help of technology

• use algebraic modelling

• determine relationships between two variables [in measurement formulas] by collecting and analysing data

• compare the graphs of linear and non-linear [measurement] relations

• describe the connections between various representations of relations

• demonstrate an understanding that straight lines represent linear relations and curves represent non-linear relations

• construct tables of values, graphs and formulas of linear and non-linear relations

• determine values for relations by using formulas


• determine the relationship between variables in measurement formulas, e.g., the radius and area of a circle form a relationships that is non-linear

• graph relationships that are determined by measurement formulas, e.g., investigate the relationship between the height and radius of a cone if the volume remains constant

• select the equations of straight lines from a given set of equations of linear and non-linear relations, e.g., C = 2πr and A = πr2

• identify the geometric significance of m and b in the equation y = mx + b through investigation, e.g., investigate the geometric significance of 2π in C=2πr

• create tables of values, plot points, graph lines, by hand and with technology





Name: Date:

Knowing Facts and Procedures Calculate the area of the given trapezoid. Show your work.

Hint: A =(a + b)h

2

Reasoning and Proving Ryan baked a rectangular chocolate cake. His sister ate part of it. Now Ryan has to cut the rest of the cake to share equally with his brother David. Show where he should make the cut(s). Explain your answer.

Communicating The T-Square Tiling Company makes ceramic floor tiles. Note: The square tiles that are shown are the same size. AB and CD have the same length. Explain how Janet could use the formula for the area of a trapezoid to convince Meredith that the inside dark areas are the same size.

A B C D Hint: A =

(a + b)h2

Making Connections Andrea’s backyard is rectangular. Its dimensions are 15.0 m by 10.0 m. Andrea’s family is making a garden from the patio doors to the corners at the back of the yard. The patio doors are 2.0 m wide. Determine the area of the garden. Show your work.

Expectation – Measurement, 7m37: Estimate and calculate the perimeter and area of an irregular two dimensional shape, e.g., trapezoid, hexagon.



Name: Date:

Knowing Facts and Procedures Calculate the area of the given circle. Show your work.

r = 2.5 cm

Hint: A = πr2

Reasoning and Proving Westview School has a track in the playground.

You want to run 2 km every day. Determine how many times you have to go around the track. Show your work.

Communicating Janice works at Buttonique – a company that makes buttons. To determine the cost of producing a new button, she needs to know the surface area of one side of the button. Explain how Janice would determine this area.

Making Connections Pax Man figure has a radius of r units.

r

Which of the following formulas could be used to determine the perimeter of Pax Man?

a) 2πr − 14+ r + r b) 0.75πr 2

c) 34

(2πr) + r + r d) 2πr − 14

Give reasons for your answer.

Expectation – Measurement, 8m47: Estimate and calculate the radius, diameter, circumference, and area of a circle, using a formula in a problem solving context.



Name: Date:

Knowing Facts and Procedures Determine the surface area of an open-topped cylinder that has a diameter of 8.0 cm and a height of 10.0 cm. Show your work.

Hint: There is only one circular face. A= πr2 + 2πrh

Reasoning and Proving A company produces labels for cans of food. A label goes completely around the can. Each can is 12.0 cm high and has a diameter of 8.0 cm. The labels are printed side by side on a long roll of paper that is the width of 1 label. label 1 label 2 label 3 label 4 etc.

Determine the number of labels that would fit on a roll of paper that is 20.0 m long. Show your work.

Communicating Jeremy says there are three formulas for the surface area of a cylinder:

1) A= πr2 + 2πrh 2) A= 2πr2 + 2πrh 3) A= 2πrh

Explain his reasoning.

Making Connections How much cardboard would it take to make a box with dimensions 40 cm x 50 cm x 30 cm? Show your work.

Expectation – Measurement and Geometry MG2.02: Solve simple problems using the formulas for the surface area of prisms and cylinders and for the volume of prisms cylinders cones and spheres.

12 cm

8 cm



Name: Date:

Proficiency Target Met. [Practise occasionally to maintain or improve your proficiency level.] □ Still Developing [Search out extra help and practise until you are ready for another opportunity to demonstrate proficiency.]

□

1. Determine the area of each shape. Show your work.

a)

Answer: ________________ b)

Answer: ________________ 2. Determine the perimeter of each shape. Show your work.

a)

Answer: ________________ b)

Answer: ________________

Expectation: Estimate and calculate the perimeter and area of an irregular two-dimensional shape, e.g., trapezoid, hexagon

4 cm

5 cm3 cm

8 cm

6 cm2 cm

2 cm

3 cm

5 cm6 cm

6 cm

12 cm

10 cm

3 cm

2 cm



Name: Date:


□

1. Determine the area of each shape. Show your work.

a)

Answer: ________________ b)

Answer: ________________ 2. Determine the perimeter of each shape. Show your work.

a)

Answer: ________________ b)

Answer: ________________

Expectation: Estimate and calculate the radius, diameter, circumference, and area of a circle, using a formula in a problem solving context

diameter = 6.5 cm

radius = 3.3 cm

radius = 2.7 cm

diameter = 7 cm



Name: Date:


□

1. Determine the surface area of each shape. Show your work.

a)

Base: 2.0 cm × 5.0 cm Height = 3.0 cm

Answer: ________________ b)

Closed on both ends h = 12 m, r = 3

Answer: ________________

2. Determine the volume of each shape. Show your work.

a) h = 4.2 cm, r = 2.2 cm

Answer: ________________

b)

h = 23 cm, r = 10 cm Answer: ________________

Expectation: Solve simple problems using the formulas for the surface area of prisms and cylinders and for the volume of prisms cylinders cones and spheres



Name: Date:

Find 3 different ways to determine the area of:

1

2

3

Expectation: Estimate and calculate the perimeter and area of an irregular two dimensional shape, e.g., trapezoid, hexagon


Extend Your Thinking Grade 7 (Answers) Possible ways students could respond.

1

A1 = 3 × 4 = 12 cm2 A2 = 3 × 4 = 12 cm2 A3 = 3 × 14 = 42 cm2 Total area = 12 + 12 + 42 = 66 cm2

2

A1 = 3 × 7 = 21 cm2 A2 = 3 × 7 = 21 cm2 A3 = 3 × 8 = 24 cm2 Total area = 21 + 21 + 24 = 66 cm2

3

Area of entire rectangle = 7 × 14 = 98 cm2 Area A1 = 4 × 8 = 32 cm2 Required area = 98 – 32 = 66 cm2



Name: Date:

Find 2 different ways to determine the area of the rectangle ABCD:

1 2

Expectation: Apply the Pythagorean relationship to numerical problems involving area and right triangles.


Extend Your Thinking Grade 8 (Answers) Possible ways students could respond. 1. A student still operating at the concrete stage may choose to use a scale drawing to approximate the

area.

The area is approximately 10 × 4.6 = 46 units2 using a scale drawing. (Note: Although the teacher may expect a student to apply a particular piece of mathematical knowledge, in a problem-solving context, the student may find some unexpected way to solve the problem.)

2.

Using Pythagoras’ relationship.

10

100

3664

68 222

=

=

+=

+=

x

x

524h

h524

h1021ABEareaand68

21∆ABEArea

=

×=∴

××=∆××=

2units4852410ABCDArea

=

×=

3.

2units 48242 ABCD rectangle

246821ABE∆

ABCD rectangle21ABE∆

=×=∴

=××=

=


Extend Your Thinking Grade 9 Applied

Name: Date:

Find 2 different ways to fine the area of this patch of pavement.

1 2

Expectation: Calculate sides in right triangles, using the Pythagorean theorem, as required in topics throughout the course, e.g., measurement.


Extend Your Thinking Grade 9 Applied (Answers) Possible ways students could respond.

1.

Use Pythagoras’ relationship to determine that h = 4 m.

2

2

m12432

m643211

=×=

=××=

A

A

Total area = 6 + 12 = 18 m2

2.


2m64321321 =××=== AAA

Total area = A1 + A2 + A3 + = 3 × 6 = 18 m2

3.

Use Pythagoras’ relationship to determine that h = 4 m. A (rectangle) = 6 × 4 = 24 m2

A (triangle) = 2m64321

=××

A = 24 – 6 = 18 m2

4.


( )

( )

( )2m18

4921

4632121)(trapezoid

=

×=

×+=

×+= hbaA



Name: Date:

Trevor travels from A to B by walking only south or east on the streets shown. The first time he follows the route indicated by the solid lines and determines that his walk was 16 blocks long. The second time Trevor walks from A to B by following streets south or east only, he follows the route indicated by the broken lines and determines that his walk was again16 blocks long. Will his trip always be 16 blocks long?

Answer Yes. Trevor will have to travel 8 blocks east and 8 blocks south, or a total of 16 blocks, no matter in what order he chooses to do the east and south parts of the trip. This concept can be connected to finding perimeter of a step shape, where sizes of the steps are not known. Perimeter is 2 × 12 + 2 × 22 = 68 cm since the 4 vertical steps on the right add to 12 and the 4 upper horizontal lengths add to 22.

A

B

N



Name: Date:

1. Shelley says that the circle with diameter 4 has a smaller area than the square with side 4

and a larger area than the square with a diagonal 4. Is this true for any number that Shelley chooses?

2. Is it always true that the circumference of a circle is more than 6 times the radius of the

circle?

Answer 1. Yes.

2. Yes. C = 2πr and 2π =& 2 × 3.1415 =& 6.283, which is more than 6. ∴ C > 6r


Is This Always True? Grade 9 Applied

Name: Date:

The largest rectangle with a perimeter 16 cm is a square. Is it always true that the largest rectangle with a given perimeter is a square?

Answer Yes. The largest number for area can be determined in a table of values. If perimeter is 16 cm, length plus width is 8 cm.

Length

(cm) Width (cm)

Area (cm2)

1 7 7 2 6 12 3 5 15 4 4 16 5 3 15 6 2 12 7 1 7

In a graph of area vs. length, the highest point on the graph identifies the area.

The highest point on my graph occurs when x = 4. This means that the length is 4. So, the width must also be 4 and it makes a square.

The largest area is 16 2cm . This happens when the length = width = 4 cm. This is a square.

TIPS: Section 2 – Integers © Queen’s Printer for Ontario, 2004 Page 1

Integers Context • The natural numbers {1, 2, 3, … } were probably the first mathematical concepts ever considered.

Natural numbers were used for counting possessions. The set of integers became necessary for counting both possessions and losses. The plus and minus signs that are used for addition and subtraction are the same signs that are used to indicate positive and negative numbers in the set of integers {… –3, –2, –1, 0, +1, +2, +3, … }. The dual roles of these signs can be confusing to students.

• Each integer has two attributes, size and sign. Understanding of integers develops when students represent integers with a variety of models. Models have representations of both size and sign. When using coloured tiles, sign is represented by colour and size is represented by number of tiles. When using number lines, sign is represented by direction and size is represented by length (or distance). After the representations in different models are understood, contextual problems can provide opportunities for students to develop conceptual understanding of operations with integers. When using coloured tiles, addition is modelled by “putting tiles together”; subtraction by “taking tiles away.” The resultant colour and number of tiles represent the sign and size of the answer. Students can truly demonstrate their understanding when they can use more than one model to explain their thinking about a solution to a problem.

• Procedural proficiency in operations with integers will be beneficial to students as they move to the more abstract concepts of algebra.

Context Connections

Temperature Golf Scores Cheque Books Balance Sheets

Money Graphs Negative (Opposite) Scientific Notation

Molecule Charge Above/Below Sea Level Stock Prices Other Connections

Manipulatives • coloured tiles • number lines

Technology • virtual tiles • The Geometer’s Sketchpad® • calculators

Other Resources http://illuminations.nctm.org/lessonplans/6-8/videotaping/ http://mathforum.org/dr.math/faq/faq.negxne.g., html http://www.mathgoodies.com/lessons/toc_vol5.shtm


Connections Across Grades Selected results of word search using the Ontario Curriculum Unit Planner Search Words: negative, integer, rational

Grade 6 Grade 7 Grade 8 Grade 9 Grade 10 • identify real-

world applications of integers (e.g., reading below-zero temperatures)

• compare, order, and represent decimals, integers, multiples, factors, and square roots

• compare and order integers (e.g., on a number line)

• represent the addition and subtraction of integers using concrete materials, drawings, and symbols

• add integers, with and without the use of manipulatives

• compare, order, and represent fractions, decimals, integers, and square roots

• understand and apply the order of operations with brackets for integers

• demonstrate an understanding of the rules applied in the multiplication and division of integers

• solve and explain multi-step problems involving fractions, decimals, integers, percents, and rational numbers

• compare and order fractions, decimals, and integers

• discover the rules for the multiplication and division of integers through patterning (e.g., 3 × [–2] can be represented by 3 groups of 2 red disks)

• add and subtract integers, with and without the use of manipulatives

• multiply and divide integers • ask “what if” questions; pose

problems involving fractions, decimals, integers, percents, and rational numbers; and investigate solutions

Academic • demonstrate facility with critical numerical

skills, including mental mathematics, estimation, operations with integers as necessary for working with equations and analytic geometry and operations with rational numbers as necessary in analytic geometry, measurement, and equation solving

Applied • demonstrate facility in operations with integers,

as necessary to support other topics of the course (e.g., polynomials, equations, analytic geometry)

• demonstrate facility in operations with percent, ratio and rate, and rational numbers, as necessary to support other topics of the course (e.g., analytic geometry, measurement)

Academic and Applied • determine the meaning of negative exponents

and of zero as an exponent from activities involving graphing, using technology, and from activities involving patterning

• evaluate numerical expressions involving natural-number exponents with rational-number bases

• substitute into and evaluate algebraic expressions involving exponents, to support other topics of the course (e.g., measurement, analytic geometry)

The expectations do not explicitly state “negative, integer, or rational”

Summary of Prior Learning and Next Steps In earlier years, students: build understanding that addition involves joining, subtraction involves taking one group away from another, repeated addition can be

represented by multiplication, and repeated subtraction can be represented by division build understanding that division can be related to multiplication, addition and multiplication are commutative whereas subtraction and

division are not commutative explore, compare, order, and represent whole numbers, decimals, and fractions using concrete materials identify real-world applications of integers use the correct order of operations to evaluate expressions with whole numbers

In Grade 7, students: order and compare integers on a number line add or subtract two integers using concrete materials, e.g., coloured tiles, then (after doing the concrete manipulation) representing the

operation using symbols by the end of Grade 7 students will be proficient in adding integers without the use of manipulatives (i.e., using symbols)

In Grade 8, students: develop and use the rules for the multiplication and division of two integers develop proficiency in addition, subtraction, multiplication, and division of integers, with and without manipulatives develop proficiency in using order of operations with brackets in applications that involve integers

In Grade 9 Applied, students: graph in all four quadrants represent very small numbers in scientific notation, e.g., 5.2 × 10–33 apply exponent rules, e.g., 5−3 × 5−4 = 5−7 which require adding, subtracting, and multiplying integers manipulate polynomial expressions, solve equations (Note: Students in Grade 8 do not have to solve equations with integer solutions.) use data which includes integers and rationals calculate finite differences (which requires proficiency with subtracting integers)

In Grade 10, students: build understanding and proficiency to support Grade 10 expectations, e.g., polynomials, equations, graphing



Suggested Instructional Strategies Helping to

Develop Understanding

Grade 7 • Use real-world applications to develop the concept of negative numbers, e.g., thermometers, distance above and

below sea level, elevators where 0 represents the ground level. • Use a variety of models (symbols, coloured tiles, directed distances, etc.) to represent integers and ensure that

students understand how to represent a positive quantity, a negative quantity, and zero in each model. +4 −2 0

Coloured Tiles Directed Distances

Kinesthetic Movement 4 steps to the right 2 steps to the left 2 steps to the right followed

by 2 steps to the left • Take integer numbers from contexts, e.g., yards gained/lost in football, then compare, order, and represent the

integers using different models. • Present an addition problem (start with the addition of two positive integers) and ask students to find the answer

using different models. • Do all possible combinations of additions ensuring that students understand how addition is represented in each

model, e.g., when using coloured tiles the addition of −3 is represented by adding 3 red tiles to the collection of tiles. • Encourage students to look for patterns, e.g., Is anything always the same or always different when two negative

numbers are added? • Write the symbolic representation of each addition using longhand notation as concepts are being developed. • Pose questions that require subtraction then determine how to represent the subtraction with different models, start

with very familiar questions like (+5) − (+2) and progress to subtracting negatives. (+2) − (−1) = 3 Start with 2 yellow tiles.

Add a zero model then remove One red tile (to subtract −1)

• Allow use of concrete materials for addition and subtraction of integers until the concepts have been consolidated. Note: Students are not required to do subtraction without manipulatives until Grade 8.

• As students practise working with integers, look for patterns, e.g., which of the following questions have the same answer? (–2) + (−3), (−2) – (−3), (−2) − (+3), (+2) − (−3), etc. Help students build understanding that subtracting a number is equivalent to adding the opposite number.

• Use manipulatives to develop the understanding that expressions like (−3) − (+2) and (−3) + (−2) are different representations that have the same value.

• By the end of Grade 7, students need to be proficient in adding integers, e.g., (−3) + (+5), without manipulatives. Grade 8 • Help students become proficient at subtracting integers without manipulatives by building on students’ Grade 7

proficiency (in adding integers symbolically) and their understanding that subtraction is the same as addition of the opposite integer.

• Teach students to read and write shorthand notation once the concepts have been consolidated and represented using longhand notation, e.g., the shorthand notation for (−3) + (−2) is − 3 − 2 which can be read as “negative 3 plus negative 2.” The expression −3 − 2 can also be interpreted as “negative 3 subtract positive 2”, but, since this interpretation leads to the same answer, and since addition is easier to envision than subtraction, the “negative 3 plus negative 2” interpretation helps most students.

• Use manipulatives and pictorial representations to introduce multiplication of a positive times a positive and a positive times a negative. Use the commutative property of multiplication, i.e., a × b = b × a, to establish that a negative times a positive is always negative.

• Use patterning to discover the rules for multiplication (and division) of a negative by a negative.

• Teach students to recognize and write divisions both horizontally and vertically, e.g., (−14) ÷ 7 and 714− .

• Highlight for students the different ways to read and interpret “−,” depending on context, e.g., interpret −2 × (−3) as the “opposite of 2 groups of negative 3”, interpret − 3 − 4 as “negative 3 plus negative 4”.

• Students should be able to explain why each of the following has the value 12: −3 (−4), −3× (−4), (−3) × (−4), −3 × (−4). Students should be able to contrast these questions to − 3 − 4 which has the value of −7.

• Extend understanding of order of operations to examples that involve integers. • By the end of Grade 8, students need to be proficient in all 4 operations (+, −, ×, ÷) without use of manipulatives. Grade 9 Applied • Diagnostically determine if some students need further practice on integer operations before those skills are needed

in connection with polynomials, exponents, and equations and provide materials for out-of-class practice as needed. • Review integer operations briefly using the algebra tiles before combining algebraic terms. • Include exponents in order of operation questions. Emphasize that the base of a power is negative only if the

negative number is contained in brackets, e.g., (−3)2 = (−3) × (−3) = 9 whereas −32 = − (3 × 3) = −9, i.e., the negative of 32.

• Develop the meaning of zero and negative exponents through patterning activities and context, e.g., bacterial growth.

• Progress from

concrete to visual to symbolic representations of integers.

• Make notes that incorporate pictorial (visual) representations of the coloured tiles.

• Help students learn to read and interpret expressions (see examples in Grade 8 strategies).

• Use Venn diagrams to illustrate the relationship between integers and other sets of numbers.

• Provide many opportunities for students to demonstrate mastery of integer skills without calculators.

• Ensure that students know how to use the +/– key on calculators.

• Do not say “two negatives make a positive” – this causes confusion since the operation is not stated.

• Make practice fun through a variety of integer games and interactive Internet activities.

• Connect prior use of manipulatives and visual representations of integers to manipulatives and visual representations of algebraic terms in Grade 9.

• Connect prior knowledge of fractions and integers to develop understanding of rationals.


A Sampling of Connections Across Strands Grade 7



• all of the expectations on page 2 are from this strand

• explain numerical information in their own words and respond to numerical information in a variety of media

• research and report on uses of measurement instruments in projects at home, in the workplace, and in the community (Note: Thermometers register integral values.)

• describe designs in terms of images that are congruent, translated, rotated, and reflected (This provides opportunities to talk about opposites, e.g., a clockwise/ counterclockwise rotation of 90° can be associated with a rotation of positive/negative 90°.)

• recognize patterns and use them to make predictions


• systematically collect, organize, and analyse data

Grade 8




• connect area to the names of integer tiles in preparation for algebra

• describe and justify a rule in a pattern

• write an algebraic expression for the nth term of a numeric sequence

• find patterns and describe them using words and algebraic expressions


• systematically collect, organize, and analyse primary data

• understand and apply the concept of the best measure of central tendency

• apply a knowledge of probability in sports and games, weather predictions, and political polling

Grade 9 Applied

Number Sense and Algebra Measurement and Geometry Analytic Geometry Relationships


• judge the reasonableness of answers produced by a calculator, a computer, or pencil and paper, using mental mathematics and estimation

• use scientific notation • determine, from the examination of

patterns, the exponent rules for multiplying and dividing monomials and the exponent rule for the power of a power, and apply these rules in expressions involving one variable

• algebraic manipulation (including measurement and other formulas)

• use algebraic modelling as one of several problem-solving strategies in various topics of the course (e.g., relations, measurement, direct and partial variation, the Pythagorean theorem, percent)

• connect area to the names of algebra tiles

• measurement formulas

• plot points on the xy-plane and use the terminology and notation of the xy-plane correctly

• graph lines by hand, using a variety of techniques (e.g., making a table of values, using intercepts, using the slope and y-intercepts)

• write the equation of a line • identify the properties of the slopes of line segments (i.e.,

direction, positive or negative rate of change, steepness, parallelism, perpendicularity) through investigations facilitated by graphing technology, where appropriate

• determine the equation of a line, given the slope and y-intercept, the slope and a point on the line, and two points on the line

• determine, through investigation, the properties of the slope and y-intercept of a linear relation

• identify y = mx + b as a standard form for the equation of a straight line, including the special cases x = a, y = b

• determine the slope of a line segment, using the formula

runrisem =

• identify the geometric significance of m and b in the equation y = mx + b through investigation

• identify, by calculating finite differences in its table of values, whether a relation is linear or non-linear

• determine relationships between two variables by collecting and analysing data

• compare the graphs of linear and non-linear relations

• describe the connections between various representations of relations

Summary or synthesis of curriculum expectations is in plain font. Verbatim curriculum expectations are in italics.


Developing Mathematical Processes Grade 7 Short-answer Questions Addressing the Different Mathematical Processes Name: Date: Knowing Facts and Procedures You are given several counters.

Represents +1 Represents –1

Make a diagram to show a representation for the following expression:

(–5) + (+3) Diagram: Answer:

Reasoning and Proving Cards are made using the digits from 1 to 9.

Red represents a negative number.

Black represents a positive number.

… represents –2

… represents +3

What card can be taken out of the following collection of cards to give a total value of +2? Red 2 Black 4 Black 1 Red 4 Black 3 Red 6 Give reasons for your answer.

Communicating Vicrum and Bob are friends. They lend and borrow money from each other constantly. Bob decided to use integers to keep track of the money. He decided that a positive integer would represent a loan from Bob to Vicrum. A negative integer would represent a loan from Vicrum to Bob. Bob wrote the following expression to represent their transactions. (+6) + (–4) + (+2) Write a story to explain the expression that Bob recorded.

Making Connections Integer Links Golf and Country Club has a golf course rated at 72 strokes. A player scores above par, par, or below par. For example, if a player uses 76 shots his score is +4 (4 over par). A golfer shot a round of 65 on Integer Links. What integer represents his par score? Give reasons for your answer.

Number Sense and Numeration – 7m21 Represent the addition and subtraction of integers using concrete materials, drawings, and symbols

2 3


Developing Mathematical Processes Grade 8 Short-answer Questions Addressing the Different Mathematical Processes Name: Date:

Knowing Facts and Procedures You are given several counters.

Represents +1 Represents –1

Make a diagram to show a representation for the following expression:

3(–2) Diagram: Answer:

Reasoning and Proving Cards are made using the digits from 1 to 9.

Red represents a negative number.

Black represents a positive number.

… represents –2

… represents +3

Group No. of Cards Card A 1 Black 7 B 2 Black 4 C 2 Red 8 D 3 Red 3

Remove either Group A, B, C or D so that the average of all the remaining cards is +1. Give reasons for your answer.

Communicating Vicrum and Bob are friends. They lend and borrow money from each other constantly. Bob decided to use integers to keep track of the money. He decided that a positive integer would represent a loan from Bob to Vicrum. A negative integer would represent a loan from Vicrum to Bob. Bob wrote the following expression to represent their transactions. 4(–2) + 3(+4) Write a story to explain the expression that Bob recorded.

Making Connections Integer Links Golf and Country Club has a golf course rated at 72 strokes. A player scores above par, par, or below par. For example, if a player uses 76 shots his score is +4 (4 over par). A golfer scores for four rounds of golf were +3, –4, –2 and +1. What integer represents the average score for the four rounds? Give reasons for your answer.

Number Sense and Numeration – 8m5 Demonstrate an understanding of the rules applied in the multiplication and division of integers.

2 3


Developing Mathematical Processes Grade 9 Applied Short-answer Questions Addressing the Different Mathematical Processes Name: Date: Knowing Facts and Procedures

Find the slope of the line segment AB. Show your work.

Reasoning and Proving A line segment joins point A to point B. Point B has co-ordinates of (–2, 5). The slope of the line segment AB is –2. Give possible co-ordinates for point A. Give reasons for your answer.

Communicating Bob and Mahood were discussing the question shown below: ( )211− They disagreed about the sign of the answer. Explain why the sign of the answer is negative.

Making Connections Integer Links Golf and Country Club has a golf course rated at 72 strokes. A player scores above par, par, or below par. For example, if a player uses 76 shots his score is +4 (4 over par). A golfer’s scores for four rounds of golf were +3, –4, –2 and –1. What integer represents the average par score for the four rounds? Show your work.

Number Sense and Algebra – NA1.03 Demonstrates facility in operations with integers, as necessary to support other topics of the course, (e.g., polynomials, equations, analytic geometry).

A(−2, 3)

B(1, −3)



Name: Date:

Proficiency Target Met [Practise occasionally to maintain or improve your proficiency level.]

Still Developing [Search out extra help and practise until you are ready for another opportunity to demonstrate proficiency.]

1 Re-write the following integers in order from lowest to highest. –2, +5, –6, +1, 0 ___ ___ ___ ___ ___ 2. Place <, > or = between each pair of integers to make a true statement. a) –5 +5 b) –6 –3 c) –9 –10 d) +5 +3 e) 5 –6 f) +3 3 3. a) State an integer that is 3 less than 1. Answer: ______ b) State an integer that is 4 more than –5. Answer: ______

Expectation: Compare and order integers (e.g., on a number line).



Name: Date:



Add.

Expectation: Add integers with or without the use of manipulatives.

a) (–4) + (–3) = b) (+5) + (–3) =

c) (–3) + (+6) = d) (4) + (+3) =

e) (–8) + (+3) = f) (+3) + (–10) =

g) (–3) + (–3) = h) (–3) + (+2) =

i) (–5) + (+1) = j) (–10) + (+20) =



Name: Date:



a) (–2) + (–7) b) (–2) – (–7)

c) (–2) × (–7) d) (–8) – (+3)

e) 4 – (–1) f) –2 + 6

g) (–8) ÷ (–1) h) 2

6−

i) –8 × (4) j) –3 – 5

k) –3 + 5 – 2 + 1 l) 10 – (–2)(+3)

m) (2 – 3)(–5 + 3) n) (–2)(3) + (–4)(–2)

o) ⎟⎠⎞

⎜⎝⎛−−

−⎟⎠⎞

⎜⎝⎛ −

520

48

Expectations: Add and subtract integers with and without the use of manipulatives; Multiplication and division of integers; Understand and apply the order of operations with brackets for integers.



Name: Date:



1. Evaluate a) (2)4 b) (–2)3 c) 3(–2)2 2. If a = –2, b = +3 and c = –1 Evaluate a) ab

b) –3abc

c) ac + 5 d) –3 – bc

e) b3 f) (–3a)2

g) a2 – c2

Expectation: Evaluate numerical expressions involving natural-number exponents with rational bases; Substitute into and evaluate algebraic expressions involving exponents, to support other topics of the course.


Extend Your Thinking Grade 7 Name: Date: Your math teacher has asked you to help a classmate determine when the sum of two integers is a negative integer. Explore the possible types of combinations for adding two integers. Take into account the size and the sign of each integer. Summarize your conclusions in a format that will be easy for your classmates to understand.

Expectation: Represent the addition and subtraction of integers using concrete materials, drawings, and symbols.


Extend Your Thinking Grade 7 (Answers) Possible ways students could respond. In each case, students investigate the sum of two positive integers, the sum of a positive and negative integer, the sum of a negative and positive integer, and the sum of two negative integers. 1. Students most comfortable with concrete materials might choose to use integer tiles.

(Yellow represents positive and red represents negative.)

Add two positive integers e.g., 2 + 3 2 + 3 = 5 All of the tiles will always be yellow so the sum will always be positive. This is not what we want.

Add a negative to a positive integer e.g., 2 + (–3)

2 + (–3) = –1 e.g., 2 + (–1)

2 + (–1) = 1 If the size of the second integer is greater than that of the first, the sum will be negative. This is what we want.

Add a positive to a negative integer e.g., –3 + 2

–3 + 2 = –1 e.g., –2 + 3

–2 + 3 = 1 If the size of the second integer is less than that of the first, the sum will be negative. This is what we want.

Add two negative integers e.g., –2 + (–3)

–2 + (–3) = – 5 All of the tiles will always be red so the sum will always be negative. This is what we want.

Summary: When adding two integers, the sum will be negative if the integer with the larger size is negative or if both integers are negative.


2. Students comfortable with drawings might choose to use integer line arrows. Add Two Positives 2 + 3 2 + 3 = 5

Since the arrows start at zero and can only move to the right, the sum will always be positive. This is not what we want.

Add a Negative and a Positive –2 + 3 –2 + 3 = 1

The positive arrow is too large for the negative one, so the sum is positive. This is not what we want.

–3 + 2 –3 + 2 = –1 The positive arrow is not large enough to take the sum into the positive part of the line. This is what we want.

2 + (–3) 2 + (–3) = –1 The negative arrow is too large for the positive one, so the sum is negative. This is what we want.

3 + (–2) 3 + (–2) = 1 The negative arrow is not large enough to take the sum into the negative part of the line. This is not what we want.

Add Two Negatives –2 + (–3) –2 + (–3) = –5

Since the arrows start at zero and can only move to the left, the sum will always be negative. This is what we want.

Summary:

Add + –

+ always +

could be– or +

– could be+ or –

always –

3. Some kinesthetic learners might choose to walk a number line on the floor.

Here walking to the right indicates a positive integer and walking to the left represents a negative integer. The diagrams in 2 show how the student would walk. The summary is the same.

0

2 3

5

–2 0 1

–3 –1 0

2 0 –1

3 0 1

0 –2 –5


4. Some students may prefer to use The Geometer’s Sketchpad®. Choose Open under File on the Menu

bar. Select Program Files\Sketchpad\Samples\Teaching Mathematics\Add_Integers. 5. Some students may quote memorized rules and apply them without understanding. This is not

recommended. positive + positive = positive and add the sizes positive + negative = sign of the integer with larger size and subtract the sizes negative + negative = negative and add the sizes

Summary: When I apply the rules, I find that the sum of two integers is a negative when you add two negative integers or you add a positive integer and negative integer where the size of the negative integer is larger.

6. A student may choose to experiment with a scientific calculator. They could track their investigation on a chart:

1st integer 2nd integer result

–2 –5 –

3 21 +

–1 –8 – Their investigation and summary would look similar to one of the ones above.

(Note: Different brands of calculators have different conventions and keys for entering negative numbers.)


Extend Your Thinking Grade 8 Name: Date: Janet says that two negatives make a positive. Investigate her claim with respect to the four basic operations with integers and summarize your conclusions.

Adding Integers Subtracting Integers

Multiplying Integers Dividing Integers

Expectation: Demonstrate an understanding of the rules applied in the multiplication and division of integers.


zero

remove two groups of –3

zero

remove three groups of –2

Extend Your Thinking Grade 8 (Answers) Possible ways students could respond. Students may not have made the connection that the sign rules for multiplication and division of integers do not apply to addition and subtraction. 1. Students most comfortable with concrete materials might choose to use integer tiles.

(Yellow represents positive and red represents negative.)

Adding Integers e.g., –2 + (–3) –2 + (–3) = –5 All of the tiles will always be red so the sum will always be negative. So Janet’s statement is inaccurate for adding integers.

Subtracting Integers e.g., –2 – (–3) –2 – (–3) = 1 This example supports Janet’s statement. e.g., –3 – (–2) –3 – (–2) = –1 This example refutes Janet’s statement. Janet’s statement is not always true for subtracting integers.

Multiplying Integers e.g., –3 × (–2) –3 × (–2) = 6 This example supports Janet’s statement. e.g., –2 × (–3) –2 × (–3) = 6 This example supports Janet’s statement. Janet’s statement is true for multiplying integers.

Dividing Integers e.g., –6 ÷ (–2) Separate into groups of –2 There are three groups. –6 ÷ (–2) = 3 This example supports Janet’s statement.

Summary: The statement “A negative and a negative make a positive” is only consistently true for multiplication and division of integers. It is never true for addition and is only sometimes true for subtraction.


2. Students comfortable with drawings, might choose to use integer line arrows. Adding Integers e.g., –2 + (–3) –2 + (–3) = – 5 Since both arrows are left-pointing, the sum will always end left of zero, indicating a negative sum. Janet’s statement is not accurate for adding integers.

Subtracting Integers Subtraction is the opposite to addition, we can model subtraction by pointing the subtracting integer’s arrow in the opposite direction. e.g., –2 – (–3) –2 – (–3) = 1 This example supports Janet’s statement. e.g., –3 – (–2) –3 – (–2) = –1 This example refutes Janet’s statement. Janet’s statement is not always accurate for subtracting integers.

Multiplying Integers e.g., –2 × (–3) –2 is the opposite of 2. We can think of –2 × (–3) as the opposite of 2 × (–3). 2 × (–3) = –6 The opposite of –6 is 6. Therefore, –2 × (–3) = 6 Since the arrows are both left-pointing, the product will always be negative, Therefore, the opposite will always be positive. This example supports Janet’s statement.

Dividing Integers e.g., –6 ÷ (–2) Find the number of groups of –2 there are in –6. Three –2 arrows are equivalent to one –6 arrow. –6 ÷ (–2) = 3 This example supports Janet’s statement.

Summary: Janet’s statement is only accurate for multiplication and division of integers. It is not accurate for addition and is only accurate for subtraction when the subtracting integer has a larger size.

–2

subtracting –3

0 1

3 –1 0

–5 –2 0

–6 –3 –0 –6 –4 –2 0


3. Some students may quote memorized rules and apply them without understanding. This is not

recommended. Addition: negative + negative = negative and add the sizes (refutes) Subtraction: “add the opposite”

negative – negative = negative + positive = sign of the integer with larger size and subtract the sizes (inconclusive)

Multiplication: the product of two integers with the same sign is a positive integer (supports) Division: the quotient of two integers with the same sign is a positive integer (supports)

4. Some students may prefer to use The Geometer’s Sketchpad®. Choose Open under File on the Menu

bar. Select Program Files\Sketchpad\Samples\Teaching Mathematics\Add_Integers. 5. A student may choose to experiment with a scientific calculator.

Their investigation and summary would look similar to one of the ones above. (Note: Different brands of calculators have different conventions or keys for entering negative numbers.)


Extend Your Thinking Grade 9 Applied Name: Date: A classmate does not understand why –32 and (–3)2 do not have the same value, but –23 and (–2)3 do have the same value. Use diagrams and/or symbols to explain what to look for when evaluating powers.

Expectation: Demonstrate facility in operations with integers as necessary to support other topics of the course.


Extend Your Thinking Grade 9 Applied (Answers) Possible ways students could respond. There are two concepts that the student will have to convey. The first concept involves notation. If an exponent is placed to the right of parentheses, as in (–3)2, the contents of the parentheses, in this case –3, are consider the base of the power. In the absence of parentheses, as is –32, the notation is indicating –1 × 32 or the opposite of 32. The second concept involves even and odd exponents in powers whose bases are negative. 1. Notation: –32 and (–3)2

–32 means –1 × 32. Since order of operations requires exponentiation prior to multiplication (BEDMAS), –1 × 32 means –1 × 3 × 3. (–3)2 means (–3) × (–3).

–32 = –1 × 3 × 3 This is the opposite to the result of three groups of 3. The opposite of 9 is –9. Therefore, –32 = –9

(–3)2 = (–3) × (–3) –3 is the opposite of 3. We can think of (–3) × (–3) as the opposite of 3 × (–3). 3 × (–3) 3 × (–3) = –9 The opposite of –9 is 9. (–3) × (–3) = 9 So, considering these examples, –32 and (–3)2 have opposite values.

0 3 6 9

–9 –6 –3 0


–23 = –1 × 2 × 2 × 2 This is the opposite to the result of two groups of two groups of 2. The opposite of 8 is –8. Therefore, –23 = –8 (–2)3 = (–2) × (–2) × (–2) First consider (–2) × (–2) Similar to case 2 above, (–2) × (–2) = 4 So, (–2) × (–2) × (–2) = (–2) × 4 –2 is the opposite of 2. We can think of (–2) × 4 as the opposite of 2 × 4. 2 × 4 = 8 The opposite of 8 is –8. Therefore, (–2)3 = –8

So, considering these examples, –23 and (–2)3 have the same value. Why is (–2)3 negative while (–3)2 is positive? Consider the following table.

(–1)1 (–1)2 (–1)3 (–1)4 (–1)5

–1 (–1) × (–1) (–1) × (–1)

× (–1) (–1) × (–1)

× (–1) × (–1)

(–1) × (–1) × (–1) × (–1)

× (–1)

opposite of –1 opposite of (–1) × (–1)

opposite of (–1) × (–1)

× (–1)

opposite of (–1) × (–1)

× (–1) × (–1)

1

opposite of 1

opposite of –1 opposite of 1

–1 1 –1

A pattern emerges. If the base is negative and the exponent odd, the power is negative. If the base is negative and the exponent is even, the power is even. Note: If the base is positive, it doesn’t matter whether the exponent is odd or even. The power is positive. So, when evaluating powers look for the following clues: • if an exponent is to the right of parentheses, the base is the number enclosed in the parentheses,

e.g., in (–3)2, –3 is the base; in (3)2, 3 is the base

• if the base is positive, regardless of the sign of the exponent, the power is positive, e.g., 32 = 9; 33 = 27.

• if the base is negative, the power is positive if the exponent is even, e.g., (–3)2 = 9, and negative if the exponent is odd, e.g., (–3)2 = –27.

• if there is a negative preceding a power, the result will be the opposite of the power, e.g., –32 = –9; –(–3)3 = 27.

(one set of two groups of 2) (second set of two groups of 2)

0 2 4 6 8


2. A discussion about notation similar to the beginning of answer 1 would appear here.

Rules for multiplication of two integers: • if the signs of both integers are the same, the product is positive.

• if the signs of the integers are opposite, the product is negative.

–23 = –1 × 2 × 2 × 2 A negative times a positive is a negative: This negative times a second positive is a negative. This negative times a third positive is a negative. So, –23 = –8 (–2)3 = (–2) × (–2) × (–2) A negative times a negative is a positive. This positive times a negative is a negative. Considering these examples, –23 and (–2)3 have the same value.

A development of a pattern for negative bases and even and odd exponents similar to answer 1 would appear here. The points to look for would also be similar.

3. A student might choose to use a file like IntegerOnlyMultiplication.gsp from The Geometer’s

Sketchpad® to carry out a development similar to answer 2. The student would not have to have memorized the integer rules for multiplication but would still be able to draw appropriate conclusions.

4. Care should be taken when using a scientific calculator to carry out an investigation similar to answer

1 or 2. Some calculators interpret the key stokes − , 3 , , 2 to mean (–3)2. Others do not. If the calculator is of the second kind, then a development similar to question 2 could be carried out. During the process of the investigation, the exponent key, yx or xy or would be used.

–32 = –1 × 3 × 3 A negative times a positive is a negative: This negative times a second positive is a negative. So, –32 = –9 (–3)2 = (–3) × (–3) A negative times a negative is a positive. So, (–3)2 = 9 Considering these examples, –32 and (–3)2 have opposite values.


Is This Always True? Grade 7 Name: Date: 1. Patti says that if you add the same negative integer five times, the sign of the sum is

negative and its size is five times the size of the original integer. For example:

–7 + (–7) + (–7) + (–7) + (–7) = –(5 × 7) [7 is the size of –7] = –35

Is Patti’s statement true for all negative integers?

2. Dina says that if you add a negative integer to a positive integer, the sum is zero or less. Is Dina’s statement true for all integers? 3. Bruce says that if you add two opposite integers, the sum is zero. Is Bruce’s statement true for all integers?

Answers 1. Yes. Adding several negative integers takes the sum farther from zero on the negative side of the

number line, causing its sign to be negative. And as we know from multiplication of whole numbers, adding a number five times is equivalent to multiplying the number by 5. Using integer tiles, 5 × (–7) is represented by five groups of negative 7 and results in 35 negative tiles which represent the product –35.

2. No. Counter example: 8 + (–5) = 3. 3. Yes. Opposite integers are the same distance from zero on the number line but on opposite sides.

When added they cancel each other out leaving the sum at zero. Using integer tiles, adding opposite integers is modelled by placing together the same number of positive tiles as negative tiles. This results in a model of 0.


Is This Always True? Grade 8 Name: Date: 1. Bert says that to subtract one integer from another, simply add to the first number the

opposite of the one you are subtracting. For example: 9 – 5 = 9 + (–5) = 4 Is Bert’s statement true for all integers? 2. Emmanuelle says that if you divide two negative integers, the quotient is a positive integer. Is Emmanuelle’s statement true for all negative integers? 3. Simon says that, since finding a square root undoes the squaring of a number, the square

root of 16 is 4; the square root of 25 is 5, and the square root of 100 is 10. Is Simon’s statement true for all perfect squares?

Answers 1. Yes. Addition is the opposite operation to subtraction (represented on the number line by motion in

the opposite direction). To compensate for motion in the opposite direction, the opposite integer must be added to get to the same spot on the number line that would have been reached if subtraction had been used. Using integer tiles, interpreting 9 – 5 as positive 9 subtract positive 5 looks like:

and leaves an answer of positive 4. Interpreting 9 – 5 as positive 9 add negative 5 looks like: and also results in an answer of positive 4 using the zero principle for the five pairs of positive/

negative tiles. Illustrate as many examples using both positive and negative integers as are needed to convince students that “To subtract an integer, you can, instead, add the opposite integer.” Since addition is easier to do mentally, it is recommended that all strings of integers be interpreted as addition questions, e.g., 3 – 8 + 1 can be interpreted as positive 3 add negative 8, add positive 1.


2. Yes. There are several ways to reason this:

i) A negative integer divided by a negative integer is the opposite of a positive integer divided by a negative integer. Since a positive integer divided by a negative integer is a negative integer, its opposite is a positive integer.

ii) Using an example to look at it another way, you could think about –10 ÷ (–2). You would think

“How many groups of –2 are there in –10?” Picturing the integer line, you would see five left pointing 2-unit arrows lined up starting at zero and ending at –10. So the quotient has to be positive five (the five arrows). This would hold true for all negative integers divided by a negative integer.

iii) A multiplication statement that corresponds to (–10) ÷ (–2) is (n) × (–2) = –10. Since we know

that we need a positive to multiply by the negative to yield an answer of negative 10, n must be positive 5.

3. No, not necessarily. Since 42 = 16 and (–4)2 = 16, the square root of 16 could be 4 or –4. There are

two square roots for 16. The symbol 16 means “the positive square root of 16”. Therefore, 16 = 4 and – 16 = –4.


Is This Always True? Grade 9 Applied Name: Date: 1. Wen-Hao says that if the first differences for a linear relation are all the same, that difference

is the slope of the line. Is Wen-Hao’s statement true? 2. Michelle says that any positive or negative integer, raised to the exponent zero equals one. Is Michelle’s statement true? 3. Bart says that if you multiply a number plus n by the same number minus n, the product will

be the square of the number minus the square of n.

For example: Following Bart’s statement, (6 + 3) × (6 – 3) would equal 62 – 32 = 36 – 9 = 27 We know that (6 + 3) × (6 – 3) = 9 × 3 = 27 using proper order of operations. Therefore, the pattern worked in this case. Is Bart’s statement true for all numbers and for all n?

Answers 1. No, not necessarily.

x y = 2x + 3 first differences

3 9

6 15 6

9 21 6

12 27 6

If the increment in x values is 1 however, then a constant first difference would be the same as the slope.

Slope for y = 2x + 3 is 2.


2. Yes. Using power rules, n

n

aa

= an – n = a0. Also, n

n

aa

= 1 since any number, except zero, divided by

itself is 1. Since the question n

n

aa

is the same, the answers must be equal. Therefore, 10 =a . Since a

can represent any integer, except zero, (any integer, except 0)0 = 1. This could also be investigated by patterning.

Power Standard

34 81

33 27

32 9

31 3

30 x

The only number that x could be in order to maintain the pattern is 1. So, 30 = 1. The same logic would apply to any integer, except zero, as the base.

3. Yes. Let x represent the number. Therefore, (x + n)(x – n) = x2 – n2

x – n

x + n

(x + n)(x – n) x – n

x n

x – n

n

x

[(x – n) + n = x]

x

x

x × x = x2

x – n

n × n = n2

[x – (x – n) = n]

÷ 3

÷ 3

÷ 3

÷ 3

TIPS: Section 2 – Fractions © Queen’s Printer for Ontario, 2004 Page 1

Fractions Context • Fractions have a variety of meanings. The fraction 5

2 can be interpreted as 2 parts of a whole that has been divided into 5 equal parts (part of a whole). This fraction also expresses 2 parts of a group of 5 (part of a set) where the elements of the set are not necessarily identical, e.g., 2 out of 5 books on a shelf. As ratios or rates, fractions are used for comparisons.

• A fraction can represent a division or a measurement. • Fractions are used daily in construction, cooking, sewing, investments, time, sports, etc. Since many

occupations require workers to think about and use fractions in many different ways, it is important to develop a good understanding of fractions. Using manipulatives and posing higher level thinking questions helps build understanding of what fractions represent.

• Understanding builds when students are challenged to use a variety of representations for the same fraction (or operation) and when students connect fractions to ratios, rates of change, percents, or decimals.

Context Connections

Cooking Music Sharing Retail/Shopping

Interest Rates Measurement Slope Time

bhA 21=

Circle Graphs Construction Formulas Other Connections

Manipulatives Technology • cubes • rulers • colour tiles • pattern blocks • geoboards • tangrams • coloured rods • grid paper • base 10 blocks

• Spreadsheet software • The Geometer’s Sketchpad® • calculators • graphing calculators • word processing software

Other Resources http://www.uwinnipeg.ca/~jameis/New%20Pages/MYR21.html http://www.standards.nctm.org/document/chapter6/numb.htm http://math.rice.edu/~lanius/proportions/rate9.html http://mmmproject.org/number.htm http://matti.usu.edu/nlvm/nav/category_g_2_t_1.html


Connections Across Grades Selected results of word search using the Ontario Curriculum Unit Planner Search Words: rational, fraction, ratio, rate, denominator, numerator, multiple, factor

Grade 6 Grade 7 Grade 8 Grade 9 Grade 10 • compare and order, and represent the

relationship between, fractions with unlike denominators using concrete materials and drawings

• select and perform computation techniques appropriate to specific problems involving unlike denominators in fractions … and determine whether the results are reasonable

• justify and verify the method chosen for calculations … fractions …

• order fractions … on any number line • explain processes and solutions with fractions

… using mathematical language • relate fractions to … rates, and ratios using

concrete materials, drawings, and symbols • demonstrate an understanding of ratio • compare and order mixed numbers and

improper fractions with unlike denominators using concrete materials, drawings, and symbols

• use skip counting to assist in solving questions about factors and denominators

• explain their thinking when solving problems involving … fractions …

• solve simple rate and ratio problems • understand and explain the characteristics of

multiples and factors and of composite and prime numbers

• identify and describe the characteristics of multiples and factors, and composite and prime numbers, to 100

• use skip counting to assist in solving questions about factors and denominators

• understand and explain operations with fractions using manipulatives

• solve and explain multi-step problems involving simple fractions …

• demonstrate an understanding of operations with fractions using manipulatives

• add and subtract fractions with simple denominators using concrete materials, drawings, and symbols

• relate the repeated addition of fractions with simple denominators to the multiplication of a fraction by a whole number

• ask “what if” questions; pose problems involving simple fractions …and investigate solutions

• solve problems involving fractions …

• compare, order, and represent… multiples, factors, …

• generate multiples and factors of given numbers

• compare, order, and represent fractions, …

• demonstrate proficiency in operations with fractions

• solve and explain multi-step problems involving fractions … and rational numbers

• demonstrate an understanding of operations with fractions

• add, subtract, multiply, and divide simple fractions

• understand the order of operations with brackets and exponents and apply the order of operations in evaluating expressions that involve fractions

• apply the order of operations (up to three operations) in evaluating expressions that involve fractions

• ask “what if” questions; pose problems involving fractions … and rational numbers; and investigate solutions

• represent composite numbers as products of prime factors

Applied • demonstrate

facility in operations with percent, ratio and rate, and rational numbers, as necessary to support other topics of the course

Academic and Applied • evaluate

numerical expressions involving natural-number exponents with rational-number bases

Applied • solve

problems involving percent, ratio, rate, and proportion by a variety of methods and models

• determine the measures of the sides and angles in right triangles, using the primary trigonometric ratios

• solve first-degree equations in one variable, including those with fractional coefficients, using an algebraic method

Summary of Prior Learning and Next Steps In earlier years, students: explore and build understanding of multiples and factors of composite and prime numbers develop an understanding of how fractions of unlike denominators can be compared using concrete materials, drawings, and symbols explore, compare, order, and represent fractions, decimals, percents, rates, and ratios using concrete materials compare and order mixed numbers and improper fractions with unlike denominators using concrete materials, drawings, and symbols

In Grade 7, students: continue to use concrete materials and drawings to build understanding of fractions use concrete materials and drawings to demonstrate understanding of addition, subtraction, multiplication, and division with fractions add and subtract fractions (including mixed and improper) using symbols develop understanding of what is meant by a fraction in its lowest terms by dividing the numerator and denominator by the same factors

In Grade 8, students: develop proficiency in comparing, ordering, and representing fractions using concrete materials, drawings, and symbols develop proficiency in operations with fractions with and without concrete materials

In Grade 9 Applied, students: develop proficiency with operations involving rationals and in using rationals to represent the slope of a linear function use rationals in context in various problems involving analytic geometry, relationships, and measurement use rationals as bases when evaluating expressions involving exponents

In Grade 10, students: use ratios in problem solving when working with similar triangles, trigonometry, and direct and partial variation (related to linear functions) develop proficiency in solving first-degree equations involving fractional coefficients


2 groups of one-half



Grade 7 • Students need to continue to build understanding about illustrating, comparing, and ordering fractions, e.g.

use zero, one-half and one as benchmarks. Ask students to state which benchmark a fraction is closer to, and to name a fraction that is even closer to the benchmark.

• Continue to build a variety of comparison techniques through patterning, e.g., 53 is less than 5

4 because

there are fewer of the same size parts, whereas 53 is larger than 8

3 because although there are the same

number of parts, one part in the first number is larger than one part in the second number – notice that common denominators or common numerators are helpful when comparing fractions.

• Encourage students to explain their thinking and ensure that they understand concepts before using rules. Students may demonstrate proficiency quickly but if they have not developed understanding, the observed mastery may disappear.

• Give students many and varied opportunities to build understanding of the addition and subtraction of fractions using concrete materials and drawings before moving to symbols. Pose questions like, How may different ways can you express five-quarters as the sum or difference of two fractions? Ask students to explain why each addition or subtraction represents the same fraction.

• Use a visual representation to help students see a variety of solutions.

sample solutions 89

81

21

47

41

43

21

45 1 +=−=+=+=

• Connect the multiplication of fractions to prior knowledge about whole numbers, e.g. If each person receives 2 pieces of pizza, how many pieces were distributed? If each person receives one-sixth of a pizza how much pizza was distributed? Each of these problems can be modelled with counters, in the first case one counter represents 1 piece of pizza whereas in the second case it represents one-sixth of a pizza. Have students discuss the different interpretation of 6 counters in each case.

• Build on their prior knowledge that a repeated addition can be expressed as a multiplication. • Encourage understanding of division by considering division problems as grouping, as repeated subtraction,

and as sharing problems:

• Use a variety of types of concrete materials. Curriculum expectations do not require students to do division or multiplication of fractions without concrete materials until Grade 8. However, students should be proficient at symbolic addition and subtraction of fractions by the end of Grade 7.

Grade 8 • Have students explain their solutions using multiple representations – both concrete and symbolic. • Only introduce rules like the “invert and multiply” division rule when the operation makes sense to students

and they have had opportunities to explain division of fractions in their own words. • Include use of alternate rules:

e.g., 25

410

84

810

21

45 ==÷=÷

By finding a common denominator, this question becomes ten (eighths) divided by 4 (eighths) which has the same numeric solution as 10 metres divided by 4 metres or simply 10 ÷ 4.

• Ensure that students are able to interpret fractional answers in context, e.g., If I had a half-cup measure, I could fill it two and a half times and I would have one and a quarter cups of flour.

• When students demonstrate understanding of the four operations with fractions, have them apply the operations to order of operations questions involving brackets, using a maximum of three operations.

• Use fractions as possible values to substitute into algebraic expressions or formulas. • By the end of Grade 8, students should have a thorough understanding of fractions. Grade 9 Applied • Extend students working knowledge of positive fractions to negative fractions, e.g., negative slopes. • Introduce the new terminology “rational number,” i.e., any number that can be expressed as a positive or

negative fraction and written as a terminating or repeating decimal number. • Use rate triangles to determine slopes. • Use rational numbers as possible values to substitute into algebraic expressions or formulas.

e.g., 21

45 ÷

“How many groups (or sets) of one-half are there in five-quarters?”

“How many times can you subtract 21 from 4

5 ?”

The drawing shows that there are 212 groups (of one-half).

• To determine equivalent

fractions or find common denominators, students must understand the concepts of factors and multiples. They need to understand that multiplying or dividing numerator and denominator by the same quantity is the same as multiplying or dividing by 1.

• Keep denominators simple so students can easily make drawings, use concrete materials, explain their solutions, and build understanding.

• When expressing a fraction in simplest form or in lowest terms, it is not necessary to change an improper fraction into a mixed fraction. Students

need to understand that 45 is

between 1 and 2 and can be represented visually as shown to the left, or shown on a number line at 4

11 .

Improper fractions rather than mixed fractions are used to express slopes of lines.

• If asked to place a number of fractions with different denominators on a number line, students may need help deciding on the number of divisions to make between units.

• Count fractions aloud on a number line, e.g.,

68

67

66

65

64

63

62

61 ,,,,,,, .

Then, reduce written versions to 3

467

65

32

21

31

61 ,,1,,,,, .


Connections Across Strands


and Numeration Measurement Geometry and Spatial Sense

Patterning and Algebra

Data Management and Probability

• all expectations from page 2 are from this strand

• generate multiples and factors of given numbers

• solve problems that involve converting between fractions, decimals, and percents

• justify the choice of method for calculations: estimation, mental computation, concrete materials, pencil and paper, algorithms (rules for calculations), or calculators

• identify relationships between and among measurement concepts

• develop fraction skills by working with 2-D figures and their areas

• make increasingly more informed and accurate measurement estimations based on an understanding of formulas and the results of investigations

• develop fraction skills by working with problems involving 3-D models

• describe designs in terms of images that are congruent, translated, rotated, and reflected.

• understand and describe patterns which involve simple fractions

• identify the favourable outcomes among the total number of possible outcomes and state the associated probability (e.g., of getting a heads in a fair coin toss)

• display data and read data on bar graphs, pictographs, and circle graphs


and Numeration Measurement Geometry and Spatial Sense Patterning and Algebra Data Management

and Probability

• all expectations from page 2 are from this strand

• solve problems that involve converting between fractions, decimals, percents, unit rates, and ratios

• apply percents in solving problems involving discounts, sales tax, commission, and simple interest

• use a calculator for questions that are beyond the proficiency expectations for operations using pencil and paper

• recognize that there is a constant relationship between the radius, diameter, and circumference of a circle, and approximate its value through investigation

• understanding and apply accurate measurement and estimation strategies

• use measurement formulas • explain the relationships

between various units of measurement

• construct and solve problems involving lines and angles

• construct line segments and angles using a variety of methods (e.g., paper folding, ruler and compass)

• describe patterns and represent symbolically


• investigate inequalities and test whether they are true or false by substituting whole number values for the variables

• use fractions to express the probability of an event

• explain the choice of intervals used in constructing bar graphs or the choice of symbols in pictographs

• construct and interpret line graphs, comparative bar graphs, circle graphs, and histograms

• identify 0 to 1 as a range from “never happens” (impossibility) to “always happens” (certainty) when investigating probability

Grade 9 Applied Number Sense and Algebra Measurement

and Geometry Analytic Geometry Relationships


• mental mathematics and estimation • use a scientific calculator effectively

for applications that arise throughout the course

• substitute into measurement formulas and solve for one variable, with and without the help of technology

• use algebraic modelling as one of several problem-solving strategies in various topics of the course (e.g., relations, measurement, direct and partial variation, the Pythagorean theorem, percent)

• solve problems involving the area of composite plane figures (e.g., combinations of rectangles, triangles, parallelograms, trapezoids, and circles)

• solve problems involving perimeter, area, surface area, volume, and capacity in applications

• identify practical situations illustrating slope (e.g., ramps, slides, staircases) and calculate the slopes of the inclines

• determine the slope of a line segment, using the formula m = rise/run

• identify the geometric significance of m and b in the equation y = mx + b through investigation

• identify the properties of the slopes of line segments (i.e., direction, positive or negative rate of change, steepness, parallelism, perpendicularity) through investigations facilitated by graphing technology, where appropriate

• use slope to graph lines and to determine the equation of a line

• demonstrate an understanding of some principles of sampling and surveying… and apply the principles in designing and carrying out experiments to investigate the relationships between variables …

• organize and analyse data, using appropriate techniques … ratios

• determine values of a linear relation by using the formula of the relation …

• calculate finite differences • plot points • describe the effect on the graph

and the formula of a relation of varying the conditions of a situation they represent …




Developing Mathematical Processes Grade 7 Short-answer Questions Addressing the Different Mathematical Processes Name: Date: Knowing Facts and Procedures Use the circles to illustrate the following addition.

65

21

31 =+

Reasoning and Proving Three friends, Ahmed, Anya and Eric, have a lemonade stand. They decided to share any profits in the following way.

Ahmed will get 32 of the total profits

Anya will get 41 of the total profits

Eric will get 61 of the total profits

What is the error the children have made? Give reasons for your answer.

Communicating Marie was helping Pat with his homework. He wrote the following incorrect solution.

52

31

21 =+

Mario showed Manuel the correct procedure

65

62

63

31

21 =+=+

Manuel then said to Mario, “I can do the questions now but I have no idea WHY I need a common denominator when adding fractions. Explain to Manuel why a common denominator must be used to add fractions.

Making Connections Vesna purchased a dozen eggs. She used 8 of the eggs. What fraction of the eggs did she use? Illustrate your answer with a diagram.

Number Sense and Numeration – 7m17 Demonstrate an understanding of operations with fractions using manipulatives.

31 =

21 =

= 65


Developing Mathematical Processes Grade 8 Short-answer Questions Addressing the Different Mathematical Processes Name: Date: Knowing Facts and Procedures Complete the diagram to illustrate:

121

41

31 =×


Two pizzas are cut as shown above. The pizzas are to be shared equally among three friends. Add more cuts to each pizza to insure that • all pieces are the same size and • the pizzas can be shared equally. What fraction of the pizzas does each person get? Give reasons for your answer.

Communicating Jason incorrectly thinks that 6 divided by one- half is 3.

Explain in words why 12216 =÷ instead of 3.

Illustrate your answer with a diagram.

Making Connections Sanjay’s father went on a short trip in his truck. The 60-litre gas tank was full when he started his trip.

He used about 54 of the gasoline during the

trip. How many litres of gas does he have left in his tank? Show your work.

Number Sense and Numeration – 8m15 Demonstrate an understanding of operations with fractions.


Developing Mathematical Processes Grade 9 Applied Short-answer Questions Addressing the Different Mathematical Processes Name: Date:

Knowing Facts and Procedures The slope of two ramps

are 1613 and 8

7 .

Which slope is largest? Show your work.

Reasoning and Proving Ahmed, Anja and Eric invested in a business. They each invested a different amount of money. Ahmed invested $30 000 Anja invested $20 000 Eric invested $10 000 The first year the business had a profit of $180 000.What should each person’s share of the profits be? Give reasons for your answer.

Communicating In shop classes students are often required to find one half of a fraction. Explain why:

a) 52

54

21 =of

b) 165

85

21 =of .

Making Connections In shop class, you are asked to cut out a round table top with a

diameter of 4334

inches. In order to draw the shape on a piece of plywood, you have to calculate the radius of the table top. What is the radius expressed as a fraction? Show your work.

Number Sense and Algebra – NA1.03 Demonstrate a facility in operations with percent, ratio and rate, and rational numbers as necessary to support other topics in the course.

$

Hint: What fraction of the total investment was made by Ahmed?

Hint: a diagram may help explain your answer



Name: Date:



1. Add and subtract. Express your answers in lowest terms.

a) 81

83+

b) 81

43−

c) 65

31+

d) 32

27−

e) 813

835 +

Expectation: Add and subtract fractions with simple denominators using concrete materials, drawings and symbols.



Name: Date:



Add, Subtract, Multiply, and Divide as indicated. Put your answers in lowest terms

a) 81

43× b)

34

94÷

c) 21 32 of d)

32

94+

e) 21

32− f)

322

433 −

g) 218

751 ÷ h)

319 ÷

i) 325

53of j)

521

513 −

Expectation: Add, subtract, multiply, and divide simple fractions.



Name: Date:



Simplify

a) 52

53 −+

b) ⎟⎠⎞

⎜⎝⎛−−

21

43

c) 15

84

5 −×

−

d) 98

32÷

e) 3

32⎟⎠⎞

⎜⎝⎛

Expectation: Demonstrate facility in operations with percent, ratio and rate, and rational numbers, as necessary, to support other topics of the course.


Extend Your Thinking Grade 7 Name: Date:

Betty cut 43 m from a piece of fabric

322 m long.

She needs to know if there is enough material left for two pieces 65 m long each.

Show three different ways she could find out.

Expectation: Add and subtract fractions with simple denominators using concrete materials, drawings, and symbols.


Extend Your Thinking Grade 7 (Answers) Possible ways students could respond. 1. A student operating at the concrete stage may use coloured strips of paper. A visual learner may draw

a picture of the lengths of fabric. The following diagrams illustrate what the first student may cut out and the second student may draw.

Total fabric cut off is 1229 with

123 or

41 left over.

There is enough fabric for the two 65 m lengths.

Using this representation of 1 m, the entire 322 m piece of fabric is represented by:

represents 1 m,

the shaded part represents 43 m,

the striped part represents 65 m.

To illustrate both fractions on the same strip, merge the markings on the two strips to get

129

43=

1210

65=

1210

65=

If

Now make equal-sized fractions by adding breaks

as shown by the dashes.

then

and

1232

1282

322 ==


2. Some students may already be comfortable using symbols and fraction rules.

65

65

43

322 −−−

= 65

65

43

38

−−−

= 1210

1210

129

1232

−−−

= 123

= 41

3. Some students using symbols may approach it differently.

43

322 −

= 43

38−

= 129

1232

−

= 1223

65

65+

= 6

10

= 1220

1220

1223

>

4. Some students may convert to decimals.

6.2322 =

691.1

75.0666.2

−

75.043=

25.0666.1691.1

−

38.065=

666.1

383.0

383.0

+

5. Some students may choose to use a scientific calculator with a fraction key.

Since the display shows 1 4, meaning 41 , there is some fabric left over indicating there is enough

for all three pieces.

(Note: Different calculators may show fractions differently.)

5 6 ) = cba 6 + 4 – 3 c

ba3 – 2 cba 2 c

ba 5( cba

There is enough fabric for all three pieces.




Extend Your Thinking Grade 8 Name: Date:

Larry has a board 213 m long.

He needs to know how many pieces 43 m long each he can cut from this board.

Show three different ways he could find out.

Expectation: Add, subtract, multiply and divide simple fractions.


Extend Your Thinking Grade 8 (Answers) Possible ways students could respond. 1. A student still operating at the concrete stage may choose to use coloured rods.

She might ask herself, How many groups of 43 are there in

213 ?

Larry can get four pieces of the required length. 2. A student still working at the drawing stage may choose to use rectangles.

Larry can cut four pieces each 43 m long.

3. A student might use symbols and repeated subtraction.

After cutting the first

43 m piece:

After cutting the second

43 m piece:

After cutting the third

43 m piece:

43

411

−

= 48 is left

43

48−

= 45 is left

43

45−

= 42 (not enough for another piece)

43

213 −

= 43

27−

= 43

414

−

= 4

11 is left Larry will get four pieces the correct length.


4. Some students could choose to use the rules for division of fractions.

43

213 ÷

= 43

27÷

= 34

27×

OR =34

27× (Some students might reduce at this stage)

= 628 =

314

= 3

14 = 324 (Some students might point out that the

32 represents two

= 324 thirds of a

43 m piece, which is

42 or

21 m.)

Larry can get four full pieces. 5. A student may choose to convert to decimals and use a calculator. 75.05.3 ÷ = 6666666666.4 Therefore, he can get four pieces the right length. 6. Some students may choose to use a scientific calculator with a fraction key.

Since the display shows 4 2 3, meaning 324 , Larry will be able to cut four boards of length

43 m from the larger board. (Note: Different calculators may show fractions differently.)

2 1

4 2 1 33 cba

÷ c

ba cba =


Extend Your Thinking Grade 9 Applied Name: Date:

Gerty has a cube-shaped box whose inside dimensions measure 1019 cm.

She needs to know the maximum number of numbered cubes, whose sides measure

531 cm, that will fit into the box.

Show three different ways she could find out.

Expectation: evaluate numerical expressions involving natural number exponents with rational number bases.


Extend Your Thinking Grade 9 Applied (Answers) Possible ways students could respond. It is assumed in all answers that the numbered cubes will be placed in the box face to face in layers and that there will be some air space in each of the three dimensions. 1. It is possible that coloured rods could be used with the 10 rod as the unit. However, this might be a bit

unwieldy given that the length of the box would have to be represented using 11 rods. Some students may still need the comfort of concrete materials though. (See answer 2.) The rods could be used to determine the number of numbered cubes that would line up in one dimension only. That number of cubes, 5, would then have to be cubed. The rods would not need to be used for this operation. 53 = 125

So, 125 numbered cubes could fit into the box. 2. A student might use a scale model in one dimension.

Cut out a strip of paper 1019 cm long. Cut out several strips

531 cm long.

Five is the maximum number of 531 cm strips that can be placed in

1019 cm or less.

So, five numbered cubes will fit along each inside edge of the box. Since the box has three dimensions, the number of numbered cubes is 53 or 125. (Note: A student may employ the logic in this answer but use diagrams of rectangles instead of cutouts.)

3. A student might use symbols and estimation.

Since 4 × 531 = 4 × 1 + 4 ×

53 = 4 +

512 = 4 +

522 =

526 (<

1019 ),

4 cubes will fit and leave enough room for 1 more cube.

Since 5 × 531 = 5 × 1 + 5 ×

53 = 5 +

515 = 5 + 3 = 8 (<

1019 ),

5 cubes will fit and the

1011810

19 =− cm left is

not enough to fit in a 6th cube.

So the maximum number of cubes for each dimension is 5.

cm10

19

cm5

31


4. A student might use a scale drawing in two dimensions.

The maximum number of cubes that will fit in the box is 125.

5. A student might use a scale drawing in three dimensions.

The student could determine that there are 5 cubes per dimension as was done in previous answers.

air

View from top of box

5 × 5 = 25 25 cubes per layer There are five layers like this. 5 × 25 = 125

5 cubes

5 cubes

5 cubes

5 cubes

5 cubes

air


5. Some students might choose to use the rules for exponents and fractions.

3

531

3

1019 ⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛ ÷

= 33

58

1091

⎟⎠⎞

⎜⎝⎛÷⎟

⎠⎞

⎜⎝⎛

= 3

3

3

3

58

1091

÷ OR 3

58

1091

⎟⎠⎞

⎜⎝⎛ ÷

= 125512

1000753571

÷ = 3

85

1091

⎟⎠⎞

⎜⎝⎛ ×

= 512125

1000753571

× = 3

1691⎟⎠⎞

⎜⎝⎛

= 4096

753571 =

3

3

1691

= 40964003183

= 4096

753571

= 40964003183

Therefore, she can fit 183 numbered cubes into the box.

(Note: Students would need to think about the practical context to realize that this method gives an answer that is too big since it assumes use of space in the box that cannot accommodate the numbered cubes.)

6. Some students might choose to use decimals.

1.91019 = 6.1

531 =

9.1 ÷ 1.6 = 5.6875 So, 5 numbered cubes can fit in one dimension. Since the box has three dimensions, the number of cubes is 53 or 125.

7. A student could choose to use a scientific calculator with a fraction key.

Since the display shows 5 11 16, meaning 16115 , Gerty can fit five along one of the edges. Since

the box has three dimensions, the number of cubes is 53 or 125.

8

1

2

5 3 1 10 1 9 cba

÷ c

ba cba = c

ba


Is This Always True? Grade 7 Name: Date: 1. Skye says that if you increase both the numerator and denominator of a fraction by the

same amount, the result will always be greater than the original fraction.

For example:

Start with 32 . Now add, let’s say 5, to the numerator and denominator.

87

5352=

++ which is greater than

32 .

Is Skye’s statement true for all proper fractions?

2. Jeff says that if you subtract a smaller fraction from a larger fraction, the result will always be

between the two original fractions. Is Jeff’s statement true for all fractions?

3. Graham says that if you add 21 to any fraction, the common denominator will always be

even. Is Graham’s statement true for all fractions?

Answers 1. Yes. When you increase the numerator and denominator equally, the difference between the two

remains the same but both have become larger. Therefore, their ratio becomes closer to 1, so the fraction is larger.

2. No. Counter example: 101

108

109

=− (not between the two original fractions)

3. Yes. If the denominator of the first fraction is odd, you will need to multiply it by 2 to get the lowest

common denominator. So the common denominator will be a multiple of two and therefore, even. If the denominator of the first fraction is even, then it will be the common denominator since two will divide evenly into every even number.


Is This Always True? Grade 8 Name: Date: 1. Nisha was playing with some fractions as follows.

• start with 21

• then add41

21

21

=× , giving a total of 43

• then add 81

21

41

=× , giving a new total of 87

She noticed that each new number to be added was quite a bit smaller than the previous one. She was curious to find out if she carried this pattern on for as long as she wanted

would the sum always be less than 10099 .

2. Jack says that if you have to divide a whole number by a mixed number whose fraction part

is 21 , it is easier to double both the whole number and the mixed number, then divide.

For example:

21436÷

972 ÷= = 8

Is Jack’s statement true for all whole numbers divided by a mixed number ending in 21 ?

Answers 1. No. The pattern of sums is such that the numerator is always one less than the denominator that was

just added. So, when 128

1 is added, the sum is 128127 which is larger than

10099 .

2. Yes. Represent the whole number by a and the mixed number by b. Then

2a ÷ 2b

= ba

22

ba

=

Doubling both numbers maintains equivalence. Doubling the whole number gives another whole number.

Doubling the mixed number whose fraction part is 21 also gives a

whole number. Dividing the 2 whole numbers is easier than dividing by the mixed number.


Is This Always True? Grade 9 Applied Name: Date: 1. Carolyn says that as you add more and more terms, the sum gets closer to 1 but is always

less than 1.

...21

21

21

21 432

+⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛+

2. Stephen says that if you subtract one negative fraction from another negative fraction, the

difference will always be negative. Is Stephen’s statement true for all fractions? 3. Vadim teacher gave him 100 fractions to square. He had to calculate

2222

21100,...,

213,

212,

211 ⎟

⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛ .

After doing a few, John noticed a pattern:

2

211 ⎟⎠⎞

⎜⎝⎛ =

412

2

212 ⎟⎠⎞

⎜⎝⎛ =

416

2

213 ⎟⎠⎞

⎜⎝⎛ =

4112

4121 +×

4132 +×

4143 +×

Is it always true that the answer is (a number) × (the number that is 1 bigger) +41 ?

Answers

1. Yes. 43

41

21

=+ , 87

81

41

21

=++ , 1615

161

81

41

21

=+++ .

As the pattern carries on, the sum is such that the numerator is always one less than the denominator

that was just added. The pattern continues as 128127,

6463,

3231 . These sums are getting closer to 1.

Each new sum will not exceed one, because the fraction that is being added each time is only one half as much that which is needed to reach 1.

2. No. Counter example: 41

43

21

=⎟⎠⎞

⎜⎝⎛−−− (which is not negative)

Is Carolyn’s statement true?


3. Yes.

In general, the number being squared could be represented by n +21 . The square is the area of a

square with n +21 on each side.

Therefore, n × n + n +

41 = n × (n + 1) +

41 , which is (a number) × (the number that is 1 bigger) +

41 .

n 21

n

21

n × n

n21

n21

41

Area = n × n + n21 + n

21 +

41

= n × n + n + 41

n × n means we are adding n, n times. n × n + n means we are adding n, one more than n times. So, n × n + n means (n + 1) × n or n × (n + 1).

TIPS for Grades 7, 8, and 9 Applied Math · TIPS for Grades 7, 8, and 9 Applied Math is designed to...

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