Rigor Breakdown

© 2012 Common Core, Inc. All rights reserved. commoncore.org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Units

Rigor BreakdownPart 3: Application

Grades P-2


N YS C O M M O N CO R E M AT H E M AT I C S C U R R I C U LU M A Story of Units

Session Objectives

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• Understand the application component of rigor called for in the Standards, as defined by guiding documents

• Examine various activities in A Story of Units that engage students in application; compare and contrast activities

• Highlight Standards for Mathematical Practice in the application activities in A Story of Units

• Recognize the balance and intensity of all three components of rigor in A Story of Units

• Articulate how the three components of rigor support and relate to each other



Application

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Early reflection:• What is application?



ApplicationDefined by the Instructional Shifts

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“Students are expected to use math and choose the appropriate concept for application even when they are not prompted to do so. Teachers provide opportunities at all grade levels for students to apply math concepts in “real world” situations. Teachers in content areas outside of math, particularly science, ensure that students are using math – at all grade levels – to make meaning of and access content.”



ApplicationDefined by the Publishers’ Criteria

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“The phrase ‘real-world problems’ …is used to establish expectations …for applications and modeling.” (page 5)

“Applications take the form of problems to be worked on individually as well as classroom activities centered on application scenarios…” (page 11)



AGENDA

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Application – Word ProblemsApplication – Real-World ProblemsApplication – Modeling

Application – Word Problems



Application through Word Problems

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• Provide a “real world” situational context, even if the problem itself is not particularly “real-world.”

Jason’s allowance is 2 fifths as much as Sarah’s. Sarah’s allowance is half of Max’s allowance. If Max earns an allowance of $20 per week, what is Jason’s allowance?

• Prompt students to consolidate their knowledge of which operation is applicable to a given situational context



Application in A Story of Units

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• The Read, Draw, Write (RDW) process for solving word problems is modeled to students by teachers, promoting perseverance in reasoning through problems.



Video Clip – Word Problems

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Reflections:• How did the teacher model the problem

solving process?•What are the impacts and advantages of

having students apply mathematical concepts in real world contexts?• How is this different from how you

address application in your classroom?• Identify Mathematical Practices.



Video Clip: Candy Problem

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This video can be found on EngageNY.org



Video Clip – Word Problems

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Reflections:• How did the teacher facilitate the RDW

problem solving process?•What are the impacts and advantages of

this example of problem solving?• How is this different from how your

school/district program currently addresses application?• In what ways were Standards for

Mathematical Practice evident?



More on Application from the Publishers’ Criteria

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“Materials in grades K-8 include an ample number of single-step and multi-step contextual problems that develop the mathematics of the grade, afford opportunities for practice, and engage students in problem solving.” (page 11)



Multi-Step Problems in the Standards

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2 2.OA.1 Two-step, addition and subtraction within 100

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3.OA.83.MD.3

Two-step, involving the four operationsTwo-step, how many more and how many less questions using the information presented in scaled bar graphs with several categories

4 4.OA.3 Multi-step, involving whole numbers and the four operations

55.MD.1 Multi-step, real world problems that include converting among

different-sized standard measurement units within a given measurement system



Student Work Sample Grade 2 – Two-Step Word Problem

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Jack has $344. Mason has $266 more than Jack. How much do Jack and Mason have altogether?



Student Work SampleGrade 3 – Two-Step Word Problem

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Sam bought 4 bags of candy. Each bag contained 30 pieces of candy. The candy was shared equally among 6 children. How many pieces of candy did each child receive?



Student Work SampleGrade 3 – Two-Step Word Problem

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Student Work SampleGrade 4 – Multi-Step Word Problem

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Laney had 1690 tokens. Mia had 380 fewer tokens than Laney. Laney gave some tokens to Mia. In the end, Mia had 3 times as many tokens as Laney. How many tokens did Laney have in the end?




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Mrs. Jones wants to give each of her 18 students a gift for the holidays. She has a budget of $144. She bought 18 identical gifts each costing $6.00. She purchased 18 printed gift boxes for $1 each. She wants to buy ribbon to tie around each box. The ribbon costs $6 per yard. If she uses 9 inches of ribbon for each box, can she afford to buy the ribbon and still keep to her budget of $144? Explain why or why not.




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Lesson Engagement – RDW Process

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Students may ask themselves these questions to guide them through the problem solving process:• “What do I see?” • “Can I draw something?” • “What can I draw?” • “What can I learn from my drawing?”

After drawing, students write a statement responding to the question.



RDW

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Meagan had $1780 and Lisa had $1910. Lisa gave some money to Meagan. In the end Meagan had twice as much money as Lisa. How much money did Lisa give to Meagan?



Lesson Engagement – RDW Process

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Reflections:•What surprised you about working

through the RDW process yourselves?



Application – Word ProblemsKey Points

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• Word problems are a form of application.• Word problems give situational contexts to develop

students’ schema of which situations call for which operations.

• The RDW process encourages visualization, reflection, and perseverance.

• Two-step problems begin in 2nd grade.• Multi-step problems begin in 4th grade.



AGENDA

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Real-World Problems in the Standards

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33.MD.73.MD.8

Relate area to multiplication and divisionPerimeters of polygons

44.MD.34.MD.7

Area and perimeter formulas for rectangles Unknown angles on a diagram

55.NF.65.NF.7

Multiplication of fractions and mixed numbersDivision of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions

55.MD.1

5.MD.5

Convert among different-sized standard measurement units within a given measurement systemVolume problems

55.G.2 Graphing points in the first quadrant of the coordinate plane,

and interpret coordinate values of points in the context of the situation.




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“Materials attend thoroughly to those places in the content standards where expectations for multi-step and real world problems are explicit.” (page 11)



Activity – Writing Real-World Problems

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Choose one standard calling for real-world problems and write your own example of a real-world problem for that standard.



Application – Real-World ProblemsKey Points

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• Standards for real-world problems are present in grades 3-5

• Real-world problems are problems that people are faced with solving in the real world • Find the amount of fence needed to surround a given area• Converting among different sized measurement units• Multiplying fractions involved in doubling or halving a

recipe



AGENDA

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Application - Modeling

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Early reflection:•What does modeling mean to you?



Modeling in Pre-Kindergarten

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Modeling of environment using basic shapes

• From Pre-Kindergarten Overview: “[Students] use basic shapes and spatial reasoning to model

objects in their environment.”



Modeling in Kindergarten

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1. Modeling real world objects using geometric shapes and figures

• K.G.5 Model shapes in the world by building shapes from components (e.g., sticks and clay balls)

and drawing shapes.

2. Modeling of early addition and subtraction situations using concrete materials

• From Kindergarten Overview: “Model simple joining and separating situations with sets of objects.”



Modeling in Grade 1

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Modeling all forms of addition and subtraction situations using concrete materials:

• From Grade 1 Overview: “Use a variety of models, including discrete objects and length-based models (e.g., cubes connected to form lengths), to model add-to, take-from, put-together, take-apart, and compare situations to develop meaning for the operations of addition and subtraction, and to develop strategies to solve arithmetic problems with these operations.”

• 1.NBT.4 and 1.NBT.6



Modeling in Grade 2

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Place Value Models Applied to Addition and Subtraction • From Grade 2 Overview: “They solve problems within 1000 by

applying their understanding of models for addition and subtraction”• 2.NBT.7 Add and subtract within 1000, using concrete models or

drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method. Understand that in adding or subtracting three-digit numbers, one adds or subtracts hundreds and hundreds, tens and tens, ones and ones; and sometimes it is necessary to compose or decompose tens or hundreds.



Modeling in Grades 3-5

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1. Equal-sized groups, array and area models for multiplication and division

3.MD.7, 4.NBT.5, 4.NBT.6, 5.NBT.6, 5.NBT.7

2. Visual fraction models Most standards in the NF domain for grades 3-5

3. Place value and number line models extended to operations with decimals and fractions

“They apply their understandings of models for decimals, decimal notation, and properties of operations to add and subtract decimals to hundredths.”



Modeling in Grades 6-12

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• Modeling real life objects with geometric figures• Modeling relationships using equations, inequalities,

and sets of equations and inequalities• Modeling relationships between two data sets using

functions• Investigating chance processes with probability

models




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“Modeling builds slowly across K-8, and applications are relatively simple in early grades.” (page 11)



Lesson Engagement – Modeling

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Lesson Engagement – Modeling

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Reflection:• Analyze the impacts and advantages of

modeling in grades P-5.



Modeling is a Mathematical Practice

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MP.4 Model with mathematics.Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. (continued…)



Modeling is a Mathematical Practice

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(continued…) Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.



Application – ModelingKey Points

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• Modeling is required at all grade levels.• Modeling builds across grade levels, primarily

concrete in grades K-3, transitioning into pictorial in grades 3-5.

• Modeling is a Mathematical Practice.




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• Interesting problems that connect mathematics to students’ environment, other disciplines, and to the mathematics itself.

• Goal is for students to gain understanding and modeling skills needed to solve problems they have never seen before.

• Concrete materials and pictorial diagrams develop students’ ability to visualize quantitative relationships, reinforcing understanding.




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• Time allotted to application varies, but is commonly 10-20 minutes of the lesson.

• The placement of an application problem may go before or after the conceptual development: • Placement before can provide important context and

structure to understanding a new concept.• Placement after gives usefulness of a just-learned concept.

• Application also appears imbedded in conceptual development, worksheets, and homework, providing opportunity to work both as a class and individually.



Key Points

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• Application involves using relevant conceptual understandings and appropriate strategies even when not prompted to do so.

• Application is called for in the Standards in three ways: single and multi-step word problems, real-world problems, modeling.

• A Story of Units provides frequent, rich opportunities for students to practice application both as a group and individually.

• Application problems are often also opportunities to nurture the Standards of Mathematical Practice.



Next Steps

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• How will you incorporate this information about application into instruction?

• How will you share this information about application with your colleagues?



A Call for Equal Intensity and Balance

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The Instructional Shifts:“Students are practicing and understanding. There is

more than a balance between these two things in the classroom – both are occurring with intensity. Teachers create opportunities for students to participate in ‘drills’ and make use of those skills through extended application of math concepts...”



A Call for Equal Intensity and Balance

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The Publishers’ Criteria:“To help students meet the expectations of the

Standards, educators will need to pursue, with equal intensity, three aspects of rigor in the major work of each grade: conceptual understanding, procedural skill and fluency, and applications.” (page 5)

“Materials and tools reflect the balances in the Standards…” (page 9)



Intensity and Balance in A Story of Units

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Intensity and Balance in A Story of Units

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Three Components of Rigor

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Reflection:• How do the three components of rigor support and relate

to each other?



Three Components of Rigor

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Publishers’ Criteria:“The three aspects of rigor are not always separate in

materials. (Conceptual understanding needs to underpin fluency work; fluency can be practiced in the context of applications; and applications can build conceptual understanding.)” (page 11)

Rigor Breakdown

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