Mathematics Curriculum Geometry · 1/2/2019 · Although there are many types of geometry, school...

Mathematics Curriculum Geometry

Buena Regional Schools District Buena, NJ

Buena Regional School District – Geometry 2

Board of Education

Debra Bell, President James Abba, Vice-President

Mark Beamer Jr. John Cressey Syd D’Angelo

Valentina DiPrimio Lynda Gazzara

Al Gazzara Barbara Meyrick

Matt Walker Edward Zebedies

Superintendent

John Destefano

Supervisor

Brian Kern

Curriculum Committee

Curriculum Approval Date

2015


Table of Contents

Page(s)

➢ Course Description………………………………………………………………………………………………………………..4 - 5

➢ Course Goals………………………………………………………………………………………………………………..…………...6

➢ Course Enduring Understandings…….…………………………………………………………………………………..……7

➢ Common Core State Standards - Mathematical Practices……………………………………………………8 – 10

➢ Common Core State Standards – Mathematical Content………………………………………………….10 – 13

➢ 21st Century Skills……………………………………………………………………………………………………………..….….14

➢ Unit Names…………………………………………………………………………………………………………………………..…14

➢ Materials……………………………………………………………………………………………………………………………..….15

➢ Technology Standards.………………………………………………………………………………………………………15 - 17

➢ Grading Policy……….…….…………………………………………………………………………………………………..……..18

➢ Unit Plans

➢ Pacing Charts

➢ Benchmarks

➢ Appendix

A. Instructional Strategies

B. Habits of Mind

C. Best Practices


Course Description:

An understanding of the attributes and relationships of geometric objects can be applied in diverse contexts—interpreting a schematic

drawing, estimating the amount of wood needed to frame a sloping roof, rendering computer graphics, or designing a sewing pattern for

the most efficient use of material.

Although there are many types of geometry, school mathematics is devoted primarily to plane Euclidean geometry, studied both

synthetically (without coordinates) and analytically (with coordinates). Euclidean geometry is characterized most importantly by the

Parallel Postulate, that through a point not on a given line there is exactly one parallel line. (Spherical geometry, in contrast, has no

parallel lines.)

During high school, students begin to formalize their geometry experiences from elementary and middle school, using more precise

definitions and developing careful proofs. Later in college some students develop Euclidean and other geometries carefully from a small set

of axioms.

The concepts of congruence, similarity, and symmetry can be understood from the perspective of geometric transformation. Fundamental

are the rigid motions: translations, rotations, reflections, and combinations of these, all of which are here assumed to preserve distance

and angles (and therefore shapes generally). Reflections and rotations each explain a particular type of symmetry, and the symmetries of

an object offer insight into its attributes—as when the reflective symmetry of an isosceles triangle assures that its base angles are

congruent.

In the approach taken here, two geometric figures are defined to be congruent if there is a sequence of rigid motions that carries one onto

the other. This is the principle of superposition. For triangles, congruence means the equality of all corresponding pairs of sides and all

corresponding pairs of angles. During the middle grades, through experiences drawing triangles from given conditions, students notice ways

to specify enough measures in a triangle to ensure that all triangles drawn with those measures are congruent. Once these triangle

congruence criteria (ASA, SAS, and SSS) are established using rigid motions, they can be used to prove theorems about triangles,

quadrilaterals, and other geometric figures.

Similarity transformations (rigid motions followed by dilations) define similarity in the same way that rigid motions define congruence, thereby formalizing the similarity ideas of "same shape" and "scale factor" developed in the middle grades. These transformations lead to the criterion for triangle similarity that two pairs of corresponding angles are congruent. The definitions of sine, cosine, and tangent for acute angles are founded on right triangles and similarity, and, with the Pythagorean

Theorem, are fundamental in many real-world and theoretical situations. The Pythagorean Theorem is generalized to non-right triangles by

the Law of Cosines. Together, the Laws of Sines and Cosines embody the triangle congruence criteria for the cases where three pieces of


information suffice to completely solve a triangle. Furthermore, these laws yield two possible solutions in the ambiguous case, illustrating

that Side-Side-Angle is not a congruence criterion.

Analytic geometry connects algebra and geometry, resulting in powerful methods of analysis and problem solving. Just as the number line

associates numbers with locations in one dimension, a pair of perpendicular axes associates pairs of numbers with locations in two

dimensions. This correspondence between numerical coordinates and geometric points allows methods from algebra to be applied to

geometry and vice versa. The solution set of an equation becomes a geometric curve, making visualization a tool for doing and

understanding algebra. Geometric shapes can be described by equations, making algebraic manipulation into a tool for geometric

understanding, modeling, and proof. Geometric transformations of the graphs of equations correspond to algebraic changes in their

equations.

Dynamic geometry environments provide students with experimental and modeling tools that allow them to investigate geometric

phenomena in much the same way as computer algebra systems allow them to experiment with algebraic phenomena.

Connections to Equations. The correspondence between numerical coordinates and geometric points allows methods from algebra to be applied

to geometry and vice versa. The solution set of an equation becomes a geometric curve, making visualization a tool for doing and

understanding algebra. Geometric shapes can be described by equations, making algebraic manipulation into a tool for geometric

understanding, modeling, and proof.

-Common Core State Standards -

Course Goals:

G-CO Congruence A. Experiment with transformations in the plane B. Understand congruence in terms of rigid motions C. Prove geometric theorems D. Make geometric constructions

G-SRT Similarity, Right Triangles, and Trigonometry

A. Understand similarity in terms of similarity transformations B. Prove theorems involving similarity C. Define trigonometric ratios and solve problems involving right triangles


D. Apply trigonometry to general triangles G-C Circles

A. Understand and apply theorems about circles B. Find arc lengths and areas of sectors of circles

G-GPE Expressing Geometric Properties with Equations

A. Translate between the geometric description and the equation for a conic section B. Use coordinates to prove simple geometric theorems algebraically

G-G MD Geometric Measurement and Dimension

A. Explain volume formulas and use them to solve problems B. Visualize relationships between two-dimensional and three-dimensional objects

G-MG Modeling with Geometry

A. Apply geometric concepts in modeling situations

Course Enduring Understandings: Ideas that have lasting value beyond the classroom. Consider, “what do we want students to understand and be able to use several years from now, after they have forgotten the details?”

Geometry is a mathematical system built on accepted facts, basic terms and definitions.

Geometry and spatial sense offer ways to interpret and reflect on our physical environment.

Visualization can help you see the relationships between two figures and help you connect the properties of real objects with two-dimensional drawings of these objects.

Some attributes of geometric figures, such as length, area, volume and angle measure, are measurable.

All areas of mathematics are connected by fundamental concepts and procedures.

Analyzing geometric relationships develops reasoning and justification skills.

Every plane figure has perimeter and area.

Every three dimensional shape has a surface area and volume.


Common Core State Standards for Mathematics Standards for Mathematical Practice (MP): 1. Make sense of problems and persevere in solving them. Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, “Does this make sense?” They can understand the approaches of others to solving complex problems and identify correspondences between different approaches.

2. Reason abstractly and quantitatively. Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize—to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents—and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects.

3. Construct viable arguments and critique the reasoning of others. Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.

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. 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.

5. Use appropriate tools strategically. Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts.

6. Attend to precision. Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions.

7. Look for and make use of structure. Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 × 8 equals the well remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property. In the expression x2 + 9x + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 – 3(x – y)2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y.

8. Look for and express regularity in repeated reasoning. Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation (y – 2)/(x – 1) = 3. Noticing the regularity in the way terms cancel when expanding (x – 1)(x + 1), (x – 1)(x2 + x + 1), and (x– 1)(x3 + x2 + x + 1) might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results.


Standards for Mathematical Content: Congruence G-CO

Experiment with transformations in the plane

1. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.

2. Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g.,

translation versus horizontal stretch).

3. Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.

4. Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.

5. Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry

software. Specify a sequence of transformations that will carry a given figure onto another.

Understand congruence in terms of rigid motions

6. Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.

7. Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.

8. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.

Prove geometric theorems

9. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior

angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant

from the segment’s endpoints.

10. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are

congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.

11. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.

Make geometric constructions

12. Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines,

including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.


13. Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle

Similarity, right triangles, and trigonometry G-SRT Understand similarity in terms of similarity transformations

1. Verify experimentally the properties of dilations given by a center and a scale factor:

a. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged.

b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor.

2. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all

corresponding pairs of sides.

3. Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.

Prove theorems involving similarity

4. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the

Pythagorean Theorem proved using triangle similarity.

5. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.

6. Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios

for acute angles.

7. Explain and use the relationship between the sine and cosine of complementary angles.

8. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.★

Apply trigonometry to general triangles

9. (+) Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.

10. (+) Prove the Laws of Sines and Cosines and use them to solve problems.

11. (+) Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying

problems, resultant forces).

Circles G-C

Understand and apply theorems about circles

1. Prove that all circles are similar.

2. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed

angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.

3. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.


4. (+) Construct a tangent line from a point outside a given circle to the circle.

Find arc lengths and areas of sectors of circles

5. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the

angle as the constant of proportionality; derive the formula for the area of a sector.

Expressing Geometric Properties with Equations G-GPE

Translate between the geometric description and the equation for a conic section

1. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.

2. Derive the equation of a parabola given a focus and directrix.

3. (+) Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant.

Use coordinates to prove simple geometric theorems algebraically

4. Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the

coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2).

5. Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point).

6. Find the point on a directed line segment between two given points that partitions the segment in a given ratio.

7. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g. using the distance formula.*

Geometric Measurement and Dimension G-MGD

Explain volume formulas and use them to solve problems

1. Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments.

2. (+) Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures.

3. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.★

Visualize relationships between two-dimensional and three-dimensional objects

4. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations

of two-dimensional objects.

Modeling with Geometry G-MG

Apply geometric concepts in modeling situations


1. Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).★

2. Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).★

3. Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).★

21st Century Skills:

A. Critical Thinking and Problem-Solving

B. Creativity and Innovation

C. Collaboration, Teamwork and Leadership

D. Cross-Cultural Understanding and Interpersonal Communication

E. Communication and Media Fluency

F. Accountability, Productivity and Ethics

Unit Names:

Unit I: Language and Tools of Geometry

Unit II: Similarity, Congruency, Proof

Unit III: Application of Angles and Segments

Unit IV: Extending to three dimensions


Materials and Resources:

Textbook: Geometry Common Core Pearson cr 2012 Auxiliary Materials: Pearson TI – 84 Graphing Calculators Exam View Smart Board Geo Boards

Technology Standards (by the end of grade 12): 8.1 Educational Technology

A. Technology Operations and Concepts

• 8.1.12.A.1 Construct a spreadsheet, enter data, and use mathematical or logical functions to manipulate data, generate charts and

graphs, and interpret the results.

• 8.1.12.A.2 Produce and edit a multi-page document for a commercial or professional audience using desktop publishing and/or graphics

software.

• 8.1.12.A.3 Participate in online courses, learning communities, social networks, or virtual worlds and recognize them as resources for

lifelong learning.

• 8.1.12.A.4 Create a personalized digital portfolio that contains a résumé, exemplary projects, and activities, which together reflect

personal and academic interests, achievements, and career aspirations.

B. Creativity and Innovation

• 8.1.12.B.1 Design and pilot a digital learning game to demonstrate knowledge and skills related to one or more content areas or a real

world situation.

C. Communication and Collaboration

• 8.1.12.C.1 Develop an innovative solution to a complex, local or global problem or issue in collaboration with peers and experts, and

present ideas for feedback in an online community

D. Digital Citizenship

• 8.1.12.D.1 Evaluate policies on unauthorized electronic access (e.g., hacking) and disclosure and on dissemination of personal

information.


• 8.1.12.D.2 Demonstrate appropriate use of copyrights as well as fair use and Creative Commons guidelines.

• 8.1.12.D.3 Compare and contrast international government policies on filters for censorship.

• 8.1.12.D.4 Explain the impact of cyber crimes on society

E. Research and Information Literacy

• 8.1.12.E.1 Develop a systematic plan of investigation with peers and experts from other countries to produce an innovative solution to

a state, national, or worldwide problem or issue.

• 8.1.12.E.2 Predict the impact on society of unethical use of digital tools, based upon research and working with peers and experts in the

field.

F. Critical Thinking, Problem Solving, and Decision-Making

• 8.1.12.F.1 Select and use specialized databases for advanced research to solve real-world problems.

• 8.1.12.F.2 Analyze the capabilities and limitations of current and emerging technology resources and assess their potential to address

educational, career, personal, and social needs.

8.2 Technology Education, Engineering, and Design

A. Nature of Technology: Creativity and Innovation

• 8.2.12.A.1 Design and create a technology product or system that improves the quality of life and identify

trade-offs, risks, and benefits.

B. Critical Thinking, Problem Solving, and Decision Making

• 8.2.12.B.1 Design and create a product that maximizes conservation and sustainability of a scarce resource,

using the design process and entrepreneurial skills throughout the design process.

• 8.2.12.B.2 Design and create a prototype for solving a global problem, documenting how the proposed design

features affect the feasibility of the prototype through the use of engineering, drawing, and other technical

methods of illustration.

• 8.2.12.B.3 Analyze the full costs, benefits, trade-offs, and risks related to the use of technologies in a potential

career path.

C. Technological Citizenship, Ethics, and Society

• 8.2.12.C.1 Analyze the ethical impact of a product, system, or environment, worldwide, and report findings in a web-based publication

that elicits further comment and analysis.

• 8.2.12.C.2 Evaluate ethical considerations regarding the sustainability of resources that are used for the design, creation, and

maintenance of a chosen product.

• 8.2.12.C.3 Evaluate the positive and negative impacts in a design by providing a digital overview of a chosen product and suggest

potential modifications to address the negative impacts.

D. Research and Information Fluency


• 8.2.12.D.1 Reverse-engineer a product to assist in designing a more eco-friendly version, using an analysis of trends and data about

renewable and sustainable materials to guide your work.

E. Communication and Collaboration

• 8.2.12.E.1 Use the design process to devise a technological product or system that addresses a global issue,

and provide documentation through drawings, data, and materials, taking the relevant cultural perspectives

into account throughout the design and development process.

F. Resources for a Technological World

• 8.2.12.F.1 Determine and use the appropriate application of resources in the design, development, and creation of a technological

product or system.

• 8.2.12.F.2 Explain how material science impacts the quality of products.

• 8.2.12.F.3 Select and utilize resources that have been modified by digital tools (e.g., CNC equipment, CAD software) in the creation of a

technological product or system.

G. The Designed World

• 8.2.12.G.1 Analyze the interactions among various technologies and collaborate to create a product or system demonstrating their

interactivity.

Grading Policy:

The grade will consist of the following: homework/ participation, tests, quizzes, benchmarks and a final exam. Foundations: Homework /Participation 33% Quizzes 33% Test / Benchmark 34%

College Prep: Homework/Participation 25% Quizzes 25% Test/ Benchmark 50% Honors: Homework/Participation 20% Quizzes 20%


Test/Benchmark 60%

Content Area: Mathematics Course: Geometry

Unit Plan Title: Unit I: Language and Tools of Geometry

Domain

Congruence Transformations, Construction, Modeling and Applying,

Overview/Rationale

This gives students the foundational tools for developing viable geometric arguments using the relationships student studied in middle school related to lines, transversals, and special angles associated with them. In this unit students will also develop rigorous definitions of three familiar congruence transformations: reflections, translations, and rotations and use these transformations to discover and prove geometric properties.

Standard(s)

Common Core State Standards for Mathematics

G.CO.1,2,3,4,5,12,13 G.GPE.4,5,6,7

Standards for Mathematical Practices

MP 1 MP 5 MP 3 MP 6 MP 4 MP 7


Technology Standard(s)

8.1.12.A.1

Interdisciplinary Standard(s)

CCSS>ELA-LITERACY.RST.9-10.3

CCSS>ELA-LITERACY.RST.9 -10.8

Essential Question(s)

What are the building blocks of geometry?

How can you describe the attributes of geometric elements and figures?

How do you describe relationships among geometric elements figures?

What is the relationship between construction and congruency?

Enduring Understandings

Geometry is a mathematical system built on accepted facts, basic terms and definitions.

A postulate or axiom is an accepted statement of fact.

You can use special geometric tools to make a figure that is congruent to an original figure without measuring.

Construction is more accurate than sketching and drawing.

In this unit plan, the following 21st Century themes and skills are addressed.

Check all that apply.

Indicate whether these skills are E-Encouraged, T-Taught, or A-Assessed in this unit by marking E, T, A on the line before the appropriate skill.

21s t Century Skil ls


21s t Century Themes

Global Awareness

A Critical Thinking & Problem Solving

Environmental Literacy E Creativity and Innovation

Health Literacy T Collaboration, Teamwork and Leadership

Civic Literacy E Cross-Cultural and Interpersonal Communication

X Financial, Economic, Business and Entrepreneurial Literacy

E Communication and Media Fluency

A Accountability, Productivity and Ethics

Instructional Plan

Pre-Assessment

Student Learning

Targets/Objectives

Student Strategies D Activities and Resources D Formative Assessment D

The students will be able

to….

• Identify, name and represent points, segments, rays, and planes

• Apply basic facts, postulates and theorems about points, lines and planes

• Word Wall • Flashcards • Graphic

Organizer-e.g.Foldables

• SmartBoard

• Direct Instruction: Teacher directed Instruction for the use of providing information or developing step-by-step skills Indirect Instruction: High level of student involvement where teacher is the facilitator/supporter/resources person.

• Independent Study Options: Teacher guided with individual/group option.

• Interactive Instruction: Cooperative learning strategies such as peer-partner learning and discussions, reporting and sharing problem solving techniques (Requires the teacher to structure the activity

• Questions

• Homework

• Guided Practice

• Small White

Boards

Exit Tickets G.GO.2

G.GO.3

G.GO.4

G.GO.5

G.GO.6

• G.GO.7


and the students to actively discuss/share.) Students have the opportunity to learn from both the teacher as well as from each other.


to….

• Find the midpoint of a segment

• Use the distance formula

• Apply the midpoint and distance formulas to problem solving situations

• Graph paper • Geometer’s

Sketch Pad

• Direct Instruction: Teacher directed Instruction for the use of providing information or developing step-by-step skills

• Indirect Instruction: High level of student involvement where teacher is the facilitator/supporter/resources person.


• Interactive Instruction: Cooperative learning strategies such as peer-partner learning and discussions, reporting and sharing problem solving techniques (Requires the teacher to structure the activity and the students to actively discuss/share.) Students have the opportunity to learn from both the teacher as well as from each other.

• Questions

• Homework

• Guided Practice

• Small White

Boards

• Exit Tickets

• Compass • Straight edge

• Direct Instruction: Teacher directed Instruction for the use of

• Questions

• Homework

• Guided Practice



to….

• Identify and demonstrate proficient use of the different tools that can be used for constructing geometric objects (protractor, compass, ruler, geometric

software) Describe why construction of geometric figures is important

• Find and compare segment lengths

• Construct midpoints and congruent segments (Using various methods)

Apply the Ruler Postulate to problem solving situations.

• Apply the Segment Addition Postulate to problem solving situations

• Geometric software

• Interactive Activities

providing information or developing step-by-step skills




• Small White

Boards

Exit Tickets


to….

• Name, classify and compare angles

• Compass • Straight edge • Geometric

software • Interactive

Activities


• Indirect Instruction: High level of student involvement where

• Questions

• Homework

• Guided Practice

• Small White

Boards


• Measure and construct angles and angle bisectors using various tools

Apply the angle

addition postulate to

problem solving

situations

teacher is the facilitator/supporter/resources person.



Exit Tickets


to….

• Identify special pairs of angles (adjacent, vertical, complementary,

supplementary) Use angle relationships to find the measure of angles.

• Apply angle relationships to problem solving situations



Activities




• Interactive Instruction: Cooperative learning strategies such as peer-partner learning and discussions, reporting and sharing problem solving

• Questions

• Homework

• Guided Practice

• Small White

Boards

Exit Tickets


techniques (Requires the teacher to structure the activity and the students to actively discuss/share.) Students have the opportunity to learn from both the teacher as well as from each other.


to….

• Construct congruent segments, congruent angles, perpendicular bisectors, angle bisectors, using a compass, straightedge, patty paper.

• Define congruency • Explain the relationship

between congruency and construction

• Use geometric software to make, points, lines, planes and complete constructions

• Make conjectures after manipulating constructions



Activities





• Questions

• Homework

• Guided Practice

• Small White

Boards

Exit Tickets

Benchmark Assessment(s)


D – Indicates differentiation

Differentiation

Strategies/Modifications:

(i.e. ESL, Special Education, ALPS)

Note IEP for Special Education; Multilingual Glossary for ESL; ALPS Enrichment Activities (See

Resources for suggestions/ideas); Remediation as needed using formative assessments; Higher

level/order problems for Honors; graphic organizers; peer helpers, chunking assignments; verbal

and written directions; allow for verbal as well as written responses; use of calculator, continued

use of manipulatives and models.

Teacher Derived Multiple Choice and Open – Ended Questions given at the end of Unit 1

Summative Written Assessments

Mid chapter, Chapter Tests and Quizzes

Summative Performance Assessments:

Constructions

Examples/Experiences

Suggested Time Frame: 9 weeks



Unit Plan Title: Unit II: Similarity, Congruency, Proof

Domain

Congruency, Similarity, Right Triangle and Trigonometry,

Overview/Rationale

This unit explores basic theorems and conjectures about triangles, including the triangle inequality conjecture, The Triangle Sum Theorem, and the Theorems regarding centers of triangles. This unit also builds on the students’ work with transformations from Unit 1 to formalize the definition of congruent triangles. Students reason to identify criteria for triangle congruence and use precise notation to describe the correspondence in congruent triangles.

Standard(s)













Global Awareness

Critical Thinking & Problem Solving

Environmental Literacy Creativity and Innovation

Health Literacy Collaboration, Teamwork and Leadership

Civic Literacy Cross-Cultural and Interpersonal Communication

Communication and Media Fluency


Financial, Economic, Business and Entrepreneurial Literacy

Accountability, Productivity and Ethics

Instructional Plan

Pre-Assessment

Student Learning Targets/Objectives Student Strategies D Activities and Resources D Formative Assessment D







Suggested Time Frame:



Unit Plan Title:

Domain

Overview/Rationale

Standard(s)













Global Awareness









Instructional Plan

Pre-Assessment










Unit Plan Title:


Domain

Overview/Rationale

Standard(s)













Global Awareness








Instructional Plan

Pre-Assessment





Unit Plan Title:

Domain

Overview/Rationale

Standard(s)



21s t Century Themes 21s t Century Skil ls

Global Awareness








Instructional Plan

Pre-Assessment




Unit Plan Title:

Domain

Overview/Rationale

Standard(s)













Global Awareness








Instructional Plan

Pre-Assessment






Unit Plan Title:

Domain

Overview/Rationale


Standard(s)













Global Awareness








Instructional Plan

Pre-Assessment



PACING GUIDE – Geometry UNIT NAME Unit I: Language and Tools of Geometry

Essential Question/ Lesson Title/Pages

# of Days

Student Learning Objectives Common Core State Standards

Standards for Math Practice

Formative/SummativeAssessment(s)


What are the building

blocks of geometry?

How can you describe

the attributes of

geometric elements

and figures?

Chapter 1

Pages 1 - 78

18 days

The students will be able to….

• Identify, name and represent points, segments, rays, and planes

• Apply basic facts, postulates and theorems about points, lines and planes

• Find the midpoint of a segment

• Use the distance formula • Apply the midpoint and

distance formulas to problem solving situations

• Identify and demonstrate proficient use of the different tools that can be used for constructing geometric objects (protractor, compass, ruler,

geometric software)

Describe why construction of geometric figures is important Name, classify and compare angles

• Measure and construct angles and angle bisectors using various tools

• Apply the angle addition

postulate to problem

solving situations

• Find and compare segment lengths

• Construct midpoints and congruent segments (Using various methods)

G.CO.1

G.GO.12

G.GPE.4

G.GPE.7

MP 1 MP 4 MP 5

• Teacher

Observations

• Questions

• Homework

• Guided Practice

• Small White

Boards

• Exit Tickets

• 2 – 3 Quizzes

• 1 Test


solving situations

What is the

relationship between construction and congruency?

Chapter 3

Pages 137 - 214

16 days

• Apply the Ruler Postulate to problem solving situations.

• Apply the Segment Addition Postulate to problem solving situations

• Construct congruent segments, congruent angles, perpendicular bisectors, angle bisectors, using a compass, straightedge, patty paper.

• Define congruency • Explain the relationship

between congruency and construction

• Use geometric software to make, points, lines, planes and complete constructions

• Make conjectures after manipulating constructions

G.CO.1

G.CO.9

G.CO.10

G.CO.12

MP 1

MP 4

MP 5

MP 6

• Teacher

Observations

• Questions

• Homework

• Guided

Practice

• Small White

Boards

• Exit Tickets

• 2 – 3 Quizzes

• 1 Test

• Performance

task


How do you describe

relationships among

geometric elements

figures

Chapter 9

Pages 541 - 610

14 days

• Find translation images of figures

• Find reflection images of figures

• To draw and identify rotation images

• Indentify congruence transformation.

• Understand dilation of figures

• Identify similarity transformation and verify properties

G.CO.2 G.CO.3 G.CO.4 G.CO.5 G.CO.6 G.CO.7 G.CO.8 G.SRT.2 G.SRT.3

MP 1 MP 4 MP 5 MP 6 MP 7

• Teacher

Observations

• Questions

• Homework

• Guided

Practice

• Small White

Boards

• Exit Tickets

• 2 – 3 Quizzes

• 1 Test

• Performance

task

Benchmark


UNIT NAME Unit II: Similarity, Congruency, Proof


# of Days

Student Learning Objectives

Common Core State

Standards


Formative/Summative Assessment(s)


How can you find the

sum of the measures

of polygon angles?

How can you classify

quadrilaterals?

Chapter 6

Sections:

6.1,6.2,6.4,6.6,6.8

12 Days

• Construct regular polygons

• Classify polygons based on their sides and angles

• Find and use the measures of the interior and exterior angles of polygons

• Prove and apply properties of parallelograms

• Use properties of parallelograms to solve problems

• Prove that a quadrilateral is a parallelogram

• Prove and apply properties of rectangles, rhombuses, and squares

• Prove that a given quadrilateral is a rectangle, square, or rhombus (no coordinate proofs)

•

G.CO.11

G.CO.12

G.CO.13

G.SRT.5

MP 1 MP 4 MP 8

• Teacher

Observations

• Questions

• Homework

• Guided Practice

• Small White

Boards

• Exit Tickets

• 2 – 3 Quizzes

• 1 Test

• Performance

task

• Benchmark


How does an

understanding of

transformations lead

to a better

understanding of

similarity?

How do you show two triangles are congruent?

How is proportional reasoning of geometric figures used to solve problems? Chapter 4 Page 215 - 280

11 days

• Use the properties of similarity transformations to verify AA

• Explain why SSS, SAS and AA are sufficient for proving triangles similar

• Prove triangles similar • Use triangle similarity to solve problems

• Use properties of similar triangles to find segment lengths

• Construct the triangle proportionality theorem

• Apply proportionality (If a line parallel to a side of a triangle intersects the other two sides, then it divides those sides proportionally and triangle angle bisector theorems

•

G.SRT.1

G.SRT.2

G.C.1

MP 1 MP 4 MP 5 MP 6 MP 7

• Teacher

Observations

• Questions

• Homework

• Guided Practice

• Small White

Boards

• Exit Tickets

• 2 – 3 Quizzes

• 1 Test

Performance task


How do you use proportions to find side lengths in similar polygons? How does an

understanding of

transformations lead

to a better

understanding of

similarity?

How do you identify

corresponding parts of

similar triangles?

How is proportional

reasoning of

geometric figures

used to solve

problems?

Chapter 7

Pages 432 - 484

11 days

• Write ratios and solve proportions

• Apply ratios and proportions to problem solving situations

• Explore properties of similar polygons using various methods

• Determine properties of similar polygons

• Apply properties of similar polygons to solve problems

• Draw fractals by iteration (Honors Only – Optional A)

• Draw a Koch Curve (Honors Only – Optional A)

• Draw a Koch Snowflake (Honors Only – Optional A)

G.SRT.1

G.SRT.2

G.C.1

MP 1 MP 3 MP 4

• Teacher

Observations

• Questions

• Homework

• Guided Practice

• Small White

Boards

• Exit Tickets

• 2 – 3 Quizzes

• 1 Test

MID TERM


UNIT NAME Unit III: Application of Angles and Segments


# of Days


Common Core State

Standards




How do you find a

side length or angle

measure in a right

triangle?

How do similar right

triangles apply to

trigonometric ratios?

How do you find a


measure in a right

triangle?


triangles apply to


How does

trigonometry model

problem solving

situations?

How do you find a


measure in a right

triangle?


triangles apply to


Chapter 8 Sections 8.1 – 8.4

12 Days

• Solve problems involving angles are depression and angles of elevation

• Use geometric software to explore trigonometric ratios in right triangles

• Find the sine, cosine and tangent of an acute angle

• Use the sine, cosine and tangent ratios to determine side lengths and angle measures in right triangles.

• Use trigonometric ratios to solve right triangles and real world problems

• To find and use relationships in similar right triangles

• Use geometric mean to find segment lengths in right triangles

• Use ratios to make indirect measurements

• Use scale drawings to solve problems

• Explore the Golden Ratio and explain how it relates to the

G.SRT.4

G.SRT.5

G.SRT.6

G.SRT.8

G.GPE.6

NQ.1

N.Q.2

G.SRT.7

G.SRT.9

G.SRT.10

G.SRT.11

N.VM.1 (+)

N.VM.4 (+)

MP 1 MP 3 MP 4

• Teacher

Observations

• Questions

• Homework

• Guided Practice

• Small White

Boards

• Exit Tickets

• 2 – 3 Quizzes

• 1 Test


Fibonacci Sequence (Honors Only)


How can you prove

relationships between

angles and arcs of a

circle?

When lines intersect a

circle or within a

circle, how do you find

the measures of

resulting angles, arcs,

and segments?

What characteristics

make a circle unique

as a geometric figure?

How can properties

and relationships from

other geometric

figures applied to

circles?

Chapter 12

Page 762 - 818

13 days

• Identify tangents, secants and chords

• Use the properties of tangents to solve problems

• Construct a tangent to a circle at a point

• Use patty paper to explore the properties of chords

• Use congruent chords, arcs and central angles to solve problems

• Apply the properties of chords including perpendicular bisectors to chords

• Construct the center of a circle

• Find the measure of an inscribed angle

• Find the measure of an angle formed by a tangent and a

chord Apply the properties of inscribed angles to problem solving situations

Construct a tangent line from a point outside a

given circle to the circle • Find measures of

angles formed by chords, secants and tangents

G.C.2

G.C.3

G.C.4

G.C.5

G.GPE.1

G.GPE.4

G.GPE.5

G.GPE.7

G.CO.11

MP 1 MP 6 MP 8

• Teacher

Observations

• Questions

• Homework

• Guided Practice

• Small White

Boards

• Exit Tickets

• 2 – 3 Quizzes

• 1 Test

Benchmark


• Find the lengths of segments associated with circles

• Use angle measures and segments to solve problems


UNIT NAME Unit IV: Extending to Three Dimensions


# of Days


Common Core State

Standards




How do you find the

area of a polygon or

find the circumference

and area of a circle?

How do perimeter

and area of similar

polygons compare?

How do you

determine the

intersection of a solid

and a plane?

How do you find the

surface area and

volume of a solid

compare?

How do the surface

areas and volumes of

similar solids

compare?

Chapter 10 Pages 616 - 667

15 days

• Find the area of triangles and quadrilaterals

• Apply area formulas to problem solving situations

• Develop and apply the formulas for the area and circumference of a circle

• Develop and apply the formulas for the area of a regular polygon

• Use the area addition postulate to find the areas of composite figures

• Use composite figures to estimate the areas of irregular shapes

• Apply the area of composite figures to problem solving situations

• Find the area and perimeter of similar figures

•

G.MG.1

G.MG.3

G.GMD.1

G.GMD.4

G.MG.2

MP 1 MP 4 MP 5 MP 6

• Teacher

Observations

• Questions

• Homework

• Guided Practice

• Small White

Boards

• Exit Tickets

• 2 – 3 Quizzes

• 1 Test


How do you

determine the

intersection of a solid

and a plane?

How do you find the

surface area and

volume of a solid

compare?

How do the surface

areas and volumes of

similar solids

compare?

Chapter 11

Page 688 - 755

19 days

• Make nets of three dimensional figures

• Classify three-dimensional figures according to their properties

• Visualize cross sections of space figures

• Use nets and cross sections to analyze three dimensional figures

Find the volume of a

prism

• Find the volume of a cylinder

• Find the volume of a pyramid

• Find the volume of a cone

• Find the volume of composite figures

• Find the volume of a sphere

• Use the volume formulas in problem solving situations

• Use density concepts in modeling situations based on area and volume

• Informally justify the volume formulas

G.GMD.1

G.GMD.3

G.GMD.4

G.MG.2

MP 1 MP 4 MP 5 MP 6

• Teacher

Observations

• Questions

• Homework

• Guided Practice

• Small White

Boards

• Exit Tickets

• 2 – 3 Quizzes

• 1 Test

• Benchmark

•


• Compare the volumes of similar Find the surface areas of prisms

• Find the surface areas of cylinders

• Find the surface areas of pyramids

• Find the surface areas of cones

• Find the surface areas of spheres

• Apply the surface area formulas to problem solving situations

• figures

Final Exam

Appendix A - Guide Effective Instructional Strategies

Marzano’s Nine Instructional Categories Divided into Specific Behaviors

General Instructional

Category

Specific Behaviors


1. Identifying

similarities and

differences

*assigning in-class and homework tasks that involve comparison and classification

*assigning in-class and homework tasks that involve metaphors and analogies

2. Summarizing

and note

taking

*asking students to generate verbal summaries *asking students to generate written summaries

*asking students to take notes

*asking students to revise their notes, correcting errors and adding information

3 Reinforcing

effort

and providing

recognition

*recognizing and celebrating progress toward learning goals throughout a unit

*recognizing and reinforcing the importance of effort

*recognizing and celebrating progress toward learning goals at the end of a unit

4. Homework

and practice

*providing specific feedback on all assigned homework *assigning homework for the purpose of students practicing skills and procedures

that have been the focus of instruction

5. Nonlinguistic

representations

*asking students to generate mental images representing content *asking students to draw pictures or pictographs representing content

*asking students to construct graphic organizers representing content

*asking students to act out content *asking students to make physical models of content

*asking students to make revisions in their mental images, pictures, pictographs,

graphic organizers, and physical models

6. Cooperative

Learning

*organizing students in cooperative groups when appropriate

*organizing students in ability groups when appropriate

7. Setting

objectives and

providing

feedback

*setting specific learning goals at the beginning of a unit *asking students to set their own learning goals at the beginning of a unit

*providing feedback on learning goals throughout the unit

*asking students to keep track of their progress on learning goals *providing summative feedback at the end of a unit

*asking students to assess themselves at the end of a unit

8. Generating

and testing

hypotheses

*engaging students in projects that involve generating and testing hypotheses through problem solving tasks

*engaging students in projects that involve generating and testing hypotheses

through decision making tasks



through investigation tasks

*engaging students in projects that involve generating and testing hypotheses through experimental inquiry tasks


through systems analysis tasks *engaging students in projects that involve generating and testing hypotheses

through invention tasks

9. Questions,

cues, and

advance

organizers

*prior to presenting new content, asking questions that help students recall what they might already know about the content

*prior to presenting new content, providing students with direct links with what

they have studied previously *prior to presenting new content, providing ways for students to organize or think

about the content

Buena Regional School District – Geometry 67 Appendix B - Habits of Minds The success of a well-prepared college student is built upon a foundation

of key habits of mind that enable students to learn content from a range of disciplines. Unfortunately, the development of key

habits of mind in high school is often overshadowed by an instructional focus on de-contextualized content and facts necessary to

pass exit examinations or simply to keep students busy and classrooms quiet. For the most part, state high-stakes standardized

tests require students to recall or recognize fragmented and isolated bits of information. Those that do contain performance tasks

are severely limited in the time the tasks can take and their breadth or depth. The tests rarely require students to apply their learning

and almost never require students to exhibit proficiency in higher forms of cognition (Marzano, Pickering, & McTighe, 1993).

Several studies of college faculty members nationwide, regardless of the selectivity of the university, expressed near-universal

agreement that most students arrive unprepared for the intellectual demands and expectations of post-secondary (Conley, 2003a).

For example, one study found that faculty reported that the primary areas in which first-year students needed further development

were critical thinking and problem solving (Lundell, Higbee, Hipp, & Copeland, 2004). The term “habits of mind” was selected

for this model to describe the intelligent behaviors necessary for college readiness and to emphasize that these behaviors need to

be developed over a period of time such that they become ways of thinking, habits in how intellectual activities are pursued. In

other words, habits of mind are patterns of intellectual behavior that lead to the development of cognitive strategies and capabilities

necessary for college-level work. The term habits of mind invokes a more disciplined approach to thinking than terms such as

“dispositions” or “thinking skills.” The term indicates intentional and practiced behaviors that become a habitual way of working

toward more thoughtful and intelligent action (Costa & Kallick, 2000). The specific habits of mind referenced in this paper are

those shown to be closely related to college success. They include the following as the most important manifestations of this way

of thinking:

Intellectual openness: The student possesses curiosity and a thirst for deeper understanding, questions the views of others when

those views are not logically supported, accepts constructive criticism, and changes personal views if warranted by the evidence.

Such open mindedness helps students understand the ways in which knowledge is constructed, broadens personal perspectives

and helps students deal with the novelty and ambiguity often encountered in the study of new subjects and new materials.

Inquisitiveness: The student engages in active inquiry and dialogue about subject matter and research questions and seeks

evidence to defend arguments, explanations, or lines of reasoning. The student does not simply accept as given any assertion that

is presented or conclusion that is reached, but asks why things are so.

Analysis: The student identifies and evaluates data, material, and sources for quality of content, validity, credibility and relevance.

The student compares and contrasts sources and findings and generates summaries and explanations of source materials.

Reasoning, argumentation, proof: The student constructs well-reasoned arguments or proofs to explain phenomena or issues,

utilizes recognized forms of reasoning to construct an argument and defend a point of view or conclusion, accepts critiques of or


COMMON RECOMMENDATIONS OF NATIONAL CURRICULUM REPORTS

*LESS whole-class, teacher-directed instruction (e.g., lecturing)

*LESS student passivity: sitting, listening, receiving, an absorbing information

*LESS presentational, one-way transmission of information from teacher to student

*LESS prizing and rewarding of silence in the classroom

*LESS classroom time devoted to fill-in-the-blank worksheets, dittos, workbooks, & other “seatwork”

challenge to assertions, and addresses critiques and challenges by providing a logical explanation or refutation, or by

acknowledging the accuracy of the critique or challenge.

Interpretation: The student analyzes competing and conflicting descriptions of an event or issue to determine the strengths and

flaws in each description and any commonalities among or distinctions between them; synthesizes the results of an analysis of

competing or conflicting descriptions of an event or issue or phenomenon into a coherent explanation; states the interpretation that

is most likely correct or is most reasonable, based on the available evidence; and presents orally or in writing an extended

description, summary, and evaluation of varied perspectives and conflicting points of view on a topic or issue.

Precision and accuracy: The student knows what type of precision is appropriate to the task and the subject area, is able to

increase precision and accuracy through successive approximations generated from a task or process that is repeated, and uses

precision appropriately to reach correct conclusions in the context of the task or subject area at hand.

Problem solving: The student develops and applies multiple strategies to solve routine problems, generate strategies to solve non-

routine problems, and applies methods of problem solving to complex problems requiring method-based problem solving. These

habits of mind are broadly representative of the foundational elements that underlie various “ways of knowing.” These are at the

heart of the intellectual endeavor of the university. They are necessary to discern truth and meaning as well as to pursue them.

They are at the heart of how post-secondary faculty members think, and how they think about their subject areas. Without the

capability to think in these ways, the entering college student either struggles mightily until these habits begin to develop or misses

out on the largest portion of what college has to offer, which is how to think about the world.

Appendix C - Best Practice: Today’s Standards for Teaching

Buena Regional School District – Geometry 69 *LESS student time spent reading textbooks and basal readers

*LESS attempts by teachers to thinly “cover” large amounts of material in every subject area

*LESS rote memorization of facts and details

*LESS emphasis on the competition and grades in school

*LESS tracking or leveling students into “ability groups”

*LESS use of pull-out special programs

*LESS use of and reliance on standardized tests

*MORE experiential, inductive, hands-on learning *MORE active learning, with all the attendant noise and movement of students doing, talking, and Collaborating

*MORE diverse roles for teachers, including coaching, demonstrating, and modeling

*MORE emphasis on higher-order thinking: learning a field’s key concepts and principles

*MORE deep study of a smaller number of topics, so that students internalize the field’s way of inquiry *MORE reading of real texts: whole books, primary sources, and nonfiction materials

*MORE responsibility transferred to students for their work: goal setting, record keeping, monitoring, sharing,

exhibiting, and evaluating *MORE choice for students (e.g., choosing their own books, writing topics, team partners, and research projects)

*MORE enacting and modeling of the principles of democracy in school

*MORE attention to affective needs and varying cognitive styles of individual students *MORE cooperative, collaborative activity; developing the classroom as an interdependent community

*MORE heterogeneous classrooms where individual needs are met through individualized activities, not

segregation of bodies

*MORE delivery of special help to students in regular classrooms *MORE varied and cooperative roles for teachers, parents, and administrators

*MORE reliance on descriptive evaluations of student growth, including observational/anecdotal records,

conference notes, and performance assessment rubrics


Best Practices ~ Math INCREASE DECREASE

TEACHING PRACTICES

Use of manipulative materials

Cooperative group work Discussion of mathematics

Questioning and making conjectures

Justification of thinking

Writing about mathematics Problem-solving approach to instruction

Content integration

Use of calculators and computers Being a facilitator of learning

Assessing learning as an integral part of instruction

PROBLEM SOLVING

Word problems with a variety of structures and solution paths

Everyday problems and applications

Problem-solving strategies (especially representational strategies) Open-ended problems and extended problem-solving projects

Investigating and formulating questions from problem situations

CREATING REPRESENTATIONS

Creating one’s own representations that make sense

Creating multiple representations of the same problem or situation

Translating between representations of the same problem or situation Representations using electronic technology

Using representations to make the abstract ideas more concrete

Using representations to build understanding of concepts through reflection

Sharing representations to communicate ideas

COMMUNICATING MATH IDEAS

Discussing mathematics

TEACHING PRACTICES

Rote practice

Rote memorization of rules and formulas Teaching by telling

Single answers and single methods to find answers

Stressing memorization instead of understanding

Repetitive written practice Use of drill worksheets

Teaching computation out of context

Reliance on paper and pencil calculations Being the dispenser of knowledge

Testing for grades only

PROBLEM SOLVING

Use of cue words to determine operation to be used

Practicing problems categorized by type

Practicing routine, one-step problems

CREATING REPRESENTATIONS

Copying conventional representations without understanding

Reliance on a few representations

Premature introduction of highly abstract representations Forms of representations as an end product or goal

COMMUNICATING MATH IDEAS

Doing fill-in-the-blank worksheets


Reading mathematics

Writing mathematics

Listening to mathematical ideas

REASONING AND PROOF

Drawing logical conclusions

Justifying answers and solution processes

Reasoning inductively and deductively

MAKING CONNECTIONS

Connecting mathematics to other subjects and to the real world Connecting topics within mathematics

Applying mathematics

NUMBERS/OPERATIONS/COMPUTATION

Developing number and operation sense

Understanding the meaning of key concepts such as place value,

fractions, decimals, ratios, proportions, and percents Various estimation strategies

Thinking strategies for basic facts

Using calculators for complex calculations

GEOMETRY/MEASUREMENT

Developing spatial sense

Actual measuring and exploring the concepts related to units of measure

Using geometry in problem solving

STATISTICS/PROBABILITY

Collecting and organizing data

Using statistical methods to describe, analyze, evaluate, and make

decisions

ALGEBRA

Recognizing and describing patterns Identifying and using functional relationships

Developing and using tables, graphs, and rules to describe situations

Using variables to express relationships

Answering questions that need only yes or no responses

Answering questions that need only numerical responses

REASONING AND PROOF

Relying on authorities (teacher, answer key)

MAKING CONNECTIONS

Learning isolated topics Developing skills out of context

NUMBERS/OPERATIONS/COMPUTATION

Early use of symbolic notation

Memorizing rules and procedures without Complex and tedious paper-and-pencil computations

GEOMETRY/MEASUREMENT

Memorizing facts and relationships Memorizing equivalencies between units or measure

Memorizing geometric formulas

STATISTICS/PROBABILITY

Memorizing formulas

ALGEBRA

Manipulating symbols

Memorizing procedures


ASSESSMENT

Making assessment an integral part of teaching

Focusing on a broad range of mathematical skills Tasks and taking a holistic view of mathematics

Developing problem situations that require applications of a number of

mathematical ideas Using multiple assessment techniques, including written, oral, and

demonstration formats

ASSESSMENT

Having assessment be simply counting correct answers on tests for the

sole purpose of assigning grades Focusing on a large number of specific and isolated skills

Using exercises or word problems requiring only one or two skills

Using only written tests

Mathematics Curriculum Geometry · 1/2/2019 · Although there are many types of geometry, school...

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Transcript of Mathematics Curriculum Geometry · 1/2/2019 · Although there are many types of geometry, school...