ReseaRch—BesT PRacTIces Putting Research into Practice€¦ · their properties. A figure is no...

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ReseaRch—BesT PRacTIces

Putting Research into Practice

Dr. Karen C. Fuson, Math Expressions Author

From Current Research: How Geometric Thinking Develops

At the visual level of thinking, figures are judged by their appearance. We say, “It is a square. I know that it is one because I see it is.” Children might say, “It is a rectangle because it looks like a box.”

At the next level, the descriptive level, figures are the bearers of their properties. A figure is no longer judged because “it looks like one” but rather because it has certain properties. For example, an equilateral triangle has such properties as three sides; all sides equal; three equal angles; and symmetry, both about a line and rotational.

At this level, language is important for describing shapes. However, at the descriptive level, properties are not yet logically ordered, so a triangle with equal sides is not necessarily one with equal angles.

At the next level, the informal deduction level, properties are logically ordered. They are deduced from one another; one property precedes or follows from another property. Students use properties that they already know to formulate definitions—for example, for squares, rectangles, and equilateral triangles—and use them to justify relationships, such as explaining why all squares are rectangles or why the sum of the angle measures of any triangle must be 180°. However, at this level, the intrinsic meaning of deduction, that is, the role of axioms, definitions, theorems, and their converses, is not understood.

UNIT 8 | Overview | 711Q

Research & Math BackgroundContents Planning

Clements, Douglas H. “Teaching and Learning Geometry”. A Research Companion to Principles and Standards for School Mathematics. 2003. NCTM: Reston, VA pp. 151–178

Beckmann, Sybilla. Mathematics for Elementary Teachers with Activity Manual Third Edition. 2011. Addison-Wesley Pearson, pp. 425–466

Van de Walle, John A., Karp, Karen M., Bay-Williams, Jennifer M. Elementary and Middle School Mathematics: Teaching Developmentally Seventh Edition 2009. Allyn & Bacon

National Council of Teachers of Mathematics, Teaching Children Mathematics (Focus Issue: Geometry and Geometric Thinking) 5.6 (Feb. 1999).

My experience as a teacher of geometry convinces me that all too often, students have not yet achieved this level of informal deduction. Consequently, they are not successful in their study of the kind of Geometry that Euclid created, which involves formal deduction.

van Hiele, Pierre M. “Developing Geometric Thinking Through Activities That Begin with Play.” Teaching Children Mathematics 5.6 (Feb. 1999).

Other Useful References

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Getting Ready To Teach Unit 8Using the Common Core Standards for Mathematical PracticeThe Common Core State Standards for Mathematical Content indicate what concepts, skills, and problem solving students should learn. The Common Core State Standards for Mathematical Practice indicate how students should demonstrate understanding. These Mathematical Practices are embedded directly into the Student and Teacher Editions for each unit in Math Expressions. As you use the teaching suggestions, you will automatically implement a teaching style that encourages students to demonstrate a thorough understanding of concepts, skills, and problems. In this program, Math Talk suggestions are a vehicle used to encourage discussion that supports all eight Mathematical Practices. See examples in Mathematical Practice 6.

Mathematical Practice 1Make sense of problems and persevere in solving them.

Students analyze and make conjectures about how to solve a problem. They plan, monitor, and check their solutions. They determine if their answers are reasonable and can justify their reasoning.

TeACHeR eDITION: examples from Unit 8

MP.1 Make Sense of Problems Check Answers Encourage students to check their answers to Problems 1–5 by substituting the solution into the original equation.

Lesson 6

MP.1 Make Sense of Problems Act It Out Students in the group can make a map in the classroom using the paths between desks or tables for the streets. Ask for volunteers to walk along the streets. Identify which paths are parallel (the students never meet), perpendicular (the students meet at a right angle), or neither (the paths cross at an acute or obtuse angle).

Lesson 7

Mathematical Practice 1 is integrated into Unit 8 in the following ways:

Analyze RelationshipsAct it Out

Write an EquationDraw a Diagram

Check AnswersUse a Different Method

UNIT 8 | Overview | 711S

ACTIVITY 2ACTIVITY 2


Mathematical Practice 2Reason abstractly and quantitatively.

Students make sense of quantities and their relationships in problem situations. They can connect diagrams and equations for a given situation. Quantitative reasoning entails attending to the meaning of quantities. In this unit, this involves using equations to find sums and differences of angle measures.

TeacheR ediTion: examples from Unit 8

MP.2 Reason Quantitatively In Exercise 4, make sure students understand that they need to draw three angles whose sum is 360°. This exercise introduces the idea that angle measures, like other measures, are additive. Students can add the measures in Exercise 5 to confirm the sum. Extend the concept by having students use Circles (TRB M28) to draw and measure four or five angles with a sum of 360°.

Lesson 3

MP.2 Reason abstractly and Quantitatively Connect Diagrams and Equations In this activity, students will use information from a diagram to write an equation. To set up their equations, they must identify the pieces of information that are given and recognize what information is unknown. Some of the angle measures may be stated in the problem while others will be labeled on the diagram.

Lesson 5


Reason Abstractly Reason Quantitatively

Reason Abstractly and QuantitativelyConnect Diagrams and Equations

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Mathematical Practice 3Construct viable arguments and critique the reasoning of others.

Students use stated assumptions, definitions, and previously established results in constructing arguments. They are able to analyze situations and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others.

Students are also able to distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is. Students can listen to or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.

Math Talk is a conversation tool by which students formulate ideas and analyze responses and engage in discourse. See also MP.6 Attend to Precision.


MP.3 Construct a Viable Argument Write the following statements on the board:

A rectangle is never a square. false

A square is always a parallelogram. true

Ask Student Pairs to decide if each statement is true or false. Students should write or state sentences that support their answer. Their sentences may include examples or counterexamples.

Lesson 8

What’s the Error? S M A L L G R O U P S

MP.3, MP.6 Construct Viable Arguments/Critique Reasoning of Others Puzzled Penguin Have students look at the figures for Exercises 9–13 on Student Book page 279.

• Puzzled Penguin says angle A in the second figure near Exercises 9–13 measures 90°. Is Puzzled Penguin correct? No; A is the common endpoint for a number of rays, so there is more than one angle A.

• What can Puzzled Penguin do to fix the error? Puzzled Penguin can name the angle with three letters: ∠IAO or ∠OAI.

Ask for volunteers to name all of the angles. Possible answer: ∠UAE, ∠UAO, ∠UAI, ∠EAO, ∠EAI, ∠OAI

Lesson 2


Construct a Viable ArgumentCritique the Reasoning of Others

Puzzled Penguin Compare Representations

UNIT 8 | Overview | 711U

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Mathematical Practice 4Model with mathematics.

Students can apply the mathematics they know to solve problems that arise in everyday life. This might be as simple as writing an equation to solve a problem. Students might draw diagrams to lead them to a solution for a problem. Students apply what they know and are comfortable making assumptions and approximations to simplify a complicated situation. They are able to identify important quantities in a practical situation and represent their relationships using such tools as diagrams, tables, graphs, and formulas.

Teacher ediTion: examples from Unit 8

MP.1, MP.4 Make Sense of Problems/Model with Mathematics Write an Equation For Problems 1–5 on Student Book page 293, students use addition to find the measures of whole angles. Ask students to carefully read each problem to determine if the information they need is marked directly on the diagram or if it can be determined from given information. In Problem 3, for example, students are told that ∠VST is a right angle. Students should then realize they know that angle’s measure (90°) even though the angle measure is not actually shown as a label on the diagram. Students should write and solve an equation to solve each problem.

Lesson 6

MP.4 Model with Mathematics Make a Model In Problem 8 on Student Book page 294, students find the angle of repose for sand. This is the angle a granular material makes with the ground when it is poured into a conical shape. Students can find their own angles of repose for granular materials like sugar or rice (30–44°), salt (32°), crushed ice (19°), oats (21°), or wheat flour (45°). Pour the material through a funnel to form a conical shape. Then measure the outside angle at the base of the cone. Subtract that measure from 180° to get the angle of repose. (Possible angle measures are given.)

Lesson 6


Draw a DiagramWrite an EquationMake a Model

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Mathematical Practice 5Use appropriate tools strategically.

Students consider the available tools and models when solving mathematical problems. Students make sound decisions about when each of these tools might be helpful. These tools might include paper and pencil, a straightedge, a ruler, or the MathBoard. They recognize both the insight to be gained from using the tool and the tool’s limitations. When making mathematical models, they are able to identify quantities in a practical situation and represent relationships using modeling tools such as diagrams, grid paper, tables, graphs, and equations.

Modeling numbers in problems and in computations is a central focus in Math Expressions lessons. Students learn and develop models to solve numerical problems and to model problem situations. Students continually use both kinds of modeling throughout the program.


MP.5 Use Appropriate Tools Concrete Model Have students make a right angle with their pencils or draw a right angle on a piece of paper.

• How can you remember what a right angle is? Rectangles have right angles; the prefix rect- and the word right sound a bit alike.

Ask students to see how many different ways they can flip or reverse their angles without changing the angle size. Make sure that they understand that only the size of the angle counts. Neither the orientation of the angles nor the lengths of the sides of the angle (here, the pencils) matter.

Lesson 1

MP.5 Use Appropriate Tools MathBoard Explain to students that an angle can be separated, or decomposed, into two or more smaller angles. Have students sketch an angle on paper or on their MathBoards. Then have students draw another ray from the vertex into the interior of the original angle. They can include labels to be able to name the original angle and the new adjacent angles formed. Students should see that, as in Activity 1, the measure of the whole angle is the sum of the measures of the smaller angles they created.

Lesson 5


Use Appropriate ToolsMathBoardUse a Straightedge

Use a RulerConcrete ModelDraw a Diagram

Use CutoutsProtractor

UNIT 8 | Overview | 711W

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Mathematical Practice 6Attend to precision.

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. They are careful about specifying units of measure to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, expressing numerical answers with a degree of precision appropriate for the problem context. Students give carefully formulated explanations to each other.

TeAcher ediTion: examples from Unit 8

MP.6 Attend to Precision Verify Solutions To verify that two given lines are perpendicular or to draw perpendicular lines themselves, students can use the corner of a piece of paper. They can also use a protractor to draw the 4 right angles.

Lesson 7

MP.6 Attend to Precision Explain a Representation Have volunteers share their completed drawings with the class. Students should explain why the shape in their drawing has line symmetry. Ask the class if they agree with the line of symmetry that each drawing shows.

Lesson 11

MATH TALK Have students discuss which triangle can be used twice to form each of the quadrilaterals as they complete Exercises 10–15 on Student Book page 308. (Alternatively, you may want to give students the figures on the Triangles and Diagonals page (TRB M18).)

Lesson 9

MATH TALKin ACTION

Explain that each 1-degree turn is 1 ___ 360 of the way around from a ray back to itself, or 1 ___ 360 of a circle.

The drawing on Student Book page 277 shows that a right angle traces a quarter turn in a circle. How many degrees is this?

Maya: We need to find how many 1-degree turns are in a quarter turn.

Charlie: I can multiply to find the answer. There are 360 degrees in a circle. To find how many degrees are in a quarter turn, I multiply 360° × 1 _ 4 = 90°.

Rachel: So, to make a right angle, you make 90 1-degree turns around a circle. Since 90 × 1 ___ 360 = 1 _ 4 , I know that 90° is one quarter of a circle.

Lesson 2


Attend to PrecisionExplain a Representation

Verify a SolutionPuzzled Penguin

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Mathematical Practice 7Look for structure.

Students analyze problems to discern a pattern or structure. They draw conclusions about the structure of the relationships they have identified.


MP.7 Look for Structure Identify Relationships In Exercises 4–15, students look at the angles of a triangle to name it. Encourage students to look at all of the angles in a triangle to make their decisions. Right triangles and obtuse triangles have acute angles also but are named by their largest angles.

Invite volunteers to share their answers to Exercise 16. Guide students to look at angles of different sizes and to compare the lengths of the sides opposite such angles.

Lesson 4

MP.7 Use Structure For Exercises 17–19, point out that the letter names of each triangle can be written in different ways. For example, the possible letter names for the triangle with vertices A, B, and C are: ∆ABC, ∆ACB, ∆BAC, ∆BCA, ∆CAB, and ∆CBA. Give students time to write the possible three-letter names for the triangle with vertices D, E, and F.

Lesson 4


Look for Structure Use StructureIdentify Relationships

UNIT 8 | Overview | 711Y

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Class Activity

Czechia Switzerland South Africa Denmark Kuwait

Sudan Guyana Paraguay Japan India

Bangladesh Jamaica Trinidad andTobago

United Statesof America

Laos

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► Math and Flags of the World

Flags are used in many different ways. Some sports teams use flags to generate team spirit, a flag might be used to start a race, or a homeowner might use a flag for decoration. States and countries also use flags as a representation of their communities. Each flag is different, both in color and design.

Use the designs on the flags to answer the questions.

1. What types of quadrilaterals are used in the Kuwait flag?

2. How many designs have no parallel lines? Name the flags.

3. How many designs have perpendicular lines? Name the flags.

4. Which designs have at least two lines of symmetry?

Name Date

rectangle, trapezoids

4; Czechia, Guyana, Japan, Bangladesh

4; Switzerland, Denmark, Kuwait, United States of America

Switzerland, Paraguay, Japan, India, Jamaica, Laos

UNIT 8 LESSON 12 Focus on Mathematical Practices 317

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Class Activity

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► Designer FlagsDesign your own flag in the space below. Your flag design should include each of the following: one triangle, one pair of parallel lines, and one 30° angle.

5. What type of triangle did you draw in your flag design? Explain how the sides of the triangle helped you classify the triangle.

6. Compare the flag design you made to the flag design that a classmate made. How are the two designs the same? How are they different? What shapes did you use that your classmate did not use?

Name Date

Answers will vary based on the type of triangle drawn.

Answers will vary.

318 UNIT 8 LESSON 12 Focus on Mathematical Practices

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Student edItIOn: LeSSOn 12 pageS 317–318

Mathematical practice 8Look for and express regularity in repeated reasoning.

Students use repeated reasoning as they analyze patterns, relationships, and calculations to generalize methods, rules, and shortcuts. As they work to solve a problem, students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results.

teacher edItIOn: examples from unit 8

Mp.8 use repeated reasoning Generalize Students should be able to use the same method to talk about other angles as parts of a circle. Some examples might be: For a complete turn around the circle, what is the measure? 360 What fraction of a circle is a 90° angle? 1 _ 4 What fraction of a circle is a 180° angle? 1 _ 2 What about a 40° angle? 1 _ 9 What about a 120° angle? 1 _ 3

Lesson 3

Mp.8 use repeated reasoning Conclude After students complete the exercises, they should be able to draw the conclusion that some quadrilaterals have exactly 1 pair of parallel lines, some have 2 pairs of parallel lines, while others have none.

Lesson 10


GeneralizeConcludeDraw Conclusions

Focus on Mathematical practices Unit 8 includes a special lesson that involves solving real world problems and incorporates all eight Mathematical Practices. In this lesson students use what they know about parallel and perpendicular lines, triangles, quadrilaterals and lines of symmetry to solve problems about flag designs.

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Getting Ready to Teach Unit 8Learning Path in the Common Core StandardsUnit 8 broadens students’ understanding of plane geometric figures including points, lines, line segments, rays, angles, and polygons. Students learn how to measure angles using a protractor and sort and classify angles and triangles.

Students also apply their understanding of geometric properties, such as parallel and perpendicular lines and congruence, to identify certain types of quadrilaterals and to explore the relationships among quadrilaterals. Students also explore the relationships between triangles and rectangles and the properties of line symmetry.

Help Students Avoid Common ErrorsMath Expressions gives students opportunities to analyze and correct errors, explaining why the reasoning was flawed.

In this unit, we use Puzzled Penguin to show typical errors that students make. Students enjoy explaining Puzzled Penguin’s errors and teaching Puzzled Penguin the correct way to measure angles and analyze geometric figures. The following common errors are presented to the students as letters from Puzzled Penguin and as problems in the Teacher Edition that were solved incorrectly by Puzzled Penguin.

→ Lesson 2: Using one letter to name an angle when several angles share that vertex

→ Lesson 5: Incorrectly solving an equation to find an unknown angle measure

→ Lesson 9: Drawing a line that is not perpendicular to the opposite side of the triangle

→ Lesson 11: Drawing a line that results in shapes which are the same size and shape, but which are not also mirror images

In addition to Puzzled Penguin, there are other suggestions listed in the Teacher Edition to help you watch for situations that may lead to common errors. As a part of the Unit Test Teacher Edition pages, you will find a common error and prescription listed for each test item.

Math Expressions VOCABULARY

As you teach the unit, emphasize

understanding of these terms.

• angle• degree• diagonal• line symmetry

See the Teacher Glossary.

UNIT 8 | Overview | 711AA


Angles

Lessons

1 2 3

Plane Figures In Lesson 1, students identify, explore the properties of, and draw plane figures including lines, line segments, rays, points, and angles. For example, students use the definition of a line to reason that although the lines below are drawn to different lengths, they are really the same length because all lines go on forever in both directions.

Allowing students to explore and compare the properties of these figures, as well as giving them an opportunity to draw them, is beneficial because it helps conceptualize what can often be abstract definitions of plane figures.

Classifying Angles In Lesson 1, students classify angles as acute, obtuse, or right by comparing angles to a right angle. The lesson presents the size of an angle as the amount of rotation, or turn, from one side of the angle to the other side. This informal exploration provides the foundation students will use as they move on to formalizing their knowledge of angles when they use a protractor to measure angles and compare the measurement to 90°.

Degrees Lesson 2 presents the concept of the degree as the unit of measurement used to measure the size of an angle. In the images below, students can see an angle with the measure of 1 degree, or 1°, and that five of these angles create an angle with the measure of 5°.

The measure of an angle is the total number of 1-degree angles that fit inside it.

This angle measures 5 degrees.

Images like the ones above are beneficial for students because they help them visualize the degree as a measurement unit. Additionally, they can more easily see that 5 ∙ 1° = 5°.

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Angles and Circles Lesson 2 also presents angles in the context of circles. An angle is shown as an interior angle of a circle with its vertex at the center of the circle. Students see, for example, that a right angle has a measure of 90° and that a 90° turn traces one quarter of a circle. In this way, students see that a 360° angle traces a complete circle and each degree is 1 ___ 360 of the way around the circle.

A right angle has a measure of 90°. A 90° turn traces one quarter of a circle. 90˚

A straight angle measures 180°. A 180° turn traces one half of a circle.

The angle below measures 360°. A 360° turn traces a complete circle.

180˚ 360˚

Measuring Angles In Lesson 2, students use a protractor to measure angles. To use a protractor accurately, students need to align the vertex of the angle carefully with the center mark on the protractor and align one ray with the zero line. Most protractors are labeled clockwise and counterclockwise. Students should choose the appropriate scale depending on the orientation of the angle.

C

A

B

In Lesson 3, students apply their understanding of circles and using a protractor to drawing interior angles of given sizes in circles.

Students connect to their understanding of fractions as they express angle measures as fractions of 360°. For example, they describe a 60° angle as being 60 ___ 360 = 1 _ 6 of the circle or as an angle that makes 60 1-degree turns. Since each of those turns is 1 ___ 360 of the circle, a 60° turn is 6 ∙ 1 ___ 360 = 60 ___ 360 = 1 _ 6 of the circle.

UNIT 8 | Overview | 711CC


Triangles

Lessons

4 5 6

Classifying Triangles In Lesson 4, students learn to identify the angles in a triangle as obtuse, right, or acute. They use these observations to classify triangles by angles as obtuse triangles, right triangles, or acute triangles.

right obtuse acute

Students also attend to the sides of a triangle to identify the sides as equal or not equal. They use the observations of the congruency of the sides to classify the triangles by sides as equilateral, isosceles, or scalene.

equilateral isosceles scalene

As students gain experience in identifying the attributes of the angles and sides in triangles, they begin to realize that triangles can be classified in different ways. For example, if an acute triangle has 0 equal sides, it is also scalene. If a right triangle has 2 equal sides, it is also isosceles. During this exploration, students also realize that some combinations of classifications are not possible. For example, an equilateral triangle cannot also be right or obtuse and a right triangle cannot have an obtuse angle.

Triangles

acute isosceles

S; r S; o

E; a

I; o

I; a

I; r

S; a

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Angle Measure Sums and Differences In Lesson 5, students learn how to add angle measures of two adjacent angles. This concept lays the foundation that students will need when they learn about angle pairs, such as complementary and supplementary angles, in future grades. For example, students see that a 30° and a 60° angle make a right angle and that a 130° and a 50° angle make a straight angle.

30˚60˚

E

GF

H

130˚ 50˚

R

QPN

right angle straight angle

Students apply this understanding of adding angles to subtracting angles to find unknown angle measures.

72˚ ?

D

CBA

For example, in the diagram above, if students know that the measure of angle ABC is 180°, they can write and solve an equation to find the unknown angle measure.

180° - 72° = x or 72° + x = 180°

Solving equations involving angles is beneficial because it helps students connect their algebraic and geometric understandings.

UNIT 8 | Overview | 711EE


Triangles and Quadrilaterals

Lessons

7 8 9 10 11

Quadrilaterals In Unit 8, students explore the attributes of quadrilaterals. Exploration of quadrilaterals is the context through which students learn about special line and line segment pairs: parallel lines and perpendicular lines.

parallel line segments perpendicular lines

Students also learn to identify adjacent sides and opposite sides of figures. Students apply this understanding to identify and classify quadrilaterals, such as parallelograms, trapezoids, rhombuses, rectangles, and squares.

parallelogram trapezoid rhombus rectangle square

Presenting students with this specialized vocabulary and having them apply it to identifying quadrilaterals is beneficial because it gives students a meaningful context through which they not only learn these ideas, but apply them as well.

Students deepen their knowledge as they explore the relationships among quadrilaterals. The category diagram below is presented as a way to represent these relationships. A category diagram is a beneficial tool because it helps students to organize their thinking.

Parallelogram

Rhombus Square Rectangle

Quadrilateral

Trapezoid

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Relating Triangles and Quadrilaterals In Lesson 9, students use the concept of a diagonal to explore the relationship between triangles and quadrilaterals. Students see that a diagonal divides a quadrilateral into two triangles which have the same size and shape. This concept forms the basis for the formulas for the areas of triangles and parallelograms. Students see that a quadrilateral can be decomposed into two triangles with the same size and shape and that two triangles with the same size and shape can be composed into a quadrilateral.

E F

GH

E F

GH

E F

GH

Students also learn how to draw perpendicular lines in triangles. This skill prepares students to identify the altitude, or height, of a figure without right angles.

Polygons Lesson 10 introduces the idea of a polygon. Polygons are closed figures with straight sides, and triangles and quadrilaterals are types of polygons. Students sort triangles and quadrilaterals by angles and by sides; for example, as polygons with right angles and as polygons with perpendicular sides. Students learn that polygons can be sorted in more than one way.

Line Symmetry To expand students’ exploration of the properties of geometric figures, Lesson 11 presents line symmetry. Students see that a line of symmetry divides a figure in half so that, if the figure is folded along the line, the two halves will match exactly. Students determine if a figure has line symmetry, and if so, identify the line or lines of symmetry. Students are asked not only to draw lines of symmetry, but also to draw the other half of symmetric figures. This constructing helps solidify their understanding of symmetry. Using graph paper further enables students to visualize symmetry in figures.

E F

GH

E F

GH

E F

GH

UNIT 8 | Overview | 711GG


Problem Solving

Lessons

2 6 7

Problem Solving Plan In Math Expressions a research-based problem-solving approach that focuses on problem types is used.

• Interpret the problem• Represent the situation• Solve the problem• Check that the answer makes sense.

Maps In Lessons 2 and 7, students apply their understanding of geometric figures to solving problems involving maps. Students solve problems involving roads represented as line segments, including parallel and perpendicular line segments, and turns represented as angles.

Adding and Subtracting Angle Measures In Lesson 6, students apply their understanding of angle measures by adding and subtracting angle measures to solve real life problems. For example, they analyze the angles in bridges and the way balls rebound in miniature golf games to solve problems. Students also apply their understanding of angle measures to help them interpret a circle graph.

CatFish

Dog

Horse 60º85º

150º

Focus on Mathematical Practices

Lesson

12

The standards for Mathematical Practice are included in every lesson of this unit. However, there is an additional lesson that focuses on all eight Mathematical Practices. In this lesson, students use what they know about geometry to solve problems involving different types of flag designs.

711HH | UNIT 8 | Overview

ReseaRch—BesT PRacTIces Putting Research into Practice€¦ · their properties. A figure is no...

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