Classify Two-Dimensional Figures - Full Lesson

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Lesson Overview

LESSON 29

Classify Two-Dimensional Figures

Learning Progression

In Grade 4 students classified two-dimensional figures based on their attributes, such as having parallel or perpendicular sides and having right, acute, or obtuse angles. In the previous Grade 5 lesson students described how attributes belonging to a larger category belong to subcategories of that category and they used simple Venn diagrams and tree diagrams to show category/subcategory relationships.

In this lesson students use their understanding of shape properties, categories, and subcategories to classify shapes into Venn diagrams and tree diagrams, including Venn diagrams that have regions that partially overlap.

In Grade 6 students will work with figures in the coordinate plane and will find the area of two-dimensional figures.

Lesson Vocabulary

• trapezoid (exclusive) a quadrilateral with exactly one pair of parallel sides.

• trapezoid (inclusive) a quadrilateral with at least one pair of parallel sides.

Review the following key terms.

• attribute any characteristic of an object or shape, such as number of sides or angles, lengths of sides, or angle measures.

• category a collection of objects grouped together based on attributes they have in common.

• hierarchy a ranking of categories based on attributes.

• polygon a two-dimensional closed figure made with three or more straight line segments that do not cross over each other.

• subcategory a category within a larger category. It shares all the same attributes as the larger category. For example, parallelograms are a subcategory of quadrilaterals.

• tree diagram a hierarchy diagram that connects categories and subcategories with lines to show how they are related.

• Venn diagram a diagram that uses overlapping ovals (or other shapes) to show how sets of numbers or objects are related.

Lesson Objectives

Content Objectives• Classify two-dimensional figures in a

Venn diagram or tree diagram based on properties of the figures.

• Draw and use Venn diagrams and tree diagrams to show the relationships among categories of two-dimensional figures.

Language Objectives• Discuss the key term attribute and its

meaning with a partner.

• Explain how Venn diagrams and tree diagrams show the relationships between categories.

• Describe relationships among two-dimensional figures shown in Venn diagrams and tree diagrams.

Prerequisite Skills

• Recognize parallel and perpendicular lines.

• Recognize right, acute, and obtuse angles.

• Recognize that triangles can be classified based on the lengths of their sides (isosceles, equilateral, scalene).

• Sort two-dimensional figures based on the kinds of sides they have and on the kinds of angles they have.

• Understand how a Venn diagram and a tree diagram show category-subcategory relationships.

CCSS FocusDomainGeometry

ClusterB. Classify two-dimensional figures into

categories based on their properties.

Standard5.G.B.4 Classify two-dimensional figures in a hierarchy based on properties.

Additional Standard5.G.B.3 (See Standards Correlations at the end of the book for full text.)

Standards for Mathematical Practice (SMP)SMPs 1, 2, 3, 4, 5, and 6 are integrated in every lesson through the Try-Discuss-Connect routine.*

In addition, this lesson particularly emphasizes the following SMPs:

5 Use appropriate tools strategically.

7 Look for and make use of structure.

* See page 305m to see how every lesson includes these SMPs.

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Lesson Pacing Guide

PERSONALIZE

i-Ready Lesson*Grade 5• Classify Two-Dimensional Figures

Independent Learning

PREPARE

Ready Prerequisite LessonGrade 4• Lesson 33 Classify Two-Dimensional Figures

RETEACH

Tools for InstructionGrade 4• Lesson 33 Attributes of Shapes

Grade 5• Lesson 29 Classify Plane Figures

REINFORCE

Math Center ActivitiesGrade 5• Lesson 29 Organize Polygons on a

Venn Diagram• Lesson 29 Organize Triangles on a

Venn Diagram

EXTEND

Enrichment ActivityGrade 5• Lesson 29 Sorting Shapes

Small Group DifferentiationTeacher Toolbox

Lesson MaterialsLesson (Required) none

Activities Per pair: geoboard, rubber bandsPer group: 1 set of index cards, each showing a parallelogram, square, or rectangle (prepared in various sizes, colors, orientations), set of 5 sticky notes with letters on them (blue X, red X, green X, blue Y, blue Z), set of 10 sticky notes with vegetable categories written on them (peppers, carrots, vegetables, potatoes, red pepper, hot peppers, seeded vegetables, green peppers, root vegetables, and sweet potatoes), large sheet of paper, markers, blank sticky notesFor display: masking tape, Venn diagram with ovals for Triangles, Acute Triangles, and Isosceles Triangles (see Session 3 Hands-On Activity for details)

Math Toolkit geoboard, rubber bands, tracing paper, grid paper, rulers

SESSION 1

Explore45–60 min

Interactive Tutorial* (Optional) Prerequisite Review: Identify Two-Dimensional Figures

Additional PracticeLesson pages 589–590

Classifying Two-Dimensional Figures • Start 5 min• Try It 10 min• Discuss It 10 min• Connect It 15 min• Close: Exit Ticket 5 min

SESSION 2

Develop45–60 min

Classifying Two-Dimensional Figures • Start 5 min• Try It 10 min• Discuss It 10 min• Model Its 5 min• Connect It 10 min• Close: Exit Ticket 5 min


Fluency Classifying Two-Dimensional Figures

SESSION 3

Develop45–60 min

Classifying Two-Dimensional Figures with Tree Diagrams• Start 5 min• Try It 10 min• Discuss It 10 min• Model Its 5 min• Connect It 10 min• Close: Exit Ticket 5 min


Fluency Classifying Two-Dimensional Figures with Tree Diagrams

SESSION 4

Refine45–60 min

Classifying Two-Dimensional Figures• Start 5 min• Example & Problems 1–3 15 min• Practice & Small Group

Differentiation 20 min• Close: Exit Ticket 5 min

Lesson Quiz or Digital Comprehension Check

Whole Class Instruction

* We continually update the Interactive Tutorials. Check the Teacher Toolbox for the most up-to-date offerings for this lesson.

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LESSON 29

Connect to Family, Community, and Language DevelopmentThe following activities and instructional supports provide opportunities to foster school, family, and community involvement and partnerships.

Connect to FamilyUse the Family Letter—which provides background information, math vocabulary, and an activity—to keep families apprised of what their child is learning and to encourage family involvement.

GoalThe goal of the Family Letter is to provide opportunities for family members to support students as they learn more about how two-dimensional figures are related and classified.

ActivityIn the Classifying Two-Dimensional Figures activity, students and family members use Venn diagrams to show how figures are related and describe the relationships using the figures’ attributes.

Math Talk at HomeEncourage students to continue to talk with their family members about attributes that can be used to describe two-dimensional figures and discuss the relationships between them. Remind students that two-dimensional figures can be classified into categories and subcategories based on their attributes.

Conversation Starters Below are additional conversation starters children can write in their Family Letter or math journal to engage family members:

• What are some ways you classify objects at work or at home?

• What are some objects you can classify by size? by shape?

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ACTIVITY CLASSIFYing Two-DimensiOnal FiguresDo this activity with your child to classify two-dimensional fi gures.

Work together with your child to describe how fi gures are classifi ed in Venn diagrams.

• Look at the fi gures in the Venn diagrams below and talk about how the fi gures are related to each other.

• Work together to describe the attributes of the fi gures. Tell what attributes the fi gures do and do not share. The words in the box describe some attributes of fi gures that you might use in your discussion.

three sidesright angleequal side lengths

equilateralacute anglediff erent side lengths

isoscelesobtuse angle

• Write category names in each oval to classify the shapes.

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Available in Spanish

Teacher Toolbox

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Classify Two-Dimensional Figures

29Dear Family,This week your child is continuing to make diagrams to classify two-dimensional figures.Your child will continue to explore how Venn diagrams and tree diagrams can be used to show relationships among categories of two-dimensional fi gures.

Figures can be grouped into categories by their attributes or properties, such as the number of their sides and angles, the length of their sides, and the measure of their angles. Your child knows all fi gures in a category share at least one attribute. While subcategories share the attribute(s) of the broader category, fi gures in subcategories have additional specifi c properties.

Your child knows how to use Venn diagrams or tree diagrams to show that one category is a subcategory of another. Now your child will use more complex diagrams to classify fi gures. The Venn diagram below shows “Triangles” as the broader category. The labeled ovals represent subcategories of triangles.

Triangles

Obtuse

Right

Acute

Isosceles

Equilateral

Invite your child to share what he or she knows about using diagrams to classify fi gures by doing the following activity together.

The category Right does not overlap Obtuse. Right triangles and obtuse triangles only share attributes of the broader category, Triangles.

The category Equilateral is nested completely inside the category Isosceles. Equilateral triangles have all the attributes of, and are a subcategory of, isosceles triangles.

Notice that Right partly overlaps Isosceles. Some, but not all, right triangles can also be classifi ed as isosceles triangles.

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Connect to Community and Cultural ResponsivenessUse these activities to connect with and leverage the diverse backgrounds and experiences of all students.

Connect to Language DevelopmentFor ELLs, use the Differentiated Instruction chart to plan and prepare for specific activities in every session.

Session 1 Use with Additional Practice, problem 3.• Discuss with students how triangles can be categorized by their

sides and angles. Encourage students to think about objects shaped like triangles. Invite volunteers to provide examples. Some examples of objects might include building blocks, tortilla chips, sails from a sailboat, and folded napkins. Have students support their answers by describing the attributes of each object.

Levels 1–3 Levels 2–4 Levels 3–5

Prepare for Session 1Use with Try It.

English Language Learners:Differentiated InstructionELL

Speaking/Reading Read the Try It problem aloud. Organize students into triads. Have each group member model one of the shapes on a geoboard. Ask them to use gestures, words, and pictures to discuss the attributes of each shape, then work together to create a Venn diagram.

Provide the following sentence starters to guide small group conversation.

• Squares are a type of .

• Rectangles are a type of .

• Rectangles and squares both are .

Speaking/Listening Read the Try It problem with students. Have them form pairs to restate the problem in their own words and determine the category words that can be used to label a Venn diagram to arrange the quadrilaterals. Provide the following sentence starters for partners to complete:

• Squares are a subcategory of .

• Rectangles are a subcategory of .

• Rectangles and squares are both subcategories of .

After partners complete their Venn diagram, ask them to explain how the diagram shows the categories ranked from most general to most specific.

Speaking/Writing Have students read and solve Try It with a partner. Have partners discuss the relationships between the categories, including describing one category as a subcategory of another. Encourage them to write their explanations using complete sentences and precise mathematical language.

Ask students to share and compare their explanations with another pair.

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LESSON 29

StartConnect to Prior KnowledgeWhy Review sorting shapes based on their properties to prepare for classifying shapes using Venn diagrams.How Have students record their groups on their papers using shape numbers. Encourage early finishers to sort in more than one way. Have students share and compare their sorts.

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Start

Grade 5 Lesson 29 Session 1 | Explore Classifying Two-Dimensional Figures

Divide the shapes into two groups. Give each group a title to explain your grouping.

2

6

1

4

3

5

Possible SolutionsGroup 1: Shapes with a right angle (1, 5, 6); Group 2: Shapes without a right angle (2, 3, 4);Group 1: Quadrilaterals (2, 3, 5, 6); Group 2: Not Quadrilaterals (1, 4)

TRY ITMake Sense of the ProblemTo support students in making sense of the problem, help them understand each shape’s name as well as the task of drawing a Venn diagram.

DISCUSS ITSupport Partner DiscussionTo reinforce the language of hierarchy, encourage students to use attributes, properties, category, and subcategory as they talk to each other.

Look for, and prompt as necessary for, understanding of:

• the shapes are rectangle, parallelogram, and square

• Venn diagrams show a subcategory with an oval entirely contained within the oval for the broader category

Common Misconception Look for students who are not comfortable categorizing the shapes from general to specific. As students present solutions, have them specify the properties of each shape.

Select and Sequence Student SolutionsOne possible order for whole class discussion:

• concrete models or drawings to support reasoning about properties

• lists or tables of properties along with a Venn diagram

• Venn diagrams with categories from parallelograms to rectangles to squares

• Venn diagrams that include the broader category of quadrilaterals

Support Whole Class DiscussionPrompt students to explain the relationships shown by their diagrams.

Ask How do [student name]’s and [student name]’s Venn diagrams show how to order the shapes from the most general category to the most specific subcategory?

Listen for The most general category is quadrilaterals because all 3 shapes have 4 sides and 4 angles. The first subcategory is parallelograms, which also have 2 pairs of parallel sides of equal length. Next are rectangles, which also have 4 right angles. The most specific subcategory is squares, which also have 4 sides of equal length.

Purpose In this session students draw on their understandings of categories and subcategories of two-dimensional shapes to arrange given quadrilaterals into a Venn diagram. They share their drawings of Venn diagrams with one another to explore how the shapes share attributes. They will look ahead to think about how Venn diagrams can use partially overlapping regions to show that two categories of shapes share properties even though neither category is a subcategory of the other.

SESSION 1 Explore

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Learning Target• Classify two-dimensional fi gures in a

hierarchy based on properties.SMP 1, 2, 3, 4, 5, 6, 7

LESSON 29

TRY IT Math Toolkit• geoboard• rubber bands• tracing paper• grid paper• rulers

DISCUSS ITAsk your partner: Can you explain that again?

Tell your partner: I started by . . .

Previously, you learned that attributes of a category are shared by its subcategories. Use what you know to try to solve the problem below.

Arrange the quadrilaterals shown below into a Venn diagram that shows a hierarchy of categories from most general to most specifi c. Label the categories represented by these shapes in your Venn diagram.

Explore Classifying Two-Dimensional FiguresSESSION 1

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Possible student work:

Sample A

Parallelograms

Rectangles

Squares

Sample B

Quadrilaterals

Parallelograms

Rectangles

Squares

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LESSON 29 EXPLORE

Lesson 29 Classify Two-Dimensional Figures

SESSION 1

CONNECT IT1 LOOK BACK

Explain how you organized the shapes in your Venn diagram.

2 LOOK AHEADIn the Venn diagram below, the red and blue ovals represent subcategories of polygons. Three polygons are placed into the ovals, with one shape in both ovals.

Polygons

a. What property is shared by the two shapes in the red oval?

b. What property is shared by the two shapes in the blue oval?

c. Use your answers to parts a and b to write subcategory labels for each oval.

d. Why is the equilateral triangle inside of both the red and blue ovals?

3 REFLECTIn a Venn diagram, what is true about two categories represented by ovals that overlap without one oval being completely inside the other?

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Possible answer: They both have three sides.

Triangles Equilateral Polygons

Possible answer: Shapes in the two categories each have some, but not all,

of the properties of the other category.

Possible answer: The parallelogram is in the outer region because the rectangle and the square have all the attributes of a parallelogram, so rectangles and squares are subcategories of parallelograms. The rectangle is next and the square is last because a square has all the attributes of a rectangle.

Possible answer: They have all sides equal in length.

An equilateral triangle has three sides and all equal sides.

CONNECT IT 1 LOOK BACK

Look for understanding that the shapes are organized in a hierarchy starting with the broader category of quadrilaterals (or parallelograms if students don’t include quadrilaterals) to the most specific subcategory of squares.

Visual ModelMake a class Venn diagram.

If . . . some students are unsure about ordering figures by properties in a Venn diagram,

Then . . . use this activity to have them act out sorting and classifying shapes.

Materials For each group: 1 set of index cards, each showing pictures of a parallelogram, square, or rectangle (prepared in various sizes, colors, orientations); For display: masking tape

• In an open floor space, use tape to outline a nested Venn diagram with regions for parallelograms, rectangles, and squares.

• Give each student an index card with a shape.

• Have the group stand in the outside of the oval for parallelograms. Ask: Look at your shape. Why do you belong in the Parallelograms region? What properties do you all share? [4 sides, 2 pairs of parallel sides, 2 pairs of sides with equal length]

• Tell students to move to the Rectangles region if they think their shape has the properties of a rectangle. Ask: What property do you have that the other parallelograms do not have? [4 right angles] Do you still share the other properties of a parallelogram? [yes]

• Tell students to move to the Squares region if they think their shape has the properties of a square. Ask: What property do you have that other rectangles do not have? [4 equal sides]

• Have students pick new cards and repeat.

2 LOOK AHEAD Point out that Venn diagrams can also show relationships between shapes that have some properties in common but neither has all the properties of the other. Students should be able to name properties the shapes share and why the equilateral triangle belongs in the region of overlap.

Close: Exit Ticket

3 REFLECTLook for understanding that the ovals for categories overlap without one being fully contained in the other when each shape shares some but not all properties of the other shape and that the oval for a subcategory is inside the oval for another category when the shape has all properties of the broader category.

Common Misconception If students do not use precision in their explanations and just say categories that overlap share properties, then have them describe how they share properties in more detail. Ask if they share all properties, some properties, or no properties. Look for students to describe that categories that overlap share one or more properties but not all properties.

Real-World ConnectionEncourage students to think about everyday places or situations in which

people can use the concept of ordering from most general to most specific. For example, think about answering the question “Where do you live?” You could answer based on your continent, country, state, city, town, neighborhood, or street name. All answers would be accurate, but the name of your continent would be the most general answer and the name of your street would be the most specific answer.


LESSON 29

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Name:


2 Look at the Venn diagram. Why is the square inside of all the ovals?

1 Think about what you know about Venn diagrams. Fill in each box. Use words, numbers, and pictures. Show as many ideas as you can.

Examples Examples Examples

What Is It? What I Know About It

Venn diagram

LESSON 29 SESSION 1

Prepare for Classifying Two-Dimensional Figures

Parallelograms

Rectangles

Squares

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Possible answer: Squares have all of the properties of both rectangles and parallelograms, so they are also rectangles and parallelograms.

Possible answers:

a drawing that uses overlapping ovals to show how categories of shapes relate

When the ovals in a Venn diagram overlap, it means that a shape that belongs in the area of overlap can be called by the category labels of both ovals.

Quadrilaterals

Rectangles

Squares

Triangles

Isosceles

Equilateral

Polygons

Hexagons Pentagons

Solutions

Support Vocabulary Development

1 Display and say the term Venn diagram. Then display the questions What is it? and What do I know about it? Have students discuss the answers with a partner. Call on volunteers to share their responses.

Display examples of Venn diagrams and point out that the ovals in the diagrams show relationships between categories.

2 Have students read the problem and work with a partner to determine why the square is in the innermost oval. Provide the following sentence starters to guide discussions:

• The square is a rectangle because . . .

• The square is a parallelogram because . . .

Supplemental Math Vocabulary• attributes

• overlap

• relationship

SESSION 1 Additional Practice



English Language Learners:Differentiated InstructionELL

Speaking/Writing Have pairs read Connect It problem 6. Ask partners to make a list of similarities and differences between the models and strategies in the Model Its as well as strategies they used in Try It.

Have students use the list to determine which model or strategy they prefer to use. Ask students to explain their selection in writing.

Speaking/Writing Read Connect It problem 6. Pair students and have them discuss the similarities and differences between the models and strategies in the Model Its. Ask students to explain to their partners which model or strategy they like best.

Provide a sentence frame to guide students during their conversation and ask them to write their ideas:

• I like the because .

After students complete their sentences, have them take turns reading them to their partners.

Speaking/Writing Read Connect It problem 6. Review the Model Its with students. Provide the following terms and have the students repeat them after you: table, circles, inner circles.

Help students use the words to tell which model or strategy they like the best. Provide the following sentence starter:

• I like using .

Have students copy the complete sentence and read it to a partner.


LESSON 29 SESSION 1

3 Solve the problem. Show your work.

Arrange the triangles below into the Venn diagram. Label each oval as a catgeory of triangles.

Triangles

4 Check your answer. Show your work.

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All of the shapes are triangles. I can sort them into right triangles and isosceles triangles. The circled shape is both a right triangle and an isosceles triangle.

Right triangles Isosceles triangles


RightTriangles

Isosceles Triangles

Prepare for Session 2Use with Connect It.

3 Assign problem 3 to provide another look at using Venn diagrams to classify two-dimensional figures.

This problem is similar to the problem about arranging quadrilaterals in a Venn diagram. In both problems, students are asked to use a Venn diagram to show relationships between the categories of shapes. The question asks students to arrange the given triangles in a Venn diagram for which the structure is provided and to label the ovals with titles that describe the categories.

Students may want to use geoboards, tracing paper, grid paper, or rulers.

Suggest that students read the problem three times, asking themselves one of the following questions each time:

• What is this problem about?

• What is the question I am trying to answer?

• What information is important?

Solution: See the completed Venn diagram on the student page. Students may also switch the positions of the left and right triangles and reverse the labels for the ovals. Medium

4 Have students check their answer in another way, such as describing the reasoning they used to arrange the figures in the diagram.


LESSON 29


LESSON 29


Develop Classifying Two-Dimensional FiguresSESSION 2

Read and try to solve the problem below.

Sort these shapes into the Venn diagram below to show the relationships among parallelograms, squares, rectangles, rhombuses, and quadrilaterals.

A B C D E F G

Use at least one shape in each region of the diagram. Classify the shapes by labeling each region with the category name.

TRY IT

DISCUSS ITAsk your partner: How did you get started?

Tell your partner: I knew . . . so I . . .

Math Toolkit• geoboard• rubber bands• tracing paper• grid paper• rulers

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Sample A

Sample B

Note: Students may choose to draw each shape in the Venn diagram.

Quadrilaterals Shapes A and G

ParallelogramsShape B

RhombusesShape E

RectanglesShapes C and F

SquaresShape D

StartConnect to Prior KnowledgeWhy Prepare students for classifying shapes into a Venn diagram by recognizing that shapes can belong to multiple categories.How Have students identify two categories to which a shape belongs and say whether one category is a subcategory of the other.


Start

Name two categories each shape belongs to. Is one category a subcategory of the other?

1 2 3

Grade 5 Lesson 29 Session 2 | Develop Classifying Two-Dimensional Figures

Possible Solutions1: triangle, equilateral triangle;2: rectangle, square;3: polygon, quadrilateral;In each case, the second category is a subcategory of the first.Develop Language

Why Reinforce understanding of the word hierarchy.

How Remind students that a hierarchy is a way of classifying objects by ranking them in a connected order based on certain properties or attributes. Provide examples of hierarchies in everyday life, such as a family tree or a simple organizational chart of your school. Have students make connections and explain the relationships represented by the hierarchy.

TRY ITMake Sense of the ProblemTo support students in making sense of the problem, point out that the blank Venn diagram in the Try It space will be used in the problem. Be sure students understand that they are both identifying the category name for each region and placing specific shapes into those regions.

Ask How many categories does the Venn diagram have regions for?

DISCUSS ITSupport Partner DiscussionEncourage students to use the term property as they discuss their solutions.

Support as needed with questions such as:

• How did you get started?

• Did you move shapes from one oval to another as you solved the problem? Why?

Common Misconception Look for students who put the rhombus in the center of the middle overlapping ovals and the square and rectangles on the outside of these ovals. Ask them to list the properties of each shape.


• concrete models or drawings to support reasoning about properties

• lists or tables of properties

• shapes drawn in the Venn diagram

• letters used to represent shapes in the Venn diagram

Purpose In this session students solve a problem that requires sorting quadrilaterals into a Venn diagram. The purpose of this problem is to have students develop strategies for sorting shapes in a more complex Venn diagram that includes partially overlapping ovals.

SESSION 2 Develop



LESSON 29 DEVELOP


Explore ways to understand classifying two-dimensional fi gures using a Venn diagram.

Sort these shapes into a Venn diagram to show the relationships among parallelograms, squares, rectangles, rhombuses, and quadrilaterals.

A B C D E F G

Use at least one shape in each region of the diagram. Classify the shapes by labeling each region with the category name.

Model ItYou can use a table to compare properties of the categories of polygons.

Property Parallelograms Quadrilaterals Rectangles Rhombuses Squares

4 sides 3 3 3 3 3

2 pairs of parallel sides 3 3 3 3

2 pairs of sides of equal length

3 3 3 3

4 sides of equal length 3 3

4 right angles 3 3

Model ItYou can show the relationships among the categories with a Venn diagram.

Two regions of the Venn diagram are labeled and three of the shapes have been classifi ed by placing them into the diagram.

Write the remaining category names and the letters for Figures C, D, E, and F in the diagram.

Quadrilaterals

Parallelograms

A

B

G

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Rhombuses:E

Rectangles:C and F

Squares:D

Support Whole Class DiscussionCompare and connect the different representations and have students identify how they are related.

Ask How does your model show the relationships among the shapes? Is there more than one correct way to place the shapes in the Venn diagram?

Listen for Students should recognize Venn diagrams that show correct category/subcategory relationships of the shapes. Representations of properties may be in lists or tables, and Venn diagrams may include letters or shapes drawn in the regions of the diagram as well as the category labels for each oval and the region of overlap. Rhombuses and rectangles may be on either side of the region of overlap at the center of the diagram.

MODEL ITSIf no student presented these models, connect them to the student models by pointing out the ways they each represent:

• each shape in the problem

• relationships between shapes

Ask How are the shapes shown in the problem represented in each Model It?

Listen for In the first Model It, the shape names are shown in a table with properties of each shape identified by Xs. The second Model It shows the shapes sorted into a Venn diagram.

For the table model, prompt students to consider a table as an aid to identify properties.

• How does the table show the properties a shape has? the properties a shape does not have?

• How might using this type of table help you make a Venn diagram?

For the Venn diagram model, prompt students to relate the Venn diagram to the table.

• How does the placement of a shape in the Venn diagram relate to the number of Xs shown for that shape in the table?

• Rectangles and Rhombuses each have four Xs: three for the same properties and one for different properties. What does this mean about the ovals for Rectangles and Rhombuses in the Venn diagram?

Deepen UnderstandingTable ModelSMP 7 Use structure.

When discussing the table model, prompt students to consider how the number of Xs in each column can help them order the shapes from most general to most specific.

Ask How do the number of Xs show which shape is the most general category?

Listen for Quadrilaterals have one X for the attribute 4 sides and no Xs for other more specific properties, so quadrilaterals are the most general category.

Ask How do the number of Xs show which shape is the most specific subcategory?

Listen for Squares are the only shape with Xs for every property. So, squares are the most specific subcategory.

Ask If you rearranged the columns from most general to most specific, what pattern would you see in the number of Xs?

Listen for The number of Xs would increase, or stay the same, from left to right.


LESSON 29


Connect ItNow you will use the problem from the previous page to help you understand how to use Venn diagrams to classify two-dimensional fi gures.

1 Look at the Quadrilaterals column in the table and the placement of shapes A and G in the Venn diagram. Why are shapes A and G placed in the outermost oval?

2 Explain how the Venn diagram supports this statement: All parallelograms are quadrilaterals, but not all quadrilaterals are parallelograms.

3 Look at the table. Which categories share all the properties listed for parallelograms?

How is this shown in the Venn diagram?

4 Look at the table. Which properties are not shared by rectangles and rhombuses?

How is this shown in the Venn diagram?

5 Look at your Venn diagram. Why is Shape D inside all the ovals?

6 REFLECTLook back at your Try It, strategies by classmates, and Model Its. Which models or strategies do you like best for classifying two-dimensional fi gures? Explain.

SESSION 2

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Possible answer: Quadrilaterals are the most general category. The only property in the table that shapes A and G share with the other shapes is the property of having 4 sides.

Possible answer: The oval for parallelograms is inside the oval for quadrilaterals. A parallelogram has all of the properties of a quadrilateral, but a quadrilateral does not always have all of the properties of a parallelogram.

Possible answer: The ovals for rectangles and rhombuses and the region for squares, are all completely inside the oval for parallelograms.

The two ovals overlap, but one is not totally inside the other.

Possible answer: The square has the most properties. Squares are a subcategory of all the other categories.

Some students may say they like using a table because it helps them

organize their thinking before they draw a Venn diagram.

rectangle, rhombus, square

4 right angles and 4 sides of equal length.

CONNECT IT• Remind students that one thing that is alike about

the representations is that they show how the shape categories relate to each other.

• Explain that on this page, students will use their knowledge of properties of shapes to understand the category relationships shown by overlapping regions in a Venn diagram.

Monitor and Confirm

1 – 3 Check for understanding that:

• quadrilaterals are the broadest category because they share only one property with other shapes

• the relationships between ovals in the Venn diagram show how the categories of shapes are related

• shapes within an oval that is completely inside another oval have all the properties of the shapes represented by the larger oval

Support Whole Class Discussion

4 Tell students that this problem will prepare them to provide the explanation required in problem 5.

Ask Look at the overlapping ovals for Rhombuses and Rectangles again. What does it mean if a shape goes in the region of overlap?

Listen for The shape is both a rectangle and a rhombus, having the properties of both rectangles and rhombuses.

5 Look for the idea that a square has the most properties, sharing all of the properties of the shapes represented by the larger ovals.

6 REFLECT Have all students focus on the strategies used to solve this problem. If time allows, have students share their preferences with a partner.

SESSION 2 Develop

Hands-On ActivitySort objects that are in overlapping categories.

If . . . students are unsure about Venn diagrams with overlapping ovals,

Then . . . use this activity to help them understand the region of overlap.

Materials For each group: set of 5 sticky notes with letters on them (blue X, red X, green X, blue Y, blue Z); large sheet of paper; markers; blank sticky notes

• Have each group draw two overlapping ovals on their paper. Distribute sets of prepared sticky notes to the groups.

• Tell groups to sort the sticky notes into their two ovals based on properties. Have then write a category name for each oval, based on their sort.

• Ask: What are the categories for your two ovals? Why? [Xs and Blue Letters; all the sticky notes show an X and/or a blue letter]

• Ask: Which sticky notes are in the region of overlap? Why? [blue Xs; blue Xs have both properties, X and blue letter]

• If time allows, give groups blank sticky notes and have them make a set of letters or shapes that can be sorted into two overlapping categories.



LESSON 29 DEVELOP


SESSION 2

Apply itUse what you just learned to solve these problems.

7 Mathematicians defi ne trapezoids in two diff erent ways:

• Trapezoids are quadrilaterals with exactly one pair of parallel sides.

• Trapezoids are quadrilaterals with at least one pair of parallel sides.

Write exactly one pair or at least one pair to show which defi nition of trapezoid was used to make each Venn diagram below.

Trapezoids

Parallelograms

RectanglesTrapezoids

Parallelograms

Rectangles

8 Sort these shapes into the Venn diagram, based on properties of having parallel sides and having perpendicular sides. Write category labels for the two ovals.

A B C D E F

9 Which defi nition in problem 7 lets you classify shape A in problem 8 as a trapezoid? Explain.

594


The definition that a trapezoid has at least one pair of parallel sides; Possible explanation: Shape A is a parallelogram with two pairs of parallel sides. The “at least one pair” Venn diagram shows that parallelograms are also trapezoids under that definition.

at least one pair exactly one pair

A B

C

DE F

Shapes with Parallel Sides Shapes with Perpendicular Sides

APPLY ITFor all problems, encourage students to think about and possibly list the properties of the shapes involved as they work with the Venn diagrams.

Problem 7 is an opportunity to draw students’ attention to the inclusive and exclusive definitions of trapezoid, explaining that either definition is acceptable but only one definition can be used in any given situation. In any discussion or problem situation dealing with trapezoids, it is important to know which definition is in effect.

7 Left: at least one pair; If a trapezoid is defined as having at least one pair of parallel sides, parallelograms and rectangles are both subcategories of trapezoids because they also have at least one pair of parallel sides. Note: this is the inclusive definition of trapezoid.

Right: exactly one pair; If a trapezoid is defined as having exactly one pair of parallel sides, trapezoids do not overlap at all with parallelograms and rectangles, because parallelograms and rectangles have more than one pair of parallel sides. Note: this is the exclusive definition of trapezoid.

8 See the completed Venn diagram on the Student Worktext page; Students may show labels for “Shapes with Parallel Sides” and “Shapes with Perpendicular Sides” on either oval.

Close: Exit Ticket9 The definition that a trapezoid has at least one

pair of parallel sides; See possible explanation on the Student Worktext page.

Students’ solution should indicate understanding of:

• the properties of a parallelogram

• the properties of a trapezoid

• the difference between the inclusive and exclusive definitions of trapezoid

Error Alert If students’ solution is exactly one pair of parallel sides, then review the meaning of parallel. Have students look at the sides of shape A again after your review and have them trace it as necessary to help determine the number of pairs of parallel sides it has.


LESSON 29


Name:


Study the Example showing how to use Venn diagrams to classify two-dimensional fi gures. Then solve problems 1–5.

ExampleYou can use a Venn diagram to show the relationships between acute, right, obtuse,isosceles, and equilateral triangles.

You can plan your Venn diagram using a table describing properties of the triangles.

Triangle Description

Acute all acute angles

Right 2 acute angles and 1 right angle

Obtuse 2 acute angles and 1 obtuse angle

Isosceles at least 2 sides of equal length

Equilateral all sides equal length

Triangles

Obtuse

Isosceles

Acute Equilateral

Right

1 Look at the Venn diagram above. Can a right triangle ever be an equilateral triangle? Explain.

LESSON 29 SESSION 2

Practice Classifying Two-Dimensional Figures

595

No; Possible explanation: Right triangles do not overlap at all with equilateral triangles. Right triangles always have a right angle and equilateral triangles never have a right angle.

Solutions

1 See possible explanation on student page. Basic


Fluency & Skills Practice Teacher Toolbox

Assign Classifying Two-Dimensional Figures

In this activity students use Venn diagrams to classify triangles and quadrilaterals. This practice sharpens students’ abilities not only to see obvious differences between shapes but also to identify subtler similarities. For example, students may note that squares and rhombuses both have 4 sides of equal length. They may also see that right triangles, rectangles, and some trapezoids have at least 1 right angle.

©Curriculum Associates, LLC Copying is permitted for classroom use.

Name:

Fluency and Skills Practice

Classifying Two-Dimensional Figures

1 Properties: 4 right angles, 4 sides of equal length

AB

CD E F

2 Properties: 3 sides, at least 1 right angle

A BC

DE F

Sort the shapes into the Venn diagram, based on given properties. Write category labels for the two ovals. Then write the letter of each shape correctly in the diagram.



LESSON 29 SESSION 2

2 Look at the Venn diagram on the previous page. Write a statement about the relationship between acute triangles and isosceles triangles.

3 Look at the Venn diagram on the previous page. Write a statement about the relationship between acute triangles and equilateral triangles.

4 Draw a Venn diagram in the space below to show the relationships among the categories of isosceles, scalene, and equilateral triangles within the broader category, Triangles.

Triangles

5 Determine whether each statement is True or False. Draw a picture to help if needed.

True False

A scalene triangle is never isosceles. � �

A right triangle is sometimes equilateral. � �

A right triangle is never isosceles. � �

A scalene triangle can be a right, obtuse, or acute triangle. � �

596

Possible answer: An isosceles triangle can be an acute triangle if all of the angles in the isosceles triangle are acute. Not all isosceles triangles have angles that are all acute.

Possible answer: An equilateral triangle is always an acute triangle because all of the angles in an equilateral triangle are acute.

Scalene

Isosceles

Equilateral

2 See two possible statements on the student page; Students need only provide one statement. Students may also say that not all acute triangles are isosceles triangles. All explanations reflect that the ovals for acute triangles and isosceles triangles overlap, but neither is fully contained within the other. Medium

3 See one possible statement on the student page; Students may also say that not all acute triangles are equilateral triangles. All explanations reflect that the oval for equilateral triangles is completely inside the acute triangles oval. Medium

4 See the Venn diagram on the student page. The oval for scalene triangles should not touch the ovals for isosceles and equilateral triangles, because those types of triangles have at least 2 sides of equal length. Medium

5 A (True); D (False); F (False); G (True) Challenge

ELL

Levels 3–5Levels 2–4Levels 1–3

English Language Learners:Differentiated Instruction

Speaking/Writing Have students read the Try It problem independently and plan their tree diagram using a table to describe the properties of the shapes.

Organize students into pairs to share and compare their tables.

Ask students to critique their partner’s work, ask questions, and point out possible mistakes. Then have partners work together to create the tree diagram.

Speaking/Writing Read the Try It problem with students. Have students form pairs to discuss and solve the problem. Ask them to create a table of the shapes and describe their properties.

Have partners work use index cards with the names of each shape on them to organize their tree diagram.

Ask each pair to determine how many categories they used to classify the shapes. Call on partners to share their categories. Then allow time for students to revise their tree diagrams based on the comments of others.

Speaking/Writing Read the Try It problem aloud to students. Display a 2 3 8 table with the headings Shapes and Properties. Ask students to identify the shapes to be classified. Record the shapes on the table. As a group, fill in the properties for trapezoid. Discuss the definition provided in the problem.

Organize students into small groups and have them copy and complete the table. Give each group seven index cards. Ask them to label each index card with the name and picture of a shape to be classified. Have group members lay the cards out on a table, think aloud about their properties, and manipulate the cards to arrange their tree diagram.

Prepare for Session 3Use with Try It.


LESSON 29

Start

Connect to Prior KnowledgeWhy Prepare students for classifying shapes into a tree diagram by recognizing a hierarchy of categories.How Have students identify attributes that shapes share and attributes that are unique.


Start

Name two attributes these shapes share and two attributes these shapes do not share.

Grade 5 Lesson 29 Session 3 | Develop Classifying Two-Dimensional Figures with Tree Diagrams

Possible SolutionsBoth shapes have four sides. Both shapes have four angles.The rectangle has four right angles, but the rhombus has no right angles.The rhombus has four sides that are all the same length. The rectangle has opposite sides the same length.

Develop LanguageWhy Analyze the antonyms inclusive and exclusive to reinforce understanding of the two definitions of trapezoid.How Discuss the everyday meanings of the words include and exclude, then display the two definitions of trapezoid. Guide students to identify which shapes are excluded by the definition used in the Try It problem (parallelograms, rhombuses, rectangles and squares).

TRY ITMake Sense of the ProblemTo support students in making sense of the problem, have them restate the two definitions of trapezoid that they have seen, and clarify which definition of trapezoid is used in this problem.

Ask What are the two different definitions of trapezoids and which definition is used in this problem?

DISCUSS ITSupport Partner DiscussionEncourage students to use the terms property and attribute as they discuss their solutions.

Support as needed with questions such as:

• How did you get started?

• How did you organize the information in the problem?

Common Misconception Look for students who include trapezoid in the path between quadrilateral and parallelogram. Ask them the definition of trapezoid and listen for the words at least one pair of parallel sides or exactly one pair of parallel sides.


• concrete models or drawings to support reasoning about attributes

• list or table of attributes

• hierarchy of categories shown in a tree diagram

• classification of shapes A and B

Purpose In this session students solve a problem that requires classifying shapes using a tree diagram. The purpose of this problem is to have students develop an understanding of the relationships among figures that may be classified into more than one category, and to display categories in a tree diagram to show hierarchy based on the properties of the figures.

SESSION 3 Develop


LESSON 29


Develop Classifying Two-Dimensional Figures with Tree Diagrams

SESSION 3

Read and try to solve the problem below.

Make a tree diagram to show the hierarchy of the following shapes based on their properties: rhombus, trapezoid, polygon, square, quadrilateral, rectangle, parallelogram. Use the exclusive defi nition of trapezoid in your diagram: a trapezoid is a quadrilateral with exactly one pair of parallel sides.

Then, classify these shapes into as many categories as possible.

TRY IT Math Toolkit• geoboard• rubber bands• tracing paper• grid paper• rulers

DISCUSS ITAsk your partner: How did you get started?

Tell your partner: At fi rst, I thought . . .

BA

597


Sample A

Sample B

Parallelogram

Quadrilateral

Polygon

Trapezoid

Square

RectangleRhombus

Shape A: rhombus, parallelogram, quadrilateral, polygonShape B: trapezoid, quadrilateral, polygon

Shape A: rhombus, parallelogram, quadrilateral, polygonShape B: trapezoid, quadrilateral, polygon

Parallelogram Square

QuadrilateralPolygon

Trapezoid

Rectangle

Rhombus


Support Whole Class DiscussionCompare and connect the different representations and have students identify how they are related.

Ask How does your tree diagram show the relationship between categories of shapes? How can you find all classifications for each shape using your diagram?

Listen for Students should recognize how tree diagrams can be used to show the hierarchy of shapes. In both models, the categories move from most general to most specific. For example, Figure A is a rhombus. Student may follow the tree diagram either left or up from rhombus to find less specific categories to which all rhombuses also belong. Listen for clarification that as you move down or to the right on the tree diagram, the classifications become more specific, and as you move up or left the classifications are more general.

MODEL ITSIf no student presented these models, connect them to the student models by pointing out the ways they each represent:

• each attribute of the listed shapes

• relationships between categories of shapes

Ask How are the shapes in the problem represented in each Model It?

Listen for In the first Model It, the shape categories are shown in a Venn diagram. In the second Model It categories of shapes are shown in a tree diagram.

For the Venn diagram model, prompt students to consider the Venn diagram as a way to show hierarchy of categories.

• How does the Venn diagram show the relationship between a category and a sub-category?

• To what other categories does a rhombus belong?

• How can using a Venn diagram help you make a tree diagram?

For the tree diagram model, prompt students to relate the tree diagram to the Venn diagram.

• How do categories of shapes in the tree diagram relate to categories in the Venn diagram?

• A trapezoid in the Venn diagram would be placed inside quadrilaterals, which is inside polygons. How is this shown in the tree diagram?


LESSON 29 DEVELOP


SESSION 3

Explore ways to understand classifying two-dimensional fi gures using a tree diagram.

Make a tree diagram to show the hierarchy of the following shapes based on their properties: rhombus, trapezoid, polygon, square, quadrilateral, rectangle, parallelogram. Use the exclusive defi nition of trapezoid in your diagram: a trapezoid is a quadrilateral with exactly one pair of parallel sides.

Then, classify these shapes into as many categories as possible.

Model ItYou can use a Venn diagram to look at the hierarchy of the shapes.

Model ItYou can show the hierarchy among shapes with a tree diagram.

This tree diagram has the most general category at the left (or top) and more specifi c categories as you go towards the right (or down). Each shape belongs in every category before it.

Write the remaining shape names in the diagram.

QuadrilateralsPolygons

Trapezoids

Rectangles

BA

598

Trapezoids

ParallelogramsQuadrilaterals

Polygons

Rhombuses Squares Rectangles

Parallelograms Squares

Rhombuses

Deepen UnderstandingTree Diagram ModelSMP 7 Use structure.

When discussing the tree diagram, prompt students to consider how the structure of a tree diagram allows specific shapes to fit in more than one category.

Ask How does the category that is farthest to the right relate to the two categories on its left?

Listen for The tree diagram moves from most general to most specific. This means the category farthest to the right, squares, is a sub-category of the categories it connects to, rectangles and rhombuses.

Ask When a shape is classified in a specific location on the tree diagram, how can you follow the branches to find all of the categories to which the shape belongs?

Listen for Because the categories move from most general to most specific, you can follow branches backwards from a shape’s classification to more general categories. For example, in the tree diagram you can move from trapezoid to quadrilateral to polygon. When a shape is classified as specifically as possible, you can’t classify it in any of the categories to the right on the tree diagram.


LESSON 29

SESSION 3 Develop

CONNECT IT• Remind students that one thing that is alike about

the representations is that they show how the categories are related.

• Explain that on this page, students will use their knowledge of properties of quadrilaterals to understand how tree diagrams show a hierarchy and organize shapes from general to specific categories based on attributes.

Monitor and Confirm

1 – 2 Check for understanding that:

• each diagram shows categories from most general to most specific

• the most specific categories are those with no regions inside them

Support Whole Class Discussion

3 – 4 Tell students that these problems will prepare them to provide the explanation required in problem 5.

Ask Where does shape A belong in the tree diagram? How can you decide what other categories shape A or a square can be classified as?

Listen for The most specific category for shape A is a rhombus. Shapes belong to all of the more general categories that are directly connected when you move left (or up) in a tree diagram. Because square is directly connected on the left to both rectangle and rhombus, a square can be classified as both a rectangle and a rhombus, as well as all of the other categories to the left of rectangle and rhombus.

5 Look for the idea that level farthest to the right or at the bottom of the tree diagram is the most specific classification and has the properties of all the categories above it.

6 REFLECT Have all students focus on the strategies used to solve this problem. If time allows, have students share their preferences with a partner.


SESSION 3

Connect ItNow you will use the problem from the previous page to help you understand how to use tree diagrams to classify two-dimensional fi gures.

1 Look at the Venn Diagram. Which part of the diagram shows the most general category? Which part of the diagram shows the most specifi c category?

2 How is the tree diagram similar to the Venn diagram? How is it diff erent?

3 How can you see all of the categories that a shape belongs to in a tree diagram? What are all of the categories that shape A belongs to?

4 Why do the branches that come off Rectangles and Rhombuses both lead to Squares?

5 Explain the relationship between the properties of categories when you move left or right (or up or down) in a tree diagram.

6 REFLECTLook back at your Try It, strategies by classmates, and Model Its. Which models or strategies do you like best for classifying two-dimensional fi gures? Explain.

599

Possible answer: The outer rectangle shows the most general category - polygons. The sections that have no sub-categories inside them are the most specific categories – trapezoids and squares.

Possible answer: Both diagrams show the hierarchy of the shapes, from most general to most specific. The Venn diagram shows this by regions, and the tree diagram shows this by branches.

Possible answer: Each shape belongs to all of the categories that come before it, so follow the branch to the left to see all of the categories. Shape A is a rhombus. It is also a rectangle, parallelogram, quadrilateral, and polygon.

Possible answer: A square belongs in both of these categories. A square is a rectangle because it is a quadrilateral with four right angles. A square is a rhombus because it is a quadrilateral with four sides that have the same length.

Possible answer: All of the properties in the level to the left (above) are also true of the level you are on. When you move to the right (downward) in a tree diagram, you are moving to a category that has all of the properties of the previous category, with additional properties.

Some students may say they like using a tree diagram because it shows how

categories of shapes are related in a linear fashion. Other students may prefer a Venn

diagram because it shows which categories are contained within other categories.

Hands-On ActivitySort objects into specific categories.

If . . . students are unsure about levels within a tree diagram,

Then . . . use this activity to help them understand that each level leads to objects with more specific properties.

Materials For each group: set of 10 sticky notes with the following words: peppers, carrots, vegetables, potatoes, red pepper, hot peppers, seeded vegetables, green peppers, root vegetables, and sweet potatoes; large sheet of paper; markers

• Distribute prepared sticky notes, papers, and markers to the groups.

• Ask: What is true of all of the words on the sticky notes? [Possible answer: They are all vegetables.] This is the top level of your tree diagram.

• Have students create a tree diagram with the sticky notes, with branches coming off of “Vegetables.”

• Emphasize that each vegetable at the bottom of the tree diagram belongs to all of the categories it is connected to above.


APPLY ITFor all problems, encourage students to think about and possibly list the properties of the shapes as they work with the tree diagrams.

Problem 7 is an opportunity to draw students’ attention to the hierarchy of triangles when classified by side lengths. An isosceles triangle is defined as having at least two sides of equal length. An equilateral triangle has all three sides equal length, making it a subcategory of isosceles triangles.

7 Cyrus is not correct; See possible explanation on Student Worktext page.

8 See the completed tree diagram on the Student Worktext page.

Close: Exit Ticket

9 equilateral, isosceles, equiangular, acute, triangle; See possible explanation on the Student Worktext page.

Students’ solutions should indicate understanding of:

• the properties of an equilateral triangle

• the properties of an acute triangle

• the properties of an isosceles triangle

• an understanding of the hierarchy shown in a tree diagram

Error Alert If students’ solution is only equilateral, then review hierarchy of shapes in a tree diagram. Have students explain why the equilateral triangle not only belongs in the category equilateral triangles, but also in all categories above to which it is connected.


LESSON 29 DEVELOP


SESSION 3

Apply itUse what you just learned to solve these problems.

7 Cyrus made the following tree diagram to show the relationship among scalene triangles, equilateral triangles, and isosceles triangles. Is his diagram correct? Explain.

8 An equilateral triangle has three sides that have the same length. An equilateral triangle may also be called an equiangular triangle because each of the three angles has the same measure, 608. Insert an additional category in the tree diagram for equiangular triangles. Explain.

9 Look at the tree diagrams in the two problems above. Shape E has all sides the same length. What are all categories of triangles to which this triangle belongs? Explain.

E

Scalene

Triangles

Isosceles

Equilateral

Triangles

Acute Right Obtuse

600

Possible answer: Cyrus’s diagram is not correct. His tree diagram shows that all isosceles triangles are equilateral, but this is not true. The diagram would be correct if Equilateral and Isosceles switch places.

Equiangular triangles have three angles that each measure 608. This makes them a subcategory of acute triangles.

equilateral, isosceles, equiangular, acute, triangle; Possible explanation: Shape E has three equal sides, making it equilateral and three equal angles, making it equiangular. All of the classifications moving upward following lines in a tree diagram are also true of a given shape.

Equiangular


LESSON 29


Solutions

1 See possible statements on the student page. Students need only provide one statement. Students may also state that rhombuses and squares are both parallograms or are both quadrilaterals. Basic


Name:


Study the Example showing how to use a tree diagram to classify two-dimensional fi gures. Then solve problems 1–5.

ExampleYou can use a tree diagram to show the relationship between quadrilaterals, parallelograms, rectangles, squares, and rhombuses.

You can arrange your tree diagram using descriptions from a table of the properties of each quadrilateral.

Figure Description

Quadrilateral 4 sides

Parallelogram 4 sides2 pairs of parallel sides

Rhombus 4 sides2 pairs of parallel sides4 sides of equal length

Rectangle 4 sides2 pairs of parallel sides4 right angles

Square 4 sides2 pairs of parallel sides4 sides of equal length4 right angles

1 Look at the tree diagram above. Write a statement about the relationship between rhombuses and squares.

Practice Classifying Two-Dimensional Figures with Tree Diagrams

LESSON 29 SESSION 3

Rectangles

Quadrilaterals

Parallelogram

Squares

Rhombuses

601

Possible answer: All squares are rhombuses. A rhombus can be a square if all four angles are right angles. Not all rhombuses are squares.

Fluency & Skills Practice Teacher Toolbox

Assign Classifying Two-Dimensional Figures with Tree Diagrams

In this activity students practice making a tree diagram and using it to classify shapes. Understanding how categories are related and how to classify shapes into more than one category based on the properties will develop students’ analytical thinking as well as their understanding of the categories and subcategories of geometric figures.

©Curriculum Associates, LLC Copying is permitted for classroom use.

Name:

Fluency and Skills Practice

Classifying Two-Dimensional Figures with Tree Diagrams

1 Organize the fi gures in the tree diagram. Write one category name in each box.

2 Name all categories from the tree diagram that apply to each fi gure.

A.

B.

C.

3 Where does an acute triangle fi t in the tree diagram above? Explain.

The two-dimensional figures below can be categorized by their properties.

polygons triangles hexagons squares

rectangles quadrilaterals equilateral triangles


2 Yes; See possible explanation on student page. Students should recognize in a tree diagram when two lines lead to the same category, this category contains all of the properties of the two categories that connect to it. Medium

3 See the tree diagram on the student page. Medium

4 See possible explanation on the student page. Medium

5 A (Always);E (Sometimes);I (Never);K (Sometimes) Challenge


Name:


LESSON 29 SESSION 3

2 Look at the tree diagram on the previous page. Can a rhombus ever be a rectangle? Explain.

3 Draw a tree diagram to show the relationship among the following categories: polygons, pentagons, quadrilaterals, parallelograms, trapezoids, rectangles, and squares. Use the inclusive defi nition of trapezoid: a quadrilateral with at least one pair of parallel sides.

4 How would your tree diagram in the previous problem be diff erent if you used the exclusive defi nition for trapezoid? Explain.

5 Determine whether each statement is always, sometimes, or never true. Use the inclusive defi nition for trapezoid.

Always Sometimes Never

A square is a parallelogram. � � �

A trapezoid is a rectangle � � �

A pentagon is a parallelogram. � � �

A trapezoid is a square. � � �

602

Yes; Possible explanation: A rhombus is a parallelogram with four equal length sides. A rectangle is a parallelogram with four right angles. A figure that has both four equal length sides and four right angles is a square, which is both a rhombus and a rectangle.

Possible answer: The exclusive definition of trapezoid states there are exactly two parallel sides. Quadrilaterals would branch into Trapezoids and Parallelograms. Nothing would be below Trapezoids, and Rectangles and Squares would be below Parallelograms.

Trapezoids

Polygons

Pentagons

Parallelograms

Rectangles

Quadrilaterals

Squares


English Language Learners:Differentiated Instruction

Speaking/Writing Have students form pairs and read Apply It problem 1. Ask students to discuss the shapes listed in the problem with their partner and how the categories of shapes are related. After they draw their Venn diagrams or tree diagrams, have students work together to think of phrases that tell about the relationship between quadrilaterals and trapezoids. Then ask them to write a sentence that describes the relationship. Have students share sentences with another set of partners and review.

Speaking/Writing Pair students. Read Apply It problem 1 aloud. Encourage partners to create a table of the shapes and their properties to plan their diagrams. Provide a bank of terms for students to use as they engage in conversations about classifying shapes. Possible terms may include classify, property, attribute, relationship, general, specific, parallel, and sides.

Listening/Speaking Pair students. Read Apply It problem 1 aloud. Have students circle the terms quadrilaterals, polygons, trapezoids, and hexagons. Encourage students to create a table to describe the properties of each shape. Have students write the name and draw a picture of each shape on an index card, and manipulate the cards to arrange their diagrams.

Prepare for Session 4Use with Apply It.ELL


LESSON 29


LESSON 29


Refine Classifying Two-Dimensional FiguresSESSION 4

Complete the Example below. Then solve problems 1–8.

EXAMPLEHugo classifi ed polygons based on properties of having exactly four sides and having all sides equal in length.

Look at how Hugo placed three fi gures in a tree diagram.

Polygons

Quadrilaterals Equilateral Shapes

Draw a shape that belongs to both branches of the tree diagram. What is the name of the shape you drew?

Solution

Apply it1 Draw a Venn diagram or a tree diagram to show the relationships

among quadrilaterals, polygons, trapezoids, and hexagons. Then write a statement about the relationship between quadrilaterals and trapezoids. Show your work.

What shape has four sides and all sides equal in length?

What is the most general category in your diagram?

PAIR/SHAREHow can this tree diagram be shown with a Venn diagram?

PAIR/SHAREWhat does your diagram show about hexagons andquadrilaterals?

603

square


Trapezoids have all the attributes of quadrilaterals.

Polygons

Quadrilaterals

Trapezoids

Hexagons

StartCheck for UnderstandingWhy Confirm understanding of using Venn diagrams or tree diagrams to classify two-dimensional figures.

How Have students make a Venn diagram or a tree diagram using any strategy they want.


Start

Grade 5 Lesson 29 Session 4 | Refi ne Classifying Two-Dimensional Figures

Draw a Venn diagram or a tree diagram to classify these shapes.

Parallelograms Polygons

Rhombuses Quadrilaterals

SolutionLook for diagrams showing the following hierarchy, from most general to most specific: Polygons, Quadrilaterals, Parallelograms, Rhombuses

Purpose In this session students solve problems involving making and interpreting Venn diagrams or tree diagrams and then discuss and confirm their answers with a partner.

Before students begin to work, use their responses to the Check for Understanding to determine those who will benefit from additional support.

As students complete the Example and problems 1–3, observe and monitor their reasoning to identify groupings for differentiated instruction.

SESSION 4 Refine

If the error is . . . Students may . . . To support understanding . . .

any order other than the correct

order

not be able to identify the properties of the given shapes.

Have students make a table with the names of the shapes as the headings of four columns. Work with students to list the properties of each of the shapes so that students can see, for example, that a rhombus has all the properties that parallelograms have plus one more.

Error Alert



LESSON 29 REFINE


2 Draw a Venn diagram or tree diagram to show the relationships among these polygons.

Polygon Description

Quadrilateral polygon with exactly 4 sides

Trapezoid quadrilateral with exactly 1 pair of parallel sides

Parallelogram quadrilateral with 2 pairs of parallel sides

3 Look at the Venn diagram below.

Triangles

Isosceles Triangles

Equilateral Triangles

ScaleneTriangles

Which statement is true?

� An equilateral triangle is always an isosceles triangle.

� An isosceles triangle is always an equilateral triangle.

� Scalene and isosceles triangles share no attributes.

� The label inside the largest oval could be Acute Triangles.

Brad chose � as the correct answer. How did he get that answer?

PAIR/SHAREHow would the diagram change if the description of the trapezoid used the defi nition quadrilateral with at least one pair of parallel sides?

PAIR/SHAREWhat would you say to Brad to help him understand his mistake?

The table contains one of the de� nitions of a trapezoid.

All triangles are three-sided polygons.

604

Possible diagram: Quadrilaterals

Parallelograms Trapezoids

Possible answer: Brad thinks an isosceles triangle has all of the attributes of an equilateral triangle.

ExampleThe drawing of a square shown is one way to solve the problem. Students could also draw a rhombus.

Look for A square or a rhombus is a shape that has four sides and all equal sides.

APPLY IT1 See possible tree diagram on the Student

Worktext page. Students may also draw a Venn diagram. Based on the relationships shown in the diagram, students may say that all trapezoids are quadrilaterals. They may also say that some quadrilaterals are trapezoids. DOK 2

Look for Polygons are the most general category in the diagram because polygons share with the other shapes only the attribute of being a closed shape with 3 or more straight sides.

2 See the possible Venn diagram on the Student Worktext page. Students use the descriptions in the table to identify the relationships between the categories and make a Venn diagram or a tree diagram. DOK 2

Look for If a trapezoid is defined as having exactly one pair of parallel sides, the oval for trapezoids does not overlap with the oval for parallelograms because parallelograms have two pairs of parallel sides.

3 A; Students could solve the problem by comparing each statement to the Venn diagram, noting that statement A is true because the oval for equilateral triangles is completely inside the oval for isosceles triangles.

Explain why the other two answer choices are not correct:

C is not correct because all triangles share at least one property: having 3 sides and 3 angles.

D is not correct because not all triangles have only acute angles. Some triangles also have right or obtuse angles. DOK 3


LESSON 29


SESSION 4

4 Draw a tree diagram to show the relationships among triangles, quadrilaterals, isosceles triangles, and polygons.

5 Use the diagram in problem 4. Write two diff erent statements that describe relationships among the shapes.

6 Could you add the two shapes below to your diagram in problem 4? If so, where would you put them? Name each shape as you explain your thinking.

605

Possible diagram:

Polygons

Quadrilaterals

Isosceles Triangles

Triangles

Answers will vary. Possible answer: Isosceles triangles have all the properties that triangles have. All triangles and quadrilaterals are polygons.

Possible answer: You could add the pentagon, but not the circle. In the tree diagram, you could add a new branch below polygons, beside quadrilaterals and triangles. You could not add the circle to the tree diagram because a circle is not a polygon.

4 See possible tree diagram on the Student Worktext page. DOK 2

5 See possible statements on the Student Worktext page. Students may also say that some polygons are quadrilaterals (or triangles or isosceles triangles). Students may say that some triangles are isosceles triangles. DOK 3

6 Pentagons, yes; circles, no; See possible explanation on the Student Worktext page. DOK 2

Error Alert Students may think that a circle is a polygon with one side, so it belongs as a separate subcategory of polygons. Review the definition of polygon, a closed figure with 3 or more straight sides.

SESSION 4 Refine

Differentiated Instruction

RETEACH EXTEND

Hands-On ActivityClassify triangles using a Venn diagram and geoboards.

Students struggling with using Venn diagrams to sort and classify

Will benefit from additional work with sorting shapes into Venn diagrams

Materials For each pair: geoboard, rubber bands; For display: Venn diagram with ovals for Triangles, Acute Triangles, and Isosceles Triangles (large enough for geoboards to be placed in each region; regions for acute and isosceles overlap)

• Have each pair make a triangle with their geoboard and rubber bands.

• Talk through the placement of the geoboard triangles in the Venn diagram. Ask: Does your geoboard belong inside the Triangles oval? inside the Acute Triangles oval? inside the Isosceles Triangles oval? Why? Listen for understanding of the properties of each pair’s triangle and how its properties determine its placement.

• For any region without a triangle in it, ask one pair to make a triangle that belongs in that region. Have them explain why the triangle belongs in the region.

• Change the label Acute Triangles to Right Triangles and repeat the activity.

Challenge ActivityLook at properties of kites.

Students who have achieved proficiency

Will benefit from deepening understanding of classifying shapes

• Show students a shape that has two pairs of adjacent sides that are equal, one pair shorter than the other. Tell students that the shape is a kite.

• Have students list properties of the kite. [4 sides, 2 pairs of equal sides, opposite sides are not equal]

• Have students explain whether the kite would go inside or outside an oval for parallelograms in a Venn diagram.



LESSON 29 REFINE


SESSION 4

7 Saul drew a Venn diagram to show how rhombuses and squares are related.He used arrows to label each region with properties of shapes inside that region.

Rhombuses Squares

2 pairs ofparallel sides

4 sides ofequal length

perpendicularsides

4 right angles

Part A Is Saul’s Venn diagram correct? If not, what mistake did he make?

Part B Describe the relationship between rhombuses and squares.

8 MATH JOURNALA regular polygon has all sides of equal length. Sami says that all squares, equilateral triangles, and pentagons can be classifi ed as regular polygons. Is Sami correct? Draw a Venn diagram and explain your thinking.

SELF CHECK Go back to the Unit 4 Opener and see what you can check off .606

No; Possible explanation: Two pairs of parallel sides is also a property of squares, so this property should point to the region where the ovals overlap. The oval for squares should be completely inside the oval for rhombuses.

Squares are subcategory of rhombuses because a square has all of the properties of a rhombus. A square also has 4 right angles. Not all rhombuses have 4 right angles.

Regular Polygons

Pentagons

Squares Equilateral Triangles

No; Possible explanation: Squares and equilateral triangles are regular polygons because their sides all have equal length. A pentagon is any 5-sided polygon. Only some pentagons are regular pentagons.

Possible diagram:

7 Part ASaul’s Venn diagram is not correct; See the possible explanation on the Student Worktext page. Students may draw a rhombus and a square to help solve the problem. Students may also say that the Venn diagram should show the oval for Squares completely inside the oval for Rhombuses because squares have all the properties of rhombuses.

Part BSee possible answer on the Student Worktext page; responses should show understanding that all squares can also be classified as rhombuses. DOK 3

Close: Exit Ticket

8 MATH JOURNAL Student responses should indicate understanding that squares and equilateral triangles can be classified as regular polygons, but not all pentagons are regular; students may draw an example of a pentagon for which not all sides are the same length.

Error Alert If students say Sami is correct, then ask them to define pentagon, square, and equilateral triangle. Note that a pentagon is a closed shape with five sides. Draw a regular pentagon and a pentagon with at least two sides of different lengths. Ask if both are still pentagons.

REINFORCE PERSONALIZE

Problems 4–8Classify two-dimensional figures.

All students will benefit from additional work with classifying two-dimensional figures by solving problems in a variety of formats.

• Have students work on their own or with a partner to solve the problems.

• Encourage students to show their work.

Provide students with opportunities to work on their personalized instruction path with i-Ready Online Instruction to:

• fill prerequisite gaps

• build up grade-level skills

SELF CHECK Have students consider whether they feel they are ready to check off any new skills on the Unit 4 Opener page.

©Curriculum Associates, LLC Copying is not permitted.606a Lesson 29 Classify Two-Dimensional Figures

LESSON 29

Lesson Quiz Teacher Toolbox

1©Curriculum Associates, LLC

Copying permitted for classroom use.Grade 5 Lesson 29 Classify Two-Dimensional Figures

Name ___________________________________________________________ Date ____________________

Lesson 29 Quiz

Solve the problems.

Use the tree diagram below to answer questions 1 – 3.

right triangles

polygons

rhombuses

trianglesquadrilaterals

parallelograms

1 Which statement describes all rhombuses?

� a polygon with three sides

� a polygon with exactly 1 pair of parallel sides

� a quadrilateral with 4 sides of equal length

� a parallelogram with 4 right angles.

2 Alex wants to add this shape to the tree diagram using the exclusive defi nition of trapezoid.

Where should he place the shape on the tree diagram? Explain.

3 Which category on the tree diagram is the least specifi c? Explain.

(1 point)

Possible answer: The trapezoid is a quadrilateral because it has 4 sides.

It is not a parallelogram because it has exactly 1 pair of parallel sides,

not 2 pairs of parallel sides. Therefore, the trapezoid goes below

quadrilaterals and to the side of parallelograms on the tree diagram.

Possible answer: Polygons are the least specific because all the other

subcategories of shapes have all the attributes of a polygon.

(2 points)

(2 points)

Tested SkillsAssesses 5.G.B.3, 5.G.B.4

Problems on this assessment form require students to be able to categorize two-dimensional figures based on properties and to interpret and Venn diagrams and tree diagrams to classify figures. Students will also need to be familiar with acute, obtuse, and right angles, parallel and perpendicular lines, equilateral and isosceles triangles, and how to sort figures based on side lengths and angle types.

Alternately, teachers may assign the Digital Comprehension Check online to assess student understanding of this material.

Error Alert Students may:

• confuse, incorrectly identify, or ignore some attributes of different figures.

• assume a relationship in one direction in a tree diagram is also true in the other direction.

• not be able to interpret or complete a tree diagram.

• not be able to interpret a Venn diagram.

• not realize that at least one also includes more than one.

Short Response Scoring Rubric

Points Expectations

2

Response contains the following:• Correct solutions and/or reasoning. (1 point)• Well-organized, clear, and concise work that

demonstrates thorough understanding of math concepts. (1 point)

1

Response contains the following:• Mostly correct solution(s).• Shows partial or good understanding of math

concepts.

0

Response contains the following:• Incorrect solution(s).• No attempt at finding a solution.• No effort to demonstrate an understanding of

mathematical concepts.

Choice Matrix Scoring Rubric

2 points 1 point 0 points

All answers are correct

1 incorrect answer 2 or more incorrect answers

©Curriculum Associates, LLC Copying is not permitted. 606bLesson 29 Classify Two-Dimensional Figures

Differentiated Instruction

RETEACH REINFORCE EXTEND

Teacher Toolbox

Enrichment ActivitiesStudents who have achieved proficiency with concepts and skills and are ready for additional challenges

Will benefit from group collaborative games and activities that extend understanding

Math Center ActivitiesStudents who require additional practice to reinforce concepts and skills and deepen understanding

Will benefit from small group collaborative games and activities (available in three versions—on-level, below-level, and above-level)

Tools for InstructionStudents who require additional support for prerequisite or on-level skills

Will benefit from activities that provide targeted skills instruction

2©Curriculum Associates, LLC

Copying permitted for classroom use.Grade 5 Lesson 29 Classify Two-Dimensional Figures

Name ___________________________________________________________ Date ____________________

Lesson 29 Quiz continued

4 Use the Venn diagram. Decide if each statement is true. Choose Always True, Sometimes True, or Never True for each statement.

AlwaysTrue

SometimesTrue

NeverTrue

An equilateral triangle is an obtuse triangle. � � �

A right triangle is an isosceles triangle. � � �

An equilateral triangle is an isosceles triangle. � � �

An isosceles triangle is an acute triangle.

� � �

An obtuse triangle is an acute triangle.

� � �

5 Classify the shapes shown below by writing the letter of each shape in the category that describes it. Shapes may fall into more than one category. All shapes may not be used.

A B C D E

At Least 1 Pair ofPerpendicular Sides

At Least 2 Sides ofEqual Length

At Least 1 Pair ofParallel Sides

Triangles

Obtuse Right

Acute

Isosceles

Equilateral

(2 points)

(2 points)

C, E B, C, E A, B, E

Solutions1 C; Students could solve the problem by using

what they know about properties to identify rhombuses as a subcategory of quadrilaterals and parallelograms with 4 sides of equal length. A is not correct because rhombuses are polygons with 4 sides, not 3 sides. B is not correct because rhombuses have 2 pairs of parallel sides, not 1 pair of parallel sides. D is not correct because rhombuses are parallelograms with opposite equal angles, but may not have 4 right angles. 1 point 5.G.B.3, DOK 2

2 See possible answer on the student page. 2 points 5.G.B.4, DOK 2

3 See possible answer on the student page. 2 points 5.G.B.3, DOK 2

4 C (Never); E (Sometimes); G (Always); K (Sometimes); O (Never) 2 points 5.G.B.3, DOK 2

5 See completed classification on the student page. 2 points 5.G.B.4, DOK 2

Classify Two-Dimensional Figures - Full Lesson

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