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5 Mathematics Teacher Resource Book 2015 Florida MAFS Table of Contents Ready® Florida MAFS Program Overview A6 Supporting the Implementation of the Florida MAFS A7 Answering the Demands of the MAFS with ReadyA8 The Standards for Mathematical Practice A9 Depth of Knowledge Level 3 Items in Ready Florida MAFSA10 Cognitive Rigor Matrix A11 Using Ready Florida MAFSA12 Teaching with Ready Florida MAFS Instruction A14 Content Emphasis in the MAFS A16 Connecting with the Ready® Teacher Toolbox A18 Using i-Ready® Diagnostic with Ready Florida MAFS A20 Using Ready Practice and Problem SolvingA22 Features of Ready Florida MAFS Instruction A24 Supporting Research A40 Correlation Charts Florida MAFS Coverage by Ready Instruction Interim Assessment Correlations A44 A48 Lesson Plans (with Answers) MAFS Emphasis Unit 1: Number and Operations in Base Ten Lesson 1 Understand Place Value 1 5 M 13 M 21 M 31 M 41 M 49 M MAFS Focus - 5.NBT.1.1 Embedded SMPs - 1–8 Lesson 2 Understand Powers of Ten MAFS Focus - 5.NBT.1.2 Embedded SMPs - 2–8 Lesson 3 Read and Write Decimals MAFS Focus - 5.NBT.1.3a Embedded SMPs - 2, 4–7 Lesson 4 Compare and Round Decimals MAFS Focus - 5.NBT.1.3b, 5.NBT.1.4 Embedded SMPs - 1, 2, 4–7 Lesson 5 Multiply Whole Numbers MAFS Focus - 5.NBT.2.5 Embedded SMPs - 1–8 Lesson 6 Divide Whole Numbers MAFS Focus - 5.NBT.2.6 Embedded SMPs - 1–5, 7 M = Lessons that have a major emphasis in the MAFS S/A = Lessons that have supporting/additional emphasis in the MAFS MAFS Emphasis Unit 1: Number and Operations in Base Ten (continued) Lesson 7 Add and Subtract Decimals 57 M 67 M 77 M MAFS Focus - 5.NBT.2.7 Embedded SMPs - 2–7 Lesson 8 Multiply Decimals MAFS Focus - 5.NBT.2.7 Embedded SMPs - 1–5, 7 Lesson 9 Divide Decimals MAFS Focus - 5.NBT.2.7 Embedded SMPs - 1–5, 7 Unit 1 Interim Assessment 89 Unit 2: Number and Operations—Fractions 92 Lesson 10 Add and Subtract Fractions 96 M 106 M 114 M 122 M 130 M 140 M 148 M 158 M 166 M MAFS Focus - 5.NF.1.1 Embedded SMPs - 1, 2, 4, 7 Lesson 11 Add and Subtract Fractions in Word Problems MAFS Focus - 5.NF.1.2 Embedded SMPs - 1–8 Lesson 12 Fractions as Division MAFS Focus - 5.NF.2.3 Embedded SMPs - 1–5, 7 Lesson 13 Understand Products of Fractions MAFS Focus - 5.NF.2.4a Embedded SMPs - 1–8 Lesson 14 Multiply Fractions Using an Area Model MAFS Focus - 5.NF.2.4b Embedded SMPs - 1–8 Lesson 15 Understand Multiplication as Scaling MAFS Focus - 5.NF.2.5a, 5.NF.2.5b Embedded SMPs - 1, 2, 4, 6, 7 Lesson 16 Multiply Fractions in Word Problems MAFS Focus - 5.NF.2.6 Embedded SMPs - 1–8 Lesson 17 Understand Division with Unit Fractions MAFS Focus - 5.NF.2.7a, 5.NF.2.7b Embedded SMPs - 1–8 Lesson 18 Divide Unit Fractions in Word Problems MAFS Focus - 5.NF.2.7c Embedded SMPs - 1–8 Unit 2 Interim Assessment M = Lessons that have a major emphasis in the MAFS S/A = Lessons that have supporting/additional emphasis in the MAFS 177 MAFS Emphasis Unit 3: Operations and Algebraic Thinking Lesson 19 Evaluate and Write Expressions 180 183 S/A 193 S/A MAFS Focus - 5.OA.1.1, 5.OA.1.2 Embedded SMPs - 1, 2, 5, 7, 8 Lesson 20 Analyze Patterns and Relationships MAFS Focus - 5.OA.2.3 Embedded SMPs - 1, 2, 7, 8 Unit 3 Interim Assessment Unit 4: Measurement and Data Lesson 21 Convert Measurement Units 203 206 208 S/A 218 S/A 228 S/A 238 M 246 M 254 M 262 M MAFS Focus - 5.MD.1.1 Embedded SMPs - 1, 2, 5–7 Lesson 22 Solve Word Problems Involving Conversions MAFS Focus - 5.MD.1.1 Embedded SMPs - 1, 2, 5–7 Lesson 23 Make Line Plots and Interpret Data MAFS Focus - 5.MD.2.2 Embedded SMPs - 1, 2, 4–7 Lesson 24 Understand Volume MAFS Focus - 5.MD.3.3a, 5.MD.3.3b Embedded SMPs - 2, 4–7 Lesson 25 Find Volume Using Unit Cubes MAFS Focus - 5.MD.3.4 Embedded SMPs - 2, 4–7 Lesson 26 Find Volume Using Formulas MAFS Focus - 5.MD.3.5a, 5.MD.3.5b Embedded SMPs - 1–8 Lesson 27 Find Volume of Composite Figures MAFS Focus - 5.MD.3.5c Embedded SMPs - 1–8 Unit 4 Interim Assessment Unit 5: Geometry Lesson 28 Understand the Coordinate Plane 271 274 277 S/A 285 S/A 295 S/A 303 S/A MAFS Focus - 5.G.1.1 Embedded SMPs - 4, 6, 7 Lesson 29 Graph Points in the Coordinate Plane MAFS Focus - 5.G.1.2 Embedded SMPs - 1, 2, 4–7 Lesson 30 Classify Two-Dimensional Figures MAFS Focus - 5.G.2.4 Embedded SMPs - 2, 3, 5–7 Lesson 31 Understand Properties of Two-Dimensional Figures MAFS Focus - 5.G.2.3 Embedded SMPs - 2, 6, 7 Unit 5 Interim Assessment M = Lessons that have a major emphasis in the MAFS S/A = Lessons that have supporting/additional emphasis in the MAFS 311 Develop Skills and Strategies Lesson 30 (Student Book pages 268–275) Classify Two-Dimensional Figures Lesson Objectives The Learning Progression •Classify two-dimensional figures in a hierarchy based on properties. This standard emphasizes that two-dimensional figures can be classified on many different levels based on their properties. •Use flow charts, Venn diagrams, and tree diagrams to show the hierarchical relationship of twodimensional figures. Prerequisite Skills •Identify two-dimensional figures. •Recognize parallel and perpendicular lines. •Recognize right, acute, and obtuse angles. Vocabulary hierarchy: a ranking of categories based on properties This standard could be used alongside 5.G.3 to demonstrate that while the figures in a certain category of two-dimensional figure have the same attributes, these figures can often be classified more specifically based on those attributes they do not share. Students should understand that two figures could be called the same thing based on one level of classification but be called two different things at the next level of classification. There are two definitions for trapezoids that are used in the United States. This lesson defines a trapezoid as a quadrilateral with at least one pair of parallel sides. In Grade 6, students will apply what they know about two-dimensional figures to help them find the area of polygons and graph polygons on the coordinate plane. Teacher Toolbox Teacher-Toolbox.com Prerequisite Skills Ready Lessons Tools for Instruction ✓ 5.G.2.4 ✓ ✓✓ Interactive Tutorials MAFS Focus 5.G.2.4 Classify two-dimensional figures in a hierarchy based on properties. STANDARDS FOR MATHEMATICAL PRACTICE: SMP 2, 3, 5, 6, 7 (see page A9 for full text) L30: Classify Two-Dimensional Figures ©Curriculum Associates, LLC Copying is not permitted. 295 Part 1: Introduction Lesson 30 At a Glance Students identify the properties of a rectangle, square, and parallelogram. Then students determine the most specific name for three polygons shown. Develop skills and strategies Lesson 30 Arrange the polygons below so that a polygon can also be called by the name of the polygon before it. Order them from left to right. A •Have students read the problem at the top of the page. B C explore it use the math you already know to solve the problem. •Review the meaning of parallel lines. [Lines that never intersect and always remain the same distance apart.] Draw examples of intersecting, parallel, and perpendicular lines on the board. Ask students to describe the lines. What are the properties of polygon A? 2 pairs of parallel and congruent sides, 4 right angles Circle all the names that polygon A can be called. quadrilateral parallelogram rectangle square What are the properties of polygon B? 2 pairs of parallel and congruent sides Circle all the names that polygon B can be called. quadrilateral •Work through Explore It as a class. As you work through the first question, have students justify why polygon A can be called a quadrilateral [a closed figure with 4 sides], a parallelogram [opposite sides are parallel], and a rectangle [4 right angles]. •Ask student pairs or groups to explain their answers for the remaining questions. 5.g.2.4 in this lesson, you will classify polygons by their properties such as sides and angles. take a look at this problem. •Tell students that this page models using properties of polygons to determine the most specific name for a polygon. •Ask students what properties all three polygons have in common. [4 sides, parallel sides] Ask what polygon this describes. [parallelogram] MaFs classify two-Dimensional Figures Step By Step •Have students use the same reasoning to justify why polygon B can be called a quadrilateral and a parallelogram and to justify why polygon C can be called a quadrilateral, a parallelogram, a rectangle, and a square. Part 1: introduction parallelogram rectangle square What are the properties of polygon C? 4 congruent sides, 4 right angles, parallel opposite sides Circle all the names that polygon C can be called. quadrilateral parallelogram rectangle square Write the most specific name for each polygon shown above. rectangle parallelogram A: B: C: square How can you order the polygons so that each can also be called by the name of the ones before it? parallelogram, rectangle, square 268 L30: Classify Two-Dimensional Figures ©Curriculum Associates, LLC Copying is not permitted. Mathematical Discourse •Which is the most specific description of a breakfast menu item: a fried egg, an egg, or a fried egg over easy? Describe how this compares to determining the most specific name for polygons A, B, and C. Responses should indicate an understanding that the description that provides the most information is the most specific description. •Why might it make more sense to call polygon C a square when you could also call it a rectangle, a parallelogram, or a quadrilateral? Calling polygon C a square gives the most information possible about the figure. 296 L30: Classify Two-Dimensional Figures ©Curriculum Associates, LLC Copying is not permitted. Part 1: Introduction Lesson 30 At a Glance Students use a Venn diagram and a flow chart to show the hierarchy of the polygons from the previous page. Step By Step •You may wish to review Venn diagrams with your class before discussing this page. Part 1: introduction Find out More When you order categories of shapes by their properties, you put them in a hierarchy. A hierarchy starts with the most general group. Then it shows how more specific groups are related. A Venn diagram can show categories and subcategories. This Venn diagram shows that parallelograms have all the properties that quadrilaterals have and some new ones. Rectangles have all the properties that parallelograms have and some new ones. Squares have all the properties that rectangles have and some new ones. Quadrilaterals •Read Find Out More as a class. •Have students look at the Venn diagram. Work from the outermost figure to the innermost. Guide students to understand that a parallelogram is a quadrilateral because it has all of the properties of a quadrilateral plus some additional properties. Ask students to name those additional properties. [opposite sides are parallel, opposite sides are congruent] •Similarly, a rectangle is a parallelogram because it has all of the properties of a parallelogram plus some more. Ask students to name those additional properties. [four right angles] •Finally, have students explain why a square is a rectangle. [A square has all of the properties of a rectangle plus all sides are congruent.] Lesson 30 Parallelograms Rectangles Squares A flow chart also can show the hierarchy of the categories and subcategories of shapes. The most general category is at the left, while the most specific is at the right. This flow chart shows the three quadrilaterals from the most general to the most specific. Quadrilaterals Parallelograms Rectangles Squares reflect 1 How are the flow chart and the Venn diagram alike? How are they different? Possible answer: they are alike because they order quadrilaterals first and squares last. they are different because the venn diagram has a larger area for the largest category, and the flow chart shows each category in same-sized boxes. L30: Classify Two-Dimensional Figures ©Curriculum Associates, LLC Copying is not permitted. 269 •Point out that the flow chart shows the same relationships. Concept Extension Explore ordering three-dimensional figures. Point out that just as two-dimensional figures can be ordered from general to most specific, so can three-dimensional figures. Show students an example of a generic prism [faces are parallelograms], a rectangular prism, and a cube. Point out that the faces of a generic prism are parallelograms, the faces of a rectangular prism are rectangles, and the faces of a cube are squares. Ask students to use what they learned about parallelograms, rectangles, and squares to order the three-dimensional figures from least to most specific. L30: Classify Two-Dimensional Figures ©Curriculum Associates, LLC Copying is not permitted. Real-World Connection Encourage students to think about everyday places or situations where people might see or talk about a hierarchy. Example: A hierarchy can be applied to answer the question “Where do you live?” For example, 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 street would be the most specific answer for where you live. 297 Part 2: Modeled Instruction Lesson 30 At a Glance Part 2: Modeled instruction Students use a table to organize the properties of triangles. Then students use a tree diagram to order the triangles in a hierarchy. Lesson 30 read the problem below. then explore different ways to understand ordering shapes in a hierarchy. Order the following triangles from the most general to the most specific: scalene triangle, isosceles triangle, and equilateral triangle. Use a tree diagram to order them. Step By Step •Read the problem at the top of the page as a class. Model it you can understand the problem by organizing the properties of the triangles in a table before ordering them in a tree diagram. •Read the first Model It. Be sure that students understand that a triangle can be called isosceles if it has 2 or 3 congruent sides. Point out that you could also say that an isosceles triangle has at least 2 congruent sides. triangle Isosceles Scalene Equilateral side Properties 2 or 3 congruent sides no congruent sides 3 congruent sides Model it you can represent the problem by starting a tree diagram of the triangles. A tree diagram can show the hierarchy of the categories and subcategories of shapes. The most general category, triangles, is the top box of the tree. The subcategories of triangles, the different kinds of triangles to be ordered, are the branches. •Read the second Model It. Ask students to explain why Triangle is the most general category. [All of the other categories are specific types of triangles.] Triangles Category Subcategories SMP Tip: Students use clear and precise language to list the properties of isosceles, scalene, and equilateral triangles (SMP 6). 270 ELL Support Point out that a tree diagram gets its name because its shape resembles a tree with many branches. L30: Classify Two-Dimensional Figures ©Curriculum Associates, LLC Copying is not permitted. Mathematical Discourse •How does ordering the triangles compare to ordering the quadrilaterals on the previous page? Responses may vary. Possible answer: You could use a list of properties when ordering the quadrilaterals, but the list of properties for the triangles is too involved. •How is a tree diagram similar to a Venn diagram? Responses may vary but should indicate an understanding that both show a hierarchy of items. Both would show the relationships between isosceles, scalene, and equilateral triangles. 298 L30: Classify Two-Dimensional Figures ©Curriculum Associates, LLC Copying is not permitted. Part 2: Guided Instruction Lesson 30 At a Glance Students revisit the problem on page 270 to complete a tree diagram to understand the relationships among isosceles, scalene, and equilateral triangles. Step By Step •Read Connect It as a class. Be sure to point out that the questions refer to the problem on page 270. •Guide students to understand that all triangles can be classified as either scalene or isosceles because scalene triangles have no congruent sides and isosceles triangles have at least 2 congruent sides. Since “at least 2” means 2 or more, equilateral triangles are also isosceles triangles. SMP Tip: Students model the hierarchical order of triangles using a tree diagram (SMP 4). Point out that they could also use a flow chart or Venn diagram to model the order. •Ask students how they would show that all equilateral triangles are isosceles triangles using a Venn diagram. [The equilateral box (or circle) should be nested inside the isosceles box.] Concept Extension Explore Venn diagrams. Point out that the Venn diagram for quadrilaterals, parallelograms, rectangles, and squares from the Introduction included space outside of the parallelogram category because there are figures that are quadrilaterals but not parallelograms. Ask students to name or draw such a shape. [kite] Explain that when drawing a Venn diagram for triangles, scalene and isosceles triangles make up the whole category. There are no triangles that do not fit in one of these two categories. L30: Classify Two-Dimensional Figures ©Curriculum Associates, LLC Copying is not permitted. Part 2: guided instruction Lesson 30 connect it now you will solve the problem from the previous page using the tree diagram and the shared properties of the triangles from the table. 2 Why is “Triangles” in the top row of the tree diagram? Possible answer: that’s the biggest category. 3 Write “Scalene” and “Isosceles” in the second row of the tree diagram. Triangles Why can their categories NOT overlap? a triangle cannot have congruent sides and be classified as scalene. Scalene Isosceles 4 Write “Equilateral” beneath “Isosceles.” Why can all equilateral triangles be classified as isosceles? Equilateral they have at least 2 congruent sides. 5 How can you use a tree diagram to order shapes? Possible answer: Place the most general category of shape at top and more specific categories of shapes beneath. try it use what you learned about ordering shapes in a hierarchy to solve this problem. 6 Complete the Venn diagram below to show the hierarchy of isosceles, scalene, and equilateral triangles. Triangles scalene Isosceles equilateral L30: Classify Two-Dimensional Figures ©Curriculum Associates, LLC Copying is not permitted. 271 Try It Solutions 6Solution: See possible Venn diagram above; Students may draw a Venn diagram with an outside category of triangles, two non-overlapping subcategories of scalene and isosceles, and a category of equilateral that is nested inside isosceles. ERROR ALERT: Students who draw equilateral such that it overlaps both scalene and isosceles may not understand what overlapping categories of a Venn diagram represent. 299 Part 3: Guided Practice Lesson 30 Part 3: guided Practice Lesson 30 study the model below. then solve problems 7–9. Lesson 30 8 Create a Venn diagram to show the ranking of the polygons Student Model Categories that can never overlap, like hexagons and quadrilaterals, are shown as separate circles. Part 3: guided Practice Create a Venn diagram to show the hierarchy of quadrilaterals, polygons, trapezoids, and hexagons. Look at how you could show your work using a Venn diagram. described in the chart. shape Trapezoid Isosceles trapezoid Parallelogram “At least 2” means 2 or more. Description quadrilateral with at least 1 pair of parallel sides trapezoid with at least 2 congruent sides quadrilateral with 2 pairs of parallel sides Possible venn diagram: Polygons Trapezoids Quadrilaterals Hexagons Isosceles Trapezoids Trapezoids Pair/share Recreate the hierarchy with a tree diagram. 9 Look at the flow chart below. Which type of triangle is the most specific? 7 Look at the tree diagram below. Write a statement about the Quadrilaterals relationship between acute triangles and equilateral triangles. Plane Figures Triangles Polygons Pentagons Hexagons Acute Pair/share Write a statement about the relationship between acute triangles and obtuse triangles. Right Obtuse Pair/share Draw one example of a polygon in each separate category of your Venn diagram. Parallelograms The flow chart is like a tree diagram. But the arrows show that the ranking moves from left to right instead of top to bottom. Which statement is true? Circle the letter of the correct answer. Equilateral Solution: Possible answer: all equilateral triangles are acute triangles. a A plane figure will always be a polygon. b All polygons are plane figures. c A polygon is always a quadrilateral, a pentagon, or a hexagon. D A hexagon is not a plane figure. Brad chose c as the correct answer. How did he get that answer? Possible answer: brad read the chart backward. Pair/share Does Brad’s answer make sense? 272 L30: Classify Two-Dimensional Figures L30: Classify Two-Dimensional Figures ©Curriculum Associates, LLC Copying is not permitted. ©Curriculum Associates, LLC Copying is not permitted. 273 At a Glance Solutions Students use tables, Venn diagrams, tree diagrams, and flow charts to classify plane figures in a hierarchy. Ex A Venn diagram illustrating the hierarchy is shown. Students may begin by creating a table or list of properties. Step By Step •Ask students to solve the problems individually and label categories and subcategories in their diagrams. •When students have completed each problem, have them Pair/Share to discuss their solutions with a partner or in a group. 7Solution: Students may say that all equilateral triangles are acute triangles. They may also say that some acute triangles are equilateral triangles. (DOK 2) 8Solution: See possible student work above; Students use the descriptions in the table to create a Venn diagram. (DOK 2) 9Solution: B; “Polygons” is a subcategory of “Plane Figures,” which means that all polygons are plane figures. Explain to students why the other two answer choices are not correct: A is not correct because circles are plane figures but are not polygons. D is not correct because hexagons are plane figures. (DOK 3) 300 L30: Classify Two-Dimensional Figures ©Curriculum Associates, LLC Copying is not permitted. Part 4: MAFS Practice Lesson 30 Part 4: MaFs Practice Lesson 30 Part 4: MaFs Practice 3 Solve the problems. Lesson 30 The word “isosceles” can be used to describe any polygon with at least 2 congruent sides. Look at the following flow chart. 1 Look at the shape below. Isosceles Trapezoids Quadrilaterals Which is a correct classification for this shape from LEAST specific to MOST specific? 2 A polygon, quadrilateral, rectangle B quadrilateral, parallelogram, square C polygon, quadrilateral, rhombus D quadrilateral, rectangle, square Parallelograms Rectangles Squares Part A Draw an example of an isosceles trapezoid. Possible isosceles trapezoid: Classify the triangles as scalene, isosceles, or obtuse. Draw the triangles in the correct box of the table. If a triangle fits more than one classification, draw it in all the boxes that apply. If none of these classifications apply, leave it outside the boxes. Part B Explain how isosceles trapezoids relate to parallelograms. Possible answer: all parallelograms are isosceles trapezoids. trapezoids have at least 2 congruent sides and at least 2 parallel sides. SCALENE ISOSCELES OBTUSE OBTUSE Part C Can you use the term isosceles to describe a rectangle? Explain your reasoning. Possible answer: yes. a rectangle has at least two congruent sides. self check Go back and see what you can check off on the Self Check on page 251. 274 L30: Classify Two-Dimensional Figures L30: Classify Two-Dimensional Figures ©Curriculum Associates, LLC Copying is not permitted. At a Glance Students classify plane figures in a hierarchy based on properties to answer questions that might appear on a mathematics test. Solutions 1Solution: A; The most general name for the shape shown is polygon. The shape has 4 sides, so it is also a quadrilateral. The shape has 2 pairs of congruent sides and 4 right angles, so it is also a rectangle. (DOK 2) ©Curriculum Associates, LLC Copying is not permitted. 275 3Part A Solution: Students draw a trapezoid that has two sides of the same length; See possible drawing above. Part B Solution: See possible student explanation above. Part C Solution: Yes; See possible student explanation above. (DOK 2) 2Solution: See above for solution; A scalene triangle has 3 sides of different lengths. An isosceles triangle has at least 2 congruent sides. An obtuse triangle has an angle greater than 90°. (DOK 2) L30: Classify Two-Dimensional Figures ©Curriculum Associates, LLC Copying is not permitted. 301 Differentiated Instruction Lesson 30 Assessment and Remediation •Ask students to draw a Venn diagram to classify the following shapes in order from least specific to most specific: parallelogram, polygon, rhombus, quadrilateral [polygon, quadrilateral, parallelogram, rhombus]. Remind students that a rhombus is a parallelogram with four congruent sides. •For students who are struggling, use the chart below to guide remediation. •After providing remediation, check students’ understanding. Ask students to draw a Venn diagram to classify the following shapes in order from least specific to most specific: parallelogram, square, rhombus, quadrilateral [quadrilateral, parallelogram, rhombus, square] •If a student is still having difficulty, use Ready Instruction, Level 4, Lesson 32. If the error is . . . Students may . . . To remediate . . . any order other than the correct one not be able to identify all of 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. Hands-On Activity Challenge Activity Build quadrilaterals that fit the given conditions. Name the figure in different ways. Materials: geoboards and geobands Challenge students to provide as many different names as they can for figures that you draw. Have students make a shape that fits conditions you supply. Ask students to name the shape they made. Label important features such as congruent sides and right angles in your drawings. •four sides, opposite sides are parallel, no right angles [parallelogram (or rhombus)] Include figures such as an equilateral triangle, a parallelogram, a square, a rectangle, and a rhombus. Have students justify why each of the names they use applies to the figure. •four sides, opposite sides are parallel, four right angles [rectangle (or square)] •four sides, opposite sides are parallel, four right angles, all sides are congruent [square] If time permits, provide conditions for students to build different triangles and have students build and name the triangles you have described. 302 L30: Classify Two-Dimensional Figures ©Curriculum Associates, LLC Copying is not permitted. Focus on Math Concepts Lesson 31 (Student Book pages 276–281) Understand Properties of Two-Dimensional Figures Lesson Objectives The Learning Progression •Understand that attributes belonging to a category of two-dimensional figures also belong to all subcategories of that category. This standard emphasizes that two-dimensional figures are categorized and subcategorized based on their attributes, and that a figure in one subcategory shares category-defining attributes with figures from the other subcategories with its category. •Use Venn diagrams, flow charts, and tree diagrams to model how properties are shared by categories of polygons. Prerequisite Skills •Identify two-dimensional figures. •Recognize parallel and perpendicular lines. •Recognize right, acute, and obtuse angles. Students should be familiar with examples of various categories of two-dimensional figures and the attributes shared by figures of any subcategories. For example, figures categorized as triangles include scalene, isosceles, equilateral, right, obtuse, and acute. While triangles in these subcategories are different, they all share the category attribute of three sides. Vocabulary Teacher Toolbox attribute: a characteristic of a polygon (e.g., the number of sides) subcategory: a category that is completely nested inside another category; it shares all the same properties as the category it is nested inside of. Teacher-Toolbox.com Prerequisite Skills Ready Lessons Tools for Instruction Interactive Tutorials ✓ ✓ 5.G.2.3 ✓ ✓ ✓✓ MAFS Focus 5.G.2.3 Understand that attributes belonging to a category of two-dimensional figures also belong to all subcategories of that category. For example, all rectangles have four right angles and squares are rectangles, so all squares have four right angles. STANDARDS FOR MATHEMATICAL PRACTICE: SMP 2, 6, 7 (see page A9 for full text) L31: Understand Properties of Two-Dimensional Figures ©Curriculum Associates, LLC Copying is not permitted. 303 Part 1: Introduction Lesson 31 At a Glance Students explore how to describe polygons. Students learn that a polygon can be categorized in more than one way. Focus on Math concepts Lesson 31 MaFs Understand Properties of Two-Dimensional Figures 5.g.2.3 how do we group polygons into categories? Step By Step Polygons are grouped into categories by their attributes, such as the number of sides or angles, the side lengths, and the angle measures. All polygons in the same category share certain attributes. Some attributes of polygons are described in the table below. •Introduce the Question at the top of the page. •Review the meanings of congruent [same size and same shape], parallel [lines that never intersect and stay the same distance apart], and perpendicular [lines that intersect at a 90 degree angle]. •Review the marks that indicate congruent sides and right angles. •Review (or introduce) the meaning of a regular polygon [all sides and angles are congruent]. Ask students to name a regular quadrilateral [square]. Polygon Attribute Description Scalene No congruent sides Example Isosceles At least 2 congruent sides Equilateral All congruent sides Regular All congruent sides Irregular At least 1 side and 1 interior angle are not congruent to the others Right At least 1 pair of perpendicular sides Parallel sides At least 1 pair of opposite sides, if extended forever, will never intersect think Can a polygon be categorized in more than 1 way? Think about how a quadrilateral is defined. It is a polygon with 4 sides. So any shape with 4 straight sides can be called both a polygon and a quadrilateral. If the quadrilateral has two pairs of parallel sides, then it can also be called a parallelogram. •Read the Think question with students. •Point out that since an isosceles triangle has at least two congruent sides and an equilateral triangle has three congruent sides, an equilateral triangle can also be classified as an isosceles triangle. Part 1: introduction shade a polygon above that can be named both a quadrilateral and parallelogram. Every parallelogram is a quadrilateral because every parallelogram has 4 sides. But not all quadrilaterals are parallelograms because not all quadrilaterals have two pairs of parallel sides. 276 L31: Understand Properties of Two-Dimensional Figures ©Curriculum Associates, LLC Copying is not permitted. Mathematical Discourse ELL Support It is important that students understand the phrase “at least” when discussing attributes of polygons. Use the phrase “at least” in sentences to help students better understand the meaning of the phrase. For example, to say that a cell phone costs at least $75 means it costs $75 or more. If you say that you need to study for at least two hours tonight, you mean two hours or more. Have students tell you what “at least two congruent sides” means. [two or more congruent sides] Ask, If a triangle has three congruent sides, can you say that it has at least two congruent sides? [yes] 304 •Is it more useful to categorize triangles by their side lengths or by their angle measures? Explain your reasoning. Responses should indicate an understanding that both categories are useful depending on the context of the situation. •In how many different ways can a polygon can categorized? Explain your reasoning. Responses may vary but should indicate an understanding that it depends on the polygon. For example, a quadrilateral can be categorized two ways, as a polygon and a quadrilateral. A parallelogram can be categorized three ways, as a polygon, quadrilateral, and parallelogram. L31: Understand Properties of Two-Dimensional Figures ©Curriculum Associates, LLC Copying is not permitted. Part 1: Introduction Lesson 31 At a Glance Students use a Venn diagram to show the relationship between isosceles, equilateral, right, and obtuse triangles. Part 1: introduction think How can we show the relationships among polygons with a diagram? A Venn diagram is a useful tool for organizing categories of polygons that share characteristics. Step By Step Triangles •Read the Think question with students. •Tell students that categories that do not overlap do not share any properties. Ask students why an equilateral triangle can never also be an obtuse triangle. [Obtuse triangles have an angle that measures greater than 90 degrees, but all the angles of an equilateral triangle measure 60 degrees.] •Point out that categories that overlap can share properties. For example, you can have a triangle that is both right and isosceles. •A category that is nested completely inside another category shares all the properties of the category it is nested inside of. Lesson 31 The Venn diagram shows me that right triangles never overlap with equilateral triangles. Obtuse Isosceles Equilateral Acute Right Notice that Right partly overlaps Isosceles. A right triangle can share all the properties of an isosceles triangle. Also notice that Right does not overlap Obtuse. A right triangle will never share all the properties of an obtuse triangle. The category for equilateral triangles is nested completely inside the category for isosceles triangles to show that it shares all of the same properties and is a subcategory of isosceles triangles. reflect 1 What do you learn from the diagram showing that the space for obtuse triangles partly overlaps the space for isosceles triangles? Possible answer: an obtuse triangle can also be an isosceles triangle. •Have students read and reply to the Reflect directive. L31: Understand Properties of Two-Dimensional Figures ©Curriculum Associates, LLC Visual Model Work with students to draw an example inside each category and overlapping categories in the Venn diagram. Include markings to indicate right angles and congruent sides as appropriate. Be sure to include an acute scalene triangle in the space labeled “triangles” but outside of any other categories. L31: Understand Properties of Two-Dimensional Figures ©Curriculum Associates, LLC Copying is not permitted. Copying is not permitted. 277 Mathematical Discourse •Can you think of another relationship that could be modeled with a Venn diagram? Responses may vary. Possible answers: whole numbers, prime numbers, and composite numbers; integers, whole numbers, and natural numbers •How could you explain to a friend why a Venn diagram is a useful tool? Responses should indicate an understanding that a Venn diagram visually shows categories and subcategories to easily determine what characteristics items do and do not share. 305 Part 2: Guided Instruction Lesson 31 At a Glance Part 2: guided instruction Students complete a Venn diagram and table of properties for quadrilaterals. explore it a venn diagram can help you understand what properties are shared by categories of polygons. Step By Step 2 The Venn diagram shows categories of quadrilaterals with different properties. Write the name of each category that fits the description. •Tell students that they will have time to work individually on the Explore It problems on this page and then share their responses in groups. You may choose to work through the first problem together as a class. A. 4 sides Quadrilaterals B. At least 1 pair of parallel sides C. 2 pairs of parallel sides D. 4 congruent sides Rhombuses •Note that a trapezoid is defined as a quadrilateral with at least one pair of parallel sides. •Take note of students who are still having difficulty and wait to see if their understanding progresses as they work in their groups during the next part of the lesson. STUDENT MISCONCEPTION ALERT: When completing the table, students may notice that categories A through D repeat the properties of the preceding category. Students may mistakenly extend this pattern to category E. Point out that category E is not nested inside category D, so it does not share all the properties of category D. Emphasize that it is nested inside categories A, B, and C, so it does share all the properties of those categories. Lesson 31 Trapezoids Parallelograms F. Squares E. 4 right angles Rectangles 3 Use the Venn diagram to fill in the table below. 278 category Properties name A 4 sides quadrilateral B 4 sides, at least 1 pair of parallel sides trapezoid C 4 sides, 2 pairs of parallel sides parallelogram D 4 sides, 2 pairs of parallel sides, 4 congruent sides rhombus E 4 sides, 2 pairs of parallel and congruent sides, 4 right angles rectangle F 4 sides, 2 pairs of parallel sides, 4 congruent sides, 4 right angles square L31: Understand Properties of Two-Dimensional Figures ©Curriculum Associates, LLC Copying is not permitted. Mathematical Discourse •What pattern do you see in the chart? Responses should indicate an understanding that a nested category repeats all the properties of the category in which it is nested. •Explain why a square is also a rectangle, a rhombus, a parallelogram, a trapezoid, and a quadrilateral. Responses may vary but should indicate an understanding that a square shares the properties of each of the other figures. 306 L31: Understand Properties of Two-Dimensional Figures ©Curriculum Associates, LLC Copying is not permitted. Part 2: Guided Instruction Lesson 31 At a Glance Students use a Venn diagram to understand how properties are shared by categories of quadrilaterals. Then students use a flow chart to answer questions about the properties of quadrilaterals. Part 2: guided instruction Lesson 31 talk about it use the venn diagram to help you understand how properties are shared by categories of quadrilaterals. yes 4 Is every property of parallelograms also a property of all rectangles? Is every property of rectangles also a property of all parallelograms? Step By Step •Organize students into pairs or groups. You may choose to work through the Talk About It problems together as a class. •Walk around to each group, listen to, and join in on discussions at different points. Use the Mathematical Discourse questions to help support or extend students’ thinking. •Ask students to draw examples of figures in each category. Discuss whether or not the figure drawn could also belong to another category. no Explain what the Venn diagram shows about the relationship between rectangles and parallelograms. Possible answer: all rectangles are parallelograms, but only some parallelograms are rectangles. classify each inference statement as true or false. if false, explain. 5 The opposite angles of any parallelogram are congruent. Therefore, the opposite angles of any rhombus are congruent. true 6 The diagonals of any square are congruent. Therefore, the diagonals of any rhombus are congruent. false; Possible explanation: not all rhombuses are squares. try it another Way the flow chart below shows another way to think about how quadrilaterals are categorized and ranked. Rectangles Quadrilaterals Trapezoids Squares Parallelograms Rhombuses use the flow chart to describe the statements as true or false: 7 The diagonals of a rectangle are congruent. Therefore, the diagonals of any SMP Tip: Students use reasoning skills to classify statements about properties of quadrilaterals as true or false (SMP 1). Have students share their reasoning with a partner. parallelogram must be congruent. true L31: Understand Properties of Two-Dimensional Figures ©Curriculum Associates, LLC •Direct the group’s attention to Try It Another Way. Have a volunteer from each group come to the board to explain the group’s solutions to problems 7 and 8. false 8 A rhombus has 2 lines of symmetry. Therefore, a square has 2 lines of symmetry. Copying is not permitted. 279 Mathematical Discourse •Luca says a four-sided plane figure with two pairs of parallel sides is a quadrilateral. Federico says it is a parallelogram. Who is correct? Explain your reasoning. Responses should indicate an understanding that both are correct, but parallelogram is a more precise name. •How could you explain that not all rectangles are squares? Responses should include the fact that not all rectangles have four congruent sides. L31: Understand Properties of Two-Dimensional Figures ©Curriculum Associates, LLC Copying is not permitted. 307 Part 3: Guided Practice Lesson 31 At a Glance Part 3: guided Practice Students demonstrate their understanding of categorizing polygons. Lesson 31 connect it talk through these problems as a class. then write your answers below. Step By Step 9 categorize: Polygons are either convex or concave. A convex polygon has all interior angles less than 180°. A triangle is an example. A concave polygon has at least 1 interior angle greater than 180°. The quadrilateral below is an example. •Discuss each Connect It problem as a class using the discussion points outlined below. 210° Categorize concave polygons, convex polygons, triangles, quadrilaterals, and rectangles in a Venn diagram. Draw an example of each polygon in the diagram. Possible venn diagram. Categorize: Polygons •You may choose to have students work in pairs to encourage sharing ideas. Concave Convex Triangles Quadrilaterals SMP Tip: Students draw a Venn diagram to model Rectangles the relationship between properties of concave polygons, convex polygons, quadrilaterals, and rectangles (SMP 4). Point out that they could also draw a flow chart or tree diagram to model the relationships. •Have students compare their Venn diagram with a partner’s. Ask a volunteer to explain any differences between their diagram and their partner’s diagram. •Ask students how they decided which were the most general categories. [Triangles, quadrilaterals, and rectangles are all types of polygons; polygons are the most general category.] •Ask students to explain why triangles can never be concave polygons. [Each of the angles of a triangle measures less than 180 degrees. A concave polygon has at least one interior angle greater than 180 degrees.] Explain: 10 explain: Nadriette said that a rectangle can never be called a trapezoid. Explain what is incorrect with Nadriette’s statement. Possible answer: since a rectangle has at least 1 pair of parallel sides, it can be called a trapezoid. 11 create: Describe the properties of a shape that is both a rectangle and a rhombus. Draw an example. a parallelogram with 4 congruent sides and 4 right angles: a square 280 L31: Understand Properties of Two-Dimensional Figures ©Curriculum Associates, LLC Copying is not permitted. Create: •This discussion gives students an opportunity to describe the properties of and draw a shape given certain criteria. •Ask students to describe what methods they can use to answer the question. [List and compare the properties of a rectangle and the properties of a rhombus, draw a Venn diagram, flow chart, or tree diagram.] •The second problem focuses on the ability to explain, in words, how two shapes are related. •Ask students to describe how rectangles are related to trapezoids based on the Venn diagram and the flow chart on the previous pages. [All parallelograms are trapezoids.] Based on those diagrams, what other figures could be classified as trapezoids? [rectangles, rhombuses, and squares] 308 L31: Understand Properties of Two-Dimensional Figures ©Curriculum Associates, LLC Copying is not permitted. Part 4: Performance Task Lesson 31 At a Glance Part 4: Performance task Students create a tree diagram showing the relationship between types of triangles. Then students use the diagram to write statements about types of triangles that are always true and sometimes true. Lesson 31 Put it together 12 Use what you just learned about classifying polygons to complete this task. a Create a tree diagram to show the following types of triangles: acute, obtuse, right, isosceles, and equilateral. Use information in the table to help you. triangle Acute Right Obtuse Scalene Isosceles Equilateral Step By Step •Direct students to complete the Put It Together task on their own. angle Properties all acute angles 2 acute angles and 1 angle exactly 90° 2 acute angles and 1 obtuse angle acute, right, or obtuse acute, right, or obtuse all acute angles Possible tree diagram: •Explain that the categories in each row of the tree diagram can be placed in any order as long as they are connected to the subcategories correctly. Triangles •As students work on their own, walk around to assess their progress and understanding, to answer their questions, and to give additional support, if needed. Acute Right Obtuse Equilateral Isosceles Scalene b Write a statement that is always true about the relationship between obtuse triangles and equilateral triangles. Possible answer: an equilateral triangle can never be classified as an obtuse triangle. •If time permits, have students share their tree diagrams with a partner. c Write a statement that is sometimes true about the relationship between acute triangles and isosceles triangles. Possible answer: an isosceles triangle can be an acute triangle. Scoring Rubrics See student facsimile page for possible student answers. L31: Understand Properties of Two-Dimensional Figures ©Curriculum Associates, LLC A Copying is not permitted. 281 Points Expectations 2 1 0 The tree diagram indicates a student’s understanding of the relationship between the properties of triangles. The tree diagram correctly connects categories and subcategories. B An effort was made to accomplish the task. The tree diagram demonstrates some understanding of the relationship between the properties of triangles, but the student’s diagram is missing categories, misplaces categories or subcategories, or incorrectly connects categories and subcategories. There is no tree diagram or the diagram shows little or no understanding of the relationship between the properties of triangles. L31: Understand Properties of Two-Dimensional Figures ©Curriculum Associates, LLC Copying is not permitted. C Points Expectations 2 The student wrote a statement about the relationship between obtuse triangles and equilateral triangles that is always true. 1 An effort was made to accomplish the task, but the statement the student wrote is not always true. 0 There is no response or the response shows no evidence of understanding the relationship between obtuse and equilateral triangles. Points Expectations 2 The student wrote a statement about the relationship between acute triangles and isosceles triangles that is sometimes true. 1 An effort was made to accomplish the task, but the statement the student wrote is not true. 0 There is no response or the response shows no evidence of understanding the relationship between acute and isosceles triangles. 309 Differentiated Instruction Lesson 31 Intervention Activity On-Level Activity Model and categorize triangles. Create a model of a Venn diagram. Materials: geoboards, strips of paper Materials: string, paper, scissors Group students into pairs. Tell students to write each of the following categories on its own strip of paper, and include a description of each: triangle, right triangle, acute triangle, obtuse triangle, scalene triangle, isosceles triangle, and equilateral triangle. Have one student display a triangle on the geoboard. Have the other student choose as many strips of paper as possible that describe the triangle shown. Then have students switch roles. Distribute string and scissors to each student. Have students draw and cut out a quadrilateral, trapezoid, parallelogram, rhombus, rectangle, and square. Tell students to label and write the properties of each figure on their cutouts. Then have students use string to make a Venn diagram for the figures, using their figures to label each section. Have students draw and cut out other quadrilaterals, write all the categories that describe them on the cutouts, and place them appropriately in their Venn diagram. If time allows, repeat for quadrilaterals, parallelograms, trapezoids, rectangles, rhombuses, and squares. Challenge Activity Justify classifications of polygons. Ask students to answer each of the following questions and to provide an explanation for each answer. •Are all rectangles squares? [No, students may draw a rectangle that does not have 4 congruent sides.] •Are all squares rectangles? [Yes, students may list all of the properties of a rectangle and note that a square has all of these properties.] •Is every quadrilateral a parallelogram or a trapezoid? [No, students may draw a quadrilateral that does not have at least one pair of parallel sides.] •Are all parallelograms rectangles? [No. Not all parallelograms have 90 degree angles.] •Is there a quadrilateral that can be classified as a quadrilateral, trapezoid, parallelogram, rectangle, and rhombus? [Yes. A square shares all of the properties of these figures.] 310 L31: Understand Properties of Two-Dimensional Figures ©Curriculum Associates, LLC Copying is not permitted. Built for the LAFS and MAFS Brand-new, not repurposed content guarantees students get the most rigorous instruction and practice out there. 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