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MathTR-Gr7-Chap03 7/22/04 5:25 PM Page 82 Fraction Operations Key Words equivalent fractions common denominator multiple Get Ready Words proper fraction improper fraction mixed number numerator denominator Curriculum Expectations MAJOR EXPECTATIONS Number Sense and Numeration 7m1, 7m11, 7m18, 7m19, 7m23, 7m26 CONTRIBUTING EXPECTATIONS Number Sense and Numeration 7m5, 7m6, 7m7, 7m16, 7m24, 7m25 Geometry and Spatial Sense 7m47 Chapter Problem A Chapter Problem is introduced in the Chapter Opener. Having students discuss their understanding of how to answer the Chapter Problem will provide you with an idea of what students currently know about this topic. You may wish to have students complete the Chapter Problem Revisits that occur throughout the chapter. These Mini-Chapter problems are particularly useful for special education students and those working at Level 1 or 2 because these revisits will assist students in doing the Chapter Problem Wrap-Up on page 111. Alternatively, you may wish to assign only the Chapter Problem Wrap-Up when students have completed Chapter 3. The Chapter Problem Wrap-Up is a summative assessment. 82 MHR • Mathematics 7: Making Connections Teacher’s Resource MathTR-Gr7-Chap03 7/22/04 5:25 PM Page 83 Planning Chart Section Suggested Timing Core Questions and Typical Assignment Teacher’s Resource Blackline Masters Chapter Opener • 10 min (optional) Assessment Tools Interventions Materials and Technology Tools Formative Assessment: Chapter Problem Get Ready • 40 min Please use the diagnostic tool on BLM GR3A to assess which questions individual students need to complete BLM GR3A Letter to Parents BLM GR3B Create a Fraction Mind Map BLM GR3C Chapter 3 Diagnostic Checklist BLM GR3D Investigate Fractions Diagnostic Assessment: BLM GR3A BLM GR3B BLM 3A Writing Fractions BLM 3B Comparing and Ordering Fractions BLM 3C Multiples BLM 3D Equivalent Fractions • • • • • • 3.1 Add Fractions Using Manipulatives • 40 min Core Questions: 1, 11, 15, 16 Typical Assignment: 1–3, 5, 7, 9–11, 13, 14, 16 [17–19 Levels 3 and 4] Master 5 Tangram Master 8 Grid Paper BLM 3.1A Pattern Block Worksheet BLM 3.1B Add Fractions Try This! Checkbric BLM 3.1C Extra Practice Formative Assessment: Chapter Problem, #10, Try This!, #16 BLM 3.1B BLM 3E Chapter Problem Revisit BLM 3F Section 3.1 Try This! • pattern blocks • pencil crayons 3.2 Subtract Fractions Using Manipulatives • 40 min Core Questions: 1–10, 11, 12, 15 Typical Assignment: 1–3, 5, 7, 8, 10, 12–15 [16, 17 for Level 4] Master 5 Tangram BLM 3.1A Pattern Block Worksheet BLM 3.2A Subtract Fractions Try This! Checkbric BLM 3.2B Extra Practice Formative Assessment: Chapter Problem, #14, Try This!, #15 BLM 3.2A BLM 3G Chapter Problem Revisit BLM 3H Section 3.2 Try This! • pattern blocks • pencil crayons 3.3 Find Common Denominators • 40 min Core Questions for Levels 1 and 2: 1–9, 12, 14 Typical Assignment: 1–3, 5, 8–11, 13, 14 [15 for Level 4] BLM 3.3A Common Denominators Try This! Checkbric BLM 3.3B Extra Practice Formative Assessment: Try This!, #14 BLM 3.3A BLM 3I Section 3.3 3.3 Try This! • paper • pencil crayons BLM 3.4A Add and Subtract Fractions Try This! Checkbric BLM 3.4B Extra Practice Formative Assessment: Chapter Problem, #12, Try This!, #20 BLM 3.4A BLM 3.5A More Fraction Problems Try This! Checkbric BLM 3.5B Extra Practice Formative Assessment: Try This!, #18 BLM 3.5A 3.4 Add and Subtract Fractions Using a Common Denominator • 40 min 3.5 More Fraction Problems • 40 min Core Questions: 1–9, 11, 13, 14 Typical Assignment: 1–11, 13–17 (12, 18, 19 for Level 4) Chapter 3 Review • 40 min pattern blocks fraction circles base ten blocks tangrams fraction strips cuisenaire rods • pattern blocks BLM 3J Section 3.5 Try This! BLM 3K Chapter 3 Wordsearch Chapter 3 Practice Test • 40 min BLM 3.6A Chapter 3 Test BLM 3.6B Chapter 3 Test Assessment Summative Assessment: BLM 3.6A BLM 3.6B Chapter Problem BLM 3.6C Chapter 3 Problem Wrap-Up Rubric BLM 3.6D Chapter 3 Mark Summary BLM 3.6E BLM Answers Summative Assessment: BLM 3.6C BLM 3L Chapter Problem Wrap-Up Chapter 3 • MHR 83 MathTR-Gr7-Chap03 7/22/04 5:25 PM Page 84 Suggested Timing 40 min Materials pattern blocks fraction circles base ten blocks tangrams fraction strips cuisenaire rods Related Resources BLM GR3A Letter to Parents BLM GR3B Create a Fraction Mind Map BLM GR3C Chapter 3 Diagnostic Checklist BLM GR3D Investigate Fractions Get Ready Words • Have students add the terms proper fraction, improper fraction, mixed number, numerator, denominator, multiples, and equivalent fractions, with definitions and examples, to their personal math dictionaries. Diagnostic Assessment Prior to starting Chapter 3, explain that the next topic is about fractions. Allow students to do a think-pair-share about fractions. Discuss with students when they have used fractions in everyday life and what they know about fractions. You may wish to brainstorm and develop a mind map that includes references to fractions. Encourage students to talk about what they know. Try to elicit ideas from all class members. Method 1: Challenge students to show how much they know about fractions. Hand out BLM GR3B Create a Fraction Mind Map. Ask students to complete the statement and then use the boxes to create a mind map about the fraction 134. While students are working, circulate and identify which skills students already have and which they need to review. Students can colour the areas in which they are comfortable. The colour will help you assess what difficulties students are having. Consider initiating conversations to confirm possible assessments. Use BLM GR3C Chapter 3 Diagnostic Checklist to assess whether students are ready to start the work on fractions. If you are using Method 1, note the amount and quality of information students provide about writing fractions, comparing and ordering fractions, multiples, and equivalent fractions. Use this assessment to decide which parts of the Get Ready section each student needs to do. Method 2: Have students discuss and then develop a journal to explain what they know about the topic and how they use 84 fractions in everyday life. Use BLM GR3C Chapter 3 Diagnostic Checklist to identify what you are looking for in each response. Reinforce the Concepts: Students who have difficulty with D, K, and L do not have a mind picture of what 134 looks like, nor have they made a connection between fractions and their own lives. Have these students work with pattern blocks and other concrete materials to help them develop a mind picture of fractions with simple denominators. BLM GR3D Investigate Fractions may assist them in this area. Teaching Suggestions Introduction Students need to have experience with the topics Writing Fractions, Comparing and Ordering Fractions, Multiples, and Writing Equivalent Fractions in order to understand and complete this chapter. For students having difficulty with understanding fractions, use a concrete approach and continually reinforce with visuals such as number lines and diagrams. Encourage students to work in pairs or to discuss difficulties as a class. MHR • Mathematics 7: Making Connections Teacher’s Resource MathTR-Gr7-Chap03 7/22/04 5:25 PM Page 85 Accommodations Language Colour-code math terms by strand (Number sense and Numeration, Measurement, etc.), and post on a math word wall (e.g., measurement terms might be colour-coded green). Visual/Perceptual/Spatial/Motor Students may find it helpful to use rectangular grids to compare and order fractions or to find equivalent fractions and fractions in lowest terms. ESL Explain to students that, in the English language, new words can be created by adding a different suffix, or ending, to a root word. For example, this chapter is called Fraction Operations. Explain that to “operate” means to do a task. An “operator” is the person who does the task (suffix -or), and an “operation” is the task (suffix -ion). Common Errors • Students may have trouble identifying the denominator from diagrams. Rx Have students verbalize the number of parts in each whole, and remind them that the total number of parts in each whole is the denominator. • Students may have trouble comparing fractions with different denominators. Rx Remind students that only equal items can be compared, thus the total number of parts (denominators) must be equal. Have students list several multiples of each denominator and circle the ones that are common. Get Ready Answers (pages 84–85) 3 1 1. a) ; 1 2 2 7 3 b) ; 1 4 4 • Students may include the number as the first multiple. 8 2 c) ; 2 3 3 2. Diagrams may vary. a) b) Rx Remind students that every multiple is the product of the number and a natural number. When listing multiples, encourage students to start with the first natural number, the number 1. c) d) • Students may forget to multiply or divide both the numerator and denominator by the same number. e) Rx Encourage students to represent the fractions visually by drawing. f) 3. Diagrams may vary. 1 1 a) b) 4 3 1 1 1 – – – 8 4 4 Interventions 2 c) 3 5 – 8 1 – 3 BLM 3A Writing Fractions, BLM 3B Comparing and Ordering Fractions, BLM 3C Multiples, and BLM 3D Equivalent Fractions provide extra practice for students who need it. 2 – 3 4. 0 1 – 2 3 – 4 1 5. a) 2, 4, 6, 8, 10 b) 4, 8, 12, 16, 20 c) 5, 10, 15, 20, 25 3 – 2 3 1– 4 2 3 6. 15 4 1 7. a) ; 12 3 2 1 b) ; 8 4 8. Answers may vary. Chapter 3 • MHR 85 MathTR-Gr7-Chap03 7/22/04 5:25 PM Page 86 Add Fractions Using Manipulatives 3.1 Suggested Timing 40 min Materials pattern blocks pencil crayons Related Resources Master 5 Tangram Master 8 Grid Paper BLM 3.1A Pattern Block Worksheet BLM 3.1B Add Fractions Try This! Checkbric BLM 3.1C Extra Practice Specific Expectations 7m1, 7m7, 7m18, 7m24, 7m25, 7m26, 7m47 Link to Get Ready The Get Ready sections Writing Fractions, Comparing and Ordering Fractions, and Equivalent Fractions provide the needed skills for this section. You may wish to have students complete these sections before starting this section. 1. Draw a diagram to represent each fraction. 1 a) 2 2 b) 3 3 c) 8 2. Draw a diagram representing an equivalent fraction for each fraction in question 1. 3. Calculate the area. 8 mm 4. Explain how these triangles are related. X 86 Mental Math 6. Estimate. 7. Estimate. 34 46 53 62 78 39 242 177 746 10. 446 214 A B Z know it is isosceles. 8. Estimate. 14 mm C 5. Draw an isosceles triangle. Show how you Y MHR • Mathematics 7: Making Connections Teacher’s Resource 9. $86.49 25.35 MathTR-Gr7-Chap03 7/22/04 5:26 PM Page 87 Near Compatible Estimation: When estimating, have students consider what pairs of numbers add to a multiple of 10. For example, estimate 76 44 19 26 53. Think: 76 26 add to about 100. 44 53 add to about 100. The answer is about 220. Either introduce this idea before students do the Warm-Up or suggest it afterwards as a possible way of estimating the answers for questions 6–8. Discover the Math Answers (pages 86–87) Teaching Suggestions Introduction As a class, discuss the section opener. Provide pairs of students with pattern blocks and have them determine some of the relationships among the pattern block pieces. Mental Math—Do the following imaging exercise with students. Read the clues. Allow sufficient time for students to visualize each clue. Then, have students draw a picture of each clue. • Think of a circle. • Think of one third of that circle. • Think of another third. • Draw what is left. 1 1 1 2. 2 trapezoids: 1; 3 rhombuses: 2 2 3 1 1 1 1 1 1 1; 6 triangles: 3 3 6 6 6 6 1 1 1 1 1; 1 trapezoid, 3 triangles: 6 6 2 6 1 1 1; 1 trapezoid, 1 rhombus, 1 triangle: 6 6 1 1 1 1 1; 1 rhombus, 4 triangles: 2 3 6 3 1 1 1 1 1; 2 rhombuses, 2 triangles: 6 6 6 6 2 1 1 1 3 6 6 3. Reflect Concrete materials and diagrams make the answer visual. I can see what fractions make a whole without always naming the fraction. Pattern blocks represent part(s) of a whole. ESL Discuss the terms “manipulatives” and “concrete materials.” Ask students to name some examples of these materials. How would students refer to manipulatives or concrete materials in their native language? Gifted and Enrichment Have students name each shape and write the value of the fractions that make it. For example, if a hexagon is 1, then 6 triangles or Discover the Math Have students work through the investigation. Discuss any questions as a class. 1 6 1 1 6 6 1 6 1 6 1 6 1. Encourage students to see how they can mix fractions. For example, if a hexagon is 1, then 1 trapezoid and 3 triangles or 12 1 6 1 6 1 6 1. Also, have them consider what might happen if a rhombus or trapezoid equals 1. Chapter 3 • MHR 87 MathTR-Gr7-Chap03 7/22/04 5:26 PM Page 88 Accommodations Visual/Perceptual/Spatial/Motor Have students place pattern block pieces on top of a hexagon to see the spacing to make one whole. Ongoing Assessment Communicate the Ideas • Check that students understand the concept of equivalent fractions. You may wish to have students explain the Key Ideas in their own words. • Ask students to explain why each fraction need not be a part of equal size. Common Errors • Students may not represent the fractions as having the same number of parts. Rx Have students place the equal pattern block pieces on top of the shape to identify equivalence. Accommodations Language/ Visual/ Perceptual/ Spatial/ Motor • Display the Key Ideas on chart paper and discuss as a class. • Provide Master 5 Tangram to help students with questions 13 and 14. Interventions • Have students use BLM 3.1A Pattern Block Worksheet to visualize the addition for question 12. • Students who need extra practice can use BLM 3.1E Extra Practice and Mathematics 7 Workbook, pages 23–25. • Use BLM 3E Chapter Problem Revisit for students who need assistance with question 10. • Use BLM 3F Section 3.1 Try This! for those students who are having difficulty with question 16. Method 1: Have students use pattern blocks to create a hexagon. Using manipulatives allows students to see the results in a concrete fashion. Method 2: Have students use BLM 3.1A Pattern Block Worksheet to identify the parts of each whole and to determine which shapes make one whole. Students can colour the shapes they are showing and write the related addition statement below the shapes. Method 3: Use BLM 3.1A Pattern Block Worksheet as a transparency. Colour the identified sections in the appropriate colour (green triangle; blue rhombus; red trapezoid). As a class, identify which shapes would complete the cartoons. Write the related addition statement below each shape. The Example illustrates how to add fractions using concrete materials and diagrams. Have students try both methods to solve the problem. Key Ideas Ongoing Assessment Try This! Use BLM 3.1B Add Fractions Try This! Checkbric for formative assessment of the material covered in this section. 88 Have students read and review the Key Ideas. Have students answer and discuss the Communicate the Ideas questions. MHR • Mathematics 7: Making Connections Teacher’s Resource MathTR-Gr7-Chap03 7/22/04 5:26 PM Page 89 Communicate the Ideas Answers (page 87) 1. Answers may vary. 1 – 6 1 – 6 2. Answers may vary. 1 – 6 1 – 6 3 – 6 1 – 6 1 – 6 ⴝ ⴝ 1 – 2 3. Answers will vary depending on method is used. Check Your Understanding Answers (pages 88–89) 1 1 4. a) b) 2 3 5 1 5. a) b) 6 2 4 1 1 1 1 4 6. a) ; 6 6 6 6 6 6 1 1 1 1 7. a) ; b) 2 3 6 2 2 1 1 2 c) ; d) 3 2 6 3 2 2 1 8. a) b) or c) 3 6 3 3 6 9. a) or 1 b) or 1 3 6 Communicate the Ideas Suggest that students consider the use of concrete materials in their answer to question 3. Use the answer to the questions to assess their readiness to start the Check Your Understanding. Check Your Understanding 2 1 1 2 b) ; 3 3 3 3 5 1 1 5 ; 6 2 3 6 1 1 1 1; 1 2 3 6 5 6 5 c) 6 10. Answers will vary. Possible answers include: 2 3 1 (whole puzzle) 5 5 1 1 2 2 11. a) 1 1 b) 1 1 3 3 3 3 1 1 c) 1 1 2 2 2 2 1 2 1 2 1 12. a) b) c) d) 1 3 3 3 3 3 3 1 1 1 1 13. a) 1 b) 1 2 2 2 2 1 1 1 c) 1 2 4 4 1 1 1 14. 4 4 2 15. a) & b) Answers may vary. Question Planning Chart Core questions for Levels 1 & 2: 1–11, 14, 15, 16 Typical assignment: 3, 5, 7, 9–11, 13, 14, 16 (17–19 for Level 4 and some Level 3s) 16. Answers may vary. 17. Diagrams may vary. 1 – 3 1 – 3 For question 15, encourage students to use www.mcgrawhill.ca/books/math7 and follow the links to a web site where they can build pattern block diagrams on-screen. 1 – 6 1 1 1 5 – + – + – = – 3 3 6 6 5 b) 1 6 18. a) Diagrams may vary. 1 – 2 1 – 2 1 – 3 1 – 6 1 – 3 5 19. 1 6 Chapter 3 • MHR 89 MathTR-Gr7-Chap03 7/22/04 5:26 PM Page 90 Subtract Fractions Using Manipulatives 3.2 Suggested Timing 40 min Materials pattern blocks pencil crayons Related Resources Master 5 Tangram BLM 3.1A Pattern Block Worksheet BLM 3.2A Subtract Fractions Try This! Checkbric BLM 3.2B Extra Practice Specific Expectations 7m5, 7m6, 7m7, 7m16, 7m18, 7m24, 7m25, 7m26, 7m47 1. Write the fraction represented in this diagram. 2. Draw a diagram to represent an equivalent fraction for question 1. 3. Add using concrete materials or diagrams. 1 1 3 6 a) 90 Mental Math 5. Classify the triangle in two ways. Give reasons for your classifications. 2450 3252 743 8. Estimate. 9. $875.86 10. 8.6 6.2 1 2 2 3 Draw a parallelogram. Explain how to find the area. 7. Estimate. 193 346 208 892 310 240 b) 4. 6. Estimate. A 60° 60° B 60° C MHR • Mathematics 7: Making Connections Teacher’s Resource 63.55 MathTR-Gr7-Chap03 7/22/04 5:26 PM Page 91 Discover the Math Answers (page 90) 1 1 1 1 1 1 1. 3 ways: 1 ; 1 ; 1 ; 2 6 6 6 6 3 2. Reflect Pattern blocks help you to manipulate the fraction represented in the question. Physically removing the blocks being subtracted shows what is remaining. Accommodations Language/Visual/Perceptual/ Spatial/Motor Students may be confused about how to take away 12. Encourage students to make as many fractions greater than 12 but less than 1 as they can using pattern blocks or BLM 3.1A Pattern Block Worksheet. Then, ask them to remove 12 by circling or highlighting a trapezoid or its equivalent. ESL Discuss the similarities and differences among the terms “illustration,” “diagram,” “sketch,” and “drawing.” Teaching Suggestions Strategies Introduction In this section, students learn how to subtract fractions using manipulatives. Discuss the section opener as a class. Provide pairs of students with pattern blocks. Have students determine what fraction of the hexagon pictured is and is not covered. This investigation uses modelling to have students subtract. Students could share and explain how they are using this strategy. Invite students to try each other’s techniques. Discover the Math The purpose of this investigation is to provide students with practice in subtracting fractions using manipulatives. Prior to doing this activity, allow students time to freely explore the pattern blocks and familiarize themselves with the fraction relationships. 1 Method 1: Have students use pattern blocks to subtract . Ideally, each 2 student should have a set of manipulatives. Remind students that a trapezoid represents 12, and to take it away, the first fraction needs to be greater than 1 trapezoid. Students may find it useful to use BLM 3.1A Pattern Block Worksheet for recording purposes. Common Errors • Students may not represent the fractions with parts of equal size. Rx Encourage students to use manipulatives to demonstrate the amount that is being removed. This will help them see that they can only subtract pattern blocks that are available. Chapter 3 • MHR 91 MathTR-Gr7-Chap03 7/22/04 5:26 PM Page 92 Journal Ask students to reflect on their preferred method for subtracting fractions (e.g., using manipulatives, using diagrams, or using subtraction sentences). Which method do they like best, and why? Ongoing Assessment • Are students able to use pattern blocks to represent subtracting fractions? • Do students use parts of equal size—and use them correctly? • Do they remove the blocks represented by the fraction that is being subtracted? • Do they correctly identify the fraction that remains? Communicate the Ideas Listen as students model verbal communication. Can they explain a variety of ways to represent a fraction? Can they show visually how to subtract fractions? Accommodations Language/Visual/Perceptual/Spatial/Motor For question 12 in Check Your Understanding, allow students to give an oral explanation or to use visuals. Encourage students to use tangrams to manipulate the values. Provide Master 5 Tangram. Interventions Students who have difficulty with subtraction may wish to visit the web site listed with question 15 on page 89. Have them “virtually” explore subtraction. Ongoing Assessment Try This! BLM 3.2A Subtract Fractions Try This! Checkbric provides formative assessment for student work in this section. Interventions • Use BLM 3G Chapter Problem Revisit to assist students who are having difficulty with question 14. • You may wish to use BLM 3H Section 3.2 Try This! for those students who need extra scaffolding for question 15. • BLM 3.2B Extra Practice and Mathematics 7 Workbook, pages 26–28, provides extra basic practice. Gifted and Enrichment Questions 16 and 17 ask students to think of two hexagons as one whole. After students complete the questions, ask them to describe the strategies they used in solving the problems. Method 3: Use BLM 3.1A Pattern Block Worksheet as a transparency. Colour the identified sections in the appropriate colour (green triangles; blue rhombuses; red trapezoid). As a class, identify when 12 could be taken away. Record the responses below each visual as a subtraction sentence. Have students use BLM 3.1A Pattern Block Worksheet to record the findings. The Example illustrates two methods for subtracting fractions: using concrete materials and using diagrams. Discuss with students what is happening in each subtraction problem and how the method shown divides the shape into sixths. Many students will find it useful to show 12 as sixths in order to subtract. Discuss how to get that equivalent fraction. Key Ideas Method 2: Students can use BLM 3.1A Pattern Block Worksheet to identify parts that are greater than 12. Have students highlight the 12, as represented by a trapezoid, that is being removed. Then, have them use a subtraction sentence to record what they have done. Have students read and review the Key Ideas section. Remind students that fractions must be represented by equal-sized parts. Communicate the Ideas Have students answer and discuss the Communicate the Ideas questions. 92 MHR • Mathematics 7: Making Connections Teacher’s Resource MathTR-Gr7-Chap03 7/22/04 5:26 PM Page 93 7. Diagrams may vary. a) b) c) 1 1 8. a) 2 3 d) 2 1 b) 3 6 9. Diagrams may vary. a) b) 10. Diagrams may vary. a) b) c) d) 11. Diagrams may vary. a) Check Your Understanding b) Question Planning Chart Core questions for Levels 1 and 2: 1–10, 11, 12, 15 Typical assignment: 1–3, 5, 7, 10, 12–15 (16, 17 for Level 4 and some Level 3s) Communicate the Ideas Answers (page 91) 1. –2 3 –2 3 –2 3 2. –2 ⴚ –1 3 6 1 1 12. 2 4 1 13. a) 1 2 1 b) 1 4 3 14. Answers may vary. Season puzzle: 1 (number 4 5 of pieces left); Colour puzzle:1 8 15. Answers may vary. Check Your Understanding Answers 1 16. a) 4 (pages 92–93) 17. a) Diagrams may vary. 1 3. a) 1 3 1 1 b) 3 6 c) 1 1 5. a) 2 6 1 1 c) 2 3 6. Diagrams may vary. a) b) 1 3 c) 1 4 4 5 2 c) 6 6 4. Diagrams may vary. a) b) 2 1 b) 3 6 1 b) 1 4 c) b) In this example, 1 is represented by a 2 whole hexagon and 13 is represented by two blue rhombi. c) The numerical answer is represented by pattern blocks twice the size as those that would represent the answer if 1 hexagon 1 whole. Chapter 3 • MHR 93 MathTR-Gr7-Chap03 7/22/04 5:26 PM Page 94 3.3 Find Common Denominators Suggested Timing 40 min Materials paper pencil crayons Related Resources BLM 3.3A Common Denominators Try This! Checkbric BLM 3.3B Extra Practice Specific Expectations 7m1, 7m7, 7m11, 7m16, 7m24, 7m25 Link to Get Ready The Get Ready section Multiples provides the needed skills for this section. You may wish to have students review this material before starting this section. Balancing Strategy: This is similar to the near compatible estimation strategy discussed in section 3.1. It involves changing the numbers to make the addition easier to compute. For example, you might take one or more from one addend and add it to the other. The final sum is maintained. Once students have practised this strategy, encourage them to look for places to use it. For example, have them add 68 57. Think: 68 2 70 70 50 120 94 Now, take that 2 away from the 7. 120 5 125 When discussing these strategies with students, have them consider what other mental strategy they might use. For example, another way to add 68 57 is to use the front-end strategy discussed in section 1.3. 60 50 110 8 7 15 110 15 125 MHR • Mathematics 7: Making Connections Teacher’s Resource MathTR-Gr7-Chap03 7/22/04 5:26 PM Page 95 1. Add using concrete materials or diagrams. 1 3 1 2 1 6 a) 2 3 b) 2. Subtract using concrete materials or diagrams. 1 2 2 3 a) 1 1 6 b) 3. Find a common denominator for: 1 2 1 3 1 2 a) and 1 4 b) and 4. What is the perimeter? 4.5 cm 5. Classify the quadrilateral. Explain your choice. P Q S R 2 cm Mental Math Teaching Suggestions 6. 49 55 7. 68 37 9. Estimate. 10. Estimate. $ 245.60 53.22 198.19 8429.6 3231.8 8. 159 33 Introduction As a class, discuss the section opener visual. Have students discuss in pairs how to determine the fraction of the rectangle that is covered. Mental Math—Have students order the following fractions from least to greatest using benchmarks and number sense, not common denominators: 4 3 2 1 , , . Students can use zero, , and 1 as benchmarks. 5 6 8 2 Discover the Math This investigation introduces students to a concrete method of finding a common denominator. Method 1: Have students do the investigation as shown in the text. Encourage them to see how the two ways of paper folding result in a common denominator. Method 2: Have students work through the investigation by drawing diagrams representing the fractions. Students then can model the fractions by shading in the appropriate sections. Have them practise the technique with some other simple fractions. Chapter 3 • MHR 95 MathTR-Gr7-Chap03 7/22/04 5:26 PM Page 96 Discover the Math Answers (page 94) 3. a) 12 4 6. 12 3 b) 3; 12 7. Reflect Folding the paper to represent the denominators of each fraction will show a multiple that both denominators have in common. Literacy Connections Have students practice using “…” in their lists of multiples. ESL In their own words students write an explanation for finding a common denominator. Ongoing Assessment • Do students understand what is meant by a common denominator? • Can they find a common denominator using one or more methods? Strategies Ask students to identify what pattern they see in the following two series of numbers: 2, 4, 6, 8 and 3, 6, 9, 12. Have them predict the next common multiple for each series. Method 3: Have students identify the first five multiples of the numbers 4 and 3. Have students circle the common multiples. The Example illustrates three methods for finding a common denominator. Reinforce that all of the methods work and are useful. Key Ideas Have students read and review the Key Ideas section. Ensure that students understand how to use each solution strategy. Communicate the Ideas Have students answer and discuss the Communicate the Ideas questions. Check Your Understanding Question Planning Chart Ongoing Assessment Core questions for Levels 1 and 2: 1–9, 12, 14 Typical assignment: 1–3, 5, 8–11, 13, 14 (15 for Level 4 and some Level 3s) Communicate the Ideas Check that students realize they can find a common denominator in several ways, including paper folding, using a diagram, and listing multiples. 96 MHR • Mathematics 7: Making Connections Teacher’s Resource MathTR-Gr7-Chap03 7/22/04 5:26 PM Page 97 Ongoing Assessment Try This! • BLM 3.3A Common Denominators Try This! Checkbric provides formative assessment for student work in this section. • Level 4 performance can be demonstrated by showing multiple ways to determine which fraction is greater. Journal Have students complete the following statement in their journals: The best way to find a common denominator is... because... Accommodations Language/Visual/Perceptual/Spatial/Motor For questions 9 and 12, allow students to demonstrate their understanding through paper folding or diagrams. For question 12, allow students to explain their understanding and process orally. Interventions • Use BLM 3I Section 3.3 Try This! for those students who are having difficulty with question 14. • Students having difficulty finding more than one common denominator may benefit from listing the multiples of the denominators until they identify two common multiples. • BLM 3.3B Extra Practice and Mathematics 7 Workbook, pages 29–30, provides additional practice. Encourage students to use paper folding and/or diagrams for at least questions 4, 7, and 8, even if they are able to find a common denominator mentally. Questions 11 and 12 are good application questions. Question 14 allows students to investigate fractions through comparing and to use their skills at finding common denominators. Communicate the Ideas Answers (page 96) 1. No. She could draw a diagram or use multiples to find a common denominator of 40. 2. 24 and 42 are both multiples of 3 and 6. Other possible common denominators include: 6, 12, 18, 30, and 36. 3. Any multiple of 6 could be used; 6 is best because it is the smallest common denominator. Check Your Understanding Answers (page 97) 4. Answers may vary; lowest common denominators b) 8 c) 12 d) 10 given a) 12 5. Answers may vary; lowest common denominators b) 21 c) 20 d) 24 given a) 15 6. Answers may vary. 4 2 3 1 a) ; 6 3 6 2 5 1 6 2 c) ; 15 3 15 5 18 3 20 5 b) ; 24 4 24 6 8 2 6 1 d) ; 12 3 12 2 7. Answers may vary. a) 6 b) 24 c) 15 d) 12 8. Answers may vary. a) 4 b) 10 9. Answers may vary. a) 15, 30 b) 12, 24 10. Answers may vary. a) 6, 12, 18 b) 8, 16, 24 11. 12, 24, 36 12. a) 12, 18 b) Answers may vary. For example, you could find common multiples of 2 and 6. 13. Answers may vary. 6 5 5 2 a) 10; , b) 8; , 10 10 8 8 14. Methods may vary. 5 2 a) b) 6 5 15. Answers may vary. For example, you could find common multiples of 2, 3, and 4. a) 12 or 24 b) 60 Chapter 3 • MHR 97 MathTR-Gr7-Chap03 7/22/04 5:26 PM Page 98 Add and Subtract Fractions Using a Common Denominator 3.4 Suggested Timing 40 min Materials pattern blocks Related Resources BLM 3.4A Add and Subtract Fractions Try This! Checkbric BLM 3.4B Extra Practice Specific Expectations 7m1, 7m6, 7m7, 7m11, 7m16, 7m18, 7m19, 7m24, 7m25, 7m26 Strategies Discuss how the strategy of model making helps students visualize a problem and arrive at a solution. Interventions Allow students to use fraction circles or diagrams to model questions concretely. 1. State two common denominators for each pair of fractions. 1 4 1 6 a) and 1 2 6. 159 33 7 8 b) and 7. $2.49 $0.24 2. Write 3 equivalent fractions for each fraction. 1 3 a) Mental Math 8. Estimate. 2.4 6.3 7.8 9.2 3 4 b) 9. 100 1 3. Use diagrams to subtract: 1 6 10. 200 145 4. Identify the numerator and denominator 3 8 of . 1 4 1 2 5. Use diagrams to add: 98 MHR • Mathematics 7: Making Connections Teacher’s Resource 64 MathTR-Gr7-Chap03 7/22/04 5:26 PM Page 99 Literacy Connections • Adding and Subtracting Fractions: Have students explain why it is not necessary to subtract the denominator. • Remembering Common Denominators: Have students brainstorm other ways that can help them to remember to use common denominators. Accommodations Language/Visual/Perceptual/Spatial/Motor Have students make a diagram of the section-opener problem and write a subtraction sentence about it. Teaching Suggestions Introduction As a class, discuss the section opener. Draw students’ attention to the photograph. Discuss how large the pizza was, what fraction was eaten, and what is left. Discover the Math The purpose of this investigation is to provide students with different methods for adding and subtracting fractions. • Example 1 illustrates the procedure for adding fractions using pattern blocks. This method reinforces the concrete understanding of adding fractions. The Example also offers an algorithm strategy of solving by using a common denominator. The solution is semiconcrete because the common denominator is determined using paper folding. L Chapter 3 • MHR 99 MathTR-Gr7-Chap03 7/22/04 5:26 PM Page 100 Strategies Encourage students to continue making models to add and subtract fractions until they are confident about visualizing fractions. Ongoing Assessment Communicate the Ideas Are students confident about what steps to follow when adding or subtracting fractions with different denominators? Common Errors • Students may try to add and subtract fractions that have different denominators. • Example 2 shows the comparable procedure for subtracting fractions. Two methods are depicted: Method 1 uses manipulatives; Method 2 uses multiples. Review the thought balloons with students. • Example 3 illustrates two strategies for adding the same fraction: using manipulatives and multiplying by a whole number. Review the thought balloon with students. Key Ideas Have students read and review the Key Ideas section. Ask them to explain in their own words why a common denominator is needed to add or subtract fractions. Communicate the Ideas Have students answer and discuss their solutions to the Communicate the Ideas questions. Rx Use pictures to explain why you can’t add or subtract fractions that have different denominators. Encourage students to draw a diagram or fold paper to find a common denominator. 100 MHR • Mathematics 7: Making Connections Teacher’s Resource MathTR-Gr7-Chap03 7/22/04 5:26 PM Page 101 Check Your Understanding Question Planning Chart Core questions for Levels 1 and 2: 1–4, 5–10,11–13, 20 Typical assignment: 1–4, 5–6, 8–10,12, 14, 15, 17, 18, 20 (21 for Level 3 and some Level 2s; 19, 22, 23 for Level 4 and some Level 3s) Communicate the Ideas Answers (page 101) 1. Diagrams may vary. Ongoing Assessment Try This! • BLM 3.4A Add and Subtract Fractions Try This! Checkbric provides formative assessment for material covered in this section. • Level 4 performance will be evident by the clarity and completeness of the explanation. Interventions • Review with students how to find common denominators. • BLM 3.4B Extra Practice and Mathematics 7 Workbook, pages 31–34, provides additional reinforcement. 2. Use 6 as a common denominator because it is the smallest common multiple 3 2 1 of both 2 and 3. 6 6 6 5 3 8 3. Answers may vary. For example, use 6 as a common denominator: 6 6 6 1 1 1 3 3 1 4. ; 6 6 6 6 6 2 Chapter 3 • MHR 101 MathTR-Gr7-Chap03 7/22/04 5:26 PM Page 102 Interventions Encourage students to identify a pattern in Question 11 on page 102. Have them visualize two halves. They might find it easier to say, “2 of one half”, “3 of one third”, etc. Gifted and Enrichment • Questions 19 and 23 will challenge students to identify common multiples from three numbers. • Question 22 will encourage students to consider order of operations with fractions. They may wish to do the Making Connections activity on page 103 in connection with this question. Check Your Understanding Answers (pages 101–103) 2 3 5 1 2 3 5. a) b) 6 6 6 4 4 4 2 3 5 1 3 4 c) d) 8 8 8 6 6 6 1 1 1 3 6. a) b) c) d) 4 8 3 8 3 4 1 4 7. a) b) or 1 c) 4 3 3 5 8. Answers may vary. 4 1 3 3 2 1 a) b) 6 6 6 4 4 4 6 5 1 7 2 5 c) d) 10 10 10 8 8 8 9. Answers may vary. 2 5 7 a) 10 10 10 6 1 10 16 c) or 1 15 15 15 15 1 3 1 10. a) 3 or 1 2 2 2 2 11. a) 1 b) They all equal one. 2 3 1 10 13 b) or 1 12 12 12 12 1 3 4 2 d) or 6 6 6 3 1 5 2 b) 5 or 1 3 3 3 c) 1 1 8 12. Answers may vary. Total number of pieces: 8 = or 1; 8 8 1 3 Number of pieces that did not fall out: 3 8 8 3 15 1 4 12 13. a) 5 or 7 b) 3 9 or 4 2 2 2 3 3 102 MHR • Mathematics 7: Making Connections Teacher’s Resource MathTR-Gr7-Chap03 7/22/04 5:26 PM Page 103 Making Connections (page 103) Review order of operations with students. Have them challenge each other with fraction questions that include brackets, multiplying, adding, and subtracting. Answers 5 12 5 6 a) b) Journal Have students answer the following questions in their journals: What do you find most interesting about this section? What is the most challenging? Check Your Understanding Answers (pages 101–103) 20. Diagrams may vary. a) ⴙ 9 16 25 25 14. ; Since is greater than 1, the 24 24 24 24 addition shows that together they shovelled more than 1 whole driveway. One or both of their statements are incorrect. 1 – 4 1 – 2 b) are not the same size. 7 16. a) 12 ⴙ 1 – 6 5 – 6 9 10 9 2 1 5 10 1 1 3 c) or , or . 12 12 12 3 6 6 12 4 2 4 3 21. Diagrams may vary. 1 4 ⴙ 9 2 1 2 1 17. is greater. , 10 5 2 5 2 2 1 5 9 5 , 3 6 6 10 6 5 2 1 5 3 3 1 18. 1 is greater. 1 , 1 , 8 3 3 8 8 8 3 13 1 19 19. a) or 1 b) 12 12 20 ⴝ 2 – 3 ⴝ 5 b) 12 3 – 4 ⴙ 15. a) Answers may vary. The sections being added b) ⴝ 1 ⴙ 1 – 2 ⴝ 1 – 4 3 1– 4 1 3 25 1 22. a) 1 or 2 3 4 12 2 2 1 11 1 b) 2 or 1 5 2 10 10 2 1 3 19 19 23. ; Since is less than 1, the 5 4 10 20 20 addition shows that together they did not clean all the windows. They should not be paid the full amount. Chapter 3 • MHR 103 MathTR-Gr7-Chap03 7/22/04 5:26 PM Page 104 3.5 More Fraction Problems Suggested Timing 40 min Related Resources BLM 3.5A More Fraction Problems Try This! Checkbric BLM 3.5B Extra Practice Specific Expectations 7m1, 7m5, 7m6, 7m7, 7m11, 7m16, 7m18, 7m19, 7m23, 7m24, 7m25, 7m26 Compatible Number Estimation: Students can use compatible numbers to help them subtract. For example, when subtracting 750 1014 766, think of the pairing 250 to make 1000. Think: 766 is 16 more than 750, so the answer will be 16 different. But, 1014 is 14 greater than 1000, so the answer will be only 2 different. The answer is 252. 1. List four multiples of 5 and 10. 5. Draw this triangle: • one side measures 5 cm • one side joins a 25° angle and a 40° angle 2. Rewrite each expression with a common denominator, and solve. 1 3 1 2 a) 3 8 1 3 b) 3. Write each repeated addition as a multiplication, and evaluate. 1 1 1 1 4 4 4 4 1 1 1 1 b) 3 3 3 3 1 4 a) Mental Math 6. Estimate. 106 77 8. Estimate. 508 250 9. 5.9 3.6 10. 78.9 45.2 4. Draw a composite shape. Show how to split it to find the area. 104 7. Estimate. 1000 498 MHR • Mathematics 7: Making Connections Teacher’s Resource MathTR-Gr7-Chap03 7/22/04 5:26 PM Page 105 Literacy Connections Have students apply the steps outlined in the Literacy Connections to the question in the Introduction. Strategies • Make a Picture or Diagram: Emphasize that a rough sketch will often work. • Look for a Pattern: Ask students how looking for a pattern might help them to solve Example 2. Common Errors • Students do not know where to begin in solving a problem. Rx Encourage students to reference their solution strategy list. Using a diagram helps make the problem more concrete. This can point toward a solution strategy. • Students may use the incorrect mathematical operation. Rx Encourage students to list words that refer to addition (e.g., sum, altogether, total) and subtraction (e.g., difference, leftover, remaining) to help them identify the necessary operation. Ongoing Assessment Are students able to identify and apply appropriate strategies? Teaching Suggestions Introduction As a class, discuss the section opener. Discuss the photograph of sandwiches that are remaining. Consider how each sandwich was cut, the number of remaining pieces, and how to combine them. Discover the Math In this investigation, students learn about different strategies that they can use for solving problems containing fractions. Method 1: Have students work in pairs to develop a strategy for solving the sandwich question. Allow students to work for 10 minutes, and then have them present their solution strategies. Develop a list of the different strategies used. Discuss the content of the thought balloon. Method 2: Have students complete each example using manipulatives. • The concrete method used in Example 1 will reinforce student understanding of adding fractions in a problem-solving format. • Example 2 needs a patterning strategy because different items are added. Discuss the content of the thought balloon, and when, in real life, students might need to use this technique. Chapter 3 • MHR 105 MathTR-Gr7-Chap03 7/22/04 5:26 PM Page 106 Communicate the Ideas Answers (page 106) 1. Answers may vary. You could draw a diagram. 1 8 1 2. Strategies may vary. 8 or 1 6 6 3 3. Answers may vary. 4. Each whole sandwich is 4 quarters, and there are 1 6 1 6 quarters in total. So, 6 or 1. There 4 4 2 1 are 1 sandwiches. 2 Ongoing Assessment Communicate the Ideas • Can students identify the strategies they use to solve problems? • Are students able to incorporate new strategies as they solve problems? Journal Ask students to choose one journal entry from this chapter that they would like to share. If you wish, give them time to write a good copy. Collect and assess the journal using Assessment Master 11 Journal Assessment Rubric. You may wish to respond personally to student thoughts and ideas as reflected in the journal. Key Ideas Have students read and review the Key Ideas section. Encourage students to detail their solution to the sandwich problem. Discuss why a different strategy should be used to check answers. Communicate the Ideas Have students answer and discuss their solutions to the Communicate the Ideas questions. Discuss why it is useful to know a number of different strategies. Check Your Understanding Question Planning Chart Core questions for Levels 1and 2: 1–4, 5–9, 11, 15, 18 Typical assignment: 1–4, 5, 7, 9, 11, 13, 15, 16, 18 (19, 20 for Level 4 and some Level 3s) Accommodations Language/Visual/Perceptual/ Spatial/Motor Question 11 is a good question to diagram. Once students understand the visual, they will be able to approach the problem. 106 Questions 13 and 14 are good communication questions. Questions 15 and 17 are good application questions. MHR • Mathematics 7: Making Connections Teacher’s Resource MathTR-Gr7-Chap03 7/22/04 5:26 PM Page 107 Ongoing Assessment Try This! • BLM 3.5A More Fraction Problems Checkbric provides formative assessment for student work in this section. • Level 4 students should be encouraged to solve the same problem in two ways, to compare their strategies, and identify the more efficient solution. Gifted and Enrichment Question 19 presents a good challenge. Stronger students will comment that the diagonals also add to 1. After doing this question, students may wish to develop their own magic squares using fractions. Interventions • BLM 3J Section 3.5 Try This! provides scaffolding for question 17. • BLM 3.5B Extra Practice and Mathematics 7 Workbook, pages 35–36, provides extra practice for those who need it. Check Your Understanding Answers (pages 106–107) 17 1 5. a) or 4 4 4 5 6. a) 12 5 1 7. or 1 4 4 13 5 8. or 1 8 8 7 1 9. or 3 2 2 17 5 b) or 2 6 6 3 b) 8 27 3 c) or 3 8 8 2 c) 5 1 15. a) 5 1 1 b) or 3 7 16. a) 5m 15 m b) 75 m2 2 17. a) 3 14 1 12 1 10. sandwiches: or 3; oranges: or 1 4 2 8 2 3 1 8 2 1 11. 1, 1 or 1 2 2 6 6 3 c) 40 m b) 7 c) 15 5 3 1 18. a) top row: ; middle row: or ; 12 12 4 1 bottom row: 12 b) Answers may vary. 19. 6 12. 18 5 13. a) blue: , purple: 16 6 3 green: or 16 8 11 b) c) 16 1 14. 5 3 2 1 , white: or , 16 16 8 13 16 Chapter 3 • MHR 107 MathTR-Gr7-Chap03 7/22/04 5:26 PM Page 108 Suggested Timing 40 min Related Resources BLM 3K Wordsearch Accommodations Visual/Perceptual/Spatial/Motor • Encourage students to use diagrams wherever possible. Pattern blocks or fraction circles will reinforce the concrete meaning of fractions. Language/Memory • Allow students to refer to personal math dictionaries, index card files, or notes. Interventions • Have students complete BLM 3K Wordsearch to review Key Words. • Have students identify problems, and then suggest they review those sections in the chapter. • Mathematics 7 Workbook, pages 37–38, provides additional reinforcement. 1 4 1 4 1 2 1. Show why 1. 7 8 Teaching Suggestions 2. Draw a diagram to represent 1 . 1 6 2 3 3. Find a common denominator for and . Show your method. 1 3 1 3 1 3 1 3 4. Evaluate in two ways. 5. Your lunch has two sandwiches cut into quarters. After lunch, two pieces are left. How many sandwiches is this? Mental Math 6. Estimate. 1024 250 Using the Practice Review Provide an opportunity for the students to discuss any questions, consider alternative strategies, and ask about questions with features they find difficult. After they complete the Chapter Review, encourage students to make a list of questions that caused them difficulty and include the related sections and teaching examples. They can use this to focus their studying for the final test on the chapter’s content. 7. Estimate: 1008 650 8. Estimate: 10 089 7486 9. 108 78 655 4 432 10. 1000 282 MHR • Mathematics 7: Making Connections Teacher’s Resource MathTR-Gr7-Chap03 7/22/04 5:27 PM Page 109 Ongoing Assessment Chapter Review This is an opportunity for students to assess themselves by completing selected questions and checking their answers. They can then revisit any questions with which they had difficulty. 15. Diagrams may vary. Make sure that the fractions are the same as those shown in question 14. 2 1 16. a) 6 6 4 1 b) 6 2 17. Diagrams may vary. Make sure that the fractions are the same as those in question 16. 1 18. Diagrams may vary. a) 3 2 1 b) or 6 3 19. Answers may vary. a) 12 b) 20 7. G 20. Answers may vary. a) 18 b) 24 1 1 1 1 1 b) 6 6 6 6 6 1 1 1 1 d) 6 6 6 6 21. Answers may vary. a) 12 b) 10 Review Answers (pages 108–109) 1. D 5. A 2. F 6. C 1 1 8. a) 3 3 1 1 1 c) 2 6 6 3. H 4. B 9. Diagrams may vary. 5 5 4 2 a) b) c) or 6 6 6 3 1 1 1 23. a) 2 3 6 1 1 1 4 2 b) or 3 6 6 6 3 represent those shown in the question. correct fractions from question 9. 1 1 1 b) 1 2 3 6 2 12. Diagrams may vary. a) 3 1 1 13. a) 2 b) 5 3 6 1 5 1 14. a) 1 b) 2 6 6 22. 12, 18 24. Diagrams may vary. Make sure that the fractions 10. Diagrams may vary. Check that they show the 11. a) 1 c) 6 5 b) 6 7 25. a) 30 7 1 26. or 2 3 3 1 13 b) or 1 12 12 6 4 1 1 27. a) red: or , blue: or , 24 24 4 6 9 5 3 white: or , grey: 24 24 8 5 10 b) or 12 24 15 5 c) or 24 8 5 1 11 5 28. or 2 muffins; or 1 apples 2 2 6 6 Chapter 3 • MHR 109 MathTR-Gr7-Chap03 7/22/04 5:27 PM Page 110 Suggested Timing 40 min Related Resources BLM 3.6A Chapter 3 Test BLM 3.6B Chapter 3 Test Assessment BLM 3.6C Chapter Problem Wrap-Up Rubric BLM 3.6D Chapter 3 Mark Summary Summative Assessment After students complete the Practice Test, you may wish to use BLM 3.6A Chapter 3 Test as a summative assessment. BLM 3.6B Chapter 3 Test Assessment will help you keep track of student achievement. Interventions BLM 3L Chapter Problem Wrap-Up provides scaffolding for the Chapter Problem Wrap-Up on page 111. Accommodations Visual/Perceptual/ Spatial/Motor • Students can use BLM 3.1A Pattern Block Worksheet to complete the Chapter Problem Wrap-Up. Language/Memory • Allow students to refer to personal math dictionaries, index card files, or notes. 110 Teaching Suggestions Using the Practice Test This practice test can be assigned as an in-class or take-home assignment. If it is used as an assessment, use the following guidelines to help you evaluate the students: • Can students add fractions using manipulatives? • Can students use pattern blocks to represent subtracting fractions? • Are students able to find a common denominator? • Are students able to add and subtract fractions using a common denominator? • Do students use different strategies to solve problems containing fractions? MHR • Mathematics 7: Making Connections Teacher’s Resource MathTR-Gr7-Chap03 7/22/04 5:27 PM Page 111 Chapter 3 Practice Test 1. B 2. D 3. B 4. C 5. A 6. a) Diagrams may vary. 2 1 1 b) 3 2 6 1 4 7. a) 4 5 5 2 10 3 b) 5 or 1 7 7 7 8. Answers may vary. a) 4 b) 15 9. 12, 24 1 1 25 10. a) or 1 b) 24 30 24 3 11. a) 1 10 b) Strategies may vary. For example, you could subtract and compare your answers. 3 2 19 14 12. or 1. Eric is correct. 5 3 15 15 13. 16 14. a) Both are correct. Both 12 and 24 are multiples of 2, 3, and 4. 23 11 b) or 1 12 12 Study Guide Use the following study guide to direct students who have difficulty with specific questions to appropriate examples to review. Question 1, 2 3 4, 5 6 7, 8, 9 10, 11, 12 13 14 Section(s) 3.1 3.2 3.1, 3.2 3.2 3.3 3.4 3.5 3.4 Refer to Example Example Example, Example Example Example Examples 1 and 2 Examples 1 and 2 Example Chapter 3 • MHR 111 MathTR-Gr7-Chap03 7/22/04 5:27 PM Page 112 This problem should be accessible to all students. Less confident students will use materials that are familiar, such as pattern blocks, whereas more able students will create their own unique pattern pieces to fit the conditions given. Some may try to incorporate interlocking tabs, as are found with jigsaw puzzles. 1. Introduce the problem. 2. Clarify the assessment criteria by reviewing BLM 3.6C Chapter 3 Problem Wrap-Up Rubric with students. 3. Remind individual students that they have worked on the chapter problem during Chapter Problem revisits throughout the chapter and that these will help them. 4. Possibly share with all students the example from the TR. 5. Allow students time to work on the problem, either individually or in a group. They should do separate puzzles and reports. Summative Assessment • Use BLM 3.6C Chapter 3 Problem Wrap-Up Rubric to assess student achievement. • Use BLM 3.6D Chapter 3 Mark Summary to summarize student work in this chapter. Level 3 Sample Response 1. a) • A rectangle divided into 6 equal sections with 1 section red, 1 section blue, 2 sections green, and the remaining 2 sections yellow. • Using pattern blocks, there is. – 1 yellow hexagon covered by 2 green triangles, – one triangle labelled red – one triangle labelled blue, – 1 blue rhombus beside the triangles is labelled green. • Using centimetre cubes or coloured tiles, there should be 6 pieces: 1 red, 1 blue, 2 green, and 2 yellow. 110a b) • I know that 13 of the puzzle is green because 13 is equal to 26, and 2 out of 6 sections are shaded green. c) • 1 3 of the puzzle is yellow because there were 2 sections left over. 2 sections out of 6 total sections is 26 or 13. 2. • 3 sections of the puzzle are red, 3 sections are blue, 4 sections are green, and the remaining 2 sections are yellow. • Since 12 of the puzzle is either red or blue, this means that 6 of the puzzle pieces are red or blue. • The fraction of green sections is double the fraction of yellow sections. The 2 yellow sections make up 16 of the puzzle. The 4 green pieces make up 13 of the puzzle. 16 is half of 13. • The sum of red and blue sections equals half the puzzle, since 14 14 12. • The sum of green and yellow sections equals half the puzzle, since 13 16 12. • The total number of blue and red pieces is equal to the total number of green and yellow pieces. Level 3 Notes • Students create appropriate puzzle pieces. • Explanations show good understanding of fractions. • May have some minor errors. MHR • Mathematics 7: Making Connections Teacher’s Resource MathTR-Gr7-Chap03 7/22/04 5:27 PM Page 113 What Distinguishes Level 2 At this response level, look for the following: – Puzzles are designed with slight errors or in the incorrect proportions – Few fraction relations provided – Puzzles are inaccurate – Explanations are simplistic or vague What Distinguishes Level 4 At this response level, look for the following: – Puzzle may represent an equivalence of 6. For example, a rectangle divided into 12 equal sections with 2 sections red, 2 sections blue, 4 sections green and the remaining 4 sections yellow. – A puzzle in equivalent ratios – Greater depth in explanation – Creating a puzzle using fractions of their own choosing – A variety of fraction relations Chapter 3 • MHR 111a