Grade 7 Mathematics. Unit 5. Operations with Fractions. Estimated Time: 24 Hours

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1 Grade 7 Mathematics Operations with Fractions Estimated Time: 24 Hours [C] Communication [CN] Connections [ME] Mental Mathematics and Estimation [PS] Problem Solving [R] Reasoning [T] Technology [V] Visualization Grade 7 Mathematics Curriculum Outcomes 59

2 Grade 7 Mathematics Curriculum Outcomes 60

3 : Operations with Fractions Overview Introduction Students will focus on developing skills and understanding the addition and subtraction of fractions. The big ideas in this unit are: Equivalent fractions represent the same quantities. The concept of equivalent fractions is very useful when comparing, ordering, simplifying, and operating with fractions. The use of manipulatives such as fraction strips and fraction circles, number lines, and pattern blocks is an effective way to model the addition and subtraction of fractions. It creates a concrete base for a traditionally difficult concept. Addition and subtraction of fractions requires common denominators. Estimation strategies for these two operations are based on using benchmarks like 0, 3, 4 Context, 2 etc. 4 The students will model, using manipulatives, the addition and subtraction of fractions. They will be encouraged to informally generalize rules for these operations that are based on their investigations. Through the use of these investigations, and guidance from the teacher, the students will discover the need to use common denominators when adding, subtracting, comparing and ordering fractions. They will discover the algorithm for adding and subtracting fractions. Once again estimation will play an important role in helping students to decide if their answers are sensible. The students will then apply these algorithms to adding and subtracting mixed numbers. Why are these concepts important? Developing a good understanding of adding and subtracting fractions will permit students to: Understand real-life situations that require fractions such as; The clock ("a quarter 'till"). Electricians (gauge/length of wires). Plumbing (thickness of pipe, diameter of pipe, length of pipe). Carpenters (thickness/length/width of wood). Engineers (just math equations). Metal fabrication (length/width/gauge of metal). Taxes/budgeting (obvious math involved). Cooking (measurements like HALF a cup...). In your car (km PER hour, km PER liter). Paying for things in general ( penny is /00 of a dollar, writing out checks.) Be ready to learn and understand future topics in math such as algebra and proportions. It isn't that they can't see the solution. It is that they can t see the problem. G. K. Chesterton ( ) Grade 7 Math Curriculum Guide 6

4 Specific Outcome Elaborations: Suggested Learning and Teaching Strategies It is expected that students will: 7N5. Demonstrate an understanding of adding and subtracting positive fractions and mixed numbers, with like and unlike denominators, concretely, pictorially and symbolically (limited to positive sums and differences). [C, CN, ME, PS, R, V] Achievement Indicators 7N5. Model addition of positive fractions, using concrete representations, and record symbolically. 7N5.2 Determine the sum of two given positive fractions with like denominators. 7N5.3 Determine a common denominator for a given set of positive fractions. 7N5.4 Simplify a given positive fraction by identifying the common factor between the numerator and denominator. Lesson 5. in the student text briefly models like fractions using pattern blocks, clocks and fraction circles. It primarily demonstrates like denominators, but includes some examples in which one of the denominators is a simple multiple of the other. Teachers will need to model several more examples using these manipulatives in order to ensure student understanding. Students should also have the opportunity to model using the manipulatives since they are hands-on experiences. Lesson 5.2 uses fraction strips and number lines to support the same indicators. Students should be able to use the models to understand fractional equivalents and how they can be useful when adding fractions and changing them to their simplest form. Using the fractions strips and number line masters in the ProGuide pp , students will combine both the fraction strips and number lines to model sums and to illustrate the concept of common denominators. Grade 7 Mathematics Curriculum Outcomes 62

5 Suggested Assessment Strategies Pencil and Paper Write an addition sentence to represent the total fraction of each hexagon that is shaded. Use an addition sentence to find the total value of the shaded hexagons in each case. A. B. Resources/Notes The national library of virtual manipulatives provides an interesting activity on adding using common denominators with various models at v/frames_asid_06_g_3_ t_.html?from=category_ g_3_t_.html C. Informal Observation An alternative, but similar activity would be to create cards with addition sentences and their equivalents in pattern blocks as used in the Pencil and Paper exercise above. Each student would receive a card with either the addition sentence, or the pattern block representation. They mix-up and match-up within the class to find their partner. Each group must then explain to another group, or to their class, why they belong together. Math Makes Sense 7 Lesson 5. Lesson 5.2 : Operations with Fractions TR: ProGuide, pp. 4 6 & pp. 7 Master 5.3, 5.8, 5.27 Master 5.0, 5., 5.4, 5.5, 5.6, 5.7, 5.9, 5.28 PM 28, PM 25 CD-ROM Masters ST: pp ST: pp Practice and HW Book pp pp. 09 Grade 7 Mathematics Curriculum Outcomes 63

6 Specific Outcome It is expected that students will: 7N5. Demonstrate an understanding of adding and subtracting positive fractions and mixed numbers, with like and unlike denominators, concretely, pictorially and symbolically (limited to positive sums and differences). [C, CN, ME, PS, R, V] (Cont d) Achievement Indicators 7N5.5 Model addition of positive fractions with unlike denominators, using concrete representations, and record symbolically. Elaborations: Suggested Learning and Teaching Strategies In the previous lessons, students used models to add using like denominators. They also modelled unlike denominators when one denominator was a multiple of the other. Lesson 5.3 develops the addition algorithm for fractions. The addition of fractions with unlike denominators that are not simple multiples of each other will require students to multiply the numerator and denominator of each fraction by the same number. Example: 7N5.6 Determine the sum of two given positive fractions with unlike denominators. Ideally, students should use the Least Common Multiple (LCM) of the unlike denominators. Through the use of benchmarks (close to 0,,) developed in 2 Unit 3, students will estimate the solution and use their estimate to verify the reasonableness of the answer obtained using the algorithm. (This elaboration is continued on the next two page spread ) Grade 7 Mathematics Curriculum Outcomes 64

7 Suggested Assessment Strategies Resources/Notes Pencil and Paper. Create three addition sentences that give the same sum as You cannot use like denominators in the sentences 2 2 you create. 2. Magic square. The sum of each row, column and diagonal in this magic square must equal. Find the missing values. Magic Square Solution A tangram is a square puzzle that is divided into seven shapes. A. Suppose piece A is. What are the values of pieces B, C, 4 D, E, F and G? B. What is the sum of A and B? C. If you subtract D from the whole puzzle, what value remains? D. Which two tangram pieces add up to the value of C? E. Invent a problem on your own and solve it. Math Makes Sense 7 Lesson 5.3 : Operations with Fractions TR: ProGuide, pp. 2 5 Master 5.4, 5.5, 5.6, 5.7, 5.20, 5.29 PM 27 CD-ROM Masters ST: pp Practice and HW Book pp. 2 4 Grade 7 Mathematics Curriculum Outcomes 65

8 Specific Outcome It is expected that students will: 7N5. Demonstrate an understanding of adding and subtracting positive fractions and mixed numbers, with like and unlike denominators, concretely, pictorially and symbolically (limited to positive sums and differences). [C, CN, ME, PS, R, V] (Cont d) Achievement Indicators 7N5.5 Model addition of positive fractions with unlike denominators, using concrete representations, and record symbolically. 7N5.6 Determine the sum of two given positive fractions with unlike denominators. Elaborations: Suggested Learning and Teaching Strategies Here is another example of adding fractions with unlike denominators. Find the sum of the fractions: Students should think 4 3 is a little bit more than a half and 6 is less than a half so the answer should be close to. Then they can use the previous algorithm to calculate: = = = 2 Finally, they should look at their answer and ask themselves if is reasonable based on their estimate of. 2 Note: When a common denominator must be found, the common denominator that is chosen should be the lowest common denominator. Simply multiplying the denominators of the fractions being adding or subtracted will not guarantee a lowest common denominator. The lowest common 3 denominator for + is 2, not Grade 7 Mathematics Curriculum Outcomes 66

9 Suggested Assessment Strategies Resources/Notes Performance Use pattern blocks to create a design on triangular grid paper (Program Master 27). Then use fraction addition to name the design. Consider the flower design illustrated in Appendix 5-A. It is possible to use several different addition sentences to name the same design. Journal. If a problem required you to add fourths and thirds, is it possible for the sum to be sixths? Why or why not? You may use an example or a diagram to help you explain your answer. 2. If a problem required you to add fourths and thirds, is it possible for the sum to be sevenths? Why or why not? You may use an example or a diagram to help you explain your answer. Interview A classmate missed yesterday s class. When solving a problem today he suggested that = 0. How would you convince him that this is not a reasonable solution? Math Makes Sense 7 Lesson 5.3 Game/Activity Refer to Appendix 5-B for the Connect Three game. Grade 7 Mathematics Curriculum Outcomes 67

10 Specific Outcome It is expected that students will: 7N5. Demonstrate an understanding of adding and subtracting positive fractions and mixed numbers, with like and unlike denominators, concretely, pictorially and symbolically (limited to positive sums and differences). [C, CN, ME, PS, R, V] (Cont d) Achievement Indicators 7N5.7 Model subtraction of positive fractions, using concrete representations, and record symbolically. Elaborations: Suggested Learning and Teaching Strategies Lesson 5.4 of the student text begins with subtraction involving unlike denominators using pattern blocks. Students will learn that addition and subtraction of fractions with unlike denominators uses the same algorithm. Teachers may wish to model several examples using fraction circles or fraction strips. For example: In this case, students must understand that they are simply removing one part of a set of equivalent quantities. This can be demonstrated by modelling 4 5 using fraction strips or fraction circles and removing one portion representing. The answer 5 is the remaining portion of N5.8 Determine the difference of two given positive fractions with like denominators. 7N5.9 Determine the difference of two given positive fractions with unlike denominators. The subtraction of fractions with unlike denominators that are not simple multiples of each other will require students to multiply the numerator and denominator of each fraction by the same number. This is identical to the algorithm used for addition. Ideally, students should use the Least Common Multiple (LCM) of the unlike denominators. Through the use of benchmarks (close to 0,,) developed in 2 Unit 3, students will estimate the solution and use their estimate to verify the reasonableness of the answer obtained using the algorithm. (This elaboration is continued on the next two page spread ) Grade 7 Mathematics Curriculum Outcomes 68

11 Suggested Assessment Strategies Resources/Notes Observation Ask students to use concrete materials or diagrams to show why the following is an incorrect procedure = = = Informal Observation Students can play the game Tic-Tac-Toe Fractions. A really useful game for adding and subtracting fractions. See ProGuide (Page V) and Master 5.8a, 5.8b and 5.8c. Math Makes Sense 7 Lesson 5.4 Lesson 5.5 : Operations with Fractions TR: ProGuide, pp & pp Master 5.2, 5.4, 5.5, 5.6, 5.7, 5.2, 5.30 Master 5.4, 5.5, 5.6, 5.7, 5.22, 5.3 CD-ROM Masters ST: pp ST: pp Practice and HW Book pp. 5 7 pp Grade 7 Mathematics Curriculum Outcomes 69

12 Specific Outcome It is expected that students will: 7N5. Demonstrate an understanding of adding and subtracting positive fractions and mixed numbers, with like and unlike denominators, concretely, pictorially and symbolically (limited to positive sums and differences). [C, CN, ME, PS, R, V] (Cont d) Achievement Indicators 7N5.7 Model subtraction of positive fractions, using concrete representations, and record symbolically. 7N5.8 Determine the difference of two given positive fractions with like denominators. Elaborations: Suggested Learning and Teaching Strategies Find the difference of the fractions: Students should think 9 4 is a little bit less than a half and 3 is a little less than a half. The difference between them should therefore be almost 0 or just a little bit more than = = = 9 = 9 Finally, they should look at their answer and ask themselves if is reasonable based on their estimate of something a little 9 bit more than 0. 7N5.9 Determine the difference of two given positive fractions with unlike denominators. Grade 7 Mathematics Curriculum Outcomes 70

13 Suggested Assessment Strategies Resources/Notes Math Makes Sense 7 Lesson 5.4 Lesson 5.5 Grade 7 Mathematics Curriculum Outcomes 7

14 Specific Outcome It is expected that students will: 7N5. Demonstrate an understanding of adding and subtracting positive fractions and mixed numbers, with like and unlike denominators, concretely, pictorially and symbolically (limited to positive sums and differences). [C, CN, ME, PS, R, V] (Cont d) Achievement Indicators 7N5.0 Model addition and subtraction of mixed numbers with like denominators, using concrete representations, and record symbolically. 7N5. Determine the sum or difference of two mixed numbers with like denominators. 7N5.2 Model addition and subtraction of mixed numbers with unlike denominators, using concrete representations, and record symbolically. 7N5.3 Determine the sum and difference of two mixed numbers with unlike denominators. Elaborations: Suggested Learning and Teaching Strategies Now that the models for addition and subtraction have been studied separately by the students, the same models and skills can now be used in the study of mixed fractions. Lessons 5.6 and 5.7 explore the subtraction of mixed numbers using fraction circles, number lines and fraction strips. Lesson 5.7 also introduces Cuisenaire rods as a model for subtracting mixed fractions. Teachers may consult the link for use of this model in the resource section of this guide. When adding and subtracting mixed fractions students may approach the problem in different ways. They may choose to keep the mixed fraction form or, they may change the mixed fractions to improper fractions. For addition: Mixed Fraction Form = = = 2 8 = 2 and 8 = 3 8 Improper Fraction Form = = = = 8,36, = 3 8 (This elaboration is continued on the next two page spread ) Grade 7 Mathematics Curriculum Outcomes 72

15 Suggested Assessment Strategies Interview Consider the following two problems: 3 and Without calculating, explain how you could determine which answer would be greater. Journal Describe at least two ways you can calculate Resources/Notes An introduction to Cuisenaire rods and their use in the study of fractions can be found at u/participants/silha/lesso ns/cuisen2.html Math Makes Sense 7 Lesson 5.6 Lesson 5.7 : Operations with Fractions TR: ProGuide, pp & pp Master 5.3, 5.4, 5.5, 5.6, 5.7, 5.23, 5.32 Master 5.3, 5.4, 5.5, 5.6, 5.7, 5.24, 5.33 PM 28 CD-ROM Masters ST: pp ST: pp Practice and HW Book pp pp Grade 7 Mathematics Curriculum Outcomes 73

16 Specific Outcome Elaborations: Suggested Learning and Teaching Strategies It is expected that students will: 7N5. Demonstrate an understanding of adding and subtracting positive fractions and mixed numbers, with like and unlike denominators, concretely, pictorially and symbolically (limited to positive sums and differences). [C, CN, ME, PS, R, V] (Cont d) Achievement Indicators 7N5.0 Model addition and subtraction of mixed numbers with like denominators, using concrete representations, and record symbolically. 7N5. Determine the sum or difference of two mixed numbers with like denominators. 7N5.2 Model addition and subtraction of mixed numbers with unlike denominators, using concrete representations, and record symbolically. 7N5.3 Determine the sum and difference of two mixed numbers with unlike denominators. For subtraction: Mixed Fraction Form = = Students will be challenged by 2 4 and therefore must think about regrouping. Students should think: and and which will allow them to calculate: = 2 Improper Fraction Form = = = = 2 (All Cont d) Grade 7 Mathematics Curriculum Outcomes 74

17 Suggested Assessment Strategies Resources/Notes An introduction to Cuisenaire rods and their use in the study of fractions can be found at u/participants/silha/lesso ns/cuisen2.html Math Makes Sense 7 Lesson 5.6 Lesson 5.7 Grade 7 Mathematics Curriculum Outcomes 75

18 Specific Outcome It is expected that students will: 7N5. Demonstrate an understanding of adding and subtracting positive fractions and mixed numbers, with like and unlike denominators, concretely, pictorially and symbolically (limited to positive sums and differences). [C, CN, ME, PS, R, V] (Cont d) Achievement Indicators 7N5.4 Simplify the solution to a given problem involving the sum or difference of two positive fractions or mixed numbers. 7N5.5 Solve a given problem involving the addition or subtraction of positive fractions or mixed numbers, and determine if the solution is reasonable. Elaborations: Suggested Learning and Teaching Strategies Throughout the sections on adding and subtracting fractions, it is necessary for students to simplify their answers. Simplified answers may be proper fractions, improper fractions or mixed numbers in simplest form depending on the context of the problem. Example: Kyra is making cookies. She has 2 bags of 4 chocolate chips. She adds 2 3 of these bags to her cookie dough. a) What fraction of the total amount of chocolate chips is left? b) Kyra then decides to add bags of butterscotch chips to 2 the dough as well. How many bags of chips does Kyra use in total to bake her cookies? For part a), students should think 2 bags is a little more 4 than two bags. Kyra then uses 2 3 bags which is a little less than two bags. Therefore she has two little bits or about half a bag left over. Then they calculate: Students must reflect 9 5 upon their answer to = determine if it is 4 3 reasonable = In this case, seven twelfths is very close to a = 2 2 half. 7 = 2 Kyra has 7 of a bag of 2 chocolate chips left. (This elaboration is continued on the next two page spread ) Grade 7 Mathematics Curriculum Outcomes 76

19 Suggested Assessment Strategies Resources/Notes Journal Is it possible to find two mixed numbers which add together to form a whole number? Explain your answer and, if possible, give an example to support your explanation. Pencil and Paper. Andrew plays guitar in a rock band. For a song that is 36 measures long he plays for 4 measures, rests for measures, plays for another 6 measures, rests for 2 4 measures and plays for the last section. How many measures are in the last section? 2. This week, Mark practised piano for 6 4 h, and talked on the phone for 3 2 h, played soccer for 4 3 h. A. How many hours did Mark spend practising piano and playing soccer? B. Hour many more hours did Mark spend playing soccer than talking on the phone? Math Makes Sense 7 Lesson 5.6 Lesson 5.7 Grade 7 Mathematics Curriculum Outcomes 77

20 Specific Outcome It is expected that students will: 7N5. Demonstrate an understanding of adding and subtracting positive fractions and mixed numbers, with like and unlike denominators, concretely, pictorially and symbolically (limited to positive sums and differences). [C, CN, ME, PS, R, V] (Cont d) Achievement Indicators 7N5.4 Simplify the solution to a given problem involving the sum or difference of two positive fractions or mixed numbers. 7N5.5 Solve a given problem involving the addition or subtraction of positive fractions or mixed numbers, and determine if the solution is reasonable. Elaborations: Suggested Learning and Teaching Strategies For part b) students should think 2 4 bags is a little more than two bags and is almost one full bag, but not quite. 2 Therefore Kyra uses a little more than 3 bags of chips in total. Then they calculate: = = = = = 2 6 = 2 and 6 = 3 6 Kyra used in total. 3 6 bags of chips Note that the final answer must be simplified. Students must reflect upon their answer to determine if it is reasonable. In this case, the answer is very close to the estimate. Grade 7 Mathematics Curriculum Outcomes 78

21 Suggested Assessment Strategies Resources/Notes Math Makes Sense 7 Lesson 5.6 Lesson 5.7 Grade 7 Mathematics Curriculum Outcomes 79

22 Grade 7 Mathematics Curriculum Outcomes 80

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