Fraction and Mixed-Number Addition and Subtraction

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1 Fraction and Mixed-Number Addition and Subtraction Objective To guide students in the use of pattern blocks to add and subtract fractions and mixed numbers. epresentations etoolkit Algorithms Practice EM Facts Workshop Game Family Letters Assessment Management Common Core State Standards Curriculum Focal Points Interactive Teacher s Lesson Guide Teaching the Lesson Ongoing Learning & Practice Differentiation Options Key Concepts and Skills Identify the whole or the ONE. [Number and Numeration Goal ] Represent fractions and mixed numbers with pattern blocks. [Number and Numeration Goal ] Identify equivalent fractions. [Number and Numeration Goal ] Model fraction and mixed-number addition and subtraction with pattern blocks. [Operations and Computation Goal ] Identify a triangle, hexagon, trapezoid, and rhombus. [Geometry Goal ] Key Activities Students model fraction and mixed-number sums and differences with pattern blocks. Playing Angle Tangle Student Reference Book, p. 0 Math Masters, p. 7 protractor straightedge Students practice measuring and estimating the measures of angles. Math Boxes Math Journal, p. 99 Students practice and maintain skills through Math Box problems. Study Link Math Masters, p. Students practice and maintain skills through Study Link activities. READINESS Exploring Fractions that Sum to One Math Masters, p. 7 Full House: An Invitation to Fractions straightedge Students divide cakes into thirds, fourths, sixths, and eighths and write number models. EXTRA PRACTICE Solving Frames-and-Arrows Problems Math Masters, p. 9 Students practice adding and subtracting fractions. Ongoing Assessment: Recognizing Student Achievement Use journal page 98. [Operations and Computation Goal ] Materials Math Journal, pp B Study Link 7 pattern blocks calculator slate overhead pattern blocks (optional) colored chalk (optional) Advance Preparation For the optional Readiness activity in Part, obtain a copy of Full House: An Invitation to Fractions by Dayle Ann Dodds (Candlewick Press, 009). Teacher s Reference Manual, Grades pp. 9,, 9, 0 9 Unit 7 Fractions and Their Uses; Chance and Probability

2 Getting Started Mathematical Practices SMP, SMP, SMP, SMP, SMP7 Content Standards.NF.a,.NF.b,.NF.c,.NF.d,.MD. Mental Math and Reflexes Write fractions and mixed numbers on the board for students to decompose on their slates in at least two different ways. Suggestions: Sample answers are given. = _ + _ ; _ = _ + _ = _ 8 + _ 8 ; _ 7 8 = _ 8 + _ 8 + _ 8 = _ + _ + _ ; _ = _ + _ 8_ 8_ = _ + _ ; _ 8 = _ + = _ + _ + _ ; _ = _ + _ 9 7_ 7_ 0 0 = _ 0 + _ 0 ; _ 7 0 = _ 0 + _ 0 + _ 0 = + _ + _ ; _ = _ + _ + _ + _ = _ + _ ; _ = + _ + 9 _ 9 = _ _ _ 9 ; _ 9 = + _ _ 9 + _ 9 Math Message If the hexagon pattern block is the whole, what fractions are represented by the trapezoid, the rhombus, and the triangle? Study Link 7 Follow-Up Ask students to share their solution strategies for Problem. Sample answer: _ of quarters = 0 quarters, or $.0; _ of 0 nickels = nickels, or 0 cents. Therefore, Rosa spent $.0 + $0.0 = $.80. Each can costs cents, so she was able to buy 8 cans with her money. Teaching the Lesson Math Message Follow-Up WHOLE-CLASS Review answers. The trapezoid represents, the rhombus represents, and the triangle represents. Ask students to use a trapezoid, a rhombus, and a triangle to form a hexagon. Ask what number model describes what they just did. + + = Tell students that in this lesson they will use pattern blocks to model fraction and mixed-number addition and subtraction problems. For all of the activities, the hexagon pattern block represents the whole, or the ONE. Adjusting the Activity ELL Use colored chalk that matches the pattern block colors to write the number model on the board: + + =. AUDITORY KINESTHETIC TACTILE VISUAL Modeling Fraction and Mixed-Number Sums WHOLE-CLASS Use pattern blocks on the overhead to work through fraction and mixed-number addition problems with the class. Suggestions: _ + = _, or + _ = _, or _ + = + = _ + = + = 7_, or + = _ + = _, or + = _ + _ = _, or + = _, or _ + _ = SOLVING Links to the Future These activities provide exposure to adding and subtracting fractions using manipulatives. Paper-and-pencil addition and subtraction of fractions is a Grade Goal. Lesson 9

3 Modeling Fraction and Mixed-Number Addition with Like Denominators Have students use pattern blocks to find the sum of fractions and mixed numbers with like denominators. Example : + =? Have students take rhombuses and combine them to form _. Example : + _ =? Have students take hexagon and triangle to represent, and triangles to represent _. Then have students combine them to form _. Modeling Fraction and Mixed-Number Addition with Unlike Denominators You can either let students devise their own methods or demonstrate the following step-by-step procedure. This process involves an extra step of trading in blocks for one kind of block. Example : _ + =? Step : Model the fractions to be added. Step : Combine the blocks to show the sum. Step : Trade for one kind of block. Step : Name the fraction for the sum. 9 Unit 7 Fractions and Their Uses; Chance and Probability

4 Example : _ + =? Step : Model the numbers to be added. Step : Combine the blocks to show the sum Step : Trade for one kind of block Step : Name the mixed number for the sum = Modeling Fraction and Mixed-Number Differences PROBLEM SOLVING WHOLE-CLASS Modeling Fraction and Mixed-Number Subtraction with Like Denominators Now have students use pattern blocks to find differences of fractions and mixed numbers with like denominators. Example : _ - =? Have students take rhombuses and then remove or cover rhombus to show the difference is. Example : _ - =? Have students take hexagons triangles and then remove or cover hexagon and triangle to show the difference is _. Modeling Fraction and Mixed-Number Subtraction with Unlike Denominators Ask students how they could use pattern blocks to solve _ _. After a few minutes, ask students to share their approaches. Subtraction can be harder to model than addition, so students methods may be awkward. Below are two approaches. Cover-Up Method Model both fractions with pattern blocks. Then put the blocks representing the smaller fraction on top of the blocks representing the larger fraction. The part of the larger fraction that remains uncovered is the difference. Lesson 9A

5 Example : _ - _ =? Step : Model the fractions with pattern blocks. Step : Cover up the larger fraction with the smaller fraction. Step : The uncovered part of the larger fraction is the difference. The uncovered part is a triangle representing. So, _ - _ =. Example : _ - =? Step : Model the mixed numbers with pattern blocks. Step : Cover up the larger mixed number with the smaller mixed number. Step : The uncovered part of the larger mixed number is the difference. The uncovered part is hexagon and triangles representing _. So, _ - = _, or. Take-Away Method Model the larger fraction with pattern blocks. Then take away blocks representing the smaller fraction, trading for blocks of the proper size if necessary. The remaining blocks represent the difference. Example : _ - _ =? Step : Model the larger fraction with pattern blocks. Trade triangles for rhombuses. Take away rhombuses. Step : Remove blocks representing the smaller fraction. The smaller number, _, can be represented by rhombuses. Since there are no rhombuses, trade triangles for rhombuses. Then remove rhombuses. (See margin.) 9B Unit 7 Fractions and Their Uses; Chance and Probability

6 Step : The block(s) that are left represent the difference. _ - _ = Date Student Page Time Fraction and Mixed-Number Sums & Differences. Use pattern blocks to find fractions that add up to whole. Draw lines to show the blocks you used. Write a number model to show that the sum of your fractions is. Sample answers: Whole hexagon Example : - =? Step : Model the larger mixed number with pattern blocks = = + + = + = + + = _ + =. Use pattern blocks to find fractions that add up to _. Draw lines to show the blocks you used. Write a number model to show that the sum of your fractions is _. Step : Remove blocks representing the smaller number. The smaller number,, can be represented by a triangle. Since there are no triangles, trade trapezoid for triangles. Then remove triangle. + _ = _ + = _ + = _ Solve. You may use pattern blocks or any other method. _, or _, or. _ - _ =. _ - =. _ - =. _ - = Math Journal, p. 98 Trade trapezoid for triangles. Step : The block(s) that are left represent the difference. =, or Pose fraction and mixed number subtraction problems for students to solve with pattern blocks. Suggestions: _ - _ = _, or _ - = _ - = - = _ - = _, or _ - _ = _ - = - _ = - _ = _ - _ = _, or Date Time Fraction and Mixed-Number Sums & Differences cont. 7. Use pattern blocks to find three different pairs of mixed numbers that add up to _. Use your Geometry Template to illustrate the mixed numbers. Write a number model to show that the sum of each pair of mixed numbers is _. a. Student Page Sample answers: + = _ Whole hexagon b. + _ = _ c. + = _ Solve. You may use pattern blocks or any other method., or _ 8. = 9. _ = 0. =. _ = Math Journal, p. 98A Lesson 9

7 Date Time Solving Fraction Number Stories. Rithik ate _ of a cheese pizza. He then ate of a veggie pizza. _ a. What fraction of a pizza did he eat in all? _ + = _ of a pizza, or _ of a pizza Sample number models are given. b. Did he eat more or less than a whole pizza? less How do you know? Sample answer: _ would be whole pizza, and _ < _.. Karina walked of a mile to school. After school, she walked _ of a mile to the store, and then of a mile back to her home. a. How far did she walk after school? b. How far did she walk in all? _ mile _ + = mile, or mile + _ + = _. Stephano is making pancakes and waffles for his guests. a. He needs _ cup of milk for the pancakes and _ cup of milk for the waffles. How much milk does he need in all? _ + _ = _ b. Stephano has _ cups of milk. Will he have any left over? If so, how much milk will be left? _ - = Try This Student Page _, or cups Yes. cup. Kumba has one dollar. He spent of the dollar on a pencil and _ 0 of the dollar on an eraser. 7 a. What fraction of the dollar did he spend? = 0 7 b. What fraction of the dollar does he have left? = 0 Math Journal, p. 98B PARTNER Solving Fraction and Mixed- Number Addition and Subtraction Problems (Math Journal, pp. 98 and 98A) Students use pattern blocks to solve fraction and mixed-number addition problems. They use pattern blocks or any other method to solve fraction and mixed-number subtraction problems. Ongoing Assessment: Recognizing Student Achievement Journal page 98 Problem Use journal page 98, Problem to assess students ability to use pattern blocks to solve fraction addition problems. Students are making adequate progress if they are able to show at least one combination of pattern blocks that adds up to and write an appropriate number model. Some students might be able to show multiple solutions to the problem. [Operations and Computation Goal ] Using a Calculator to Add and Subtract Fractions WHOLE-CLASS Have students practice entering fraction addition and subtraction problems on their calculators. Example : _ + = _ On a TI-: n d + n d On a Casio fx-: + Date Math Boxes. Circle of all the triangles. Mark Xs on _ of all the triangles.. Plot and label each point on the coordinate grid. A (,0) B (,) C (,) D (,) E (,) Sample answer:. A bag contains red blocks, blue blocks, 7 green blocks, and orange blocks. You put your hand in the bag and, without looking, pull out a block. About what fraction of the time would you expect to get a blue block? 0, or 0 0 C D Student Page B E A 9 Time Math Journal, p. 99. Insert parentheses to make these number sentences true. ( ( ( ( a = 0. b = 0. c. 9. = d =. Draw and label a angle. Sample answer: ) ) ) ) T R Y acute This angle is an (acute or obtuse) angle. 9. If centimeter on a map represents 0 kilometers, then a. cm represent 0 km. b. 9. cm represent 9 km. c. cm represent 0 km. d.. cm represent km. e. 0. cm represent km. 0 Example : - = On a TI-: n d - On a Casio fx-: - Solving Fraction Number Stories (Math Journal, p. 98B) n d PARTNER Have students solve number stories involving fractions and mixed numbers. They can use pattern blocks or any other method to solve. Ongoing Learning & Practice Playing Angle Tangle (Student Reference Book, p. 0; Math Masters, p. 7) PARTNER Students play Angle Tangle to practice measuring and estimating the measures of angles. See Lesson - for additional information. 9 Unit 7 Fractions and Their Uses; Chance and Probability

8 Math Boxes (Math Journal, p. 99) INDEPENDENT Mixed Practice Math Boxes in this lesson are paired with Math Boxes in Lesson 7-7. The skill in Problem previews Unit 8 content. Study Link (Math Masters, p. ) INDEPENDENT Home Connection Students solve fraction addition and subtraction problems. Name Date Time STUDY LINK Fractions Study Link Master less Sample answer: = 0.8. Jake has _ of a dollar. Maxwell has 0 of a dollar. Do they have more or less than $.00 in all?. Jillian draws a line segment inches long. Then she makes the _ line segment _ inches longer. How long is the line segment now? in.. A pizza was cut into slices. Benjamin ate of the pizza and Dana ate. What fraction of the pizza was left?. Rafael drew a line segment 7_ 8 inches long. Then he erased _ 8 inch. How long is the line _ segment now? 8 inches. Two hexagons together are one whole. Draw line segments to divide each whole into trapezoids, rhombuses, and triangles. Write a number model to show how the parts add up to the whole. Sample answers: in. 7 8 in.? 7 inches 8 in = + _ + _ = _ + _ = Differentiation Options READINESS Exploring Fractions that Sum to One (Math Masters, p. 7) SMALL-GROUP Min Practice 8 9. of = 7. = 9_ 0 of _ 8 of = 9. = of Math Masters, p. Literature Link To explore the addition of fractions with like denominators, read Full House: An Invitation to Fractions by Dayle Ann Dodds (Candlewick Press, 009). Record the number model to show how the cake was divided among the people. Have students divide the cakes and write number models to show how one cake can be evenly divided among different numbers of people. EXTRA PRACTICE Solving Frames-and-Arrows Problems (Math Masters, p. 9) INDEPENDENT Min Name Date Time Dividing Cakes Use a straightedge to divide the cakes below among different numbers of people. Make sure each person gets an equal share. Write a number model to show what you did.. Six people Teaching Master =. Three people + + = To provide practice adding and subtracting fractions, have students solve Frames-and-Arrows problems. Use Math Masters, page 9 to create problems to meet the needs of individual students or have students create and solve their own problems. Planning Ahead In preparation for Lesson 7-, direct students to remove Activity Sheets and from the back of the journal and cut the Fraction Cards apart. Have students write their initials on the cards for identification.. Four people = =. Eight people Math Masters, p. 7 Lesson 97

9 Name Date Time STUDY LINK Fractions 7. Jake has _ of a dollar. Maxwell has _ 0 of a dollar. Do they have more or less than $.00 in all?. Jillian draws a line segment _ inches long. Then she makes the line segment _ inches longer. How long is the line segment now? inches in. in.. A pizza was cut into slices. Benjamin ate of the pizza and Dana ate _. What fraction of the pizza was left?. Rafael drew a line segment _ 7 8 inches long. Then he erased _ 8 inch. How long is the line segment now? inches? 7 8 in. 8 in.. Two hexagons together are one whole. Draw line segments to divide each whole into trapezoids, rhombuses, and triangles. Write a number model to show how the parts add up to the whole. Practice Copyright Wright Group/McGraw-Hill _. of = 7. = _ 9 0 of _ 8 of = 9. = _ of

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