Multiplying Fractions by Whole Numbers

Transcription

1 Multiplying Fractions by Whole Numbers Objective To apply and extend previous understandings of multiplication to multiply a fraction by a whole number. 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 Use a number line to represent a fraction. [Number and Numeration Goal ] Understand a fraction a_ b as a multiple of _ b. [Number and Numeration Goal ] Determine between which two whole numbers a fraction lies. [Number and Numeration Goal ] Solve number stories involving multiplication of a fraction by a whole number. [Operations and Computation Goal ] Write equations to model number stories. [Patterns, Functions, and Algebra Goal ] Key Activities Students use a number line as a visual fraction model to represent a fraction a_ b multiplied by a whole number n as the product n ( a (n a) b ) or b. They solve number stories involving multiplication of a fraction by a whole number by using visual fraction models and equations to represent the problems. Ongoing Assessment: Informing Instruction See page D. Key Vocabulary multiple equation Materials Math Journal, pp. A E Study Link half-sheets of paper calculator (optional) Advance Preparation Math Boxes a Math Journal, p. F Students practice and maintain skills through Math Box problems. Ongoing Assessment: Recognizing Student Achievement Use Math Boxes, Problem. [Operations and Computation Goal ] Study Link a Math Masters, p. A Students practice and maintain skills through Study Link activities. READINESS Skip Counting to Show Multiples of Unit Fractions Math Masters, p. B calculator Students use calculators to skip count by unit fractions. ENRICHMENT Visual Models for Multiplying a Fraction by a Whole Number Student Reference Book, p. Math Masters, pp. C and D Students explore alternative visual fraction models for multiplying a fraction by a whole number. EXTRA PRACTICE -Minute Math -Minute Math, pp. and Students practice multiplying fractions by whole numbers. Teacher s Reference Manual, Grades pp., A Unit Fractions and Their Uses; Chance and Probability

2 Getting Started Mental Math and Reflexes Have students name the next three multiples in a sequence. Suggestions:,,,...,,,,,...,,,,,...,,,,,...,,,,,...,,,,,...,,,,,...,, ;,;,;...,;,;,,,,...,, Math Message Name the next three multiples in each sequence.,,,,,,,, _, _, _ Study Link Follow-Up Have small groups compare the results of the penny toss experiment. Ask volunteers to share their answers for Problem. Have students indicate thumbs-up if they agree. Teaching the Lesson Math Message Follow-Up WHOLE-CLASS DISCUSSION Ask students how they determined the next three multiples in each sequence. Possible strategies: Think of the problem as skip counting by _ s. To get the next multiple, add _ to the previous fraction. For example, _ + _ = _ ; _ + _ = _ ; _ + _ = _ ; and so on. Think in terms of equal groups. For example, group of _ is _ ; groups of _ is ; groups of _ is _ ; groups of _ is _ ; and so on. Tell students that in this lesson they will use their understanding of multiples to multiply fractions by whole numbers. Using a Visual Fraction Model to Multiply a Unit Fraction by a Whole Number (Math Journal, pp. A) PARTNER Adjusting the Activity Provide students with calculators to assist with skip counting. See the Part Readiness activity for additional information. AUDITORY KINESTHETIC TACTILE VISUAL NOTE In Third Grade Everyday Mathematics children participated in skip-counting activities to help them memorize the multiplication facts. While completing these activities, they were finding multiples. A multiple of a number is the product of a counting number and the number itself. Date a Time Multiples of Unit Fractions For Problems, fill in the blanks to complete an equation describing the number line... Equation: _ = _ Equation: _ = _, or _ Student Page Draw the number line below on the board or overhead.. Equation: _ = _, or _ For Problems, use the number line to help you multiply the fraction by the whole number.. Equation: _ = _, or _ Have volunteers explain how they could use the number line and their understanding of multiples to help them solve the problem.. Equation: _ =. Equation: _ =, or, or _ Math Journal, p. A -_EMCS_S_MJ_G_U_.indd A // : AM Lesson a B

3 Date a Time An Algorithm for Multiplying a Fraction by a Whole Number Example : Equation: _ = _ Student Page One way is to visualize jumps or hops on the number line, starting at. The fraction tells the size of the jump; the whole number tells the number of jumps. Thus, _ is jumps, each _ unit long. You end up at _. So, _ = _, or _. Example : Equation: _ = _ Write an equation to describe each number line.. a. _ = _ b.. a. b. _ = _ = _ = _. Study the pairs of number lines above. Use the patterns you see to describe a way to multiply a fraction by a whole number. Sample answer: If I take the whole number, multiply it by the numerator of the fraction and then write the product over the denominator, that is my answer. Math Journal, p. B -_EMCS_S_MJ_G_U_.indd B B // : PM Partners complete journal page A. Tell students that an equation is a number sentence with an equals sign, such as _ = _. As you circulate and assist, pose questions such as the following: Which number in the equation tells you the size of the jump? The first fraction Which number in the equation tells you the number of jumps? The whole number Can you name the products in Problems and as mixed numbers? _ = _ ; _ = _ Using a Visual Fraction Model to Multiply Any Fraction by a Whole Number (Math Journal, pp. B and C) WHOLE-CLASS Date a Time Multiplying Fractions by Whole Numbers Use number lines to help you solve the problems. _. _ =. _ = _, or. _ = _. _ = _, or _., or _, or _ = _ Student Page Sample answer: Have partners study the examples at the top of journal page B. On a half-sheet of paper, students should record any similarities and differences they see between the equations modeled on the number lines. Expect students to share observations such as the following: Both equations involve multiplication of a fraction by a whole number. Both equations have the same product. The factors in the equations are different, but _ is a multiple of _ and is a multiple of. It takes more jumps of _ to get to _ than it does jumps of _ because the jumps of _ are smaller than the jumps of _. The whole number factor in _ = _ is twice as much as the whole number factor in _ = _. The fraction factor in _ = _ is half as much as the fraction factor in _ = _.., or _ = _ Sample answer: C -_EMCS_S_MJ_G_U_.indd C Math Journal, p. C // : PM C Unit Fractions and Their Uses; Chance and Probability

4 Have partners complete Problems and on journal page B by writing a multiplication equation to describe each number line. When students have completed Problem, bring the class together to discuss the algorithm for multiplication of a fraction by a whole number. The pattern can be expressed as: n a b = _ (n a). b Have students complete journal page C for additional practice multiplying fractions by whole numbers. Encourage students to use the pattern they discovered on journal page B to check their answers. Solving Number Stories (Math Journal, pp. D and E) Pose the following number story: PARTNER When Carlos goes to the gym, he exercises for _ of an hour and burns about calories. Last week he went to the gym times. How many hours did Carlos spend at the gym last week? Date a Time Solving Number Stories Suma and her sister Puja are making blueberry-wheat muffins for breakfast. The recipe lists the following ingredients: cup flour egg _ cup whole-wheat flour _ cup skim milk teaspoons baking powder _ cup honey _ cup blueberries _ cup cooking oil _ teaspoon salt _ teaspoon cinnamon Use the list of recipe ingredients to help you solve the number stories below. For each problem, write an equation to show what you did.. The sisters decided to double the recipe. Student Page a. How many cups of whole-wheat flour do they need now? _, or cup(s) Equation: _ = _ b. How many cups of blueberries do they need now? _, or _, or _ cup(s) Equation: _ = _ c. How many cups of honey do they need now? _, or _ cup(s) Equation: _ = _. Suma and Puja decide to make muffins instead of. a. How many teaspoons of salt do they need now? _, or teaspoon(s) Equation: _ = _ b. How many teaspoons of cinnamon do they need now?, or _, or _ _ teaspoon(s) Equation: = c. How many cups of skim milk do they need now? _, or cup(s) Equation: _ = _ Math Journal, p. D -_EMCS_S_MJ_G_U_.indd D // : PM Ongoing Assessment: Informing Instruction Watch for students who are distracted by the extra in the number story. Encourage them to eliminate irrelevant information by determining exactly what they want to find out, what information they already know, and what they might need to know in order to solve the problem. On a half-sheet of paper, have students draw a visual fraction model to represent the number story. Expect drawings such as the following: Then have students write a multiplication equation to represent the problem. _ = Ask students to determine between which two whole numbers of hours the product lies. and hours Have them explain their strategy for finding the answer. Possible strategies: Use the number line drawn to represent the number story. Note that the product lies between _, or, and _, or. The fraction _ can be renamed as the mixed number _ by dividing the numerator,, by the denominator, : R. The quotient,, is the whole number part of the mixed number. The remainder,, is the numerator of the fraction part of the mixed number. It tells how many fourths are left over after making as many wholes as possible. NOTE In Lesson -, students used number models to model number stories. A number model is a number sentence or part of a number sentence. A number model can include an equal sign, but it is not required. An equation is a number sentence with an equal sign. See Section. in the Teacher s Reference Manual for more information. Lesson a D

5 Have partners complete journal pages D and E. Encourage students to use visual fraction models, such as number lines, to help them solve the problems. When reviewing answers, pose questions such as the following: Which of the products on journal page D can you rename as whole numbers? Problem a: _ = ; Problem a: _ = ; Problem c: _ = Between which two whole numbers does the product in Problem b lie? and In Problem, how did you decide which whole number you would multiply the recipe ingredients by? Sample answer: The recipe makes muffins. If the sisters want muffins they will need to quadruple the recipe because =. How did you solve Problem? Sample answer: Let the letter a stand for the number of meetings Cole would need to attend and write the equation a _ = _. Use the algorithm for multiplying a fraction by a whole number and think: What number times will give me? =, so _ = _. Cole will need to attend meetings. In Problem, between which two whole-number distances does the distance _ miles lie? Between and miles Allow time for students to share and solve the number stories they wrote for Problem. For each problem, pose questions such as the following: Between which two whole numbers does the answer lie? Can you use a visual fraction model or an equation to represent the problem? Student Page Date Time a Solving Number Stories continued The Hillside Elementary School walking club meets every Monday after school. The table below shows how far some students walked at their last meeting. Student Miles Katie Mahpara Nikhil Cole Maria Jack Use the information in the table to solve the number stories.. a. If Katie walks the same distance at every meeting, how far will she walk after meetings? b. After meetings? _, or _ miles miles c. After meetings, Katie will have walked between. Circle the best answer. and miles and miles and miles. a. If Jack walks the same distance at every, or _, or _ meeting, how far will he walk after meetings? miles b. After meetings, Jack will have walked between. Circle the best answer. and miles and miles and miles. a. If Mahpara walks the same distance at every, or meeting, how far will she walk after meetings? miles b. After meetings, Mahpara will have walked between. Circle the best answer. and miles and miles and miles Try This. If Cole walks the same distance at every meeting and wants to walk a total of _ miles, how many meetings will he need to attend? meetings _. Make up your own multiplication number story about Nikhil or Maria. Answers vary., or _ Math Journal, p. E -_EMCS_S_MJ_G_U_.indd E // : AM E Unit Fractions and Their Uses; Chance and Probability

6 Ongoing Learning & Practice Math Boxes a (Math Journal, p. F) INDEPENDENT Date a Math Boxes. Karen used square feet of her back yard for a garden. Vegetables fill _ of her garden space. Tomato plants fill _ of the space taken up by vegetables. How many square feet are used for tomatoes? square feet Student Page Time. Multiply. Use a paper-and-pencil algorithm., = Mixed Practice Math Boxes in this lesson are paired with Math Boxes in Lessons - and -. The skill in Problem previews Unit content. Ongoing Assessment: Recognizing Student Achievement Math Boxes Problem Use Math Boxes, Problem to assess students ability to solve mixed-number addition problems. Students are making adequate progress if they are able to solve Problem a, which involves mixed numbers with like denominators. Some students may be able to solve Problem b, which involves mixed numbers with unlike denominators, by using equivalent mixed numbers with like denominators, using manipulatives, or drawing pictures. [Operations and Computation Goal ]. a. Lukasz drew a line segment that was _ inches long. Then he extended it another _ inches. How long is the line segment now? _ inches b. Sybil drew a line segment _ inches long. Then she extended it another _ inches. How long is the line segment now? _ inches. Complete the table and write the rule. Rule: -. in out _EMCS_S_MJ_G_U_.indd F Math Journal, p. F. Write an equivalent fraction, decimal, or whole number. Decimal Fraction a... b.. c. d... Complete. a. in.= ft in. b. ft = in. c. in. = ft in. d. ft = yd ft e. _ yd = ft in. // : PM Study Link a (Math Masters, p. A) INDEPENDENT Home Connection Students use number lines to multiply fractions by whole numbers. Study Link Master Name Date Time a Multiplying Fractions by Whole Numbers Use the number lines to help you solve the problems. _. _, or =. _ = _, or _, or _. _ = _, or Write a multiplication equation to represent the problem and then solve.. Rahsaan needs to make batches of granola bars. A batch calls for _ cup of honey. How much honey does he need? Equation: _ = _, or _ cups. Joe swims _ of a mile days a week. How far does he swim every week? Equation: _ = _, or miles How far would he swim if he swam every day of the week? Equation: _ = _, or _, or _ miles Practice. a. List the factor pairs of. and b. Is a prime number? yes. a. List the factor pairs of. and ; and b. Is a prime number? no Math Masters, p. A A A-D_EMCS_B_MM_G_U_.indd A // : PM Lesson a F

7 Name Date Time a Skip Counting by a Unit Fraction. Use your calculator to count by _ s. Complete the table below. s s s s s s s s s. Use your calculator to count by _ s. Complete the table below. s s s s s s s s s. Use your calculator to count by _ s. Complete the table below. s s s s s s s s s _. Use your calculator to count by _ s. Complete the table below. s s s s s s s s s. Use your calculator to count by _ s. Complete the table below. s s s s s s s s s. How is skip counting by _ s on your calculator from to nine _ s the same as finding the product _? Sample answer: When you skip count by _ from nine times, you are finding nine groups of _. This is the same as _. Math Masters, p. B A-D_EMCS_B_MM_G_U_.indd B Teaching Master _ // : AM Differentiation Options READINESS Skip Counting to Show Multiples of Unit Fractions (Math Masters, p. B) SMALL-GROUP Min To explore multiples of unit fractions, have students skip count on the calculator. Remind students that when you skip count by a number, your counts are the multiples of that number. Review the steps for counting by s on the calculator. Students can program their calculator using the following steps: TI-:. Press On/Off and Clear simultaneously. This clears your calculator display and memory.. Press Op + Op. This tells the calculator to count up by s.. Press. This is the starting number. Casio fx-:. Press. This clears your calculator display and memory.. Press. This tells the calculator to count by s.. Press. This tells the calculator to count up.. Press. This is the starting number. Now the calculator is ready to count by s. Without clearing their calculators, have students press the Op key or the key. Press the Op key or the key repeatedly as the students count together by s. Next have students skip count by the unit fraction _. You may first need to remind students of the steps to enter a fraction on their calculators. To enter _ : On a TI-: n d. On a Casio fx-:. Have students skip count by unit fractions to complete the tables on Math Masters, page B. Afterward, discuss how Problem highlights the concept that a fraction such as _ means the same thing as ( _ ). In general, a_ = b a ( _ b ). G Unit Fractions and Their Uses; Chance and Probability

8 ENRICHMENT SMALL-GROUP Visual Models for Multiplying a Fraction by a Whole Number (Student Reference Book, p. ; Math Masters, pp. C and D) Min To extend students understanding of fraction multiplication, have them explore two different models: addition and area. Begin by having students read Student Reference Book, page. Discuss the example provided for each model as a group. Have students complete Math Masters, pages C and D. For page C, encourage the groups to discuss how each number in the problem was represented in the model. The whole number is the number of rectangles drawn. The denominator of the fraction is the number of equal parts each rectangle is divided into. The numerator of the fraction is the number of parts of each rectangle that are shaded. Name Date Time a Addition Model for Multiplying Draw models for each product. Then add the fractions to find the product. Sample shading is given in models.. _ = _ + _ = _. _ = _ + _ + _ = _. _ = _ + _ = _. _ = Teaching Master _ + _ + _ + _ = _ SMALL-GROUP EXTRA PRACTICE -Minute Math Min A-D_EMCS_B_MM_G_U_.indd C Math Masters, p. C // : PM To offer students more experience with multiplying fractions by whole numbers, see -Minute Math, pages and. Teaching Master Name Date Time a Area Model for Multiplying For each problem, divide the model into strips, and then shade a fraction of the area to find the product. _. _ Sample shading is of square units = square unit(s) _ given in models. So, _ =.. _ of square units = _ square unit(s) So, _ = _.. _ of square units = _ square unit(s) So, _ = _. _. _ of square units = square unit(s) _ So, _ =. Math Masters, p. D A-D_EMCS_B_MM_G_U_.indd D // : AM Lesson a H

9 Name Date Time a Multiplying Fractions by Whole Numbers Use the number lines to help you solve the problems.. _ =. _ =. _ = Write a multiplication equation to represent the problem and then solve.. Rahsaan needs to make batches of granola bars. A batch calls for _ cup of honey. Copyright Wright Group/McGraw-Hill How much honey does he need? Equation:. Joe swims _ of a mile days a week. How far does he swim every week? Equation: How far would he swim if he swam every day of the week? Equation: Practice. a. List the factor pairs of. b. Is a prime number?. a. List the factor pairs of. b. Is a prime number? A

10 Name Date Time a Skip Counting by a Unit Fraction. Use your calculator to count by _ s. Complete the table below. s _ s _ s _ s _ s _ s _ s _ s _ s. Use your calculator to count by _ s. Complete the table below. s _ s _ s _ s _ s _ s _ s _ s _ s. Use your calculator to count by _ s. Complete the table below. s _ s _ s _ s _ s _ s _ s _ s _ s. Use your calculator to count by _ s. Complete the table below. s _ s _ s _ s _ s _ s _ s _ s _ s _. Use your calculator to count by _ s. Complete the table below. s _ s _ s _ s _ s _ s _ s _ s _ s. How is skip counting by _ s on your calculator from to nine _ s the same as finding the product _? Copyright Wright Group/McGraw-Hill B

11 Name Date Time a Addition Model for Multiplying Draw models for each product. Then add the fractions to find the product.. _ =. _ =. _ = Copyright Wright Group/McGraw-Hill. _ = C

12 Name Date Time a Area Model for Multiplying For each problem, divide the model into strips, and then shade a fraction of the area to find the product. _. of square units = square unit(s) So, _ =. _. of square units = square unit(s) So, _ =. _. of square units = square unit(s) So, _ =. _. of square units = square unit(s) So, _ =. Copyright Wright Group/McGraw-Hill D