Multiplication and Division Number Stories

Transcription

1 Multiplication and Division Number Stories Objective To provide practice solving multiplication and division number stories by using diagrams to organize information. 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 Describe the inverse relationship between multiplication and division. [Operations and Computation Goal 3] Solve multiplication and division number stories. [Operations and Computation Goals 3 and 4] Use repeated addition, skip counting, and arrays to model multiplication. [Operations and Computation Goal 7] Write number models to represent multiplication and division number stories. [Patterns, Functions, and Algebra Goal 2] Write number models containing grouping symbols. [Patterns, Functions, and Algebra Goal 3] Key Activities Students use Multiplication/Division Diagrams to organize information in number stories and write number models for the problems. Ongoing Assessment: Informing Instruction See page 403. Ongoing Assessment: Recognizing Student Achievement Use journal pages 138 and 139. [Patterns, Functions, and Algebra Goal 2] Key Vocabulary Multiplication/Division Diagram Materials Math Journal 1, pp. 138 and 139 Math Masters, p. 437 (optional) transparency of Math Masters, p. 436 (optional) slate counters (optional) Multiplication/Division Facts Table (optional) Advance Preparation Solving Extended Facts Math Journal 1, p. 140 Students solve extended multiplication facts. Math Boxes Math Journal 1, p. 137 Students practice and maintain skills through Math Box problems. Study Link Math Masters, p. 177 Students practice and maintain skills through Study Link activities. READINESS Playing Division Arrays Student Reference Book, p. 240 Math Masters, p. 470 per partnership: 1 each of number cards 6 18 (from the Everything Math Deck, if available), 1 six-sided die counters Students review the relationship between multiplication and division using a concrete model. ENRICHMENT Writing Multiplication/Division Number Stories Students apply their understanding of the inverse relationship between multiplication and division. Teacher s Reference Manual, Grades 4 6 pp. 19, , Unit 6 Division; Map Reference Frames; Measures of Angles

2 Getting Started Mental Math and Reflexes Pose fact pairs to highlight the inverse relationship between multiplication and division. Suggestions: 2 5 = / 5 = = / 4 = = / 7 = = / 6 = = / 8 = = / 6 = = / 4 = = / 9 = = / 8 = 9 Math Message There are 6 rows of chairs. There are 4 chairs in each row. How many chairs are there in all? 1 Teaching the Lesson Math Message Follow-Up (Math Masters, pp. 436 and 437) WHOLE-CLASS Interactive whiteboard-ready epresentations are available at to help you teach the lesson. Algebraic Thinking Have students share their answers and strategies. Such strategies may include using counters; drawing pictures; making arrays; counting by 1s, 4s, or 6s; repeatedly adding 4s or 6s; doubling; or knowing that 4 6 = 24 or 6 4 = 24. (See margin.) Review the Multiplication/Division Diagram as a way of keeping track of the information in the number story. Draw the following diagram on the board (or use a transparency of Math Masters, page 436), and record the given information. Use a variable to represent the unknown. 4 chairs in each row rows chairs per row total chairs 6 4 t 6 rows In this problem, the number of groups and the number of objects in each group are known. The total number of objects is sought. Point out that multiplication can be used to find the total number of objects. Write below the diagram. Ask students to give a number model that represents the number story using a letter for the unknown, and record it on the board. 6 4 = t Now erase the variable in the diagram and replace it with the answer. Write and ask students to give a number model with the answer in place of the unknown. Record the summary number model on the board. 6 4 = 24 Have students use the relationship between multiplication and division to check if their answers make sense. There are 6 rows of chairs with 4 chairs in each row. There are 24 total chairs = 4 Tell students that in this lesson they will use Multiplication/ Division Diagrams to help them solve division number stories. Array sketched using Math Masters, page 437 Lesson 401

3 Solving Division Number Stories (Math Masters, p. 436) WHOLE-CLASS Links to the Future In Unit 12 of Fourth Grade Everyday Mathematics students solve rate problems. Rate problems, like the problems students will encounter in this lesson, often contain the word per, which means in each or for each. rows chairs per row total chairs Algebraic Thinking Because the format of the Multiplication/ Division Diagram is identical for multiplication and division, the diagram also helps reinforce the inverse relationship between the two operations. Pose the following problem: Jenna and Salim are setting up chairs for the class play. There are 32 chairs. Their teacher asked them to arrange the chairs so that there are 4 rows with the same number of chairs in each row. How many chairs should Jenna and Salim put in each row? Explain that this is an equal-sharing problem. In equal-sharing problems, the number of groups (rows) and the total number of objects (chairs) are known. The number of objects in each group (chairs per row) is sought. Draw a Multiplication/Division Diagram on the board, or use a transparency of Math Masters, page 436. (See margin.) Guide students through the following steps: Step 1: Decide what you need to find out. How many chairs will be in each row? Write a variable for the unknown in the chairs per row column. Step 2: Identify the data (information) you need to solve the problem. The total number of chairs to be shared and the number of rows that will share them Write the data in the appropriate columns. Step 3: Decide what to do to find the answer. Divide. 32 chairs must be shared (divided) equally by 4 rows. Write a number model to represent the problem using a variable for the unknown. 32 / 4 = c Step 4: Do the computation. Encourage students to use counters, use a Multiplication/Division Facts Table, or draw an array. Replace the variable in the diagram with the answer to the computation. 8 rows chairs per row total chairs 4 c 8 32, Adjusting the Activity Have students use a Multiplication/Division Facts Table for division. For example, to find the answer to 32 / 4, move across the 4s row to the cell containing 32. Then move up the column containing 32 to find the answer, 8, at the top. Tell students they can think of the arrow (see margin) from the 4 to the 32 as a representation of the question: How many 4s are in 32? A U D I T O R Y K I N E S T H E T I C T A C T I L E V I S U A L 402 Unit 6 Division; Map Reference Frames; Measures of Angles

4 Step 5: Below the diagram, write the answer with the unit and write a number model to summarize the number story. 8 chairs per row 32 / 4 = 8 Ask students to use the relationship between multiplication and division to check if their answers make sense. 32 chairs are arranged in 4 rows. There are 8 chairs in each row. 4 8 = 32 Ongoing Assessment: Informing Instruction Watch for students who recognize the inverse relationship between multiplication and division and solve the problem by thinking 4 times what number equals 32? Help them write a number model that represents their thinking: 4 c = 32. Now pose an equal-grouping problem in which the number of objects per group (chairs per row) and the total number of objects (chairs) are known and the number of groups (rows) is sought. The school custodian is setting up chairs for a parent meeting. There are 43 chairs. The principal asked the custodian to place 7 chairs in each row. How many rows of chairs will there be? Guide students through Steps 1 5. rows chairs per row total chairs r / 7 = r 6 rows (1 chair left over) 43 / 7 6 R1 Have students use the relationship between multiplication and division to check if their answers make sense. 43 chairs are arranged with 7 chairs in each row. There are 6 rows of 7 chairs with 1 chair left over. (6 7) + 1 = 43 NOTE When the result of division is expressed as a quotient and non-zero remainder, the authors prefer to use an arrow rather than an equal sign, as in R1. The arrow is read as is, yields, or results in. This is because 43 7 = 6 R1 is not a proper number sentence. However, this is a subtle usage that need not be required of students. Date LESSON Time Multiplication/Division Number Stories Fill in each Multiplication/Division Diagram. Then write a number model with a variable for the unknown. Be sure to include a unit with your answer. Solve the problem and write a summary number model. 1. The profit from the Maple Street lemonade stand was $36. Four girls will share this amount equally. What will each girl s share be? girls dollars per girl total dollars 4 d $ = d $ = 9 2. Sheila has 57 pictures to put in her photo album. She can put 6 pictures on each page. How many pages will be completely filled up when she is finished? Sample number models are given. pages pictures per page total pictures p pages Student Page 57 6 = p R3 3. Reuben walks a total of 35 blocks going to and from school each week. He always walks the same route. How many blocks does he walk each day? days blocks per day total blocks 5 b = b 7 blocks 35 5 = 7 178A 178B Math Journal 1, p. 138 Lesson 403

5 Date Time LESSON Multiplication/Division Number Stories cont. 4. Hassan is helping his teacher put 8 centimeter cubes into each paper cup for a math project. How many paper cups can he fill if there are 79 cubes? paper cups cubes per paper cup total cubes p cups Try This 79 8 = p R7 5. Mr. Henning s fourth-grade class is planning a field trip to see a play. The bus will cost $100, and the tickets will cost $125. The 25 students will share the total cost equally. How much will each student pay for the field trip? students dollars per student total dollars 25 d 225 Student Page = d $ = 9 6. Last year, Martina sold 73 boxes of cookies for her club. This year, she sold three times as many. If she collected $876 this year, how much did each box cost? Sample number models are given. boxes dollars per box total dollars 219 d 876 Number model(s) with unknown: 73 3 = 219; = d $4 Summary number model(s): 73 3 = 219; = 4 Math Journal 1, p. 139 Solving Number Stories (Math Journal 1, pp. 138 and 139) INDEPENDENT Students solve Problems 1 4 on their own. Encourage them to work with a partner, as necessary, to solve Problems 5 and 6. Ongoing Assessment: Recognizing Student Achievement PROBLEM SOLVING Journal pages 138 and 139 Problems 1 4 Use journal pages 138 and 139, Problems 1 4 to assess students ability to write number models to represent number stories. Students are making adequate progress if the number models accurately reflect the content of the multiplication and division number stories and include the correct use of symbols. Some students may be able to explain the connection between the Multiplication/Division Diagrams and the number models. [Patterns, Functions, and Algebra Goal 2] 2 Ongoing Learning & Practice Solving Extended Facts (Math Journal 1, p. 140) INDEPENDENT Students solve extended multiplication facts. Some problems require students to find the product; others ask for one of the factors. Date LESSON Student Page Time Extended Multiplication Facts = = = = = = = 4, = 5, = 4, = 5, = 45, = 56, = = = = = = 1, = 3, = = 3, = 1, = 36, = 18, = = = = 2, = 4, = 2, = = = 4, = ,000 = 48, = 240,000 Math Journal 1, p Math Boxes (Math Journal 1, p. 137) INDEPENDENT Mixed Practice Math Boxes in this lesson are paired with Math Boxes in Lesson 6-3. The skill in Problem 5 previews Unit 7 content. Writing/Reasoning Have students write a response to the following: Explain how you rounded the number in Problem 2 to the nearest ten million. Sample answer: 409,381,886 is between 400,000,000 and 410,000,000. The number halfway between the lower number and the higher number is 405,000, ,381,886 is closer to the higher number, so I rounded to 410,000,000. Study Link (Math Masters, p. 177) INDEPENDENT Home Connection Students use Multiplication/Division Diagrams to help them solve problems. 404 Unit 6 Division; Map Reference Frames; Measures of Angles

6 3 Differentiation Options Date LESSON Math Boxes Student Page Time 1. Measure each line segment to the nearest millimeter. READINESS Playing Division Arrays (Student Reference Book, p. 240; Math Masters, p. 470) PARTNER 5 15 Min To explore the relationship between multiplication and division using a concrete model, have students play Division Arrays. Consider having students record rounds of play on Math Masters, page 470. a. R About 11 cm b. C 5 mm About 8 cm 7 mm 2. Round 409,381,886 to the nearest a. hundred. 409,381,900 b. ten thousand. 409,380,000 c. ten million. 410,000,000 d. hundred million. 400,000,000 S S 3. Multiply. Use a paper-and-pencil method. 2,494 = ENRICHMENT PARTNER Writing Multiplication/Division Number Stories ELL 5 15 Min 4. Complete. a = 100 b = c. 1,000 = 10 3 d. 10 to the ninth power = 1,000,000, Circle 5_ 6 of the squares Algebraic Thinking To apply students understanding of the inverse relationship between multiplication and division, have them write and solve multiplication and division number stories. Have them record a number model using a letter for the unknown and a summary number model. Some students may be interested in writing and solving multistep problems and problems that involve distances, intervals of time, liquid volumes, masses of objects, or money. Stories may look similar to the following: Math Journal 1, p Yataiva divided 450 pennies equally among the tellers at a bank. One roll has 50 pennies, and each teller got 3 rolls. How many tellers are there? (450 50) 3 = t; 3 tellers; (450 50) 3 = 3 To support English language learners, provide an opportunity for students to share and revise their writing. For example: Read problems aloud or have students read their own problems aloud. Have partners read and comment on each other s drafts. Have students edit problems as a whole-class activity. Name Date Time STUDY LINK Study Link Master Multiplication/Division Number Stories Fill in each diagram. Write a number model with a variable for the unknown. Then solve and write a summary number model. Sample number models are given. 1. Trung wants to rearrange his collection of 72 animals on a shelf in his room. How many equal rows of 9 animals can he make? rows animals per row total animals r = r 8 rows 72 9 = 8 2. An average porcupine has about 30,000 quills. About how many quills would 4 porcupines have? porcupines quills per porcupine total quills 4 30,000 t 30,000 4 = t 120,000 quills 4 30,000 = 120, There are 168 calculators for the students at Madison School. A box holds 8 calculators. How many boxes are needed to hold all of the calculators? 178A 178B boxes calculators per box total calculators b = b 21 boxes = 21 Practice = = Math Masters, p. 177 Lesson 405

7 Name Date Time STUDY LINK Multiplication/Division Number Stories Fill in each diagram. Write a number model with a variable for the unknown. Then solve and write a summary number model. 178A 178B 1. Trung wants to rearrange his collection of 72 animals on a shelf in his room. How many equal rows of 9 animals can he make? rows animals per row total animals 2. An average porcupine has about 30,000 quills. About how many quills would 4 porcupines have? porcupines quills per porcupine total quills Copyright Wright Group/McGraw-Hill 3. There are 168 calculators for the students at Madison School. A box holds 8 calculators. How many boxes are needed to hold all of the calculators? boxes calculators per box total calculators Practice 4. = =

8 Name Date Time Multiplication/Division Diagrams per in all Number model: per in all Number model: per total Number model: Copyright Wright Group/McGraw-Hill 436