exploring arrays and facts - mcgraw-hill education · • use multiplication facts to estimate the...

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Explorations

Lesson 4�8 283

Advance PreparationTape fact platters (see Planning Ahead, page 282) to the board. You may also draw fact platters on the board.

Teacher’s Reference Manual, Grades 1–3 pp. 92, 186–189, 196–199

Key Concepts and Skills• Generate multiplication fact families.

[Operations and Computation Goal 3]

• Use multiplication facts to estimate the number of dots in a large array. [Operations and Computation Goal 3]

• Use arrays to solve a multidigit multiplication problem. [Operations and Computation Goal 4]

• Explore the inverse relationship between multiplication and division. [Patterns, Functions, and Algebra Goal 2]

Key ActivitiesExploration A: Children estimate the number of dots in an array. Then they find the exact number using calculators.Exploration B: Children find the smallest number of chairs that can be arranged in equal rows to satisfy certain conditions. Exploration C: Children use a fact platter to generate sets of multiplication and division facts within fact families.

Ongoing Assessment: Informing Instruction See page 285.

MaterialsHome Link 4�7Exploration A: Per group:Math Journal 1, p. 9222 square pattern blocks � calculatorExploration B: Per group: Math Journal 1, p. 93Math Masters, p. 106at least 35 countersExploration C: Per partnership:Math Masters, p. 107fact platters

Playing Beat the Calculator (Multiplication)Math Journal 1, p. 156Student Reference Book, p. 279per group: calculatorChildren practice multiplication skills.

Ongoing Assessment: Recognizing Student Achievement Use an Exit Slip (Math Masters, page 398). [Operations and Computation Goal 3]

Math Boxes 4�8Math Journal 1, p. 94

Home Link 4�8Math Masters, p. 108

ENRICHMENTExploring with Arraysper partnership: counters, paper

EXTRA PRACTICE Minute Math +Minute Math +®, p. 89

Teaching the Lesson

132

4

Ongoing Learning & PracticeOngoing Learning & Practice

Differentiation Options

Exploring Arrays and Facts

Objectives To provide opportunities for children to estimate the number of dots in a large array, solve a problem involving factors of whole numbers, and practice multiplication facts.t

��

eToolkitePresentations Interactive Teacher’s

Lesson Guide

Algorithms Practice

EM FactsWorkshop Game™

AssessmentManagement

Family Letters

CurriculumFocal Points

Common Core State Standards

283_EMCS_T_TLG1_G3_U04_L08_576809.indd 283 9/29/11 11:51 AM

284 Unit 4 Multiplication and Division

Exploration A: How Many Dots?LESSON

4�8

Date Time

Materials □ square pattern blocks

□ calculator

1. Estimate how many dots are in the array at the right.

About dots

Make another estimate. Follow these steps.

2. Cover part of the array with a square pattern block. About howmany dots does one block cover?

25–30 dots

3. Cover the array. Use as many square pattern blocks as you can. Do not go over the borders of the array. How many blocks did you use?

20 blocks

4. Use the information in Steps 2 and 3 to estimate the total

number of dots in the array. About 500–600 dots

5. Find the exact number of dots in the array. Use a calculator to help you. Total number of dots = 504

Follow-Up

Describe how you found the exact number of dots. Sample answer: I counted the number of dots in one column and one row. ThenI used my calculator to multiply.

Try This

Answers vary.

79-101_EMCS_S_SMJ_G3_U04_576353.indd 92 1/26/11 9:11 AM

Math Journal 1, p. 92

Student Page

1 Teaching the Lesson

� Math Message Follow-Up WHOLE-CLASSDISCUSSION

There are 18 dots in the array. Expect that few children will count the dots. Most children will probably use the strategy from Lesson 4-2, finding the number of rows 3 rows and the number of dots per row. 6 dots per row Multiply to find the total number of dots. 3 × 6 = 18The explorations in this lesson use arrays and multiplication facts.

� Exploration A: Estimating SMALL-GROUP ACTIVITY

a Number of Dots(Math Journal 1, p. 92)

� First children use any method to estimate the number of dots in the array on journal page 92.

� Then they make a second estimate with the help of square pattern blocks: They count the number of dots that are covered by one square block.

� Next they tile the array with square blocks and use the total number of blocks to estimate the number of dots in the whole array.

Possible solution strategy: Multiply the number of blocks covering the array by the number of dots covered by one block. Then, add the approximate number of dots left uncovered.

The Try This problem asks children to use a calculator to find the exact number of dots in the array and to explain their strategy. Possible solution strategy: Multiply the number of rows by the number of dots per row.

SOLVING

Getting Started

Math Message How many dots are in this array? Be prepared to explain how you got your answer.

Home Link 4�7 Follow-Up Before turning in their work, have children compare their answers with a partner.

PROBLEMPPPPPPPPPPPPPPPPPPPPROPROPROPROPROPROPROPROPROPROPROPROPROPROPROPROPRPP BBBBBBBBBBB EEMLEBLEBLELEBLLELLLBLEBLEBLEBLEBLEBLEBLEBLEBLEMMMMMMMMMMMMMMLLLLBLBLBLBBLLBLBLBLBLBLBLBLBL MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMEEEEEEEEEEEEEEEEEEEEEEEEEEEEEPROBLEMSOLVING

LELELELELELEMMMMMMMMLROROROROROROROROROROROROROBLBLBLBLBLBLBLBLBLBLBLBLBLROBLBLBLBLBLBLBLBLLLBL MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMEEEEEEEEEEEGGGGGLLLLLLLLLLLLVINVVINVINVINVINNNNVINVINVINVINVINVINGGGGGGGGGGGOLOOOLOLOLOLOLOLOOSOSOSOOSOSOSOSOOSOOSOSOSOOSOOOOO VINVVINGGGGGGGGGGLLLLLLLLLVINVINVINVINVINVINVINVINVINVINVIVINVINV NGGGGGGGGGGGGSOSOSOSOLSOSOSOOSOLSOSOSOSOSOSOSO GGGGGGGGGGGLLLLLVVVVVVVLLLLLLLLLLVVVVVVVVVSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSS GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGNNNNNNNNNNNNNNNNNNNNNNNNNNNIIIIIIIIIIIIIIIIIIIIVVVVVVVVVVVVVVVVVVVVVVLLLLLLVVVVVVVVVLLVVVVVVVVLLLLLLLLVVVVVLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLOOOOOOOOOOOOOOOOOOOSOLVING

Mental Math and Reflexes Pose multiplication facts. Suggestions:

1 × 5 5 2 × 5 10 10 × 5 50 2 × 4 8 3 × 4 12 5 × 4 20 3 × 6 18 4 × 6 24 6 × 6 36

Mathematical PracticesSMP1, SMP2, SMP3, SMP4, SMP5, SMP6, SMP7Content Standards3.OA.1, 3.OA.7, 3.OA.9, 3.MD.6, 3.MD.7a

284-287_EMCS_T_TLG1_G3_U04_L08_576809.indd 284 3/9/12 9:14 AM

Exploration B: Setting Up ChairsLESSON

4� 8

Date Time

1. Record the answer to the problem about setting up chairs from Math Masters,page 106.

There were chairs in the room.

2. Circle dots below to show how you set up the chairs for each of the clues.

25

Rows of 2 Rows of 3 Rows of 4 Rows of 5

1 left over 1 left over 1 left over 0 left over


Student Page

Lesson 4�8 285

Name Date Time

Materials � Math Journal 1, page 93

� 35 pennies or other counters

Problem A teacher was setting up the chairs in his classroom forparent night. He wanted them to be in rows with thesame number of chairs in each row. Use the clues to findout how many chairs were in the room.

Clues

1. When the teacher tried to set up the chairs in rows of 2, there was 1 leftover chair.

2. He also had 1 leftover chair when he tried to set them up in rows of 3.

3. One of his students suggested that he try rows of 4. There was still 1 leftover chair.

4. Finally, he tried rows of 5. This worked fine. There were no leftoverchairs.

5. The number of chairs in the room was the smallest possible numberthat fits all of the clues.

What to Do

1. Experiment with counters to build rows. Hint: Could the answer be 5 chairs? 10 chairs (2 rows of 5 chairs each)? 15 chairs (3 rows of 5 chairs each)?

2. When you have found the answer, record it on journal page 93.Next, circle dots to show the chairs in rows of 2, 3, and 4, each with1 chair left over. Then show the chairs in rows of 5.

Exploration B: Setting Up ChairsLESSON

4�8

Math Masters, p. 106

Teaching Master

Adjusting the Activity

Ongoing Assessment: Informing InstructionWatch for children who do not have a strategy for solving the Try This problem. Remind them of the strategy they used to find the total number of dots in the Math Message.

� Exploration B: Setting Up Chairs SMALL-GROUP ACTIVITY

(Math Journal 1, p. 93; Math Masters, p. 106)

Children solve the following problem using counters.

A teacher is setting up chairs in equal rows for parent night. If he tries to arrange them in equal rows of 2, 3, or 4, there is always 1 leftover chair. If he tries to arrange them in equal rows of 5, he can make equal rows. What is the smallest number of chairs that can be arranged in equal rows of 5 but not in equal rows of 2, 3, or 4? 25 chairs After children have solved the problem, they record their work on journal page 93.

� Exploration C: Practicing PARTNER ACTIVITY

Multiplication and Division Facts with a Fact Platter(Math Masters, p. 107)

As partners work together at the board,

� One partner writes any factor from 1 through 10 in the center of the fact platter.

� The other partner multiplies the factor in the center by each factor on the platter and writes the products on the board around the rim of the platter.

� Partners help and check each other.

� They also write at least one division fact for each multiplication fact on the board.

� Then they erase the board, trade jobs, and keep going until each person has had several turns picking a factor for the center of the platter.

Have children use numbers 1–5 as the center factor. Have children use factors larger than 5.

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

PROBLEMBBBBBBBBBBOOOOOOOOOOOBBBBBBBBBBBBBBBBBBBBBBBBBBB MMMMMEEEEMMMBLELBLELBLLLLBLEBLEBLEBLEBLEBLEBLEEEEMMMMMMMMMMMMMMOOOOOOOOOOOBBBBBBLBLBBLBLLBLLLLPROPROPROPROPROPROPROPROPROPROPPRPRPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPROROOROROROOROOPPPPPPP MMMMMMMMMMMMMMMMMMMMMMMEEEEEEEEEEEEEEEEEELELELEEEEEEEEEELLLLLLLLLLLLLLLLLLLLLLLLRRRRRRRRRRRRRRRRPROBLEMSOLVING

BBBBBBBBBBBBBBBBBBBB EEEEMMMMMMMOOOOOOOOOBBBBBBBBOOROROROOROROROROROROROO LELELELEEEEEELEEMMMMMMMMMMMMLEMLLLLLLLLLLLLLLLLLLLLLLRRRRRRRRRRRGGGLLLLLLLLLLLVINVINVINVINVINNNNVINVINVINNVINVINVINVINV GGGGGGGGGGGOLOOOLOOLOLOLOO VINVINVINVLLLLLLLLLVINVINVINVINVINVINVINVINVINVINVINVINVINVINNGGGGGGGGGGGOLOLOLOLOLOLOLOLOOO VVVVLLLLLLLLLLVVVVVVVVVVOOSOSOOSOSOSOSOSOSOSOOSOSOSOSOSOOOOOSOSOSOSOSSOOSOSOSOSOSOSOSOSOSOSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSS VVVVVVVVVVVVVVVVVVVVVVLLLLLLVVVVVVVVLLVVVVVVVVLLLLLLLLVVVVVLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLSSSSSSSSSSSSSSSSSSSSSS GGGGGGGGGGGGGGGGOOOOOOOOOOOOOOOOOOO GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGNNNNNNNNNNNNNNNNNNNNNNNNNNNIIIIIIIIIIIIIIIIIIISOLVING

Adjusting the Activity Have children find another number of chairs that fits all of the clues, except for being the smallest number. 85 chairsAUDITORY � KINESTHETIC � TACTILE � VISUAL

EM3cuG3TLG1_284-287_U04L08.indd 285 12/22/10 1:50 PM

286 Unit 4 Multiplication and Division

Name Date Time

LESSON

4�8

Materials � fact platter

� chalk

� eraser

Work with a partner at the board.

1. Take turns. On your turn, pick a number from1 to 10. Write it in the middle with amultiplication sign.

2. Your partner takes that number and multipliesit by each number on the circle, writing theproducts along the outside of the platter.

3. Check the products together. Make corrections.

4. Then, each of you writes one division fact forevery multiplication fact around the circle.

42 ÷ 6 = 75. Check each other’s work.

6. Erase the board. Trade roles. Start again.

7. Keep going until each of you has had several turns picking anumber for the center.

2

8 19

5

6�

74

3

10

6

42

30

Exploration C: Fact Platters


Teaching Master

5. 56,937Which digit is in the tens place? Which digit is in the thousands place? Which digit is in the hundreds place? Which digit is in the ones place? 7

9

6

3

3. Fill in the circle for the best answer.The perimeter of the square is

A. 12 cm

B. 16 cm

C. 8 cm

D. 20 cm

Date Time

94

2. Write three things that you thinkare very likely to happen.

will start on time tomorrow.home after school. Schoolhave lunch today. I will goSample answer: I will

4. Complete the Fact Triangle. Writethe fact family.

� �

� �

� �

� � 3721732121372173

6. Use �, �, or �.

3,065 3,605

23,605 20,365

32,605 23,605

50,007 50,700�

�

�

�

1. Make equal groups.

30 days make weeks

with days left over.

56 pennies make quarters

with pennies left over.62

24

Math BoxesLESSON

4� 8

150 151 55

18 19 13 20

4 cm

4 cm

•

3

21

�,�

7

73


Student Page

A fact platter

2 Ongoing Learning & Practice

� Playing Beat the Calculator SMALL-GROUP ACTIVITY

(Student Reference Book, p. 279; Math Journal 1, p. 156)

Children develop automaticity with multiplication facts by playing Beat the Calculator. Have children record a check mark on the Fact Power Table (journal page 156) for each fact that they solve correctly when playing the role of the Brain. For Fact Power Table directions, see Lesson 4-5. For game directions, see page 279 in the Student Reference Book.

Ongoing Assessment: Exit Slip

�Recognizing Student AchievementUse an Exit Slip (Math Masters, page 398) to assess children’s progress toward demonstrating automaticity with multiplication facts through 10 × 10. Children record the facts from the Fact Power Table for which they have earned at least one check mark. Children are making adequate progress if they record at least 5 facts from the unshaded portion of the Fact Power Table. Some children may record more than 5 facts. [Operations and Computation Goal 3]

� Math Boxes 4�8 INDEPENDENTACTIVITY

(Math Journal 1, p. 94)

Mixed Practice Math Boxes in this lesson are paired with Math Boxes in Lesson 4-10. The skills in Problems 5 and 6 preview Unit 5 content.

Writing/Reasoning Have children write an answer to the following: For Problem 1, show how you found one of your answers. Explain your thinking. Sample answer: I know

there are 7 days in 1 week. I made an array with 7 dots in each row. I made 4 rows. That was 28 days. 2 days were left over.


Lesson 4�8 287

HOME LINK

0�0

Name Date Time

FamilyNote

HOME LINK

4�8

Make a dot inside each small square in one row. Then fill in the blanks.

1. Number of rows:

Number of squares in a row:

Number model: � �

Area: square units

2. Number of rows:

Number of squares in a row:


Area: square units

Mark the dots to show each array. Then fill in the blanks.

3. Make a 4-by-8 array.


4. Make a 9-by-5 array.

Number model: � � 4559

3284

424276

76

353557

57

Your child uses the same procedure for finding the area of a rectangle that is used for findingthe number of dots in an array. For Problem 3 it does not matter whether your child drawsan array with 4 rows of 8 dots or 8 rows of 4 dots. What is important is that the array has two sides that have 4 dots and two sides that have 8 dots. The same concept is true for Problem 4.

Please return this Home Link to school tomorrow.64 65

Arrays and Areas


Home Link Master

� Home Link 4�8 INDEPENDENTACTIVITY

(Math Masters, p. 108)

Home Connection Children write number models and find the areas of rectangles, using grids and unit squares.

3 Differentiation Options

ENRICHMENT PARTNER ACTIVITY

� Exploring with Arrays 15–30 Min

To further explore arrays, have children use arrays to prove that the sum of two odd numbers is always an even number. Pose the following problem to the children:

Use arrays to show that the sum of two odd numbers is always an even number. Explain. Show your work on paper. Use counters to help.

If children have difficulty getting started, you might suggest that they make arrays with 2 rows.

EXTRA PRACTICE SMALL-GROUP ACTIVITY

� Minute Math+ 5–15 Min

To offer children more experience with multiplication and division, see the following page in Minute Math+:

Number Stories: page 89.

Planning Ahead

For Part 1 in Lesson 4-9, collect a variety of maps.

For the Readiness activity in Lesson 4-9, have children bring in toy animals.


exploring arrays and facts - mcgraw-hill education · • use multiplication facts to estimate the...

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