equal shares and equal groups - em … shares and equal groups objective to review division as equal...
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254 Unit 4 Multiplication and Division
Advance PreparationSee the Unit Organizer for literature links.Prepare an area for an Arrays Exhibit. Write on the board:
pennies in all children pennies per child pennies remaining
For the optional Readiness activity in Part 3, obtain a copy of The Doorbell Rang by Pat Hutchins (HarperCollins Publishers, 1989).
Key Concepts and Skills• Use basic facts to solve division problems.
[Operations and Computation Goal 3]
• Use equal sharing and equal grouping to model division. [Operations and Computation Goal 6]
Key ActivitiesChildren practice division by solving equal-sharing and equal-grouping number stories. They form arrays with counters and draw pictures of arrays.
Ongoing Assessment: Recognizing Student Achievement Use journal page 84. [Measurement and Reference Frames Goal 4]
MaterialsMath Journal 1, p. 84Home Link 4�2per partnership: 30 pennies or counters
Playing Division ArraysStudent Reference Book, p. 282Math Masters, p. 418 (optional)per group: 1 each of number cards 6–18 (from the Everything Math Deck, if available), 18 counters, 1 six-sided dieChildren practice making arrays.
Math Boxes 4�3Math Journal 1, p. 85Children practice and maintain skillsthrough Math Box problems.
Home Link 4�3Math Masters, p. 90Children practice and maintain skillsthrough Home Link activities.
READINESS
Making Equal Groups of CookiesMath Masters, p. 91The Doorbell RangChildren explore solving equal-group problems.
ENRICHMENTModeling Division with Base-10 Blocksper group: base-10 blocks, 4 pieces of paper, half-sheet of paperChildren model equal sharing with large numbers using base-10 blocks.
Teaching the Lesson Ongoing Learning & Practice
132
4
Differentiation Options
Equal Shares andEqual Groups
Objective To review division as equal sharing and equal grouping.e
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eToolkitePresentations Interactive Teacher’s
Lesson Guide
Algorithms Practice
EM FactsWorkshop Game™
AssessmentManagement
Family Letters
CurriculumFocal Points
Common Core State Standards
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Lesson 4�3 255
NOTE The Math Message problem is an example of an equal-sharing situation. The number of groups and the total number of objects are known. The number of objects in each group is to be found.
Leah
Matthew
14 Î
14 pennies 2 children 7 pennies per child 0 pennies remaining
Leah and Matthew share 14 pennies equally.
1 Teaching the Lesson
� Math Message Follow-Up WHOLE-CLASSDISCUSSION
Counters are a useful tool for solving equal-sharing and equal-grouping division problems. Have partners act out the Math Message problem with 14 pennies. Ask volunteers to share their solution strategies. Some examples are listed below.
Strategy 1
Count out the pennies, 1 or 2 at a time, until each child has the same number of pennies or 1 penny is left over. For example, “One for you, one for me. Two for you, two for me....”
Strategy 2
Think in terms of multiplication: 2 × ? = 14.
Strategy 3
Use the basic division fact 14 ÷ 2 = 7. If 2 children share 14 pennies, then each child will get 7 pennies.
Draw a picture on the board to illustrate the solution. Record the solution as shown in the margin.
If children begin to use the terms even and odd to describe the number of pennies, encourage them to do so.
Getting Started
Math Message Leah and Matthew share 14 pennies equally. How many pennies does each child get?
Home Link 4�2 Follow-Up Have children share array illustrations they have found. Begin an Arrays Exhibit. Encourage children to keep looking for pictures of arrays.
Mental Math and ReflexesChildren count from 1 to 30, clapping at the intervals indicated below.
Every 5th number 1, 2, 3, 4, 5 (clap); 6, 7, 8, 9, 10 (clap); 11, 12, 13, 14, 15 (clap); and so on to 30
Every 3rd number 1, 2, 3 (clap); 4, 5, 6 (clap); 7, 8, 9 (clap); and so on to 30 Every 4th number 1, 2, 3, 4 (clap); 5, 6, 7, 8 (clap); 9, 10, 11, 12 (clap); and so on to 30
Mathematical PracticesSMP1, SMP2, SMP3, SMP4, SMP5, SMP6, SMP7Content Standards3.OA.1, 3.OA.2, 3.OA.3, 3.OA.4, 3.OA.6, 3.OA.7
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256 Unit 4 Multiplication and Division
NOTE The problem below is an example of an equal-grouping situation: The number of objects per group and the total number of objects are known. The number of equal groups is to be found. It is not important for children to be able to distinguish between equal-sharing and equal-grouping number stories; it is important, however, that they have experience with both types.
30 pennies 6 children 5 pennies per child 0 pennies remaining
30 pennies arranged in 6 equal groups
Leah
Matthew
15 Î
15 pennies 2 children 7 pennies per child 1 penny remaining
Leah and Matthew share 15 pennies equally.
� Solving Equal-Sharing WHOLE-CLASS ACTIVITY
Number Stories with CountersRepeat the Math Message activity with 15 pennies. Again, have children share strategies as you draw a picture and record the solution as shown in the margin. Discuss what happens: When pennies are shared equally, some pennies might be left over.
Pose other equal-sharing stories. Encourage children to use counters or draw pictures. Record the pictures and solutions on the board.
● 12 pennies, 3 children, 4 pennies per child, 0 remaining
● 15 pennies, 4 children, 3 pennies per child, 3 remaining
● 20 pennies, 6 children, 3 pennies per child, 2 remaining
● 18 band members are lined up in 3 equal rows. How many members are in each row? 6 members How many are left over? 0 members
Science Link Astronomers estimate that there are about 70 sextillion—7 with 22 zeros after it—stars in the
universe. To help children understand how big a number this is, explain that if all the people in the United States shared the stars equally, each person would have more than 230 trillion stars.
� Solving Equal-Grouping PARTNER ACTIVITY
Number StoriesPose the following problem to the class. Encourage children to use counters to solve it. Each child has 5 pennies. There are 30 pennies total. How many children have pennies? 6 children Have volunteers share their solution strategies. Some examples are listed below:
Strategy 1
Count out the pennies, 5 at a time to each child, until all of the pennies are distributed or it is not possible to make another group of 5. There are 6 groups of 5 pennies. There are no pennies left over.
Strategy 2
Think in terms of multiplication: 5 × ? = 30.
Strategy 3
Use the basic division fact 30 ÷ 5 = 6. If there are 30 pennies and each child has 5, then there must be 6 children.
Draw a picture on the board or use pennies on the overhead to illustrate the solution. Record the solution as shown in the margin.
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Adjusting the Activity
Division PracticeLESSON
4�3
Date Time
Use counters to find the answers. Fill in the blanks.16¢ shared equally 1. by 2 people:
8 ¢ per person
0 ¢ remaining
25¢ shared equally 4. How many people
get 5¢?
5 people
0 ¢ remaining
30 stamps shared equally 7. by 10 people:
3 stamps per person
0 stamps remaining
2. by 3 people:
5 ¢ per person
1 ¢ remaining
5. How many people get 3¢?
8 people
1 ¢ remaining
8. by 5 people:
6 stamps per person
0 stamps remaining
3. by 4 people:
4 ¢ per person
0 ¢ remaining
6. How many peopleget 6¢?
4 people
1 ¢ remaining
9. 21 days 7 days per week
3 weeks
0 days remaining
You may use the picture to help you solve.
10. Geri has 15 feet of ribbon. She needs 3 feet of ribbon for one bow.
How many bows can she make? 5 bows
15 feet
Try This
You may draw a picture or use counters to help you solve. 11. The animal hospital makes leashes for dogs. They can
make 3 leashes from 18 feet of rope. How many feet of rope does it take to make 1 leash? 6 feet
84
�
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Math Journal 1, p. 84
Student Page
Lesson 4�3 257
20 legs, 4 legs per horse, 5 horses
20 pennies 6 children 3 pennies per child 2 pennies remaining
20 pennies arranged into 6 equal groups with 2 pennies remaining
JournalPage 84 �
12 feet, 4 equal pieces, 3 feet each piece
As necessary, pose other equal-grouping stories. Encourage children to use counters or draw pictures. Record the pictures and solutions on the board.
● Each child must have 3 pennies. There are 20 pennies. How many children can have 3 pennies? 6 children with 2 pennies remaining
● 24 pennies, 8 pennies per child, 3 children, 0 pennies remaining
● 18 pennies, 4 pennies per child, 4 children, 2 pennies remaining
● 15 pennies, 6 pennies per child, 2 children, 3 pennies remaining
● 16 pennies, 4 pennies per child, 4 children, 0 pennies remaining
● Tyrone has 12 feet of rope, which he will cut into 4 equal pieces. How long will each piece of rope be? 3 feet long
● Anna was trying to determine how many horses were in a large stall. She could see only the horses’ legs. Anna counted 20 legs. How many horses were there? 5 horses
In solving the final two problems, children may find that drawing a picture is more helpful than making an array with counters.
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
� Solving Division Number PARTNER ACTIVITY
Stories(Math Journal 1, p. 84)
Children use counters or pictures, as necessary, to solve equal-sharing and equal-grouping division problems. (Problems 1–3, 7, 8, and 11 are equal-sharing situations. Problems 4–6, 9, and 10 are equal-grouping situations.)
Ongoing Assessment: Journal Page 84 �
Recognizing Student AchievementUse the recorded time at the top of journal page 84 to assess children’s progress toward telling time to the nearest minute and writing time in digital notation. Children are making adequate progress if they are able to write the time to the nearest 5 minutes. Some children may be able to write time to the nearest minute; some may be able to record the seconds. [Measurement and Reference Frames Goal 4]
ELL
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Adjusting the Activity
Links to the Future
258 Unit 4 Multiplication and Division
5. Complete the numbermodel for the 4 by 4 array.
How many rows?
How many dots in each row?
� � 16444 dots
4 rows
3. Use �, �, or �.
9 � 9 13 � 5
13 � 4 11 � 5
11 � 4 13 � 8�
�
�
Date Time
2. Corinne wants new tires for herbicycle. They cost $41.10 each,with tax included. Estimate abouthow much money she will need.
about $Number model:
40 � 40 � 80
80.00
6. Write the number that is 100 more.
76
300
471
8,634
5,925 6,0258,734571400176
1. On the centimeter grid below,draw a shape with an area of 10square centimeters.
Math BoxesLESSON
4� 3
154–156
1350 51
191
64 65 18 19
Sample answer:
Unit
Sample answer:
150 151
4. Find the perimeter. Fill in the circlefor the best answer.
A 14 cm B 6 cm
C 7 cm D 12 cm
2 cm
2 cm
2 cm
2 cm
3 cm
3 cm
Math Journal 1, p. 85
Student Page
Games
Number card Die Array formed Leftovers? Score
10 2 no 10
9 2 yes 4
14 3 yes 4
Division ArraysMaterials � number cards 6–18 (1 of each)
� 1 six-sided die� 18 counters
Players 2 to 4Skill Division (with remainder) for equal groupingObject of the game To have the highest total score.Directions
1. Shuffle the cards. Place the deck number-side downon the table.
2. Players take turns. When it is your turn, draw a cardand take the number of counters shown on the card.You will use the counters to make an array.◆ Now roll the die. The number on the die is the
number of equal rows you must have in your array.◆ Make an array with the counters.◆ Your score is the number of counters in one row.
If there are no leftover counters, your score isdouble the number of counters in one row.
3. Players keep track of their scores. The player withthe highest total score at the end of 5 rounds wins.
Student Reference Book, p. 282
Student Page
The equal-sharing activities in this lesson are an early exposure to division concepts. These concepts will be revisited throughout third grade. Using and explaining strategies to divide a multidigit whole number by a 1-digit whole number is a Grade 4 Goal.
2 Ongoing Learning & Practice
� Playing Division Arrays SMALL-GROUP ACTIVITY
(Student Reference Book, p. 282; Math Masters, p. 418)
Children practice making arrays by playing Division Arrays. Have children read page 282 in the Student Reference Book for game directions.
Have children make arrays on square grid paper (Math Masters, page 418).
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
� Math Boxes 4�3 INDEPENDENTACTIVITY
(Math Journal 1, p. 85)
Mixed Practice Math Boxes in this lesson are paired with Math Boxes in Lesson 4-1. The skill in Problem 6 previews Unit 5 content.
� Home Link 4�3 INDEPENDENTACTIVITY
(Math Masters, p. 90)
Home Connection Children practice equal-sharing and equal-grouping division with number stories.
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Name Date Time
Division with CountersHOME LINK
4�3
Your child is beginning to use division to solve number stories. A first step is to use counters,such as uncooked macaroni or pennies, to represent each problem. This helps your child to understand the meaning of division.
Please return this Home Link to school tomorrow.
FamilyNote
73 74
Show someone at home how to do division using pennies, uncookedmacaroni, or other counters.
1. 25 counters are shared equally 2. 25 counters are shared equallyby 5 people. by 10 people.
counters per person counters per person
counters remaining counters remaining
3. 31 days in January 4. 22 children7 days per week 5 children per team
weeks in January teams
days remaining children remaining
5. Mrs. Blair has 34 pencils to give to her 15 students. How manypencils can she give each student?
pencils pencils left over
6. Caleb shared 22 jelly beans with his sister. How many jelly beansdid each child get?
jelly beans jelly beans left over011
42
2344
5025
UnitWrite these problems on the back of this page. Solve. Show your work.
7. � 614 � 37 8. 23 � 8 �
9. 123 � 678 � 80131577
Practice
Math Masters, p. 90
Home Link Master
LESSON
4�3
Name Date Time
Equal Groups of Cookies
1. Draw circles to show how many cookies are on each plate at the beginning of the story.
2. How many cookies did Ma make altogether?
3. Tom and Hannah arrive. Now 4 children share the cookies. Draw a picture to show how they share the cookies.
4. Draw a picture to show how 6 children share.
5. At the end, there are 12 children. If each child gets 3 cookies, how many cookies do they have altogether? Draw a picture to show how you found your answer.
There are 36 cookies in all.
12
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Math Masters, p. 91
Teaching Master
Lesson 4�3 259
3 Differentiation Options
READINESS INDEPENDENTACTIVITY
� Making Equal Groups 5–15 Min
of Cookies(Math Masters, p. 91)
To explore solving equal-group problems, have children read the book The Doorbell Rang. After they read the book together, have them complete Math Masters, page 91. Have children describe each new situation using words like share, split, divide, equal, same, and discuss what they have learned about making equal groups.
ENRICHMENT SMALL-GROUPACTIVITY
� Modeling Division with 15–30 Min
Base-10 BlocksTo further explore the concept of equal groups, have children model problems with base-10 blocks. Pose the following problem: You need to divide 653 paper clips into 4 boxes. How many paper clips will be in each box? Children use base-10 blocks to represent the paper clips. They use four pieces of paper to represent the boxes and distribute the blocks equally on the pieces of paper. Children will need to make trades: longs for flats and cubes for longs. They may use base-10 block shorthand to record solutions on a half-sheet of paper. There will be 163 paper clips in each box with 1 paper clip left over.
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