Division
from Multiplication
Objective To guide children as they review multiplication/
division fact families, and investigate the relationship between
d
multiplication and division.
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Practice
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Workshop
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Teaching the Lesson
Key Concepts and Skills
• Use manipulatives to solve equal-shares
number stories. Family
Letters
Assessment
Management
Common
Core State
Standards
Ongoing Learning & Practice
1 2
4 3
[Operations and Computation Goal 4]
• Investigate the connections between
equal groups and equal shares. [Operations and Computation Goal 4]
• Use the commutative property (turn-around
rule) to write number models. [Patterns, Functions, and Algebra Goal 3]
Key Activities
Children review the turn-around rule for
multiplication and learn that there is no
turn-around rule for division; children use
Fact Triangles to generate multiplication and
division facts; children write and solve related
division facts for given multiplication facts.
Playing Addition Card Draw
Curriculum
Focal Points
Differentiation Options
READINESS
Math Journal 2, p. 300
Math Masters, p. 446
per partnership: 4 each of number
cards 1–10 and 1 each of number
cards 11–20 (from the Everything
Math Deck, if available), slate or
scratch paper
Children practice adding three numbers.
Building Arrays with Equal Shares
Math Boxes 12 5
Math Masters, p. 394
Children complete “What’s My Rule?” tables
involving multiplication and division facts.
Math Journal 2, p. 301
Children practice and maintain skills
through Math Box problems.
Math Masters, p. 393
counters
Children explore division using a concrete
model to build arrays.
EXTRA PRACTICE
Completing “What’s My Rule?” Tables
ELL SUPPORT
Home Link 12 5
Building a Math Word Bank
Math Masters, pp. 391 and 392
Children practice and maintain skills
through Home Link activities.
Differentiation Handbook, p. 143
Children add the terms equal groups,
equal shares, and division to their Math
Word Banks.
Ongoing Assessment:
Recognizing Student Achievement
Use the Math Message. [Operations and Computation Goal 4]
Materials
Math Journal 2, pp. 280 and 299
Home Link 124
Math Masters, p. 415
transparency of Math Masters, p. 444
(optional) Fact Triangles (Math Journal 2,
Activity Sheets 7–10)
Advance Preparation
For the Using Fact Triangles for Division activity in Part 1, you might want to make an overhead transparency
of Math Masters, page 444.
Teacher’s Reference Manual, Grades 1– 3 pp. 194–198
890
Unit 12 Year-End Reviews and Extensions
Interactive
Teacher’s
Lesson Guide
Mathematical Practices
SMP1, SMP2, SMP4, SMP5, SMP6, SMP8
Content Standards
Getting Started
2.OA.2
Mental Math and Reflexes
Math Message
Have children record addition fact sums on their
slates. For example:
On an Exit Slip (Math Masters,
page 415) write the fact family for
this Fact Triangle and then draw an
array for one of the multiplication
facts.
Sum-equals-10
6 + 4 10
Harder Facts
7 + 3 10
8 + 5 13
8 + 2 10
9 + 6 15
7 + 4 11
•
8
×, ÷
4
2
Home Link 12 4 Follow-Up
Doubles +1 and Doubles +2
5 + 7 12
Poll the class to see how much time children spent
practicing with the Fact Triangles. What is the
median practice time?
6 + 7 13
7 + 9 16
1 Teaching the Lesson
Math Message Follow-Up
WHOLE-CLASS
ACTIVITY
SOLVING
Ongoing Assessment:
Recognizing Student Achievement
Math
Message
Use the Math Message to assess children’s progress toward using arrays to
model multiplication. Children are making adequate progress if they can draw
an array for 1 of the multiplication facts. Some children may be able to
draw/compare arrays for both multiplication facts.
[Operations and Computation Goal 4]
List the facts in the fact family on the board as children name
them. (See margin.) Some children may forget to name the
division facts because they have used Fact Triangles mainly to
practice multiplication facts. Review the turn-around shortcut for
multiplication. To illustrate that there is no turn-around rule for
division, pose the following problems:
2
4
8
8
Æ
Æ
÷
÷
1. Four granola bars are shared equally by 2 children. How many
bars does each child get? 2 bars Summarize with a number
model: 4 ÷ 2 = 2.
2. Two granola bars are shared equally by 4 children. How many
bars does each child get? _12 of a bar To act out the problem,
represent the two bars with two identical rectangular pieces of
paper. Ask a volunteer to show how to share the bars equally
among 4 children. Tear each piece of paper in half. Each child
gets _12 of a granola bar. Summarize with a number model:
2 ÷ 4 = _12 .
4
2
2
4
=
=
=
=
8
8
4
2
Adjusting
the Activity
ELL
Show the children the 2, 2, 4 Fact Triangle.
1
Point out that _2 is not part of the family.
AUDITORY
KINESTHETIC
TACTILE
VISUAL
Therefore, 4 ÷ 2 is not equal to 2 ÷ 4.
Lesson 12 5
891
Using Fact Triangles
PARTNER
ACTIVITY
for Division Facts
(Math Journal 2, Activity Sheets 7–10; Math Masters, p. 444)
Algebraic Thinking Remind children that one partner can cover
any of the three numbers on a Fact Triangle before showing it to
the other partner.
×,÷
3
4
The multiplication problem 3 × 4 = ?
●
To generate a multiplication problem, cover the top number
(by the dot).
●
To generate a division problem, cover either of the bottom
numbers.
Draw a large Fact Triangle on the board or use an overhead
transparency of Math Masters, page 444. Fill it in to show a
division problem. For example, write 20 at the top, 5 and ? at the
bottom, and 20 ÷ 5 = ? under the Fact Triangle. Remind children
that a good way to solve a division problem is to think multiplication.
For example, to solve 20 ÷ 5 = ?, think:
5 times what number equals 20?
12
•
⫻,⫼
20
×, ÷
3
?
5
This shows 20 ÷ 5 = ?
Since 5 × 4 = 20, then 20 ÷ 5 = 4.
The division problem 12 ÷ 3 = ?
Repeat this routine. For each division problem, write a related
multiplication number model.
12
Examples:
⫻, ⫼
14 ÷ 2 = ? Think: 2 × ? = 14
4
The division problem 12 ÷ 4 = ?
15 ÷ 3 = ? Think: 3 × ? = 15
80 ÷ 10 = ? Think: 10 × ? = 80
One of the advantages of Fact Triangles is that they remind
children to do division by asking: By what number do I multiply?
For each division problem, the child receives two visual messages—
a division message and a related multiplication message.
Links to the Future
This is a beginning exposure to multiplication/division facts. Demonstrating
automaticity with multiplication facts through 10 × 10 is a Grade 3 Goal.
Demonstrating proficiency with the related division facts is a Grade 4 Goal.
892
Unit 12 Year-End Reviews and Extensions
Student Page
Solving Related Multiplication/
PARTNER
ACTIVITY
Date
Time
LESSON
Related Multiplication and Division Facts
12 5
Division Fact Problems
Solve each multiplication fact. Use the fact triangles to help you.
(Math Journal 2, pp. 280 and 299)
1. 3 × 7 =
Then use the three numbers to write two division facts.
Go over the example on journal page 299 before asking partners to
do the other problems. Note that some of the harder multiplication
facts have been selected.
Because children are not expected to have memorized all of these
facts, they can use their Product Tables (journal page 280) to complete
the problems.
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
21
÷ 7
÷ 3
=
27
9
3
=
3. 3 × 9 =
27
27
Adjusting the Activity
21
21
÷
÷
5. 4 × 8 =
32
32
÷
÷
=
24
24
÷
÷
4. 4 × 7 =
=
32
8
4
=
35
7
5
=
3
9
28
28
÷
÷
6. 4 × 9 =
=
4
8
36
36
÷
÷
24
8
3
=
28
4
7
=
36
9
4
=
40
8
5
=
3
8
=
7
4
=
4
9
=
V I S U A L
7. 5 × 7 =
35
35
2 Ongoing Learning & Practice
Playing Addition Card Draw
2. 3 × 8 =
3
7
÷
÷
8. 5 × 8 =
=
5
7
40
40
÷
÷
5
8
=
Math Journal 2, p. 299
289_312_EMCS_S_SMJ_G2_U12_57640X.indd 299
3/3/11 7:44 AM
PARTNER
ACTIVITY
(Math Journal 2, p. 300; Math Masters, p. 446)
This game provides practice with adding three numbers. You may
want to explain the rules of the game, as given on journal
page 300.
Student Page
Date
Game Master
Time
LESSON
12 5
Name
□
□
□
□
Time
1 2
4 3
Addition Card Draw Score Sheet
Addition Card Draw Directions
Materials
Date
score sheet from Math Masters, p. 446
4 each of number cards 1–10
Game 1
1 each of the number cards 11–20
1st turn:
+
slate or scratch paper
Players
2
Skill
Add 3 numbers
Game 2
1st turn:
+
=
2nd turn:
+
Object of the Game To get the higher total
+
=
3rd turn:
Directions
+
Shuffle the cards and place the deck with the numbers facing down.
Take turns.
+
+
=
+
=
+
=
+
=
+
=
+
=
2nd turn:
+
3rd turn:
+
=
Total:
+
Total:
1. Draw the top 3 cards from the deck.
2. Record the numbers on the score sheet. Put the 3 cards in a
separate pile.
Game 4
Game 3
3. Find the sum. Use your slate or paper to do the computation.
1st turn:
1st turn:
After 3 turns:
+
4. Check your partner’s work. Use a calculator.
+
=
2nd turn:
5. Find the total of the 3 answers. Write the total on the score sheet.
+
The player with the higher total wins.
+
=
3rd turn:
+
Math Journal 2, p. 300
+
3rd turn:
+
=
Total:
289_312_EMCS_S_SMJ_G2_U12_57640X.indd 300
+
2nd turn:
+
Total:
Math Masters, p. 446
3/3/11 7:44 AM
377-405_EMCS_B_MM_G2_U12_576949.indd 446
3/8/11 1:31 PM
Lesson 12 5
893
Student Page
Date
Math Boxes 12 5
Time
LESSON
12 5
7
10 more 100 less
378
268
4,689
4,789 4,799
40,870 40,880 40,770
1,899
1,999 2,009
How many minutes per hour?
368
1
_
3. Shade 2 of the shape. Write
the equivalent fraction.
24
How many weeks per year?
about 52
2
in
out
1
10
10
60
100
8
25
250
Rule
×10
80
Answers vary.
:
11
10
35 .
12
40
2
9
20 minutes
later will be
6 : 55 .
3
8
60 -
4
7
6
101
6. Solve.
1
Writing/Reasoning Have children draw, write, or
verbalize their answer to the following: How did
- 23 = 17 and
you find the missing numbers in
60 = 28 in Problem 6? Sample answer: For the first
problem, I added 23 and 17. In the second problem, I subtracted
28 from 60.
86
4. Fill in the table.
6
6
Mixed Practice Math Boxes in this lesson are linked with
Math Boxes in Lessons 12-7 and 12-8.
60
How many hours per day?
3
1
_
or _
5. The time is
(Math Journal 2, p. 301)
2. How many days per week?
1. Write the number that is
6
5
49 =
Home Link 12 5
Unit
32
70
(Math Masters, pp. 391 and 392)
= 28
- 21
26
Math Journal 2, p. 301
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3/3/11 9:56 AM
Home Connection Children solve pairs of related division
and multiplication problems. Pictures of Fact Triangles
are shown to help children solve the problems.
Home Link Master
Home Link Master
Name
INDEPENDENT
ACTIVITY
- 23 = 17
54 = 80 -
15 minutes earlier was
6 : 20 .
Date
Time
Name
12 5 ×, ÷ Facts Practice
HOME LINK
Date
12 5 ×, ÷ Facts Practice
HOME LINK
Family
Note
INDEPENDENT
ACTIVITY
Math Boxes
In this lesson, your child has connected multiplication and division facts by using Fact
Triangles and completing fact families. A good way to solve division problems is to think in
terms of multiplication. For example, to divide 20 by 5, ask yourself: 5 times what number
equals 20? Since 5 × 4 = 20, 20 ÷ 5 = 4.
7. 20 ÷ 5 =
38
Please return this Home Link to school tomorrow.
4
8. 30 ÷ 6 =
20
Think:
5 × ? = 20
5
5
30
Think:
6 × ? = 30
×, ÷
Solve these division facts. Think multiplication.
Time
continued
×, ÷
6
4
5
Use the Fact Triangles to help you.
1. 14 ÷ 2 =
7
2. 24 ÷ 4 =
14
Think:
2 × ? = 14
3. 21 ÷ 3 =
7
7
4. 18 ÷ 6 =
21
Think:
3 × ? = 21
5. 28 ÷ 7 =
7
4
6. 16 ÷ 4 =
28
Think:
7 × ? = 28
×, ÷
4
13. 54 ÷ 9 =
×, ÷
894
81
×, ÷
9
9
Math Masters, p. 392
Math Masters, p. 391
377-405_EMCS_B_MM_G2_U12_576949.indd 391
9
9
9
Think:
9 × ? = 81
×, ÷
6
4
14. 81 ÷ 9 =
54
Think:
9 × ? = 54
63
×, ÷
7
7
6
8
9
Think:
7 × ? = 63
×, ÷
6
16
4
12. 63 ÷ 7 =
42
3
4
Think:
4 × ? = 16
×, ÷
7
18
32
×, ÷
4
7
7
Think:
6 × ? = 42
8
Think:
4 × ? = 32
×, ÷
5
11. 42 ÷ 6 =
6
10. 32 ÷ 4 =
35
6
3
Think:
6 × ? = 18
×, ÷
3
×, ÷
4
7
Think:
5 × ? = 35
24
Think:
4 × ? = 24
×, ÷
2
9. 35 ÷ 5 =
6
3/1/11 4:32 PM
Unit 12 Year-End Reviews and Extensions
377-405_EMCS_B_MM_G2_U12_576949.indd 392
3/1/11 4:32 PM
Teaching Master
Name
3 Differentiation Options
Date
LESSON
Time
Equal Rows
12 5
Use counters to build arrays for each problem. Find the number in
each row for the arrays.
READINESS
Building Arrays with
INDEPENDENT
ACTIVITY
1. Use 6 counters. Build an array that has 2 rows. How many are in
each row?
Write a number model to show how you found the number in
each row. Sample answers:
5–15 Min
Equal Shares
3
in each row?
Write a number model to show how you found the number in
each row.
To explore division using a concrete model, have children build
arrays. When children have completed the page, have them
describe how building the arrays helped them solve the problems.
Their number models for finding the number in each row could be
a missing-factor number model or a division number model.
Number Model:
Tables
4
in each row?
Write a number model to show how you found the number in
each row.
Number Model:
4 + 4 + 4 + 4 = 16; 16 = 4 × ?
Try This
counters. Build an array that has
rows. How many
are in each row?
INDEPENDENT
ACTIVITY
5–15 Min
4 + 4 + 4 = 12; 12 = 4 × ?
3. Use 16 counters. Build an array that has 4 rows. How many are
4. Use
Completing “What’s My Rule?”
2 + 2 + 2 = 6; 6 = 2 × ?
Number Model:
2. Use 12 counters. Build an array that has 4 rows. How many are
(Math Masters, p. 393)
EXTRA PRACTICE
3
Write a number model to show how you found the number in
each row.
Answers vary.
Number Model:
Math Masters, p. 393
377-405_EMCS_B_MM_G2_U12_576949.indd 393
3/1/11 4:32 PM
(Math Masters, p. 394)
Algebraic Thinking Children complete “What’s My Rule?” tables
in which some input and some output numbers are missing. These
problems provide children with another way to look at the
relationship between multiplication and division facts.
ELL SUPPORT
Building a Math Word Bank
SMALL-GROUP
ACTIVITY
5–15 Min
(Differentiation Handbook, p. 143)
To provide language support for division, have children use the
Math Word Bank Template found on Differentiation Handbook,
page 143. Ask children to write the terms equal groups, equal
shares, and division; draw pictures representing the terms; and
write other related words. See the Differentiation Handbook for
more information.
Teaching Master
Name
Date
LESSON
Time
“What’s My Rule?”
12 5
Complete the tables in Problems 1–3.
1.
2.
Rule
×2
3.
Rule
×10
Rule
×5
in
out
in
out
in
out
3
6
10
2
20
40
0
0
15
7
14
5
50
10
50
8
16
7
70
8
40
6
12
10
100
20
100
5
4
3
Complete the table
and write the rule.
Complete the table
and write the rule.
Write a rule of your
own. Fill in the table.
4.
5.
6.
Rule
Rule
×4
×2
in
out
in
out
1
2
2
8
2
4
3
12
3
6
5
20
5
10
6
24
8
16
10
40
Answers vary.
Rule
in
out
Math Masters, p. 394
377-405_EMCS_B_MM_G2_U12_576949.indd 394
3/1/11 4:32 PM
Lesson 12 5
895