Factors of a
Whole Number
Objective To guide children as they identify whole-number
ffactors of whole numbers.
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Game™
Teaching the Lesson
Key Concepts and Skills
• Use multiplication facts to solve problems. [Operations and Computation Goal 3]
• Use multiplication facts to find
whole-number factors of a whole number. [Operations and Computation Goal 3]
• Use arrays to model whole-number
factors of a whole number. [Operations and Computation Goal 6]
Key Activities
Children identify factors of whole numbers,
reinforcing the link between multiplication
and division; they play Factor Bingo to
practice identifying factors.
Ongoing Assessment:
Recognizing Student Achievement
Use Mental Math and Reflexes. [Number and Numeration Goal 2]
Key Vocabulary
factors
Materials
Math Journal 2, p. 219
Student Reference Book, pp. 285 and 286
Home Link 95
Math Masters, p. 448
transparency of Math Masters, p. 448
(optional) per group: 4 each of number
cards 2– 9 (from the Everything Math Deck,
if available) pennies or other counters slate half-sheet of paper
742
Unit 9
Multiplication and Division
Family
Letters
Assessment
Management
Common
Core State
Standards
Ongoing Learning & Practice
Using the Partial-Products
Algorithm
Math Journal 2, p. 220
Math Masters, pp. 273 and 274
(optional)
base-10 blocks (optional)
Children practice the partial-products
algorithm.
Ongoing Assessment:
Informing Instruction See page 746.
Math Boxes 9 6
Math Journal 2, p. 221
Children practice and maintain skills
through Math Box problems.
Home Link 9 6
Math Masters, p. 286
Children practice and maintain skills
through Home Link activities.
Curriculum
Focal Points
Interactive
Teacher’s
Lesson Guide
Differentiation Options
READINESS
Playing Array Bingo
Student Reference Book, p. 273
Math Masters, p. 442
per partnership: paper clips or envelopes scissors
Children explore fractions using an
array model.
ENRICHMENT
Playing Finding Factors
Math Masters, p. 287
per partnership: 2 different-colored counters,
2 different-colored crayons
Children apply their understanding of factors.
ELL SUPPORT
Building a Math Word Bank
Differentiation Handbook, p. 132
Children add the term factor to their
Math Word Banks.
Mathematical Practices
SMP1, SMP2, SMP4, SMP5, SMP6, SMP7
Content Standards
Getting Started
Mental Math and Reflexes
3.OA.2, 3.OA.3, 3.OA.5, 3.OA.7
Math Message
Have children find fractions of whole numbers, using
counters if necessary. They record their answers
on half-sheets of paper. Share strategies after solving problems.
Suggestions:
1 of 20 10
_
2
1
_
of 12 3
4
1 of 18 6
_
3
2
_
of 21 14
3
2 of 45 30
_
Home Link 9 5 Follow-Up
3
5
_
of 16 10
8
Ongoing Assessment:
Recognizing Student Achievement
You want to pack 24 bottles of juice into full
cartons. Each carton holds 4 bottles. Can you pack
all 24 bottles into cartons so none are left over?
Have children share strategies for making estimates
and solving the problems.
Mental Math
and
Reflexes
Use Mental Math and Reflexes to assess children’s progress toward using
concrete materials to model common fractions. Children are making adequate
progress if they successfully complete Levels
and
problems. Some
children may complete Level
problems successfully.
[Number and Numeration Goal 2]
1 Teaching the Lesson
Math Message Follow-Up
WHOLE-CLASS
DISCUSSION
ELL
Have children model the solution by making an array with
counters or by drawing it on slates. For example, if they make
rows of 4 counters, each row represents a full carton. There will be
6 rows. Arrays help children visualize factors as the rows and
columns that make up the array for a number. Building arrays is
a good beginning for work in later grades with prime, composite,
and square numbers.
6 full cartons with 4 bottles each
Ask: How many full cartons are there? 6 Are there any leftover
bottles? No Explain to children that one way to think about the
problem is to ask, “How many 4s are there in 24?”
Pose the same problem to the class, but vary the number of bottles
that a full carton will hold: 6 bottles? 9 bottles? 10 bottles?
12 bottles? Have children model each situation with an array
of counters. For 6 bottles, there are 4 rows of 6 counters. For
9 bottles, 2 rows have 9 counters, but the last row will be short
3 counters, so this number does not work.
Ask: What other sizes of cartons can be used to pack 24 bottles so
none are left over? Help children see that 24 bottles can be packed
into full cartons that hold 1, 2, 3, 4, 6, 8, 12, or 24 bottles. The
numbers 1, 2, 3, 4, 6, 8, 12, and 24 are called factors of 24. To
support English language learners, write factor on the board.
Ask volunteers to come to the board to draw different arrays
representing 24.
Lesson 9 6
743
NOTE Although a factor may be a whole
number, decimal, or a fraction, the lessons in
Third Grade Everyday Mathematics involve
whole-number factors only.
Ask children to describe the factors of 24 in their own words.
Record their key ideas on the board. Possible responses: The
factors of 24 are whole numbers that can be multiplied to get 24;
they are the numbers of rows or columns in arrays for 24; they are
whole numbers that will divide 24 without leaving a remainder.
Identifying Factors of a
WHOLE-CLASS
ACTIVITY
Whole Number
Pose a problem similar to the Math Message, and ask children to
solve it. They may use counters or drawings. Record their solutions
in a table on the board. List the numbers and factors in the table
in sequential order so the children can better identify patterns.
For example: There are 15 bottles.
●
Can they fill cartons that hold 3 bottles each? Yes Is 3 a factor
of 15? Yes
●
Can they fill cartons that hold 4 bottles each? No Is 4 a factor
of 15? No
●
What other sizes of cartons could you use? 1, 5, 15 Are these
also factors of 15? Yes
Repeat with other whole numbers; however, omit mention of
cartons and bottles. Ask children to identify factors of numbers
and record the answers in the table. Arrays might help children
get mental images of factors.
Number
Factors
15
1, 3, 5, 15
16
1, 2, 4, 8, 16
17
1, 17
18
1, 2, 3, 6, 9, 18
19
1, 19
20
1, 2, 4, 5, 10, 20
Ask children to make observations about the table. Guide children
with questions such as the following:
744
Unit 9 Multiplication and Division
●
Which numbers have exactly 2 factors? 17 and 19
●
What is another whole number not in the table that has only
2 factors? Sample answers: 3, 5, 7, 11
●
Which numbers have 2 as a factor? 16, 18, and 20
●
Which numbers in the table have the most factors? 18 and 20
●
Which numbers have an even number of factors? 15, 17, 18,
19, and 20
●
Which number has an odd number of factors? 16
●
Which number is a factor of all whole numbers? 1
●
Is every whole number a factor of itself? Yes
Student Page
Date
LESSON
Adjusting the Activity
ELL
9 6
䉬
Time
Factor Bingo Game Mat
To support English language learners, point out the difference between
a number with 2 factors and a number with 2 as a factor.
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
Introducing Factor Bingo
V I S U A L
SMALL-GROUP
ACTIVITY
(Math Journal 2, p. 219; Math Masters, p. 448;
Student Reference Book, pp. 285 and 286)
Discuss the rules for Factor Bingo on pages 285 and 286 in the
Student Reference Book. You might want to make an overhead
transparency of Math Masters, page 448 and play a demonstration
game with the class.
Playing Factor Bingo
SMALL-GROUP
ACTIVITY
Write any of the numbers
2 through 90 onto the grid
above.
2
3
5
6
7
8
9 10
11
12
13
14 15
4
16
17
18
19 20
21
22
23
24 25
26
27
28
29 30
You may use a number
only once.
31
32
33
34 35
36
37
38
39 40
41
42
43
44 45
46
47
48
49 50
To help you keep track
of the numbers you use,
circle them in the list.
51
52
53
54 55
56
57
58
59 60
61
62
63
64 65
66
67
68
69 70
71
72
73
74 75
76
77
78
79 80
81
82
83
84 85
86
87
88
89 90
Math Journal 2, p. 219
(Math Journal 2, p. 219; Math Masters, p. 448)
Make sure the children understand the rules before completing
their boards. Children use the Factor Bingo Game Mat on journal
page 219 or make up a new one on Math Masters, page 448. After
several games, some children may discover that certain numbers
have more factors than others. Their chances of winning are then
enhanced by a judicious choice and placement of numbers. When
children complete play, hold a discussion about good board
numbers (ones that have several factors between 2 and 9) and
impossible board numbers (numbers that do not have factors
between 2 and 9).
Math Journal 2, p. 219 is identical to
Math Masters, p. 448.
Adjusting the Activity
Have children remove the 2 and 5 cards from their decks. Distribute
additional blank copies of Math Masters, page 448 and ask children to complete
a new game mat. Have them discuss why they selected the numbers and
locations they did.
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
Lesson 9 6
745
Student Page
Date
Time
LESSON
9 6
2 Ongoing Learning & Practice
Using the Partial-Products Algorithm
Multiply. Show your work. Compare your answers with your partner’s
answers. If you disagree, discuss your strategies with each other. Then,
try the problem again.
1.
2.
68
× 2
3.
Using the Partial-Products
96
× 5
5 [90s] ∑ 450
5 [6s] ∑ + 30
450 + 30∑ 480
2 [60s] ∑ 120
2 [8s] ∑ + 16
120 + 16∑ 136
4.
47
× 4
Algorithm
(Math Journal 2, p. 220; Math Masters, pp. 273 and 274)
Children use the partial-products algorithm to find products
of 1-digit numbers and multidigit numbers on journal page 220.
85
× 9
9 [80s] ∑ 720
9 [5s] ∑ + 45
720 + 45∑ 765
4 [40s] ∑ 160
4 [7s] ∑ + 28
160 + 28∑ 188
INDEPENDENT
ACTIVITY
Ongoing Assessment: Informing Instruction
5.
6.
231
× 6
508
× 5
5 [500s] ∑ 2,500
5 [8s] ∑ + 40
2,540
6 [200s] ∑ 1,200
6 [30s] ∑
180
6 [1s] ∑ + 6
1,386
Math Journal 2, p. 220
204-239_EMCS_S_MJ2_G3_U09_576418.indd 220
3/29/11 5:52 PM
NOTE The partial-products algorithm is
an application of the Distributive Property
of Multiplication over Addition. At this time,
children are not expected to use the term
distributive property. They should, however,
begin to develop an understanding of the
property as they use the partial-products
algorithm.
Math Boxes 9 6
Mixed Practice Math Boxes in this lesson are paired
with Math Boxes in Lesson 9-8. The skill in Problem 6
previews Unit 10 content.
Writing/Reasoning Have children write an answer to the
following: Explain how you decided if the game in Problem
5 was fair. Sample answer: A 6-sided die has an equal
number of even and odd numbers, so players have an equal chance
of rolling a winning number.
Time
LESSON
96
1.
Math Boxes
Estimate. Malachi sold 19 boxes
of candy for $2.50 a box.
About how much money should
he have?
2.
Solve.
(9 × 9) – (43 + 9) =
35
1,091
$50.00
About
Home Link 9 6
29
= (5,600 ÷ 80) ÷ 2
= 963 + (567 – 439)
191
1
_
3
1
_
3
1
_
3
1
_
3
>
=
<
<
16
4.
Use your Fraction Cards.
Write >, <, or = to make the
number sentence true.
Use bills and coins.
Share $63 equally among
9 people.
1
_
4
Number model:
$63 ÷ 9 = ? or 9 × ? = $63
4
_
12
7
_
Each person gets $
8
7
6
You and a friend are playing a
game with a 6-sided die. You win if
an odd number is rolled. Your friend
wins if an even number is rolled.
Do you think this game is fair?
Circle one.
yes
.
4
_
31 32
5.
73
6.
no
Measure this line segment.
It is about
It is about
long.
2_14
5_1
2
inches long.
centimeters
137–139
143–145
Math Journal 2, p. 221
204-239_EMCS_S_MJ2_G3_U09_576418.indd 221
746
INDEPENDENT
ACTIVITY
(Math Masters, p. 286)
Sample answer:
20 × $2.50 = $50.00
Number model:
3.
INDEPENDENT
ACTIVITY
(Math Journal 2, p. 221)
Student Page
Date
Watch for children who are unsure of using the partial-products algorithm. Have
them use the Array Grid (Math Masters, pages 273 and 274) and base-10 blocks
to model the problems. Then have them write number models for each partial
product. For example:
29
×4
4 [20]s
→
80
4 [9]s
→ + 36
80 + 36 → 116
Unit 9 Multiplication and Division
8/25/11 8:14 AM
Home Connection Children find all possible arrangements
of 18 chairs in equal rows. They list all of the whole-number
factors of 18 and tell someone at home how finding the
chair arrangements can help them list the factors.
Home Link Master
Name
3 Differentiation Options
Date
HOME LINK
Time
96
Arrays and Factors
Family
Note
Discuss with your child all the ways to arrange 18 chairs in equal rows. Then help your child
use this information to list the factors of 18 (pairs of numbers whose product is 18).
䉬
64–67
Please return this Home Link to school tomorrow.
READINESS
Playing Array Bingo
PARTNER
ACTIVITY
Work with someone at home.
The third-grade class is putting on a play. Children have invited
18 people. Gilda and Harvey are in charge of arranging the 18 chairs.
They want to arrange them in rows with the same number of chairs
in each row, with no chairs left over.
15–30 Min
(Student Reference Book, p. 273; Math Masters, p. 442)
Yes or no:
Can they arrange
the chairs in …
To explore factors using an array model, have children play
Array Bingo. Children cut apart the Array Bingo cards on
Math Masters, page 442. Discuss the rules for Array Bingo on
page 273 in the Student Reference Book, and have children play in
pairs. When finished, have children clip their Array Bingo cards
together (or place them in an envelope) to store in their tool kits.
1 row?
ENRICHMENT
Playing Finding Factors
yes
2 rows? yes
3 rows? yes
no
4 rows?
no
5 rows?
yes
6 rows?
no
7 rows?
no
8 rows?
9 rows? yes
no
10 rows?
18 rows? yes
PARTNER
ACTIVITY
List all the factors of the number
18. (Hint: 18 has exactly 6 factors.)
If yes,
how many chairs
in each row?
18
9
6
3
2
1
1
9
chairs
chairs
2
6
18
3
chairs
How does knowing all the ways
to arrange 18 chairs in equal
rows help you find the factors
of 18? Tell someone at home.
chairs
Sample answer:
chairs
When 18 chairs are
chairs
arranged in equal rows,
chairs
the number of rows
chairs
and the number of chairs
chairs
in each row are factors
chairs
of 18.
chairs
15–30 Min
Math Masters, p. 286
(Math Masters, p. 287)
To apply their understanding of factors, have children play
Finding Factors on Math Masters, page 287. When they have
completed a few rounds of the game, have children discuss
their strategies.
ELL SUPPORT
Building a Math Word Bank
SMALL-GROUP
ACTIVITY
5–15 Min
(Differentiation Handbook, p. 132)
To provide language support for multiplication, have children use
the Word Bank template on Differentiation Handbook, page 132.
Ask children to write the term factor, draw a picture representing
the word, and write other related words. See the Differentiation
Handbook for more information.
Teaching Master
Name
LESSON
96
䉬
Date
Time
Finding Factors
Materials
䊐 2 different-colored counters, 2 different-colored crayons
䊐 Finding Factors gameboard (see below)
Players
2
Object of the Game To shade five products in a row, column, or diagonal
Directions
1. Player A places a counter on one of the factors in the Factor Strip at
the bottom of the gameboard.
2. Player B places a second counter on one of the factors in the Factor
Strip. (Two counters can cover the same factor.) Now that two
factors are covered, Player B wins the square that is the product of
the two factors. Player B shades this square with his or her color.
3. Player A moves either one of the counters to a new factor on the
Factor Strip. If the product of the two covered factors has not been
shaded, Player A shades this square with his or her color and wins
the square.
4. Play continues until 5 squares in a row, column, or diagonal are
shaded in the same color.
p
g
py g
4
10
20
30
45
64
3
9
18
2
8
16
27
40
56
1
7
15
25
36
54
28
42
63
5
12
21
32
48
72
6
14
24
35
49
81
Factor Strip
1
2
3
4
5
6
7
8
9
Math Masters, p. 287
Lesson 9 6
747