Grades 4-5
October 9, 2013
Factors and Multiples
& Distributive Property

Tile Activity
• Count out 12 tiles.
• Arrange them so that they form a rectangle.

Tile Activity
• Count out 12 tiles.
• Arrange them so that they form a rectangle.
• What multiplication sentence could you write about your rectangle?
• Where in your figure do you see the numbers in your multiplication sentence?

Tile Activity
Suppose you made this rectangle.

Tile Activity
What multiplication sentence could you write about your rectangle?

Tile Activity
What multiplication sentence could you write about your rectangle?
3 x 4 = 12 (for example)

Tile Activity
Where in your figure do you see the numbers in the math sentence?
3 x 4 = 12

Tile Activity
3 x 4 = 12
3
1 2 3 4
5 6 7 8
9 10 11 12
4

Tile Activity
3
1 2 3 4
5 6 7 8
9 10 11 12
We say that the numbers 3 and 4 are factors of 12. The number 12 is a multiple of 3 and a multiple of 4.
4

Tile Activity
• Can you write a division sentence about your rectangle?
• Where are the factors and multiples in your number sentence?

Tile Activity
12 ÷ 3 = 4, for example.

Tile Activity
12 ÷ 3
3
1 2 3 4
5 6 7 8
9 10 11 12
We say that the numbers
3 and 4 are factors of 12.
The number 12 is a multiple of 3 and a multiple of 4.
4

Tile Activity
• What other rectangle can you make with 12 tiles?
• What factors and multiples does your new rectangle illustrate?

Tile Activity
• Use your 12 tiles make a rectangle with a length of 7.
• Does it work?
• Is 7 a factor of 12?
• Is 12 a multiple of 7?
• What if you try to use your 12 tiles to make a rectangle with a length of 5? Do you think it will work? Why?

Tile Activity
• What are ALL the rectangles you can make with 12 tiles?
• What are ALL the factors of 12?

Tile Activity
• On another day, do a similar activity with 20 tiles, 24, 30, 36, 48. Try the same activity with
13 tiles.

Factor and Multiple Games Online
1) HoodaMath – Factor Feeder www.hoodamath.com/games/factorfeeder.html
2) nrich Factors and Multiples Game www.nrich.maths.org/5468
3) BBC.UK – The Legend of Dick and Dom www.bbc.co.uk/bitesize/ks2/maths/number/factors
_multiples/play

More Factor and Multiple Games Online
4) Johnnie’s Math Page – Ghost Multiples, Egg
Multiples, and more www.jmathpage.com/JIMSMultiplicationfactorsand multiples.html
5) Math Playground - Pumpkin Multiples and more www.mathplayground.com/multiples.html
6) Count On- Octopus factors www.counton.org/games/map-numbers/octopus

• How can you help your students build fluency with factors and multiples?

Distributive Property of Multiplication over Addition
Let’s begin with some mental math. Solve these problems and remember what strategy you used:
8 x 24
3 x 56
9 x 53

Distributive Property of Multiplication over Addition
Did your strategy incorporate the distributive property?
8 x 24
3 x 56
9 x 53

Distributive Property of Multiplication over Addition
Can you apply the distributive property to this problem?
32 x 15
Video 5.2 from Number Talks: Helping Children
5 th grade teacher Lee Ann Davidson

Video 5.2
How did the students break the numbers apart?
Is this the distributive property?

Introducing the Distributive Property
Meaningfully
• Using tiles, show how you would model:
4 x 3 – Choose one color of tile.
4 x 8 – Choose a different color of tile.
4 x (3 + 8)
In your representation, how is the four distributed over the three and eight?
How could you use these examples to build students’ understanding of the distributive property?

4 x 3
4 x 3

(4 x 3) + (4 + 8)
(4 x 3) + (4 x 8)

3 + 8

4(3 + 8)
4(3 + 8)

Modeling with base 10 blocks
Use base 10 blocks to model the following:
3 x 14
What mental math strategy could be represented by your base 10 blocks?

3 x 14
3 x 14
3 x 14 = (3 x 10) + (3 x 4)
Now show 7 x 22 with base 10 blocks.

7 x 22 =
7 x 22

7 x 22
7 x 22 = ( 7 x 20 + (7 x 2) = 140 + 14 = 154

7 x 22 =
Or: 7 x 22

7 x 22
7 x 22 = (7 x 10) + (7x 12) = 70 + 84 = 154

Try 11 x 12

11 x 12

11 x 12

11 x 12
11 x 12 = (10 x 10) + (10 x 1) + (2 x 10) + ( 2 x 1)
= 100 + 10 + 20 + 2 = 132

3 Methods for Multiplying
11 x 12
11 x 12
12 22
120
132
110
132
Which one would you use?
How do the other methods work?
11 x 12
2
20
10
100
132

11 x 12
11
X 12
22
110
132
110 + 22 = 132

11 x 12
11
X 12
22
110
132
110 + 22 = 132

11 x 12
120 + 12 = 132
11
X 12
12
120
132

11 x 12
120 + 12 = 132
11
X 12
12
120
132

11 x 12
100 + 10 + 20 + 2 = 132
11
X 12
2
20
10
100
132

11 x 12
100 + 10 + 20 + 2 = 132
11
X 12
2
20
10
100
132

• How can you help your students make sense of the distributive property of multiplication over addition?

Using the Distributive Property to Divide
• 84 ÷ 6
• 6 x ? = 84

Using the Distributive Property to Divide
• 84 ÷ 6
• 6 x ? = 84
• 60 + 24 = 84

Using the Distributive Property to Divide
• 84 ÷ 6
• 6 x ? = 84
• 60 + 24 = 84
• 60 ÷ 6 = 10
• 24 ÷ 6 = 4

Using the Distributive Property to Divide
• 84 ÷ 6
• 6 x ? = 84
• 60 + 24 = 84
• 60 ÷ 6 = 10
• 24 ÷ 6 = 4
• 84 ÷ 6 = 10 + 4 = 14

• 144 ÷ 9
Try this problem!

Two strategies for 144 ÷ 9
144 ÷ 9
9 x ? = 144
144 = 72 + 72
72 ÷ 9 = 8
72 ÷ 9 = 8
144 ÷ 9 = 8 + 8 = 16
16
144 ÷ 9
9 x ? = 144
144 = 90 + 54
90 ÷ 9 = 10
54 ÷ 9 = 6
144 ÷ 9 = 10 + 6 =

Two online videos
Khan Academy – an explanation of the distributive property with a helpful visual model. https://www.khanacademy.org/math/arithmetic/ order-of-operations/ditributive_property
Learnzillion – Approaches the distributive property as a strategy for difficult facts.
www.learnzillion.com/lessons/966-use-thedistributive-property-of-multiplicaion-to-solveunfamiliar-facts