Algebraic Thinking: Properties of Whole Numbers CCSS 3.OA.5 - Free Printable

Let's solve each problem step by step using the properties of whole numbers as indicated in the worksheet. The properties mentioned are:

1. Identity Property – Adding 0 or multiplying by 1 doesn't change a number.
2. Zero Property – Any number multiplied by 0 is 0.
3. Commutative Property – Order doesn’t matter for addition and multiplication:
$ a + b = b + a $, $ a \times b = b \times a $
4. Associative Property – Grouping doesn’t matter for addition and multiplication:
$ (a + b) + c = a + (b + c) $, $ (a \times b) \times c = a \times (b \times c) $
5. Distributive Property – Multiplication distributes over addition:
$ a \times (b + c) = a \times b + a \times c $

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1. Using the IDENTITY and ZERO property

#### 1.1 $ 3 + 0 = ? $
- Identity Property of Addition: Adding 0 to any number gives the same number.
- So, $ 3 + 0 = 3 $

✔ Answer: 3

#### 1.2 $ 14 \times 0 = ? $
- Zero Property of Multiplication: Any number times 0 is 0.
- So, $ 14 \times 0 = 0 $

✔ Answer: 0

#### 1.3 $ 18 \times 1 = ? $
- Identity Property of Multiplication: Any number times 1 is itself.
- So, $ 18 \times 1 = 18 $

✔ Answer: 18

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2. Using Commutative Property

#### 2.1 $ 5 + 8 = 8 + ? $
- Commutative Property: $ a + b = b + a $
- So, $ 5 + 8 = 8 + 5 $

✔ Answer: 5

#### 2.2 $ 9 \times 6 = 6 \times ? $
- Commutative Property: $ a \times b = b \times a $
- So, $ 9 \times 6 = 6 \times 9 $

✔ Answer: 9

#### 2.3 $ 3 + 8 = ? + 3 $
- $ 3 + 8 = 8 + 3 $ → So the missing number is 8

✔ Answer: 8

#### 2.4 $ 4 \times 7 = ? \times 4 $
- $ 4 \times 7 = 7 \times 4 $ → So the missing number is 7

✔ Answer: 7

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3. Using Associative Property

#### 3.1 $ 46 + 3 + 2 = 2 + (46 + ?) $
We need to make sure the grouping matches the associative property.

Left side: $ 46 + 3 + 2 $
Right side: $ 2 + (46 + ?) $

To match associative property:
$ (46 + 3) + 2 = 46 + (3 + 2) $, but here it’s written as $ 2 + (46 + ?) $

So we want: $ 46 + 3 + 2 = 2 + (46 + 3) $

Thus, the missing number is 3

✔ Answer: 3

#### 3.2 $ 4 \times 2 \times 5 = 5 \times (? \times 2) $
We have: $ 4 \times 2 \times 5 $

Using associative property:
$ (4 \times 2) \times 5 = 4 \times (2 \times 5) $, but here it's $ 5 \times (? \times 2) $

So we want: $ 5 \times (4 \times 2) $

Thus, the missing number is 4

✔ Answer: 4

#### 3.3 $ 4 \times 5 \times 2 = ? \times (5 \times 2) $
We can write this as:
$ (4 \times 5) \times 2 = 4 \times (5 \times 2) $

So, $ ? = 4 $

✔ Answer: 4

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4. Using Distributive Property

#### 4.1 $ 3 \times (2 + 5) = ? $
Use distributive property:
$ 3 \times (2 + 5) = 3 \times 2 + 3 \times 5 = 6 + 15 = 21 $

✔ Answer: 21

#### 4.2 $ 4 \times (3 + 6) = ? $
$ 4 \times (3 + 6) = 4 \times 3 + 4 \times 6 = 12 + 24 = 36 $

✔ Answer: 36

#### 4.3 $ 6 \times (4 + 3) = ? $
$ 6 \times (4 + 3) = 6 \times 4 + 6 \times 3 = 24 + 18 = 42 $

✔ Answer: 42

#### 4.4 $ 9 \times (4 + 2) = ? $
$ 9 \times (4 + 2) = 9 \times 4 + 9 \times 2 = 36 + 18 = 54 $

✔ Answer: 54

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5. Using Distributive Property to break into smaller problems

We need to rewrite each multiplication using distributive property.

#### 5.1 $ 12 \times 6 $
Break 12 into $ 10 + 2 $:
$ 12 \times 6 = (10 + 2) \times 6 = 10 \times 6 + 2 \times 6 = 60 + 12 = 72 $

✔ Answer: $ (10 \times 6) + (2 \times 6) $

#### 5.2 $ 8 \times 7 $
Break 8 into $ 5 + 3 $, or $ 4 + 4 $. Let's use $ 5 + 3 $:
$ 8 \times 7 = (5 + 3) \times 7 = 5 \times 7 + 3 \times 7 = 35 + 21 = 56 $

✔ Answer: $ (5 \times 7) + (3 \times 7) $

#### 5.3 $ 7 \times 9 $
Break 7 into $ 5 + 2 $:
$ 7 \times 9 = (5 + 2) \times 9 = 5 \times 9 + 2 \times 9 = 45 + 18 = 63 $

✔ Answer: $ (5 \times 9) + (2 \times 9) $

#### 5.4 $ 4 \times 8 $
Break 8 into $ 5 + 3 $:
$ 4 \times 8 = 4 \times (5 + 3) = 4 \times 5 + 4 \times 3 = 20 + 12 = 32 $

✔ Answer: $ (4 \times 5) + (4 \times 3) $

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✔ Final Answers Summary:

1.
- 1.1: 3
- 1.2: 0
- 1.3: 18

2.
- 2.1: 5
- 2.2: 9
- 2.3: 8
- 2.4: 7

3.
- 3.1: 3
- 3.2: 4
- 3.3: 4

4.
- 4.1: 21
- 4.2: 36
- 4.3: 42
- 4.4: 54

5.
- 5.1: $ (10 \times 6) + (2 \times 6) $
- 5.2: $ (5 \times 7) + (3 \times 7) $
- 5.3: $ (5 \times 9) + (2 \times 9) $
- 5.4: $ (4 \times 5) + (4 \times 3) $

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