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Properties of Operations

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Properties of Operations

Complete each question. Show working where asked. Remember: the commutative property changes the order, the associative property changes the grouping, and the distributive property multiplies across a sum or difference.

Recognising the Properties

1.Match each property to its description.

Commutative property
Associative property
Distributive property

The grouping of the numbers changes, but the result stays the same.
The order of the numbers changes, but the result stays the same.
Multiply a number by each term inside brackets, then add or subtract.

2.Which statement correctly describes the commutative property?

It applies to addition and multiplication.
It applies to subtraction and division only.
It changes the grouping of numbers.
It means multiplying across brackets.

Commutative Property

3.Complete each equation by reversing the order of the numbers: 11 × 6 = ___ × ___; 14 + 9 = ___ + ___; 8 × 7 = ___ × ___; 13 + 5 = ___ + ___.

4.Show that 7 × 9 and 9 × 7 have the same value. Write the calculation for each product.

5.In your own words, explain why switching the order of factors does not change their product. Give one example.

Associative Property

6.Fill the gaps so each equation shows a change in grouping only: (6 + 2) + 9 = 6 + (___ + ___); (4 × 3) × 5 = ___ × (3 × ___); (7 + 8) + 1 = ___ + (8 + ___); (2 × 9) × 4 = 2 × (___ × ___).

7.Evaluate both sides of (5 + 4) + 8 = 5 + (4 + 8). What do you notice?

8.Use the grids to count the objects represented by each expression: (2 × 4) × 3 and 2 × (4 × 3). What is the total?

(2 × 4) × 3

2 × (4 × 3)

Distributive Property

9.Complete each equation using the distributive property: 4 × (6 + 3) = (4 × ___) + (4 × ___); 7 × (5 + 2) = (___ × 5) + (___ × 2); 3 × (9 + 4) = (3 × ___) + (___ × 4); 6 × (8 + 2) = (___ × ___) + (___ × ___).

10.Use the distributive property to calculate 8 × (10 + 4). Show the expanded calculation and answer.

11.Select every expression equal to 9 × (4 + 3).

(9 × 4) + (9 × 3)
9 × 7
(9 + 4) × (9 + 3)
36 + 27
63

Mixed Practice

12.Identify the property illustrated by each equation: 16 + 5 = 5 + 16; (3 × 7) × 2 = 3 × (7 × 2); 6 × (8 + 1) = (6 × 8) + (6 × 1); 12 × 4 = 4 × 12.

13.Use operation properties to find 4 × 25 × 6 efficiently. Show how you regroup the factors.

14.Calculate 7 × 18 by writing 18 as 20 − 2 and using the distributive property.

15.Write one true equation of your own to demonstrate each: commutative property, associative property, and distributive property.

Everyday Applications

16.Hana buys 5 packs of felt pens at $4 per pack and 5 packs of notebooks at $3 per pack. Use the distributive property to work out the total cost in New Zealand dollars.

17.Give one reason operation properties are useful when solving calculations. Include a short numerical example.

18.Describe a real-life situation where changing the order of two groups being added would keep the total unchanged.