Basic 9 · Mathematics · Number Operations
The Associative and Distributive Properties
In this topic you will use the associative property of addition and multiplication, apply the distributive property and combine these properties to solve problems.
By the end of these notes you should be able to:
- Use the associative property
- Apply the distributive property
- Combine number properties to solve problems
What you already know
- Commutative property
Copy everything that has a blue line on its left into your notebook. The other parts are for reading.
Lesson 1
- Strand
- Number
- Sub-strand
- Number Operations
- Indicator
- B9.1.2.1.3 – Use the associative property of addition and multiplication
Introduction
17 + 8 + 2 looks harder than it is. By grouping the numbers cleverly, you can make it much easier.
1. What Is the Associative Property?
Associative property: Changing the grouping of numbers does not change the result.
| Operation | Example | Associative? |
|---|---|---|
| addition | (2 + 3) + 4 = 2 + (3 + 4) | yes |
| multiplication | (2 × 3) × 4 = 2 × (3 × 4) | yes |
| subtraction | (9 − 4) − 2 ≠ 9 − (4 − 2) | no |
| division | (16 ÷ 4) ÷ 2 ≠ 16 ÷ (4 ÷ 2) | no |
In algebra, (a + b) + c = a + (b + c), and (a × b) × c = a × (b × c). The commutative property changes the order of the numbers, while the associative property changes the brackets, or grouping. The two are often used together.
2. Using the Property
Making tens
Calculate 17 + 8 + 2.
- Group the numbers that make 10: 17 + (8 + 2).
- 17 + 10.
Answer: 27
Multiplication
Calculate 25 × (4 × 19).
- Regroup: (25 × 4) × 19.
- 100 × 19.
Answer: 1,900
With fractions
Calculate ⅔ + (⅓ + ¾).
- Regroup: (⅔ + ⅓) + ¾.
- 1 + ¾.
Answer: 1¾
3. Why Grouping Matters
Subtraction and division are not associative. For (9 − 4) − 2, the answer is 3. For 9 − (4 − 2), the answer is 7. This is why brackets matter, and why we work from left to right when a calculation has only subtraction and division. A careless change of brackets could cost marks.
4. In Daily Life
When a shopper adds up prices, they may group those that make a round number. GH₵12.50 + GH₵7.50 + GH₵35 is easier if you first add 12.50 and 7.50, which gives 20. Grouping also helps in volume calculations. The volume of a box measuring 5 cm × 7 cm × 2 cm can be found as (5 × 2) × 7 = 70 cm³. Looking for helpful groupings is a habit of good mathematicians.
New words
- associative
- Grouping does not change the result.
- grouping
- Putting numbers together.
- bracket
- A symbol showing what is done first.
- volume
- The space inside an object.
Summary
- Addition and multiplication are associative.
- Subtraction and division are not.
- Regroup to make calculations easier.
Class activity: Clever Grouping
- Write ten numbers on cards.
- Group them to make easy totals.
- Compare your methods.
You need: Bottle tops, Ghana cedi play notes, Number cards
Exercises
Answer in your exercise book first, then check your answers.
1. (2 + 3) + 5 = 2 + (3 + 5) shows the… property.
- A. associative
- B. commutative
- C. distributive
- D. identity
Show answer
Answer: associativeGrouping.2. 4 × (25 × 7) =
- A. 700
- B. 175
- C. 280
- D. 1,000
Show answer
Answer: 700100 × 7.3. Division is associative.
- A. True
- B. False
Show answer
Answer: FalseCounterexample.4. (a + b) + c = a + (b + ______).
Show answer
Answer: cAlgebra.5. Calculate 46 + 19 + 54 mentally.
Show answer
Answer: 119100 + 19.6. Find the volume of a box 25 cm × 3 cm × 4 cm.
Show answer
Answer: 300 cm³(25 × 4) × 3.
Lesson 2
- Strand
- Number
- Sub-strand
- Number Operations
- Indicator
- B9.1.2.1.4 – Use the distributive property in solving problems
Introduction
7 × 13 is hard for some people. 7 × 10 plus 7 × 3 is easy. This trick is the distributive property.
1. What Is the Distributive Property?
Distributive property: Multiplying a sum is the same as multiplying each part and then adding.
In algebra, a(b + c) = ab + ac, and a(b − c) = ab − ac. Think of a teacher giving 3 pens and 2 books to each of 5 learners. The total items are 5 × (3 + 2) = 25, or 5 × 3 + 5 × 2 = 15 + 10 = 25.
2. Using the Property in Number
Breaking apart
Calculate 7 × 13.
- 7 × (10 + 3).
- 70 + 21.
Answer: 91
Near a round number
Calculate 12 × 98.
- 12 × (100 − 2).
- 1,200 − 24.
Answer: 1,176
Factorising
Calculate 37 × 6 + 37 × 4.
- 37 × (6 + 4).
- 37 × 10.
Answer: 370
3. Using the Property in Algebra
| Expression | Result |
|---|---|
| 3(2x + 5) | 6x + 15 |
| 4(a − 3) | 4a − 12 |
| −2(x + 7) | −2x − 14 |
| 6x + 9 | 3(2x + 3) |
| 5y − 10 | 5(y − 2) |
Expanding removes brackets, and factorising puts them in. Be careful with negative signs: −2(x − 3) = −2x + 6. Learners often multiply only the first term. Each term in the bracket must be multiplied.
4. Area Model
A rectangle of length 13 and width 7 can be split into two parts: 7 × 10 and 7 × 3. The total area is 70 + 21 = 91. This picture explains why the distributive property works. It also leads to how we multiply larger numbers, and later to expanding (x + 2)(x + 3) in algebra.
5. Where It Is Used
Long multiplication depends on this property. For 23 × 45, you find 23 × 40 and 23 × 5 and add them. Bills use it too. If five workers earn a wage of GH₵50 and a bonus of GH₵10, the total is 5(50 + 10) = GH₵300.
New words
- distributive
- Spreading multiplication over a sum.
- expand
- To remove brackets.
- factorise
- To put in brackets.
- term
- A part of an expression.
Summary
- a(b + c) = ab + ac.
- Break apart numbers near round numbers.
- Expand and factorise with care over signs.
Class activity: Area Models
- Draw area models for three products.
- Write the matching calculation.
- Check with a calculator.
You need: Bottle tops, Ghana cedi play notes, Number cards
Exercises
Answer in your exercise book first, then check your answers.
1. 3(x + 4) =
- A. 3x + 12
- B. 3x + 4
- C. x + 12
- D. 7x
Show answer
Answer: 3x + 12Multiply each term.2. 15 × 102 =
- A. 1,530
- B. 1,500
- C. 1,520
- D. 1,503
Show answer
Answer: 1,5301,500 + 30.3. a(b + c) = ab + ac.
- A. True
- B. False
Show answer
Answer: TrueProperty.4. Removing brackets is called ______.
Show answer
Answer: expandingTerm.5. Factorise 8x + 12.
Show answer
Answer: 4(2x + 3)Common factor.6. Calculate 56 × 7 + 56 × 3 using the property.
Show answer
Answer: 56056 × 10.
Lesson 3
- Strand
- Number
- Sub-strand
- Number Operations
- Indicator
- B9.1.2.1.4 – Use the distributive property and associative property of addition and multiplication in solving problems
Introduction
Each property is a tool. Skilled mathematicians use several at once.
1. Review of Properties
| Property | Rule |
|---|---|
| commutative | a + b = b + a; ab = ba |
| associative | (a + b) + c = a + (b + c) |
| distributive | a(b + c) = ab + ac |
| identity | a + 0 = a; a × 1 = a |
| inverse | a + (−a) = 0; a × 1/a = 1 |
2. Using Several Properties
Combining
Calculate 4 × 17 × 25 + 3 × 100.
- Commutative: 4 × 25 × 17.
- Associative: (4 × 25) × 17 = 1,700.
- 3 × 100 = 300.
- 1,700 + 300.
Answer: 2,000
Algebra
Simplify 2(3x + 4) + 5(x − 1).
- Distributive: 6x + 8 + 5x − 5.
- Commutative: 6x + 5x + 8 − 5.
- Collect like terms.
Answer: 11x + 3
3. Solving Equations
Properties explain the steps in solving equations. To solve 3(x + 2) = 18, use the distributive property: 3x + 6 = 18. Subtract 6, using the inverse of addition: 3x = 12. Divide by 3: x = 4. You may check by substitution: 3(4 + 2) = 18. Each step has a reason, and knowing this reason helps you to avoid mistakes.
Word problem
A school buys 12 tables at GH₵85 and 12 chairs at GH₵15. Find the total cost.
- 12 × 85 + 12 × 15.
- 12 × (85 + 15).
- 12 × 100.
Answer: GH₵1,200
4. Justifying Your Steps
In mathematics, it is good practice to give reasons for your steps. Writing "by the distributive property" shows that you understand why the step is valid. This prepares you for mathematical proof, which is studied at senior high school. Looking for structure before you calculate also saves time. Before you begin a long calculation, ask whether there is a shorter way. Often, there is.
New words
- identity
- A number that leaves another unchanged.
- inverse
- A number that undoes another.
- substitution
- Replacing a letter by a number.
- justify
- To give reasons.
Summary
- Use several properties together to simplify work.
- Properties explain the steps in equations.
- Look for structure before you calculate.
Class activity: Property Detective
- Solve five expressions.
- Name each property you use.
- Compare your solutions.
You need: Bottle tops, Ghana cedi play notes, Number cards
Exercises
Answer in your exercise book first, then check your answers.
1. a + 0 = a shows the…
- A. identity property
- B. inverse property
- C. distributive property
- D. order
Show answer
Answer: identity propertyProperty.2. 2(x + 3) + x =
- A. 3x + 6
- B. 2x + 6
- C. 3x + 3
- D. 6x
Show answer
Answer: 3x + 6Expand.3. Properties explain the steps in solving equations.
- A. True
- B. False
Show answer
Answer: TrueReasoning.4. Replacing a letter by a number is ______.
Show answer
Answer: substitutionTerm.5. Solve 2(x + 5) = 16.
Show answer
Answer: x = 3Expand.6. Simplify 3(2y − 1) − 2(y + 4).
Show answer
Answer: 4y − 11Careful with signs.
Homework
- Solve five equations, and name the property used in each step.