Number · Number Operations · Lesson 2
Mental Mathematics Strategies and Number Properties
Good mental mathematicians are not people who calculate faster — they are people who change a hard sum into an easy one before they start. In this lesson you will learn halving and doubling, the distributive property and other number properties that let you work out sums like 28 × 5, 18 × 6 and 8 × 99 in your head.
By the end you will be able to
- ✔State and use the commutative, associative and distributive properties of multiplication.
- ✔Use halving and doubling to multiply two numbers mentally.
- ✔Use the distributive property to split (or compensate) one factor and multiply mentally.
- ✔Choose an efficient mental strategy for a given multiplication.
Key words
- Mental mathematics
- Working out answers in your head, using number facts and clever strategies rather than long written methods.
- Halving and doubling
- Halving one factor and doubling the other; the product stays the same, e.g. 28 × 5 = 14 × 10.
- Distributive property
- a × (b + c) = a × b + a × c. Multiplying a sum is the same as multiplying each part and adding.
- Commutative property
- The order of multiplying does not matter: 4 × 25 = 25 × 4.
- Associative property
- The grouping does not matter: (7 × 4) × 25 = 7 × (4 × 25).
- Compensation
- Rounding a number to a friendly one, calculating, then adjusting: 8 × 99 = 8 × 100 − 8.
Curriculum reference
Content standard: B7.1.2.1 Apply mental mathematics strategies and number properties used to solve problems.
Indicator: Apply mental mathematics strategies and number properties used to perform calculations.
Starter
The kelewele seller's quick maths
At Osu night market, Maame Akua sells kelewele at GH₵5 a pack. A group of 28 students from a nearby SHS buys one pack each. Before you have even found your calculator she says, "GH₵140, please!" She did not multiply 28 by 5 the long way. She thought: half of 28 is 14, double 5 is 10, and 14 × 10 = 140. Today you will learn her tricks.
1. The number properties behind the tricks
Every mental strategy in this lesson works because of a few properties (rules) that are always true for multiplication. You have been using some of them since primary school without naming them. Knowing them by name helps you explain why your shortcut gives the right answer — and examiners often ask for that.
| Property | Rule | Example |
|---|---|---|
| Commutative | a × b = b × a | 25 × 7 = 7 × 25 |
| Associative | (a × b) × c = a × (b × c) | (7 × 4) × 25 = 7 × (4 × 25) = 7 × 100 |
| Distributive | a × (b + c) = a × b + a × c | 7 × (10 + 5) = 70 + 35 |
| Distributive (subtraction) | a × (b − c) = a × b − a × c | 6 × (20 − 2) = 120 − 12 |
| Identity | a × 1 = a | 365 × 1 = 365 |
| Zero | a × 0 = 0 | 4,875 × 0 = 0 |
The big idea is this: friendly numbers such as 10, 100, 1,000, 25 × 4 = 100 and 125 × 8 = 1,000 are easy to work with. The properties allow you to rearrange, split or adjust numbers until friendly numbers appear.
2. Halving and doubling
If you halve one factor and double the other, the product does not change. Why? Halving divides by 2 and doubling multiplies by 2, and those two actions cancel out. Picture a rectangle of squares 28 long and 5 tall. Cut it in half and stack the two halves: you get a rectangle 14 long and 10 tall. The squares have only been moved, so the total is still the same.
▶ Cut the 28 × 5 strip in half and stack the halves. Nothing is lost: 28 × 5 = 14 × 10 = 140.
Halving and doubling is most useful when one number is even and the other is a 5, 25, 50 or 125, because doubling those gives 10, 50, 100 or 250. You can repeat the trick: 125 × 4 = 250 × 2 = 500 × 1 = 500. And 16 × 25 = 8 × 50 = 4 × 100 = 400.
3. Splitting and compensating with the distributive property
The distributive property lets you split a hard factor into friendly parts. To find 7 × 15, think of 15 as 10 + 5. Then 7 × 15 = 7 × 10 + 7 × 5 = 70 + 35 = 105. In a picture, the 7 by 15 rectangle is cut into a 7 by 10 piece and a 7 by 5 piece.
▶ The area model: split 15 into 10 + 5, find each part, then add the parts.
Sometimes it is easier to go past a friendly number and come back. This is called compensation. For 18 × 6, think of 18 as 20 − 2: (20 − 2) × 6 = 20 × 6 − 2 × 6 = 120 − 12 = 108. For 8 × 99, think of 99 as 100 − 1: 8 × 100 − 8 × 1 = 800 − 8 = 792. Numbers ending in 8 or 9 are good candidates for compensation.
- 1Look at the factors. Is one of them close to a friendly number (10, 20, 50, 100)?
- 2Write it as a sum (15 = 10 + 5) or a difference (99 = 100 − 1).
- 3Multiply EACH part by the other factor.
- 4Add the parts (for a sum) or subtract them (for a difference).
4. Regrouping to find friendly products
The commutative and associative properties let you change the order and grouping of factors so that friendly pairs meet. For 28 × 25, split 28 into 7 × 4. Then 28 × 25 = 7 × 4 × 25 = 7 × (4 × 25) = 7 × 100 = 700. For 4 × 37 × 25, reorder to (4 × 25) × 37 = 100 × 37 = 3,700.
There is rarely only one good strategy. 28 × 25 could also be done by halving and doubling (14 × 50 = 7 × 100 = 700) or by splitting (28 × 25 = 20 × 25 + 8 × 25 = 500 + 200 = 700). Choose the path with the fewest, easiest steps, and use a second method to check when the answer matters.
| If you see… | Try… | Example |
|---|---|---|
| An even number × 5, 25, 50 or 125 | Halving and doubling | 36 × 5 = 18 × 10 = 180 |
| A factor like 13, 15, 21, 32 | Split with the distributive property | 6 × 13 = 60 + 18 = 78 |
| A factor ending in 8 or 9 | Compensate | 19 × 7 = 140 − 7 = 133 |
| A 4 or a multiple of 4 with 25 | Regroup to make 100 | 36 × 25 = 9 × 100 = 900 |
Worked examples
Try each one yourself first, then open the solution and compare every step.
Example 1
Halving and doubling once
Show the solution
- Step 128 is even and 5 doubles to 10, so use halving and doubling.
- Step 2Halve 28: 14. Double 5: 10.
- Step 314 × 10 = 140.
Answer: 140
Example 2
Halving and doubling twice
Show the solution
- Step 1Double 125 and halve 4: 125 × 4 = 250 × 2.
- Step 2Do it again: 250 × 2 = 500 × 1.
- Step 3500 × 1 = 500. (Or think (125 × 2) × 2 = 250 × 2 = 500.)
Answer: 500
Example 3
Splitting with the distributive property
Show the solution
- Step 1Split 15 into 10 + 5: 7 × 15 = 7 × (10 + 5).
- Step 2Multiply each part: 7 × 10 = 70 and 7 × 5 = 35.
- Step 3Add: 70 + 35 = 105.
Answer: 105
Example 4
Compensating
Show the solution
- Step 118 = 20 − 2, so 18 × 6 = (20 × 6) − (2 × 6) = 120 − 12 = 108.
- Step 299 = 100 − 1, so 8 × 99 = (8 × 100) − (8 × 1) = 800 − 8 = 792.
Answer: 18 × 6 = 108 and 8 × 99 = 792
Example 5
Regrouping to make 100
Show the solution
- Step 1Total = 28 × 25.
- Step 2Split 28 = 7 × 4, so 28 × 25 = 7 × (4 × 25).
- Step 34 × 25 = 100, so the total is 7 × 100 = 700.
- Step 4Check by splitting: 20 × 25 + 8 × 25 = 500 + 200 = 700 ✓
Answer: GH₵700
Summary
- Mental strategies turn hard calculations into easy ones by making friendly numbers (10, 100, 1,000).
- Halving and doubling: halve one factor, double the other — the product stays the same.
- Distributive property: a × (b + c) = ab + ac and a × (b − c) = ab − ac. Multiply EVERY part.
- Compensation: round to a friendly number, multiply, then adjust (8 × 99 = 800 − 8).
- Commutative and associative properties let you reorder and regroup, e.g. 28 × 25 = 7 × (4 × 25) = 700.
Practice: 15 questions
5 easy, 5 medium, 5 hard. Each one is marked straight away, with a full worked solution.
You can practise without an account, but sign up free or sign in to save your progress.