Basic 7 · Mathematics · Number Operations
Mental Mathematics Strategies and Number Properties
In this topic you will apply mental strategies such as compensation, doubling and halving and breaking numbers apart, use the properties of numbers to calculate quickly, and solve word problems mentally.
By the end of these notes you should be able to:
- Use mental strategies to add, subtract, multiply and divide quickly
- Apply the commutative, associative, distributive and identity properties in calculations
- Solve word problems using mental mathematics
What you already know
- Multiplication and division facts
- Multiplying by powers of ten
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
- B7.1.2.1.2 – Apply mental mathematics strategies and number properties used to perform calculations
Introduction
A trader at Makola market adds prices in her head faster than many people with calculators. She uses mental strategies that you can learn too.
1. Breaking Numbers Apart (Partitioning)
Add mentally
Calculate 456 + 238.
- Break 238 into 200 + 30 + 8.
- 456 + 200 = 656; 656 + 30 = 686; 686 + 8 = 694.
Answer: 694
2. Compensation
Round one number to make the calculation easy, then adjust. To add 397 + 245, add 400 + 245 = 645, then subtract the extra 3 to get 642. To subtract 523 − 198, subtract 200 to get 323, then add back 2 to get 325.
3. Doubling and Halving
Multiply mentally
Calculate 25 × 48.
- Halve 48 to get 24 and double 25 to get 50: 50 × 24.
- Halve again: 100 × 12.
Answer: 1,200
4. Using Multiples of 10
To multiply by 9, multiply by 10 and subtract the number once: 9 × 68 = 680 − 68 = 612. To multiply by 11, multiply by 10 and add the number: 11 × 34 = 340 + 34 = 374. To multiply by 5, multiply by 10 and halve: 5 × 86 = 860 ÷ 2 = 430.
5. Choosing a Strategy
| Strategy | Good for | Example |
|---|---|---|
| partitioning | adding and multiplying | 7 × 46 = 7 × 40 + 7 × 6 |
| compensation | numbers close to a round number | 499 + 136 |
| doubling and halving | multiplying by 5, 25, 50 | 25 × 36 |
| multiples of 10 | multiplying by 9, 11, 99 | 99 × 7 |
New words
- mental mathematics
- Calculating in your head without writing.
- partitioning
- Breaking a number into parts.
- compensation
- Adjusting after rounding to make a calculation easy.
- strategy
- A planned method for solving a problem.
Summary
- Partitioning, compensation, doubling and halving and using multiples of 10 make calculations easier.
- Choose a strategy that suits the numbers.
- Mental strategies save time and help check written answers.
Class activity: Mental Maths Challenge
- The teacher calls out calculations using cedi play notes as prices.
- Learners solve them mentally and explain the strategy used.
- Compare strategies to find the quickest for each problem.
You need: Bottle tops, Ghana cedi play notes, Number cards
Exercises
Answer in your exercise book first, then check your answers.
1. 398 + 150 using compensation is…
- A. 548
- B. 538
- C. 558
- D. 448
Show answer
Answer: 548400 + 150 − 2.2. 25 × 32 =
- A. 800
- B. 750
- C. 850
- D. 700
Show answer
Answer: 80050 × 16 = 100 × 8.3. 9 × 54 = 540 − 54.
- A. True
- B. False
Show answer
Answer: TrueMultiply by 10, subtract once.4. Breaking a number into parts is called ______.
Show answer
Answer: partitioninge.g. 238 = 200 + 30 + 8.5. Calculate 11 × 47 mentally and show your strategy.
Show answer
Answer: 470 + 47 = 517Multiply by 10, add once.6. Use compensation to calculate 1,000 − 387.
Show answer
Answer: 1,000 − 400 = 600, then add back 13 = 613.Round and adjust.
Lesson 2
- Strand
- Number
- Sub-strand
- Number Operations
- Indicator
- B7.1.2.1.2 – Apply mental mathematics strategies and number properties used to perform calculations
Introduction
Some rules are true for all numbers. These rules, called number properties, let us rearrange calculations to make them easier.
1. The Commutative Property
For addition and multiplication, the order of numbers does not change the answer: 8 + 15 = 15 + 8 and 6 × 9 = 9 × 6. This is the commutative property. It does not work for subtraction or division: 10 − 4 ≠ 4 − 10.
2. The Associative Property
For addition and multiplication, the way numbers are grouped does not change the answer: (3 + 7) + 9 = 3 + (7 + 9) and (2 × 5) × 13 = 2 × (5 × 13). Grouping helps mental calculation: 4 × 17 × 25 = (4 × 25) × 17 = 100 × 17 = 1,700.
3. The Distributive Property
Multiplication can be distributed over addition or subtraction: a × (b + c) = a × b + a × c. For example, 6 × 47 = 6 × (40 + 7) = 240 + 42 = 282. Also, 8 × 98 = 8 × (100 − 2) = 800 − 16 = 784.
4. Identity and Zero Properties
| Property | Rule | Example |
|---|---|---|
| additive identity | a + 0 = a | 56 + 0 = 56 |
| multiplicative identity | a × 1 = a | 89 × 1 = 89 |
| zero property | a × 0 = 0 | 1,234 × 0 = 0 |
5. Using Properties Together
Calculate using properties
Calculate 25 × 13 × 4 + 0.
- Commutative: 25 × 4 × 13.
- Associative: (25 × 4) × 13 = 100 × 13 = 1,300.
- Additive identity: 1,300 + 0 = 1,300.
Answer: 1,300
New words
- commutative property
- Changing the order does not change the answer.
- associative property
- Changing the grouping does not change the answer.
- distributive property
- a × (b + c) = a × b + a × c.
- identity
- A number that leaves another unchanged (0 for addition, 1 for multiplication).
Summary
- Addition and multiplication are commutative and associative; subtraction and division are not.
- The distributive property splits multiplication over addition or subtraction.
- 0 is the additive identity and 1 is the multiplicative identity.
Class activity: Property Cards
- Match calculation cards to the property they show.
- Use a property to solve each calculation mentally.
- Create two new examples for each property.
You need: Bottle tops, Ghana cedi play notes, Number cards
Exercises
Answer in your exercise book first, then check your answers.
1. 7 × 12 = 12 × 7 shows the… property.
- A. commutative
- B. associative
- C. distributive
- D. zero
Show answer
Answer: commutativeOrder changed.2. 9 × 102 = 9 × 100 + 9 × 2 uses the… property.
- A. distributive
- B. commutative
- C. identity
- D. associative
Show answer
Answer: distributiveSplitting.3. Subtraction is commutative.
- A. True
- B. False
Show answer
Answer: False10 − 4 ≠ 4 − 10.4. The multiplicative identity is ______.
Show answer
Answer: 1a × 1 = a.5. Use the distributive property to calculate 7 × 99.
Show answer
Answer: 7 × (100 − 1) = 700 − 7 = 693Distributive.6. Calculate 2 × 37 × 50 mentally and name the properties used.
Show answer
Answer: (2 × 50) × 37 = 100 × 37 = 3,700; commutative and associative.Grouping.
Lesson 3
- Strand
- Number
- Sub-strand
- Number Operations
- Indicator
- B7.1.2.1.3 – Apply mental mathematics strategies to solve word problems
Introduction
Every day we solve problems in our heads: How much change will I get? How many bags are needed? Mental strategies make these problems quick and easy.
1. Steps for Word Problems
- Read the problem carefully and identify what is asked.
- Pick out the important numbers.
- Decide the operation: add, subtract, multiply or divide.
- Choose a mental strategy.
- Calculate and check that the answer makes sense.
2. Worked Problems
Change at the shop
Akua buys a bag of rice for GH₵198 and cooking oil for GH₵47. She pays with GH₵300. How much change does she get?
- Total cost: 198 + 47 = 200 + 47 − 2 = 245.
- Change: 300 − 245 = 55.
Answer: GH₵55
Groups for a trip
312 learners are going on an excursion. Each bus carries 52 learners. How many buses are needed?
- 312 ÷ 52: think 52 × 6 = 312 (50 × 6 = 300, 2 × 6 = 12).
Answer: 6 buses
3. Multi-Step Problems
Farm produce
A farmer harvests 48 baskets of tomatoes. He sells each basket for GH₵25 and spends GH₵350 on transport. How much money does he have left?
- Sales: 48 × 25 = 24 × 50 = 12 × 100 = 1,200.
- Money left: 1,200 − 350 = 850.
Answer: GH₵850
4. Checking Answers
Always check that the answer is reasonable. If Akua paid GH₵300 and her change was GH₵155, the total would be only GH₵145, which is less than the price of the rice alone. So the answer would be wrong. Estimating and checking prevent such mistakes.
New words
- word problem
- A mathematics question written as a real-life story.
- operation
- Addition, subtraction, multiplication or division.
- multi-step problem
- A problem that needs more than one calculation.
- change
- Money given back when you pay more than the cost.
Summary
- Read, identify numbers, choose the operation and strategy, then calculate.
- Multi-step problems need more than one calculation.
- Check that answers are reasonable in the real situation.
Class activity: Problem Creators
- Write three word problems about your community using cedi amounts.
- Exchange with another pair and solve mentally.
- Explain your strategies.
You need: Bottle tops, Ghana cedi play notes, Number cards
Exercises
Answer in your exercise book first, then check your answers.
1. Kofi has GH₵100 and spends GH₵63. His change is…
- A. GH₵37
- B. GH₵47
- C. GH₵33
- D. GH₵163
Show answer
Answer: GH₵37100 − 63.2. 15 boxes each hold 24 eggs. How many eggs are there?
- A. 360
- B. 350
- C. 340
- D. 380
Show answer
Answer: 36015 × 24 = 30 × 12.3. A multi-step problem needs more than one calculation.
- A. True
- B. False
Show answer
Answer: TrueSeveral steps.4. Money given back after paying is called ______.
Show answer
Answer: changee.g. GH₵55.5. A trader buys 12 crates of drinks at GH₵95 each. What is the total cost?
Show answer
Answer: GH₵1,140 (12 × 95 = 12 × 100 − 12 × 5 = 1,200 − 60).Compensation.6. A school has 864 learners in 24 classes of equal size. How many learners are in each class? Show a mental strategy.
Show answer
Answer: 36; 24 × 36 = 864 (24 × 30 = 720, 24 × 6 = 144).Division by multiplication.
Homework
- Solve five shopping problems from your home mentally and write the strategies used.