Integers and Powers
Section: Number | Syllabus: Cambridge Primary Mathematics (0845)
Estimating, Adding and Subtracting Integers
- An integer is a positive or negative whole number, or zero.
- Examples: ..., −3, −2, −1, 0, 1, 2, 3, ...
- Integers do not include fractions or decimals.
- A number line helps us understand how integers are added and subtracted.
Adding and Subtracting on a Number Line
- Adding a positive number means moving right.
- Subtracting a positive number means moving left.
- Adding a negative number means moving left.
- Subtracting a negative number means moving right.
For example:
- −3 + 15 means start at −3 and move 15 places right.
−3 + 15 = 12: adding 15 moves 15 places to the right.
Subtracting a Negative Number
Subtracting a negative number moves you towards the right on the number line.
- −4 − (−9)
- Subtracting −9 is the same as adding 9.
- −4 − (−9) = −4 + 9 = 5
−4 − (−9) = 5: subtracting negative 9 moves 9 places to the right.
So remember:
Subtracting a negative = adding the positive
Worked Example: Adding and Subtracting Integers
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Question: Work out −8 + 15.
- Step 1: Start at −8 on the number line.
- Step 2: Adding 15 means move 15 places right.
- Step 3: You land on 7.
- Answer: −8 + 15 = 7
-
Question: Work out 6 − 11.
- Step 1: Start at 6.
- Step 2: Subtracting 11 means move 11 places left.
- Step 3: You land on −5.
- Answer: 6 − 11 = −5
Estimating Integer Calculations
- An estimate gives an answer that is close to the exact answer.
- To estimate, round each number to a nearby value that is easy to work with - often the nearest 10, 100 or 1000, depending on the size of the numbers involved.
- Estimate first to check whether an exact answer is reasonable.
- For example, −48 + 21 is close to −50 + 20 = −30.
- The exact answer is −27, which is close to the estimate.
Worked Example: Estimating Integer Calculations
Question: Estimate the answer to −79 + 32.Estimate the answer to −79 + 32.
Common Mistakes
MistakeThinking that subtracting a negative means moving left
FixSubtracting a negative means moving right: 6 − (−3) = 6 + 3 = 9.
MistakeForgetting the negative sign in an answer
FixUse a number line to check which side of zero your answer should be on.
MistakeTreating an estimate as the exact answer
FixAn estimate is only a close value. Give the exact answer when the question asks for it.
Order of Operations and Brackets
- Brackets are worked out first.
- Then multiplication and division, working from left to right.
- Then addition and subtraction, working from left to right.
Brackets → Multiplication/Division → Addition/Subtraction
Why Does the Order Matter?
Consider:
3 + 4 × 2
- Multiplication comes before addition.
- 4 × 2 = 8
- 3 + 8 = 11
Now add brackets:
(3 + 4) × 2
- The brackets tell us to calculate 3 + 4 first.
- 3 + 4 = 7
- 7 × 2 = 14
The brackets change which operation is performed first, so they can change the answer.
Multiplication and Division: Left to Right
When a calculation has only multiplication and division, work through it from left to right - doing the operations in the wrong order can change the answer.
20 ÷ 4 × 5
- Working left to right: 20 ÷ 4 = 5, then 5 × 5 = 25
- Doing the multiplication first by mistake gives 4 × 5 = 20, then 20 ÷ 20 = 1 - the wrong answer
Using the Laws of Arithmetic
- The commutative law means that changing the order of numbers in addition or multiplication does not change the answer.
- 3 + 7 = 7 + 3
- 4 × 5 = 5 × 4
- The associative law means that changing the grouping of numbers in addition or multiplication does not change the answer.
- (2 + 3) + 4 = 2 + (3 + 4)
- (2 × 3) × 4 = 2 × (3 × 4)
These laws can help you rearrange or group numbers to make calculations easier.
Worked Example: Simplifying a Calculation
-
Question: Work out 6 + 3 × 4.
- Step 1: Multiplication comes first: 3 × 4 = 12.
- Step 2: Add 6: 6 + 12 = 18.
- Answer: 18
-
Question: Work out (6 + 3) × 4.
- Step 1: Work out the brackets: 6 + 3 = 9.
- Step 2: Multiply: 9 × 4 = 36.
- Answer: 36
Common Mistakes
MistakeCalculating from left to right without considering the order of operations
FixMultiplication and division come before addition and subtraction.
MistakeIgnoring brackets
FixBrackets tell you which calculation must be done first.
MistakeApplying the commutative law to subtraction or division
Fix8 − 3 is not the same as 3 − 8, and 12 ÷ 4 is not the same as 4 ÷ 12.
Estimating and Multiplying Whole Numbers
- Before multiplying, an estimate can help you predict the size of the answer.
- Whole numbers up to 10 000 can be multiplied by 1-digit or 2-digit whole numbers.
- Rounding a number to a convenient value makes estimation easier.
Worked Example: Estimating a Multiplication
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Question: Estimate 4892 × 7, then find the exact answer.
- Step 1: Round 4892 to 5000.
- Step 2: Calculate 5000 × 7 = 35 000.
- Estimate: approximately 35 000.
Exact Calculation
Now calculate the exact answer:
4892 × 7 = 34 244
The exact answer, 34 244, is reasonably close to the estimate of 35 000, so the answer is sensible.
Multiplying by a Two-Digit Number
When multiplying by a two-digit number, split the number into tens and ones.
36 × 24 = (36 × 20) + (36 × 4)
- 36 × 20 = 720
- 36 × 4 = 144
- 720 + 144 = 864
This works because 24 = 20 + 4.
The whole rectangle is 36 × 24 - splitting it into 36 × 20 and 36 × 4 shows why the two parts add up to the same total
Common Mistakes
MistakeTreating an estimate as the exact answer
FixKeep the estimate and exact answer separate.
MistakeForgetting that a two-digit number contains tens and ones
FixSplit 24 into 20 + 4 before multiplying if this makes the calculation easier.
Estimating and Dividing Whole Numbers
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