Free Sample › Cheatsheet

Year 7 Number Cheatsheet

Year 7 · Number · Australian Curriculum v9

What You Need to Know

Index notation (powers) is a shorthand way to show repeated multiplication (e.g. 2³ = 2 × 2 × 2)
HCF (Highest Common Factor) is the largest number that divides two or more numbers exactly
LCM (Least Common Multiple) is the smallest number that is a multiple of two or more numbers
Square numbers and square roots are inverse operations (√25 = 5 because 5² = 25)
The associative law shows that (a + b) + c = a + (b + c) and (a × b) × c = a × (b × c)
The commutative law shows that a + b = b + a and a × b = b × a
The distributive law shows that a(b + c) = ab + ac
Integer operations follow sign rules: same signs give positive, different signs give negative
Ratios compare quantities; rates compare quantities with different units
Percentages can be used to calculate profit and loss in financial contexts

Concept	Rule or Fact
Index Notation	aⁿ = a × a × a × ... (n times); a is the base, n is the exponent
Powers of 10	10¹ = 10, 10² = 100, 10³ = 1000, 10⁴ = 10000
Square Numbers	1² = 1, 2² = 4, 3² = 9, 4² = 16, 5² = 25, 6² = 36, 7² = 49, 8² = 64, 9² = 81, 10² = 100
Square Roots	√4 = 2, √9 = 3, √16 = 4, √25 = 5, √36 = 6, √49 = 7, √64 = 8, √81 = 9, √100 = 10
Integer Addition	Positive + Positive = Positive \| Negative + Negative = Negative \| Different signs: subtract, use sign of larger absolute value
Integer Subtraction	a − b = a + (−b); change the sign of the second number and add
Integer Multiplication/Division	Positive × Positive = Positive \| Negative × Negative = Positive \| Positive × Negative = Negative
Associative Law	(a + b) + c = a + (b + c) and (a × b) × c = a × (b × c)
Commutative Law	a + b = b + a and a × b = b × a
Distributive Law	a(b + c) = ab + ac and a(b − c) = ab − ac
Ratio	Compares two or more quantities (e.g. 3:4 means 3 parts to 4 parts)
Rate	Compares quantities with different units (e.g. 60 km/h, $5 per litre)
Best Buy	Compare unit prices (price per unit) to find the cheapest option
Profit %	Profit % = (Profit ÷ Cost Price) × 100
Loss %	Loss % = (Loss ÷ Cost Price) × 100

Example 1: Index Notation

Write 2 × 2 × 2 × 2 in index notation and calculate the value

Example 2: Finding HCF and LCM

Find the HCF and LCM of 12 and 18

Example 3: Integer Operations

Calculate −5 + 8 − (−3)

Example 4: Ratio and Rate

The ratio of boys to girls in a class is 3:4. If there are 12 girls, how many boys are there?

Ratio: 3:4 means 3 boys for every 4 girls
If 4 girls corresponds to 3 boys, then 12 girls corresponds to 3 × (12 ÷ 4) = 3 × 3 = 9 boys
Answer: 9 boys

Mistake: Reading 3⁴ as 3 × 4 = 12 instead of 3 × 3 × 3 × 3 = 81. Fix: Index notation means repeated multiplication, not regular multiplication.
Mistake: Calculating −5 + 8 as −13 instead of 3. Fix: When signs are different, subtract the absolute values and use the sign of the larger absolute value.
Mistake: Confusing HCF and LCM. Fix: HCF is the highest factor they share; LCM is the lowest multiple they both have. HCF is usually smaller, LCM is usually larger.
Mistake: In the distributive law, forgetting to multiply both terms inside the brackets. For 2(3 + 5), calculating 2 × 3 + 5 = 11 instead of 2 × 3 + 2 × 5 = 16. Fix: Distribute the multiplier to every term inside the brackets.
Mistake: Comparing ratios 3:4 and 6:9 as unequal. Fix: Simplify both ratios: 3:4 stays 3:4, and 6:9 simplifies to 2:3. They are not equal.

Question 1: Calculate 5²

5 × 5 = 25

Question 2: What is √64?

8 (because 8 × 8 = 64)

Question 3: Calculate −7 + (−5)

Both negative: −7 + (−5) = −12

Question 4: Find the HCF of 20 and 30

Factors of 20: 1, 2, 4, 5, 10, 20. Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30. HCF = 10

Question 5: Using the distributive law, expand 4(x + 3)

4(x + 3) = 4x + 4(3) = 4x + 12

Index notation: A shorthand way to express repeated multiplication using a base and exponent
Base: The number being multiplied in index notation
Exponent (Power): The number that tells you how many times to multiply the base
Square number: A number that is the product of a whole number multiplied by itself
Square root: A number that when multiplied by itself gives the original number
Factor: A number that divides exactly into another number with no remainder
Multiple: A number that results from multiplying a number by a whole number
HCF (Highest Common Factor): The largest number that divides two or more numbers exactly
LCM (Least Common Multiple): The smallest number that is a multiple of two or more numbers
Associative law: Grouping numbers differently in addition or multiplication gives the same result
Commutative law: The order of numbers in addition or multiplication does not matter
Distributive law: Multiplying a number by a sum equals the sum of the individual products
Integer: A whole number that can be positive, negative, or zero
Ratio: A comparison of two or more quantities of the same kind
Rate: A comparison of two quantities with different units

Want all 66 cheat sheets across every year level and topic?