Integers, squares and powers, fractions, decimals and percentages, place value, rounding and estimating, ratios, rates and modelling (with Year 8 rates, percentage change and money models). Compare and calculate with integers; squares, roots, prime factors, HCF and LCM; equivalent fractions and conversions; the four operations with fractions and percentages; then negative fractions and decimals.
Numbers below zero: which is less, and how adding and subtracting move along the line.
On a number line, the number further left is less. So −7 is less than −2, even though 7 is more than 2.
Start at the first number and move along the line. Adding a negative moves left; subtracting moves the opposite way to adding, so subtracting a negative moves right.
a × b is a jumps of b from zero. A negative number of jumps goes the opposite way, so a negative times a negative is positive. Division undoes it with the same signs.
Square numbers and square roots, exponent notation, prime factor trees, and the highest common factor and lowest common multiple.
A square number is the area of a square: 6² = 6 × 6 = 36. The square root undoes it: √36 = 6, the side of a square with area 36.
43 is not a perfect square, but it sits between 36 and 49, so √43 is between 6 and 7. Closer to 49 or to 36? Square a decimal to find out: 6.6² ≈ 43.6.
2 × 2 × 2 × 2 is written 2⁴: the base 2 is multiplied by itself, and the exponent 4 counts how many times. 2⁴ = 16, not 8.
Split a number into factor pairs until every branch is a prime: 36 = 2 × 2 × 3 × 3 = 2² × 3². Every tree for 36 ends with the same primes.
Put two prime factorisations side by side. The primes they share multiply to the highest common factor; all the primes (shared ones once) multiply to the lowest common multiple.
Equivalent fractions, fractions as decimals and percentages, their places on the number line, the four operations with fractions and percentages, choosing a strategy, and negative fractions and decimals.
Fractions are equal when they shade the same amount. Multiply or divide the top and the bottom by the same number: 2/3 = 4/6 = 8/12. Dividing by the highest common factor gives the simplest form.
One amount, three ways to write it. A percentage is out of 100, so 3/4 = 75/100 = 75% = 0.75. Divide the top by the bottom for a decimal; put a percentage over 100 and simplify for a fraction.
Split each unit into equal parts and count from zero. A negative fraction or decimal counts to the left, so −1.5 sits halfway between −2 and −1.
Parts can be counted together only when they are the same size. Cut both fractions into the same parts first: 1/3 + 1/4 = 4/12 + 3/12 = 7/12. The lowest common multiple of the bottoms is the smallest choice.
a/b × c/d is c/d of a/b: multiply the tops and multiply the bottoms. Multiplying by a fraction less than 1 makes a number smaller. Dividing asks how many fit: 3 ÷ 1/4 = 12.
15% of $80 is 0.15 × 80 = $12. Take it off for a discount ($68, or 80 × 0.85 in one step), add it on for an increase ($92, or 80 × 1.15). And 18 out of 24 is 18 ÷ 24 = 75%.
The same calculation can be quick or slow. Change the order (4 × 2.7 × 25 = 100 × 2.7), change the form (0.75 × 36 = 3/4 of 36), or split a number (12 × 2.5 = 24 + 6). Then check the answer is about right.
The sign rules for integers work for every fraction and decimal: work out the sign, then the size. Choose an efficient order: multiplying can be done in any order (25 × 4 first), and a hard product can be split (6 × 2.5 = 6 × 2 + 6 × 0.5).
Place value columns as powers of 10 and expanded notation, rounding decimals to a given accuracy, and estimating to check an answer.
Each place value column is a power of 10: the thousands column is 10³, the ones column is 10⁰ = 1. So 4052 = 4 × 10³ + 5 × 10¹ + 2 × 10⁰.
3.746 to 2 decimal places is 3.75: look at the next digit, 6, and round up. How much to round depends on what the number is for.
Round each number to its first digit, then calculate: 38 × 52 ≈ 40 × 50 = 2000. If the calculator says 197.6 or 19 760, something slipped.
Comparing quantities as a ratio, equivalent and simplest ratios, and sharing a total in a ratio.
2 red counters and 3 blue: the ratio of red to blue is 2 : 3 (part to part); red to all the counters is 2 : 5 (part to whole). Order matters.
2 : 3 = 4 : 6 = 6 : 9: multiply or divide both numbers by the same number. 12 : 18 simplifies to 2 : 3 by dividing both by 6.
Share $60 in the ratio 2 : 3. Add the parts (5), find one part ($60 ÷ 5 = $12), then multiply: $24 and $36.
Comparing two quantities with different units (km per hour, dollars per kilogram), converting rates, and using them to solve problems.
A rate compares two quantities with different units: 60 km per hour, $4 per kg, 80 beats per minute. The unit rate is how much for 1 of the second quantity: divide the first by the second. A ratio compares quantities in the same units. Speeds convert with 1 km/h = 1000 m ÷ 3600 s.
Use a rate to find a total (rate × amount), or the amount (total ÷ rate). For speed: distance = speed × time and time = distance ÷ speed. Show the units at every step, and check the answer makes sense.
Using fractions, percentages and ratios to model money and measurement problems, and justifying the choices.
Discounts, markups and profit: formulate the problem, choose fractions, percentages or ratios, solve, and say what the answer means in dollars.
Mixes, scale drawings and models: say what is known and unknown, choose the ratio, solve, and answer in the units of the situation.
An increase of p% multiplies by (1 + p/100); a decrease multiplies by (1 − p/100). One change after another means multiplying the multipliers, so up 20% then down 20% is × 1.2 × 0.8 = × 0.96: a 4% fall overall, not back to the start.
Model a money problem by choosing what to calculate, then check the answer makes sense. Pay is the rate times the hours. Income tax is worked out in brackets: each part of the income is taxed at its own rate, so for incomes above $18 200 the rate on the last dollar (the marginal rate) is higher than the tax as a share of all the income (the effective rate).
Model a practical problem: say what you need, decide whether it is a ratio (parts of the same kind, like cordial to water) or a rate (two different units, like m² per litre), set it up, solve, then interpret the answer in the situation, such as rounding up to whole tins.
Every model makes assumptions: a steady speed with no stops, a fare with no starting charge, an exchange rate with no fees. When a model's answer and real life differ, look for the assumption that does not hold, then refine the model, for example by adding a fixed starting amount.
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