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Core skills · Level 8 of 12

Roots, identities and compound interest

These are the skills everyday life actually runs on: percentages and discounts, interest on a loan, ratios in a recipe, and solving for an unknown. Small slips here grow into big gaps later, so the practice keeps asking why. You check your answers, find the mistakes in worked examples, and see each rule on a number line or a grid.

Practise level 812 topics explained

Rational numbers

Fractions with signs: compare, add, multiply and find reciprocals.

Key idea

Rational numbers are fractions that can be positive or negative. All the fraction rules still work, and the sign rules from integers decide the sign.

How to do it

  1. To compare: give them the same bottom number. The one further right on the number line is greater.
  2. To add with the same bottom: add the tops (with their signs), keep the bottom.
  3. To multiply: multiply tops and bottoms, then use the sign rule.
  4. The reciprocal flips the fraction and keeps the sign.
Practise rational numbers

Linear equations

Equations with x on both sides, brackets and word problems.

Key idea

Keep the balance: do the same thing to both sides until x is alone.

How to do it

  1. Get all the x terms on one side, by taking the smaller one away from both sides.
  2. Get the numbers on the other side.
  3. Divide by the number in front of x.
  4. For word problems: call the unknown x, write the equation, solve, then check.
Practise linear equations

Squares & square roots

Square numbers as real squares, and finding the side again.

Key idea

A square number is the number of tiles in a square. The square root is the length of its side.

How to do it

  1. n² = n × n.
  2. √ asks: which number times itself gives this?
  3. Perfect squares only end in 0, 1, 4, 5, 6 or 9.
  4. 1 + 3 + 5 + … (n odd numbers) = n².
Practise squares & square roots

Cubes & cube roots

Cube numbers as real cubes, and finding the side again.

Key idea

A cube number is the number of small cubes in a big cube. The cube root is the length of its side.

How to do it

  1. n³ = n × n × n.
  2. ∛ asks: which number times itself, three times, gives this?
  3. The volume of a cube = side³.
Practise cubes & cube roots

Negative powers & standard form

Powers below zero, the zero power, and very small numbers.

Key idea

A negative power means “one divided by”: a⁻ⁿ = 1 ÷ aⁿ. Any number (except 0) to the power 0 is 1.

How to do it

  1. a⁻ⁿ = 1 ÷ aⁿ.
  2. a⁰ = 1.
  3. The same laws still work: multiply → add the powers, divide → subtract them.
  4. Very small numbers in standard form have a negative power of 10.
Practise negative powers & standard form

Algebraic identities

(a + b)², (a − b)², (a + b)(a − b) with area pictures, and factorising.

Key idea

(a + b)² is a real square made of four pieces: a², b², and two rectangles of ab. That is why (a + b)² = a² + 2ab + b².

How to do it

  1. (a + b)² = a² + 2ab + b².
  2. (a − b)² = a² − 2ab + b².
  3. (a + b)(a − b) = a² − b².
  4. (x + a)(x + b) = x² + (a + b)x + ab.
Practise algebraic identities

Discount, tax & compound interest

Sales, GST, percentage change and interest on interest.

Key idea

Most money questions are “percent of” questions. Compound interest also earns interest on last year’s interest.

How to do it

  1. Discount = discount % of the marked price. Price paid = marked price − discount.
  2. Tax such as GST is added on: price + tax % of the price.
  3. Percentage change = change ÷ original × 100.
  4. Compound interest: amount = P × (1 + R/100) for each year.
Practise discount, tax & compound interest

Direct & inverse proportion

More means more, or more means less: unitary method problems.

Key idea

In direct proportion both things grow together. In inverse proportion, when one grows the other shrinks, and their product stays the same.

How to do it

  1. Ask: if one goes up, does the other go up (direct) or down (inverse)?
  2. Direct: find the value for 1, then multiply.
  3. Inverse: multiply to find the total work, then divide.
Practise direct & inverse proportion

Mensuration

Circles, trapeziums, cuboids and cylinders.

Key idea

Each shape has its own formula. Perimeter and circumference are lengths; area is squares; volume is cubes.

How to do it

  1. Circle: circumference = 2πr, area = πr² (use π = 22/7).
  2. Trapezium: area = ½ × (sum of the parallel sides) × height.
  3. Cuboid: volume = l × w × h. Surface area = 2(lw + wh + hl).
  4. Cylinder: volume = πr²h.
Practise mensuration

Quadrilaterals & polygons

Angle sums, regular polygons and kinds of quadrilaterals.

Key idea

The inside angles of a polygon with n sides add up to (n − 2) × 180°. The outside angles always add up to 360°.

How to do it

  1. A quadrilateral's angles add up to 360°.
  2. Any polygon: (n − 2) × 180°.
  3. A regular polygon has equal angles: divide the total by n.
  4. The exterior angles of any polygon add up to 360°.
Practise quadrilaterals & polygons

Graphs & coordinates

Find points on a grid and read straight-line graphs.

Key idea

A point on a grid is written (x, y): first how far along, then how far up.

How to do it

  1. Always start at 0, the origin.
  2. The first number (x) is how far ALONG.
  3. The second number (y) is how far UP.
  4. On a line graph, find the value on the bottom axis, go up to the line, then across.
Practise graphs & coordinates

Probability

Chances with dice, coins, spinners and bags of marbles.

Key idea

Probability = the number of ways something can happen ÷ the total number of possible outcomes.

How to do it

  1. Count all the possible outcomes: that is the bottom number.
  2. Count the outcomes you want: that is the top number.
  3. Probability is between 0 (impossible) and 1 (certain).
Practise probability