Teacher guide: Area, volume, circles and Pythagoras

The idea in plain words

Area counts unit squares: cm², m². A rectangle is rows of squares, so its area is length × width.

A parallelogram is a rectangle pushed over. Cut the triangle off one end along the perpendicular height and move it to the other end: it becomes a rectangle with the same base and height. So $A = bh$. Two copies of a triangle make a parallelogram with the same base and height, so a triangle is half: $A = \tfrac{1}{2}bh$. Sliding the top along a line parallel to the base changes the shape but not the area.

Real problems need three moves: choose the base and the height at a right angle to it (ignore slanted sides), convert every length to one unit, and split a shape into rectangles and triangles (or subtract a missing part).

Why it matters

Paint, carpet, tiles, land and fabric are bought by area. The same "base times height" idea gives volume next, and the area of a circle and trapezium in Year 8.

What students get wrong

• Using the slanted side as the height. Base 7, slanted side 5, height 4 worked out as 7 × 5. Find the ∟.
• Forgetting the half for a triangle. ½ × 8 × 5 given as 40. Two triangles make that parallelogram.
• Mixing units. 50 cm × 2 m = 100. Change to 0.5 m × 2 m = 1 m².
• Writing cm for an area. Area counts squares: cm².

Year 8: composite and irregular shapes

A composite shape is cut into pieces you know (or seen as a rectangle with pieces taken away); each piece uses its own lengths, and a missing side is the difference of two others. An irregular shape is estimated on a grid: full squares, plus half the part squares, times one square's area; a finer grid is closer. A curved perimeter is estimated with straight pieces (always a little short). The common slips: giving a piece the whole shape's height, and counting part squares as whole.

The words

base, perpendicular height (at a right angle to the base), parallelogram, composite shape (made of simpler shapes), square units (cm², m²; 1 m² = 10 000 cm²), irregular shape, estimate.

The idea in plain words

Volume counts unit cubes: cm³, m³. A box fills up in layers. Each layer covers the base, so it holds as many cubes as the base has squares; the height counts the layers. So $V = A \times h$, and for a box $V = l \times w \times h$.

Every right prism works the same way: it has the same cross-section all the way along, like a loaf of bread. Volume = area of the cross-section × length. A triangular prism's cross-section is a triangle, so halve first.

Capacity is how much a container holds: 1 cm³ holds 1 mL, and 1000 cm³ = 1 L.

Why it matters

Water tanks, concrete, packaging and medicine doses are volumes and capacities. The cross-section idea carries straight on to cylinders in Year 9.

What students get wrong

• Counting only the outside. A 4 by 3 by 2 box counted as 26 visible squares. It is full: 2 layers of 12.
• Forgetting the half for a triangular prism. Tent end 2 m by 1.5 m worked out as 3 m². It is a triangle: 1.5 m².
• Calling the face it stands on the cross-section. A tent stands on a rectangle, but its slices are triangles.
• Dividing by 100 for litres. 1000 mL = 1 L.

Year 8: capacity and prism problems

1 cm³ holds 1 mL, 1000 cm³ holds 1 L and 1 m³ holds 1000 L; a cubic metre is 100 × 100 × 100 cm³, not 100. For any right prism V = A × h (the end's area times the length), and a missing length is V ÷ A. The common slips: converting a cubic unit with the length factor once, and using length × width × height on a triangular prism.

The words

volume, cubic units (cm³, m³), prism, cross-section, capacity, millilitre (mL), litre (L), kilolitre (kL), right prism.

The idea in plain words

A circle is all the points the same distance from its centre: the radius $r$. The diameter goes right across through the centre, so $d = 2r$. A chord joins any two points (the diameter is the longest); the circumference $C$ is the distance all the way round.

Roll any circle out flat for one turn and the track is just over 3 diameters. For every circle, $C \div d$ is the same number, π (pi) $= 3.14159\ldots$, whose digits never end or settle into a repeating pattern. So $C = \pi d = 2\pi r$, and backwards $d = C \div \pi$. Use $\pi \approx 3.14$ or $\tfrac{22}{7}$, or keep π for an exact answer ($10\pi$ cm).

Why it matters

Wheels, tracks, pipes and anything round. π comes back in the area of a circle (Year 8), cylinders (Year 9) and well beyond.

What students get wrong

• Radius for diameter. A wheel of radius 30 cm given C = π × 30. Double first: C = π × 60.
• π as exactly 3.14 or 22/7. Both are close; π never ends.
• Multiplying by π to go backwards. The diameter is smaller than the circumference: divide.
• A bigger π for a bigger circle. C and d grow together, so C ÷ d stays π.

Year 8: the area of a circle

Slices of a circle laid top to tail make a near-rectangle πr long and r high, so A = πr²; it is between 2r² and 4r². To find the radius from an area, divide by π and take the square root. Parts of circles (semicircles, quarters, rings, tracks) are fractions of circles added to or taken from other shapes. The common slips: 2πr for the area, and stopping at r².

The words

centre, radius, diameter, chord, circumference, π (pi), sector, semicircle.

The idea in plain words

In a right-angled triangle, the side opposite the right angle is the hypotenuse $c$, the longest side. Draw a square on each side: the two smaller squares together have exactly the area of the biggest. That is Pythagoras' theorem, $a^2 + b^2 = c^2$.

To find the hypotenuse, square the two shorter sides, add, and take the square root. To find a shorter side, subtract: $b^2 = c^2 - a^2$. Check that the hypotenuse is still the longest.

It also works backwards as a test. With $c$ the longest side: $a^2 + b^2 = c^2$ means right-angled; smaller means obtuse; bigger means acute. Whole numbers that fit, like 3-4-5 and 5-12-13, are Pythagorean triples.

Why it matters

Ladders, ramps, screens, diagonal paths and checking a corner is square all use it. It leads to distance on a graph, trigonometry and 3D measurement.

What students get wrong

• Adding the sides. 6 + 8 = 14 for the hypotenuse. The squares add: 36 + 64 = 100, c = 10.
• Stopping at c². 25 is an area; the side is √25 = 5.
• Adding for a shorter side. That gives a side longer than the hypotenuse. Subtract.
• Testing with the wrong side as c. c must be the longest side.

The words

hypotenuse, right-angled triangle, Pythagoras' theorem, Pythagorean triple.

The idea in plain words

Every place keeps its own clock, set by its time zone: an offset from UTC (Coordinated Universal Time). Sydney is UTC+10, or UTC+11 during daylight saving; London is UTC+0, or +1 in its summer.

A city with a bigger offset is ahead (its clock is later); a smaller offset is behind. To change a time, find the difference between the offsets, add or take it away, and change the day when you cross midnight. For a flight, add the flight time to the departure time (in the departure zone), then change to the arrival zone. 24-hour time avoids mixing up am and pm.

Why it matters

Calls to family overseas, online meetings with other states, flights and sports broadcasts all cross time zones. It is also a good workout in adding and taking away times past midnight.

What students get wrong

• Adding for a place that is behind. A smaller offset means an earlier time.
• The right hour, the wrong day. Past 24:00 is the next day; below 00:00 is the day before.
• Adding the flight time but not changing zones. The answer is still on the departure clock; change it to the arrival zone.
• Daylight saving. Offsets change through the year, and not every state uses it: check before planning.

The words

time zone, UTC, offset, ahead, behind, 24-hour time, duration, daylight saving, International Date Line.