lesson
Keep both parts of the unit
1 · Learn
A compound unit combines two measurements. Speed in kilometres per hour is distance ÷ time; unit price in pounds per item is cost ÷ number of items. Density in grams per cubic centimetre is mass ÷ volume. The word 'per' means for each one of the second unit.
A uniform model block has mass 240 g and volume 80 cm³. Its density is 240 ÷ 80 = 3 g/cm³. Each cubic centimetre has mass 3 g in this model. A larger piece of the same material can have more mass without having greater density.
Rearrange density = mass/volume: mass = density × volume and volume = mass/density, using positive volume and density here. Convert units before calculating. Since 1 kg = 1,000 g, 0.54 kg is 540 g. Since 1 m³ = 1,000,000 cm³, 1 g/cm³ equals 1,000 kg/m³.
2 · Worked example
A uniform block has mass 0.54 kg and volume 200 cm³. Convert 0.54 kg to 540 g, then divide: 540/200 = 2.7 g/cm³. A 50 cm³ piece of the same material has mass 2.7 × 50 = 135 g. Check that 135/50 returns 2.7.
3 · Your turn
A uniform block has mass 360 g and volume 120 cm³. Find its density. Find the mass of a 40 cm³ piece of the same material. Separately, a journey covers 90 km in 1.5 hours: find its average speed.
Check your answer
Density 3 g/cm³; mass 120 g; average speed 60 km/h. Density and speed divide different quantities, so their units cannot be interchanged.
4 · Apply your learning
Draw a table comparing a 40 cm³ piece and a 120 cm³ piece of this material. Include mass and density. Explain which quantities triple and which stay constant; these are supplied model data, not a request to handle an unknown substance.
Adult guidance and safety
Ask for the reasoning and a check, not only a numerical result. Supply accurate diagrams, coordinate grids and data displays where required. Use varied examples and revisit prerequisite gaps; these short teaching models are not a completed assessment programme.
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Check a calculation · Year 5 onward
Use this only when your teacher or adult allows a calculator for the task. Show your thinking first. It does not replace mental or written arithmetic and does not mark your lesson answer.
Powers, roots and trigonometry · Year 7 onward
Use only when the task allows a calculator. Choose one operation at a time and record your working. Trigonometry here always uses degrees (DEG), not radians. Inverse sine, cosine and tangent return an angle, not a reciprocal. Keep exact fractions and π in answers when requested.
Approximate numerical results use up to 12 significant digits. Inverse sine returns −90° to 90°, inverse cosine 0° to 180°, and inverse tangent −90° to 90°; these are principal values, not every possible solution of a trigonometric equation.
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