National 5 Mathematics: Both Papers

Ten questions spread across both National 5 papers: the surds, indices, quadratics and rearrangement that have to be automatic when there is no calculator, and the trigonometry, sectors, similar shapes and compound percentages that carry the longer problems when there is one.

Every wrong option here is the number a specific mistake produces, and the explanation names it. Similar shapes is the clearest case: an area ratio of 4 gives a linear scale factor of 2, not 4, and the option that uses the area ratio directly on a length is the single most common error on the topic. Compound depreciation is the other: taking 30 per cent off a van in one go is not the same as taking 15 per cent off twice, and it always produces a figure that is too low, because the second year's loss comes off a smaller amount than the first. In the area-of-a-triangle question one option is what you get by forgetting the half, and in the sector question one is the arc length rather than the area.

The statistics question is there for a distinction that costs marks every year: the interquartile range is not the range. One measures the spread of the middle half, the other the distance between the two most extreme values, and both numbers are available from the same list.

Marks at National 5 go to method as well as to answers, so each explanation sets the working out line by line rather than stating the result — a clearly reasoned attempt that ends in the wrong number still scores, and reading how the right one was reached is what makes the next one faster.

Every question here was written for this set. Past papers and marking instructions are published by Qualifications Scotland and are their copyright; nothing is reproduced from them.

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• Simplify surds by finding square factors, and apply the power and product rules for indices in the right order
• Solve a quadratic by factorising, and change the subject of a formula that contains a squared term
• Find the equation of a line from a point and a gradient without mistaking the point for the intercept
• Use the half-a-b-sine-C formula for the area of a triangle, and the sector formula rather than the arc formula
• Take the square root of an area ratio before applying it to a length
• Work out a compound percentage change over several years, and say why treating it as a simple one is always wrong
• Find quartiles from a list and give the interquartile range rather than the range

A van is bought for £18,000. It loses 15 per cent of its value in each of the first two years. What is it worth at the end of the second year? — one of ten questions written for this set, alongside surds, indices, a quadratic, a straight line, the area of a triangle, a sector, similar shapes and the interquartile range.