By-Hand Skills Sheet – IGCSE Mathematics 0580 Extended

What you need before you start. A sharp HB pencil, a 30 cm ruler and an eraser. Nothing on this sheet needs a calculator, and the two graph sections are deliberately arithmetic you can do in your head.

Section 1 · Drawing and using a graph (E2.9–E2.11, E3.2)

The graph questions are worth a lot of marks and almost all of them are drawing marks. Every value below can be worked out without a calculator.

1.1 Complete the table of values for y = x² − 3x − 1, then plot the points on the grid and draw a smooth curve through them. [2 + 3]

x	−1	0	1	2	3	4
y	3	−1	−3		−1

Squaring a negative gives a positive: (−1)² = 1, so y = 1 + 3 − 1 = 3. Sign errors here wreck the whole curve.

How the marks are awarded 2 marks for the table, one per correct value. P2 for all six points plotted correctly (P1 if four or five are right), then C1 for a smooth curve through them. That last mark is lost by joining the points with straight lines, and by drawing a flat-bottomed “U” instead of a curve that turns at one point. Plot with a small neat cross, not a fat dot — a dot 2 mm wide is 2 mm of doubt.

Marking points
The two missing values are y = −3 at x = 2 and y = 3 at x = 4.
The curve is symmetrical about x = 1.5, halfway between the two −3 values. Its lowest point is at (1.5, −3.25) — below both of them, which is why a flat bottom is wrong.
Draw the curve in one confident sweep, turning your wrist, not in short strokes. Turn the page if that helps.
1.2: the curve crosses the x-axis at about x = −0.3 and x = 3.3. Those are the solutions of x² − 3x − 1 = 0.
1.2: the line y = 2 meets the curve at about x = −0.8 and x = 3.8. Those solve x² − 3x − 1 = 2, that is x² − 3x − 3 = 0.
1.3: the tangent at x = 3 touches at (3, −1). Reading a triangle off it: 4.5 up for 1.5 across, so the gradient is 4.5 ÷ 1.5 = 3.
(The exact gradient there is 3, so a good tangent gets it exactly. Anything from about 2.5 to 3.5 would be accepted.)

1.2 On the same grid, draw the line y = 2. Use your graph to solve (a) x² − 3x − 1 = 0 and (b) x² − 3x − 3 = 0. [1 + 2 + 2]

For (a) you do not draw anything new — y = 0 is the x-axis. For (b), rearrange until the left-hand side matches the curve you have already drawn, and whatever is left over is the line.

How the marks are awarded B1 for a ruled horizontal line at y = 2. Then 2 marks for each pair of solutions, one per value, read off to 1 decimal place and allowed a tolerance either side. You must give both solutions — a quadratic that crosses twice has two answers, and giving one scores half. Mark the crossing points on the graph; it shows the examiner where the numbers came from.

1.3 By drawing a tangent, estimate the gradient of the curve at the point where x = 3. [3]

A tangent touches at one point and does not cross. Draw it long — a short tangent gives a tiny triangle and a badly inaccurate gradient.

How the marks are awarded B1 for a ruled tangent touching at x = 3. M1 for a gradient triangle with its values read off the axis scales and shown on the graph. A1 for the value, inside the accepted range. Two things lose marks here every year: measuring the triangle in centimetres instead of in graph units, and drawing a chord (a line that cuts the curve twice) instead of a tangent. Note the x and y scales on this grid are the same size per unit, but on many exam grids they are not — always read the numbers off the axes.

Section 2 · Statistical diagrams (E9.5, E9.6)

Two diagrams Cambridge asks you to draw by hand every year, and they are almost pure drawing marks.

2.1 The table shows the revision time and test score for ten students. Plot a scatter diagram, draw a line of best fit, and use it to estimate the score for a student who revises for 5.5 hours. [3 + 1 + 1]

hours, h	1	2	2	3	4	4	5	6	7	8
score, s	22	30	35	41	45	52	55	62	68	75

Work out the mean of each row first — both come out exactly, without a calculator — and plot that point too. A line of best fit should pass through it.

How the marks are awarded P3 for all ten points plotted correctly (P2 for eight or nine, P1 for six or seven). B1 for a single ruled straight line of best fit with roughly as many points above it as below. B1 for the reading, with dashed lines drawn on the graph showing where you took it from — without those lines the reading mark is often not given. Do not force the line through the origin, and do not join the dots.

Marking points
Mean hours = 42 ÷ 10 = 4.2. Mean score = 485 ÷ 10 = 48.5. Plot (4.2, 48.5) and put your ruler on it.
One ruled line, drawn right across the data, with about five points each side. Not a curve, not a freehand line, not several tries.
Reading at h = 5.5 gives about 58. Anything from roughly 55 to 61 is a good line read correctly.
Dashed lines up from 5.5 and across to the score axis. That is the working, and it is worth a mark.
The correlation is strong positive: as revision time goes up, so does the score. If asked to describe it, use both words.
Beware extrapolation: this line would predict a score of 90 for 10 hours, which is beyond the data and beyond what you should claim.

2.2 80 people were timed. Complete the cumulative frequency column, plot the cumulative frequency curve, and use it to estimate the median, the lower quartile and the upper quartile. [1 + 3 + 3]

time t (min)	0 < t ≤ 10	10 < t ≤ 20	20 < t ≤ 30	30 < t ≤ 40	40 < t ≤ 50	50 < t ≤ 60
frequency	6	14	24	20	12	4
cum. freq.	6	20

Plot each cumulative frequency against the upper end of its class — 6 at t = 10, 20 at t = 20, and so on. Plotting at the midpoint is the single commonest error in this question.

How the marks are awarded B1 for the cumulative frequency column. P2 for the six points at the correct upper class boundaries (P1 for four or five) and C1 for a smooth increasing curve through them starting at (0, 0). Then 1 mark each for the median, LQ and UQ, with dashed lines drawn across at cf = 40, 20 and 60. Read at n⁄2, n⁄4 and 3n⁄4 — that is 40, 20 and 60 out of 80. Using n + 1 is a discrete-data rule and is not what this question wants.

Marking points
Cumulative frequencies: 6, 20, 44, 64, 76, 80. The last one must equal the total, 80. If it does not, you have added up wrongly — check before plotting.
Plotted at t = 10, 20, 30, 40, 50, 60, and the curve starts at (0, 0) because nobody took a negative time.
A cumulative frequency curve never goes down. If yours does, a value is wrong.
Median at cf = 40 → about 28 minutes. LQ at cf = 20 → 20 minutes. UQ at cf = 60 → about 38 minutes.
Interquartile range = 38 − 20 = 18 minutes. It measures spread and, unlike the range, it ignores the extremes.
To answer “how many took more than 45 minutes”, read the cf at 45 (about 70) and subtract from 80 → about 10. Always check whether the question wants the count below or above.

How examiners award the marks on this sheet

Read this before you start, and again before you hand the paper in.

Rule every straight line. Axes, lines of best fit, tangents. Freehand straight lines lose accuracy marks even when the number you read off happens to be right.
Plot with a small cross, not a blob. A fat dot is several millimetres of uncertainty and examiners have a tolerance measured in half-squares.
Curves are smooth and drawn in one go. Never join plotted points with straight segments, and never flatten the bottom of a parabola. If the curve looks hairy, erase it and draw it again in one movement.
Show where you read a value from. Dashed lines across to the axis on a cumulative frequency curve, a gradient triangle on a tangent, dashed lines on a scatter graph. These carry marks of their own and cost you ten seconds.
Let the mark allocation tell you how much to draw. A 5-mark question wants five separate visible things. Count what is on the page against the marks before you move on.

Cambridge IGCSE Mathematics 0580 Extended · by-hand skills · 5 exercises · E2.9–E2.11 and E3.2 graphs, E9.5 and E9.6 statistical diagrams · Print at 100% on A4 and complete in pencil.