Linear law in Additional Math 0606: common mistakes and examiner advice

Linear law questions ask you to turn a non-linear relationship into a straight line, plot it, and use the gradient and intercept. The reports show that students know the idea. The marks go missing in the transformation, in which values are plotted, and in finishing the question. This guide collects what Cambridge examiners say students get wrong, based on recent reports.

Summarised in our own words from a reading of the examiner reports.

Mistakes that cost marks

• Plotting the wrong values. A common source of error in all methods was to assume the coordinates were values of x and y, not x² and lg y. Others did not transform the data at all and plotted the given points.
• Mixing up y and lg y. Writing "y =" instead of "ln y =" when giving the line, and, after stating Y = 1.4X + 2.2, stopping without replacing Y with ln y. A common error was to use 2 rather than 2².
• Drawing the line. Joining plotted points with a freehand line, with a series of straight lines from one point to the next, or with a curve, rather than a single straight line. Others drew lines that passed through the origin, or plotted values of y/10 or used base 10 logarithms in a way that did not suit the scaled axes.
• The gradient. Some realised they needed the gradient but did not know what it represented. Others reversed the x and y coordinates, used the given data instead of the transformed data to find the gradient, used points not on the line (such as (0.8, 3.4)), or used the midpoint of the given coordinates. Reports note that most candidates found the gradient correctly.
• The intercept. Slips when finding the y-intercept, and incorrect substitution when finding c (using 2.25² or 4.75² rather than 2.25 or 4.47, for example).
• Matching to the constants. Confusing A and b, incorrect matching of the gradient to b, and incorrect use of lg A.
• Not finishing the rearrangement. Leaving the answer as y = (1/3) e^(−1.5x² + 19.2) in one part, when the rearranged form was asked for, and incorrectly writing e^(−1.5x² + 19.2) as e^(−1.5x²) + e^19.2. In another, a final answer such as y = e^(−2x³) + e^15.5 came from using an incorrect operation in the last step.
• Solving for x. Square-rooting instead of squaring in the last step, cubing each term as a final step, omitting the negative root, and truncating values (±3.24) instead of rounding to 3 significant figures. Many made algebraic errors when making x² the subject.
• Ignoring the graph. Many did not realise they could use their straight-line graph, and tried to use values of A and b from an earlier part, which were usually incorrect.
• Reading the question. The equation of the line of best fit was often omitted, an answer was written as an inequality such as 4.6 ≤ T ≤ 7 instead of a number of units of time, and in another question the two critical values (±2.5) were reduced to 2.5.
• Accuracy. Giving answers to less accuracy than required, and using ln rather than lg. Reports say the latter still earned method marks in later parts.

What examiners advise

• Write the transformed equation in the form Y = mX + c, and say what X and Y are in terms of the original variables.
• Plot the transformed data, then draw a single straight line through the points.
• Use points on your line to find the gradient, and say what the gradient and intercept represent.
• Use the line when asked to, and give the equation of the line of best fit.
• Complete the question: replace Y and X with the original variables, and give the answer in the form asked.
• Give answers to the accuracy required, or better, and round only at the end.

Habits that help

• Write a small table of X and Y values next to the question before you plot anything.
• Check your finished line with one more given point, to see if it fits.
• Before you give a final answer, re-read the question to see what form it asks for.

Questions students ask

What do I plot in a linear law question?

The transformed data, such as x² and lg y, not the given values. A common error in all methods was to assume the coordinates were x and y. Say what X and Y are in terms of the original variables.

How do I find the gradient and intercept?

Use two points on your drawn line, not the given data, and say what each represents. Students reversed the x and y coordinates or used given data instead of transformed data.

How should I draw the line?

As a single straight line through the points. Freehand lines, a series of straight lines joining points, or curves lose marks.

How do I write the final equation?

Replace Y and X with the original variables. Students wrote "y =" instead of "ln y =", stopped at Y = 1.4X + 2.2, or wrote e^(−1.5x² + 19.2) as e^(−1.5x²) + e^19.2.

Can I use my graph to answer a later part?

Yes, when asked to. Many did not realise they could use their straight-line graph and tried to use values of A and b from an earlier part, which were usually incorrect.