From Numbers to Notions: Why Constants Are Hard at A‑Level

Check out this modulus question below:

and compare it to this one:

Why is the first image a lot easier to digest than the second?

Constants feel ambiguous

Students are used to numbers like 3 or −5 doing very concrete jobs. Letters, however, are mentally categorised as “variables”, so when something like a or b is described as a constant, it feels contradictory.

In our example:

Given that a and b are positive constants…

Many students instinctively ask:

• “Can I choose values for them?”

• “Are they changing?”

• “Why don’t they just tell me what they are?”

This uncertainty often leads to hesitation or incorrect assumptions.

Constants break overgeneralized rules

Students often learn transformations as templates:

• “Inside brackets affects the x-direction”

• “Outside brackets affects the y-direction”

But when constants appear inside absolute values, students struggle to interpret what they actually do.

For our equation y= mod(2x-a) many students incorrectly treat this as:

• a vertical stretch by 2 and

• a shift left by a

They miss the fact that the factor of 2 changes how far the graph shifts, meaning the vertex occurs at x=a/2. This is conceptually demanding and easy to misread.

Constants hide structure

With numbers, students can “check” their work visually or numerically. With constants, that safety net disappears.

In the question provided, students must:

• identify the vertex algebraically

• find intercepts symbolically

• reason about relative position, not numerical location

This shifts responsibility from calculation to interpretation, which many students are not yet fluent in.

Absolute value compounds difficulty

Absolute value graphs already require piecewise thinking. When combined with constants, students must coordinate:

• symmetry

• gradients

• intercepts

• domain splitting

For example, to find where y= mod(2x-a)+b meets the x‑axis, students must reason that mod(2x-a)+b=0 has no solution if b>0.

This feels unintuitive because students are so conditioned to “solve equations” that the idea of no intersection due to a constant shift feels abstract.

So what strategies can students take to overcome this constants challenge?

Treat constants as “locked numbers”

A powerful mental model is to describe constants as:

“Numbers you’re not allowed to choose, but which don’t change.”

I would encourage you to:

• circle constants in a question

• say what would happen if they were large or small

• discuss direction and position without needing exact values

Use 'think-aloud' transformation chains

Instead of jumping straight to sketching, model a verbal chain:

• Start with y= mod x

• Stretch vertically by scale factor 2

• Translate by a/2 to the right

• Translate by b upwards

Slowing down this reasoning helps students disentangle constants from procedures.

Emphasize conditions not answers

Questions like:

• “Will this graph cross the x‑axis?”

• “Is that always true for any a>0?”

push students toward general reasoning, which is exactly what constants demand.

This is especially effective with absolute value and logarithmic graphs.

Try strategic substitution

Before sketching, try to:

• temporarily try a=4, b=1

• sketch the graph

• then generalize what changed

This bridges concrete understanding and abstract reasoning without undermining the mathematics.

Normalise uncertainty

Finally, be certain that:

• struggling with constants is normal

• discomfort often signals conceptual growth

• A‑level maths expects reasoning beyond numbers

Conclusion:

Constants are not difficult because they are complicated — they are difficult because they force students to think relationally rather than procedurally. Questions like the one I have shared are effective precisely because they expose whether students truly understand structure, not just technique.