Arij Asad Maths

For lessons and independent practice

From textbook skills to admissions problems

Two practical sequences for choosing a starting point, repairing a gap and applying the idea in an unfamiliar problem.

Original teaching resource by Arij Asad

Plan around the reasoning you can see

Ask the student to keep their working, including abandoned attempts. Choose the next task from the first uncertain step: a definition, an algebraic move, a graph or a decision about cases.

The timings below are flexible starting points for a 45-minute session. Shorten or extend each stage to fit the student.

Sequence 1

Algebra to TMUA

Before the lesson: ask the student to try the 20-minute algebra diagnostic. Bring their working, not just their final answers.

  1. Review one decision 5 minutes

    Choose one uncertain answer. Ask: “What did you know at this point?” and “Which condition must still be true?” Note whether the gap is a method, a sign, a domain restriction or the meaning of “and”.

  2. Repair the foundation 10 minutes

    Use the question’s matched topic guide or textbook chapter. Ask the student to explain one worked example, close it, and reproduce the reasoning. Choose a few relevant exercises from their own textbook; use the linked SolutionBank only after an attempt.

  3. Check transfer 10 minutes

    Try the paired follow-up without the model solution. Change one condition verbally and ask what would change in the argument. A remembered answer is less useful than a reason that survives this change.

  4. Return to a mixed problem 15 minutes

    Choose an algebra or graph question from the TMUA archive that fits the student’s current preparation. Hide its topic label. Ask the student to name the clue that suggested their method.

  5. Set the next practice 5 minutes

    Write one specific habit to use next time, then assign a small mixed set. At the next lesson, retry one question from a clean page before looking at the old solution.

Get the question sheets and teacher solutions

Sequence 2

Calculus and graph sketches to STEP

For a student who already knows differentiation and basic integration, use a short problem to connect graph shape, signs and area before moving to a longer STEP question.

  1. Sketch before integrating 10 minutes

    Find the roots and stationary points in the original problem below. Ask the student to predict which parts of the graph contribute positive or negative signed area.

  2. Calculate and interpret 10 minutes

    Calculate the signed integral and total area separately. Have the student explain why the two answers differ, using their sketch and a sentence as well as equations.

  3. Generalise with a parameter 15 minutes

    Try the extension. Predict the power of the parameter in the area before integrating, then check the prediction by substitution or direct calculation.

  4. Explain and transfer 10 minutes

    Write a complete solution with the interval and sign choices stated. Use the guide to writing a strong STEP solution, then choose a suitable calculus question from the STEP archive for further practice.

Original practice

Signed integral or total area?

Let f(x)=x3−3xf(x)=x^3-3x on [−3,3][-\sqrt3,\sqrt3]. Sketch the graph, find its stationary points, evaluate its signed integral over this interval, and find the total area between the graph and the xx-axis.

Show a hint

Factor f(x)f(x), then differentiate. Use the sign of the graph on each side of zero before interpreting the integral.

Show the full solution

Factor f(x)=x(x2−3)f(x)=x(x^2-3), so the roots are −3,0,3-\sqrt3,0,\sqrt3. The derivative is f′(x)=3x2−3f^{\prime}(x)=3x^2-3, giving stationary points (−1,2)(-1,2) and (1,−2)(1,-2). The first is a local maximum and the second a local minimum.

The function is odd, so the signed integral over the symmetric interval is 00. The graph is positive on (−3,0)(-\sqrt3,0) and negative on (0,3)(0,\sqrt3).

The total area is therefore −2∫03(x3−3x) dx=−2[x44−3x22]03=92.-2\int_0^{\sqrt3}(x^3-3x)\,\mathrm{d}x=-2\left[\frac{x^4}{4}-\frac{3x^2}{2}\right]_0^{\sqrt3}=\frac92. The areas add even though the signed integrals cancel.

Extension

Predict the effect of a parameter

Let a>0a>0 and fa(x)=x3−3a2xf_a(x)=x^3-3a^2x. Predict, then calculate, the total area between this graph and the xx-axis on [−3a,3a][-\sqrt3a,\sqrt3a].

Show the full solution

The roots are −3a,0,3a-\sqrt3a,0,\sqrt3a. Setting x=atx=at gives fa(at)=a3(t3−3t)f_a(at)=a^3(t^3-3t) and dx=a dt\mathrm{d}x=a\,\mathrm{d}t. Thus widths scale by aa, heights by a3a^3, and area by a4a^4.

The total area is 9a4/29a^4/2. Directly, −2∫03a(x3−3a2x) dx=−2[x44−3a2x22]03a=9a42.-2\int_0^{\sqrt3a}(x^3-3a^2x)\,\mathrm{d}x=-2\left[\frac{x^4}{4}-\frac{3a^2x^2}{2}\right]_0^{\sqrt3a}=\frac{9a^4}{2}. The signed integral remains zero by odd symmetry.

Keep the practice useful

Finish with a short explanation the student can use again: what the clue was, which method it suggested and how they checked the result. Choose subsequent questions from their work rather than from a fixed score threshold.

The diagnostic and examples here are original resources by Arij Asad. Linked textbooks, SolutionBank materials and past papers belong to their credited publishers and creators. Use the current official specification when choosing admissions-test practice.