Senior Secondary Mathematics | Making Calculus Problems Easier: An IB Mathematics Tutor Explains the Power of “Changing the Representation”

[Senior Secondary Mathematics / Calculus] “I know the formulas, so why do I still not know where to begin when I face an unfamiliar problem?” If your child encounters this situation in senior secondary mathematics, I would suggest that parents do not immediately put it down to carelessness or simply ask them to do more practice papers. Sometimes, what is worth examining is not whether they can carry out the calculations, but how they choose their first step.

Students who have learned calculus may think of differentiation when they encounter a “shortest distance” problem: first write down an expression for the distance, then differentiate it. This is a valid approach. However, before putting pen to paper, it is worth looking at the structure of the diagram. The same problem may have a much simpler solution.

The following example, taken from one of my teaching videos, illustrates this way of thinking: “redraw the diagram without changing the problem.” Parents do not need to understand calculus to follow the main idea.

Imagine a coordinate plane where point A is at [ ], and point C is at [ ]. We need to travel from A to a point B on the x-axis, and then from B to C. Point B can move freely along the x-axis. The question is: where should B be placed so that the total length of the two paths is as short as possible?

Let the coordinates of B be [ ]. Using the distance formula, the total distance is:

For students who have learned calculus, they could now differentiate the expression and determine when the minimum occurs. There is nothing wrong with this method. However, before dealing with two square roots, I would first ask: “Can we redraw this diagram without changing the distances we need to compare?”

Think of the x-axis as a mirror. Reflect point C across it to obtain C′ below the axis. Since B lies on the mirror line, no matter where B moves along the x-axis, the distance from B to C is equal to the distance from B to C′.

Therefore, the original route, AB + BC, can be replaced by AB + BC′, which has exactly the same total length. It may look as though we have simply moved a point below the axis, but the problem has now been reframed: we are looking for the shortest route from A to C′ via B.

The shortest distance between two points is a straight line. When B is the point where the line AC′ intersects the x-axis, A, B and C′ lie on the same straight line. This broken path then has the shortest possible length, and this intersection point does lie within the range in which B is allowed to move.

The key point is not to memorise that “reflection should be used whenever there is a shortest-distance problem.” What matters is being able to explain two things: after the reflection, why does the total length remain unchanged? And does the point B found by straightening the path satisfy the original restrictions?

After redrawing the diagram, the horizontal distance between A and C′ is 10, while the vertical distance is 20. Therefore, the minimum total length of the original two-part route is:

How can we find the shortest route when we must first travel to the x-axis before going to another point? Reflect point C as though it were in a mirror, and the route can be straightened. What first appears to be a difficult calculus problem becomes a question about distance and straight lines.

If we also need to find the position of B, we only need to consider the line AC′. Its equation is [ ]. When it intersects the x-axis, [ ], so [ ]. Therefore, B should be chosen as [ ].

The problem has not suddenly become easier. Rather, we have found a representation that makes the relationships easier to see. Calculus can solve this problem, but geometric reflection makes the reason why this route is shortest immediately visible in the diagram.

If a student can only say, “Reflect C below the axis, then draw a straight line,” after reading the solution, they may have memorised the steps without necessarily understanding the reason. I would suggest asking the following three questions to see what they have understood.

“What remains unchanged after the reflection?” The key is that BC and BC′ have the same length, so we have not quietly replaced the original problem with a different distance problem.

“Why can’t we simply draw a line directly from A to C?” The original problem requires the route to touch the x-axis. The line segment joining A and C does not reach the x-axis, so this condition cannot be ignored.

“What if B is only allowed to move along a short section of the x-axis—can we still use the same method?” We must first check whether the intersection point of AC′ and the x-axis falls within the permitted range. If it does not, we cannot simply take the length of AC′ as an attainable answer; the problem must be reconsidered.

These questions are not intended to make parents replace the teacher or explain the entire method themselves. Instead, they help children make their understanding of “I get it” more concrete. Being able to explain why a method works, and knowing when it cannot be applied directly, is the kind of understanding worth developing.

I would not want students to take away the message that “calculus is bad, but tricks are important.” What is truly worth practising is the habit of observing before calculating: What are the restrictions in the problem? What relationships remain unchanged? Is there another way to draw or represent the situation that makes the problem clearer?

In senior secondary mathematics, computational fluency and flexible method selection can coexist. The next time your child gets stuck on an unfamiliar problem, instead of only asking, “Which step do you not understand?”, parents can also ask: “Why did you choose this method? If you redraw the diagram, might you see another route?”

Written by: Ken Sir

Founder of Mathetactics Education and an IB Mathematics tutor, specialising in IB and A-Level Mathematics. He completed his studies at the University of Hong Kong on a three-year full-tuition scholarship, has authored several senior secondary mathematics reference books, and was invited by Longman to deliver public examination analysis seminars.

Original Article: HKET TOPick https://topick.hket.com/article/4213037

English translation provided by AI.

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