Where to Land

Why do this problem?

This problem requires students to apply Pythagoras’ Theorem in context. They will derive an expression to maximise rather than an equation to solve. There are opportunities for trial and error or drawing and interpreting graphs, and/or using graphing or spreadsheet software. Very advanced students might approach the problem using calculus.

The ideas in this problem can also be related to the bending of light rays as they meet a different substance.

Possible approach

You might start by displaying the graphic in the question and introducing the problem using the edge cases of minimum and/or maximum swimming. Students could then spend some time trying particular landing points. Some students may need to be reminded of the speed, distance, time equation and the relevant rearrangements. Combining the class’s results might be sufficient to find the best integer value.

You could compare methods all together at this point and generalise to find a ‘total time’ formula. You could use the formula to use a trial and improvement method like interval splitting, or spreadsheet software like Excel.

Alternatively, you could ask the students to graph their formula, either by hand or using graphing software they are familiar with. By hand, they could begin by plotting the integer points that they have already found. How do we interpret the graph to find our final solution? There is an opportunity here to discuss the level of accuracy we might expect from hand drawn graphs.

Key questions

How long does it take if Chris swims straight to the river bank?

If you choose a landing point, can you find how long the journey will take?

Can we use algebra to represent any landing point?

Is there a way to visualise lots of different possibilities for x all at once?

Possible support

Garden Shed offers practice using Pythagoras’ Theorem in a different context. A structured spreadsheet could be made that allowed students to enter distances and see a ‘total time’ output. This would shift the focus of the task toward effective use of trial and error.

Possible extension

What happens if you change the speeds Chris can swim and run?

Capable students might use calculus to solve the problem. Ladder and Cube also uses Pythagoras’ Theorem and could include elements of numerical analysis.

For a similar problem involving a circular swimming pool see To Swim or to Run?