This is not a fancy lesson by any standards. But it reflects a typical lesson of Higher maths. The lesson lasts for 50 minutes but by the time the pupils arrive we usually get around 45 minutes class time.
As we are nearing exam time it is important for pupils to understand how long a question should take them so when I set a starter question I try to give an idea of how long the question should take.
I set the class this starter question:
This is a 7 mark question and should take approximately 9 minutes.
Most of the class managed to complete the question in the given time. However, a few of the class fell into the trap – when working out the stationary points there are two solutions. Only one of these stationary points lies within the given interval. The pupils that didn’t notice had double the work to do as they had to find the corresponding y-coordinate and find the nature. It was an excellent talking point about really reading a question fully, including all the little details that may end up being crucial.
This is the learning outcome for our current topic is to be able to determine the optimal solution for a given problem.
I was planning to give the pupils this example and work through it with them on the board then give the pupils some more questions to practice.
A rectangle has breadth x cm.
It has a perimeter of 16cm.
a) Find an expression for the height of the rectangle in terms of x
b) Find an expression for the area of the rectangle in terms of x
c) Calculate the maximum area.
Then I had a change of heart. I considered the problem and realised that the pupils had all the skills needed to solve the problem – they just didn’t know it.
So I challenged each group to answer it – with nudging from me if required.
At first most of the pupils just stared at the problem saying they didn’t know where to start. Then the penny dropped – the question involves area and perimeter of a rectangle. Simple. Again, lots of interesting discussion points arose. One group kept querying how a rectangle could have stationary points. This led to a discussion about what an expression for area A(x) actually means and how we could plot this on a graph.
Granted this was a relatively simple optimization problem to start with but I was really pleased to see pupils using their prior knowledge to solve a problem without needing me to be standing at the front of the room showing them step by step what to do. I like lessons where I don’t do most of the talking.