Trial and Improvement

In this post I want to address a common misconception that comes up when teaching trial and improvement. Here's an example which illustrates the problem:

These trials tell us that the solution is between 2.1 and 2.2. The question requires an answer to one decimal place, so our final solution will be either 2.1 or 2.2.

At this point it looks like our solution will be 2.1, because x = 2.1 gives us the closest answer to five. But this is a cubic function, not a linear function, so this is an incorrect assumption (and a common misconception amongst students).

We must now check the midpoint of 2.1 and 2.2:

Let’s draw our last three trials on a number line:

We can see that the solution must lie between 2.15 and 2.2.  All values between 2.15 and 2.2 round to 2.2, therefore our final answer must be 2.2. 

You can see here that it was absolutely necessary to check x = 2.15. And not just because you'd lose a mark in a GCSE for failing to check it! You could plot the function, as shown below (use Desmos - it's fantastic!) and zoom in on the solution to help your pupils understand this point.

I got this example from a discussion on TES.  There is a similar example in this blog post.

Resources
While I'm on the subject of trial and improvement, let's take a quick look at a few teaching resources:

• Teachitmaths has a great PowerPoint explaining the method.
• I particularly like the way this worksheet by jhofmannmaths is set out.
• Justmaths.co.uk has a Whodunnit activity and exam questions.
• Mr Barton Maths has a 'Win a Million' game.
• Teachmathematics.net has a good Excel activity.
• Mr Collins writes about this animal zoo resource in his blog.
• Finally, an extension worksheet on linear interpolation.