Resourceaholic: Favourite Problems

19 June 2015

Favourite Problems

On Tuesday 23rd June I'm hosting a #mathscpdchat entitled "Problem solving: what are 'good' problems? Where can you find them?".

What's a Problem?
A problem, in the mathematical sense, can be defined as "an inquiry starting from given conditions to investigate or demonstrate a fact, result, or law". The word originated in the late 14th century,'a difficult question proposed for solution', from Old French problème, and directly from Latin problema, from Greek problema 'a task, that which is proposed, a question' (source: etymonline).

A problem from solvemymaths.com

I like this quote from Jennifer Piggot, "A problem is something that you probably will not immediately see how to tackle or which once started will challenge your thinking. This means that at some point you are likely to get stuck and this is OK. In fact it is important that you get stuck so that we can talk about possible ways of getting unstuck" (source: nrich).

My Favourite Problems
During next week's Twitter chat I'd like to see examples of your favourite problems. These might be interesting questions that you use when teaching specific topics - for example do you have a great question on standard form? Straight line graphs? Sequences? Remember that we're focussing on problems as opposed to resources, activities or worksheets.

Here's one of my favourite problems:

Source: Chris Smith's Newsletter, Issue 51

This is the sort of problem that could take minutes or hours to solve. It's certainly suitable for bright GCSE and A level students to have a go at. At first look it's one of those 'where on earth do I start?' questions. Draw in a few lines and a 'Eureka!' moment may come. We end up with a fairly straightforward problem involving Pythagoras and a quadratic equation (I say straightforward, but I'm assuming that the problem-solver knows Pythagoras' Theorem and knows how to solve quadratic equations... Prerequisite knowledge is a feature of many geometrical problems).

Eventually I aim to pull together a set of good problems for each topic on the curriculum - I've made a start with this Pythagoras Problem Set.

Problems don't have to be highly complex. Any question that makes students stop and think is valuable. For example this week I was looking for a starter on coordinates for my Year 7 class. I could have gone for a standard plotting activity - I believe that ongoing procedural practice is very important - but instead I've gone for the question below:

From Squares and Coordinates (Don Steward)

It's an accessible problem (the shape is a square) but I think my students will find it challenging. In fact I'm worried that I'll be met with a chorus of "can't do it, Miss" and then I'll lose their attention. I think this is a common fear of maths teachers - we want to give our students problems that will teach them to be independent and resilient mathematicians, but we worry that they'll lose focus if they can't work out how to get started. This is something we can discuss in the Twitter chat on Tuesday.

Finally, here's a problem that I've used a number of times when teaching surds - I love this one: