Visual Representations

When students think about presenting their solution to a mathematical problem visually I get very excited because I find those visual solutions hard to generate myself, I am getting better it, but I don't see the mathematics in that way when I am solving a problem generally, I get too wrapped up in the equations in the formal mathematical notation.  The reason I get excited about those visual representations for a problem is that they often are much easier than the formal mathematics and they lead to some great mathematics.

The other day I did a task with my class that I got from Professor Peter Sullivan from Monash University who I had the pleasure of working with once per term for three previous years in my work as a numeracy coach.  The question is as follow

 On a train the probability has a backpack is 0.7 and the probability that someone has a laptop is 0.6.  How many people might be on the train and how many of them might have both a backpack and a laptop?

Show your answer in two different ways.

I like this question on a range of levels

• It is not immediately obvious how many people might be on the train just by looking at the probabilities, you need to reason it how and have a fairly flexible understanding of number.
• There is a range of answers to the question but the range is not infinite, Given their number of people on the train there are is definitely both a maximum and a minimum that can have both.
• The problem can be attacked in a lot of different ways, it lends itself to multiple solutions
• The quantity of reasoning in the mathematics needs to be quite high, there is a lot to think about, model and justify

One of the groups working on this task came up with the solution shown below, it has been altered from the original (which was just dots showing the same information) but the same thinking was there.  They first started by looking at the 0.7 and 0.6 and converting them to 70% and 60% which they then turned into 70/100 and 60/100 which simplified to 7/10 and 6/10.  From this they reasoned that ten was the minimum number that could be on the train. From here they attacked the main part of the problem by first  assuming that there was as much sharing as possible, making sure everyone with a computer also has a backpack.  This gives six people having both items, this is shown by the image below on the left.  They next thought about orgainsing the same but with as little sharing as possible, by making sure everyone who didn't have a backpack needed to have a computer.  This scenario gave the image on the right below, an overlap of three. Hence they gave a range of answers as being between three and six out of ten people.

The animation below shows they range of answers with more clarity

What occurred to me as I was documenting their thinking on the board is how nicely this representation led into the intended purpose of the lesson, Venn diagrams and two-way tables. Simply by circling the laptops in one colour and circling the laptops in another I had generated a fairly accurate Venn diagram as shown below.  You could clearly see in each case how many were in the intersection of the two sets, how many belonged to only one of the two sets and how many were outside of those two sets.

What I really liked about this is now easily it made the transition to Venn diagrams and two-way tables. from our initial starting point of probability of simple events.