In addition to her already fantastic blog, Fawn Nguyen has been blogging about Math Talks that she is doing with her classes. A Math Talk is simply an opportunity for students to think, and share their thinking about mathematics, generally without paper and pencil.

We’ve been doing a lot of work around proportional reasoning in our board and in the province. My latest favourite question is one generated from a team of teachers that were working on some problems to use early in the grade 9 course.

**Would you rather have $500 or the value of a stack of quarters as tall as you?**

The question seems simple enough but as I’m using it with various groups, both teachers and students, I’m amazed at the wide variety of strategies used to when thinking about this problem. It is a nice opportunity to get at ideas of proportional reasoning, multiplicative thinking and unitizing.

When I’ve posed this problems with groups I usually make them guess first based on gut instinct without any time to think.

After some time to think and then share with a partner I have them share their ways of thinking about this:

- a roll of quarters is worth $10 and about 2.5 inches high. So that’s $20 for 5 inches…..$40 for 10 inches, $240 for 60 inches
- 6 quarters is about 1cm high, so that is $1.50 for 1cm, $3.00 for 2cm, If I’m 180cm tall, that is about $270
- A roll of quarters $10, so you would need to stack up 50 rolls to make $500, and I’m definitely shorter than 50 rolls of quarters…
- my foot is about 3 rolls of quarter long and I am about 7 of my feet tall, so $30 x 7 = $210

What I love about this discussion is the wide variety of “units” that people use – based on their schema. Some work in inches, some in centimetres, some (mainly those with retail experience) talk about rolls of quarters and some use personal referents.

I also love the amount of gesturing that occurs when talking about this problem. I think we underestimate the power of gesturing when thinking and talking about mathematics – in particular when it comes to proportional reasoning.

After having time to think about the problem, I have had some ask whether $500 is too large of a value to use since most people seem to estimate their “quarter worth” to be around $250. I point out that they didn’t realize this until they had time to think about the problem so I’m not sure it matters. In fact I don’t want it to be too close because the problem is not about exactness but rather about some general thinking about proportions.

Inevitably some start thinking about how tall you would have to be to make the quarters a better decision and the discussion continues.

If you are looking for more ideas for Math Talks check out Fawn’s blog!