I always like doing Number Talks in classrooms but my experience with these has mainly been with elementary students. So I jumped a the chance of trying one out with high school students last week. These were Grade 9 students in the new de-streamed math course here in Ontario. As the students were going to be working on a task looking at car depreciation, their teacher wanted to kick-start their thinking on finding percentages of an amount. This is the number talk that we gave them:
Find:
50% of $64
10% of $64
20% of $64
5% of $64
15% of $64
We asked the questions one at a time and students wrote their answers on individual white boards which they then showed us.
Everyone was happy enough with the first question: some said you just find half of $64, others said you divide it in 2. I was able to illustrate this using a simple bar model:
The second question caused a bit more thought. Whilst some wrote $6.40, others wrote $6.4. One or two students quietly told me that they thought it was a bit more than $6. I asked the students who had written $6.4 'How many dollars and how many cents is this?' There were some who had the misconception that this meant six dollars and four cents. (This is often an issue if students are calculating a percentage on their calculator).
Returning to my bar model, I asked how I could split it to get 10% partitions.
Once they told me ten parts, I could then emphasise that 10% is our 'Golden Percentage' as it is friendly to calculate and useful for creating other percentages. We could also split the $64 into $60 and $4 to show that 10% of each of these was $6 and 40 cents respectively. This helped address the '$6.04' misconception.
The third and fourth questions were then solved very quickly and successfully which just left the 15% of $64. Most saw this as '10% + 5%' but one or two saw it as '3 times 5%' and one person did '25% – 10%' which I have not seen before.
All of this allowed us to emphasise that part of being a good mathematician is being able to split things up and then put them back together (or, to be technical, decomposing and recomposing)
We then split randomly grouped the students into groups of three and assigned them to vertical whiteboards to work out 8% of $64 in any way they could think of. We knew that this is something they would have learned in earlier grades, we expected them to use their calculators to do 0.08×64
Instead we got a variety of different methods that connected to the Number Talk:
Using a ratio
Find 20%, subtract from 100% to get 80%, ÷10 to get 8%
Find 1% then multiply by 8 to get 8%
Find 5% then add three lots of 1%
Find 10%, ÷5 to get 2% then take this off 10%
It was great to see all these decompositions and I can only think that this strengthened the students' number sense. At the same time, we also drew their attention to the fact that sometimes the numbers might not be so friendly, in which case multiplying by 0.08 might be the better option.
All of this prepped them perfectly for the car depreciation task.
More examples of Number Talks can be found here.
