This Week in Math, Week 2
Apr. 1st, 2010 10:59 pm![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
velvetpageThe Big Idea:
Fractions aren't about quantity so much as proportion.
The Problem:
Prove that 3/8 is greater than 1/4 but less than 1/2. Use at least two different representations to prove this.
Most kids used a number line first, bringing in their work from last week on equivalent fractions. A couple used simple arrays instead. A few got nowhere because they used three different representations for the three fractions; with their permission, we used that as a key teaching in the lesson: that comparing fractions is always easier when the wholes are the same size and shape.
The fun came when I changed the focus of the problem somewhat today. The problem became: is 7/9 greater than or less than 7/8?
First I instructed them to come up with a prediction. I made note of the kids who told me that 7/8 was greater because ninths mean smaller pieces than eighths; those are the kids who grasp the proportionality of fractions. But I didn't challenge the more common prediction that 7/9 was the greater number because nine was greater than eight. I simply told them to write down their prediction and their reason for it, reminded them of the lesson learned during the previous problem (that the whole must always be the same size and it helps if it's the same shape) and let them go at it.
I got a lot of number lines again. I suggested to a few kids that they use what they knew about multiplication and division; for example, could they draw a rectangle that was divisible by both eight and nine using graph paper? I guided a few kids through the process of figuring out how many squares were coloured in the 7/9 array versus the 7/8 array, and reaffirmed that I was proud of them for recognizing that their prediction had been wrong, and writing their revised answer. We briefly reviewed how a mathematical proof is the exact same process as the scientific method: develop a hypothesis, develop a method for testing that hypothesis, work through the method, compare your answer with your hypothesis, and conclude that you were either right or wrong.
I encouraged kids who clearly got it to use the best math language they could and to explain their pattern rule: as the denominator gets bigger, the pieces get smaller.
All of the kids who originally believed that 7/9 was bigger ended up revising their hypothesis and understanding the pattern rule by the end of the lesson. I have a few kids I need to see for guided math on Tuesday because they were helping with the raffle or otherwise unavailable for part of the lesson, though.
Next Week in This Week in Math: The Math of School Raffles. We did the raffle today, and I kept all the tickets for each item in a numbered brown paper bag for the express purpose of probability experiments next week. There were a few poor sports commenting on how someone (i.e. me - no, I didn't let them get away with it) must have cheated, since they put twenty tickets in one box and somebody else won anyway. I'm looking forward to examining the probabilities involved in that one.
The following week, we'll be doing coin tosses, which will lead directly into Pascal's Triangle.
Fractions aren't about quantity so much as proportion.
The Problem:
Prove that 3/8 is greater than 1/4 but less than 1/2. Use at least two different representations to prove this.
Most kids used a number line first, bringing in their work from last week on equivalent fractions. A couple used simple arrays instead. A few got nowhere because they used three different representations for the three fractions; with their permission, we used that as a key teaching in the lesson: that comparing fractions is always easier when the wholes are the same size and shape.
The fun came when I changed the focus of the problem somewhat today. The problem became: is 7/9 greater than or less than 7/8?
First I instructed them to come up with a prediction. I made note of the kids who told me that 7/8 was greater because ninths mean smaller pieces than eighths; those are the kids who grasp the proportionality of fractions. But I didn't challenge the more common prediction that 7/9 was the greater number because nine was greater than eight. I simply told them to write down their prediction and their reason for it, reminded them of the lesson learned during the previous problem (that the whole must always be the same size and it helps if it's the same shape) and let them go at it.
I got a lot of number lines again. I suggested to a few kids that they use what they knew about multiplication and division; for example, could they draw a rectangle that was divisible by both eight and nine using graph paper? I guided a few kids through the process of figuring out how many squares were coloured in the 7/9 array versus the 7/8 array, and reaffirmed that I was proud of them for recognizing that their prediction had been wrong, and writing their revised answer. We briefly reviewed how a mathematical proof is the exact same process as the scientific method: develop a hypothesis, develop a method for testing that hypothesis, work through the method, compare your answer with your hypothesis, and conclude that you were either right or wrong.
I encouraged kids who clearly got it to use the best math language they could and to explain their pattern rule: as the denominator gets bigger, the pieces get smaller.
All of the kids who originally believed that 7/9 was bigger ended up revising their hypothesis and understanding the pattern rule by the end of the lesson. I have a few kids I need to see for guided math on Tuesday because they were helping with the raffle or otherwise unavailable for part of the lesson, though.
Next Week in This Week in Math: The Math of School Raffles. We did the raffle today, and I kept all the tickets for each item in a numbered brown paper bag for the express purpose of probability experiments next week. There were a few poor sports commenting on how someone (i.e. me - no, I didn't let them get away with it) must have cheated, since they put twenty tickets in one box and somebody else won anyway. I'm looking forward to examining the probabilities involved in that one.
The following week, we'll be doing coin tosses, which will lead directly into Pascal's Triangle.