## Monday, January 21, 2013

### Proportional Reasoning And Other Boring Words

Last week I did some proportion questions with my students. I found that some methods for solving these questions were preferred to others. I'll highlight these methods plus a few other things I noticed but first, here's a quick question I wanted to test out with some readers.

How would you solve this question:

To make 10 pancakes, a recipe calls for 16 Tbsp of sugar.
How much sugar is used for 17 pancakes?

Good. Got it? Save your solution until end.

Here's the lesson I did with students and some observations.

Proportional Reasoning Lesson1

This is the pre-activity from MARS, thanks to Fawn Ngyuen for citing this resource.

First:
Results:
• Often students found the price of one pancake (Unit Price) and then x 10.
• Less frequently students did 10 ÷ 4 (Scale Factor) then x 6.
• Doubling and halving was quite popular.
• One student did cross multiplication "because that's the way I learned it."
Then:
Results:
• the increase in size was easier than the decrease.
• doubling and halving was used for some of the .75 and the 2.5
• No cross multiplication was used.
And finally:
Results:
• some adding of 15 to get 31 was used.2
• this question, more so than any other, was left blank.
Solutions were shared in small groups and each group presented the solution they felt was most successful. As a class we sorted, critiqued and supported the solutions presented. They fell into 4 categories.
• Scale Factor
• Unit Price
• Doubling and Halving
• And the rarely used Cross Multiplication
Sharing (as a whole class) the pros and cons of each method was the most rewarding. My biggest issue was actually reading the MARS timelines for each phase of the lesson. I was way off! The ideal situation as advised by the MARS document calls for,
• 15 min. working on the sheet individually. (took me 35)
• 15 min. producing small group solutions (took me 40+)
• 20 min. Class discussion (took 40)
I also have to mention that I put this on the screen as a Minds On:
Kudos to the brilliant 101qs for this perfect 'Minds On' task. One of my favourite 101qs. Questions were (in order from popular to least)
1. How much for five cans?
2. How much is one can?
3. What happened to the missing can?
4. Where are they shopping?
Almost exclusively they all calculated 1 can, then multiplied by 5. (Unit Price Method) One student calculated 1 can and subtracted that from 6.

Ok. Now try:
A picture has dimensions 16 height x 10 width. A proportional enlargement is made. The width is now 17, what is the height? Write down a solution that comes to mind. My question to you is: Did you solve #1 and #2 differently? My students do. Does the pseudo-context used change the method of solution?

1. [I have to say the word 'Proportional' in combination with 'Reasoning' is a tough sell for students. We need a snazzier name. Rates and Ratios is only slightly better. Is Scale enough?]
2. [I should submit that for a math mistake]

1. I did solve #1 and #2 differently. In #1, I calculated the tablespoons of sugar/pancake by dividing 16 by 10. Then, I multiplied this number, 1.6, by 17.

In #2, I determined the scale factor by comparing the widths and multiplied the height, 16, by this number 1.7.

I suspect this is what your students did.

Interesting that both questions can be modeled by 16/10 = x/17 but that the context does change my method of solution. In #1, I'm being pushed towards per pancake thinking. I'm looking for a vertical relationship in the proportion above. In #2, I feel the need to compare length with length and width with width. This time, I'm looking for a relationship btwn the numbers horizontally.

I wonder to what extent the numbers involved influence this as well. For example, let's say it takes 7 tablespoons of flour to make 4 pancakes and I want to know how many tablespoons of flour I need to make 12 pancakes. As in #1, I could determine the number of tablespoons of flour per pancake (1.75 isn't too bad, and I could make this uglier to make my point) but it's easier to look for a second relationship and multiply 7 by 3.

Similarly, I can elicit two different strategies by asking students in Grade 3 to solve each of the following:
#1 I have 48 stickers. I give 36 away. How many do I have left?
#2 I would like to collect all 48 stickers in a set. I have 36. How many more do I need?

Each can be solved by calculating 48 – 36. The first suggests removal whereas the second suggests adding on.

Together, these two contexts demonstrate how there is flexibility in the way in which we solve these problems.

Chris

2. The comparison to grade 3 is one. I appreciate the example and I'm sure other teachers could find examples that work for high school. The push towards "per pancake" also happens with the Juice Can Grocery example.

"...these two contexts demonstrate how there is flexibility in the way in which we solve these problems."

Yes! We spent a good amount of time analyzing 4 methods and deciding which worked 'best' and which was preferred by students. I hope that the flexibility came through. Thanks for the contribution Chris.

3. Did you first teach them multiple solution methods, and then they decided to use the unit price method? Or is this the first lesson on the topic of ratios, and the unit price method was their natural inclination to solve these problems?

1. The second. This is the pre-activity. They were ask to complete it without me having instructed them on any of the methods. Unit price was one of the methods they naturally chose, but not the only one.