Proportions

I’m doing a long term sub job at my old school in 6th grade, math & science. Next week is the last week of school – I’m happy and sad about it. I will miss talking to my friends there, but I’m glad I’m not going to be teaching full time anymore. But this post is about proportions!

Chapter 14 is Proportions. I went through the chapter and decided how I wanted to teach it, starting with defining a proportion as 2 equivalent ratios, with an = between them. Why do some kids want to write the ratios without the = ? Anyhow, next we worked on solving the proportions by finding relationships, vertically or horizontally. (I was so surprised when an algebra student told me he was simplifying his proportion horizontally. This was probably 7 – 9 years ago, and I ALWAYS simplified vertically, like simplifying a fraction. When he told me he was doing it horizontally I’m sure I thought, “You aren’t allowed to do that!” His dad had told him. I do remember I said OK, then wondered if it always worked. I was so good at following and teaching the procedures then.) Anyhow, now I teach the kids to look vertically or horizontally for a relationship. (But not diagonally! I tell them when you’re crossing out the numbers to rewrite the simplified numbers, you’ll also put a line through the = if you’re doing it diagonally – which now makes it “not equal”. I hope that makes sense to them.)

The next day we were going to solve proportions that didn’t have a convenient relationship. The book has the Cross Product Property with the example using a, b, c, & d, showing how the property works and that’s supposed to explain everything. In reality I think most teachers show the kids how to cross multiply and there isn’t a lot of discussion of WHY we can use the Cross Product Property – and the example with only variables is NOT helpful. So I decided we were going to solve the proportions by making common numerators or common denominators. We’d already discussed how proportions are equivalent ratios, so now we would just need to make them equivalent.

For example: 2/3=x/4 can be rewritten with a common denominator of 12. (I can’t figure out how to copy in a fraction from Word, so I’m just being lazy and using the /.) So now the proportion is 8/12 = 3x/12. Our denominators are the same. What do we need to do to make our numerators the same? We need to solve the equation 8 = 3x!

Here’s another example: 4/5 = 9/x. This was probably the 3rd example we did in class. Can we make common denominators? NO? What can we do? A student said we should make common numerators! (What? That’s not how we solve fractions….) So we did. We said the new proportion was 36/45 = 36/4x. So now the numerators are the same, how do we make the denominators the same? Solve the equation 45 = 4x!

We did 4 problems together then I rewrote what we’d done as a summary page, just showing the original proportion, the proportion with equal numerators or denominators, and the problem we had to solve at the end. I asked the kids to look at the original proportion, and the problem we had to solve at the end – thinking quietly for one minute, then talking to their groups. It was so awesome, it was totally quiet, then when I said “Okay, you can talk to your groups” it just exploded. It seemed like they couldn’t wait to share what they saw. What they saw was they could get the “problem” by “cross multiplying”. What?? I had them explain each one to me, while I used different colored highlighters, highlighting the diagonal number in the proportion, then drawing an arrow to that term in the problem. It was a very colorful page!

Fast forward to the review day and we talked about WHY they could cross multiply to solve a proportion. I wanted them to be able to tell me that it was because if we rewrote the proportion with equal numerators or denominators the piece we needed to solve in that proportion would be the problem we’d solve by cross multiplying. We could do it because cross multiplying was a short-cut to getting common numerators or denominators. I also asked this question on their test. I think around 75% get it. Now can they keep it?

I was really happy that I didn’t just teach them HOW to do it, they figured out that part on their own. Mostly!

We also solved problems from situations (word problems). I want them to write a “word ratio”, then the proportion, then solve it and give me a word in the answer. So not just x = 8, but x = 8 penguins. I did one problem with them, talking about each part, defining the Word Ratio as “the way to organize your proportion”, then gave them another problem to do and walked around checking that they had the word ratio and the proportion. How they solve it is their choice. I’m walking around and see one girl making Ratio Tables (which the class is AMAZING at by the way, great job Mrs. Marien!) instead of writing a proportion. You can definitely solve the problem this way, but I wanted to see that she understood how to write a proportion! I walk around, check each child’s work, then go back and do 2 more problems with the class (calling on students for each step). We’re done, here’s 7 more for you to do…check with your tables as you go.

It is so interesting to me that I can do 3 problems with the class, then give them 7 more to do, and as I walk around I’ll see kids doing whatever they want to get the answer. It’s not all about the ANSWER!! I’m walking around checking, helping, making sure they’re writing an = between the ratios and writing their word ratios. One boy was on his last problem when I got to his table….yes, he was this far along because he hadn’t written a single word ratio. Not for my examples or for the other 7. He didn’t feel like it, he could do it without them. I told him on the test I’m going to ask him to write word ratios, and they help him organize his numbers in the proportion. (It’s actually helpful! Don’t just do it because I’m going to test you on it, do it so you get the problem right!) Another boy – next table – isn’t writing any proportion, he’s just solving the problems using division – they only thing on his paper is his arithmetic. He’s my boy who thinks the goal is to get the answers ASAP, then be finished and bug other kids. And the answers don’t have to make sense. But I digress…

Anyhow, I think I did a good job teaching proportions without telling them HOW to do it. They can look at the proportion and see their own relationship, write a new one with common denominators, cross multiply to solve everything…it’s their choice.

They didn’t ALL do well on the test; my boy without any word ratios got either a 64 or 61%. But, one class had 5-100%’s and 5-99%’s (they got 1/2 off something), with 66% getting 90% or better. The other class only had 1-100%, but they had 56% with 90% or higher (yes, my no word ratio boy is in that class). One thing I need to do is connect how they can write a proportion from the Ratio Table. Many probably have already connected it, but we did a problem the other day with a Ratio Table and I thought they’d see they could go out of the table and write the problem as a proportion – but “that’s not how you do a Ratio Table”! Hopefully I can connect this before school is out (on Friday!!!) – actually I connect this, hopefully the kids can connect this if I make a point to show them! As the goal is for the kids to see how all the math flows together – what they do isn’t a bunch of unconnected concepts!!!