Yesterday was our first day into a unit on similarity with the ninth grade students.
The issue that comes up every year is that students like to cross multiply, but are incredibly mechanical in their understanding of why they do so. They don't like fractions that aren't simplified, and can usually simplify them well. They bring up the fact that multiplication of numerator and denominator by the same number is equivalent to multiplying by one. They seem to have very little understanding of how this relates to units and unit conversions as well.
I changed my approach this year to be much less review of how to solve proportions. I wanted to get at the aspects of measurement that are inherent to math problems involving similarity. I wanted to get them to ask themselves a bit more about why they took the steps they took in solving proportions in the process.
I started with a couple simple problems in the warm-up. Here was one:
I took pictures of two students' work, put them side by side, and asked the class which one they thought was a better answer to the question:
The resulting vote and conversation was especially spirited, particularly for a class that normally rejects whole class discussion. We talked about the ideas of approximate and exact answers, a couple of students pointing out that substitution of the approximate answers would result in a false statement in the equation.
After this, I showed them another picture and asked if the LEGO pieces in this picture would go together:
Every hand went up.
I then showed them the bricks, which I had made on our school's new 3D printer:
Pause for groans. Some key things were said in response to my 'playing dumb' question of why the two bricks won't fit together. One student even directly said that they looked similar to each other, but that they weren't the same size. I wanted them to have in the back of the heads that I was going to be pushing them to always think about figure with the same shape, different size.
We then made it to the second task of the warm-up activity. I asked them to estimate (and subsequently measure) the ratio of one of my heads in this image to the next:
I developed the following points:
- When communicating ratios to another person, begin explicit and clear about order is extremely important.
- Despite the different units, these ratios are all communicating the same relationship from one head to the next. This relationship is even more obvious when we write the ratio as a fraction instead of using the colon notation.
- The approximate values of this fraction are all roughly the same. We don't need to convert units either for this to happen - the units divide themselves out in the fraction.
I went on to define a proportion and reviewed the idea of cross products. They were a bit surprised when I showed them that cross products were equal for equivalent fractions. Part of this was because they saw me equate 2/5 and 4/10 and immediately said they were equal because one simplified into the other. I gave them 2/5 and 354453764/886359410 and they were a bit more willing to see that cross products can be a slicker way to check equality.
One more point that I made was that a proportion with a variable in it was really a question. If we are saying two fractions are equal to each other, and one (or more) of the fractions has a variable, what does that mean about the value of the variable? It led to a bit more conversation about the reasons for cross multiplication as a method of solving proportions, and I was satisfying then leaving students to work through some more review problems on their own.
The final piece we talked about whole group was this open ended question:
They were able to come up with some, but struggled to make ratios that were more than simple multiples. This was surprising, as their mental calculation skills are generally quite strong. As shown in the example, I gave them one way to see how to come up with an arbitrary set of lengths that fit the requirement.
I then showed them this question:
Some of the students realized (and explained eloquently) that they could divide the length by 7 and find the length of a single 'unit', and then multiple that unit by 3 or 4 to get the length. Explanations for why this worked didn't really materialize. I introduced the algebraic approach, and students saw it as an explanation, but seemed to be fine with just remembering it as a method rather than as a rationale.
The more that I teach proportions and similarity, the more I feel compelled to have students ground the concepts in measurements. Making measurements, especially by hand, is not something they typically do on a day-to-day basis, so there's a bit of a novelty factor there. These conversations about measurements, units, and fractions were authentic - there was a need to talk about these ideas in the context I established, and the students did a great job of feeling and then filling that need during the class. Nothing we did was a particularly real world task though. What made this real was my attempts to first frame the skills that we needed to review in the context of a need for those skills. I try to do this often, and I'd like to mark this as a success story.