After the first few days of basic skill practice, lots of pre-assessment, getting down routines, and such – it was time to begin our first unit, called “Stretching and Shrinking”. This unit’s central idea is on proportionality and ratio, with some ideas embedded on the properties of geometric figures.
Ratio has been a very tough concept to teach over the past couple of years. Sometimes we write it as a fraction, but it’s not exactly a fraction, except when it represents a part-to-whole comparison. It can represent a part-to-part comparison that can be scaled up and down. I actually have a hard time framing all of the essential questions and big ideas related to ratio and proportion.. I’ve tried several times to write them and have gotten stuck. No wonder students struggle.
Usually this unit is started by engaging students in a graphing activity and having them fill out a table of measurements, then guiding them through calculating the ratios in the measurements, and then having them write down the definitions of “similar” and “ratio”.
My colleagues and I designed a little different intro task this time. We presented students with lots of drawings of “Wumps”, cute little cartoon characters.
I put the students into small groups and presented them with their task: the Wumps on the page represented three families. Geometrically similar Wumps belonged to the same family, which meant they have the same shape, but could come in different sizes. They had to sort the Wumps into the families and then justify why they belonged in each family. They could use rulers, protractors, pencil, and paper to get some measurements to support their ideas.
Students got to work, and after 20 minutes, I had a lot of papers that had Wumps sorted into three families: the “Short and Fat” family, the “Tall and Thin” family, and the “Regular and Normal” family. I walked around and prompted groups to tell me why they made their decisions. I challenged them. “Pug is not fat. He is actually very thin as I can see he’s only 7 units wide. So why are you telling me he’s fat?” They pushed back, saying they could see that of course he was fat because his head was so wide.
A few groups noticed that the Wumps’ noses were different shapes. The “Fat and Short” family, in particular, all had square noses. I encouraged them, saying that square was a very mathematical concept and they should go with it. What makes squares special? They responded that all four sides were equal. I suggested they could look into the side lengths of the other Wumps’ noses and see how they compared.
One group noticed that the ears were more or less pointy depending on which family they were in. A young man said that the tall, thin family had ears “like nails”, whereas the short, fat family had ears “like houses”. I suggested perhaps they could measure the pointiness of the ears with a tool. “A protractor!” they said, and I told them to please organize their measurements into a table so they could easily look for patterns. I realized then that this structure was what was missing from the other groups, and it was why they had a hard time staying on the task. They weren’t making tables or organizing their information at all, and so they could not see patterns or begin to understand ratios if the information was scattered about everywhere – or just in their heads. The class was almost finished, so I collected all of their materials and closed things up by having students write in their planners.
Overnight, I looked at some of their work. I decided to start the activity the next day by showing the class some of their classmates’ thinking and then see if we could come up with suggestions or extensions to that thinking. The next day, students finished a quiz and then I displayed some work from yesterday that was very typical of the papers I got from groups:
We examined the work and found that a couple of Wumps were misplaced, and I asked students if they noticed the same properties in the Wumps – the short and fat, tall and skinny, and “regular” Wumps. I focused on the “tall and thin” statement, noting that the smallest “thin” Wump was not actually tall, it was short – but it was just tall in comparison to its width. I suggested to the students that the “tall and thin” description was focusing on two key measurements on the Wump – what were they? The students responded “Length and Width!” I noted that the table at the top had enough space that students could gather the length and width of the Wumps and organize the measurements to see how they compared. I modeled how they could make a table if the height and width were what they wanted to focus on.
I displayed another sample of student work.
I praised this students’ work, using some mathematical justification for her statements. I asked the class if this was a little more information to describe “short and fat” versus “regular” Wumps, and they agreed. I suggested that I would like to see the numbers that made them come to the conclusion. If they were focusing on the width and height of the Wumps, could they organize those in the table to show us the comparisons? I didn’t say that this student was really starting to hit on the idea of ratio, which was exciting. That multiplicative comparison was the same for all Wumps in the same family and that was the BIG idea of this lesson. I hoped her group would run with the idea and I hoped other groups could get to this place too.
I showed one more example.
I asked the class if anyone else had noticed the nose shapes in the families, and many students agreed that they had. The square noses, super skinny noses, and “rectangular” noses were dominant features on the Wumps. I asked students what measurements they would need to describe a “square” nose, and they responded that they would need the side lengths. I asked what they would use to describe a “super skinny” rectangle, and they responded “Length and Width!” I suggested that this group might have hit on another set of measurements that could be used to describe the Wumps, and they should go ahead and collect all of the measurements, organize them into a table, and see if they could see a pattern.
Finally, I mentioned the group that had noticed the pointiness of the ears and asked the class if they noticed any other angles that distinguished one family of Wumps from the others. Some students mentioned that the smiles were different in each family, and I said that was an interesting observation and perhaps they could investigate the angles or lengths of the smiles, collect and organize the measurements, and see what patterns came out.
I put the students back to work. Unfortunately, some groups were still not collecting data and seemed stuck at just the visual observation of the Wumps. Other groups were comparing irrelevant measurements such as the area of the ear or the width only. I knew they could see a pattern with the area of part of the shape, but it was a much more difficult pattern to see with the knowledge they have. I let them go with it, but was not sure how to process what they found. Some groups counted the dots for the length and width, and this made for an interesting dilemma – the number of dots is not the same as the length – it’s always length + 1 because the “zero” dot is counted. The ratios would not look quite right when number of dots was used. Happily, some groups did collect a table of data on measurements such as nose dimensions, arm dimensions, smile angles, or head height vs. width. I knew we could process these as a class successfully, so I saved a lot of papers to share with the class on Tuesday when we get back from the long weekend.
For a first attempt at working in groups on a challenging open-ended math problem, I thought the students did remarkably well. I’d like to see the shyer students participate more and do more of the thinking. I’d like them to get to the point that their first reaction is to start gathering numbers – quantitative information – and organizing it and looking for patterns. It took quite a bit of prompting for students to do that today, but they eventually saw the power of it and came to some interesting realizations – some of which I hadn’t noticed before. I also saw some fascinating misconceptions (such as the one about counting dots instead of counting length) that give me a heads-up on where to work with them next.