The expectation regarding student depth of understanding of equivalent ratios in middle school has risen significantly since the advent of the Common Core. A few years ago, students just scaled up or scaled down ratios to find equivalent ratios. Now, they are expected to utilize ratio tables, tape diagrams, and double number lines to solve fairly complex problems. I think, though, that oftentimes students don’t see the connections between these ideas. I think that a lot of the time, they see them as completely disparate concepts, just one more thing to know about ratios. As we wrapped up today’s lesson on double number lines (the last of the three representations for my class), I wanted to explore this idea. Do they see the big picture, that these are all different ways to represent equivalent ratios and that the whole point of the representation is to visualize things so that one can solve problems? I gave my students this exit ticket to find out a little bit more about their thinking.
Their responses were illuminating, as always. Student responses to the first question seemed to fall into two groups: those who saw the big picture (like the one shown) and those who responded with three ways to write a ratio. I think those who were mistaken on this question zeroed in on the word three and went straight for the ways to write a ratio without closely reading the question. However, I also think that if they had made the connections between the representations, they would not have honed on that word. So, I have some more work to do.
Student responses to the second and third question showed that most of them are still figuring out how to think about their thinking. I am reading Making Thinking Visible and am hoping to gain some insights into how to develop stronger metacognitio in my students. .