We are working on proportional relationships right now. We've been working on proportions and unit rates, but now it's time to move on to the next level of understanding. I'm trying to get students to see the connections between what we have learned about unit rates, and tables, graphs and equations.
The last couple of days I've been doing a series of partner activities about proportional relationships. The first activity was a breeze for the kids....find the unit rate. The first time around all of the kids had the same unit rate so they could check each other's work. Then we did another activity where they kids had different unit rates and had to compare and choose the better deal.
Then I had the kids graph the unit rates. I don't know why I'd never thought to do this before, but it was a great addition to this activity. On one graph, I had the kids graph both of their points (such as 12 books for $48 and 6 books for $24). Then they graph the unit rate (1 book for $4). It was a good chance for kids to make the connection about all of the points in a proportional relationship falling on a straight line.
With the second activity, I decided to save some time and show some graphs on Desmos instead of having kids graph them. It may have been a split second decision, but it worked out great! In one class, the Desmos graphs brought up a major misunderstanding that I wouldn't have realized otherwise....and Desmos made it so much easier to clear up. We were looking at this graph below, and my students though that the green line was a lower unit rate because the line was shorter...that was all they could see.
Luckily, with Desmos it was so easy to take care of this misconception. I quickly changed from the tables I was using to equations, as shown below.
As students watched me type the equation and saw the lines become longer, it was like I could see the light bulbs come on, and within a couple of questions they could explain not only which unit rate was better but what they had misunderstood. This was definitely a case of technology supporting understanding, in an immediate way that other tools couldn't have done...this is technology at its best. Simple, but awesome!