MPM1D1 - Day 6 More Cup Stacking

I modified the Visual Pattern we did yesterday slightly to see how quickly my students would be able to spot the differences and come up with a solution. Again I asked them to find the forty-third step and a rule/equation to find the number of squares in any step. Here's what we started with:

Most groups started by making a table of values. Those that didn't weren't really sure where to start and so I suggested a table. I was happy to see many groups showing the first differences (though we didn't call them that) in their tables.

 I asked what was the same and what was different compared to yesterday's pattern. It was great hear things like "The constant is different" or "It's going up by the same amount". We talked a little about how these showed up in the equations.

Next we moved onto more cup stacking. The goal for today was to change how we stacked the cups and how that changed the equation and graph. I asked how many cups would be needed (stacked inside one another) to reach R's height. All groups saw that the height of the stack was changing by the lip of the cup for each additional cup. A couple of groups struggled with the initial value. They thought it should be the height of a cup rather than the body of the cup. Every group did manage to come up with an equation but struggled to solve the 2-step equation needed to find the number of cups (not surprising since we have done much equation solving yet).

Once they were done finding the number of cups needed to get to R's height they went back to their seats and plotted Height vs. Number of Cups. We talked about how the graph was different from the one they made yesterday. This led to a need for some terminology (partial vs. direct variation) so we wrote a note about graphs. The note included dependent vs. independent variables, continuous vs. discrete data, lines of best fit and interpolation vs. extrapolation. I was hoping to get into partial and direct variation and slope but we ran out of time.

I gave some homework on plotting points on the Cartesian Plane and identifying whether variables were dependent or independent.

At the end of class I had a student come to me and tell me that he was feeling lost. He said he was able to follow what his group members were saying but he wasn't sure he'd be able to come up with the numbers on his own. He told me that he did well in math last year but wasn't feeling very confident. We chatted for a bit and he agreed to come in tomorrow at lunch so we can go over a few things. I'm curious to see if his issue is related to skill or confidence.

MPM1D1 - Day 5 Cup Stacking

We did our first visual pattern today. I started with this one:

I told them that the images represent the first three steps in a pattern then asked if they could find the number of squares in the forty third step. I also asked if they could they find a rule or equation to represent the number of squares in any step (the n^th step). Normally I ask for an equation and I think that is often intimidating at first. This time I focused on the rule, which we could then be turned into an equation. They worked at the problem in groups at board. They did a great job. We spent a bit of time talking about what makes an equation.

Then we moved on to Cup Stacking. I held up a styrofoam cup and asked what they noticed and what they wondered. There were some great observations but they were fairly quiet when I asked what they wondered. So I posed the question "How many cups would be needed to make a stack to my height?". This led to a discussion about how the cups were going to be stacked. Normally when I do this activity I tell the class that I want the cups stacked inside of one another. The class really wanted to stack the cups one on top of the other as shown below, so we started there. I figured we could do it both ways and discuss direct vs. partial variations.

Most groups came up with a solution pretty quickly. Once they finished I asked if they could come up with an equation that related the number of cups to the height of the stack. I had a few blank looks and reminded them of the visual pattern we did at the beginning of the class. That was enough to get them going.

We had a bit time left so they created graphs showing the height of a stack of cups vs. the number of cups. Tomorrow we'll see how stacking the cups inside one another compares.