Before beginning today's activity I wanted to clear up some issues that I had seen in Friday's homework around multiplying binomials that had a constant in front. I asked them each to expand and simplify the one shown below and then we looked at a number of ways of tackling it.
Then, on to today's activity: stacking cup systems! On their desks were 10 of each type of cup and this handout, which I copied on ledger paper.
Then I gave them their task for the day: given Styrofoam cups starting on the ground and red cups starting on the desk, determine what equal number of both types of cups would produce the same height.
They started collecting data, some groups more precisely than others. It was interesting to see that some added 2 cups, measured, added another 2 cups, measured, etc., while others added 1 cup at a time but did not use all 10 cups.
I circulated and helped them ensure that they were measuring vertical height, not along the side of the cup. My conversations with several groups about their data meant that they had to start collecting data again. For example:
S: "Each cup makes the height go up by 1 cm, so at 10 cups, the height is 17 cm."
Me, measuring 10 cups: "Is that 17 cm?"
S: "No, that's 19.5 cm..."
There were a lot of conversations about what the numbers all meant. Rate of change was interesting to talk about, especially for groups that had added 2 cups at a time. The fact that the model says that there is a y-intercept when in reality we know that 0 cups have a height of 0 cm was another avenue for discussion. One group got as far as graphing, but they had neglected the fact that the Styrofoam cups needed to start on the ground so they had to look at their models again.
Tomorrow we will have a quiz. Here is the homework I gave them today. After the quiz, they will keep working on their cup stacking systems so that they can then test their results!
Today's plan: warm-up, one more cup stacking system, then move on to volume and surface area of pyramids.
The warm-up was this Daily Desmos, where they have to determine the equation of the line and we check it on Desmos:
I had them work in table groups on the big whiteboards. Here is some of their work:
Next, I had them go around the room looking at other groups' solutions and making comments on the work. Some fixed errors while others wrote comments like "where did this come from?" and "show your work". It was interesting. One group had a blank board (they were a little off-task) and they actually took the time when students were looking at each other's work to find the slope. We went over some of the errors and cleared up some misconceptions. I also had them go to Desmos on their phones to check their equations.
Next we went back to the cups. I put up the equations for the Styrofoam and red cups on the board. I assigned them random groups to answer this question: if the red cups started on the desk and the Styrofoam cups started on the floor, when would both stacks have the same number of cups and reach the same height? Many groups answered a different question - they made the height of each stack 76 cm (the height of a desk) and solved for the number of cups.
I let them struggle with this a little longer after explaining the difference between my question and the one they answered. I had my red cups travelling around the room with me as props to help me explain what I wanted them to find.
One group thought they had it solved so I pulled out the cups and we tried it out - and one stack came up short. Back to the drawing (white) board.
Eventually I had to stop the whole class and talk about what we knew and what we were trying to find. We went over what each of the numbers in the equation represented and whether it would change if you started the stack at a different height. We got to this point together and I let them continue in their groups:
Here is what they got:
The last picture, above, was for a group that calculated that they would need 65 Styrofoam cups to reach the desk so they were changing their Styrofoam cup equation by subtracting 65 from 7 - great strategy, but they did not finish : (
The consensus was about 133 cups and this is what it looked like:
Very close!
We still had a few minutes left and they clearly needed a specific task so I gave them this question to do as an exit card:
Based on the results, I think they understand how to solve these equations (yay!). So much for pyramids today. That will now happen on Monday.
Today's warm-up was this visual pattern:
Most students saw this growing the same way:
We discussed the work they had done yesterday and they seemed to be on the right track. So we returned to this question:
We reviewed the equation relating the number of cups to the height for the Styrofoam cups, emphasizing what each variable represented (I actually wrote out the words and did not use x and y). I then gave each group 5 red cups and asked them to come up with a model. It took some work but they started measuring and figuring out the rate of change. Once most groups had an equation I took one group's data (they had a constant rate of change) and we analyzed it together.
We then set up the system to find out when we would have the same number of cups and they would reach the same height. The question they were supposed to be answering had the red cups starting on the desk, but I didn't think their models would good enough for that. We got a result of 10 cups that should give the same height and this is what it looked like:
Hmmm. Clearly not the same height. What could have caused this? The algebra we did to solve the system of equations was correct, so the model for the red cups must be the culprit (we were consistent with our model for the Styrofoam cups on Monday). I asked for other groups' models for the red cups(written in brown, above):
h = 0.3n + 10.5
h = 1.2n + 2
h = 0.5n + 10.5
So many issues of values that were simply not reasonable. Could the red cup really be 2 cm without the lip? Was the lip only 0.3 cm? Back to the drawing board, as they say. With only 3 minutes left in class I asked them to try collecting data again. I am not giving up on this so there will be more tomorrow!
I was away in Kingston (Ontario) yesterday to participate in a "Think Tank" - we were sharing our experiences around spiraling with activities and seeing what teachers in other school boards in eastern Ontario have been doing.
My fantastic substitute teacher did her best with my 2P crew, but they were not very cooperative. They did an estimation 180 as a warm-up and then started on some more cup stacking questions, found here. You can tell that I stole much of this from Dan Meyer - here is his blog post about it.
Next they had to figure this out:
And they were going to test out their answers, but I think that's when things fell apart. Sometimes working with concrete materials becomes a distraction. However, I will have a conversation with the class this morning and we will get things back on track - cups and all!
Today's warm-up was a counting circle. We started at 27x - 52 and added -2x + 5. Some students were intimidated at first, but quickly realized that they needed to deal with like terms separately.
The picture you see is of Vector from Despicable Me. One of my Calculus & Vectors students drew it, framed it and wants it to stay permanently on the whiteboard!
I started today's activity by standing, holding a stack of Styrofoam cups and asked my students what questions came to mind. It did not take long before someone asked "How many cups would it take to reach your height?". Boom! I displayed their random groups for today and explained their task to them. Each group got 10 cups and had to determine how many cups it would take to reach my height. They picked up the handout and started measuring.
Someone asked how tall I am - I said they could measure me. Two groups did, but were not very accurate, as it turns out. Most students were not being at all precise with their measurements. They said the lip was about 1 cm and 10 cups were 20 cm tall and thought that was good enough. It took some prompting to get them to be more accurate and reason through whether the rate of change made sense (some had RoC of 1 then 1.6, then 1.2...). They all eventually got to an equation, but then all but one group whose equation was correct put my height in for the number of cups! I asked them to talk about what each variable represented, but they did not see any issues with their work. Here are their results:
The closest group was 5 cups off - in previous years they were much closer. We talked about how they should have been using their equation to calculate the number of cups versus what they had actually calculated. We also discussed the sources of error.
To close out the class I showed them a short stack of red cups that look something like this
and asked if they would need more or fewer of these cups to reach my height.They reasoned through this really well and even showed that with only 7 cups, the Styrofoam cup stack would already be taller.
