We began with our counting circle. Today we started at 62 and counted down by 13. We stopped after the first student to talk strategy. I thought it was really interesting that he said he just knew the answer, but when pushed a little more he said that he subtracted 10 and then subtracted 3. Bingo! Anyone who didn't initially have a strategy (beyond counting on fingers) now had something to grasp on to. We continued for a few students and I stopped them at student J. and asked what number we would be at when we reached student S. I let them think for a bit then asked for strategy. Someone said that they could multiply. I asked what they would multiply and they said 13 times the number of people. I switched to another student and asked how many students there were from J. to S. and we counted 10. We worked out 13 times 10 (sometime students are reluctant to participate but I wait them out) and I loved that someone said that we could just add the 130 to 42 because they were both negative and we should be at -172 when we reached S. We continued and got there (yay!). Some students are still learning about how to break up numbers to make adding or subtracting easier, but I believe that they are developing this important skill with each week's counting circle.
On to the main activity: Speedy Squares. A little background... another teacher and I created this activity at our first lesson study meeting. I had not used it yet, but really wanted to so our group of four teachers co-planned it last week. Here is the setup:
The answers they had for how long it would take were very interesting. Hours, no, days! In hindsight, I should have had them all estimate how long it would take so that we could compare at the end. Asking how many cubes they would need led to some good discussion. One student said that since the perimeter was about 100, the area would definitely be more than 100. We now had an estimate that was too low. Another said that we would need 2600 squares. I asked what dimensions would give an area of 2600 and we came up with 26 by 100 - so now we had an estimate that was too high. One student was convinced that it was 236 (he did 10 x 10 twice plus 6 x 6). I let another student use a calculator to find 26 x 26 so we now knew that we would need 676 cubes. Actually, each group would need 676 cubes. How many is that? "A lot!" they all said. I showed them my 7 neat stacks of 200 cubes and said that we didn't have enough. What could we do? It took a little bit of circling around ideas to get to a student saying we could build smaller squares and then multiply. The idea here was that we could collect data on the time needed to build smaller squares and then extrapolate (yes, they did use that word!). Okay - now they were ready to go. I showed them the random groups for today, explained the role of each person in the group and set them to work. Here is the handout.
I circulated and made sure that each person had a job and that they understood how to build their squares. The more consistent they were, the better their data was likely to be (I emphasized that they could not switch builders along the way).
There were good conversations about which variable was independent and which was dependent. They are so accustomed to time being the independent variable that they had to think to ensure they understood that in this case time was dependent. They also talked about whether this was linear or quadratic data and were able to justify to each other why it was quadratic.
They used graphing calculators to do a quadratic regression and used their equation to find the time it would take to build a 26 by 26 square. Their results were interesting - ranging from about 20 minutes to about 40 minutes. We could have talked more about whether this meant one group was building twice as fast as the other and how a small change could have a big impact when you got to a side length of 26 - perhaps next time. Instead, I set up part 2:
I told them about James May who, along with around 1000 volunteers, built a real house out of Lego. Here is the link with lots of fun pictures. That got everyone's attention. We talked a bit about the factors that went into determining how long it would take to build a house out of Lego - they brought up the number of people and the length of the work day. I said that they were going to use the data they had already collected to help them figure out how long it would take them to build a house out of blocks. We will pursue that more tomorrow. I did also show them the Lego My House website (link) which determines how many blocks you need based on square footage. We briefly discussed what square footage means and did one example using the website. The results are quite something!
That was all the time we had for today... more tomorrow. I would like to say that my students were FANTASTIC! They were engaged, on-task and really implementing the mathematics they have been learning. It was truly a pleasure to have visitors.