Day 71
We started with more Estimation 180 today. We did about 5 of them. This one (day 160) was the most difficult for them:
The guesses ranged from 5500 to 2.5 million. All of which are far from the correct answer. Clearly my students need to build a few more Millennium Falcons!
We started today's cup stacking with the Styrofoam cups. This was basically a repeat of what we did yesterday, but everyone was working on the same method of stacking.
They got right to work. I gave each group 6 cups and suggested they would need a ruler. They measured and collected data and wrote an equation and made a scatter plot. I circulated and looked at their data.
One group had 1 cup with a height of 8.5 cm, 2 cups with a height of 17 cm, etc. I stacked cups and asked if the height was actually 17 cm.
I could tell which groups hadn't measured very well - they said the lip of the cup was 1 cm. When I showed them 6 cups with a ruler next to them and asked what the height was, it was apparent that they needed to be more precise with their measurements.
It was really interesting to see their equations. The lip of the cup was about 1.3 cm and the rest of the cup was about 7 cm, so the equation should have been
height = 7 + 1.3(# of cups), but they had equations with y-intercepts (height-intercepts) of around 8. They had taken the first value in the height column as the "fixed value" without realizing that it also had the height of the lip. So we came up with this table which caused issues because 0 cups have a height of 0 cm.
They saw how to fix their equations and came up with 127.5 cups to reach my height. We tested this out and they were exactly right!
Next I asked them to work on the relationship when they stacked the red cups this way:
Only one group had finished this yesterday (they got to stack the cups a different way). This is the data they came up with, along with the pattern in the # of cups and the pattern within the pattern.
They had to continue the pattern up to 10 rows then plot their data. They saw that it was quadratic so the next step was to input the data in the graphing calculator to get its equation.
Now the equation related # of rows and # of cups, not height. To figure out the # of cups needed to reach my height they needed to figure out how many rows first. They knew the height of one cup from yesterday's work.
Sadly, we didn't get a chance to test this one out today. And I am doing in-school PD all day tomorrow so we likely won't get there.
Day 72
Tomorrow is tying cup stacking to systems of equations. They will start with this, stolen from Dan Meyer:
Hopefully that will help them with the next question:
I will leave the cups out for my substitute teacher to get them to test out their answers!
That will be followed by some solving systems questions (in context) using both substitution (for equations in y = mx + b form) and elimination (for equations in Ax + By = C form).
Day 70
Today we started cup stacking! But first we did some Estimation 180. It was good to get back to it.
I have done a version of Dan Meyer's cup stacking activity for several years, but this time I started with "cup stacking wide open" - they could stack the cups any way they liked! I started by putting one red Solo cup on the floor next to me and asked them what questions came to mind. Fairly quickly someone said "How many cups would it take to reach your height?" - bingo! I put them into random groups and handed each group a sheet for them to draw a diagram of how they wanted to stack the cups, along with a guess that was too low, one that was too high and an actual guess. Once they had that done each group got a handout and 10 cups. I also wrote my height on the board.
They had fun initially stacking the cups and having them fall which made a lot of noise. Good noise, though. You know, the trying something and having to adjust because it didn't work kind of noise. Only one group chose to stack the cups one inside the other; all of the other groups stacked the cups like this:
They collected data which they graphed then came up with an equation to represent the relationship between the height and the number of cups. They were good at this because they understood that the height was dependent on the number of cups and that each cup measured the same. They came up with their calculation for the number of cups to reach my height then moved on to a different way of stacking the cups.
They tried a lot of things. Here is a sample:
Many of these options were deemed unbuildable (I just made that word up) so the one in the last picture was what most groups started tackling. It is complicated because they are looking at the relationship between the number of rows and the number of cups, not the height. The height is related to the number of rows, but it is definitely not as straightforward as the first stacking method. They will continue working on that tomorrow.
We spent the end of the period trying to see how close their predictions were to the actual number of cups needed to reach my height. But the towers kept falling down! The vents were blowing cold air and the draft seemed to be everywhere so we spent a lot of time getting nowhere. Eventually, with the help of someone holding the tower part way up, they got the 16 cups stacked and saw that the top of my head was somewhere between cup 15 and cup 16 exactly as they had predicted. Yay!
Day 69
Today's plan was to finish up the Shell Center matching activity then start Cup Stacking. But we never got beyond the matching activity. Some groups did finish, but for others it was like pulling teeth. They even asked me how I had so much energy first thing in the morning. If I didn't then we would be doomed! It took a lot of work to get them to associate the different representations, but they made progress. Here are some of the finished products:
And tomorrow we will do cup stacking!
Day 68
I organized desks in rows for the students who needed to continue working on their tests, while the others worked in groups on a matching activity from Shell Center. I really like this activity and have done it for a few years now. It involves connecting linear and quadratic expressions, words, tables and area models. Here is some sample student work:
And zooming in a little:
I also asked them to add a graph to each set. Only one group started on this, so I think they will all need a little more time to work on this activity tomorrow.
Day 67
Day 2 of cycle 3 test was as expected. A lot of "I quit", following by me saying "No" and asking a few questions to get them going again. Most finished, or did all that they could, and a few will continue on Monday.
I have been thinking about the tree activity since meeting up with Alex & Sheri on Wednesday. One of the things we discussed was that we wanted more activities that covered multiple curriculum expectations. This activity has a definite focus on the measurement expectation, but I think we can tie a lot more to it:
* Growth of the tree over time = linear relationship
* Growth of different species of trees, one starting out taller than the other = system of equations
* Area of root bed (diameter is approximately equal to the height) vs. height? = quadratic relationship
* Finding height of the tree = similar triangles / trigonometry
* Number of board-feet of lumber vs. diameter = quadratic relationship
I plan on doing this activity earlier in the semester next time around, with all these added parts. Comments and suggestions welcome!
Day 66
Today was test day. The fact that the building was shaking during much of the period due to paving going on behind the school, did not help students focus well. However, we got through day 1. No one finished the test today, but they have tomorrow to work on it too. I hope some will spend some time tonight reviewing concepts with which they struggled. I will continue in my role of cheerleader - encouraging them, prompting as necessary and generally keeping them moving forward.
I had the good fortune of spending today working with Alex Overwijk and Sheri Walker. We are all teaching grade 10 applied math this semester and have been spiraling through the curriculum doing activities most days. Saying that this has been a great experience is an understatement. I have no doubt that my students have benefited from learning this way (even if they may not always realize it). I would not teach this course ever again in any kind of "traditional" way.
Today we spent time going over the summative prep activity, which Alex did last semester and blogged about here. We talked about how to make it work, fixed up some of the pictures and created a 6th theme. We also spent time looking at Graphing Stories, Daily Desmos, Mathalicious and other cool sites before working on a new activity together. We came upon this activity by the awesome Robert Kaplinsky (@RobertKaplinsky). Hot dogs + linear + quadratic = cool! We liberally stole the idea and adapted to mini-marshmallows so that our students can actually collect the data themselves, without getting sick (hopefully). Together we came up with some good questions to follow up the data collection/graphing/finding equations part of the activity. I really love the way we bounced ideas off each other and thought about the activity in different ways. And I think what we ended up with will be good
Throughout the day we had conversations about what we are looking for in activities, what makes them "good". We talked about the effect teaching this way may have on our students - about how they will retain more of what they learn and that they will hopefully have a more positive attitude toward math. And we all know that this is a journey where the finish line is always moving so we will never be "done". I'm just really fortunate to be making the journey in such good company.
.jpg)
.jpg)
