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:

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*= m

*x*+ b form) and elimination (for equations in A

*x*+ B

*y*= C form).