Monday, March 14, 2016

Day 120: desmos graph pi day & quadratic formula

Students estimated the amount of green marshmallows in a pint class. They referred to rows and array ideas to multiply for their estimates. I like that some noted the wider opening at the top. 

I handed back the trimester 2 final and would like a parent or guardian signature on it and back by Friday. After hearing 1 estimation reasoning I reviewed the answers grading scale and how the 36 points were allocated to which problems. 

Some kids asked why we weren't eating pie in class like Mrs. Demailly's class and I basically said I don't believe in eating in class. I asked students to brainstorm everything they knew about circles and pi. I also asked them to write down any questions they had for 3 minutes.

I told students that we hadn't discussed circles as it was a 7th grade standard, but I wanted to do a pi day crash course and assume they didn't learn it last year or remember it. That's why the brainstorm was so important, to make all background knowledge public.

I asked students to trace a circle from a variety of lids and measure the diameter and circumference and tell me their data in that order. I reminded students if they used centimeters, they needed centimeters for both measurements. Same with inches. I noted students may have trouble measuring around the circle so each table had 2 strings for use.

I originally wanted data submitted via cell phone on Google forms but when I shared the form link it was editable by students which caused haywire. I just got the data manually from 10 pairs of students the rest of the periods.

I copy and pasted their data from the spreadsheet into a Desmos graph. I asked students what they noticed. They said it was a positive association. Some said it was linear. They said it looked like a scatterplot. I said that because it was linear we could make a line of best fit by typing y1~mx1. I asked them what they noticed and they said that it was a proportional relationship because it went through the origin. I asked them what else we could say and they noticed the m or growth factor or slope of the line was 3.04 (in the first class it was 3.4... eeek!). Students said it should be 3.14. They said there must have been human error.

Then I went to the board and asked the most important equation we've learned all year: y = mx+b. I asked them what the b value was, they said 0. The m was 3.04. Then I asked them in our table and our graph, what does the x axis represent? They said diameter. So I subbed D for X, to get y=3.04D. Then I asked what the y axis represented. They said it represented circumference, so I substituted C into it, getting C=3.04D or C=3.14D or with the symbol C=pi*diameter.

I asked students how we could solve for pi, and some students knew to divide by the diameter on both sides. I mentioned that cultures from Arab, Greek, Indian and Asian all separately discovered this fact about circles, that any circles circumference divided by its diameter equals the same number every time: pi.

Then they traced and cut out a circle. They folded it into fourths and we derived the area formula. If you haven't done this, it's basically rearranging the pieces of a circle into a parallelogram, whose area is base times height. I asked students what the height was in relation to the circle, and 1 or 2 students could see it was the radius, so I subbed that in. I asked them what part of the circle was the base, and they said half of the circumference. After subbing this in, along with C's formula, 2 pi R, I asked students if it matters what order you multiply in and how we could simplify it. They told me the rest, getting pi r squared, the area formula for a circle.

A student asked me this question and I didn't know the answer. I'll share this tomorrow.
Circle brainstorm.
Proving the area of a circle
More brainstorm.
I should have shown this top left illustration, showing the growth triangle with pi labeled.
In accelerated I put students into new random seats. I passed back their finals and went over the answers. Then students worked on identifying a, b, and c values in a quadratic equation in standard form. Then they substituted the values in carefully to the quadratic formula and found the x intercepts or lack thereof of a quadratic. There were some great conversations, and Nicholas demonstrated his prior knowledge below:
I like the precision of showing the minus sign as -1(-3) when subbing b=-3 in for -b. We also discussed the radicals were the exact answer while the simplified rounded decimal was the approximation.
Here Moreen and Tiffany explained how the used the quadratic equation.
This made my day when a student from 6th period gave me this gift: a knitted pi symbol!
Here are the 2 problems that Nuri and Robert will present tomorrow.

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