This estimation was one of my favorites of the year. Today students estimated the value of the complete cent sign on Day 150 of estimation 180. It's an incomplete cent sign made of pennies. I was impressed when at least one student in a few of the classes looked at the vertical line of pennies was along 5 tiles. They counted 8 tiles in it, and estimated it was 1/5 of the whole line. Therefore 8 * 5 is 40. Then they said there was 20 in the curved top part, and it was 1/3 of the curve. 20 * 3 is 60, so 60 plus 40 is 100. I thought that was awesome thinking. Students also saw it as a reflection so they doubled what they saw and then added more.
Circles are a 7th grade standard and I feel students need practice with it, especially if I expect them to be successful with finding the volume of cylinders and cones. So, here's the plan:
Start by seeing what students notice about the following image (image not mine, attributed to godfathers.com) :
- **UPDATE**Students noticed so much from this image. They thought it was Al Capone. (It was actually a pizza chain caleld "Godfather's Pizza" I randomly found by Googling pizza sizes
- Students noticed the number of slices increased by 2 slices as the size increased (!!)
- all of the numbers are even
- the circles all touch at the same point on the left
- the image is half red and half black
- the inches don't grow at a constant rate. Mini to small is 4 more inches, so is large to jumbo. The other 2 increase by 2 inches
- a medium is twice the slices and size as a mini
- concluding that when you think pizza, think circles, and think they are measured by their DIAMETER
- Hopefully students notice that the size of the pizza is measured in inches. What part of the pizza is measured in inches? The diameter!
- Draw a circle with a diameter of 3 inches. If we know that, what else do we know about the circle?
- Solve for circumference, then show this Geogebra circumference visual proof: https://tube.geogebra.org/student/m19655
- Then ask students to find the area of that same circle.
- Then show students this area visual and discuss the questions that are asked: http://tube.geogebra.org/m/1845061
- Reveal answers with this visual: http://tube.geogebra.org/m/1844529
- Then find the volume of a cylinder with that same circular base, given a height.
- Then mathography, and then assessment.
We didn't get much time to practice finding volume of cylinders but we calculated circumference and saw how it was represented on the coordinate plane in the first Geogebra applet. Then we investigated the 7 questions here http://tube.geogebra.org/m/1845061. Unfortunately it labeled the circumference but students realized it was similar to this first applet as the first outer layer of the circle peeled off. Thank you Kate Nowak for tweeting it out:
Well this is super cool and I have never seen it before. For proof of circle area formula fans. https://t.co/dLfJiGvkZi— Kate Nowak (@k8nowak) May 11, 2016
Students enjoyed the animation and I feel that triangles are a bit more accessible to students then parallelograms.
My TA's hung up the Wheel of Theodorus projects from yesterday and many students were gazing at them into lunch. I'm glad to see the students put a lot into it and like to see how other people customized it and showed their work.
In accelerated we discussed types of data displays. Some thought possible a scatterplot with their golf data. Then students realized that was 2 different variables. I did see one student graph the distances with the shot out of 10 as the x axis.
Students reminded me of the 5 important numbers of a box plot. I also asked what other types of ways can we evaluate the data besides the median?
These are the situations we did not discuss yet. Any ideas how to spice of volume for these?