Tuesday, May 10, 2016

Day 156: SA & Vol of Cylinder & Inverse Functions

Students estimated a roll of dimes. Some students said it was as many pennies because of the thickness so 50 dimes. They multiplied that by 10, to get 500 cents.

The subject matter was surface area of a cylinder as well as the volume. I took the opportunity to review what we discussed on March 14th, pi day. I asked students what we did after they traced a circle and were given a piece of string. They remembered they measured the outside of the circle, the perimeter, also known as the circumference. Then they measured the diameter and compared. They saw then that the circumference was about 3 times the length of the diameter, which ended up being pi or approximately 3.14. Then I asked them what we figured out after folding, cutting the circle, and rearranging it. They said the area formula which was pi times radius squared.
Re-establishing background knowledge and showing how to sketch a cylinder.
Then they worked on the surface area. They had to first interpret the lateral surface area as the shape of a rectangle and multiply 52 by 40 to get 2080. I discussed the mental math strategy of multiplying by 4 which is the same as doubling twice. Then students realized the circumference of the base was 52 inches. Some realized they needed to reverse the circumference formula to find the diameter by dividing 52 by pi or 3.14. This resulted in about 16.6 inches. They then divided by 2 to get the radius, and then found the area of the circle. They doubled this result because there were 2 circular bases and added that to the lateral surface area they found.
Lots of steps and labeling.
I love Darren's idea with his composition book. Fold the page after finishing that page to instantly know what page to flip to the next day. Love this ingenuity.
Had a mostly civil argument with a student about approximating irrational numbers. The student thought that for approximating 51 it was 2 away from 49, and 51 was 13 away from the higher perfect square 64. The 2 somehow meant .2 in the answer.
Through carefully chosen examples, I showed that the square root of 66 would have a large denominator with the wider range between the perfect squares. 66 is 2 away from 64, and 64 is 17 away from 81, so it's 8 2/17. When you convert 2/17 to a decimal it gets you .11 which rounds to .1, not 0.2.
More work.
In accelerated students were introduced to inverse functions. First they examined this function machine and described the pattern in words first. It was multiply the input by 3 and then add for. The equation was f(x)=3x+4. Then in part c they described the inverse of the function, which students said was basically reversing the order of operations and changing to the inverse operation each time. So this was subtract 4 and then divide by 3. They understood the input would be subtracted by 4, and the whole term would then be divided by 3.
Using inverse function notation.
Students came up and showed how they wrote the inverses of each of these functions.
After class a student wanted to review how to graph a quadratic equation as well as solve a system of linear and quadratic functions algebraically.
In periods 2 to 5 I collected the Wheel of Theodorus project. Some of them turned out astounding so I will be posting photos of those tomorrow. These were the requirements of the poster which I used to grade them.

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