## Monday, January 25, 2016

### Day 88: Scale factors & exponential decay (negative exponents)

Today's estimation was Day 81 of Estimation 180, estimating the length of the small tape measurer. I like this sequence because it builds on the previous days. Also, coming up is estimating length of books. I'm especially looking forward to estimating length of songs.

Today I started class by asking students what activity they did with the cut up shapes on Friday. Students remembered we saw which of the figures were similar to the original shape. They remembered that shapes D and G were not similar. So, I asked them to think pair share about what must be true for two shapes to be considered similar after only about 3-5 students raised their hand in each class. They came up with:

 I like how one student knew a regular tape measure had about 25 feet, so since this one was small it was about half of it.
Then students found the scale factor between 2 shapes.
 They saw that the sides of Q were all multiplied by 2.
Here we had a discussion about a new shape, shape R with a base of 25. They had to figure out the scale factor.

 Students talked about how 10 times 2 was 20, and 10 times 1/2 was 5, and 20 and 5 was 25.
In 5th period we had a richer discussion where students found the vertical height of shape R when the vertical height of shape P was 6. They saw that 6 times 2.5 was 15. And they compared it to shape Q and said 3 times 5 is 15.

In 2nd period students had confusion about how to multiply a whole number by a fraction, and I'm sure there were students confused in other classes. Some tried cross multiplying. I discussed how cross multiplying is a strategy for solving proportions. What some of them meant by that was cross canceling or cross simplifying, which is a valid step to have an easier multiplication problem. Unfortunately, some students even confused multiplying fractions with dividing fractions by thinking you had to invert one of the fractions. Nope.

 2 strategies for multiplying fractions
The last problem was predicting which type of fraction scale factors would make a figure smaller or bigger. They reasoned that a fraction greater than 1, or when the numerator was greater than the denominator, would make it greater than 1. They said that a fraction less than 1 would make it smaller. I asked them if a scale factor of -1 would make it smaller. Some thought about for a bit, but some students raised their hand and quickly added the fraction is less than 1 but greater than 0. I discussed how precision of language is so important.

 I forgot to take a picture before erasing but didn't erase this students reasoning completely. She reasoned a tape measure is about 3 inches around. It would go around about 50 revolutions. 50 times 3 is 150 inches. Then she divided by 12, got 12.5 feet and rounded down to 12 feet.
In accelerated students worked on an exponential decay with half life. Students also worked with finding an exponential function when x=-1. They have some experience with negative exponents, but this is the first time they saw it in a real context. They eventually see that raising a fraction to an exponent of -1 is the same as taking the reciprocal of it.

 When the x is zero y is 100. So, 100 is the y-intercept or A value in the exponential. To get y when x is -1, you divide 100 by 1/2 or raise 1/2 to the -1 to get 2, then multiply by 2, technically the same operation.
 After a Mathography where a student said she likes writing with her 3D pen, I was curious what she had made with it. I don't know if she made this especially for me, but she gave me this rainbow with clouds. Pretty cool!