## Tuesday, September 29, 2015

### Day 21: Comparing Expressions & More Rate of Change

Today's estimation was a box of hanging folders. I liked that some students said that they thought the hanging folder was twice as thick as a file folder so the box must be half of a file folder box (1/2 of 100 = 50). Since students overestimated, I took the opportunity to discuss different ways of interpreting percent errors that are 100% or over. Some students freeze up and just write what every the quotient is, as the percent. For example, if the get 50/25 and get 2, they put 2% instead of 200%. Also talked about getting an equivalent fraction with 100 as a denominator. I wanted students to work on their precision of language and not say multiply by 4, but multiply by 4/4, a Giant One. I showed that multiplying by 4 is actually multiplying by 4/1 which changes the fraction.

I dug into students background knowledge of how to compare 58 and 62. What symbol goes between them? They said less than. I asked how they remember the symbol. Some said alligator eats the bigger one and pacman wants to eat more. I took that opportunity to show students my prior blog post, How Inequality Symbols are taught in Shanghai. I asked students to interpret how students learn inequalities using a visual model. Once again, I honored different cultures, and the cultures that some of their parents have.

I then instructed students to build two expressions on the comparison mat. Some students felt better drawing it, but I pushed them to build it, and collaborate on how to properly write unsimplified expressions of each side. They want to avoid the difficulty and simplify then write the expression. I want them to stretch and challenge themselves.

We then reviewed it using the virtual algebra tiles on the projector. They told me how to write the original expressions and told me how to simplify. They saw the relationship between the symbols and the model. Some students remembered to remove the balanced set of x tiles at the end. This can be a confusing point. I thought of an analogy on the spot. Here it is:

Put up 5 fingers on your left hand and 4 on your right. 5 is greater than 4, so my left hand is greater, or more fingers up, then my right, correct? Pretend that x=2, and we are going to put down 2 fingers on each side, or remove x from both sides. I now have 3 and 2. Is the left hand still greater than the right hand? Yes. I think this clicked for a lot of students.

Students then looked at diagrams and described what simplifying moves the person was doing: removing zero pairs, flipping tiles from negative to positive region, and removing balanced sets. They got a half sheet and simplified it and wrote expressions to compare them. We will go over that tomorrow.

Later this week I will mention the Eating Contest analogy, which I will go into more detail later.

In accelerated, we went back to lesson 2.2.2. This was a labor intensive lesson. They had to accurately graph 3 racers in a tricycle race and answer questions about it. A lot of students did not get to the piecewise graph that dealt with domain. Some groups struggled with students working at different paces. I empathize, because it's very hard to be patient when someone in your group is not as focused as the rest and not able to listen, talk, and write at the same time. We will talk about that tomorrow, and I will talk to certain individual students.