## Wednesday, February 17, 2016

### Day 102: Slope of Proportional Relationships & Different Reps of Quadratics

Todays estimation was how many pages in the book Charlotte's Web. I liked that students compared it to their composition notebook with 100 pages and said the book was about 2 composition notebooks tall, so about 200 pages. Other students compared it to the CPM textbook on their desk, and the book Anne Frank that they're reading. I like that some also used the pencil as a size reference.

We started a new routine of going over one essential homework problem from the night before. I don't like using class time for homework, but I think if I limit it to one problem it won't take up that much time and I am hoping that students see the benefit of homework and will be more motivated to do it, seeing it as practice for assessments, even if it is worth zero percent of their grade. Once again though, I require evidence of homework completed to do corrections and retakes, so it's in their best interest to do it.

The homework problem was great because it involved two graphs that appeared to have the same slope, but one graphs x axis was scaled by 1/2, so it changed the slope. This ended up being a great preview for reading how they scaled their axes in the activity.

The lesson in the textbook was about expected amounts of voters from certain populations. Ms. Demailly and I thought it was a bit dry, so we decided to take one of the sub-problems ideas and expand it out. I would definitely like feedback on the following task that we discussed together and linked is a Google Slides document with comments enabled:

How fast do you write your name?

1. Legibly write your first name as many times as you can in 30 seconds.
2. Predict how many times you would write your name in 4 minutes if you continue at this rate. Show your work and explain your reasoning.
3. Complete this table:
 Times you write your name Number of minutes 1/2 4
4. Is the relationship proportional? Why or why not?

5. Create a properly scaled graph, graph both of your data points, and connect the points. Should it go through the origin? Are minutes your independent (x) or dependent (y) variable?

6. Find the slope of your line and write the equation of it in y=mx+b form.

7. How can you use your equation to predict how many times you’d write your name in 15 minutes?

8. How can you use your equation to predict how long it would take you to write your name 100 times?

9. Draw a line of someone who writes their name slower than you. What’s the equation of their line and how does the equation show that they are slower?

10. I saw a student’s equation and immediately knew they cheated. Their equation was y=20/3x+9. How did I know they cheated?

No classes got to problems 7 through 10. Students were immediately invested when they saw it as a competition and a way that they could see how many times they could write their name in a minute.

First I asked them how they could do this, and after realizing it would be hard to start and stop a stopwatch when you began to write your name and when you were done, they realized you should write your name multiple times and find the rate. I purposefully had students write their name for 30 seconds only, or half a minute. Then I asked them if you continued at the same rate, how many times would you write your name in 4 minutes?

Students had many ideas, some went up by 30 second chunks, others doubled the times they wrote it in half a minute, then multiplied that by 4. Some students multiplied by 8 because there are 8 30 second chunks in 4 minutes. One student mentioned they were scaling up, and 4 divided by 1/2 is 8, so multiply by 8. All of these strategies were evidence that the relationship was proportional. They reasoned you wrote your name zero times in zero minutes, you could double one quantity and double another, so it must be proportional, IF you continue to write your name at the same rate.

 After discussing and deciding that time in minutes was the independent x variable, then I made sure they labeled their 2 data points with their coordinates.

I picked a student in each class that hadn't found the slope yet so that we could demonstrate with them. I asked how we find slope between two points. They said draw a growth triangle, and then label the sides. Some points weren't located on lattice points, so it was hard to get an answer. So, I sketched their slope triangle on the board, and had students think pair share how could I get the exact height and base of the growth triangle.

Students realized you could subtract the y coordinates to get the vertical change, and the x coordinates for the horizontal change. We will formalize this later with the slope formula, but I'm glad they discovered it on their own.

The real AHA moment was when we wrote the slope as a fraction. I asked what we could do with it. Some said double both quantities to get rid of the decimal, then simplify. They said, wait a second, that's how many times you write your name in 1 minute. So, I asked how to write an equation, and since it was proportional and went through the origin, b was zero, and the slope was their unit rate.

 Here you can see students said to multiply 63/3.5 by 2/2 to get 126/7. When you divide, you get 18/1.
 This slope was 147/3.5, which ended up being 42/1.
 Here is some sample student work. A properly scaled axes, labeled coordinates, and a graph.
 Here you can see all of this students calculations for finding the slope, and his equation of y=22x.
 Here you can see the slope of 18/1.
In accelerated we reviewed four homework problems on factoring. Then students worked on a task called Water Balloon Contest. Here you can see a students work of plugging in x values into -x^2+17x-42 to graph the parabola.

I was impressed that some students knew that if you knew the x intercepts, you would know the x coordinate of the vertex because it would be halfway between them. I am going to have that student share that to the class tomorrow. Students will color code each graph, answer the questions, and share how they figured out features of the parabola. This is a great lesson because it uses all of the representations. For 1 person, then equation is given. The other the graph is given. For another, a situation is given. Finally, an incomplete table is given where students will see a pattern.