## Wednesday, March 2, 2016

### Day 113: (2x)^3, 2x^3, Dividing Power #'s & Marbleslides Parabola! (VIDEO)

Today's estimation was the clearance height of a parking garage. A majority of students said the clearance height minus Mr. Stadel's height was about the size of Mr. Stadel's head, which most students thought was a foot tall. We discussed how some students added 11 inches to 6 foot 4 inches and what type of regrouping was necessary.
 Reasoning about 6 foot 15 inches.
We started class with reviewing how to factor and simplify (x^3y^2)^2. Then we generalized that to raise a power number to another power you multiply the exponents. I wrote how to properly say twenty to the third to the eighth. Most students assumed that 20 was the base, while two or 3 students in each class disagreed and that 20^3 was the base, and it was because it was multiplied by itself after that 8 times. We then discussed the differences between that expression and 20^3*20^8. Students reasoned that in the latter expression, the two bases were the same and being multiplied so you add the exponents.
 Bridging the gap between yesterday and today's lesson.
During the last 10 minutes of class we reviewed problems 8-63 to 8-65. The first dealt with the difference between (2x)^3 and 2x^3. Once again, the bases here were different. I liked how Leo mentioned that if you substitute 1 into each equation it proves why they are not equivalent. Jumping from (2x)^3 to 2^3x^3 seemed to be a big jump for most students.
Then they worked on dividing 5^7 by 5^3. CPM suggests that a student sees a Giant One in their work after factoring or expanding the exponents. Between this and 2 to the fourth divided by 2 squared, students reasoned that when you divide you can subtract the exponents if they have the same base.

3 to the 4th divided by 3 to the 5th was an awesome conversation. Some students said it was 3^-1 and some said 1/3. I asked if they are the same? Can they be? Must they be? I asked students if 3/3 was equal to 1/1. They agreed. So if one of those 3/3's was 1/1 as a giant one, you'd have a 1 in your numerator. I wrote 3 to the -1 as 3/1 and set it equal to 1/3. I asked students what they noticed. They said it made it flip, invert, or do the reciprocal. This was a big concept that we will continue to talk about.
 First encounter with negative exponents.
 More class work.
In accelerated I introduced the following outline for how I wanted the class period to unfold:
1. In pairs you will login to one of your google accounts. The blog post will later be posted to this persons blog.
2. First you will install a Chrome extension called "Snagit".
3. Then you will work through Desmos Marbleslides Parabolas at student.desmos.com .
4. You will start recording either Challenge #1 or Challenge #2, start to finish, narrating how you are trying to solve the challenge.
5. After you've recorded it, go to your blog at Blogger. Create a new math blog and a blog post. Click the movie icon to the right of the photo icon and click on My Youtube Videos and select your video.
6. Finally, view your blog post. Copy the link to it and click send a tweet from the sidebar on my blog.
I think I am going to make a screen cast of that process that students can watch and so they can test to make sure that the audio is working on their screen cast first, and I would get a higher success rate than 3 partners out of 8 successfully recording and posting their screen casts to a blog post.

I haven't seen my students have this much fun in a long time. It got really loud in class. My colleague next door had to come in and ask us to control our volume and also heard a student tell another to shut up. I obviously don't agree with that language and students can forget their tact and get real excited when they have to share a laptop.

I intended for it to be a 2 to 1 ratio for laptops because I wanted students to work together to solve the fix it marbleslides, the predict and verify, as well as the challenge slides. Our principal and the technology committee came by to check it out and absolutely loved the amount of engagement.
 Ability to anonymize the names. Students noticed and wanted to know who their random name was.
 Some students decided to add horizontal and vertical lines to act as walls and ground floor to control the marbles.

 Some students misinterpreted the changing of the x coordinate of the vertex as changing the y intercept. Some students also thought it may get thinner.

Since I was having students do a screen cast we did not have any whole class discussions. From their short responses though I can target topics that I would like to discuss further such as:

• When they predicted what would happen when you change the a value from 0.2 to -4, some students only said it would be negative, or go upside down. Some omitted that the parabola would also get steeper and decrease quicker. Some students said it would be skinnier, which works.
• The next prediction is when students change the -3 in (x-3)^2 to 2 and predict what happens. Some students simply said it would move to the left, while others gave the exact coordinates of the vertex being (-2,1). Some students thought it would change the x coordinate from 3 to 2, which is a misconception.
• In the 3rd prediction, they change the k value, or y coordinate of the vertex value from 1 to a -2. A lot of students correctly said how the vertex's coordinates changed to (3,-2). Some students incorrectly thought the y-intercept changed to -2. Some said the vertex's y-intercept changed, which is mixing up two concepts and imprecise language.
• When students had to correct an equation to get the stars, one pair decided not to take it seriously, which I will talk to them about. Some students correctly described how they would change the equation to move the vertex. A lot also knew to make the a value a negative number. **Only one partnership discussed that you would also have to make the value more negative to make it wider** which is a great point to bring up. Some students described how they'd changed it, but did not give reasons.

And I have to include the 3 videos. They're all great. Markos and Zoe used only one parabola.

Nicholas and Robert used an interesting looking parabola in their silent video.

And Nuri and Annabel do a great job explaining their ideas and the execution of their plan. Annabel exclaims how much fun she was having.