Wednesday, January 27, 2016

Day 89: rigid T's & dilations. Negative exponents & y=a*b^x from a graph

Today's estimation was a tape measure me that was slightly bigger than yesterday. My favorite reasoning was if was 1 and a half times bigger so we got to do mini number talk on how they multiplied a whole number by a mixed number. 

Here are the success goals for my first class 2nd period. The goals improved as the day progressed. I did not mention reflections as a goal but students still used it as a transformation successfully which pleased me.

I'm taking an online course from west Ed called formative assessment insights and part of the class is picking a lesson and standard and developing building blocks, learning goals and success criteria prior to teaching the lesson. They suggest developing success criteria with the students. It was great. Although it took some time it was time well spent and lots of students wrote it down and participated. 

I like that we had the opportunity to talk about converting meters to centimeters and centimeters to meters in this estimation.
For each learning goal, I had students talk with their elbow partner and group about what the success criteria should be. For translating, I asked what are you SPECIFICALLY doing to the coordinates to move horizontally? What about vertically?

For the second goal, rotating, I said to accurately describe a rotation, there are three requirements. Discuss then tell me. They came up with these 3: how many degrees (90, 180, 270, and we discussed why 360 doesn't matter because the shape ends up right where it started), about what point, and what direction. I asked them to be specific, is there only "one direction?" Pause.... laughter. The students understood the pun and appreciated. They laughed, smiled, as I did.

In this class I got to give them an improvised joke / pun about the direction of a rotation. 5th period got the joke too.
In the last two periods I remembered to include the word scale factor when discussing multiplying both coordinate by the same number, the scale factor.

I had students independently and silently setup their graphs and graph rectangles A and B. Once it was setup they discussed whether the two figures were similar and why. Then they devised steps to transform rectangle A into rectangle B. Halfway through their work I gave them a hint, trying to relate working backwards with equations, with working backwards with their transformation steps. I reminded them that they could start with their end result and undo their dilation. Christian did that here:

Christian undid the dilation by dividing all of B's coordinates by 2. Students were surprised that it ended up overlapped, and not inside the bigger figure.
This group decided to dilate by 2 first, then translate, rotate, and then I believe they reflected across the y-axis. Ethan showed his work here:

I like how the steps are labeled here.
In third period, Mikaela and her group rotated first, then translated to the MIDDLE of the bigger figure, and then dilated. A handful of students understood that it had to be in the middle before dilating anticipating that it would expand out.
Clear diagram with easy to follow instructions.
Here you can see the success criteria and the steps easy to follow.
In accelerated students picked up where they left off. They were making sense of raising to the zero power, to the -1, and finally to the -2. They realized that 1/2 to the -1 was 1/2 divided by 1/2, then divided by 1/2 again. Then they realized that it was the same as taking the reciprocal. Then they reasoned that raising to the negative 2nd power is taking the reciprocal of your base, then raising it to the positive 2nd power.

In the upper right hand corner we discussed their first graph that they had to write an exponential equation of. Students saw that from writing a table of the two points, they could find the multiplier, 7/5, which is your base, or b. The a value, in y=a*b^x is the y intercept, or whatever y is when x is 0. Groups and students within their groups were working at many different paces in this activity.

Examples and generalizing the rules.
The last part of the lesson is a super important learning log. Make up an exponential equation with two coordinates and explain how you find the equation of it. I want students to take pictures of these eventually, blog about it, and then tweet it out so their classmates can see it.
Harrison made a super complicated example, but his explanation seems pretty good. It will be easier to read when I have them type it up in a blog entry.

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