Thursday, November 19, 2015

Day 55: Pattern->Table->Graph & More Elimination

During my prep period I started grading students peer to peer feedback post its where I asked them to give a poster 2 stars and a wish. The 2 stars would be positive and math specific feedback. The I wish would be a statement or question that would help the group improve their work. The student on the left said he wishes the "graph was shown in a bigger, clearer way." The student on the write is much more specific and wishes "your intervals on your graph were by something smaller than 10 so the coordinates can be seen more clear." WOW! He really understood the point of being specific in his constructive feedback!!!

I loved today's estimation. Students estimated a tall cylindrical vase. I participated in 2nd period and estimated you could fill 3 cans full of water into the vase. That ended up being close. A lot of students thought it was only double. Only a few noticed the difference in the diameters of the bases.

Class started with observing the similarities and differences between tile patterns 1, 2, and 3. Students said they were all growing by 4 tiles and in different places. They all had a different number of tiles in Figure 0. Then they told me the equations of each pattern. This prompted them to observe any connections between the tile patterns and the equations. They noticed they all had 4x and grew by 4 tiles. They also had plus 1, 2, or 3, which were the different amounts of tiles in Figure 0.

Then they completed the tables based on the tile patterns. Then I had them space out their x axis by 5 square units so that they could see the relationships between the tile pattern graphs. After they color coded the growth to each line, we had a class discussion. Instead of going through parts a, b, and c explicitly, I asked students to tell me what they observed and noticed about the graphs.

They said the first 3 graphs were 1 apart. I asked why that could be. Students noticed that the 3 lines had different Figure 0 amounts that were 1 apart. They also noticed that they were parallel and grew by the same amount. A few students noticed that tile pattern 4 intersected the other 3 lines and only grew by 2. It also shared a point with pattern 3 because they both had 3 tiles in Figure 0.

I asked students how you could figure out the number of tiles in Figure 0 by looking at a graph and not the pattern. They said look at the y axis. I asked if that point had any significance or if we can call it something. They told me it's the y-intercept. So, students made the connection that Figure 0 is the y-intercept. I told some of the classes that that's the beauty of looking at an equation and being able to know where one point is right away.

Then students investigated a graph of 3 tile patterns.

In accelerated I handed back their assessments. We had a nice discussion about common mistakes with negative exponents 4^-3. I asked how we could rewrite it. They told me 1/4^3. So then I asked what is 1/4^-3? Davin had a nice explanation. He said make the 1 in the numerator a 4^0. Then subtract the exponents. 0-(-3) is 3 so it's just 4^3 or reciprocal of 1/4^3. Here is his reasoning:

Students practiced the elimination and substitution method. Then they were presented with a problem that needed to be multiplied by 2 on both sides to get it to cancel. They were also challenged with a problem hat interpreted a no solution. Students reported it meant the two equations were parallel because there is no point of intersection.

The final challenge was a system where they figured each equation would be multiplied by different numbers to get a common multiple that would cancel. I heard groups discuss this problem, but we didn't have enough time for a student to demonstrate it.

The last 10 minutes were student volunteers coming to the board to show how they solved each problem for closure.

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