Thursday, February 18, 2016

Day 103: Equation of Trend Line & Quadratic Reps

Today's estimation was the pages in To Kill A Mockingbird. Students thought it was about one and a half times as long as Charlotte's web. There's a LOT of book page estimating coming up, so I feel I'm going to skip some so I can squeeze in some Which One Doesn't Belong Warmups.

We discussed a great homework problem where 4 students got 4 different answers for the slope of a line. They got 3/4, -3/4, 4/3, and 3/4. I started with asking students what the answer couldn't be. They saw, probably with help from the Slope Dude Says video, that it was a line with negative slope, so it couldn't have a positive slope. Then students described where they saw the lattice points and pointed out that the vertical change was negative 3, and the horizontal change was positive 4.

Then we reviewed the topic from yesterday because it was a bit rushed at the end. I asked a random student who I thought may not have calculated their slope yesterday for their 2 coordinates for number of times they wrote their name in 1/2 minute and their prediction for 4 minutes. I asked how we could find slope? They said draw a triangle. Then I reminded them that we could count squares, but my example had no axes drawn, so how could we do it more efficiently? They said that you can subtract the y coordinates to find the vertical change, and subtract the x coordinates to get the horizontal change. After getting an equivalent ratio without decimals, then dividing, they saw their slope was their unit rate, for how many times they wrote their name in 1 minute.
Students reminded me to find 4 minutes, you multiply 10 by 8 because there are 8 1/2 minute chunks in 4 minutes.
Then, I introduced them to the slope formula, that they had used without really knowing it. I told them they would see it again next year in Algebra I.

I then introduced the Learning goal which was to make a scatterplot, write an equation of the line of best fit, and use it. The success criteria was to use what you know about m and b to write the equation in y=mx+b form. Use substitution to make a prediction. I think I may have given too much info to use substitution, but they had to articulate what number they were substituting for what variable, which was good later.

The main problem was a 30 mile charity bike ride. A girl, Aurelie, had been training, and wanted to predict how long she would take to finish the ride. They convinced me that the dependent variable was the quantity we wanted to predict, so that was on the y axis. I related it back to Newton's revenge where we wanted to predict Yao Ming's reach because we knew his height.

Here Dominic had a nice graph, showed how he got the slope. We talked about the substitution, but he didn't show his work for that yet.

Here the substitution was shown.
The really cool part about this lessons progression, is that you make a prediction with your line of best fit first, and it's not that accurate. Then when students write an equation and use substitution, they get a super accurate prediction that is within the range they thought it was on the graph. A lot of a hah moments.
My new favorite strategy: showing a dashed line to show exactly how you used your line of best fit to make a prediction.

Modeling the substitution work.

An easier substitution.
More substitution.
In accelerated students finished their graphs of the 4 people in the water balloon throwing competition. Before they continued, we reviewed 4 homework problems. Two of these type are on the assessment. The first one was difference of squares and students saw that both terms are squares, so you put the square roots in each factor and have one plus and one minus sign, HENCE "difference" of squares.

I forgot to label the second official method, but it's called perfect square trinomials. It's basically doing a diamond with mental math. Problem c was also a perfect square trinomial but with a positive middle term. The final one was a difference of squares problem that had to be factored completely first. The student divided all terms by 5, factored, and left it at x squared minus 9. Another student raised their hand and said that's not right because they forgot to put the 5 outside the parentheses. I complimented this student for politely showing how you respond to an incorrect answer.
Homework review.
Students found the x intercepts and the coordinates of the vertices of each parabola. There were great conversations about how they found it. I made a point to talk about the situation to graph where the water balloon was launched from the 10 yard line and landed at the 16 reaching a height of 27. They interpreted that as x intercepts of 10 and 16 and the y coordinate of the vertex was 27. I asked them how we could find the x coordinate. Another student raised their hand and said it's halfway between the x intercepts because the vertex splits the parabola in half. I was impressed. Therefore, there's a distance of 6 between them, half of that is 3, so 10 plus 3 is 13, so 13 was the vertex. Here are their results:

They also showed me how they got 8.5 for the x coordinate of the vertex, as well as plugging it in for x to the equation to get the y coordinate of 30.25.
We also discussed proper way to write domain and range.

To finish off, I was super impressed with this students work for the graph where only 3 points were known. He went ahead and did some math work and got an equation. I said well, let me put it in Desmos and see if it's right! And it was. We got some oooohhhhs and aahhhhhs when we saw this. I had to take a picture of how he did it. I haven't taught him any of this but he literally wrote a system of 3 quadratic equations. You can see he subtracted a pair of them to eliminate the C variable. Then he eliminated the b variable, to solve for a. He plugged a into one of them to solve for b, and then solved for c. He got y=-3x^2+78x-480. I was floored and it's probably the most excited I've been by what a student did through their own determination. I did not teach this student any of these methods. He learned them in Kumon, and I am super impressed that he applied his techniques properly and in the context of our CPM textbook!

Systems of quadratics!!! (not an Algebra I topic, to my knowledge....)

Confirmation with Desmos... very impressed.

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