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.|
|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.|
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.
|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.|
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.|