As you can see in the picture, I asked students to volunteer how you can subtract 425 from 1000. I was delighted when students offered repeated subtraction, 1000-400=600, then 600-25 is 575. Then some offered counting up, like counting back change at a register. A student said add 75 to 425 to get 500, then add 500 more to get 1000. That's 500+75 which is the difference, or the answer.

Some students offered the old fashioned way of crossing out and regrouping after they knew what the answer was. I asked if it's easy to make an error that way. They agreed it was. I then wanted to know them what exactly the 9, 9, and 0 meant. It was 900 + 90 + 10. A lot of "ohhhhs." Another student offered 1000-500 to get 500. Then since you subtracted 75 to much, add 75 back on. I tried to stress that when you put -500 and +75 together you get your -425. I finally suggested 1 of my favorite strategies that no students mentioned, which I call take one give one or sometimes with money, "take a penny, give back a penny." I asked what's 1000 -1 everyone? 999. Then you subtract. I commented that notice I regrouped NOTHING here. Then what do I do after subtracting? Add the 1 back on. I circled the -1 and +1, and showed we basically just used a zero pair, relating to our work with simplifying.

In class, students simplified equations, determining their answers (x=-3, x=0, and x=any solution). With 15 minutes to go in class, for closure, different student volunteers operated my computer with the virtual tiles while other volunteers. Students aren't used to seeing an x tile and no tiles on the other side, but some picked it up quickly. If you see nothing in math, it still has a number, zero, because you can add zero to anything. Some students thought infinite solutions problem was no solution or 0. If they thought 0, I asked them to pretend -x was 0. if -x=-x, is it true if x=0? Yes. Try it for another.

Finally, there was a great problem, -x+2=4. Students were able to tell me using their background knowledge how to interpret -x=2. I took a picture of it. Basically, students said to multiply or divide by -1 on both sides, which involves physically turning over all tiles both sides. I challenged students to show me another way. They said to add balanced sets of positive x and then add balanced sets of -2 to make zero pairs on the right. It was great.

In accelerated, we reviewed how to interpret y=3 and x=4 equations. Basically, I asked them what does y=3 mean? It means that the y-coordinate is 3. So I said give me a couple points that are on y=3. They gave me let's say (1,3) and (4,3). I asked them to calculate the slope. Even when they saw 0/3, some students said undefined. So we continued to discuss this. Then graphing it, and relating it to rise over run. If you have a pizza, and you ate 0/3 of it, you ate no pizza. It's possible.

For x=4, same routine.They calculated it, and there was still a bit of uncertainty. Some once again said undefined, which is correct. How can we prove it? They related it to rise over run. The graph rises towards infinity, or any number, and it does not run, or move horizontally left and right. We then summarized with y=3 is a horizontal line, and x=4 is a vertical line.

We did not get to start the next lesson, because many had not finished from yesterday. So we analyzed a volunteers scatter plot after reviewing why time was the dependent variable for the biking data and distance biked was the independent variable. They then wrote equations of the lines of best fit and then confirmed their predictions with it. Instead of starting the section I had planned, they worked on the last problem of the section instead.

Oh. 1 more thing. On a students assessment they don't understand the easier way to graph this slope but mathematically it is 100% correct. I thought it showed a lot of ingenuity.

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