I wanted to use the ideas of a “thinking classroom” today for partitioning line segments. Due to covid and keeping students separated, I had each student work on their own. This was great for seeing what each student could do and forcing each student to do their own work. However, it was not great for getting all students to help each other and move through more problems. More about that later. It was a two-hour late start so we only had 35 minutes of class today.

I started off with this question:

If H = (0,0) and W = (0,12), where would you place S between H and W so that HS = 2 SW?

I also said they could think of it like… H is Home and W is work, and I wanted them to find Starbucks if the distance from Home to Starbucks was twice the distance from Work to Starbucks.

Then I went around class checking on each student as they got solutions. When they got the correct answer I would adjust the problem in some way.

1st adjustment – Make W = (0,27) or (0,21) different for each student until a student is solid with this kind of adjustment.

2nd adjustment – Make W = (9,6) or (12,3) different for each student until they got it.

3rd adjustment – Make W = (15,10) and HS = 4 SW. So change the proportion as well.

4th adjustment – Make W = (x,y) and HS = k SW. Ask them if can find a formula for S using x,y, and k.

I was really impressed with my top students in the class. 4-5 kids in each 10th grade geometry class got to the 4th adjustment and 2 students per class came up with a correct formula. What a great accomplishment.

The middle of the road students made it to the 2nd or 3rd adjustment and since I gave them multiple problems at each stage were able to practice a few problems at each level. I feel like they got some good practice in on the problems.

The students who never made it past the first adjustment are my main concern. Everyone was able to solve the first problem with minimal assistance. Because I didn’t have students in groups, I was running all over the classroom trying to check 3-4 times more problems than I would if students were in groups. I didn’t have enough time to stop and help this less productive group of students as much as I would like to.

I may try the same thing again tomorrow but also move H away from the origin. I like this method of doing these problems because I think that the process of trying to figure them out is more important that using a formula I give them to just find the answer.

If I show them how to use a formula to find these answers, then I feel like I’m robbing the more driven students the opportunity of figuring the problems out in order to help some students get a correct answer without knowing what they are doing. These same students will not likely remember the formula or when to use it in the future without the understanding of why it works.

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