- Anticipating. Select a groupworthy task and anticipate how students will approach it (successes and common mistakes).
- Monitoring. Check in on groups to see what ideas are working and which aren't. I have made this much easier by downloading the Google Drive app to my iPhone, and taking photos directly into a folder for that day's lesson, ready to be displayed on my large display screen later.
- Selecting. Basically, once I take a picture of a piece of student work that is most likely part of the closure discussion. I also take a mental or post it note about certain sticking points that students argued and convinced each other on.
- Sequencing. In this case, the files are usually queued up in the correct order of the folder by the order I took the photos in, but in the case of the lesson below I sequenced the two tables of a system of equations, a graph with the points plotted, and the equations written with substitution (or Equal values method) used to solve the solution to the system algebraically.
- Connecting. This happens by questions I ask during the closure. I asked how does two tables help us solve this problem? Where does it show us the answer? (where the same two coordinates are). Then when looking at the graph: how does the graph show us the solution? (the point of intersection where the two lines meet). How does the solution show up in the algebra? (After solving for x, that's the x coordinate of the solution, and when substituted or checked the y value is the y coordinate of the point of intersection).

The groupworthy task is a problem called Chubby Bunny. A cat weighs 5 pounds but gains 3 pounds per week. A bunny weighs 19 pounds and gains 1 pound per week. When will they be the same weight at the same time? This lesson 5.2.3 from CPM's Core Connections Course 3 asks students to solve the problem multiple ways showing different representations. Some students went straight to the algebraic method. Some went straight the the graph or table. Students realized to make an accurate graph it would be wise to make a table to see how far up and to the right their graph would go and what intervals to choose.

1st period:

This student used labeled horizontal tables with the values labeled as years and weight.

This student also has their graph completed with the point of intersection's coordinates labeled.

As you can see, this student used the Equal Values method. They may not hear this method later, so I've tried to tell students that it can also be called substitution, since whatever y is equal to is being substituted into the other equations y variable. This student forgot to omit the y= part in the first line, but thereafter did.

The next problem dealt with two schools, one that was shrinking and the other that was growing. Students had to make the jump to writing a negative sign in front of their growth and strictly using an algebraic method since the values were too large to put in a table and/or graph.

Neatly labeled vertical tables.

Properly scaled graph with two lines.

Steps clearly shown to solve the system.

The next problem solved: 20 years the two high schools will have the same population.

Vertical tables.

Color coded graphs.

Part of the lesson is asking students what x=7 represents. (The same weight in 7 years) I then ask how you could see what that weight is. When they are not sure, I ask them how they can be sure that x=7 is correct? (Check your solution, which leads to finding out what y equals in each equation)

The main points of the closure discussion were as follows: How did making tables help you find the answer? Students said it was when the weeks and weight were the same amount at the same time, 7 weeks and 26 pounds. Some students thought the answer might be 20 pounds because the bunny was 20 pounds after 1 week, and the cat was 20 pounds after 5 weeks. Their peers told them that it had to be the same weight after the same amount of weeks also. I prompted students to circle the solution in each table.

Then I asked the same question about the graph. Students said it was when their two graphes crossed, and the solution was at the point of intersection (7,26). Once again I asked what each number represented in the coordinates.

Finally, students described how they combined two equations into one equation. They then described subtracting x from both sides and so on. Like I said, students needed to be prompted and reminded to check their solution afterwards even though they knew their y value using the prior representations.

So, this is how I am running most of my lessons now. Circulating, finding common misconceptions, sequencing the order of taking photos of student sample work, prompting them to explain their work during the closure, or asking a volunteer to explain their work. Students absolutely love getting a photo of their work taken because they know the whole class will eventually see it during the closure (last 10 to 12 minutes)

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