This lesson works on two things. One is using models to help students both make sense of the problem as well as to help explain their reasoning. The other is to compare a weak and a strong argument and to practice strengthening weak arguments. The warm up uses the visual scaffold for the border problem. I model one solution and ask them to explain the solution that I have shown. This helps them see the connection between the diagram and the math that has to be done. They learn the value of using a diagram.
To work on stregthening arguments I display one of their arguments from a previous lesson (Punk Prank Payback) on the document camera and ask them if it would be enough to convince another person, especially one who doesn't understand. They have time to work together to come with additional evidence or explanation that would strengthen the argument. This not only improves their effectiveness in peer instruction, but also their argumentation skills.
For this warm up I made a poster modeling one way of solving the border problem from their homework. I ask students to try to explain my solution by figuring out why I drew the diagram as I did and what each part represents. I also ask them to figure out where I got the numbers in the numeric expression. As we go over this problem I point to each section of the border and have them explain that it represents the number of tiles that would be in that part of the border. I ask them how we know there are 12 tiles there and they should explain that ten will fit along the upper side of the 10 by 10 square plus the two more in the corners. I then refer to the expression and ask where the numbers came from so that they connect the mathematical with the physical model. If they don't ask or suggest, I ask if the expression could be written in a different way. Normally, I would expect them to ask if it could be written 2x12+2x10. Rather than answering the question for them I would ask the class where in the diagram they see 2x12 and 2x10, to which they respond that there are two sides of each. This helps them view the diagram as a useful tool for making sense of the problem and also encourages them to figure things out on their own rather than turning to the teacher for all the answers.
Next I ask them to look at their own solution and explain it to their math family, suggesting that they draw a diagram if they haven't already. Then I ask them to explain their solution clearly enough so that I can model it on the board. See the pictures of their models in the resources.
I present a problem from the homework Perimeter and Area Arguments from an earlier lesson (Punk Prank Payback) in which students had to write an argument for or against my statement. They were supposed to explain their reasoning, but this is a huge weak area for them. I preface this by pointing out that if they dissagree with my statement then their argument has to be strong enough to convince me, a person who misunderstands. Most of them have given one sentence responses with no explanation.
In my first statement I say that I am putting new tile in my kitchen which measures 8 by 9 feet and that I am going to ask the sales associate for 34 square feet of tile. The response I have chosen is "I am against this because you ordered too little." You can choose any response that is incomplete. I allow students to work together to provide more evidence and explanation then ask that they share it. I hope to hear things like, "You added up all the sides, which will give you the perimeter, not the area.", "You need to find the area by multiplying the two dimensions."I add all suggestions one at a time asking at each point if it strengthens the argument. I also want to give credit to all students who came up with that, so I am sure to ask who came up with the same point before moving on and asking for more suggestions. We keep going until the class is satisfied that the argument is convincing.
I have them practice the process one or two more times taking care to choose responses that are correct, but weak. I don't want students to be distracted by correcting math errors.
For the remainder of class students continue working on the homework. Many of them likely did not draw diagrams and I want to encourage them to do so now. I tell them they will find drawing such a model so helpful that I don't think the extra credit problem will be so tricky, so it is now required.