7th Grade Math : Unit #6 - Proportionality on a graph : Lesson #7

Clarify & Correct arguments

Objective: SWBAT determine the strength and accuracy of an argument by checking the mathematical evidence.
Standards: 4.NF.A.2 6.RP.A.1 MP3 MP6
Subject(s): Math
60 minutes
1 Intro & Rationale - 0 minutes

This lesson is all in the details and attention to precision. I expect them to struggle because they are not used to critiquing the work of others, especially when it is presented in a written argument. Taking some time with this helps students develop a more critical eye so they can really double check their work and the work of others. When students disagree in their conclusion they often don't know what to do to and the argument never gets off the ground. This lesson teaches them how to test and double check the evidence. The mistakes are authentic, because they came from student work. Most of the mistakes are not language based, so ELL students are more likely to find them and be able to point them out because they will be more focused on the numbers. The lesson may have to be split into 2 days depending upon how readily students will engage in this type of lesson.

2 Warm up - 20 minutes

warm up evaluate and strengthen argument.docx This warm shows students two arguments about the same true claim. They only get to see one at a time and I ask if they think it is a strong or weak argument. Both are strong and convincing. One of them sites simplified ratios as well as the graphic evidence and the other only discusses simplified ratios. The second, however, goes into more detail and explains the simplified ratios better.

When students are asked what would make each argument even stronger, they generally think they should be combined. Some also suggest making a graph along with the argument.

I ask them to try to show the graph that would support this claim from the information in the arguments. When they have drawn the graph I ask how confident they are that everyone's graph looks the same (very) and that if they were all able to draw the graphs accurately from the description that is a good sign of a strong argument.

3 Exploration - 34 minutes

The first part of the exploration is the clarification part. I show two arguments Is it enough.docx with correct claims, but weak evidence that needs clarification. Both refer to graphic evidence and it follows really well from the warm up, because they are trying to picture it in their minds as they read the argument.

I show the first argument: Alissa and Austin have the same black to white proportions because they are both on the same line. I ask if they think there is room for doubt or if the argument is enough. I have a graph ready so I can play dumb, depending on their response. If they say it is fine I draw the line shown in Room for doubt 1.docx , which is crooked. "so I can say that all of these use the same proportions too?" They need to clarify that the line is straight and goes through zero. Otherwise I continue the same "playing dumb" with Room for doubt 2.docx and Room for doubt 3.docx .

Once I show the second argument, which is similar to the first, they should be able to clarify it correctly. This helps them both think about the detail needed for clear argumentation and also forces them to clarify their new knowledge of graphing proportional relationships.

The second part of the exploration is the correction part. Critique and correct argument.docx I show them four incorrect arguments and ask them to work together in their math family groups to find what is incorrect then rewrite the argument. These ones are really confusing and don't all make sense. It is hard to figure out how someone might have made a mistake, so I ask them just to pick one and work an figuring out how to fix it together. It is important to remind them that in order to make the necessary corrections they need to go back to the original data.

If students have trouble getting started on this I would just pick one for them and ask them to double check the numbers and the math in order to find the mistake. Helping students find and correct the mistakes.docx Spending time looking for and correcting mistakes improves their ability to help each other. I finish by asking a couple of groups to share their corrections. I ask them to bring up the original data so they can refer to it as they explain how they found the mistakes and show where they found the information they needed to correct them. I also ask them to point out anything that was done correctly. Sometimes they will find that the math was done correctly, but the incorrect numbers were used. This often prevents kids from finding the mistakes because they don't look back at the data.

Building a lesson on students' mistakes
Student Led Inquiry

This lesson was based on mistakes that my students had made in an earlier lesson. At this point my students had gotten used to looking at mistakes and figuring out where they had gone wrong. I like using my students ideas to plan lessons. Not only does it send the message that their thinking is important and valuable, but it directly addresses their needs. I also wanted them to understand that they are not done when they get to a final answer and I wanted to give them an opportunity to revise.

However, this was a really difficult task for my students. The first problem they had was interpreting the claim. Some of them are written with imprecise language that makes them difficult to follow. They also had trouble deciding what part of it was wrong.  Also, the original data that was used to present the claim was on a separate sheet of paper so they had to switch back and forth.

If I were to do this lesson again I would:

  • Separate it into two lessons - one for stregthening arguments and one for correcting.
  • Make the warm up a reminder of the correct answers, so students would more easily recognize what the mistakes were.
  • Have the original data or the correct claims on the same page as the mistakes (and I might do one mistake at a time).

I also might give them more specific questions to help them first identify the mistake:

  • What math did the person do?
  • Where did they get their numbers?
  • What is confusing about it?
  • What looks right about it?
  • What question would you ask this person who made the mistake?

Then I might ask them how they would respond to the person who made the mistake to help them understand.