This is a reteaching/intervention lesson in which I regroup students strategically for peer instruction. I make sure every students who is struggling has access to 2 or more students who can help them. After the assessment I can see who is struggling either with the idea of area and perimeter or with calculations. Often students have trouble with the concept that there are different types of measurements, one dimensional and two dimensional. Because they lack understanding they often try to simply memorize. I try to give them contextual experiences to better their understanding. Because we live in an agricultural community most of my students are familiar with raising farm animals.
This activity uses different manipulatives to represent chickens and goats and uses graph paper to draw diagrams of the animal yards. The graph paper helps students visualize and model the two different types of measurements. Having students trace the straight line used to represent the fence reinforces the idea that perimeter "surrounds" the figure and having students use a highlighter to show where the grass is that the goats are eating helps connect the idea of area to the idea of "covering".
Graph paper is passed out to students and they are asked to use it to model the problems from their homework, which would have been difficult for them to solve without modeling. I expect there to be a lot of disagreement in their answers, so I tell them to sketch a diagram on the grid paper for the problems they disagree on, using one grid line to represent one foot. Students instantly see how much easier the problems are to understand and solve with the diagram and, hopefully, will begin using diagram models on their own. This helps students develop their tool box.
Each student has at least one piece of graph paper and a cup with two different types of manipulatives representing chickens and goats. Students place some, but not all, of their "chickens" on their graph paper and draw a rectangular yard on the grid lines surrounding them so they can't get out. Next I ask them to figure out how much fencing they would need to enclose the yard if one grid line equalled one foot. I tell them to share their strategy with their math family. I circulate to check in with some of the struggling students to see that they are sharing their strategy, paying attention to other strategies that are shared, and getting the help they need from their peers. If I see dots along the outer edges of their rectangle I can tell they are counting all the way around. If I see numbers along the edges I know they are looking for alternatives to counting. If a student adds only two sides I ask if the chickens can get out and then let another student take over the explanation. If a student is multiplying the numbers I ask if we have multiple sides of the same length and again let another student take over. It is important to ask the question and hand it off to another student to encourage the peer instruction.
We repeat the process with the rest of their chickens.
For the goats we ask a different question, "how much grass do the goats eat?" Having students highlight where the grass is in the pen helps them see that they are covering squares. This helps them see the difference between perimeter and area calculations and units. Some students will count the squares and others will multiply. It is important to let the struggling students count the squares if they need to, because this will reinforce the two different types of measures (counting lines and counting squares). I ask these groups to clarify why multiplying works to find the total number of squares of grass.
The whole class discussion is important for students to see the connections between the different strategies for counting lines and counting squares. It is important for the struggling students to see that counting is correct. Often they end up feeling that their strategy that made so much sense to them was incorrect, while in fact it is not. These students need to understand that it is not their concept that needs to be replaced, but their strategy, or their means to the same end. They really only need to see the connection between their counting strategy and the more efficient adding or multiplying strategy.
This discussion highlights these connections. I model a similar problem with chickens and goats on the projector and I ask students to share different strategies. As they share I ask these questions.
Does this strategy get to the correct solution?
Why does this strategy work? Why do both of these strategies work?
How is this strategy the same as another strategy?
Why is this strategy better than another?
These help them see the connections and benefits of each strategy.