Making Sense of Area Formulas

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Objective

SWBAT understand the derivation of the formula for the area of a rectangle.

Big Idea

There are multiple strategies for calculating area and the formula represents the simplest most efficient method.

Intro & Rationale

My students come to me with a very weak understanding of area and perimeter. They mix up the two algorithms, get the units wrong, and have no understanding of why the formulas work.

I teach this lesson so students can see the relationship between various strategies for calculating the area of a rectangle. Many of my students use the very unsophisticated strategy of counting up all the individual squares inside the rectangle. While this shows me that they understand the concept of what area is, it is a very inefficient and limiting strategy, which they are unable to use without a grid. Several of my students have become very confused at being told that they are not calculating the area correctly. They either think their understanding of area is wrong or they don't understand how they can get the right answer if they are doing it wrong.

This lesson shows them that they are right, but it also shows them why the other strategies including the formula, LxW, are also right with the added benefit of being simpler, faster, and more efficient.

Warm Up

20 minutes

Today's warm up is of a similar form as the last lesson (sorting out perimeter) in that students are given a rectangle with it's dimensions labeled (18 ft. by 6 ft.). They are given a half sheet of graph paper and asked to draw the figure on the graph paper. Three equations showing the area are given and students are asked to describe what each number represents in the figure they drew on the graph paper.

As I circulate I am looking for students who are struggling. I might direct them to talk to their math family groups and find out what they are doing. I expect that once they draw their rectangle on the graph paper they might ignore it, thinking they are done with that problem. I will remind them that it holds all the answers for them. I ask what 108 represent in their diagram, "where do they see it?"

As I see students using the "useful vocabulary" I may bring this to the attention of the class. "Christina, which of the vocabulary terms did you use for number one?"

Exploration

34 minutes

I ask students to write down their explanation about making sense of the repeated addition strategy (the backside of the warm up). Then I have them share at their math family group and then with the class. They may have moved on to this part if they finished early, but they can't go any further,because I haven't given them the problems to work on yet. I want to make sure they use the strategy they are asked to. I also want to circulate to see if anyone is drawing the rectangle on graph paper. I purposely give them a really small piece of graph paper to discourage this. See the possible responses document for more questioning ideas.

I want to support the students in their learning at their level, but I definitely want students to see the benefit of the multiplication formula. Otherwise they are doing the math the hard way. I would want to finish up by asking students what is the most efficient way to find the area of a rectangle. Which one will take the fewest steps and be the fasted way to the solution? I may want to give them an example with really big dimensions like 200 ft. by 40 ft. Do we want to add forty 200s or two hundred 40s? Or is it faster to do 200 x 40?