The purpose of this activity is to familiarize students with the idea of expressing the same area in different ways. Additionally, this activity emphasizes the geometric foundation for combining like terms, for example, why is equivalent to . Students look for and make use of structure as they connect different ways of representing the area of figures (MP7).

The goal of this discussion is to see how the same value, in this case the area of a figure, can be represented in multiple mathematical ways. Display the correct matches and resolve any questions students have. Focus on rationales for why equivalent expressions are equivalent. Here are some questions for discussion:

• “In general, when does it make sense to add and when does it make sense to multiply?” (You add when you are combining like measures and expressing the amount in the whole thing, like two lengths to get a total length, or two areas to get a total area. You multiply when you want to use the length and width to express the area.)
• “How can the length and width of Rectangle F be represented in different ways?” (We can write the length and width as and , or we can combine them to get and .)
• “How can the area of Rectangle F be represented in different ways?” (It can be 6 separate by squares or , or it can be the entire length multiplied by the entire width, which is .)
• “Can you use these rectangles to explain why isn’t equivalent to ?” (One is a square with side lengths , and the other is two squares, one with side length and one with side length 1. From the picture, we can tell the areas are not the same.)

One purpose of this activity is to practice seeing the total length of a segment as the sum of its pieces, and using the whole side length of a rectangle to express its area. Another is to generate examples of equivalent expressions and to understand why they are equivalent based on an understanding of area and the distributive property.

Monitor for students who write different but equivalent expressions to represent the area of a rectangle. For example, , , and for Figure B.

As students represent the same situations with equivalent expressions, they reason abstractly and quantitatively (MP2).

The purpose of this discussion is to explore equivalent expressions representing the same things. Invite previously-selected students to share different but equivalent expressions for length and area. (Reassure students that it doesn’t matter which side is considered the length or the width.) If any students express confusion over a response, encourage the student who came up with the response to explain. For example, a student might wonder why can be expressed as .

Display the table from the Task Statement for all to see, and write any equivalent expressions that are mentioned next to each other. For example, for the area of Figure A, you might write , , and . Show in the figure how you can see (the product of one side and the other whole side) and (the area of each smaller rectangle, added up). Emphasize that all of these expressions correctly represent the area of the whole figure, and highlight use of the distributive property where it occurs.