Lesson 3

The purpose of this Math Talk is to elicit strategies and understandings students have for expressing total lengths and areas, given some dimensions. These understandings help students develop fluency and will be helpful later in this lesson when students will need to be able to combine like terms to express a total length and express an area of a rectangle in different ways using the distributive property. In this activity, students have an opportunity to notice and make use of structure (MP7) because there are opportunities to invoke the distributive property and to replace expressions with equivalent expressions.

Display one problem at a time. Give students quiet think time for each problem and ask them to give a signal when they have an answer and a strategy. Keep all problems displayed throughout the talk. Follow with a whole-class discussion.

Ask students to share their strategies for each problem. Record and display their responses for all to see. To involve more students in the conversation, consider asking:

• “Who can restate \(\underline{\hspace{.5in}}\)’s reasoning in a different way?”
• “Did anyone have the same strategy but would explain it differently?”
• “Did anyone solve the problem in a different way?”
• “Does anyone want to add on to \(\underline{\hspace{.5in}}\)’s strategy?”
• “Do you agree or disagree? Why?”

The purpose of this activity is to familiarize students with the idea of expressing the same area in different ways. Another is to emphasize the geometric foundation for combining like terms, for example, why \(n+n+n\) is equivalent to \(3n\).

Ensure students notice in the task statement that every shape that looks like a square is a square.

Display the correct matches and resolve any questions students have. Focus on rationales for why equivalent expressions are equivalent. Possible 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 \(n+n+n\) and \(n+n\), or we can combine them to get \(3n\) and \(2n\).)
• “How can the area of rectangle F be represented in different ways?” (It can be 6 separate \(n\) by \(n\) squares or \(6n^2\), or it can be the entire length multiplied by the entire width which is \((3n)\boldcdot(2n)\).)
• "Can you use these rectangles to explain why \((n+1)^2\) isn’t equivalent to \(n^2+1^2\)?" (One is a square with side lengths \(n+1\) and the other is two squares, one with side length \(n\) 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 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, \(2(t+5+t)\), \(2t+10+2t\), and \(4t+10\) for figure B.

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 \(a+a+a\) can be expressed as \(3a\).

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 \(5(a+4)\), \((4+a) \boldcdot 5\), and \(5a+20\). Show in the figure how you can see \(5(a+4)\) (the product of one side and the other whole side) and \(5a+20\) (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.