This activity is to prepare students for the Warm-up in the associated Algebra 1 lesson, where they will need to explain why a diagram can be represented by two different expressions. As students use what they know about the area of rectangles to write expressions, they are looking for and making use of structure (MP7).

The goal of this discussion is to see the expressions as equivalent. Here are some questions for discussion:

• “How would the values of the two expressions compare if is 2?” (They would each equal 56.)
• “If is 10?” (They would each equal 112.)
• “If is 1,000?” (They would each equal 7,042.)
• “We call the two expressions ‘equivalent expressions.’ What does that mean? How would you define it?” (Equivalent expressions have the same value for all values of the variables.)

Highlight that the two expressions and are equivalent expressions. This means they have the same value for any value of . Even though we can’t check all possible values of , we know they are equivalent because they represent the same area, and because they are equivalent by the distributive property.

The purpose of this activity is to show that the area of a rectangle is the same as the sum of the area of its parts. The work in this activity is similar to the work in the associated Algebra 1 lesson, except that it is more scaffolded, and it focuses on operations with numbers instead of with variables. As students write expressions and calculate the areas of rectangles and partitioned rectangles, they look for and make use of structure (MP7).

• Add the parts of the sides of the rectangles together and then multiply
• Find the area of each smaller rectangle and then add
• Represent the length and width of the rectangle with expressions and use the distributive property

These practice exercises offer an opportunity to practice rewriting expressions using the distributive property. There may be more work here than can reasonably be completed in the available time. If that is the case, consider assigning a subset of items that would be most beneficial for your students to practice.

Note that one of the questions gives an expression like for the area of a rectangle, and asks for a possible length and width. While 3 and may be the most likely response, another example of a valid response would be and .

Students make sense of problems and persevere in solving them when they determine equivalent expressions (MP1).