The purpose of this activity is for students to analyze different ways of decomposing a gridded rectangle to find the total number of squares in a rectangle. For example, students see that the area of a rectangle that is 3 units by 6 units can be found adding and and relate that strategy to the expression . The area can also be found by decomposing the rectangle into two halves or finding twice, which is represented by .

The reasoning here allows students to visually make sense of strategies for multiplication that are based on the associative and distributive properties of multiplication. The focus is not on naming the properties, but rather on interpreting the expressions and relating them to the quantities in the diagrams (MP7).

This activity uses MLR2 Collect and Display. Advances: conversing, reading, writing.

• “Are there any other words or phrases that are important to include on our display?”
• As students share responses, update the display, by adding (or replacing) language, diagrams, or annotations. 
• Remind students to borrow language from the display as needed.

In this activity, students are given expressions that represent strategies for finding the area of rectangles. The strategies are based on the distributive property and the associative property of multiplication. Students interpret the expressions by marking or shading area diagrams and connect each expression to the product of two factors (MP2). For instance, students see that to find the value of is to find the value of or .

• “What are the two factors you can multiply to find the area of Noah’s rectangle?” (7 and 3) 
• “How are those numbers related to the expression that he wrote: ?”  ( is 21. Finding , which is 15, then adding , which is 6, also gives 21. )
• “Where do you see the two factors in his expression?” (The 7 is 5 and 2 combined. The 3 is in the and .)
• Repeat the line of questioning with Priya’s rectangle and expression.