Lesson 10

The purpose of this How Many Do You See is for students to use grouping strategies to describe the quantities they see.

• Groups of 2
• “How many do you see? How do you see them?”
• Flash the image.
• 30 seconds: quiet think time

• Display the image.
• “Discuss your thinking with your partner.”
• 1 minute: partner discussion
• Record responses.
• Repeat for each image.
• Keep the images displayed for the launch of the next activity.

• “How can we use amounts that we can see quickly to find the total number of squares?” (We can look for repetition of the number of squares that we can easily see. We can add to or multiply the number of squares we can easily see.)
• Consider asking:
  • “Who can restate the way _____ saw the squares in different words?”
  • “Did anyone see the squares the same way but would explain it differently?”
  • “Does anyone want to add an observation to the way _____ saw the squares?”

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, they see that the area of a rectangle that is 3 units by 6 units can be found adding \(3 \times 5\) and \(3 \times 1\) and relate that strategy to the expression \((3 \times 5) + (3 \times 1)\). The area can also be found by decomposing the rectangle into two halves or finding \(3 \times 3\) twice, which is represented by \(2 \times (3 \times 3)\).

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

• Groups of 2
• Display the first problem.
• “Take a minute to make sense of how Andre and Elena found the area of a rectangle.”
• 1 minute: quiet think time

• “Work with your partner to discuss how their strategies are alike and different, and how the numbers in each of their expressions relate to their diagrams.”
• 5–7 minutes: partner work time
• Share responses.
• “Work with your partner to complete the second problem.”
• 3–5 minutes: partner work time

MLR2 Collect and Display

• Circulate, listen for and collect ways that students decompose the rectangle and the language students use to describe the strategies they used. Listen for: decomposed, smaller parts, smaller rectangles, \(2 \times 2 \times 9\), \(2 \times 18\), \(4 \times 5 + 4 \times 4\), and \(5 \times 4 + 4 \times 4\).
• Record students’ diagrams, words, and phrases on a visual display and update it throughout the lesson.

• “How did you find the area of the rectangle?”
• “How could you use a product you already know to find the area of the rectangle? How could you show your strategy on the rectangle?”

• “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, they see that to find the value of \(2 \times (2 \times 6)\) is to find the value of \(4 \times 6\) or \(6 \times 4\).

• Groups of 2
• “Take a minute to read the directions of the activity. Then, talk to your partner about what you are asked to do.”
• 1 minute: quiet think time
• 1 minute: partner discussion
• Answer any clarifying questions from students.
• Give students access to colored pencils, crayons, or markers.

• “Mark or shade each diagram to represent how each student found the area.”
• 3–5 minutes: independent work time
• “Share with your partner how you used the rectangles to show each expression.”
• 3–5 minutes: partner discussion

• “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: \((5\times 3)+(2 \times 3)\)?”  (\(7 \times 3\) is 21. Finding \(5\times3\), which is 15, then adding \(2\times3\), 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 \(5 \times 3\) and \(2 \times 3\).)
• Repeat the line of questioning with Priya’s rectangle and expression.