Lesson 7

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

When students use grouping to find the total in a multiple of tens, they look for and make use of structure (MP7).

• 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.

• “What expressions could we record for the different ways that students saw the tens?” (\(6 \times 10\), because some students saw 6 groups of 10. \(3 \times (10 \times 2)\), because some students saw 2 rows of 10, then multiplied by 3. \((3 \times 10) \times 2\), because some students multiplied 3 times 10 for each column, then multiplied by 2.)
• Consider asking:
  • “Who can restate in different words the way _____ saw the tens?”
  • “Did anyone see the tens the same way but would explain it differently?”
  • “Does anyone want to add an observation to the way _____ saw the tens?”

The purpose of this activity is for students to solidify what they have learned about the relationship between multiplication and division. Students start by creating a drawing of equal groups. They then get a drawing created by another student in their group and write a division situation to match it. Then, they pass their paper and use the drawing of equal groups and the situation to write a multiplication equation. In the final round of this “carousel” structure, students write a division equation to match the other representations. 

When students relate drawings, situations, and equations they reason abstractly and quantitatively (MP2). As students look through each other’s work, they add to the representations and can defend different points of view. Students are able to critique the work of others and construct viable arguments (MP3).

Students work on the same box on a graphic organizer as the other students in their group, so if they struggle, encourage them to talk to their group. Remind students that what they are creating should match what has already been filled in.

• Groups of 4
• Give each student a recording sheet.
• “In the first box on your sheet, create a drawing that shows equal groups of objects. This drawing will be used by other students in your group to fill in the other boxes.”
• 3 minutes: independent work time

• “Pass your paper to your right. In Box 2, write a description of a division situation that matches the drawing you were just passed.”
• 3 minutes: independent work time
• “Pass your paper to your right. In Box 3, write a multiplication equation that matches the drawing and division situation you just received. Use a symbol for the unknown quantity.”
• 2 minutes: independent work time
• “Pass your paper to your right. In Box 4, write a division equation that matches the drawing, division situation, and multiplication equation you just received. Use a symbol for the unknown quantity.”
• 2 minutes: independent work time
• “Pass your paper one more time. You should have your original drawing back.”
• “Talk to your group about which box was the most difficult for you to fill in. Share ideas about what helped you most during this activity.”
• 5 minutes: small-group discussion

• “What strategies were shared in your group?” (When I wasn’t sure about writing a situation, I looked back at the drawing and tried to imagine something I could be dividing that looks like the drawing. When I was writing an equation it helped me to imagine the situation happening.)
• “As you look at your paper, what are some connections you notice between multiplication and division?” (I can use both multiplication and division to represent the same drawing or situation. The multiplication equations are all missing a factor, but the division equations are all missing the quotient.)

The purpose of this activity is for students to represent and solve problems involving equal groups. Students can solve the problem first or write the equation first, depending on the order that makes the most sense to them. Students write equations with a symbol standing for the unknown quantity to represent each problem, but can write either a multiplication equation or a division equation. A multiplication equation and a division equation that represent the same problem are highlighted in the synthesis.

• Groups of 2
• “These situations are all about things that you could find on a desk or around a desk. What are some things that you could find on a desk or around a desk?”
• 30 seconds: quiet think time
• Share responses. 

• “Read through each situation and write an equation with a symbol that represents the unknown quantity for each situation. Then, solve and determine the unknown number in each equation. You can solve the problem first or write an equation first depending on what order makes the most sense to you. Be prepared to explain your reasoning.”
• 7–10 minutes: independent work time
• Monitor for students who write a division equation and a multiplication equation for the same situation to share during the synthesis.
• “Now, share your equations and your solutions with your partner. Take turns sharing your equations and solutions.”
• 3–5 minutes: partner discussion

If students don’t find a solution to the problems, consider asking: “What is this problem about?” and “How could you represent the problem?”

• Have students share a division equation and a multiplication equation that were written to represent the same division situation and display for all to see.
• Discuss differences in equations students wrote.
• Consider asking:
  • “How does each number in the equations represent the situation?”
  • “_____ wrote _____ and _____ wrote _____ to represent the same problem. How are those equations the same and different?” (One of the equations is a division equation, but the other equation is a multiplication equation with an unknown factor. They used different symbols for the unknown amount. Both symbols in the equation represent the missing _____ in the situation.)
• Have students share strategies they used to solve the problem.