Lesson 10

The purpose of this warm-up is to elicit different strategies for counting objects arranged in groups of 2, which will be useful when students multiply by 2 in a later activity. While students may notice and wonder many things about these images, flexible ways of seeing the groups and strategies for finding the total number of objects are the important discussion points.

When students see the socks are grouped by 2 and use that to find the total, they are looking for and making use of structure (MP7).

• Groups of 2
• Display the image.
• “What do you notice? What do you wonder?”
• 1 minute: quiet think time

• “Discuss your thinking with your partner.”
• 1 minute: partner discussion
• Share and record responses.

• “How does this problem relate to what we know about multiplication?” (There are equal groups of socks, we can say there are 6 groups of 2.)

The purpose of this activity is for students to build on the work they have done with scaled picture graphs to use the tape diagram as a new representation of multiplication. The scale of the picture graph will be used to help students think about a category of the graph as a situation involving equal groups.

To add movement to this activity, students could find someone in the class who represented a different category than they did or represented the same category in a different way. When they find a person, they can describe what is the same and what is different about their representations.

• Groups of 2
• Display the picture graph and tape diagram.
• “What do you notice? What do you wonder?” (The key says each square represents 2 signs. The diagram shows 3 groups of 2. I wonder whether they represent the same thing.)
• 1 minute: quiet think time
• 1 minute: partner discussion time
• “How does the diagram show the speed limit signs that Elena saw on the way home?”
• Share responses.

• “Now independently represent the data from another category in the graph with your own drawing or diagram.”
• 1-2 minutes: independent work time
• “Share how you represented the data in your drawing or diagram with your partner.” 
• 2–3 minutes: partner discussion
• Monitor for students who create a tape diagram to represent one of the other categories to use during the synthesis.

• Have students share different ways they represented a category in the graph.
• Display a student created tape diagram or make a quick sketch of one to represent the street signs Elena saw on the way home.
• “Which category does this tape diagram represent? How do you know?”
• “How is the diagram the same as the scaled picture graph?” (Each picture and each part of the tape diagram represents 2 signs.)
• “How is the diagram different than the scaled picture graph?” (In the graph, you have to read the key to know that each picture shows two signs. In the diagram, each part is labeled with a 2.)

The purpose of this activity is for students to connect situations involving equal groups to drawings and tape diagrams. A sorting task gives students opportunities to analyze representations, statements, and structures closely and make connections (MP2, MP7). Students explain why two cards match and have opportunities to critique and question their peers' reasoning (MP3). When explaining, students have opportunities to revise their language to make their explanations more precise and clear (MP6). After sorting and describing their sort, students notice that all of the representations reinforce the meaning of multiplication as a way to express equal groups.

Students will spend all of the next lesson working with expressions. Keep the equal groups cards for the next lesson.

• Groups of 2
• Give each group of 2 students a set of cards.
• “This set of cards includes drawings, situations, and diagrams. Take a little time to think about how some of the cards could match.”
• 1 minute: quiet think time
• “Find the cards that match. Work with your partner to justify your choices. Be ready to explain your reasoning.”
• 5 minutes: partner work time
• Have students share the matches they made and how they know those cards go together.
• Listen for the language students use to describe their match. If students only reference the numbers that match, consider asking:
  • “What do you mean when you say _____?”
  • “How could you use the words ‘equal groups’ to explain?”

• “Now you’re going to create a drawing or diagram to represent two different situations.”
• 3–5 minutes: independent work time
• Monitor for the students who draw equal groups and students who draw a tape diagram.

If students create representations that do not match the number of groups or size of the groups in the situations, consider asking:

• “How did you represent the situation?”
• “How could you show the groups in the situation? How could you show the objects in each group?”

• For each situation in the last 2 problems, display 2–3 student representations, at least one drawing of equal groups and one tape diagram. Leave them displayed.
• “What do all these representations have in common?” (They all show equal groups. They all have groups. Each group has the same amount.)
• “Where are the 4 bags and 2 strawberries in each drawing or diagram?”
• “Where are the 4 hands and 5 fingers in each drawing or diagram?”