Warm-up: Notice and Wonder: Compare Representations (10 minutes)
The purpose of this warm-up is to elicit the idea that number lines and tape diagrams can be used to represent the same relationships between numbers, which will be useful when students use tape diagrams and number lines in a later activity to interpret and solve story problems. While students may notice and wonder many things about these representations, the important discussion points are the way both representations show quantities and how they could be used to represent addition.
- 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.
- “What equation do you think best matches these diagrams?” (\(5+4=9\))
- “Where do we see the 5 in both diagrams?” (the 5 on the number line is the distance from 0 to the point at 5)
- “Where do you see the 4 in both diagrams?” (the 4 on the number line is the 4 spaces that were moved to the right)
Activity 1: Card Sort: Represent Stories (20 minutes)
The purpose of this activity is for students to make connections between different representations of story problems. Students match stories, equations, number lines, and tape diagrams (MP2, MP7). The synthesis focuses on how the representations are the same and different. Students recognize some representations help to make sense of the story, while others help to show their thinking when finding solutions.
Advances: Speaking, Conversing
Materials to Copy
- Story Problems Card Sort (stories, equations, number lines, diagrams)
- Create a set of cards from the blackline master for each group of 3.
- Groups of 3
- Give each group a set of cards (A–P).
- “We have been representing addition and subtraction on the number line.”
- “Today we are going to think about how we can use diagrams to make sense of stories and use number lines to show our thinking.”
- “Lin and her friends were making connecting cube trains. These stories are about all the ways they built their trains.”
- “Match each story to a diagram, a number line, and an equation.”
- 10 minutes: small-group work time
- Display cards A, E, G, O.
- “How are these representations of the same problem the same? How are they different?” (I can see the whole length of the train in the diagram, but I only see a point at 37 on the number line and I don’t see anything but the numbers in the equation. They all show 37 as one part and they all show addition.)
Activity 2: All Kinds of Representations (15 minutes)
The purpose of this activity is for students to solve story problems using any strategies and representations that make sense to them. In previous lessons, students have used number lines, tape diagrams, and equations to represent situations and solve problems. They will have the opportunity to apply their understanding and make choices about how to make sense of problems and show their thinking. In the synthesis, students reflect on which representations are most helpful to them and make connections across representations (MP2). This question will likely elicit a variety of responses from students which can emphasize the point that students should use representations that make sense to them.
Supports accessibility for: Conceptual Processing, Memory
Materials to Copy
- Number Line to 100
- Groups of 2
- Give students access to the blackline master.
- “While doing the card sort, you had a chance to read stories and find matching representations.”
- “Now you are going to solve story problems and represent your thinking in a way that makes sense to you.”
- “There are blank number lines and diagrams that you can use if it helps.”
- “You and your partner will read each problem together and then solve on your own.”
- “Be prepared to explain your thinking.”
- 8 minutes: partner work time
- As students work, consider asking: "What is the same or different about the way you showed your thinking and the way your partner showed theirs?"
- Monitor for students who use a number line and tape diagram for the same problem.
Solve each problem. Show your thinking. Use a number line or diagram if it helps.
- Clare started with 24 cubes and added on some more. Clare made a train with 42 cubes. How many cubes did Clare add on?
- Andre had 37 cubes. Then he added 39 more to make the train longer. How many cubes did Andre use?
- Mai wanted her train to be 55 cubes long. So far she has 47. How many more cubes does Mai need?
- Invite previously selected students display their work side by side for all to see.
- “What was the same or different between _____ and _____ representations?”
- Help students make connections across representations by highlighting the places where they see the solution in each.
- “Which representations do you find most helpful?”
“Today you solved problems and compared representations. You have learned how you can use diagrams to make sense of story problems and how number lines can help you show your thinking.”
“We've spent a lot of time looking at addition and subtraction on the number line during this unit. Tell a partner 1 way the number line helped you understand a strategy or someone else’s thinking.”