Tape Diagrams and Equations
5 min
Teacher Prep
Setup
Narrative
Students recall tape diagram representations of addition and multiplication relationships.
In these materials, when multiplication is used to represent equal groups, such as "5 groups of 2," the factors are written in the same order as when described verbally: the first factor is the number of groups and the second is the number in each group (or size of each group). But students do not have to follow that convention. They may use their understanding of the commutative property of multiplication to represent relationships in ways that make sense to them.
Launch
Give students 2 minutes of quiet think time, followed by a whole-class discussion.
Student Task
-
Here are two diagrams. One represents 2+5=7. The other represents 5⋅2=10. Which is which? Label each diagram with the value that represents the total.
-
Draw a diagram that represents each equation.
4+3=7
4⋅3=12
Sample Response
- Diagram A: 5⋅2=10, Diagram B: 2+5=7
Students write 10 in the box for Diagram A and 7 in the box for Diagram B. - Sample response:
4+3=7
4⋅3=12
Activity Synthesis (Teacher Notes)
Invite students to share their responses, diagrams, and rationales. The purpose of the discussion is to give students an opportunity to articulate how operations can be represented by tape diagrams. Consider asking:
- “Where do you see the 5 in the first diagram?” (There are 5 equal parts represented by 5 same-size boxes.)
- “Why is there only one 2 in the second diagram? (2 is only added once in the equation 2+5=7.)
- “How did you find the value of the total to write in the first diagram?” (Compute 2+2+2+2+2 or 5⋅2 or find the total value in the multiplication equation that the diagram represents.)
- “Explain how you knew what the diagrams for 4+3=7 and 4⋅3=12 should look like.”
- “How are the representations of 4⋅3 alike? How are they different?” (They all show the same total, 12. Some represent 4⋅3 as 4 groups of size 3, while others represent it as 3 groups of size 4.)
Standards
- 6.EE.6·Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.
- 6.EE.B.6·Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.
10 min
15 min
Knowledge Components
All skills for this lesson
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Tape Diagrams and Equations
5 min
Teacher Prep
Setup
Narrative
Students recall tape diagram representations of addition and multiplication relationships.
In these materials, when multiplication is used to represent equal groups, such as "5 groups of 2," the factors are written in the same order as when described verbally: the first factor is the number of groups and the second is the number in each group (or size of each group). But students do not have to follow that convention. They may use their understanding of the commutative property of multiplication to represent relationships in ways that make sense to them.
Launch
Give students 2 minutes of quiet think time, followed by a whole-class discussion.
Student Task
-
Here are two diagrams. One represents 2+5=7. The other represents 5⋅2=10. Which is which? Label each diagram with the value that represents the total.
-
Draw a diagram that represents each equation.
4+3=7
4⋅3=12
Sample Response
- Diagram A: 5⋅2=10, Diagram B: 2+5=7
Students write 10 in the box for Diagram A and 7 in the box for Diagram B. - Sample response:
4+3=7
4⋅3=12
Activity Synthesis (Teacher Notes)
Invite students to share their responses, diagrams, and rationales. The purpose of the discussion is to give students an opportunity to articulate how operations can be represented by tape diagrams. Consider asking:
- “Where do you see the 5 in the first diagram?” (There are 5 equal parts represented by 5 same-size boxes.)
- “Why is there only one 2 in the second diagram? (2 is only added once in the equation 2+5=7.)
- “How did you find the value of the total to write in the first diagram?” (Compute 2+2+2+2+2 or 5⋅2 or find the total value in the multiplication equation that the diagram represents.)
- “Explain how you knew what the diagrams for 4+3=7 and 4⋅3=12 should look like.”
- “How are the representations of 4⋅3 alike? How are they different?” (They all show the same total, 12. Some represent 4⋅3 as 4 groups of size 3, while others represent it as 3 groups of size 4.)
Standards
- 6.EE.6·Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.
- 6.EE.B.6·Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.