Your learner may have previously used bar models and tape diagrams to show relationships between numbers. In this lesson, they will build on that knowledge by modeling and solving equations with tape diagrams. By drawing their own diagrams and exploring equations with variables, they will gain a deeper understanding of solving problems with visual models.
Before beginning the lesson, your learner should know how to identify the parts of an equation and reason about unknown values.
Key Concepts for Equations with Tape Diagrams
Here are a few concepts that are important to know for this lesson:
Tape Diagrams: Tape diagrams serve as visual representations of the relationships between quantities. In a tape diagram, a single line or "tape" represents the entirety of a quantity, with segments of the tape representing the parts of the total quantity.
Variables: Until now, your learner has probably worked with equations that have numbers but no variables. Variables are symbols (written as letters) that represent unknown quantities or values. Solving an equation means finding the value of the variable.
Tape Diagrams vs Bar Models: While tape diagrams and part-whole bar models share similarities, they differ in their visual representation. Part-whole bar models typically consist of two bars: one representing the total quantity and the other depicting its individual parts. On the other hand, a tape diagram uses a single tape (or bar) that illustrates the parts. It is implied that the length of the tape represents the whole quantity.
Note: In this lesson, equations will be represented using tape diagrams. However, your learner may choose to represent equations as part-whole bar models if it is more helpful for their understanding.
Teaching Plan
The following activities will help your learner become confident with modeling equations with tape diagrams. Remember to go at a pace that is comfortable for your learner.
Examples and visuals to support the lesson:
1. Revisiting Tape Diagrams
In this activity, we will revisit the concept of using tape diagrams to visually represent addition and multiplication problems.
Show examples of tape diagrams and corresponding equations to your learner. Encourage them to match the provided tape diagrams with the given equations.
Ask questions to ensure they grasp the connection between the diagrams and equations. For instance, inquire how they determined which equation matches with with diagram, and how each number in the equation relates to the tape diagram.
Next, provide equations for your learner to draw their own tape diagrams. Prompt them to explain their thought process in drawing the tape diagrams.
Engage in a discussion about whether there could be multiple ways to represent the same equation using tape diagrams.
Skill Check
I can use tape diagrams to represent addition and multiplication problems.
2. Exploring Equations with Variables
Once your learner is comfortable making tape diagrams for equations, introduce them to equations with variables. At this stage, it's okay if your learner doesn't know the term "variable." They just need to know that the letter represents an unknown number.
Provide various tape diagrams to match with equations, now including variables in the tape diagrams and equations.
If necessary, remind your learner that the variable is simply sitting in place of an unknown number. While finding the value of the unknown number is not the focus of this activity, your learner may take the initiative to do so.
After your learner matches the equations and tape diagrams, have them explain their methods for matching them.
Discuss how they recognize when a diagram represents addition or multiplication.
Focus their attention on the tape diagrams that match with multiple equations and have them brainstorm reasons why this is possible. For example, they might mention that 4 • x (4 times x) can also be written as x + x + x + x, or that the equations 4 • x = 12 and 12 = 4 • x represent the same relationship, just written differently.
Skill Check
I can draw tape diagrams for equations with unknown values.
3. Reasoning about Unknown Values
In this activity, your learner will draw tape diagrams to match given equations, then reason about the unknown value that makes the equation true.
Provide your learner with equations such as 18 = 3 + x and 18 = 3 • y.
Have them draw tape diagrams to model them and use the tape diagram to find the value of the unknown number (the variable).
Encourage your learner to explain their reasoning and why they know the value they found is correct.
One way to prove that their answer is correct is by redrawing the tape diagram with the solution in place of the variable.
Skill Check
I can use tape diagrams to find unknown values in equations.
4. Review and Discuss
Wrap up with a discussion to gauge your learner's understanding of the lesson. Consider asking some of the following questions:
“Why are tape diagrams useful to visualize a relationship?” (Sample response: You can see the way quantities are related.)
“Where in the tape diagram do we see the equal sign that is in the equation it represents?” (The fact that the sum of the parts has the same value as the whole; the numbers and letters in the boxes add up to the total shown for the whole rectangle.)
“Why can a diagram be represented by more than one equation?” (Because more than one operation can be used; for example, the same diagram can be represented by an addition or a subtraction equation. Or because when two expressions are equal, it doesn’t matter how they are arranged around the equal sign.)
“Describe some ways to represent the relationship 23 + x = 132.” (Tape diagram with two unequal parts, other equivalent equations like x = 132 - 23.)
“Describe some ways to represent the relationship 5 • x = 230” (Tape diagram with 5 equal parts, other equivalent equations like x = 230 ÷ 5).
In this lesson plan, your learner explored tape diagrams, equations with variables, and the process of reasoning about unknown values. Through practical activities and discussions, they have gained insight into the usefulness of tape diagrams for representing and solving equations. Thanks to your guidance, your learner can now confidently use tape diagrams as visual tools in their problem-solving journey.
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Skill Check