In this lesson plan, we will explore strategies for solving equations with hanger diagrams. Your learner will be introduced to hanger diagrams as a tool for modeling balanced equations. They will then learn how to use the hanger diagrams to solve addition and multiplication equations.
Key Concepts
Here are a few key concepts that are helpful to know for this lesson:
Hanger Diagrams: Learners use hanger diagrams to visually grasp the concept of balance in equations, where the total weight or quantity on one side equals that on the other side. They can resemble a hanger or balance scale with objects or symbols that represent values placed on either side.
Balanced Equations: In the context of equations, "balanced" refers to a state where both sides of the equation are equal. This means that the expressions or values on the left side of the equation are identical in quantity or weight to those on the right side.
Solving Equations: Solving an equation involves finding the value of the variable that makes the equation balanced or true.
Addition Equations: Addition equations can be solved using subtraction. For example, to find the value of x in the equation x + 5 = 7, we can subtract 5 on both sides.
Multiplication Equations: Multiplication equations can be solved using division. To find the value of x in the equation 3x = 12, we can divide 3 from both sides.
Teaching Plan
The following activities will teach your learner strategies for solving equations with hanger diagrams. Remember to go at a pace that is comfortable for your learner.
Examples and visuals to support the lesson:
1. Exploring Balanced and Unbalanced Hangers
In this activity, your learner will reason about balanced and unbalanced situations using hangers with equal and unequal weights.
Display the photo of socks on hangers and ask your learner to describe what they notice and what they wonder.
Use the word “balanced” to describe the hanger on the left and “unbalanced” to describe the hanger on the right.
Explain to your learner that the hanger on the left is balanced because the two pink socks have equal weight, and the hanger on the right is unbalanced because one blue sock is heavier than the other.
Skill Check
I can recognize balanced and unbalanced hangers.
2. Reasoning About Abstract Shapes on Hangers
This activity will help your learner reason about the weights of abstract shapes on hangers and understand balanced and unbalanced situations.
Provide your learner with an image of a hanger with unbalanced shapes and an image with balanced shapes.
Ask your learner to share some things that must be true, could be true, and cannot possibly be true about the diagrams. Encourage them to explain their reasoning.
Discuss how the hanger diagrams work by explaining that when the diagram is balanced, there is equal weight on each side. For example, in a balanced diagram, one triangle weighs the same as three squares. Conversely, in an unbalanced diagram, one side is heavier than the other. For instance, in an unbalanced diagram, one triangle is heavier than one square.
Skill Check
I know that a balanced hanger has equal weight on each side, but an unbalanced hanger has different weights on each side.
3. Using Balanced Hanger Diagrams to Solve Equations
In this activity, your learner will use balanced hanger diagrams to find the value of variables in addition and multiplication equations.
Provide your learner with sets of hanger diagrams and equations, explaining that each square labeled with a 1 weighs 1 unit, and each shape labeled with a letter has an unknown weight.
Demonstrate how the hanger diagrams can be used to find the value of the variable. For addition equations, show how shapes can be removed from each side to find the variable's value. For multiplication equations, explain how grouping the shapes on each side can reveal the value of the variable.
Ask your learner to explain how they can determine from looking at a hanger that it can be represented by an equation involving addition or multiplication. Encourage them to discuss the moves that can be done to ensure the hanger stays balanced.
Skill Check
I can match balanced hanger diagrams to equations and reason about the value of the variable.
4. Writing and Solving Equations with Hanger Diagrams
This activity will help your learner write and solve equations using balanced hanger diagrams.
Provide your learner with diagrams of balanced hangers representing addition and multiplication equations, with each object labeled with its weight. Some weights are labeled with numbers, but some are unknown and labeled with a variable.
Have your learner write an equation to represent one of the balanced hangers and find the weight of the shape with a letter (i.e., find the value of the variable). Encourage them to use both the diagram and the equation to reason about their answer.
For added challenge, provide problems that involve fractions or result in a solution that is a fraction.
Skill Check
I can use balanced hangers to write and solve equations.
5. Comparing Strategies for Equations
Use this activity to check your learner's understanding of solving equations using hanger diagrams.
Provide your learner with two equations - an addition equation and a multiplication equation. Ask them to draw a hanger to match each equation and solve each equation by finding the unknown value on the hanger.
Have your learner compare the strategies used for each equation and discuss how they are alike and different.
Skill Check
I can explain and compare my strategies for solving addition and multiplication equations with hanger diagrams.
In this lesson plan, your learner explored hanger diagrams, providing them with a tangible way to understand and reason about weight and balance. They then used hanger diagrams to understand and solve addition and multiplication equations. With these skills, your learner will be prepared to solve a variety of algebraic equations.
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Skill Check