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Unit 8, Lesson 15

Solving Equations with Absolute Values

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Integrated Math 1

15.1

Warm-up

5 mins

Where Does 30 Go?

Standards Alignment

Building On

Addressing

Instructional Routines

None

Materials

None

Activity Narrative

In this activity, students look at how adding or subtracting a value in different places in an absolute value function changes the graph of the function. First, they match a function to its graph, then to a description based on a familiar context. Monitor for students who:

Check coordinate pairs or input values.
Reason about transforming the graph or finding distance.

Launch

Arrange students in groups of 2.

Tell students to close their books or devices (or to keep them closed). Display the graph for all to see. Give students 1 minute of quiet think time, and ask them to be prepared to share at least one thing they notice and one thing they wonder. Record and display responses without editing or commentary for all to see. If possible, record the relevant reasoning on or near the graph.

If the connection to absolute value does not come up during the conversation, ask students to discuss this idea.

Activity

None

Student Task Statement

Graph of a v-shape with vertex at 30 comma 0.

Which of these functions is represented by this graph? Explain your reasoning.
One of these functions takes a guess as an input, , and outputs the distance between the guess and 30 as the output. Which function is it? Explain your reasoning.
Write a function that takes a guess as an input and outputs the distance between the guess and -8.

Activity Synthesis

The purpose of the discussion is to connect the different representations to the function involving absolute value.

Invite students to share their solutions to the first two questions, including their reasoning. If possible, include at least 1 group that checked values to find the answer and 1 group that reasoned about the structure of the function.

Then invite a group to share their answer for the last question. Ask students what the graph of this function would look like (a V shape with the point at ).

15.2

Activity

15 mins

Using Distance to Solve Absolute Value Equations

Instructional Routines

None

Materials

None

Activity Narrative

In this activity, students explicitly make the connection between absolute value and distance, then solve and write equations involving absolute value using the distance connection. Students should be able to reason about the solutions by drawing or imagining a number line.

15.3

Activity

15 mins

Using Cases to Solve Absolute Value Equation

Instructional Routines

None

Materials

None

Activity Narrative

In this activity, students use the piecewise definition of absolute value to solve equations. In particular, they write two new linear equations based on inequality cases from the piecewise definition, then solve them. This method is offered for students who feel more comfortable solving equations in this way, but it should not be considered superior to reasoning about distance on a number line.

Lesson Synthesis

Display the equation

. Ask students

“How can this equation be interpreted as a distance on the number line?” (The distance between and 6 is 5.)
“What two equations can you write using the piecewise definition that can be used to solve this equation?” ( and )
“Which method would you use to solve this equation?” (I would use distance on a number line because it is easier for me to understand.)

If time allows, ask students to think of situations in which it is important to be a certain distance from something, but it does not matter which side of the thing we are on. For example, we should stand 7 feet away from a fire when using a fire extinguisher.

to access Cool-down.

Student Lesson Summary

The statement is true and can be interpreted as "The distance between 3 and 5 is 2." This is why it is also true to write .

The idea can also be used to solve equations in which a distance is known to a point. For example, tells us that the distance between and 4 is 6. On a number line, it might look like this.

A number line labeled x, 4, and x showing the distance between either x and 4 is 6.

What values of make this statement true? Both and make the equation true, so they are solutions to the absolute value equation.

Another way to solve an equation involving an absolute value is to look at the piecewise meaning.

We can take the equation and split it into two cases.

If , then the equation becomes . We can solve this using what we learned about linear equations. Adding 4 to each side, we can rewrite the equation as . We can check that this is fine for this case because the case begins with "If . . ." and ends with a value for that is greater than or equal to 4.
If , then the equation becomes . We can distribute the -1 on the left to get . From there, we can subtract 4 from each side and multiply each side of the result by -1 to get . We can also check that this is fine for this case because the case begins with "If . . ." and ends with a value for that is less than 4.

Putting the results of the cases together, we again get that the solutions are -2 and 10.

Whether you prefer reasoning about distances on a number line or using the piecewise definition to write equations in different cases, the solutions should be the same.

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Launch

Arrange students in groups of 2.

Remind students that one way to understand absolute value is using a piecewise definition. Display the function:

To show how this definition can be applied to situations where the inside is not just , ask students,

“What does this piecewise function look like if we replace every with ?” ()
“What does it look like if we solve the inequalities for ?” ()

Adjust the displayed function to incorporate the new understanding:

Read through Andre’s solution together to help students understand how to use two cases to solve an absolute value equation. Tell students that they do not need to include all of the reasoning every time they solve an equation involving absolute value. They should use the cases to write two new equations based on the piecewise definition and solve them, then check their answers in the original equations involving absolute value. For example, this work is sufficient to show when using Andre’s method:

Read the instructions for the four equations students should solve.

Tell groups to work on one of the questions together. Then each partner should select two to do individually and then check their partner’s solution.

Activity

None

Student Task Statement

Andre solves the equation by first getting the absolute value by itself on one side of the equation.

Then he uses the piecewise definition to split the equation into two cases.

If , then , so the equation becomes , which can be solved to . This fits with the initial condition because .
If , then , so the equation becomes , which can be solved to . This fits with the initial condition because .

Solve 1 of these equations using Andre's method of using cases. Solve a different equation by reasoning about distances on a number line. Solve the other 2 equations using either method.

Student Response

to access Student Response.

Building on Student Thinking

Students may write new equations without parentheses, which particularly affects the new equation that has a negative in front. Ask students to look closely at the two new equations Andre wrote and to check their solutions.

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Lesson 15

Solving Equations with Absolute Values

Warm-up

Where Does 30 Go?

Standards Alignment

HSA-CED.A.1

HSF-BF.B.3

Instructional Routines

Materials

Activity Narrative

Launch

Activity

Student Task Statement

Activity Synthesis

Activity

Using Distance to Solve Absolute Value Equations

Instructional Routines

Materials

Activity Narrative

Activity

Using Cases to Solve Absolute Value Equation

Instructional Routines

Materials

Activity Narrative

Lesson Synthesis

Student Lesson Summary

Have feedback on the curriculum?

Student Response

Building on Student Thinking

Are You Ready for More?

Launch

Activity

Student Task Statement

Student Response

Building on Student Thinking

Are You Ready for More?

Activity Synthesis

Launch

Activity

Student Task Statement

Student Response

Building on Student Thinking

Are You Ready for More?

Activity Synthesis

Standards Alignment

HSA-CED.A.1

Standards Alignment

HSA-CED.A.1

HSA-REI.A.1

HSF-IF.C