# Lesson 14

Absolute Value Meaning

These materials, when encountered before Algebra 1, Unit 4, Lesson 14 support success in that lesson.

## 14.1: Math Talk: Closest to Zero (10 minutes)

### Warm-up

The purpose of this Math Talk is to elicit strategies and understandings students have about the distance from 0 on the number line. These understandings help students develop fluency and will be helpful later in this lesson when students will need to be able to think about distance from 0 for various rational numbers.

### Launch

Display one problem at a time. Give students quiet think time for each problem and ask them to give a signal when they have an answer and a strategy. Keep all problems displayed throughout the talk. Follow with a whole-class discussion.

### Student Facing

For each pair of values, decide mentally which one has a value that is closer to 0.

$$\text{-}\frac{4}{5}$$ or $$\frac{5}{4}$$

$$\frac{1}{12}$$ or $$\frac{1}{14}$$

-0.001 or 0.0001

1.3 or $$\frac{4}{3}$$

### Activity Synthesis

Ask students to share their strategies for each problem. Record and display their responses for all to see. To involve more students in the conversation, consider asking:

• “Who can restate $$\underline{\hspace{.5in}}$$’s reasoning in a different way?”
• “Did anyone have the same strategy but would explain it differently?”
• “Did anyone solve the problem in a different way?”
• “Does anyone want to add on to $$\underline{\hspace{.5in}}$$’s strategy?”
• “Do you agree or disagree? Why?”

## 14.2: What is Absolute Value? (10 minutes)

### Activity

In this activity students recall the notation for absolute value presented in grade 6 by explaining the meaning of absolute values in a situation and then comparing values using absolute value. In the associated Algebra 1 lesson, students build on the understanding from grade 6 by thinking of the absolute value function as a piecewise function.

### Student Facing

1. One of the lowest places in Europe is 23 feet below sea level. We can use “-23 feet” to describe its elevation, and “$$|\text{-}23|$$ feet” to describe its vertical distance from sea level. In the context of elevation, what would each of the following numbers describe?
1. 37 feet
2. $$|37|$$ feet
3. -6 feet
4. $$|\text{-}6|$$ feet
2. Water freezes at 0 degrees Celsius. For each pair of temperatures, which temperature is closest to the freezing point of water? Use absolute value to explain your reasoning.
1. $$14^\circ \text{C}$$ or $$\text{-} 15^\circ \text{C}$$
2. $$\text{-} 5^\circ \text{C}$$ or $$\text{-} 2^\circ \text{C}$$
3. $$\text{-} 12^\circ \text{C}$$ or $$18^\circ \text{C}$$

### Activity Synthesis

The purpose of the discussion is for students to recall their understanding of absolute value and its meaning in context. Ask students,

• “What does the equation $$|\text{-} 4| = 4$$ tell you?” (It tells me that -4 is 4 away from 0 on the number line.)
• “Clare is complaining that she cannot pay off her credit card this month. When you ask why, she replies with $$|\text{-}245| > 175$$. How do you think this inequality makes sense as a reason?” (It probably means that Clare owes $245 on her credit card, but only has$175. Since the amount owed on the credit card is greater than the amount of money she has, she cannot pay off the credit card.)
• “Is the rule $$y = |x|$$ a function? Explain your reasoning.” (Yes, because each input has exactly 1 output.)
• “Mai wants to write the meaning of absolute value in some notes and writes $$|x| = x$$. Is this equation true for every value of $$x$$?” (No. If $$x$$ is negative, this is not true. For example, $$|\text{-}5| = \text{-}5$$ is not a true equation.)

## 14.3: Absolute Value Expressions (20 minutes)

### Activity

In this activity, students get more familiar with how to use the absolute value by learning its place in order of operations. In the associated Algebra 1 lesson, students examine absolute value as a piecewise function. This activity helps familiarize students with absolute value as an operation before they understand it as a function.

### Launch

Tell students, “In terms of order of operations, absolute value is treated as having a parentheses within the absolute value. For example, to find the value for $$3|2 - 8|+4$$, first think of the expression as $$3|(2 - 8)|+4$$. This means that the subtraction should be done first $$3|\text{-}6|+4$$, then the absolute value $$3 \boldcdot 6 + 4$$, then the multiplication $$18 + 4$$, and last the addition. In this example, the value is 22.”

### Student Facing

Find the value of these expressions.

1. $$|10-12|$$
2. $$|\text{-}16 + 5|$$
3. $$3|9 - 4|$$
4. $$|8 - 10| - 16$$
5. $$2|\text{-}13 - 2| + 18$$
6. $$9 - |19 + 3|$$

### Student Response

• “Tyler thinks that finding an absolute value just means ‘to make it positive.’ For the first question, Tyler writes $$10 + 12$$ then finds the value to be 22. What is wrong with Tyler’s thinking?” (First, absolute value means the distance from 0 on the number line, not ‘make it positive.’ Then, Tyler tried to use some distributive process for absolute value which does not result in the correct value.)
• “Jada thinks that the parentheses inside of the absolute value means you can distribute into the absolute value. She solved the third question by this method, so her first step was to rewrite the expression as $$|27 - 12|$$. Does this example work? What if the value on the outside of the absolute value was -3 instead of 3? Find the value of $$\text{-}3|9 - 4|$$ by finding the absolute value before multiplying, then again by distributing first. Do you get the same value?” (Yes, distributing works and results in the same value for the third question. However, if the number being multiplied is negative, it does not work. Doing the absolute value first results in the value -15, but distributing first results in the value +15.)