what is the absolute value of –6? use the number line to help answer the question.

If y'all're educational activity math to students who are ready to larn well-nigh accented value, typically around Class half dozen, here's an overview of the topic, along with two lessons to introduce and develop the concept with your students.

What Does Absolute Value Mean?

Absolute value describes the distance from zero that a number is on the number line, without because direction. The absolute value of a number is never negative. Take a look at some examples.

• The absolute value of 5 is five. The distance from 5 to 0 is 5 units.

• The accented value of –5 is 5. The distance from –5 to 0 is 5 units.

• The absolute value of 2 + (–vii) is v. When representing the sum on a number line, the resulting point is 5 units from nada.

• The accented value of 0 is 0. (This is why nosotros don't say that the absolute value of a number is positive. Zero is neither negative nor positive.)

Accented Value Examples and Equations

The about common way to stand for the absolute value of a number or expression is to environs it with the accented value symbol: two vertical straight lines.

• |6| = 6 means "the absolute value of 6 is 6."
• |–6| = 6 means "the absolute value of –half dozen is 6."
• |–ii – 10| means "the absolute value of the expression –2 minus x."
• –|x| means "the negative of the accented value of x."

The number line is not just a way to show distance from zero; information technology's also a useful manner to graph equalities and inequalities that contain expressions with accented value.

Consider the equation |ten| = 2. To show ten on the number line, you need to bear witness every number whose absolute value is ii. At that place are exactly 2 places where that happens: at two and at –2:

At present consider |x| > 2. To show ten on the number line, you lot need to show every number whose absolute value is greater than two. When you graph this on a number line, apply open dots at –two and 2 to signal that those numbers are non part of the graph:

In full general, you get two sets of values for any inequality |x| > k or |x| ≥ thousand , where k is any number .

Now consider |ten| ≤ 2. Yous are looking for numbers whose absolute values are less than or equal to 2. This is true for any number betwixt 0 and 2, including both 0 and 2. Information technology is as well true for all of the opposite numbers betwixt –2 and 0. When you graph this on a number line, the closed dots at –two and 2 indicate that those numbers are included. This is due to the inequality using ≤ (less than or equal to) instead of < (less than).

In general, you lot only get one fix of values for any inequality |x| < thou or |x| ≤ k , where thou is any number .

One mode to think well-nigh information technology is, you're still getting two sets of values (the "negative" set and the "positive" gear up), simply because they meet at zero, they converge into ane set. These inequalities tin can be rewritten without absolute value signs by writing the expression within the absolute value as falling betwixt two numbers:

• You can rewrite |ten| < ii like this: –2 < 10 < 2
• You lot can rewrite |x| ≤ 4 like this: –4 ≤ x ≤ iv
• You can rewrite |10 + 6| < 25 like this: –25 < x + 6 < 25

Lesson 1: Introducing the Concept

Materials: A number line and colored dots that the entire class tin see

Grooming: If y'all don't have a commercially prepared number line, depict one either on the chalkboard or on a long (preferably thin) canvass of paper. If educational activity remotely, share an accented value number line that the entire class tin can see. Include at to the lowest degree –twenty to 20.

Standards:

• Interpret statements of inequality as the relative position of numbers on a number line. (half dozen.NS.C.seven.A)
• Create and interpret statements of society for rational numbers in real-world contexts. (half-dozen.NS.C.7.B)

Prerequisite Skills and Concepts: Students demand to exist familiar with the inequality symbols and how to make and use a number line. They also demand to exist able to compute with negative numbers.

• Stand and so that the door is to your right.
• Ask: About how far am I from the door?Students should respond with an estimated number of feet.
• Ask: If I were blindfolded, how would you tell me where the door is?Students should respond with both the estimated number of feet and a direction. If students provide dissimilar units, even nonstandard ones like "steps" or "arm lengths," phone call attention to the differences and encourage them.
• Now, stand so that the door is to your left.
• Ask: About how far am I from the door now?Students should respond with an estimated number of anxiety.
• Enquire: If I were blindfolded, how would you tell me where the door is?Students should respond with both the estimated length and a direction.
• Say: When I asked how far I was from the door, y'all gave me a number of feet, and information technology didn't matter which fashion I was facing. But, when I asked where the door was in relation to my position, you lot gave me a direction also as a number, and and then it did matter which manner I was facing. Today, we'll discuss absolute value. Absolute value tells you how far a number is from zero. But it doesn't tell you which way to go! It doesn't tell you what direction a number is from cipher.
• Point to vi on the number line.
• Enquire: What number is this? How far from nada is information technology?Go along this questioning with –6, 0, xiv, and –14, and so forth, until you are certain that students can differentiate between a directed distance and an accented distance. Reinforce the response to each altitude question by saying, " The absolute value of [bare] is [blank]" and writing |___| = ___.
• Divide the course into 2 teams. Squad i is the Signed Numbers Team, and Team ii is the Accented Value Squad.
• Ask: Can someone on Squad i enquire a question that would require a signed number—or a direction—equally its reply?

  You are looking for questions such as: What was the temperature in degrees Fahrenheit? How do I go to Lake Erie from here?If appropriate for yous and your students, compare the different ways that signed numbers and management appear in student answers. For example, a temperature value merely has "direction" because degrees in Fahrenheit include negative and positive values, then a sign is needed for clarity. Directions to Lake Erie would require a variety of signed numbers: where to travel due east (positive) vs. west (negative) and where to travel north (positive) vs. southward (negative).

• Ask: Tin someone on Team 2 inquire the same question so that it does not require a signed number—or a direction—as its answer?

  You are looking for questions such equally: How far above nil degrees Fahrenheit was the temperature? How far north is Lake Erie from here?

  Have the teams alternate going first, thinking of questions that require a signed or absolute number to answer, and so having the other team revise the question. Encourage students to be creative with their questions.

• Bring the total course together over again. Place colored dots at –vi and +3 on the number line.
• Enquire: How would you compare these ii numbers?Arm-twist student responses and contrast differences in precision, order, and vocabulary. Ask students to both say and write the comparison: –6 < +iii or –6 < three.
• Ask: How would you compare the accented values of these two numbers?Enquire students to both say and write the comparison. |–6| > |+iii| or |–vi| > |3|.

  Discuss why the management of the comparison symbol inverse, using the large number line to illustrate what the students and you say. Practise several more examples until you are satisfied that students can compare both signed numbers and accented values.

Lesson two: Developing the Concept

Materials: Index cards or digital "cards" that can be distributed among the class

Standards:

• Sympathize the absolute value of a rational number as its distance from 0 on the number line. (half dozen.NS.C.vii.C)

Preparation: Make cards for I Accept…Who Has?

• Say: Retrieve that absolute value is the altitude that a number is from 0, no affair which direction.
• Enquire: Can someone write an equation that means "24 is the absolute value of the number that is vi less than x?"The equation, 24 = |10 – 6|, represents the situation. You may need to repeat the equation several times, slowly, every bit students try to parse it out.
• Ask: What can be the value of the expression inside the absolute value symbols?It is natural to evidence that ten – 6 tin can have a value of 24. Aid students run across that the expression can also have a value of –24. If necessary, remind them of your previous give-and-take about directed distance from nil every bit opposed to absolute distance from zero.
• Ask: If the expression tin have a value of 24 or –24, what values can x have?Take students endeavor to notice possible values for x themselves at beginning. And so have them compare what they found, and facilitate a word around different strategies they used.If 10 = thirty, then x – vi = 24. If x = –xviii, so x – half dozen = –24. There are two possible values for 10: 30 and –xviii.
• Repeat the last 3 questions using a variety of absolute value expressions:

  |13 – x| = 14 (Solution: x = –1 or 10 = 27) |25 + ten| = 25 (Solution: ten = 0 or 10 = –50) 42 = |210| (Solution: 10 = 21 or ten = –21) i = |x/36| (Solution: ten = 36 or x = –36) 0 = |36/10| (Solution: There is no value for x that satisfies this equation.)

Wrap-Up and Assessment Game

• Enquire students to write and share their ain definitions and real-life examples of absolute value situations.
• Play I Take...Who Has? Make up a set up of 15 index cards with accented value equations and fifteen index cards containing values for the variable. If index cards aren't bachelor, or y'all're adapting this for remote learning, create a way for the xxx equations beneath to exist distributed equally as as possible among your students.

Absolute Value Cards	Variable Value Cards
|10 + 5| = 20	x = 15
|5 – x| = 30	x = –25
|x + 6| = 41	x = 35
|–27 – x| = twenty	x = –47
–7 + |x| = 0	10 = –7
|25 – x| = xviii	x = 7
|x + –5| = 38	x = 43
|37 – x| = lxx	x = –33
114 – |x| = 7	x = 107
|–x + 100| = 21	ten = 121
–|1 + x| = -80	x = 79
|10| = 81	x = –81
|10 + 3| = 84	10 = 81
|25 + ten| = 62	x = –87
|x – 26| = 11	x = 37

Each Absolute Value Carte du jour listed has two values for ten. These values overlap so that each Variable Value Card satisfies two of the given accented value equations (the start and second values satisfy the kickoff equation, the second and third values satisfy the second equation, and and then on, until the terminal and first values satisfy the last equation).

Distribute the cards or equations equally. Be sure they've all been distributed. Cull a student to say "I have" and and so read the value or equation on their card. Then have the pupil say "Who has a friction match for my card?" Whatever student with a match should say "I Accept…Who Has…," and the game gain until all cards have been read. You might have students stand when the game starts and sit as they offer a response. To keep all engaged, offer a advantage for successful completion of the game, encouraging challenges to suspect responses.

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