Absolute Values (slideshow used with a number line)

Learning Objectives
• Students will be able to find and order absolute values of rational
numbers.
• Students will also be able to apply absolute values in real-world
contexts.
2

Four Parts
1. A Light Review
a) A Review of the Number Line and Representations of Number-Based
Values (.xlsx or .pdf)
2. Finding and Ordering of Absolute Values of Rational Numbers
3. Applying Absolute Values in the Real World
4. Quick Review
3

Learning Resources
• A Review of the Number Line and Representations of Number-
Based Values (as .xlsx or .pdf)
• Absolute Values (slideshow) (.pptx or .pdf)
• Video: “Absolute Values” (.mp4)
4

Definitions
• Absolute numbers: Positive whole numbers
• Absolute values: The magnitude or distance of a number from
zero on a number line, irrespective of its sign (whether negative or
positive)
• Decimals: The presentation of non-whole numbers by placing the
partial part to the right of a decimal point and the whole number
part to the left
• Fractions: Parts of a whole represented as a numerator and
denominator like ¼
8

Definitions (cont.)
• Integers: Positive and negative natural numbers
• Magnitude: Size or extent of a phenomena, a measure, a quantity,
a value, an amount
• Negative values: Numbers less than zero, often indicating losses
(often indicated by a – negative sign in front of the number or
value)
• Odds: The probability of a particular outcome, as in for or against
a particular outcome (expressed as x:y)
• Percentages: Values representing a proportion or ratio expressed
as a fraction of 100 (usually indicated by a % sign)
9

Definitions (cont.)
• Positive values: Numbers greater than zero, often indicating gains
(sometimes indicated by a + positive sign in front of the number or
value)
• Rational number: Any number that can be represented as a ratio,
a fraction, a division statement
• Reducing, simplifying: Representing fractional values in their
“lowest terms” for easy representation, easy calculations
10

Finding and
Ordering of
Absolute Values
of Rational
Numbers
12

Absolute Values
• The absolute value of a number is its distance from zero on the
number line, regardless of whether it is positive or negative.
• The absolute value may include coverage on the number line
involving both the negative and the positive side (and overlapping
the zero). For example, if x = -2 to +2, x has an absolute value of 4
or |x| = 4.
• Because the value represents a “distance” (viewed spatially), the
absolute value of any number will always be non-negative (either
positive or zero).
13

Starting Points
• The starting points in terms of rational numbers can be any value
in any number of representational forms.
• Remember to reduce and simplify the amount before trying to
place it in a number line.
14

Practice
• Place the following on your
(imaginary) number line. Use
whichever representation you
are most comfortable with…but
take a look at the other
equivalencies (where available)
across each row as well.
15

16
Whole #s Decimals Fractions Percentag
es
Odds Symbols Words Combinations
10 10.0 10/1 1000% --- X (roman
numeral)
Ten, a perfect 10 (9 + 1), (1 x 10),
(10 * 1)
--- 0.02 or .02 2/100 2% 2:98 --- Two percent (100% - 98%),
(1% + 1%)
--- 3.1415… --- --- --- π Pi Ratio btwn.
circumference
and diameter of
a circle
--- 0.07142857142
857
1/14 7% 1:13 --- One to 13 odds;
7 percent
chance
---
-
$12,000,001.11
-
$12,000,00
1 and
11/100
-
$1,200,000
,111%
--- --- Negative twelve
million and one
dollars and 11
cents

Calculating Magnitude (Size) with a Number
Line
• Absolute values are the sum total of the amount of units covered
in the number line.
• For example, a car travels from -7 to 8 on the number line. What is
the absolute value (the number of units between the two)? If you
said |15|, you are correct.
18

Why Absolute Values are Always Positive
• A car travels from -1 to -9. What is the absolute value? Why is the
absolute value |8|? Why isn’t it |-8|?
• Hint: It has something to do with the spatiality of the number
line…the fact that magnitude measures are units on the number
line.
19

Applying
Absolute Values
in the Real World
20

About Magnitudes
• Absolute values deal with magnitudes: the “size” or “extent” or
“measure” of a thing. [Remember, absolute values are defined as
units or steps from 0.]
• Absolute values deal with measures, such as distances, weights,
volumes, widths, heights, and costs.
• Magnitudes / absolute values deal with the scales of a thing.
Scales offer a range of values to measure a particular
phenomena. These relate to relative size.
• Absolute values help reason through the amounts of difference, of
change.
21

Driving Distances
• When a navigating GPS tool calculates driving distances between
two cities, it uses |absolute values| on the connecting roads to
indicate the actual mileage.
• It does not offer an “as the crow flies” sort of straight-line
calculation.
• Absolute values are about hyper precision.
• If a car drives north 20 miles and then south 2 miles and then east
10 miles, the calculation has to be additive: 20 + 2 + 10 = |32
miles|. The driving south two miles ds not mean that the mileage
is 18 + 10 = 28 miles (which would be incorrect).
23

Practice
• Route 1 to El Dorado requires
driving due south for 40 miles and
then west for 30 miles.
• Route 2 to El Dorado requires
driving southwest for 40 miles and
then east for 5 miles and then
south again for 10 miles and then
east for 2 miles.
• Which route saves on mileage in
terms of absolute mileage values?
24

Manufacturing Tolerances
• Absolute values are used to describe the “tolerances” or “margins
of error” that may be acceptable in manufacturing.
• A machine part may be able to be “off” the desired surface
thickness by only |.0001 mm| or whatever.
25

Temperature Changes
• It is a white-hot Kansas summer. The heat is a searing 110°
(degrees) Fahrenheit. A sudden storm blows in, resulting in a drop
of temperature to 70° Fahrenheit. What is the magnitude
(absolute value) of the degree drop?
• It is an unusually blue-cold Kansas winter. The temperature
started at -15° Fahrenheit. By the afternoon, it is at 2° Fahrenheit.
(What is the distance between these two rational numbers on a
number line?) What is the absolute value of the temperature
degree gain? Why? Is your absolute value negative or positive,
and why?
26

Home Buying
• You are looking to buy a home.
The home seller has marked up
the home $50,000 but then
says they are lowering the price
by $10,000. What is the
|absolute value| of the markup
at the current price?
27

Home Buying (cont.)
• You are looking to buy a home. You have a down payment of 20% of the
home’s value or $40,000. You have closing costs of $3,000 total before
any pay-downs. You have homeowner’s insurance to pay to the tune of
$3,000. You’re paying $500 for a home inspection. The HOA bill will
come due shortly, and that is $500 a year.
• The home seller is tossing in a cash contribution of $6500 for a
thorough interior paint job and to do some cosmetic fixes discovered
during the home inspection. The mortgage loan company is
contributing $500 to help with closing costs.
• The absolute total value to achieve the home buying is “|x|”. What is |x|?
• The out-of-pocket costs for you is y. What is the |y|?
28

Quick Review
• What is a number line? Why does a number line have positive and
negative numbers? What is the power of 0 in a number line?
• Define “rational numbers.” Are there more rational numbers in the
world or irrational ones? Why?
• Why are numerical values expressible as some mix of the
following: whole numbers, decimals, fractions, percentages,
odds, symbols, words, and other mixed combinations?
• What is an |absolute value|? How does an |absolute value|
indicate magnitude or size? Scale? Relative size? Why is an
absolute value always positive, never negative?
30

Quick Review (cont.)
• What are some ways to use |absolute values| to understand
magnitude (size) out in the world?
• In driving distances?
• In calculating price differences?
• In manufacturing workplaces to understand tolerances?
• In calculating the change in temperatures over time?
• How do absolute values help a person not fall into some bad math
traps?
31

Questions in the
Chat?
Please post any questions you may
have in the chat.
32

Next Up
• A number line can also be used to understand multiples, factors,
and exponents. These relate to multiplication as applied to the
integers, negative and positive.
• Make sure that you have the addition and subtraction down from
this lesson of |absolute values| …and the number line logic.
33

Notes
• The number line illustration was created by Rod Pierce and shared
by Hakunamenta on Creative Commons with Creative Commons
licensure release in 2012. The link is
https://commons.wikimedia.org/wiki/File:Number-line.svg.
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There is a link to the original source on Wikipedia: https://en.wikipedia.org/wiki/Integer

Absolute Values (slideshow used with a number line)

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