1. Absolute Value and Comparing Rational Numbers
Lesson 3.2.2
Have answer key from lesson plan with you while using PowerPoint.
See notes on each slide for teaching points and additional answers.

2. This “distance” is called the absolute value.
Prior…
Yesterday, we learned about Integers and how to graph them on a horizontal and vertical number line.
We also learned about opposites.
Two numbers are opposites
if they are the same distance from zero on a number line but are on opposite sides of zero.
This “distance” is called the absolute value.

3. Today… We will review how to graph rational numbers on a number line.
We are also going to learn about Absolute Value.

4. Let’s Get Started… Graphing and comparing rational numbers Section 2
absolute value
elevation
distance
magnitude

5. (Step 1) -¾ lies between 0 and -1 ¾ lies between 0 and 1
Locate and graph ¾ and its opposite on a number line.
Step 3   
If the number is positive…
Look at the numerator and count over that many tick marks going RIGHT.
Step 1
Determine between which two numbers the rational number lies.
Step 2
Look at the denominator and then divide the segment into that many equal parts.
If the number is negative…
Look at the numerator and count over that many tick marks going LEFT.
The opposite of ¾ is -¾.
Explain how to graph a fraction on a number line using this example.
The problem tells you to graph ¾ and its opposite.
The opposite of ¾ is – ¾. So, those are the two dots that are going to be graphed.
-¾ lies between 0 and -1
¾ lies between 0 and 1 -1 1

6. (Step 2) -¾ and ¾ both have a denominator of 4
Locate and graph ¾ and its opposite on a number line.
Step 3   
If the number is positive…
Look at the numerator and count over that many tick marks going RIGHT.
Step 1
Determine between which two numbers the rational number lies.
Step 2
Look at the denominator and then divide the segment into that many equal parts.
If the number is negative…
Look at the numerator and count over that many tick marks going LEFT.
-¾ and ¾ both have a denominator of 4
4 equal spaces
4 equal spaces -1 1

7. (Step 3) ¾ is positive -¾ is negative go LEFT to the ¾ -¾
Locate and graph ¾ and its opposite on a number line.
Step 3   
If the number is positive…
Look at the numerator and count over that many tick marks going RIGHT.
Step 1
Determine between which two numbers the rational number lies.
Step 2
Look at the denominator and then divide the segment into that many equal parts.
If the number is negative…
Look at the numerator and count over that many tick marks going LEFT.
¾ is positive
go RIGHT to the 3rd tick mark -1 1
-¾ is negative go LEFT to the
3rd tick mark -¾
Maybe remind them that ¾ means that we have “3 out of 4 parts”.
Fill in the rest of the fractions for this number line!

8. You Try
1) Write the length of each segment as a fraction and a decimal.
Fraction: _____ Decimal: _____ ?

9. Answer
1) Write the length of each segment as a fraction and a decimal.
Fraction: _____ Decimal: _____
1/5 .2 ?

10. You Try
Identify the position of Point A and Point B on the number line
as a fraction and a decimal.
Point A Point B
Fraction: _____ Fraction: _____
Decimal: _____ Decimal: _____ B A
Which is greater?
Compare using an inequality symbol.
_____ _____

11. Answer Identify the position of Point A and Point B on the number line
as a fraction and a decimal.
Point A Point B
Fraction: Fraction:
Decimal: Decimal: B A
Which is greater? Compare using an inequality symbol.
>

12. Improper Fractions
On this number line, each whole unit is divided into ____’s.
An improper fraction is a fraction where the numerator is bigger than (or equal to) the denominator.
How would you write each improper fraction on this number line as a whole or mixed number?
Walk students through converting 3/3, 4/3, 5/3 and 6/3 to a whole number or mixed number. Have them write it in their student notes.
3/3 = 3 divided by 3 = 1
4/3 = 4 divided by 3 = 1 1/3
5/3 = 5 divided by 3 = 1 2/3
6/3 = 6 divided by 3 = 2
On this number line, each whole unit is divided into third’

13. You Try
Using the skills that you have obtained in Chapter 3, represent the below rational numbers on the provided number line.
Be sure and label your fractions and decimals!
Then order from least to greatest.
Go over different strategies to graph these rational numbers.
To graph 4.4… Students may want to think of it as 4/10’s and divide the segment located between 4-5 into 10 equal parts, then plot by counting over to the fourth tick mark (moving right) . They also may realize that .4 is just under half and can estimate where the point should go.
To graph -2.8… Students can divide the segment located between -2 and -3 into 10 equal parts, then plot point by counting over to the eighth tick mark (moving left). They can also estimate by finding half of the segment and realizing that -.8 is to the left of that.
To graph 3 ¼… Students can divide the segment located between 3-4 into 4 equal parts, then plot.
To graph -14/3…turn to a mixed number first.
To graph -1/5… Students can divide the segment located between 0 and -1 into 5 equal parts, then plot.

14. Answer
Using the skills that you have obtained in Chapter 3, represent the below rational numbers on the provided number line.
Be sure and label your fractions and decimals!
Then order from least to greatest.
Go over different strategies to graph these rational numbers.
To graph 4.4… Students may want to think of it as 4/10’s and divide the segment located between 4-5 into 10 equal parts, then plot by counting over to the fourth tick mark (moving right) . They also may realize that .4 is just under half and can estimate where the point should go.
To graph -2.8… Students can divide the segment located between -2 and -3 into 10 equal parts, then plot point by counting over to the eighth tick mark (moving left). They can also estimate by finding half of the segment and realizing that -.8 is to the left of that.
To graph 3 ¼… Students can divide the segment located between 3-4 into 4 equal parts, then plot.
To graph -14/3…turn to a mixed number first.
To graph -1/5… Students can divide the segment located between 0 and -1 into 5 equal parts, then plot.

15. Moving On… Graphing and comparing rational numbers Section 2
absolute value
elevation
distance
magnitude

16. Absolute Value
The Absolute Value is the distance a number is from zero on a number line.
Distance is always stated as a positive number,
so… the absolute value of a number is always positive.
Sometimes distance is called the magnitude because in mathematics,
magnitude refers to the size or length of something. Again, always positive.
The absolute value of -3 = 3
The absolute value of 3 = 3
and
3 units
Positives !

17. The symbol for absolute value is represented by two bars, | |.
| x | = x |-x| = x
Examples Read like this
| -2 | = 2 The absolute value of -2 is 2.
| 10 | = 10 The absolute value of 10 is 10.

18. You Try
Directions: Complete the following chart. Look at example for guidance.
Answers can be found in lesson plan.
Stress that absolute value is the numbers distance from zero on number line.

19. Answer
Directions: Complete the following chart. Look at example for guidance.

20. Let’s do an Exploratory Activity… This will help you understand
absolute value, distance and magnitude
in a Real-World context……
But, before we get started…
Let’s talk a second about elevation. Then we will begin the activity.
Proceed to next slide and go over the word “ELEVATION” before passing out the exploratory activity sheet.

21. Elevation think of it as your Integer.
Elevation is Positive
200 feet
Elevation
When you see the word ELEVATION,
think of it as your Integer.
It can be positive (+) or negative (-)
depending on the situation.
Elevation is 0
Sea Level
Elevation is Negative
- 20 feet
Review “elevation” before passing out the Exploratory Activity sheets.
Vocabulary
The elevation of a geographic location is its height above or below a fixed reference point, most commonly the Earth's sea level.

22. Part 1 Exploratory Do Part 1 of the Exploratory Activity.
Then STOP so that we can review
before you continue to Part 2.
Pass out the Exploratory Activity. Give students a few minutes to work on Part 1. Then proceed to next slide to review Part 1.
Students need to have their graphs corrected before proceeding to Part 2.

23. Exploratory
Discuss deciding upon an appropriate scale for the number line. Ask why intervals of ones or twos will not work. They should say…”because there are not enough tick marks to plot 38 and -30”.
Ask why intervals of 5 make sense. They should say “because a lot of the numbers in the story problem are multiples of 5 and you will still be able to graph all the numbers”.
Have students come to the board to graph and label the different situations. Pictures may be drawn as well. Tell the students to keep the pictures simple (like stickmen). Discuss as students graph the situations. If they graphed it at the wrong location, make sure it is fixed before proceeding to next item.

24. Exploratory Vocabulary
Call on students to answer.
Vocabulary
The elevation of a geographic location is its height above or below a fixed reference point, most commonly the Earth's sea level.
Think of it as your Integer. It can be positive (+) or negative (-) depending on the situation.

25. Exploratory
Call upon students to answer. Have them explain why. For example, -30 is the least because it is below zero and farther down the number line.

26. Exploratory
Call up on students to determine 2 inequality statements for the situation using the elevations.
-5>-15
-15<-5

27. Exploratory
Call upon students to determine 2 inequality statements for the situation using the elevations.
-15>-30
-30<-15

28. Exploratory
Part 2
Think-Pair-Share or Elbow Partners would be great for Part 2.
Give students a few minutes and then proceed to next slide to review and teach.

29. Exploratory
Let’s determine these distances and then see how we can represent it using Absolute Value !
“travel” indicates distance which means that it needs to be positive number.
f) 5 feet; The distance to the surface can be represented using absolute value |-5| = 5
15 feet; The distance to the surface can be represented using absolute value |-15| = 15
30 feet; |30| = 30
Distance or any anything that is measuring the magnitude of something is always stated as a positive (+)!

30. Exploratory
Some may have counted the intervals to determine the distance.
Others may have added the distance above zero to the distance below zero. Both methods are fine.
15 feet; |-5| + |10| = 5+10 = 15
25 feet; |-15| + |10| = = 25
68 feet; |-30| + |38| = = 68
45 feet; |30| + |-15| = = 45
What methods did you discover when trying to determine the distance between objects where…
One object is above 0 and the other object is below 0?

31. Exploratory When dealing with distance or magnitude…
Distance from the captain to diver 1 < Distance from the ocean surface to the floor.
When diver #1 swims to -10 feet, his distance below the surface = the captain’s distance above the surface. They are opposites!
When dealing with distance or magnitude…
use positive (+) numbers.
In other words, use the absolute values.

32. RULE – Finding Distances
When finding distances between two points on a number line:
If the numbers are on the same side of 0…
Subtract the absolute values of the two numbers.
Sea Turtle Diver #1
|-25| |-5| = 25 – 5 = 20 feet
If the numbers are on “opposite” sides of 0…
Add the absolute values of the two numbers.
Sea Turtle Captain
|-25| |10| = = 35 feet
On the first one (sea turtle and diver 1) Prove that it is true by counting the intervals between the points.
On the second one (sea turtle and captain) count the intervals but reference 0. For example… can say that there are the 5 intervals below zero and 2 intervals above 0.
Ask then about from sea turtle and surface of the ocean. Which rule would you follow?
Answer… both actually work, but simpler to just take the absolute value of the number. |-25| = 25

33. You Try!
Use a number line and absolute value to find the distance between each pair of numbers listed below.
1) -7 and -2 2) and 5 3) and 0
1) Plot the points. The points are on same side of 0, so subtract the absolute values.
|-7| - |-2| = 7 – 2 = 5
Then count intervals to prove distance.
Plot the points. The points are on opposite sides of 0, so add the absolute values.
|-7| + |5| = = 12
Then count intervals to prove distance. Be sure to reference zero… seven intervals to the left of zero and 5 intervals to the right of zero.
Plot points.
|-7| = 7

34. Answer
Use a number line and absolute value to find the distance between each pair of numbers listed below.
1) -7 and -2 2) and 5 3) and 0
|-7| - |-2| = 7 – 2 = 5
|-7| + |5| = = 12
|-7| = 7
For each one, plot the points. Discuss if the dots are on the same side of 0 or different sides and do we add or subtract.
Then count intervals to prove distance. Be sure to reference zero. Example: seven intervals to the left of zero and 5 intervals to the right of zero.

35. In this situation, it is the amount of debt!
When “Less” is “More”!
Jessica and Amy are both overdrawn on their checking accounts.
Graph and label the two account balances on the number line below.
Order the rational numbers from least to greatest. ______, ______
2) Use absolute value to determine who is in more debt?
Jessica
Amy
Account Balance
-$4.75
-$2.25
no debt!
Model for students.
Order from least to greatest: , -2.25
Demonstrate & discuss graphing and labeling balances.
2) Jessica is more in debt.
Jessica The distance from to 0.
|-4.75| = 4.75 in debt
Amy The distance from to 0.
|-2.25|= 2.25 in debt
Balance Absolute Value
Jessica ___________ = ______________ debt
Amy ___________ = ______________ debt
Magnitude
In this situation, it is the amount of debt!

36. RULE - magnitude It makes sense…
When dealing with NEGATIVE numbers…
The SMALLER the negative number, the GREATER the magnitude (or absolute value).
Jessica
Amy
no debt!
- 4.75
|- 4.75| = $4.75
- 2.25
|- 2.25| = $2.25
It makes sense…
Jessica is more in debt because her distance from $0 is greater.

37. You Try!
Malachi has a balance of -$18. Miggle is more in debt than Malachi.
Draw a diagram that shows the range of numbers that could be Miggle’s balance. Let $0 represent no debt.
Which of the following could be Miggle’s balance?
Circle all that apply.
-$ $ $ $11 -$80
See if you can figure out a way to represent multiple possible answers on your diagram?
You don’t need this diagram to be as detailed as a number line.
Keep it simple! $0
This does not need to be detailed like a number line. It is a quick way for students to draw out a situation to help them answer a question.
Students should place a tick mark somewhere to the left of $0 and label it -$18.
Then draw an arrow to the left starting at -18 to show that Miggle’s debt can be any of those amounts.
Look at the student notes answer key in the lesson plan to see diagram.
b) -19, -20, -80
Miggle

38. Answer
Malachi has a balance of -$18. Miggle is more in debt than Malachi.
Draw a diagram that shows the range of numbers that could be Miggle’s balance. Let $0 represent no debt.
Which of the following could be Miggle’s balance?
Circle all that apply.
-$ $ $ $11 -$80
See if you can figure out a way to represent multiple possible answers on your diagram?
You don’t need this diagram to be as detailed as a number line.
Keep it simple!
This does not need to be detailed like a number line. It is a quick way for students to draw out a situation to help them answer a question.
Students should place a tick mark somewhere to the left of $0 and label it -$18.
Then draw an arrow to the left starting at -18 to show that Miggle’s debt can be any of those amounts.
Look at the student notes answer key in the lesson plan to see diagram.
b) -19, -20, -80
Miggle

39. Closure
Today we learned that absolute value is the distance a number is from _____ on a number line?
Absolute Value is always stated as a _______ number?
What symbol is used for absolute value?
In our Exploratory Activity, we used absolute value to measure distances above and below sea level. Distances are always stated as a ______ number?
In mathematics, what “m” word is sometimes used to describe
size or length of something?
Is it stated as a positive or negative
number?
The temperature went from -10° C to 20° C.
How can you use what we leaned today to describe the
change?
zero, positive, ||
Positive
Magnitude; positive
|-10| + |20| = = 30 degrees Celsius change.

40. Closure What does each tick mark represent and how can you tell? ? A
What are two equivalent fractions that represent Point A. ? A
Each tick mark is 1/10th because between integer there are 10 equal parts.
-5/10 and -½

41. Classwork I’ll give you _______ minutes!
When time is up, pull up lesson plan and display the answers.
Go over as many as you can.
I’ll give you _______ minutes!