Ordering and Absolute Value of Rational Numbers

Lesson 2

Teacher:       

Date: 

Subject/Grade: 6^th grade mathematics

Length: 85 min

CC/ES Goal/ Objective:

6.NS.7 – Understand ordering and absolute value of rational numbers

6.NS.7a – Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram.

6.NS.7b – Write, interpret, and explain statements of order for rational numbers in real-world contexts.

6.NS.7c - Understand the absolute value of a rational number as its distance from 0 on the number line.

Lesson Objective: The learner can explain orally or in writing why the absolute value of a number, x, is equal to the absolute value of its opposite, -x.  The learner will be able to give and understand the standard definition of absolute value as a number’s distance from zero on a number line orally, graphically, or in writing.  The learner can demonstrate graphically on a number line why the absolute value of a number, -x, is equal to the absolute value of its opposite, x.

Vocabulary: Number line, rational numbers, opposites, absolute value, positive, negative, greater than, less than, reference point, origin

Materials: Document projector, white boards, computer and smart board hook-up, blank sheet of paper, pencil, Warm-up worksheet, Group scenarios worksheet, Formative assessment problems, Quiz problems

1.     Get Student Attention – Warm up (5 min)

Post warm-up to smartboard (see attachment A) and have students complete all three problems.  Upon completion have class share their answers and justify why they chose their specific numbers to represent the water, the boy, and the fish in the photograph. 

For an extension have students to try to generate an inequality statement with a partner to represent a comparison between the positions of the boy and the fish, in relation to the water.

2. State Objective (5 min)

Write the lesson objective to be displayed on the smart board and read it to the class. 

The learner can explain orally or in writing why the absolute value of a number, x, is equal to the absolute value of its opposite, -x.  The learner will be able to give and understand the standard definition of absolute value as a number’s distance from zero on a number line orally, graphically, or in writing.  The learner can demonstrate graphically on a number line why the absolute value of a number, -x, is equal to the absolute value of its opposite, x.

Remind students the real world applications of rational numbers and their opposites are numerous and populate a wide variety of fields. Tell students that learning to interpret the absolute value of a rational number will allow them to compare and make value judgements of rational numbers in real-life situations.

3. Activate Prior Knowledge (10 min)

Notes:

·      Students should understand values of negative numbers, i.e., -1/2 is smaller than -1/4.

·      Students should split the distance between numbers to show the place of the rational number. Students may struggle breaking the number line into pieces when placing the rational numbers. For example: -2/3 is larger than -1, but -2/3 is smaller than -1/2. When placing -2/3 on the number line, it will be to the left of -1/2. 

·      Opposite numbers are the same distance from 0 on the number line.  When added together, the sum is 0.

Students have already studied the concept of a rational number as a point on the number line.  Have class discussion to review definition and prior understandings related to absolute value.  Use document projector to remind students the absolute value of a number (x or -x) is just the value of the numeral, ignoring the sign, where positive numbers are written as numbers alone and negative numbers are written with a subtraction sign preceding the numeral. The absolute value of a number explains the distance the number is from zero on a number line.  Model the notation used to represent the calculation of absolute value, lxl = x or l-xl = x, for example |-5| = 5.  Remind students that one way to describe a number’s distance from zero on a number line is by calculating its absolute value. 

Review inequality statement vocabulary and remind students that they can use inequality statements to compare rational numbers in real-life situations.  Ask students to write down as many real-life situations involving the comparison of rational numbers as they can in 2 minutes. After ask students to pair up and share with a partner.  Circulate room while students work and discuss.

4. Present Content (10 min)

Using the document projector give real-life examples of comparing and ordering rational numbers.  Make sure to model absolute value notation and the use of number lines why giving explanations of examples and taking questions/facilitating discussion.  Examples of comparing rational numbers:

·      Owing money

·      Sports statistics

·      Measurement of altitude

·      Temperature measurement

·      Height measurement

5. Provide Learning Guidance (25-30 min)

Assign students to groups of 3 or 4.  Distribute one scenario from the “Group Scenarios” worksheet (see attachment B) to each group. Read over the scenarios and make sure each group understands what’s being asked.  Have each group of students follow the directions on their scenario and work through each real-life connection to rational numbers.  Circulate around to groups and ask, “how are you determining is the numbers are negative or positive? How can a numbers absolute value help to write an inequality statement?  How would these numbers be ordered on a number line?”  Remind students to use number lines and absolute value to help in solving the problems. 

When a group completes a scenario give them a new scenario.  After each group has had a chance to complete all six scenarios, assign each group one of the scenarios to present to the class.  Have each group discuss and justify their inequality statements to the using number lines, absolute value, diagrams, pictures, etc.

6. Elicit Performance (7 min)

Have students return to desks.  Project two ordering and absolute value of rational numbers problems onto the smartboard (see attachment C).  Using a blank sheet of paper, have students individually work out problems and justify their answers using number lines, absolute value, diagrams, etc.

7. Provide Feedback (3 min)

Using the document projector provide the solutions and explanations for both problems, making sure to model number line construction and absolute value notation throughout explanations.  Use the thumbs up, thumbs down method to gage how the class is performing.  If 80% or more of class correctly completes the problems the teacher may move on to assessment. 

If the class is still struggling, teacher should provide two new ordering and absolute value problems and again provide feedback on the given problems using the document projector. Again teacher checks for 80% class threshold and continues to model examples and take questions until at least 80% of class completes problems successfully.

8. Assessment (10 min)

In class quiz (see attachment D). Assessment will be will be turned in at the end of class, graded for correctness overnight and returned to the students the next day.

9. Retention and Transfer (5 min)

Teacher will Cask class and display questions using document projector:

Retention

“What is an absolute value and how can it be used in the real-world situations? Provide an example, use number lines, absolute value reasoning, diagrams, etc to justify your example”

Transfer

“How can absolute value be useful when measuring the distance between a negative and a positive point on a number line?”

Have students write their thoughts down on exit ticket paper or talk in small groups.

Attachment:  A

Warm-Up

1. Describe the position of the

  fish in comparison to the surface

  of the water.

2. Describe the position of the

 boy in comparison to the surface

 of the water.

3. What number would you use to

 represent the position of each of the

 parts of the photograph below?

 Explain.

            a. Water

            b. Boy

            C. Fish

Image Source:  http://iamdutchman.wordpress.com/2012/11/19/fish-poo/funny-man-fishing-fish-stick/

Attachment:  B

Group Scenarios

Owing Money

1.     Gabby, Tony, and Anne owe their aunt money for an upcoming field trip. Gabby owes $5.25, Tony owes $3.50, and Anne owes $5.75.

1. Who has the most debt?
2. Write an inequality statement comparing the three amounts of debt.

Golf Scores

2.     Kyle, Mekhi, and Cameron go golfing. The golf course has a par of 70.  Kyle ends the day at 67 giving him a score of -3. Mekhi gets a 71 giving him a score of +1. Cameron gets a 69 giving him a score of -1.

1. Who got the best score (Remember the best score is the lowest score)?
2. Write an inequality statement comparing the three golf scores.

Football Yardage

3.     The Ravens started their drive on the 0 yard line. Their running back ran for a loss of 3 yards. Then the wide receiver caught a pass for a 10 yard gain. The team then got a 5 yard penalty from the offensive lineman. Finally, the running back ran for a 15 yard gain.

a.    Draw a number line and locate each of the 5 events on the number line.

b.    Write an inequality statement representing each of the 5 points on your number line.

Altitude Below Sea Level

4.     Some sites around the United States have an altitude below sea level. Badwater, California in Death Valley is 282 ft below sea level.  New Orleans, Louisiana is 7 ft below sea level. Niland, California is 141 ft below sea level.

a.    Which site is the closest to sea level?

b.    Write an inequality statement comparing the distances to sea level.

Temperature

5.     The record lows for each month of the year for Huntington, West Virginia are listed below.

27°F, -2°F, 15°F, -24°F, -21°F, 39°F, 46°F, 43°F, 16°F, 29°F, 4°F, -14°F

a.    What is the coldest recorded temperature for Huntington, West Virginia?

b.    Write an inequality statement comparing the record lows for Huntington, West Virginia.

Height

6.     To ride a rollercoaster at Hershey Park you must be 48 inches or taller.  Sydnee is 48  inches, Brian is 48  inches, and Eunice is 48  inches.

a.    Are they all tall enough to ride the rollercoaster?

b.    Write an inequality statement arranging the heights from tallest to shortest.

Attachment:  C

Mathematics 6

Items to Support Formative Assessment

Unit 1: The Number System

6.NS.7 Understand ordering and absolute value of rational numbers.
6.NS.7b. Write, interpret, and explain statements of order for rational numbers in real-world contexts.

6.NS.7b Short Task
Create a real-world problem that could be represented by -14 < 4.  

Possible Solutions could include accurate statements about temperature, account balances, sea level, gains and losses, etc.

6.NS.7b Short Item
A scuba diver is swimming  feet below the surface of the water.  A coral reef is located  feet below the surface of the water. 

·       Write the number that represent the locations of the scuba diver and coral reef.

·       Compare the numbers using an inequality.

·       Sketch this scenario and describe the locations of the diver, the reef, and the surface of the water compared to one another.

Possible Solution:
The diver is  and the reef is .

 <   or   >

Students may choose to sketch on a number line. Responses could include statements about the reef being farther from the surface of the water (or deeper) than the diver.  The diver is closer to the surface than the reef.  The surface of the water is zero.

 6.NS.7b Short Item
Jill’s family bought a large pizza for $11.99, a salad for 5.99, and a $2 drink.  Molly’s family bought two calzones for $7.99 each and an order of wings for $8.99.

Compare the cost of each family’s order using an inequality.

Solution:  
Jill’s family spent $19.98 and Molly’s family spent $24.97.

$19.98 < $24.97 or  $24.97 > $19.98

6.NS.7b Item
a. -15 degrees Celsius is colder than -5 degrees Celsius.  Write this fact as an inequality.

Solution: -15 < -5

b. Draw a diagram of this inequality statement.
Possible Solution:  

6.NS.7b

Tanaysha and Derick were asked to write an inequality statement for the following temperatures (in degrees Fahrenheit):

                                                      -3, -7,  -1

Derrick wrote the inequality  -7 > -3 > -1. Tanaysha’s wrote the inequality   -1 > -3 > -7.

Who do you think is correct? Use a number line to help support your answer.

Attachment:  D

In-class Quiz                                                                                     Name___________

                                                                                                            Date____________

Name a positive or negative number to represent each solution and represent it on a number line.

1)    Saving $15

2)    12 feet below sea level

3)    What is the opposite of -6

Use a number line to find the absolute value of each integer. 

4)    │-7 │

5)     │4 │