THE NEWARK PUBLIC SCHOOLS

THE OFFICE OF MATHEMATICS

Grade 6 The Number System 6.NS.5 & Review

2012 COMMON CORE STATE STANDARDS ALIGNED MODULES

The Number System 6.NS.5

MATH TASKS

Apply and extend previous understandings of numbers to the system of rational numbers.

Goal:

Prerequisite Skills:

Thus far, students have been using positive whole numbers in a variety of ways: to compare, measure, label, and quantify. In this module, students will extend their understanding of rational numbers, including negative values.

 Understand equivalence of and translations between positive rational numbers in fraction and decimal form  Use a number line to locate, order, and compare positive rational numbers  Extend numbers to the left of zero

Essential Questions: How does the understanding of the properties of number (fractions, decimals (that have a finite or a repeating decimal representation), and percents as different representations of rational numbers) extend to addition, subtraction, multiplication and division of rational numbers?

Office of Mathematics

THE NEWARK PUBLIC SCHOOLS

Embedded Mathematical Practice(s) MP.1 Make sense of problems and persevere in solving them Lesson 5 MP.2 Reason abstractly and quantitatively Review of the Number System MP.3 Construct viable arguments and critique the reasoning of others MP.4 Model with mathematics MP.5 Use appropriate tools strategically Lesson 4 MP.6 Attend to precision Review of the Number System MP.7 Look for and make use of structure MP.8 Look for and express regularity in repeated reasoning

Lesson 3 Golden Problem

Lesson 2 Golf Scores 6.NS.5 (extended) Representing positive and negative numbers in the real world.

Lesson Structure: Introductory Task Guided Practice Collaborative Work Journal Questions Skill Building Homework

Lesson 1 Warmer in Miami 6.NS.5 Describing positive and negative numbers as quantities. Page 2 of 19

Lesson 1: Multiple Representation Framework CONCRETE REPRESENTATIONS 

2-color coin counters to represent negatives and positives



Number Lines



Thermometers and other equally partitioned tools PICTORIAL REPRESENTATIONS



Number Lines (Horizontal)



Number Lines (Vertical)



Distance / Vector Model

Adding Integers Addition is modeled as putting a second vector’s tail at the first vector’s head and finding where the second vector’s head extends to.

3 + -4 = -1

Subtracting Integers Subtraction can be thought of as comparing the two vectors p, and q, by putting both tails together (starting each from zero) and asking the question: “How would one extend a vector from the head of p to the head of q?” The length and direction of that vector would be the result of the subtraction.

3 - -4 = 7



ABSTRACT REPRESENTATIONS p – q = p + (-q) Applying Properties of Numbers p - -q = p + q

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6.NS.5 Lesson 1 - “Warmer in Miami” Introductory Task

Guided Practice

Collaborative Work

Homework

Assessment

Introductory Task 6.NS.5 Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation. Warmer in Miami

On the same winter morning, the temperature is −28° F in Anchorage, Alaska and 65° F in Miami, Florida. How many degrees warmer was it in Miami than in Anchorage on that morning?

Focus Question(s) How can the number line be used to help students develop an understanding of negative and positive numbers?

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6.NS.5 Lesson 1 - “Warmer in Miami” Introductory Task

Guided Practice

Collaborative Work

Homework

Assessment

6.NS.5 Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation.

1. Arrange the follow rational numbers on the number line such as: -3, 5, ¾, 1.2, -.8, -6.1, .5, -2/5, -7/4 .

2. Write a number between -2 and 2. Create equal partitions between these two numbers and then name the numbers in between.

3. If a = -6 explain the meaning of each value below then place the values on the number line using precise spacing. a. a

b. -a

c. a+1

d. a-1

e. 2a

f. a + -a

Focus Question(s) How does a number’s proximity to zero help to explain its size in comparison to other numbers? How does this understanding apply when two numbers are equally as close to zero? Page 5 of 19

6.NS.5 Lesson 1 - “Warmer in Miami” Introductory Task

Guided Practice

Collaborative Work

Homework

0

b

Assessment

a

1. On the number line above, the numbers a and b are the same distance from 0. What is a + b? Explain how you know.

0

b

1

a

2. A number line is shown above. The numbers 0 and 1 are marked on the line, as are two other numbers a and b. Which of the following numbers is negative? Choose all that apply. Explain your reasoning. 1) a−1 2) a−2 3) −b 4) a+b 5) a−b 6) ab+1 Journal Question The temperature last night was -18o . On which number line does

best represent -18? Explain your choice.

A. -20

-10

0

10

20

-20

-10

0

10

20

-20

-10

0

10

20

-20

-10

0

10

20

B.

C.

D.

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6.NS.5 Lesson 1 - “Warmer in Miami” Introductory Task

Guided Practice

Collaborative Work

Homework

Assessment

1. Determine the number indicated on the number line below.

10

26

2. Determine the number indicated on the number line below.

-20

12

3. Determine the number indicated on the number line below.

-1

3

4. Determine the number indicated on the number line below.

-2

-1

5. Determine the number indicated on the number line below.

-2.6

-2.5 Page 7 of 19

6.NS.5 (EXTENDED) Lesson 2 - Golf Scores Introductory Task

Guided Practice

Collaborative Work

Homework

Assessment

6.NS.5 Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation.

Introductory Task Marilynn’s class played a game that is scored a little like golf. In golf, the score is based on a number called par. Each player gets a score that is either at, above, or below par. The lower the score the better, so it’s best to get a score below par. For Marilynn’s game, par was 54. The players on Marilynn’s team had the following scores in terms of par: -3, +9, -10, -1, +15, -7, +4, and -6. What was the winning score? What was the worst score? What is the team’s overall score?

Focus Question(s) How can the number line be used to add and compare negative and positive numbers?

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6.N5.5 (EXTENDED) Lesson 2 - Golf Scores Introductory Task

Guided Practice

Collaborative Work

Homework

Assessment

1. Write the integer that best represents the following scenarios: a. A deposit of $35 into a bank account b. 11 more than -12 c. A car goes 15 mph faster each second for 5 seconds d. A gain of 8 pounds e. A $0.25 rise in the price of gasoline f. A temperature of 5o below 0 g. A bucket leaks 3 ml of water per second for 12 seconds h. A loss of 8 pounds i. 25 feet below sea level j. 12 fewer than -11 2. In Buffalo, NY, the temperature was -14oF in the morning. If the temperature dropped 7oF, what is the temperature now? 3. A submarine was situated 800 feet below sea level. If it ascends 250 feet, what is its new position? 4. A submarine was situated 450 feet below sea level. If it descends 300 feet, what is its new position? 5. During the football game, Justin caught three passes. One was for a touchdown and went 52 yards. The second was for a first down and was for 17 yards. The third was a screen pass that did not work so well and ended up with a gain of -10 yards. What was the total yardage gained by Justin on the pass plays? 6. Louis was doing a science experiment about temperature. He first measured the temperature of some water and found it was 17°C. Then he put the water in the freezer and recorded the temperature two hours later. It had fallen to -11°C. What was the change in temperature in two hours? What was the average rate of temperature change per hour?

Focus Question(s) How is the subtraction of integers different from the addition of integers? Are there times when it is similar?

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6.N5.5 (EXTENDED) Lesson 2 - Golf Scores Introductory Task

Guided Practice

Collaborative Work

Homework

Assessment

1. Today was a strange weather day. The temperature started out at 9 oC in the morning and went to -13oC at noon. It stayed at that temperature for six hours and then rose 7oC. How far above the freezing point (0oC) was the temperature at 6pm? 2. A submarine dove 836 ft. It rose at a rate of 22 feet per minute. What was the depth of the submarine after 12 minutes? 3. Mary has $267 in her checking account. She writes checks for $33, $65, and $112. What is the balance in her account now? 4. Denver, Colorado is called, “The Mile High City” because its elevation is 5280 feet above sea level. Someone tells you that the elevation of Death Valley, California is -282 feet. 1. Is Death Valley located above or below sea level? Explain. 2. How many feet higher is Denver than Death Valley? 3. What would your elevation be if you were standing near the ocean? 5. A stunt plane ended a maneuver 60 meters below the top of a skyscraper, after starting 70 meters below the top of that same skyscraper. Which integer represents the stunt plane's change in altitude? 6. The temperature of a beef patty was -4 degrees when taken out of a freezer. The patty was immediately warmed in a frying pan and after 4 minutes, its temperature was 86 oC. Find the increase in the temperature of the patty during the 4-minute interval. Given that the temperature of the beef patty increased at a constant rate, calculate the temperature of the beef patty after 1 minute in the frying pan. 7. In the Sahara Desert one day it was 136°F. In the Gobi Desert a temperature of -50°F was recorded. What is the difference between these two temperatures? 8. Metal mercury at room temperature is a liquid. Its melting point is -39°C. The freezing point of alcohol is -114°C. How much warmer is the melting point of mercury than the freezing point of alcohol? 9. Katherine is very interested in cryogenics (the science of very low temperatures). With the help of her science teacher she is doing an experiment on the effect of low temperatures on bacteria. She cools one sample of bacteria to a temperature of -51°C and another to -76°C. What was the temperature difference in the two experiments?

Journal Question A football team gained 6 yards on a first down, lost 15 yards on the second down, and gained 12 yards on the third down. How many yards do they need to gain on the fourth down to have a 10 yard gain from their starting position? Explain how you arrived at your answer. Page 10 of 19

6.N5.5 (EXTENDED) Lesson 2 - Golf Scores Introductory Task

Guided Practice

Collaborative Work

Homework

Assessment

PROBLEM SOLVING 1

In Detroit, Michigan, the temperature was -15° F in the morning. If the temperature dropped another 10° F, what is the temperature now?

2

On the number line, order these values: .55, 1/3, 1.1, -2, -1/2, 0, .05.

3

What happens to the number values as you move down (left) on the number line? What does that tell you about moving up (right) on the number line?

4

Where can you place parentheses ( ) in the expression to give the value of 7? 5+ 4 - 3 x 2

5

Five 10th grade students were in a 5 mile cross country race. Sandy came in third with a time of 33 minutes. Three of the runners had times of -5, -3, and +4 minutes relative to Sandy’s time. The fifth place racer’s time was 12 minutes slower that the winner’s. What was each runner’s time?

6

A fly that started 20 centimeters above the windowsill is now 50 centimeters below the windowsill. What integer represents the fly's change in altitude?

7

One day in July, the temperature at ground level at the airport was 90o. A pilot reported the temperature at 10,000 feet was 50o. How much did the temperature drop per 1000 feet?

8

The temperature at noon on a winter day was 8o C. At midnight, the temperature had dropped 15o. What was the temperature at midnight?

9

In 2001, Standard Register reported a net loss of $43.9 million. In 2002, the company had a net income of $33 million. How much more money did the company make in 2002 than in 2001?

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Lesson 3 – Golden Problem Introductory Task

Guided Practice

Collaborative Work

Homework

Assessment

Use the transaction register to answer the questions below. A transaction register is used to record money deposits and withdrawals from a checking account. It shows how much money Mandy, a college student, had in her account as well as the 4 checks she has written so far. Check No. 1 2 3 4

Date 9/04 9/07 9/13 9/16 9/24

Description of Transaction allowance from parents college bookstore — textbooks graphing calculator bus pass Charlie’s Pizza

Payment

Deposit $500

Balance $500

$291 $99 $150 $12

1. Subtract each withdrawal to find the balance after each check was written. If Mandy spends more than $500, record that amount as a negative number.

2. Which check did Mandy write that made her account overdrawn?

3. Mandy called home and asked for a loan. On 9/26, her parents loaned her $500. Mandy didn’t make any other deposits or withdrawals between 9/24 and 9/26. After depositing the check from her parents, what was the balance in her register?

4. After her parents let her borrow the $500 from Exercise 3, Mandy wants to spend $300 on clothes and $150 on decorations for her dorm room. Does she have enough money in the bank? Express her balance with an integer if she buys these items.

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Golden Problem Rubric: 3-Point Response 

The student correctly indicates the following:

Check No. 1 2 3 4 o o o

Date 9/04 9/07 9/13 9/16 9/24

Description of Transaction allowance from parents college bookstore — textbooks graphing calculator bus pass Charlie’s Pizza

Payment $291 $99 $150 $12

Deposit $500

Balance $500 209 110 -40 -52

Check #3 caused the account to be overdrawn After depositing the $500, Mandy now has $448.00 Mandy does not have enough money in her account for $450 worth of additional purchases. She needs an additional $2.

2-Point Response  



The student shows correct work but does not provide the correct answer. OR The student commits a significant error but provides a correct response based on their incorrect work with clear explanations. OR The student provides the correct response and shows correct work but fails to provide a clear explanation for each part.

1-Point Response 

The student only begins to provide a solution

0-Point Response 



The response demonstrates insufficient understanding of the problem’s essential mathematical concepts. The procedures, if any, contain major errors. There may be no explanation of the required solutions, or the explanation may not be understandable. How decisions were made may not be readily understandable. OR The student shows no work or justification.

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New Vocabulary

Rational Number:

Any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero.

Integer:

Positive and negative whole numbers including zero.

Page 14 of 19

Lesson 3 - Review The items developed for this section can be used within the 5th instructional week to provide additional practice and review prior to the post assessment and during as when deemed appropriate by the teacher. (Items 1 – 8: 6.NS.1) 1. 3 people share

pound of chocolate. How much of a pound of chocolate does each person get?

2. Manny has yard of fabric to make book covers. Each book is made from yard of fabric. How many book covers can Manny make? 3. Michelle has ¾ cup of pancake mix. Each pancake uses 1/8 cup of pancake mix. How many pancakes can Manny make? 4.

A rectangular strip of land has an area of 5/6 square yards. The strip is 2/3 yards long. How wide, in yards, is the strip of land?

5.

Look at the number sentence below:

7  6

6.

=7

5 1   6 12

7. How many ¾ cup servings are in 4 ½ cups of chocolate ice cream? Show or explain how you found your answer.

8. Look at the number sentence below:

2 1   3 6 a) Write a story problem that can be solved using this number sentence. b) Solve your story problem. c) Use the model below to show or explain your answer.

Page 15 of 19

9. Which picture models

1 1  ? (6.NS.1) 2 8

A.

C.

B.

D.

Items 10 – 12: 6.NS.5 10. During the 2011 Travelers’ Golf Championship, Fredrik Jacobson won with a score of -20. This means his score was 20 strokes below par. John Daly had a score of 1 which means he was 1 stroke above par. 

What does a score of 0 mean?

11. The elevation of Mount Washington in New Hampshire is 6,288 feet. This means that Mount Washington is 6,288 feet above sea level. The elevation of Death Valley is -282 feet. What does an elevation of -282 feet mean?

12. Mark an X on the number line that shows the location of the opposite of Point A.

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13. Shade the thermometer to show -8°F. (6.NS.5)

14. Look at the coordinate plane below. (5.G.1)



What are the coordinates of Point A?



Graph the point (3, -5) and label it Point B.

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15. Which number on the number line is the same distance from both Point A and Point B? (6.NS.5) Show or explain how you found your answer.

16. If 6 is 30% of a value, what is that value? (6.RP.3c)

17. On a coordinate plane, a line segment is drawn between Point A (-3, 4) and Point B (3, 4). What are the coordinates of the point on the line segment that is half way between Points A and B? (5.G.1)

Page 18 of 19

18. If the point (3,-5) is reflected across the x-axis, what are the coordinates of the new point? Show or explain how you found your answer. You may use the coordinate plane below to help you find the answer. (5.G.1)

19. Use the information below to answer the questions. (6.NS.5) State

Record Low temperature (°F)

Connecticut -32 Maine -48 Massachusetts -35 New Hampshire -47 Rhode Island -25 Vermont -50 Which state had the coldest temperature? Show or explain how you found your answer.

20. A whale is swimming at a depth of -25 feet. A submarine is located at a depth of -40 feet. Sea level is 0 feet. Which is deeper, the whale or the submarine? Show or explain how you know your answer is correct. (6.NS.5)

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