1

UNIT 1 NUMBER SYSTEMS

2

Comparing and Ordering Integers

What are different types of numbers?

Natural Numbers (Counting)- ________________________________________________________________

Whole Numbers- __________________________________________________________________________

Integers- _________________________________________________________________________________

Inverse- __________________________________________________________________________________

1. Place ONLY the integers on the number line.

[a] 3 [b] 5 [c] 2

1 [d] -1 [e] 0 [f] -4 [g]

4

35

Determine the integer value of each of the following.

2. $5.00 off the original price

3. 2 degrees above zero

4. an 8 yard gain

5. a 4.5 yard loss

6. a $25 deposit

7. a $15.00 withdrawal

Comparing Numbers

To show a number is greater than another number we use _________________.

To show a number is less than another number we use ___________________.

To show a number is equal to another number we use ____________________.

Example: Compare 12 ____ 4

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

3

Comparing and Ordering Integers

Compare each of the following to make the inequality true.

8.

[a] 5 ____ 6 [b] 10 ____ 8 [c] −6 ____ 6

[d] 7 ____ 6 [e] 4 _____ 5 [f] -10 ____ 0

9. Determine the additive inverse for each of the following.

[a] 3 [b] 5 [c] 899 [d] 900 [e] 9

10. Order the following integers from least to greatest:

{-3, -500, 43, 1, 0, -73, 300}

11. The temperature in Alaska at noon is -12°. Use the number line to answer the following questions

[a] How many degrees warmer is 9°?

[b] How many degrees colder is 15°?

[c] At midnight, the temperature had dropped 5°, what is the

temperature now?

[d] How many degrees would the temperature at noon have to

increase to get to 0°?

12. Which of the following is the largest integer?

a) 300 b) 1 c) 250 d) 0

13. Find the sum of each.

[a] 10 + 10 [b] -11 + 11 [c] 5 + 5

-7°

-8°

-9°

-10°

-11°

-12°

-13°

-14°

-15°

-16°

-17°

4

Comparing and Ordering Integers

Write an integer for each situation.

1. an 8-yard gain

2. 2° above zero

3. a loss of 15 pounds

4. 20 ft. below sea level

5. Compare each of the following to make the inequalities true.

a. 3 ____ 4

b. 7 ____ 10

c. 1 ____ 15

d. 9 ____ 10

6. Order the integers in this set from least to greatest:

{-3, 5, -7, -2, 0}

7. Name the additive inverse of each integer.

a. -7 b. 23 c. -400

8. Find the sum of each.

a. 1 + 1 b. -7 + 7 c. 100 + -100

9. A stock opened at $7 per share on Monday.

a. The stock’s value increased $3 on Monday, what is the value now?

b. On Tuesday, the value of the stock decreased by $5. What is the stock’s value now?

c. By the end of the week, the value of the stock decreased by $9 from its original value. What is

the value at closing on Friday? Use a number line to justify your answer.

5

Order of Operations

Order of Operations- _______________________________________________________________________

How can I remember this?

Examples:

1. 3432

Practice:

2. 32615

3. 611

362

Simplify the following:

1. 5 – 2 + 7

2. 2 + (3 − 2)

3. 12 + 3 ∙ 2

4. 261222

5. 24 ÷ 2 ∙ 6

6. 10 + 8 ÷ 2

6

Order of Operations

Simplify the following:

1. 412

265

2. 2812

3. 735)43(

4. (8 − 4) ÷ 2

5. 42 + (2 − 3)2

6. 32 − 52

7. )24(1242

8. 23)22( 3

9. 3 ∙ (5 − 2)

10. 56 (7 ∙ 2) + 1

7

Absolute Value

What is an absolute value?

- Absolute Value is the distance from zero on a number line

- Distance is ALWAYS positive

- The symbol for absolute value is _______________.

Example:

1. What is the distance from -3 to 0?

2. What is the absolute value of -3?

3. What is the distance from 3 to 0?

4. What is the absolute value of 3?

Try the following on your own:

5. |9|

6. |−1|

7. |−3| + |−1|

8. |−1| + |−3|

9. |3 − 1|

10. |3| + |−1|

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

8

Absolute Value

Practice:

Compare the following to complete the inequalities.

1. |−5| _____ |−6|

2. |−6| ______ |6|

3. 7 _____ -6

4. 9 ______ 110

Order the integers in each set from least to greatest.

5. {−3, 4, |−2|, |5|, 0}

6. {−1, −4, |−4|, |0|, 5}

7. Decide if each of the following is always true, sometimes true, or never true for all integer values of x.

[a] |𝑥| = 𝑥 [b] |−𝑥| = 𝑥 [c] −|𝑥| = 𝑥 [d] |𝑥| = |−𝑥|

8. Jack rides his bike north 15 miles from his home. Alex rides his bike 10 miles south from the same

house.

a. How far is Jack from his home?

b. How far is Alex from the house?

c. How far away is Jack from Alex?

9. A friend does not know how to order integers. Explain on the lines below how you would order the

following set of integers: {-3, 5, 1, |−4|, -10}

__________________________________________________________________________________________

__________________________________________________________________________________________

__________________________________________________________________________________________

9

Absolute Value

1. The absolute value of two numbers that are additive inverses will __________ be the same.

a) always b) sometimes c) never

2. Which number added to negative three gives the sum of zero?

a) 0 b) 3

1 c) 3 d) 30

3. Which two numbers have an absolute value of 1?

Compare the following inequalities.

4. 415 _____ -13 5. 4 _____ 5 6. |−12|_____ 4

7. In a golf tournament, the lowest score is the leader. The chart to the right

shows the scores of five different golfers after a tournament.

a. How many strokes is Woods ahead of Mickelson?

b. Which two golfers have scores that are additive inverses?

c. How many strokes is Goosen ahead of Singh?

d. Put the golfers in order from best score to worst score (least to greatest):

Determine if the equation is True or False. If false, explain why.

8. 35 + |−35| = 35 − 35

9. 400 − |−25| = 400 − 25

10. 100 + |−1| = 100 + 1

Player Score

T. Woods -10

V. Singh +3

E. Els +10

P. Mickelson -4

R. Goosen -1

10

Adding Integers

What is the result of an addition problem? ____________________

Addition with a number line:

Examples:

1) −1 + 3 2) −1 + (−3)

Steps: 1. Start at -1 on the number line Steps: 1. Start at -1 on the number line

2. Move 3 spaces in the positive direction 2. Move 3 spaces in the negative direction

Alternate Method (song):

Same Signs Different Signs

Add and Keep Subtract- Keep the sign of the larger number

then you’ll be exact!!!

Let’s try these together!

1. −2 + 4

2. (- 5) + (3)

3. −2 + (−5)

4. −3 + (7)

5. −6 + 7

6. −3 + −4 + 4

7. −5 + (−1) + 6

8. 2 + (−5) + 5

9. The temperature in Vermont is recorded at -12°. At the same time, the temperature in New York is 15°

warmer. What is the temperature in New York?

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

11

Adding Integers

Adding Larger Integers:

Keep in mind: If you are adding a positive, the number should get bigger (move in positive direction)

If you are adding a negative, the number should get smaller (move in negative direction)

Practice:

1. -3 + -4

2. -10 + 80

3. 5 + -9

4. -2 + 7

5. 10 + -1

6. -8 + 10

7. -4 + (-4) +20

8. -5 + -5 + -5

9. 10 + -4 + 5

10. 73 + (−13)

11. −120 + 20

12. −120 + (−20)

13. −47 + (−3)

14. −78 + 80

15. 38 + (−24) + 14

16. A submarine is 350 feet below sea level, over the course of the next three hours, the submarine rose 120

feet. What is the submarine’s distance below sea level?

17. An elevator starts on the ground floor. If it goes up 3 floors, then down 2 floors, and finally up 6 floors,

what floor is it on?

18. The temperature in city A is -35°. If the temperature in city B is the additive inverse of -35°, how much

warmer is city B?

19. The sum of -7 and what number is 2?

12

Adding Integers

Find the sum of the following expressions.

1. -15 + 10

2. 20 + (-8)

3. -5 + (-5)

4. 8 + (-4) + 6

5. -3 + -6 + 4

6. -2 + -1 + -9

REVIEW (MULTIPLE CHOICE):

7. Which of the following integers represents

the greatest negative integer?

a) -4 b) -1 c) 400 d) -400

8. Which of the following integers is the

smallest?

a) -7 b) -2 c) 800 d) -100

9. Which of the following integers represents

the distance from -3 to 5?

a) 5 b) 2 c) 8 d) -8

10. The absolute value of a number is:

a) always positive b) always negative

c) sometimes negative d) never positive

11. The absolute value of a number is:

a) always positive b) always negative

c) sometimes negative d) never positive

12. Which of the following must be equal to

|−𝑥|?

a) −|𝑥| b) |𝑥|

c) −|−𝑥| d) 𝑥

13

Subtracting Integers

The result of a subtraction problem is called: _______

Subtraction with a number line:

To subtract an integer, add its opposite (inverse)

𝑎 − 𝑏 = 𝑎 + (−𝑏)

or

𝑎 − (−𝑏) = 𝑎 + (𝑏)

Examples:

1) −𝟏 − 𝟑 2) −𝟏 − (−𝟑)

Steps: Steps: 1. Re-write the problem using addition

1. Start at -1 on the number line 2. Start at -1 on the number line

2. Move 3 spaces to the ______ 3. Move 3 spaces to the _______

Try these!

1. −2 − 4

2. 5 − (−3)

3. −2 − (−5)

4. −3 − (7)

5. −6 − 7

6. −3 − (−4) − 4

7. −5 − (−1) + 6

8. 2 − (−5) − 5

9. The temperature in Chicago is 38°. It is 40° colder in North Dakota. What is the temperature in North

Dakota?

10. The temperature in Maine is -21°. At the same time, the temperature in Texas is 79°. What is the

difference in the two temperatures?

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

14

Subtracting Integers

Keep in mind: When you subtract a negative you are really adding. Ex. – (–10) = +10

Practice:

1. −30 − 20

2. 40 − −10

3. 73 − (−13)

4. −120 − 20

5. −120 − (−20)

6. −47 − (−3)

7. −78 − 80

8. 38 − (−24) + 14

9. The temperature in San Jose is recorded at 82°. At the same time, the temperature in Seattle is 95°

colder. What is the temperature in Seattle? (Draw a picture of a thermometer to help.)

10. The temperature in Michigan is -6°. At the same time, the temperature in New Mexico is 94°. What is

the difference in the two temperatures? (Draw a picture of a thermometer to help.)

11. Michael is 8 years old. His sister Anna is 7 years older than him, and his brother Rocco is 11 years

younger than his sister. How old is his brother?

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Subtracting Integers

1. 7 – 10

2. -15 – (-1)

3. 20 – 32

4. -9 – (-6)

5. -3 – 5 + 8

6. -3 – 3 – 3

7. Write 8 – 2 as many ways as you can. 1)_______________ 2) _____________ 3)______________

Think of anymore?

8. Ryan has $75 in his bank account. He withdraws $48, and then deposits $12. What is Ryan’s new

balance?

9. Your friend is having trouble simplifying 20 − (−38). Write an explanation to help your friend solve

the problem.

____________________________________________________________________________________

____________________________________________________________________________________

____________________________________________________________________________________

____________________________________________________________________________________

10. Revie! Complete the following.

a. −|−3| + |4|

b. |−4| + |3|

c. |−4 + 3|

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Commutative/Word Problems

Commutative Property- _____________________________________________________________________

Example: Use the commutative property to simplify:

1. 30 + 45 + −30

2. −53 + 7 + 53

Practice

1. 21 + 47 + (−47) + −4

2. 34 + 21 + −34

3. (−8

5) + (−72) + (

8

5)

4. 83 + −83 + 27 + −27

5. −20 + 90 + −30 + (−90)

6. 432 + 68 + 11 + −500

Word Problems

Positive Number Negative Number

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Commutative/Word Problems

Practice:

1. Joey owes his friend $10. He pays back $4, and then borrows another $17. How much money does

Joey owe his friend?

2. A submarine is 800 feet below sea level. Over the course of the next few hours, the submarine ascends

200 feet, descends 400 feet, ascends 200 feet and descends 900 feet. How far below sea level is the

submarine?

3. The temperature at midnight is recorded at -11°. Over the next ten hours, the temperature increased

13°. What is the temperature after this ten hour period?

4. An elevator started on the 9th floor goes up 2 floors, then down 5 floors, then up 3 floors, then down 6

floors. On what floor is the elevator now?

5. The temperature in Anchorage, Alaska is recorded at -17°. At the same time, the temperature in Los

Angeles, California is 97° warmer than in Anchorage. What is the temperature in Los Angeles?

6. In Buffalo, New York, the temperature was -14oF in the morning. If the temperature dropped 7oF at

12:00pm, what is the temperature now?

7. A submarine was situated 750 feet below sea level. If it descends (goes down) 200 feet, what is its new

position?

18

Commutative/Word Problems

Review!

Write an integer to represent each situation:

1. Five dollars off the original price

2. two degrees above zero

Compare the following Inequalities.

3. 9 _____ 9 4. )9(8 _____ 17

Evaluate each of the following.

5. 5 – 2

6. 7 – (-9)

7. 82315

Order these integers from least to greatest.

8. {−1, −4, |−4|, |0|, 5}

9. Maggie deposits $35 in the bank. She then withdraws $10 on Monday, deposits $15 on Tuesday, and

then withdraws $14 on Wednesday. How much does Maggie have left in the bank?

10. A stock opens at $450 per share on Monday. The chart displays the change over the course of the next

few days. What is the value of the stock per share at closing on Friday?

Day Change

Monday +$21

Tuesday -$13

Wednesday -$8

Thursday +$15

Friday -$6

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Word Problems

Complete the following word problems!

1. In the Sahara Desert one day it was 136oF. In the Gobi Desert a temperature of -50oF was recorded.

What is the difference between these two temperatures?

2. Mt. Everest, the highest elevation in Asia, is 20,320 feet above sea level. The Dead Sea, the lowest

elevation, is 282 below sea level. What is the difference between these two elevations?

3. A runner jogs 14 miles in one direction. He then turns around and jogs 18 miles in the opposite

direction.

a. How far is the runner from his starting position?

b. How far did the runner jog in total?

4. A scuba diver is 180 feet below sea level. She ascends 32 feet, and then descends 48 feet. What is her

current depth?

5. An explorer jumps out of a plane and parachutes into a cave. He jumped out of the plane at 300 feet

above sea level, and lands at the bottom of the cave, which is 900 feet below sea level.

a. How far was the explorer’s jump?

b. Once in the cave, the explorer continues deeper into the cave. If he climbs to the lowest point in the

cave, and records the depth at 1524 feet below sea level, how far down did he climb from where he

landed?

6. A roller coaster at Six Flags has a largest drop of -276 feet. A roller coaster at Dorney Park has a

largest drop of -239 feet. How much bigger is the drop at the roller coaster at Six Flags?

20

Multiplying and Dividing Integers

What is the result for a multiplication problem called? ______________________

What is the result for a division problem called? ______________________

Rules:

1) Count the negative signs

Odd number of negative signs - Answer Negative

Even number of negative signs - Answer Positive

2) Multiply or Divide

OR use this chart!

Examples:

1. −5 ∙ −2

2. −8 ∙ 4

3. −25 ÷ 5

4. 25 ÷ −5

Practice:

1. (7)(0)

2. 8

0

3. −24

8

4. (−1)4

5. −1 ∙ −3 ∙ −4 ∙ 2

6. −15 ∙ −2

Any number multiplied by zero is _______

Any number divided by zero is ___________

21

Multiplying and Dividing Integers

7. (−3)(10)

8. 56 ÷ −7

9. −2 ∙ 1 ∙ −3

12. −2 ∙ −6 ∙ 2 ∙ −1

13. (−1)3

14. (−1)246

10. −81

−9

11. −12 ÷ −4

15. 28

−7

16. −2 (−3)

Extended Response:

17. One night in January, the temperature in Alaska is −16℉. The next day, the temperature is half of

what it was the night before. What is the temperature?

18. The volunteer club raked leaves at several senior citizens’ homes in the neighborhood. If each group

of three students could remove 8 cubic meters of leaves in one hour, find an integer to represent the

number of cubic meters of leaves 12 students could remove in 3 hours?

19. During the fourth quarter, the Patriots were penalized 3 times for the same amount for a total of 45

yards. Write a division sentence to represent this equation. Then find the number of yards for each

penalty.

22

Multiplying and Dividing Integers

Compute the following expressions.

1. (−9)(−8)

2. (−15)(−3)

3. 535

4. (16)(−4)

5. 4

16

6. (−3)(−1)

7. 5

20

8. (−1)(−1)(−1)

9. −8 ∙ 0 ∙ 2

10. −21 ÷ 7

11. 0

)8(

12. 8

)32(

23

Evaluating Expressions

Algebraic expression – _____________________________________________________________________

Evaluate – _______________________________________________________________________________

Variable - ________________________________________________________________________________

Substitution Property – ______________________________________________________________________

STEPS:

1) Write the original problem.

2) Rewrite the expression with the values of each variable substituted in (use parentheses to substitute).

3) Simplify by using order of operations. SHOW ALL WORK!

Examples: Evaluate each expression if n = 4, p = 3, and t = 6

1) 3𝑛 + 𝑝 2) pt 22 3) 3𝑝 – 𝑛 + 4

Practice: Evaluate each expression if n = 2, p = 4 and t = 3

1) 5𝑛 + 𝑝 2) -2.4t 3) 3(𝑝 – 𝑛) + 4 4) 𝑝 ÷ (𝑡 − 1)

5)

t

np 6) tpn 7)

)13(

)4( 2

t

p 8) 𝑝 − 𝑛𝑡

USE THE GIVEN FORMULA TO EVALUATE:

9) Drew drove to Chicago at an average rate of 50 mph. The trip took him 17 hours. How far did Drew drive?

𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒 = 𝑅𝑎𝑡𝑒 𝑥 𝑇𝑖𝑚𝑒

24

Evaluating Expressions

Evaluate each expression if x = 2, y = 3 and z = 5

1. 2𝑥 + 𝑧

2. 𝑧 – 2𝑥

3. 3𝑥 – 𝑦 + 3

4. z

xy5

5. 2xy

6. 23x

7. 53

42

y

x

8. zx 26

9. John is deciding whether he wants to install a rectangular pool or a cylindrical pool in his back yard. The pool

company shows him two models, one a rectangular prism and one a cylinder, that are the same price. He wants to

determine which pool would hold a larger volume of water.

a. The rectangular prism pool has a width of 10 feet, a length of 20 feet, and a height of 5 feet. Using

the formula for volume of a rectangular prism, determine the volume of this model.

𝑉 = 𝑙𝑤ℎ

b. The cylindrical pool has a radius of 10 feet, and a height of 4 feet. Using the formula for volume of

a cylinder, estimate the volume of this model. (use 3 as an estimate for π at the very end of the

problem)

𝑉 = 𝜋𝑟2ℎ

c. Which pool has a greater volume? By how much?

25

Introduction to Fractions and Decimals

Vocabulary:

1) Decimal: _____________________________________________________________________

2) Fraction: _____________________________________________________________________

3) Difference: _____________________________________________________________________

4) Product: _______________________________________________________________________

5) Quotient: ______________________________________________________________________

6) Rounding: Replacing a number with another number that is approximately equal but shorter.

Decimals:

Place Value

Word Bank

8 , 5 0 3 , 7 8 9 . 5 1 4

Rules for rounding:

Step 1- Find the place value you are rounding to and underline it.

Step 2- Look at the number to the right and think “five or more add one more – four or less let it rest”

Step 3- Write the appropriate number and drop all numbers to the right.

Example: Round 87.657689to the nearest hundredth. “Five or more? Yes Add one more.

87.66 is the answer! Guided Practice: Round the following:

1. 87.657 to the nearest tenth _________________ 2. 22.423 to the nearest hundredth ______________

3. 1.359 to the nearest whole number _________ 4. 1.99 to the nearest tenth __________________

Fill in the chart with the

appropriate place value

Ten-thousands Tenths Thousands Tens

Millions And Hundreds Thousandths

Ones Hundredths Hundred-thousands

26

Introduction to Fractions and Decimals

ROUNDING

Round the following decimals to the nearest tenth:

8. 3.19 9. 4.921 10. 5.909 11. 89.985 12. 12.487

13. 5. 479 14. 72.134 15. 41.295 16. 9.987 17. 1.05

Round the following decimals to the nearest hundredth:

18. 3.297 19. 8.9294 20. 75.989 21. 8.495 22. 18.783

Round the following decimals to the nearest thousandth:

23. 3.2978 24. 2.4234 25. 52.0091 26. 18.1236 27. 21.7253

Comparing Decimals Write <, >, or =.

28. 0.15 ____ 0.7 29. 0.38 ____ 0.24

30. 0.04 ____ 0.9 31. 1.258 ____ 1.9 32. 0.8992 ____ 0.9

Fractions:

Converting Decimals to Fractions.

METHOD 1:

METHOD 2:

WHAT ARE EQUIVALENT FRACTIONS? GIVE AN EXAMPLE.

__________________________________________________________________________________________

__________________________________________________________________________________________

________________________________________________________________________

Given the decimal convert to a fraction.

33. 0.25 34. 0.3 35. 5.75 36. 3.0 37. 3.8

27

Introduction to Fractions and Decimals

Round the following:

1. 8.652 to the nearest tenth ___________________________

2. 9.3734 to the nearest hundredth ___________________________

3. 0.854 to the nearest whole number ___________________________

4. Which is greater 0.42 or 0.419998? ______________ Explain your reasoning.

__________________________________________________________________________________________

__________________________________________________________________________________________

__________________________________________________________________________________________

Compare the following decimals. Write <, >, or =.

5. 0.2 ____ 0.44 6. 0.258 ____ 0.25 7). 0.1 ____ 0.101

State and explain if the following fractions are equivalent.

8. 2

5 and

10

25 9.

3

8 and

9

16 10.

2

4 and

2000

40000

28

Converting Rational Numbers to Decimals

Vocabulary:

1) Convert: ________________________________________________________________________________

2) Rational Numbers: ________________________________________________________________________

Previous Knowledge:

Write each as a decimal: 1

4= ____

1

2= _____

3

4 = ______

4

4= ______

TYPES OF FRACTIONS TYPES OF DECIMALS PROPER FRACTIONS

TERMINATING DECIMALS

IMPROPER FRACTIONS

NON-TERMINATING DECIMALS

MIXED NUMBERS

REPEATING DECIMALS

FRACTIONS TO REMEMBER:

1

4 = .25

2

4 =

12 = .5

3

4 = .75

1

5 = .2

2

5 = .4

3

5 = .6

4

5 = .8

1

3 = . 3̅

2

3 = . 6̅

1

8 = .125

3

8 = .375

5

8 = .625

7

8 = .875

1

9 = . 1̅

2

9 = . 2̅

4

9 = . 4̅

5

9 = . 5 ̅

7

9 = . 7̅

8

9 = . 8̅

1

10 = .1

3

10 = .3

7

10 = .7

9

10 = .9

Make connections!

What would be: a) 𝟖1

4 ________ b) 12

2

5 _______ c) 4

1

3 __________

D) 234 _______ E) 7

1

2 _______ *F) 5.33333…. ________ F) 6.8_______

29

Converting Rational Numbers to Decimals

Any rational number can convert to a decimal using long division; know that the decimal form of a rational

number terminates in zeroes or eventually repeats.

Guided Practice:

Convert the fraction into a decimal using long division and determine if the decimal is repeating or terminating.

1) 2

5= ____ 2) −

1

3= _____ 3) 8

5

6 = ______

4) 1

9= ____ 5) −

3

8= _____

Try These: Remember to show all work

Write each fraction as a decimal using long division and determine if the decimal is repeating or terminating.

1) 1

8= ____ 2)

2

3= _____ 3) −

1

4 = ______ 4) 4

5

9 _____

5) −1

11= ____ 6)

3

5= _____ 7) 7

7

10= ______ 8)

11

12= _____

9) Write a fraction that is equivalent to a terminating decimal between 0.5 and 0.75.

Think:

10) Fractions in simplest form that have denominators of 2, 4, 5, 8, and 10 produce terminating decimals.

Fractions with denominators of 3, 6, 9 and 11 produce repeating decimals. What causes the difference? Explain.

__________________ decimal __________________ decimal __________________ decimal

__________________ decimal __________________ decimal

_____________ decimal ____________ decimal _____________ decimal _____________ decimal

_____________ decimal _____________ decimal _____________ decimal _____________ decimal

30

Converting Rational Numbers to Decimals

Convert the following fractions to a decimal using long division. State whether it terminates or repeats.

1) 4

9 = 2) −

1

6 = 3) 4

4

25 =

4) 5

8 5) −

1

2 6) 2

6

11

Word Problems:

7) Use the table that shows decimal and fraction

equivalents. Which fraction represents 8.0 ?

A) 5

4 B)

99

80 C)

6

5 D)

9

8

8) Zoe went to lunch with a friend. After tax, her bill was $12.05. Which mixed number represents this amount

in simplest form?

A) 2

112 B)

20

112 C)

10

512 D)

100

512

Convert the following mentally.

9) 4

32 10)

3

15 11) 5.75 12) 3..2

Decimal Fraction

3.0

9

3

4.0

9

4

5.0

9

5

6.0

9

6

31

Converting Mixed Numbers and Improper Fractions

Vocabulary:

1) Improper Fraction: ____________________________________________________________________

2) Mixed Number: ______________________________________________________________________

Converting Mixed Number to Improper Fractions:

Rules:

1) Multiply the denominator and the integer

2) Add the product to the numerator

3) Take that sum and place it over the ORIGINAL denominator

Model Problem:

1) 24

5 = Step 1) 5 ∙ 2 = 10 Step 2) 10 + 4 = 14 Step 3)

14

5

24

5 2

4

5 2

4

5

14

5

Examples:

1) 42

3 = 2) 2

1

2 = 3) 5

5

6 =

4) 83

4 = 5) 6

3

10 = 6) 7

4

7 =

To convert an improper fraction to a mixed number:

Rules:

1) Divide the numerator by the denominator

2) Write down the integer

3) Write the remainder over the original denominator

Model Problem: 14

5 14 ÷ 5 = 2 remainder 4 2

4

5

+ 10

32

Converting Mixed Numbers and Improper Fractions

Examples:

7) 12

5 8)

68

3 9)

100

6

10) 14

6 11)

66

10 12)

987

2

13) Sally is baking a birthday cake for her mom. The recipe calls for 9

2 cups of flour and

4

3 cups of oil.

However, her measuring cup is only labeled in mixed numbers.

How much flour would 9

2 cups be as a mixed number?

How much oil would 4

3 cups be as a mixed number?

14) Steven is putting away test tubes in science class. He has 50 test tubes and 12 will fit on each rack.

How many rack racks will Steven fill? Write the answer as a mixed number.

15) Chrissy ordered 4 pizzas cut into 8 slices each. 10 slices were eaten. Write the remaining number of pies

as a mixed number.

16) Cole was 18

5 miles away from Betty. How many miles is that as a mixed number?

33

Converting Mixed Numbers and Improper Fractions

Convert the following into an improper fraction:

1. 35

6 = 2. 7

8

9 = 3. 4

4

7 =

Convert the following into a mixed number:

4. 16

3 5.

24

5 6.

6587

2

Word Problems:

7. Marty drove his Lamborghini DeLorean 53

4 miles to get to the clock tower.

How far is that as an improper fraction?

8. Spot the dog ran away from home. A neighbor called and said that he was found in their backyard 16

5 miles

away. How far is that as a mixed number?

Convert mentally to a decimal.

9. 2

112 10.

10

21

34

Comparing and Ordering Rational Numbers

METHOD 1

1. CONVERT ALL NUMBERS TO DECIMALS (ALL TO

THE SAME PLACE)

2. COMPARE OR ORDER

EX: 3

10 ,

1

4 ,

3

8 , 0.5 , 0.7

METHOD 2

1. CONVERT ALL NUMBERS TO FRACTIONS WITH A

COMMON DENOMINATOR

2. COMPARE OR ORDER

EX: 3

10 ,

1

4 ,

3

8 , 0.5 , 0.7

COMPARE:

1. 0.6 0.525 2. 3

4

3

8 3. 0.8

17

20 4. 3

5

8 3.625

5. 0. 6̅ 0.6 6. −45

− 79 7. −2 5

8 −3.6 8. 4

7

8 3.9

ORDER FROM LEAST TO GREATEST AND PLOT THE GIVEN SET OF NUMBERS ON THE NUMBER LINE:

9. 7

10 ,

3

4 , −

3

8 , 0.25 , 0.9 10. 2

58

, − 14

, − 1 14 , 0.25 , −1.75

11. 2

3 ,

3

2 , −

1

3 , 0. 3̅ , 0.3 12.

6

5 ,

4

5 , 2

1

4 , √9 , 𝜋

FRACTIONS AND DECIMALS-COMPARE AND ORDER DAY 3 HOMEWORK

35

Comparing and Ordering Rational Numbers

COMPARE:

1. 0.7 0.60 2. 3

20

7

40 3. 0.4

9

20 4. 3

1

4 3.3

5. −0.5 − 0.7 6. −49

− 59 7. 0.7

15

20 8. 6

1

8 6.12

9. 0. 4̅ 0.4 10. −3

12 − 1

4 11. −0.75 −0.7 12. −4 1

9 −4.5

ORDER THE GIVEN SET OF NUMBERS FROM LEAST TO GREATEST:

13. 7

9 ,

3

4 , −

1

8 , −0.5 , 0.1 14.

7

8 , −

5

4 , 5

3

8 , √225 , − 11

15. 2

5 ,

4

5 , −

2

5 , √25 16.

7

5 , 1

1

5 , −

5

6 , − √4

36

Comparing and Ordering Rational Numbers

PLOT THE GIVEN SET OF NUMBERS ON THE NUMBER LINE:

1. 13

10 , 1

4 , − 1

2 , 2.25 , 1.9 2. 2

45

, − 34

, 1 14

, −2 , √4

3. ANGELA ATE 3

8 OF A BAG OF CHIPS. ALEX ATE

1

4 OF THE BAG.

[A] WHO ATE MORE?

[B] WHAT FRACTION OF THE BAG REMAINS?

COMPARE THE FOLLOWING:

4. . 0. 5̅ 0.5 5. . 21

4 2.3 6. −

37

− 47

ORDER THE FOLLOWING RATIONAL NUMBERS FROM GREATEST TO LEAST:

7. 𝟐14

, − 710

, 2 18 , − √169 , − 11, .7

37

Adding, Subtracting, Multiplying, Dividing Decimals

Adding/ Subtracting Decimals:

Rules:

1) Line up the decimals

2) Add or subtract

Find the Sum or Difference:

1) 49.2 + (-5.63) 2) 9.4 – 4.08 3) 16.2 + 24.9

4) 195.62 – 35.1 5) 12.6 + 2.7 + 108.67 6) 9.001 – (-2.4)

2.3 0.13 93.95

7) - 0.4 8) + 3.87 9) - 45.2

Multiplying Decimals:

Rules:

1) Ignore the decimals in the numbers

2) Multiple the given numbers as if they were whole numbers

3) Count the places after the decimal in each number

4) Count that number of places from the right side in your answer

Examples:

Find each product:

1) 1.02 2) - 58 3) -4.15

x 3.6 x - 2.1 x 2. 6

4) (8.7)(0.45) 5) (12.15)(3.5) 6) (0.91)(2.7)

7) An apple costs $.60. How much will it cost to purchase a dozen apples?

38

Adding, Subtracting, Multiplying, Dividing Decimals

Dividing Decimals:

Rules:

1) Rewrite each division problem as long division.

2) Do not start dividing until you change the outside number to a whole number

3) Move the inside decimal the same amount of places as you did the outside number

4) Write the decimal up into the answer

5) Divide the two numbers as whole numbers to find the quotient

Examples:

Find each quotient:

1) 7.74 ÷ 1.8 2) 19.2 ÷ 3.2 3) -83.7 ÷ -2.7

4) 300

−75 5)

300

7.5 6)

300

0.75

Look at the denominators of each of the above three problems. Notice that the decimal point moves one place to

the left from one exercise to the next. Describe what happens to the quotients.

__________________________________________________________________________________________

__________________________________________________________________________________________

__________________________________________________________________________________________

Word Problems:

7) Peanuts costs $1.75 per jar. How many jars can you buy with $14?

8) You spend $13.92 for fabric. Each yard costs $4.35. How many yards of fabric do you buy?

9) After digging up lilac bushes in a garden, a landscape architect uses sod to cover the ground. The sod costs

$2.25 per yard. He pays $31.50. How much sod does he buy?

9) A car travels 360.25 miles. It uses 13.1 gallons of gas. How many miles per gallon of gas does the car get?

39

Adding, Subtracting, Multiplying, Dividing Decimals

Evaluate

1) 4.6 + 8.79 2) - 8.7 – 2.03 3) 14.8 + 29.07

4) (4.6)(3.9) 5) (-1.8)(0.7) 6) (2.33)(3.56)

7) 4.85 ÷ 0.1 8) 29.75 ÷ 0.7 9) −4.74

−0.06

Breakfast Menu

11) Alicia paid $1.32 for a bag of potato chips. The chips cost $0.55 per pound. How much does the bag of

potato chips weigh?

12) Nina and three friends ate lunch at a café. They decided to split the bill evenly. The total bill was $17.84.

How much was each person’s share?

1 Egg $1.75

Toast $0.99

Bacon $.50

Milk $2.25

10) a) If Michael orders 1 egg, toast, bacon, and milk, how much

will it cost?

b) Michael gives a $10 bill to the cashier. How much change will

he receive?

40

Adding and Subtracting Factions and Mixed Numbers

ADD/SUBTRACT FRACTIONS

RULES:

1) FIND A COMMON DENOMINATOR

2) ADD OR SUBTRACT NUMERATORS

3) KEEP THE DENOMINATOR THE SAME

4) SIMPLIFY FRACTION INTO LOWEST TERMS

FIND THE SUM OR DIFFERENCE WITHOUT USING A CALCULATOR: (USE A SEPARATE PIECE OF PAPER)

1) 1

8+

5

8 2)

5

6−

1

6 3)

7

10− (−

1

10)

4) - 62

5+ 1

4

5 5) 7

2

3− 1

1

6 6) 6

2

5+ 1

4

10

7) 1

7+

5

9 8)

6

7−

9

49 9)

8

25−

9

10

10) 52

5+ 4

4

9 11) −8

2

3− 9

1

6 12) 16

2

9+ 1

7

10

13) 1

16+

1

32 14)

6

18−

1

3 15)

12

5−

9

15

16) −12

3+ 3

7

9 17) 8

1

3− 3

1

12 18) 11

2

5+ 1

11

10

41

Adding and Subtracting Factions and Mixed Numbers

WORD PROBLEMS:

19) TO MAKE LEMONADE, YOU USE 31

3 CUPS OF LEMON CONCENTRATE AND 1

1

3 CUPS OF WATER. HOW MANY CUPS

OF LEMONADE DO YOU MAKE?

20) YOU HAVE 4 CUPS OF FLOUR AND YOU NEED TO USE 13

4 CUPS OF FLOUR FOR A COOKIE RECIPE. HOW MUCH

FLOUR WILL YOU HAVE LEFT?

21) MARK DRIVES HIS CAR 371

2 MILES WEST ON THE LONG ISLAND EXPRESSWAY, HOWEVER HE DROVE PAST HIS

EXIT. HE TURNS AROUND AND GOES EAST FOR 41

3 MILES TO HIS EXIT. HAD HE NOT MISSED HIS EXIT, HOW FAR

WOULD MARK HAVE HAD TO TRAVEL ON THE HIGHWAY?

22) JESSICA’S CAR HAS A GAS TANK THAT HOLDS 183

5 GALLONS OF GAS. JESSICA KNOWS THAT THE TANK ONLY

HAS 31

3 GALLONS OF GAS LEFT IN IT. HOW MUCH GAS WOULD IT TAKE TO FILL UP THE GAS TANK?

23) MEGAN WENT TO A YANKEE GAME IN MAY. IT TOOK HER 31

2 HOURS TO GET TO THE GAME. THE GAME HAD A

11

3 HOUR RAIN DELAY BEFORE STARTING. AFTER THE DELAY, THE GAME TOOK 2

5

6 HOURS. IT THEN TOOK MEGAN

21

3 HOURS TO GET HOME. HOW LONG WAS IT FROM THE TIME MEGAN LEFT HER HOME UNTIL THE TIME SHE GOT

BACK HOME?

42

Adding and Subtracting Factions and Mixed Numbers

FIND THE SUM OR DIFFERENCE WITHOUT A CALCULATOR:

1) 51

3+ 3

2

3 2) 14

1

2− 7

1

5 3) 9 − (−5

5

6) 4) 9

1

6+ 6

1

4

5) 7

8+

4

5 6) −

1

12−

5

6 7) 2

4

5+

3

5 8) 6

1

3+ 2

4

7

WORD PROBLEMS:

9) ON SATURDAY, YOU HIKED 43

8 MILES. ON SUNDAY, YOU HIKED 3

1

2 MILES. HOW FAR DID YOU HIKE DURING THE

WEEKEND?

10) THE GAS TANK IN YOUR FAMILY’S CAR WAS 7

8 FULL WHEN YOU LEFT YOUR HOUSE. WHEN YOU ARRIVED AT

YOUR DESTINATION, THE TANK WAS 1

4 FULL. WHAT FRACTION OF A TANK OF GAS DID YOU USE DURING THE TRIP?

11) SAWYER ROWED 2

3 MILE. MAGGIE ROWED

8

10 MILE. WHO ROWED FURTHER? HOW MUCH FURTHER?

43

Multiplying and Dividing Factions and Mixed Numbers

IMPORTANT VOCABULARY:

COMPLEX FRACTIONS: _______________________________________________________________________

RECIPROCAL: ______________________________________________________________________________

MULTIPLYING FRACTIONS BY HAND

RULES:

1. CONVERT MIXED NUMBERS TO IMPROPER FRACTIONS (IF NECESSARY)

2. SIMPLIFY EACH FRACTION

3. MULTIPLY STRAIGHT ACROSS

FIND EACH PRODUCT:

1) 1

2∙

2

3 2) -

1

4∙

2

5 3) −2

1

2∙ −1

3

5

DIVIDING FRACTIONS BY HAND (WITHOUT A CALCULATOR)

RULES:

1. CONVERT MIXED NUMBERS TO IMPROPER FRACTIONS (IF NECESSARY)

2. CHANGE TO MULTIPLICATION OF THE RECIPROCAL

3. FOLLOW MULTIPLICATION STEPS

FIND EACH QUOTIENT:

4) 1

4 ÷ (−

3

8) 5) −

3

5 ÷ −

2

3 6)

5

16 ÷ 2

1

2

FIND THE PRODUCT OR QUOTIENT:

7) 1

7×

5

9 8)

6

7÷

9

50 9)

8

11∙

9

10

44

Multiplying and Dividing Factions and Mixed Numbers

WORD PROBLEMS

**IN WORD PROBLEMS, WHEN DEALING WITH FRACTIONS THE WORD ‘OF’ MEANS TO MULTIPLY**

EXAMPLE 1: 3

5 OF 8 EXAMPLE 2:

2

3 OF

9

10

PROBLEM SOLVING

10) IF 4

5 OF THE 12,000 PEOPLE AT THE MET GAME ARE WEARING METS HATS. HOW MANY PEOPLE ARE WEARING

MET HATS?

11) ANTHONY 4

5 OF AN APPLE PIE. THE NEXT DAY, KYLE ATE

1

2 OF THE REMAINING PIE. HOW MUCH DID KYLE

EAT?

12) ERIN MAKES 21

2 CUPS OF PUDDING. HOW MANY

1

3 CUP SERVINGS CAN SHE GET FROM THE PUDDING?

13) DEVEN CAN RUN 1

6 MILE IN 2 MINUTES. HOW LONG SHOULD IT TAKE TO RUN 2 MILES?

45

Multiplying and Dividing Factions and Mixed Numbers

FIND THE PRODUCT OR QUOTIENT WITHOUT A CALCULATOR – SHOW ALL WORK

1) 4

5 ÷

1

10 2) −

4

5 ÷

1

5 3)

1

9 ÷

5

6

4) 31

3 ∙

1

3 5) 4

1

6 ∙ −2

2

5 6) 6

1

2 ÷ 1

1

2

7) 33

8 ÷ 1

1

4 8) −

3

5 ÷ −

5

3 **9)

1

2

3

4

WORD PROBLEMS

10) JOANNE HAS 131

2 YARDS OF MATERIAL TO MAKE COSTUMES. EACH COMPLETE COSTUME REQUIRES 1

1

2 YARDS

FOR THE TOP AND 3

4 YARD FOR THE BOTTOM. HOW MANY COMPLETE COSTUMES CAN SHE MAKE?

11) ON THE FIRST DAY THE APPLE iPHONE 5S WAS RELEASED, THE LOCAL STORE HAD 200 IN STOCK. BY 8:00 PM,

THE STORE HAD SOLD 3

5 OF THEIR STOCK.

[A] HOW MANY iPHONES WERE LEFT IN THE STORE’S STOCK?

[B] IF EACH iPHONE SOLD FOR $500, HOW MUCH MONEY DID THE APPLE STORE MAKE IN iPHONE SALES?

46

Complex Fractions

Vocabulary

Proper Fraction _______________________________________________________________

Improper Fraction_____________________________________________________________

Mixed Number________________________________________________________________

Complex Fraction - a fraction where the numerator, denominator, or both contain a fraction.

Change the following mixed numbers to an improper fraction.

a) 4

12 b)

5

45 c)

10

38

Review

Solve each of the following and reduce your answers to lowest terms.

1. 5

2

3

1 2. )

2

13(

5

26 3. )

4

3()

3

1(

Guided Practice 4.

4

310

9

5. 3

5

2

2

1

47

Complex Fractions

Try These

6.

5

23

2

7.

6

18

1

8.

5

23

1

9.

15

62

4

11

10. 5

2210 11.

9

41

6

12

Solve each of the following questions and be sure to show all work.

12. Luke walked his dog 4

3mile every day. It takes him

3

1 hour to walk that distance. How fast does he walk

in miles per hour?

13. The length of a kangaroo’s leap can be up to 2

16 times its height. If a kangaroo is

2

17 feet tall, how

far can it jump?

14. Susan threw the javelin 3

276 meters for her first throw and

4

372 meters for her second throw. How

much longer was her first throw than her second throw?

*15. Mr. McCahill wants to make a shelf with boards that are 3

11 feet long. If he has an 18 foot board, how

many pieces can he cut from the big board?

48

Complex Fractions & Review

Evaluate. Use separate paper if you need more room.

1) )4

31(

6

57 2)

2

1

8

1 3)

7

6

3

12

4)

10

82

3

5) 3

5

2

6)

6

54

1

Thinking Question – Take one step at a time!

7)

3

14

1

3

2

8)

2

11

3

15

9) Sandy is having trouble with his assignment. His shown work is as follows:

3

4

4

3

However, her answer did not match the answer that his teacher gives him. What is Sandy’s mistake? Find the

correct answer.

____________________________________________________________________________________

____________________________________________________________________________________

____________________________________________________________________________________

112

12

3

4

4

3 x