2.1A Homework: Additive Inverse in Context and Chip/Tile Model*
Name:
Period:
State the set(s) of number each number belongs to:
1. -0
3.
2.
Whole Number
Rational Number
Integer
Real Number
Rational Number
Real Number
16
Natural Number
Whole Number
Integer
Rational Number
Real Number
4.
-4.2
Rational Number
Real Number
Draw a chip/tile model for each sum. Record the sum.
At this point, drawing the models remains important for most students. (Models may vary.)
For the following exercises, use the key below:
5. 7 + (–4)
3
7. 4 + 6
10
9. –4 + (–2)
–6
11. 7 + (–6)
1
6. –3 + 8
5
8. –5 + 8
3
10. –6 + (–3)
–9
12. 4 + (–8)
–4
13. Suppose a + b = c. Can c be less than zero (c < 0)? Explain.
a. What must you know about a and b relative to each other? a and/or b must be < 0. If only one is <
0, then its absolute value must be > than the other to make the sum negative.
b. What must you know about a and b relative to c? The sum of a and b must be negative. If only
one of the two addends is negative, then its absolute value must be > the absolute value of c.
For #14–16, use your prior knowledge and tile exploration to simplify the expression.
14.
-42 + (-65) + 17
15. 71 + (-9) + 23
-90
16. -33 + (-725) + 33
85
-725
What value for ? makes the number sentence true? Show your work.
17. -66 + ? = 10
18. -63 + -81 + ? = -63
76
19. ? + 186 = -12
81
-198
Simplify the expressions then replace ___ with =, > , or < to make a true statement.
20. (-349 + 275) > (97 + -184)
-74
-87
21. (144 + -186) >
- 42
(-32 + -98)
-130
For the question below, show your work and write your answer in a complete sentence.
22. Carl purchased stock in the Dependable Equipment Company. The price per share of the stock fell by
$4.30 in the first month, rose by $2.50 in the second month, and rose by $2.60 in the third month. Carl
sold the stock for $22 per share at the end of the third month. Did he gain or lose money?
He gained $0.80
23. When evaluating the expression x + 3, you discover the result is negative. What must be true about x?
Explain.
x has to be less than -3.
Spiral Review:
24. If an integer is equal to its absolute value, then the integer must be a __________ integer or zero.
positive
25. If an integer is equal to the opposite of its absolute value, then the integer must be a __________
integer or zero.
negative
26. Explain why there is no number that can replace to make the equation | |
true.
The absolute value of n means the distance n is from zero. Distance is always positive, so | | cannot
be a negative number.
2.1B Homework: Add Integers Using a Number Line*
Name:
Period:
Circle the operation you are going to perform. Model the following expressions using a number
line and write the answer.
1.
12 + 5 = 17
2.
–7 + 5 = –2
3. 9 + (–9) = 0
6. –15 + (–4) –19
7. 7 + (–16) –9
4. –11 + (–5) = –16
5. 8 + (–13) = –5
8. You borrowed $25 from your Mom to buy clothes. A week later you repaid her $13.
a. Write an addition expression to show how much money YOU owe. (–25) + 13
Focus on the term “expression.” Help students understand that a numerical expression is a
translation of the situation in words to a situation in numbers and symbols. (These exercises do
not involve equations. Why? You’re not asserting two expressions are equal. You are simply
expressing a situation numerically.)
b. Label the number line and model the expression. Circle the answer to the expression.
–12
c. Explain your expression and model.
Answers will vary.
9. Write a context for the following addition expression: 25 + (-15)
Answers will vary.
a.
Model the context using the number line. Be sure to label the number line. Circle the answer.
--5
b.
0
5
10
15
20
25
Simplify your expression and answer in a complete sentence
-15 + 25 or 25 + -15
10
10. Sally lives in Minnesota, where the winter temperature is often below zero. One day, the thermometer
read 10 degrees below zero, but it rapidly rose 18 degrees in one hour as a warm front moved in.
Then it dropped 18 degrees as night fell. What is the resulting temperature? Draw a chip model or
number line to represent this scenario. -10 + 18 + (-18) = -10
Diagrams/number lines will vary
Spiral Review: Evaluate if a = -2, b = 4, and c = -7. Show all work.
11. a + b
12. a + c
13. c + b + a
2.1C Homework: Integer Addition with Applications *
Name:
Period:
Circle the operation. Find the sum. Models are optional.
1. 14 + (–14) 0
9. 62 + (–57) + 91 96
2. -23 + 31 8
10. –42 + (–35) –77
3. –3 + (–3) –6
11. –51 + 81 30
4. 0 + (-57) -57
12. 1256 + (-2017) –761
5. 37 + 24 + 19 80
13. –531 + (–252) –783
6. -17 + (-4) + (-11) -32
14. 43 + (–45) –2
7. 41 + (-32) 9
15. –47 + 68 21
8. 37 + (-33) + (-42) -38
16. –91 + (–91) –182
Write an addition expression to represent the word problem and simplify. Write your answer in
context using a complete sentence. Include a chip model or number line to solve the problem.
17. The Golf Tournament shows that Todd’s score was 1 on his first round and –3 on his second round.
What was his score for the first two rounds?
1 + -3 = -2; Todd’s final score was -2.
18. A stunt plane started a maneuver 35 feet below the top of a skyscraper. If he traveled an additional 60
feet down, where was the stunt plane compared to the top of the skyscraper?
-35 + -60 = -95; The stunt plane is 95 feet below the top of the skyscraper.
Write an addition expression to represent the word problem and simplify. Write your answer in
context using a complete sentence. Models are optional.
19. Ocean water freezes at about -2.5 degrees Celsius. Fresh water freezes at 0 degrees Celsius.
Antifreeze, a liquid used in the radiators of cars, freezes at -64 degrees Celsius. Imagine that the
temperature has dropped to the freezing point for ocean water. How many degrees more must the
temperature drop for the antifreeze to turn solid? Teacher Note: Some students may try to set up an
equation.
64 + - 2.5 = 61.5
61.5 degrees
20. Which of the following expressions are equivalent to -13 + 7? Select all that apply. Show your work.
a. 142 + (-148)
b. -62 + 19 + 49
c. (-46.15) + 75.5 + (-35.35)
d. 7 + (-13)
21. Write two different integer addition expressions with three terms resulting in the sum of -14.
Student answers will vary.
Spiral Review: Simplify the following.
22. 4(3) + 8 ÷ 4
24.
23. 4(3.7 + 2)2
14
158
129.96
25. 36 – 27 ÷ 9 ÷ 1
33
2.1D Homework: Model Subtraction of Integers*
Name:
Period:
Circle the operation you are going to perform. Explain in words what needs to be done and draw a
chip/tile model for each and state the difference.
1. –4 – (–3) = –1
2. 8 – 3 = 5
3. –5 – 8 = –13
4. 7 – (–7) = 14
5. –4 – (–5) = 1
6. 3 – 10 = –7
Circle the operation. Model the following
problems using the number line. Find the
difference.
Circle the operation. Draw your own a chip/tile
model and number line model. Find the
difference for each.
7. 4 – 17 –13
12. 7 – (–3) 10
8. 5 – (–10) 15
13. 8 – (–4) 12
9. –7 – 6 –13
14. 5 – 12 –7
10. –8 – (–9) 1
11. 9 – 4 5
15. 6 – 18 –12
16. 14 – 3 11
17. On Tuesday, Joe made withdrawals of $25, $45, and $75 from his savings account. On the same day,
his twin sister Marge made withdrawals of $35, $55, and $65 from her savings account.
a. Find the total amount Joe withdrew.
$145
b. Find the total amount Marge withdrew.
$155
c. Marge and Joe’s brother also withdrew money from his savings account on Tuesday. He made
three withdrawals and withdrew $10 more than Marge did. What are three possible amounts he
could have withdrawn? Explain.
Answers vary.
$165
18. Explain how you could use a number line to show that -4 + 3 and 3 + (-4) have the same value.
Number lines will vary, but students’ answers should both result in -1.
19. Which number line model represents the sum of
(
)? D
For #20–21, draw a chip model or number line and solve the problem. Answer in a complete
sentence.
20. On a winter day, the temperature dropped from -3 degrees Celsius to -11 degrees Celsius. Find the
change in temperature.
The change in temperature was 8 degrees.
Models or number lines may vary.
21. A parachutist jumped from an airplane flying at an altitude of 1100m, dropped 200m in the first 25
seconds, and then dropped 35 m in the next 35 seconds. What was the altitude of the parachutist 60
seconds after jumping?
The altitude of the parachutist was 865 meters 60 seconds after jumping.
For #22–23, write an expression and solve the problem. Answer in a complete sentence
22. Arianna receives an allowance every 2 weeks that includes $20 for school lunches. During the past 4
weeks, she spent $7.50, $8.25, $5.25, and $8.75 on lunches. How much did Arianna have left from
the money allowed for lunches for the 4 weeks?
40 – (7.50 + 8.25 + 5.25 + 8.75) = $10.25
Arianna had $10.25 left.
23. Steve had $65.10 in his checking account on June 1. He wrote two checks in June, one for $42.99.
Steve forgot to write down the amount of the other check. At the end of the month, he received a
notice that his account was overdrawn by $22.11. What was the amount of Steve’s second check?
65.10 – 42.99 – 22.11 remaining
22.11 + 22.11 = $44.22 for 2nd check
Steve’s second check was $44.22.
Spiral Review:
24. Plot the following points on the same number line.
2.5, , 3, -4, -2.6, 0.25, -1.2
-5
-4
-3
-2
-1
0
1
2
3
4
5
2.1E Homework: Subtraction and Integers/Review*
Name:
Period:
Write an equivalent addition expression for each subtraction expression, and then simplify the
expression.
Subtraction Expression
Addition Expression
Simplify
1. –71 – 35
–71 + –35
–106
2. 63 – 11
63 + –11
52
3. 12 – (–37)
12 + 37
49
4. –16 – (–8)
–16 + 8
-8
5. 34 – 10
34 + –10
24
6. –42 – 17
–42 + –17
–59
7. –8 – (–13)
–8 + 13
5
8. 5 – (–19)
5 + 19
24
Circle the operation you are going to perform. Find the indicated sum or difference.
9.
3 – (–9) = 12
10.
–4 + (–1) = –5
11.
–3 – (–7) = 4
12.
8–5=3
13.
4 – 9 = –5
14.
5 – (–2) = 7
15.
–3 – 4 = –7
16.
–5 + (–2) = –7
17.
3 – (–1) = 4
18.
2 – 6 = –4
19.
–5 + 10 = 5
20.
–2 – (–9) = 7
Use a chip or number line model to solve each of the following. Write a numerical expression that
models your solution and then write a complete sentence stating your answer.
Student models and expressions will vary.
21. The temperature rose 13˚F between noon and
5 p.m. and then fell 7˚F from 5 p.m. to 10 p.m.
If the temperature at noon was 75˚F, what
would the temperature be at 10 p.m.?
75 + 13 – 7
At 10 p.m. the temperature is 81°.
22. Ricardo’s grandmother flew from Buenos Aires,
Argentina to Minnesota to visit some friends.
When Ricardo’s grandmother left Buenos Aires
the temperature was 84°. When she arrived in
Minnesota, it was –7°. What was the
temperature change for Ricardo’s
grandmother?
84 – (–7)
She experienced a 91° change.
23. Paula was standing on top of a cliff 35 feet
above sea level. She watched her friend Juan
jump from the cliff to a depth of 12 feet into the
water. How far apart were the two friends?
24. The Broncos got possession of the football on
the 20 yard line. They ran for an 8 yard gain.
The next play was a 3 yard loss. What was
their field position after the two plays?
35 – (–12)
The two friends were 47 feet apart.
20 + 8 + (–3)
The Broncos were on the 25 yard line.
25. For two months, David fed his cat Diet Chow brand cat food. For the next two months, he fed his cat
Kitty Diet brand food. The table shows the cat’s change in weight over 4 months.
Cat’s Weight Change (oz)
Diet Chow, Month 1
-8
Diet Chow, Month 2
-18
Kitty Diet, Month 3
3
Kitty Diet, Month 4
-19
Which brand of cat food resulted in the greatest weight loss for David’s cat? Explain.
Diet Chow: -26 ounces
Diet Chow resulted in the greatest weight loss.
Kitty Diet: -16 ounces
26. When evaluating the expression x – 5, the result is a positive number. What does this tell you about
the value of x? Explain.
x must be greater than 5.
27. How would your answer for #26 change if you wanted the result to be a whole number? Explain.
x must be 5 or any whole number greater than 5.
Section 2.1 Review
Name:
Period:
Read all the directions carefully. You must show ALL of your work to receive credit!
Circle the operation. Draw your own chip/tile model and number line. Find the difference for each.
1. 12 – 5 7
2. –3 – 7 –10
3. –8 – 3
5. –2 – (–6)
–11
4
4. –9 – (–4)
–5
6. -10 – (-6)
-4
7. Write a context problem requiring the expression -10 + 4. Then solve and answer in a complete
sentence.
Answers will vary.
8. Talia was estimating the difference between two positive numbers x and y (where x > y). First she
rounded x up by a small amount. Then she rounded y down by the same amount. Finally, she
subtracted the rounded values. Which of the following statements is correct? Justify your answer(s).
The difference of her estimate is larger than x – y. Justifications may vary.
There is not enough information to compare x – y with her estimate
a. Her estimate is smaller than x – y
d. Her estimate equals y – x
b. Her estimate is larger than x – y
c. Her estimate equals x – y
For the following problems
e.
Her estimate is 0.
a. Draw a chip/tile model.
b. Draw a number line model.
c. State the number of zero pairs,
d. Find the sum/difference.
9. 5 + (-2)
a.
b.
c.
d.
10. -2 + (-3)
a.
b.
c.
d.
11. -3 + 5
a.
b.
c.
d.
12. -4 – 6
a.
b.
c.
d.
13. -6 – (-3)
a.
b.
c.
d.
14. 7 – (-4)
a.
b.
c.
d.
Use a chip or number line model to solve each of the following. Write a numerical expression that
models your solution and then write a complete sentence stating your answer. Student models
will vary.
15. Chloe had $45 in her bank account. After she
went shopping, she looked at her account
again and she had –$12. How much did she
spend shopping?
45 – (–12)
Chloe spent $57.
16. There is a Shaved Ice Shack on South Street.
Both Mary and Lina live on South Street.
Mary lives 5 blocks west of the shack and Lina
lives 3 blocks east of it. How far apart do they
live?
5 – (–3)
They live 8 blocks apart.
17. Eva went miniature golfing with Taylor. She
shot 4 under par and Taylor shot 2 under par.
By how many strokes did Eva beat Taylor?
18. Mitchell owes his mom $18. He’s been
helping around the house a lot, so his mom
decided to forgive $12 of his debt. How much
does he owe now?
–4 – (–2)
Eva won by 2 strokes (–2); in golf you want a
lower score.
–18 - (-12)
Mitchell owes $6.
19. On the first play of a possession, Drew Brees
was sacked 4 yards behind the line of
scrimmage. He then threw a pass for a 12
yard gain. Did the Saints get a first down? (A
team must move forward 10 yards to get a first
down.)
20. Joe had $2. He found a quarter, but he lost a
dollar in the vending machine. How much
money does he have now?
2 + 0.25 + (–1) = 1.25
Joe has $1.25
–4 + 12
Drew Brees' team has made a gain of 8 yards.
This is not enough for a first down.
Simplify each expression. Show all necessary work. A model is not required.
21. –12 + 7 = –5
22. 3 – 8 = –5
23. 9 – (–11) = 20
24. 8 – (–2) = 10
25. –8 + 10 = 2
26. –5 – 21 = –26
27. 8 + (–10) = –2
28. 9 + (–1) = 8
2.2A Homework: Multiply Integers
Name:
Period:
TEACHER NOTE: It may be helpful to have student write in words what the expression means before
drawing the models and writing it as repeated addition or subtraction
1.
Draw a chip/tile model for each multiplication. Rewrite as an addition or subtraction expression.
Simplify
b)
a)
4+4
8
-7 + (-7) + (-7)
-21
d)
c)
-5 – 5
-10
-(-1) – (-1) – (-1) – (-1) – (-1) – (-1) – (-1)
7
e)
f)
-(-7) – (-7)
14
-4 + (-4)
-8
g)
h)
3+3+3+3+3+3
18
Take away 5
-5
2. What rules have you noticed when multiplying integers? Explain any rules you have observed from
above. Explain why each rule works. You should reference how multiplication relates to
addition/subtraction and how/why you can use zero pairs.
a.
__Positive____________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
b.
__Negative__________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
c.
__Negative__________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
d.
__Positive___________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
3. Using your rules from #2, simplify the following expressions:
a.
b.
-168
c.
13(-4)
-120
d.
(-2)(5)(10)
-52
e.
3(-8)(-4)
-100
f.
96
4. There is a product, xy, such that x ≠ 0 and y > 0
a. If xy < 0, what must be true about x?
x must be less than 0
b.
If xy > 0, what must be true about x?
x must be greater than 0
c.
For xy + 1 to be negative, what must be true about x?
x must be less than -1
(-1)(-3)(-6)
-18
Write a multiplication expression for each situation, simplify, and write the answer in a complete
sentence.
5. Kim borrowed $12 from 6 different friends. How much money does she owe her friends altogether?
6(-12) = -72
Kim borrowed $72 from her friends.
NOTE: -12(6) would not be a correct expression
6. The temperature rose 8 degrees Fahrenheit an hour for 7 hours. What was the rise in temperature
after 7 hours?
7(8) = 56
The temperature rose 56⁰ F in 7 hours.
7. Jim was deep sea diving off the coast of La Jolla last weekend. He descended 3 feet every minute.
How many feet did he descend in 10 minutes?
10(-3) = -30
Jim descended 30 feet in 10 minutes.
8. For each set of clues, find the pair of numbers that satisfies both clues.
a. Clue #1: The sum of two number is 44.
Clue #2: The difference between the two numbers is 12.
16, 28
b. Clue #1: The difference between two numbers is 15.
Clue #2: The greater number multiplied by the lesser number is -50.
10, -5
Spiral Review: Simplify the following:
9.
11.
2 + (-3) -1
|
|
-5
10. 2 – (-3)
5
12. 8(-7) + 3(-2)
-62
Using your integer rules, what value of the variable makes the statement true?
13. 2 – x = -6
x=8
14. 7 – (-x) = 14
x=7
15. 4(x) = -20
x = -5
16. (-7)(x) = 49
x = -7
2.2B Homework: Rules and Structure for Division of Integers/Application*
Name:
Period:
Find each quotient.
1. 36 ÷ (2) ÷ 3
2. 52 ÷ (4)
6
13
6
3. 72 ÷ (3) ÷ 4
4. -39 ÷ (13)
3
5. 280 ÷ (4) ÷ 5
6. 63 ÷ 9
7
7.
36 ÷ (3)
12
8.
60 ÷ (15)
4
9.
190 ÷ (5) ÷ 19
10. -140 ÷ 20
-7
14
2
11.There are three numbers a, b, and c that are non-zero rational numbers such that
. Given this
equation, use the letters below to complete four true equations. Note: you may use the letters more than once
and not all the letters will be used.
Letters to choose from:
Equation #1:
a, b, c, -a, -b, -c
Equation #2:
Equation #3:
Equation #4:
(-b)
(-a, b or a, -b)
(-c)
(-c, b or c, -b)
Write a division expression for each situation. Answer the question in a complete sentence.
Remember to pay close attention to the signs used in your expression.
TEACHER NOTE: It may be helpful to remind students to underline key words.
12. Keith borrowed a total of $30 by borrowing the same amount of money from 5 different friends. How much
money does Keith owe each friend?
30 ÷ 5
Keith owes each friend $6.00.
13. The temperature fell 12ºF over 4 hours. What was the average change in temperature per hour?
12 ÷ 4
The temperature changed 3○F each hour.
14. Max lost 24 pounds in 8 weeks on his new weight-loss plan. What was his average change in weight per
week?
24 ÷ 8
Max lost an average of 3 pounds each week.
15. Siegfried borrowed $4 a day until he had borrowed a total of $88. For how many days did he borrow
money?
88 ÷ 4
Siegfried borrowed money for 22 days.
Write an expression and solve. Answer in a complete sentence.
16. Ellie withdrew $20 at a time from her bank account and withdrew a total of $140. Leslie withdrew $45 at a
time from her bank account and withdrew a total of $270. Who made the greater number of withdrawals?
Justify your answer.
Ellie: 7
Leslie: 6
Ellie made more withdrawals.
17. Rachel went to Sequoia National Park with her family. She hiked down a trail at a constant rate for 10
minutes. Her change in elevation was -200 feet. Rachel hiked down the next 20 minutes at a different
rate. Her change in elevation for this part was -300 feet. During which portion of the hike did she walk
down at a faster rate? Explain your reasoning.
First part: -20
Second part: -15
The first portion, she walked at a constant rate of 20 feet per minute.
18. Kiley divided an integer, x, by -3 and got 8. She divided 8 by an integer, y, and got -4. Find the quotient of
integer x and integer y.
x = -24
y = -2
19. A perfect score on a test with 45 questions is 180. Each question is worth the same number of points.
a. How many points is each question on the test worth?
Each question is worth 4 points.
b.
Francis got a score of 80 on the test. Write a division sentence using negative numbers where the
quotient represents the number of questions Francis answered incorrectly.
Francis answered 25 questions incorrectly.
20. A video game player receives $50 for every correct answer and pays $45 every time he gets a question
incorrect. After a new game of 30 questions, he misses 16. What was his final total?
(14 × 50) + (16 × (45)) = 700 + (720) = 20;
He lost $20.
Section 2.2 Review
Name:
Period:
Draw a chip/tile model for each multiplication. Rewrite as an addition or subtraction expression.
Simplify.
2.
1.
-6 + (-6)
-12
2 + 2+ 2
6
4.
3.
-2
-4 – 4
-8
-(-3) – (-3)
6
5.
6.
-(5) – (5)
-10
-1 + (-1)
-2
Write an expression to represent each word problem. Simplify the expression then answer the
question in a complete sentence.
7. Alicia owes $6 to each of 4 friends. How much money does she owe?
4  (6) = 24;
Alicia owes $24 to her friends.
8. Maggie owes the candy store $35. Each of 5 friends will help her pay off her debt. How much will each
friend pay if they each give the same amount?
35 ÷ 5 = 7;
Each of Maggie's friends pays $7.
9.
An oven temperature dropped 135○ in 15 minutes. How many degrees per minute did the temperature
drop?
135 ÷ 15 = 9;
The temperature dropped an average of 9 degrees per minute.
10.
You go into a business partnership with three other friends. Your business loses $65,000. You agree to
share the loss equally. How much has each of the four people lost?
65,000 ÷ 4 = 16,250
11.
Lila bought 4 pairs of jeans for $33 each. How much will Lila have to pay for the jeans (before tax)?
(4)(33)
12.
Each person loses $16,250.
Lila will have to pay $132 for the jeans.
Karla borrowed $5 each from 4 different friends. How much money does Karla owe her friends
altogether?
4(5)
Karla owes $20 to her friends.
Note: it is not the same as 5(-4)
Simplify the following:
13. 5(3) (-2)
30
2
16. 30 ÷ 5 ÷ (-3)
19. 15 20
22. 68 ÷ (2)
14. 15 ÷ (5)
300
34
17. 25(3)(-1)
20.
-3
15. (5) (4) ÷ (-2)
-75
1
23. (6) (10)
18. 94(2)
21.
60
10
188
5
24. -144 ÷ (-3) ÷ 3
16