COMPACTED MATHEMATICS: CHAPTER 3
INTEGERS IN SPORTS
TOPICS COVERED:
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Introduction to integers
Opposite of a number and absolute value
Adding integers
Subtracting integers
Multiplying and dividing integers
Integer Labs
Survival Guide to Integers Project
Activity 3-1: Introduction to Integers
Name:
The number line can be used to represent the set of integers. Look carefully at the number line
below and the definitions that follow.
Definitions
The number line goes on forever in both directions.
This is indicated by the arrows.
Whole numbers greater than zero are called positive
integers. These numbers are to the right of zero on the
number line.
Whole numbers less than zero are called negative
integers. These numbers are to the left of zero on the
number line.
The integer zero is neutral. It is neither positive nor
negative.
The sign of an integer is either positive (+) or negative (-),
except zero, which has no sign.
Two integers are opposites if they are each the same
distance away from zero, but on opposite sides of the
number line. One will have a positive sign, the other a
negative sign. In the number line above, +3 and -3 are
labeled as opposites.
Activity 3-2: Introduction to Integers
Name:
Definitions:
Integers – the whole numbers and their opposites (positive counting numbers, negative counting
numbers, and zero)
Opposite of a number – a number and its opposite are the same distance from zero on the number
line
Example: 7 and 7 are opposites
Absolute value – the number of units a number is from zero on the number line without regard to
the direction
Example: the absolute value of 6 is 6
The sign for absolute value is two parallel lines: 6 = 6
1-10. Place the correct letter corresponding to each integer on the number line below.
Place the corresponding letter above the correct place in the number line below:
-10
0
+10
A.
5
B.
2
C.
7
D.
4
E. 9
F.
1
G.
6
H.
3
I.
0
J. 6
Write an integer to represent each situation.
11.
lost $72
12. gained 8 yards
13. fell 16 degrees
Name the opposite of each integer.
14.
26
15.
83
16.
100
Compare the following integers. Write <, >, or =.
17.
5 ___ 8
18.
12 ___ 13
19.
10 ___ 21
20.
7 ___ 11
Find the absolute value of the following numbers.
21.
11
22.
6
23.
55
24.
0
25.
28
26.
203
27.
75
28.
3
Activity 3-3: Introduction to Integers
Name:
1. List the following temperatures from greatest to least.
A
The temperature was 25 degrees Fahrenheit below zero.
B
The pool temperature was 78 degrees Fahrenheit.
C
Water freezes at 32 degrees Fahrenheit.
D
The low temperature in December is -3 degrees Fahrenheit.
E
The temperature in the refrigerator was 34 degrees Fahrenheit.
Think of the days of the week as integers. Let today be 0, and let days in the past be negative
and days in the future be positive.
2.
If today is Tuesday, what integer stands for last Sunday?
3.
If today is Wednesday, what integer stands for next Saturday?
4.
If today is Friday, what integer stands for last Saturday?
5.
If today is Monday, what integer stands for next Monday?
Circle the number that is greater.
6.
7.
4 or 13
33 or  41
10.
2 or  4
11.
Write true or false.
14.
3  7
17.
5  5
9 or  7
8.
0 or -4
9.
0 or 7
12.
5 or 5
13.
1 or 11
15.
9  1
16.
6  2
18.
8  8
19.
5  6
Write an integer to represent each situation.
20. moving backwards 4 spaces on a game board
21. going up 3 flights in an elevator
22. a 5-point penalty in a game
23. a $1 increase in your allowance
Order from least to greatest.
24.
{6, 3,1, 1, 5, 7, 0,9}
25.
{2, 1,3, 4, 6,13, 8, 2}
Activity 3-4: History of Negative Numbers
Name:
For a long time, negative solutions to problems were considered "false" because they couldn't be
found in the real world (in the sense that one cannot have a negative number of, for example,
seeds).
The abstract concept was recognized as early as 100BC – 50BC. The Chinese discussed methods
for finding the areas of figures; red rods were used to denote positive, black for negative. They
were able to solve equations involving negative numbers. At around the same time in ancient
India, sometime between 200BC and 200AD, they carried out calculations with negative numbers,
using a "+" as a negative sign. These are the earliest known uses of negative numbers.
In Egypt, Diophantus in the 3rd century AD referred to the equation equivalent to 4x + 20 = 0 (the
solution would be negative) in Arithmetica, saying that the equation was absurd, indicating that no
concept of negative numbers existed in the ancient Mediterranean.
During the 7th century, negative numbers were in use in India to represent debts. The Indian
mathematician Brahmagupta discusses the use of negative numbers. He also finds negative
solutions and gives rules regarding operations involving negative numbers and zero. He called
positive numbers "fortunes", zero a "cipher", and negative numbers a "debt".
From the 8th century, the Islamic world learnt about negative numbers from Arabic translations of
Brahmagupta's works, and by about 1000 AD, Arab mathematicians had realized the use of
negative numbers for debt.
Knowledge of negative numbers eventually reached Europe through Latin translations of Arabic
and Indian works.
European mathematicians however, for the most part, resisted the concept of negative numbers
until the 17th century, although Fibonacci allowed negative solutions in financial problems where
they could be interpreted as debits and later as losses. At the same time, the Chinese were
indicating negative numbers by drawing a diagonal stroke through the right-most non-zero digit.
The first use of negative numbers in a European work was by Chuquet during the 15th century. He
used them as exponents, but referred to them as “absurd numbers”.
The English mathematician Francis Maseres wrote in 1759 that negative numbers "darken the very
whole doctrines of the equations and make dark of the things which are in their nature excessively
obvious and simple". He came to the conclusion that negative numbers did not exist.
Negative numbers were not well-understood until modern times. As recently as the 18th century,
the Swiss mathematician Leonhard Euler believed that negative numbers were greater than
infinity, and it was common practice to ignore any negative results returned by equations on the
assumption that they were meaningless.
Taken from Wikipedia (en.wikipedia.org)
Activity 3-5: Adding Integers with Same Sign
Name:
Find each sum. White counters are positive. Black counters are negative.
1. 5   3 
2. 4   7 
a. How many counters are there? _______
a. How many counters are there? ________
b. Do the counters represent positive
b. Do the counters represent positive
or negative integers? _________________
c. 5   3   _________
or negative integers? _________________
c.
4   7   ________
Model each addition problem on the number line to find each sum.
3. 4   2   _______
4. 5   5   _______
5. 3   6   _______
6. 7   5   _______
Find each sum.
7. 7   1  _______
9. 36   17   _______
8. 5   4   _______
10. 51   42  _______
11. 98  126  _______
12. 20   75   _______
13. 350   250   _______
14. 110   1,200   _______
Solve.
15. A construction crew is digging a hole. On the first day, they dug a hole 3
feet deep. On the second day, they dug 2 more feet. On the third day,
they dug 4 more feet. Write a sum of negative numbers to represent this
situation. Find the total sum and explain how it is related to the problem.
________________________________________________________________________________________
Activity 3-6: Adding Integers with Same Sign
Name:
Solve.
1. A grocery sells green apples and red apples. On Monday, the store put
500 of each kind of apple on display. That day, the store sold 42 red
apples and 57 green apples. On Tuesday, the store sold 87 red apples
and 75 green apples. On Wednesday, the store sold 29 red apples and
38 green apples.
a. Write an addition expression using negative integers to show the
number of red apples the store sold.
____________________________________________________________________________________
b. Write an addition expression using negative integers to show the
number of green apples the store sold.
____________________________________________________________________________________
c. Did the store have more red apples or green apples left over?
Explain.
____________________________________________________________________________________
____________________________________________________________________________________
2. A hotel has 18 floors. The hotel owner believes the number 13 is
unlucky. The first 12 floors are numbered from 1 to 12. Floor 13 is
numbered 14, and the remaining floors are numbered from 15 to 19. The
hotel manager starts on the top floor of the apartment building. He rides
the elevator two floors down. The doors open and a hotel guest gets in.
They ride the elevator three floors down. The hotel guest gets off the
elevator. The hotel manager rides the elevator the remaining floors
down to the first floor.
a. Write an addition expression using negative integers to show the
number of floors the hotel manager rode down in the elevator.
____________________________________________________________________________________
b. On what floor did the hotel guest get off the elevator? Explain.
____________________________________________________________________________________
____________________________________________________________________________________
Activity 3-7: Adding Integers with Same Sign
Name:
Find each sum. White counters are positive. Black counters are
negative. The first one is done for you.
2. 4   6  
1. 5  2 
7
a. How many counters are there? _______
a. How many counters are there? _______
b. Do the counters represent positive
b. Do the counters represent positive
positive
or negative numbers? ______________
c.
or negative numbers? _________________
7
5  2  ______________
c.
4   6   _______
Model each addition problem on the number line to find each sum. The
first one is done for you.
5
3. 3   2   ________
4. 5   1  ________
5. 4   3   ________
6. 1   6   ________
Find each sum. The first one is done for you.
4
7. 3   1  ________
8. 6   2   _______
9. 12   7   _______
10. 20   15   _______
Solve.
11. The table shows how much money Hannah withdrew in 3 days.
Day
Dollars
Day 1
Day 2
Day 3
5
1
2
Find the total amount Hannah withdrew. ___________________________
Activity 3-8: Adding Integers with Different Signs
Name:
Show the addition on the number line. Find the sum.
1. 2  (3) _________________
2. 3  4 _________________
Find each sum.
3.  4  9
_______________
7. 5  (7)
_______________
11. 50  (7)
_______________
4. 7  (8)
_______________
8. 9  (5)
_______________
12. 27  (6)
_______________
5. 2  1
6. 6  (9)
_______________
9. (1)  9
________________
10. 9  (7)
_______________
13. 1  (30)
________________
14. 15  (25)
_______________
________________
Solve.
15. The temperature outside dropped 13°F in 7 hours. The final temperature
was 2°F. What was the starting temperature?
________________________________________________________________________________________
16. A football team gains 8 yards in one play, then loses 5 yards in the next.
What is the team’s total yardage for the two plays?
________________________________________________________________________________________
17. Matt is playing a game. He gains 7 points, loses 10 points, gains
2 points, and then loses 8 points. What is his final score?
________________________________________________________________________________________
18. A stock gained 2 points on Monday, lost 5 points on Tuesday, lost
1 point on Wednesday, gained 4 points on Thursday, and lost 6 points
on Friday.
a. Was the net change for the week positive or negative? ___________________________
b. How much was the gain or loss? ___________________________
Activity 3-9: Adding Integers with Different Signs
Name:
Tell whether each sum will be positive or negative. Then find
each sum.
1. 3  ( 7)
2. 14  (9)
_______________
5. 11  (5)
3. 12  5
_______________
6. 7  8
_______________
4. 3  8
_______________
________________
7. 8  7
_______________
8. 2  3
_______________
________________
9. If two integers have the same sign, what is the sign of their sum?
________________________________________________________________________________________
10. When adding two integers with different signs, how do you find the
sign?
________________________________________________________________________________________
Evaluate a  b for the given values.
11. a  9, b  24
12. a  17, b  7
_______________________
14. a  15, b  15
_______________________
________________________
15. a  20, b  20
________________________
13. a  32, b  19
________________________
16. a  30, b  12
________________________
Solve.
17. The high temperature for the day dropped 7F between Monday and
Tuesday, rose 9F on Wednesday, dropped 2F on Thursday, and
dropped 5F on Friday. What was the total change in the daily high
temperature from Monday to Friday?
________________________________________________________________________________________
18. Karen deposited $25 in the bank on Monday, $50 on Wednesday and
$15 on Friday. On Saturday, she took out $40. Karen’s original balance
was $100. What is her balance now?
________________________________________________________________________________________
19. Lance and Rita were tied in a game. Then Lance got these scores:
19, 7, 3, 11, 5. Rita got these scores: 25, 9, 5, 9, 8. Who had the
higher score? How much higher was that higher score?
________________________________________________________________________________________
Activity 3-10: Adding Integers with Different Signs
Name:
Show the addition on the number line. Then write the sum. The first
one is done for you.
1. 2  (3)
2. 3  (4)
_______________________________________
1
________________________________________
Find each sum. The first one is done for you.
3. 4  (9)
4. 7  (8)
_______________________
5
6. 5  7
_______________________
9. 2  (7)
_______________________
________________________
7. 9  (5)
________________________
10. 6  (4)
________________________
5. 2  1
________________________
8. 1  9
________________________
11. 15  9
________________________
Solve. The first one is done for you.
12. The temperature dropped 12F in 8 hours. If the final temperature was
7°F, what was the starting temperature?
________________________________________________________________________________________
5F
13. At 3 P.M., the temperature was 9F. By 11 P.M., it had dropped 31F.
What was the temperature at 11 P.M.?
________________________________________________________________________________________
14. A submarine submerged at a depth of 40 feet dives 57 feet more. What
is the new depth of the submarine?
________________________________________________________________________________________
15. An airplane cruising at 20,000 feet drops 2,500 feet in altitude. What is
the airplane’s new altitude?
________________________________________________________________________________________
Activity 3-11: Addition of Integers
Add or subtract.
1.
2  8
Name:
2.
8  4
3.
6  3
4.
6  4
5.
1  7
6.
8  3
7.
2  6
8.
6  9
9.
5  7
10.
4  7
11.
4  7
12.
4  7
13.
2  1  3
14.
0  5
15.
3  2  1
16.
5  5
17.
6 1
18.
6  1
Some of the sixth grade teachers decide to try out for the Dallas Cowboys. They each are allowed
one rushing attempt against the Cowboys defense. The table below summarizes the results of their
attempts:
Johnsen
Atkins
Hoag
+18
8
19
Underwood
+24
Loewen
+2
Buckmaster
26
Snow
Mangham
+37
Landry
+6
13
Use the table above to answer the following addition problems.
19.
Mangham + Buckmaster
20.
Underwood + Johnsen
21.
Snow + Atkins
22.
Hoag + Landry
23.
Atkins + Mangham
24.
Snow + Landry
25.
Loewen + Underwood
26.
Johnsen + Buckmaster
27.
Snow + Hoag
28.
Landry + Johnsen
29.
Underwood + Mangham
30.
Atkins + Buckmaster
31.
Hoag + Atkins + Snow
32.
Hoag + Landry + Loewen
33.
Buckmaster + Atkins
34.
Johnsen + Hoag
35.
Place the teachers in order from the worst
carry (smallest) to the best carry (largest).
Compare. Write <, >, or =.
5  6 _____ 6  5
36.
37.
8  10 _____  3  6
4  9 _____  8  5
39.
20  12 _____  12  4
38.
Activity 3-12: Addition of Integers on a Number Line
Name:
Below are several rushing attempts in a football game. Plot the attempts on the number lines to
determine to total amount of yardage.
1. a gain of 3 yards and then a gain of 4 yards (3 + 4)
-10
-5
0
5
10
5
10
2. a loss of 5 yards and then a gain of 7 yards ( 5  7 )
-10
-5
0
3. a loss of six yards and then another loss of 2 yards ( 6  2 )
-10
-5
0
5
10
5
10
5
10
4. a gain of 8 yards and then a loss of 9 yards ( 8  9 )
-10
-5
0
5. a loss of 3 yards and then a loss of 1 yard ( 3  1)
-10
-5
0
6. a gain of 7 yards and then a loss of 7 yards ( 7  7 )
-10
-5
0
5
10
Activity 3-13: Subtracting Integers
Name:
Show the subtraction on the number line. Find the difference.
1. 2  3
2. 5  (1)
_______________________________________
________________________________________
Find the difference.
3. 6  4
_______________
7. 18  (18)
_______________
11. 9  15
_______________
4. 7  (12)
_______________
8. 23  (23)
_______________
12. 12 14
_______________
5. 12  16
_______________
9. 10  (9)
_______________
13. 22  (8)
_______________
6. 5  (19)
________________
10. 29  (13)
________________
14. 16  (11)
________________
Solve.
15. Monday’s high temperature was 6C. The low temperature was 3C.
What was the difference between the high and low temperatures?
________________________________________________________________________________________
16. The temperature in Minneapolis changed from 7F at 6 A.M. to 7F at
noon. How much did the temperature increase?
________________________________________________________________________________________
17. Friday’s high temperature was 1C. The low temperature was 5C.
What was the difference between the high and low temperatures?
________________________________________________________________________________________
18. The temperature changed from 5C at 6 P.M. to 2C at midnight. How
much did the temperature decrease?
________________________________________________________________________________________
19. The daytime high temperature on the moon can reach 130C. The
nighttime low temperature can get as low as 110C. What is the
difference between the high and low temperature?
________________________________________________________________________________________
Activity 3-14: Subtracting Integers
Name:
For each set of values find x  y. Answer the questions that follow.
1. x  14, y  2
2. x  11, y  11
_______________________
4. x  9, y  9
________________________
5. x  9, y  20
_______________________
7. x  9, y  11
________________________
6. x  0, y  9
________________________
8. x  1, y  1
_______________________
3. x  8, y  15
________________________
9. x  5, y  5
________________________
________________________
10. If x and y are both positive, when is x  y negative? ______________________________________
11. If x and y are both negative, when is x  y positive? ______________________________________
Solve.
12. The temperature changed from 7F at 6 P.M. to 5F at midnight. What
was the difference between the high and low temperatures? What was
the average change in temperature per hour?
________________________________________________________________________________________
13. The lowest point in the Pacific Ocean is about 11,000 meters. The
lowest point in the Atlantic Ocean is about 8,600 meters. Which ocean
has the lower point? How much lower?
________________________________________________________________________________________
14. At 11,560 feet above sea level, Climax, Colorado is the highest town in
the United States. The lowest town is Calipatria, California at 185 feet
below sea level. Express both of these distances as integers and tell which is
closer to sea level. How much closer to sea level is the town that is closer?
__________________________________________________________
Use the table for 15–16.
Temperatures at a Ski Resort
Day
High
Low
Saturday
8F
3F
Sunday
6F
2F
15. On which day was the difference in temperature greater? _________________________________
16. How much greater was the difference one day than the other? ____________________________
Activity 3-15: Subtracting Integers
Name:
Show the subtraction on the number line. Then write the difference.
The first one is done for you.
1. 3  8
2. 5  (1)
_______________________________________
–5
________________________________________
Find each difference. The first one is done for you.
3. 3  4
–7
_______________________
6. 8  8
_______________________
9. 8  1
_______________________
4. 7  (2)
________________________
7. 5  (5)
________________________
10. 7  (9)
________________________
5. 12  6
________________________
8. 1  (2)
________________________
11. 3  8
________________________
Solve. The first one is done for you.
12. The daytime temperature on the planet Mercury can reach 430C.
The nighttime temperature can drop to 180C. What is the difference
between these temperatures?
610C
________________________________________________________________________________________
13. An ice cream company made a profit of $24,000 in 2011. The same
company had a loss of $11,000 in 2012. What is the difference between
the company’s financial results for 2011 and 2012?
________________________________________________________________________________________
14. The high temperature on Saturday day was 6F. The low temperature
was 3F. What was the difference between the high and low
temperatures for the day?
________________________________________________________________________________________
Activity 3-16: Subtraction of Integers
Name:
An integer and its opposite are the same distance from 0 on a number line. The integers 5 and -5
are opposites. The sum of an integer and its opposite is 0. To subtract an integer add its opposite.
t  69
Example 1:
m  10  12
t  6  9
Example 2:
t  3
m  10  12
m2
Add or subtract.
1.
2  8
2.
8  (4)
3.
6  3
4.
6  4
5.
1  7
6.
38
7.
2  6
8.
6  9
9.
5  (7)
10.
4  (7)
11.
4  7
12.
4  7
13.
2  (1)  (3)
14.
8  8
15.
2  3  1
16.
5  (5)
17.
6 1
18.
6  1
In hockey, each player is given a plus/minus rating. This rating is based on how many goals are
scored by their team while the player is on the ice minus how many goals are scored by the
opposing team while the player is on the ice. A high number is good and a low number is bad.
Here are the best and worst plus/minus ratings for 2009-2010:
1
Jeff Schultz – WSH
+50
874
Ryan Potulny – EDM
21
2
Alex Ovechkin – WSH
+45
875
Kyle Okposo – NYI
22
3
Mike Green – WSH
+39
876
Steve Staios – EDM
27
4
Nicklas Backstrom – WSH
+37
877
Shawn Horcoff – EDM
29
5
Daniel Sedin – VAN
+36
878
Rod Brind'Amour – CAR
29
6
Alexander Semin - WSH
+36
879
Patrick O'Sullivan – EDM
35
Use the table above to answer the following subtraction problems.
19.
Schultz – Okposo
20.
Staios – Green
21.
Sedin – Ovechkin
22.
O’Sullivan – Semin
23.
Potulny – Backstrom
24.
Brind’Amour – Horcoff
25.
Green – O’Sullivan
26.
Semin – Schultz
27.
Staois – Brind’Amour
28.
Potulny – Schultz
29.
Semin – Sedin – Schultz
30.
Backstrom – Green
31.
Horcoff - Ovechkin
32.
Ovechkin – O’Sullivan
33.
Okposo – Staios
34.
Potulny – Brind’Amour
Activity 3-17: Subtraction of Integers
Name:
Subtracting integers is often the hardest of the four basic operations for students. Sometimes
students try to take a shortcut and they don’t change the signs to “add the opposite.” The problem
can be easy to miss when you don’t change these signs.
Here are some other explanations to help you remember why we can change the subtracting
problem to an addition problem.
PARTY #1: This is a positive party. It is filled with positive people. What could you do to make
this party less positive?
 One option would be to make some of the positive people go home. This means you are
subtracting positive people.
 A second option would be to bring in some negative people. This means you are adding
negative people.
Therefore you have accomplished the same thing two different ways.
Subtracting positives is the same as adding negatives.
PARTY #2: This is a negative party. It is filled with negative people. What could you do to make
this party less negative (more positive)?
 One option would be to make some of the negative people go home. This means you are
subtracting negative people.
 A second option would be to bring in some positive people. This means you are adding
positive people.
Therefore you have accomplished the same thing two different ways.
Subtracting negatives is the same as adding positives.
Activity 3-18: Subtraction of Integers on a Number Line
Name:
1. 7 – 2
-10
-5
0
5
10
-5
0
5
10
-5
0
5
10
-5
0
5
10
-5
0
5
10
-5
0
5
10
2. 4 – 6
-10
3. 6  1
-10
4. 5  3
-10
5. 3  4
-10
6. 2  5
-10
Activity 3-19: Applying Addition and Subtraction of Integers
Name:
Write an expression to represent the situation. Then solve by finding
the value of the expression.
1. Owen is fishing from a dock. He starts with the bait 2 feet below the
surface of the water. He reels out the bait 19 feet, then reels it back in 7
feet. What is the final position of the bait relative to the surface of the
water?
________________________________________________________________________________________
2. Rita earned 45 points on a test. She lost 8 points, earned 53 points, then
lost 6 more points. What is Rita’s final score on the test?
________________________________________________________________________________________
Find the value of each expression.
3. 7  12  15
4. 5  9  13
_______________________________________
5. 40  33  11
________________________________________
6. 57  63  10
_______________________________________
7. 21  17  25  65
_______________________________________
________________________________________
8. 12  19  5  2
________________________________________
Compare the expressions. Write ,  or .
9  1  16
9. 15  3  7
10. 31  4  6
17  22  5
Solve.
11. Anna and Maya are competing in a dance tournament where dance
moves are worth a certain number of points. If a dance move is done
correctly, the dancer earns points. If a dance move is done incorrectly,
the dancer loses points. Anna currently has 225 points.
a. Before her dance routine ends, Anna earns 75 points and loses 30 points.
Write and solve an expression to find Anna’s final score.
____________________________________________________________________________________
b. Maya’s final score is 298. Which dancer has the greater final score?
Activity 3-20: Applying Addition and Subtraction of Integers
Name:
Write an expression to represent the situation. Then solve by finding
the value of the expression.
1. Jana is doing an experiment. She is on a dock that is 10 feet above the
surface of the water. Jana drops the weighted end of a fishing line
35 feet below the surface of the water. She reels out the line 29 feet, and
then reels it back in 7 feet. What is the final distance between Jana and
the end of the fishing line?
________________________________________________________________________________________
2. Kirsten and Gigi are riding in hot air balloons. They start 500 feet above
the ground. Kirsten’s balloon rises 225 feet, falls 105 feet, and then rises
445 feet. Every time Kirsten’s balloon travels up or down, Gigi’s balloon
travels 15 feet farther in the same direction. Then both balloons stop
moving so a photographer on the ground can take a picture.
a. Find Kirsten’s final position relative to the ground.
____________________________________________________________________________________
b. Is Kirsten or Gigi closer to the ground when the photographer takes
the picture?
____________________________________________________________________________________
3. In a ring-toss game, players get points for the number of rings they can
toss and land on a colored stake. They earn 20 points for landing on a
red stake and 30 points for landing on a blue stake. They lose 10 points
each time they miss. The table shows the number of rings tossed by
David and Jon during the game.
a. Write and evaluate an expression that represents David’s total score.
Player
Red
Blue
Miss
David
2
3
3
Jon
3
2
2
____________________________________________________________________________________
b. Who scored more points during the game?
____________________________________________________________________________________
Activity 3-21: Applying Addition and Subtraction of Integers
Name:
Write an expression to represent the situation. Then solve by finding
the value of the expression. The first one is done for you.
1. Jeremy is fishing from a dock. He starts with the bait 2 feet below the
surface of the water. He lowers the bait 9 feet, then raises it 3 feet. What
is the final position of the bait relative to the surface of the water?
2  9  3  8; 8 feet below the surface of the water
________________________________________________________________________________________
2. Rita earned 20 points on a quiz. She lost 5 points for poor penmanship,
then earned 10 points of extra credit. What is Rita’s final score on the
quiz?
________________________________________________________________________________________
Find the value of each expression. The first one is done for you.
3. 7  1  5
4. 5  9  10
_______________________________________
1
5. 40  30  10
________________________________________
6. 2  8  19
_______________________________________
7. 12  14  6
________________________________________
8. 50  60  10
_______________________________________
________________________________________
Compare the expressions. Write , , or .
10  11  30
9. 20  5  10
10. –10  40 – 5
25  15  3
Solve.
11. Angela is competing in a dance competition. If a dance move is done
correctly, the dancer earns points. If a dance move is done incorrectly,
the dancer loses points. Angela currently has 200 points. Angela then
loses 30 points and earns 70 points. Write and evaluate an expression
to find Angela’s final score.
________________________________________________________________________________________
Activity 3-22: Integer Word Problems
Name:
Write the expression for each word problem and then solve.
Jerry Jones has overdrawn his account by $15. There is $10
1. service charge for an overdrawn account. If he deposits $60,
what is his new balance?
The outside temperature at noon was 9 degrees Fahrenheit.
2. The temperature dropped 15 degrees during the afternoon.
What was the new temperature?
The temperature was 10 degrees below zero and dropped 24
3.
degrees. What is the new temperature?
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
The football team lost 4 yards on one play and gained 9 yards
on the next play. What is the total change in yards?
The temperature in Tahiti is 27 degrees Celsius. The
temperature in Siberia is 33 degrees Celsius. What is the
difference in temperatures?
Horatio Hornswoggle was born in 57 B.C. and died in 16 A.D.
How old was Horatio when he died?
You have a bank account balance of $357 and then write a
check for $486. What is your new balance?
A mountain climber is at an altitude of 4572 meters and, at the
same time, a submarine commander is at 609 meters. What
is the difference in altitudes?
The Roman Empire was established in 509 B.C. and fell 985
years later. In what year did the Empire fall?
A scuba diver is at an altitude of 12 meters and a shark is at
an altitude of 31 meters. What is the difference in altitudes?
A submarine descended 32 feet below the surface of the
ocean. It then rose 15 feet to look at a shark. Write an
expression and solve to find the submarines current depth.
In January, the temperature at Mt. Everest averages 36C .
It can drop as low as 60C . In July, the average summit
temperature is 17 degrees Celsius warmer. What is the
average temperature at the summit of Mt. Everest in July?
What is the difference in elevation between Mt. McKinley
(+20,320 feet) and Mt. Everest (+29,035 feet)?
Find the difference in elevation between Death Valley ( 282
feet) and the Dead Sea ( 1348 feet).
The highest ever recorded temperature on earth was 136F in
Africa and the lowest was 129F in Antarctica. What is the
difference of these temperatures recorded on Earth?
The temperature in Mrs. Cagle’s room was 14 F yesterday,
but it rose 8F today. What is the new temperature today?
The boiling point of water is 212 F and 460F is its
absolute lowest temperature. Find the difference between
these two temperatures.
Activity 3-23: More Negatives
Name:
A negative sign signifies the opposite of an integer. For example, the opposite of 4 is 4 . The
opposite of 4 would be (4) . As we have learned from subtracting and our discussions of
subtraction (4) is equal to 4.
Simplify each expression.
(8)
1.
2.
(27)
3.
 36
4.
45
5.
 14
6.
0
7.
(12)
8.
( 57)
9.
(20)
10.
 51
11.
 25
12.
 (16)
Match the integer expression with the verbal expression.
 12
13.
(A) the opposite of negative twelve
14.
12
15.
16.
 12
(12)
17.
12
(B) the absolute value of twelve
(C) the opposite of the absolute value of negative twelve
(D) the absolute value of negative twelve
(E) the opposite of the absolute value of twelve
Solve and explain.
18. Is there a least positive integer? Explain.
19. Is there a greatest positive integer? Explain.
20. Is there a smallest integer that is negative? Explain.
21. Is there a largest integer that is negative? Explain.
Write always, never, or sometimes.
22. The sum of two negative integers is negative…
23. The sum of a positive integer and a negative integer is positive…
24. The sum of 0 and a negative integer is positive…
25. Zero minus a positive integer is negative…
26. The difference of two negative integers is negative…
Temperature on Pluto =
370F
Temperature on the moon
during the day = 417F
Temperature on Mercury =
950F
Temperature on the moon
during the night = 299F
Temperature on Earth = 59F
Temperature at moon’s poles
is constantly 141 F
Using the table above, write and solve five word problems involving the concepts we have learned
about integers. At least three of the problems should involve addition or subtraction.
Activity 3-24: Master’s Golf Results
Name:
2010 PGA Tour Masters Results
Place
Name
4th
Round
Score
1
2
3
4
6
10
Phil Mickelson
Lee Westwood
Anthony Kim
Tiger Woods
Fred Couples
Ian Poulter
-5
-1
-7
-3
-2
+1
Final
Score
Place
-16
-13
-12
-11
-11
-5
18
26
36
38
42
45
Name
4th
Round
Score
Final
Score
Ernie Els
Kenny Perry
Lucas Glover
Retief Goosen
Zach Johnson
Sergio Garcia
-4
+2
+2
+1
+3
+6
-1
+1
+4
+6
+7
+10
In golf, the goal is to get the lowest score possible. A score of “E” is equivalent to a 0. Use the
table to answer the following questions.
1. List the 12 players above in order from best to worst based on their 4th round score. If there is
a tie, the player with the better final score should come first.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13-24. Determine the absolute value of the final score for each player.
Phil Mickelson
Lee Westwood
Anthony Kim
Tiger Woods
Fred Couples
Ian Poulter
Ernie Els
Kenny Perry
Lucas Glover
Retief Goosen
Zach Johnson
Sergio Garcia
Determine the sum of the following groups of players’ final scores.
25.
Woods + Goosen
26.
Perry + Couples
27.
Garcia + Kim
28.
Johnson + Els + Garcia
29.
Mickelson + Poulter
30.
Woods + Kim + Glover
31.
Westwood + Els
32.
Goosen + Couples + Els
Determine the difference of the following groups of players’ final scores.
33.
Woods – Goosen
34.
Perry – Couples
35.
Mickelson – Westwood
36.
Kim – Woods – Els
37.
Poulter – Couples
38.
Glover – Garcia
39.
Johnson – Els
40.
Goosen – Garcia – Woods
Activity 3-25: Addition and Subtraction of Integers
Solve each equation.
x  7  (5)
1.
Name:
2.
10  9  n
3.
w  12  (5)
4.
t  13  (3)
5.
10  12  z
6.
7  8  k
7.
m  11  (6)
8.
0  (21)  b
9.
13  (11)  h
10.
f  52  52
11.
6  5  (4)  t
12.
4  (5)  6  m
13.
k  3  8  (9)
14.
a  6  (2)  (1)
15.
10  (5)  6  n
16.
c  8  8  (10)
17.
36  (28)  (16)  24  y
18.
x  31  19  (15)  (6)
Solve each equation.
4  1  f
19.
20.
h  5  (7)
21.
z  9 12
22.
a  765  (34)
23.
652  (57)  b
24.
c  346  865
25.
d  136  (158)
26.
x  342  (456)
27.
y  684  (379)
28.
b  658  867
29.
657  899  t
30.
3004  (1007)  r
31.
21  24  b
32.
15  (86)  a
Tell if each of the subtraction sentences would always, sometimes, or never be true. Support
your answer with examples.
33. positive – positive = positive
34. negative – positive = negative
35. negative – negative = positive
36. positive – negative = negative
negative – positive = positive
38. positive – positive = negative
37.
Activity 3-26: Square Game
Name:
Directions: Players take turns joining any two dots next to each other. Diagonals are not allowed. When a
player makes a square, the player's initials go in the box. When all the squares are completed, add up all the
integers in your boxes. Then subtract this total from 25. The player with the highest score is the winner.
ROUND 1
PLAYER 1:
-3
2
4
-6
-2
1
7
-4
3
-1
5
3
6
2
-5
3
-4
1
4
-3
6
-1
2
5
-4
TOTAL OF ALL BOXES: ____________
Now subtract this total from 25:
PLAYER 2:
25 -_____ = _______ (final score)
TOTAL OF ALL BOXES: ____________
Now subtract this total from 25:
25 -_____ = _______ (final score)
ROUND 2
-3
2
4
-6
-2
1
7
-4
3
-1
5
3
6
2
-5
3
-4
1
4
-3
6
-1
2
5
-4
Activity 3-27: Positive 4
Name:
In two minutes name as many sums of integers that yield a positive 4 as you can. You may loop
pairs of integers that are next to each other, either horizontally, vertically, or diagonally.
-4
8
-3
7
-2
4
-7
5
-1
9
-4
7
1
-8
2
-4
5
-5
1
-7
6
-4
8
-5
-9
2
-5
7
-3
8
-8
2
-3
6
-5
4
5
-1
2
-4
4
-6
5
-4
9
-1
4
-7
-7
6
-1
8
-3
2
-1
4
-3
6
-7
3
3
-2
8
-5
7
-9
4
-3
7
-2
5
-5
-8
6
-4
3
-7
2
-9
6
-2
1
-8
5
2
-4
6
-2
5
-1
7
-5
5
-6
9
-3
-6
9
-2
8
-1
7
-2
3
-3
9
-1
6
4
-3
2
-9
7
-3
6
-5
7
-8
3
-2
In two minutes name as many sums of integers that yield a positive 4 as you can. You may loop
pairs of integers that are next to each other, either horizontally, vertically, or diagonally.
-4
8
-3
7
-2
4
-7
5
-1
9
-4
7
1
-8
2
-4
5
-5
1
-7
6
-4
8
-5
-9
2
-5
7
-3
8
-8
2
-3
6
-5
4
5
-1
2
-4
4
-6
5
-4
9
-1
4
-7
-7
6
-1
8
-3
2
-1
4
-3
6
-7
3
3
-2
8
-5
7
-9
4
-3
7
-2
5
-5
-8
6
-4
3
-7
2
-9
6
-2
1
-8
5
2
-4
6
-2
5
-1
7
-5
5
-6
9
-3
-6
9
-2
8
-1
7
-2
3
-3
9
-1
6
4
-3
2
-9
7
-3
6
-5
7
-8
3
-2
Activity 3-28: Adding and Subtracting Integers
Name:
Integer Operation Game
Using a deck of cards, pull out two cards. Add the two cards together using these rules:
 Reds are negative and blacks are positive
 Jacks are 11, Queens are 12, Kings are 13, and Aces are 1.
Activity 3-29: Multiplication and Division of Integers
Name:
The Official Kissing Rules help you remember answer signs on multiplying or dividing problems.
A boy sees a girl he likes. (+)
The boy does kiss her. (+)
The boy is happy! (+)
A boy sees a girl he does not like. (-)
The boy does not kiss her. (-)
The boy is happy! (+)
+
+
+
-
-
+
A boy sees a girl he likes. (+)
The boy does not kiss her. (-)
The boy is not happy. (-)
A boy sees a girl he does not like. (-)
The boy does kiss her. (+)
The boy is not happy. (-)
When multiplying/dividing two positives or
two negatives, the answer is positive.
Solve each equation.
m  2(8)
1.
+
-
-
-
+
-
When multiplying/dividing one negative and
one positive, the answer is negative.
2.
t  3(4)
3.
x  8(4)
4.
p  (5)(5)
5.
r  (12)(5)
6.
w  (4)2
7.
e  12(13)
8.
v  14(3)
9.
n  (14)  5
10.
h  (12)2
11.
d  7  8
12.
b  9(10)
Evaluate each expression if m  6 , n  3 , and p  4 .
13.
4m
14.
np
15.
2mn
16.
2m 2
17.
5np
18.
10mp
19.
12np
20.
mnp
21.
p2
Solve each equation.
22.
f  16  4
23.
v  100 10
24.
m  28  7
25.
g  52  4
26.
d  125  25
27.
q  32  16
28.
e  120  12
29.
t  45  9
30.
p  33  3
31.
z  36 12
32.
d  200  25
33.
c  88 11
Evaluate each expression if e  36 , f  4 , and g  3 .
e
e
34.
35.
g2
f
37.
e2
f
40.
e2
fg
38.
48
g
41.
 100
f
36.
e
fg
39.
eg
f
42.
e2
g2
Activity 3-30: Multiplying Integers
Name:
Find each product.
1. 4(20)
_______________________
4. (13)(3)
_______________________
7. (9)(21)
_______________________
10. 10(8)
_______________________
2. 6(12)
________________________
5. (10)(0)
________________________
8. 11(1)
________________________
11. 9(6)
________________________
3. (8)(5)
________________________
6. (5)(16)
________________________
9. 18(4)
________________________
12. 7(7)
________________________
Write a mathematical expression to represent each situation. Then find
the value of the expression to solve the problem.
13. You play a game where you score 6 points on the first turn and on each
of the next 3 turns. What is your score after those 4 turns?
________________________________________________________________________________________
14. The outdoor temperature declines 3 degrees each hour for 5 hours.
What is the change in temperature at the end of those 5 hours?
________________________________________________________________________________________
15. You have $200 in a savings account. Each week for 8 weeks, you take
out $18 for spending money. How much money is in your account at the
end of 8 weeks?
________________________________________________________________________________________
16. The outdoor temperature was 8 degrees at midnight. The temperature
declined 5 degrees during each of the next 3 hours. What was the
temperature at 3 A.M.?
________________________________________________________________________________________
17. The price of a stock was $325 a share. The price of the stock went down
$25 each week for 6 weeks. What was the price of that stock at the end
of 6 weeks?
________________________________________________________________________________________
Activity 3-31: Multiplying Integers
Name:
Find each product.
1. (14)(7)
2. (24)(5)
_______________________
3. 12(12)
________________________
4. 15(9)(1)
6. 3(6)(2)
5. 2(3)(4)
_______________________
________________________
________________________
8. 6(60)(4)
7. 40(78)(0)
_______________________
________________________
9. 24(7)(7)
________________________
________________________
Write a mathematical expression to represent each situation. Then find
the value of the expression to solve the problem.
10. A football team loses 4 yards on each of three plays. Then they
complete a pass for 9 yards. What is the change in yardage after those
four plays?
________________________________________________________________________________________
11. You have $220 in your savings account. You take $35 from your
account each week for four weeks. How much is left in your account at
the end of the four weeks?
________________________________________________________________________________________
12. A submarine is at 125 feet in the ocean. The submarine makes three
dives of 50 feet each. At what level is the submarine after the three
dives?
________________________________________________________________________________________
Find each product. Use a pattern to complete the sentences.
13. 1(1) __________
14. 1(1)(1) __________
16. 1(1) (1)(1)(1) __________
15. 1(1)(1)(1) _________
17. 1(1)(1)(1)(1)(1) __________
18. When multiplying integers, if there is an odd number of negative factors,
then the product is _________________.
If there is an even number of negative factors, then the product is
_________________.
Activity 3-32: Multiplying Integers
Name:
Find each product. The first one is done for you.
1. 3(2)
6
_______________________
4. (4)(7)
_______________________
7. 10(5)
_______________________
10. 1(1)
_______________________
2. 5(0)
________________________
5. (3)(4)
________________________
8. 2(9)
________________________
11. 2(6)
________________________
3. (1)(8)
________________________
6. (6)(6)
________________________
9. 7(10)
________________________
12. 2(2)
________________________
Write a mathematical expression to represent each situation. Then find
the value of the expression to solve the problem. The first one is done
for you.
13. You play a game where you score 3 points on the first 5 turns. What is
your score after those 5 turns?
5(3)  15; 15 points
________________________________________________________________________________________
14. The outdoor temperature gets 1 degree colder each hour for 3 hours.
What is the change in temperature at the end of those 3 hours?
________________________________________________________________________________________
15. A football team loses 4 yards on each of 2 plays. What is the change in
yardage after those 2 plays?
________________________________________________________________________________________
16. You take $9 out of your savings account each week for 7 weeks. At the
end of 7 weeks, what is the change in the amount in your savings
account?
________________________________________________________________________________________
17. The price of a stock went down $5 each week for 5 weeks. What was the
change in the price of that stock at the end of 5 weeks?
________________________________________________________________________________________
Activity 3-33: Multiplying Integers
Name:
Complete the table below using your knowledge of integers as well as noticing the pattern that the
table creates.
5
15
4
12
3
-5
-4
-3
-2
-1
0
3
6
9
2
6
1
3
0
0
x
-1
-2
-3
-4
-5
0
1
2
3
12
15
4
5
Activity 3-34: Multiplying Integers
Name:
The multiplication table below contains 42 mistakes. Shade in each box that contains a mistake. You will end up with a famous
farming expression.
X
2
-4
-9
6
-3
6
-12 -27 -18
3
8
-1
4
-8
-2
-6
7
-5
9
-7
9
-24
-3
12
-24
6
-18 -21 -15
27
-21
81
63
9
-18 -36 -81
54
-27
72
9
36
-72 -18
54
-6
12
54
-36
18
-48
-6
24
48
-36 -42 -30 -54 -42
5
-10 -20 -45
30
-15
40
5
20
-40 -10
30
25
45
35
-7
14
-28 -63 -42
21
-56
-7
28
-56
-42 -49 -35
63
-49
-24
12
14
63
35
45
Activity 3-35: Dividing Integers
Name:
Find each quotient.
1. 7 84
_________________
4. 28  7
_________________
2. 38   2 
_________________
5. 121  (11)
_________________
3. 27 81
_________________
6. 35  4
_________________
Simplify.
7. ( 6  4)  2
_________________
8. 5(8)  4
_________________
9. 6(2)  4( 3)
_________________
Write a mathematical expression for each phrase.
10. thirty-two divided by the opposite of 4
________________________________________________________________________________________
11. the quotient of the opposite of 30 and 6, plus the opposite of 8
________________________________________________________________________________________
12. the quotient of 12 and the opposite of 3 plus the product of the opposite
of 14 and 4
________________________________________________________________________________________
Solve. Show your work.
13. A high school athletic department bought 40 soccer uniforms at a cost of
$3,000. After soccer season, they returned some of the uniforms but
only received $40 per uniform. What was the difference between what
they paid for each uniform and what they got for each return?
________________________________________________________________________________________
14. A commuter has $245 in his commuter savings account. This account
changes by $15 each week he buys a ticket.
a. If the account changed by $240, for how many weeks of tickets
did the commuter buy?
____________________________________________________________________________________
b. If the commuter wants to buy 20 weeks of tickets, how much must
he add to his account?
____________________________________________________________________________________
Activity 3-36: Dividing Integers
Simplify.
8
1. 
 ( 12)
2
_________________
Name:
2.
6 15  7

3
2
3. 3  2(4  7)  9
_________________
_________________
The integers from 3 to 3 can be used in the blanks below. Which of
these integers produces a positive, even integer for the expression?
Show your work for those that do.
4. 
8
 4 (______)  2
2
_____________________________________
6. ______ 
2
3
_____________________________________
5.
(_____) 3

4
2
_____________________________________
1
 1 
7. 


2
 _____ 
_____________________________________
Solve. Show your work.
8. In a sports competition, Alyssa was penalized 16 points. She received
the same number of penalty points in each of 4 events. How many
points was she penalized in each event?
________________________________________________________________________________________
9. The surface temperature of a deep, spring-fed lake is 70F. The lake
temperature drops 2F for each yard below the lake surface until a depth
of 6 yards is reached. From 6 yards to 15 yards deep, the temperature is
constant. From 15 yards down to the spring source, the temperature
increases 3F per foot until the spring source is reached at 20 yards
below the surface.
a. What is the temperature at 10 yards below the surface?
____________________________________________________________________________________
b. What is the temperature at 50 feet below the surface?
____________________________________________________________________________________
____________________________________________________________________________________
c. Write an expression for finding the lake temperature at the spring
source.
____________________________________________________________________________________
Activity 3-37: Dividing Integers
Name:
Find the quotient. The first one is done for you.
2. 27   3 
1. 3 15
_________________
5
3.
_________________
28
7
_________________
Compare the quotients. Write , , or .
4. 4 16

16 4
5. 11  77

77  11
6.
48
6

48
6
Write a mathematical expression for the written expression. Then
solve. The first one is done for you.
7. the opposite of 45 divided by 5
45 5  9
_____________________________________
9. negative 38 divided by positive 19
_____________________________________
8. fifty-five over negative eleven
_____________________________________
10. negative four divided by negative two
_____________________________________
Solve. Show your work. The first one is done for you.
11. Four investors lost 24 percent of their combined investment in a
company. On average, how much did each investor lose?
24
 4  6; On average, each investor lost 6%.
________________________________________________________________________________________
_______________________________________________________________________________________________________________________________
12. The temperature in a potter’s kiln dropped 760 degrees in 4 hours. On
average, how much did the temperature drop per hour?
_______________________________________________________________________________________________________________________________
_______________________________________________________________________________________________________________________________
13. The value of a car decreased by $5,100 over 3 years. On average, how
much did its value decrease each year?
_______________________________________________________________________________________________________________________________
_______________________________________________________________________________________________________________________________
Activity 3-38: Applying Integer Operations
1. (3)(2)  8
2. (18)  3  (5)(2)
________________________
________________________
4. 24  (6)(2)  7
5. 4(8)  3
________________________
________________________
Name:
3. 7(3)  6
________________________
6. (9)(0)  (8)(5)
________________________
Compare. Write , , or .
7. (5)(8)  3
(6)(7)  1
8. (8)(4)  16  (4)
(9)(3)  15  (3)
Write an expression to represent each situation. Then find the value of
the expression to solve the problem.
9. Dave owns 15 shares of ABC Mining stock. On Monday, the value of
each share rose $2, but on Tuesday the value fell $5. What is the change
in the value of Dave’s shares?
________________________________________________________________________________________
10. To travel the Erie Canal, a boat must go through locks that raise or lower
the boat. Traveling east, a boat would have to be lowered 12 feet at
Amsterdam, 11 feet at Tribes Hill, and 8 feet at Randall. By how much
does the elevation of the boat change between Amsterdam and Randall?
________________________________________________________________________________________
11. The Gazelle football team made 5 plays in a row where they gained
3 yards on each play. Then they had 2 plays in a row where they lost 12
yards on each play. What is the total change in their position from where
they started?
________________________________________________________________________________________
12. On Saturday, Mrs. Armour bought 7 pairs of socks for $3 each, and a
sweater for her dog for $12. Then she found a $5 bill on the sidewalk.
Over the course of Saturday, what was the change in the amount of
money Mrs. Armour had?
Activity 3-39: Applying Integer Operations
Name:
Complete the table to answer questions 1–4.
You Own
Company
Monday
Tuesday
Wednesday Net Gain or Loss
1.
5 shares
ABC
$2
$5
$1
2.
2 shares
DEF
$8
$7
$10
3.
8 shares
GHI
$2
$9
$6
4.
7 shares
JKL
$5
$12
$3
5. What expression shows your net gain or loss on GHI Company?
________________________________________________________________________________________
6. How much value did you gain or lose overall? ___________________________________
Write an expression to represent each situation. Then, find the value of
the expression to solve the problem.
7. A submarine cruised below the surface of the water. During a training
exercise, it made 4 dives, each time descending 45 feet more. Then it
rose 112 feet. What is the change in the submarine’s position?
________________________________________________________________________________________
8. A teacher wanted to prevent students from guessing answers on a
multiple-choice test. The teacher graded 5 points for a correct answer, 0
points for no answer, and 2 points for a wrong answer. Giselle answered
17 questions correctly, left 3 blank, and had 5 wrong answers. She also got
8 out of 10 possible points for extra credit. What was her final score?
________________________________________________________________________________________
9. Hugh wrote six checks from his account in the following amounts: $20,
$20, $12, $20, $12, and $42. He also made a deposit of $57 and was
charged a $15 service fee by the bank. What is the change in Hugh’s
account balance?
________________________________________________________________________________________
10. a.
Without finding the product, what is the sign of this
product? Explain how you know.
(4)(1)(2)(6)(3)(5)(2)(2)
________________________________________________________________________________________
b. Find the product. _____________________________________________________________________
Activity 3-40: Applying Integer Operations
Name:
Find the value of each expression. Show your work. The first one is
done for you.
1. 15  (6)(2)
2. (5)(3)  18
 15  (12)
Multiply
________________________________________
________________________________________
3
Add.
________________________________________
________________________________________
3. 42  (6)  23
4. 52  45  (9)
________________________________________
________________________________________
________________________________________
________________________________________
Write an expression to represent each situation. Then find the value of
the expression to solve the problem. The first one is done for you.
5. Mr. Carlisle paid his utility bills last weekend. He paid $50 to the phone
company, $112 to the power company, and $46 to the water company.
After he paid those bills, what was the change in the total amount of
money that Mr. Carlisle had?
(50)  (112)  (46)  208; He had $208 less.
________________________________________________________________________________________
6. Over 5 straight plays, a football team gained 8 yards, lost 4 yards, gained
7 yards, gained 3 yards, and lost 11 yards. What is the team’s position
now compared to their starting position?
________________________________________________________________________________________
7. At the grocery store, Mrs. Knight bought 4 pounds of apples for $2 per
pound and 2 heads of lettuce for $1 each. She had a coupon for $3 off the
price of the apples. After her purchases, what was the change in the
amount of money that Mrs. Knight had?
________________________________________________________________________________________
8. The depth of the water in a water tank changes every time someone in
the Harrison family takes a bath or does laundry. A bath lowers the water
level by 4 inches. Washing a load of laundry lowers the level by 2 inches.
On Monday the Harrisons took 3 baths and washed 4 loads of laundry.
By how much did the water level in the water tank change?
________________________________________________________________________________________
Activity 3-41: Negative times a Negative is WHAT?
Name:
Why is it when you multiply two negative numbers you get a positive number? Good question!
The First Answer
Some people think of a negative as meaning “not”. So if I say, “I am not going to the store,” that is
sort of the negative version of “I am going to the store.”
So what do two “nots” mean? Consider this sentence: “You may tell me NOT to go to the store, but
I’m NOT going to do what you say!” By negating your negation, I am insisting that I will go to the
store.
Two “nots” cancel each other out, just like two negatives.
The Second Answer
Let’s use negatives with money. A green chip is worth $5. A red chip means that I owe you $5. So
if you lose $5, you can represent that by giving up a green chip or by picking up a red chip. So a
green chip is +$5 and a red chip is -$5.
If you gain three green chips, what happens? 3 times $5 equals a $15 gain.
If you gain three red chips, what happens? 3 times -$5 equals a $15 loss.
What if you lose three green chips? You just lost $15. -3 times $5 equals a $15 loss.
What is you lose three red chips? You just gained $15. -3 times -$5 equals a $15 gain.
The Third Answer
How about proving it with a pattern?
So….
3 5
25
1 5
05
1 5
2  5
+15
+10
+5
0
-5
-10
3  5
2  5
1 5
0  5
1 5
2  5
-15
-10
-5
0
+5
+10
Activity 3-42: Multiplying and Dividing Integers
Name:
Solve each equation.
1.
x  6  8
2.
y  12  4
3.
x  9  (11)
4.
y  (7)(17)
5.
14(4)  h
6.
15(10)  k
7.
(10)(8)(2)  r
8.
(3)(3)(10)  t
9.
w  (12)(1)(6)
10.
y  (20)(5)(5)
11.
x  (4)(16)(6)
12.
n  (16)(9)(2)
Evaluate each expression if x=-5 and y=-6.
3y
13.
14.
8x
17.
15x
18.
19 y
15.
4 y
16.
12x
19.
6xy
20.
4xy
Divide.
21.
16  4
22.
27  3
23.
25  (5)
24.
63  (9)
25.
15  ( 3)
26.
14  (7)
27.
56  (8)
28.
72  8
29.
21  (7)
Solve each equation.
150
x
30.
25
31.
k
98
14
32.
x
312
24
33.
208
t
26
34.
180
n
15
35.
z
930
30
36.
189
p
21
37.
288
d
18
38.
b
396
36
Evaluate each expression if x  8 and y  12 .
39.
x2
40.
x  ( 4)
41.
36  y
42.
0 y
43.
y
6
44.
x
4
45.
144
y
46.
136
x
At noon on Friday, the temperature was 0 degrees. Six hours later the
47. temperature was -18 degrees. On average, what was the temperature
change per hour?
Mangham Architecture has monthly profits of $1200, $755, -$450, $210,
48.
and -$640 over 5 months. What was the average profit for those months?
Activity 3-43: All Integer Operations
Name:
Solve.
1.
9  (13)
2.
2(25)
3.
(6  17)  20
4.
50  30
5.
56  (8)
6.
(5  6)  87
7.
32  37
8.
(15  3)  14
9.
(13  2)  12
10.
(10  5)(2)
11.
(3  4)  7
12.
(5  30)(3)
13.
(9  6)  4
14.
(30  22)  6
15.
(8  8)  8
16.
(20  4)  11
17.
(28  10)  7
18.
12  36
19.
(13  12)(4)
20.
(4  6)  8
21.
(64  2)  2
22.
5  20
23.
30  2
24.
(40  50)  9
25.
9  (19)
26.
7  11
27.
(42  7)  6
28.
7  11
29.
60  5
30.
(12  18)  15
The symbols , , x, and  can be used only once in each number sentence below. Remember the
correct order of operations!
31.
+6
33.
10
35.
30
37.
(30
39.
41.
43.
45.
-3
(5
2=0
32.
5) = 9
34.
-6
(-4
[(-6
-3)
-1] = 28
36.
-6)
(-3
-1) = 20
38.
(-3
5
-5
(5
-5) = 9
40.
-3
(-4
4)
(4
-4) = -8
42.
-3)2
(-3
3)2 = 36
44.
(3
-3
-2)
-6
-1
-7 = -2
(-10
-1)2 = -54
(-2
8)
(-6
(-8
2)2
2
1
5) = 6
(5
6) = -1
-2)
-3 = 12
I am an integer. When you add -1 to me, the sum is the opposite of the
difference when you subtract -5 from me. What integer am I?
46. Find two integers having a product of negative 15 and a sum of positive 2.
47. Find two integers having a product of negative 30 and a sum of negative 1.
48. Find two integers having a product of positive 27 and a sum of negative 12.
49. Find two integers having a product of negative 64 and a sum of positive 12.
50. Find two integers having a product of positive 40 and a sum of negative 13.
-4 = -9
-2
2 = -4
Activity 3-44: Absolute Value
Complete the table below.
x
1.
4
2.
3
3.
2
4.
1
Name:
x 2
x
5.
2x
2
6.
1
7.
2
8.
3
9.
4
10.
When x is negative, its absolute value is….
11.
x is negative always, sometimes or never?
12.
x  2 is positive always, sometimes or never?
13.
x is less than 2 x always, sometimes or never?
14.
2 x is greater than x  2 always, sometimes or never?
Kyle has four integer cards. Two cards show positive integers and two cards show negative integers.
-9
8
4
-5
15. What is the sum of all four cards?
16. What is the largest sum Kyle can make with two cards?
17. What is the smallest sum Kyle can make with two cards?
18. What is the smallest sum that Kyle can make with three cards?
19. What is the largest difference Kyle can make with two cards?
20. What is the smallest difference Kyle can make with two cards?
21. What is the difference closest in value to 10 that Kyle can make with two cards?
22. What is the largest product Kyle can make with two cards?
23. What is the smallest product Kyle can make with two cards?
24. What is the largest product Kyle can make with three cards?
25. What is the smallest quotient Kyle can make with two cards?
Activity 3-45: Survival Guide to Integers
Name:
Choose one of the following topics: Weather (Temperature), Money, Golf, Time (Years), Elevations
and Altitudes, Game/Video Game Scores, Football, or Physical Science (Atoms and Molecules).
Then pick a more specific theme such as “Jeopardy!” under the main topic of Games or “Scuba Diving”
under the topic Elevations and Altitudes. Check with Mr. Mangham if you have another topic you
wish to use which is not on this list.
Your Survival Guide will consist of 8 pages (2 folded pieces of construction paper). The goal is to
teach integers to students who have not learned about them yet. The following details what
information should be included on each page.
Page 1: Title Page – Title, Pictures, Theme
 Your title must include the words “Survival Guide to Integers”
(10 points)
Page 2: Introduction to Integers
 State at least three places of where we use negative numbers in real life (include
specific examples of how they would be used in each)
 Give definitions and examples for these words:
o Integer (provide examples of integers and numbers that are not integers)
o Opposite of a number
o Absolute value
(20 points)
ADDITION – Pages 3 and 4 – Make sure to include a variety of samples (positive plus negative
where there are more positives, positive plus negative where there are more negatives, negative plus
negative, etc.)
Page 3: Addition of integers
 Teach how to add integers using both:
o Yellow and red chips (introduce zero pairs)
o Number lines
 Explain in words what is happening
 Provide specific examples of each
Page 4: Addition of integers
 Teach how to add integers in mathematical expressions (without chips or a number
line) by providing specific examples
 Write 4 word problems involving adding integers and relating to your theme. Do
not solve. Your problems must include a mixture of negative and positive numbers
and must make logical sense.
(20 points)
SUBTRACTION – Pages 5 and 6 – Make sure to include a variety of samples which show all the
different possibilities for subtraction problems
Page 5: Subtraction of integers
 Teach how to subtract integers using both:
o Yellow and red chips (make sure to include zero pair problems)
o Number lines
 Explain in words what is happening
 Provide specific examples of each
Page 6: Subtraction of integers
 Teach how to subtract integers in mathematical expressions (without chips or a
number line) by providing specific examples
 Write 4 word problems involving subtracting integers and relating to your theme.
Do not solve. Your problems must include a mixture of negative and positive
numbers and must make logical sense.
(30 points)
MULTIPLICATION AND DIVISION – Pages 7 and 8
Page 7: Rules for multiplying and dividing integers
 Create your own graphic to demonstrate “The Official Kissing Rules”
 Your graphic should relate to your theme in some way
 Teach (explain) how the rules work and how they apply to problems
 Provide specific examples with numbers
Page 8: Multiplying and dividing integers
 Write 5 problems which involve a mixture of multiplication and division of
integers. You do not need any word problems.
 Write 5 problems which involve integers and order of operations. You must
include at least one multiply or divide in each. Also include other operations
(addition, subtraction), parenthesis, exponents, square roots, etc.
(20 points)
The following, in order, will play a major part in your overall grade:
1) Each topic above is completed with mathematical accuracy
2) Each topic is well explained (i.e. pretend you are teaching someone who has never seen a negative
number before)
3) A wide variety of examples are given (combinations of positive and negative numbers)
4) Your overall use of a theme
5) Neatness, Colorful, Easy-to-follow
Want another example instead of the Kissing Rules? How about this one:
Good things happen to good people, this is good
Good things happen to bad people, this is bad
Bad things happen to good people, this is bad
Bad things happen to bad people, this is good
SURVIVAL GUIDE TO INTEGERS GRADING RUBRIC
NAME: _______________________________________
Cover and
Theme
Intro to
Integers
Addition
Subtraction
Multiplication/
Division
TOTAL
Contains Theme
Possible Points
5
Says Survival Guide to Integers
3
Neat and interesting
2
What are integers (definition/examples)
6
Where used in real-life
6
Opposite definition/examples
4
Absolute value definition/examples
4
Add with chips (zero pairs)
3
Add on number line
3
Add mathematically
3
Written explanation
4
Wide variety of examples
4
Four word problems with +/- integers
3
Subtract with chips (zero pairs)
4
Subtract on number line
4
Subtract mathematically
6
Written explanation
6
Wide variety of examples
6
Four word problems with +/- integers
4
Kissing Rule table with theme
4
Apply rules in examples
3
Written explanation
4
5 problems
3
5 order of operation problems
3
10 correct answers listed
3
100
Your score
Activity 3-46: How Do I Learn
Name:
In the space provided write an “A” if you agree or a “D” if you disagree.
1.
I prefer reading a story rather than listening to someone tell it.
2.
I would rather watch television than listen to the radio/IPod.
3.
I remember faces better than names.
4.
I like classrooms with lots of posters and pictures around the room.
5.
The appearance of my handwriting is important to me.
6.
I think more often in pictures.
7.
I am distracted by visual disorder or movement.
8.
I have difficulty remembering directions that were told to me.
9.
I would rather watch athletic events than participate in them.
10. I tend to organize my thoughts by writing them down.
11. My facial expression is a good indicator of my emotions.
12. I tend to remember names better than faces.
13. I would enjoy taking part in dramatic events like plays.
14. I tend to sub vocalize and think in sounds.
15. I am easily distracted by sounds.
16. I easily forget what I read unless I talk about it.
17. I would rather listen to the radio/IPod than watch TV.
18. My handwriting is not very good.
19. When faced with a problem, I tend to talk it through.
20. I express my emotions verbally.
21. I would rather be in a group discussion than read about a topic.
22. I prefer talking on the phone rather than writing a letter/email to someone.
23. I would rather participate in athletic events than watch them.
24. I prefer going to museums when I can touch exhibits.
25. My handwriting gets worse when the space becomes smaller.
26. My mental pictures are usually accompanied by movement.
27.
I like being outdoors and doing things like biking, camping, swimming,
hiking, etc.
28. I remember best what was done rather than what was seen or talked about.
29.
When faced with a problem, I often select the solution involving the greatest
activity.
30. I like to make models or other hand crafted items.
31. I would rather do experiments than read about them.
32. My body language is a good indicator of my emotions.
33.
I have difficulty remembering verbal directions if I have not done the
activity before.
SCORING:
Total number of A responses in questions 1-11
Total number of A responses in questions 12-22
Total number of A responses in questions 23-33
The first number is your visual score. If this number is much higher than your other two you are a
visual learner: These learners need to see the teacher's body language and facial expression to fully
understand the content of a lesson. They tend to prefer sitting at the front of the classroom to avoid
visual obstructions (e.g. people's heads). They may think in pictures and learn best from visual
displays including: diagrams, illustrated text books, overhead transparencies, videos, flipcharts and
hand-outs. During a lecture or classroom discussion, visual learners often prefer to take detailed notes
to absorb the information.
Visual Learner Characteristics
Visual learners are those who learn through seeing things. Look over the characteristics below to see if
they sound familiar. A visual learner:
Is good at spelling but forgets names.
Needs quiet study time.
Has to think awhile before understanding lecture.
Is good at spelling.
Likes colors & fashion.
Dreams in color.
Understands/likes charts.
Is good with sign language.
Learning Suggestions for Visual Learners
Draw a map of events in history or draw scientific process.
Make outlines of everything!
Copy what’s on the board.
Ask the teacher to diagram.
Diagram sentences!
Take notes, make lists.
Watch videos.
Color code words, research notes.
Outline reading.
Use flashcards.
Use highlighters, circle words, underline.
Best Test Type for Visual Learners:
Diagramming, reading maps, essays (if you’ve studied using an outline), showing a process
Worst test type:
Listen and respond tests
The second number is your auditory score. If this number is much higher than your other two you are
an auditory learner: They learn best through verbal lectures, discussions, talking things through and
listening to what others have to say. Auditory learners interpret the underlying meanings of speech
through listening to tone of voice, pitch, speed and other nuances. Written information may have little
meaning until it is heard. These learners often benefit from reading text aloud and using a tape
recorder.
Auditory Learner Characteristics
Auditory learners are those who learn best through hearing things. Look over these traits to see if they
sound familiar to you. You may be an auditory learner if you are someone who:
Likes to read to self out loud.
Is not afraid to speak in class.
Likes oral reports
Is good at explaining.
Remembers names.
Notices sound effects in movies.
Enjoys music.
Is good at grammar and foreign language.
Reads slowly.
Follows spoken directions well.
Can’t keep quiet for long periods.
Enjoys acting, being on stage.
Is good in study groups.
Auditory Learners Can Benefit from:
Using word association to remember facts and lines.
Recording lectures.
Watching videos.
Repeating facts with eyes closed.
Participating in group discussions.
Using audiotapes for language practice.
Taping notes after writing them.
Worst test type:
Reading passages and writing answers about them in a timed test.
Best test type:
Auditory Learners are good at writing responses to lectures they’ve heard. They’re also good at oral
exams.
The third number is your tactile/kinesthetic score. If this number is much higher than your other two
you are a tactile/kinesthetic learner: Tactile/Kinesthetic persons learn best through a hands-on
approach, actively exploring the physical world around them. They may find it hard to sit still for long
periods and may become distracted by their need for activity and exploration.
Kinesthetic Learner Characteristics
Kinesthetic learners are those who learn through experiencing/doing things. Look over these traits to
see if they sound familiar to you. You may be a kinesthetic learner if you are someone who:
Is good at sports.
Can’t sit still for long.
Is not great at spelling.
Does not have great handwriting.
Likes science lab.
Studies with loud music on.
Likes adventure books, movies.
Likes role playing.
Takes breaks when studying.
Builds models.
Is involved in martial arts, dance.
Is fidgety during lectures.
Kinesthetic Learners Can Benefit from:
Studying in short blocks.
Taking lab classes.
Role playing.
Taking field trips, visiting museums.
Studying with others.
Using memory games.
Using flash cards to memorize.
Worst Test Type:
Long tests, essays.
Best Test Type:
Short definitions, fill-ins, multiple choice.
Activity 3-47: Integer Overview
Name:
Integer – all whole numbers and their opposites (or positive and negative counting numbers and zero)
Absolute value – the distance a number is from zero. The absolute value of -8 is 8. The absolute
value of 11 is 11.  8  8
11  11
Opposite of a number – To find the opposite, simply change the sign. A number and its opposite add
up to zero. The opposite of 5 is -5. The opposite of -12 is 12.
two positives
two negatives
one positive and one
negative
Adding Integers
Add the numbers like usual
Adding a positive and a positive will create more positives.
Add the numbers, put a negative sign in front of the answer
Adding a negative and a negative will create more negatives.
When adding a positive and a negative, some positives and
negatives will combine and cancel each other out.
Ask: Do I have more positives or more negatives? The answer will
determine the sign of the final answer. Then ask, how many more
positives do I have than negatives (or how many more negatives do
I have than positives?). This will determine the correct number to
go with the sign.
Subtracting Integers
Subtracting is the opposite of addition. Thus, the easy way to
subtract a number is to simply add its opposite.
all
Ex. 6 - -3 = 6 + +3 = 9
Ex. -12 – 7 = -12 + -7 = -19
Multiplying/Dividing Integers
When multiplying and dividing, determine the number as with normal multiplication and
division. Use the table below, The Kissing Rules, to determine the sign.
This
x this = this
Phrase to remember….
A girl sees a boy see likes (+). She does kiss him (+).
+
+
+
She is happy (+).
A girl sees a boy she likes (+). She does not kiss him (-).
+
She is sad (-).
A boy sees a girl he doesn’t like (-). He does kiss her (+).
+
He is sad (-).
A boy sees a girl he doesn’t like (-). He does not kiss her (-).
+
He is happy (+).
Activity 3-S: Chapter 3 Summary
Name:
ADDING INTEGERS
When adding two positive integers, add the numbers together. Your answer is always going to be
positive. Ex. 5 + 6 = 11
When adding two negative integers, add the two numbers together and place a negative sign in front of
your answer. Ex. -6 + -3 = -9
When adding a positive and a negative integer, first ask yourself, “Are there more positives or
negatives?” If there are more positives, you answer is going to be positive. If there are more negatives,
then you answer is going to be negative. Then ask yourself, “How many more (negatives/positives)
are there than (positives/negatives)?”
Ex. -8 + 6 Are there more negatives or positives? Negatives. How many more? 8 – 6 = 2. So there
are 2 more negatives than positives and your final answer is -2.
SUBTRACTING INTEGERS
The easiest way to subtract integers is to always turn the expression into an addition problem.
To subtract an integer add its opposite. An integer and its opposite are the same distance from 0 on a
number line so the integers 5 and -5 are opposites.
Then follow the rules above for adding.
Ex. -7 - -8 Change the problem to -7 + +8. There are more positives than negatives. How many more?
One. So your answer is +1.
MULTIPLYING/DIVIDING INTEGERS
When multiplying or dividing integers, multiply and divide just like you would for positive numbers
to get the number part of the answer. Then use the table below to determine the correct sign.
First Sign
Second Sign
Answer Sign
+
+
+
+
When multiplying two positives or two negatives,
the answer is positive.
First Sign
Second Sign
Answer Sign
+
+
When multiplying one negative and one positive,
the answer is negative.
Activity 3-POW (Guess and Check): Farmer Ann
Name:
Farmer Ann was chosen to go into town and spend the entire amount of $100 of a tax return on exactly
100 farm animals – some cattle, some sheep, and some horses. She discovered that sheep cost $0.50
each, cattle cost $1.00 each, and horses cost $10.00 each. This purchase, to buy exactly 100 animals
for $100 and have some of each kind of animal, proved more difficult that she had thought it would be.
Your job is to find a solution to Farmer Ann’s dilemma.
A good method to solve this type of problem is to use guess and check. There are several possible
ways and the one we are going to use is to guess some total of animals that add up to 100. From those
three numbers determine what your overall price is. Based on whether it was too high or too low, you
can now switch out some animals and make another guess.
Make at least 10 guesses to determine a correct answer to this problem. For each guess you must have
a total of 100 animals. There is more than one right answer, so keep guessing even if you guess
correctly early on.
Cattle ($1)
Guess 1
Guess 2
Guess 3
Guess 4
Guess 5
Guess 6
Guess 7
Guess 8
Guess 9
Guess 10
Guess 11
Guess 12
Guess 13
Guess 14
Guess 15
Sheep ($0.50)
Horses ($10)
Total $$$
10 all-time hottest
temperatures
Obtained from state temperature
records, NCDC
1. Death Valley, California
134
2. Lake Havasu, Arizona
128
3. Laughlin, Nevada
125
4. Lakewood, New Mexico
122
5. Alton, Kansas
121
6. Steele, North Dakota
121
7. Ozark, Arkansas
120
8. Tipton, Oklahoma
120
9. Seymour, Texas
120
10. Usta, South Dakota
120
1. Prospect Creek, Alaska
-80
2. Rogers Pass, Montana
-70
3. Peters Sink, Utah
-69
4. Riverside, Wyoming
-66
5. Maybell, Colorado
-61
6. Tower, Minnesota
-60
7. Parshall, North Dakota
-60
8. Island Park Dam, Idaho
-60
9. McIntosh, South Dakota
-58
10. Couderay, Wisconsin
-55
PRESS CONTROL & CLICK ON THE LINK
http://www.homeschoolmath.net/online/integers.php
http://www.gamequarium.com/integers.html
http://www.kn.pacbell.com/wired/fil/pages/listintegersbr.html
http://www.interactivemaths.net/taxonomy/term/99
http://teacherweb.com/MA/KingPhilipMiddleSchool/MrsMcGovern-7thGradeMathematics/ap3.st
m
http://www.squidoo.com/integers
http://www.quia.com/cb/64603.html
http://classroom.jc-schools.net/basic/math-integ.html
http://cte.jhu.edu/techacademy/web/2000/heal/siteslist.htm
(Go down to Algebra and Pre-Algebra sites and many integer links are listed.)
http://www.learningwave.com/chapters/workingfront/integersfront.html
http://www.shodor.org/interactivate/activities/ArithmeticFour/
http://www.onlinemathlearning.com/integer-games.html
http://www.mathgoodies.com/lessons/vol5/subtraction.html
http://www.learningshortcuts.com/s62/60/chapterA1.html
http://www.mathguide.com/cgi-bin/quizmasters/IntegersSubtract.cgi