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Lesson 2-1: Introduction to Integers How are integers used in real life situations? The summer of 1999 was unusually dry in parts of the United States. In the graph, a value of -8 represents 8 inches below the normal rainfall.
a. What does a value of -7 represent?
b. Which city was farthest from its normal rainfall?
c. How could you represent 5 inches above normal rainfall?
Key Terms 1. Integers – The set of whole ______________ and ______________ numbers including zero.
Ex. An integer can be any one of the following :... ‐4, ‐3, ‐2, ‐1, 0, +1, +2, +3, +4, ...
1. Positive Number: ________________________________________________
2. Negative Number: ____________________________________________
Objectives: Understand the meaning of negative numbers. Compare and/or order any real numbers. (A1.1.1.1.1)
Zero is neither positive or negative
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Real World: Integers can be used to represent real world data When numbers are used to represent physical quantities (altitudes, temperatures, and amounts of money are examples), it may be necessary to distinguish between positive and negative quantities. The symbols + and -- are used for this purpose. The altitude of Mount Whitney is 14,495 ft above sea level (+14,495 ft).
Quick Check! 1. Write an integer to describe the situation: a) 70 below zero ________ b) Owing $50 _________ c) 150m above sea level ________
d) Going up 4 floors on an elevator _______ e) Move back 4 spaces ___________
f) Five shots under par ___________________
2. Write a number to represent the temperature shown by the thermometer.
5°C
The thermometer shows _____ degrees Celsius below zero, or _____. 0
-5°C
Negative (-) Positive (+)
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Plotting Integers on a Number Line 3. Plot: ___________________________________________________________ On a number line, negative numbers extend to the left of zero, and positive numbers extend to the right of zero. The numbers below are examples of integers. Integers are numbers that can be written without fractions or decimals.
Ex. Put the integers on the number line: 3, -4, 2, 5, -2, 1, -5 4. Opposite Numbers – numbers that are the ___________________ from zero
in the________________ direction.
Ex: +7 and -7 are opposites
Quick Check!
1. Graph each set of numbers on a number line. a. -4, -6, 5 b. 3, -3, -2
2. Write the Opposite of each Integer
a) + 34 = _______ b) - 12 = ________ c) -2 = _______
d) +64 = _______ e) -843 = _______
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5 -1
-3 -5
-5 2
-4 -2
-2 2
0 -2
-1 3
-3 3
5. Comparing Integers: Numbers are arranged from least to greatest.
Ex. Put the correct sign between the pairs of integers. *If you are having trouble, think of the numbers in terms of money. Positive numbers represent the money you have. Negative numbers represent money you owe. Write an inequality using the numbers in each sentence. Use the symbols < or >.
a. -4° is colder than 2°. b. -6 is greater than -10
Would you rather… Have Owe
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Ex 2. The final round scores of the top ten finishers in the 2000 World Championship LPGA tournament were -4, -14, -1, +1, +2, +5, 0, +3, -10, and -2. Order the scores from least to greatest. Then graph each integer on a number line.
Quick Check! 1. Put the integers in order from least to greatest.
A. -4, +6, -2, +1, 0 _____________
B. -2, -4, -1, -13 _____________
C. -6, +6, -3, +3, 0 _____________
D. -22, -36, -1, 0 _____________ 2. The table shows the record low temperatures in °F for selected states. Order the temperatures from least to greatest.
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6. Absolute Value – The distance from _______ to a number. Not _______ or _______.
Absolute value is written with 2 bars around the number. ex. l3l, l-4l
Ex 1:
Ex 2:
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Lesson 2-2: Adding Integers Ways to Add Integers:
1. Picture: Using lines
2. Number Line When the number is positive, count to the right. When the number is negative, count to the left.
Integer Addition Lab 1
You can use a thermometer to model addition of integers.
1. Suppose the temperature starts at 10°F and increases 30° during the day. Complete the addition statement to show the new temperature.
10° + 30° = ____________
2. Suppose the temperature starts at -10°F and drops 30° overnight. Complete the addition statement to show the new temperature.
-10° + (-30°) = _________ What do you notice about the answers and where you moved on the thermometer?
ZERO PAIRS: one positive cancels out 1 negative.
Means +2 (positive 2)
Means ‐3 (negative 3)
Objectives: Understand addition of positive and negative numbers using models, representation and abstract methods. (A1.1.1.2)
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Adding Integers with the Same Sign
Question: Were your predictions about adding numbers with the same sign on the previous page correct?
A) 2 Positive Numbers
1. Draw a picture
B) 2 Negative Numbers
5 + 2 = ___
‐3 + ‐2 = ___
2. Use a number line
See: Both positive Think: Same direction Do: Combine
See: Both positive Think: Same Direction Do: Go right
1. Draw a picture
2. Use a number line See: Both negative Think: Same Direction Do: Go left
See: Both negative Think: Same Direction Do: Combine
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Integer Addition Rules
1. ____________________ Sign – If the signs are the same, ______________the numbers and put the sign of the numbers in front of your answer. Ex. Both Positive: 9 + 5 = ______ Both Negative: -9 + -5 = ______
Quick Check!
i. -3 + -5 = ___________ ii. 4 + 7 = __________
iii. 3 + 4 = _________ iv. -6 + -7 = _________
v. 5 + 9 = __________ vi. -7 + -4 = ___________
In football, forward progress is represented by a positive integer. Being pushed back is represented by a negative integer. Suppose on the first play a team loses 5 yards and on the second play they lose 2 yards.
a. What integer represents the total yardage on the two plays?
b. Write an addition sentence that describes this situation.
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Integer Addition Lab 2
You can use a thermometer to model addition of integers.
1. Suppose the temperature starts at -50°F and increases 40° during the day. Complete the addition statement to show the new temperature.
-50° + 40° = ____________
2. Suppose the temperature starts at 40°F and drops 70° overnight. Complete the addition statement to show the new temperature.
40° + (-70°) = _________ What do you notice about the answers and where you moved on the thermometer?
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Circle the larger number.
Adding Integers with Different Signs Ex. 1
1. Draw a picture.
2. Now use a number line to find answer
Ex. 2
1. Draw a picture.
2. Now use a number line to find answer Question: Were your predictions about adding numbers with the same sign on the previous page correct?
- 2 + 5 =
6 + (-4) =
See: Different signs Think: Opposite Direction Do: Remove zero pairs
See: Different signs Think: Opposite Direction Do: Start at first number, move direction of second number
See: Different signs Think: Opposite Direction Do: Remove zero pairs
See: Different signs Think: Opposite Direction Do: Start at first number, move direction of second number
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2. ______________________Signs – If the signs are different, ____________________
the _____________________ from the larger number and put the ____________________ of the _________ number in front of your answer.
Example: -9 + 5 = ________ ___________
Quick Check! Add Using a Number Line.
1. +3 + -5 =
State whether each sum is positive or negative. Explain your reasoning. a. -4 + (-5) b. 12 + (-2) c. -11 + 9 d. 15 + 10
Add.
a) 3 + -5 = ________ ___________
b) 3 + (-4) = ________ ___________
c) -4 + 7 = = ________ ___________
d) -6 + 7 = = ________ ___________
e) 5 + -9 = = ________ ___________
f) 15 + -9 = = ________ ___________
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More Integer Addition Rules 3. The sum of ____ and any integer is the ______________________.
Ex. 14 + 0 = _________
-14 + 0 = _________ 4. Two numbers with the same absolute value but different signs are called opposites. An integer and its opposite are also called ________________________________.
The sum of any number and its additive inverse is ____. 6 + (-6) = 0
The sum of two ________________ integers is ____.
9 + -9 = ________ Quick Check!
1. a. -3 + 0= ___________ b. 4 + 0 = __________
c. -3 + 3 = _________ d. 7 + -7 = _________
2. What is the additive inverse of the following:
a. 2? b. -6? c. 14? D. -20?
3. Give an example of two integers that are additive inverses.
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Ex. 1 9 + (- 3) + (- 9) Ex. 2 -4 + 6 + (-3) + 9 Quick Check!
1. A score of 0 is called even par. Two under par is written as -2. Two over par is written as +2. Suppose a player shot 4 under par, 2 over par, even par, and 3 under par in four rounds of a tournament. What was the player’s final score?
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Lesson 1-7: Subtracting Integers Ways to Subtract Integers:
1. Picture: Using lines
2. Number line
Change the problem to addition sentence. When the number is positive, count to the right. When the number is negative, count to the left.
Subtracting Integers, it is as easy as adding.... in fact you have to change it into an addition problem.
Intro: Hailey owed $200 on her credit card. She made another purchase of $20 how much does she now owe?
=
Same: leave the first number alone.
Change the minus sign to an ______________ sign.
Opposite; Change the 2nd number to its ______________________.
a) If negative, change it to a _____________. b) If the number is ___________, change it to a negative.
So Hailey now owes _____________.
Objectives: Understand subtraction of positive and negative numbers using models, representation and abstract methods. (A1.1.1.2)
When you see 2 Negatives in the middle, cover with addition symbol
Means +2 (positive 2)
Means ‐3 (negative 3)
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Subtracting Integers:
Example 1.
Example 2.
5 ‐ 12 = ___
5 ‐ ‐1 = ___
1. Draw a picture
2. Use a number line
See: Subtracting a positive Think: equals adding a negative Do: Different Signs‐ cross out zero pairs.
See: Subtracting a positive Think: equals adding a negative Do: Different Signs‐ Start at first number, move direction of second number
1. Draw a picture
2. Use a number line
See: Subtracting a negative Think: equals adding a positive Do: Same Signs‐ combine
See: Subtracting a negative Think: equals adding a positive Do: Both positive‐ Go right
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Example 4. ‐4 ‐ ‐2 = ___
1. Draw a picture
2. Use a number line
See: Subtracting a positive Think: equals adding a negative Do: Same Signs‐ combine
See: Subtracting a positive Think: equals adding a negative Do: Both negative‐ Go left
1. Draw a picture
3. Use a number line
See: Subtracting a negative Think: equals adding a positive Do: Different Signs‐ cross out zero pairs
See: Subtracting a negative Think: equals adding a positive Do: Different Signs‐ Start at first number, move direction of second number
Example 3.
‐6 ‐ 1 = ___
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Remember: SAME, CHANGE, OPPOSITE How are addition and subtraction of integers related? Ex. 1 You can use a number line to subtract integers. The model below shows how to find 6 - 8.
Step 1: Start at 0. Move 6 units right to show positive 6. Step 2: From there, move 8 units left to subtract positive 8.
a. What is 6 - 8?
b. What direction do you move to c. What addition sentence is also
indicate subtracting a positive integer? modeled by the number line above?
Ex 2. When you subtract -3 - 5, the result is the same as adding -3 + (-5).
Subtracting Adding Its an Integer Additive Inverse
Show Changes:
Same Change Opposite Answer
1. -12 -- 50
2. 10 -- -90
3. 25 -- 34
4. -100 -- 37
2‐2=___
2‐1=___
2‐0=___
2‐(‐1)=___
2+____=____
2+____=____
2+____=____
2+____=____
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Quick Check!
1. Use models and pictures to find each difference. a. 9 - 4 b. -4 - (-3) c. -7 - 4
2. Use a number line to find each difference.
a. -7 - (-2) b. -6 - (-1) c. -8 - (-5)
3. Write the additive inverse for each subtraction problem. a. -9 - 3 b. -6 - (-4) c. 5 - (-4) 4. Find the difference. Show changes with a colored pencil.
a. -2 - (-8) b. 7 - (-10) c. 8 - 11 5. Evaluate each expression if x = 10, y = -4, and z = -15. Show your substitution.
a. x - (-10) b. y – x c. x + y - z
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Lesson 1-8: Multiply and Divide Integers
Activity Lab
Look at patterns of integer multiplication:
1. 6 X -18 = 2. (-2 X 5) X 3 =
3. (4 X -6) X 7 = 4. (8 X 9) (2 X -1) = 5. Each problem above has ___________negative factor(s).
6. Each product is .
7. -17 X -3 = 8. (-6 X 5) X -8 =
9. (9 X -2) (-3 X 1) = 10. (-4 X -7) (2 X 2) = 11. Each problem above has _____________ negative factor(s).
12. Each product is .
13. (-5 X -4) X -7= 14. -3 X (-8 X -2) =
15. (-3 X -4) (-2 X 8) = 16. (-10 X 1) (-7 X -1) = 17. Each problem above has __________ negative factor(s).
18. Each product is .
19. (-3 X -5) (-2 X -2)= 20. (-9 X -1) (-4 X -3) =
21. (-6 X -1) (-4 X -11) = 22. (-20 X -5) (-3 X -6) =
23. Each problem above has __________ negative factor(s).
24. Each product is .
25. Draw a conclusion about the product of positive and negative integers.
Objectives: Understand multiplication of positive and negative numbers using patterns and abstract methods. (A1.1.1.2)
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Real Life Connections: How are the signs of factors and products related: The temperature drops 7°C for each 1 kilometer increase in altitude. A drop of 7°C is represented by -7. So, the temperature change equals the altitude times -7. The table shows the change in temperature for several altitudes.
a. Suppose the altitude is 4 kilometers. Write an expression to find the temperature change.
b. Use the pattern in the table to find 4(-7). Multiplying 1 Positive and 1 Negative Number:
Multiplication is repeated addition. So, 3(-7) means that -7 is used as an addend 3 times.
3(-7) = (-7) + (-7) + (-7) = - 21
Multiplying 2 Negative Numbers:
The product of two positive integers is positive. What is the sign of the product of two negative integers? Use a pattern to find (-4)(-2).
(-4)(2) = -8 (-4)(1) = -4 (-4)(0) = 0 (-4)(-1) = 4
(-4)(-2) = (-4)(-3) =
Each product is 4 more than the previous product
+4
+4
+4
+4
One positive and one negative factor = _______ product.
2 negative factors = _____________ product.
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Rules: Same Signs: ______________ ex. 4 · 2 = 8 -4 · -2 = 8 Different Signs: ____________ ex. -4 · 2 = -8 4 · -2 = -8 Quick Check!
1. State whether each product is positive or negative:
a. 6 · 2 b. -7 · -2 c. -4 · 12 d. 4 · -6
2. Give an example of three integers whose product is negative.
3. Find the product.
a. 6 · 2 = b. -7 · -2 = c. -4 · 12 =
d. 4 · -6 = e. -10 · 5 = f. -8 · -2 =
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Dividing Integers Follow the same rules as multiplying integers: Same Signs: ______________ ex. 8 ÷ 2 = 4 -8 ÷ -4 = 2 Different Signs: ____________ ex. -8 ÷ 2 = -4 8 ÷ -4 = -2
Ex. Find the Quotient: a. 18 ÷ 2 = b. -20 ÷ -2 = c. -48 ÷ 12 =
AVERAGE (MEAN): Division is used in statistics to find the average, or mean of a set of data. To find the mean of a set of numbers, find the sum of the numbers and then divide by the number in the set.
Ex. Rachel had test scores of 84, 90, 89, and 93. Find the average (mean) of her test scores.
On your Own!
1. Find the average (mean) of -2, 8, 5, -9, -12, and -2.
Find the sum of the test scores.
Divide by the number of scores.
Objectives: Understand division of positive and negative numbers using patterns and abstract methods. (A1.1.1.2)
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Quick Check!
1. State whether each quotient is positive or negative:
b. 16 ÷ 2 b. -70 ÷ -2 c. -48 ÷ 12 d. 24 ÷ -6
2. Find the Quotient:
a. 16 ÷ 2 = b. -70 ÷ -2 = c. -48 ÷ 12 =
d. 24 ÷ -6 = e. -10 ÷ 5 = f. -8 ÷ -2 =
3. Explain how to find the average of a set of numbers. 4. In their first five games, the Jefferson Middle School basketball team scored
46, 52, 49, 53, and 45 points. What was their average number of points per game?
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Lesson 1-9: Applications with Integers Ex. 1: Find the profit or loss for each month.
b. October
+ ( ) = There was a of for October.
c. November
+ ( ) =
There was a of for November.
d. December
+ ( ) = There was a of for December.
Objectives: To use integers in real life situations.
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Ex. 2 Ballooning: Hot air balloons generally descend at a rate of 200 to 400 feet per minute. A balloon descends 235 feet per minute for 4 min. Write an integer to express the balloon’s total movement.
Understand 1. What are you being asked to do?
2. Which word describes the direction of the balloon?
3. Will the integer be positive or negative? Plan and Carry Out
4. Each minute the balloon descends how many feet?
5. How many minutes is the balloon descending?
6. Write an expression to tell the balloon’s movement.
7. The balloon moves _________________________________
Ex 3: Sarah’s savings account had $125 in it before she deposited her $255 paycheck. She then wrote the following checks: $20 for a parking ticket, $35 for her electric bill, $111 for her phone bill, $65 for her cable bill, and $89 for her new cell phone. Does Sarah have enough money left to buy a $50 DVD player? Explain.
Income and Expenses for Flower Mania
Month Income Expenses
Jan. $11,917 –$14,803
Feb. $12,739 –$9,482
Mar. $11,775 –$10,954
Apr. $13,620 –$15,149
Quick Check:
1. Find the profit or loss for Flower Mania for January and for March.
January March
2. Hiking: You are at the highest point of Lost Mine Trail. The elevation is 6,850 feet. You hike down the trail to an elevation of 5,600 feet. What is your change in elevation?
Understand a. What are you being asked to do? ______________ _______________________________________________________________________
Lesson 1-10: Coordinate Plane Key Terms:
1. Coordinate Plane: _____________________________________________________
_____________________________________________________
2. __________________: horizontal axis
3. __________________: vertical axis
4. __________________: place where the axes intersect; (0,0)
5. __________________: quadrants
6. __________________: ordered pair
Objectives: Graph points on a 4 quadrant coordinate plane.
Coordinate Plane: On a coordinate plane there are 2 axes:
1. ____________________________ - horizontal axis
2. ____________________________ - vertical axis
The axes intersect at the ______________________________________.
There are _______ quadrants on the coordinate plane.
An _______________________ identifies the _____________________ of a point.
These numbers are the ___________________________ of the point.
( 1 , 2)
Quick Check:
Ordered Pair x-coordinate y-coordinate
( -3, -5)
(-2 , 4)
(6, -5)
___ ‐ coordinate ___ ‐ coordinate
Axes:
Review:
o A graph has both _________________ and ________________ numbers.
o There are _________ quadrants on a coordinate plane.
o The ____ - axis is the vertical axis.
The ____ - axis is the horizontal axis.
X-axis:
o Numbers to the right of the y-axis
are _______________.
Numbers to the left of the y-axis
are ________________.
Y-axis:
o Numbers above of the x-axis are
_______________.
Numbers below of the x-axis are
________________.
The origin (where the 2 axis intersect) is _____________.
Quadrants:
Rules for Quadrants: o The x-and y- coordinate will always be positive in Quadrant _____.
o The x- coordinate will be _________ and y- coordinate will be
______________ in Quadrant 2.
o The x-and y- coordinate will always be negative in Quadrant _____.
o The x- coordinate will be _________ and y- coordinate will be
______________ in
Quadrant 4.
Quadrant X Y
1
2
3
4
Quick Check: Name the quadrant the ordered pairs are in:
1. ( -3 , 2 ) _______________ 2. ( 2 , -3 ) _______________
3. ( 3 , 3 ) _______________ 4. ( -4 , -4 ) _______________
(____ , ____) (____ , ____)
(____ , ____) (____ , ____)
Ordered Pairs:
A point on a graph has an ____ - coordinate and a _____- coordinate. The first number in an ordered pair is the ____ - coordinate.
o To find the x-coordinate move ____________ or _______________ from the origin ( 0 , 0 ).
The second number in an ordered pair is the ____ - coordinate. o To find the y-coordinate move ________ or ____________ from the x-
coordinate number. When the point is found on the graph, make a dot to symbolize the
ordered pair.
