chapter 1 integers
Copy and complete the statement using
CHAPTER 1 INTEGERS
< or >. 1. 12 ? 14
2. 36 ? 26
3. −2 ? −5
4. −15 ? −8
5. 13 ? −10
6. −20 ? 19
Natural Numbers
1.1
The first numbers developed were the Natural Numbers, also called the Counting Numbers.
What is a natural number?
1, 2, 3, 4, 5, ...
What is a whole number? What is an integer? How can you use integers to represent the velocity and the speed of an object?
The three dots, (...), means that these numbers continue forever: there is no largest counting number..
Think of counting objects as you put them in a container those are the counting numbers.
Whole Numbers Adding zero to the Counting Numbers gives us the Whole Numbers. 0, 1, 2, 3, 4, ...
Integers Adding the negative numbers to the whole numbers yields the Integers. ...-3, -2, -1, 0, 1, 2, 3, ...
Counting numbers were developed more than 35,000 years ago.
In this case, "..." at the left and right, means that the sequence continues in both directions forever.
It took 34,000 more years to invent zero.
There is no largest integer...nor is there a smallest integer.
This the oldest known use of zero (the dot), about 1500 years ago. It was found in Cambodia and the dot is for the zero in the year 605.
http://www.smithsonianmag.com/history/originnumber-zero-180953392/?no-ist
1
Work with a partner. You are gliding to the ground wearing a parachute. The table shows your height above the ground at different times.
Integers on the Number Line
a. Describe the pattern in the table. How many feet do you move each
Negative Integers
Zero
-5 -4 -3 -2 -1
0
Numbers to the left of zero are less than zero
second? After how many seconds will you land on the ground?
Positive Integers
1
Zero is neither positive or negative
2
3
4
5
Numbers to the right of zero are greater than zero
b. What integer represents your speed? Give the units.
c. Do you think your velocity should be represented by a positive or negative integer? Explain your reasoning.
`
d. What integer represents your velocity? Give the units.
Work with a partner. You release a group of balloons. The table shows the height of the balloons above the ground at different times.
Work with a partner. The table shows the height of a firework’s parachute above the ground at different times
a. Describe the pattern in the table. How many feet do the balloons move each second? After how many seconds will the balloons be at a height of 40 feet?
b. What integer represents the speed of the balloons? Give the units.
c. Do you think the velocity of the balloons should be represented by a positive or negative integer? Explain your reasoning.
a. Describe the pattern in the table. How many feet does the parachute move each second?
b. What integer represents the speed of the parachute? What integer represents the velocity? How are these integers similar in their relation to 0 on a number line?
d. What integer represents the velocity of the balloons? Give the units.
• Examples are shown below.
Absolute Value of Integers The absolute value is the distance a number is from zero on the number line, regardless of direction. Distance and absolute value are always non-negative (positive or zero).
4
This is read, "the absolute value of 4".
Math Practice
• Communication: Give examples of velocities of two objects (A and B), where Velocity A > Velocity B but Speed A < Speed B.
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 What is the 4 ? Click to Reveal
4
2
Copy and complete the statement using , or =.
Find the absolute value. 1.
2.
3.
4.
5.
6.
7.
8.
The freezing point is the temperature at which a liquid becomes a solid.
5. ∣−2∣
−1
6. −7
∣6∣
7. ∣10∣
11
8. 9
∣−9∣
9. Is the freezing point of airplane fuel or candle wax closer to the freezing point of water? Explain your reasoning.
a. Which substance in the table has the lowest freezing point?
b. Is the freezing point of mercury or butter closer to the freezing point of water, 0°C?
Exit Ticket: The freezing point of vinegar is −2°C. Is the freezing point
Add.
of vinegar or honey closer to the freezing point of water? Explain your
1. 10 + 12
2. 14 + 28
3. 26 + 32 + 19
4. 47 + 35 + 68
5. 12 + 33 + 59 + 18
6. 82 + 13 + 29 + 97
reasoning.
3
1.2
Adding Integers Is the sum of two integers positive, negative,
First, let's add a positive and negative integer: 4 + (-5) = ?
or zero? How can you tell?
Before we start, we must understand that -5 means to move 5 spaces to the left of 0, as shown below. -5
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 Now, add the -5 to the end of the 4. -5 4 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 Therefore, 4 + (-5) = -1.
Adding Integers
Adding Integers Addition is commutative, so we get the same answer for (-5) + 4 = ?
Now let's add two negative integers: (-4) + (-5) = ? Like the last example, we start off going to the left 4 spaces, but then we continue to the left 5 spaces. -5
+4
-4
-5 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 Therefore, (-5) + 4 = -1.
Therefore, (-4) + (-5) = -9. Don't memorize "rules" to add positive & negative numbers.. Instead, sketch the number line to perform each addition problem. That way you know that you are correct every time.
Addition Rules: same sign = add different signs = sub.
4
Find −2 + (−4). Use a number line to check your answer.
Add. 1. 9 + 4
2. −5 + (−1)
3. −8 + 13
4. 12 + (−8)
5. −10 + 6
6. −14 + 14
Add. 1. 7 + 13
a. Find 5 + (−10). 2. −8 + (−5)
3. −20 + (−15)
b. Find −3 + 7.
c. Find −12 + 12.
The list shows four bank account
Add.
transactions in July. Find the change C
4. −2 + 11
5. 9 + (−10)
6. −31 + 31
in the account balance.
7. WHAT IF? In the Check problem, the deposit amounts are $30 and $40. Find the change C in the account balance.
5
• What do you know about the sum A + B? Explain your reasoning.
Subtract. 1. 45 − 11
2. 87 − 23
3. 91 − 14
4. 76 − 69
5. 87 − 29 − 13
6. 65 − 52 − 11
• Two integers have different signs. Their sum is −8. What are possible values for the two integers?
Subtracting Integers
1.3
8-2=6 How are adding integers and subtracting integers related? What are the rules for subtracting integers?
Since 8 is positive, we need to travel to the right 8 steps. Next, instead of moving to the right 2 spaces, as in addition, subtraction moves in the opposite direction, which means we move 2 spaces to the left. subtract 2 8 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 Our answer is 6.
Subtracting Integers How would we show 3 - 8?
Adding & Subtracting Integers How would we show (-4) - 6?
As in the last example, we would start off going right 3 and then left 8.
Remember that (-4) is shown by going 4 spaces to the left. Subtracting 6, take us 6 more spaces to the left.
subtract 8 3
subtract +6 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
-4
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 Therefore, 3 - 8 = -5
Therefore, (-4) - 6 = -10
6
Adding & Subtracting Integers The most potentially confusing case is subtracting a negative integer.
Subtract. 1. 5 − 8
2. 12 − (−3)
3. −5 − (−6)
4. −4 − 6
5. 9 − 16
6. −7 − 10
How would we show 1 - (-6)? First, we would go 1 space to the right, because of the 1. Adding -6 spaces would take us to the left, but we must do the opposite since we are subtracting, moving 6 steps to the right.
1
subtract -6
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 Therefore, 1 - (-6) = 7
Subtraction Rule: KEEP - FLIP - CHANGE Keep the first number the same, flip the subtraction to addition, change the second number to the opposite sign
a. Find 3 − 12.
Subtract. 1. 8 − 3
2. 9 − 17
4. −14 − 9
5. 9 − (−8)
3. −3 − 3
b. Find −8 − (−13).
6. −12 − (−12)
c. Find 5 − (−4).
7
Evaluate −7 − (−12) − 14.
Evaluate the expression. 7. −9 − 16 − 8
8. −4 − 20 − 9
9. 0 − 9 − (−5)
10. −8 − (−6) − 0
11. 15 − (−20) − 20
12. −14 − 9 − 36
13. The highest elevation in Mexico is 5700 meters, on Pico de Orizaba.The lowest elevation in Mexico is −10 meters, in Laguna Salada.Find the range of elevations in Mexico.
Which continent has the greater range of elevations?
Is Subtraction Commutative?
Add. 1. 9 + 9
Subtraction would be commutative if a - b = b - a
2. −7 + (−7)
Is that true? Test that with some numbers and discuss with you group.
4 - 5 = -1 3. −3 + (−3) + (−3)
4. 5 + 5 + 5
Answer
5-4=1 Since -1 ≠ 5. 1, 6then + 6 + 46 -+56 ≠ 5 - 4
6. −4 + (−4) + (−4) + (−4)
In general, a - b ≠ b - a Subtraction is not commutative.
8
Multiplication
1.4 negative, or zero? How can you tell?
Multiplication can be indicated by putting a dot between two numbers, or by putting the numbers into parentheses. (We won't generally use "x" to indicate multiplication since that letter is used a lot in algebra for variables.) So multiplying 3 times 2 will be written as: 3·2 or
Math Practice
Is the product of two integers positive,
MP.7: Look fo structure.
Ask: How is m to addition?
Note: answer shown on the
(3)(2)
Multiplication
Multiplication is Commutative
Multiplication is repeated addition.
Since addition is commutative...
So, to find the product of 3 · 2 we would add the number 2 to itself three times:
And multiplication is just repeated addition, multiplication is commutative:
3·2=2+2+2 3·2=2+2+2=6 +2
+2
+2
+2
+2
+2
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 +3
2·3=3+3=6
+3
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
Multiplication is Commutative a·b=b·a Adding a number "a" to itself "b" times yields the same result as adding a number "b" to itself "a" times. Adding "a" to itself "b" times yields "ab". +a +a +a ab Adding "b" to itself "a" times yields "ba" which is the same as "ab". +b
+b ab
Multiplying Negative Integers It works the same way when you multiply a negative number by a positive number. So, 3 · (-2) just indicates to add -2 to itself three times: 3 · (-2) = (-2) + (-2) + (-2) = -6
-2
-2
-2
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 Would (-3) · 2 give us the same answer of -6? Explain your answer.
9
Multiplying Negative Integers Since multiplication is commutative,
Multiplying Positive & Negative Integers
(-3) · 2 = 2 · (-3) = (-3) + (-3) = -6
When multiplying two numbers, if both are positive, the answer is positive.
So, this just becomes adding -3 to itself 2 times.
If one is negative and the other is positive, the answer is negative.
-3
-3 How about if both numbers being multiplied are negative?
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
Multiplying a positive and negative integer results in a negative integer: (a)(-b) = (-a)(b) = -ab
Multiplying Positive & Negative Integers How do we interpret (-3)(-2)?
• “Today we learned that a negative integer multiplied by a negative integer is positive. Be sure to mentally check all of your steps so that you are not confusing anything.”
Since, (3)(-2) means to add -2 to itself 3 times We could interpret (-3)(-2) to mean to subtract (-2) from itself three times. We already learned that addition and subtraction are inverses, so subtracting -2 is the same as adding 2. So, (-3)(-2) indicates to add 2 to itself 3 times. So, (-3)(-2) = (3)(2) = 6
Multiply. 1. 7 • 2
2. 9(−7)
3. −6(8)
4. −8(−10)
5. 6 • (−5)
6. −5 • (−12)
10
Multiply. a. 3(−4)
Multiply. b. −7 ⋅ 4
a. Evaluate (−2)2.
1. 5 ⋅ 5
2. 4(11)
3. −1(−9)
4. −7 ⋅ (−8)
5. 12 ⋅ (−2)
6. 4(−6)
7. −10(−6)(0)
8. −7 ⋅ (−5) ⋅ (−4)
Evaluate the expression. 9. (−3)2
10. (−2)3
11. −72
12. −63
b. Evaluate −52.
c. Evaluate (−4)3.
The bar graph shows the number of taxis a company has in service. The number of taxis decreases by the same amount each year for 4 years. Find the total change in the number of taxis.
13. A manatee population decreases by 15 manatees each year for 3 years. Find the total change in the manatee population.
11
Multiply.
• Write the number −4 on the board. Ask students to write each of the
1. −10 • 5
2. −5 • (−6)
3. 8 • (−9)
4. −15 • 7
5. −22 • (−8)
6. -32 • (−4)
following and then share their responses. • “Write a multiplication problem that has −4 as one of the factors, and has a negative product.”
• “Write a second multiplication problem that has −4 as one of the factors, and has a positive product.”
1.5 Is the quotient of two integers positive, negative, or zero? How can you tell?
Divide.
Divide.
1. 12 ÷ (−3)
2. −32 ÷ 4
3. −56 ÷ (−8)
4.
5.
6.
a. 75 ÷ (−25)
b.
12
Evaluate 10 − x2 ÷ y when x = 8 and y = −4.
Divide. 1. 14 ÷ 2
2. −32 ÷ (−4)
3. −40 ÷ (−8)
4. 0 ÷ (−6)
5.
6.
Evaluate the expression when a = −18 and b = −6.
You measure the height of the tide using the support beams of
7. a ÷ b
8.
9.
a pier. Your measurements are shown in the picture. What is the mean hourly change in the height?
10. The height of the tide at the Bay of Fundy in New Brunswick decreases 36 feet in 6 hours. What is the mean hourly change in the height?
13
CHAPTER 1 INTEGERS
< or >. 1. 12 ? 14
2. 36 ? 26
3. −2 ? −5
4. −15 ? −8
5. 13 ? −10
6. −20 ? 19
Natural Numbers
1.1
The first numbers developed were the Natural Numbers, also called the Counting Numbers.
What is a natural number?
1, 2, 3, 4, 5, ...
What is a whole number? What is an integer? How can you use integers to represent the velocity and the speed of an object?
The three dots, (...), means that these numbers continue forever: there is no largest counting number..
Think of counting objects as you put them in a container those are the counting numbers.
Whole Numbers Adding zero to the Counting Numbers gives us the Whole Numbers. 0, 1, 2, 3, 4, ...
Integers Adding the negative numbers to the whole numbers yields the Integers. ...-3, -2, -1, 0, 1, 2, 3, ...
Counting numbers were developed more than 35,000 years ago.
In this case, "..." at the left and right, means that the sequence continues in both directions forever.
It took 34,000 more years to invent zero.
There is no largest integer...nor is there a smallest integer.
This the oldest known use of zero (the dot), about 1500 years ago. It was found in Cambodia and the dot is for the zero in the year 605.
http://www.smithsonianmag.com/history/originnumber-zero-180953392/?no-ist
1
Work with a partner. You are gliding to the ground wearing a parachute. The table shows your height above the ground at different times.
Integers on the Number Line
a. Describe the pattern in the table. How many feet do you move each
Negative Integers
Zero
-5 -4 -3 -2 -1
0
Numbers to the left of zero are less than zero
second? After how many seconds will you land on the ground?
Positive Integers
1
Zero is neither positive or negative
2
3
4
5
Numbers to the right of zero are greater than zero
b. What integer represents your speed? Give the units.
c. Do you think your velocity should be represented by a positive or negative integer? Explain your reasoning.
`
d. What integer represents your velocity? Give the units.
Work with a partner. You release a group of balloons. The table shows the height of the balloons above the ground at different times.
Work with a partner. The table shows the height of a firework’s parachute above the ground at different times
a. Describe the pattern in the table. How many feet do the balloons move each second? After how many seconds will the balloons be at a height of 40 feet?
b. What integer represents the speed of the balloons? Give the units.
c. Do you think the velocity of the balloons should be represented by a positive or negative integer? Explain your reasoning.
a. Describe the pattern in the table. How many feet does the parachute move each second?
b. What integer represents the speed of the parachute? What integer represents the velocity? How are these integers similar in their relation to 0 on a number line?
d. What integer represents the velocity of the balloons? Give the units.
• Examples are shown below.
Absolute Value of Integers The absolute value is the distance a number is from zero on the number line, regardless of direction. Distance and absolute value are always non-negative (positive or zero).
4
This is read, "the absolute value of 4".
Math Practice
• Communication: Give examples of velocities of two objects (A and B), where Velocity A > Velocity B but Speed A < Speed B.
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 What is the 4 ? Click to Reveal
4
2
Copy and complete the statement using , or =.
Find the absolute value. 1.
2.
3.
4.
5.
6.
7.
8.
The freezing point is the temperature at which a liquid becomes a solid.
5. ∣−2∣
−1
6. −7
∣6∣
7. ∣10∣
11
8. 9
∣−9∣
9. Is the freezing point of airplane fuel or candle wax closer to the freezing point of water? Explain your reasoning.
a. Which substance in the table has the lowest freezing point?
b. Is the freezing point of mercury or butter closer to the freezing point of water, 0°C?
Exit Ticket: The freezing point of vinegar is −2°C. Is the freezing point
Add.
of vinegar or honey closer to the freezing point of water? Explain your
1. 10 + 12
2. 14 + 28
3. 26 + 32 + 19
4. 47 + 35 + 68
5. 12 + 33 + 59 + 18
6. 82 + 13 + 29 + 97
reasoning.
3
1.2
Adding Integers Is the sum of two integers positive, negative,
First, let's add a positive and negative integer: 4 + (-5) = ?
or zero? How can you tell?
Before we start, we must understand that -5 means to move 5 spaces to the left of 0, as shown below. -5
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 Now, add the -5 to the end of the 4. -5 4 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 Therefore, 4 + (-5) = -1.
Adding Integers
Adding Integers Addition is commutative, so we get the same answer for (-5) + 4 = ?
Now let's add two negative integers: (-4) + (-5) = ? Like the last example, we start off going to the left 4 spaces, but then we continue to the left 5 spaces. -5
+4
-4
-5 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 Therefore, (-5) + 4 = -1.
Therefore, (-4) + (-5) = -9. Don't memorize "rules" to add positive & negative numbers.. Instead, sketch the number line to perform each addition problem. That way you know that you are correct every time.
Addition Rules: same sign = add different signs = sub.
4
Find −2 + (−4). Use a number line to check your answer.
Add. 1. 9 + 4
2. −5 + (−1)
3. −8 + 13
4. 12 + (−8)
5. −10 + 6
6. −14 + 14
Add. 1. 7 + 13
a. Find 5 + (−10). 2. −8 + (−5)
3. −20 + (−15)
b. Find −3 + 7.
c. Find −12 + 12.
The list shows four bank account
Add.
transactions in July. Find the change C
4. −2 + 11
5. 9 + (−10)
6. −31 + 31
in the account balance.
7. WHAT IF? In the Check problem, the deposit amounts are $30 and $40. Find the change C in the account balance.
5
• What do you know about the sum A + B? Explain your reasoning.
Subtract. 1. 45 − 11
2. 87 − 23
3. 91 − 14
4. 76 − 69
5. 87 − 29 − 13
6. 65 − 52 − 11
• Two integers have different signs. Their sum is −8. What are possible values for the two integers?
Subtracting Integers
1.3
8-2=6 How are adding integers and subtracting integers related? What are the rules for subtracting integers?
Since 8 is positive, we need to travel to the right 8 steps. Next, instead of moving to the right 2 spaces, as in addition, subtraction moves in the opposite direction, which means we move 2 spaces to the left. subtract 2 8 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 Our answer is 6.
Subtracting Integers How would we show 3 - 8?
Adding & Subtracting Integers How would we show (-4) - 6?
As in the last example, we would start off going right 3 and then left 8.
Remember that (-4) is shown by going 4 spaces to the left. Subtracting 6, take us 6 more spaces to the left.
subtract 8 3
subtract +6 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
-4
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 Therefore, 3 - 8 = -5
Therefore, (-4) - 6 = -10
6
Adding & Subtracting Integers The most potentially confusing case is subtracting a negative integer.
Subtract. 1. 5 − 8
2. 12 − (−3)
3. −5 − (−6)
4. −4 − 6
5. 9 − 16
6. −7 − 10
How would we show 1 - (-6)? First, we would go 1 space to the right, because of the 1. Adding -6 spaces would take us to the left, but we must do the opposite since we are subtracting, moving 6 steps to the right.
1
subtract -6
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 Therefore, 1 - (-6) = 7
Subtraction Rule: KEEP - FLIP - CHANGE Keep the first number the same, flip the subtraction to addition, change the second number to the opposite sign
a. Find 3 − 12.
Subtract. 1. 8 − 3
2. 9 − 17
4. −14 − 9
5. 9 − (−8)
3. −3 − 3
b. Find −8 − (−13).
6. −12 − (−12)
c. Find 5 − (−4).
7
Evaluate −7 − (−12) − 14.
Evaluate the expression. 7. −9 − 16 − 8
8. −4 − 20 − 9
9. 0 − 9 − (−5)
10. −8 − (−6) − 0
11. 15 − (−20) − 20
12. −14 − 9 − 36
13. The highest elevation in Mexico is 5700 meters, on Pico de Orizaba.The lowest elevation in Mexico is −10 meters, in Laguna Salada.Find the range of elevations in Mexico.
Which continent has the greater range of elevations?
Is Subtraction Commutative?
Add. 1. 9 + 9
Subtraction would be commutative if a - b = b - a
2. −7 + (−7)
Is that true? Test that with some numbers and discuss with you group.
4 - 5 = -1 3. −3 + (−3) + (−3)
4. 5 + 5 + 5
Answer
5-4=1 Since -1 ≠ 5. 1, 6then + 6 + 46 -+56 ≠ 5 - 4
6. −4 + (−4) + (−4) + (−4)
In general, a - b ≠ b - a Subtraction is not commutative.
8
Multiplication
1.4 negative, or zero? How can you tell?
Multiplication can be indicated by putting a dot between two numbers, or by putting the numbers into parentheses. (We won't generally use "x" to indicate multiplication since that letter is used a lot in algebra for variables.) So multiplying 3 times 2 will be written as: 3·2 or
Math Practice
Is the product of two integers positive,
MP.7: Look fo structure.
Ask: How is m to addition?
Note: answer shown on the
(3)(2)
Multiplication
Multiplication is Commutative
Multiplication is repeated addition.
Since addition is commutative...
So, to find the product of 3 · 2 we would add the number 2 to itself three times:
And multiplication is just repeated addition, multiplication is commutative:
3·2=2+2+2 3·2=2+2+2=6 +2
+2
+2
+2
+2
+2
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 +3
2·3=3+3=6
+3
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
Multiplication is Commutative a·b=b·a Adding a number "a" to itself "b" times yields the same result as adding a number "b" to itself "a" times. Adding "a" to itself "b" times yields "ab". +a +a +a ab Adding "b" to itself "a" times yields "ba" which is the same as "ab". +b
+b ab
Multiplying Negative Integers It works the same way when you multiply a negative number by a positive number. So, 3 · (-2) just indicates to add -2 to itself three times: 3 · (-2) = (-2) + (-2) + (-2) = -6
-2
-2
-2
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 Would (-3) · 2 give us the same answer of -6? Explain your answer.
9
Multiplying Negative Integers Since multiplication is commutative,
Multiplying Positive & Negative Integers
(-3) · 2 = 2 · (-3) = (-3) + (-3) = -6
When multiplying two numbers, if both are positive, the answer is positive.
So, this just becomes adding -3 to itself 2 times.
If one is negative and the other is positive, the answer is negative.
-3
-3 How about if both numbers being multiplied are negative?
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
Multiplying a positive and negative integer results in a negative integer: (a)(-b) = (-a)(b) = -ab
Multiplying Positive & Negative Integers How do we interpret (-3)(-2)?
• “Today we learned that a negative integer multiplied by a negative integer is positive. Be sure to mentally check all of your steps so that you are not confusing anything.”
Since, (3)(-2) means to add -2 to itself 3 times We could interpret (-3)(-2) to mean to subtract (-2) from itself three times. We already learned that addition and subtraction are inverses, so subtracting -2 is the same as adding 2. So, (-3)(-2) indicates to add 2 to itself 3 times. So, (-3)(-2) = (3)(2) = 6
Multiply. 1. 7 • 2
2. 9(−7)
3. −6(8)
4. −8(−10)
5. 6 • (−5)
6. −5 • (−12)
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Multiply. a. 3(−4)
Multiply. b. −7 ⋅ 4
a. Evaluate (−2)2.
1. 5 ⋅ 5
2. 4(11)
3. −1(−9)
4. −7 ⋅ (−8)
5. 12 ⋅ (−2)
6. 4(−6)
7. −10(−6)(0)
8. −7 ⋅ (−5) ⋅ (−4)
Evaluate the expression. 9. (−3)2
10. (−2)3
11. −72
12. −63
b. Evaluate −52.
c. Evaluate (−4)3.
The bar graph shows the number of taxis a company has in service. The number of taxis decreases by the same amount each year for 4 years. Find the total change in the number of taxis.
13. A manatee population decreases by 15 manatees each year for 3 years. Find the total change in the manatee population.
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Multiply.
• Write the number −4 on the board. Ask students to write each of the
1. −10 • 5
2. −5 • (−6)
3. 8 • (−9)
4. −15 • 7
5. −22 • (−8)
6. -32 • (−4)
following and then share their responses. • “Write a multiplication problem that has −4 as one of the factors, and has a negative product.”
• “Write a second multiplication problem that has −4 as one of the factors, and has a positive product.”
1.5 Is the quotient of two integers positive, negative, or zero? How can you tell?
Divide.
Divide.
1. 12 ÷ (−3)
2. −32 ÷ 4
3. −56 ÷ (−8)
4.
5.
6.
a. 75 ÷ (−25)
b.
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Evaluate 10 − x2 ÷ y when x = 8 and y = −4.
Divide. 1. 14 ÷ 2
2. −32 ÷ (−4)
3. −40 ÷ (−8)
4. 0 ÷ (−6)
5.
6.
Evaluate the expression when a = −18 and b = −6.
You measure the height of the tide using the support beams of
7. a ÷ b
8.
9.
a pier. Your measurements are shown in the picture. What is the mean hourly change in the height?
10. The height of the tide at the Bay of Fundy in New Brunswick decreases 36 feet in 6 hours. What is the mean hourly change in the height?
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