Notes for Wednesday 9-25
Know this Multiplication can be represented three ways.
4x3
4(3)
𝟒∙𝟑
3a
Understanding multiplication rules of rational numbers. PROOF!!!!
Looking at patterns
a.
5x5
b.
5 x -2
5x4
4 x -2
5x3
3 x -2
5x2
2 x -2
5x1
1 x -2
5x0
0 x -2
5 x -1
-1 x -2
5 x -2
-2 x -2
5 x -3
-3 x -2
BIG IDEAS:
1. Positive x Negative = _____
Example. 6 x -3
2. Negative x Positive = _____
Example. -3 x 6
This can be said because multiplication is ______________________.
3. Negative x Negative = _____
4. Positive x Positive =_____
MORE PROOF!!
Multiplication
Reading the expression
Repeated Addition
sum
3 ∙ (-5)
5 ∙ (-4)
-2 ∙ (-8)
-4 ∙ (-10)
BIG IDEAS:
1. Positive x Negative = _________
Example. 6 x -3
2. Negative x Positive = __________
Example. -3 x 6
This can be said because multiplication is ______________________.
3. Negative x Negative = _________
4. Positive x Positive = __________
Using the Distributive Property as Proof of the rules of multiplication
What is the Distributive Property?
a(b + c) = ab + ac
a(b – c) = ab – ac
Well what does that mean?
3(4 + 5) = 3(4) + 3(5)
2(10 - 5) = 2(10) - 2(5)
We will need to recall additive inverses/opposites before we use the distributive
property to prove the rules for multiplication.
-5 + 5 = 0
3 + (-3) = 0
Use the distributive property and additive inverses to prove the product of 4(-5).
4(5 + -5) = 4(0)
Use the distributive property and additive inverses to prove the product of 3(-8).
3(8 + -8) = 3(0)
Use the distributive property and additive inverses to prove the product of -4(-5).
-4(5 + -5) = 4(0)
Use the distributive property and additive inverses to prove the product of -3(-8).
-3(8 + -8) = -3(0)
Looking at fact families. Discovering the rules for Division.
If 3 x 5 = 15 then 15 ÷ 3 = 5 and 15 ÷ 5 = 3
If (-3)(8) = -24 then -24 ÷ 8 = -3 and -24 ÷ (-3) = 8
If (-9)(-4) = 36 then 36 ÷ (-9) = -4 and 36 ÷ (-4) = -9
Thus proving that the rules for division are the same as those for multiplication.
1. Positive ÷ Negative = _____
3. Negative ÷ Negative = ______
2. Negative ÷ Positive = _____
4. Positive ÷ Positive =_____
Don’t get them mixed up with adding and subtracting rational numbers.
Here’s a way to remember them! Use the TIC-TAC-TOE chart!
Find each product or quotient. “Do together”
1.  7  3
2.
5. - 12
6.
84
9. (-5)(-4)(-1)
19 + (-28)
3.
12(-8)
4. 15 – (-17)
−45
5
7. -83– 31
10. 6(-1)(-10)
11.
8.
40
−8
12.
0
−3
2
0
Independent Practice
Find each product or quotient.
1. 9  23
2.
5.
6.
18
−6
9 + (-23)
3.
12(-3)
4. 35 – (-7)
−35
−7
7. -12 – 3
9. (-2)(-4)(9)(-1)
10.
12. -8 –(- 9)
13.
−
0
5
32
8
8.
11. 3(-1)(-10)
14.
8
0
−72
12
Exit Pass for 9-25
1. Fill in the multiplication/division TIC-TAC-TOE Chart
Find each product or quotient.
2. -9(-3)
3.
6. -9 – 12
7. -2 + 16
12
−4
4. -1(-3)(4)(-2)
8. 12 – 15
5. -18 ÷ -6
9. 7( 4)
Exit Pass for 9-25
1. Fill in the multiplication/division TIC-TAC-TOE Chart
Find each product or quotient.
2. -9(-3)
3.
6. -9 – 12
7. -2 + 16
12
−4
4. -1(-3)(4)(-2)
8. 12 – 15
5. -18 ÷ -6
9. 7( 4)
Admit Slip for 9 -26
Find each product or quotient.
1. 8(-4)
2. −
5. 4  2
6.
24
6
−10
−2
3.
0
−3
4.
−3
0
4.
−3
0
4.
−3
0
7. (-2)(-1)(5)(4)(-3)
Admit Slip for 9-26
Find each product or quotient.
1. 8(-4)
2. −
5. 4  2
6.
24
6
−10
−2
3.
0
−3
7. (-2)(-1)(5)(4)(-3)
Admit Slip for 9-26
Find each product or quotient.
1. 8(-4)
2. −
5. 4  2
6.
24
6
−10
−2
3.
0
−3
7. (-2)(-1)(5)(4)(-3)
Notes for Thursday 9-26
Interpret products of rational numbers by describing real-world contexts. You may
need to extend some of the number lines by writing the numbers in.
1.
iTunes sells 4 iPhone apps at the cost of $2 per app (4 x 2 = 8).
Addition problem:____________________ Multiplication problem:______________
2.
You spend $3 each on 4 bottles of Gatorade. (4 x -3 = -12).
Addition problem:____________________ Multiplication problem:______________
Addition problem:____________________ Multiplication problem:______________
3.
Your brother owes $3 to each of 3 friends, This is a loss for the brother.
(-3 x 3 = -9).
Addition problem:____________________ Multiplication problem:______________
4.
You tell 3 of your friends not to worry about paying you the $6 each that they
owe you. This is a gain for the friends. (-3 x -6 = 18) Think of this as 3 friends getting
to keep $6 each.
Multiplication problem:______________
5. The temperature dropped 2°F each hour for 4 hours. What was the total change in
temperature?
Addition problem:____________________ Multiplication problem:______________
6. The football team had two losses of 5 yards each.
Addition problem:____________________ Multiplication problem:______________
7. You sell 2 bags of cotton candy at a cost of $3.00 each.
Addition problem:____________________ Multiplication problem:______________
8. You spend $4.00 each time you order nachos at the football game. You order
nachos twice.
Addition problem:____________________ Multiplication problem:______________
9. You forgive (which means this is the opposite of getting any of your money back
from four people) 4 debts of $3 each.
Multiplication problem:______________
Interpret quotients of rational numbers by describing real-world contexts. You may
need to extend some of the number lines by writing the numbers in.
1. The temperature decreased at a constant rate from 0°F to -35°F in 5 hours. What
was the rate of descent? In other words, find the change per hour.
2. A submarine descended (went down) 3 feet per minute. If the ocean is 20,000 feet
deep, how long will it take the submarine to descend from the surface to the ocean
floor?
3. Each weekend, Ms. Brown spends the same amount of money on dinners out with
friends. In the past 7 weekends, she has spent $210 on dining out. From this outing,
what integer represents the change in her wallet after taking her friends out each
weekend?
4. Charla has an account balance of -32 dollars. She wants to pay off her debt in four
equal installments. What debt would she pay off each month?
5. Thomas wants to swim to -10 feet. He starts at the surface and it takes him 5 seconds
to swim to that depth. Write an integer to describe the distance he descended each
second?
6. Over the summer in Albany, New York, the depth of the local reservoir decreased
18 inches. If the water depth decreased by 2 inches each week, how many weeks did it
take for this change to occur?
7. A submarine descends to a depth of 32 feet below sea level. The submarine took 4
seconds to make its descent. If the submarine traveled the same distance each second,
what integer represents its change in elevation each second?
8. A football team lost a total of 20 yards over 4 plays. On average, what integer
represents the football team’s loss on each play?
9. A car gets 25 miles per gallon. How many gallons are needed to travel 275 miles?
10. During the morning, the temperature changed a total of -27 degrees. If the
temperature decreased by 4.5 degrees every hour, how many hours did it take for the
temperature to change?
Exit Pass for 9-26
1. While playing poker, Jean lost $56. He played for seven hours. What integer
represents his loss each hour?
2. Over the course of 4 months, Spiffy the dog lost 24 pounds. What integer
represents his loss per month?
3. It took a submarine 20 seconds to drop to 140 feet below sea level from the surface.
What was the rate of the descent?
4. A submarine descended (went down) 5 feet per minute. If the ocean is 20,000 feet
deep, how long will it take the submarine to descend from the surface to the ocean
floor?
5. Over the summer in Albany, New York, the depth of the local reservoir decreased
16 inches. If the water depth decreased by 2 inches each week, how many weeks did it
take for this change to occur?
Homework for 9-26
1. While playing poker, Jean lost $28. He played for seven hours. What integer
represents his loss each hour?
2. A hiker climbed to an altitude of 60 feet. He climbed 2 feet every minute. How many
minutes did it take him to climb to 60 feet?
3. Over the course of 4 months, Spiffy the dog lost 36 pounds. What integer
represents his loss per month?
4. During low tide in North Carolina, the beachfront in some places is about 350 feet
from the ocean to the homes. High tide can change the width of the beach at a rate of
-17 feet per hour. The width of the beachfront changed -102 feet from low tide to
high tide. How many hours passed for the change to happen? Do not use 350 feet. It
is not needed to answer the question.
5. It took a submarine 20 seconds to drop to 100 feet below sea level from the
surface. What was the rate of the descent?
6. The temperature decreased at a constant rate from 0°F to -40°F in 5 hours. What
was the rate of descent? In other words, find the change per hour.
7. A submarine descended (went down) 4 feet per minute. If the ocean is 20,000 feet
deep, how long will it take the submarine to descend from the surface to the ocean
floor?
8.
Charla has an account balance of -60 dollars. She wants to pay off her debt in
four equal installments. What debt would she pay off each month?
9. Thomas wants to swim to -35 feet. He starts at the surface and it takes him 5 seconds
to swim to that depth. Write an integer to describe the distance he descended each
second?
10. Over the summer in Albany, New York, the depth of the local reservoir decreased
24 inches. If the water depth decreased by 2 inches each week, how many weeks did it
take for this change to occur?
Admit Slip for 9-27
1. While playing poker, Jean lost $56. He played for seven hours. What integer
represents his loss each hour?
2. Over the course of 4 months, Spiffy the dog lost 24 pounds. What integer
represents his loss per month?
3. It took a submarine 20 seconds to drop to 140 feet below sea level from the surface.
What was the rate of the descent?
4. A submarine descended (went down) 5 feet per minute. If the ocean is 20,000 feet
deep, how long will it take the submarine to descend from the surface to the ocean
floor?
5. Over the summer in Albany, New York, the depth of the local reservoir decreased
16 inches. If the water depth decreased by 2 inches each week, how many weeks did it
take for this change to occur?
6. Katherine is very interested in cryogenics (the science of very low temperatures).
With the help of her science teacher she is doing an experiment on the effect of low
temperatures on bacteria. She cools one sample of bacteria to a temperature of -72°C
and another to -86°C. Write a numerical expression that represents the difference in
the temperatures. Then solve.
7. During the football game, Justin caught three passes. One was for a touchdown and
went 28 yards. The other was for a first down and was for 5 yards. The other was on a
screen pass that did not work so well and ended up a loss of 18 yards. Write a
numerical expression that would represent the total yardage gained by Justin during
the game. Then solve.
8. It is -15°F in Rantoul and it is 95°F in Honolulu. You decide to fly from Rantoul to
Honolulu. Write a numerical expression that represents the temperature difference
between the two cities. Then solve.
9. A submarine was situated 638 feet below sea level. It then descends 384 feet.
Write a numerical expression to represent its new position. Then solve.
10. A cliff diver dives from a ledge that is 65 feet above the surface of the water. The
diver reaches a depth of 25 feet under the surface of the water before returning to the
surface. Write an expression that represents the distance between the diver’s high
point and low point. Then solve.