8th-Grade Math
“Distance Learning 2.0”
April 27 - May 21, 2020�(last 4 weeks of the school year)
How it will work
MATH class setup on Schoology
MATH class on Schoology: In each weekly folder
SLOPE
Ratios and Proportions
Warm Up: ANSWERS
Find an equivalent ratio for each:
Warm Up: ANSWERS
Find an equivalent ratio for each:
Warm Up: ANSWERS
Find an equivalent ratio for each:
Warm Up: ANSWERS
Find an equivalent ratio for each:
Warm Up: ANSWERS
Find an equivalent ratio for each:
x , y
= (12, 35)
– ( 2, 5)
Δx = 10, Δy = 30
SLOPE = Δy / Δx
= 30 / 10
SLOPE = 3
Warm Up: ANSWERS
(6) How can you tell if a line has a positive slope or a negative slope?
Positive slope: vertical amount increases; line goes UP as you go left → right
Negative slope: vertical amount decreases; line goes DOWN as you go left → right
POSITIVE SLOPE NEGATIVE SLOPE
Some of our favorite animals:
Some of our favorite animals:
Some of our favorite animals:
Some of our favorite animals:
Some of our favorite animals:
Some of our favorite animals:
Some of our favorite animals:
Some of our favorite animals:
Today’s Lesson Objectives:
I Will Be Able To:
Proportions: ratios of two things / how they go together
Proportion of students to workbooks in my class = 1 / 2
Proportion of green M&Ms to red M&Ms = 3 / 2
Proportions: ratios of two things / how they go together
Proportion of students to workbooks in my class = 1 / 2
Proportion of green M&Ms to red M&Ms = 3 / 2
Proportions: ratios of two things / how they go together
Proportion of students to workbooks in my class = 1 / 2�I could have 50 students and 100 books, or 30 students and 60 books, ….��
�Proportion of green M&Ms to red M&Ms = 3 / 2�You could get 3 green/ 2 red, or 6 green/4 red, or 15 green/10 red, or …
50
30
1
2
100
60
Proportions: ratios of two things / how they go together
Proportion of students to workbooks in my class = 1 / 2�I could have 50 students and 100 books, or 30 students and 60 books, ….��
�Proportion of green M&Ms to red M&Ms = 3 / 2�You could get 3 green/ 2 red, or 6 green/4 red, or 15 green/10 red, or …
50
30
1
2
100
60
You don’t really know how many of each�you just know the ratio of the two quantities
Why are we talking about this?
Why are we talking about this?
Because SLOPE works exactly the same way!
Why are we talking about this?
Because SLOPE works exactly the same way!
�What is slope the ratio of?
Why are we talking about this?
Because SLOPE works exactly the same way!
�What is slope the ratio of?
The ratio of UP to ACROSS
What is slope the ratio of?
The ratio of UP to ACROSS��
SLOPE is:
- The ratio of how much it goes up to how much it goes across
Slope = 5
up:across = 50:10
SLOPE is:
- The ratio of how much it goes up to how much it goes across
Slope = 5
up:across = 50:10
up:across = 5:1
Slope = 5
up:across = 25:5
SLOPE is:
- The ratio of how much it goes up to how much it goes across
Slope = 5
up:across = 50:10
up:across = 5:1
Slope = 5
up:across = 25:5
Slope = 5
up:across = 10:2
That’s why it doesn’t matter which points you pick - you always get the same answer
(0, –9)
(1, –5)
(3, 3)
(4, 7)
Up 4
Across 1�
Slope = ratio = 4/1
(0, –9)
(1, –5)
(3, 3)
(4, 7)
Up 4
Across 1�
Slope = ratio = 4/1
Up 12
Across 3�
Slope = ratio = 12/3� = 4/1
(0, –9)
(1, –5)
(3, 3)
(4, 7)
(0, –9)
(1, –5)
(3, 3)
(4, 7)
SLOPE = Δy / Δy
= 4 / 1
SLOPE = 4 .
x , y
= (4, 7)
– (3, 3)
Δx = 1, Δy = 4
SLOPE = Δy / Δy
= 12 / 3
SLOPE = 4 .
x , y
= (3, 3)
– (0, –9)
Δx = 3, Δy = 12
You Try:
(0, –9)
(1, –5)
(3, 3)
(4, 7)
SLOPE = Δy / Δy
= 4 / 1
SLOPE = 4 .
x , y
= (4, 7)
– (3, 3)
Δx = 1, Δy = 4
SLOPE = Δy / Δy
= 12 / 3
SLOPE = 4 .
x , y
= (3, 3)
– (0, –9)
Δx = 3, Δy = 12
It doesn’t matter what two points you pick, the ratio (up:across) will always be the same!
12/3 or � 8/2 or
4/1 or ...
SLOPE IS CONSTANT
Always the same ratio
That’s why the line is straight!
What is slope the ratio of?
The ratio of UP to ACROSS
What is slope the ratio of?
The ratio of UP to ACROSS
Y direction
X direction
X
Y
What is slope the ratio of?
The ratio of UP to ACROSS
Y direction
X direction
X
Y
Ratio of how far UP to how far ACROSS
UP: difference of y-coordinates Δy�
ACROSS: difference of x-coordinates Δx
RATIO: UP to ACROSS� = Δy / Δy
What is slope the ratio of?
The ratio of UP to ACROSS��If the line goes up by 3 then it goes across by 2
What is slope the ratio of?
The ratio of UP to ACROSS��If the line goes up by 3 then it goes across by 2 �If the line goes up by 6 then it goes across by ???
What is slope the ratio of?
The ratio of UP to ACROSS��If the line goes up by 3 then it goes across by 2 �If the line goes up by 6 then it goes across by 4
What is slope the ratio of?
The ratio of UP to ACROSS��If the line goes up by 3 then it goes across by 2 �If the line goes up by 6 then it goes across by 4 �If the line goes up by 15 then it goes across by ???
What is slope the ratio of?
The ratio of UP to ACROSS��If the line goes up by 3 then it goes across by 2 �If the line goes up by 6 then it goes across by 4 �If the line goes up by 15 then it goes across by 10
What is slope the ratio of?
The ratio of UP to ACROSS��If the line goes up by 3 then it goes across by 2 �If the line goes up by 6 then it goes across by 4 �If the line goes up by 15 then it goes across by 10
“Y increases”
What is slope the ratio of?
The ratio of UP to ACROSS��If the line goes up by 3 then it goes across by 2 �If the line goes up by 6 then it goes across by 4 �If the line goes up by 15 then it goes across by 10
“Y increases”
“X increases”
What is slope the ratio of?
The ratio of UP to ACROSS��If Y increases by 3 then X increases by 2 �If Y increases by 6 then X increases by 4 �If Y increases by 15 then X increases by 10
“Y increases”
“X increases”
���If Y increases by 3 then X increases by 2 �If Y increases by 6 then X increases by 4 �If Y increases by 15 then X increases by 10
�
(10,15)
(4,6)
(2,3)
That’s what slope is.
The ratio of how much y goes up (Δy)� to how much x goes up (Δx)��If Y increases by 3 then X increases by 2 �If Y increases by 6 then X increases by 4 �If Y increases by 15 then X increases by 10
� Δy/Δx = 3/2 = 6/4 = 15/10 (= 3/2)
(10,15)
(4,6)
(2,3)
SLOPE = 3/2
Slope = the ratio of UP to ACROSS��If Y increases by 3 then X increases by 2 �If Y increases by 6 then X increases by 4 �If Y increases by 15 then X increases by 10
Can use it to model proportional relationships
Slope = the ratio of UP to ACROSS��If Y increases by 3 then X increases by 2 �If Y increases by 6 then X increases by 4 �If Y increases by 15 then X increases by 10
�
Can use it to model proportional relationships
Slope = the ratio of UP to ACROSS��If Y increases by 3 then X increases by 2 �If Y increases by 6 then X increases by 4 �If Y increases by 15 then X increases by 10
�
Let Y represent one of the quantities in your proportion�Let X represent the other quantity in your proportion
Can use it to model proportional relationships
Slope = the ratio of UP to ACROSS��If Y increases by 3 then X increases by 2 �If Y increases by 6 then X increases by 4 �If Y increases by 15 then X increases by 10
�Proportion of green M&Ms to red M&Ms = 3 / 2�You could get 3 green/ 2 red, or 6 green/4 red, or 15 green/10 red, or …
Can use it to model proportional relationships
Slope = the ratio of UP to ACROSS��If Y increases by 3 then X increases by 2 �If Y increases by 6 then X increases by 4 �If Y increases by 15 then X increases by 10
�Proportion or green M&Ms to red M&Ms = 3 / 2�You could get 3 green/ 2 red, or 6 green/4 red, or 15 green/10 red, or …
“GREEN M&Ms increase”
“RED M&Ms increase”
Proportion of green M&Ms to red M&Ms = 3 / 2
Every time they put 3 more green M&Ms in the bag, they then put in 2 red ones
�If Y increases by 3 then X increases by 2 �If Y increases by 6 then X increases by 4 �If Y increases by 15 then X increases by 10 ��Graph RED M&M (Y) increase & GREEN M&M (X) increase ... proportional rate is 3/2 … graph with a slope = 3/2
(10,15)
(4,6)
(2,3)
Y = GREEN M&Ms
X = RED M&Ms
What if (UP / ACROSS) ratios are not the same?
What if (UP / ACROSS) ratios are not the same?�→ NOT a straight line … non-linear
UP / ACROSS = 2/1
UP / ACROSS = 4/1
UP / ACROSS = 1/3
What if (UP / ACROSS) ratios are not the same?�→ NOT a straight line … non-linear
UP / ACROSS = slower (0.5 /1 = ½ )
UP / ACROSS = a little faster (2/2 = 1)
UP / ACROSS = even faster (4.5/3 = 1.5)
Proportion of yellow M&Ms to blue M&Ms = 3 / 1
Every time they put 3 more yellow M&Ms in the bag, they put in 1 blue one
Y = YELLOW M&Ms
X = BLUE M&Ms
Proportion of yellow M&Ms to blue M&Ms = 3 / 1
Every time they put 3 more yellow M&Ms in the bag, they put in 1 blue one
�If Y increases by 3 then X increases by 1 �If Y increases by 6 then X increases by 2 �If Y increases by 9 then X increases by 9 ��
(3,9)
(2,6)
(1,3)
Y = YELLOW M&Ms
X = BLUE M&Ms
(4,12)
Proportion of yellow M&Ms to blue M&Ms = 3 / 1
Every time they put 3 more yellow M&Ms in the bag, they put in 1 blue one
�If Y increases by 3 then X increases by 1 �If Y increases by 6 then X increases by 2 �If Y increases by 9 then X increases by 9 ��Increase in yellow (Y) by 3 = increase blue (X) by 1� ... proportional rate is 3 / 1 … graph with a slope = 3
(3,9)
(2,6)
(1,3)
Y = YELLOW M&Ms
X = BLUE M&Ms
(4,12)
Proportion of yellow M&Ms to blue M&Ms = 3 / 1��If Y increases by 3 then X increases by 1 �If Y increases by 6 then X increases by 2 �If Y increases by 9 then X increases by 9 �
(3,9)
(2,6)
(1,3)
Y = YELLOW M&Ms
X = BLUE M&Ms
(4,12)
Proportion of yellow M&Ms to blue M&Ms = 3 / 1��If Y increases by 3 then X increases by 1 �If Y increases by 6 then X increases by 2 �If Y increases by 9 then X increases by 9�� y = 3x�
(3,9)
(2,6)
(1,3)
Y = YELLOW M&Ms
X = BLUE M&Ms
(4,12)
Proportion of yellow M&Ms to blue M&Ms = 3 / 1��If Y increases by 3 then X increases by 1 �If Y increases by 6 then X increases by 2 �If Y increases by 9 then X increases by 9�� y = 3x� when x goes up by 1, y goes up by 3 times as much�
(3,9)
(2,6)
(1,3)
Y = YELLOW M&Ms
X = BLUE M&Ms
(4,12)
Proportion of yellow M&Ms to blue M&Ms = 3 / 1��If Y increases by 3 then X increases by 1 �If Y increases by 6 then X increases by 2 �If Y increases by 9 then X increases by 9�� y = 3x� when x goes up by 1, � y goes up by 3 times as much�� y = mx (that’s why this is the slope!)�
(3,9)
(2,6)
(1,3)
Y = YELLOW M&Ms
X = BLUE M&Ms
(4,12)
Suppose you start with 5 yellow M&Ms already in the bag
Y = YELLOW M&Ms
X = BLUE M&Ms
Suppose you start with 5 yellow M&Ms already in the bag
When you have 0 blue ones you already have 5 yellow ones
Y = YELLOW M&Ms
X = BLUE M&Ms
(0,5)
Start off with 5 yellow M&Ms in the bag���
Y = YELLOW M&Ms
X = BLUE M&Ms
(0,5)
Start off with 5 yellow M&Ms in the bag��Then ratio is the same as before …�Every time you add 1 blue one to the bag (X)�You add 3 yellow ones to the bag (Y)�
Y = YELLOW M&Ms
X = BLUE M&Ms
(0,5)
Start off with 5 yellow M&Ms in the bag��Then ratio is the same as before …�Every time you add 1 blue one to the bag (X)�You add 3 yellow ones to the bag (Y)��Start at 5�The go up/across by a ratio of 3/1�
Slope still = 3�
Y = YELLOW M&Ms
X = BLUE M&Ms
(0,5)
SLOPE = 3
Start at 5�Then go up/across by a ratio of 3/1�(when x goes up by 1, y goes up by 3 times as much)��
Y = YELLOW M&Ms
X = BLUE M&Ms
(0,5)
SLOPE = 3
Start at 5
� y = 3x + 5� ��
Y = YELLOW M&Ms
X = BLUE M&Ms
(0,5)
SLOPE = 3
Start at 5
� y = 3x + 5
Then go up/across by a ratio of 3/1� (increase y by 3 every time you increase x by 1)
Y = YELLOW M&Ms
X = BLUE M&Ms
(0,5)
SLOPE = 3
Start at 5 (y intercept = starting amount)
� y = mx + b
Then go up/across by a ratio of 3/1� (increase y by 3 every time you increase x by 1)� Slope = how much it goes up from there� how much the ratio of y/x increase is
Y = YELLOW M&Ms
X = BLUE M&Ms
(0,5)
SLOPE = 3
Start at 5 (y intercept = starting amount)
� y = mx + b y = 3x + 5
Then go up/across by a ratio of 3/1� (increase y by 3 every time you increase x by 1)� Slope = how much it goes up from there� how much the ratio of y/x increase is
Y = YELLOW M&Ms
X = BLUE M&Ms
(0,5)
SLOPE = 3
Let’s try one more example ...
Let’s try one more example ...
If I start of with $20 in my pocket, then add $5 for every 1 hour I babysit.�What graph and equation model that?
Let’s try one more example ...
If I start of with $20 in my pocket, then add $5 for every 1 hour I babysit.�What graph and equation model that? (y = $, x = hours)
Let’s try one more example ...
If I start of with $20 in my pocket, then add $5 for every 1 hour I babysit.�What graph and equation model that? (y = $, x = hours)
Start at $20 (y intercept = starting amount)
Let’s try one more example ...
If I start of with $20 in my pocket, then add $5 for every 1 hour I babysit.�What graph and equation model that? (y = $, x = hours)
Then go up/across by a ratio of 5/1� (up $5 for every 1 hour)�� Slope = how much it goes up from there� how much the ratio of y/x increase is
Start at $20 (y intercept = starting amount)
SLOPE = 5
(0,20)
(1,25)
(2,30)
(3,35)
Let’s try one more example ...
If I start of with $20 in my pocket, then add $5 for every 1 hour I babysit.�What graph and equation model that? (y = $, x = hours)
y = 5x + 20
Then go up/across by a ratio of 5/1� (up $5 for every 1 hour)�� Slope = how much it goes up from there� how much the ratio of y/x increase is
Start at $20 (y intercept = starting amount)
SLOPE = 5
(0,20)
(1,25)
(2,30)
(3,35)
How much money will I have after 7 hours of babysitting?
If I start of with $20 in my pocket, then add $5 for every 1 hour I babysit.�(y = $, x = hours)
y = 5x + 20
How much money will I have after 7 hours of babysitting?
If I start of with $20 in my pocket, then add $5 for every 1 hour I babysit.�(y = $, x = hours)
y = 5x + 20
(7,55)
How much money will I have after 7 hours of babysitting?
If I start of with $20 in my pocket, then add $5 for every 1 hour I babysit.�(y = $, x = hours)
y = 5x + 20
x = 7 hours
y = 5x + 20
y = 5(7) + 20
y = 35 + 20
y = 55 $55
(7,55)
We’re Done!
We’re Done!
Try the “homework” problems posted on Schoology (due Weds.4/28 @ 10am).
Answers will be posted on Schoology after 10:00am on Weds.
If you have questions:� email me at jay.legenhausen@mnps.org (or click “Contact” on my website)
I will host a office hours at 10:00am Weds. 4/28/2020 to review the answers�for anyone who wants to join in. Join via Teams on Schoology.