4-1 Slope / 3 days
4-2 Slope Intercept Equation / 3 days
4-3 Point Slope Equation / 4 days
4.1 - 4.3 Quiz / 1 day
4-4 Parallel/Perpendicular Lines / 3 days
4-5 Scatter Plots / 4 days
Test Review / 1 day
Test / 1 day
Cumulative Review / 1 day
Total days in Unit 4 – Analyzing Linear Equations = 21 days
Review Question
How did we find the slope of the equations in the previous unit?
The change in y, over the change in x. Today, we will try to figure out why that works.
Discussion
How would you compare these two sled riding hills? The first hill is steeper
In math, we don’t use “steep”. Instead we use the word slope.
What is causing the first hill to be steeper than the second?
The amount that the y value changes is bigger than the amount that the x value changes. Show this in the drawings above. So to calculate how steep a line is (slope), we must compare the changes in y and x.
How do you find the change in height (y)? Subtract
How do you find the change in horizontal distance (x)? Subtract
How would you compare the slopes (steepness) of line 1 and 2 below?
The steepness is the same but the direction is different. Therefore, lines 1 and 2 have the same numeric slope but line 2 is negative. Notice the negative doesn’t have anything to do with the steepness of the line, but purely the direction of the line.
A positive slope creates a line that goes up/right.
A negative slope creates a line that goes down/right.
Slope – steepness and direction of a line
SWBAT calculate the slope of a line base on a graph
Example 1: Find the slope of the line.
m = 2/6; Show that the slope is the same whether you go top to bottom -2/-6 or bottom to top 2/6
Example 2: Find the slope of the line.
You can choose any two points on the line. We will get the same answer no matter what. (m = -6/4)
Example 3: Draw a line that has a slope of .
You Try!
1. Draw a line that has a slope of . 2. Draw a line that has a slope of .
3. Draw a line that has a slope of 4. 4. Draw a line that has a slope of -3.
What did we learn today?
Find the slope of the line.
1. 2.
3. 4.
5. 6.
7. 8.
9. Change in y: -8 in 10. Change in x: 3 feet
Change in x: 5 in Change in y: 4 feet
11. Change in y: 8 feet 12. Change in y: -300 cm
Change in x: -2 feet Change in x: -4 m
13. Draw a line that has a slope of . 14. Draw a line that has a slope of .
15. Draw a line that has a slope of 5. 16. Draw a line that has a slope of -7.
Review Question
What does slope mean? Steepness and direction of a line
How do you find slope? Compare the changes in y’s and x’s
How do you define direction? Positive – up/right, Negative – down/right
Discussion
So to calculate how steep a line is (slope), we must compare the changes in y and x.
How do you find the change in height (y)? Subtract
How do you find the change in horizontal distance (x)? Subtract
Remember we just did this… y = __x + __. This was the value that we put into the first blank.
SWBAT calculate the slope given two points
Definition
Slope: * You can use y2 –y1. We are just trying to simplify the process.
A positive slope creates a line that goes up/right.
A negative slope creates a line that goes down/right.
Example 1: Find the slope between (4, 12) (2, 1).
What direction does the line go? Up/Right
Graph the two points to confirm answer.
Example 2: Find the slope between (-4, 1) (2, 2).
What direction does the line go? Up/Right
Graph the two points to confirm answer.
Example 3: Find the slope between (2, 1) (-1, 8).
What direction does the line go? Down/Right
Graph the two points to confirm answer.
You Try!
Calculate the slope and direction of the line. Then graph.
1. (4, 8) (3, 2) m = 6/1 2. (1, 5) (7, 4) m = 1/-6
3. (-1, 2) (1, -4) m = 6/-2 4. (-2, 1) (3, 8) m = 7/5
What did we learn today?
For each problem:
a. Find the slope.
b. Describe the line as up/right or down/right.
c. Graph the two points.
1. (6, 8) (2, 7) 2. (8, 8) (6, 1)
3. (2, 6) (3, 1) 4. (-4, -8) (1, 4)
5. (10, 5) (-4, 1) 6. (-2, 1) (3, -5)
7. (1, -5) (-4, 6) 8. (-4, -2) (-3, -5)
9. (10, 8) (-2, -7) 10. (8, 9) (3, 2)
Find the slope of the line.
11. 12.
13. Draw a line that has a slope of . 14. Draw a line that has a slope of -6.
Review Question
What does slope mean? Steepness and direction of a line
How do you find slope? Compare the changes in y’s and x’s
How do you define direction? Positive – up/right, Negative – down/right
Discussion
We know that if the change in y is bigger than the change in x the line is steep.
We also know that if the change in x is bigger than the change in y the line is flat.
What if the change in y’s and x’s are the same? It would give us a slope of 1
What kind of line would that give us? “Average”
We will say any line with a slope greater than one is steep and less than one is flat.
We understand different scales. For example, we understand the grading scale: 0% to 100%. We are going to try to understand the scale for slopes. It is going to be a bit confusing but let’s try.
Estimate the slope of each of the lines starting with the middle line and work your way left and right.
m = 4/0 m = 4/2 = 2 m = 2/2 = 1 m = 1/4 = .25 m = 0/4 = 0
Notice that a horizontal line’s slope is zero. If you go “half way up” from horizontal, the slope is just one. You would think that a vertical line would be two. But it is not.
How steep is a vertical line? It is so steep that we can’t put a number on it. It is undefined.
How steep is a horizontal line? It is so flat that it is zero.
SWBAT calculate the slope of horizontal and vertical lines
Definitions
Horizontal Line – slope of zero
Vertical Line – undefined slope
Example 1: Find the slope between (4, 5) (4, 8). Then graph.
The slope is undefined. It is a vertical line.
Example 2: Find the slope between (5, 3) (2, 3). Then graph.
The slope is zero. It is a horizontal line.
You Try!
Calculate the slope and direction. Then describe the steepness of the line (flat, steep, average, horizontal, or vertical). Then graph.
1. (6, 8) (4, 8) m = 0, Horizontal 2. (6, 2) (3, 4) m = -2/3, Flat
3. (7, 5) (7, 4) m = undefined, Vertical 4. (3, 3) (4, 8) m = 5/1, Steep
What did we learn today?
Calculate the slope and direction. Then describe the steepness of the line (flat, steep, average, horizontal, or vertical). Then graph.
1. (6, 2) (2, 1) 2. (4, 2) (3, 1)
3. (1, 6) (3, 8) 4. (2, 4) (2, 1)
5. (4, 6) (-3, 6) 6. (2, 0) (0, 8)
7. (8, 3) (2, 4) 8. (-4, 1) (0, 2)
9. (-1, 6) (4, 6) 10. (3, 6) (3, 8)
Find the slope of the line.
11. 12.
Review Question
What does slope mean? Steepness and direction of a line
Discussion
We are going to learn the slope intercept equation today.
What two things do you think we need to know to use this equation? Slope, intercept
What does intercept mean? The place where a line touches an axis.
What does the y-intercept mean? The place where line touches the y axis.
What is the y-intercept?
1. 2. 3. (5, 2) (0, -1)
y-int = -1
y-int = -3 y-int = 1
SWBAT write an equation of a line using the slope intercept equation
Definitions
x-intercept – place where line touches the x-axis
y-intercept – place where line touches the y-axis
y = mx + b (Slope intercept equation)
m = slope, b = y-intercept
Example 1: Write an equation of a line with a slope of -2 and a y-intercept of 8.
y = mx + b
y = -2x + 8
Example 2: Write an equation of a line that goes through the point (0, -4) and has a slope of .
y = mx + b
y = 1/3x – 4
Example 3: Write an equation of a line that goes through the points (0, 3) and (5, 1).
What two things do we need to know in order to write an equation of a line? Slope, Intercept
y = mx + b
y = -2/5x + 3
You Try!
Write an equation with the following conditions.
1. m = 4, y-intercept = -2 y = 4x – 2
2. Horizontal line that touches the y-axis at -3. y = 0x – 3; y = -3
3. (0, -1) (3, -2) y = -1/3x – 1
4. Vertical line that touches the x-axis at 4. x = 4
5. Write an equation of a line that goes through the point (0, 2) and has a slope of . y = 1/5x + 2
What did we learn today?
Write an equation of the line with the given conditions.
1. Slope: 2, y-intercept: -6
2. (0, -3) (1, 5)
3. Horizontal line that touches the y-axis at 2.
4. Slope: , y-intercept: 3
5. (0, 4) (3, -1)
6. Slope: , y-intercept: 0
7. Slope: -1, y-intercept: -6
8. A line that goes through the point (0, -3) and has a slope of .
9. Slope: 0.5, y-intercept: 7.5
10. Vertical line that touches the x-axis at -1.
Write an equation of the line shown in each graph.
11. 12.
Review Question
What is the slope intercept equation? y = mx + b
What does each letter represent?
y is y
m is slope
x is x
b is the y-intercept
Discussion
Why is this equation so easy? You just plug in the slope and y-intercept.
How could this equation help us? Graphing
SWBAT graph an equation of a line using the slope intercept equation
Example 1: Graph: y = 3x + 1
How would you graph this line in previous units? T-Charts
How could you graph this line using the slope intercept equation?
Start at (0, 1) because that is the y-intercept then go up three and over one using the slope.
Which way is easier? Slope Intercept
Example 2: Graph: y + 2x = -4
What is different about this equation? It’s not in the slope intercept form.
After some manipulation: y = -2x – 4
To graph start at (0, -4) because that is the y-intercept then go down two and over one using the slope.
Example 3: Graph:
To graph start at (0, -2) because that is the y-intercept then go down three and over four using the slope.
Example 4: Graph: 2x – 5y = -8
What is different about this equation? It’s not in the slope intercept form.
After some manipulation:
To graph start at (0, 8/5) because that is the y-intercept then go up two and over five using the slope.
You Try!
Graph each line.
1. y = 2x – 5 Start at (0, -5) then go up 2 over 1
2. Start at (0, 5) then go down 2 over 5
3. y – 4x = 1 Start at (0, 1) then go up 4 over 1
4. x = -2 Vertical line at x = 2
5. 4x – 3y = 5 Start at (0, -5/3) then go up 4 over 3
6. y = 5 Horizontal line at y = 5
What did we learn today?
Graph each equation.
1. y = 3x + 1 2. y = x – 2
3. y = -4x + 1 4. y = -x + 2
5. 6.
7. y + 3x = -2 8. y – 2x = -3
9. y = 2x + 3 10. y = -5x + 1
11. y + x = 3 12.
13. y = 5 14. x = -4
Write the equation of the line. Then graph.
15. Horizontal line that touches the y axis at 2.
16. Slope: , y-intercept: 2
17. (0, -1) (3, 4)
18. A line that goes through the point (0, 5) and has a slope of .
19. Write an equation of a line that passes through the origin with slope 3.