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Unit 4 – Analyzing Linear Equations Length of section

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

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Review QuestionHow 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)? SubtractHow 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 graphExample 1: Find the slope of the line.

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Section 4-1: Slope (Day 1) (CCSS: F.IF.6, F.LE.1.b)

y

x

y x

2 1

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.

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2

6

2

7

Section 4-1 Homework (Day 1)

5 1

2

6

3 ft

24 in

2 4

6 5

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.

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3 in

3 in

Review QuestionWhat does slope mean? Steepness and direction of a lineHow do you find slope? Compare the changes in y’s and x’sHow 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)? SubtractHow do you find the change in horizontal distance (x)? SubtractRemember 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/RightGraph the two points to confirm answer.

Example 2: Find the slope between (-4, 1) (2, 2).

What direction does the line go? Up/RightGraph the two points to confirm answer.

Example 3: Find the slope between (2, 1) (-1, 8).

What direction does the line go? Down/RightGraph 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/-63. (-1, 2) (1, -4) m = 6/-2 4. (-2, 1) (3, 8) m = 7/5

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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.

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Section 4-1 Homework (Day

4 ft

36 in


Review QuestionWhat does slope mean? Steepness and direction of a lineHow do you find slope? Compare the changes in y’s and x’sHow 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 1What 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

DefinitionsHorizontal Line – slope of zero

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2

2

4

2 4 1

x

y

y

x x

y

Vertical Line – undefined slopeExample 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, Flat3. (7, 5) (7, 4) m = undefined, Vertical 4. (3, 3) (4, 8) m = 5/1, Steep


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.

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10

12


Review QuestionWhat 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

Definitionsx-intercept – place where line touches the x-axisy-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 + by = -2x + 8

Example 2: Write an equation of a line that goes through the point (0, -4) and has a slope of .

y = mx + by = 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 + by = -2/5x + 3

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Section 4-2: Slope Intercept Equation (Day 1) (CCSS: A.SSE.2, A.CED.2, A.REI.10, F.IF.7, F.IF.7.a, F.IF.9, F.BF.3, F.BF.4.a, F.LE.5)

(-5, 2)

(0, -2)

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


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)


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.

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Review QuestionWhat is the slope intercept equation? y = mx + bWhat does each letter represent?y is ym is slopex is xb 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 + 1How would you graph this line in previous units? T-ChartsHow 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 = -4What 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.

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Section 4-2: Slope Intercept Equation (Day 2) (CCSS: A.SSE.2, A.CED.2, A.REI.10, F.IF.7, F.IF.7.a, F.IF.9, F.BF.3,

F.BF.4.a, F.LE.5)

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 = -8What 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 14. x = -2 Vertical line at x = 25. 4x – 3y = 5 Start at (0, -5/3) then go up 4 over 36. y = 5 Horizontal line at y = 5


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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.


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.

20. (3, 9) (0, 6)

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Review QuestionWhat is the slope intercept equation? y = mx + bWhat does each letter represent?y is y, m is slope, x is x, b is the y-intercept

Discussion Why do we learn about linear equations?The writing of linear equations is used often in computer programming.

Think about the automated system in a parking garage. A computer reads your ticket. Then it calculates how long you were parked. Finally, it calculates how much you need to pay.

What type of calculations/equations does it use? First, it must subtract the time you entered from the time you left the garage. Then it calculates how much you will pay.

Does the amount you pay increase each minute you are parked? No, it increases based on increments of time

Is this a linear relationship? NoIf it is not linear, then what would it look like?

Notice this is also applicable to fantasy football. A player gets 1 point for every ten yards he rushes for. Therefore, he gets 1 point for 10 yards up to 19 yards. Once he gets to 20 yards, it jumps to 2 points.

This is what we will be doing today. We will take a real life situation and try to write an Algebraic equation for it.

SWBAT write a linear equation given a real life situation

Example 1: At a bowling alley, it costs $3 to rent shoes plus $2 per game.Define variables. Then write an equation to model the total cost

Let T = total costLet g = number of games

T = 2g + 3

Notice that $3 is the starting point and $2 is what it is going up by. These represent the y-int and slope.

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Section 4-2: Slope Intercept Equation (Day 3) (CCSS: A.SSE.2, A.CED.2, A.REI.10, F.IF.7, F.IF.7.a, F.IF.9, F.BF.3,

F.BF.4.a, F.LE.5)

How much does it cost for 4 games?T = 2(4) + 3T = 8 + 3T = $11

How many games did I play if it cost $15? 15 = 2g + 3-3 - 3 12 = 2g 2 26 = g


1. Define variables. Then write an appropriate equation for the given amount.

a. Jimmy rented a bike for $10 plus $2 per hour. What is the total cost?

b. An auto repair shop charges $50 plus $25 per hour. What is the total cost?

c. A candle is 6 inches tall and burns at a rate of inch per hour. What is the total height?

d. The temperature is 20° and is expected to fall 2° each hour during the night. What is the current temperature?

2. Your grade is a 78% and increases 3% for each assignment you turn in.

a. Define variables. Then write an appropriate equation for your current grade.

b. How many assignments do you have to do in order to get an 87%?

c. How many assignments do you have to do in order to get at least an 85%?

d. What will your grade be if you do 7 assignments?

1. Define variables. Then write an appropriate equation.

a. You have $100 and plan to save $5 per week. What is your total savings?

b. You lost 10 pounds and plan to lose 3 pounds per month. What is your total weight loss?

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Section 4-2 In-Class Assignment (Day 3)


c. You make $25000 per year and expect to get a raise of $3000 per year. What is your current salary?

2. Johnny bench presses 125 pounds. His bench press increases 5 pounds per week.

a. Define variables. Then write an appropriate equation to represent how much Johnny can bench press.

b. How much can he bench after 4 weeks?

c. How many weeks will it take for him to be able to bench 180 pounds?

3. You are $9000 in debt. You plan to pay back $1200 per year.

a. Define variables. Then write an appropriate equation to represent how much you currently owe?

b. How much will you owe after 3 years?

c. How many years have you been paying if you still owe $3000?

d. How many years will it take to be out of debt?

4. Graph each equation.

a. y = 4x + 3 b. y = -4x – 1

c. d. y + 5x = 3

5. Write the equation of the line. Then graph.

a. Horizontal line that touches the y-axis at -4.

b. Slope: , y-intercept: -3

c. (0, 5) (1, -3)

d. Vertical line that touches the x-axis at 7.

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Review QuestionWhat is the slope intercept equation? y = mx + bWhen do we use the slope intercept equation?When we are given a slope and y intercept. That is why it is called the slope intercept equation.

Discussion How would you write an equation of a line that passes through (-3, 1) and has a slope of -2? You can’t right now

Why can’t we use the slope intercept equation? Don’t know the intercept

Can you guess what the name of the equation we need? Point Slope; because they give us a point and a slope

SWBAT write a linear equation using the point slope equation

DefinitionPoint Slope equation: y – y1 = m(x – x1)

y = yx = xx1, y1 = coordinates of the pointm = slope

Example 1: Write an equation of a line that passes through (-3, 1) and has a slope of -2.y – 1 = -2(x + 1)y – 1 = -2x – 2 y = -2x – 1

Example 2: Write an equation of a line that passes through (1, 2) and (3, 6).What do we need to use the point slope equation? Point, slope

y – 2 = 2(x – 1) or y – 6 = 2(x – 3) y – 2 = 2x – 2 y – 6 = 2x – 6 y = 2x y = 2x

Notice we can use either point.Notice we can use point slope when we have two points.

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Section 4-3: Point Slope Equation (Day 1) (CCSS: A.SSE.2, A.CED.2, A.REI.10, F.IF.7, F.IF.7.a, F.IF.9, F.BF.3,

F.LE.5)

You Try!Write an equation of a line in slope intercept form with the following conditions.

1. Passes through (-3, 2), m = 4 y = 4x + 12. Passes through (2, 4) and (3, -1) y = -5x + 14

3. m = , y-intercept = -2 y = -1/3x – 2

4. Passes through (2, -1), m = -3 y = 3x – 7


Write an equation of a line in slope intercept form with the following conditions. Then graph.

1. m = 2, (2, 3) 2. m = -3, (-2, 1) 3. (5, 2) (4, 5) y = 2x – 1 y = -3x – 5 y = -3x + 17

4. m = , y-intercept = 2 5. (4, 2) (5, 4) 6. m = -2, (-1, -5)

y = -2/3x + 2 y = 2x – 6 y = -2x – 7

7. m = , y-intercept = 4 8. m = -4, (-1, -2) 9. (-2, 5) (-4, 9)

y = 1/5x + 4 y = -4x – 6 y = -2x + 1

10. m = 1, y-intercept = -2 11. (5, -2) (3, -5) 12. m = , (2, -1)

y = x – 2 y = 3/2x – 19/2 y = 1/4x – 3/2

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Review Question What is the slope intercept equation? y = mx + bWhat is the point slope equation? y – y1 = m(x – x1)How do you know when to use each one? Look at the name of the equation

Discussion When do we use the slope intercept equation? When we have a slope and y-interceptWhen do we use the point slope equation? When we have a point and slope or when we have 2 points

SWBAT write a linear equation using the point slope equation and slope intercept equation

Example 1: Given y = -2x + 5. Find the slope and y-intercept. Then graph.m = -2y-intercept = 5Graph: Start at (0, 5) then go down 2 and over 1.

Example 2: Given a line passes through (2, -4) and has a slope of 3. Find the slope and y-intercept. Then graph.y + 4 = 3(x – 2)y + 4 = 3x – 6 y = 3x – 10 m = 3y-intercept = -10Graph: Start at (0, -10) then go up three and over one.

You Try!For each problem do the following:

a. Write the equation in slope intercept formb. Find the slopec. Find the y-interceptd. Graph

1. y = -2x + 3 y = -2x + 3, m = -2, y-int = 32. Passes through (-1, 3), m = 2 y = 2x + 5, m = 2, y-int = 53. x = 4 m = undefined, y-int = DNE4. Passes through (3, 8) and (4, 5) y = -3x + 17, m = -3, y-int = 175. y = -2 y = -2, m = 0, y-int = -2


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Section 4-3: Point Slope Equation (Day 2) (CCSS: A.SSE.2, A.CED.2, A.REI.10, F.IF.7, F.IF.7.a, F.IF.9, F.BF.3, F.LE.5)

For each problem do the following:a. Write the equation in the slope intercept formb. Find the slopec. Find the y-interceptd. Graph

1. y = 4x + 1 2. (6, 2) (4, 8)

3. m = -3, y-intercept = 2 4. y = 3

5. x = - 4 6. m = -3, (2, -1)

7. y = -2x + 5 8. (2, -5) (-2, 7)

9. m = 4, (-3, 2) 10. (2, 5) (4, 5)

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Review QuestionWhat is the slope intercept equation? y = mx + bWhat is the point slope equation? y – y1 = m(x – x1)How do you know when to use each one? Look at the name of the equation

Discussion What do we know about the y-intercept? It’s where a line touches the y-axis.What do we know about the value of ‘x’ at a y-intercept? It is 0.

What do we know about the x-intercept? It’s where a line touches the x-axis.What do we know about the value of ‘y’ at an x-intercept? It is 0.How could you find the x-intercept? Put 0 in for ‘y’; then solve for ‘x’

SWBAT write a linear equation and identify the key components of the line

Example 1: Given y = 3x + 9.Find the slope, y-intercept and x-intercept. Then graph.

m = 3

y-intercept = -3

x-intercept: 0 = 3x+ 9 -9 = 3x -3 = x

Graph: Start at (0, 9) then go up 3 and over 1.

Example 2: Write an equation of a line that passes through (1, -3) and has a m = -2. Find the slope, y-intercept and x-intercept. Then graph.

y + 3 = -2(x – 1)y + 3 = -2x + 2y = -2x – 1

m = -2

y-intercept = -1

x-intercept: 0 = -2x – 1 1 = -2x -1/2 = x

Graph: Start at (0, -1) then go down 2 and over 1.

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You Try!For each problem do the following:

a. Write the equation in slope intercept formb. Find the slopec. Find the y-interceptd. Find the x-intercepte. Graph

1. Passes through (-3, 2), m = -2 y = -2x – 4, m = -2, y-int = -4, x-int = -22. y + 4x = 12 y = -4x + 12, m = -4, y-int = 12, x-int = 3

3. y = ?, m= und, x-int = -1/3, y-int = DNE

4. Passes through (3, 4) and (5, 6) y = 2x – 2, m = 2, y-int = -2, x-int = 15. m = 3, y-intercept = -1 y = 3x – 1, m = 3, y-int = -1, x-int = 1/3


For each problem do the following:a. Write the equation in the slope intercept formb. Find the slopec. Find the y-interceptd. Find the x-intercepte. Graph

1. y = -2x + 4 2 (1, 5) (2, 3)y = -2x + 4, m = -2, y-int = 4, x-int = 2 y = 2x + 3, m= 2, y-int = 3, x-int = -3/2

3. m = 2, y-intercept = -2 4. y = -5y = 2x – 2, m = 2, y-int = - 2, x-int = 1 y = - 5, m = 0, y-int = -5, x-int = none

5. x = 2 6. m = 3, (-2, 3)x = 2, m = undefined, y-int = none, x-int = 2 y = 3x + 9, m = 3, y-int = 9, x-int = -3

7. 8. (4, -6) (3, -2)

y = 1/4x + 1/3, m = 1/4, y-int = 1/3, x-int = -4/3 y = -4x + 10, m = -4, y-int = 10, x-int = 10/4

9. m = -2, (3, -5) 10. (-2, 5) (4, 5) y = -2x + 1, m = -2, y-int = 1, x-int = 1/2 y = 5, m = 0, y-int = 5, x-int = none

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Review QuestionWhat is the slope intercept equation? y = mx + bWhat is the point slope equation? y – y1 = m(x – x1)How do you know when to use each one? Look at the name of the equation

DiscussionToday’s activity is a great time to emphasize skills such as working together, time management, neatness, and organization.

SWBAT write a linear equation and identify the key components of the line

ActivityGive each group one set of problems (one set = 1 problem). Give each group about 3 minutes to complete the problem. Then rotate the problems around the class giving the groups 3 minutes with each set of problems.

For problems 1-9, write the equation in slope intercept form, state the slope, y-intercept, x-intercept. Then graph the equation.

1. m = -3, y-intercept = 122. (-3, -8) (-1, -8)3. m = 2, (2, -4)4. 2(4x + 5y) = 3(3y – 4x) – 4 5. y = -4x – 10 6. y = 57. m = 3, y-intercept = -28. -7x = 359. m = -2, (2, -5)

10. Johnny saved $6500 to pay rent. His rent is $500 per month. a. Write an equation to model this situation.b. How much money will Johnny have left after 5 months?c. How many months will it take until Johnny runs out of money?

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Names ____________________________________________________________

1. y = 2. y = m = m = y-int = y-int = x-int = x-int = graph: graph:

3. y = 4. y =m = m = y-int = y-int =x-int = x-int =graph: graph:


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9. y = 10. m = a.y-int = b.x-int = c.graph:

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Review QuestionWhat is the slope of a horizontal line? 0What is the slope of a vertical line? Undefined

Discussion What does parallel mean? Don’t touchWhat makes two lines parallel? Same slopes

SWBAT write an equation of a line that is parallel to another line

Example 1: Write an equation of a line in slope intercept form that is parallel to y = 4x + 2 and goes through the point (4, 1). Let’s take a look at what they are asking first.

Now let’s try to write the equation. Remember we need a point and a slope in order to write an equation of a line. What is the slope of our new line? 4y – 1 = 4(x – 4)y – 1 = 4x – 16y = 4x – 15

Example 2: Write an equation of a line in slope intercept form that is parallel to y + 2x = 3 and goes through the point (-2, 5). Let’s take a look at what they are asking first.

Now let’s try to write the equation. Remember we need a point and a slope in order to write an equation of a line. What is the slope of our new line? -2; Because the original equation would be rewritten as y = -2x + 3

y – 5 = -2(x + 2)y – 5 = -2x – 4 y = -2x + 1

You always ask when can we use this. This can helpful when you are building cement steps. The forms to the steps must be parallel to each other in order to ensure that the steps are flat.

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Section 4-4: Parallel/Perpendicular Lines (Day 1) (CCSS: Prepares for G.GPE.5*)

You Try!Write an equation of a line in slope intercept form that is parallel to each line and goes through the given point. 1. y = 2x – 3; (-1, 3) y = 2x + 52. y + 3x = 4; (2, -3) y = -3x + 3 3. x = 3; (2, 4) x = 24. 3x + 2y = 5; (1, 5) y = -3/2x + 6 1/25. y = 5; (-1, 2) y = 2


Write an equation of a line in slope intercept form that passes through the given point and is parallel to the given line. Then state the slope, y-intercept, and x-intercept. Then graph the original line and new line.

1. y = 3x + 5; (3, 7) 2. y = -x – 1 ; (1, -5)y = 3x + 2, m = 3, y-int = 5, x-int = 2/3 y = -1x – 1, m = -1, y-int = -1, x-int = -4

3. y + 4x = 3; (-2, -5) 4. y = 4; (2, -3)y = -4x – 13, m = -4, y-int = -13, x-int = 13/-4 y = -3, m = 0, y-int = -3, x-int = none

5. ; (2, 3) 6. 2y + 5x = -3; (-2, 1)

y = 1/2x + 2, m = 1/2, y-int = 2, x-int = -4 y = -5/2x – 4, m = -5/2, y-int = -4, x-int = -8/5

7. x = -3; (1, -5) 8. y = -2x – 1; (1, 2)x = 1, m = undefined, y-int = none, x-int = 1 y = -2x + 4, m = -2, y-int = 4, x-int = 2

9. y = 4x; (1, 2) 10. y = -2x – 5; (0, 3)y = 4x – 2, m = 4, y-int = -2, x-int = 1/2 y = -2x + 3, m = -2, y-int = 3, x-int = 3/2

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Review QuestionWhat is the slope of a horizontal line? 0What is the slope of a vertical line? Undefined

Discussion What does perpendicular mean? Intersect at a right angle What makes two lines perpendicular? Opposite reciprocal slopesIt should be obvious that the signs on the slope are opposite (+ and -) because perpendicular lines go in opposite directions. But they just can’t have opposite signs. This would not create a right angle. (see 1st picture) Their steepness needs to be opposite (reciprocal) as well. This will create a 90° intersection. (see

2nd picture) Therefore perpendicular lines have slopes that are opposite reciprocals.

SWBAT write an equation of a line that is perpendicular to another line

Example 1: Let’s make sure we know how to find perpendicular slopes (opposite reciprocals):

a. m = , Perpendicular Slope = ? -3/2

b. m = , Perpendicular Slope = ? 4

c. m = 8, Perpendicular Slope = ? -1/8

d. Horizontal Line, Perpendicular Slope = ? Undefined

e. Vertical Line, Perpendicular Slope = Zero

Example 2: Write an equation of a line in slope intercept form that is perpendicular to y = 4x – 2 and goes through the point (5, -2). Let’s take a look at what they are asking first.

Now let’s try to write the equation. Remember we need a point and a slope in order to write an equation of a line. What is the slope of our new line? -1/4

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Example 3: Write an equation of a line in slope intercept form that is perpendicular to and

goes through the point (-3, 2). What is the slope of our new line? 2; Because the original equation would be rewritten as y = -1/2x + 3y – 2 = 2(x + 3)y – 2 = 2x = 6y = 2x + 8

You Try!Write an equation of a line in slope intercept form that goes through the given point and is parallel or perpendicular to the original line.

1. y = 3x + 1; (1, -2); Perpendicular y = -1/3x – 1 2/3

2. ; (-2, 3); Perpendicular y = -3x – 3

3. y = 5; (1, -1); Perpendicular x = 14. 2x + 5y = 1; (-1, 4); Parallel y = 2/5x + 3 2/55. x = 5; (-1, 2); Perpendicular y = 2


Write an equation of a line in slope intercept form that passes through the given point and is perpendicular to the given line. Then state the slope, y-intercept, and x-intercept. Then graph the original line and new line.

1. ; (2, -3) 2. y = 3x – 1; (1, -3)

y = -3x + 3, m = -3, y-int = 3, x-int = 1 y = -1/3x – 2 2/3, m = -1/3, y-int = - 2 2/3, x-int = - 8

3. y + 2x = -1; (-2, -4) 4. y = 4; (-1, 3)y = 1/2x – 3, m = 1/2, y-int = -3, x-int = 6 x = 1, m = undefined, y-int = none, x-int = -1

175


Write an equation of a line in slope intercept form that goes through the given point and is parallel or perpendicular to the original line. Then state the slope, y-intercept, and x-intercept.Then state the slope, y-intercept, and x-intercept.

5. y = 3x – 4; (3, 1); Parallel 6. ; (-3, 1); Perpendicular

y = 3x – 8, m = 3, y-int = -8, x-int = 8/3 y = 7x + 22, m = 7, y-int = 22, x-int = -22/7

7. x = -1; (3, -2); Perpendicular 8. 2x – y = 5; (1, -4); Parallel y = - 2, m = 0, y-int = -2, x-int = none y = 2x – 5, m = 2, y-int = -6, x-int = 3

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Review QuestionWhat makes two lines parallel? Same slopesWhat makes two lines perpendicular? Opposite reciprocal slopes

Discussion How do you get better at something? PracticeTherefore, we are going to practice writing equations today. We are going to have many days like this during the school year. In order for you to be successful, you need to take advantage of the time and ask questions from your classmates and teachers.

SWBAT write an equation of a line that is perpendicular/parallel to another line

Example 1: Write an equation of a line in slope intercept form that is perpendicular to y = 3x – 2 and goes through the point (-3, 2).

You Try!Write an equation of a line in slope intercept form that goes through the given point and is parallel or perpendicular to the original line. Then state the slope, y-intercept, and x-intercept.

1. y = 3x + 3; (1, -2); Parallel y = 3x – 5, m = 3, y-int = -5, x-int = 5/3

2. ; (4, -1); Perpendicular y = -3x + 11, m = -3, y-int = 11, x-int = 11/3

3. y = 2; (4, -1); Parallel y = -1, m = 0, y-int = -1, x-int = none4. y – 4x = 1; (-1, -2); Perpendicular y = -1/4x – 2 1/4, m = -3, y-int = -2 1/4, x-int = 9/25. y = -4; (-1, 6); Perpendicular x = -1, m = und, y-int = none, x-int = -16. 3y + 2x = 5; (-3, 2); Parallel y = -2/3x, m = -2/3, y-int = 0, x-int = 0


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Write the equation in slope intercept form then state the slope, y-intercept, x-intercept, and graph.

1. Write an equation of a line in slope intercept form that passes through (2, 5) and (3, 2).y = -3x + 1, m = -3, y-int = 1, x-int = 1/3

2. Write an equation of a line in the slope intercept form that has a slope of and passes through (2, -4).

y = 1/3x – 4 2/3, m = 1/3, y-int = 4 2/3, x-int = 14

3. Write an equation of a line in slope intercept form that is parallel to y = -2x + 5 and passes through (5, -7).y = -2x + 3, m = -2, y-int = 3, x-int = 3/2

4. Write an equation of a line in slope intercept form that is perpendicular to -2y – 8x = 2 and passes through (8, -4).y = 1/4x – 6, m = 1/4, y-int = -6, x-int = 24

5. Write an equation of a horizontal line that passes through the point (-2, 5).y = 5, m = 0, y-int = 5, x-int = none

6. Write an equation of a line in slope intercept form that is perpendicular to the x-axis and passes through (-7, -11).x = 7, m = undefined, y-int = none, x-int = -7

7. Write an equation of a line in slope intercept form that is parallel to y = 2 and passes through (-4, 5).y = 5, m = 0, y-int = 5, x-int = none

8. Write an equation of a line in slope intercept form that has a slope of and passes through (0, -6).

y = -2/3x – 6, m = -2/3, y-int = -6, x-int = -9

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Review QuestionHow are perpendicular lines related? Their slopes are opposite reciprocals.

Discussion Let’s say that we collected some data on the number of hours the students in our class studied and the grade they received. When plotted on a graph, would the data points be in a straight line? NoWhat are some things that would cause the data to be erratic? Difficulty of material, student

SWBAT read and create scatter plots

DefinitionScatter plot – graph that shows the relationship of two sets of data

To find how many pieces of data are in a scatter plot, just count the dots!

Example 1: What was the highest score in the class? 95%How many students studied for one hour? 4How many students scored above 75%? 12

Example 2: Draw a scatter plot of the following data set. Time spent at the mall and amount of money left.

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Time(minutes) Money

10 $20030 $15540 $14570 $8580 $8090 $5

Section 4-5: Scatter Plots (Day 1) (CCSS: S.ID.6.a, S.ID.6.b, S.ID.6.c, S.ID.7, S.ID.8, S.ID.9)

Time Studying vs. Test Score

6065707580859095

100

0 1 2 3 4

Time studying (Hours)

Test

Sco

re (P

erce

nt)

Test score

Things to remember when creating a scatter plot:1. Choose good starting and ending points for each axis2. Choose sensible scales3. Time always goes on the x-axis

You Try!1. Create a scatter plot based on the following sets of data.

2. Create a scatter plot based on the following sets of data.

DiscussionHow does this relate to our study of lines? The data is almost in a line pattern.Notice the data sets are sort of in a line pattern. They are not perfectly linear as that requires the ‘x’ and ‘y’ values to increase or decrease at a constant rate. Tomorrow we will try to summarize these data sets by drawing a line that best fits data sets.


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Time(hours)

Money Earned

1 $82 $154 $355 $456 $509 $85

Age of Car

(years)

Value of Car

1 $15,0002 $12,5003 $10,0006 $7,50010 $5,00015 $375

1. The following data sets represent the amount of time students spend playing Playstation and their average grades:

Time (hours) 1 2 3 4 5 6 7 8

Average Grade 96 98 85 83 78 81 68 65

a. Draw a scatter plot based on the data sets.

b. Predict the average grade for a student that plays for 15 hours.

c. Predict what the time would be when a student started to fail.

2. The following data sets represent the amount of time driving in a car and how far you traveled.

Time (hours) 1 2 3 4 5 6 7 8

Distance Traveled 45 105 140 210 270 325 385 420


b. Predict the total distance traveled after 12 hours.

c. Predict the amount of time necessary to drive 800 miles.

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Review QuestionWhy couldn’t you use a pie graph or bar graph for the two homework problems? Two sets of dataWhen should we use a scatter plot? When we are graphing two sets of data

Discussion What is the difference between the two scatter plots in each of your two homework problems? DirectionWhat is causing this to happen? Data; slope

SWBAT identify positive/negative relationshipsSWBAT draw a best fit line

DefinitionsPositive Relationship/Slope - up/right, x increases/y increases

Example: Driving Time and Distance What would happen if x and y were decreasing? It would still be positive. Think of a slope of a negative over a negative. It is still a positive slope.

This would not be a perfectly linear relationship. What would cause this data not to be in a “perfect” line? Traffic, pace of driving, red lights

Can someone give me another example of a positive relationship?

Negative Relationship/Slope - down/right, x increases/y decreases or vice versa

Example: Time and Battery Life on IphoneThis would not be a perfectly linear relationship. What would cause this data not to be in a “perfect” line? Sleep mode, using the web, taking pictures

Can someone give me another example of a negative relationship?

What type of relationship exists in homework examples 1 and 2? #1 negative, #2 positive

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x

y

x

y

No Relationship - scattered

Example: Example: hair color, grades

Can someone give me another example of a scattered relationship?

You Try!Determine whether a scatter plot of the data for the following might show a positive, negative, or no

relationship. 1. Time spent in the gym and your strength. Positive2. The amount of songs on your iPod and the amount of space left. Negative3. Total text messages and your bill. No relationship (if you have unlimited plan)

DefinitionBest fit line – line that summarizes the data set

Things to remember:1. Follow the basic direction of the data2. Same amount of points above and below the line3. Draw line through as many points as possible

Example 1: Draw a best fit line.

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x

y

Time Studying vs. Test Score

6065707580859095

100

0 1 2 3 4

Time studying (Hours)

Test

Sco

re (P

erce

nt)

Test score

Example 2: Draw a best fit line.

You Try!Draw a best fit line for each of your two homework problems.


1. Determine whether a scatter plot of the data for the following might show a positive, negative, or no relationship.

a. Age of a car and value of the car. b. The size of a family and the weekly grocery bill. c. The size of a car and the cost. d. A person’s weight and percent body fat. e. Time spent playing video games and time spent on outdoor activity.

2. Draw a best fit line for each graph.

a.

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b.

3. The following data set represents the average salary (in thousands) for people who have a four year college degree.

Year 1999 2000 2001 2002 2003 2004 2005 2006

Salary(Thousands) 52 55 58 62 62 66 67 69


b. Draw a best fit line.

c. What type of relationship exists between the two sets of data?

d. Predict the average salary in 2010.

e. Predict the year in which salaries will be $100,000.

4. The following data set represents the miles traveled and how much gas is left.


b. Draw a best fit line.

c. What type of relationship exists?

d. How many gallons should be left after you travel 300 miles?

e. How far did you travel if you have 12 gallons left?

f. What factors cause the points not to be in a straight line?

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Miles Traveled 25 100 110 140 220

Gallons of gas 14 10 8 7 4


Review QuestionExplain how to draw a best fit line. 1. Follow the basic direction of the data2. Same amount of points above and below the line3. Draw line through as many points as possible

Discussion What two things do you need to write an equation of a line? Slope, y-interceptWhat would the equation of this best fit line be?

y = __x + __The y-intercept is 250. Then calculate the slope by using two points on the best fit line. (40, 150) and (70, 90)

y = -2x + 250

SWBAT write the equation of the best fit line

Example 1: Days and money saved.Draw a scatter plot and best fit line. Then write the equation of the line.

DAYS Money Saved

2 2012 3517 3825 6535 7540 80

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y = __x + __The y-intercept is 15. Then calculate the slope by using two points on the best fit line.(0, 15) and (35, 75)

You Try!Write the equation of the best fit line for problems 3 and 4 from last night’s homework.


1. Determine whether a scatter plot of the data for the following might show a positive, negative, or no relationship.

a. The height of a person and their shoe size. b. The amount a student talks in class and their grade. c. The amount of shots a basketball player takes and the amount of shots they make. d. The color of someone’s shoes and their grade.

2. Write the equation of the best fit line.a.

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b.

3. The following data set represents the grade a person is in and their IQ.

Grade 5 6 7 8 9 10 11 12

IQ 75 78 85 100 102 120 125 140

a. Draw a scatter plot and a best fit line based on the data set.

b. Write the equation of the best fit line.

c. What do the slope and y-intercept represent in this equation?

d. What type of relationship exists between the two sets of data?

e. Predict the IQ after two years of college.

4. The following data set represents the amount of songs on your IPod and the amount of space left.

a. Draw a scatter plot and best fit line based on the data set.

b. Write the equation of the best fit line.

c. What do the slope and y-intercept represent in this equation?

d. What type of relationship exists between the two sets of data?

e. Predict the amount of space that would be left with 1200 songs.

f. What factors would cause the points not to be in a straight line?

g. How would the best fit line and slope change if we added the point 1700 songs, 4.0 gigs?

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Songs 100 250 300 400 650 750 1000

Space Left (Gigs) 7.5 7.1 6.7 6.3 5.4 5.1 4.3


Review QuestionHow do you write the equation of the best fit line?Use y = mx + b. Find the y-intercept (b). Then locate two points on the line. Then find the slope (m) between the two points.

Discussion Scatter plots and best fit lines are used in engineering. When engineers are designing roadways they must calculate how many lanes of traffic and traffic lights are needed. In order to do this, they collect data. They collect data on how many cars are added to the roads for different size housing plans. This data is then graphed on a scatter plot.

Once the data is graphed, a best fit line and equation are developed. The engineers use this equation the next time someone wants to put in a housing plan. They enter the amount of new homes into the best fit equation and get a value for how many new cars the development will add to the current roadways.The engineers use this information to figure out how many new lanes and traffic lights will be needed.

SWBAT will make up a data set that represents a positive/negative relationship

ActivityIf you truly understand something, then you can talk freely about it. Specifically, you should be able to come up with your own explanations about the topic. This is what we will be doing today.

1. Make up a data set (at least 10 points) that has a positive relationship. Then do the following:

a. Write a sentence describing your data set.b. List your data set in a table. c. Make a scatter plot.d. Draw a best fit line.e. Find the equation of the best fit line.

2. Make up a data set (at least 10 points) that has a negative relationship. Then do the following:

a. Write a sentence describing your data set.b. List your data set in a table. c. Make a scatter plot.d. Draw a best fit line.e. Find the equation of the best fit line.


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Review QuestionWhat does a graph of a positive relationship look like? Up/rightWhat does a graph of a negative relationship look like? Down/right

SWBAT study for the Unit 4 Test

DiscussionHow do you study for a test? The students either flip through their notebooks at home or do not study at all. So today we are going to study in class.

How should you study for a test? The students should start by listing the topics.

What topics are on the test? List them on the board- Slope- Slope Intercept Equation- Point Slope Equation- Parallel and Perpendicular- Scatter Plots

How could you study these topics? Do practice problems; study the topics that you are weak on

Practice Problems Have the students do the following problems. They can do them on the dry erase boards or as an assignment. Have students place dry erase boards on the chalk trough. Have one of the groups explain their solution.

For problems 1-6, find the slope, y-intercept, x-intercept, and graph.

1. Write an equation of a line in slope intercept form that passes through (5, 8) and (4, 3).y = 5x – 7, m = 5, y-int = -7, x-int = 7/5

2. Write an equation of a line in slope intercept form that has a slope of -3 and passes through (-4, 3). y = -3x – 9, m = -3, y-int = -9, x-int = -3

3. Write an equation of a line slope intercept form that is parallel to y = 3x – 7 and passes through (7, -2). y = 3x – 23, m = 3, y-int = -23, x-int = 23/3

4. Write an equation of a line in slope intercept form that is perpendicular to y + 4x = 5 and passes through (-3, -3). y = 1/4x – 2 1/4, m = 1/4, y-int = 2 1/4, x-int = 9

5. Write an equation of a line in slope intercept form that is parallel to x = 3 and passes through (2, 6).x = 2, m = undefined, y-int = none, x-int = 2

6. Write an equation of a line in slope intercept form that is perpendicular to y-axis and passes through the origin. y = 0, m = 0, y-int = 0, x-int = All Reals

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Unit 4 Review

7. Given the following data sets:a. Draw a scatter plot.b. Does a positive or negative relationship exist? +c. Write the equation of the best fit line. y = 1.5x + 10d. How many people should the cinemas expect if there are 345 cars in the lot? 600 e. What factors cause this data not to be perfectly linear? (Hint: my mom can drive

one way to the movies) Different amounts of people/car.f. How could this information help the owner of the cinema? Less cars = more people riding

together = more families = more family movies

# of cars in movie cinemas lot

# of people in the movie theatre

75 135100 168125 220150 305175 315200 425

8. After you do the review problems, pick out one or two topics that you are weak on and find three problems from your notes or homework and do them.

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Unit 4 Standardized Test Review

1. What are the slope and y-intercept represented in the graph? a. m = -1, b = -2 b. m = 1, b = -2 c. m = 1, b = 2 d. m = -1, b = 2

2. What is an equation for the line that passes through the coordinates (2, 0) and (0, 3)?a. y = -3/2x + 3 b. y = -2/3x + 2 c. y = -3/2x – 3 d. y = -2/3x – 2

3. Which of the following is the equation of a line with a slope of 0 and passes through the point (4, 6)?a. x = 4 b. x = -4 c. y = 6 d. y = -6

4. What are the slope and y-intercept of the line 7x – 3y = 4?a. m = 7/3, b = -4/3 b. m = -7/3, b = 4/3 c. m = -7/3, b = -4/3 d. m = 7/3, b = 4/3

5. Johnny’s restaurant sells hamburgers. The amount charged for a hamburger (h) is based on the cost for a plain hamburger plus an additional charge for each topping (t) as shown in the equation: h = .60t + 5. What does the number .60 represent in the equation?

a. The number of toppingsb. The cost of a plain hamburgerc. The additional cost for each toppingd. The cost of a hamburger with one topping

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The following problem requires a detailed explanation of the solution. This should include all calculations and explanations.

6. Joey collected data on how long he studies and what his test grade is.

a. Based on these two data points, write an equation of a line in slope intercept form.

Minutes Studying Grade

0 555 65

b. What is the slope, y-intercept, and x-intercept of the line?

c. Based on the following data set, draw a scatter plot and best fit line.

Minutes Studying Grade

0 555 65

10 7215 7020 8525 98

d. Write an equation for the best fit line.

e. What type of relationship exists between the two sets of data?

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UNIT 4 CUMULATIVE REVIEW

SWBAT do a cumulative review

DiscussionWhat does cumulative mean?All of the material up to this point.

Our goal is to remember as much mathematics as we can by the end of the year. The best way to do this is to take time and review after each unit. So today we will take time and look back on the first three units.

Does anyone remember what the first four units were about? Let’s figure it out together.1. Pre-Algebra2. Solving Linear Equations3. Functions4. Analyzing Linear Equations

Things to Remember:1. Reinforce test taking strategies: guess/check, eliminate possibilities, work backwards, and estimating.2. Reinforce the importance of retaining information from previous units.3. Reinforce connections being made among units.

In-Class Assignment

1. What set of numbers does belong?

a. Counting b. Whole c. Integers d. Rationals

2. 2(4x + 3) = 8x + 6 is an example of what property?a. Commutative b. Associative c. Distributive d. Identity

3. What is the value of -5 + 12 ?a. 17 b. -7 c. 7 d. -11

4. What is the value of -8 – 12 ? a. -20 b. 20 c. 4 d. -4

5. What is the value of 12 – 8.2 ? a. 20.2 b. 3.8 c. 4.8 d. 21

6. What is the value of (-2.5)(.34) ? a. -1.7 b. -85 c. -8.5 d. -.85

7. What is the value of -1.488 ÷ .24 ? a. -6.2 b. -62 c. -.062 d. -6

8. What is the value of ?

a. 17/18 b. 16/18 c. 43/18 d. 35/18

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9. What is the value of ?

a. 35/3 b. 38/3 c. 1/2 d. 12/3 10. 33

a. 3 b. 6 c. 9 d. 27 11. = a. 21 b. 29 c. 220.5 d. 87

12. = a. 16 b. 2 c. 6 d. 4

13. 103 ÷ (-2 • 5) – 2 a. -96 b. -98 c. -100 d. -102

14. 10(2x) – 2(4x) = a. 12x b. -28x c. 28x d. 14x

15. (x + 5) – (3x + 10) = a. x + 10 b. 4x + 6 c. -2x – 5 d. -2x + 15

16. 4(2x + 12) + 4 = 3x + 52 + 5x a. -25/3 b. 22/5 c. Empty Set d. Reals

17.

a. 35 b. -35 c. 5 d. 1

18. -3(2x + 7) = -6x + 11 a. 2 b. 1 c. Empty Set d. Reals

19.

a. 21 b. -11 c. 20 d. 14

20. Solve for y: -2(3a + y) = -5b

a. b. y = 5b – 2a c. d.

21. Solve y = -x – 2; given a domain of {-2, 0, 5} a. -2, 4, 5 b. 0, -2, -7 c. 0, 2, 5 d. 1, 2, 3

22. y = 2x – 4 a. b. c. d.

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23. y = 3 a. b. c. d.

24. If f(x) = 4x – 5, find f(3). a. 7 b. 8 c. 9 d. 10

25. Which equation is not a linear equation?

a. b. c. d.

26. Which equation is not a function?

a. b. c. d.

27. Write an equation for the following relation: (2, 10) (6, 8) (10, 6).

a. b. c. d.

28. Write an equation of a line that passes through the points (2, 10) and (7, 20).

a. y = -2x + 6 b. y = 4x + 12 c. y = 2x + 6 d. y = 2x – 11

29. Write an equation of a line that is perpendicular to and passes through the point (-2, 4).

a. y = -3x + 6 b. y = -3x – 2 c. y = 3x + 6 d. y = 2x – 11

30. Write an equation of a line that is parallel to y + 4x = -5 and passes through the point (5, -3). a. y = -4x + 17 b. y = 4x + 12 c. y = -4x + 6 d. y = 2x – 11

31. Write an equation of a line that is parallel to the y-axis and passes through the point (-2, -5). a. x = -4 b. x = -2 c. y = -4 d. y = -2

32. Write an equation of a line that passes through the point (-2, 4) and has a m = 3. a. y = 3x – 2 b. y = 3x – 10 c. y = 3x + 10 d. y = 3x

33. Write an equation of a line that has m = -2 and a y-intercept of -3. a. y = 2x – 3 b. y = -2x – 10 c. y = -2x – 3 d. y = -2x

34. What is the x-intercept of the line y = 3x + 9? a. 9 b. 3 c. -3 d. 0

35. Which of the following equations could be the best fit line for sets of data that include time traveled and miles traveled? a. y = 45x b. y = -45x c. y = 45x + 25 d. y = 45

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