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ContentsSlope and Equations of Lines................................................................................................................................................3

“I Can” Do Math (Expressions & Equations)..........................................................................................................................4

“I Can” Do Math (Expressions & Equations)..........................................................................................................................5

M4 L15 The Slope of a Non-Vertical Line Notes...................................................................................................................6

M4 L15 The Slope of a Non-Vertical Line Classwork 1.......................................................................................................8

M4 L15 The Slope of a Non-Vertical Line Classwork 2.....................................................................................................10

M4 L15 The Slope of a Non-Vertical Line Exit Ticket 10-24-2018........................................................................................12

M4 L16 Slope Formula Notes.............................................................................................................................................13

M4 L16 Slope (Computation) Classwork..........................................................................................................................15

M4 L16 Slope (Computation) Exit Ticket.........................................................................................................................16

M4 L17 Part 1 Rewrite Standard Form Notes....................................................................................................................17

M4 L17 Part 2 Rewrite Standard Form Notes....................................................................................................................18

M4 L17 Part 2 Rewrite Standard Form Classwork...........................................................................................................20

M4 L17 Rewrite Standard Form Exit Ticket.....................................................................................................................22

M4 L18 Graphing a line from the slope-intercept form.....................................................................................................23

M4 L18 Graphing a line from the slope-intercept form CW............................................................................................25

M4 L18 Graphing a line from the slope-intercept form Exit Ticket.................................................................................28

M4 L19 Graphing by finding the x and y-intercepts...........................................................................................................29

M4 L19 Classwork............................................................................................................................................................31

M4 L19 Graphing by finding the x and y-intercepts Exit Ticket.......................................................................................33

M4 L20 Create Equation from a Graph Notes....................................................................................................................34

M4 L20 Classwork 1.........................................................................................................................................................36

M4 L20 Create Equation from a Graph Exit Ticket..........................................................................................................38

M4 L21 Using Slope to Form an Equation Notes................................................................................................................39

M4 L21 Using Slope to Form an Equation CW.................................................................................................................41

M4 L21 Using Slope to Form an Equation Exit Ticket......................................................................................................43

M4 L22 Constant Rates Revisited Notes............................................................................................................................44

M4 L22 Constant Rates Revisited CW 11/9/2018.....................................................................................................46

M4 L22 Constant Rates Revisited Exit Ticket...................................................................................................................47

Standards: 8.EE.B.5 & 8.EE.B.6Module 4

Topic CLesson 15-23

Slope and Equations of Lines

“I Can” Do Math (Expressions & Equations)

I can understand the connections between proportional relationships, lines and linear equations.

8.EE.B.5 I can graph proportional relationships, interpreting the unit rate as the slope of the graph.

8.EE.B.5 I can use a table, an equation or graph to decide the unit rate of a proportional relationship.

8.EE.B.5 I can use the unit rate of a graphed proportional unit rate to compare different proportional relationships.

“I Can” Do Math (Expressions & Equations)

I can understand the connections between proportional relationships, lines and linear equations.

8.EE.B.6 I can use similar triangles to explain why the slope m is the same between two points on a non-vertical line in a coordinate plane.

8.EE.B.6 I can explain that an equation in the form of y=mx will represent the graph of a proportional relationship with a slope of m and y-intercept of 0.

8.EE.B.6 I can explain that an equation in the form of y=mx + b represents the

graph of a linear relationship with a slope of m and a y-intercept of b.

M4 L15 The Slope of a Non-Vertical Line NotesNotebook p.56

8.EE.B.5: Graph proportional relationships interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed.

Learning Target: ____________________________________________________________________________________________________________________________________________

Positive and Negative Slope

Slope is a number that describes the ____________________________________ of a line. Example 1 and 2: A line that goes through the origin.

① ②

① The slope is ____________ . ② The slope is ______________ .

Example 3: A line that doesn’t go through the origin.

③ Big Ideas:

1. We describe slope as an __________or a __________.

2.

③ The slope is ________________ .

M4 L15 The Slope of a Non-Vertical Line Notes Notebook p.57

Let’s look at one of the proportional relationships that we graphed in Lesson 11.

Elias mows a lawn at a constant rate. Suppose he mowed a 35-square-foot lawn in 2.5 minutes.

What is the slope of this line?

What is the unit rate of mowing the lawn?

When we graph proportional relationships, the ___________________ is interpreted as the _______________ of the graph of the line, which is why

slope is referred to as the __________________.

t (time in minutes)Linear Equation:

y= 352.5t

y (area in square feet)

0 y= 352.5

(0) 0

1 y= 352.5

(1 ) 352.5

=14

2 y= 352.5

(2 ) 702.5

=28

3 y= 352.5

(3 ) 1052.5

=42

4 y= 352.5

(4 ) 1402.5

=56

M4 L15 The Slope of a Non-Vertical Line Classwork 110/24/2018

Opening Exercise

Graph A Graph B

a. Which graph is steeper?

b. Write directions that explain how to move from one point on the graph to the other for both Graph A and Graph B.

Opening Exercise

Pair 1:

Graph A Graph B

c. Which graph is steeper?

d. Write directions that explain how to move from one point on the graph to the other for both Graph A and Graph B.

e. Write the directions from part (b) as ratios, and then compare the ratios. How does this relate to which graph was steeper in part (a)?

M4 L15 The Slope of a Non-Vertical Line Classwork 210/24/2018

1. What is the slope of this non-vertical line?


3. Which of the lines in Exercises 1 and 2 is steeper? Compare the slopes of each of the lines. Is there a relationship between steepness and slope?

M4 L15 The Slope of a Non-Vertical Line Classwork 2 10/24/2018




M4 L15 The Slope of a Non-Vertical Line Exit Ticket 10-24-2018

Name:________________________ Cohort: _____



M4 L16 Slope Formula Notes Notebook p.58 8.EE.B.6: Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane.

Learning Target: ____________________________________________________________________________________________________________________________________________

Miranda Leung, 10/24/18,

I can explain why the slope m is the same between any 2 points on a line using similar triangles.


Key idea:Use m to represent slopeGraphs are read from left to rightThe slope of a line is always the same regardless of what two ordered pairs are selected.

M4 L16 Slope (Computation) Notes Notebook p.59

_________________________________________________________________________________________________

_________________________________________________________________________________________________

M4 L16 Slope (Computation) Classwork 10/25/2018 Online Google Form

https://docs.google.com/forms/d/e/1FAIpQLSfXxrEt5pfc5yGtRzEukoRl5mzc0jzu-TIqmYuVHKFFY3IaVw/viewform?usp=sf_link

M4 L16 Slope (Computation) Exit Ticket 10/25/2018

Name:________________________ Cohort: _____

M4 L17 Part 1 Rewrite Standard Form NotesNotebook p.60

8.EE.B.6: Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane.

Learning Target: ____________________________________________________________________________________________________________________________________________

No guided notes. Check notebook p.60 and 61.


I can rewrite the standard form of an equation into slope-intercept form.

M4 L17 Part 2 Rewrite Standard Form NotesNotebook p.62


Learning Target: ____________________________________________________________________________________________________________________________________________

Example 1:You have $20in savings at the bank. Each week, you add $2 to your savings. Let y represent the total amount of money you have saved at the end of x weeks. Write an equation to represent this situation, and identify the slope of the equation. What does that number represent?

Example 2:A friend is training for a marathon. She can run 4 miles in28 minutes. Assume she runs at a constant rate. Write an equation to represent the total distance, y , your friend can run in x minutes. Identify the slope of the equation. What does that number represent?

Example 3:Four boxes of pencils cost $5. Write an equation that represents the total cost, y , for x boxes of pencils. What is the slope of the equation? What does that number represent?


I can rewrite the standard form of an equation into slope-intercept form.

M4 L17 Rewrite Standard Form Notes Notebook p.63

Example 4:

Solve the following equation for y : −4 x+8 y=24. Then, answer the questions that follow.

a. Based on your transformed equation, what is the slope of the linear equation −4 x+8 y=24?

b. Complete the table to find solutions to the linear equation.

x Transformed Linear Equation: y

-2

0

2

4

c. Graph the points on the coordinate plane.

d. Find the slope between any two points.

e. Note the location (ordered pair) that describes where the line intersects the y-axis.

M4 L17 Part 2 Rewrite Standard Form Classwork10/30/2018

1. Solve the following equation for y : 9 x+3 y=21. Then, answer the questions that follow.

a. Based on your transformed equation, what is the slope of the linear equation 9 x+3 y=21?



-1

0

1

2




2. Solve the following equation for y : 2 x+3 y=−6. Then, answer the questions that follow.

a. Based on your transformed equation, what is the slope of the linear equation 2 x+3 y=−6?



-6

-3

0

3




M4 L17 Rewrite Standard Form Exit Ticket10/30/2018

1. Solve the following equation for y : 5 x− y=4. Then, answer the questions that follow.

a. Based on your transformed equation, what is the slope of the linear equation 5 x− y=4?



-1

0

1

2

c. Graph the points on the coordinate plane. d. Find the slope between any two points.


M4 L18 Graphing a line from the slope-intercept form Notebook p.64


Learning Target: ____________________________________________________________________________________________________________________________________________

Example 1

Graph the equation y=23x+1. Name the slope and y-intercept point.

Example 2

Graph the equation y=−34x−2. Name the slope and y-intercept point.


I can graph the equation y=mx+b form using the information about slope and y-intercept point.

Example 3

Graph the equation y=4 x−7. Name the slope and y-intercept point.

You do:

2. Graph the equation y=52x−4.

a. Name the slope and the y-intercept point.

b. Graph the known point, and then use the slope to find a second point before drawing the line.

M4 L18 Graphing a line from the slope-intercept form CW10/31/2018

Opening ExerciseExamine each of the graphs and their equations. Identify the coordinates of the point where the line intersects the y-axis. Describe the relationship between the point and the equation y=mx+b .

a. y=12x+3 b. y=−3x+7

c. y=−23x−2 d. y=5 x−4

M4 L18 Classwork 10/31/2018

1. Graph the equation y=−3x+6.



2. The equation y=1 x+0 can be simplified to y=x . Graph the equation y=x .



3. Graph the equation y= 45x−5.



M4 L18 Graphing a line from the slope-intercept form Exit Ticket 10/31/2018

Name:__________________ Cohort:_______Exit Ticket

Ms.Leung said that the graphs of the equations below are incorrect. Find the student’s errors, and correctly graph the equations.

1. Student graph ofy=12x+4 :

Error:

Correct graph of the equation:

M4 L19 Graphing by finding the x and y-intercepts Notebook p.66


Learning Target: ____________________________________________________________________________________________________________________________________________


I can graph the equation in standard form Ax+By=C using intercepts.

Goal: Graph using x and y intercepts. Notebook p.67

Steps: Simply replace the symbols x and y with zero, one at a time, and solve.Graph the equation: 2 x+3 y=9.

①Replace x with zero, and solve for y to determine the _____-intercept point.2 (0 )+3 y=9

The y-intercept point is at¿¿.

②Replace y with zero, and solve for x to determine the _____-intercept point.2 x+3 (0 )=9

The x-intercept point is at ¿¿.

③Plot and Graph! Now that we know the intercepts, we can place those two points on the graph and connect them to graph the linear equation 2 x+3 y=9.

Summary:Graphing using ___________________ is an efficient way of graphing linear equations that are in _______________ form. Graphing using the __________ and ¿¿ point is the most efficient way of graphing linear equations that are in ______________ form. Creating a table and finding solutions is another way that we learned to graph linear equations. All three methods work, but some methods will save time depending on the form of the equation.


1. Graph the equation −3 x+8 y=24

This equation is in _______________________ form. I will graph using intercepts / slope and y-intercepts.

①Replace ¿¿ with _______, and solve for y to determine the _____-intercept point.


②Replace ¿¿ with _______, and solve for x to determine the _____-intercept point.


③Plot and Graph!


2. Graphthe equation x−6 y=15using intercepts .This equation is in _______________________ form. I will graph using intercepts / slope and y-intercepts.

①Replace ¿¿ with _______, and solve for y to determine the _____-intercept point.


②Replace ¿¿ with _______, and solve for x to determine the _____-intercept point.


③Plot and Graph!

M4 L19 Graphing by finding the x and y-intercepts Exit Ticket 11/1/2018 Name:__________________ Cohort:_______

Use your notes and classwork to guide you for step 1, 2 and 3.

M4 L20 Create Equation from a Graph NotesNotebook p.68


Learning Target: ____________________________________________________________________________________________________________________________________________

Given a line, we want to be able to write the equation that represents it. Which form of linear equation do you think will be most valuable for this task?

Standard Form or Slope – Intercept Form

It is because we can easily identify the ___________ and _________________ from both the equation and the graph.

Write the equation that represents the line shown.

Example 1 – Write equation in Slope-Intercept Form:

① Identify the y-intercept point.

② Find the slope.

③ Create the equation in slope-intercept form: y = mx + b


I can write the equation of a line by finding the y-intercept and slope from the graph.

Example 2 – Write equation in standard form: Notebook p.69


② Find the slope.

③ Create y = mx + b

④ Rewrite to standard form Ax + By = C

① ③

② ④

Example 3 – Different scale on the graph:


② Find the slope.



① ③

② ④

M4 L20 Classwork 1 11/5/2018

1. Write the equation that represents the line shown.


② Find the slope.



① ③

② ④


①

②

③

④

M4 L20 Classwork 2 11/5/2018


① ③

② ④



② Find the slope.



①

②

③

④

M4 L20 Create Equation from a Graph Exit Ticket11/6/2018

Name:__________________ Cohort:_______

1a. Write an equation in slope-intercept form that represents the line shown.

1b. Rewrite the slope-intercept form in standard form.

M4 L21 Using Slope to Form an Equation NotesNotebook p.70


Learning Target: ____________________________________________________________________________________________________________________________________________

Example 1

Let a line l be given in the coordinate plane. What linear equation is the graph of line l?

We can pick two points to determine the slope, but the precise location of the y-intercept point cannot be determined from the graph.

① Find the slope using the slope formula: Find two points, label x1y1 x2y2

Use the slope formula

Solve to find the slope

② Find the y-intercept point by substitution:

Pick one point (x, y)

Substitute in y = mx + b


Do you think it matters which point we choose to substitute into the equation? That is, will we get a different equation if we use the point compared to ? No, because there can be only one line with a given slope that goes through a point.


Recall what it means for a point to be on a line; the point is a solution to the equation. In the equation , is a solution, and is the slope. Can we find the value of ? Explain.Yes. We can substitute one of the points and the slope into the equation and solve for


I can determine the equation of a line from a graph by using information about slope and a point.

Example 2 Notebook p.71

Let a line l be given in the coordinate plane. What linear equation is the graph of line l?

We learned that we need to know the slope, so we must identify two points we can use to calculate the slope. Then we can use the slope and a point to determine the equation of the line.

① Find the slope using the slope formula: Find two points, label x1y1 x2y2

Use the slope formula

Solve to find the slope

② Find the y-intercept point by substitution:

Pick one point (x, y)

Substitute in y = mx + b


Do you think it matters which point we choose to substitute into the equation? That is, will we get a different equation if we use the point compared to ? No, because there can be only one line with a given slope that goes through a point.


Recall what it means for a point to be on a line; the point is a solution to the equation. In the equation , is a solution, and is the slope. Can we find the value of ? Explain.Yes. We can substitute one of the points and the slope into the equation and solve for

M4 L21 Using Slope to Form an Equation CW11/7/2018

Exercises1. Write the equation for the line l shown in the figure.

2. Write the equation for the line l shown in

the figure.

3. Write the equation for the line l shown in the figure.

4. Determine the equation of the line that goes through points (−4 ,5 ) and (2 ,3 ).

M4 L21 Using Slope to Form an Equation Exit Ticket11/7/2018

Name:_______________________________ Cohort:_______

1. Write the equation for the line l shown in the figure below. SHOW ALL YOUR WORK.

M4 L22 Constant Rates Revisited NotesNotebook p.72

8.EE.B.5: Graph proportional relationships interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed.


Learning Target: ____________________________________________________________________________________________________________________________________________

Example 1: Cristian paints a wall at a constant rate of 2 square feet per minute. Assume he paints an area y , in square feet, after x minutes.

a. Express this situation as a linear equation in two variables.

b. Sketch the graph of the linear equation.

c. Using the graph or the equation, determine the total area he paints after 8 minutes, 1 12

hours, and 2 hours. Note that

the units are in minutes and hours.


I can use what I know about slope to determine which of the two has a greater rate of change.

2. The figure below represents Nathan’s constant rate of walking. Notebook p.73

a. Nicole just finished a

5

-mile walkathon. It took

her 1.4hours. Assume she walks at a constant rate. Let yrepresent the distance Nicole walks in xhours. Describe Nicole’s walking at a constant rate as a linear equation in two variables.

b. Who walks at a greater speed? Explain.

3. The graph below represents the constant rate of watts of energy produced from a single solar panel produced by Company A.

Company B offers a solar panel that produces energy at an average rate of 325 watts in 2.6hours. Assuming solar panels produce energy at a constant

rate, determine which company produces more efficient solar panels (solar panels that produce more energy per hour).

a. If we let y represent the energy produced by a solar panel made by Company B in xminutes, then the constant rate is

3252.6

= yx

We need to compare the slope of the line for Company A with the slope in the equation that represents the rate for Company B.

The slope of the line representing Company A is _______________________.

The slope of the line representing Company B is ¿¿.

Since ¿¿, Company _________ produces the more efficient solar panel.

M4 L22 Constant Rates Revisited CW11/9/2018

1.

a. Lucio can build 3 birdhouses in 5 days. Assuming he builds birdhouses at a constant rate, write the linear equation that represents the situation.

b. The figure represents Alex’s constant rate of building the same kind of birdhouses. Who builds birdhouses faster? Explain.

2. Train A can travel a distance of 500 miles in 8 hours. Assuming the train travels at a constant rate, write the linear equation that represents the situation.a. The figure represents the constant rate of travel for Train B.

b. Which train is faster? Explain.

M4 L22 Constant Rates Revisited Exit Ticket11/9/2018

Name:_______________________________ Cohort:_______

1. Water flows out of Pipe A at a constant rate. Pipe A can fill 3 buckets of the same size in 14 minutes. Write a linear equation that represents the situation.

The figure below represents the rate at which Pipe B can fill the same-sized buckets.

Which pipe fills buckets faster? Explain.

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