From: 21st Century Lessons: A Boston Teachers Union Initiative and Corey Cheever. Use this Common Core State Standards aligned lesson to engage middle school math students with learning about identifying the slope of a line, and graphing a line with a given slope. The "Do Now" will remind students about the order of operations when dealing with negative numbers and fraction bars. Then, the students will see a demonstration of positive, negative, zero, and undefined slope. During the exploration, students will find slope by definition (rise/run), and the practice will turn towards the slope formula. Finally, the homework assignment investigates slope with regards to geometry. Find this linear equation lesson and companion worksheets - all free - on Share My Lesson: http://www.sharemylesson.com/teaching-resource/the-slope-of-a-line-50033011/

1. 21st Century Lessons
The Slope of a Line
Primary Lesson Designer(s):
Corey Cheever
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21st Century Lessons – Teacher Preparation
• Spend AT LEAST 30 minutes studying the
Lesson Overview, Teacher Notes on each
slide, and accompanying worksheets.
• Set up your projector and test this PowerPoint file to make
sure all animations, media, etc. work properly.
Please do the following as you prepare to deliver this lesson:
• Feel free to customize this file to match the language and
routines in your classroom.

5. 5
Lesson Objective Students will be able to identify the slope of a line, and graph
a line with a given slope.
Lesson Description The Do Now will remind students about the order of
operations when dealing with negative numbers and fraction
bars. Then, the students will see a demonstration of positive,
negative, zero, and undefined slope. During the exploration,
students will find slope by definition (rise/run), and the
practice will turn towards the slope formula. Finally, the
homework assignment investigates slope with regards to
geometry.
Lesson Overview (1 of 4)

6. 6
Lesson Vocabulary Slope
Lattice Point
Vertical
Horizontal
Materials Graph Paper
Pencil
Ruler
Common Core
State Standard
CCSS.MATH.CONTENT.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; derive the equation y = mx for a line through the
origin and the equation y = mx + b for a line intercepting the
vertical axis.
Lesson Overview (2 of 4)

7. 7
Scaffolding During the practice, all questions are covered on the power
point slide. The teacher may choose to have the students work
first, then go over the answers, alternate between the students
seeing the answers and trying on their own, or working with
the students side by side.
Enrichment The last question of the practice asks students to give a
conjecture about slope and collinear points. This is a great
opportunity for students to experience a low threshold, high
ceiling question. This concept is also further explored in the
homework.
Online Resources for
Absent Students
https://www.engageny.org/resource/grade-8-mathematics-
module-5
Lesson Overview (3 of 4)

8. 8
Lesson Overview (4 of 4)
Before and After Before: Students need to have a strong understanding of adding
and subtracting integers, the order of operations, and plotting
points on a coordinate grid.
After: Students will be able to find the slope of any line, and graph
a line with a given slope. This prepares them to deal with functions
in the form: y = mx + b.
Topic Background Slope is an overlapping concept throughout each grade in the
Common Core standards. With regards to functions, it is very
important to have a deep understanding of slope before starting to
discuss linear functions. Slope can be found everywhere in the real
world, and this lessons includes multiple examples of slope in
everyday life.

9. Warm Up
OBJECTIVE: Students will be able to define slope and find the slope of a line
given two points on the line or given the graph of the line.
LANGUAGE OBJECTIVE: SWBAT define and describe the following words -
Slope, Lattice Point, Horizontal, Vertical.
Agenda
9
Evaluate each of the following expressions.
1.
𝟕−𝟑
𝟓−𝟑
= 2.
𝟒−(−𝟐)
𝟒−𝟑
=
3.
𝟒−𝟏𝟎
𝟐−(−𝟏)
= 4.
−𝟐−𝟑
−𝟏−(−𝟒)
=
6
-2
2
2
4

1
6


3
6
3
5

10. Agenda:
1) Warm Up – Arithmetic Review (Individual)
2) Launch – What is slope? (Class)
3) Explore – Which points to pick? (Partner)
4) Summary – The Slope Formula (Class)
5) Practice – Applying our new knowledge (Small Group)
6) Assessment – Exit Slip (Individual)
10
OBJECTIVE: Students will be able to define slope and find the slope of a line given two points
on the line or given the graph of the line.
LANGUAGE OBJECTIVE: SWBAT define and describe the following words - Slope, Lattice Point,
Horizontal, Vertical.

11. Launch – What is slope?
Agenda
11
Our friend Luis is riding his bike. He goes up two different hills. Which hill will
be harder for Luis to pedal up?
HILL #1 HILL #2

12. Launch – What is slope?
Agenda
12
Later on, Luis is going down two hills. Which hill will he gain more speed on?
HILL #3 HILL #4

13. Launch – What is slope?
Agenda
13
The measure of how hard it is for Luis to pedal up the hill, or how much speed
Luis gains down hill is known as slope. Here is the formal definition for slope.
Slope of a line – The ratio of
vertical distance between two
points and horizontal distance
between the same two points.
Here is an easy way to remember slope:
Vertical
(rise)
Horizontal
(run)

14. Launch – What is slope?
Agenda
14
The harder it is to pedal up hill for Luis, the larger the slope.
SLOPE = POSITIVE

15. Launch – What is slope?
Agenda
15
The more speed Luis gains going downhill, the more negative the slope is
SLOPE = Negative

16. Launch – What is slope?
Agenda
16
When Luis is riding on flat ground, there is no slope, or a slope of zero.
SLOPE = 0

17. Launch – What is slope?
Agenda
17
Luis can’t ride his bike straight up or down, so the slope is undefined.
SLOPE is undefined

18. Launch – What is slope?
Agenda
18
Here are some real world examples of slope. A wheelchair ramp can have a
slope of 1/12 at the most. This means for every 1 foot a person goes up, they
travel 12 feet across. This is a small positive slope.
12 feet
1 foot

19. Launch – What is slope?
Agenda
19
The steepest stairs in the world can be found in Huashan mountain, in China.
The slope here would be a large, positive number.

20. Launch – What is slope?
Agenda
20
The “steeps” of San Fransico include some of the biggest hills in a U.S. city. The
slope shown in the picture to the left would be negative.

21. Launch – What is slope?
Agenda
21
Let’s find the exact slope of a line on a coordinate axis.
Step 1 – Pick two points on the line that
that have integer coordinates.
These are called lattice points.
Step 2 – Draw a vertical line from the
LEFT dot to the same height as the
RIGHT dot.
Count the distance. This is the RISE.
2
1
Step 3 – Draw a straight across to
the right dot.
Count the distance. This is the RUN.
Step 4 – Divide the RISE by the RUN
to get the SLOPE.
SLOPE = RISE/RUN = 2/1 = 2

22. Launch – What is slope?
Agenda
22
Let’s find the exact slope of a line on a coordinate axis.
VERY IMPORTANT:
Remember, always go from left to
right.
If your vertical line goes down, your
RISE is negative.
If you can’t remember, think of Luis
Riding his bike!
-2
1
SLOPE = RISE/RUN = -2/1 = -2

23. Explore – Which Two Points to Pick?
Agenda
23
Here is the main question:
Does it matter which two
points we pick to find the
slope of a line?

24. Explore – Which Two Points to Pick?
Agenda
24
Remember, similar triangles are
triangles whose corresponding angles
are congruent.
Also, the ratios of corresponding sides of
similar triangles are equal!
3
5
6
10
We will need to remember what the term similar triangles means to answer our
question. Here is a definition that you will need to use during the exploration.

25. Explore – Which Two Points to Pick?
Agenda
25
?
5
16
10Here are two more similar triangles.
Can you find the missing side on this triangle?!
10
16
5
?

10
16
5
8


26. Explore – Calculating Slope
Agenda
26
Work with your partner. We will
investigate the slope of a line a
little more closely.
You will get a worksheet and a
ruler. You should:
-Remember rise/run goes left to
right
-Plot the points carefully
-Be careful with positives and
negatives!
1-Partners
2-Share Out
3-Discussion
In 10 minutes you will be asked to stop to discuss!
Click on the timer!

27. Explore – Which points to pick?
Agenda
27

28. Explore – Student Share Out
Agenda
28
Discussion - (5 Min)
Why were we able to find the length of
the missing side?
Click here to see an interactive display!
Did anyone find the missing side?

29. Summary
29
Agenda
No matter what two points we pick, we will create a triangle will its two sides in
the same ratio. Therefore, we can pick any two points we want!
(𝑥1, 𝑦1)
(𝑥2, 𝑦2)
The vertical distance, or the RISE
is the difference between the
two y values: 𝑦2 − 𝑦1.
The horizontal distance, or the
RUN is the difference between
the two x values: 𝑥2 − 𝑥1.
So, the slope is:
𝑆𝐿𝑂𝑃𝐸 =
𝑦2 − 𝑦1
𝑥2 − 𝑥1
𝑦2 − 𝑦1
𝑥2 − 𝑥1

30. Summary
30
Agenda
This is known as the slope formula.
(𝑥1, 𝑦1)
(𝑥2, 𝑦2)
Given two points with coordinates 𝑥1, 𝑦1 and
𝑥2, 𝑦2 the slope of the line that goes through the
two points is given by:
𝑚 =
𝑦2 − 𝑦1
𝑥2 − 𝑥1
Note that an “m” is usually used to
represent slope.

31. Practice
31
Agenda
1. Find the slope between the two lines using the slope formula.
a) (3, 5) and (-2, 3)
The slope is 2/5.
Every time the line rises two units, it goes to the right five units!

32. Practice
32
Agenda
1. Find the slope between the two lines using the slope formula.
b) (4, 24) and (6, 36)
The slope is 6.
Every time the line rises 6 units, it goes to the right one unit!

33. Practice
33
Agenda
1. Find the slope between the two lines using the slope formula.
c) (2, 1) and (2, -4)
The slope is undefined.
The line goes straight up and down.

34. Practice
34
Agenda
1. Find the slope between the two lines using the slope formula.
d) (3.2, -2) and (3, -1)
The slope is -5.
Every time the line goes DOWN 5 units, it goes to the right one unit.

35. Practice
35
Agenda
2a. Find the slope of the line.
1. Pick two lattice points:
(0, 2) and (4, 4).
2. Use the slope formula:
The slope of this line is ½. Every time the line goes up one
unit, it goes to the right by two.

36. Practice
36
Agenda
2b. Find the slope of the line.
1. Pick two lattice points:
(0, 2) and (1, -1).
The slope of this line is -3. Every time the line goes DOWN
three units, it goes to the right by one.
2. Let’s do this one by counting:
1
-3
Rise = -3
Run = 1
Slope = Rise/Run = -3/1 = -3

37. Practice
37
Agenda
3a. Find missing side of the triangle.
The slope of the line is 2/3.
So, this triangle must have sides in
the ratio of 2/3.
The length of the missing side is 4!
4

38. Practice
38
Agenda
3b. Find the missing side of the triangle.
The slope of the line is -5/2.
So, this triangle must have sides in
the ratio of -5/2.
The length of a triangle cannot be
negative, so the length of the
missing side must be 8!
8

39. Practice – Challenge Problem
39
Agenda
The red triangle is
8 𝑏𝑦 3.
The blue triangle is
5 𝑏𝑦 2.
8
3
≠
5
2
Therefore, the triangles
are not similar.
Since the triangles are not
similar, the larger shape is
actually a quadrilateral, not a
triangle!

40. Assessment – Exit Ticket
40
Agenda

41. Homework –
41
Agenda

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Teacher Development and Advancement
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