share my lesson: the slope of a line

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.

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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)

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Lesson Vocabulary SlopeLattice PointVerticalHorizontal

Materials Graph PaperPencilRuler

Common Core State Standard

CCSS.MATH.CONTENT.8.EE.B.6Use 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)

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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)

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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.

Warm UpOBJECTIVE: 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

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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

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1

6

3

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3

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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)

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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.

Launch – What is slope?

Agenda

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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

Launch – What is slope?

Agenda

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Later on, Luis is going down two hills. Which hill will he gain more speed on?

HILL #3 HILL #4

Launch – What is slope?

Agenda

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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)

Launch – What is slope?

Agenda

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The harder it is to pedal up hill for Luis, the larger the slope.

SLOPE = POSITIVE

Launch – What is slope?

Agenda

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The more speed Luis gains going downhill, the more negative the slope is

SLOPE = Negative

Launch – What is slope?

Agenda

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When Luis is riding on flat ground, there is no slope, or a slope of zero.

SLOPE = 0

Launch – What is slope?

Agenda

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Luis can’t ride his bike straight up or down, so the slope is undefined.

SLOPE is undefined

Launch – What is slope?

Agenda

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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

Launch – What is slope?

Agenda

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The steepest stairs in the world can be found in Huashan mountain, in China. The slope here would be a large, positive number.

Launch – What is slope?

Agenda

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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.

Launch – What is slope?

Agenda

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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.

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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

Launch – What is slope?

Agenda

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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

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SLOPE = RISE/RUN = -2/1 = -2

Explore – Which Two Points to Pick?

Agenda

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Here is the main question:

Does it matter which two points we pick to find the

slope of a line?

Explore – Which Two Points to Pick?

Agenda

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Remember, similar triangles are triangles whose corresponding angles are congruent.

Also, the ratios of corresponding sides of similar triangles are equal!

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5

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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.

Explore – Which Two Points to Pick?

Agenda

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?

5

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10Here are two more similar triangles.Can you find the missing side on this triangle?!

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16

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?

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16

5

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Explore – Calculating Slope

Agenda

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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!

Explore – Which points to pick?

Agenda

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Explore – Student Share Out

Agenda

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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?

Summary

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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 − 𝑦1𝑥2 − 𝑥1

𝑦2 − 𝑦1

𝑥2 − 𝑥1

Summary

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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 − 𝑦1𝑥2 − 𝑥1

Note that an “m” is usually used to represent slope.

Practice

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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!

Practice

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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!

Practice

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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.

Practice

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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.

Practice

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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.

Practice

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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:

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-3

Rise = -3

Run = 1

Slope = Rise/Run = -3/1 = -3

Practice

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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!

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Practice

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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!

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Practice – Challenge Problem

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Agenda

The red triangle is 8 𝑏𝑦 3.

The blue triangle is 5 𝑏𝑦 2.

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3≠5

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Therefore, the triangles are not similar.

Since the triangles are not similar, the larger shape is actually a quadrilateral, not a triangle!

Assessment – Exit Ticket

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Agenda

Homework –

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Agenda

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