SlopeSlope

Teacher: Mr. ValdiviaMr. ValdiviaEmail – mvaldivia@dadeschools.netGrade Level: 88 Subject: MathematicsMathematicsNGSSS: MA.8.A.1.2MA.8.A.1.2 Interpret the slope…when graphing a linear equation for a real-world problem. AlsoAlso MA.8.A.1.1MA.8.A.1.1

Opening QuestionsOpening Questions

Have you ever been skiing?

What is the difference between a beginner’s trail, often called the bunny slope, and the expert’s trail, often called the black diamond slope?

Which one of the two would have a steeper slope? Can we calculate the slope of a trail?

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Rise: The vertical change when the slope of a line is expressed as the ratio rise/run.

Run: The horizontal change when the slope of a line is expressed as the ratio rise/run.

Slope: A measure of steepness (or constant rate of change) of a line on a graph.

Vocabulary

Distancia Vertical: El cambio vertical cuando la pendiente de una linea se expresa como la razon distancia vertical/distancia horizontal

Distancia Horizontal: El cambio horizontal cuando la pendiente de una linea se expresa como la razon distancia vertical/distancia horizontal.

Pendiente: Medida de la inclinacion (o la tasa de cambio constante) de una linea en una grafica. Razon de la distancia vertical a la distancia horizontal.

Vocabulario

The constant rate of change of a line is called the slope of the line.

The rise is the difference of the y-values of two points on a line.

The run is the difference in the x-values of two points on a line.

The slope of a line is the ratio of rise to run for any two points on the line.

Y is the dependent variable.

X is the independent variable.

(5, 4)

(1, 2) Then count horizontally to a second point to find the run.

Count vertically from any one point on the line to find the rise.

Finding the Slope of a Line

slope = =24

12

The slope of the line is . 12

–2–4

slope = = 12

(3, 2)

(–1, –2)

Then count horizontally to a second point to find the run.

Count vertically from any one point on the line to find the rise.

Finding the Slope of a Line

slope = = 144

The slope of the line is 1.

–4–4slope = = 1

If you know any two points on a line, or two solutions of a linear equation, you can find the slope of the line without graphing. The slope m of a line through the points (x1, y1) and (x2, y2) is as follows:

yy2 2 –– yy1 1 xx2 2 –– xx11 m =

Find the slope of the line that passes through (–2, –3) and (4, 6).

Finding Slope, Given Two Points

Let (x1, y1) be (–2, –3) and (x2, y2) be (4, 6).

6 – (–3)4 – (–2) Substitute 6 for y2, –3 for

y1, 4 for x2, and –2 for x1.

96=

The slope of the line that passes through (–

2, –3) and (4, 6) is . 32

=y2 – y1

x2 – x1

32=

Find the slope of the line that passes through (–4, –6) and (2, 3).

Finding Slope, Given Two Points

Let (x1, y1) be (–4, –6) and (x2, y2) be (2, 3).

3 – (–6)2 – (–4)

Substitute 3 for y2, –6 for y1, 2 for x2, and –4 for x1.

96=

The slope of the line that passes through (–

4, –6) and (2, 3) is . 32

=y2 – y1

x2 – x1

32=

Example Problem

The table shows the total cost of gas per gallon. Use the data to make a graph. Find the slope of the line and explain what it shows.

Graph the data.

Cost of GasCost of Gas

GallonsGallons CostCost

00 00

33 66

66 1212

6

9

9

12

6 0

3

3 x

y

Gallons

Cost of Gas

Cost

Example Problem Continued

Find the slope of the line:

The slope of the line is 2. This means that for every gallon of gas, you will pay another $2.

=y2 – y1

x2 – x1

6 3=

12 – 6 6 – 3

= 2

The slope of a line my be positive, negative, zero, or undefined.

The slope of a line my be positive, negative, zero, or undefined.

Lesson Quiz: Part I

1. (4, 3) and (–1, 1)

2. (–1, 5) and (4, 2)

25

53

AssessmentAssessment

Find the slope of the line passing through each pair of points.

Lesson Quiz: Part II

3. The table shows how much money Susan earned as a house painter for one afternoon. Use the data to make a graph. Find the slope of the line and explain what it shows.

x

y

64 2 8 10 12 14 0

1020

30

4050

60

7080

The slope of the line is 7. This means Susan earned $7 for each hour worked.

AssessmentAssessment

2. The table shows the number of hours a student works and her earnings. Identify the slope of the line and explain what it shows.

A. The slope of the line is 15. This means that the student earns $15 for every hour that she works. B. The slope of the line is 30. This means that the student earns $30 for every hour that she works.

QuiQuizzStudent ResponseStudent Response

Class activityClass activity

RemediatiRemediationon

• Teacher will use the classroom floor tiles (1’x1’) as grid and create an x and y axis with blue tape. Teacher will draw lines with red tape for students to find slope.• Teacher will place two points on the gridded floor and students will find the slope of the points by using appropriate formula.

Group ProjectGroup Project

Students will log on to www.thatquiz.org Under Integers, click on Algebra. On the left hand side of the screen check Slope and change length to complete 20 exercises.

Students will pair in groups of 4 and create a ski resort to include 10 slopes for visitors. Students will label and color code the slopes beginner, intermediate, and advanced for skiers based on their calculations of slope. Students must also include lifts for transportation.