Slopes and Lines

Mr. Epstein

Y = mX + b

X

Y

How to Get Your Dream Car!

In 3 years, do you want to buy this:

Or this?

Agenda• Objectives for the Week

– Learn about the “slope” of a line– Learn formulas for lines– Learn properties of lines– Learn how to solve problems involving lines

• Objectives for the Day – Learn about slope

• Activities for the Day– Introduction to slopes: definitions and examples– Work Sheets done in pairs– If time, play “Who Wants to be a Millionaire” game

Example: Save $10 a week

Week Savings1 $102 $203 $304 $40..

1 2 3 4

40302010

Slope Definition: The change in Y divided by the change in X between 2 points.

Note(s): Sometimes called: Rate of change or “rise over run”. Abbreviated using the letter “m”.

Week

Savings

Example: Save $10 a week

1 2 3 4

40302010

m = change in y --------------------- = change in x

(40-10) 30---------- = ---- = 10(4-1) 3

Week

Savings

Slope Definition: The change in Y divided by the change in X between 2 points.

Note(s): Sometimes called: Rate of change or “rise over run”. Abbreviated using the letter “m”.

Example: Save $10 a week

1 2 3 4 Week

Savings

40302010

You can calculate the slope using two points, p1=(x1,y1) and p2=(x2,y2), using:m = y2 –y1

----------- x2 – x1

m = change in y --------------------- = change in x

(40-10) 30---------- = ---- = 10(4-1) 3 P2=(4,40)

P1= (1,10)

x1x2

y2 y1

y2x2

y1x1

Example: Car value drops

2001 2006Year

Value

$15,000

You can calculate the slope using two points, p1=(x1,y1) and p2=(x2,y2), using:m = y2 –y1

----------- x2 – x1

p1 = (2001,25000)p2 = (2006,15000) 15000 - 25000m = ---------------------------- 2006 - 2001 = -10000 ------------- 5

= -2000 (car drops in value $2000/year)

$25,000

Pair Activity

• There’s a challenge problem on the back if you get done with problems 1 & 2. But don’t worry if you don’t get to it!

• Both students in pair should complete and discuss problem 1 before moving to problem 2.

• You need to get done with problems 1 & 2! Raise your hand if you have a question!

Who Wants to be a Millionaire?

A: 1

C: 2

B: What’s the point?

D: 0

50:50

Question:

How many different points are needed to define a line?

10987654321

$1 Million$500,000$250,000$100,000$50,000$25,000$10,000$5,000$1,000$100

A: -1

C: Is he really gonna give us $1,000,000?

B: 3

D: 1

50:50

10987654321

$1 Million$500,000$250,000$100,000$50,000$25,000$10,000$5,000$1,000$100

Question:

What is the slope of the following line:

P2=(3,3)

P1=(0,0)

A: Up

C: Down

B: Not enough info

D: Negative slopes don’t exist

50:50

10987654321

$1 Million$500,000$250,000$100,000$50,000$25,000$10,000$5,000$1,000$100

Question:

Which way do lines with negative slope go when moving left to right?

A: Need 2 points to solve

C: 0

B: undefined

D: ∞ (infinity)

50:50

10987654321

$1 Million$500,000$250,000$100,000$50,000$25,000$10,000$5,000$1,000$100

Question:

What is the slope of the following vertical line?

A: -1/2

C: -5/2

B: -2

D: 2

50:50

10987654321

$1 Million$500,000$250,000$100,000$50,000$25,000$10,000$5,000$1,000$100

Question:

If p1=(1,3) and p2=(5,-5), what is the slope of the line connecting them?

A: 1

C: 2

B: 3

D: Can’t tell

50:50

10987654321

$1 Million$500,000$250,000$100,000$50,000$25,000$10,000$5,000$1,000$100

Question:

Which slope is the smallest?

Hint: Pay attention to sign of slope!

1

2

3

A: -6

C: -60

B: 1/6

D: 180

50:50

10987654321

$1 Million$500,000$250,000$100,000$50,000$25,000$10,000$5,000$1,000$100

Question:

The Storm Runner Roller Coaster drops 180 ft. If the horizontal distance from the peak to the base is 30 ft, what is the slope? (Hint: slope=change in y ÷ change in x. )

A: 3

C: 1

B: 2

D: All slopes are equal

50:50

10987654321

$1 Million$500,000$250,000$100,000$50,000$25,000$10,000$5,000$1,000$100

Question:

Which line has the greatest slope?

123

A: 5/11

C: 11/5

B: 5/6

D: -6/11

50:50

10987654321

$1 Million$500,000$250,000$100,000$50,000$25,000$10,000$5,000$1,000$100

Question:

As of Saturday, the Yankees record was 5 wins and 6 losses. If the games played is the X-axis (independent variable) and the wins is the Y-axis (dependent variable), what is the slope?

A: -1

C: a/b

B: b/a

D: Not enough info

50:50

10987654321

$1 Million$500,000$250,000$100,000$50,000$25,000$10,000$5,000$1,000$100

Question:

For any point p1=(a,b), where a and b are real numbers such that a ≠ b, what is the slope between this point and the point p2=(b,a)?

NOTE: Slope = (b-a)/(a-b) = -1

Congratulations!

You’re NOT a Millionaire

But,You have learned a lot about Slopes!!!!!