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Slopes and Lines Mr. Epstein. Y. Y = mX + b. X. How to Get Your Dream Car!. In 3 years, do you want to buy this:. Or this?. 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 - PowerPoint PPT Presentation
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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!!!!!