Slope and Rate of Change

Slope and Rate of Change Lesson

(slope = rate of change)

Four Types of Slope

• Positive: up and to the right OR down and to the left

• Negative: up and to the left OR down and do the right

• Zero: no vertical change, horizontal line 

• Undefined: no horizontal change, vertical line

Slope on a Graph

•  $\frac{rise}{run}$  

•  $\frac{change\ in\ y}{change\ in\ x}$  

•  $\frac{\Delta y}{\Delta x}$  

• Answer should be a fraction, improper fraction or whole number, no mixed numbers or decimals

More About Zero and Undefined Slopes

Horizontal lines have no vertical change, so when you do

 $\frac{rise}{run}$   you get 0 over a number.  Zero over any number will always be zero.



Vertical lines have no horizontal change, so when you do $\frac{rise}{run}$  you get a number over 0.  Because you cannot divide by zero, we say this is undefined.

Multiple Choice

What is the slope of the line?

• $m=\frac{3}{2}$

• $m\ =\ \frac{2}{3}$

• $m\ =\ -\frac{3}{2}$

• $m\ =\ -\frac{2}{3}$

Multiple Choice

What is the slope of this line?

• Zero

• -2

• undefined

• 8

Slope from a Table

• Find the change in the y column

• Find the change in the x column

• slope/rate of change =  $\frac{change\ in\ y}{change\ in\ x}$  

Multiple Choice

What is the slope shown on the table?

• m = -4

• m = $\frac{2}{3}$

• m = $\frac{3}{2}$

• m = $-\frac{2}{3}$

Slope from Two Ordered Pairs

• Use the slope formula

•  $\frac{y_2-y_1}{x_2-x_1}$  

• It doesn't matter which ordered pair is which

• This strategy can also be used for values in a table or point on a graph

Multiple Choice

Find the slope of the line that contains (3, -4) and (-1, 6)

• $m\ =\ 1$

• $m=\frac{5}{2}$

• $m=-\frac{5}{2}$

• $m=-1$

Slope/Rate of Change in Word Problems

• Marci travels 65 miles per hour

• Daniel earns $14 per hour

• Sheila bakes 3 cookies per person

• Ground beef costs $4.50 per pound

• ______per______ indicates a rate of change

• Could also be "each hat costs $14" or "car rental is $50 a day", etc.

• 
  

Ordering Pairs in a Word Problem

Hansel and Gretel are walking through the woods dropping breadcrumbs to help them find their way home. After ten minutes, their breadbasket has 120 crumbs left. After thirteen minutes, the basket has 96 crumbs left.  

What is the rate of change (breadcrumbs per minute)? How many breadcrumbs did they start with?

Ordering Pairs in a Word Problem

Hansel and Gretel are walking through the woods dropping breadcrumbs to help them find their way home. After ten minutes, their breadbasket has 120 crumbs left. After thirteen minutes, the basket has 96 crumbs left.  

This gives us the ordered pairs (10, 120) and (13, 96)

Ordering Pairs in a Word Problem

Hansel and Gretel are walking through the woods dropping breadcrumbs to help them find their way home. After ten minutes, their breadbasket has 120 crumbs left. After thirteen minutes, the basket has 96 crumbs left.  

 $\frac{96-120}{13-10}=-\frac{24}{3}=-8\$  

They drop 8 breadcrumbs per minute.

Ordering Pairs in a Word Problem

Hansel and Gretel are walking through the woods dropping breadcrumbs to help them find their way home. After ten minutes, their breadbasket has 120 crumbs left. After thirteen minutes, the basket has 96 crumbs left.  

 y = mx + b      m = -8     (x, y) = (10, 120)  

120 = -8(10) + b       120 = -80 + b     200 = b
They started with 200 breadcrumbs

(could have used either ordered pair)

Multiple Choice

Jack is running away from the Giant, and he has a pretty good head start. After five seconds, he is 43 feet away from the Giant. After ten seconds, Jack is 58 feet away.

Find the rate of change.

• 3 feet per second

• 5.8 feet per second

• 8.6 feet per second

Multiple Choice

Jack is running away from the Giant, and he has a pretty good head start. After five seconds, he is 43 feet away from the Giant. After ten seconds, Jack is 58 feet away.

How much of a head start did Jack get?

• 43 feet

• 28 feet

• 48 feet

• 38 feet