Dudley can type 378 words in 3 minutes. How many
words can Dudley type per minute?
1
3
What if he could type 42 words in minutes, how many
words per minute is that?
Mark reduced the size of a rectangle to a height of 2
in. What is the new width if it was originally 24 in
wide and 12 in tall?
Rate: A ratio that compares two quantities measured in
different units.
Example: Miles Per Seconds
3 𝑚𝑖
1,700 𝑠𝑒𝑐𝑠
Unit Rate: A rate in which the second quantity in the
comparison is one unit.
60 𝑚𝑖𝑛𝑠
1 ℎ𝑟
Example: Minutes Per Hour
Always put the unit number 1 in the bottom right
proportion box.
Example:
20 𝑚𝑖𝑙
1
ℎ𝑜𝑢𝑟
3
=
? 𝑚𝑖𝑙
𝟏 𝒉𝒐𝒖𝒓
Exploring Rates
While remodeling her kitchen, Angela is repainting.
She estimates that she paints 55 square feet every
half-hour. How many square feet does Angela paint
per hour?
Area Painted
(sq ft)
55
Time (hr)
1
2
1
1
1
2
2
1
2
2
Josh
1
hikes
2
Exploring Rates
1
4
mile every 15 minutes, or hour.
1
6
1
Lisa hikes
3
mile every 10 minutes, or hour. How far do they each hike
in 1 hour? 2 hours?
Josh
Distance (mi)
Time (hr)
1
2
1
4
1
2
3
4
1
2
1
3
1
6
1
3
1
2
1
2
Lisa
Distance (mi)
Time (hr)
Who is going faster?
Proportional Relationship: One in which the rate of
change is constant , or one in which the ratio of one
quantity to the other is constant.
Proportion: A statement that two rates or ratios are
equivalent.
Example:
6 𝑚𝑖
2 ℎ𝑟
=
3 𝑚𝑖
1 ℎ𝑟
; the unit rate is 3 mph.
Constant
Variable
Coefficient
Constant: A Value that does not change.
Example:
3x + 2 = 14
Example 2: If you made $9 per hour of work.
Constant of Proportionality: A constant ratio of two
variables related proportionally. (UNIT RATE)
Example: Miles Per Hour
Miles
Hour
20
1
?
2
60
3
?
5
k=
𝑦
𝑥
or y = kx
where
𝑦 = 𝑟𝑖𝑠𝑒
𝑥 = 𝑟𝑢𝑛
Lets Look at the Formula:
k=
𝒚
𝒙
How do we get that to look like this?
y = kx
Multiply x to both sides (y and k)
𝒚𝒙
kx =
𝒙
K = Constant of Proportionality
Y = Rise
X = Run
Y
X
Find the constant of proportionality k. Then write an
equation for the relationship between x and y.
a.
b.
X
Y
2
10
4
20
6
30
8
40
k=
y=
Y
X
2
8
4
16
6
24
8
32
k=
y=
Find the constant of proportionality k. Then write an
equation for the relationship between x and y.
a.
b.
X
Y
2
14
4
28
6
42
8
56
k=
y=
Y
X
7.5
3
15
6
22.5
9
35
14
k=
y=
Let’s Do a Worksheet (or 2)
I’m going to help you set up some problems and walk
you through the rest, but most of this worksheet will
be done on your own. What you don’t finish is
homework and will be due tomorrow. If you don’t
turn it in, you will not be able to correct it for more
points.
Find the constant of
proportionality k. Then
write an equation for the
relationship between x and
y. Then graph the equation.
k=
y=
Y
40
30
20
10
X
2
X
Y
3 5 7 9
9 15 21 27
4
6
8
10
Independent Variable (X): You may freely choose a
value for x.
Dependent Variable (Y): A variable that depends on
one or more other variables.
The Coordinate Plane
Rise Over Run: How you calculate the slope of a line.
Example: k =
𝑟𝑖𝑠𝑒
𝑟𝑢𝑛
or y =
2
x
3
+4
Rise (y): The vertical change of the line on the graph.
Run (x): The horizontal change of the line on the graph.
**Hint**: If the slope is 3x (not a fraction) then you
make it a fraction and just put the slope over 1.
Example: y = 3x +2 is really y =
𝟑
x
𝟏
+2
Independent Variable (X): You may freely choose a
value for x.
Dependent Variable (Y): A variable that depends on
one or more other variables.
Positive Slope: Ex. 1x
Y
Negative Slope: Ex. -1x
Y
X
X
y = mx + b
m = Slope =
𝑦 = 𝑟𝑖𝑠𝑒
𝑥 = 𝑟𝑢𝑛
b = Y-Intercept (Where it crosses the yaxis).
Example:
2x
X
X is the independent variable
Y is the dependent variable
Y depends on X:
y = 2x
=2( )
Y
-2
-1
0
1
2
The variable y is not independent since it depends
on the number chosen for x.
Let’s Graph It: y = 2x
Y
X
Y
-2
-1
0
1
2
-4
-2
0
2
4
X
A.
B.
C.
D.
Undefined Slope
The slope of a vertical line. A vertical line has undefined slope
because all points on the line have the same x-coordinate.
y-axis
x-axis
Zero Slope
The slope of a horizontal line. A horizontal line has slope 0
because all its points have the same y-coordinate.
y-axis
x-axis
Graph the following Equation: y = 4x
Graph the following Equation: y = 2x + 3
Graph the following Equation: y = 2x - 2
Graph the following Equation: y =
𝟑
x
𝟐
Graph the following Equation: y =
𝟐
x
𝟑
+1
Graph the following Equation: y = -x + 2
Graph the following Equation: y = -
𝟏
x
𝟒
-2
Worksheet # 1:
y=
𝟏
x
𝟒
-1
Worksheet # 2:
y=-x+2
Worksheet # 5:
y = -3x - 3
Worksheet # 6:
y=4
Y-intercept = - 4
slope = 2
Write the equation:
________________
Y-intercept = -3
slope = -
𝟏
𝟒
Write the equation:
________________
Y-intercept = 5
slope =
𝟑
𝟐
Write the equation:
________________
Bell Ringer
12-8-14
A garden snail moves
rate?
1
1
6
1
3
foot in hours. What is the unit
Feet
Hours
1
3
2
3
3
3
=1