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Name______________________________________ Period ___________
Name______________________________________ Period ___________
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Lesson 1 – rates
Vocabulary and Examples o Rate__________________________________________________
______________________________________________________
______________________________________________________
____________________________________________________
o And remember, a ratio is just a comparison of two numbers by
_____________________. Or, to put it simply, a ratio is basically a
_____________________.
This is a ratio: This is a rate: Students and tables are different units.
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o Unit Rate_____________________________________________
______________________________________________________
______________________________________________________
______________________________________________________
To find the unit rate, _________________________ both parts of the rate by
the _____________________________.
6𝑠𝑡𝑢𝑑𝑒𝑛𝑡𝑠2𝑡𝑎𝑏𝑙𝑒𝑠
÷ 2÷ 2
=
The unit rate can also be described as ______ students per table, or ______
students/table.
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o Unit Price ____________________________________________
______________________________________________________
______________________________________________________
______________________________________________________
______________________________________________________
For example, if it costs $20 for four books, then how much does it cost for one book?
Whenever you’re making a unit price ratio, put the money on top.
Divide both parts of the rate by the denominator. The unit price is __________________________-
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Sample Problems
1) Dream Stream offers 4 months of digital music streaming for $60. Mathster Music offers 6 months of digital music streaming for $75. Which streaming service is the better buy?
2) After 3.5 hours, Pasha had traveled 217 miles. If she travels at a constant rate of speed, how far will she have traveled after four hours?
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Work area for Self-check Quiz
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Lesson 2 – Complex Fractions and Unit
rates Vocabulary o Complex Fraction______________________________________
______________________________________________________
______________________________________________________
______________________________________________________
______________________________________________________
o Complex fractions are simplified when the numerator and denominator
are ___________________ (_______________ are
________________ and __________________ whole numbers.)
Simplifying Complex Fractions
o Remember, a fraction is also just a _____________________problem.
o ____________________ the numerator by the denominator to simplify
the complex fraction.
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Example
Dividing Fractions
Instead of dividing the fractions, _______________________ by the
____________________________.
Cross-Cancel If You Can
This reads as _______________ divided by
____________.
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Finding Unit Rates
When you have to find unit rates involving complex fractions, just treat the
complex fraction as a ________________ _________________.
Example
(Copy down the notes for solving this problem here.)
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Changing Percents to Fractions
Percent means “per 100” or “divided by 100.” Take the given percent and
_________________by 100.
Example
The tax rate is %. Express this rate as a fraction in simplest form.
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Sample Problems
1) Simplify the complex fraction: /012
2) Change the following percent into a fraction: 60 45%
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Lesson 3 – Convert Unit rates
Vocabulary o Unit Ratio_____________________________________________
______________________________________________________
______________________________________________________
______________________________________________________
______________________________________________________
o Dimensional Analysis___________________________________
______________________________________________________
______________________________________________________
______________________________________________________
______________________________________________________
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Common Relationships of Measure
The Process
• Like units will ____________________ when one is in the numerator
and the other is in the denominator.
• Set up your fractions in such a way that the units you don’t want
_____________out, leaving the units you do want.
Example A skydiver is falling at about 176 feet per second. How many feet per minute is he falling?
We want to get rid of __________________ (because we are converting to
minutes) so we’ll place seconds in the _________________ of the second
fraction.
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(Show example here)
So, the skydiver falls ______________________ feet per minute.
Multi-Step Example Sometimes we have to make several conversions in order to get to the desired
rate. You can set up the ____________________ all at once to achieve this.
A pipe is leaking 1.5 cups of oil per day. About how many gallons of oil per week is the pipe leaking?
In this example we want to change from cups to gallons and we want to change from days to weeks.
_________ gallon = _________ cups. (Show example here)
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Sample Problems
1) Water weighs 8.34 pounds per gallon. How many ounces per gallon is the weight of water?
2) Lorenzo rides his bike at a rate of 5 yards per second. About how many miles per hour can Lorenzo ride his bike?
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Work area for Self-check Quiz
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Lesson 4 – Proportional and nonproportional
Relationships
Vocabulary Proportional _______________________________________________
__________________________________________________________
__________________________________________________________
______________________________________________
Nonproportional ___________________________________________
__________________________________________________________
__________________________________________________________
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Equivalent Ratios ___________________________________________
__________________________________________________________
__________________________________________________________
Example 1 Determine if the table of values is proportional.
Time (hours) 2 4 6 8
Pages read 50 100 150 200
If you divide the pages read by the time (in hours) for each column, you get:
• 50 ÷ 2 = ______ pages/hr
• 100 ÷ 4 = ______ pages/hr
• 150 ÷ 6 = ______ pages/hr
• 200 ÷ 8 = ______ pages/hr
Given that the unit rate is the ________________ for every column, the table
of values is ____________________________.
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Example 2 It costs $10 per hour to play laser tag, plus a $5 entry fee. Is the number of hours you can play laser tag proportional to the total cost? First, make a table of values.
Hours of Laser Tag
Total Cost
Next, divide the columns.
• ______ ÷ ______ = $________ per hour
• ______ ÷ ______ = $________ per hour
• ______ ÷ ______ = $________ per hour
The number of hours you can play laser tag is __________
______________________ to the total cost, because the __________
___________ is not the same in each column.
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Sample Problems
1) 2)
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Lesson 5 – Graph Proportional Relationships
Vocabulary
Coordinate Plane ___________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
Quadrants _________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
Ordered Pair ______________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
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x-coordinate ______________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
y-coordinate _______________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
The Coordinate Plane
Graph point A at the ordered pair (-4,2)
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Identify Proportional Relationships • In the last lesson, we learned to identify proportional relationships by
________________ across columns to see if the result was the same for
every entry.
• We can also identify proportional relationships with
____________________.
• If the graph of two quantities is a ______________ _____________
that travels through the ________________ [point (0,0)], then the two
quantities are _______________________.
Example 1
The slowest mammal in the world is the tree sloth. It moves at a speed of 6 feet
per minute. Determine whether the number of feet the sloth moves is
proportional to the number of minutes it moves by graphing on the coordinate
plane.
First, make a table of values.
Time (min)
Distance (ft)
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Next, graph the ordered pairs on the coordinate plane.
The line passes through the ______________ (because in 0 minutes the sloth
would travel 0 feet) and the line is _________________ which means the
relationship _____ ________________________.
Dist
ance
(ft)
Time (min)
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Example 2
The cost of renting a video game is shown in the table. Determine whether the
cost is proportional to the number of games rented by graphing on the
coordinate plane.
Notice that when we extend the line the meet the y-axis, it doesn’t cross at the
____________. This means the cost of the video games is _______
____________________ to the number of video games rented.
We can also double check by comparing two of the entries.
≠
Number of games
Cost
($)
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Sample Problems
1) 2)
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Lesson 6 – Solve Proportional Relationships
Vocabulary
Proportion________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
Cross Products_____________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
68=34
This is a proportion because the two fractions are __________________.
The cross products are:
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Using Proportions to Write Equations
• Since cross products of proportions are _____________
____________, you can make an _________________ by setting the
cross products equal to each other.
• ________________ for the missing quantity.
Example 1
625
=𝑑30
Example 2
The ratio of girls to boys at Middle Earth High School is 2:3. If there are 600
students at MEHS, then how many are girls?
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Using Unit Rates
• _____________________ to find the unit rate.
• Write an _____________________ to express the relationship.
• ________________________ the given value into the equation.
Example
Mrs. Baker paid $2.50 for 5 pounds of bananas. Write an equation relating the
cost c to the total number of pounds p of bananas. How much would Mrs. Baker
pay for 8 pounds of bananas?
𝑚𝑜𝑛𝑒𝑦𝑝𝑜𝑢𝑛𝑑𝑠
=
The cost is __________ times the number of pounds.
c =
c =
c =
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Sample Problems
1) 2)
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Work area for Self-check Quiz
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Lesson 7 – Constant Rate of Change
Vocabulary
Rate of change_____________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
Constant rate of change______________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
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Using a Table to Determine the Constant Rate of Change
Basically, you can find the unit rate to determine the constant rate of change.
𝑐ℎ𝑎𝑛𝑔𝑒𝑖𝑛𝑚𝑜𝑛𝑒𝑦𝑐ℎ𝑎𝑛𝑔𝑒𝑖𝑛𝑐𝑎𝑟𝑠
=
The money earned increases by ______ per car washed. _________ per car is
the unit rate, and is a ________________ ___________ of
_____________.
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Use a Graph to Determine the Constant Rate of Change
• If your graph is _________________, then it represents a constant rate
of change (___________________ means it’s a straight line.)
• To find the rate of change, select two ____-coordinates and subtract
them. Subtract the corresponding ____-coordinates. Divide the change in
y by the change in x, and you’ll have the constant rate of change (also
known as __________.)
EFGHIJKHLKMJN(P1QPR)EFGHIJKHFTUVN(W1QWR)
=
The constant rate of change (or unit rate, or slope) is ______ miles for
every hour.
Take two coordinates and subtract them. Let’s take (6,240) and (2,80)
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Sample Problems
1) 2)
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Work area for Self-check Quiz
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Lesson 8 – Slope Vocabulary
Slope_____________________________________________________
__________________________________________________________
__________________________________________________________
Slope_____________________________________________________
__________________________________________________________
__________________________________________________________
Slope_____________________________________________________
__________________________________________________________
__________________________________________________________
Rise_____________________________________________________
__________________________________________________________
__________________________________________________________
Run_____________________________________________________
__________________________________________________________
__________________________________________________________
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Example 1 First, find the slope using the change in y divided by the change in x. Then, find the slope using rise over run.
𝑠𝑙𝑜𝑝𝑒 =∆𝑦∆𝑥
=(𝑦Z − 𝑦\)(𝑥Z − 𝑥\)
=
Now we can try rise over run. Find a point on the graph where it crosses the
gridlines clearly. Count up for rise and over for run.
𝑠𝑙𝑜𝑝𝑒 =𝑟𝑖𝑠𝑒𝑟𝑢𝑛
=
First, let’s identify two points by their coordinates. We can use (1,2) and (5,10). Any two points will work, just make sure they are on the line.
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Sample Problems
1) 2)Find the slope:
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Work area for Self-check Quiz
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Lesson 9 – Direct Variation
Vocabulary
Direct variation____________________________________________
__________________________________________________________
__________________________________________________________
Constant of variation________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
Constant of proportionality__________________________________
__________________________________________________________
__________________________________________________________
𝑦𝑥= 𝑘𝑦 = 𝑘𝑥
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Find the Constant of Proportionality from a Graph
• If the data forms a line, then the rate of change is constant. The constant of
proportionality (k) is y÷x.
• To find the constant of proportionality, _________________ the y-
coordinate by its corresponding x-coordinate.
𝑘 =𝑦𝑥=
𝑘 =𝑦𝑥=
𝑘 =𝑦𝑥=
𝑘 =𝑦𝑥=
Each ordered pair gives us a k of _________, which means that the pool fills at a rate of __________ inches every minute.
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Finding the Constant of Proportionality in the Equation of a Line
• Using __________________, we can rewrite the formula for k so that it
resembles the equation of a _______________.
• Lines typically follow the formula y=mx+b, where m is the slope and b is
the y-intercept.
• However, with direct variation, the line always passes through the
__________ (0,0) so there isn’t a b value (it would be zero.)
• Therefore, y=kx resembles y=mx, and the _____________ would be
the same as the constant of proportionality.
Rewrite With Algebra
𝑦𝑥= 𝑘
To isolate the y, we want to do the inverse operation. The opposite of dividing by x is multiplying by x. Repeat on the other side as well.
The x’s cancel.
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Example
The distance y traveled in miles by the Chang family in x hours is represented by
the equation y=55x. Identify the constant of proportionality. Then explain what
it represents.
The equation of this line, y=______x is very similar in format to the equation
for the constant of proportionality, y=_______x.
The constant of proportionality, or k, would be ______. This means the Chang
family traveled ___________ miles per hour.
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Determine Direct Variation
Not all situations with a constant rate of change are proportional relationships.
Not all linear functions are direct variations.
We can divide each ordered pair (y÷x) to see if we have a constant of proportionality. 111 =
192 =
273 =
354 =
Notice that we have a linear relationship, but the line does not pass through the ____________, so there is no direct variation (for direct variation to exist, the line must pass through the origin.)
These values are _______ identical, so there is no __________ ratio.
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Sample Problems
1) 2)
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Work area for Self-check Quiz
