Chapter 1 - Ratios and Proportional Reasoning

transcript

Name______________________________________ Period ___________

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.

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.

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 __________________________-

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?

Work area for Self-check Quiz

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.

Example

Dividing Fractions

Instead of dividing the fractions, _______________________ by the

____________________________.

Cross-Cancel If You Can

This reads as _______________ divided by

____________.

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.)

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.

Sample Problems

1) Simplify the complex fraction: /012

2) Change the following percent into a fraction: 60 45%

Lesson 3 – Convert Unit rates

Vocabulary o Unit Ratio_____________________________________________

______________________________________________________

o Dimensional Analysis___________________________________

______________________________________________________

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.

(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)

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?

Lesson 4 – Proportional and nonproportional

Relationships

Vocabulary Proportional _______________________________________________

__________________________________________________________

______________________________________________

Nonproportional ___________________________________________

__________________________________________________________

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 ____________________________.

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

The number of hours you can play laser tag is __________

______________________ to the total cost, because the __________

___________ is not the same in each column.

Sample Problems

Lesson 5 – Graph Proportional Relationships

Vocabulary

Coordinate Plane ___________________________________________

__________________________________________________________

Quadrants _________________________________________________

__________________________________________________________

Ordered Pair ______________________________________________

__________________________________________________________

x-coordinate ______________________________________________

__________________________________________________________

y-coordinate _______________________________________________

__________________________________________________________

The Coordinate Plane

Graph point A at the ordered pair (-4,2)

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)

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 _____ ________________________.

Time (min)

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

Sample Problems

Lesson 6 – Solve Proportional Relationships

Vocabulary

Proportion________________________________________________

__________________________________________________________

Cross Products_____________________________________________

__________________________________________________________

This is a proportion because the two fractions are __________________.

The cross products are:

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

=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?

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.

Sample Problems

Lesson 7 – Constant Rate of Change

Vocabulary

Rate of change_____________________________________________

__________________________________________________________

Constant rate of change______________________________________

__________________________________________________________

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

_____________.

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)

Sample Problems

Lesson 8 – Slope Vocabulary

Slope_____________________________________________________

__________________________________________________________

Slope_____________________________________________________

__________________________________________________________

Slope_____________________________________________________

__________________________________________________________

Rise_____________________________________________________

__________________________________________________________

Run_____________________________________________________

__________________________________________________________

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.

Sample Problems

1) 2)Find the slope:

Lesson 9 – Direct Variation

Vocabulary

Direct variation____________________________________________

__________________________________________________________

Constant of variation________________________________________

__________________________________________________________

Constant of proportionality__________________________________

__________________________________________________________

𝑦𝑥= 𝑘𝑦 = 𝑘𝑥

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=

Each ordered pair gives us a k of _________, which means that the pool fills at a rate of __________ inches every minute.

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.

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.

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 =

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.

Sample Problems

Chapter 1 - Ratios and Proportional Reasoning

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