Unit Rates and Proportions: Apply
Try This
Choose an activity you enjoy, such as walking, shooting basketball, or reading, etc. Time yourself for 1 minute to learn how much you can accomplish in that amount of time. You can also estimate what you think you could accomplish in 1 minute. For example, you might walk 120 mile, shoot 5 baskets, or read 2 pages. Now calculate your unit rate. Describe your unit rate with words, a table, a graph, and an equation. Unit Rates and Proportions: Apply
Targeted Activity
Level 1
It takes about 40 oranges to make a gallon of orange juice. The table lists the number of quarts, pints, cups, tablespoons, and teaspoons in a gallon. Find the number of oranges needed to make each measure of orange juice to complete the table:
Measure
Number Equivalent to One Gallon
Number of Oranges Required to Make 1 Measure of Orange Juice
gallon
1 gallon = 1 gallon
40 oranges make 1 gallon
quart
4 quarts = 1 gallon
____ oranges make 1 quart
pint
8 pints = 1 gallon
____ oranges make 1 pint
cup
16 cups = 1 gallon
____ oranges make 1 cup
tablespoon
256 tablespoons = 1 gallon
____ oranges make 1 tablespoon
teaspoon
768 teaspoons = 1 gallon
____ oranges make 1 teaspoon
Unit Rates and Proportions: Apply
Show What You Know
The Dog Walker graph shows how much you can earn as a dog walker. The horizontal axis shows the number of hours worked, and the vertical axis shows the amount of money earned for working those hours. Use the graph to answer the questions.
What is the hourly rate you could earn for walking dogs?
How much do you earn if you work 5 hours?
How much do you earn if you work 20 hours?
If you babysit for 4 hours, you will earn $30. Is the hourly rate for babysitting greater or less than the rate for dog walking? By how much?
Find the minimum wage for your state. Draw a graph to represent your state’s minimum wage and compare it to the rates you can earn by babysitting and walking dogs. How does it compare?
Dog Walker Graph
A coordinate plane contains a straight diagonal line that slants upward, from bottom left to upper right.
The x-axis is titled Hours Worked and ranges from 0 to 20 in increments of 5. The y-axis is titled Dollars Earned and ranges from 0 to 80 in increments of 20. The line begins at left parenthesis 0 comma 0 right parenthesis and passes through the following four closed points: left parenthesis 4 comma 20 right parenthesis, left parenthesis 8 comma 40 right parenthesis, left parenthesis 12 comma 60 right parenthesis, and left parenthesis 16 comma 80 right parenthesis. X-values are approximate.
Reveal Answer
Level 2
Last week Ian drove 136 miles and used 8 gallons of gas, Isela drove 144 miles and used 4.5 gallons of gas, and Laura drove 84 miles and used 3 gallons of gas. Determine the rates. Whose car got the most miles per gallon?
Last week Ian drove 136 miles and used 8 gallons of gas, Isela drove 144 miles and used 4.5 gallons of gas, and Laura drove 84 miles and used 3 gallons of gas. Determine the rates. Whose car got the most miles per gallon?
Reveal Answer
Level 3
Review the contents of the lessons in the unit. Then go back to the Check-In and Practice exercises and try them again.Check-In
Question 1
Dean wrote the equation y=13x to show the relationship between the number of hours, x , and the number of miles he can ride on his bike, y . He used his equation to calculate how many miles he could ride his bike in 5 hours:
yyy=13x=13×5=65
Dean determined that he could ride 65 miles in 5 hours.
What are the two quantities being related in this problem? How are they related?
What is another way Dean could have shown the relationship between the two quantities in this problem besides an equation?
How did Dean demonstrate his ability to reason abstractly and quantitatively? Reasoning Abstractly and Quantitatively
To be successful in math, it is important to be able to reason both abstractly and quantitatively.
To reason is to understand and work through a problem logically. Something is abstract if it is not a concrete, physical object. Therefore, to reason abstractly in math means to understand and work through a math problem using symbols as opposed to physical objects.
To reason quantitatively means to understand and work with the relationships between different quantities to solve problems.
Reasoning Abstractly and Quantitatively in the Real World
There are many fields in which reasoning abstractly and quantitatively is important. For example, computer programmers write code to represent objects or actions that they want a program to take. The code is made up of a set of abstract symbols that the programmers have to understand and work with. Can you think of other jobs where reasoning abstractly is a major part of the work?
As you work toward becoming a math student who can reason abstractly and quantitatively, here are three areas to work on:
Practice turning real scenarios into symbols.
Write an equation, make a table, or make a graph to represent the scenario described in the problem. Use the equation/table/graph to help solve the problem.
Practice explaining the meaning of the abstract symbols used in a problem.
When you are in the middle of a problem, check to make sure you can explain what all the variables you are using represent and how they are related.
Make sure you understand the relationship between two quantities in a problem and are able to represent that relationship in multiple ways.
Try to represent the relationship between two related quantities using a graph, a table, and an equation. How are all these representations of the same relationship similar? How are they different?
Example
Maryam’s little sister loves to do puzzles. After timing her one morning, Maryam realizes her sister does two puzzles every five minutes. Maryam wonders how many puzzles her sister could do in an hour. To solve this problem, she pulls out two puzzles at a time from the cabinet while counting by fives (to represent five minutes) until she gets to sixty. She determines that her sister could do 24 puzzles in one hour. What could Maryam do to reason through this problem more abstractly and quantitatively?
Solution
Maryam did a great job coming up with the answer using the puzzles as a physical prop to help her count. However, sometimes it is more time consuming or actually impossible to solve a problem using real physical objects.
Maryam could have focused on the relationship between the two quantities of time and number of puzzles. Her sister does 2 puzzles in 5 minutes. The relationship between number of puzzles and time is a proportional relationship.
Maryam could write an equation, make a table, or make a graph to represent the relationship between the two quantities and help solve her problem.
What if Maryam wrote an equation? First she could determine the unit rate. How many puzzles does her sister do in one minute? Her sister does 2 puzzles in 5 minutes, so she does 25 of a puzzle per minute. Maryam could write the equation y=25x where x is the number of minutes and y is the number of puzzles to represent the relationship between the number of minutes and the number of puzzles.
Maryam could then use the equation to determine the number of puzzles her sister could do in one hour, or sixty minutes. She can substitute 60 for x in her equation.
yyy===25×60120524
Using the equation, she gets the same answer that her sister could do 24 puzzles in one hour. One benefit to using an equation is that Maryam can now quickly determine how many puzzles her sister could do in any number of minutes.
Reveal Answer
Question 2
Odval is comparing the price of gas at two gas stations near her home. The gas station across the street from her (Gas Station A) is selling gas for $4.12 per gallon. Her brother tells her that he just got 5 gallons of gas from the gas station down the street (Gas Station B) for $19.95.
Odval knows that for each gas station, the relationship between the number of gallons of gas and the price is a proportional relationship. She knows the unit rate at Gas Station A is $4.12 per gallon. At Gas Station B, the unit rate is $19.955=$3.99 per gallon.
Odval decides to create a table and a graph to represent the proportional relationship between gallons of gas and price for each gas station.
Gallons of Gas Price ($) at Gas Station A Price ($) at Gas Station B
1 4.12 3.99
2 8.24 7.98
3 12.36 11.97
4 16.48 15.96
5 20.60 19.95
A coordinate plane shows gallons of gas on the x-axis ranging from 0 to 6 in increments of 1. The y-axis shows price in dollars ranging from 0 to 24 in increments of 2.
A solid line with an arrow at one end is labeled A and a dotted line with an arrow at one end is labeled B. The solid line extends from left parenthesis 0 comma 0 right parenthesis to approximately left parenthesis 6 comma 24.7 right parenthesis. The dotted line extends from left parenthesis 0 comma 0 right parenthesis to left parenthesis 6 comma 23.9 right parenthesis.
Odval feels that both her table and graph demonstrate that Gas Station B has a better deal on gas right now.
Interpret Odval’s table and graph. How do both the table and the graph show the relationship between the number of gallons of gas and the price?
How did Odval demonstrate her ability to reason abstractly and quantitatively?
The table and graph both show the relationship between the number of gallons of gas and the price at each gas station. The table lists the number of gallons of gas and the corresponding price at each gas station. The graph plots the number of gallons of gas on the x-axis and the price on the y-axis, with a line representing the relationship.
Odval demonstrated her ability to reason abstractly and quantitatively by using the unit rates to determine the price per gallon at each gas station. She then created a table and graph to represent the relationship between gallons of gas and price, showing that Gas Station B had a better deal on gas.