LESSON 1: Put It on the Plane • M3-125
LEARNING
GOALS
• Write and solve two-step equations to solve real-world problems.
• Use multiple representations to reason about quantities and analyze problem situations.
• Identify independent and dependent variables. • Interpret negative
solutions to problem situations.
You have solved two-step equations algebraically. How can graphs of linear equations be used to solve equations?
WARM UP
Angela charges $35 an hour for tutoring services plus a $5 travel fee if she has to go to the student’s house.
1. Name the quantities that are changing in this problem situation.
2. Name the quantities that remain constant.
3. Write an equation for the amount Angela charges, assuming she must travel to the student’s house. 4. If Angela made $75, how many hours did she tutor?
Put It on the
Plane
Representing Equations with Tables
and Graphs
1
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Getting Started
It’s All Greek to Me
Ms. Jackson translates books for a living. She decides to change her fees to keep up with the cost of living. She will charge an initial fee of $325 to manage each project and $25 per page of translated text. Ms. Jackson does not consider partial pages in her fees.
1. Name the quantities that change in this problem situation.
2. Name the quantities that remain constant.
LESSON 1: Put It on the Plane • M3-127
Graphing Linear Equations
AC T I V I T Y
1.1
You can represent a problem situation in many ways. You used verbal descriptions to represent the relationship between the number of pages Ms. Jackson translates and her total fees. Let’s consider how to represent the relationship with tables and graphs.
1. Complete the table that shows the various projects that Ms. Jackson has managed recently.
When representing a relationship as a graph, you need to ensure that the bounds of your graph are an appropriate size to surround the data from the table.
2. What is the least number of pages that Ms. Jackson could translate? What is the greatest number of pages that Ms. Jackson has translated recently?
3. What are the least and greatest amounts of money that Ms. Jackson has earned?
4. Consider the ranges of values in the table to choose lower and upper bounds for the x- and y-axes. Write the lower and upper bound values in the table shown for each quantity.
Variable Quantity Lower Bound Upper Bound Interval Pages Translated Earnings
5. Calculate the difference between the upper and lower bounds for each quantity and the number of tick marks that you have on each axis. Then, choose an appropriate interval for each axis and write these in the table.
Number of Pages Total Fees for the Project (dollars) 1 2 400 425 10 1150 2100 92
The interval is the number you are counting by on a given axis.
Sometimes, you have to adjust your bounds based on the interval you choose. Just make sure that your data remains visible within the bounds that you choose.
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6. Use the bounds and intervals to label each axis. Then, create a graph of the data from the table.
x y
Pages Translated
Total Fees (dollars)
7. Are the data continuous or discrete? Explain your reasoning.
8. Describe the relationship between the two quantities represented in the graph.
9. Write a linear equation to represent this situation. Make sure to define your variables.
When you first analyzed this situation, you listed two quantities that remain constant in this scenario: $325 and $25 per page.
10. Refer to your equation and graph to answer each question.
a. Where is $325 represented in the graph and in the equation?
b. Where is $25 per page represented in the graph and in the equation? Draw a line through the points on your graph to model the relationship.
LESSON 1: Put It on the Plane • M3-129 How would a solution to the equation appear on the graph?
11. Is there a proportional relationship between the number of pages translated and Ms. Jackson’s earnings? Justify your answer using the table, equation, and graph.
12. Use the graph to answer each question. Explain your reasoning.
a. Approximately how much money would Ms. Jackson earn if she translated 57 pages?
b. Approximately how many pages would Ms. Jackson need to translate to earn $750?
13. For each translating project Ms. Jackson completed this month, determine if her pay was correct. If it is not, state the amount she should have received. Explain each answer in terms of the equation and the graph.
a. Ms. Jackson translated a 23-page technical manual for Technicians Reference Guide Inc. She received a check for $900.
b. Ms. Jackson translated a 42-page year-end report for Sanchez and Johnson Law Office. She received a check for $1050.
c. Ms. Jackson translated a 35-page product specification document for Storage Pros. She received a check for $2075.
You have represented the situation with Ms. Jackson’s book translation business multiple ways: as a scenario, in a table, with an equation, and on a graph. These representations are useful for analyzing the situation in different ways.
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Interpreting Situations
in More Than One Quadrant
AC T I V I T Y
1.2
El Capitan is a 3000-foot vertical rock formation in Yosemite National Park in California. The granite cliff is one of the most popular challenges for experienced rock climbers. On July 3, 2008, Hans Florine and Yuji Hirayama scaled El Capitan in a record time of 2 hours 43 minutes and 33 seconds.
1. On average, about how fast in feet per minute did the record holders climb?
Two new climbers want to attempt to break the record by climbing El Capitan in 2 hours and 30 minutes.
2. If these climbers are to reach their goal, on average, how fast in feet per minute will they have to climb?
You want to watch the climbers attempt to break the record for climbing El Capitan. On the morning of the climb, you arrive late at 11:30 A.M. When you arrive, the climbers are exactly halfway to the top.
3. How many feet high are the climbers?
What are the two quantities that are changing in this problem situation?
LESSON 1: Put It on the Plane • M3-131
Pay attention to the units of measure!
4. Assuming they are climbing at the average rate needed, how many feet up the cliff will the climbers be:
a. in two more minutes?
b. in a quarter of an hour?
c. in one hour?
5. Consider the quantities in this scenario.
a. Which quantity depends on the other?
b. Identify and define the independent and dependent variables with their units of measure for this situation.
c. Write an equation for calculating the value of the
dependent variable when the value of the independent variable is given.
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6. Use your equation to determine how long after 11:30 A.M. it
will take the climbers to reach the top at 3000 feet. What time would the climbers reach the top?
7. Use your equation to determine when the climbers are 1400 feet up the cliff. What does this answer mean in terms of the problem situation?
8. Use your equation to determine how high up the cliff the climbers were:
a. two minutes before 11:30 A.M.
b. a half hour before 11:30 A.M.
9. Use your equation to determine how many minutes before 11:30 A.M. the climbers started to climb. What time of day
