Grade 6 Mathematics · Unit 9: Relationships Between Two VariablesStandard 6.AT.11

Lesson 9.2Analyze Graphs of Relationships Between Two Variables

Start hereWords, worked example, and sentence starters

Name Date Period

Learning target I can use a table and a graph to determine and analyze the relationship between two variable quantities.

Start hereWords for this lesson

Read each word before you begin. Say it out loud.

WordWhat it meansExample
graphSpanish: gráfica A picture on a coordinate plane that shows how two quantities are related. points that rise in a line
table of valuesSpanish: tabla de valores A list of matched pairs of values that shows a relationship using numbers. months 1, 2, 4 beside cost 24.95, 49.90, 99.80 — one row per pair.
ordered pairSpanish: par ordenado Two numbers written as (x, y) that name one point on a graph. The independent variable comes first. (3, 12)
axisSpanish: eje One of the two number lines that frame a graph. Each axis is labeled with a quantity from the problem. x-axis and y-axis
coordinatesSpanish: coordenadas The numbers in an ordered pair that tell how far to move along each axis to reach a point. x-coordinate, y-coordinate
Rate of changeSpanish: Tasa de cambio How much y changes each time x goes up by one. Up $4 for each hour.

How it worksWorked exampleturning a table into points on a graph

These numbers are not on your problems. The steps are. Follow them with your own numbers.

A rental site charges $7 per movie. Make a table for 1, 2, and 3 movies, then list the points to plot.

  1. Independent: number of movies (x). Dependent: total cost (y).
  2. Multiply each x by the rate: 1 → 7 · 2 → 14 · 3 → 21
  3. Write each row as an ordered pair (x, y): (1, 7), (2, 14), (3, 21).
  4. Plot each pair: move right along the x-axis first, then up to the y value.

Answer: Points to plot: (1, 7), (2, 14), and (3, 21). They rise in a straight path.

Now you trySame steps, your turn

The steps are the same as the worked example. The numbers are yours. Do the work.

A subway car holds 40 riders. Make a table for 1, 2, and 4 cars, then list the points to plot.

  1. Independent: (x). Dependent: (y).
  2. Multiply each x by riders: 1 → · 2 → · 4 →
  3. Write each row as an ordered pair: ( __ , __ ), ( __ , __ ), ( __ , __ ).
  4. Plot each pair: move first, then to the y value.
Answer:three ordered pairs to plot

Say it and write itSentence starters

Finish each sentence out loud with a partner. Then use them in your writing.

  • The independent variable is , so it goes on the axis.
  • The ordered pair ( , ) means that .
  • To plot a point I move first, then .
  • As x increases, y , so the points .

Word bank graphordered paircoordinatesaxisplotpointtableinputoutputrateincreasesorigin

Watch outA common mistake

Reversing the coordinates of a point — writing (cost, tickets) instead of (tickets, cost). The independent variable always comes first in the ordered pair.

Grade 6 Mathematics · Unit 9: Relationships Between Two VariablesStandard 6.AT.11

Lesson 9.2Analyze Graphs of Relationships Between Two Variables

Version ASupported practice

Mastery check☐ Exceeds☐ Meets target☐ Needs practice
Name Date Period

Learning target I can use a table and a graph to determine and analyze the relationship between two variable quantities.

  1. 1Circle the letter of the best answer. Show how you know.

    Football tickets cost $45 each. What is the total cost of 3 tickets?

    1. A$45
    2. B$48
    3. C$90
    4. D$135

    Hint What is the cost of ONE ticket?

    Show your work
  2. 2Circle the letter of the best answer. Show how you know.

    Which ordered pair belongs on the ticket graph for 2 tickets?

    1. A(2, 90)
    2. B(2, 45)
    3. C(2, 47)
    4. D(90, 2)

    Hint Which quantity goes first in the ordered pair?

    Show your work
  3. 3Circle the letter of the best answer. Show how you know.

    On the ticket graph, what should the two axes be labeled?

    1. ANumber of tickets and price of one ticket
    2. BNumber of tickets and total cost
    3. CTotal cost and $45
    4. DNumber of games and number of tickets

    Hint Which TWO quantities change in this situation?

    Show your work
  4. 4Circle the letter of the best answer. Show how you know.

    The Sandia Peak tram travels 1,200 feet each minute. How far has it traveled after 3 minutes?

    1. A1,200 feet
    2. B1,203 feet
    3. C2,400 feet
    4. D3,600 feet

    Hint How far does the tram go in ONE minute?

    Show your work
  5. 5Circle the letter of the best answer. Show how you know.

    A table shows a train traveling 48 kilometers every hour: 1 hour, 48 km; 5 hours, 240 km. How many kilometers does it travel in 2 hours?

    1. A48 km
    2. B50 km
    3. C96 km
    4. D144 km

    Hint How far in ONE hour?

    Show your work
  6. 6Circle the letter of the best answer. Show how you know.

    As the number of tickets purchased increases, the total cost…

    1. Adecreases
    2. Bstays the same
    3. Ccannot be determined
    4. Dincreases

    Hint Compare the totals for 1, 2, and 3 tickets.

    Show your work

Explain your thinkingFind a card you called a match and your partner did not. Pick one ordered pair from it and test it in cost = 45 × tickets — what does the point on the graph tell you that the table does not?

Sentence starter The ordered pair ___ does / does not match because 45 × ___ = ___.

How did it go? ☐ 4 · I can teach it☐ 3 · I've got it☐ 2 · I need a hint☐ 1 · I need help
Grade 6 Mathematics · Unit 9: Relationships Between Two VariablesStandard 6.AT.11

Lesson 9.2Analyze Graphs of Relationships Between Two Variables

Version BCore practice

Mastery check☐ Exceeds☐ Meets target☐ Needs practice
Name Date Period

Learning target I can use a table and a graph to determine and analyze the relationship between two variable quantities.

Part 1Understand the ideamodels and meaning
  1. 1Circle the letter of the best answer. Show how you know.

    The ticket table and ticket graph are two representations. Which statement about them is true?

    1. AThe graph shows a different relationship than the table
    2. BOnly the table can show that cost increases with tickets
    3. COnly the graph is a real mathematical representation
    4. DBoth show the same relationship — the table with numerical values, the graph in a more visual way
    Show your work
Part 2Apply itreal-world problems
  1. 2Circle the letter of the best answer. Show how you know.

    The Sandia Peak tram travels 1,200 feet each minute and the ascent takes 15 minutes. How long is the whole ascent in feet?

    1. A1,215 feet
    2. B12,000 feet
    3. C16,800 feet
    4. D18,000 feet
    Show your work
  2. 3Circle the letter of the best answer. Show how you know.

    A tram starts at 0 feet and travels 1,200 feet each minute. On its graph, minutes are on the horizontal axis and feet on the vertical axis. Which ordered pair lies on this graph?

    1. A(5, 6000)
    2. B(5, 1200)
    3. C(5, 5000)
    4. D(6000, 5)
    Show your work
  3. 4Circle the letter of the best answer. Show how you know.

    A table pairs hours with kilometers: (1, 48), (2, 96), (5, 240). How many kilometers are traveled each hour?

    1. A24
    2. B48
    3. C96
    4. D240
    Show your work
  4. 5Circle the letter of the best answer. Show how you know.

    Why is the line on the football ticket graph dashed instead of solid?

    1. AThe graph is unfinished
    2. BDashed lines always mean the cost is decreasing
    3. CThe ticket price changes between points
    4. DYou can only buy whole numbers of tickets, so the points between whole numbers do not make sense
    Show your work
  5. 6Circle the letter of the best answer. Show how you know.

    Using the ticket table, what are the coordinates of the point for 4 tickets?

    1. A(4, 180)
    2. B(4, 160)
    3. C(4, 49)
    4. D(180, 4)
    Show your work

Explain your thinkingFind a card you called a match and your partner did not. Pick one ordered pair from it and test it in cost = 45 × tickets — what does the point on the graph tell you that the table does not?

How did it go? ☐ 4 · I can teach it☐ 3 · I've got it☐ 2 · I need a hint☐ 1 · I need help
Grade 6 Mathematics · Unit 9: Relationships Between Two VariablesStandard 6.AT.11

Lesson 9.2Analyze Graphs of Relationships Between Two Variables

ChallengeExtension

Mastery check☐ Exceeds☐ Meets target☐ Needs practice
Name Date Period

Learning target I can use a table and a graph to determine and analyze the relationship between two variable quantities.

Part 1Understand the ideamodels and meaning
  1. 1Plot and label each point on the grid.

    Plot the ticket-cost relationship: one point for each number of tickets.

    012345604590135180225270TicketsTotal cost ($)
    Show your thinking
Part 2Apply itreal-world problems
  1. 2Circle the letter of the best answer. Show how you know.

    Nakai's savings account has a principal of $750 and earns 3.5% simple interest each year. How much interest does Nakai earn in one year?

    1. A$3.50
    2. B$26.25
    3. C$75.00
    4. D$262.50
    0%50%100%
    Work
  2. 3Circle the letter of the best answer. Show how you know.

    Nakai earns $26.25 in simple interest every year. On the graph of years vs. interest earned, which point represents 4 years?

    1. A(4, 105)
    2. B(4, 26.25)
    3. C(4, 78.75)
    4. D(4, 750)
    Show your work
  3. 4Circle the letter of the best answer. Show how you know.

    After 10 minutes, the Sandia Peak tram has traveled 12,000 feet. The whole ascent is 18,000 feet. How much farther must it travel to reach the top?

    1. A1,200 feet
    2. B5,000 feet
    3. C6,000 feet
    4. D12,000 feet
    Show your work
  4. 5Circle the letter of the best answer. Show how you know.

    A second tram travels 900 feet per minute. If both trams are graphed on the same axes (minutes vs. feet), how do the graphs compare?

    1. AThe 900 ft/min line rises more steeply
    2. BThe two lines are identical
    3. CThe Sandia Peak line (1,200 ft/min) rises more steeply than the 900 ft/min line
    4. DThe 900 ft/min line goes downward
    Show your work

Explain your thinkingFind a card you called a match and your partner did not. Pick one ordered pair from it and test it in cost = 45 × tickets — what does the point on the graph tell you that the table does not?

How did it go? ☐ 4 · I can teach it☐ 3 · I've got it☐ 2 · I need a hint☐ 1 · I need help