Grade 6 Mathematics · Unit 3: Ratios & RatesStandard 6.AT.3a

Lesson 3.4Determine Equivalent Ratios Using Graphs

Start hereWords, worked example, and sentence starters

Name Date Period

Learning target I can graph the values from a ratio table as points on the coordinate plane.

Start hereWords for this lesson

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

WordWhat it meansExample
Coordinate planeSpanish: Plano cartesiano A grid with a line going across and a line going up to plot points. x across, y up
Ordered pairSpanish: Par ordenado Two numbers (x, y) that tell where a point is on a grid. (2, 14)
Linear patternSpanish: Patrón lineal Points that make a straight line on a grid. (1,2), (2,4), (3,6) all line up
ProportionalSpanish: Proporcional Two amounts that grow together at the same rate. A straight line from (0,0) through (2,6) and (4,12)
OriginSpanish: Origen The point (0, 0) where the two grid lines cross. 0 pages, 0 stickers
Ratio tableSpanish: Tabla de razones A table of equal ratios that shows a pattern. Cups of juice: 2, 4, 6 | Cups of water: 3, 6, 9

How it worksWorked exampleturning a ratio into points on a graph

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

A sticker sheet holds 7 stickers per page. Let x = pages and y = stickers.

  1. Make a table. 1 page → 7 2 pages → 14 3 pages → 21
  2. Write each row as an ordered pair. (1, 7) (2, 14) (3, 21)
  3. Plot the points and add (0, 0) — zero pages hold zero stickers.
  4. Draw one straight line through the origin and all the points.

Answer: The points (0, 0), (1, 7), (2, 14), (3, 21) lie on one straight line.

Now you trySame steps, your turn

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

A theater has 9 seats in every row. Let x = rows and y = seats.

  1. Make a table. 1 row → 2 rows → 3 rows →
  2. Write each as an ordered pair. ( 1 , ) ( 2 , ) ( 3 , )
  3. What does the origin ( 0 , 0 ) mean here?
  4. Find the point for 5 rows. ( 5 , )
Answer:the ordered pairs and the point for 5 rows

Say it and write itSentence starters

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

  • The point ( , ) means .
  • The origin means .
  • I found y by multiplying x by .
  • The points form a straight line because .

Word bank coordinate planeordered pairoriginx-axisy-axisplotpointstraight linetablemultiplyacrossup

Watch outA common mistake

A common mistake in Graph Ratio Tables is writing the ordered pair in the wrong order — putting the second table value first instead of matching (x, y) to the table's column order. For example, if a recipe table shows 2 cups of sugar for every 4 cups of flour, a student might plot (4, 2) instead of (2, 4), landing the point in a completely different spot on the grid. Before you plot, check: which quantity goes on the x-axis, and does your ordered pair start with that number?

Grade 6 Mathematics · Unit 3: Ratios & RatesStandard 6.AT.3a

Lesson 3.4Determine Equivalent Ratios Using Graphs

Version ASupported practice

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

Learning target I can graph the values from a ratio table as points on the coordinate plane.

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

    A recipe uses 1 cup of sugar for every 2 cups of flour. Plot the ordered pairs: (1, 2), (2, 4), (3, 6).

    012345012345678910Cups of SugarCups of Flour

    Hint Each pair is (sugar, flour): the first number tells you how far RIGHT, the second how far UP.

    Show your thinking
Part 2Apply itreal-world problems
  1. 2Circle the letter of the best answer. Show how you know.

    A ratio table shows (2, 6), (4, 12), and (6, 18). Which ordered pair comes next if the pattern continues?

    1. A(7, 20)
    2. B(8, 24)
    3. C(8, 22)
    4. D(10, 24)

    Hint Watch each coordinate separately: x goes 2, 4, 6, … and y goes 6, 12, 18, …

    Show your work
  2. 3Circle the letter of the best answer. Show how you know.

    When you graph equivalent ratios, the points will always form what shape?

    1. AA curved line
    2. BA straight line through the origin
    3. CA zigzag
    4. DA circle

    Hint Think about what happened when you plotted equivalent-ratio points earlier in this lesson.

    Show your work
  3. 4Circle the letter of the best answer. Show how you know.

    A ratio table shows (2, 8), (3, 12), and (5, 20). What is the y-value when x = 7?

    1. A21
    2. B24
    3. C28
    4. D35

    Hint Compare each pair: 2 → 8, 3 → 12, 5 → 20. What is done to x to get y every time?

    My work — one step per lineWhat I did to both sides
    1. 1LookWhat is done to the variable?
    2. 2UndoDo the opposite — both sides.
    3. 3SimplifyOne side at a time.
    4. 4CheckPut it back in.
  4. 5Circle the letter of the best answer. Show how you know.

    Which point would NOT be on the graph of the ratio 1:5?

    1. A(2, 10)
    2. B(3, 10)
    3. C(4, 20)
    4. D(5, 25)

    Hint On the 1:5 graph, every y-value is exactly 5 times its x-value.

    Show your work
Part 3Explain and correctreasoning
  1. 6Find the mistake. Explain it, then show the correct work.

    Level 1 — Spot the Common Mistake

    1. 1Read the ratioA recipe uses 1 cup of sugar for every 2 cups of flour. Plot the point for 3 cups of sugar.
    2. 2Find the flour3 cups of sugar means 3 + 2 = 5 cups of flour.
    3. 3Write the ordered pair(3, 5)

    Which step has the mistake? Explain what went wrong, then show the correct work.

    Hint Look at Step 2. The student added 2 to the sugar instead of multiplying by 2.

    Correct work
    Correct answer:

Explain your thinkingWhat pattern do the plotted points make? How does the graph show the ratio of 1 sundae to 2 ounces of sauce?

Sentence starter Every time the sundaes go up by ___, the ounces go up by ___, so the points line up in a ___ starting from ___.

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 3: Ratios & RatesStandard 6.AT.3a

Lesson 3.4Determine Equivalent Ratios Using Graphs

Version BCore practice

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

Learning target I can graph the values from a ratio table as points on the coordinate plane.

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

    Chef Reyes uses 3 cups of flour for every 1 cup of sugar. Plot the ordered pairs from the ratio table: (1,3), (2,6), (3,9), (4,12).

    012345603691215Cups of SugarCups of Flour
    Show your thinking
  2. 2Complete the table. Show how you found each value.

    Use the graph pattern y = 5x to complete the ratio table, then identify what the ordered pairs would be.

    xyOrdered Pair
    15(1, 5)
    2
    4
    6
    Show your work
Part 2Apply itreal-world problems
  1. 3Circle the letter of the best answer. Show how you know.

    Points from a ratio table are plotted at (1, 4), (2, 8), and (3, 12). What is the ratio of y to x for each point?

    1. A1:4
    2. B2:8
    3. C3:4
    4. D4:1
    Show your work
  2. 4Circle the letter of the best answer. Show how you know.

    Two students graph ratio tables. Student A plots (1,2), (2,4), (3,6). Student B plots (1,3), (2,6), (3,9). Whose line is steeper?

    1. AStudent A
    2. BSame steepness
    3. CCannot tell
    4. DStudent B
    Show your work
Part 3Explain and correctreasoning
  1. 5Find the mistake. Explain it, then show the correct work.

    On Level — Spot the Common Mistake

    1. 1Read the ratioA recipe uses 4 cups of water for every 1 cup of rice (rule: y = 4x). A student is plotting points for the line.
    2. 2List the points(1, 4), (2, 8), (3, 12)
    3. 3Add another pointThe student also plots (4, 14) and says it belongs on the same line.
    4. 4ConclusionAll four points form a straight line.

    Which step has the mistake? Explain what went wrong, then show the correct work.

    Correct work
    Correct answer:

Explain your thinkingWhat pattern do the plotted points make? How does the graph show the ratio of 1 sundae to 2 ounces of sauce?

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 3: Ratios & RatesStandard 6.AT.3a

Lesson 3.4Determine Equivalent Ratios Using Graphs

ChallengeExtension

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

Learning target I can graph the values from a ratio table as points on the coordinate plane.

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

    Two recipes are graphed: Recipe A uses 2 cups of mix per 1 serving, and Recipe B uses 3 cups of mix per 2 servings. Plot both sets of points and determine which recipe uses more mix per serving.

    0123456024681012ServingsCups of Mix
    Show your thinking
Part 2Explain and correctreasoning
  1. 2Find the mistake. Explain it, then show the correct work.

    Find Jada's Mistake

    1. 1Read the ratio tableCookies: 2, 4, 6 | Cups of milk: 1, 2, 3
    2. 2Write ordered pairs(1, 2), (2, 4), (3, 6)
    3. 3Plot the pointsPlotted (1,2), (2,4), (3,6) on the grid
    4. 4ConclusionThe x-axis is cookies and y-axis is cups of milk

    Which step has the mistake? Explain what went wrong, then show the correct work.

    Correct work
    Correct answer:
  2. 3Answer in complete sentences.

    A student plots the points (1, 3), (2, 6), (3, 9), and (4, 15) from a ratio table. She says they form a straight line because they go up. Is she correct? Explain how to check.

    Write your answer

Explain your thinkingWhat pattern do the plotted points make? How does the graph show the ratio of 1 sundae to 2 ounces of sauce?

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