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

Lesson 9.3Write Equations to Represent Relationships Between Two Variables

Set B · Version ASecond form · supported practice

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

Learning target I can write an equation from a table or a graph to model the relationship between the dependent and independent variables.

Why This Matters Miguel is interested in joining a recreation center. Recreation World charges the monthly fee shown on the flyer. How can Miguel determine the cost of being a member of Recreation World for any number of months?

RememberWrite Equations to Represent Relationships Between Two Variables

Writing Two-Variable Equations. Formula: y = kx | Dependent = (Unit Rate) × Independent A table can only list some values. An equation captures the whole pattern at once, so it works for any number of months or hours.

  1. Identify the independent variable (x) and dependent variable (y).
  2. Find the constant unit rate (k = y ÷ x).
  3. Write the equation relating the two variables: y = kx (or y = x + c)

Words: equation (ecuación)variable (variable)rate (tasa)represent (representar)define the variables (definir las variables)

Part 1Understand the ideamodels and meaning
  1. 1Write the name of the correct group on each line.

    Sort each equation by what it really says.

    Groups: Says 'y is 4 times x' Says something different
    • y = 4x
    • y = x × 4
    • x = y ÷ 4
    • y = x + 4
    • x = 4y
    • y = 4 ÷ x

    Hint Test with a value: if x = 2 and y is 4 times x, then y must be 8. Which equations agree?

  2. 2Complete the table. Show how you found each value.

    A canoe rents for $15 per hour: c = 15h. Complete the table.

    Hours (h)SubstituteCost in dollars (c)
    2
    5
    8

    Hint Replace h with the number of hours, then multiply by the unit rate 15.

    Show your work
Part 2Apply itreal-world problems
  1. 3Circle the letter of the best answer. Show how you know.

    In the ordered pair (x, y), which value is graphed along the horizontal axis?

    1. Ay
    2. BBoth of them
    3. CNeither one
    4. Dx
    Show your work
  2. 4Circle the letter of the best answer. Show how you know.

    A table pairs x = 1, 2, 3 with y = 3, 6, 9. Which ordered pair is in the table?

    1. A(2, 6)
    2. B(2, 3)
    3. C(3, 2)
    4. D(6, 2)
    Show your work
  3. 5Circle the letter of the best answer. Show how you know.

    On a graph of hours worked and money earned, which axis shows the independent variable?

    1. AThe vertical axis
    2. BThe horizontal axis
    3. CEither axis works
    4. DNeither axis
    Show your work

Explain your thinkingBoth equations charge by a rate. What do m and h each stand for in their own situation, and why would c = 24.95m be wrong for the canoe?

Sentence starter In ___, ___ stands for ___, so the rate ___ means ___.

3-Point Exit Checklist: ☐ 1. Units & Labels Verified ☐ 2. Mathematical Reasoning Stated ☐ 3. Calculations Double-Checked
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 9.3Write Equations to Represent Relationships Between Two Variables

Set B · Version BSecond form · core practice

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

Learning target I can write an equation from a table or a graph to model the relationship between the dependent and independent variables.

Why This Matters Miguel is interested in joining a recreation center. Recreation World charges the monthly fee shown on the flyer. How can Miguel determine the cost of being a member of Recreation World for any number of months?

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

    A table shows x = 0, 1, 2, 3 and y = 5, 7, 9, 11. How much does y change each time x increases by 1?

    1. AIt increases by 1
    2. BIt increases by 2
    3. CIt increases by 5
    4. DIt increases by 7
    Show your work
  2. 2Circle the letter of the best answer. Show how you know.

    Diamond is considering using an online music service. She can either purchase songs one at a time or pay to stream songs. Each choice has its own relationship between the number of songs and the total cost. In Diamond's situation, what should be the independent variable in both equations?

    1. Athe total cost
    2. Bthe price per song
    3. Cthe type of service (purchase or stream)
    4. Dthe number of songs
    Show your work
  3. 3Circle the letter of the best answer. Show how you know.

    Diamond is considering using an online music service. She can either purchase songs one at a time or pay to stream songs. Each choice has its own relationship between the number of songs and the total cost. Why does Diamond need TWO separate equations instead of one equation for both options?

    1. APurchasing and streaming likely follow different cost rules (such as a per-song price vs. a flat rate), so each relationship between songs and cost is different
    2. BThe number of songs is different for each option
    3. COne equation can only have one variable
    4. DStreaming has no relationship between songs and cost
    Show your work
  4. 4Circle the letter of the best answer. Show how you know.

    Diamond is considering using an online music service. She can either purchase songs one at a time or pay to stream songs. Each choice has its own relationship between the number of songs and the total cost. Once Diamond has both equations, how does she use them to decide which option is better?

    1. AAdd the two equations together to find one combined cost
    2. BPick whichever equation has the smaller numbers written in it, without solving anything
    3. CSubstitute the same number of songs into both equations and compare the two total costs
    4. DGraph only one equation and assume the other follows the same line
    Show your work
  5. 5Circle the letter of the best answer. Show how you know.

    Recreation World charges $24.95 per month. Let m = number of months and c = total cost. Which equation represents the relationship?

    1. Ac = 24.95m
    2. Bc = 24.95 + m
    3. Cm = 24.95c
    4. Dc = m ÷ 24.95
    Show your work

Explain your thinkingBoth equations charge by a rate. What do m and h each stand for in their own situation, and why would c = 24.95m be wrong for the canoe?

3-Point Exit Checklist: ☐ 1. Units & Labels Verified ☐ 2. Mathematical Reasoning Stated ☐ 3. Calculations Double-Checked
How did it go? ☐ 4 · I can teach it☐ 3 · I've got it☐ 2 · I need a hint☐ 1 · I need help