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

Lesson 9.39.3 Small Group · Group 2

Start hereWords, worked example, and sentence starters

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, explain what each part stands for, and build one for a situation I have not seen before.

Why This Matters This is your challenge small group for Lesson 9.3. You can already get the answer — so in here the answer is the starting point. We ask when the method holds, when it breaks, and how you would convince someone who disagrees.

Start hereWords for this lesson

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

WordWhat it meansExample
equationSpanish: ecuación A mathematical sentence with an equals sign that shows two amounts are the same. y = 5x
variableSpanish: variable A letter that represents a quantity that can change, or vary. x, y, c, h
rateSpanish: tasa A fixed amount of one quantity for every one unit of the other quantity. $3 for each pound
representSpanish: representar To show a relationship using a tool such as a table, a graph, or an equation. The same membership shown as a table, a graph, AND c = 24.95m.
define the variablesSpanish: definir las variables To state exactly what quantity each letter in an equation stands for. c = total cost
ConstantSpanish: Constante A number in an equation that never changes. the 5 in y = 5x

How it worksWorked example

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

Write an equation for the pairs (1, 6), (2, 12), (5, 30).

  1. 6 ÷ 1 = 6, 12 ÷ 2 = 6, 30 ÷ 5 = 6.
  2. The rate is 6 every time.
  3. Check: 6 × 5 = 30.

Answer: y = 6x

Now you trySame steps, your turn

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

A canoe rents for $15 per hour. Write an equation for the total cost of a canoe rental.

Answer:

Say it and write itSentence starters

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

  • In my equation, x stands for and y stands for .
  • The constant rate is because every y value is .
  • My equation is y = , which means .
  • I checked my equation by substituting for x.

Word bank equationvariablerateconstantdefinesubstitutetimespertablerowrelatecheck

Watch out!A common mistake

Writing the equation with addition

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

Lesson 9.39.3 Small Group · Group 2

Group 2Challenge 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, explain what each part stands for, and build one for a situation I have not seen before.

Why This Matters This is your challenge small group for Lesson 9.3. You can already get the answer — so in here the answer is the starting point. We ask when the method holds, when it breaks, and how you would convince someone who disagrees.

Part 1Understand the ideamodels and meaning
  1. 1Complete each step, then write the final answer.

    A plumber charges a $40 visit fee plus $25 for each hour. Let h = hours and c = total cost. Write an equation. What does a 3-hour job cost?

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

    Table A: when x = 1, 2, 3, y = 7, 14, 21. Table B: when x = 1, 2, 3, y = 7, 9, 11. Which equation matches Table B?

    1. Ay = 2x + 5
    2. By = 7x
    3. Cy = x + 6
    4. Dy = 9x
    Show your work
Part 2Apply itreal-world problems
  1. 3Complete each step, then write the final answer.
    Games (g)Total cost (c)
    1$10
    2$16
    3$22
    Look at how the cost changes.

    A bowling alley charges $10 for 1 game, $16 for 2 games and $22 for 3 games. Let g = games and c = total cost. Write an equation.

    1. 1How much does each game add?
    2. 2Test the shortcut c = 10g with 2 games. Does it work?
    3. 31 game costs $10, and $6 pays for the game. What is the fixed fee?
    4. 4Write the equation: rate times games, plus the fee.
    Final answer:
  2. 4Complete each step, then write the final answer.
    0123451234567891011121314151617181920Toppings (t)Total cost (c)(0, 5)(1, 8)(2, 11)(4, 17)
    The line starts at $5, not at 0.

    A graph shows the cost of a pizza. It goes through (0, 5), (1, 8), (2, 11) and (4, 17). Let t = toppings and c = total cost. Write an equation.

    1. 1What does the point (0, 5) mean?
    2. 2How much does each topping add?
    3. 3Write the equation.
    4. 41 topping costs $8. Why is the equation not c = 8t?
    Final answer:
  3. 5Complete each step, then write the final answer.
    Time (hours)217Total cost ($)34?119÷ ?÷ ?× ?× ?
    Go to 1 first, then scale up.

    A table shows 2 hours cost $34, 5 hours cost $85 and 7 hours cost $119. Let h = hours and c = total cost. Complete the equation: c = ___h.

    Show your work
    Final answer:
  4. 6Complete each step, then write the final answer.

    A gym charges $35 for 1 month, $55 for 2 months and $75 for 3 months. Let m = months and c = total cost. Complete the equation: c = 20m + ___.

    Show your work
    Final answer:

Explain your thinkingHow can you tell from a table whether the equation is c = rate × h or c = rate × h + fee? Use one row to prove it.

ChallengeWrite your own problem

Write a new word problem that uses this lesson's idea. Then solve it and show every step.

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