Mission 22 Unit 9 6.EE.C.9

Dependent Variables

6.EE.C.9 · Unit 10
Today's objective: Use variables to represent two quantities that change together.
Need a hint?
Re-read the problem and underline the numbers and the question. Pick one representation (model, table, or equation), show your steps, and check that your answer makes sense for the situation.

The school store sells custom spirit bracelets for $3 each. There is also a $5 setup fee for each order. The student council needs to figure out the total cost for any number of bracelets so they can plan the budget for spirit week. Your team must identify the independent and dependent variables, write an equation, and predict costs for different order sizes: 10, 25, 50, and 100 bracelets.

Input → Rule → Output INDEPENDENT Number of bracelets (b) c = 3b + 5 Rule / Equation DEPENDENT Total cost in dollars (c) Prediction Table Bracelets (b) Total Cost (c = 3b + 5) 10 3(10) + 5 = $35 25 3(25) + 5 = $80 50 3(50) + 5 = $155 100 3(100) + 5 = $305 Spirit Bracelets

The Problem

How much will it cost to order bracelets for spirit week? The school store charges $3 per bracelet plus a $5 setup fee per order. The student council has a budget of $200. Your team needs to: (1) identify which variable is independent and which is dependent, (2) write an equation, (3) predict costs for 10, 25, 50, and 100 bracelets, and (4) find the maximum number of bracelets they can afford.

Visual Model: Input-Output Machine

RULE MACHINE c = 3b + 5 Multiply by 3, then add 5 (cost per bracelet) + (setup fee) INPUT: b # of bracelets OUTPUT: c Total cost ($) b = 10 3(10) + 5 c = $35 b = 25 3(25) + 5 c = $80

Step-by-Step Investigation Guide

  1. Name the variables. What two quantities change in this problem? The number of bracelets and the total cost. Which one do YOU choose? Which one DEPENDS on your choice? Why is the number of bracelets the independent variable?
  2. Write the equation. If each bracelet costs $3 and there is a $5 fee, then: Total cost = 3 times (number of bracelets) + 5, or c = 3b + 5. What does the 3 represent? What does the 5 represent?
  3. Build a table. Use b = 10, 25, 50, and 100. Substitute each value into c = 3b + 5 and calculate. As b gets bigger, what happens to c? Does it increase by the same amount each time?
  4. Find the budget limit. The council has $200. Solve: 200 = 3b + 5. What is the maximum number of bracelets? If the answer is not a whole number, should you round up or down? Why?
  5. Interpret your results. Explain what each part of the equation means in the real-world context. Why does the cost start at $5 even with zero bracelets? What would the equation look like if there were no setup fee?

Language Support: Key Vocabulary

Independent Variable
The variable you choose or control. It goes on the x-axis. Here: number of bracelets (b).
Dependent Variable
The variable that changes because of your choice. It goes on the y-axis. Here: total cost (c).
Equation
A math sentence with an equal sign that shows how two things are related. Example: c = 3b + 5.
Substitute
Replace a variable with a number. Example: if b = 10, then c = 3(10) + 5 = 35.
Rate
How much one quantity changes for each unit of another. Here: $3 per bracelet.
Initial Value
The starting amount before any change happens. Here: $5 setup fee.
Sentence Frames:

"The independent variable is ___ because it is the value we ___."

"The dependent variable is ___ because it changes when ___ changes."

"When b = ___, the total cost c = 3(___) + 5 = ___."

Multiple Representations

Bracelets (b) Calculation Total Cost (c)
0 3(0) + 5 $5
10 3(10) + 5 $35
25 3(25) + 5 $80
50 3(50) + 5 $155
65 3(65) + 5 $200
100 3(100) + 5 $305
Bracelets (b) Cost ($) $0 $50 $100 $150 $200 $300 0 10 25 50 (10, $35) (25, $80) (50, $155) Budget: $200

Notice the line is straight because the cost increases at a constant rate ($3 per bracelet). The line crosses the y-axis at $5 (the setup fee).

In words: "The total cost depends on the number of bracelets ordered. For every additional bracelet, the cost goes up by $3. Even before buying any bracelets, there is a $5 setup fee. So if the student council orders 65 bracelets, they will spend exactly $200, which is their entire budget."

Team Roles

Facilitator
Read the scenario aloud. Guide the team to first name the variables before jumping to calculations. Keep the timer visible.
Model Builder
Draw the input-output machine and the coordinate graph. Plot the points from the table on graph paper.
Precision Checker
Verify each substitution: does 3(25)+5 really equal 80? Check that the budget equation is solved correctly. Confirm labels on axes.
Reporter
Write the final recommendation: "The council can order at most ___ bracelets because ___." Prepare to explain what independent and dependent mean.

Timed Lab Phases

Ready
Click a phase, then press Start.
03:00
  • Read the scenario. Underline the two quantities that change.
  • Assign roles.
  • Discuss: Which variable do you control? Which one depends?
  • Checkpoint: Can you name the independent and dependent variable?
  • Write the equation c = 3b + 5.
  • Build the table with b = 0, 10, 25, 50, 100.
  • Model Builder: draw the coordinate graph.
  • Checkpoint: Does your table have at least 5 rows? Does the graph show a straight line?
  • Solve 200 = 3b + 5 to find the maximum bracelets.
  • Explain what happens if the order is 66 bracelets instead of 65.
  • Write your budget recommendation.
  • Checkpoint: Does your answer make sense in context? (You cannot order partial bracelets.)
  • Reporter: practice explaining independent vs. dependent.
  • Team: quiz each other on the defense questions.
  • Make sure you can explain what the 3 and the 5 represent.
  • Checkpoint: Can everyone point to the graph and explain the budget line?

Challenge

Extension: A second vendor charges $2.50 per bracelet but has a $20 setup fee. Write the equation for Vendor B. For what number of bracelets does Vendor B become cheaper than Vendor A?
What If...?
  • What if the setup fee were $0? How would the equation and graph change?
  • What if the price per bracelet doubled to $6? How would the graph look different?

Real-Life Connection: Ride-share apps use the same idea: a base fare (initial value) plus a rate per mile. The total fare depends on the distance. Identifying which variable depends on which helps you compare services and make smart choices.

Defense Preparation

  1. Which is the independent variable and which is dependent? How do you know? "The independent variable is ___ because we choose it. The dependent variable is ___ because it changes based on ___."
  2. What does the 3 mean in c = 3b + 5? What does the 5 mean? "The 3 represents ___ and the 5 represents ___."
  3. How did you find the maximum number of bracelets for the $200 budget? "We solved the equation 200 = 3b + 5 by first ___ and then ___."
  4. How does the graph show the relationship between bracelets and cost? "The graph shows a straight line that starts at ___ and goes up by ___ for each bracelet."
Rubric Quick-Check:
  • Variables correctly identified as independent and dependent
  • Equation c = 3b + 5 written and used
  • Table with at least 4 input-output pairs
  • Graph with labeled axes and plotted points
  • Budget question solved and explained

Exit Product

Submit a one-page Lab Report that includes:
  • Variables identified and labeled (independent: b, dependent: c)
  • Equation: c = 3b + 5
  • Completed table with at least 5 rows
  • Coordinate graph with labeled axes and budget line
  • Written answer: maximum bracelets within $200 budget
Self-Assessment:
  • I can identify independent and dependent variables in a real situation
  • I can write an equation that shows how the variables are related
  • I can use a table and graph to show the relationship
  • I can solve the equation to answer a budget question

Work Space

Variables:



Table and Graph:



Solution and Defense: