Unit 9 Projects — Teacher Answer Key
Two-Variable Relationships | 6.EE.9 | y = kx, tables, graphs
For teachers only — this page shows fully worked solutions using each project's default input values. Students who enter different (valid) values will get different numbers, but the same reasoning process applies. Use these as your grading reference and model responses.
Standards: 6.EE.9 — identify independent and dependent variables, write equations y = kx, analyze relationships using tables and graphs.
Version A — Streaming Channel Growth Lab
Phase 1 Identify Your Variables
Independent & Dependent Variables · 6.EE.9
Growth rate k = 50 subscribers/week
Starting subscribers = 0
y = 50x
Independent variable (x): weeks — the teacher/student controls how many weeks have passed.
Dependent variable (y): subscribers — the total subscriber count depends on how many weeks have passed.
Reasoning: The channel gains 50 subscribers every week, so k = 50. After x weeks, total subscribers y = 50 × x.
Sample 4 / 3 / 2 Rubric Scoring — Identifying Variables
4 (Expert): Student names both variables correctly and explains that "weeks" is controlled (independent) while "subscribers" changes as a result — referencing k in context.3 (Proficient): Both variables correctly identified with correct labels.
2 (Developing): Only one variable correct, or labels are swapped.
Phase 2 Build Your Data Table
Tables of Values · 6.EE.9
k = 50
x = 0 through 5
Table for y = 50x
| Weeks (x) | Equation | Subscribers (y) |
|---|---|---|
| 0 | y = 50 × 0 | 0 |
| 1 | y = 50 × 1 | 50 |
| 2 | y = 50 × 2 | 100 |
| 3 | y = 50 × 3 | 150 |
| 4 | y = 50 × 4 | 200 |
| 5 | y = 50 × 5 | 250 |
Each y value increases by 50 — the constant rate of change k.
Sample 4 / 3 / 2 Rubric Scoring — Table & Graph
4 (Expert): All six table values correct; student notes the constant +50 increase and connects it to the proportional relationship.3 (Proficient): All table values correct.
2 (Developing): At least 4 values correct, or minor arithmetic error in one row.
Phase 3 Read Specific Values & View the Graph
Reading the Graph · 6.EE.9
k = 50
Find y when x = 7
Find x when y = 300
Week 7 → 350 subscribers
Find y (forward): y = 50 × 7 = 350 subscribers
Find x (inverse): x = 300 ÷ 50 = 6.00 weeks to reach 300 subscribers
On the graph: the line passes through the origin (0, 0) and rises 50 units for every 1 unit rightward. Week 7 point is at (7, 350).
Sample 4 / 3 / 2 Rubric Scoring — Table & Graph
4 (Expert): Both readings correct (y given x and x given y); student explains the inverse operation x = y ÷ k.3 (Proficient): At least one reading correct with correct setup shown.
2 (Developing): One reading attempted; setup shown even if arithmetic is off.
Phase 4 Channel Showdown
Compare Relationships · 6.EE.9
Your channel k = 50
Rival channel k = 35
Compare at week x = 10
Your channel wins!
Your channel (y = 50x): y = 50 × 10 = 500 subscribers
Rival channel (y = 35x): y = 35 × 10 = 350 subscribers
Your channel leads by 150 subscribers at week 10. The higher k = 50 produces a steeper line on the graph.
Quick Check — Exact Answer
If y = 30x, what is y when x = 4?
y = 30 × 4 = 120 subscribers
The calculator accepts only 120 as correct (from
checkQuick()).Sample 4 / 3 / 2 Rubric Scoring — Writing the Equation
4 (Expert): Correct winner identified; student explains that the larger k means faster growth and a steeper graph line.3 (Proficient): Correct totals for both channels; correct winner named.
2 (Developing): One channel computed correctly; comparison attempted.
Deliverable Growth Report — Sample Expert Response
"My channel's independent variable is weeks (x) and my dependent variable is subscribers (y). My equation is y = 50x, which means I gain 50 new subscribers every week, starting from 0. After 7 weeks I will have 350 subscribers, and my channel will hit 300 subscribers partway through week 6 (exactly at week 6.00). My channel grows faster than the rival because my growth rate k = 50 is higher than the rival's k = 35 — on a graph, my line is steeper and pulls ahead by 150 subscribers at week 10."
Version B — Phone Plan Showdown
Phase 1 Identify Your Variables
Independent & Dependent Variables · 6.EE.9
Plan A monthly cost k = $25
Starting cost at month 0 = $0
y = 25x
Independent variable (x): months — the number of months subscribed is controlled.
Dependent variable (y): total cost ($) — depends on how many months are paid.
Every month you pay $25, so k = 25. After x months, total cost y = 25 × x.
Sample 4 / 3 / 2 Rubric Scoring — Identifying Variables
4 (Expert): Both variables named correctly; student justifies that months is chosen (independent) and cost accumulates as a result (dependent), with k identified as the monthly rate.3 (Proficient): Both variables correctly identified and labeled.
2 (Developing): One label correct, or variables are swapped.
Phase 2 Build a Cost Table
Tables of Values · 6.EE.9
Plan A k = $25/month
x = 1 through 6
Plan A Cost Table (y = 25x)
| Month (x) | Equation | Total Cost (y) |
|---|---|---|
| 1 | y = 25 × 1 | $25.00 |
| 2 | y = 25 × 2 | $50.00 |
| 3 | y = 25 × 3 | $75.00 |
| 4 | y = 25 × 4 | $100.00 |
| 5 | y = 25 × 5 | $125.00 |
| 6 | y = 25 × 6 | $150.00 |
Cost increases by $25 each month — the constant rate of change k.
Sample 4 / 3 / 2 Rubric Scoring — Table & Comparison
4 (Expert): All six cost values correct; student notes the constant $25 increase and connects it to the proportional relationship y = 25x.3 (Proficient): All values correct.
2 (Developing): At least 4 values correct or minor arithmetic error.
Phase 3 Compare Plan A vs. Plan B
Compare Two Relationships · 6.EE.9
Plan A k = $25/month
Plan B k = $18/month
Compare at month x = 6
Plan B is cheaper
Plan A (y = 25x): y = 25 × 6 = $150.00
Plan B (y = 18x): y = 18 × 6 = $108.00
Plan B saves the family $42.00 at month 6. On a graph, Plan B has a flatter (lower) line because its k is smaller.
Sample 4 / 3 / 2 Rubric Scoring — Table & Comparison
4 (Expert): Both totals correct; savings amount calculated; student explains the graph interpretation (lower k = flatter line = less cost).3 (Proficient): Both totals correct and correct plan identified as cheaper.
2 (Developing): One plan computed correctly; comparison attempted.
Phase 4 Make Your Recommendation
Decision & Equation Check · 6.EE.9
Plan A k = $25/month
Plan B k = $18/month
12-month totals
Plan B wins at 12 months!
Plan A: y = 25 × 12 = $300.00
Plan B: y = 18 × 12 = $216.00
Plan B saves $84.00 over a full year. On a graph, Plan B has the lower (flatter) line throughout all 12 months.
Quick Check — Exact Answer
If y = 12x, what is y when x = 5?
y = 12 × 5 = $60
The calculator accepts only 60 as correct (from
checkQuick()).Sample 4 / 3 / 2 Rubric Scoring — Writing the Equation & Communication
4 (Expert): Both 12-month totals correct; Plan B recommended with dollar-amount savings stated; graph interpretation included.3 (Proficient): Both totals correct; Plan B correctly recommended.
2 (Developing): One total correct; recommendation attempted with partial support.
Deliverable Plan Recommendation — Sample Expert Response
"The independent variable is months (x) and the dependent variable is total cost in dollars (y). Plan A's equation is y = 25x and Plan B's equation is y = 18x. After 6 months, Plan A costs $150 and Plan B costs only $108 — Plan B is already $42 cheaper. Over a full year (12 months), Plan A costs $300 versus Plan B's $216, so Plan B saves the family $84 annually. I recommend Plan B because its lower monthly rate k = 18 produces a flatter cost line and significant long-term savings."