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LEVEL 2 · ENRICHMENT

Relationships Between Variables

Standard: 6.EE.C.9 (Independent & Dependent Variables, Tables, Graphs, Equations)

Directions: Solve each problem, showing tables and equations. Connect the rule, the table, and the situation. For "Explain" prompts, write complete sentences. Finish with the Stretch Problem.

Challenge Problems

1. A gym charges a $20 sign-up fee plus $15 per month. Write an equation for total cost (y) after x months, then find the cost after 6 months. MULTI-STEP

2. A table shows: x = 1 → y = 7, x = 2 → y = 10, x = 3 → y = 13. Find the rule connecting x and y. REASONING

Hint: y goes up by 3 each time, and there is a starting value.

3. Two plans: Plan A is y = 5x; Plan B is y = 3x + 10. For what value of x do they cost the same? REASONING

4. Complete the table for y = 2x + 3, then list the ordered pairs you would plot. MULTI-STEP

x 0 1 2 3
y

5. A car travels 60 miles per hour. Write the equation, then explain why this graph would be a straight line through the origin. EXPLAIN

6. Compare these two rules: y = x + 10 and y = 10x. Which grows faster as x increases? Explain. EXPLAIN

7. A pool drains at 8 gallons per minute, starting with 200 gallons. Write an equation for water left (y) after x minutes. After how many minutes is it empty? REAL-WORLD

8. Design your own real-world relationship. Define x and y, write an equation, and make a 4-row table. OPEN-ENDED

x
y

9. A graph passes through (0, 2), (1, 4), (2, 6). Write the equation of the rule. REASONING

10. A phone plan's cost is c = 0.05t + 30, where t = texts. If the bill is $45, how many texts were sent? MULTI-STEP

★ Stretch Problem

Two friends save money. Maria starts with $50 and adds $10 each week. Jordan starts with $20 and adds $16 each week. (a) Write an equation for each. (b) After how many weeks will they have the same amount? (c) Who will have more money in week 10, and by how much? Justify using a table or equations.

Answer Key

  1. y = 15x + 20; at x = 6: 15(6) + 20 = $110
  2. y = 3x + 4
  3. 5x = 3x + 10 → 2x = 10 → x = 5
  4. y: 3, 5, 7, 9; pairs (0,3), (1,5), (2,7), (3,9)
  5. y = 60x; line through origin because at x = 0, y = 0 and y increases by the same 60 each hour (constant rate)
  6. y = 10x grows faster; it multiplies x by 10 each step, while y = x + 10 only adds 1 per step
  7. y = 200 − 8x; empty when 200 − 8x = 0 → x = 25 minutes
  8. Answers vary; equation must match table consistently
  9. y = 2x + 2
  10. 45 = 0.05t + 30 → 0.05t = 15 → t = 300 texts
  11. Stretch: (a) Maria m = 10w + 50, Jordan j = 16w + 20; (b) 10w + 50 = 16w + 20 → 6w = 30 → w = 5 weeks; (c) week 10: Maria $150, Jordan $180, Jordan has $30 more