Neft Teacher · Learn It

Unit 9 · Standard 6.AT.11

Write Equations to Represent Relationships Between Two Variables

Lesson 9-3 · Learn It — how the math works

Why we're learning this

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?

✱ Math Workbench

Learn It

From a pattern in the table to an equation

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.

Picture itSee what the math looks like before you read about it

A flyer for Recreation World shows a monthly membership fee of $24.95. Below it is a table: 1 month, $24.95. 2 months, $49.90. 3 months, $74.85. 4 months, $99.80. 10 months, $249.50.
Where you see this in real lifeA flyer for Recreation World shows a monthly membership fee of $24.95. Below it is a table: 1 month, $24.95. 2 months, $49.90. 3 months, $74.85. 4 months, $99.80. 10 months, $249.50.

Watch me solve itI do — read one step at a time

The problem looks like this
Table: 1 month costs $24.95, 2 months $49.90, 3 months $74.85, and 10 months $249.50; the rule is c = 24.95m.Months (m)12310mTotal cost (c)$24.95$49.90$74.85$249.5024.95m
  1. The quantities are the number of months of membership and the total cost. The total cost depends on the number of months.
  2. In the table, each month adds $24.95: 1 month is $24.95, 2 months are $49.90, 3 months are $74.85.
  3. The pattern in words: the total cost is $24.95 times the number of months.
  4. — let m = number of months and c = total cost in dollars — and write the : c = 24.95m.
  5. Now ANY number of months works: for 10 months,c = 24.95 × 10 = 249.50

Watch out — mistakes students make here

  • $224.55 is the cost of 9 months — one month short.
  • $60 is the cost of 4 hours, not 5.
  • $80 would need a rate of $16 per hour — the canoe is $15 per hour.

Try it with meWe do — you answer first, then check

Same steps, new numbers.

  1. A canoe rents for $15 per hour. Which quantity is independent — hours or total cost?
  2. The hours of rental is independent; the total cost depends on it.
  3. Say the pattern in words: the total cost equals $15 times the number of hours.
  4. Let h = hours rented and c = total cost. What matches?
    Check this step
    c = 15h.
  5. Check it: a 3-hour rental is 15 × 3 = 45 dollars, and a 5-hour rental is 15 × 5 = 75 dollars.15 × 3 = 45 dollars

On your ownYou do — solve this one by yourself

Before you start, remember:

  • For each situation below, say the pattern in words first: total = × amount.
  • Then define your and write the .

Solve: Using c = 24.95m, what is the total cost of 10 months of membership?

My answer

Stuck? Open one hint at a time.

Hint 1

Replace m with 10.

Hint 2

Multiplying by 10 moves the decimal point one place.

Hint 3

24.95 × 10 = ?

Check my answer
✅ Answer: $249.50
Substitute m = 10: c = 24.95 × 10 = 249.50.

Turn & TalkSay it out loud with a partner

Recreation World charges $24.95 per month, and the table shows the cost growing by exactly $24.95 every row. Why is the c = 24.95m correct, but c = 24.95 + m would be wrong?

Try starting with:

  • c = 24.95m is correct because each month ___ another $24.95.c = 24.95m es correcto porque cada mes ___ otros $24.95.
  • c = 24.95 + m would give the wrong total for 2 months because it would equal $___, not $49.90.c = 24.95 + m daría un total incorrecto para 2 meses porque sería igual a $___, no $49.90.
Use these wordsequationratemultiply