Neft Teacher · Learn It

Unit 9 · Standard 6.AT.11

Analyze Graphs of Relationships Between Two Variables

Lesson 9-2 · Learn It — how the math works

Why we're learning this

The ticket shows the cost to attend a college football game. How does the total cost of tickets relate to the number of tickets purchased?

✱ Math Workbench

Learn It

Two pictures of the same relationship

A table shows a relationship with numbers. A graph shows the same relationship with points. Reading either one tells you how the dependent variable responds to the independent variable.

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

A ticket to a college football game costs $45. Next to the ticket is a table: 1 ticket, $45. 2 tickets, $90. 3 tickets, $135. Beside the table is a graph with points climbing up and to the right.
Where you see this in real lifeA ticket to a college football game costs $45. Next to the ticket is a table: 1 ticket, $45. 2 tickets, $90. 3 tickets, $135. Beside the table is a graph with points climbing up and to the right.

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

The problem looks like this
A coordinate grid with (1, 45), (2, 90), (3, 135) plotted.−100−100−50−505050100100 x y 0 (1, 45)(2, 90)(3, 135)
  1. Each ticket costs $45, so the total cost is 45 times the number of tickets: 1 ticket is $45, 2 tickets are $90, 3 tickets are $135.
  2. The number of tickets is the independent variable, so it goes first in each : (1, 45), (2, 90), (3, 135).
  3. I label one 'Number of tickets' and the other 'Total cost ($)', then plot each pair as a point.
  4. The points rise to the right: as the number of tickets increases, the total cost increases.

Watch out — mistakes students make here

  • The numbers in a coordinate point aren't a time measurement here.
  • 12,000 feet is the distance after 10 minutes, not the full 15.
  • 16,800 feet is the distance after 14 minutes — one minute short of the top.

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

Same steps, new numbers.

  1. The tram travels 1,200 feet every minute. Which quantity is independent — time or distance?
  2. Time is independent, so our pairs look like (minutes, feet): (1, 1200), (2, 2400), (5, 6000).
  3. The ascent takes 15 minutes, so the last point is (15, 18000) — because 15 times 1,200 is 18,000 feet.
  4. Each point means: after THIS many minutes, the tram has traveled THIS many feet.

On your ownYou do — solve this one by yourself

Before you start, remember:

  • For each situation below, decide what one point on the represents.
  • Say it as a sentence: 'The point (___, ___) means ___ .'

Solve: The Sandia Peak tram travels 1,200 feet each minute and the ascent takes 15 minutes. How long is the whole ascent in feet?

My answer

Stuck? Open one hint at a time.

Hint 1

How many minutes does the full ascent take?

Hint 2

Each of those 15 minutes covers 1,200 feet.

Hint 3

Multiply 1,200 by 15.

Check my answer
✅ Answer: 18,000 feet
The total distance is 15 × 1,200 = 18,000 feet.

Turn & TalkSay it out loud with a partner

Football tickets cost $45 each: 1 ticket is $45, 2 tickets are $90, 3 tickets are $135. When you write the for 3 tickets, which number goes first — 3 or 135 — and why?

Try starting with:

  • The ordered pair is (___, ___) because ___ comes first.El par ordenado es (___, ___) porque ___ va primero.
  • If I wrote (135, 3) instead, that would mean ___.Si escribiera (135, 3) en cambio, eso significaría ___.
Use these wordsordered pairindependent variableaxis