## Stairs

## Opening

# Stairs

Discuss the following with your classmates.

- In which set of stairs is the height of your location on the stairs proportional to the number of stairs you have climbed?

Discuss the following with your classmates.

- In which set of stairs is the height of your location on the stairs proportional to the number of stairs you have climbed?

Determine whether a situation represents a proportional relationship.

A girl is walking along a road and counting the number of steps she takes as she walks.

*The distance she covers is proportional to the number of steps she takes.*

- When is this statement true? If it is never true, explain why.

What would have to be true about the girl’s steps for the statement to be true?

Water is dripping rapidly from a faucet into a tub.

You plug the drain to see how quickly the water accumulates in the tub.

*The volume of water in the tub is proportional to the time that has passed since you plugged the drain.*

- When is this statement true? If it is never true, explain why.

What would have to be true about the drips of water for the statement to be true?

You are riding a Ferris wheel at an amusement park.

*Your height from the ground is proportional to the amount of time that has passed since the Ferris wheel started moving.*

- When is this statement true? If it is never true, explain why.

Sketch a picture of a Ferris wheel. Indicate the distance between each Ferris wheel rider and the ground.

At an amusement park, a tram moves along a track.

*The total distance the tram travels is proportional to the time that it travels.*

- When is this statement true? If it is never true, explain why.

There is a stack of paper.

*The height of the stack of paper is proportional to the number of pieces of paper in the stack.*

- When is this statement true? If it is never true, explain why.

Prepare a presentation that explains and analyzes your findings.

- How did you determine whether a situation represents a proportional relationship?
- What do the situations that represent proportional relationships have in common?
- What is true about the situations that do not represent proportional relationships?

Create your own problem similar to the Work Time problems. Come up with a situation and write a statement that makes a claim about proportional relationships in the situation. Trade problems with another student.

Take notes about your classmates’ explanations and strategies for determining whether a situation represents a proportional relationship.

As your classmates present, ask questions such as:

- How did you know that the relationship is proportional?
- Is there a particular word that you looked for to help you determine whether a situation is proportional?
- What do you think is interesting about this situation?
- Did you draw a picture to help you? If so, can you explain your picture?
- In the problem you wrote, what are the variables in the situation?
- How would you describe the unit rate (for example, miles per hour)?

Write a summary about what makes a situation proportional.

Check your summary.

- Do you explain that in order for a situation to represent a proportional relationship, some value in the situation must be constant?
- Do you discuss the fact that there are some relationships that can never be proportional, and provide an example?

Write a reflection about the ideas discussed in class today. Use the sentence starter below if you find it to be helpful.

**One thing I learned about proportional relationships is …**