Defining The Constant Of Proportionality

Defining The Constant Of Proportionality

Introduction to the Constant of Proportionality

Opening

Introduction to the Constant of Proportionality

Look at the table you completed comparing the dimensions of the real Empire State Building with those of the miniature building.

  • What is the constant ratio in the table?

The constant ratio in a proportional relationship is called the constant of proportionality.

You can use the constant of proportionality to write a formula that represents the relationship between the quantities that vary.

  • Let m be the height of the miniature.
  • Let r be the height of the real building.

The formula that represents the proportional relationship between the dimensions of the real building and the miniature building is r = 20m.

The 20 in the formula is the constant of proportionality.

The formula for a proportional relationship has the form

y = kx

where k is the constant of proportionality.

INTERACTIVE: Introducting the Constant of Proportionality

Math Mission

Opening

For a proportional relationship, find the constant of proportionality based on a table of values, and write a formula.

Height of the Miniature

Work Time

Height of the Miniature

The height of a miniature New York City building is 120 the height of the real building. This describes the relationship between the miniature, 1, to the real building, 20.

Let m be the height of the miniature building.

Let r be the height of the real building.

  • How would you write a formula using 120 as the constant of proportionality?

Hint:

After you write your formula, try substituting in some numbers for the variables to check that the formula makes sense.

Driving to the Park

Work Time

Driving to the Park

Look at the table you completed in the last lesson that shows the relationship between the time Mr. Lee drove and the distance he traveled.

  • Is there a constant of proportionality? If not, explain why not.
  • If so, give the values of both constants of proportionality.
  • If this is a proportional relationship, use one of the values to write a formula that represents the proportional relationship between time and distance traveled.

INTERACTIVE: Driving to the Park

Hint:

First write your formula. Then try substituting numbers for the variables to check that the formula makes sense.

Temperature at the Park

Work Time

Temperature at the Park

Look at the table about the temperature at the amusement park on the day the Lee family visited.

  • Is there a constant of proportionality? If not, explain why not.
  • If so, give the values of both constants of proportionality.
  • If this is a proportional relationship, use one of the values to write a formula that represents the proportional relationship between time and temperature.

Prepare a Presentation

Work Time

Prepare a Presentation

Explain how you can use a table of values that shows a proportional relationship to help you write a formula that represents the relationship. Justify your thinking with your work.

Challenge Problem

What is the constant of proportionality in this equation? Explain your answer.

38=yx

 

Make Connections

Performance Task

Ways of Thinking: Make Connections

As your classmates present, take notes about proportional relationships and the constant of proportionality.

Hint:

As your classmates present, ask questions such as:

  • Are there always two ways to write the relationship between two quantities?
  • Is there a third way to express the relationship?
  • Did you test your formula? How?
  • Can you express your formula using words instead of letters? Does it makes sense?
  • Where can you find the constant of proportionality in a table?

Proportional Relationships

Formative Assessment

Summary of the Math: Proportional Relationships

Read and Discuss

The relationship between two varying quantities can be expressed by a constant ratio. For example, the ratio of tricycle wheels to tricycles is 3 : 1.

You can write: wt=31

This relationship is an example of a proportional relationship, because even if the total number of tricycle wheels or the total number of tricycles varies, the ratio between the tricycle wheels and the tricycles will remain the same. You say: “The number of wheels is proportional to the number of tricycles.”

You can write: w = 3t

You can also express this relationship as the ratio of tricycles to wheels: the ratio is 1 to 3, or 1 : 3. You say: “The number of tricycles is proportional to the number of wheels.”

You can write: tw=13 or t=13w

Hint:

Can you:

  • Write a formula using the constant of proportionality?
  • Calculate the constant of proportionality from a table of values?

Reflect On Your Work

Work Time

Reflection

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

One thing that still confuses me about the constant of proportionality is  …