Defining quantities and variables for a model

About this lesson

Integrated Math I › Relationships between quantities

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Defining quantities and variables for a model

Warm-up

Three quick questions from earlier lessons.

Discover

You want to know which of two pizzas is the better deal. What should you compare?

  1. The price per square inch
  2. The prices only

Key idea

A model starts with well-chosen quantities.

Before writing an equation, decide which quantities matter, give each a name, a unit and, if needed, a time period ("n = number of texts sent in one month"), and use rates such as miles per gallon or dollars per hour to compare situations of different sizes.

Worked example

A taxi charges a fixed fee plus a price per mile.

m = miles driven on one ride; C = cost of the ride in dollars
Quantities.
C = fee + (price per mile) · m
Model.

Your turn, guided

Your turn: a runner covers 10 km in 50 minutes. Pace in minutes per km?

55
50 ÷ 10.

Practice ★

  1. A phone plan costs 37 dollars a month plus 15 cents per text. Maya wants a model for her monthly bill. Which is the best definition of the variable?
    1. Let n be the number of texts Maya sends in one month
    2. Let n be texts
    3. Let n be the cost of one text
  2. A phone plan costs 30 dollars a month plus 6 cents per text. Maya wants a model for her monthly bill. Which is the best definition of the variable?
    1. Let n be the number of texts Maya sends in one month
    2. Let n be texts
    3. Let n be the cost of one text
  1. A runner's pace is the number of minutes she needs for each kilometer. She runs 2 km in 8 minutes. What is her pace, in minutes per kilometer?

Watch out!

Practice ★★

  1. Car A used 11 gallons of fuel on a trip and car B used 14 gallons on a different trip. To compare how fuel-efficient the two cars are, which quantity is most useful?
    1. Miles traveled per gallon of fuel, for each car
    2. The number of gallons each car used
    3. The total miles each car has ever driven
  2. Car A used 12 gallons of fuel on a trip and car B used 15 gallons on a different trip. To compare how fuel-efficient the two cars are, which quantity is most useful?
    1. Miles traveled per gallon of fuel, for each car
    2. The number of gallons each car used
    3. The total miles each car has ever driven
  1. Boss problem. A club makes t-shirts. Each shirt costs 8 dollars to make, plus a one-time setup fee of 50 dollars. Each shirt sells for 14 dollars. Let n be the number of shirts made and sold.

Exit ticket

Three questions to check you have got it.

  1. A phone plan costs 37 dollars a month plus 15 cents per text. Maya wants a model for her monthly bill. Which is the best definition of the variable?
    1. Let n be the number of texts Maya sends in one month
    2. Let n be texts
    3. Let n be the cost of one text
  2. A phone plan costs 30 dollars a month plus 6 cents per text. Maya wants a model for her monthly bill. Which is the best definition of the variable?
    1. Let n be the number of texts Maya sends in one month
    2. Let n be texts
    3. Let n be the cost of one text
  3. A phone plan costs 38 dollars a month plus 12 cents per text. Maya wants a model for her monthly bill. Which is the best definition of the variable?
    1. Let n be the number of texts Maya sends in one month
    2. Let n be texts
    3. Let n be the cost of one text

Summary card

Defining quantities

Name each quantity with its unit and period; compare with rates.

Watch out: a vague variable ("x = money") leads to a wrong model.

End of the lesson

Next lesson: Converting units of measurement Back to the chapter

Answers: Defining quantities and variables for a model

A

  1. Let n be the number of texts Maya sends in one month
  2. Let n be the number of texts Maya sends in one month

B

  1. 44

C

  1. Miles traveled per gallon of fuel, for each car
  2. Miles traveled per gallon of fuel, for each car

D

  1. 172172

E

  1. Let n be the number of texts Maya sends in one month
  2. Let n be the number of texts Maya sends in one month
  3. Let n be the number of texts Maya sends in one month

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