Relationships between quantities

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1. 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.

2. Converting units of measurement

Warm-up

Three quick questions from earlier lessons.

Discover

A table is 2 meters long. About how many centimeters is that?

  1. 200
  2. 0.02

Key idea

Smaller units: more of them.

To convert, use the rate between the units (1 ft = 12 in, 1 yd = 3 ft, 1 m = 100 cm, 1 kg = 1000 g). Converting to a smaller unit, multiply; to a larger unit, divide.

Worked example

Convert 54 inches to feet.

54÷12=4.554 \div 12 = 4.5 ft
Feet are larger: divide.

Your turn, guided

Your turn: 3 kg in grams.

30003000
× 1000.

Practice ★

  1. How many inches are in 5 feet? (1 foot = 12 inches)
  2. How many inches are in 2 feet? (1 foot = 12 inches)
  1. How many meters are 600 centimeters? (1 m = 100 cm)

Watch out!

Practice ★★

  1. A rope is 11 yards and 2 feet long. How long is it in feet? (1 yard = 3 feet)
  2. A rope is 10 yards and 2 feet long. How long is it in feet? (1 yard = 3 feet)

Exit ticket

Three questions to check you have got it.

  1. How many inches are in 5 feet? (1 foot = 12 inches)
  2. How many inches are in 2 feet? (1 foot = 12 inches)
  3. How many inches are in 7 feet? (1 foot = 12 inches)

Summary card

Converting units

To a smaller unit: multiply; to a larger unit: divide.

Watch out: check that the answer makes sense in size.

3. Percent error

Warm-up

Three quick questions from earlier lessons.

Discover

A 50 cm board is measured as 52 cm. Which describes the error?

  1. A 4% error
  2. A 2% error

Key idea

Percent error compares the error with the actual value.

Percent error =∣measured−actual∣actual×100= \dfrac{|\text{measured} - \text{actual}|}{\text{actual}} \times 100. A tolerance of p%p\% allows values within p%p\% of the target, above or below.

Worked example

A 400 g bag with at most 5% error.

2020 g
5% of 400.
From 380 g to 420 g
Range.

Your turn, guided

Your turn: actual 40, measured 38. The percent error.

55
2 ÷ 40 × 100.

Practice ★

  1. A board is really 36 cm long, but Ana measures 40 cm. What is her percent error?
  2. A board is really 73 cm long, but Ana measures 76 cm. What is her percent error?
  1. A box really holds 400 cups, but a label says 520. What is the percent error?

Watch out!

Practice ★★

  1. A bag should hold 600 g of rice, with at most 7% error. What is the smallest allowed mass (g)?
  2. A bag should hold 400 g of rice, with at most 8% error. What is the smallest allowed mass (g)?

Exit ticket

Three questions to check you have got it.

  1. A board is really 36 cm long, but Ana measures 40 cm. What is her percent error?
  2. A board is really 73 cm long, but Ana measures 76 cm. What is her percent error?
  3. A board is really 52 cm long, but Ana measures 53 cm. What is her percent error?

Summary card

Percent error

Error ÷ actual × 100.

Watch out: divide by the actual value, not the measured one.

Answers: Relationships between quantities

1. 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

2. Converting units of measurement

A

  1. 6060
  2. 2424

B

  1. 66

C

  1. 3535
  2. 3232

D

  1. 6060
  2. 2424
  3. 8484

3. Percent error

A

  1. 1009\frac{100}{9}
  2. 30073\frac{300}{73}

B

  1. 3030

C

  1. 558558
  2. 368368

D

  1. 1009\frac{100}{9}
  2. 30073\frac{300}{73}
  3. 2513\frac{25}{13}