Copyright © 2011 Pearson Education, Inc. Approaches to Problem Solving Discussion Paragraph In your past mathematics classes, it might have seemed that

Copyright © 2011 Pearson Education, Inc.

Approaches to Problem SolvingDiscussion Paragraph

In your past mathematics classes, it might have seemed that mathematical problems involved only numbers and symbols. But the mathematical that you encounter in other classes, in jobs, and in daily life are almost always posed in words. That is why Chapter 1 focused on logic and critical thinking. These skills help you find the key ideas buried in problems that are posed in words. This chapter we begin a study of quantitative problem solving, in which the problems involve words and numbers.

Copyright © 2011 Pearson Education, Inc. Slide 2-3

Unit 2A

The Problem-Solving Power of Units

Global Melting p.80


The Power of Units

There is no single best approach to solving quantitative problem.

You will learn some strategies and guidelines to help you solve problems.


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Units

The units of a quantity describe what is being measured or counted.

Read kilowatts hours as “kilowatt-hours.”

hyphenMultiplication

Read ft ft ft, or ft3, as

“cubic feet” or “feet cubed.”

cube or cubicRaising to third power

Read ft ft, or ft2, as

“square feet” or “feet squared.”

squareRaising to second power

Read miles hours as “miles per hour.”

perDivision

ExampleKey word or symbol

Operation

2-AIdentifying UnitsCN 1a-c

1. Identify the units of the answer for each of the following cases.

a. The price you paid for gasoline, found by dividing its total cost in dollars by the number of gallons of gas that you bought.

b. The area of a circle, found with the formula пr², where the radius r is measured in cm.

c. A volume found by multiplying an area measured in acres by a depth measured in ft.


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Problems to try

Your average speed on a long walk, found by dividing distance traveled in miles by time elapsed in hours.

The unit price of oranges, found by dividing the price in dollars by the weight in pounds.

The cost of a piece of carpet, found by dividing its price in dollars by its area in square yards.

The flow rate of a river in which 5000 cubic feet of water flow past a particular location every second.


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Unit Conversions

Convert a distance of 9 feet into inches.

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Everyday Problems

Everyday problems can be solved with unit conversion.

There are many different ways to write the conversion of 1.

The KEY is to use the correct form.


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Conversion Factors

1in. 12

ft 1or 1

ft 1

in. 12or ft 1in. 12

A conversion factor is a statement of equality that is used to convert between units.

Some conversion factors:

1hr 24

day 1or 1

day 1

hr 24or day 1hr 42

2-AFeet to InchesCN 2a-c

2. a. Convert a distance of 7 feet into inches.

b.Convert 24 feet to inches

c.Convert 24 feet to yards


2-AInches to FeetCN 3a-d

3. a. Convert a length of 102 inches to feet

b. Now try Minutes to Seconds

c. Convert 25 minutes to seconds

d. Convert 2.5 hours to seconds


2-AUsing a Chain of ConversionsCN 4

4. How many seconds are there in one day?


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Using a Chain of Conversions

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1 yd = 3 ft

1 yd2 = 1 yd × 1 yd

= 3 ft × 3 ft

= 9 ft2

Conversions with Units Raised to Powers

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Hint

We take special care when converting units raised to powers.

We may not know the conversion factor between square yards, but we know that 1 yard = 3 ft, so we can replace it to say

1 yd x 1yd = 1 square yard, so 3 ft x 3 ft = 9 square feet


2-ACarpeting a RoomCN 5

You want to carpet a room that measures 10 feet by 12 feet, making an area of 120 square feet. But carpet is usually sold by the square yard rather than by the square foot.

5. How many square yards of carpet do you need?


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Hint

Follow the same idea with cubic yards as you do square yards.

1yd x 1 yd x 1 yd = 1 cubic yard, so 3ft x 3ft x 3ft = 27 cubic feet.


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Cubic Units

How many cubic yards of soil are needed to fill a planter that is 20 feet long by 3 feet wide by 4 feet tall?

The volume is 20 ft × 3 ft × 4 ft = 240 ft3

1 yd = 3 ft, so (1 yd)3 = (3 ft)3 = 27 ft3

2-ACubic Units: Purchasing Garden SoilCN 6

You are preparing a vegetable garden that is 40 feet long and 16 feet wide, and you need enough sold to fill it6 to a depth of 1 foot. The landscape supply store sells soil by the cubic yard.

6. How much soil should you order?


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Currency Conversions

You return from a trip to Europe with 120 euros. How many dollars do you have?

$1.256120 euros $150.72

1 euro

2-APrice ConversionCN 7

At a French department store, the price for a pair of jeans is 45 euros.

7. What is the price in US dollars? (use Table 2.1

for exchange rates)


2-ABuying CurrencyCN 8

You are on holiday in Mexico and need cash.

How many pesos can you buy with $100? (use Table 2.1)


2-A

Let’s try

You arrive in London with $400. How many pounds can you buy?

As you leave Paris, you convert 4500 euros to dollars. How many dollars do you receive?

You return from Mexico with 3000 pesos. How much are they worth in U.S. dollars?


2-A

Exchange Rates

Current exchange rates are constantly changing, but you can always get current rates by typing “exchange rates” into any search engine.

Find the reciprocals of the numbers in the Dollars per Foreign column of Table 2.1.

FYI: Banks offer better rates, some places charge fees.


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Problem Solving with Units

Units can help us decide HOW to solve

Same units – use addition and subtraction ONLY

Different units – usae multiplication and division.


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1. Identify the units involved in the problem. Use them to

decide how to approach the problem; and determine what units to expect in the answer.

2. Perform any operations on both the numbers and their associated units.

Numbers with different units cannot be added or subtracted.

Combine different units through multiplication, division, or raising to powers.

Replace division with multiplication by the reciprocal.3. Make sure your answer is in the units you

expected.

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Using units to Check Answers

Checking answers is one of the most important reasons for using units.

You should always check your method and your calculations as well as your units!!!


2-ADistance Time and SpeedCN 9

Speed in miles per hour is distance / time

You can always rearrange the equation…

distance = speed x time time = distance / speed

9. A car is traveling 25 miles every half-hour. How fast is it going?


2-A

Let’s try

An airliner travels 45 miles in 5 minutes. What is its speed in miles per hour?

Competition speed skydivers have reached record speeds of 614 miles per hour. At this speed, how many feet would you fall every second?


2-A


$12,50050 acres $625,000

1 acre


You are buying 50 acres of farm land at a cost of $12,500 per acre. What is the total cost?

The answer should be in dollars. We multiply the acreage by the cost per acre:

2-ABuying Farm LandCN 10

10. You are buying 30 acres of farm land at $12,000 per acre. What is the total cost

* an acre was originally defined as the amount of land a pair of oxen could plow in a day.


2-AExam CheckCN 11

You are a grader for a math course. An exam questions reads: “Eli purchased 5 pounds of apples at a price of 50 cents per pound. How much did he pay for apples?”

On the paper you are grading, a student has written: “50/5 = 10. He paid 10 cents.”

11. Write a note to the student explaining what went wrong.


2-A

Use unit conversions to answer

An airline travels 45 miles in 5 minutes. What is its speed in miles per hour?

What is the total cost of 1.2 cubic yards of soil if it sells for $245 per cubic yard?

A hose fills a hot tub at a rate of 3.2 gallons per minutes. How many hours will it take to fill a 300 gallon-hot tub?


2-A

More unit conversions

Suppose you earn $8.50 per hour and work 24 eight-hour days in a month. How much do you earn in that month?

The median salary for the New York Yankees in 2008 was $1,875,00. Assuming a 160-game season, express this salary in dollars per game.

If you sleep an average of 8 hours each night, how many hours do you sleep in a year?


2-AGas MileageCN 12

Start analyzing a problem by looking at its units.

Your destination is 90 miles away and your fuel gauge shows that your gas tank is only one-quarter full. You know that your tank holds 12 gallons of gas and that your car averages about 25 miles per gallon.

12. Do you need to stop for gas?


2-A

Let’s try

You plan to take a 2000 mile trip in your car, which averages 32 miles per gallon.

How many gallons of gasoline should you expect to use?

Would a car that has only half the gas mileage(16 miles per gallon) require twice as much gasoline for the same trip?

Explain


2-AMelting Ice CN 13


We can find the correct answer

simply by working with

units.

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Homework 2A DP: Problem Solving Discussion CN: 1-13 QQ: 1-10 P.92: 1-12 1 web

87. S. American Adventure 88. Polar Ice Melting

1 world 89. Are the units clear 90. Units on the Highway 91. False Advertising


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Copyright © 2011 Pearson Education, Inc. Approaches to Problem Solving Discussion Paragraph In your past mathematics classes, it might have seemed that