# High School: Number and Quantity

### Quantities HSN-Q.A.2

2. Define appropriate quantities for the purpose of descriptive modeling.

You've assigned your math class a set of word problems. One of the questions ends with, "How many people voted for candidate Jones?" Fingers fly over calculators as your students try to determine the correct answer. After a minute or two, Tommy raises his hand and announces his answer: 4,602.28

Everything inside of you wants to scream, "Are you out of your mind, Tommy boy? A closer answer would be 4 because at least it would be possible! How on earth could a candidate receive 28 hundredths of a vote?"

Okay, breathe. Poor Tommy has made the kind of error that drives math teachers like you to the brink of insanity. He's given an answer that makes absolutely no sense.

Whatever else math is or isn't, it is always logical. Answers should always make sense. Math frequently relies on general knowledge—concepts that we assume are known by everyone over the age of about seven—along with a healthy dose of common sense (not you, Thomas Paine).

The problem, of course, is that people have a tendency to ignore both common sense and the knowledge that they possess. Particularly when calculators are involved, there's the tendency to run with the answer you're given. That ignores one very basic principle: the calculator answers the question you ask, not the one you intended to ask. Every answer, every step along the way, should always be reality-based. Make sure your students know that.

#### Drills

1. Walter has decided to drive from Long Island New York to Orlando to visit his family. He knows the trip is about 1,000 miles. About how many hours can he expect to drive?

20

The national speed limit is 55 mph. If we assume an average of about 50 mph—allowing for traffic, we get exactly 20 hours of driving (1,000/50). Even if we had allowed for an average speed of 60 mph, we would have gotten just under 17 hours, so 20 would still have been the logical choice. Remember, Walter is planning to drive, not ride a bicycle or fly.

2. Dominic is painting his living room. He's done the math, and he knows that if he works alone, the job should take 6 hours. His two brothers decide to help, and work at the same rate as Dominic. Which is a reasonable estimate for how long the job should take?

2 hours

Dominic is getting help painting the room. That means the job should take less time than it took without help. If three of them each do the same amount of work (logical, since they're working at the same rate), they can split the 6-hour job between the 3 workers, and each works for 2 hours.

3. The Willis Tower in Chicago (formerly known as the Sears Tower) was the tallest building in the world for 25 years. Its Skydeck is on Floor 103. Which would be a good estimate for the height above the ground at which the Skydeck rests?

1000 feet

The actual answer is 1353 feet, but let's use a little logic. Right now, wherever you are, look up. You know your height, and you can see how far above your head the ceiling is. How high would you say the ceiling is—8 feet? Maybe 10? Figure that some people are over 6 feet tall, so a ceiling needs to be at least 7 feet above the ground, and public places are larger than private homes. So let's go with about a 10-foot high guesstimate for the height of each story. Multiplying 10 feet per story by 103 stories gives us 1030 feet, so choice (D) makes the most sense.

4. Nick is in the seventh grade. He works part time as a caddy and babysits his little cousin. He has determined that it's only 4 more years until he can drive, and he wants to save for a car. He knows that it will be a used car, and that it won't cost more than \$8,000. About how much should he plan to save each month?

\$150

There are 12 months in each year, so Nick has 48—we'll say 50 for the purposes of estimation—months to save. When you divide \$8,000 by 50 months, you get \$160 per month. Besides, the other answers really don't make much sense. Nick wants to drive as a senior in high school, not wait until he's a senior citizen. As far as choice (D) goes, he's in seventh grade. Where does a 12 year old get \$400 a month?

5. Against all the peer pressure in the world, you love the music of Johann Sebastian Bach. There's no shame in that. You do lots of research into his life, and discover that he died in 1750 when he was in his sixties. Which of the following is a logical answer to, "When was he born?"

1688

The actual answer is 1685. But let's look at the question logically. First, we want him born before he dies, which eliminates choice (A). Take 1750 and subtract 60, and we have 1690 as the latest year in which he could be born. If we instead subtract 70, we'll get 1680, the earliest year in which he could have been born. The only of the choices in that range is 1688.

6. There are about 300 million people in the United States. Of these 300 million people, only about 175 million vote. Which percentage best approximates the ratio of voters to total population? (Don't worry about who's a citizen and who isn't.)

60%

If we actually divide 175 million by 300 million, we get 58.3%, but we don't even have to do that. If we know that 150 is half of 300, we can already tell the percentage will have to be above 50%, eliminating (A). Since 200 million out of 300 million would be two thirds (or 66.7%), we know it has to be lower than that percentage. That means (B) is our only option.

7. A pack of gum costs \$0.95. If it has 16 sticks of gum, how much would one individual stick of gum cost?

\$0.06

If we divide the total price (\$0.95) by the number of sticks of gum in the package (16), we'll have the price for each individual stick. The value of (A) is the closest to the actual value per stick, but there's a problem: we can't have fractions of a cent. That means (A) is incorrect, and so is (B). The difference between (C) and (D) is the units. The right answer is (C) because (D) assumes the pack costs 0.95¢, not \$0.95.

8. A field that's 200 feet by 200 feet can hold a certain number of people. If we give each person exactly 5.3 square feet of personal space, what's a legitimate estimation of how many people the field can hold?

7,500

If we actually consult a calculator and do the math, we get the answer (A). It's wrong because we can't have 0.2 of a person, unless we're talking limbs but in that case, we've got bigger problems than fitting them on a field. We can definitely fit more than 1, and 100,000 is clearly impossible. Our closest option, then, is (B) because it's a whole number, and closest to the actual calculated number.

9. A student writes that planes travel at elevations of around 350,000 feet. What is the most likely explanation for this error?

Incorrect decimal placement

Most commercial airplanes often travel at elevations of between 30,000 to 40,000 feet. That means (D) is out. An answer of 350,000 feet seems to be an error in the number of zeros rather than an error in the actual units. (Using "10,668,000,000 micrometers" or "1,944 and a half giraffes" would be considered incorrect units, but feet seem appropriate.) That means the correct answer is (B).

10. Michelle drives her car a distance of 129 miles. If she drives at an average speed of 42 miles an hour, about how long will it take her to reach her destination?