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# High School: Number and Quantity

### Quantities HSN-Q.A.1

1. Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.

Word problems don't typically provide helpful hints on using probability to win the lotto or anything useful in real life. No, they tend to read more like this:

Juan, Harry, Liam, Hans, and Tom are going out to lunch. Juan contributes 35 pesos, Liam contributes 9 Euros, Harry contributes 500 Indian Rupees, Hans contributes 18 Dutch Guilders, and Tom contributes \$10. How much tip should they leave?

Okay, perhaps we exaggerated just a little, but you get the point—the question is silly. (For starters, why don't they all have the same currency if they're in the same country regardless of where they're from?) But it does make a good point. In order to solve problems of this sort, we need to have everyone dealing with the same units. We're talking to you, Juan, Harry, Liam, Hans, and Tom.

Students should know that the easiest way to change from one unit of measurement to another is through the use of proportions. For example, in the problem above, it would help a whole lot to know that one Dutch Guilder is worth about \$0.57. We could set up a proportion:

With a little arithmetic (or the unparalleled magic of a calculator), we know that Hans contributed about \$10.26. How nice of him.

Which unit of measurement do we use? It's hard to generalize, since it depends on the situation. For example, if we're measuring the distance between cities in the U.S., miles are probably the way to go. But if you're setting up a scale drawing of where you want to put your new bedroom furniture, the odds are good that feet or even inches will make more sense. (Tell your students not to mix up their units; otherwise, they might end up with new bedroom furniture that won't fit through the doorway!)

Speaking of distances, if students ever find themselves lost (and without the GPS on their smartphones, for some reason), knowing how to use old-fashioned maps might prove useful. If they need to travel "half an inch on the map," you might want to point out the legend at the bottom of the map. That's the part that converts that half inch into 50 miles, or whatever ratio they choose.

#### Drills

1. Which would be the most appropriate unit of measurement for a chef buying meat for his restaurant for the next week?

Pounds

Feet, seriously? Okay, even though the odds are good that his restaurant serves steaks measured in ounces, he doesn't want to buy them that way. Assuming his restaurant sells enough steaks to stay in business, he'll serve quite a few of those steaks. But, on the other hand, a ton is a lot of meat—like 2,000 pounds! He would have to sell a whole bunch of steaks to go through even one ton of meat. Pounds are the only units that makes sense.

2. As Margaret packs for her trip to Fiji, she looks up the average daily temperature for July and finds that it's about 14 degrees Celsius. She knows that F = 1.8C + 32, where F represents degrees in Fahrenheit and C represents degrees Celsius. About what temperature should she expect in degrees Fahrenheit?

57

No, that's not a silly answer. Since Fiji is in the Southern hemisphere, July is winter there. margaret took that 14 degrees and substituted it in for the C. She multiplied it by 1.8, and then added the 32. (She didn't add first because she was paying attention when the teacher taught PEMDAS.) Now Margaret's only problem is packing all the clothes she knows she needs to bring without going over the baggage weight for the plane.

3. Max can read about 150 words per minute. Unfortunately for Max, he has waited until this weekend to do all his reading. The book he has to read is 310 pages long. If a typical paperback has about 350 words per page, how many hours should he set aside for this reading?

12 hours

We can multiply 310 pages by 350 words a page to get about 108,500 words total. Divide that by 150 words a minute, and we're talking about 723 minutes. Of course, that doesn't help much. So let's divide minutes by 60 to determine the number of hours. Yeah, hours make much more sense. Max needs just over 12 hours to do his reading. Of course that's reading time, not sleeping time or eating time or texting time or showering time. Looks like a long weekend for Max.

4. In which of the following situations is anyone likely to need 10,000 digits of pi?

Working on the Space Shuttle for NASA

For most of the other projects, a more convenient approximation of pi (like 3.14 or 227) would do, particularly in the first two. But NASA requires a pretty high degree of accuracy; being off by a tiny little bit could result in an incorrect trajectory for a space shuttle, which would be very, very bad.

5. Alexandra is trying to campaign for a \$1 raise in her allowance. She has made up a PowerPoint to show her parents, and wants to make a bar graph of the allowances of the 35 kids in her homeroom. Their allowances range from \$3 per week to \$8.50 per week. Which unit would make the sense for each box of the bar graph to represent?

\$1

Choosing increments of \$10 wouldn't show anything—all her classmates make between \$0 and \$10. Using only \$0.25 probably wouldn't make her case, since mom or dad might argue in favor of a raise of only a quarter. Increments of \$0.01 would drive both Alexandra and her parents crazy. That kid making \$8.50 a week would need an awful lot of boxes.

6. Albert Einstein is driving down a country road at the speed of light. Just kidding. He's actually going an average of 70 miles an hour. He started driving at 3 PM exactly and arrived at his destination at 5:27 PM. If we need to calculate the distance from his starting point to his destination, our answer will be in which units?

Miles

We need a unit of distance, or length. That automatically eliminates answers (B) and (C), since they're units of time and speed. We're already using miles, and we don't know the conversion factor (even though it takes two seconds to look it up on Google). We might as well stick with miles.

7. We want to calculate the speed at which a ball rolls down a hill. It takes 2.3 minutes for the 1-pound ball to roll down a hill sloped at a 53° angle. If the distance from the top of the hill to the bottom is 72 feet, our answer will be in which units?

Feet per minute

What are we looking for again? Oh, right. Speed, or how long it takes for something to travel a certain distance. That means we want to use units of length and time, or in this particular question, feet and minutes. We aren't interested in (A), which has units of weight, or (D), where angles are involved. But how do we calculate speed? We know speed is distance over time (just like it's miles per hour or inches per second). In this case, we'll put feet over minutes to get (C) as our answer.

8. Given a line with slope of 200, what would be the ideal intervals in the coordinate plane for both the x and y-axes? In other words, how many units per gradation should there be for both the x and y-axes to see the line as best as possible?

1 for x, 100 for y

We know that the slopes of lines are given as  What that means is the numerator tells us how many units up the line goes, and the denominator tells us how many units sideways the line goes. In our case, the line goes up 200 units for every 1 unit. That means we want our x-axis to be relatively small, but our y-axis can be in the hundreds. Answers (A) and (D) are the same, essentially, and (B) is the opposite of what we want. That means (C) is ideal.

9. Your friend wants you to measure his height. Which units should you use?

Inches

Let's consider our options. We can rule out (D) automatically, since that's a unit of weight. Your friend might weigh a certain number of kilograms, but he's certainly not that many kilograms tall. Nanometers would be ridiculous to use (a nanometer is 10-9 meters!), and yards wouldn't be accurate enough. That leaves (C) as the best unit to use.

10. It takes Casper (the friendly ghost) 62 minutes to fly from his haunted house to another one. The haunted house is 47 miles away from his haunted house. In that case, how long would it take him to fly to Wendy's house, which is 71 miles away from his?