Lesson 1.3 · Unit 1 · N-Q.1, N-Q.2, N-Q.3
Units as algebra: converting by canceling on the page
A number without a unit is usually not an answer. Units also carry information the numbers do not: they tell you whether you have set a problem up correctly, because if the units of your result are wrong then the arithmetic was wrong too, whatever the calculator says.
- A conversion factor is a fraction equal to 1. Since \( 1 \text{ mile} = 5280 \text{ feet} \), both \( \frac{5280 \text{ ft}}{1 \text{ mi}} \) and \( \frac{1 \text{ mi}}{5280 \text{ ft}} \) equal 1, so multiplying by either changes the units without changing the quantity.
- Choose the orientation that cancels. Put the unit you want to remove in the position opposite to where it currently sits, so it divides out.
- Write the units on the page and cross them off. This is the whole method. If the units that remain are the ones you wanted, the setup is right.
- Chain as many factors as you need in one line rather than converting in separate stages, which reduces rounding and reduces mistakes.
- A rate is a quotient of two quantities and its unit says so: miles per hour, dollars per pound, people per square mile. Read the slash as the word per.
- Interpret a rate as a sentence. "The slope is 3.5" says nothing; "the cost rises by 3.5 dollars for each additional pound" says everything.
- Choose units that make the numbers readable. A national budget in dollars and a bacterium's length in miles are both technically correct and both useless.
Where students lose marks: multiplying when they should divide, because they guessed rather than canceled. Never decide by feel whether to multiply or divide by 5280. Write the factor as a fraction, check which unit cancels, and let the algebra decide.
The problem. A car is traveling at 88 feet per second. Convert this to miles per hour, and interpret the answer.
Step one: write what you have as a fraction with its units. \[ \frac{88 \text{ ft}}{1 \text{ s}} \] Writing it as a fraction is what makes the canceling visible.
Step two: list the conversions you will need. Feet must become miles, and seconds must become hours. \( 1 \text{ mi} = 5280 \text{ ft} \), and \( 1 \text{ h} = 3600 \text{ s} \), since \( 60 \times 60 = 3600 \).
Step three: orient the first factor to cancel feet. Feet is currently on the top, so the conversion factor needs feet on the bottom: \( \frac{1 \text{ mi}}{5280 \text{ ft}} \).
Step four: orient the second factor to cancel seconds. Seconds is currently on the bottom, so the factor needs seconds on the top: \( \frac{3600 \text{ s}}{1 \text{ h}} \).
Step five: write the whole chain and cancel. \[ \frac{88 \text{ ft}}{1 \text{ s}} \times \frac{1 \text{ mi}}{5280 \text{ ft}} \times \frac{3600 \text{ s}}{1 \text{ h}} \] Feet appears once on top and once on the bottom, so it cancels. Seconds appears once on the bottom and once on top, so it cancels. What remains is miles on top and hours on the bottom, which is exactly the unit asked for. The setup is confirmed before any arithmetic.
Step six: do the arithmetic. \[ \frac{88 \times 3600}{5280} = \frac{316{,}800}{5280} = 60 \]
Step seven: state the answer with its unit. 60 miles per hour.
Step eight: interpret it and sanity check. The car covers 60 miles in each hour of travel at this speed. As a check, 88 feet per second is a little under 90, and 60 miles per hour is a common highway speed, so the magnitude is plausible. Had the factors been flipped, the units would have come out as \( \frac{\text{ft}^2}{\text{mi} \cdot \text{s}^2} \) or similar nonsense, which is the signal to stop and re-orient rather than to keep going.
Problems 1 to 5 are routine. Problems 6 to 10 are multi-step or ask for reasoning.
- Convert 5 feet to inches.
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\( 5 \text{ ft} \times \frac{12 \text{ in}}{1 \text{ ft}} \), feet cancels. 60 inches
- Convert 2.5 hours to minutes.
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\( 2.5 \times 60 \). 150 minutes
- A recipe uses 3 cups per batch. How many cups for 7 batches?
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\( 7 \text{ batches} \times \frac{3 \text{ cups}}{1 \text{ batch}} \). 21 cups
- State the unit of the rate "distance divided by time".
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A unit of distance per unit of time, such as miles per hour or meters per second
- Convert 120 minutes to seconds.
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\( 120 \times 60 \). 7,200 seconds
- A printer produces 22 pages per minute. How many pages in 45 minutes, and how long to
print 1,000 pages?
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For the first: \( 45 \text{ min} \times \frac{22 \text{ pages}}{1 \text{ min}} = 990 \) pages. For the second, flip the rate so pages cancels: \( 1000 \text{ pages} \times \frac{1 \text{ min}}{22 \text{ pages}} = 45.45\ldots \) minutes, about 45.5 minutes. 990 pages; about 45.5 minutes
- Convert 30 miles per hour to feet per second.
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\[ \frac{30 \text{ mi}}{1 \text{ h}} \times \frac{5280 \text{ ft}}{1 \text{ mi}} \times \frac{1 \text{ h}}{3600 \text{ s}} \] Miles cancels and hours cancels, leaving feet per second. \( \frac{30 \times 5280}{3600} = \frac{158{,}400}{3600} = 44 \). 44 feet per second
- A tank fills at 12 gallons per minute. Express this in gallons per hour and explain why
the number gets larger.
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\( \frac{12 \text{ gal}}{1 \text{ min}} \times \frac{60 \text{ min}}{1 \text{ h}} = 720 \) gallons per hour. The number grows because an hour is a much longer period than a minute, so the same rate delivers 60 times as much in it. The quantity of water per unit time has not changed; only the unit it is reported in has. 720 gallons per hour; the time unit is 60 times larger
- A student converts 3 hours to seconds and gets 180. Diagnose the error.
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They converted hours to minutes and stopped. \( 3 \text{ h} \times \frac{60 \text{ min}}{1 \text{ h}} = 180 \) minutes, and a second conversion is still needed: \( 180 \text{ min} \times \frac{60 \text{ s}}{1 \text{ min}} = 10{,}800 \) seconds. Writing the units on the page would have caught it immediately, because the units after the first step read "minutes" rather than "seconds". 10,800 seconds; they stopped after converting to minutes
- A car uses fuel at 28 miles per gallon and fuel costs $3.60 per gallon. Find the cost per
mile, and explain how the units confirm the setup.
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The target unit is dollars per mile, so dollars must end on top and miles on the bottom. Start with the price, which already has dollars on top, and multiply by the reciprocal of the fuel economy so that gallons cancels and miles lands on the bottom: \[ \frac{3.60 \text{ dollars}}{1 \text{ gal}} \times \frac{1 \text{ gal}}{28 \text{ mi}} = \frac{3.60}{28} \frac{\text{dollars}}{\text{mi}} \] Gallons cancels, leaving dollars per mile as required. \( 3.60 \div 28 = 0.12857\ldots \), so about $0.129 per mile, or roughly 12.9 cents. Note that multiplying the two figures instead would have given units of dollars times miles per gallon squared, which is meaningless, and that is the signal that the setup was wrong. About $0.129 per mile; gallons cancels to leave dollars per mile