Multiplying as scaling

When you reach for it

The school office has a copy machine that can make a picture bigger or smaller. You type in a size setting, written as a fraction, and every length in the copy is that fraction of the same length in the original. Yemi has a picture 12 inches wide. With the setting \displaystyle \frac{3}{4} the copy is \displaystyle \frac{3}{4} \times 12 inches wide. With the setting \displaystyle \frac{5}{3} it is \displaystyle \frac{5}{3} \times 12 inches wide.

Before she presses the button, Yemi wants to know one thing: will the copy be bigger or smaller than her picture? She can tell without multiplying at all. The answer sits in the setting, the number she multiplies by.

This is multiplication as scaling, which means resizing. In a product such as \displaystyle \frac{3}{4} \times 12, think of 12 as the thing being resized and \displaystyle \frac{3}{4} as the scale factor, the amount it is resized by. Comparing the scale factor with 1 tells you how the product compares with 12. You already know how to work out a fraction of a number (A fraction times a whole number) and a fraction of a fraction (A fraction times a fraction). This lesson is about judging the size of the answer before you find it.

The lesson runs over three days. Day 1 reads a product as a resizing and compares it with the number being resized. Day 2 gives the rule for every case, a factor less than 1, equal to 1 or more than 1, and shows why it is true. Day 3 shows that a fraction such as \displaystyle \frac{4}{4} is a factor of exactly 1, which explains why equivalent fractions have the same value, and uses the rule to compare and order products.

All the numbers in this lesson are positive.

The procedure

Day 1: multiplying as resizing

Start with the setting \displaystyle \frac{3}{4}. The copy is \displaystyle \frac{3}{4} \times 12 inches wide, which is \displaystyle \frac{3}{4} of 12. To find it, cut 12 into 4 equal parts. Each part is 3, and three of them make 9:

Notice something before the arithmetic. Three fourths is only part of one whole, so three fourths of 12 is only part of 12. Taking a part of something leaves you with less than all of it. So the copy had to come out narrower than 12 inches, whatever the exact number turned out to be.

Now the setting \displaystyle \frac{5}{3}. Five thirds is three thirds and two more thirds, so \displaystyle \frac{5}{3} = 1\frac{2}{3}. That means \displaystyle \frac{5}{3} of 12 is one whole 12 and then two thirds of 12 more. Two thirds of 12 is 8, since each third is 4:

The rule for a fraction times a whole number gives the same answer: \displaystyle \frac{5 \times 12}{3} = \frac{60}{3} = 20. And again the size was clear in advance. A factor of more than one whole gives all of 12 and then some, so the copy had to come out wider than 12 inches.

This is the habit for Day 1. Look at a product as one number being resized by the other. A whole number factor already worked this way: 3 \times 12 is three times as long as 12 (times as many). A fraction factor works the same way. \displaystyle \frac{5}{3} \times 12 is five thirds as long as 12, and \displaystyle \frac{3}{4} \times 12 is three fourths as long as 12.

Which number is the scale factor is up to you. By the commutative property of multiplication (properties of multiplication), \displaystyle 12 \times \frac{3}{4} = \frac{3}{4} \times 12, so you may read either factor as the one doing the resizing. On a copy machine it is natural to call the setting the scale factor, but the mathematics does not care about the order.

To tell whether a fraction is more or less than 1, compare its top with its bottom. When the top is smaller, as in \displaystyle \frac{3}{4}, there are not enough pieces to make a whole, so the fraction is less than 1. When the top is bigger, as in \displaystyle \frac{5}{3}, there are more than enough, so the fraction is more than 1. When they are equal, as in \displaystyle \frac{4}{4}, the fraction is exactly 1.

Day 2: the rule for every case

Day 1 had two cases. A third case finishes the picture. For any positive number b:

• If the factor a is less than 1, then a \times b is less than b.
• If the factor a is equal to 1, then a \times b is equal to b.
• If the factor a is more than 1, then a \times b is more than b.

In symbols, with < read “is less than” and > read “is greater than”:

Why, with area. Draw a rectangle 12 long and 1 tall. Its area is 1 \times 12 = 12. Now draw rectangles that are also 12 long, but \displaystyle \frac{3}{4} tall and \displaystyle \frac{5}{3} tall. Their areas are \displaystyle \frac{3}{4} \times 12 and \displaystyle \frac{5}{3} \times 12, because area is length times width, fraction sides included (A fraction times a fraction, Day 3).

The rectangle \displaystyle \frac{3}{4} tall is shorter than the dashed one, so it fits inside it with a strip left over on top. It has less area than 12. The rectangle \displaystyle \frac{5}{3} tall is taller than the dashed one, so it holds the whole dashed rectangle and a strip more. It has more area than 12. The rectangle exactly 1 tall is the dashed one, with area exactly 12.

Why, with numbers. Split a factor more than 1 into 1 and the extra part. For \displaystyle \frac{5}{3} = 1 + \frac{2}{3}, the distributive property (multiplying a sum by multiplying each part and adding) gives

The extra part, \displaystyle \frac{2}{3} \times 12, is a positive amount added on, so the product is more than 12. For a factor less than 1, find what it is missing from a whole. Since \displaystyle \frac{3}{4} + \frac{1}{4} = 1,

That is 9 + 3 = 12. The product \displaystyle \frac{3}{4} \times 12 falls short of 12 by the positive amount \displaystyle \frac{1}{4} \times 12 = 3, so it is less than 12. The same two steps work for every factor and every positive b, which is why the rule has no exceptions among positive numbers.

On a number line, a factor less than 1 moves b toward 0, and a factor more than 1 moves it away from 0. A factor of exactly 1 leaves it where it is.

How far it moves. The farther the factor is from 1, the bigger the change. Compare \displaystyle \frac{2}{3} \times 12 = 8 with \displaystyle \frac{3}{4} \times 12 = 9: both are less than 12, but \displaystyle \frac{2}{3} is farther below 1 than \displaystyle \frac{3}{4}, so it shrinks 12 more. In general, for the same positive b, a bigger factor gives a bigger product, because the bigger factor is the smaller one plus a positive extra, and that extra times b is added on.

Comparing with either factor. Each factor can be read as the scale factor for the other. So the product can be compared with a by looking at b, just as it is compared with b by looking at a. When both factors are less than 1, the product is less than both of them. That is why every part of a part in A fraction times a fraction came out smaller than the part it started from.

Day 3: a factor of exactly 1, and comparing products

Alejandro sets the machine to \displaystyle \frac{4}{4}. Four fourths is one whole, so every length in the copy is all of the same length in the original, and the copy is the same size. In numbers, \displaystyle \frac{4}{4} \times 12 = \frac{48}{4} = 12. Any fraction with the same top and bottom, such as \displaystyle \frac{2}{2}, \displaystyle \frac{7}{7} or \displaystyle \frac{100}{100}, is a disguised 1:

This explains a fact you have used for a long time. To make an equivalent fraction, you multiply the top and the bottom by the same number (Making an equivalent fraction by multiplying):

Read the middle as a product of two fractions. By the rule for multiplying fractions, \displaystyle \frac{2 \times 4}{3 \times 4} is \displaystyle \frac{2}{3} \times \frac{4}{4}. So multiplying the top and the bottom by 4 is multiplying the whole fraction by \displaystyle \frac{4}{4}, which is multiplying by 1. By Day 2, a factor of 1 leaves the size alone. That is why \displaystyle \frac{2}{3} and \displaystyle \frac{8}{12} name the same amount and sit at the same point on the number line (Equivalent fractions).

Comparing a product with a mixed number. Is \displaystyle \frac{7}{8} \times 2\frac{1}{2} more or less than \displaystyle 2\frac{1}{2}? The factor \displaystyle \frac{7}{8} has a top smaller than its bottom, so it is less than 1, and the product is less than \displaystyle 2\frac{1}{2}. No arithmetic was needed.

You can also say how much less. The factor \displaystyle \frac{7}{8} is short of 1 by \displaystyle \frac{1}{8}, so the product is short of \displaystyle 2\frac{1}{2} by \displaystyle \frac{1}{8} of \displaystyle 2\frac{1}{2}. To check, write \displaystyle 2\frac{1}{2} as \displaystyle \frac{5}{2} (Mixed numbers and improper fractions: rewriting and comparing):

Since \displaystyle 2\frac{1}{2} = 2\frac{8}{16}, the product is indeed a little less, by \displaystyle \frac{5}{16}, and \displaystyle \frac{5}{16} is \displaystyle \frac{1}{8} \times \frac{5}{2}.

Ordering several products. When several products all resize the same number, they line up in the same order as their factors, because a bigger factor gives a bigger product. So ordering the products comes down to ordering the factors, and the factor 1 marks the spot where the product equals the number itself.

Worked examples

Examples 1 and 2 go with Day 1, Examples 3 to 5 with Day 2, and Examples 6 and 7 with Day 3.

Boundaries

The rule needs the number being resized to be positive. If it is 0, every product is 0: \displaystyle \frac{3}{4} \times 0 = 0 and \displaystyle \frac{5}{3} \times 0 = 0, so a factor less than 1 does not make the product less, and a factor more than 1 does not make it more. A copy of a picture with no width has no width at any setting.

The factor may be 0, which is less than 1, and then the product is 0, less than any positive b, as the rule says. Negative numbers are left out. When the number being resized is negative, the comparison turns around: half of a negative number is closer to 0, so it is more than the number, not less. That belongs to a later course on Integers.

The rule compares a product with one of its own factors. When two products resize different numbers by different factors, it may not settle which is bigger. For \displaystyle \frac{3}{4} \times 10 and \displaystyle \frac{6}{5} \times 9, the first is less than 10 and the second is more than 9, which does not decide between them. You have to find them: \displaystyle 7\frac{1}{2} and \displaystyle 10\frac{4}{5}.

“Less than” and “more than” say which way the size moves, not how far. To know how far, find the part of the number that is added on or left out, as on Day 2, or work out the product.

Where this goes next

The next lesson in this unit, Multiplying mixed numbers, multiplies numbers like \displaystyle 2\frac{1}{2}, and the rule gives a quick check on every answer there: a mixed number factor is more than 1, so the product must be more than the other factor. Dividing by a fraction turns into multiplying by its reciprocal (Dividing by multiplying by the reciprocal), and the copy machine already hints at why: a copy at \displaystyle \frac{3}{4} and then a copy of that copy at \displaystyle \frac{4}{3} gives back the original size.

A percent works as a scale factor too: 75\% of a number is less than the number and 150\% of it is more (Finding a percent of a quantity). Later, the copy machine becomes a scale drawing, where one scale factor multiplies every length in a figure (Scale factor multiplies every length), and a dilation, which resizes a whole shape toward or away from one fixed point.

Exercises

Exercises 1 to 8 go with Day 1, 9 to 17 with Day 2, 18 to 23 with Day 3, and the reasoning group draws on all three days.

Day 1: multiplying as resizing

For Exercises 1 to 3, say first whether the copy is bigger or smaller than the original, then find its size.

1. A picture 10 inches wide is copied at the setting \displaystyle \frac{2}{5}.
2. A picture 10 inches wide is copied at the setting \displaystyle \frac{7}{5}.
3. A picture 6 inches tall is copied at the setting \displaystyle \frac{3}{2}.
4. Which of these settings shrink a picture: \displaystyle \frac{5}{8}, \displaystyle \frac{9}{7}, \displaystyle \frac{1}{3}, \displaystyle \frac{11}{10}, \displaystyle \frac{99}{100}?
5. Alejandro’s copy came out larger than his original. Which of these could have been his setting: \displaystyle \frac{4}{9}, \displaystyle \frac{10}{9}, \displaystyle \frac{2}{2}, \displaystyle \frac{13}{12}?
6. Without multiplying, is \displaystyle \frac{5}{6} \times 24 more or less than 24? Then find it.
7. Without multiplying, is \displaystyle \frac{9}{4} \times 16 more or less than 16? Is it more or less than 2 \times 16? Then find it.
8. Jisoo copies a poster 30 inches wide at the setting \displaystyle \frac{3}{10}. Will the copy fit on a sheet 11 inches wide? Decide, then find the width of the copy.

Day 2: the rule for every case

For Exercises 9 to 13, write <, = or > in the blank without finding the product.

9. \displaystyle \frac{7}{9} \times 45 \;\_\_\; 45
10. \displaystyle \frac{12}{11} \times 8 \;\_\_\; 8
11. \displaystyle \frac{6}{6} \times \frac{3}{7} \;\_\_\; \frac{3}{7}
12. \displaystyle \frac{4}{3} \times \frac{2}{9} \;\_\_\; \frac{2}{9}
13. \displaystyle \frac{1}{10} \times 5\frac{2}{3} \;\_\_\; 5\frac{2}{3}
14. Without multiplying, compare \displaystyle \frac{3}{5} \times \frac{7}{8} with \displaystyle \frac{3}{5} and with \displaystyle \frac{7}{8}. Then find the product and check.
15. Without multiplying, compare \displaystyle \frac{5}{2} \times \frac{2}{7} with \displaystyle \frac{5}{2} and with \displaystyle \frac{2}{7}. Then find the product and check.
16. Use a rectangle 8 long to show that \displaystyle \frac{7}{4} \times 8 is more than 8, and find how much more.
17. Without multiplying, which is greater, \displaystyle \frac{5}{7} \times 63 or \displaystyle 63 \times \frac{8}{7}? Then find both.

Day 3: a factor of exactly 1, and comparing products

18. Show that \displaystyle \frac{4}{9} = \frac{20}{45} by multiplying \displaystyle \frac{4}{9} by a fraction equal to 1. Which fraction?
19. Which fraction equal to 1 turns \displaystyle \frac{7}{12} into \displaystyle \frac{21}{36}?
20. Without finding the products, put these in order from least to greatest: \displaystyle \frac{3}{5} \times 40, \displaystyle \frac{7}{5} \times 40, \displaystyle \frac{5}{5} \times 40, \displaystyle \frac{1}{8} \times 40. Then find each to check.
21. Is \displaystyle \frac{5}{6} \times 4\frac{1}{2} more or less than \displaystyle 4\frac{1}{2}? Decide, then find it.
22. Is \displaystyle \frac{9}{8} \times 1\frac{3}{5} more or less than \displaystyle 1\frac{3}{5}? Decide, then find it.
23. Jisoo copies a picture 9 inches wide at the setting \displaystyle \frac{4}{3}. Then she copies the copy at the setting \displaystyle \frac{3}{4}. How wide is each copy? Explain the final width using a factor of 1.

Reasoning

24. Yemi says, “Multiplying always makes a number bigger.” Give an example where it does not, and explain exactly when multiplying a positive number makes it bigger, smaller, or keeps it the same.
25. Explain why multiplying a number by \displaystyle \frac{n}{n} cannot change its size, and use that to explain why \displaystyle \frac{2}{5} and \displaystyle \frac{6}{15} sit at the same point on the number line.