A unit of the course: the story, then chapter by chapter — sections, numbered lessons, a source or the numbers to read, three checks each — a review per chapter, and the wrap-up at the end.
Drawn scene: the curved lanes of a running track with a picnic table in front holding a pizza cut in eighths, a wooden ruler and a tape measure
8Unit
Fractions
Number
Half a sandwich. Three quarters of a cup of milk. A 3/8-inch wrench. One third of a lap around the track. Fractions show up every day, on rulers, in recipes, on gas gauges and in sports scores. Most of the numbers you have used so far were whole numbers, but the world does not come in wholes. This unit is about the numbers in between.
You will start by learning what a fraction really is: equal parts of a whole, and also a number with its own spot on the number line. You will find different names for the same amount, decide which of two fractions is bigger, and handle amounts that are more than one whole. Then you will add, subtract and multiply fractions the way a cook, a carpenter or a runner needs to.
Finally you will divide with fractions and see why 3 ÷ 1/4 is 12, not 3/4. By the end you will be able to solve real problems, check your own answers, and explain why each step works. Every lesson starts with a situation you can picture, shows the work with real numbers, and then states the rule.
How we figured it out
c. 1650 BCE
Egyptian scribes write most fractions as sums of unit fractions, such as 1/2 + 1/4
c. 300 BCE
Euclid's Elements reasons about quantities as parts and multiples of one another
c. 100 CE
China's Nine Chapters on the Mathematical Art gives rules for adding and dividing fractions
628
Brahmagupta in India writes fractions with the numerator placed above the denominator
1202
Fibonacci's Liber Abaci brings Hindu-Arabic numerals and the fraction bar to European readers
1585
Simon Stevin's De Thiende shows how to write and compute with decimal fractions
1659
Johann Rahn's algebra book uses the ÷ sign for division
1795
France adopts the metric system, built on tenths, hundredths and thousandths
2010
Illinois adopts learning standards that place fractions on the number line in grade 3
Today
You use fractions to measure, cook, build and share, and to check your own reasoning
15
Chapter
What a Fraction Is
Fractions
Big questionHow can one number describe a slice of pizza, a mark on a ruler and a point on a number line?
The story
Three Ways to Say One Half
A ruler, a pizza and a running track look nothing alike, but by the end of the day Maya sees they share one idea.
Maya starts her Tuesday in the wood shop at her Chicago school. Her teacher hands her a board and says to mark it at 1/2 inch. Maya looks at the ruler. Between 0 and 1 there is one longer tick mark, right in the middle. That mark splits the inch into two equal pieces. She makes her mark there. The ruler is her first fraction of the day.
At lunch the cafeteria serves a round pizza cut into 8 equal slices. Maya and her friend Dev each take 2 slices. Together they have eaten 4 of the 8 slices, or 4/8 of the pizza. Dev says, "That is half the pizza." Maya counts the slices left. Four are left, and four are gone. He is right. 4/8 is another name for 1/2.
In gym class the coach sends the students around the track. "One half lap," she calls. The track is 400 meters around, so half a lap is 200 meters. Maya runs from the start line to the far side and stops. She has covered 1/2 of the whole track.
On the bus home Maya thinks about the three halves. A half inch on a ruler is tiny. Half a pizza is a big lunch. Half a track is 200 meters. The wholes are different sizes, so the halves are different sizes too. But every half is the same idea: one of two equal parts of a whole.
That is what a fraction does. It tells how many equal parts a whole is cut into, and how many of those parts you mean. Once you know the whole, the fraction tells you the rest. This chapter is about reading that message and using it.
Talk about itWhat is the whole in each of Maya's three situations, and why does knowing the whole matter before you can say what 1/2 means?
Section 1
Parts of a Whole
15.1
Equal Parts Make Fractions
Main ideaA fraction names equal parts of a whole: the bottom number counts the equal parts, the top number counts how many you mean.
Picture a pan of brownies. If you cut it into 4 pieces of different sizes, you cannot call any piece "a fourth." A needs equal parts. Cut the pan into 4 equal pieces instead. Now each piece is 1/4, one fourth, of the pan. The 4 on the bottom says the whole was cut into 4 equal parts.
A fraction has two numbers. The bottom number is the . It tells how many equal parts make the whole. The top number is the . It tells how many of those parts you are talking about. If 3 of the 4 brownie pieces get eaten, the fraction eaten is 3/4. Read it as "three fourths."
Two mistakes are common. The first is counting parts that are not equal. The second is putting the wrong count on the bottom. Suppose a strip is cut into 5 equal parts and 2 are shaded. The shaded fraction is 2/5, not 2/3. The denominator counts all 5 parts, not the 3 that are unshaded.
Words to know
fraction
a number that names equal parts of a whole, written as one number over another, like 3/4
numerator
the top number of a fraction; it counts how many equal parts you mean
denominator
the bottom number of a fraction; it tells how many equal parts make one whole
equal parts
pieces that are all exactly the same size
Check yourself
1. A pan is cut into 6 equal pieces, and 5 pieces are eaten. What fraction of the pan was eaten?
Why: The whole has 6 equal parts, so 6 is the denominator. 5 parts were eaten, so the numerator is 5: 5/6.
2. What does the denominator of a fraction tell you?
Why: The denominator counts the equal parts in one whole. In 3/4, the 4 means the whole was cut into 4 equal parts.
3. A rectangle is cut into 8 equal parts, and 3 are shaded. What fraction is NOT shaded?
Why: 8 − 3 = 5 parts are unshaded, out of 8 equal parts in all, so 5/8. The denominator stays 8; it counts every part.
15.2
Unit Fractions Build the Rest
Main ideaA unit fraction has 1 on top, and every other fraction is a stack of unit fractions counted up.
Cut a granola bar into 8 equal pieces. One piece is 1/8 of the bar. A fraction with 1 on top is a . It is the basic piece. Three pieces make 3/8, and 3/8 = 1/8 + 1/8 + 1/8. You count eighths the way you count apples: one eighth, two eighths, three eighths.
Now compare bars cut different ways. A bar cut into 4 equal pieces has big pieces. A bar cut into 12 equal pieces has small pieces. So 1/12 is smaller than 1/4. The bigger the denominator, the smaller each unit fraction is. Many students think 1/12 is bigger because 12 is bigger. Remember that 12 pieces means more cuts and smaller pieces.
Counting unit fractions can go past one . Four fourths make one whole bar: 4/4 = 1. Keep counting and you get 5/4, five fourths, which is a whole bar and one more piece. There is nothing wrong with a fraction like 5/4. It just means you counted more pieces than one whole holds.
Words to know
unit fraction
a fraction with 1 as its numerator, like 1/3 or 1/8; it names one equal part
whole
the complete object or amount that a fraction is a part of
Check yourself
1. Which of these is a unit fraction?
Why: A unit fraction has a numerator of 1. Only 1/5 has 1 on top.
2. 5/6 is made of how many pieces of 1/6?
Why: 5/6 = 1/6 + 1/6 + 1/6 + 1/6 + 1/6, which is 5 sixths. The numerator counts the unit fractions.
3. Which is smaller, 1/3 or 1/10, and why?
Why: Tenths come from cutting the whole into 10 equal pieces, thirds from only 3. More pieces means smaller pieces, so 1/10 is smaller than 1/3.
15.3
Fractions on a Number Line
Main ideaA fraction is a number with its own place on the number line, found by cutting the space between whole numbers into equal jumps.
A is a straight line with 0, 1, 2 and so on marked in order. Think of the space from 0 to 1 as one lap of a track. Cut that space into 4 equal lengths. Each length is 1/4 of the way. The first mark after 0 is 1/4, the next is 2/4, then 3/4, and then 4/4, which is the same point as 1.
To find 5/8, cut the space from 0 to 1 into 8 equal jumps. Count 5 jumps from 0 and stop. That point is 5/8. A common mistake is to count the tick marks instead of the jumps. From 0 to 1 in eighths there are 9 marks but only 8 spaces. Another mistake is calling the mark at 0 "1/8." The mark at 0 is 0. Count the jumps you take, not the marks you pass.
A ruler is a number line you can hold. The inch marks are the whole numbers. Between them are marks for halves, fourths and eighths of an inch. To find 3/4 inch, start at 0 and count 3 quarter-inch spaces. That is the same number 3/4 that names three slices of a pizza cut into 4. Same fraction, different whole.
Words to know
number line
a line with numbers placed in order at equal spaces, used to show where a number belongs
interval
the space between two marks on a number line
Check yourself
1. A number line from 0 to 1 is split into 5 equal spaces. What fraction is at the third tick mark after 0?
Why: Each space is 1/5. Three jumps of 1/5 from 0 land on 3/5.
2. A student splits 0 to 1 into fourths but counts the mark at 0 as "1/4." Where will she wrongly land when she looks for 3/4?
Why: She calls 0 "1/4," the real 1/4 "2/4," and the real 2/4 "3/4." So she stops at the point that is really 2/4, one jump short.
3. On a ruler marked in fourths, 3/4 inch is how many quarter-inch spaces from 0?
Why: 3/4 means 3 jumps of 1/4. Count three quarter-inch spaces from 0 to reach 3/4 inch.
Section 2
Equivalent Fractions
15.4
Same Amount, Different Names
Main ideaEquivalent fractions name the same amount using different-size parts, so they sit at the same point on the number line.
Take two chocolate bars of the same size. Break the first into 2 equal pieces and the second into 4 equal pieces. Eat 1 piece of the first bar and 2 pieces of the second. You ate the same amount of chocolate both times: 1/2 of a bar and 2/4 of a bar. Fractions that name the same amount are .
A number line shows it too. Mark 0 to 1 in halves. Then mark the same line in fourths. The mark for 1/2 and the mark for 2/4 land on the exact same point. Mark it in eighths and 4/8 lands there as well. So 1/2 = 2/4 = 4/8, and also 3/6. Each , whether bar or line, tells the same story.
One mistake is thinking 2/4 must be more than 1/2 because its numbers are bigger. Bigger numbers on both top and bottom can mean smaller pieces, not more chocolate. Another mistake is calling 2/3 and 3/4 equivalent because each is one piece short of a whole. They are not. On a line cut into twelfths, 2/3 is at 8/12 and 3/4 is at 9/12. Different points, different numbers.
Words to know
equivalent fractions
fractions that name the same amount, like 1/2 and 2/4
model
a picture or object, like a bar, a circle or a number line, that shows what a fraction means
Check yourself
1. Which fraction is equivalent to 1/2?
Why: 3 is half of 6, so 3 of 6 equal parts is half the whole. 3/6 lands on the same point as 1/2.
2. Fill in the blank: 2/3 = ?/6
Why: Each third splits into 2 sixths, so 2 thirds become 2 × 2 = 4 sixths. 2/3 = 4/6.
3. Why is 3/4 NOT equivalent to 2/3?
Why: Rename both in twelfths: 3/4 = 9/12 and 2/3 = 8/12. Since 9/12 and 8/12 are different, the fractions are not equivalent.
15.5
Multiply to Find Equivalents
Main ideaMultiplying the numerator and the denominator by the same number makes an equivalent fraction, because you are multiplying by 1.
Start with 3/4 of a pan of brownies. Cut every fourth into 2 smaller pieces. Now the pan has 4 × 2 = 8 pieces, and the shaded part has 3 × 2 = 6 pieces. So 3/4 = 6/8. Cut every fourth into 3 pieces instead and you get 3/4 = 9/12. To a fraction into an equivalent one, multiply the top and the bottom by the same number.
Why does this work? Multiplying the top and bottom by 2 is the same as multiplying the fraction by 2/2. And 2/2 equals 1. Multiplying any number by 1 does not change it. Try 2/5 × 3/3: the top becomes 2 × 3 = 6 and the bottom becomes 5 × 3 = 15, so 2/5 = 6/15. Same amount, smaller pieces.
A common mistake is to add the same number to the top and bottom. 1/2 does not equal 2/3, even though 1 + 1 = 2 and 2 + 1 = 3. Adding a piece changes the amount. The rule works backwards too. Divide the top and bottom of 6/8 by 2 and you get 3/4. When no number but 1 divides both parts, the fraction is in .
Words to know
multiply
to combine equal groups; 3 × 4 means 3 groups of 4, which is 12
simplest form
a fraction whose numerator and denominator share no common factor except 1, like 3/4
Check yourself
1. What number do you multiply the numerator and denominator of 2/3 by to get 8/12?
Why: 2 × 4 = 8 and 3 × 4 = 12. Multiplying top and bottom by 4 turns 2/3 into 8/12.
2. Which fraction is equivalent to 3/5?
Why: 3 × 2 = 6 and 5 × 2 = 10, so 3/5 = 6/10. 4/6 comes from adding 1 to each part, which changes the amount.
3. What is 8/12 in simplest form?
Why: Divide top and bottom by 4: 8 ÷ 4 = 2 and 12 ÷ 4 = 3. 4/6 is equivalent but not simplest, since 2 still divides both.
15.6
Whole Numbers as Fractions
Main ideaAny whole number can be written as a fraction, like 3 = 3/1 = 6/2 = 12/4, because a whole holds a full set of equal parts.
Cut a pizza into 4 equal slices. All 4 slices together are 4/4, which is pizza. Two pizzas cut the same way give 8 slices, or 8/4. Eight fourths is 2 whole pizzas. Any fraction whose numerator is a multiple of its denominator is a whole number.
A can also have a denominator of 1. The fraction 3/1 means 3 wholes, each left in 1 piece. So 3 = 3/1. It also equals 6/2, 9/3 and 12/4. On a number line marked in fourths, count 12 jumps of 1/4 from 0. You land exactly on 3.
Watch out for two mix-ups. First, 3/1 is not the same as 1/3. The fraction 3/1 is 3 wholes, while 1/3 is less than 1. Second, 5/5 equals 1, not 5. Five fifths fill one whole and stop. To check, ask how many full sets of the denominator the numerator holds. In 10/5, the 10 holds two sets of 5, so 10/5 = 2.
Words to know
whole number
one of the counting numbers and zero: 0, 1, 2, 3 and so on
one whole
a complete object or set; as a fraction, any fraction with the same top and bottom, like 4/4
Check yourself
1. What whole number equals 12/3?
Why: 12 ÷ 3 = 4, because twelve thirds make four full sets of 3 thirds, and each set is one whole.
2. Which fraction equals 5?
Why: 10 ÷ 2 = 5, so 10/2 = 5. Notice 5/5 equals 1, not 5.
3. How many fourths are in 3 wholes?
Why: Each whole holds 4 fourths, so 3 wholes hold 3 × 4 = 12 fourths. 3 = 12/4.
Section 3
Comparing Fractions
15.7
Same Bottom or Same Top
Main ideaWith the same denominator, more parts means a bigger fraction; with the same numerator, a bigger denominator means a smaller fraction.
Two pans of cornbread are cut into 8 equal pieces each. One pan has 5 pieces left and the other has 3. Which pan has more? The pieces are the same size, so just the counts. Since 5 is more than 3, 5/8 is greater than 3/8. When denominators match, the bigger numerator wins.
Now take 2/3 and 2/5. Both say "2 pieces," but the pieces are not the same size. Thirds come from 3 cuts of the whole, and fifths from 5 cuts, so thirds are bigger. Two big pieces beat two small pieces: 2/3 is 2/5. The common mistake is to pick 2/5 because 5 is the bigger number. With the same numerator, the smaller denominator gives the bigger fraction.
We write comparisons with symbols. 5/8 > 3/8 means five eighths is greater than three eighths. 2/5 < 2/3 means two fifths is less than two thirds. One rule matters before any comparison: the wholes must be the same size. Half of a small pizza is not more than 1/3 of a giant pizza. Compare fractions only when they refer to the same whole.
Words to know
compare
to decide which of two numbers is bigger, or whether they are equal
greater than
bigger than; written with the symbol >, as in 5/8 > 3/8
Check yourself
1. Which is greater, 5/9 or 7/9, and why?
Why: Both are ninths, so the pieces are the same size. 7 pieces is more than 5 pieces, so 7/9 > 5/9.
2. Which is greater, 3/4 or 3/10?
Why: Both have 3 pieces, but a fourth is bigger than a tenth. Three big pieces beat three small ones, so 3/4 > 3/10.
3. Put 2/7, 2/3 and 2/5 in order from least to greatest.
Why: Same numerator, so the biggest denominator is the smallest fraction. Sevenths are smallest, then fifths, then thirds: 2/7 < 2/5 < 2/3.
15.8
Benchmarks: 0, One Half and 1
Main ideaCompare a fraction to 0, 1/2 or 1 first; that quick check often settles which fraction is bigger.
Which is bigger, 3/8 or 5/9? The denominators and numerators are all different. Use a , a familiar number to measure against. Try 1/2. of 8 is 4, and 3 is less than 4, so 3/8 is less than 1/2. Half of 9 is 4 1/2, and 5 is more than that, so 5/9 is more than 1/2. So 5/9 > 3/8.
The benchmark 1 works for fractions close to a whole. Compare 7/8 and 9/10. Each is one piece short of 1. But 7/8 is missing an eighth, and 9/10 is missing only a tenth. A tenth is smaller than an eighth, so 9/10 is closer to 1. That makes 9/10 the bigger fraction.
The benchmark 0 works for tiny fractions. 1/12 is much closer to 0 than 1/3 is. Money gives good practice. A quarter is 1/4 of a dollar, so 3 quarters are 3/4 of a dollar. That is more than 1/2 of a dollar, because 2 quarters already make a half. Always ask: is this fraction near 0, near 1/2 or near 1?
Words to know
benchmark
a familiar number, like 0, 1/2 or 1, that you compare other numbers against
half
one of two equal parts; 1/2 of any amount
Check yourself
1. Which fraction is greater than 1/2?
Why: Half of 8 is 4, and 5 > 4, so 5/8 is more than 1/2. The others have numerators below half their denominators.
2. Which is greater, 5/6 or 7/8?
Why: 5/6 is missing 1/6 to reach 1, and 7/8 is missing 1/8. Since 1/8 is the smaller gap, 7/8 is closer to 1 and bigger.
3. A dime is what fraction of a dollar?
Why: 10 dimes make 100 cents, one dollar, so a dime is 1 of 10 equal parts: 1/10.
Section 4
More Than One Whole
15.9
Fractions Greater Than One
Main ideaWhen the numerator is bigger than the denominator, the fraction is more than one whole.
A pizza shop cuts every pizza into 4 equal slices. After a party, 7 slices are left on the counter. That is 7/4 of a pizza. Four slices fill one whole pizza box, and 3 more slices are left over. So 7/4 is one whole and 3/4 more. A has a numerator bigger than its denominator. Some books call it an , but nothing is wrong with it.
On a number line, count 7 jumps of 1/4 from 0. You pass 4/4, which is 1, and keep going three more jumps. You stop between 1 and 2. Count 8 jumps and you land exactly on 2, because 8/4 = 2. Fractions greater than one live past the 1 mark, out with the bigger numbers.
A common mistake is reading 7/4 as "7 pizzas cut into 4 pieces." It means 7 pieces that are each 1/4 of a pizza. To decide if a fraction is more than 1, compare the top to the bottom. In 7/4, 7 > 4, so it is more than one. In 3/4, 3 < 4, so it is less than one. In 4/4, the two match, so it is exactly 1.
Words to know
fraction greater than one
a fraction whose numerator is bigger than its denominator, like 7/4
improper fraction
another name for a fraction greater than or equal to one, like 5/3 or 4/4
Check yourself
1. Which fraction is greater than 1?
Why: 5 > 3, so 5/3 is more than one whole. 9/9 is exactly 1, and the others are less than 1.
2. 9/4 is between which two whole numbers on the number line?
Why: 8/4 = 2 and 12/4 = 3. Since 9 is between 8 and 12, 9/4 is between 2 and 3.
3. How many whole units are in 10/5?
Why: 5 fifths make 1 whole, and 10 fifths make 2 sets of 5, so 10/5 = 2.
15.10
Mixed Numbers
Main ideaA mixed number is a whole number plus a fraction, and you can rewrite it as a single fraction and back again.
A recipe calls for 2 1/3 cups of flour. That is a : the 2 plus the fraction 1/3. To write it as one fraction, turn the wholes into thirds. Two wholes are 2 × 3 = 6 thirds. Add the extra 1/3 and you have 7/3. A quick way: multiply the whole by the denominator, then add the numerator, and keep the denominator. 2 × 3 + 1 = 7, so 2 1/3 = 7/3.
Going the other way, take 11/4. Ask how many groups of 4 fit in 11. Two groups use 8 fourths, and 11 − 8 = 3 fourths are left. So 11/4 = 2 3/4. You can check on a number line: 11 jumps of 1/4 pass 2 and stop 3 jumps later.
Two mistakes are common. Some students read 2 1/3 as 2 times 1/3, but a mixed number means 2 plus 1/3. Others turn 11/4 into 2 1/4 by dropping the leftover count. Check by going back: 2 × 4 + 1 = 9, not 11, so 2 1/4 is wrong. The correct 2 3/4 gives 2 × 4 + 3 = 11.
Words to know
mixed number
a whole number and a fraction written together, like 2 1/3, meaning 2 plus 1/3
whole part
the whole-number piece of a mixed number; in 2 1/3 the whole part is 2
Check yourself
1. What is 3 1/2 written as a single fraction?
Why: 3 wholes are 3 × 2 = 6 halves. Add the extra half: 6 + 1 = 7 halves, so 3 1/2 = 7/2.
2. What is 14/3 written as a mixed number?
Why: Four groups of 3 use 12 thirds, and 14 − 12 = 2 thirds are left. So 14/3 = 4 2/3.
3. Which fraction is equal to 1 3/4?
Why: 1 × 4 + 3 = 7, so 1 3/4 = 7/4. The choice 13/4 comes from gluing the digits together instead of multiplying.
Chapter review
What a Fraction Is
0 / 8
1. A shape is cut into 10 equal parts, and 7 parts are shaded. What fraction is shaded?
Why: The whole has 10 equal parts, so the denominator is 10. Seven parts are shaded, so the fraction is 7/10.
2. Which fraction is equivalent to 2/5?
Why: Multiply top and bottom of 2/5 by 2: 4/10. Changing only the denominator, as in 2/10, changes the amount.
3. Which is smaller, 1/6 or 1/8, and why?
Why: Cutting a whole into 8 pieces makes smaller pieces than cutting it into 6. So 1/8 < 1/6.
4. Which fraction is between 1 and 2 on the number line?
Why: 3/2 = 1 1/2, which is between 1 and 2. 4/2 is exactly 2, 5/2 is 2 1/2, and 1/2 is less than 1.
5. Which comparison is true?
Why: With the same numerator, the smaller denominator is bigger: thirds are bigger than ninths, so 2/3 > 2/9. The other three statements are backwards.
6. What is 5 1/4 written as a single fraction?
Why: 5 × 4 = 20 fourths, plus 1 more fourth is 21 fourths: 21/4.
7. Which fraction equals 4?
Why: 12 ÷ 3 = 4, so 12/3 = 4. Remember that 4/4 is 1, not 4.
8. Which fraction is greater than 1/2?
Why: Half of 12 is 6, and 7 > 6, so 7/12 is more than 1/2. In each other choice the numerator is less than half the denominator.
Send it to your teacher
16
Chapter
Adding, Subtracting and Multiplying Fractions
Fractions
Big questionHow do you combine, take away or scale amounts when those amounts are fractions?
The story
Twice the Pancakes
A Saturday breakfast for eight people turns a simple recipe into a puzzle about fourths, halves and cups.
Jonah's dad makes pancakes every Saturday from a card that serves 4 people. Today two cousins and their parents are coming, so 8 people will eat. "Double everything," Dad says, and hands Jonah the card. The first line reads: 3/4 cup of milk.
Jonah thinks out loud. "Two times 3/4. That is 3/4 plus 3/4." He adds the tops and gets 6. Then he adds the bottoms and gets 8. "Six eighths," he says, and reaches for the measuring cup. Dad stops him. "Is 6/8 more than one cup or less?" Jonah looks at the cup. The 6/8 line sits below the 1-cup line.
That cannot be right. One 3/4 cup of milk is already most of a cup, and Jonah needs two of them. Two of something has to be more than one of it. He tries again, this time keeping the fourths as fourths: 3 fourths plus 3 fourths is 6 fourths. Four fourths make a cup, so 6 fourths is 1 cup and 2/4 more. That is 1 1/2 cups. The cup line agrees.
The flour line says 1 1/2 cups. Doubled, that is 3 cups. Then Jonah finds the problem cooks know well: the 1-cup scoop is in the dishwasher, and only the 1/2-cup scoop is clean. He has to figure out how many half-cup scoops make 3 cups. He counts by halves and gets 6 scoops.
By the time the batter is mixed, Jonah has added fractions, doubled a mixed number and counted halves into wholes. He also learned why his first answer was wrong. This chapter walks through each of those moves so you can trust your answer before you pour.
Talk about itJonah's first answer was 6/8 cup. How could he tell, without measuring, that it had to be wrong?
Section 1
Same-Size Pieces
16.1
Adding With Like Denominators
Main ideaWhen denominators match, add the numerators and keep the denominator, because you are counting pieces of the same size.
Ava eats 3/8 of a pizza and Ben eats 2/8. How much did they eat together? The slices are all eighths, the same size. Three eighths plus two eighths is five eighths, the same way 3 apples plus 2 apples is 5 apples. So 3/8 + 2/8 = 5/8. Fractions with add by counting the numerators. The denominator does not change.
A number line shows the . Start at 3/8 and jump 2 more eighths to land on 5/8. Sometimes the sum passes 1. Try 5/6 + 4/6. Five sixths plus four sixths is nine sixths, or 9/6. Six sixths make 1 whole, and 3 sixths are left. So 9/6 = 1 3/6, which is 1 1/2.
The most common mistake is adding the denominators too: 3/8 + 2/8 = 5/16. Here is why that is wrong. 5/16 is smaller than 3/8, because 3/8 = 6/16. Adding more pizza cannot leave you with less pizza. The denominator names the size of the pieces, and the piece size does not change when you add more pieces.
Words to know
like denominators
the same denominator in two or more fractions, like 3/8 and 2/8
sum
the answer to an addition problem; the sum of 3/8 and 2/8 is 5/8
Check yourself
1. What is 2/5 + 2/5?
Why: Two fifths plus two fifths is four fifths. Add the numerators, 2 + 2 = 4, and keep the denominator 5.
2. What is 5/6 + 3/6?
Why: 5 + 3 = 8 sixths, or 8/6. Six sixths make 1, and 2/6 = 1/3 is left, so the sum is 1 1/3.
3. A student writes 3/8 + 2/8 = 5/16. Why is that wrong?
Why: 3/8 equals 6/16, so 5/16 is less than what Ava ate alone. The pieces stay eighths; the sum is 5/8.
16.2
Subtracting With Like Denominators
Main ideaWhen denominators match, subtract the numerators and keep the denominator, then check by adding back.
A water bottle is marked in tenths. It is 7/10 full. Lena drinks 3/10 of the bottle on her walk to school. How much is left? Seven tenths take away three tenths leaves four tenths: 7/10 − 3/10 = 4/10. The is 4/10, which is the same as 2/5. Like addition, you work only with the numerators.
Sometimes you subtract from a whole. A pan of brownies is full, and someone eats 3/8 of it. To subtract, the whole 1 as 8/8, since 8 eighths fill the pan. Then 8/8 − 3/8 = 5/8 of the pan remains. Renaming 1 as a fraction with the right denominator makes the subtraction easy.
Two mistakes show up often. One is subtracting the denominators as well, which gives a denominator of 0 and makes no sense. Another is forgetting to rename 1. Always check by adding back. If 1 − 3/8 = 5/8, then 5/8 + 3/8 should be 8/8, which is 1. It is, so the answer is right.
Words to know
difference
the answer to a subtraction problem; the difference of 7/10 and 3/10 is 4/10
rename
to write a number as an equivalent fraction, like writing 1 as 8/8
Check yourself
1. What is 9/12 − 4/12?
Why: 9 − 4 = 5 twelfths. The denominator stays 12, so the difference is 5/12.
2. What is 1 − 2/5?
Why: Rename 1 as 5/5. Then 5/5 − 2/5 = 3/5. Check: 3/5 + 2/5 = 5/5 = 1.
3. Which addition checks that 8/9 − 5/9 = 3/9 is correct?
Why: Adding the answer back to what was taken away should give the start: 3/9 + 5/9 = 8/9. It does.
16.3
Mixed Numbers With Like Denominators
Main ideaAdd or subtract the whole parts and the fraction parts separately, then regroup any fraction that is more than one.
A carpenter joins two boards end to end: one is 2 3/4 feet long and the other is 1 2/4 feet. Add the wholes: 2 + 1 = 3. Add the fractions: 3/4 + 2/4 = 5/4. But 5/4 is more than one whole, so it as 1 1/4. Now add that 1 to the 3 wholes: 3 + 1 1/4 = 4 1/4 feet.
Subtraction can need regrouping the other way. Try 3 1/4 − 1 3/4. You cannot take 3/4 from 1/4. So 3 1/4 by borrowing one whole: 3 1/4 = 2 + 4/4 + 1/4 = 2 5/4. Now subtract: 2 5/4 − 1 3/4 = 1 2/4, which is 1 1/2. Another route is to turn both into fourths: 13/4 − 7/4 = 6/4 = 1 1/2. Same answer.
The common mistake is to subtract the smaller fraction from the bigger one no matter where it sits: 3/4 − 1/4 = 2/4, giving 2 2/4. That is wrong. Check by adding: 1 3/4 + 2 2/4 = 4 1/4, not 3 1/4. The true answer 1 1/2 checks: 1 3/4 + 1 2/4 = 2 5/4 = 3 1/4.
Words to know
regroup
to trade pieces for a whole, or a whole for pieces, like turning 5/4 into 1 1/4
rename
to write a number a different way without changing its value, like 3 1/4 = 2 5/4
Why: Rename 4 1/3 as 3 4/3. Then 3 4/3 − 1 2/3 = 2 2/3. Or 13/3 − 5/3 = 8/3 = 2 2/3.
3. To subtract 5 1/6 − 2 5/6, how should you rename 5 1/6 first?
Why: Borrow one whole as 6/6: 5 1/6 = 4 + 6/6 + 1/6 = 4 7/6. Now 7/6 − 5/6 works.
Section 2
Different-Size Pieces
16.4
Finding a Common Denominator
Main ideaTo add or subtract fractions with different denominators, first rename them so they share the same denominator.
Try 1/2 + 1/3. Halves and thirds are different sizes, so you cannot just count them. You need pieces that fit both. A is a number that both denominators divide into. For 2 and 3, that number is 6. Rename: 1/2 = 3/6 and 1/3 = 2/6. Now both fractions are sixths, and you can add.
How do you find one? List the of each denominator. Multiples of 2: 2, 4, 6, 8. Multiples of 3: 3, 6, 9. The first number on both lists is 6. For 4 and 6, the multiples are 4, 8, 12 and 6, 12, so 12 works. Multiplying the denominators, 4 × 6 = 24, also works, but 12 is smaller and easier.
Often one denominator already divides the other. For 3/4 + 1/8, use 8. Rename 3/4 as 6/8, and leave 1/8 alone. The big mistake is changing only the denominator: writing 1/2 as 1/6. That shrinks the amount. Whatever you multiply the bottom by, multiply the top by the same number.
Words to know
common denominator
a denominator shared by two or more fractions after renaming, like 6 for 1/2 and 1/3
multiple
a number you get by multiplying; 6, 12 and 18 are multiples of 6
Check yourself
1. What is a common denominator for 1/3 and 1/4?
Why: Multiples of 3: 3, 6, 9, 12. Multiples of 4: 4, 8, 12. Both lists reach 12, so 12 is a common denominator.
2. Rename 2/3 with a denominator of 12.
Why: 3 × 4 = 12, so multiply the top by 4 too: 2 × 4 = 8. 2/3 = 8/12.
3. What is the smallest common denominator for 5/6 and 1/4?
Why: Multiples of 6: 6, 12. Multiples of 4: 4, 8, 12. The smallest shared multiple is 12. 24 also works but is not the smallest.
16.5
Adding Unlike Fractions
Main ideaRename the fractions with a common denominator, then add the numerators and keep that denominator.
Priya walks 1/2 mile to the library and then 1/3 mile to the park. How far did she walk? The fractions have , so rename them in sixths: 1/2 = 3/6 and 1/3 = 2/6. Now add: 3/6 + 2/6 = 5/6. She walked 5/6 of a mile.
Try one that passes a whole. A recipe needs 3/4 cup of flour for the batter and 5/8 cup for the topping. Use eighths: 3/4 = 6/8. Then 6/8 + 5/8 = 11/8. Eight eighths make 1 cup, and 3/8 is left, so the recipe needs 1 3/8 cups of flour in all.
The common mistake is adding across: 1/2 + 1/3 = 2/5. You can catch it with an . 1/2 plus a little more must be more than 1/2. But 2/5 is less than 1/2, since half of 5 is 2 1/2. So 2/5 cannot be the sum. Estimating first tells you what size answer to expect.
Words to know
unlike denominators
different denominators in two fractions, like 1/2 and 1/3
estimate
a quick, rough answer used to check whether an exact answer makes sense
Check yourself
1. What is 1/4 + 2/3?
Why: Rename in twelfths: 1/4 = 3/12 and 2/3 = 8/12. Then 3/12 + 8/12 = 11/12.
2. What is 1/2 + 3/8?
Why: 1/2 = 4/8, so 4/8 + 3/8 = 7/8. The denominator stays 8.
3. What is 5/6 + 1/2?
Why: 1/2 = 3/6, so 5/6 + 3/6 = 8/6. That is 1 whole and 2/6, which is 1 1/3.
16.6
Subtracting Unlike Fractions
Main ideaRename with a common denominator, subtract the numerators, and check that the answer is reasonable.
A ribbon is 3/4 yard long. Nia cuts off 1/3 yard for a bow. How much is left? Fourths and thirds need a common denominator, and 12 works. Rename: 3/4 = 9/12 and 1/3 = 4/12. Subtract: 9/12 − 4/12 = 5/12. There is 5/12 yard of ribbon left.
Mixed numbers work the same way. Take 2 1/2 − 3/4. Rename in fourths: 2 1/2 = 2 2/4. You cannot take 3/4 from 2/4, so borrow: 2 2/4 = 1 6/4. Now 1 6/4 − 3/4 = 1 3/4. Or write 2 1/2 as 5/2 = 10/4, and 10/4 − 3/4 = 7/4 = 1 3/4. Both roads reach 1 3/4.
Always : 5/12 + 4/12 = 9/12 = 3/4, so the ribbon answer is right. Ask whether the answer is . Since 1/3 is more than 1/4, taking 1/3 from 3/4 must leave less than 1/2. 5/12 is less than 6/12, so it fits. The usual mistakes are subtracting denominators, or renaming the bottom but not the top.
Words to know
check by adding
to test a subtraction answer by adding it back to the amount you took away
reasonable
close to what an estimate says the answer should be
Check yourself
1. What is 5/6 − 1/3?
Why: 1/3 = 2/6, so 5/6 − 2/6 = 3/6, which is 1/2. The choice 2/3 comes from subtracting 5 − 1 without renaming.
Why: 2 1/4 = 9/4 = 27/12 and 2/3 = 8/12. Then 27/12 − 8/12 = 19/12 = 1 7/12.
Section 3
Multiplying Fractions
16.7
A Fraction Times a Whole Number
Main ideaMultiplying a fraction by a whole number is repeated addition: 3 × 2/5 means three jumps of 2/5, which is 6/5.
Each day Sofia runs 2/5 of a mile. In 3 days she runs 3 × 2/5. That is 2/5 + 2/5 + 2/5, three jumps of 2/5 on a number line. Count the fifths: 2 + 2 + 2 = 6 fifths, so the is 6/5, which is 1 1/5 miles. The rule: multiply the numerator by the whole number and keep the denominator.
The word also means multiply. Suppose 2/5 of a class of 15 students ride the bus. Split 15 into 5 equal groups: 15 ÷ 5 = 3 students in each fifth. Two fifths are 2 × 3 = 6 students. The rule gives the same answer: 2/5 × 15 = 30/5 = 6.
A common mistake is multiplying both the top and the bottom: 3 × 2/5 = 6/15. But 6/15 equals 2/5, so nothing got bigger. Only the numerator is multiplied. Try 4 × 3/4. That is 12/4, and 12 fourths make exactly 3 wholes. Four servings of 3/4 cup is 3 cups.
Words to know
product
the answer to a multiplication problem; the product of 3 and 2/5 is 6/5
of
in math, a word that means multiply, as in 2/5 of 15 = 2/5 × 15
Check yourself
1. What is 4 × 2/3?
Why: 4 × 2 = 8 thirds, or 8/3. Six thirds make 2, and 2/3 is left, so 8/3 = 2 2/3.
2. What is 3/4 of 20?
Why: One fourth of 20 is 20 ÷ 4 = 5. Three fourths are 3 × 5 = 15. Also 3/4 × 20 = 60/4 = 15.
3. What is 5 × 3/10?
Why: 5 × 3 = 15 tenths, or 15/10. Ten tenths make 1, and 5/10 = 1/2 is left, so the product is 1 1/2.
16.8
A Fraction Times a Fraction
Main ideaTo multiply two fractions, multiply the numerators and multiply the denominators; an area model shows why.
What is 1/2 of 3/4 of a brownie pan? Draw the pan as a rectangle. Cut it into 4 columns and shade 3 of them for 3/4. Now cut the pan into 2 rows and take the top row for 1/2. The part shaded both ways is 3 small pieces out of 8 in the whole pan. So 1/2 × 3/4 = 3/8. This picture is an .
The model gives a rule. The rows times the columns give the total pieces, so denominators multiply: 2 × 4 = 8. The shaded rows times the shaded columns give the double-shaded pieces, so numerators multiply: 1 × 3 = 3. Try 2/3 × 4/5. Numerators: 2 × 4 = 8. Denominators: 3 × 5 = 15. The product is 8/15.
Some students look for a common denominator before multiplying. You do not need one; each keeps its own denominator. Also, simplify at the end when you can. 2/3 × 3/4 = 6/12, and 6/12 = 1/2. A quick check: half of 3/4 should be a bit less than half of 1, and 3/8 is a bit less than 4/8. It fits.
Words to know
area model
a rectangle cut into rows and columns that shows a fraction of a fraction
factor
a number being multiplied; in 2/3 × 4/5 the factors are 2/3 and 4/5
Check yourself
1. What is 1/3 × 3/5?
Why: Numerators: 1 × 3 = 3. Denominators: 3 × 5 = 15. So the product is 3/15, which simplifies to 1/5.
2. What is 2/3 × 5/6?
Why: 2 × 5 = 10 and 3 × 6 = 18, so the product is 10/18. Divide both by 2: 5/9.
3. An area model has 4 rows and 5 columns. The shaded part covers 2 rows and 3 columns. Which product does it show?
Why: 2/4 × 3/5: the whole has 4 × 5 = 20 pieces, and 2 × 3 = 6 are shaded twice. The product is 6/20, or 3/10.
16.9
Bigger or Smaller? Scaling
Main ideaMultiplying by a fraction less than 1 makes a number smaller; multiplying by a number greater than 1 makes it bigger.
Multiplying does not always make things bigger. Start with 8. Multiply by 1/2 and you get 4, which is smaller. Multiply by 3/2 and you get 24/2 = 12, which is bigger. Multiply by 1 and you get 8, the same. Multiplying by a number is : the factor tells you whether the result grows, shrinks or stays put.
You can predict the size before you compute. 3/4 × 2/3 must be less than 2/3, because 3/4 is . And 5/4 × 2/3 must be more than 2/3, because 5/4 is more than 1. Check: 3/4 × 2/3 = 6/12 = 1/2, which is less than 2/3. Then 5/4 × 2/3 = 10/12 = 5/6, which is more than 2/3.
The common mistake is to assume a product is always larger than both factors. That is true for whole numbers bigger than 1, but not for fractions. If you eat 2/3 of a 12-inch sub, you eat 2/3 × 12 = 8 inches, less than the whole sub. When one factor is less than 1, expect the answer to be smaller than the other factor.
Words to know
scaling
multiplying a number to make it bigger or smaller by a certain factor
less than 1
a number between 0 and 1, like 3/4; multiplying by it shrinks a number
Check yourself
1. Without computing, what can you say about 7/8 × 5?
Why: 7/8 is less than 1, so multiplying 5 by it gives less than 5. In fact 7/8 × 5 = 35/8 = 4 3/8.
2. Which product is greater than 2/3?
Why: Only 4/3 is greater than 1, so only 4/3 × 2/3 = 8/9 is more than 2/3. The others shrink 2/3 or leave it the same.
3. What is 12 × 3/2?
Why: 12 × 3 = 36 halves, and 36/2 = 18. Since 3/2 is more than 1, the answer is bigger than 12.
16.10
Recipes and Measurements
Main ideaRead the problem, decide whether to add, subtract or multiply, compute carefully, and then ask whether the answer makes sense.
A cookie recipe uses 2/3 cup of sugar. Making three batches means 3 × 2/3 = 6/3 = 2 cups of sugar. Making half a batch means 1/2 × 2/3 = 2/6 = 1/3 cup. In a , "three batches" and "half a batch" both mean multiply. Ask what is being scaled and by how much.
Measurements often need adding. A 5/8-inch board is glued on top of a 3/4-inch board. The total thickness is 5/8 + 3/4. Rename 3/4 as 6/8: 5/8 + 6/8 = 11/8 = 1 3/8 inches. Cutting a piece off means subtracting. Combining pieces means adding. Making copies means multiplying.
The most common mistake is picking the operation from a single word instead of the situation. "Half" can mean multiply by 1/2 or subtract 1/2, depending on the story. After computing, check that the answer is : two batches need more sugar, not less; a glued stack is thicker than either board alone.
Words to know
word problem
a math problem told as a story that you have to turn into numbers and operations
reasonable
an answer that is the right size for the story it comes from
Check yourself
1. A recipe calls for 3/4 cup of milk. How much milk is needed for a double recipe?
Why: 2 × 3/4 = 6/4. Four fourths make 1 cup, and 2/4 = 1/2 is left, so 1 1/2 cups.
2. A rope is 2 1/2 feet long. You cut off 3/4 foot. How much rope is left?
Why: 3 × 1 = 3 and 5 × 2 = 10, so the product is 3/10.
7. Which product is less than 3/4?
Why: 5/6 is less than 1, so 3/4 × 5/6 = 15/24 = 5/8 is smaller than 3/4. The other factors are 1 or more.
8. A recipe uses 1/4 cup of oil. How much oil is needed for 5 batches?
Why: 5 × 1/4 = 5/4. Four fourths make 1 cup, and 1/4 is left, so 1 1/4 cups.
Send it to your teacher
17
Chapter
Dividing With Fractions
Fractions
Big questionWhat does it mean to divide when one of the numbers is a fraction, and how can you tell if your answer makes sense?
The story
Bows From Three Yards
A spool of ribbon, a pair of scissors and a question that looks like division but feels like counting.
Lena is wrapping gifts for a school fundraiser. She has 3 yards of red ribbon, and each bow takes 1/4 yard. She wants to know how many bows she can make before she starts cutting. "That is 3 divided by 1/4," she says, and her older brother Theo looks up from his phone.
"Division makes things smaller," Theo says. "So the answer is less than 3. Maybe 3/4." Lena is not so sure. Three fourths of a bow does not sound like enough for a fundraiser. She lays the ribbon along a yardstick. Every yard holds 4 quarter-yard pieces. She marks them off: 4 in the first yard, 4 in the second, 4 in the third.
Twelve pieces. Twelve bows from 3 yards of ribbon. Lena writes it down: 3 ÷ 1/4 = 12. Dividing by 1/4 asked how many fourths fit inside 3, and the answer was bigger than 3, not smaller. Theo puts down his phone. "So dividing by a fraction can make the number grow?" It can, when the fraction is less than 1.
Then Lena finds a short strip of gold ribbon, only 1/4 yard, and she wants to put a bit of gold on each of 3 gift tags. Now she must split a small piece into 3 equal parts. Each tag gets 1/4 ÷ 3. She pictures the whole yard cut into fourths, then each fourth cut into 3. That makes 12 pieces per yard, so each tag gets 1/12 yard.
Same numbers, 3 and 1/4, but two very different questions. One asks how many small pieces fit in a big amount. The other asks how to share a small piece. This chapter teaches both, along with a third idea: a fraction like 3/4 is itself a division, 3 shared among 4.
Talk about itTheo said division always makes things smaller. When is that true, and when is it not? Use Lena's ribbon to explain.
Section 1
Dividing by a Unit Fraction
17.1
How Many Fourths in Three?
Main ideaDividing a whole number by a unit fraction asks how many of those small pieces fit inside the whole number.
When you 3 by 1/4, you are asking, "How many fourths are in 3?" Picture 3 whole sandwiches, each cut into 4 equal pieces. The first sandwich gives 4 fourths, the second gives 4 more, and the third gives 4 more. That is 4 + 4 + 4 = 12 fourths. So 3 ÷ 1/4 = 12. The answer, 12, is the .
The picture gives a rule. Every whole holds as many pieces as the denominator says. So a whole number divided by a unit fraction equals the whole number times the denominator. 5 ÷ 1/3 = 5 × 3 = 15, because 5 wholes each hold 3 thirds. 2 ÷ 1/6 = 2 × 6 = 12.
The common mistake is to write 3 ÷ 1/4 = 3/4. That answers a different question, 3 ÷ 4, which shares 3 among 4 people. To check a quotient, multiply it by the divisor: 12 × 1/4 = 12/4 = 3. That brings you back to 3, so 12 is right.
Words to know
divide
to split an amount into equal groups, or to find how many equal groups fit inside it
quotient
the answer to a division problem; in 3 ÷ 1/4 = 12, the quotient is 12
Check yourself
1. What is 2 ÷ 1/5?
Why: Each whole holds 5 fifths, so 2 wholes hold 2 × 5 = 10 fifths. Check: 10 × 1/5 = 2.
2. What is 4 ÷ 1/3?
Why: 4 wholes each hold 3 thirds: 4 × 3 = 12. Check: 12 × 1/3 = 4.
3. Which question does 6 ÷ 1/2 ask?
Why: Dividing by 1/2 asks how many halves fit in 6. The answer is 12, since each whole holds 2 halves.
17.2
Pictures on the Number Line
Main ideaA number line shows division by a unit fraction as counting equal jumps from 0 to the whole number.
To show 2 ÷ 1/3 on a number line, draw the line from 0 to 2 and mark every third. Now count the jumps of 1/3 it takes to go from 0 to 2. There are 3 jumps in the first whole and 3 in the second, so 6 jumps. Each is one third, and 6 of them reach 2. So 2 ÷ 1/3 = 6.
The same picture works for any unit fraction. Show 5 ÷ 1/2 by marking halves from 0 to 5. Two jumps of 1/2 make each whole, so 5 wholes take 10 jumps. The pattern is always the same: jumps = whole number × denominator. That is the rule from the last lesson, drawn instead of written.
The mistake to watch for is counting instead of jumps. From 0 to 2 in thirds there are 7 tick marks, counting the ones at 0 and 2. But there are only 6 spaces between them. The quotient counts the jumps, not the marks. Put your finger at 0 and count each hop.
Words to know
jump
one move of a set length along a number line, like a hop of 1/3
tick mark
a short line drawn on a number line to show where a number sits
Check yourself
1. On a number line, how many jumps of 1/2 does it take to go from 0 to 3?
Why: Each whole takes 2 jumps of 1/2, so 3 wholes take 3 × 2 = 6 jumps. 3 ÷ 1/2 = 6.
2. A student counts 9 tick marks from 0 to 2 on a line marked in fourths and says 2 ÷ 1/4 = 9. What is the correct quotient?
Why: Nine marks have only 8 spaces between them. 2 wholes hold 2 × 4 = 8 fourths, so 2 ÷ 1/4 = 8.
3. It takes 15 jumps of 1/5 to reach 3. Which division sentence does that show?
Why: The jumps count how many fifths are in 3, which is the question 3 ÷ 1/5. The answer is 15.
17.3
Check With Multiplication
Main ideaDivision and multiplication undo each other: if 3 ÷ 1/4 = 12, then 12 × 1/4 must equal 3.
Division and multiplication are . One undoes the other. If 12 bows of 1/4 yard came from 3 yards, then 12 bows times 1/4 yard should give back 3 yards. Test it: 12 × 1/4 = 12/4 = 3. It works, so the division was right. This is the fastest for any quotient.
Try a store problem. A shop has 5 pounds of rice to pack into bags that hold 1/2 pound each. How many bags? 5 ÷ 1/2 = 5 × 2 = 10 bags. Check: 10 bags × 1/2 pound = 10/2 = 5 pounds. The check lands on the starting amount, so 10 is correct.
Now see what the check does to a wrong answer. Suppose a student writes 5 ÷ 1/2 = 5/2, which is 2 1/2. Check it: 2 1/2 × 1/2 = 5/2 × 1/2 = 5/4 = 1 1/4 pounds. That is not 5 pounds. The check fails, so 5/2 is wrong. A check that fails is useful; it tells you to look again.
Words to know
inverse operations
two operations that undo each other, like multiplying by 1/4 and dividing by 1/4
check
a second calculation that tests whether an answer is right
Check yourself
1. Which multiplication checks that 4 ÷ 1/6 = 24?
Why: Quotient times divisor should give the dividend: 24 × 1/6 = 24/6 = 4. It does.
2. What is 5 ÷ 1/2, and how do you know?
Why: 5 wholes hold 5 × 2 = 10 halves. The check 10 × 1/2 = 5 lands on the starting number.
3. How many 1/3-pound bags can be filled from 7 pounds of beans?
Main ideaDividing a unit fraction by a whole number splits a small piece into even smaller equal pieces.
Half a pan of brownies is left, and 3 friends want to . Each gets 1/2 ÷ 3. Cut the half into 3 equal strips. How big is one strip compared to the whole pan? If the other half were cut the same way, the whole pan would have 6 strips. So each strip is 1/6 of the pan: 1/2 ÷ 3 = 1/6.
The picture gives a rule for any . Cutting each of the d pieces into n smaller pieces makes d × n pieces in the whole. So 1/d ÷ n = 1/(d × n). For 1/4 ÷ 2, cut each fourth in half to get 8 pieces: 1/4 ÷ 2 = 1/8. For 1/3 ÷ 5, cut each third into 5 to get 15 pieces: 1/3 ÷ 5 = 1/15.
The common mistake is to flip the fraction and write 1/2 ÷ 3 = 3/2. But 3/2 is more than one whole pan, and each friend gets only part of a half. The answer must be smaller than 1/2. Check by multiplying: 3 friends × 1/6 each = 3/6 = 1/2, the amount that was shared. Correct.
Words to know
share equally
to split an amount so every person gets the same size piece
unit fraction
a fraction with 1 on top, like 1/2 or 1/4
Check yourself
1. What is 1/3 ÷ 4?
Why: Cut each third into 4 pieces: 3 × 4 = 12 pieces in the whole, so each is 1/12. Check: 4 × 1/12 = 4/12 = 1/3.
2. A family shares 1/5 of a garden equally between 2 children. What fraction of the garden does each child get?
Why: Cutting each fifth in half makes 10 pieces, so each child gets 1/10. Check: 2 × 1/10 = 2/10 = 1/5.
3. Which check shows that 1/2 ÷ 4 = 1/8 is correct?
Why: Four shares of 1/8 should make the 1/2 that was shared: 4 × 1/8 = 4/8 = 1/2. It does.
17.5
Which Kind of Problem Is It?
Main ideaDecide whether a story counts how many small pieces fit in a whole (whole ÷ fraction) or splits a small piece (fraction ÷ whole).
Two stories can use the same numbers and mean different things. Story one: "3 yards of ribbon, cut into 1/4-yard bows." Story two: "1/4 yard of ribbon, shared by 3 tags." The first is 3 ÷ 1/4 = 12, counting pieces. The second is 1/4 ÷ 3 = 1/12, splitting a piece. The is the amount being split, and the is what you split it by.
Ask two questions to sort out a story. First, what amount is being cut up? That is the dividend, and it goes first. Second, should the answer be more or fewer than what you started with? Fitting small pieces into a big amount gives a big count. Sharing a small piece gives an even smaller piece.
The common mistake is writing the numbers in the order they appear in the story. "1/2 pound of clay shared by 5 students" is 1/2 ÷ 5, not 5 ÷ 1/2. Each student gets 1/10 pound, a small amount, which fits the story. If you had written 5 ÷ 1/2 = 10, each student would get 10 pounds from half a pound, which makes no sense.
Words to know
dividend
the amount being divided; in 3 ÷ 1/4, the dividend is 3
divisor
the number you divide by; in 3 ÷ 1/4, the divisor is 1/4
Check yourself
1. "1/2 pound of clay is shared equally by 5 students." Which expression matches the story?
Why: The clay, 1/2 pound, is the amount being split, so it is the dividend: 1/2 ÷ 5 = 1/10 pound each.
2. "5 pounds of clay are packed into 1/2-pound bags." Which expression matches the story?
Why: The 5 pounds are being split into 1/2-pound pieces, so 5 ÷ 1/2 = 10 bags.
3. In the table above, as the divisor gets smaller, the quotient:
Why: 6 ÷ 1/2 = 12, 6 ÷ 1/3 = 18 and 6 ÷ 1/6 = 36. Smaller pieces fit more times, so the quotient grows.
Section 3
Fractions Are Division
17.6
Three Divided by Four
Main ideaA fraction is a division in disguise: 3 ÷ 4 = 3/4, because 3 wholes shared 4 ways gives each person 3/4.
Three subs are shared by 4 friends. How much does each friend get? Cut every sub into 4 equal pieces. Each friend takes 1 piece from each sub: 1/4 of the first, 1/4 of the second, 1/4 of the third. That is 3 fourths, or 3/4 of a sub. So 3 ÷ 4 = 3/4. The means "divided by."
This works for any two whole numbers. The numerator is the dividend and the denominator is the divisor. So 5 ÷ 8 = 5/8, and 1 ÷ 2 = 1/2. To check, multiply the by the number of people: 4 × 3/4 = 12/4 = 3 subs. All 3 subs are accounted for.
The common mistake is flipping the order: writing 3 ÷ 4 = 4/3. But 4/3 is more than one whole sub, and 4 people sharing only 3 subs must each get less than a whole. Ask, "Is there more than one item per person?" If not, the answer is less than 1, so the numerator must be smaller than the denominator.
Words to know
fraction bar
the line between numerator and denominator; it means "divided by"
quotient
the result of a division; 3 ÷ 4 has the quotient 3/4
Check yourself
1. What is 2 ÷ 5 written as a fraction?
Why: The dividend goes on top: 2 ÷ 5 = 2/5. Check: 5 × 2/5 = 10/5 = 2.
2. What is 7 ÷ 8 written as a fraction?
Why: 7 wholes shared 8 ways gives 7/8 each, which is less than 1 because there are fewer wholes than people.
3. Three granola bars are shared equally by 5 hikers. How much does each hiker get?
Main ideaWhen the dividend is bigger than the divisor, the fraction answer is greater than one, and you can write it as a mixed number.
Seven brownies are shared by 2 people. Each gets 7 ÷ 2 = 7/2. Since 7 is more than 2, each person gets more than one brownie. Hand out 3 whole brownies each, using 6. Split the last brownie in half. Each person ends with 3 1/2 brownies. So 7/2 = 3 1/2, a .
Try 11 ÷ 4. As a fraction it is 11/4. In whole-number division, 11 ÷ 4 = 2 with a of 3. The 2 is the whole brownies each person gets. The remainder 3 is the 3 brownies still to share among 4 people, and 3 ÷ 4 = 3/4. So 11 ÷ 4 = 2 3/4.
Two mistakes are common. One is writing the remainder as 2 R3 and stopping, which does not say how much each person gets. Another is writing 2 3/11, putting the dividend under the remainder. The remainder is shared among the divisor, so it goes over 4. Check: 4 × 2 3/4 = 8 + 3 = 11.
Words to know
remainder
the amount left over after dividing whole numbers; 11 ÷ 4 = 2 with remainder 3
mixed number
a whole number and a fraction together, like 2 3/4
Check yourself
1. What is 9 ÷ 4 written as a mixed number?
Why: 9 ÷ 4 = 2 with remainder 1. The 1 left over is shared 4 ways: 1/4. So 2 1/4. Check: 4 × 2 1/4 = 9.
2. What is 14 ÷ 3 written as a mixed number?
Why: 14 ÷ 3 = 4 with remainder 2, and 2 shared by 3 is 2/3. So 4 2/3. Check: 3 × 4 2/3 = 12 + 2 = 14.
3. Five pizzas are shared equally by 4 people. How much does each person get?
Why: 5 ÷ 4 = 5/4 = 1 1/4. Each person gets a whole pizza, and the fifth pizza is split 4 ways.
Section 4
Reasoning and Data
17.8
Is the Answer Reasonable?
Main ideaBefore you compute, predict whether the answer should be bigger or smaller than the number you started with.
Dividing by a number less than 1 gives more than you started with. 6 ÷ 1/2 = 12, because small halves fit into 6 many times. Dividing by a number greater than 1 gives less: 6 ÷ 2 = 3. And dividing by 1 changes nothing. Before you work a problem, which of these you expect.
The same thinking works when the dividend is a fraction. 1/3 ÷ 5 shares a third among 5 people, so each gets less than 1/3. The answer, 1/15, is smaller, as expected. If you had gotten 5/3, the prediction would tell you to look again, since 5/3 is bigger than 1/3.
Students who think division always shrinks a number get stuck on 7 ÷ 1/3. Instead, : thirds are small, so many of them fit in 7, about 20. The exact answer is 21. When the estimate and the answer are close, you can trust the work. When they are far apart, one of them is wrong.
Words to know
predict
to say what you expect the answer to look like before you work it out
estimate
a rough answer that is close enough to check the exact one
Check yourself
1. Without computing, what can you say about 8 ÷ 1/4?
Why: Dividing by a number less than 1 gives more than you start with. In fact 8 ÷ 1/4 = 32.
2. Which division has an answer less than 1?
Why: 1/4 ÷ 2 = 1/8, less than 1. The others are 8, 8 and 2, all more than 1.
3. A student says 5 ÷ 1/3 = 5/3. Is that reasonable?
Why: Thirds are small, so many fit in 5: 5 × 3 = 15. The check 15 × 1/3 = 5 confirms it. 5/3 is even less than 2, far too small.
17.9
Line Plots With Fractions
Main ideaA line plot stacks an X above each measurement, so you can see fractional data at a glance and compare values.
A class grows bean plants and measures each one after two weeks to the nearest 1/4 inch. To show the , they make a : a number line marked in fourths, with one X above the plant’s height for each plant. If 3 plants are 1 inch tall, there are 3 Xs stacked above 1.
A line plot answers questions fast. The tallest stack shows the most common height. The leftmost X is the shortest plant and the rightmost is the tallest. To find how much taller the tallest is than the shortest, subtract. If the shortest is 3/4 inch and the tallest is 1 1/2 inches, the difference is 1 2/4 − 3/4 = 6/4 − 3/4 = 3/4 inch.
You can also add up a stack. Two plants at 1 1/2 inches have a total height of 2 × 1 1/2 = 3 inches. The common mistake is to count tick marks or empty spots as data. Only the Xs count. A mark with no X above it means no plant had that height.
Words to know
line plot
a number line with an X stacked above each value in a set of data
data
a set of measurements or counts collected about something
Check yourself
1. In the bean plant table, how many plants measured exactly 1 inch?
Why: The row for 1 inch shows 3, meaning 3 Xs stacked above 1 on the line plot.
2. How much taller is the tallest plant (1 1/2 inches) than the shortest (3/4 inch)?
3. What is the total height of the two plants that measured 1 1/2 inches?
Why: 2 × 1 1/2 = 1 1/2 + 1 1/2 = 3 inches.
17.10
Sharing the Total Evenly
Main ideaAdd all the measurements on a line plot, then divide by the number of items to find what each would get if shared equally.
Four friends bring juice to a picnic: 1/2 quart, 1/2 quart, 1 1/2 quarts and 1 1/2 quarts. If they pour it all together and share equally, how much does each get? First find the : 1/2 + 1/2 = 1, and 1 1/2 + 1 1/2 = 3, so 1 + 3 = 4 quarts. Then divide by 4 friends: 4 ÷ 4 = 1 quart each.
Now try 3 bags of trail mix weighing 1/4, 3/4 and 1/2 pound. Rename in fourths and add: 1/4 + 3/4 + 2/4 = 6/4 = 1 1/2 pounds. Divide by 3 bags: 6/4 ÷ 3. Six fourths shared 3 ways is 2 fourths each, so each is 2/4 = 1/2 pound. Check: 3 × 1/2 = 1 1/2 pounds.
The common mistake is dividing by the number of different values instead of the number of items. If 5 plants have only 2 different heights, you still divide by 5, because there are 5 plants. Count the Xs on the line plot, not the tick marks that have Xs.
Words to know
total
the sum of all the values added together
equal share
the amount each item or person gets when a total is split evenly
Check yourself
1. Juice bottles hold 1/2, 1/2, 1 and 2 quarts. If the juice is shared equally by the 4 people who brought it, how much does each get?
Why: 36 into 129 is 3, since 36 × 3 = 108, leaving 21. Bring down 6 to make 216, and 36 × 6 = 216. The quotient is 36.
2. (Unit 6) What is 10 − 2 × 3?
Why: Multiply first: 2 × 3 = 6, then 10 − 6 = 4. Subtracting first gives 8 × 3 = 24, which breaks the rule.
3. (Unit 7) What is 600,000 − 248,317?
Why: Check by adding: 248,317 + 351,683 = 600,000, because 317 + 683 = 1,000 and 248,000 + 351,000 + 1,000 = 600,000.
4. (Unit 6) What number goes in the box? ? × 7 = 63
Why: 7 × 9 = 63, so the missing factor is 9. Subtracting 63 − 7 = 56 is the mistake.
5. (Unit 7) Which number is the least?
Why: All start with 87. Compare the hundreds: 6, 5, 6, 5. Of 87,506 and 87,560, the tens digit 0 is less than 6. So 87,506 is least.
Send it to your teacher
Write it
A recipe calls for 3/4 cup of sugar. You want to make 2 1/2 batches. Find how much sugar you need, then explain each step so a classmate could follow it. Before you finish, predict whether the answer should be more or less than 3/4 cup, and check your work a second way.
State your final answer clearly, with units, at the start.
Show each step with the actual numbers, including how you renamed any mixed number.
Say why each step works, not just what you did.
Check the answer by estimating or by using a different method, and say whether it is reasonable.
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