A fraction of a set
Three of the eight colored pencils in a box are red. Nothing has been cut into pieces, so how can a fraction say how much of the box is red?
Where you are 4 in A fraction as part of a whole
Why this exists
In Area and length models of a fraction the whole was one thing, a cake or a strip, cut into equal parts. Many wholes are not one thing. Vaiva’s box holds 8 colored pencils, and 3 of them are red. Rokas’s gumball machine holds 6 gumballs, and 5 are green. Milda has a bucket of 12 seashells. Nobody cuts a pencil or a gumball. Still there is a question to answer: how much of the box is red?
A fraction answers it. The whole is a set, a group of objects counted together, and the fraction names part of the set by counting objects. Each object does the job that one equal piece did in a cake. Three of the 8 pencils are red, so three eighths of the pencils are red.
This is the third picture of a fraction, beside the area picture and the length picture. It is also the picture behind sharing a set into equal groups, which ties fractions to division.
Definition
A set is a group of separate objects counted together as one whole: the pencils in a box, the gumballs in a machine, the seashells in a bucket.
If a set has b objects and a of them are picked out, those objects are
of the set. The bottom number, the denominator, counts every object in the set. The top number, the numerator, counts the objects picked out (see What the numerator and the denominator each tell you).
A set can also be arranged in equal groups, groups with the same number of objects in each. If a set is in b equal groups, each group is \displaystyle \frac{1}{b} of the set, and a of the groups are \displaystyle \frac{a}{b} of the set.
Two rules come with it.
- Each object counts as one, whatever its size. In a set, “equal shares” (see Equal shares and naming the whole) means one object each, or one equal group each.
- The whole is named. “Three eighths” alone says nothing; “three eighths of the pencils in Vaiva’s box” does.
In this lesson the sets have 2, 3, 4, 6 or 8 objects, or are set out in that many equal groups, so the fractions are halves, thirds, fourths, sixths and eighths.
Intuition
The lesson runs over three days. Part 1 counts objects. Part 2 counts equal groups. Part 3 asks why the whole matters and why the objects need not be the same size.
Part 1: each object is one part
Look at Vaiva’s box of pencils in Example 1. First count all the objects in the set: 8 pencils. So each pencil is one eighth of the set. Then count the red ones: 3. Point at them and count by name, “one eighth, two eighths, three eighths”, as in Counting by unit fractions. Three of the eighths are red: \displaystyle \frac{3}{8} of the pencils.
The pencils that are not red are a fraction too, \displaystyle \frac{5}{8}, and the red ones with the others are all 8 eighths, the whole box. Where the red pencils sit does not matter. Spread out, jumbled in a machine or lined up, 3 of 8 is 3 of 8.
Part 2: equal groups
Now set Milda’s 12 seashells out in 4 rows of 3, as in Example 3. Look at the rows instead of the shells. There are 4 rows, and each has the same number of shells, so the rows are 4 equal parts of the set. One row is \displaystyle \frac{1}{4} of the seashells.
That row holds 3 seashells, yet it is one fourth. The two numbers answer different questions: 3 says how many shells are in the part, and \displaystyle \frac{1}{4} says what part of the set it is. The same row can be named either way.
How many objects are in one equal group is a sharing question. Twelve shells shared into 4 equal rows gives 12 \div 4 = 3 in each row, which is What division means read as a fraction: one fourth of 12 is 3. Taking several groups of a number, such as \displaystyle \frac{3}{4} of 16, is worked out in A fraction of a whole number. Here a picture is always there to count.
Part 3: the whole, and objects of different sizes
The whole still matters. Half of a set of 8 pencils is 4 pencils, and half of a set of 2 pencils is 1. The fraction says how the set was split and how much of it was taken. How many objects that is depends on the set.
And here a set differs from a shape. In the area model the parts must be the same size: in the rectangle on the right of Figure 1 the four pieces are not equal, so the wide piece is not one fourth of the rectangle. In a set, each object counts as one. Milda’s set on the left has 4 seashells, 1 big and 3 small, so the big shell is 1 of 4 shells, \displaystyle \frac{1}{4} of the seashells. It is not one fourth of the shell material; it may well be more than all the small ones together. A fraction of a set counts objects. It says nothing about how big they are.
Results
A fraction of a set counts objects. In a set of b objects, a of them are \displaystyle \frac{a}{b} of the set.
The objects picked out and the others make the whole set. If a of the b objects are picked, then b - a are not, and \displaystyle \frac{b}{b} is the whole set.
Equal groups are parts too. A set in b equal groups has each group equal to \displaystyle \frac{1}{b} of the set, and a groups equal to \displaystyle \frac{a}{b} of it.
One unit fraction of a set is a sharing. The number of objects in \displaystyle \frac{1}{b} of a set is the number of objects divided by b, when the set shares into b equal groups with none left over.
The same fraction of a bigger set is more objects. \displaystyle \frac{1}{2} of 8 pencils is more pencils than \displaystyle \frac{1}{2} of 2 pencils.
Size does not count. Each object is one part of the set, whether it is big or small.
Why it is true
The picked objects and the others make the set. Every object in the set is either picked or not, and none is counted twice. So the two counts add to b, the number of objects in the set.
Equal groups. If b groups have the same number of objects and together hold the whole set, then each group is one of b equal parts of the set. That is what \displaystyle \frac{1}{b} means.
A unit fraction is a sharing. Sharing a set into b equal groups and asking how many land in each group is division (see What division means). Each group is \displaystyle \frac{1}{b} of the set, so the share is the number of objects in \displaystyle \frac{1}{b} of it.
Size does not count. The question a set fraction answers is “how many of the objects?”. Each object adds 1 to that count whatever its size, so every object is one equal part of the count.
Worked examples
Have counters, a handful of pencils or shells, and paper on hand. Counting answers most of these.
Part 1: red pencils in a box
Vaiva’s box holds 8 colored pencils, and the 3 shaded in Figure 2 are red. What fraction of the pencils are red? What fraction are not red?
The whole is the set of pencils in the box. Count them: 8. Each pencil counts as one, so each is one eighth of the set, \displaystyle \frac{1}{8}.
Count the red pencils: 3. Three of the eighths are red, so \displaystyle \frac{3}{8} of the pencils are red, three eighths.
The others: 8 - 3 = 5 pencils, so \displaystyle \frac{5}{8} of the pencils are not red.
Check: 3 + 5 = 8, so the red pencils and the others are all eight eighths, the whole box. The red pencils are not side by side in the box, and they do not have to be: any 3 of the 8 pencils are \displaystyle \frac{3}{8} of them.
Part 1: green gumballs in a machine
Rokas’s gumball machine (Figure 3) holds 6 gumballs, jumbled together. Five are green and 1 is orange. What fraction of the gumballs are green? What fraction are orange? Then his sister drops in 2 more orange gumballs. What fractions are green and orange now?
The whole is the set of gumballs in the machine: 6 gumballs, so each is one sixth. The gumballs sit in no rows and nothing is cut, but that does not matter: each gumball is one of 6.
Five are green, so \displaystyle \frac{5}{6} of the gumballs are green. One is orange, so \displaystyle \frac{1}{6} are orange. Check: 5 + 1 = 6.
After his sister adds 2 orange gumballs, the set has changed. It now has 6 + 2 = 8 gumballs, so each gumball is one eighth. Five are still green, so \displaystyle \frac{5}{8} are green, and 1 + 2 = 3 are orange, so \displaystyle \frac{3}{8} are orange. Check: 5 + 3 = 8.
The green gumballs did not change, but their fraction did, from \displaystyle \frac{5}{6} to \displaystyle \frac{5}{8}, because the whole set got bigger.
Part 2: seashells in rows
Milda sets out her 12 seashells in 4 rows of 3 (Figure 4). What fraction of the seashells is one row? What fraction are three rows, and how many seashells is that?
Look at the rows. There are 4 of them, and each holds 3 seashells, so they are 4 equal groups that together hold all 12 shells. Each row is one of 4 equal parts of the set: one row is \displaystyle \frac{1}{4} of the seashells.
Three rows are three of those fourths, so they are \displaystyle \frac{3}{4} of the seashells. Count the shells in them: 3 + 3 + 3 = 9 seashells.
So the same part of the set has two descriptions: 9 seashells, a count of objects, and \displaystyle \frac{3}{4} of the seashells, a fraction of the set. The fraction uses the rows as the parts, not the single shells.
Part 3: half of two sets
Vaiva has 6 pencils, and half of them are red. Rokas has 2 pencils, and half of them are red. They have the same fraction of red pencils. Do they have the same number of red pencils?
SolutionHalf means the set is split into 2 equal groups and one group is taken.
Vaiva’s 6 pencils in 2 equal groups: 6 \div 2 = 3, so half of her pencils is 3 pencils.
Rokas’s 2 pencils in 2 equal groups: 2 \div 2 = 1, so half of his pencils is 1 pencil.
No. Vaiva has 3 red pencils and Rokas has 1. The fraction \displaystyle \frac{1}{2} is the same, but it is half of a different whole each time. A bigger set has bigger equal groups, so the same fraction of it is more objects.
Part 3: a long pencil and three short ones
Vaiva’s set of 4 pencils has one new pencil 12 centimeters long and three short ones, each 4 centimeters long. What fraction of her pencils is the long pencil? She lays all four end to end (Figure 5). What fraction of that whole length is the long pencil?
As a set, the whole is the 4 pencils, and each pencil counts as one, long or short. The long pencil is 1 of the 4 pencils, so it is \displaystyle \frac{1}{4} of the pencils.
As a length, the whole is the line of pencils:
centimeters. The long pencil is 12 centimeters, and the three short ones together are also 4 + 4 + 4 = 12 centimeters. So the line is two equal lengths of 12 centimeters, and the long pencil is one of them: \displaystyle \frac{1}{2} of the length.
Both answers are right, because they are fractions of different wholes. The long pencil is \displaystyle \frac{1}{4} of the set of pencils and \displaystyle \frac{1}{2} of their length. When a question says “of the pencils”, it is counting pencils; when it says “of the length”, it is measuring.
Part 3: a bag, a cup, and both together
Rokas has a bag with 3 green and 3 orange gumballs, and a cup with 2 green gumballs. What fraction of the gumballs in the bag are green? In the cup? What fraction of all his gumballs are green?
SolutionName the whole each time.
The bag: 3 + 3 = 6 gumballs, 3 of them green, so \displaystyle \frac{3}{6} of the gumballs in the bag are green.
The cup: 2 gumballs, both green, so \displaystyle \frac{2}{2} of the gumballs in the cup are green. Two halves is the whole cup.
All of them: 6 + 2 = 8 gumballs, and 3 + 2 = 5 green ones, so \displaystyle \frac{5}{8} of all his gumballs are green. The other 3 are orange, \displaystyle \frac{3}{8}, and 5 + 3 = 8.
The same green gumballs are \displaystyle \frac{3}{6} of one set, \displaystyle \frac{2}{2} of another and \displaystyle \frac{5}{8} of a third. A fraction of a set means nothing until the set is named.
Boundaries
The objects must be whole and separate. A set fraction counts objects, so each object counts once and in full. Half a pencil is not an object of the set; once a pencil is broken, the count has changed.
The groups must be equal. A set of 12 shells in 4 rows of different lengths is not in fourths, and one of those rows is not \displaystyle \frac{1}{4} of the shells.
A set must share evenly to take a unit fraction of it. One fourth of a set of 5 shells cannot be taken without breaking a shell, since 5 shells do not make 4 equal groups of whole shells. Sharing whole things into parts is A fraction as division.
A set fraction counts, it does not measure. It says how many of the objects, not how much material, weight or length they have (Example 5).
The whole must be named, and it changes when objects are added or taken away (Example 2).
The bottom number is never zero. A set with no objects has nothing to count, so a fraction with 0 on the bottom names nothing.
At most the whole set, in this lesson. The top number is never more than the number of objects, and \displaystyle \frac{8}{8} of a set is the whole set. Counting past one whole needs a second whole, as in Fractions greater than one (improper fractions).
Where this goes next
Sharing a set and reading the share as a fraction leads to A fraction as division, where \displaystyle \frac{3}{4} also means 3 things shared among 4. Taking several equal groups of a number, such as \displaystyle \frac{3}{4} of 16 cars, is A fraction of a whole number, and a fraction of a measured amount such as minutes or miles is Fraction of a quantity.
A set also gives a second look at equal fractions. In Example 6 the bag of 6 gumballs was \displaystyle \frac{3}{6} green, and the green ones were also one of 2 equal groups, half the bag. One amount with two names is the subject of Seeing equivalent fractions in pictures and on the number line, told there with shapes.
Exercises
Part 1: each object is one part
Use Figure 6 for Exercises 1 to 4. Say what fraction of each set is shaded.
- Set A.
- Set B.
- Set C.
- Set D.
- Vaiva’s pencil cup holds 6 pencils, and 4 of them are sharpened. What fraction of the pencils are sharpened? What fraction are not?
- A bag holds 8 gumballs: 2 green, 3 orange and the rest yellow. What fraction of the gumballs is each color?
- Milda finds 3 seashells, and 1 of them is white. What fraction of her seashells is white? Then she finds 1 more seashell, and it is not white. What fraction of her seashells is white now?
- Rokas fills a cup with 4 gumballs so that \displaystyle \frac{3}{4} of them are green. How many are not green, and what fraction of the gumballs is that?
Part 2: equal groups
- Use set P in Figure 7. How many equal rows are there, and what fraction of the set is one row? What fraction of the set is shaded, and how many objects is that?
- Use set Q in Figure 7. Answer the same questions as in Exercise 9.
- Vaiva sets out 6 pencils in 2 rows of 3. What fraction of the pencils is one row? What fraction of the pencils is one pencil?
- Milda puts her 18 seashells into 3 buckets, 6 in each. What fraction of the seashells is in one bucket? In two buckets?
- Vaiva puts 8 pencils into boxes, 2 in each box. How many boxes does she fill, and what fraction of the pencils is in one box?
- Rokas lines up 24 gumballs in 8 rows of 3. What fraction of the gumballs is one row? What fraction are 3 rows, and how many gumballs is that?
- Rokas has 8 gumballs in 4 equal rows. He moves them into 2 equal rows instead. What fraction of the gumballs is one row now, and how many gumballs are in it?
Part 3: the whole, and objects of different sizes
- Half of Vaiva’s 4 pencils are red, and half of Rokas’s 8 pencils are red. How many red pencils does each have?
- Milda has a set of 3 seashells, 2 of them white. Rokas has a set of 6 seashells, 2 of them white. What fraction of each set is white? Whose set is more white?
- A set of 4 seashells has 1 tiny shell and 3 big ones. What fraction of the seashells is the tiny one? Is the tiny shell one fourth of all the shell there is?
- A bag holds 3 green gumballs and 1 orange one. A cup holds 2 green and 2 orange. What fraction of the gumballs in the bag are green? In the cup? In the bag and cup together?
- Rokas has 6 gumballs: 1 big green one and 5 small orange ones. What fraction of his gumballs are green?
- Milda’s seashells are set out in 3 equal rows, and one row has 2 seashells. How many seashells does she have? What fraction of them are in 2 rows, and how many seashells is that?
Reasoning
- Is \displaystyle \frac{1}{4} of a set always 1 object? Explain, with an example.
- A rectangle is cut into 4 pieces of different sizes, and the biggest piece is not \displaystyle \frac{1}{4} of it. A set of 4 seashells of different sizes has its biggest shell equal to \displaystyle \frac{1}{4} of the seashells. Explain why both statements are right.
- Can \displaystyle \frac{1}{2} of one set be more objects than \displaystyle \frac{3}{4} of another? Explain with sets you draw.
- Can you take \displaystyle \frac{1}{4} of a set of 5 seashells without breaking one? Explain, and give the smallest set larger than 5 shells that you can take one fourth of.
Answers
- \displaystyle \frac{1}{4}. The set has 4 objects and 1 is shaded.
- \displaystyle \frac{7}{8}. The set has 8 objects and 7 are shaded.
- \displaystyle \frac{2}{6}. The set has 6 objects and 2 are shaded. They need not sit side by side.
- \displaystyle \frac{2}{3}. The set has 3 objects and 2 are shaded.
- \displaystyle \frac{4}{6} sharpened and \displaystyle \frac{2}{6} not, since 6 - 4 = 2.
- \displaystyle \frac{2}{8} green, \displaystyle \frac{3}{8} orange and \displaystyle \frac{3}{8} yellow. The yellow ones are 8 - 2 - 3 = 3, and 2 + 3 + 3 = 8.
- \displaystyle \frac{1}{3} at first, then \displaystyle \frac{1}{4}. Her set grows to 3 + 1 = 4 seashells and still has 1 white one.
- 1 gumball, \displaystyle \frac{1}{4} of them. Three of the 4 are green, and 4 - 3 = 1 is not.
- 4 equal rows, so one row is \displaystyle \frac{1}{4} of the set. Three rows are shaded, \displaystyle \frac{3}{4} of the set, and 2 + 2 + 2 = 6 objects.
- 3 equal rows, so one row is \displaystyle \frac{1}{3} of the set. Two rows are shaded, \displaystyle \frac{2}{3} of the set, and 4 + 4 = 8 objects.
- One row is \displaystyle \frac{1}{2} of the pencils, since there are 2 equal rows. One pencil is \displaystyle \frac{1}{6} of them, since there are 6 pencils.
- \displaystyle \frac{1}{3} of the seashells in one bucket, since the 3 buckets hold equal groups that make up the whole set. Two buckets hold \displaystyle \frac{2}{3} of them.
- 4 boxes, since 2 + 2 + 2 + 2 = 8. One box holds \displaystyle \frac{1}{4} of the pencils.
- One row is \displaystyle \frac{1}{8} of the gumballs, since there are 8 equal rows. Three rows are \displaystyle \frac{3}{8} of them, 3 + 3 + 3 = 9 gumballs.
- \displaystyle \frac{1}{2} of the gumballs, with 8 \div 2 = 4 gumballs in the row. (Before, one row was \displaystyle \frac{1}{4}, with 8 \div 4 = 2 gumballs.)
- Vaiva has 2 and Rokas has 4. Half of 4 is 4 \div 2 = 2, and half of 8 is 8 \div 2 = 4.
- Milda’s set is \displaystyle \frac{2}{3} white and Rokas’s is \displaystyle \frac{2}{6} white. Milda’s set is more white. Both have 2 white shells, but Milda’s are 2 of only 3 shells, more than half her set, while Rokas’s are 2 of 6, less than half of his, since half of 6 is 3.
- \displaystyle \frac{1}{4} of the seashells, since it is 1 of 4 shells. No: it is far less than one fourth of the shell material, because the three big shells hold more. A set fraction counts shells, not how much shell there is.
- \displaystyle \frac{3}{4} of the bag, \displaystyle \frac{2}{4} of the cup, and \displaystyle \frac{5}{8} of both together: the two hold 4 + 4 = 8 gumballs, and 3 + 2 = 5 are green.
- \displaystyle \frac{1}{6}. The big gumball counts as one of 6 gumballs, like any other.
- 6 seashells, since 2 + 2 + 2 = 6. Two rows are \displaystyle \frac{2}{3} of them, 2 + 2 = 4 seashells.
- No. One fourth of a set is one of 4 equal groups, and a group is 1 object only when the set has 4 objects. Milda’s 12 seashells in 4 rows of 3 have 3 shells in each fourth.
- A fraction of a shape counts equal pieces of its area, so unequal pieces are not fourths. A fraction of a set counts objects, and each shell counts as one whatever its size, so 1 of 4 shells is \displaystyle \frac{1}{4} of the shells. The two fractions are about different wholes.
- Yes. Half of a set of 8 is 4 objects; three fourths of a set of 4 is 3 objects; and 4 is more than 3. A bigger set can make a smaller fraction hold more objects.
- No. One fourth needs 4 equal groups of whole shells, and 5 shells cannot be split that way: 4 groups of 1 leave 1 over. The smallest larger set is 8 shells, 4 groups of 2.