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Unit fractions as the counting unit of every fraction: one equal part named, then counted.
Paper packet. Every task here also exists on screen, where it is checked automatically; answers written on paper are not assessed by Nydus. When you are back at a device, enter your answers there.
In this lesson you name one equal part of a whole and learn to count it. A unit fraction such as 1/6 is the size of one part; every other fraction is a count of those parts, like 3/6. Where a fraction sits on a line and which of two fractions is greater are the next two lessons' work.
You can share a whole fairly. When a sandwich is cut into two equal pieces, each person gets one half. When a pan of cornbread is cut into four equal pieces, each piece is one fourth. You have also split shapes into equal parts and checked that the parts really are the same size. This lesson takes one of those equal parts and treats it as a unit: something we can count, just as we count ones or tens.
| Term | What it means |
|---|---|
| Whole | The one thing, or the one amount, that is being shared into parts. |
| Equal parts | Parts of the same whole that are all exactly the same size. |
| Unit fraction | One equal part of a whole, written with 1 on top, such as 1/6. |
| Numerator | The top number: how many equal parts are being counted. |
| Denominator | The bottom number: how many equal parts make the whole. |
Cut a whole into 6 equal parts. Each part is one sixth of the whole, written 1/6. A fraction with 1 on top is called a unit fraction, because it names one single part, and unit means one. Every other fraction is built by counting unit fractions. Three sixths, 3/6, is three of the 1/6 parts. Six sixths, 6/6, is every part, so it is the whole again. The bottom number tells you the size of the unit; the top number tells you how many units you have.
Another way: picture
A bar in six equal parts. Shade one part and it shows 1/6. Shade three parts and it shows 3/6. Shade all six and it shows 6/6, the whole bar.
Before you can count anything, you have to know what you are counting. The bottom number of a fraction does that job. In 1/4, the 4 says the whole was cut into four equal parts, so each part is one fourth. In 1/8, the 8 says the whole was cut into eight equal parts, so each part is one eighth. The bottom number is a description of the part, not a count of how many parts you have.
Think of a granola bar shared by four friends. Each friend gets 1/4 of the bar. Now share the same bar among eight friends. Each friend gets 1/8. Eight is a bigger number than four, but each friend gets less, because the same bar had to stretch further. That is why the bottom number can trick you. A bigger bottom number means more parts in the whole, and more parts in the same whole means each part is smaller.
The parts must be equal for the name to make sense. If you cut a sandwich into three pieces and one piece is huge, those pieces are not thirds. A third is a part that would fit exactly three times into the whole. So whenever you name a unit fraction, ask yourself two questions. What is the whole? Are the parts all the same size? Only when both answers are clear can you say what one part is called.
Once you know the size of one part, you can count parts of that size. If one part is 1/5, then two of those parts are 2/5, three of them are 3/5, and four of them are 4/5. The bottom number stays 5 the whole time, because every part is still one fifth. Only the top number changes, because only the count changes.
It helps to say the fraction aloud as a count with a unit, the way you would say three feet or three apples. Three fifths means three parts, each one a fifth. When you say it that way, you will not be tempted to change the fifths into something else just because you counted more of them. Counting pieces never changes the kind of piece.
When you have counted every part, the top and bottom numbers match. Five fifths, 5/5, is all five parts, so it is one whole. Four fourths is one whole, and ten tenths is one whole. This gives you a quick check. If the top number is smaller than the bottom number, you have less than the whole. If they match, you have exactly the whole.
A drawing is a tool for thinking, and it works best when it is careful. Start with a rectangle for the whole. Mark it into equal parts, using the bottom number to decide how many. For fourths, draw three lines so there are four parts of the same width. Then shade the parts the top number asks for.
Check the drawing before you trust it. Count the parts to make sure there are as many as the bottom number says. Look at their widths: if one part is clearly wider, the parts are not equal and the picture does not show fourths. Then count the shaded parts and compare with the top number. A drawing that passes all three checks shows the fraction honestly.
It does not matter which parts you shade. Shading the first three fourths, the last three fourths, or three fourths scattered along the bar all show 3/4, because the parts are all the same size. What matters is how many parts are shaded and how many equal parts make the whole.
Because every fraction is a count of unit fractions, you can build one by repeated adding. One eighth and one eighth make two eighths. Another eighth makes three eighths. Written out, 1/8 + 1/8 + 1/8 = 3/8. You are adding three units that are all the same size, so you add the counts and keep the size.
A common slip is to add the bottom numbers too, writing 1/8 + 1/8 = 2/16. Test that with a picture. Two eighths of a bar cover a quarter of the bar. Two sixteenths cover only an eighth. The amounts are different, so 2/16 cannot be right. The bottom number is not a count you add; it is the name of the unit, and the unit did not change.
You can also work backward. If you are told you have 5/6, you know you have five parts, each one sixth. To show it, draw sixths and shade five of them. To say how much is missing from the whole, count the parts left unshaded: one sixth. Every fraction question in this unit comes back to the same two facts: the size of a part and the count of parts.
After you name a fraction, check it in three quick ways. First, read the bottom number and ask whether it matches the number of equal parts in the whole. Second, read the top number and ask whether it matches the parts you counted or shaded. Third, compare the fraction with one whole: is the top number smaller than, equal to, or bigger than the bottom number, and does that match what you see?
Say your answer in words as well as in numbers. If you wrote 2/5, say two fifths and picture two parts out of five equal parts. If the picture in your head does not match the question, look again. Many mistakes come from switching the two numbers, writing 5/2 for two fifths. Saying the parts aloud catches that, because five halves sounds like a lot more than two parts out of five.
Finally, think about whether your answer is reasonable. One part of a whole cut into many pieces should be small. If you name one of twelve equal parts and get a fraction that sounds big, something has gone wrong. Unit fractions with large bottom numbers are small amounts, and that sense of size helps you catch an error before it spreads into the next step.
A fraction always belongs to a whole. One half of a small pizza and one half of a large pizza are both 1/2, yet the large half is more pizza. The fraction is the same because each is one of two equal parts of its own whole. The amounts differ because the wholes differ.
So when you compare two fractions, first check that they are parts of the same whole, or of wholes that are the same size. A child who gets 1/4 of a long bread loaf may get more bread than a child who gets 1/2 of a short roll. That does not mean 1/4 is bigger than 1/2. It means the wholes were not the same, so the fractions were never being compared fairly.
This is why careful questions say what the whole is: a 12-foot bed, a pan of cornbread, a bar on the page. Before you name or compare, find that whole and keep it fixed. Every rule in this unit, from counting unit fractions to placing them on a number line, assumes the whole stays the same from the start of the problem to the end.
A class garden bed is 12 feet long. The class wants to split it into equal plots along its length, one plot for each group. With 4 groups, each plot is 1/4 of the bed. Because the whole is 12 feet, one fourth is 12 ÷ 4 = 3 feet. Each group gets 3 feet of soil, and four plots of 3 feet use all 12 feet.
Two more groups join, so now 6 groups share the same bed. Each plot is 1/6 of the bed, and 12 ÷ 6 = 2 feet. The new unit fraction has a bigger bottom number, and each plot is smaller, exactly as the fraction predicts. The groups who had 3 feet now have 2.
Later, one group finishes early and lends its plot to the group next door. That group now works 2/6 of the bed: two plots, each one sixth. In feet, that is 2 × 2 = 4 feet. Notice what the fraction did and did not tell us. It told us how much of the bed each group used. It did not tell us whether every plot gets the same sunlight, which the class would have to check by watching the garden.
1/10 is a smaller part than 1/4. The whole was shared among more parts, so each part is smaller. The bottom number counts how many ways the whole was split, not how much you get. When two unit fractions are compared, the one with the smaller bottom number is the bigger part.
Read the bottom number.
8
It says how many equal parts make the whole.
Cut the bar into equal parts.
8 equal parts
Each part is 1/8 of the bar.
Read the top number.
3
It says how many of those parts to take.
Shade that many parts.
3 parts shaded
Any three parts will do, since all are equal.
Name the shaded amount.
3/8
Three parts, each one eighth of the bar.
Name the unit.
1/5
Each piece is one fifth of the same whole.
Count the pieces.
4 pieces
The unit is taken four times.
Add the counts.
1 + 1 + 1 + 1 = 4
The pieces are all the same size, so their counts add.
Keep the unit.
fifths
Counting pieces does not change their size.
Write the fraction.
4/5
Four units, each one fifth.
Compare with one whole.
4/5 < 5/5 = 1
One more fifth would complete the whole.
Name the whole.
12-foot bed
The bed is the thing being shared.
Name one plot as a unit fraction.
1/6 of the bed
Six equal plots make the whole bed.
Find the length of one plot.
12 ÷ 6 = 2 feet
Six equal plots share the 12 feet.
Count the joined plots.
2 plots
One group now uses two of the plots.
Write the joined amount.
2/6 of the bed
Two units, each one sixth.
Find its length.
2 × 2 = 4 feet
Each plot is 2 feet long.
Check against the whole.
6 × 2 = 12 feet
Six plots of 2 feet rebuild the bed.
Name the unit.
1/7
Each piece is one seventh of the whole.
Count the pieces.
3 pieces
The unit is taken three times.
Write the fraction.
Compare with one whole.
A sheet of card is cut into equal parts, and one part is 1/9 of it. How many equal parts is the sheet of card cut into?
parts equal parts
A bar is cut into 8 equal parts and 2 of them are shaded. Complete the fraction that is shaded.
Name one part.
1/unit
The number of equal parts in the whole bar names the part size.
Count the shaded parts.
count parts
Each shaded part is one unit.
Write the count over the part size.
shaded count on top, part size below
Counting parts changes the top number only.
How much is 1/8 taken 5 times? Write the answer as {{n}}/{{m}}.
n/m
A pan of cornbread is cut into 7 equal pieces. Sam eats 2 pieces. What fraction of the pan does Sam eat?
n/m of the pan
A melon is cut into equal parts, and one part is 1/4 of it. How many equal parts is the melon cut into?
parts equal parts
How much is 1/7 taken 6 times? Write the answer as {{n}}/{{m}}.
n/m
A whole is cut into 9 equal parts. Which fraction names just one of those parts?
Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.
A garden bed 12 feet long is split into 4 equal plots. One group uses 3 plots. What fraction of the bed is that, and how many feet long?
n/m of the bed, feet feet
You can name a unit fraction and count unit fractions to make other fractions. Without looking: what is one of eight equal parts called, and how much is 1/5 taken three times?
16. Your turn: 1/7 taken three times, step 3
3/7
Three units, each one seventh.
16. Your turn: 1/7 taken three times, step 4
3/7 < 7/7
Four more sevenths would complete the whole.