math.wiki
Fractions, ratio and percent › Fractions › Unit fractions › Counting by unit fractions
Open on the map
Grade 3

Counting by unit fractions

One fourth, two fourths, three fourths, four fourths. What comes next, and where is the count when it gets there?

You need
What the numerator and the denominator each tell youGrade 3
Unlocks
Fractions greater than one (improper fractions)Grade 3
Where you are 2 in Unit fractions
Fractions, ratio and percent3
Fractions15
Unit fractions2
Counting by unit fractions
Decomposing a fraction into unit fractions

When you reach for it

Counting is the first thing anyone does with whole numbers: one, two, three. Fractions can be counted the same way, once you know what is being counted. In What the numerator and the denominator each tell you a fraction was copies of one piece, the unit fraction: \displaystyle \frac{3}{8} is 3 copies of \displaystyle \frac{1}{8}. Counting by a unit fraction takes those copies one at a time, out loud: one fourth, two fourths, three fourths, four fourths, five fourths, and on.

You reach for this count whenever something grows or shrinks by equal pieces: a relay team covering a swimming lane a third at a time, a rider passing quarter-mile markers on a bike path, a child hopping along a hopscotch court of equal boxes. The count says where you are after any number of pieces. It also shows two things that are hard to see any other way. The count reaches one whole at a place you can predict: three thirds, four fourths, six sixths. And it does not stop there. Five fourths of a mile is a real place on a bike path, just past the first mile.

It is the same move as Skip counting by twos or tens, with a fraction as the step.

The procedure

The lesson runs over three days. Part 1 counts up to one whole. Part 2 keeps counting past the whole and names where the count is. Part 3 counts backward.

Part 1: counting up to one whole

Start with the step. The unit fraction is the size of one step, and it comes from the whole: one length of a swimming lane split into 3 equal parts gives steps of \displaystyle \frac{1}{3} of a length. Every step in the count is that same size.

Then count the steps, saying each count with its piece name:

\frac{1}{3}, \quad \frac{2}{3}, \quad \frac{3}{3}.

Only the numerator changes. It goes up by 1 at each step, because each step adds one more copy of \displaystyle \frac{1}{3}. The denominator stays 3 the whole time, because every step is the same size, a third of the same length. A count whose denominator changed partway would be counting pieces of different sizes, which is like counting “one inch, two inches, three feet”.

The count lands on the whole after exactly as many steps as the denominator says. The denominator 3 means that 3 equal parts make the whole length, so after 3 steps every part is covered. Three thirds is one whole length:

\frac{3}{3} = 1.

The same holds for every unit fraction. Counting by \displaystyle \frac{1}{b}, the count \displaystyle \frac{b}{b} is one whole, because b copies of \displaystyle \frac{1}{b} of the same whole fill all b of its equal parts. Two halves make one, four fourths make one, six sixths make one, eight eighths make one.

Part 2: counting past one whole

Nothing stops the count at one whole. A bike path with a marker every quarter mile (a quarter is a fourth) has its fourth marker at one mile, and the next marker is one more quarter mile along: five fourths of a mile. The rule is the same as before, numerator up by 1, denominator fixed:

\frac{4}{4}, \quad \frac{5}{4}, \quad \frac{6}{4}, \quad \frac{7}{4}, \quad \frac{8}{4}.

A count like \displaystyle \frac{5}{4} has a numerator bigger than its denominator, and that is fine. It means 5 copies of \displaystyle \frac{1}{4}, and a path longer than a mile has room for them. The reading “5 out of 4” makes no sense, but the reading “5 copies of \displaystyle \frac{1}{4}” does. Amounts like this are Fractions greater than one (improper fractions), the next lesson.

The wholes keep turning up at regular places. Every 4 steps fill one more mile, so \displaystyle \frac{4}{4} is 1 mile and \displaystyle \frac{8}{4} is 2 miles. Those landmarks say where any count is. The count \displaystyle \frac{7}{4} is 7 - 4 = 3 steps past \displaystyle \frac{4}{4}, so seven fourths of a mile is one whole mile and three fourths of a mile more. Keep both names. “Seven fourths” is the count; “one whole and three fourths more” says where the count is. They name the same place.

To go the other way, from a description to a count, count the copies. One whole is 4 fourths, and three fourths more is 3 more fourths, so the count is 4 + 3 = 7 fourths, \displaystyle \frac{7}{4}.

Part 3: counting backward

Counting back undoes counting on. Each step back takes away one copy of the unit fraction, so the numerator goes down by 1 and the denominator stays the same:

\frac{8}{6}, \quad \frac{7}{6}, \quad \frac{6}{6}, \quad \frac{5}{6}, \quad \dots, \quad \frac{1}{6}, \quad \frac{0}{6}.

Going back, the count passes the same landmarks. \displaystyle \frac{6}{6} is one whole, so stepping back from \displaystyle \frac{8}{6} reaches one whole after 8 - 6 = 2 steps. The count ends at \displaystyle \frac{0}{6}, no sixths at all, which is 0: back at the start. A count of pieces cannot go below none, so the backward count stops at 0.

Counting also measures how far apart two counts are. From \displaystyle \frac{2}{4} to \displaystyle \frac{7}{4} is 7 - 2 = 5 steps of one fourth, because each step changes the numerator by exactly 1.

Worked examples

Example 1

Part 1: a relay in thirds of a length

In a practice drill, Cristobal’s relay team shares one length of a swimming lane. Flags on the side split the length into 3 equal parts. Each swimmer swims one part and then tags the next swimmer (Figure 1). Count the part of the length covered after each swimmer. After how many swimmers has the team covered one whole length?

\displaystyle \frac{1}{3}
\displaystyle \frac{2}{3}
\displaystyle \frac{3}{3}
 one length
Figure 1Each swimmer covers one third of the length. Three thirds is the whole length.
Solution

The length is split into 3 equal parts, so each part is \displaystyle \frac{1}{3} of the length, and each swimmer adds one copy of \displaystyle \frac{1}{3}.

After the first swimmer the count is one third, \displaystyle \frac{1}{3}. After the second it is two thirds, \displaystyle \frac{2}{3}. After the third it is three thirds, \displaystyle \frac{3}{3}.

Three thirds is the whole length, because 3 swimmers have covered all 3 equal parts. So \displaystyle \frac{3}{3} = 1: the team covers one whole length after 3 swimmers, the number in the denominator.

The denominator stayed 3 all the way, because every swimmer swam the same size of part. Only the numerator, the count of swimmers so far, went up.

Example 2

Part 1: one mile by halves, fourths and eighths

Three bike paths are each one mile long. One has a marker every half mile, one every quarter mile, and one every eighth of a mile. Anouska rides each path from the start to the end and counts the markers she passes, by halves, by fourths and by eighths. How many counts does each path take, and which count lands on the end of the mile?

halves
\displaystyle \frac{2}{2}
 fourths
\displaystyle \frac{4}{4}
 eighths
\displaystyle \frac{8}{8}
Figure 2One mile counted by halves, by fourths and by eighths. Smaller steps take more counts to reach the whole.
Solution

On the first path each step is \displaystyle \frac{1}{2} of a mile. She counts \displaystyle \frac{1}{2}, \frac{2}{2}. The mile is 2 equal halves, so after 2 counts she is at the end: \displaystyle \frac{2}{2} = 1.

On the second path each step is \displaystyle \frac{1}{4} of a mile. She counts \displaystyle \frac{1}{4}, \frac{2}{4}, \frac{3}{4}, \frac{4}{4}. That is 4 counts, and \displaystyle \frac{4}{4} = 1.

On the third path each step is \displaystyle \frac{1}{8} of a mile. She counts \displaystyle \frac{1}{8} up to \displaystyle \frac{8}{8}. That is 8 counts, and \displaystyle \frac{8}{8} = 1.

All three counts end at the same place, one mile (Figure 2). Smaller steps take more counts to get there: each eighth is half of a fourth, so the mile takes twice as many eighths as fourths. In every case the count that lands on the whole has the same numerator and denominator, because it takes exactly as many steps as there are equal parts.

Example 3

Part 2: quarter miles along the bike path

Anouska’s favorite bike path has a marker every quarter mile from the start. She rides from the start past 7 markers, counting each one by fourths (Figure 3). Write her count. At which marker is she one mile from the start? Name the place where she stops in two ways.

0
\displaystyle \frac{1}{4}
\displaystyle \frac{2}{4}
\displaystyle \frac{3}{4}
\displaystyle \frac{4}{4}
\displaystyle \frac{5}{4}
\displaystyle \frac{6}{4}
\displaystyle \frac{7}{4}
\displaystyle \frac{8}{4}
 1 mile 2 miles
Figure 3Anouska’s bike path. The count passes one whole mile at \displaystyle \frac{4}{4} and keeps going.
Solution

Each marker is one more quarter mile, one more copy of \displaystyle \frac{1}{4}, so her count is

\frac{1}{4}, \quad \frac{2}{4}, \quad \frac{3}{4}, \quad \frac{4}{4}, \quad \frac{5}{4}, \quad \frac{6}{4}, \quad \frac{7}{4}.

A mile is 4 quarter miles, so the fourth marker, \displaystyle \frac{4}{4}, is one whole mile from the start.

She did not stop there. The fifth, sixth and seventh markers are \displaystyle \frac{5}{4}, \displaystyle \frac{6}{4} and \displaystyle \frac{7}{4}, all past the first mile. She stops at \displaystyle \frac{7}{4} of a mile, seven fourths.

To say where that is, count from the landmark: from \displaystyle \frac{4}{4} to \displaystyle \frac{7}{4} is 7 - 4 = 3 more steps of a fourth. So seven fourths of a mile is one whole mile and three fourths of a mile more.

Check against the next landmark: one more marker would be \displaystyle \frac{8}{4}, two whole miles, so \displaystyle \frac{7}{4} is one fourth of a mile short of 2 miles. That fits “one mile and three fourths more”.

Example 4

Part 2: the relay keeps going

Cristobal’s team swims a longer drill. Each swimmer still swims \displaystyle \frac{1}{3} of a length, and at the wall the next swimmer turns and starts the next length. How far has the team swum after 5 swimmers? Name it in two ways. How many more swimmers does it take to finish 2 whole lengths?

Solution

Count by thirds, one count per swimmer:

\frac{1}{3}, \quad \frac{2}{3}, \quad \frac{3}{3}, \quad \frac{4}{3}, \quad \frac{5}{3}.

The third swimmer finishes the first length, \displaystyle \frac{3}{3} = 1, exactly as in Example 1. The fourth and fifth swimmers are into the second length. After 5 swimmers the team has swum \displaystyle \frac{5}{3} of a length, five thirds. That is 5 - 3 = 2 steps past the first whole length, so it is one whole length and two thirds of a length more.

Each length takes 3 swimmers. The second length is finished 3 counts after \displaystyle \frac{3}{3}, at \displaystyle \frac{6}{3}, which is 2 whole lengths. The team is at \displaystyle \frac{5}{3}, so it needs 6 - 5 = 1 more swimmer.

The wholes come every 3 counts: \displaystyle \frac{3}{3} is 1 length, \displaystyle \frac{6}{3} is 2 lengths, and the next would be \displaystyle \frac{9}{3}, 3 lengths.

Example 5

Part 2: from a description to a count

Fermin chalks a hopscotch court along the sidewalk, which is laid in equal concrete squares with a crack between each. All the boxes are the same length, and 6 boxes fill exactly one sidewalk square, so each box is \displaystyle \frac{1}{6} of a square long. His court fills one whole square and runs 2 boxes into the next one (Figure 4). Counting by sixths of a square, how long is the court?

one whole square 2 more boxes
Figure 4Fermin’s court. Six boxes fill one sidewalk square, and the court runs 2 boxes into the next.
Solution

One whole square is 6 boxes, the count \displaystyle \frac{6}{6}. The court goes 2 boxes further, so the count goes on 2 more steps: \displaystyle \frac{7}{6}, then \displaystyle \frac{8}{6}.

The court is \displaystyle \frac{8}{6} of a square long, eight sixths. Counting the copies gives the same answer: 6 + 2 = 8 boxes, each \displaystyle \frac{1}{6} of a square.

Both names describe the court. “Eight sixths of a square” is the count. “One whole square and two sixths more” says where the count is. The square is the whole here; the same 8 boxes on a sidewalk with longer squares would be a different fraction.

Example 6

Part 3: hopping back by sixths

Fermin stands at the far end of his court, \displaystyle \frac{8}{6} of a square from the start. He hops back one box at a time until he reaches the start, saying the count at each landing (Figure 5). Write his count. After how many hops is he exactly one square from the start? How many hops does he take in all?

0
\displaystyle \frac{1}{6}
\displaystyle \frac{2}{6}
\displaystyle \frac{3}{6}
\displaystyle \frac{4}{6}
\displaystyle \frac{5}{6}
\displaystyle \frac{6}{6}
\displaystyle \frac{7}{6}
\displaystyle \frac{8}{6}
Figure 5Hopping back one box at a time takes one sixth away with each hop.
Solution

Each hop back takes one copy of \displaystyle \frac{1}{6} away, so the numerator goes down by 1 and the denominator stays 6. Starting from \displaystyle \frac{8}{6}, his landings are

\frac{7}{6}, \quad \frac{6}{6}, \quad \frac{5}{6}, \quad \frac{4}{6}, \quad \frac{3}{6}, \quad \frac{2}{6}, \quad \frac{1}{6}, \quad \frac{0}{6}.

His second landing is \displaystyle \frac{6}{6}, one whole square from the start. That takes 8 - 6 = 2 hops.

He reaches the start when the count is \displaystyle \frac{0}{6}, no sixths, which is 0. The numerator went from 8 down to 0 one step at a time, so he took 8 hops, one for each box.

Check: counting forward from the start to the far end took 8 steps in Example 5’s count, and going back passes the same 8 boxes.

Example 7

Part 3: forward and back along the path

Anouska starts at the \displaystyle \frac{3}{4} marker on her bike path (Figure 3). She rides forward past 5 markers, then turns around and rides back past 3 markers. At which marker does she turn, and at which does she stop? Name each place with its count and in words. How far apart are the marker she started at and the marker she stopped at?

Solution

Forward from \displaystyle \frac{3}{4}, each marker adds one fourth: \displaystyle \frac{4}{4}, \frac{5}{4}, \frac{6}{4}, \frac{7}{4}, \frac{8}{4}. She turns at \displaystyle \frac{8}{4}. That is two sets of four fourths, so it is exactly 2 whole miles from the start.

Back from \displaystyle \frac{8}{4}, each marker takes one fourth away: \displaystyle \frac{7}{4}, \frac{6}{4}, \frac{5}{4}. She stops at \displaystyle \frac{5}{4}, five fourths of a mile, which is 5 - 4 = 1 step past the first mile: one whole mile and one fourth of a mile more.

She started at \displaystyle \frac{3}{4} and stopped at \displaystyle \frac{5}{4}. The numerators differ by 5 - 3 = 2, so the two markers are 2 quarter miles apart, \displaystyle \frac{2}{4} of a mile. Check another way: she went 5 markers forward and 3 back, and 5 forward then 3 back leaves her 2 markers forward of where she began.

Boundaries

One step size the whole way. A count by unit fractions works only when every step is a copy of the same unit fraction of the same whole. Counting “one third, two thirds” and then switching to fourths partway gives a list of words that names no place.

The whole must be named. “Five fourths” is five fourths of something: of a mile, of a length, of a square. The same count means a different distance when the whole changes (Example 5).

The denominator is never zero. A whole cannot be cut into 0 equal parts, so there is no step \displaystyle \frac{1}{0} to count by.

Counting back stops at zero. Each step back removes a piece, and once no pieces are left, at \displaystyle \frac{0}{b} = 0, there is nothing more to take away.

Past one whole is allowed. A count may go on past \displaystyle \frac{b}{b} for as long as there is room. What stops working there is the reading “out of”: \displaystyle \frac{5}{4} is not 5 out of 4, but it is 5 copies of \displaystyle \frac{1}{4}.

Where this goes next

The counts past one whole, like \displaystyle \frac{5}{4} and \displaystyle \frac{7}{3}, are the subject of the next lesson, Fractions greater than one (improper fractions). The landmarks the count keeps passing, \displaystyle \frac{3}{3}, \displaystyle \frac{6}{3}, \displaystyle \frac{8}{4}, are whole numbers written as fractions, in Whole numbers written as fractions. The shorter written form of “one whole and three fourths more” is a mixed number, which comes later.

The markers on Anouska’s bike path were put there for her. Making those markers yourself, by cutting the stretch from 0 to 1 into equal parts on a number line, and then finding where a fraction sits, is Locating a fraction on the number line.

Writing a count out as its copies joined by plus signs is Decomposing a fraction into unit fractions. And counting on 2 fourths from \displaystyle \frac{3}{4} is the first step toward Adding and subtracting fractions with like denominators.

Exercises

Part 1: counting up to one whole

  1. Count by fourths from \displaystyle \frac{1}{4} until you reach one whole. How many counts did it take?
  2. Count by halves from \displaystyle \frac{1}{2} until you reach one whole.
  3. A relay team shares one length of a lane, and each swimmer swims \displaystyle \frac{1}{6} of the length. How much of the length has the team covered after 4 swimmers? How many more swimmers does it take to finish the length?
  4. How many eighths make one whole? Write the count that names one whole.
  5. Cristobal has swum \displaystyle \frac{2}{3} of a length, counting by thirds. What is his next count, and what is special about it?
  6. Fill in the missing counts: \displaystyle \frac{1}{8}, \frac{2}{8}, \square, \frac{4}{8}, \square, \square, \frac{7}{8}, \square.
  7. Counting from 0, which reaches one whole in fewer counts: counting by thirds or counting by sixths? How many counts does each take?

Part 2: counting past one whole

  1. Count by thirds from \displaystyle \frac{1}{3} to \displaystyle \frac{7}{3}. Which of your counts are a whole number of wholes?
  2. Fermin rides along Anouska’s bike path, with a marker every quarter mile, and stops at the sixth marker. Name where he stops with a count and in words.
  3. Count by halves from \displaystyle \frac{1}{2} to \displaystyle \frac{5}{2}. Which counts are whole miles on a path with a marker every half mile? Say where \displaystyle \frac{5}{2} is in words.
  4. In another relay drill each swimmer swims \displaystyle \frac{1}{4} of a length. How far has the team swum after 9 swimmers? Name it with a count and in words.
  5. Counting by eighths, which count is one whole and five eighths more?
  6. On another sidewalk, 3 hopscotch boxes fill one square. A court fills two whole squares and runs 2 boxes into a third. Counting by thirds of a square, how long is the court?
  7. Between which two whole numbers is the count \displaystyle \frac{11}{6}? How far is it from each?

Part 3: counting backward

  1. Count back by fourths from \displaystyle \frac{6}{4} to 0. Which count names one whole?
  2. Count back by thirds from \displaystyle \frac{7}{3}. How many steps back does it take to reach one whole?
  3. Anouska is at the \displaystyle \frac{9}{4} marker of a bike path with a marker every quarter mile, and rides back past 3 markers. Where is she? Say it with a count and in words.
  4. Count back by halves from \displaystyle \frac{3}{2} to 0. How many steps does it take?
  5. How many steps of one sixth does it take to count from \displaystyle \frac{4}{6} to \displaystyle \frac{10}{6}? How much is that many sixths?

Reasoning

  1. Explain why b copies of \displaystyle \frac{1}{b} always make exactly one whole, whatever b is.
  2. Counting by thirds, Fermin says “three thirds” and later “six thirds”. Explain why each of those counts is a whole number of wholes, and say which whole numbers they are.
  3. Counting by sixths, what is the count just after one whole? Explain why it is more than one.
  4. Is a count whose numerator is bigger than its denominator always past one whole? Explain, with an example.
  5. Counting back by \displaystyle \frac{1}{b}, where do you land one step back from \displaystyle \frac{1}{b}? Why can the count not go on past there?
Answers
  1. \displaystyle \frac{1}{4}, \frac{2}{4}, \frac{3}{4}, \frac{4}{4}. It took 4 counts, and \displaystyle \frac{4}{4} = 1.
  2. \displaystyle \frac{1}{2}, \frac{2}{2}. Two halves make the whole, \displaystyle \frac{2}{2} = 1.
  3. \displaystyle \frac{4}{6} of the length, four sixths. The length is 6 sixths, so it takes 6 - 4 = 2 more swimmers, ending at \displaystyle \frac{6}{6}.
  4. 8 eighths make one whole, and the count is \displaystyle \frac{8}{8}.
  5. \displaystyle \frac{3}{3}. Three thirds is one whole length: he has covered all 3 equal parts.
  6. \displaystyle \frac{3}{8}, then \displaystyle \frac{5}{8} and \displaystyle \frac{6}{8}, then \displaystyle \frac{8}{8}, one whole. The numerator goes up by 1 each count.
  7. Thirds: 3 counts, ending at \displaystyle \frac{3}{3}. Sixths take 6 counts, ending at \displaystyle \frac{6}{6}. Thirds are bigger steps, so they take fewer.
  8. \displaystyle \frac{1}{3}, \frac{2}{3}, \frac{3}{3}, \frac{4}{3}, \frac{5}{3}, \frac{6}{3}, \frac{7}{3}. The whole numbers of wholes are \displaystyle \frac{3}{3} = 1 and \displaystyle \frac{6}{3} = 2.
  9. \displaystyle \frac{6}{4} of a mile, six fourths. That is 6 - 4 = 2 steps past the first mile: one whole mile and two fourths of a mile more.
  10. \displaystyle \frac{1}{2}, \frac{2}{2}, \frac{3}{2}, \frac{4}{2}, \frac{5}{2}. Whole miles are \displaystyle \frac{2}{2} = 1 and \displaystyle \frac{4}{2} = 2. \displaystyle \frac{5}{2} is one step past \displaystyle \frac{4}{2}: two whole miles and one half of a mile more.
  11. \displaystyle \frac{9}{4} of a length, nine fourths. Wholes come at \displaystyle \frac{4}{4} and \displaystyle \frac{8}{4}, and 9 - 8 = 1, so it is two whole lengths and one fourth of a length more.
  12. \displaystyle \frac{13}{8}. One whole is 8 eighths, and 8 + 5 = 13 eighths.
  13. \displaystyle \frac{8}{3} of a square. Two whole squares are 3 + 3 = 6 thirds, and 2 boxes more make 6 + 2 = 8 thirds.
  14. Between 1 and 2. One whole is \displaystyle \frac{6}{6} and two wholes are \displaystyle \frac{12}{6}. \displaystyle \frac{11}{6} is 11 - 6 = 5 sixths past 1 and 12 - 11 = 1 sixth short of 2.
  15. \displaystyle \frac{6}{4}, \frac{5}{4}, \frac{4}{4}, \frac{3}{4}, \frac{2}{4}, \frac{1}{4}, \frac{0}{4} = 0. One whole is \displaystyle \frac{4}{4}.
  16. \displaystyle \frac{7}{3}, \frac{6}{3}, \frac{5}{3}, \frac{4}{3}, \frac{3}{3}. It takes 7 - 3 = 4 steps back to reach \displaystyle \frac{3}{3}, one whole.
  17. \displaystyle \frac{8}{4}, \frac{7}{4}, \frac{6}{4}: she is at \displaystyle \frac{6}{4}, one whole mile and two fourths of a mile more from the start.
  18. \displaystyle \frac{3}{2}, \frac{2}{2}, \frac{1}{2}, \frac{0}{2} = 0. It takes 3 steps.
  19. 10 - 4 = 6 steps. Six sixths is \displaystyle \frac{6}{6}, one whole.
  20. The denominator b means b equal parts make the whole, and \displaystyle \frac{1}{b} is one of them. So b copies of \displaystyle \frac{1}{b} of that same whole are all the parts, nothing missing and nothing extra: one whole.
  21. Three thirds is 3 copies of \displaystyle \frac{1}{3}, all 3 parts of one whole, so \displaystyle \frac{3}{3} = 1. Six thirds is another 3 thirds on top of that, a second whole, so \displaystyle \frac{6}{3} = 2.
  22. \displaystyle \frac{7}{6}. One whole is \displaystyle \frac{6}{6}, and \displaystyle \frac{7}{6} is one more sixth past it, so it is one whole and one sixth more.
  23. Yes. One whole is the count \displaystyle \frac{b}{b}, and a bigger numerator means more than b copies of \displaystyle \frac{1}{b}, so the count has gone past the whole. For example, \displaystyle \frac{5}{4} is one step past \displaystyle \frac{4}{4}.
  24. At \displaystyle \frac{0}{b} = 0, no pieces at all. Each step back takes away a piece, and at 0 there is no piece left to take.
kg1128 · updated 2026-10-21 · CC-BY-SA-4.0