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Grade 3 · 3.NF.A.2 · 3.NF.A.2A

Locating a fraction on the number line

Where on a line does the number three fourths sit, and how do you find that spot when nobody has marked it for you?

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Fractions greater than one (improper fractions)Grade 3
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Finding a fraction between two fractions
Locating a fraction on the number line

When you reach for it

Until now a fraction has been pieces of a whole: slices of a pie, wedges of a cheese wheel, as in Area and length models of a fraction and Fractions greater than one (improper fractions). In Counting by unit fractions the count ran along a bike path whose markers were already there. This lesson makes the markers yourself, and uses them to find the place where a fraction sits on the number line.

On a number line every number is a point. 0 is where the line starts, 1 is one unit to the right, 2 is one more unit on. A fraction is a number too, so it has a point of its own. The point \displaystyle \frac{3}{4} sits between 0 and 1, three fourths of the way from 0 to 1. The point \displaystyle \frac{5}{3} sits between 1 and 2. Seeing a fraction as one point on the same line as the whole numbers is the big idea here: it is not only a piece of something, it is a number with a place.

You reach for this whenever a fraction measures a distance from a start along something straight: how far along a nature trail a bench stands, where a clip goes on a clothesline, where a float stopped on a parade route. It also pays off later. Once fractions are points on one line, the one farther to the right is the greater one, and the space between two points is a length.

The procedure

The lesson runs over four days. Part 1 cuts the stretch from 0 to 1 into equal parts. Part 2 finds a fraction less than one by jumping from 0. Part 3 carries the jumps past 1. Part 4 reads a point that someone else has marked.

Part 1: cutting the stretch from 0 to 1 into equal parts

The stretch of line from 0 to 1 is one whole, one unit of length. To show fourths, cut that stretch into 4 equal parts with short marks across the line, called tick marks. Each part then has length \displaystyle \frac{1}{4}, for the same reason a pie cut into 4 equal pieces gives fourths: 4 equal parts fill the whole, so each is one fourth of it. In general, cutting the stretch from 0 to 1 into b equal parts makes each part \displaystyle \frac{1}{b} long.

The parts must be equal. If one space is wider than another, the spaces are not fourths, however many there are.

The denominator counts parts, the spaces between ticks, never the ticks themselves. Cutting the stretch into 4 parts takes 3 ticks between 0 and 1, since each tick inside splits one space into two. Counting the ticks at 0 and 1 too gives 5. Neither number is 4; the spaces are.

Equal parts can be made by halving. Find the middle of the stretch and mark it: 2 halves. Find the middle of each half: 4 fourths. Find the middle of each fourth: 8 eighths. Thirds take two ticks placed so the three spaces match. Sixths come from cutting each third in half, or each half into three, because 3 \times 2 = 6 and 2 \times 3 = 6.

Part 2: a fraction is the point at the end of its jumps

With the stretch from 0 to 1 cut into b equal parts, start at 0 and jump along one part at a time. Each jump has length \displaystyle \frac{1}{b}. After a jumps you stand on a tick, and that tick is the point \displaystyle \frac{a}{b}:

\frac{a}{b} \text{ is the point reached by } a \text{ jumps of } \frac{1}{b} \text{ from } 0.

Two things follow. The point is a place, not a piece. The fraction \displaystyle \frac{3}{4} is the spot where the third jump ends, and the length of line from 0 up to that spot is \displaystyle \frac{3}{4} of the unit. The third part itself is the space between \displaystyle \frac{2}{4} and \displaystyle \frac{3}{4}; the point \displaystyle \frac{3}{4} is its right end. And every jump starts the count from 0: the numerator is the number of jumps from 0, not from the tick before.

At the ends, \displaystyle \frac{0}{b} is no jumps at all, the point 0, and \displaystyle \frac{b}{b} is b jumps, which land on 1. That second fact gets its own lesson next, Whole numbers written as fractions.

Part 3: past 1, every whole cut the same way

The line does not stop at 1, and the jumps do not have to either. To place \displaystyle \frac{5}{3} the line must reach past 1, say to 2, and the stretch from 1 to 2 must be cut into thirds as well. Its parts are \displaystyle \frac{1}{3} long, the same as before, because the stretch from 1 to 2 is one unit long, just like the stretch from 0 to 1.

Keep jumping from 0: three jumps reach \displaystyle \frac{3}{3}, which is the point 1, and the jumps go on to \displaystyle \frac{4}{3}, \displaystyle \frac{5}{3} and \displaystyle \frac{6}{3}, which is the point 2. The whole numbers are landmarks. With b parts in each unit, 1 is b jumps from 0 and 2 is b + b jumps, so a fraction’s numerator, set against those counts, says which two whole numbers it sits between and how far past the lower one it is.

Part 4: reading a point someone else marked

The same reasoning runs backward. A line has ticks and a dot, but only a few labels. Find the stretch from 0 to 1. Count the parts in it: that is the denominator. Count the parts from 0 to the dot: that is the numerator.

When the line is labeled only at 0 and 2, find 1 first. Both units are cut the same way, so 1 is the tick with half of the parts on each side of it. The denominator is the number of parts between 0 and 1, because the unit is always the stretch from 0 to 1, never the whole line as drawn.

Worked examples

Example 1

Part 1: a clothesline in sixths

Galit strings a clothesline exactly 1 yard long between two posts. She thinks of it as a number line, with 0 at the left post and 1 at the right post. She wants a clip at every sixth of a yard, and she hangs a towel from the clip \displaystyle \frac{5}{6} of a yard from the left post. How can she cut the yard into 6 equal parts? How many clips go between the posts, and which one holds the towel?

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}
 1 yard
Figure 1Galit’s clothesline. Five jumps of \displaystyle \frac{1}{6} of a yard from the post end at the clip, \displaystyle \frac{5}{6}.
Solution

Sixths come from two cuts she can make in turn. First she finds the middle of the line, which splits the yard into 2 equal halves. Then she cuts each half into 3 equal parts. That gives 2 \times 3 = 6 equal parts, and each part is \displaystyle \frac{1}{6} of a yard, since 6 of them fill the yard.

The 6 parts need 6 - 1 = 5 clips between the posts, one at each tick inside (Figure 1). Going from the left post, the clips are at \displaystyle \frac{1}{6}, \displaystyle \frac{2}{6}, \displaystyle \frac{3}{6}, \displaystyle \frac{4}{6} and \displaystyle \frac{5}{6} of a yard, and the right post is \displaystyle \frac{6}{6}, the end of the yard.

The towel’s spot is 5 jumps of \displaystyle \frac{1}{6} from 0, so it is the fifth clip from the left. The fraction names where the fifth jump ends. The length of line from the left post to the towel is \displaystyle \frac{5}{6} of a yard, and the towel is 6 - 5 = 1 part, \displaystyle \frac{1}{6} of a yard, short of the right post.

Example 2

Part 1: eighths by halving

Sulo draws a number line from 0 to 1 on the board and wants it cut into eighths. Show how halving gets there. How many ticks are there between 0 and 1 when he is done, and how long is each part?

halves 0 1 fourths 0 1 eighths 0
\displaystyle \frac{1}{8}
\displaystyle \frac{2}{8}
\displaystyle \frac{3}{8}
\displaystyle \frac{4}{8}
\displaystyle \frac{5}{8}
\displaystyle \frac{6}{8}
\displaystyle \frac{7}{8}
\displaystyle \frac{8}{8}
Figure 2Halving the parts again and again. Two halves become four fourths, and four fourths become eight eighths.
Solution

Halving once puts 1 tick at the middle and makes 2 equal parts. Halving each of those adds 2 more ticks and makes 2 \times 2 = 4 equal parts. Halving each of those adds 4 more ticks and makes 4 \times 2 = 8 equal parts (Figure 2).

The ticks between 0 and 1 number 1 + 2 + 4 = 7. That matches the rule from Part 1: 8 parts need 8 - 1 = 7 ticks inside. Each part is \displaystyle \frac{1}{8} of the unit, because 8 equal parts fill the stretch from 0 to 1.

Labeling the ticks by jumps from 0 gives \displaystyle \frac{1}{8}, \displaystyle \frac{2}{8}, up to \displaystyle \frac{7}{8}, with 1 at \displaystyle \frac{8}{8}. The middle tick, the first one Sulo drew, is 4 jumps from 0, the point \displaystyle \frac{4}{8}. It is also the point \displaystyle \frac{1}{2} from the first halving. One point with two fraction names is the subject of Seeing equivalent fractions in pictures and on the number line, later in this unit.

Example 3

Part 2: a bench on the nature trail

A nature trail is 1 mile long, from the trailhead to a lookout at the far end. Think of it as a number line with the trailhead at 0 and the lookout at 1. Ilana is putting in a bench \displaystyle \frac{3}{4} of a mile from the trailhead. Locate the bench on the line. How far is the bench from the lookout?

0
\displaystyle \frac{1}{4}
\displaystyle \frac{2}{4}
\displaystyle \frac{3}{4}
\displaystyle \frac{4}{4}
 bench 1 mile
Figure 3The bench is at the end of the third jump, \displaystyle \frac{3}{4} of a mile from the trailhead.
Solution

The denominator 4 says to cut the mile into 4 equal parts. Find the middle of the line, then the middle of each half: 3 ticks, 4 parts, each \displaystyle \frac{1}{4} of a mile.

The numerator 3 says to take 3 jumps of \displaystyle \frac{1}{4} from 0. The first jump lands on \displaystyle \frac{1}{4}, the second on \displaystyle \frac{2}{4}, the third on \displaystyle \frac{3}{4}. The bench goes at the end of the third jump, the third tick after the trailhead (Figure 3).

The bench is a point, and the walk from the trailhead to it covers 3 parts of the trail. The third part of the trail is the stretch from \displaystyle \frac{2}{4} to \displaystyle \frac{3}{4}; the bench stands at the far end of it.

From the bench to the lookout at \displaystyle \frac{4}{4} is 4 - 3 = 1 more part. The bench is \displaystyle \frac{1}{4} of a mile from the lookout.

Example 4

Part 2: two water stands on a fun run

Ilana’s school fun run is 1 mile long, from the start at 0 to the finish at 1. Water stands go at \displaystyle \frac{3}{8} of a mile and at \displaystyle \frac{7}{8} of a mile. Describe where each stand is on the line. How far apart are the two stands?

Solution

Both fractions are eighths, so one cut serves both: halve the mile three times, as in Example 2, to get 8 equal parts with 7 ticks between 0 and 1.

The first stand is 3 jumps of \displaystyle \frac{1}{8} from 0, the third tick after the start. It is 8 - 3 = 5 parts from the end, \displaystyle \frac{5}{8} of a mile.

The second stand is 7 jumps from 0, the seventh tick, the last one before the end. It is 8 - 7 = 1 part from the end, \displaystyle \frac{1}{8} of a mile.

Between them lie the parts from the third tick to the seventh, 7 - 3 = 4 parts. The stands are \displaystyle \frac{4}{8} of a mile apart. Each jump is the same length, so counting parts between two ticks measures the distance between them.

Example 5

Part 3: a float stops at five thirds of a mile

Sulo’s town holds a parade along a route 2 miles long. A float breaks down \displaystyle \frac{5}{3} of a mile from the start. Draw the route as a number line from 0 to 2 and locate the float. Between which two whole miles is it, and how far is it from each?

0
\displaystyle \frac{1}{3}
\displaystyle \frac{2}{3}
\displaystyle \frac{3}{3}
\displaystyle \frac{4}{3}
\displaystyle \frac{5}{3}
\displaystyle \frac{6}{3}
 1 2
Figure 4Both miles are cut into thirds the same way, so the jumps carry on past 1 and land on \displaystyle \frac{5}{3}.
Solution

The denominator 3 says each part is \displaystyle \frac{1}{3} of a mile, so each unit of the line is cut into 3 equal parts. That means both units: the stretch from 0 to 1 and the stretch from 1 to 2. They are both 1 mile long, so cutting each into 3 equal parts gives parts of the same size, \displaystyle \frac{1}{3} of a mile. There are 3 + 3 = 6 parts in all.

Now jump from 0, one part at a time:

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

The third jump lands on \displaystyle \frac{3}{3}, the point 1. The float is at the end of the fifth jump, which is 5 - 3 = 2 ticks past 1 (Figure 4).

So the float is between 1 and 2 miles, \displaystyle \frac{2}{3} of a mile past the 1 mile mark. The point 2 is \displaystyle \frac{6}{3}, 6 jumps from 0, so the float is 6 - 5 = 1 part, \displaystyle \frac{1}{3} of a mile, short of the end. Check: 2 + 1 = 3 parts make the mile from 1 to 2.

Example 6

Part 3: past two on a longer trail

The long loop of the nature trail is 3 miles, with a marker post every quarter mile from the trailhead. A spring sits \displaystyle \frac{9}{4} of a mile from the trailhead. Which marker is it at, and between which two whole miles?

Solution

Each mile is cut into 4 equal parts, and the three miles are cut the same way, so every part is \displaystyle \frac{1}{4} of a mile and there are 4 + 4 + 4 = 12 parts from 0 to 3.

The whole miles are landmarks. Mile 1 is 4 jumps from 0, the point \displaystyle \frac{4}{4}. Mile 2 is 4 + 4 = 8 jumps, the point \displaystyle \frac{8}{4}. Mile 3 is 12 jumps, the point \displaystyle \frac{12}{4}.

The spring is 9 jumps from 0, so it is at the ninth marker after the trailhead. Nine is past 8 and short of 12, so the spring is between mile 2 and mile 3. It is 9 - 8 = 1 part past the 2 mile post, \displaystyle \frac{1}{4} of a mile, and 12 - 9 = 3 parts short of the end, \displaystyle \frac{3}{4} of a mile.

Example 7

Part 4: reading a dot between 0 and 1

Galit draws a number line labeled only at 0 and 1 (Figure 5). Equal tick marks cut the stretch between them, and a dot sits on one tick. What fraction does the dot stand for?

0 1
Figure 5Count the parts between 0 and 1, not the tick marks.
Solution

Start with the unit, the stretch from 0 to 1. Count the ticks between 0 and 1: there are 7. Seven ticks inside make 7 + 1 = 8 spaces, so the stretch is cut into 8 equal parts and each part is \displaystyle \frac{1}{8}. The denominator is 8, the number of parts.

Now count the parts from 0 to the dot: 5. The dot is at the end of 5 jumps of \displaystyle \frac{1}{8}, so it stands for \displaystyle \frac{5}{8}.

Check by counting from the other end: from the dot to 1 is 8 - 5 = 3 parts, and 5 + 3 = 8 parts make the whole unit.

Example 8

Part 4: a line labeled only at 0 and 2

Sulo draws a line labeled only at 0 and 2, cut into equal parts by tick marks, with a dot on one tick (Figure 6). What fraction does the dot stand for?

0 2
Figure 6Only 0 and 2 are labeled. Finding 1 comes first.
Solution

The unit is the stretch from 0 to 1, and 1 is not labeled, so find it first. There are 5 ticks between 0 and 2, making 5 + 1 = 6 equal parts. The stretch from 0 to 1 and the stretch from 1 to 2 are the same length and cut the same way, so each holds half of the parts: 6 \div 2 = 3. The point 1 is the third tick after 0.

So each unit is cut into 3 equal parts, and each part is \displaystyle \frac{1}{3}. The denominator is 3, not 6: 6 counts the parts in two units.

The dot is 4 parts from 0, so it stands for \displaystyle \frac{4}{3}. That is 4 - 3 = 1 part past the point 1, so \displaystyle \frac{4}{3} is between 1 and 2, one third past 1.

Boundaries

The parts must be equal. A tick at \displaystyle \frac{1}{4} is one fourth of the way from 0 to 1 only if all four spaces are the same length. Unequal spaces name no fraction, however many there are.

The unit is the stretch from 0 to 1. Whatever length of line is drawn, and whichever numbers are labeled, the denominator counts the parts in one unit (Example 8).

Every unit is cut the same way. Past 1, each stretch from one whole number to the next is cut into the same number of equal parts as the stretch from 0 to 1, so every jump is the same \displaystyle \frac{1}{b}.

Jumps are counted from 0. The point \displaystyle \frac{a}{b} is a jumps from 0. The numerator is not a count of tick marks, and not a count from the nearest whole number.

This line starts at 0. All the fractions here are 0 or more. Fractions on the other side of 0 are in Negative fractions and decimals on the number line.

A cut shows only its own denominator. A line cut into fourths has no tick for \displaystyle \frac{1}{3}. Every fraction has a point, but to see it the line must be cut into parts of that size.

The denominator is never zero. The stretch from 0 to 1 cannot be cut into 0 parts, so there is no jump \displaystyle \frac{1}{0}.

Where this goes next

The next lesson, Whole numbers written as fractions, looks at the ticks where the jumps land on a whole number, such as \displaystyle \frac{3}{3} at the point 1. Later in this unit, one point turns out to have many fraction names, like \displaystyle \frac{4}{8} and \displaystyle \frac{1}{2} in Example 2: that is Seeing equivalent fractions in pictures and on the number line. With fractions as points, comparing two of them with the same denominator is seeing which one is farther from 0, in Comparing fractions with the same denominator or numerator.

Further on, Finding a fraction between two fractions shows that between any two of these points there is always another, and Negative fractions and decimals on the number line carries the line past 0 to the left. “One whole and two thirds more”, the place of \displaystyle \frac{5}{3}, has the short written form of a mixed number.

Exercises

Part 1: cutting the stretch from 0 to 1

  1. Describe how to cut the stretch from 0 to 1 into fourths by halving. How many ticks are there between 0 and 1, and what are their labels?
  2. A number line from 0 to 1 is cut into thirds. Label every tick from 0 to 1.
  3. Galit cuts a clothesline 1 yard long into 8 equal parts with a clip at each tick between the posts. How long is each part, and how many clips are there?
  4. Ilana draws 5 equally spaced ticks between 0 and 1. How many parts is the stretch cut into, and how long is each part?
  5. A line from 0 to 1 is cut into thirds. How can you turn the thirds into sixths? Say why the new parts are sixths.

Part 2: fractions from 0 to 1

  1. Locate \displaystyle \frac{2}{3} on a number line from 0 to 1. Describe the cut, the jumps, and how far the point is from 1.
  2. Locate \displaystyle \frac{1}{2} on a number line from 0 to 1.
  3. A nature trail 1 mile long has a marker every eighth of a mile. A rest stop is \displaystyle \frac{5}{8} of a mile from the trailhead. Which marker after the trailhead is it, and how far is it from the end of the trail?
  4. Galit hangs a shirt from the clip at \displaystyle \frac{1}{6} of a yard and a sheet from the clip at \displaystyle \frac{4}{6} of a yard on her clothesline. How far apart are they?
  5. Where on the line is \displaystyle \frac{0}{4}? Explain with jumps.
  6. A line from 0 to 1 is cut into eighths. The third part is the space between which two points? Which of those points is \displaystyle \frac{3}{8}?

Part 3: past 1

  1. A line from 0 to 2 has each unit cut into fourths. Label every tick after 0. Which ticks are the points 1 and 2?
  2. Locate \displaystyle \frac{3}{2} on a line from 0 to 2.
  3. Sulo’s parade route is 2 miles long with a flag every sixth of a mile. A balloon is tied at \displaystyle \frac{10}{6} of a mile. Which flag after the start is it? Between which two whole miles is it, and how far is it from the end?
  4. Between which two whole numbers is \displaystyle \frac{13}{8}? How far is it past the lower one, and how far short of the higher one?
  5. A trail 3 miles long has a marker every third of a mile. Where is the marker at \displaystyle \frac{7}{3} of a mile: which marker after the start, and between which two whole miles?
  6. A line from 0 to 2 has each unit cut into sixths. How many ticks are there after 0, up to and including 2? Which of them are the points 1 and 2, and what fractions name them?

Part 4: reading a marked point

  1. A line labeled at 0 and 1 is cut into 4 equal parts. A dot sits on the first tick after 0. What fraction is the dot?
  2. A line labeled at 0 and 1 has 7 equally spaced ticks between them. A dot sits on the third tick after 0. What fraction is the dot?
  3. A line labeled only at 0 and 2 is cut into 8 equal parts. A dot sits on the fifth tick after 0. What fraction is the dot?
  4. A line labeled at 0 and 1 is cut into eighths, and the same ticks carry on past 1. A dot sits on the third tick after 1. What fraction is the dot?
  5. A line labeled only at 0 and 2 is cut into 16 equal parts. A dot sits on the fourteenth tick after 0. What fraction is the dot, and how far is it past 1?

Reasoning

  1. To place \displaystyle \frac{5}{4}, why must the stretch from 1 to 2 be cut into the same number of equal parts as the stretch from 0 to 1?
  2. Explain why cutting the stretch from 0 to 1 into b equal parts takes b - 1 ticks between 0 and 1, and why the denominator is b and not b - 1.
  3. The point \displaystyle \frac{3}{4} is a single spot on the line, yet it is called three fourths. Three fourths of what? Explain using the jumps.
Answers
  1. Find the middle of the stretch, then the middle of each half. That gives 3 ticks between 0 and 1 and 4 equal parts. The ticks are \displaystyle \frac{1}{4}, \displaystyle \frac{2}{4} and \displaystyle \frac{3}{4}, and 1 is \displaystyle \frac{4}{4}.
  2. Two ticks make 3 equal parts. The labels are 0, \displaystyle \frac{1}{3}, \displaystyle \frac{2}{3} and \displaystyle \frac{3}{3}, which is 1.
  3. Each part is \displaystyle \frac{1}{8} of a yard. Eight parts need 8 - 1 = 7 clips between the posts.
  4. 5 + 1 = 6 parts, so each part is \displaystyle \frac{1}{6} of the unit.
  5. Cut each third in half. That makes 3 \times 2 = 6 equal parts in the stretch from 0 to 1, and 6 equal parts filling one unit are sixths.
  6. Cut the stretch into 3 equal parts with 2 ticks. Take 2 jumps of \displaystyle \frac{1}{3} from 0: \displaystyle \frac{2}{3} is the second tick, 3 - 2 = 1 part, \displaystyle \frac{1}{3}, short of 1.
  7. One tick at the middle cuts the stretch into 2 equal parts. One jump of \displaystyle \frac{1}{2} from 0 lands on that middle tick, the point \displaystyle \frac{1}{2}.
  8. The fifth marker, 5 jumps of \displaystyle \frac{1}{8} from 0. It is 8 - 5 = 3 parts from the end, \displaystyle \frac{3}{8} of a mile.
  9. From the first clip to the fourth is 4 - 1 = 3 parts, so they are \displaystyle \frac{3}{6} of a yard apart.
  10. At 0. The numerator 0 means no jumps, so you stay where you started: \displaystyle \frac{0}{4} = 0.
  11. Between \displaystyle \frac{2}{8} and \displaystyle \frac{3}{8}. The point \displaystyle \frac{3}{8} is its right end, where the third jump lands.
  12. \displaystyle \frac{1}{4}, \frac{2}{4}, \frac{3}{4}, \frac{4}{4}, \frac{5}{4}, \frac{6}{4}, \frac{7}{4}, \frac{8}{4}. The point 1 is \displaystyle \frac{4}{4} and the point 2 is \displaystyle \frac{8}{4}.
  13. Cut each unit in half, so the ticks are \displaystyle \frac{1}{2}, \displaystyle \frac{2}{2} (the point 1), \displaystyle \frac{3}{2}, \displaystyle \frac{4}{2} (the point 2). Three jumps land on \displaystyle \frac{3}{2}, the middle of the stretch from 1 to 2.
  14. The tenth flag. The point 1 is \displaystyle \frac{6}{6}, and 10 - 6 = 4, so the balloon is between 1 and 2 miles, four sixths past 1. The end is \displaystyle \frac{12}{6}, so it is 12 - 10 = 2 parts, \displaystyle \frac{2}{6} of a mile, from the end.
  15. Between 1 and 2. The point 1 is \displaystyle \frac{8}{8} and 2 is \displaystyle \frac{16}{8}, so \displaystyle \frac{13}{8} is 13 - 8 = 5 eighths past 1 and 16 - 13 = 3 eighths short of 2.
  16. The seventh marker. Mile 2 is \displaystyle \frac{6}{3} and mile 3 is \displaystyle \frac{9}{3}, so it is between mile 2 and mile 3, 7 - 6 = 1 part, \displaystyle \frac{1}{3} of a mile, past mile 2.
  17. 6 + 6 = 12 ticks. The sixth is the point 1, \displaystyle \frac{6}{6}, and the twelfth is the point 2, \displaystyle \frac{12}{6}.
  18. \displaystyle \frac{1}{4}: 4 parts in the unit, and the dot is 1 part from 0.
  19. \displaystyle \frac{3}{8}: 7 ticks make 7 + 1 = 8 parts, and the dot is 3 parts from 0.
  20. \displaystyle \frac{5}{4}. Each unit holds 8 \div 2 = 4 parts, so each part is \displaystyle \frac{1}{4} and 1 is the fourth tick. The dot is 5 parts from 0.
  21. \displaystyle \frac{11}{8}. The point 1 is 8 jumps from 0, and 3 more make 8 + 3 = 11 jumps.
  22. \displaystyle \frac{14}{8}. Each unit holds 16 \div 2 = 8 parts, so the parts are eighths and 1 is the eighth tick. The dot is 14 - 8 = 6 parts past 1, \displaystyle \frac{6}{8}.
  23. The jumps from 0 to \displaystyle \frac{5}{4} must all be \displaystyle \frac{1}{4} long. Past 1, a part is \displaystyle \frac{1}{4} only if the unit from 1 to 2 is cut into 4 equal parts, like the unit from 0 to 1. Cut any other way, the fifth jump would be a different length.
  24. Each tick inside splits one space into two, so each tick adds one part to the 1 space you start with: b - 1 ticks give b parts. The denominator names the size of a part, one of b equal parts of the unit, so it counts parts, which is b.
  25. Three fourths of the unit, the length from 0 to 1. The point is where 3 jumps of \displaystyle \frac{1}{4} end, so the line from 0 to the point is 3 of the 4 equal parts of the unit long, and the fraction names that distance from 0.
kg1128 · updated 2026-10-21 · CC-BY-SA-4.0