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Where a fraction sits: jumps set by the bottom number, counted by the top; fractions past one; which side of a half; and why a bigger bottom number means a smaller fraction.
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 a fraction stops being a shaded shape and becomes a number with a place of its own. You will cut a line into equal jumps, count along it to land on a fraction, go past 1 to fractions like 5/4, work out which side of the middle a fraction sits without drawing anything, and see why cutting a whole into more pieces makes each piece smaller.
You can name a unit fraction such as 1/4 and count unit fractions to make fractions such as 3/4. You have also used number lines for whole numbers, where each jump from one mark to the next is one. This lesson puts the two ideas together. A fraction is not only a shaded part of a shape; it is a number with its own place on the line, between the whole numbers.
| Term | What it means |
|---|---|
| Number line | A straight line with equally spaced marks that shows numbers in order. |
| Interval | The stretch of line between two marks, such as from 0 to 1. |
| Jump | A move along the line from one mark to the next equal mark. |
| Unit fraction jump | One jump of size 1/d, when the gap from 0 to 1 is cut into d parts. |
| Benchmark | A number you know well, such as 0, 1/2 or 1, used to judge others. |
On a number line, the whole is the distance from 0 to 1. Cut that distance into equal parts and each part is a unit fraction. If the gap from 0 to 1 is cut into 4 equal parts, each jump is 1/4. Start at 0 and make 3 jumps, and you land on 3/4. The bottom number sets the size of a jump; the top number counts how many jumps you make. The point where you land is the fraction.
Another way: picture
A line from 0 to 1 with marks at 1/4, 2/4 and 3/4. Three equal jumps from 0 land on the third mark, which is 3/4.
Every fraction on a number line starts with the interval from 0 to 1. That interval is the whole. To show fourths, cut it into four equal parts by drawing three marks between 0 and 1. The marks must be evenly spaced, because each jump is supposed to be the same size. If one jump is longer than the others, the line no longer shows fourths.
Count the jumps, not the marks. Between 0 and 1 there are three new marks but four jumps. Learners often count the marks and decide the line shows thirds. To avoid this, put your finger on 0 and count each move: one, two, three, four. The fourth move lands on 1. Four jumps fill the whole, so each jump is 1/4.
The same idea works for any bottom number. For sixths, make six equal jumps from 0 to 1. For eighths, make eight. A bigger bottom number means more jumps squeezed into the same distance, so each jump is shorter. You can see on the line that 1/8 lands closer to 0 than 1/4 does.
Once the line is cut into equal jumps, finding a fraction is counting. To find 5/6, cut the gap from 0 to 1 into six equal jumps, then start at 0 and count five jumps. Label the point where you stop 5/6. To find 2/6, count two jumps. Every mark between 0 and 1 has a fraction name: 1/6, 2/6, 3/6, 4/6 and 5/6.
The first jump lands on 1/6, not on 0. Zero is where you start, before any jump has been made. If you call the first mark 0 or start counting at 0, every label after it will be wrong by one jump. A quick check is to look at 1. It must be the sixth jump, so its fraction name is 6/6.
You can also read a line that is already labeled. If a mark is the third of five equal jumps from 0, it is 3/5. The number of jumps from 0 to the mark goes on top. The number of equal jumps from 0 to 1 goes on the bottom. Keep those two numbers in their own jobs and the fraction will be right.
The number line does not stop at 1. Keep making jumps of the same size and you move past it. With jumps of 1/4, the fifth jump lands on 5/4, the sixth on 6/4, and the eighth on 8/4. Each jump is still one fourth, so the bottom number stays 4 while the top number keeps counting.
A fraction whose top number is bigger than its bottom number is more than one whole. 5/4 is four fourths to reach 1 and one fourth more. On the line it sits one jump past 1. 8/4 is eight jumps, which is two groups of four, so it lands exactly on 2. Whole numbers have fraction names too: 1 is 4/4 and 2 is 8/4 when the jumps are fourths.
When you place a fraction past one, find the whole number first. For 7/3, three jumps of a third reach 1 and six jumps reach 2. One more jump gives 7/3, so it sits one third past 2. Counting in groups of the bottom number gets you close quickly, and then you count the jumps left over.
One half is the middle of the gap from 0 to 1, and it is a useful landmark. A fraction is exactly one half when its top number is half of its bottom number: 2/4, 3/6, 4/8 and 5/10 all sit in the middle. A fraction is more than one half when double its top number is bigger than its bottom number. It is less than one half when double its top number is smaller.
Try 3/8. Double the top number: 3 × 2 = 6. Six is less than eight, so 3/8 sits left of the middle. Try 5/8. Double 5 is 10, which is more than 8, so 5/8 sits right of the middle. You did not need to draw anything. The doubling test tells you which side of the halfway mark a fraction lands on.
Landmarks help you check your work. If you place 7/8 just after 0, the landmark tells you something is wrong, because 7/8 is more than one half and should be close to 1. Before you mark a fraction carefully, decide roughly where it belongs: near 0, near the middle, or near 1. Then count the jumps and make sure the exact answer agrees with your estimate.
Compare two runners on the same path. One has gone 2/3 of the way and the other 2/5. Both have made two jumps, so the count is the same. The difference is the size of a jump. The first path is cut into three equal jumps and the second into five. Five jumps squeezed into the same path are shorter than three, so two of the fifths cover less ground. The runner at 2/3 is further along.
This is the fact that makes fractions feel backwards at first. With whole numbers, 5 is more than 3. With unit fractions, 1/5 is less than 1/3, because the whole was shared among more parts. The number line makes it easy to see: the mark for 1/5 sits closer to 0 than the mark for 1/3.
When you check an answer about two fractions with the same top number, picture the two lines. The one cut into more jumps has the shorter jumps, so the same count of them lands nearer to 0. Saying why, and not only which, is what stops the rule from being used the wrong way round.
Two different fractions can name the same point. Cut the gap from 0 to 1 into halves and the middle mark is 1/2. Cut the same gap into fourths and the middle mark is 2/4. Cut it into eighths and the middle mark is 4/8. These are three names for one place on the line, because each counts the same distance from 0 in a different size of jump.
You can see why by stacking the lines. Draw a line in halves, and underneath it a line of the same length in fourths. Every half-jump on the top line matches exactly two fourth-jumps on the bottom line. So one half and two fourths end at the same point. Fractions that share a point are called equivalent, and you will work with them in their own lesson.
Whole numbers also collect many names. On a line marked in thirds, the point 1 is also 3/3, and the point 2 is also 6/3. On a line marked in fifths, 1 is 5/5. When a question asks you to mark 6/3, you do not need a new mark: count six jumps of a third and you arrive at 2. Recognizing these shared points saves time and helps you check your counting, because the whole numbers are places you already know.
A school fun run follows a path exactly one mile long. The organizers want water stations spaced evenly along the path, and they decide to cut the mile into 4 equal stretches. Each stretch is 1/4 mile. The first station is 1/4 mile from the start, the second is at 2/4 mile, and the third is at 3/4 mile. The finish line is at 4/4 mile, which is the whole mile.
A runner stops to tie a shoe after passing three stations. Counting jumps from the start, the runner is at 3/4 mile, so there is 1/4 mile left. Another runner says she is halfway. On the number line, halfway is 2/4 mile, the second station. If she has passed only one station, she is at 1/4 mile and is not yet halfway.
Next year the organizers plan a two-mile run with stations every 1/4 mile. The jumps stay the same size, so the line just keeps going past 1. The station after the 1-mile mark is at 5/4 miles, and the last station before the finish is at 7/4 miles. The fractions tell the runners exactly how far they have come and how far is left. A runner at 6/4 miles has run one and a half miles, with half a mile to go.
Between 0 and 1 there is always one fewer new mark than there are jumps. Counting the marks makes fourths look like thirds. Start at 0 and count each move instead: the jump that lands on 1 tells you the bottom number. Then check the first mark: it is one jump from 0, so it must be the unit fraction, never zero and never the whole.
Name the whole.
0 to 1
The interval from 0 to 1 is one whole.
Read the bottom number.
4
It says how many equal jumps fill the whole.
Cut the interval into equal jumps.
jumps of 1/4
Four equal jumps reach 1.
Read the top number.
3
It says how many jumps to make.
Count three jumps from 0.
1/4, 2/4, 3/4
The third jump lands on 3/4.
Find the jump size.
1/3
Three equal jumps fill each whole.
Count jumps to reach 1.
3 jumps → 3/3 = 1
A full set of thirds makes one whole.
Count jumps to reach 2.
6 jumps → 6/3 = 2
Two full sets make two wholes.
Find the jumps left over.
7 − 6 = 1
The top number asks for seven jumps.
Make the last jump.
one jump past 2
Each jump is still one third.
Label the point.
7/3
Seven jumps of one third land one third past 2.
Compare the counts.
2 jumps and 2 jumps
Both runners have made the same number of jumps.
Find the first jump size.
1/3 of the path
The first path is cut into three jumps.
Find the second jump size.
1/5 of the path
The second path is cut into five jumps.
Compare the jump sizes.
1/3 > 1/5
Fewer jumps in the same path are longer.
Compare the distances.
2/3 > 2/5
Two long jumps go further than two short ones.
Check with the half landmark.
2/3 > 1/2 > 2/5
Double 2 is 4: more than 3, less than 5.
State the answer.
the runner at 2/3
That runner is further from the start.
Find the jump size.
1/6
Six equal jumps fill the gap from 0 to 1.
Count the jumps.
5 jumps
The top number asks for five.
Label the point.
Check with the half landmark.
The line from 0 to 1 is cut into 5 equal jumps. Mark 2/5 on it.
0 |——————————| 1
Mark the position with a cross, then write the value:
A line from 0 to 1 is cut into 9 equal jumps. Complete the steps that place 8/9.
Find the jump size.
1/size
The number of equal jumps from 0 to 1 sets the jump size.
Count the jumps from 0.
jumps jumps
The top number counts the jumps.
Label the landing point.
jumps made on top, jump size below
The landing point is the fraction.
The line runs from 0 to 2 in jumps of 1/3. Mark 4/3 on it. It is bigger than 1.
0 |——————————| 2
Mark the position with a cross, then write the value:
Without drawing anything: is 2/8 to the left or the right of the middle of the line from 0 to 1?
The line from 0 to 1 is cut into 4 equal parts. Counting from 0, which fraction sits at mark number 1? Write it as {{n}}/{{m}}.
n/m
The line from 0 to 2 is cut into jumps of 1/7. Mark 7/7 on it.
0 |——————————| 2
Mark the position with a cross, then write the value:
A line from 0 to 1 has 7 equal jumps. Starting at 0, make 3 jumps. Mark where you land.
0 |——————————| 1
Mark the position with a cross, then write the value:
A one-mile fun run has water stations every 1/3 mile. A runner has just passed station 1. Mark how far along the mile the runner is.
0 |——————————| 1
Mark the position with a cross, then write the value:
Two runners are on the same path. One is 2/3 of the way along; the other is 2/4 of the way. Who is further along?
Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.
A two-mile run has water stations every 1/3 mile. One station is 5/3 miles from the start. Mark it on the line from 0 to 2.
0 |——————————| 2
Mark the position with a cross, then write the value:
You can put any fraction on a number line and say roughly where it sits without drawing one. Without looking: where does 7/4 go, and which is further along the path, 3/5 or 3/8?
16. Your turn: mark 5/6, step 3
5/6
Five jumps of one sixth land one jump short of 1.
16. Your turn: mark 5/6, step 4
5 × 2 = 10 > 6
So 5/6 sits right of the middle.