MISSION · Number sense
Fractions on a Number Line
A fraction is a place on the line, not just a piece of something
LEVEL 1 · EXPLORE
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LEVEL 2 · LEARN
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The idea in one sentence
Cut the gap from 0 to 1 into equal steps, and a fraction tells you how many steps along to stop — which makes it a number with a place, like 3 or 17.
Why the pizza picture runs out
Fractions usually arrive as parts of a shape: half a pizza, a quarter of a chocolate bar. That picture is fine for 3/4 and quietly breaks for 7/4, because nobody has seven quarters of one pizza. It also makes fractions feel like a different species from whole numbers — something you do to an object rather than something you can count with.
The number line fixes both. On a line, 3/4 and 7/4 and 2 are all just places, and they can be compared, ordered, and added like any other numbers.
Building it
- Mark 0 and 1.
- The denominator says how many equal steps to cut that gap into. Fourths means four.
- The numerator says how many steps to take from 0.
So 3/4 is three steps of size one-quarter. And 7/4 is seven steps of the same size, which runs past 1 and lands three-quarters of the way to 2.
The step itself — 1/4 — is the unit fraction. Every other fraction with that denominator is a whole number of those steps, which is exactly how whole numbers are built from 1.
Equal steps is the whole rule
The most common wrecked number line has four tick marks that aren’t evenly spaced, or has the gap from 0 to 1 divided into four while the gap from 1 to 2 is divided into three. The size of a step is fixed by the denominator, not by how much room is left on the page.
Reading past 1
\[\frac{7}{4} = \frac{4}{4} + \frac{3}{4} = 1\frac{3}{4}\]Walking seven quarter-steps means passing the 1 mark after four of them and continuing three more. That’s what a mixed number records: whole steps completed, then what’s left.
Benchmarks do most of the work
Three landmarks — 0, 1/2, and 1 — settle most comparisons without any calculation.
- 7/8 is one step short of 1, so it’s just under 1.
- 5/4 is one step past 1, so it’s just over.
- 5/12 is a bit under 1/2, because half of 12 is 6.
Which is closer to 1, 7/8 or 5/4? One is 1/8 below, the other is 1/4 above, and eighths are smaller steps than quarters, so 7/8 wins. No common denominators required.
Common mix-ups
- Uneven tick marks, which make the fraction mean nothing.
- Counting tick marks instead of steps, so 1/4 gets placed on the first mark after 0 when the marks are drawn at 0, 1/4, 1/2 — right answer by luck, wrong method.
- Believing a bigger denominator means a bigger number. More cuts means smaller steps, so 1/8 < 1/3.
- Treating 7/4 as an error. An improper fraction is an ordinary number that happens to be more than 1.
Check yourself
Between which whole numbers does 7/4 sit? And which is closer to 1 — 7/8 or 5/4?
7/4 is one whole plus 3/4, so it sits between 1 and 2. And 7/8 is 1/8 away from 1 while 5/4 is 1/4 away, so 7/8 is closer.
LEVEL 3 · REFLECT
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