# Counting Spaces, Not Ticks

> **How do you find 3/4 on a number line?** Split the line from 0 to 1 into 4 equal spaces, then count 3 spaces from 0. The tick at the end of the third space is 3/4. Count spaces, not tick marks: 4 equal spaces need 5 tick marks.

- Big idea: The denominator tells how many equal spaces split 0 to 1. The numerator counts the spaces you hop from 0, not the tick marks.
- Common confusion: A fraction’s place on a number line is found by counting tick marks, so the tick at 0 gets counted as “one”, or the number of ticks is taken as the denominator.
- It often sounds like this: “There are 5 marks between 0 and 1, so these must be fifths.”
- Family: Fractions Have Size (slice 9 of 20)
- Try it yourself (about 3 minutes, free): https://slices.org/slices/counting-spaces-not-ticks

## In plain words

On a number line, the whole is the distance from 0 to 1. The denominator tells how many equal spaces to split it into: for fourths, 4 equal spaces.

Four equal spaces need five tick marks: one at 0, one at 1 and three in between. There is always one more tick than there are spaces, which is why counting ticks can mislead.

The numerator counts spaces from 0. To find 3/4, start at 0 and make three hops of one fourth each. You land on the tick at the end of the third space.

The tick at 0 is where you start, not your first hop. Counting it as “one” lands you on 2/4 instead of 3/4. Taking the number of ticks as the denominator turns fourths into fifths.

This works on any line. On a line from 0 to 2 marked in thirds, 2/3 is still two spaces from 0 and comes before 1, because the whole is still the distance from 0 to 1.

## A worked example

Picture a one-mile walk with a marker post at the start, at the finish and at every quarter mile. There are 5 posts but only 4 equal stretches of path. After walking 3 stretches you have gone 3/4 of a mile, and you are standing at the fourth post, not the third.

## Why this confusion is common

Children learn to count objects one at a time, and tick marks look like objects to count. But a fraction on a number line is a distance from 0, and distances are made of spaces. Because there is always one more tick than there are spaces, counting ticks gives answers that are one off, and the total number of ticks can be mistaken for the denominator, so that five ticks get read as fifths.

## Try this away from the screen

Lay a strip of masking tape on the floor as a number line from 0 to 1 and mark it into 4 equal spaces. Stand on 0 and step one space at a time, saying “one fourth, two fourths, three fourths”. Then ask: how many steps did you take, and how many marks have you stood on, counting the one at the start?

## Helpful ways to talk about it

- Instead of “Count the lines to find the fraction.”, try: “How many equal spaces are there from 0 to 1? How many spaces from 0 is this point?”
- Instead of “Start at 0 and count: one, two, three.”, try: “Put your finger on 0 and hop: one space, two spaces, three spaces. Where did you land?”

## Practice questions

1. Place 2/3 on this number line. **Answer:** place the point at 2/3. The line goes to 2, but the whole is still 0 to 1, split into 3 equal spaces. Two hops of 1/3 from 0 land on 2/3, before 1.
2. A line from 0 to 1 has 7 equally spaced ticks, including the ones at 0 and 1. What number is at the first tick after 0? **Answer:** 1/6. 7 tick marks from 0 to 1 make 6 equal spaces, because there is always one more tick than spaces. Each space is 1/6, so the first tick after 0 is 1/6.
3. What number is at point P? **Answer:** 3/5. Count the spaces, not the ticks: 0 to 1 is split into 5 equal spaces, so each is 1/5. P is 3 spaces from 0, so P is 3/5.
4. On a ruler, the distance from 0 to 1 is split into 8 equal spaces. How many tick marks is that, counting 0 and 1? **Answer:** 9 tick marks. There is a tick at 0 and one at the end of every space. 8 spaces end at 8 ticks, and the tick at 0 makes 9. There is always one more tick than spaces.

## Questions people ask

### Why does a number line split into fourths have 5 tick marks?

Because there is a tick at each end. Four equal spaces sit between five ticks: at 0, 1/4, 2/4, 3/4 and 1.

### How do you find the denominator on a number line?

Count the equal spaces between 0 and 1, not the tick marks. If a line from 0 to 1 has 6 equally spaced ticks, counting both ends, it has 5 spaces, so each space is 1/5.

### Where is 2/3 on a number line that goes from 0 to 2?

Split each whole into 3 equal spaces and count 2 spaces from 0. 2/3 comes before 1. A longer line does not change the whole, which is still the distance from 0 to 1.

## Curriculum connections (US Common Core)

- 3.NF.A.2.a: Represent a fraction 1/b on a number line diagram by defining the interval from 0 to 1 as the whole and partitioning it into b equal parts. Recognize that each part has size 1/b and that the endpoint of the part based at 0 locates the number 1/b on the number line.
- 3.NF.A.2.b: Represent a fraction a/b on a number line diagram by marking off a lengths 1/b from 0. Recognize that the resulting interval has size a/b and that its endpoint locates the number a/b on the number line.

## Related slices

- [The Bottom Number](https://slices.org/slices/the-bottom-number): What the bottom number of a fraction counts: all the equal parts that make one whole.
- [One Number, Not Two](https://slices.org/slices/one-number-not-two): Why 2/5 is one number between 0 and 1, not a 2 and a 5.
- [More Than One Whole](https://slices.org/slices/more-than-one-whole): Why 6/4 makes sense: a fraction can equal one whole or go beyond it.
- [Compare Without Guessing](https://slices.org/slices/compare-without-guessing): How to compare fractions like 3/8 and 4/7 by asking which side of one half each one is on.