Task 14
Fractions on a Number Line
Use rods to build equal intervals, then withdraw the rods and reason about fraction magnitude between 0 and 1.
Mathematical focus
A fraction on a number line represents a distance from zero, not merely a tick mark.
The interval from 0 to 1 represents one whole. To show fifths, thirds or quarters, that whole must be divided into the corresponding number of equal intervals.
The rods help pupils build and see those equal intervals before the rods are withdrawn and the number line becomes abstract.
I do — Build fifths
Place an orange rod on a Show Me board. Draw a line exactly the same length, remove the rod, and label the ends 0 and 1.
For this activity, orange represents one whole.
Build a train of red rods along the line. Five reds are equivalent to orange, so one red represents one-fifth of the whole.
Mark the end of each red rod, remove the rods, and label the number line:
0 ⅕ ⅖ ⅗ ⅘ 1
Trace from zero to one of the fractions. Emphasise that ⅗ is the whole distance from 0 to ⅗, not the little mark itself.
We do — Change the whole
Now choose a blue rod to represent one whole. Draw a new 0–1 line the length of blue.
Ask: Which rod can help us divide this whole into thirds?
Three light-green rods are equivalent to blue. Use them to make three equal intervals, then remove the rods and label:
0 ⅓ ⅔ 1
Teacher prompt: Is one-third always the same physical length? No. Its size depends on the size of the whole.
Your turn — Build the number lines
Create each number line on your Show Me board. Label the ends 0 and 1.
Use rods to help you make equal intervals, then remove the rods and label the fractions.
- Halves: choose a suitable rod to represent the whole.
- Quarters: use tan as the whole.
- Sixths: use dark green as the whole.
For each line, be ready to explain how you know the intervals are equal.
Show answers
- Halves: any whole that can be split into two equal rod lengths works.
- Quarters with tan: red rods partition tan into four equal intervals.
- Sixths with dark green: white rods partition dark green into six equal intervals.
Feedback / teacher prompts
- How did the rods guarantee equal intervals?
- What changed when the whole changed?
- What stayed mathematically the same?
True or False? — The tick-mark trap
A number line runs from 0 to 1. There are four tick marks between 0 and 1.
True or False: “There are four tick marks, so the line has been divided into quarters.”
Explain your answer. Sketch the line if it helps.
Show answers
False. Four internal tick marks create five equal intervals, so the line shows fifths, not quarters.
The denominator counts equal intervals, not internal marks.
Feedback / teacher prompts
This is an important misconception check. Fractions have magnitude: the marks show where intervals begin and end; they are not themselves the fractional parts.
Your turn — Take the rods away
Draw a fresh number line and label only the endpoints 0 and 1.
Do not use smaller rods to partition the line.
Estimate and mark:
- ½
- ¼
- ¾
- ⅓
- ⅕
Compare with a partner. Which were easiest to place? Which required the most estimation?
Feedback / teacher prompts
- How are pupils using benchmark fractions such as ½?
- Can they reason about relative magnitude without reconstructing every subdivision?
- This is the exit strategy: the rods establish the structure, then the number line stands on its own.
Go further
Ask partners to create an unlabelled fraction number line for one another, or draw several 0–1 lines of different physical lengths and divide each into the same fraction.
Key question: How can one-half, one-third or one-quarter have different physical lengths while still representing the same fraction?
The answer returns us to the central idea: the size of a fraction depends on the whole.
Rehearse the teaching
For a task worth practising, work in pairs. One person teaches while the other responds as a pupil using the rods. Model enough to get the pupil started, then stop explaining and question their thinking. Swap roles.
Reflect: What was harder to explain than you expected? Which wording or question worked well?