Task 11
White Isn’t Always One
Change the rod that represents one whole and see how every other value changes while the relationships between the rods remain fixed.
Mathematical focus
Cuisenaire rods do not have fixed numerical values.
The value of a rod depends on which rod has been chosen to represent one whole. Changing the whole changes the numerical value of every other rod, but the physical lengths — and therefore the relationships between the rods — stay the same.
Keep the recording relational. Write w = ½r rather than simply saying “white is a half”. Ask repeatedly: “A half of what?”
We do — Red is one whole
Put out a red rod and define it as one whole:
r = 1
Which rod is half the length of red? White. Two whites are equivalent to one red, so record:
w = ½r
Now use the rods to describe some other lengths in relation to red:
g = 3⁄2r p = 2r y = 5⁄2r d = 3r
Notice that choosing a whole does not restrict us to fractions less than one. Some rods represent proper fractions, some represent the whole, and some represent values greater than one.
We do — Change the whole
Now put out a light green rod and define this as one whole:
g = 1
Nothing about the physical rods has changed, but their numerical values have.
w = ⅓g r = ⅔g p = 4⁄3g y = 5⁄3g d = 2g
Ask: Did any rod change length? Why, then, did its numerical value change?
The whole changed. That is the central idea.
Your turn — Build the table
For each row, the rod named on the left represents one whole.
Predict first. Then use the rods to work out each relationship.
Record every answer relationally: for example, if red is the whole, record white as ½r, not simply ½.
| Whole | w | r | g | p | y | d |
|---|---|---|---|---|---|---|
| r = 1 | ||||||
| g = 1 | ||||||
| p = 1 |
Feedback / teacher prompts
Additional challenge prompt: when pink is the whole, g = ¾p. When light green is the whole, p = 4⁄3g. What changed? What stayed the same?
- What changed when the whole changed? What stayed the same?
- Did any rod actually change length?
- Why can the same rod have different numerical values?
- Which entries are less than one? Which are greater than one?
- What does the denominator tell us about the relationship between the rod and the chosen whole?
- Compare g = ¾p with p = 4⁄3g. What has been reversed?
What could you change?
Choose a different rod to represent one whole; extend the table to black, tan, blue and orange; give pupils two rods and ask them to describe the relationship in both directions; give a fractional relationship and ask pupils to find a possible pair of rods; or ask pupils to predict before using the rods to check.
Teacher note: This task deliberately follows work in which white may have represented 1. The title challenges the idea that any colour has a permanent numerical value.
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?