Task 9
Representing Fractions
Choose a whole and describe every other rod through its relationship to that whole.
We do — Choose the whole
Let the light green rod represent one whole.
White is one-third of light green because three whites make the whole:
w = ⅓g
Red is two-thirds of light green:
r = ⅔g
Continue beyond one whole. Yellow is five-thirds of light green:
y = ⁵⁄₃g
Key idea: we still have not assigned fixed number values to the rods. We choose one rod as the whole and describe the others in relation to it.
We do together — Change the whole
Now let pink represent one whole.
w = ¼p
Build five quarters:
y = ⁵⁄₄p
Your turn — Build and record
For each question, identify the whole, build the relationship, identify the required rod and record it on your Show Me board.
Which single rod is:
- one-half of dark green?
- two-thirds of dark green?
- three-quarters of tan?
- five-quarters of pink?
- seven-fifths of yellow?
- four-thirds of dark green?
- three-halves of pink?
- five-thirds of light green?
Show answers
- g = ½d
- p = ⅔d
- d = ¾t
- y = ⁵⁄₄p
- k = ⁷⁄₅y
- t = ⁴⁄₃d
- d = ³⁄₂p
- y = ⁵⁄₃g
Feedback / teacher prompts
- What does the denominator tell us to look for?
- What does the numerator tell us?
- What changes when the answer is greater than the whole?
Your turn — Describe every rod
Take the yellow rod to represent one whole.
Express every other rod as a fraction of yellow:
- in words;
- using colour letters and symbols.
Organise your results on your Show Me board.
Show answers
Feedback / teacher prompts
- What happens once we move beyond yellow?
- What do you notice about yellow and orange?
Your turn — Find them all
Using only two rods, find all the possible ways to represent:
- ⅓
- ⅔
- ¾
- ³⁄₂
Write an equation for every relationship you find.
Can you convince somebody that you have found them all?
Show answers
⅓: w = ⅓g; r = ⅓d; g = ⅓b
⅔: r = ⅔g; p = ⅔d; d = ⅔b
¾: g = ¾p; d = ¾t
³⁄₂: g = ³⁄₂r; d = ³⁄₂p; b = ³⁄₂d
Feedback / teacher prompts
- How did you search systematically?
- What changes in each representation? What stays the same?
- Why can the same fraction be represented by different pairs of rods?
What could you change?
Change the whole, specify a fraction and ask for every possible pair, give two rods and ask for their fractional relationship, include fractions greater than one, ask pupils to predict first, or create true/false fraction statements for a partner.
Exit strategy: use the rods to establish conceptual meaning, then connect deliberately to the formal notation and methods pupils will eventually use without the rods.
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?