Task 8
Odds and Evens
Move from physical definitions to conjecture, generalisation and proof.
What makes a rod odd?
Build r + w + r. It is equivalent to yellow. Build g + w + g. It is equivalent to black.
A rod is odd when its length can be represented by two equivalent lengths with one white left over. White itself is also odd.
Find the odd rods: w, g, y, k, b. Build their staircase and notice the common difference of red.
What makes a rod even?
An even length can be represented by two equivalent lengths joined together.
r = w + w p = r + r
Find the even rods: r, p, d, t, o. Their staircase also has a common difference of red.
Model one proof — Even + Even
Take red and pink.
r = w + w
p = r + r
So r + p = w + w + r + r. Rearrange:
(w + r) + (w + r)
We have two equivalent trains, so the result is even. Generalise: any two even lengths can be rearranged into two identical trains.
Your turn — Can you prove it?
Work with a partner. Choose one statement:
- Odd + Even is always Odd
- Odd + Odd is always Even
Then:
- Test the statement using rods.
- Find another example.
- Rearrange the rods to show why it works.
- Record your argument on your Show Me board.
- Explain why it must work for every possible example, not just the ones you tested.
Be ready to convince the rest of us.
Show answers
Odd + Even: odd is two equivalent parts plus one white; even is two equivalent parts. Together they can be rearranged into two equivalent trains plus one white, so the result is odd.
Odd + Odd: each odd length is two equivalent parts plus one white. Together there are two of every part and two whites, which can be rearranged into two identical trains, so the result is even.
Feedback / teacher prompts
- At what point did testing examples stop being enough?
- What makes the argument general?
- What is the difference between an example and a proof?
What could you change?
Ask pupils to create a parity conjecture of their own, test it, and then try to prove or disprove it using 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?