Task 3
Equivalence
Different-looking constructions can have exactly the same length.
We do
Build r + y. Which single rod has the same length? Black.
Although the constructions look different, their lengths are the same. Introduce the word equivalent and record:
r + y = k
Repeat with r + g = y and w + p = y.
Your turn — True or False?
Build each statement. Decide whether it is true or false.
If it is false, change one part of the statement to make it true.
- r + y = k
- r + p = k
- g + p = k
- w + y = k
- g + y = t
- p + y = b
- w + r + p = k
- r + r + g = k
Show answers
- True
- False — r + p = d
- True
- False — w + y = d
- True
- True
- True
- True
Feedback / teacher prompts
- Which statements could you decide without physically checking?
- How did the rods help you correct false statements?
- Did anybody correct a false statement differently?
Finding equivalent trains
Show a blue rod. Use red and light green; which other rod completes an equivalent train?
r + g + p = b
How many different arrangements of red, light green and pink are equivalent to blue? Build them and ask pupils to convince you they have found them all.
Teacher note: This makes the commutative property of addition physically visible.
Your turn — How many ways?
Make a train using exactly two rods which is equivalent in length to:
- the tan rod
- the blue rod
Find as many different answers as you can. Record each as an equation.
Can you organise your answers so that you know none are missing?
Feedback / teacher prompts
- Does reversing the order create something mathematically new?
- How did you search systematically?
Your turn — Three-rod trains
Make a train of exactly three rods equivalent to:
- orange
- black
- dark green
Find more than one solution where possible. Record each construction using colour letters.
Choose one target and try to find as many different three-rod trains as you can.
Your turn — Build a pattern
Choose a target rod and find all the trains that are equivalent to it.
Organise the trains so that somebody else can understand how you searched.
Can you convince your partner that you have found them all?
Feedback / teacher prompts
- What counts as a genuinely different train?
- What changes if you require exactly two or three rods?
- What changes if repeated colours are allowed or forbidden?
What could you change?
Change the target rod, specify the number of rods, restrict colours, require or forbid repeats, give a false equation to correct, or ask pupils to find and justify all possible solutions.
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?