Five Wheels-and-Handlebars Cards: Which Rule Counts?
Ms. Okafor's group is working on classification cards. She sets out five picture cards: a bicycle with two wheels, a car with four wheels, a tricycle with three wheels, a wagon with four wheels and a long pulling handle at the front, and a scooter with two wheels. The task sheet says: 'One picture does not belong. Cross it out and tell why.' Sam crosses out the car and writes, 'It is the only one that costs a lot of money.' Elena crosses out the tricycle and writes, 'It is the only one with an odd number of wheels.' Ms. Okafor asks the group to judge the two answers using only what can be seen in the pictures - nothing that requires knowing prices, engines, or how the machines are used - and then to test what happens when a sixth card, a unicycle with one wheel, is added to the set.
1. Judge Sam's rule and Elena's rule by the standard Ms. Okafor set: it must use only what the picture shows.
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Elena's rule passes; Sam's rule fails the standard.
Sam's reason depends on price, and price is not drawn anywhere on a picture card. Two people looking at the same card cannot see 'expensive.' A rule like that cannot be checked, so it cannot be the intended answer, no matter how true it happens to be in real life. Picture-classification questions are settled by visible features - shape, count, color, parts, position.
Elena's rule is countable: two, four, three, four, two. Exactly one card shows an odd number, so it isolates the tricycle and leaves a group of four that genuinely share a feature. That is the shape of a strong classification rule: count it on every card, and get exactly four on one side and one on the other.
2. The task sheet is reprinted with the wagon as the intended odd card. Write a rule that uses only visible features and isolates the wagon, and explain why your rule works.
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One workable rule: 'It is the only one with a long pulling handle at the front.'
Check each card against the pictures: the bicycle, tricycle, and scooter have short handlebars a rider grips while riding; the car has a steering wheel inside; the wagon has a long handle that sticks out at the front for someone else to pull. That handle is drawn on the card, so the rule can be checked by looking.
The lesson is that the odd card is not chosen first and justified afterward. The rule comes first, and the odd card is whatever the rule leaves alone. That is also why a group can produce several correct-looking answers to the same card set - and why they should say which rule they are using. Without naming the rule, 'the wagon' and 'the tricycle' are just two opinions.
3. A unicycle card (one wheel) is added. Does Elena's odd-number-of-wheels rule still isolate a single card? What does that show about classification rules?
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No - the rule now isolates two cards.
The wheels become 2, 4, 3, 4, 2, 1. The tricycle (3) and the unicycle (1) are both odd, so nothing is singled out, and the rule stops working even though not one word of it changed.
This is the point of the sixth card. A classification rule is only as good as the set it is applied to. Adding one card can destroy a rule that looked airtight, and the solver has to notice that and switch to a rule that still splits the set four-to-one - for example, 'the only one with pedals,' which would leave the bicycle and the tricycle, and therefore also fail. Notice that the failure is informative: it tells you the original rule depended on the set, not on the tricycle itself.