Imagine you are living thousands of years ago in a small agricultural settlement along the banks of the Saraswati river. You have a herd of cattle. Every morning, they go out into the dense forests to graze, and every evening, they return. How do you ensure that a calf has not wandered off?
Step-by-step solution
Idea: You do not need number words to compare two collections. If every object in one collection can be paired with exactly one object in the other, the two collections have the same size. This pairing is where the natural numbers 1, 2, 3, … come from.
- In the morning, as each animal walks out, drop one pebble into a clay pot. The pot now holds exactly as many pebbles as there are animals out grazing.½ mark
- In the evening, as each animal comes back, take one pebble out of the pot. Each returning animal is matched with one pebble.½ mark
- If the pot is empty at the end, every pebble found its animal: no calf is missing.½ mark
- If some pebbles are left, each leftover pebble stands for an animal that has not returned, so you know how many to go and look for. Matching objects one to one like this is called one-to-one correspondence.½ mark
Check: Say 5 animals leave: 5 pebbles go in. If only 4 return, 4 pebbles come out and 1 is left, showing exactly one animal is missing ✓.
Answer to write in the exam
Morning: one pebble put in a pot for each animal that goes out.
Evening: one pebble taken out for each animal that returns.
Pot empty ⇒ no animal missing.
Pebbles left ⇒ number of pebbles left = number of animals missing.
∴ The herd is checked by one-to-one correspondence (one pebble ↔ one animal).
Common mistakes that cost marks
- Thinking you must know number words or symbols to keep count. Pairing objects one to one works without them.
- Removing pebbles in a different way from the way they were added (for example one pebble per pair of animals). The matching must be one pebble for one animal, both morning and evening.
- Saying ‘count the cattle’ without explaining how; the question wants the method of matching.
How this can come in the exam
A shepherd ties one knot in a rope for every sheep that goes out and unties one knot for every sheep that comes back. At night 3 knots are still tied. This means
- 3 sheep went out
- 3 sheep are missing
- 3 sheep came back
- no sheep is missing
Show answer
(B) 3 sheep are missing
Each untied knot was matched with a returning sheep. The 3 knots left have no sheep to match, so 3 sheep have not returned.
What is meant by one-to-one correspondence? Give one everyday example.
Show answer
Two collections are in one-to-one correspondence when every object of one is paired with exactly one object of the other, with nothing left over on either side (1 mark). Example: in a full class each student sits on exactly one chair and no chair is empty, so the number of students equals the number of chairs, even without counting (1 mark).Try one yourself
A trader drops one bead into a bowl for every bag he loads on a cart. At the market he removes one bead for every bag he unloads. After unloading, 2 beads remain. What does this tell him?
Show answer
Two bags were loaded but not unloaded: either 2 bags are still on the cart or 2 bags were lost on the way.
More questions like this
- A merchant in the port city of Lothal is exchanging bags of spices for copper ingots. He receives 15 ingots for every 2 bags of spices. If he brings 12 bags of spices to the market, how many copper ingots will he leave with?
- Look at the sequence of numbers on one column of the Ishango bone: 11, 13, 17, 19. What do these numbers have in common? List the next three numbers that fit this pattern.
- We know that Natural Numbers are closed under addition (the sum of any two natural numbers is always a natural number). Are they closed under subtraction? Provide a couple of examples to justify your answer.
- Ancient Indians used the joints of their fingers to count, a practice still seen today. Each finger has 3 joints, and the thumb is used to count them. How many can you count on one hand? How does this relate to the ancient base-12 counting systems?
- Brahmagupta did not stop at zero. He realised that if subtraction of a number from itself can result in zero (5 − 5 = 0), then what would happen if we subtracted a larger number from a smaller one (3 − 5 = □)?