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Chapter 3 · Period 1 of 17 · 50 min

Counting Before Symbols: Natural Numbers and One-to-One Correspondence

The Dawn of Mathematics: The Human Need to Count · Inductive

This unit opens the chapter at its entry point — the practical human need to count — so the herder-and-pebble scenario is the conceptual anchor, not an illustration. Watch for students who conflate 'one-to-one correspondence' with 'counting aloud'; press them to say what correspondence means when no symbols are available. Exercise Set 3.1 Q1, p.43 is straightforward, but some students will set up the ratio inverted — monitor that step.

0-10Open with the herder-and-pebble scenario: each student pairs one pebble to one finger to model one-to-one correspondence, then the class pools observations to define N = {1, 2, 3, 4, …} and articulate why counting is the earliest mathematics.
10-25Students work in pairs on Exercise Set 3.1 Q1, p.43 — the Lothal merchant proportion — writing their proportional reasoning explicitly. Share two contrasting approaches on the board and discuss why both reach the same answer.
25-40Class discussion on Exercise Set 3.1 Q3, p.43: students produce examples to test closure of N under subtraction, then state and justify their conclusion. Record on board: N is closed under addition but not subtraction.
40-50Quick individual reflection: students write one sentence explaining what one-to-one correspondence achieves that mere labelling does not, and share with a neighbour before the teacher draws out the key idea.

Homework

Exercise Set 3.1 Q2, p.43 — identify what the Ishango bone column 11, 13, 17, 19 has in common and list the next three numbers fitting the pattern.

Assessment

Competency (C-1.1). Understands and applies the successive construction of the real number system from natural numbers through integers, rationals, irrationals to reals.

Learning outcome. Students can describe the role of one-to-one correspondence in defining the set of natural numbers and verify closure properties of N under addition and subtraction with justifying examples.

A herder uses pebbles to count his cows by one-to-one correspondence. He starts with 12 pebbles (one per cow) and the herd grows by 5 more cows. He then sells 15 cows. Which of the following correctly describes what happens when the herder tries to use natural numbers to record the final count?

  • A He can record the intermediate steps (12 and 17) using natural numbers because addition stays in N, but the final step 17 – 15 = 2 also stays in N; however, if he had sold all 17 cows instead, the subtraction 17 – 17 = 0 would leave N.
  • B He cannot record the result using a natural number, because 12 + 5 – 15 = 2 but natural numbers are closed under addition yet not under subtraction when the result would be zero or negative.
  • C He needs 2 pebbles and the system works because all natural numbers are closed under every arithmetic operation.
  • D He needs 2 pebbles, because 12 + 5 – 15 = 2, and natural numbers are closed under both addition and subtraction.

A. He can record the intermediate steps (12 and 17) using natural numbers because addition stays in N, but the final step 17 – 15 = 2 also stays in N; however, if he had sold all 17 cows instead, the subtraction 17 – 17 = 0 would leave N.

What each wrong option reveals

  • B Correctly identifies that N is not closed under subtraction when the result would be zero or negative, but misstates that 2 is not a natural number — 2 is in N; the closure failure only arises when the result is 0 or negative.
  • C Claims all four operations close N, ignoring that subtraction and division can exit N.
  • D Correctly computes 12 + 5 – 15 = 2 but falsely claims N is closed under subtraction in all cases, ignoring that subtraction can exit N.
Chapter 3 · 17 periods of 50 min

Also saved as 10 periods or 14 periods.

  • 2Bones, Tally Marks and the Earliest Number Records
    A History Written in Bone
  • 3Powers of Ten and the Indian Number Heritage
    The Indian Context: Trade and Astronomy
  • 4Zero as a Number: Brahmagupta's Rules and the Bakhśhālī Manuscript
    Zero as a Number
  • 5Śhūnyatā to Mathematics: The Philosophy Behind Zero
    From Philosophy to Mathematics: The Concept of Śhūnyatā
  • 6Integers: Debts, Fortunes and the Number Line
    Integers: Expanding the Horizon
  • 7Brahmagupta's Laws for Integer Arithmetic
    The Arithmetic of Integers
  • 8Rational Numbers: Definition, Equality and the Four Operations
    Filling the Spaces: Fractions and Rational Numbers
  • 9Distributivity and Rational Equations
    Filling the Spaces: Fractions and Rational Numbers
  • 10Placing Rational Numbers on the Number Line and Absolute Value
    Representation of Rational Numbers on the Number Line
  • 11Density of the Rationals and the Question of Completeness
    The Density of Rational Numbers
  • 12Irrational Numbers and the Proof That √2 Cannot Be a Fraction
    Irrational Numbers and the Proof That √2 Cannot Be a…
  • 13Constructing √2 with Ruler and Compass and Generalising to √n
    Construction of Length √2
  • 14π, Mādhava's Infinite Series and the Irrationality of π
    The Story of Pi (π) and Mādhava's Infinite Series
  • 15Real Numbers, Decimal Expansions and Converting Decimals to Fractions
    Real Numbers, Decimal Expansions and Converting…
  • 16Cyclic Numbers, Irrational Decimals and the Structure of the Real Line
    Cyclic Numbers, Irrational Decimals and the Structure…
  • 17The Number System as a Whole: Connections, Patterns and Open Questions
    Synthesis
Ganita Manjari · Class 9 · 14 chapters

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