WebLab STEM

Mathematics · 1. Number, ratio and numerical reasoning

Numerical foundations for problem solving

Build number sense, connect fractions–ratios–percentages and use factorisation instead of guesswork.

40 minutesContent version: 2.1
Start the interactive pathwayOpen the virtual experiment

Driving question

How can we recognise a numerical structure before choosing an operation?

Curriculum coverage

Vietnam Mathematics 6–9; Cambridge IGCSE Number.

Model scope and limitations

A calculator supports arithmetic but does not replace model choice, sign checks or order-of-magnitude estimates.

Learning objectives

  • Use prime factors to find HCF and LCM.
  • Convert accurately among fractions, decimals and percentages.
  • Solve direct/inverse proportion and estimate plausibly.

Prerequisite knowledge

  • Four operations and order of operations.

Core knowledge

Prime factors

Every integer above 1 has a unique prime factorisation apart from order.

Ratio

A ratio compares like quantities; convert to common units before simplifying.

Percentage change

% change = (new − original)/original × 100%; the denominator is the original value.

Modular arithmetic & Chinese Remainder Theorem

a ≡ b (mod m) iff m divides a − b. Fermat’s Little Theorem: a^(p−1) ≡ 1 (mod p). Chinese Remainder Theorem solves systems of congruences modulo M = ∏ m_i, foundational for RSA cryptography.

Eulerian, Hamiltonian cycles & Planar graphs

1. Eulerian cycles exist iff all vertex degrees are even; paths exist iff exactly two vertices are odd. 2. Hamiltonian cycles visit each vertex once (Dirac’s theorem: deg(v) ≥ n/2). 3. Planar graphs obey Euler’s formula V − E + F = 2 and bound E ≤ 3V − 6.

Worked example

84 = 2²×3×7 and 126 = 2×3²×7. HCF uses lower powers: 42; LCM uses higher powers: 252.

Misconceptions to avoid

  • A 20% rise followed by a 20% fall does not return to the start because the bases differ.
  • 1 is not prime because it has only one positive divisor.
  • An Eulerian path does not imply an Eulerian cycle; cycles strictly require ALL vertices to have even degrees.

Virtual experiment procedure: Number theory and factors

  1. Predict magnitude and sign before calculating.
  2. Represent the problem with a ratio diagram, table or factorisation.
  3. Calculate exactly, then verify using estimation.

Safety and cautions

  • Avoid intermediate rounding unless required; distinguish exact and approximate values.

Evidence to collect

  • A solution showing model, calculation, units and a plausibility check.
Open the virtual experiment
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<iframe src="https://visuallab.win/math-lab/?module=number_theory_k6" width="100%" height="560" frameborder="0" allow="fullscreen" loading="lazy"></iframe>

Diagnostic check

1. What is the HCF of 18 and 24?

  1. 6
  2. 72
  3. 3
Show answer and explanation

Correct answer: 6

18 = 2×3² and 24 = 2³×3, so the common part is 2×3 = 6.

2. 0.375 equals which simplest fraction?

  1. 3/8
  2. 37/5
  3. 3/5
Show answer and explanation

Correct answer: 3/8

0.375 = 375/1000 = 3/8.

3. When does a connected undirected graph have an Euler cycle?

  1. Every vertex has even degree
  2. Exactly two odd vertices
  3. No cycles exist
Show answer and explanation

Correct answer: Every vertex has even degree

By Euler-Hierholzer theorem, an Euler cycle exists iff every vertex has an even degree.

Application practice

1. A price of 200 rises by 15% to?

  1. 230
  2. 215
  3. 170
Show answer and explanation

Correct answer: 230

200 × 1.15 = 230.

2. A 2:3 ratio totals 40. The smaller part is?

  1. 16
  2. 20
  3. 24
Show answer and explanation

Correct answer: 16

There are 5 parts of 8; the smaller is 2×8 = 16.

3. What is the LCM of 12 and 18?

  1. 36
  2. 6
  3. 216
Show answer and explanation

Correct answer: 36

12 = 2²×3 and 18 = 2×3², so LCM = 2²×3² = 36.

4. By Fermat’s Little Theorem, what is the remainder of 2^6 divided by 7?

  1. 1
  2. 2
  3. 4
Show answer and explanation

Correct answer: 1

Since 7 is prime and gcd(2,7)=1, 2^(7−1) = 2^6 ≡ 1 (mod 7).

5. A convex polyhedron has 8 vertices and 12 edges. By Euler’s formula, how many faces does it have?

  1. 6 faces
  2. 8 faces
  3. 10 faces
Show answer and explanation

Correct answer: 6 faces

Euler’s formula V − E + F = 2 ⇔ 8 − 12 + F = 2 ⇔ F = 6 faces (e.g. cube).

Academic references

  1. Cambridge IGCSE Mathematics 0580 — official syllabus and resources
  2. Cambridge AS & A Level Mathematics 9709 — official syllabus
  3. OpenStax Precalculus 2e — functions, geometry and probability

This lesson is maintained against the cited sources below. Simulations are learning models and do not replace supervised physical-laboratory safety procedures.

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WebLab STEM
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Đức Tiến — Control & Automation Engineer