IB Math Revision Start free
Themed maths · 23 November

Fibonacci Day maths for IB Maths SL and HL

23 November is Fibonacci Day: written month first, 11/23 reads 1, 1, 2, 3. This ready-to-teach lesson has a 5-minute starter, a 35-minute main activity on the golden ratio, golden rectangles and Fibonacci identities, an HL extension on induction and Binet’s formula, and full worked answers.

Level
IB Maths SL and HL (AA and AI)
Time
40 minutes, plus a 10-minute extension
Topics
Sequences and recursive definitions; The golden ratio and quadratic equations; The limit of a sequence of ratios; Sigma notation and identities; Proof by induction and Binet’s formula (HL)
Equipment
The starter is non-calculator. A GDC helps in tasks A and B.

Download student sheet (PDF)Answers (PDF)

Suggested timings

PartTimeWhat
Starter5 minQuick questions on the board
Main: task A11 minThe limit of the ratios
Main: task B12 minGolden rectangles and the spiral
Main: task C12 minFibonacci identities
Extension10 minFast finishers or homework

Starter (5 minutes)

No GDC. Throughout, F1 = F2 = 1 and Fn+2 = Fn+1 + Fn.

  1. Write down F3 to F10.
  2. Solve x2 = x + 1, giving exact answers.
  3. Work out F1 + F2 + … + F6. Compare your answer with F8.
  4. Work out 21 ÷ 13 to 3 decimal places.

Main activity (35 minutes)

Task A: The limit of the ratios (11 min)

Let rn = Fn+1 ÷ Fn.

  1. Find r1, r2, …, r8, to 3 decimal places where needed. Describe what happens.
  2. Divide Fn+2 = Fn+1 + Fn by Fn+1 to show that rn+1 = 1 + 1/rn. Assuming rn tends to a limit φ, find φ exactly and to 3 decimal places.
  3. Use φ2 = φ + 1 to write φ3 and φ4 in the form aφ + b. What do you notice about the numbers?

Task B: Golden rectangles and the spiral (12 min)

A golden rectangle has its sides in the ratio φ : 1, where φ = (1 + √5)/2.

  1. A 1 by 1 square is cut from a golden rectangle measuring φ by 1, leaving a 1 by (φ − 1) rectangle. Show that this smaller rectangle is also golden.
  2. A picture frame is a golden rectangle with shorter side 30 cm. Find the longer side to 3 significant figures.
  3. Squares with sides 1, 1, 2, 3, 5, 8 and 13 cm fit together to make one rectangle. Find the total area of the squares, and show it equals the area of a 13 cm by 21 cm rectangle.
  4. A quarter circle is drawn in each square of part (c), with radius equal to the square’s side, to make a spiral. Find the length of the spiral, exactly and to 3 significant figures.

Task C: Fibonacci identities (12 min)

Spot each pattern, check it, then use it.

  1. Find F1 + F2 + … + F10 and show it equals F12 − 1.
  2. Work out Fn−1Fn+1 − Fn2 for n = 2, 3, 4, 5 and 6. Suggest a general result.
  3. Every third Fibonacci number is even. How many of F1, F2, …, F30 are even?

Extension (10 minutes)

HL: proof by induction and Binet’s formula. Let ψ = (1 − √5)/2.

  1. Binet’s formula says Fn = (φn − ψn)/√5. Check that it gives F1 = 1 and F2 = 1, then use it (with a GDC) to find F20.
  2. Prove by induction that F1 + F2 + … + Fn = Fn+2 − 1 for every positive integer n.

For teachers

Teacher notes and full worked answers
  • Fibonacci Day is 23 November because 11/23, written month first, reads 1, 1, 2, 3.
  • In task A (b), insist on the reason for rejecting (1 − √5)/2: every ratio is positive.
  • Task B (d) makes a good display: students draw the squares on squared paper and add the arcs with compasses.
  • Extension 1 explains task A: |ψ| < 1, so ψn → 0 and Fn+1/Fn → φ.

Starter

  1. 2, 3, 5, 8, 13, 21, 34, 55
    • Add the two previous terms each time: 1 + 1 = 2, 1 + 2 = 3, 2 + 3 = 5, and so on up to 21 + 34 = 55.
  2. x = (1 + √5)/2 or (1 − √5)/2
    • x2 − x − 1 = 0, so x = (1 ± √(1 + 4))/2 = (1 ± √5)/2.
  3. 20, which is F8 − 1
    • 1 + 1 + 2 + 3 + 5 + 8 = 20
    • F8 = 21, so the sum is one less than F8.
  4. 1.615
    • 21 ÷ 13 = 1.6153…

Task A: The limit of the ratios

  1. 1, 2, 1.5, 1.667, 1.6, 1.625, 1.615, 1.619: the ratios settle down, alternately above and below about 1.618
    • 1/1, 2/1, 3/2, 5/3, 8/5, 13/8, 21/13, 34/21
    • r8 = 34 ÷ 21 = 1.619 (3 d.p.)
  2. φ = (1 + √5)/2 ≈ 1.618
    • Fn+2/Fn+1 = 1 + Fn/Fn+1, that is rn+1 = 1 + 1/rn.
    • In the limit φ = 1 + 1/φ, so φ2 = φ + 1 and φ = (1 ± √5)/2.
    • Every ratio is positive, so φ = (1 + √5)/2 = 1.618 (3 d.p.).
  3. φ3 = 2φ + 1, φ4 = 3φ + 2: Fibonacci numbers again
    • φ3 = φ × φ2 = φ2 + φ = (φ + 1) + φ = 2φ + 1.
    • φ4 = φ × φ3 = 2φ2 + φ = 2(φ + 1) + φ = 3φ + 2.
    • In general φn = Fnφ + Fn−1.

Task B: Golden rectangles and the spiral

  1. 1 ÷ (φ − 1) = φ, because φ(φ − 1) = 1
    • φ(φ − 1) = φ2 − φ = (φ + 1) − φ = 1.
    • So 1 ÷ (φ − 1) = φ: the long side is φ times the short side.
  2. 48.5 cm
    • 30φ = 30 × 1.6180… = 48.54…
    • 48.5 cm (3 s.f.)
  3. 273 cm2 = 13 × 21
    • 1 + 1 + 4 + 9 + 25 + 64 + 169 = 273
    • 13 × 21 = 273. In general F12 + … + Fn2 = FnFn+1.
  4. 33π/2 ≈ 51.8 cm
    • Each quarter circle has length ¼ × 2πr = πr/2.
    • Total = π/2 × (1 + 1 + 2 + 3 + 5 + 8 + 13) = π/2 × 33 = 33π/2 = 51.83…

Task C: Fibonacci identities

  1. 143 = 144 − 1
    • 1 + 1 + 2 + 3 + 5 + 8 + 13 + 21 + 34 + 55 = 143
    • F11 = 89 and F12 = 144, so F12 − 1 = 143.
  2. 1, −1, 1, −1, 1: the result is (−1)n
    • n = 2: 1 × 2 − 1 = 1. n = 3: 1 × 3 − 4 = −1. n = 4: 2 × 5 − 9 = 1.
    • n = 5: 3 × 8 − 25 = −1. n = 6: 5 × 13 − 64 = 1.
    • So Fn−1Fn+1 − Fn2 = (−1)n.
  3. 10
    • The pattern of odd (O) and even (E) is O, O, E, O, O, E, …, because O + O = E, O + E = O and E + O = O.
    • So Fn is even exactly when n is a multiple of 3: n = 3, 6, …, 30, which is 10 numbers.

Extension

  1. F20 = 6765
    • n = 1: (φ − ψ)/√5 = √5/√5 = 1.
    • n = 2: (φ2 − ψ2)/√5 = (φ − ψ)(φ + ψ)/√5 = √5 × 1/√5 = 1.
    • n = 20: (φ20 − ψ20)/√5 = 6765.
  2. Proof
    • Base case: F1 = 1 = F3 − 1.
    • Assume the sum of the first k terms is Fk+2 − 1. Adding Fk+1 gives Fk+2 + Fk+1 − 1 = Fk+3 − 1.
    • True for n = 1, and true for n = k + 1 whenever it is true for n = k, so true for every positive integer n.

The same answers are in the answers PDF. Every answer was recomputed by computer before publishing.

Practise the topics

National Mathematics Day maths · e Day maths · Valentine’s Day maths · All themed maths

More for lessons: Weekly starters · Free standard lessons · Worksheet builder · Competition maths