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Themed maths · February

Women in maths: mathematician cards for IB Maths SL and HL

A ready-to-teach lesson for February: three ‘mathematician cards’, each built on maths that carries a woman mathematician’s name or that she is known for. A 5-minute starter, a 35-minute main activity (one card per task), an HL extension and full worked answers.

Level
IB Maths SL and HL (AA and AI)
Time
40 minutes, plus a 10-minute extension
Topics
Prime numbers and proof; Rational functions, asymptotes and points of inflection; Definite integrals and improper area; Sectors and radians; Volumes of revolution (HL extension)
Equipment
The starter is non-calculator. A GDC is needed for task B.

Download student sheet (PDF)Answers (PDF)

Suggested timings

PartTimeWhat
Starter5 minQuick questions on the board
Main: task A10 minCard 1: Sophie Germain primes
Main: task B13 minCard 2: the witch of Agnesi
Main: task C12 minCard 3: Nightingale’s rose diagram
Extension10 minFast finishers or homework

Starter (5 minutes)

No GDC.

  1. 23 is prime. Is 2 × 23 + 1 prime?
  2. Find the area of a sector of radius 6 and angle π/3, in terms of π.
  3. What happens to 8/(x2 + 4) as x gets very large?
  4. Find the gradient of y = 1/(x2 + 1) at x = 1.

Main activity (35 minutes)

Task A: Card 1: Sophie Germain primes (10 min)

A prime p is a Sophie Germain prime when 2p + 1 is also prime. For example, 11 is one, because 23 is prime.

  1. List the Sophie Germain primes less than 50. How many are there?
  2. Prove that every Sophie Germain prime bigger than 3 leaves remainder 5 when divided by 6.
  3. Start at 2 and keep applying p → 2p + 1. How many primes do you get in a row? Which number breaks the chain, and why is it not prime?

Task B: Card 2: the witch of Agnesi (13 min)

The curve y = a3/(x2 + a2) is known as the witch of Agnesi, after Maria Gaetana Agnesi. Take a = 2: y = 8/(x2 + 4).

  1. Write down the y-intercept and the equation of the horizontal asymptote.
  2. Find the coordinates of the points of inflection, giving x exactly.
  3. Use your GDC to find ∫−1010 8/(x2 + 4) dx to 3 significant figures, then try larger limits. Suggest the exact area between the whole curve and the x-axis.

Task C: Card 3: Nightingale’s rose diagram (12 min)

Florence Nightingale is known for polar area diagrams (‘rose diagrams’). Each month is a sector with the same angle, and the area of the sector shows the number. Use 1 mm2 for each case, and these made-up numbers.

  1. A diagram shows 12 months as 12 equal sectors. Find the angle of each sector in radians.
  2. One month had 150 cases. Find the radius of its sector, to 3 significant figures.
  3. Another month had 4 times as many cases. By what factor is its radius bigger?
  4. Explain why drawing the radius in proportion to the number would mislead. If the radius were doubled, by what factor would the area grow?

Extension (10 minutes)

Extension 1 is for everyone; extension 2 is HL (volumes of revolution).

  1. Start at 89 and keep applying p → 2p + 1. How many primes do you get in a row, and which number breaks the chain?
  2. (HL) The region between y = 8/(x2 + 4) and the x-axis is rotated 2π about the x-axis. Use your GDC with wide limits to suggest the exact volume, given that it is a multiple of π2.

For teachers

Teacher notes and full worked answers
  • Print the three tasks as cards and rotate groups between them.
  • The cards name each mathematician only through the maths named after her or that she is known for: invite students to research one of them for homework from a reliable source.
  • Card 2 is a good GDC task for AI students; AA students can find the points of inflection by hand.

Starter

  1. Yes: 47 is prime
    • 47 is not divisible by 2, 3 or 5, and 72 = 49 > 47, so 47 is prime.
  2. 6π
    • ½ × 62 × π/3 = 6π
  3. It tends to 0
    • The denominator grows without limit, so the fraction tends to 0.
  4. −1/2
    • dy/dx = −2x/(x2 + 1)2 = −2/4 = −1/2

Task A: Card 1: Sophie Germain primes

  1. 2, 3, 5, 11, 23, 29, 41: seven
    • 2 → 5, 3 → 7, 5 → 11, 11 → 23, 23 → 47, 29 → 59, 41 → 83 are all prime.
    • 7 → 15, 13 → 27, 17 → 35, 19 → 39, 31 → 63, 37 → 75, 43 → 87, 47 → 95 are not.
  2. Proof
    • A prime p > 3 is not divisible by 2 or 3, so p = 6k + 1 or 6k + 5.
    • If p = 6k + 1 then 2p + 1 = 12k + 3 = 3(4k + 1), a multiple of 3 bigger than 3, so not prime.
    • So p = 6k + 5.
  3. 5 primes (2, 5, 11, 23, 47); 95 = 5 × 19
    • 2 → 5 → 11 → 23 → 47 → 95
    • 95 = 5 × 19 is not prime.

Task B: Card 2: the witch of Agnesi

  1. (0, 2); y = 0
    • x = 0 gives y = 8/4 = 2.
    • As x → ±∞, y → 0.
  2. (±2/√3, 3/2)
    • y′ = −16x(x2 + 4)−2 and y″ = 16(3x2 − 4)/(x2 + 4)3.
    • y″ = 0 when x2 = 4/3, so x = ±2/√3, and y″ changes sign there.
    • y = 8/(4/3 + 4) = 8/(16/3) = 3/2
  3. 11.0; the area approaches 4π
    • ∫−1010 = 8 arctan 5 = 10.98…, which is 11.0.
    • With limits ±100 it is 12.49…, and with ±1000, 12.55…
    • The values approach 4π = 12.566… (exactly, 8 × ½ × π).

Task C: Card 3: Nightingale’s rose diagram

  1. π/6
    • 2π ÷ 12 = π/6
  2. 23.9 mm
    • ½r2 × π/6 = 150, so r2 = 1800/π = 572.9…
    • r = 23.93…, which is 23.9 mm.
  3. 2
    • Area is proportional to r2, so 4 times the area needs √4 = 2 times the radius.
  4. 4: the eye compares areas, so a doubled number would look four times as big
    • Area ∝ r2: doubling the radius multiplies the area by 4.

Extension

  1. 6 primes (89, 179, 359, 719, 1439, 2879); 5759 = 13 × 443
    • 89, 179, 359, 719, 1439 and 2879 are all prime (check with your GDC).
    • 2 × 2879 + 1 = 5759 = 13 × 443.
  2. 4π2
    • π∫ 64/(x2 + 4)2 dx over wide limits gets close to 39.478…
    • 39.478… ÷ π2 = 4.000…, so the volume is 4π2.

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

Practise the topics

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