The Maths Teacher's Bag

HUMAN

TEACHING GUIDE · OPÉRATION DINDON · JUNE 2026

◆◆◆

THE MATHS
TEACHER'S BAG
12 Objects · 12 Formulae · 90 Seconds
No Programme Change Required

◆

◆ WHAT THIS DOCUMENT IS — AND WHAT IT IS NOT

This document requires no programme change. No extra session. No coordination with other teachers. No authorisation. No special training.

The teacher writes the formula on the board exactly as they always have. Then they place the object on the desk. 90 seconds. The lesson continues.

That is all. It is enough. And it is what some teachers already do — without naming it, without claiming it, without waiting for a reform to ask them to.

◆◆◆

CARDS

12

DURATION

90 sec.

TOTAL BUDGET

€20-30

WATERMARK

HUMAN

Amine RAITI — Infrastructure Architect & SRE
Former engineering school professor · Teaching since 2006
Public document · CC BY-NC-SA 4.0 · Opération Dindon · June 2026
Dedicated to all the teachers who answered differently.

HUMAN

HOW TO USE THIS DOCUMENT — THE GESTURE IN 3 STEPS

THE TEACHER WRITES · THE TEACHER PLACES · THE LESSON CONTINUES

1

THE TEACHER WRITES

They write the formula on the board as usual. Nothing changes in the lesson. The formula is there — sin(θ), πr², a²+b²=c².

2

THE TEACHER PLACES

They pull the object from the bag and place it on the desk — or pass it around. 90 seconds. They show the link between the formula and the object. One sentence is enough.

3

THE LESSON CONTINUES

The object stays on the desk or goes back in the bag. The lesson resumes exactly where it was. Nothing was sacrificed. A seed was planted.

◆ WHEN TO BRING OUT THE OBJECT — THE THREE NATURAL MOMENTS

Moment A — Just after writing the formula, before the first numerical example. The teacher writes the formula, turns around, pulls out the object: "Before we calculate — here is what this looks like in the real world." Then they put it away and resume.

Moment B — When a student asks "what is this good for?" The teacher pulls out the object: "Good question. Look." 90 seconds. Question answered. It is no longer hanging in the room.

Moment C — At the start of the lesson, before writing anything. The teacher places the object on the desk without a word. Students look. Then they write the formula. Curiosity is already there.

◆ HOW TO READ EACH CARD

🔢 The formula
Clearly visible. The starting point.

📦 The object
What to bring. Cost. Where to find it.

✋ The steps
What to do. 3 steps maximum.

❓ The questions
What to ask the class.

👷 The trade
Who uses this in real life.

⏱ The moment
When in the lesson.

◆ ONE RULE ONLY

The object does not replace the lesson. It does not supplement it. It gives it a physical meaning for 90 seconds. That is all. That is enough. The student who has seen the screw turn and the carriage advance will never again look at sin(θ) as an abstract symbol.

HUMAN

01

ARITHMETIC · MULTIPLICATION

a × b = b × a

◆

📦 The object

Wooden lolly sticks × 100
€1 Supermarket / craft shop

✋ The manipulation

1.Arrange 3 rows of 4 sticks.

2.Flip: 4 rows of 3. Count the total.

3.Notice: same total — different arrangement.

❓ Questions for the class

Q1: Do 3 × 4 and 4 × 3 look the same physically?
Q2: Can you find an operation where the order of the gesture would change the physical result?

👷 In real life

BRICKLAYER / TILER
Laying tiles in 3 rows of 4 or 4 rows of 3 — same surface, same number, same cost.

Connection: Technology (grid layout) · Art (repeating patterns)

⏱ Moment in the lesson: Just after writing a × b = b × a on the board, before the first numerical example.

HUMAN

02

ARITHMETIC · FRACTIONS

¼ + ²⁄₄ = ¾

◆

📦 The object

An orange (or a chocolate bar)
€1 Supermarket

✋ The manipulation

1.Cut the orange into 4 equal quarters.

2.Place 1 quarter, then 2 quarters side by side.

3.Count: 3 quarters out of 4 — without calculating, just by looking.

❓ Questions for the class

Q1: How do you add ½ an orange and ⅓ an orange without calculating? What common division can you find?
Q2: Why can't you add ⅓ and ¼ directly without changing the division?

👷 In real life

COOK / PASTRY CHEF
Scaling a recipe from 4 to 6 servings: each ingredient × 6/4 = 3/2. Fractions are the tools of portioning.

Connection: Music (note values: crotchet = 1/4, minim = 1/2, semibreve = 1)

⏱ Moment in the lesson: When a student asks why fractions with different denominators cannot be added directly.

HUMAN

03

GEOMETRY · DEFINITION OF π

π = C ÷ D

◆

📦 The object

String + 3 tins of different sizes
€0 Recycling — each student brings a tin from home

✋ The manipulation

1.Wrap string around a tin. Mark, unroll, measure: that is C.

2.Measure diameter D with a ruler.

3.Calculate C ÷ D. Repeat with all 3 tins. Compare results.

❓ Questions for the class

Q1: Why do we always get the same number regardless of the tin size?
Q2: Would it work with an oval object? Why not?

👷 In real life

BOILERMAKER / PLUMBER
Calculating the length of a pipe bend before cutting: length = π × diameter × (angle/360). Without π, the cut is wrong.

Connection: Physics (wheels, pulleys) · Technology (circular gears)

⏱ Moment in the lesson: At lesson opening, before writing anything — place the 3 tins on the desk in silence.

HUMAN

04

GEOMETRY · PYTHAGORAS

a² + b² = c²

◆

📦 The object

Squared paper + scissors + dried beans
€0 School supplies + kitchen cupboard

✋ The manipulation

1.Cut 3 squares with sides of 3, 4 and 5 grid squares.

2.Fill the 3-square and 4-square with dried beans.

3.Pour all beans into the 5-square: it fills exactly.

❓ Questions for the class

Q1: Why do the two small squares fill the large one exactly?
Q2: Does it work with sides 5, 12 and 13? Calculate before checking.

👷 In real life

CARPENTER / TILER
The 3-4-5 rule: measure 3m on one wall, 4m on the other — if the diagonal is 5m, the angle is right. Every carpenter uses this without naming Pythagoras.

Connection: Construction (squareness check) · Navigation (GPS distance between two points)

⏱ Moment in the lesson: Just after writing a²+b²=c² on the board — before any numerical example.

HUMAN

05

TRIGONOMETRY · SINE AND COSINE

y = R · sin(θ)

◆

📦 The object

3 screws M6/M8/M10 20cm + nuts · cardboard crank-slider mechanism + paper fasteners
€3-5 AliExpress for screws · School supplies for crank

✋ The manipulation

1.Thread nut onto each screw. Turn 10 times — measure displacement. The ratio turns/distance is the pitch.

2.Compare the 3 screws: M6 (1mm pitch) → 10mm. M8 (1.25mm) → 12.5mm. M10 (1.5mm) → 15mm.

3.With the cardboard crank: turn the handle, watch the rod rise and fall. Measure position at 0°, 45°, 90°, 135°, 180°. Plot the curve: that is the sinusoid.

❓ Questions for the class

Q1: Why does the rod rise more slowly at the top than in the middle?
Q2: At what angle is the rod exactly at mid-height? Check with the formula.

👷 In real life

MECHANIC / ELECTRICIAN
AC mains is a sinusoid: V(t) = 325 · sin(2π·50·t). Each rotor revolution = one cycle. A piston displacement = R·sin(θ).

Connection: Physics (oscillations, waves) · Music (pure tones = sinusoids)

⏱ Moment in the lesson: When the class sees sin(θ) for the first time — place the screws and crank before explaining the unit circle.

HUMAN

06

TRIGONOMETRY · PARABOLA

f = D² ÷ (16 · d)

◆

📦 The object

Small salvaged satellite dish + marble
€0-5 Car boot sale / recycling centre

✋ The manipulation

1.Place the dish open-face upward.

2.Release a marble from different points on the rim.

3.Observe: it always rolls to the same point — the focal point. That is where the LNB goes.

❓ Questions for the class

Q1: Why do all the marbles converge on the same point regardless of starting position?
Q2: What would happen with a square surface? With a hemispherical surface?

👷 In real life

SATELLITE TECHNICIAN / RADAR ENGINEER
Positioning the LNB at the exact focal point to receive the satellite signal. 1 cm off = signal lost. The formula is the precision tool.

Connection: Physics (parabolic mirrors, solar furnaces) · Astronomy (Cassegrain telescopes)

⏱ Moment in the lesson: At the start of the quadratic curves chapter — before writing the equation y = ax².

HUMAN

07

FUNCTIONS · PROPORTIONALITY

y = k · x

◆

📦 The object

Kitchen scales + dried beans (50g minimum)
€10-15 Discount store / supermarket for the scales

✋ The manipulation

1.Weigh 10 beans, record. Weigh 20, 30, 50, 100. Plot the graph.

2.Notice: the line passes through the origin. That is y = k·x. k = mass of one bean.

3.Add a fixed weight to the pan: the line no longer passes through the origin. That is y = k·x + b.

❓ Questions for the class

Q1: How can you find the mass of a single bean without weighing it alone?
Q2: What does b represent physically when you add the fixed weight?

👷 In real life

PHARMACIST / ELECTRICIAN
Pharmacist: weight-proportional dosage — 5mg/kg × 60kg = 300mg. Electrician: Ohm's law V = R·I — voltage is proportional to current. k = R.

Connection: Economics (fixed cost + variable cost) · Physics (Hooke's law)

⏱ Moment in the lesson: Just before introducing the gradient and y-intercept.

HUMAN

08

ALGEBRA · EQUATIONS

2x + 3 = 11 → x = 4

◆

📦 The object

Kitchen scales + opaque bags + pebbles
€0 Recycling — garden pebbles, food bags

✋ The manipulation

1.Place 2 identical bags (unknown contents) + 3 visible pebbles on one side.

2.Place 11 pebbles on the other side. The scales balance.

3.Remove 3 pebbles from both sides. Divide by 2: each bag contains 4 pebbles.

❓ Questions for the class

Q1: Can you solve it without opening the bags? How?
Q2: What happens if you remove pebbles from one side only?

👷 In real life

ACCOUNTANT / SHOPKEEPER
I sold x items at €2 with €3 fixed costs and took in €11. How many items? The scales are elementary bookkeeping.

Connection: Chemistry (balancing a reaction) · Physics (equilibrium of forces)

⏱ Moment in the lesson: When introducing the rule 'what you do to one side you do to the other' — the scales show why it is physical necessity, not arbitrary rule.

HUMAN

09

STATISTICS · MEAN

x̄ = Σxᵢ / n

◆

📦 The object

Tape measure — class heights
€2 DIY shop / already available

✋ The manipulation

1.Measure each student's height. Record on the board.

2.Calculate the sum, divide by the number of students.

3.Find the median: line everyone up by height, take the middle one.

❓ Questions for the class

Q1: Does the mean height correspond to anyone in the class?
Q2: Why is the median sometimes different from the mean?

👷 In real life

DOCTOR / NURSE
The growth chart compares a child's height to the average for their age group. Percentiles are the statistics of public health.

Connection: Biology (growth, BMI) · Economics (income distribution)

⏱ Moment in the lesson: As an introduction to the statistics chapter — the data is real, students are invested.

HUMAN

10

PROBABILITY

P(A) = favourable cases / total cases

◆

📦 The object

6-sided dice + tally chart
€0 Board games already at school

✋ The manipulation

1.Roll the die 60 times. Record each result on the board.

2.Calculate the frequency of each face (count / 60).

3.Compare with theoretical probabilities (1/6 ≈ 0.167). Observe convergence.

❓ Questions for the class

Q1: After 5 consecutive sixes, has the probability of getting six on the next roll changed?
Q2: How many rolls does it take for frequencies to really resemble probabilities?

👷 In real life

INSURER / ACTUARY
Insurance premiums are calculated from the probability of a claim. An insurer who misjudges probabilities goes bust within 5 years.

Connection: Science (experimental protocol) · Games (poker strategy, backgammon)

⏱ Moment in the lesson: At lesson opening — roll the die before writing the definition of probability.

HUMAN

11

SEQUENCES · EXPONENTIAL GROWTH

uₙ = u₀ × rⁿ

◆

📦 The object

One A4 sheet of paper
€0 Already in the classroom

✋ The manipulation

1.Fold in half, then in half again. Count layers at each fold (limit: 7 folds).

2.Calculate: initial thickness 0.1mm. After 20 folds → 100m. After 42 → Moon distance.

3.Note the physical impossibility of the 8th fold — the physical limit of a mathematical phenomenon.

❓ Questions for the class

Q1: From which fold would the stack exceed your height?
Q2: Why can a sheet of paper not be folded more than 7 times?

👷 In real life

BIOLOGIST / BANKER
Bacterial division: one bacterium → 2 → 4 → 8... In 24h at 20min/generation, one becomes 4 billion. Banker: compound interest — €1,000 at 5% over 40 years = €7,040.

Connection: Biology (cell division) · Computing (algorithmic complexity O(2ⁿ))

⏱ Moment in the lesson: When introducing geometric sequences — fold the paper before writing the formula.

HUMAN

12

SEQUENCES · FIBONACCI AND GOLDEN RATIO

Fₙ = Fₙ₋₁ + Fₙ₋₂ · φ ≈ 1.618

◆

📦 The object

An artichoke or a sunflower
€1 Market / grocery shop

✋ The manipulation

1.Count the spirals in the artichoke clockwise.

2.Count anticlockwise.

3.Notice: two consecutive Fibonacci numbers (8 and 13, or 13 and 21). Calculate the ratio.

❓ Questions for the class

Q1: Why does nature use this sequence to arrange its seeds?
Q2: Build a rectangle whose sides are in the ratio φ — does it look pleasing?

👷 In real life

ARCHITECT / DESIGNER / PHOTOGRAPHER
The proportions of a classical façade, the 16/9 screen ratio (≈ φ²), the rule of thirds in photography — the golden ratio is the natural aesthetic of composition.

Connection: Art (composition, proportions) · Biology (phyllotaxis)

⏱ Moment in the lesson: As an introduction to sequences — place the artichoke on the desk without a word. Let curiosity settle.

HUMAN

SUMMARY TABLE · THE 12 OBJECTS AND THEIR BUDGET

ALL OF SECONDARY MATHS · FOR LESS THAN €30

◆

N°

FORMULA

OBJECT

COST

TRADE

01

a × b = b × a

Lolly sticks × 100

€1

Bricklayer / Tiler

02

¼ + ²⁄₄ = ¾

Orange / chocolate bar

€1

Cook / Pastry chef

03

π = C ÷ D

String + 3 tins

€0

Boilermaker / Plumber

04

a² + b² = c²

Squared paper + dried beans

€0

Carpenter / Tiler

05

y = R · sin(θ)

Screws M6/M8/M10 + crank

€3-5

Mechanic / Electrician

06

f = D² ÷ (16·d)

Satellite dish + marble

€0-5

Satellite tech / Radar engineer

07

y = k · x

Kitchen scales + dried beans

€10-15

Pharmacist / Electrician

08

2x + 3 = 11

Scales + opaque bags + pebbles

€0

Accountant / Shopkeeper

09

x̄ = Σxᵢ / n

Tape measure — class heights

€2

Doctor / Nurse

10

P(A) = favourable / total

6-sided dice

€0

Insurer / Actuary

11

uₙ = u₀ × rⁿ

A4 sheet of paper

€0

Biologist / Banker

12

Fₙ = Fₙ₋₁ + Fₙ₋₂ · φ≈1.618

Artichoke or sunflower

€1

Architect / Designer

◆◆◆

These 12 objects cover the entire secondary mathematics curriculum.
They fit in a bag. They cost less than €30 in total.
They each take 90 seconds.
And they finally answer the question every student has asked at least once:

"Sir, what is this good for?"

◆◆◆

NEMO SUPRA LEGEM EST