Tales in notation

Everyday things — a story, the news, a lesson for life, a room, a letter, a recipe — told three times: in words, with more maths, and with as much notation and as few words as possible. Slide between them and watch the words become signs.

Read it in words first. Then move the slider and read the same thing again with more maths. At the last level, check: can you still hear the story in the signs?

Draft: written by an AI helper and not yet checked by a teacher.

A storyThe late bus
Words Every morning I leave home at 7:35 and the bus comes at 7:40. The stop is five minutes away, so I always arrive just in time.
More maths Every morning: I leave at $7 : 35$ , the bus comes at $7 : 40$ , the stop is $5$ min away, so I arrive at $7 : 40$ , just in time.
Max maths $L = 7 : 35, B = 7 : 40, B - L = 5^{'} \Rightarrow$ in time, $\forall$ morning.
The newsA week of rain
Words This week it rained on four of the seven days: Monday twelve millimetres, Tuesday eight, Thursday twenty and Sunday four.
More maths Rain this week: $4$ of $7$ days. Mon $12$ mm, Tue $8$ , Thu $20$ , Sun $4$ .
Max maths $R = {Mon, Tue, Thu, Sun}$ , $∣ R ∣ = 4$ of $7$ . $r = (12, 8, 20, 4)$ mm.
A lesson for lifeSmall steps
Words If you get a little better every day, the days add up. One per cent better each day is not much on its own.
More maths A little better every day adds up. $1%$ a day is small on its own.
Max maths $\forall$ day: better by $1%$ . One day: $\times 1.01$ , small.
A descriptionMy room
Words My room is four metres long and three metres wide, so it has twelve square metres of floor. The ceiling is two and a half metres high.
More maths My room: $4 m \times 3 m = 12 m^{2}$ of floor, height $2.5$ m.
Max maths room $= 4 \times 3 \times 2.5$ m. floor $= 12 m^{2}$ .
A letterTo a friend
Words Dear friend, it has been two years since we last met, and I think of you more often than I write, which is not often enough.
More maths Dear friend: $2$ years since we met. Thoughts of you ( $f$ ) $>$ letters to you ( $n$ ), and $n ≪$ enough.
Max maths friend: $Δ t = 2$ y. $f > n ≪$ enough.
A recipeBread
Words Bread needs four things: flour, water, salt and yeast. For every five hundred grams of flour, take three hundred and fifty grams of water, ten grams of salt and five grams of yeast.
More maths Bread $=$ flour $+$ water $+$ salt $+$ yeast. Per $500$ g flour: $350$ g water, $10$ g salt, $5$ g yeast. Water/flour $= 0.7$ .
Max maths bread $= f (F, W, S, Y)$ ( $F$ flour, $W$ water, $S$ salt, $Y$ yeast). $F : W : S : Y = 500 : 350 : 10 : 5$ , $W / F = 0.7$ .
A storyMarket day
Words On market day I went out with fifty coins in my pocket and a short list: apples, eggs and a bunch of herbs.
More maths Market day: $50$ coins in my pocket, a list of $3$ things: apples, eggs, herbs.
Max maths $c_{0} = 50$ . list $= {A, E, H}$ (apples, eggs, herbs).
The newsThe match
Words The home team beat the visitors three to one on Saturday. Kick-off was at 16:00, in front of about four thousand people.
More maths Saturday, $16 : 00$ : home $3$ , visitors $1$ , before $\approx 4000$ people.
Max maths Sat $16 : 00$ : $H = 3, V = 1$ ( $H$ home, $V$ visitors). crowd $\approx 4000$ .
A lesson for lifeThe slow tree
Words A seed you plant today gives no fruit tomorrow. A fruit tree takes about five years before its first harvest, and then it bears for fifty.
More maths Plant a seed today $\Rightarrow$ fruit tomorrow $= 0$ . A fruit tree: first harvest after $\approx 5$ years, then fruit for $50$ .
Max maths seed $(t_{0}) \Rightarrow$ fruit $(t_{0} + 1) = 0$ . tree: fruit $= 0$ for $t < 5$ y; $> 0$ for $5 \leq t \leq 55$ .
A descriptionMy bicycle
Words My bicycle has two wheels, each seventy centimetres across, so one turn of a wheel carries me about two metres and twenty centimetres.
More maths My bicycle: $2$ wheels, diameter $70$ cm. One turn $= π \times 0.7 \approx 2.2$ m.
Max maths wheels: $n = 2$ , $d = 70$ cm, $C = π d \approx 2.2$ m.
A letterFrom the road
Words Dear all, I am writing from the third day of the journey. We left at dawn on Monday and have covered about four hundred and fifty kilometres so far, a hundred and fifty a day.
More maths Dear all: day $3$ of the journey. Left Monday at dawn; $\approx 450$ km so far $= 150$ km a day.
Max maths all: $t = 3$ d. $x (3) \approx 450$ km, $v ˉ = 150$ km/d.
A recipeRice
Words Rice wants three things: rice, water and salt, and one thing more, which is time. For every cup of rice take a cup and a half of water and half a teaspoon of salt.
More maths Rice $=$ rice $+$ water $+$ salt $+$ time. Per $1$ cup rice: $1.5$ cups water, $\frac{1}{2}$ tsp salt. Water/rice $= 1.5$ .
Max maths rice $= f (R, W, S, t)$ ( $R$ rice, $W$ water, $S$ salt, $t$ time). $R : W : S = 1 : 1.5 : \frac{1}{2}$ (cup, cup, tsp).
A lesson for lifeReading a definition
Words A paper says: a whole number n is called even if there is a whole number k with n equal to twice k. Read slowly. The word being defined is even, and everything after the word if is the test.
More maths A paper says: a whole number $n$ is called even if there is a whole number $k$ with $n = 2 k$ . Read slowly: the name is even; the test follows the word if.
Max maths $E := {n \in Z ∣ \exists k \in Z : n = 2 k}$
A descriptionA result in a tiny abstract
Words Result. Fix the length of the fence around a rectangular garden, say twenty metres. Among all rectangles with that fence, the square has the largest area.
More maths Result. Fix the fence $P = 20$ m around a rectangular garden. Among all rectangles with $2 (a + b) = P$ , the square has the largest area $A = ab$ .
Max maths $P = 20$ . ${a \times b ∣ 2 (a + b) = P}$ : $max A = ab$ at $a = b$ .
A storyTwelve towels
Words I had twelve wet towels and one washing line. The line holds six towels, and a towel in the sun dries in two hours.
More maths $12$ wet towels, $1$ line. The line holds $6$ ; a towel dries in $2$ h.
Max maths Towels $n$ , lines $L$ , time $t$ : $6$ per line, $2$ h per round.
The newsThe bath that took too long
Words News from the bathroom: a bath that should have filled in fifteen minutes took almost nineteen. The bath holds a hundred and fifty litres and the tap gives ten litres every minute, so fifteen minutes was the plan.
More maths Bath news: it should fill in $15$ min, but took almost $19$ . The bath holds $150$ L and the tap gives $10$ L/min, so the plan was $150 \div 10 = 15$ min.
Max maths Volume $V$ , tap $r$ , leak $ℓ$ (litres, minutes): $V = 150, r = 10, ℓ = 2$ . Plan: $\frac{V}{r} = 15$ .
A descriptionOur staircase
Words Our staircase has twelve steps. Each step goes up eighteen centimetres and is twenty-five centimetres deep.
More maths Twelve steps; each rises $18$ cm and is $25$ cm deep.
Max maths Steps $n$ , rise $h$ , depth $d$ (cm): $n = 12, h = 18, d = 25$ .
A recipePancakes for twelve
Words Batter for four people uses two eggs, two hundred grams of flour and five hundred millilitres of milk. It makes eight pancakes.
More maths Batter for $4$ people: $2$ eggs, $200$ g flour, $500$ ml milk, making $8$ pancakes.
Max maths For $4$ people: eggs $E$ , flour $F$ , milk $M$ : $(E, F, M) = (2, 200, 500)$ , pancakes $8$ .
A storyThe queue at the tap
Words Four neighbours queue at one tap. The tap fills a ten-litre bucket in two minutes. Three neighbours carry one bucket each; the fourth has three buckets and stands at the front.
More maths $4$ neighbours, $1$ tap, one bucket of $10$ L takes $2$ min. Three have $1$ bucket; one has $3$ buckets and stands first.
Max maths $4$ people, a bucket $= 2^{'}$ ; order $A = (3, 1, 1, 1)$ and order $B = (1, 1, 1, 3)$ buckets.
A lesson for lifePut it away now
Words Every day I leave about six things lying around: a cup, a sock, a book, a pen and so on. Putting one thing away takes me fifteen seconds.
More maths Each day I leave $6$ things around. Putting one away takes $15$ s.
Max maths Things per day $n$ , cost of putting away now $c$ (seconds): $n = 6, c = 15$ .
A letterNew curtains
Words Dear friend, I have made curtains for my window! The window is one hundred and twenty centimetres wide. A curtain looks better if it is gathered, so I made it one and a half times as wide as the window: one hundred and eighty centimetres.
More maths Dear friend, I have made curtains! The window is $120$ cm wide. A gathered curtain looks better, so I used $1.5 \times 120 = 180$ cm.
Max maths Dear friend! Window $w$ , fullness $k$ : $w = 120, k = 1.5$ , so $k w = 1.5 \times 120 = 180$ .
A storyThe slow clock
Words The clock in our kitchen loses four minutes every day. On Sunday I set it exactly right.
More maths Our kitchen clock loses $4$ min a day. On Sunday I set it right.
Max maths Slowness $s$ (minutes) after $d$ days: $s (d) = 4 d$ .
A lesson for lifeEnough for everyone
Words Dinner for four: one big flatbread pizza cut into eight slices, so two slices each. Each person gets one quarter of the pizza.
More maths Dinner for $4$ : one pizza in $8$ slices, so $8 \div 4 = 2$ slices each, which is $\frac{1}{4}$ of the pizza.
Max maths Pizza of $8$ slices, people $n = 4$ : share $\frac{1}{n} = \frac{1}{4}$ , that is $\frac{8}{4} = 2$ slices.
A descriptionThe wobbly table
Words My kitchen table wobbles. It has four legs, and one of them does not touch the floor. A table with three legs never wobbles, because three points always lie in one flat plane, but a fourth point may not.
More maths My table has $4$ legs, and one does not touch the floor. A table with $3$ legs never wobbles: $3$ points always lie in one plane, but the fourth may not.
Max maths Legs $n$ , plane $Π$ : $n = 3 \Rightarrow \exists! Π$ (no wobble). With $n = 4$ not always.
The newsThe night of the rain
Words Last night it rained for three hours without stopping. The rain gauge in the yard showed sixty millimetres by morning, and a normal month here gives about ninety.
More maths Last night it rained for $3$ hours. The gauge showed $60 mm$ by morning; a normal month gives about $90 mm$ .
Max maths rain $R$ , month $M$ : $t = 3 h, R = 60 mm, M \approx 90 mm$ .
A storyThe frost night
Words At ten o'clock the thermometer in the garden said four degrees Celsius. I had just planted a row of young lettuces, and I knew that frost, which starts below zero, would kill them.
More maths At $22 : 00$ the thermometer said $4^{\circ} C$ . My lettuces die in frost, that is, below $0^{\circ} C$ .
Max maths $22 : 00$ , temperature $T = 4^{\circ} C$ ; frost $F \Leftrightarrow T < 0$ .
A descriptionInside the hive
Words Inside a hive, honey is stored in wax cells, and every cell is a hexagon, a shape with six equal sides. Hexagons fit together with no gaps, so no wax is wasted.
More maths Honey is stored in wax cells, and each cell is a hexagon: $6$ equal sides. Hexagons fit together with no gaps, so no wax is wasted.
Max maths cell: hexagon $H$ , $6$ sides.
A recipeSprouts in a jar
Words You need fifty grams of dry mung beans, a clean jar, a piece of cloth and water. This is the easiest garden there is, because it grows indoors.
More maths You need $50 g$ of dry mung beans, a jar, a cloth and water. It is a garden that grows indoors.
Max maths beans $B_{0} = 50 g$ .
A lesson for lifeMore sheep, fewer days
Words A flock of one hundred and twenty sheep lives on one field. Each sheep eats two kilograms of grass a day, so the flock eats two hundred and forty kilograms a day.
More maths A flock of $120$ sheep, $2 kg$ of grass per sheep per day: the flock eats $120 \times 2 = 240 kg$ a day.
Max maths sheep $s$ , days $d$ : $s = 120, 2 s = 240 kg$ a day.
A letterThe swallows have gone
Words Dear friend, the swallows left this morning. The wires by the window, full all summer, are empty, and the house is quieter than I expected.
More maths Dear friend, the swallows left this morning. The wires by the window are empty.
Max maths wire, swallows: $n = 0$ .
The newsThe hot week
Words This was the hottest week of the summer. The afternoon temperatures on its seven days were thirty-one, thirty-three, thirty-five, thirty-seven, thirty-six, thirty-four and thirty-two degrees Celsius.
More maths The hottest week of the summer. Afternoon temperatures: $31, 33, 35, 37, 36, 34, 32$ degrees.
Max maths $d = 1, \dots, 7$ : $T_{d} = 31, 33, 35, 37, 36, 34, 32^{\circ} C$ .
A storyCounting the apples
Words On harvest day I picked apples from twelve trees. I had counted on fifty kilograms from each tree, which would make six hundred kilograms.
More maths Harvest day: $12$ trees, and I counted on $50 kg$ from each, that is $12 \times 50 = 600 kg$ .
Max maths $12 \times 50 = 600 kg$ .
A lesson for lifeThe pond and the plant
Words A little water plant grows on a pond. Each day it doubles its cover of the water, and on day thirty it covers the whole pond.
More maths A water plant doubles its cover of the pond each day and covers all of it on day $30$ .
Max maths cover $C (d)$ : $C (d + 1) = 2 C (d), C (30) = 1$ .
A descriptionThe spider's web
Words On a cold morning the garden fence is hung with webs, each one covered in tiny drops of dew. A spider has eight legs, not six like an insect, and she builds her web with them in the dark.
More maths A cold morning: dew on the webs. A spider has $8$ legs, an insect only $6$ .
Max maths legs: $8 > 6$ .
A storyA minute to spare
Words The test had twenty questions and forty minutes, so every question could have two minutes. I worked steadily and wrote my last answer after thirty-eight minutes.
More maths $20$ questions, $40$ min, so $40 \div 20 = 2$ min each. I wrote my last answer at minute $38$ .
Max maths Questions $Q$ , time $T$ min: $Q = 20, T = 40$ , and $T \div Q = 2$ . Last answer at $38$ .
The newsWon by less than a metre
Words The school sprint final was run on Friday afternoon over a hundred metres, with four runners on the track. The winner crossed the line in twelve and a half seconds.
More maths Friday final, $100$ m, $4$ runners. Winner: $12.5$ s.
Max maths Distance $d$ m, winner's time $W$ s: $d = 100, W = 12.5$ . $4$ runners.
A descriptionThe chess clock
Words A chess clock is really two clocks in one box. Each player has a button on their side, and only one of the two clocks runs at a time.
More maths Two clocks in one box; each player has a button; only $1$ of the $2$ clocks runs at a time.
Max maths Buttons: $2$ ; clock running: $1$ .
A lesson for lifeTen more each day
Words On the first day I could skip ten times before the rope caught my feet. My friend said: add ten more each day, and no more than that.
More maths Day $1$ : $s_{1} = 10$ skips. Rule: $10$ more each day, no more.
Max maths Skips on day $n$ : $s_{n}$ . $s_{1} = 10, s_{n + 1} = s_{n} + 10$ .
A letterFrom the school library
Words Dear reader, thank you for borrowing three books from the school library. Together they have three hundred and sixty pages: one of one hundred and twenty, one of one hundred and fifty, and one of ninety.
More maths Dear reader, you borrowed $3$ books: $120 + 150 + 90 = 360$ pages.
Max maths $120 + 150 + 90 = 360$ pages.
A storyThe spelling bee
Words Thirty-two children entered the school spelling bee. In every round each child gets one word, and half of the children go through to the next round.
More maths $32$ children; in each round every child gets $1$ word and half go through.
Max maths $n_{0} = 32$ children; $n_{k}$ remain after round $k$ : $n_{k + 1} = \frac{n _{k}}{2}$ .
The newsA relay for the whole class
Words A class of thirty children organised its own relay tournament. The teacher split the class into five teams of six.
More maths $30$ children, $5$ teams of $6$ : $30 \div 6 = 5$ .
Max maths $30 \div 6 = 5$ teams.
A lesson for lifeHow high is the kite?
Words To measure how high a kite flies without climbing anything, all you need is its string and a friend.
More maths Find the kite's height $h$ without climbing.
Max maths Height $h$ , no climbing.
A recipeA race game for two
Words A race game for two, ready in ten minutes. You need a sheet of paper with twenty squares drawn in a row, two dice, and one counter each.
More maths Serves $2$ , ready in $10$ min. You need $20$ squares in a row, $2$ dice, $1$ counter each.
Max maths $2$ players, $10$ min, $20$ squares, $2$ dice.
A recipeA recipe for learning
Words A recipe for learning anything: fifteen minutes a day, five days a week, for four weeks.
More maths A recipe for learning anything: $15$ min a day, $5$ days a week, $4$ weeks.
Max maths $1 5^{'} \times 5 \times 4$ .
The newsThe ferry timetable
Words The ferry crosses the water every forty minutes. The first boat leaves at six in the morning and the last at ten at night, which makes twenty-five sailings a day.
More maths The ferry crosses every $40$ min. Boat number $k$ leaves at $T_{k} = 6 : 00 + 40 k$ min. First boat $6 : 00$ , last $22 : 00$ : $(22 - 6) \times 60 = 960$ min, $960 \div 40 = 24$ gaps, so $25$ sailings a day.
Max maths $T_{k} = 6 : 00 + 40 k$ min, $k = 0, \dots, 24$ : $T_{24} = 22 : 00$ , so $25$ sailings.
A storyThe night bus
Words The night bus leaves at half past ten in the evening and arrives at a quarter past six in the morning. That is seven hours and forty-five minutes, and the driver stops three times for fifteen minutes each.
More maths Leaves at $22 : 30$ , arrives at $6 : 15$ : $90 + 375 = 465$ min, that is $7$ h $45$ min. Three stops: $3 \times 15 = 45$ min.
Max maths $22 : 30 \to 6 : 15$ is $90 + 375 = 46 5^{'}$ . Stops: $3 \times 15 = 4 5^{'}$ .
A recipePacked lunch for a long walk
Words Serves three walkers. The walk is eighteen kilometres and you cover four and a half kilometres every hour, so it takes four hours.
More maths Serves $3$ walkers. The walk is $18$ km at $4.5$ km/h: $18 \div 4.5 = 4$ h.
Max maths $n = 3$ walkers. $18 \div 4.5 = 4$ h.
A descriptionHow fast is the train?
Words The train is two hundred metres long, and it takes eight seconds to pass a signal post beside the track. So in one second it travels twenty-five metres.
More maths The train is $200$ m long and passes a post in $8$ s. Speed: $v = \frac{d}{t} = 200 \div 8 = 25$ m/s.
Max maths $d = 200 m$ and $t = 8 s$ : $v = \frac{d}{t} = 25 m/s$ .
A letterA letter after the flight
Words Dear friend, I have landed safely. We took off at ten in the evening by your clock and flew for seven hours.
More maths Dear friend, I have landed. We took off at $22 : 00$ by your clock and flew for $7$ h.
Max maths departure $22 : 00$ , flight $7$ h.
A lesson for lifeThe hour that never happens
Words Once a year, in spring, the clocks jump from two o'clock straight to three. The hour between them never happens.
More maths Once a year, in spring, the clocks jump $2 : 00 \to 3 : 00$ . The hour between them never happens.
Max maths hour between $h$ : $2 : 00 \to 3 : 00 \Rightarrow ∄ h$ .
A storyThe calendar square
Words A child tore this month's page off the wall calendar. Mum said: draw a square around nine days, three across and three down, and add them up. Tell me only the total, and I will tell you the day in the middle.
More maths A child tore this month off the calendar. Mum: draw a square of $3 \times 3$ around $9$ days, add them up and tell me only the total $S$ , and I will name the middle day.
Max maths square $3 \times 3$ , total $S$ , middle day $n$ .
A descriptionReading a map scale
Words In the corner of the map it says one to fifty thousand. That means one centimetre on the paper is fifty thousand centimetres on the ground, which is five hundred metres.
More maths Scale $1 : 50000$ : $1$ cm on the map is $50000$ cm on the ground, that is $500$ m.
Max maths $1 : 50000 \Rightarrow 1 cm = 500 m$ .
A recipeRecipe for a hill climb
Words Ingredients: one hill six hundred metres high, one pair of good legs that climb three hundred metres every hour, and one sunset at seven in the evening. The climb takes six hundred divided by three hundred, which is two hours.
More maths Ingredients: a hill $h = 600$ m high, legs that climb $v = 300$ m/h, a sunset at $19 : 00$ . The climb: $t = \frac{h}{v} = 600 \div 300 = 2$ h.
Max maths $h = 600 m$ , $v = 300 m/h$ : $t = \frac{h}{v} = 2 h$ .
The newsFuel for the long drive
Words A delivery driver planned a journey of four hundred and fifty kilometres each way. Her van uses eight litres of fuel for every hundred kilometres, and its tank holds fifty litres.
More maths A driver planned $450$ km each way. Her van uses $8$ L for every $100$ km, and the tank holds $50$ L.
Max maths $d = 450 km$ , use $\frac{8}{100} L/km$ , tank $50 L$ .
A lesson for lifeCut and choose
Words Two children have one cake and both want a fair share. Here is the oldest fair rule there is: one cuts, the other chooses.
More maths Two children, one cake: $1$ cake in all. One cuts, the other chooses.
Max maths Cake $K = 1$ ; cutter $C$ , chooser $H$ ; pieces $a$ and $1 - a$ .
A storyTwelve for eight
Words I had twelve cards and my friend had eight. Two places in my album were empty, and the two rare cards that fitted them were in my friend's pile.
More maths I had $12$ cards, my friend $8$ . My album had two gaps, and the two rare cards were his.
Max maths Cards: me $m = 12$ , friend $f = 8$ . I need $2$ rare cards $r$ .
The newsTwo discounts, one surprise
Words Shoppers were puzzled this week by a shop sign: twenty per cent off, and then ten per cent off that. Many believed the shop was giving thirty per cent off.
More maths Sign: $20%$ off, then $10%$ off again. Many thought: $30%$ off.
Max maths Price $P = 100$ ; discounts $20%$ , then $10%$ .
A storyMeeting in the middle
Words A seller wanted fifty tokens for a hat. The buyer offered thirty.
More maths The seller asked $50$ tokens; the buyer offered $30$ .
Max maths Asked $A = 50$ , offered $O = 30$ .
A letterA letter to a neighbour
Words Dear neighbour, thank you for lending me your drill for three days. My shelves are straight at last.
More maths Dear neighbour, thank you for lending me your drill for $3$ days. My shelves are straight.
Max maths Thank you: borrowed $d = 3$ days.
A descriptionThe bakery that belongs to its bakers
Words A small bakery belongs to its five workers. It has no boss: each worker owns an equal part, and each has one vote.
More maths A bakery belongs to its $5$ workers. No boss; $1$ vote each.
Max maths $n = 5$ workers, $1$ vote each.
The newsWashing up by lottery
Words Four housemates draw one name from a hat every week to decide who washes up. This week one of them complained: the washing up has been mine three weeks in a row!
More maths $4$ housemates; each week one name is drawn from a hat. One complained: $3$ weeks in a row!
Max maths $n = 4$ people, one name a week. Complaint: $3$ weeks running.
A lesson for lifeInterest on interest
Words Put one hundred tokens in a savings box that pays ten per cent a year. After one year you have one hundred and ten.
More maths $100$ tokens at $10%$ a year: after one year $100 \times 1.1 = 110$ .
Max maths Money after $n$ years: $A_{n} = 100 \cdot 1. 1^{n}$ , so $A_{1} = 110$ .
A recipePunch for everyone
Words This punch needs three parts fruit juice, one part lemon juice and four parts water. For six friends, each gets four hundred millilitres, so you need two thousand four hundred millilitres in all.
More maths Juice $J$ , lemon juice $L$ and water $W$ are in the ratio $J : L : W = 3 : 1 : 4$ . For $6$ friends with $400$ ml each: $V = 6 \times 400 = 2400$ ml.
Max maths $J : L : W = 3 : 1 : 4$ ; $n = 6$ friends, $V = 6 \times 400 = 2400$ ml.
A storyHalf a ride
Words Three friends shared a taxi home. The whole ride cost sixty tokens, and the first friend got out halfway.
More maths The fare is $60$ tokens; the first friend got out halfway.
Max maths Fare $F = 60$ ; halves $\frac{F}{2} = 30$ .
A lesson for lifeWhy a kettle takes its time
Words Put one litre of water, which weighs a kilogram, in a kettle at twenty degrees. To warm it by one degree you need about four point two kilojoules of energy, and this number is the same for every degree.
More maths One kilogram of water, $m = 1$ kg, at $20^{\circ} C$ . One degree warmer needs about $4.2$ kJ, the same for every degree: $Q = m c Δ T$ with $c \approx 4.2$ kJ per kg per degree.
Max maths Water $m = 1$ , start $T_{0} = 20^{\circ} C$ , heat $Q = m c Δ T$ , $c \approx 4.2$ .
A descriptionThe moon's new faces
Words The moon shows a new face every night, but the sun always lights the same half of it. What changes is how much of that lit half we can see from the earth. One whole cycle, from one new moon to the next, takes about twenty-nine and a half days.
More maths The sun always lights half of the moon; what we see of it changes. One cycle is $P \approx 29.5$ days.
Max maths Lit half fixed; cycle $P \approx 29.5$ days.
A storyThe wall that clapped back
Words In front of a big wall I clapped my hands once. Half a second later, the wall clapped back. I wanted to know how far away it was.
More maths I clapped once at a big wall and it clapped back $t = 0.5$ s later. How far is the wall?
Max maths Echo: $t = 0.5$ s, distance $d = v t /2$ .
A letterA letter about a shadow
Words Dear friend, I stood a one-metre stick upright in flat ground and marked the end of its shadow every hour. In the morning the shadow was long, towards midday it got shorter and shorter, and then it grew long again.
More maths Dear friend, I stood a $1$ m stick upright and marked its shadow every hour: long, shorter, shortest at midday, then long again.
Max maths Stick $h = 1$ , shadow $s (t)$ ; noon means $s (t) = s_{m i n}$ .
The newsOne ice cube, five times the chill
Words A thirty-gram ice cube was dropped into a glass holding two hundred and fifty grams of juice at twenty degrees. A few minutes later the drink had cooled to about nine degrees.
More maths A $30$ g ice cube in $250$ g of juice at $20^{\circ} C$ : a few minutes later the drink was at about $9^{\circ} C$ .
Max maths Juice $m_{j} = 250 g$ , ice $m_{i} = 30 g$ , start $T_{0} = 20^{\circ} C$ , end $T \approx 9^{\circ} C$ .
A recipeA spring balance
Words You need: a soft spring from an old pen, a thread, a ruler and ten small weights that are all alike, each of ten grams.
More maths You need: a soft spring, a thread, a ruler and $10$ weights that are alike, each $10$ g.
Max maths Number of weights $n = 10$ , each $m_{0} = 10 g$ .
A storyBreaking a magnet
Words I used to think a magnet was two pieces stuck together: a north end and a south end. So I took a magnetised needle ten centimetres long and snapped it in the middle, to get one north on its own and one south on its own.
More maths I thought a magnet was two pieces stuck together. I snapped a $10$ cm needle in the middle to get a lone north $N$ and a lone south $S$ .
Max maths Needle $L = 10 cm$ ; snap: $1 \to 2$ magnets.
A descriptionThe pendulum
Words A pendulum is a weight hanging on a string and swinging to and fro. One swing there and back is called a period. A one-metre string with a small weight takes about two seconds for it: thirty swings a minute.
More maths A weight on a string. One swing there and back is the period $T$ . For $L = 1$ m: $T \approx 2$ s, so $60/2 = 30$ swings a minute.
Max maths Period $T$ ; $L = 1 m$ gives $T \approx 2 s$ and $60/2 = 30$ a minute.
A recipeA rainbow from a hose
Words You need: a garden hose with a spray nozzle (or a spray bottle), a sunny day, and a sun that is low in the sky: morning or late afternoon.
More maths You need: a hose with a spray nozzle, a sunny day, and a low sun (morning or late afternoon).
Max maths Hose; low sun, height $s < 4 2^{\circ}$ .
A storyA stone in the well
Words It was a deep well. I dropped a pebble and counted until I heard the splash: two point one seconds.
More maths A deep well. I dropped a pebble and counted: the splash came after $T = 2.1$ s.
Max maths Splash: $T = 2.1 s$ .
A storyTwo dice and a seven
Words My brother and I play with two dice. Each throw, we add the dots. He always picks twelve and I pick seven, and whichever total comes up first wins.
More maths Two dice; each throw I add the dots and call the total $S$ . He picks $S = 12$ , I pick $S = 7$ . First to appear wins.
Max maths $S$ the sum, $A = {S = 12}, B = {S = 7}$ .
A descriptionA deck of cards
Words A deck has fifty-two cards: four suits, thirteen cards in each. One suit is hearts, so the hearts form a pile of thirteen. There are also four aces, one in each suit.
More maths A deck: $52 = 4 \times 13$ cards. Hearts $H$ with $∣ H ∣ = 13$ , aces $K$ with $∣ K ∣ = 4$ , one in each suit.
Max maths $H$ hearts, $K$ aces: $52 = 4 \times 13, ∣ H ∣ = 13, ∣ K ∣ = 4$ .
The newsThe queue that shrank
Words Good news at the ticket counter this morning: a queue of eight people, which had been there since opening, has disappeared. The counter finishes with one person every two minutes.
More maths Good news at the ticket counter: the queue of $8$ people has gone. The counter finishes with $1$ person every $2$ min.
Max maths Queue $L$ ; leaving $s = \frac{1}{2}$ , arriving $a = \frac{1}{4}$ (people per min).
A storyThe locked chest
Words At home there is a small chest with three dials, each showing the digits zero to nine. That makes a thousand possible codes. Trying one every five seconds would take up to about an hour and twenty minutes.
More maths A chest, $3$ dials, digits $0$ to $9$ : $1 0^{3} = 1000$ codes. At $5$ s per try that is up to $5000$ s, about $83$ min.
Max maths $(a, b, c) \in {0, \dots, 9}^{3}$ : $1000 \times 5 s = 5000 s \approx 83 min$ .
A lesson for lifeRoads and junctions
Words A map can be drawn as dots and lines. The dots are junctions and the lines are roads. Take four junctions called A, B, C and D, and five roads: A to B, B to C, C to D, D to A, and A to C.
More maths A map as a graph: junctions $V = {A, B, C, D}$ , roads $E = {A B, B C, C D, D A, A C}$ , so $∣ V ∣ = 4$ and $∣ E ∣ = 5$ .
Max maths $G = (V, E)$ with $V = {A, B, C, D}$ and $E = {A B, B C, C D, D A, A C}$ .
A storyWho tells the truth?
Words Three children play a riddle game. Each one either always tells the truth or always lies. The first says: the second is lying. The second says: the third is lying. The third says: the first two are both lying.
More maths Three children $A$ , $B$ , $C$ , each always truthful ( $t = 1$ ) or always lying ( $t = 0$ ). $A$ says ' $B$ lies'; $B$ says ' $C$ lies'; $C$ says ' $A$ and $B$ both lie'.
Max maths $t_{A}, t_{B}, t_{C} \in {0, 1}$ ; $t_{A} = 1 \Leftrightarrow t_{B} = 0$ ; $t_{B} = 1 \Leftrightarrow t_{C} = 0$ ; $t_{C} = 1 \Leftrightarrow (t_{A} = 0 \land t_{B} = 0)$ .
A lesson for lifeThe coin decides
Words A fair coin lands on heads or on tails, and each has a chance of one half. When you cannot choose between two things, tossing it is a good way to decide.
More maths A fair coin: $P (H) = P (T) = \frac{1}{2}$ . Two things, one toss.
Max maths $P (H) = P (T) = \frac{1}{2}$ .
A letterA letter about odd and even
Words Dear friend, I found a trick today. Take any two odd numbers and add them: the answer is always even. Three plus five is eight, seven plus nine is sixteen.
More maths Dear friend, I found a trick. For odd $a$ and $b$ , $a + b$ is even: $3 + 5 = 8$ and $7 + 9 = 16$ .
Max maths $a \equiv b \equiv 1 (mod 2) \Rightarrow a + b \equiv 0 (mod 2)$ and $3 + 5 = 8, 7 + 9 = 16$ .
A recipeSorting a row of cards
Words Ingredients: five cards numbered one to five, face up in a row in the order four, two, five, one, three; a table; patience.
More maths Ingredients: $n = 5$ cards in the row $(4, 2, 5, 1, 3)$ .
Max maths $n = 5, (4, 2, 5, 1, 3)$ .
A storySocks in the dark
Words The light is out and I must dress in the dark. The drawer holds ten black socks and ten white socks, all mixed. I need two socks of the same colour; I do not mind which.
More maths Dark. The drawer: $10$ black ( $B$ ) and $10$ white ( $W$ ), $∣ B ∣ + ∣ W ∣ = 20$ . I need $2$ of the same colour.
Max maths $∣ B ∣ = ∣ W ∣ = 10$ ; least $n$ to be sure of two of one colour.
A lesson for lifeThe sleep bank
Words Think of sleep as a bank. Every night you put hours in, and every day you spend them. The numbers here are only an example: how much sleep a person needs depends on the person, so ask a grown-up or a doctor about yours.
More maths Sleep as a bank: in at night, out by day. Example numbers only; needs differ from person to person.
Max maths Sleep: night $↑$ , day $↓$ (example numbers).
A descriptionA pulse at the wrist
Words Lay two fingers, not the thumb, which has a pulse of its own, gently on the inside of your wrist, just below the thumb. The small tap you feel is your heart pushing blood along.
More maths Two fingers on the wrist, below the thumb. The small tap is the heart pushing blood.
Max maths Wrist: pulse $c$ taps in $15$ s.
A storyThe water jug
Words Every morning I fill a two-litre jug and put it on the table. My glass holds a quarter of a litre, so eight glasses empty the jug. This is just my jug, not a rule: how much water a person should drink depends on the person and on the day.
More maths Jug: $2000$ mL. Glass: $250$ mL. So $2000 \div 250 = 8$ glasses. (Just my jug, not a rule.)
Max maths Jug $J = 2000$ , glass $g = 250$ (mL): $J = 8 g$ .
A letterThe door chart
Words Dear Grandma, we have started a growth chart on the kitchen door. On every birthday someone draws a pencil line level with the top of my head and writes the date beside it. These heights are only examples.
More maths Dear Grandma, we have a growth chart on the kitchen door: a pencil line and a date on every birthday. (Heights are examples.)
Max maths Door chart: $t$ years, $h$ cm (examples).
A descriptionThe plate
Words Picture a round plate, twenty-four centimetres across. One common way to picture a balanced meal is to cut the plate into three parts. This is only a picture to think with, not a diet. If you need special food, ask a doctor.
More maths A round plate $24$ cm across, cut in three. Just a picture, not a diet; for special food, ask a doctor.
Max maths Plate $24$ cm, three parts (a picture, not a diet).
The newsLift out of order
Words The lift in a tall block of flats was out of order for twelve days. The people on the fifth floor, the top one, had to use the stairs.
More maths Lift out of order for $12$ days; the people on floor $5$ used the stairs.
Max maths Lift out of order: $12$ days, floor $5$ .
A recipeFruit salad
Words A fruit salad for four. For the vitamin C we use rounded example numbers, because real fruit varies a lot: oranges have fifty milligrams in every hundred grams, strawberries sixty, apples five.
More maths A fruit salad for $4$ . Vitamin C per $100$ g (rounded examples only): orange $50$ , strawberry $60$ , apple $5$ mg.
Max maths Salad for $4$ ; mg per $100$ g: orange $50$ , strawberry $60$ , apple $5$ .
A storyThe fever hill
Words I had a cold, and there was a thermometer in the house. Every four hours someone wrote my temperature on a chart. This is one made-up example. When a fever needs a doctor is for a grown-up to decide.
More maths A cold, and a thermometer. Every $4$ h my temperature went on a chart (a made-up example; a grown-up decides about a doctor).
Max maths $T (t)$ in degrees, every $4$ h (example).
A lesson for lifeA billion beats
Words Nobody can count every beat of a heart, but anybody can estimate. A resting heart beats about seventy times a minute. This is an example: it differs from person to person and from animal to animal.
More maths Estimating the beats of a heart. At rest, $r = 70$ a minute (an example).
Max maths $N = r \times t$ , with $r = 70$ (example).
The newsFeet and height
Words A class of children measured each other and found a rule of thumb: the length of a foot is about fifteen hundredths of a person's height. A rule of thumb is a rough guide, not a law.
More maths A class found a rule of thumb: foot length $f \approx 0.15 h$ , where $h$ is the height. (Rough, not a law.)
Max maths Rule: $f \approx 0.15 h$ .
A storyThe head start
Words A tortoise and a hare agreed to race along a path 120 metres long. The tortoise walks two metres a minute and the hare runs six metres a minute.
More maths Track $120$ m. Tortoise: $2$ m/min. Hare: $6$ m/min.
Max maths Track $L = 120$ m; $v_{T} = 2$ , $v_{H} = 6$ m/min.
The newsRumour at the pond
Words POND NEWS. At midnight a small frog croaked a rumour: "the pond will dry up". It was not true. Every frog that heard it told two new frogs in the following hour.
More maths Midnight: a frog croaks a false rumour. Each frog that hears it tells $2$ new frogs in the next hour.
Max maths Rumour $R$ : each frog that knows it tells $2$ new ones per hour.
A lesson for lifeTrust takes time
Words Trust is built like a wall, one brick a day. Each day you keep your word, one brick is added.
More maths Trust grows by one brick for each day I keep my word.
Max maths Trust $T (d) = d$ , with $d$ honest days.
A descriptionThe rope
Words A rope is made of five thin strands twisted together. One strand alone can carry at most twenty kilograms.
More maths Rope: $5$ strands. One strand carries at most $20$ kg.
Max maths Strands $n = 5$ , each $w \leq 20$ kg.
A recipeA recipe for a habit
Words Ingredients: one small habit that takes ten minutes, one place, one fixed time of day, and thirty days.
More maths Ingredients: a habit of $10$ min, a place, a fixed time, $30$ days.
Max maths Habit $h = 10$ min, $d = 30$ days.
The newsAgreement on the seesaw
Words PLAYGROUND NEWS. A goat and a cat reached an agreement on the seesaw today, after a week of arguing.
More maths Goat $G$ and cat $C$ agree on the seesaw after a week of arguing.
Max maths Goat $G = 30$ and cat $C = 5$ kg.
A storySharpening the axe
Words I was cutting logs for the winter, eight hours a day. With my blunt axe I cut ten logs an hour.
More maths A day of $8$ h, a blunt axe: $10$ logs an hour.
Max maths Day $T = 8$ h, blunt $r_{b} = 10$ .
A letterTo one who practises
Words Dear one, you told me practice feels like nothing: twenty minutes a day and it seems that nothing changes. I am writing to tell you what that nothing hides.
More maths Twenty minutes a day, and it seems nothing changes.
Max maths $20$ min a day.
A descriptionThe honey jar
Words A jar holds 640 grams of honey. A wise bear decides: every day I take half of what is left in the jar.
More maths Jar: $J_{0} = 640$ g. Each day the bear takes half of what is left.
Max maths $J_{0} = 640$ , $J_{k} = J_{k - 1} /2$
A lesson for lifeLearning from a miss
Words I am learning to throw a ball into a basket. My first throw misses by a hundred centimetres, far too much to the right.
More maths First throw: error $e_{0} = 100$ cm.
Max maths $e_{0} = 100$

Next: challenges to understand such texts and to write your own, from words to maximum maths.

Tales in notation

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