everywhy: every why, taken apart

Curriculum series

• Ontario Grade 4 Math 33 of 33 ideas have a lesson
• Ontario Grade 8 Math 14 of 40 ideas have a lesson

Geometry · 几何

• Why does 3-4-5 work? Four identical right triangles, one big square, two arrangements: the hole in the middle turns from c² into a² + b² — the Pythagorean theorem appears on its own.
• 为什么是 3-4-5？ 四个一样的直角三角形，在同一个大正方形里换个摆法：中间的洞从 c² 变成 a² + b²——勾股定理自己浮现。
• 圆的面积，藏着一个小秘密 把圆切成小块，拼成长方形，亲眼看看 πr² 是怎么来的。
• A circle's area hides a little secret The same proof, told for kids — scroll to cut, zip, and rebuild the circle.
• τ — one full turn Sweep a radius through one full turn and watch both 2πr and πr² appear.
• Why is a circle's area πr²? Slice the circle into wedges, rearrange them into a rectangle, and watch πr² fall out.

Physics · 物理

• 描述水流的方程，会失效吗？ 调节水流箭头、拉伸三维涡旋，弄懂纳维–斯托克斯千禧年难题究竟在问什么。
• Can water’s equations break? Explore moving arrows and a stretching 3D swirl, then discover what the Navier–Stokes million-dollar question actually asks.
• 月亮为什么有圆缺？ 把月相拆成三个零件：太阳照亮半个月球、月球绕地球转、我们从不同角度看见这半个亮面。
• 铁做的船，为什么不沉？ 同一块 1 千克的铁：实心时只能排开 127 克水，追不上自重——沉；捏成碗，排开的水涨到 1000 克，浮力追上重力的那一刻，船稳住了。
• Why does a chopstick look bent in water? Take seeing apart into three pieces: light leaves the tip, bends at the surface, and your eye traces it straight back — reassemble them and the bend appears.
• 筷子插进水里，为什么弯了？ 把「看见」拆成三个零件：光从筷尖出发、在水面拐弯、眼睛沿直线反推——拼回去，弯筷子自己浮现。

Math · 数学

• Don’t miss the bus: planning with time Plan a Saturday on a fridge whiteboard: fit activities before the bus, work out when to leave, and check Max’s math. 2:50 plus 20 minutes is not 2:70, because the clock trades at 60 minutes, not 100. It’s 3:10, and the 3:05 bus is gone.
• Zoom from a kilometre to a millimetre One continuous zoom from a whole town (1 km) down to a grain of sugar (1 mm). Every click is ×10, so three clicks is ×1000: kilo means a thousand and milli means a thousandth, the same trade as place value. Then the same three clicks turn a kilogram into grams and a litre into millilitres.
• Flip and slide: fit the shadow A puzzle cartridge: move a flag onto its shadow using only slides and flips, inside a move budget. Slides change where it is, flips change which way it faces, and nothing ever changes its size. Some shadows can’t be reached, and you have to say why.
• Treasure grid: across first, then up Dig for treasure on a grid map. The clue (2, 5) means 2 across, then 5 up. Swap the numbers and you dig in the wrong place. Then write clues for a parrot who reads them literally, and race Captain Crab to find a hidden chest.
• Is a square a rectangle? The trial The Court of Shapes puts seven suspects on trial with a square-corner tool, parallel rails and a fold test. Pip says a square can’t be a rectangle because its sides are all the same. The rulebook only asks for four square corners.
• Does "unlikely" mean "never"? Madame Odds says nobody ever lands on the gold slice. Spin her wheel once, ten times, a thousand times: unlikely still happens if you spin enough. Then put spinners and bags of marbles on the chance line from impossible to certain, test each one, and build spinners to order at the carnival booth.
• How fast are you? Sort your own reaction times Play a 10-round tap test and get your own reaction times. They come out in a jumble. Split each one into tens and ones, file the ones digit on its tens stem, and a stem-and-leaf plot sorts them. Tilt it: it is a bar graph you built by sorting, with every number still in it.
• Can a graph lie about pets? Theo drew Room 4's favourite pets as a line graph, and the line says pet love is falling. Be the editor of the class paper: sort survey answers into words and numbers, run the shuffle test, then stamp which graph goes to print. Categories get bars; a line only joins things that come in order.
• Each crate holds 6. How many in 4 crates? Run the juice factory line: set the dials, press Run, and watch the crates fill. The new helper Dot reads 'each crate holds 6, fill 4 crates' as 6 + 4 = 10 bottles, and two crates stay empty. Each and per mean times: 4 crates × 6 each = 24. Then flip it around to find the crates, the crate size, the minutes, and the price.
• How do you find the number that went in? Number machines changed a secret number: +5 then ×2 made 26. Build an undo machine to run it backwards. Every + is undone by −, every × by ÷, and the last machine has to be undone first, like taking off shoes before socks. Then use undo to catch the wrong answers in Sam's homework.
• What makes something a times question? A frog pond to play in: put frogs on lily pads, line the pads up as rows, turn the rows around. The pond notices what you make: 3 + 4 + 2 isn't a times question because the groups aren't equal, 4 rows of 3 is a rectangle, 5 × 1 doesn't get any bigger, and 6 empty pads hold 0 frogs.
• What does borrowing really do in 52 − 27? A comic about Mo's snack stall, where 10 loonies trade for one $10 bill. Mo can't pay 7 loonies out of 2, so he trades a ten at Gran's counter: 4 tens and 12 loonies, still $52. Carrying and borrowing are the same trade, and trading never changes how much money there is.
• Is 47 closer to 40 or 50? You're the tram conductor: every passenger walks to the nearest station, on lines with stations every 10, 100 and 1000, plus a night line of decimals. Chip the robot just chops the last digits off, so 47 walks 7 stops back to 40 instead of 3 to 50. Beat Chip and find the one digit that decides.
• Why does a loop inside a loop multiply? A garden robot runs code: one step after another, two robots at once, a repeat loop, then a loop inside a loop. Repeat 3 around repeat 4 looks like 3 + 4 = 7 flowers. Run it: 12, because the inner loop runs 4 times on every lap.
• How many answers can □ + 3 < 10 have? □ + 3 = 10 has one answer, 7, so lots of kids say □ + 3 < 10 is 7 too. Test every number on a number line and a whole stretch lights up: 0, 1, 2, 3, 4, 5 and 6. And 7 makes it equal, not less.
• What does the letter stand for? Each bag holds b beads, and Mia has 3 × b. Lots of kids say b means “beads”. Open the bags and b turns out to be a number: how many are in each bag. Change it and the total changes.
• Shapes, table, graph: one pattern? A tile train grows 2, 5, 8, 11. Write it as a table, plot it as a graph, and it’s still the same pattern. The trap: plotting the “+3” instead of the totals. The +3 is the climb between dots.
• Does a pattern repeat or grow? A repeating pattern goes round the same core; a growing pattern adds by a rule. Step 5 of a tile tower has 11 tiles, so is step 10 double, 22? Build it and it’s 21.
• What does the = sign really mean? In 8 + 4 = □ + 5, lots of kids write 12. Put both sides on a balance and 12 tips it over: = doesn’t mean “the answer comes next”, it means “the same as”. The box is 7.
• Why is area rows × columns? Cover a rectangle with square tiles and count them: 3 rows of 4 is 12. Two pens with the same 16-unit fence hold 12 and 16 tiles, so the fence doesn’t tell you the area.
• What happens to the leftovers? Deal 13 cookies to 4 friends and 1 is left over. Is it thrown away? Cut it into 4 pieces and everyone gets 3 and 1/4: a remainder is just a share that isn’t whole yet.
• What is 3 × 1/4? 3 × 2 is three hops of 2. So 3 × 1/4 is three hops of a quarter: lay three quarter pieces end to end and you get 3/4, not 3/12.
• Is a 3 always worth three? Ten cubes snap into a rod, ten rods into a flat, ten flats into a block. Unpack the three 3s in 333 and see the same digit is worth 300, 30 or 3, depending on its seat.
• What makes an angle bigger? Two arms and a corner. Is the angle with longer arms bigger? Turn the arm, stretch it, and lay one angle on top of another: an angle measures the turn, not the arms.
• Why is 1/2 the same as 0.5? Snap a chocolate bar into 10 squares and the first digit after the point counts them. Bust the trap that 1/2 is 0.2: half the bar is 5 squares, so 1/2 = 5/10 = 0.5.
• Can a fraction be bigger than 1? A frog hops along a number line in quarters: 1/4, 2/4, 3/4, 4/4… and then 5/4. Bust the trap that fractions stop at 1, and count past it by halves, thirds and tenths.
• Why isn't the average just the middle one? Five friends read 1, 3, 3, 5 and 8 books. Level their towers of cubes, tall ones giving to short ones, and find the mean: 4, which isn't the middle one and isn't anyone's tower.
• Why isn't ×10 just "add a 0"? Make 10 copies of 34 and watch every digit slide one place left. Then try the add-a-0 trick on 0.3 and see it fail: the 0 only fills an empty seat.
• Why is 1/8 smaller than 1/3? Share one snack bar with 3 friends, then with 8. Line the pieces up on a ruler and bust the trap: a bigger bottom number means more sharers, so a smaller share.
• a³ − b³ 为什么等于 (a − b)(a² + ab + b²)？ 从大立方体上挖掉一个小立方体，把剩下的切成三块、转一转、排成一排：它们共用同一条边 a − b，公式自己拼出来。
• Why is a³ − b³ = (a − b)(a² + ab + b²)? Dig a small cube out of a big one, cut the leftover into three boxes, and line them up: they all share one edge, a − b — and the formula builds itself.
• Why does 7 × 13 equal 70 + 21? Multiplication is a rectangle. Cut 13 into 10 + 3, pull the pieces apart, and a(b + c) = ab + ac appears on its own — then test yourself on the minus-sign and square-root traps.
• Radicals Unpacked — The Radical Lab Pair factors, uncover hidden like terms, and build exact answers. A hands-on square-root and cube-root lab for Grade 8–9 enrichment.
• Fraction Snack Lab: Meet the Two Numbers Split one whole snack into equal pieces and discover how the numerator counts chosen pieces while the denominator counts all the pieces.
• Why can different fractions be equal? Cut the same whole into smaller pieces and see why 1/2, 2/4, and 4/8 represent exactly the same amount.
• 为什么不同分数可以相等？ 把同一个整体越切越细，看见 1/2、2/4 和 4/8 为什么表示完全相同的量。

Computing · 计算机

• 计算机为什么只用 0 和 1 就能数数？ 一台弹珠翻板计数机：每块翻板只有倒左和倒右两种姿势，碰到 1 就翻回 0、把弹珠往下传。看第 16 颗弹珠把 1111 一路翻成 10000。

Biology · 生物

• Why does a raisin swell in water? A kitchen lab notebook for diffusion and osmosis. Drop dye into water and watch particles nobody steers spread out, put in a membrane only water can cross, soak a plant cell and a red blood cell in salty water, then run a potato-strip lab that measures what is inside a potato. Water moves toward the saltier side, and it costs no energy.
• Take a cell apart: a 3D cell you can explore Turn, slice and take apart a 3D cell. Tap each part to see its job, turn an animal cell into a plant cell, unfold a mitochondrion, follow a protein out of the cell, then solve cases by tapping the right part. Every part has a job, and its shape gives the job away.