∞ Grade 5
For curious 10 and 11 year olds. The grade 5 course with the reasons left in, puzzles and patterns the course never reaches, and where numbers show up in the world. Three tracks: Sharpen (school math, and why it works), Around the corner (new ideas), Connections (where it shows up). One sheet a night, ten to fifteen minutes, and a starred problem if you want more.
Sharpen the grade 5 course, with the why
Place value, decimals, long multiplication, remainders, fractions (equal, added, multiplied, bigger than one, divided), volume, coordinates, brackets, rules, shapes and estimating.
Ten times bigger, ten times smaller
Every digit is worth ten times the one on its right. Slide the digits and you can multiply or divide by 10, 100 or 1000 without calculating.
What lives between 0.3 and 0.4?
Zoom in on the number line and there is always room for more. Tenths, hundredths, thousandths, and why "0.35 is bigger than 0.4" is false.
Why long multiplication works
23 × 47 is the area of a rectangle cut into four pieces. The method you were taught is those four pieces, added up.
What to do with the remainder
50 ÷ 6 can be 8, 9, 8 remainder 2, or 8 and a third. The story decides which answer is right.
The same fraction in disguise
1/2, 2/4 and 50/100 are one number with three names. Cut the pieces smaller and nothing changes but the label.
Why you cannot add thirds to quarters
Not until the pieces are the same size. Cut both into twelfths and the sum is easy.
Half of a third
Multiplying fractions is taking a part of a part. One picture shows why you multiply the tops and multiply the bottoms.
Fractions bigger than one
Seven quarters is one and three quarters. Move between the two ways of writing it, and find where each sits on the number line.
How many halves are in three?
Dividing by a half makes the answer bigger. That stops being strange once division means "how many fit".
Where does the decimal point go?
0.3 × 0.2 is not 0.6. Tenths of tenths are hundredths, and one square picture shows it.
Counting cubes without counting
Volume is layers. Count one layer, multiply by the number of layers, and you have length × width × height.
Treasure maps: coordinates
Two numbers pin down a point. Along first, then up. Plot shapes, slide them, and find the missing corner.
Brackets say what goes first
2 + 3 × 4 and (2 + 3) × 4 are different sums. Brackets are how you tell a reader which one you mean.
What's my rule?
A machine takes a number in and gives a number out. Find the rule from a table, then run it backwards.
Is a square a rectangle?
Yes, and here is why. Shapes belong to families, and a family is defined by a rule, not by a look.
Good enough answers
Round first, then calculate. Estimating tells you whether an exact answer is sensible, and often it is all you need.
Around the corner new ideas
Proofs you can see, dot patterns, prime fingerprints, old number systems, 6174, palindromes, tangrams, tilings, cube nets, corners and edges, counting routes and handshakes, coin streaks, logic grids, Latin squares, the fake coin, river crossings, a game you cannot lose, endless halving, paper folding and mazes.
Odd plus odd is always even
Not because you checked a hundred examples. Draw odd numbers as pairs with one left over and you can see it must be true.
Square numbers and triangle numbers
Some numbers make squares of dots, some make triangles. Two neighbouring triangles always make a square.
Every number's fingerprint
Break a number into factors until you cannot break any more. Whichever way you start, you end with the same primes.
Numbers before zero
Roman numerals, Egyptian strokes, Babylonian sixties. Try adding in them, and see why our ten digits won.
The number 6174
Take four digits, sort them big and small, subtract, repeat. You always arrive at the same number.
Reverse and add
Take a number, reverse it, add. Do it again. Most numbers turn into a palindrome fast. One small number has never been seen to.
Seven pieces, one area
The tangram's seven pieces make a square, a cat, a boat. The shape changes; the area cannot.
Shapes that tile
Squares tile a floor and so do triangles and hexagons. Regular pentagons do not, and the angles say why.
Eleven ways to unfold a cube
Six squares joined edge to edge. Some fold into a cube and some do not. There are exactly eleven that do.
Corners, edges and faces
Count them on a cube, a pyramid, a prism. Corners plus faces minus edges is always 2.
How many ways home?
Walk only right and up on a grid of streets. Count the routes to each corner and a pattern appears.
Everyone shakes hands
Ten people, each shakes hands with every other. How many handshakes? Count it badly, then count it well.
Three shirts, four trousers
Twelve outfits, and a tree diagram shows every one. The same number of choices on every branch means you can multiply.
Heads, heads, heads, heads
Flip a coin fifty times and write it down. Real luck has longer streaks than people expect, and that is how fakes get caught.
Who owns the fish?
Four children, four pets, a handful of clues. A grid of ticks and crosses turns guessing into certainty.
Every row, every column, once
A Latin square puts each symbol once in every row and column. Build them, count the small ones, and meet the idea behind Sudoku.
Sorting with circles
Multiples of 2 in one circle, multiples of 3 in the other. What lives in the overlap, and what is left outside?
Find the fake coin
Nine coins, one lighter, a balance with no weights. Two weighings are enough if you split into three piles, not two.
The wolf, the goat and the cabbage
A boat that holds two, and things that must not be left together. Draw every position as a dot and the puzzle becomes a map.
The race to 21
Take turns adding 1, 2 or 3. Whoever says 21 wins. Work backwards from the end and, if you go first, you can always win.
Half, and half again
Walk half way to the wall, then half of what is left, for ever. The pieces add up to exactly one.
Fold a paper to the Moon
Each fold doubles the thickness. Seven folds is hard; forty-two would reach the Moon. Doubling is faster than it looks.
How to escape any maze
Keep one hand on the wall. It works for most mazes, and the ones where it fails show why.
The area of your hand
Draw round your hand on squared paper and count. Whole squares, part squares, and how to get a good answer for any shape.
Connections time zones · maps · shopping · races · heartbeats · paper · blocks · flags
The same math, found in the wild.
What time is it in Tokyo?
The Earth turns once in 24 hours, so the world is cut into 24 time zones. Adding and subtracting hours, across midnight.
One centimetre is one kilometre
A map is the world shrunk by the same amount everywhere. Read the scale, measure with a ruler, and plan a walk.
Which pack is the better buy?
Six for $4 or ten for $6? Find the price of one, and the shelf stops fooling you.
Won by a hundredth of a second
Races are timed to hundredths. Read the stopwatch, order the finishers, and see how far a sprinter runs in 0.01 s.
A billion heartbeats
How many times has your heart beaten so far? Multiply by minutes, hours, days and years, and meet a very big number.
Why A4 is the shape it is
Cut A4 in half and you get A5, the same shape, smaller. Only one shape of rectangle can do that.
Building with blocks
How many blocks to build a wall, a tower, a hollow house? Volume, layers, and counting what you cannot see.
Fractions and symmetry in flags
Halves, thirds and quarters, mirror lines and half turns. The world's flags are a geometry lesson in colour.
Demos play with it
Eight new interactive demos go with this grade, from sliding digits to a game you can always win, alongside the other grades' demos. All of them live in the demo gallery.