Sums

Further than you think

How Far Away Is the Moon?

The Moon is 384,000 km away — far enough to fit every other planet in between.

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Bigger than your brain wants

How Big Is the Sun?

The Sun is so big that a million Earths would fit inside it.

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Putting together

What Does Adding Do?

Adding puts two groups together into one bigger total.

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A recipe with no judgement

What Makes Something an Algorithm?

An algorithm is a sequence of steps precise enough that a machine can follow it without judgement.

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Counting squares, quickly

Why Do We Multiply to Find Area?

Area is the number of squares covering a shape, and multiplying rows by columns counts them fast.

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Counting without counting

Why Are Rows and Columns Useful?

An array arranges things in rows and columns so you can multiply instead of count.

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Levelling everyone off

What Does an Average Actually Tell You?

An average is what everyone would get if the total were shared equally, which can hide a lot.

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The small things add up first

Why Does the Money Run Out Before the Month?

Budgets go wrong because small, frequent spending adds up faster than the large purchases you notice.

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More orders than atoms nearby

Has Anyone Ever Had This Hand Before?

A pack of cards has so many possible orders that a well-shuffled deck has probably never existed before.

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Counting up, not taking away

How Do You Work Out Your Change?

Change is easiest worked out by counting up from the price to the note you handed over.

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Interest on the interest

Why Does Saved Money Grow Faster Each Year?

Compound interest pays you on last year's interest as well, so growth speeds up every year.

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One thing, one word

How Does Counting Work?

Counting works by pairing each object with one number word, in a fixed order.

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Count a little, multiply a lot

How Do You Count a Huge Crowd?

Crowds are counted by counting one small area and multiplying by the number of areas.

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More ways to get there

Why Is Seven the Commonest Roll?

Seven comes up most on two dice because more combinations add up to it than to any other total.

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Taking a slice off the top

How Do Shops Work Out a Sale Price?

A discount takes a percentage off the old price, so the saving depends on what the price was.

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Fair shares

How Does Division Work?

Division shares a total into equal groups and tells you how much each group gets.

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Two of the same

What Does Doubling Mean?

Doubling means adding a number to itself, and it grows shockingly fast.

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Balance, not 'the answer'

What Does the Equals Sign Really Mean?

The equals sign means both sides balance, which is why 7 = 4 + 3 is perfectly correct.

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Close enough on purpose

What Is Estimating?

Estimating gives you a fast, roughly right answer instead of a slow exact one.

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Comparing

Which Number Is Bigger?

A bigger number is simply one you reach later when counting.

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Small steps that stop being small

Why Does Doubling Get Out of Hand?

Repeated doubling grows slowly at first and then explodes, because each step is bigger than everything before it.

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One giant number spoils it

Why Is the Average Sometimes a Lie?

An average is dragged by extreme values, while the middle value ignores them.

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Making any amount

Why Do Coins Have Different Values?

Coin values are chosen so you can make any amount with only a few coins.

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Many equal groups

What Does Multiplying Do?

Multiplying is repeated adding of equal groups, done in a single step.

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Numbers as places

What Is a Number Line?

A number line turns numbers into positions, so arithmetic becomes movement.

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Out of a hundred, always

What Does Per Cent Actually Mean?

Per cent means out of a hundred, which lets you compare quantities of different sizes.

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The fence and the field

How Are Area and Perimeter Different?

Perimeter is the distance round the edge and area is the space inside, and one does not fix the other.

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Not everything scales

How Do You Double a Recipe?

Doubling a recipe doubles the ingredients but not the pan, the time, or the temperature.

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The bit left over

What Is a Remainder?

A remainder is what is left after making as many equal groups as possible.

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Close enough, on purpose

Why Do We Round Numbers?

Rounding deliberately gives up small accuracy to get a number that is easier to think with.

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Faster than one at a time

Counting in Jumps

Skip counting means counting in equal jumps instead of one at a time.

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One triangle, three questions

How Long Will the Journey Take?

Speed, distance and time form one relationship you can rearrange to answer any of the three.

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Sharing, with an awkward remainder

How Do You Split a Bill Fairly?

Splitting a bill is division until there is a remainder, and then it is a decision.

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What is left

What Does Taking Away Do?

Subtraction finds what is left, or how far apart two numbers are.

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Counting without losing your place

Why Count in Groups of Five?

Tally marks bundle counts into fives so you can add up groups instead of recounting one by one.

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Shortcuts, not memory

What Patterns Hide in the Times Tables?

The times tables hide patterns that turn dozens of separate facts into a handful of shortcuts.

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Ten per cent, then adjust

How Do People Work Out a Tip in Their Head?

A tip is worked out by finding ten per cent and adjusting up or down, not by exact sums.

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The number under the price

Which Packet Is Actually Cheaper?

Unit price divides cost by amount so packets of different sizes can be compared.

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Stacking layers of squares

How Much Does a Box Actually Hold?

Volume is the number of unit cubes that fit inside a space, found by stacking layers of area.

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A number your eye reads first

Why Does Everything Cost £4.99?

Prices end in 99 because people read the first digit and round the rest away.

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Physics run forward

How Can Anyone Predict Tomorrow?

A forecast measures today's air everywhere and calculates it forward using known physics.

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Four clocks and some arithmetic

How Does a Phone Know Where It Is?

Nothing tracks your phone: it times signals from several satellites and works out the only place it could be.

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