Square numbers, cube numbers and calculating them

Two orders come into the same builders' yard on the same morning, and both of them say the number eight. The first is a patio. Eight slabs along one edge, eight along the other, filled in flat. The second is a storage block. Eight crates along the edge, eight across, and eight high, stacked solid. Same number both times. The patio needs sixty-four slabs. The block needs five hundred and twelve crates. Send the wrong one and you are out by four hundred and forty-eight.

So, the two shapes first. Square numbers and cube numbers are named after real shapes, and once you can see the shapes you stop guessing.

Start with the patio. Four slabs along one edge, four along the other. That is four rows of four. Four times four is sixteen, and they make a perfect square. So sixteen is a square number. Nine is a square number too, because three times three gives it.

Now the crates. Four along the edge, four across, and four high. Four times four is sixteen, and sixteen times four is sixty-four. Sixty-four crates, stacked into a perfect cube. So sixty-four is a cube number: a whole number multiplied by itself, then by itself again.

Here is the phrase to carry into the exam hall. Flat needs two. Box needs three. A flat square multiplies two of the same number together. A solid box multiplies three.

Now build the two side by side, from the same starting numbers. One, two, three, four, five. Square them and you get one, four, nine, sixteen, twenty-five. Cube the same five and you get one, eight, twenty-seven, sixty-four, a hundred and twenty-five.

Look at how fast the second row runs away. Squares grow because you are filling a flat area. Cubes grow because you are filling a solid volume, so every step adds a whole new layer on top. That is why the two lists pull apart so quickly after the start.

Your turn, and this one is a single step. One of these three numbers is a cube number. Eight, nine, or ten. Which one, and what multiplication proves it?

Have a think. I'll wait.

The answer is eight. Two times two is four, and four times two is eight. Three twos multiplied together, so eight is a cube number. Nine is a square number, three times three. Ten is neither. And notice you proved it with a multiplication, not by remembering a list.

Now, calculating them. Spotting one is a different job from working one out from scratch.

Work out twelve squared. Squared means twelve times twelve. Ten twelves is a hundred and twenty, and two more twelves is twenty-four. Add those and you get a hundred and forty-four. So twelve squared is a hundred and forty-four. Now four cubed. Four times four times four. Four times four is sixteen. Sixteen times four is sixty-four. So four cubed is sixty-four. Watch the order there: square it first, then multiply by the number one more time. That keeps three copies. Never two, never four.

Do enough of those and the common ones stick on their own. The squares up to fifteen run one, four, nine, sixteen, twenty-five, thirty-six, forty-nine, sixty-four, eighty-one, a hundred, a hundred and twenty-one, a hundred and forty-four, a hundred and sixty-nine, a hundred and ninety-six, two hundred and twenty-five. The cubes are a far shorter list. One, eight, twenty-seven, sixty-four, a hundred and twenty-five. That is one to five cubed. Add ten cubed, which is a thousand, and you have the ones worth knowing without stopping to work them out.

One flag before we move on. Negative four, squared. Squaring means multiply the number by itself, so it is negative four times negative four. Two negatives multiplied give a positive. The answer is sixteen, not negative sixteen. Same sign rule as always, applied to the number that is actually there.

Time for a bigger question. Sorting the two lists apart, which is exactly where this topic goes wrong.

Six numbers. Nine, twenty-seven, thirty-six, sixty-four, ninety, and a hundred and twenty-five. Put each one into square, cube, or neither. And one of them is going to sit in two places at once. Which one?

Pause here and work through all six of them, one multiplication at a time. I'll wait.

The squares are nine, thirty-six and sixty-four. Three times three is nine. Six times six is thirty-six. Eight times eight is sixty-four. The cubes are twenty-seven, sixty-four and a hundred and twenty-five. Three threes multiplied give twenty-seven. Three fours give sixty-four. Three fives give a hundred and twenty-five. Ninety is neither. No whole number times itself gives ninety, and none does it three times either.

And there is the number sitting in two places. Sixty-four is a square number and a cube number. Eight times eight is sixty-four, so it is square. Four times four times four is sixty-four, so it is cube. Reciting one memorised list will only ever find it in one of them.

Now the exam side. What examiners actually see go wrong here, in their own words.

First, the format. These questions usually ask you to list every square number or every cube number inside a range. Here is one. List all the two-digit square numbers. Two-digit means from ten up to ninety-nine.

Pause and write them down. I'll wait.

The answer is sixteen, twenty-five, thirty-six, forty-nine, sixty-four and eighty-one. Six of them. Here is the method that finds them. Walk up from the start. One times one is one, too small. Two times two is four, too small. Three times three is nine, still one digit. Four times four is sixteen, and you are in. Keep going to nine times nine, which is eighty-one. Ten times ten is a hundred, three digits, so you stop.

This is the part examiners report on. Their words, from a Foundation paper report:

A small majority of students were able to identify all five 3-digit cube numbers but many also listed 64 and 1000. A significant number of students did not understand the concept of a cube number.

Read what actually went wrong. Sixty-four and a thousand are genuine cube numbers. Four times four times four, and ten times ten times ten. They were marked wrong because they have two digits and four digits, not three. The range is half the question, and it is the half that gets skipped.

Then read the second sentence. A significant number did not understand the concept at all. That is why every cube number here arrived with its multiplication attached. Say the multiplication out loud and the concept cannot slip.

The other error they report is the two lists crossing over. Again, their words:

Although many students listed square numbers, a common error was listing cube numbers instead of powers of 3.

Squares written where cubes belong, and cubes written where something else belongs. Flat needs two. Box needs three. Check which of the two the question asked for before you write a single number down.

Right. Both lists, side by side.

A square number is a whole number multiplied by itself. Four times four is sixteen. A cube number is a whole number multiplied by itself, then by itself again. Four times four times four is sixty-four. Flat needs two. Box needs three. The squares run one, four, nine, sixteen, twenty-five, and on up to two hundred and twenty-five at fifteen. The cubes run one, eight, twenty-seven, sixty-four, a hundred and twenty-five, and a thousand at ten. Sixty-four sits in both lists. Squaring a negative number gives a positive answer. And when a question sets a range, check the range on every number before you write it down.

Next in this chapter: working the opposite way, starting from a number like a hundred and forty-four and finding what was multiplied to make it.

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