Squares: the second power · Martinとコード

What a number to the power of two means, how to calculate it, and why a square of numbers is useful.

A square made of numbers

The second power is a shortcut. Instead of 7 · 7 you write 7² and read it “seven squared”. It means: multiply the number by itself. That is all.

Why “to the power of two”? Because the number stands in the multiplication twice. 7² = 7 · 7 = 49.

And why a square? Picture a square with side 7 cm. Its area is 7 · 7 = 49 cm². The second power is the area of a square. That is why English calls it “square”.

You will need it all year. Pythagoras, formulae, the area of a circle — that little number up top shows up everywhere. If you know the squares up to twenty by heart, half of year 8 gets easier.

Today you will learn to read, write and calculate a power. And you will watch out for one trap that almost everyone falls into.

Adam is putting photos on a board. He wants a square: the same number of rows and columns. He has 5 rows of 5 photos. “How many do I need?” he counts on his fingers. Ema looks over his shoulder: “5 squared. Twenty-five.” Adam tries a bigger square, 8 × 8. Ema answers before he can reach for his phone: “64.” Adam does not believe it: “You’ve got that in your head?” — “I have. Squares up to twenty. It took me a week and since then maths has been faster.” That evening Adam writes a table on paper.

💡

Write the squares from 1² to 20² on paper and stick them above your desk. In a week you will know them by heart — and this year you will use them a hundred times.

How to calculate n²

The notation n² means n · n. The small number on top is the exponent and it says how many times the number stands in the multiplication. The bottom number is the base.

The steps are always the same:

Read the notation: 12² = “twelve squared”.
Write it as multiplication: 12 · 12.
Multiply: 144.

It works for every number. A decimal: 0.5² = 0.5 · 0.5 = 0.25. A fraction: (1/2)² = 1/4. With decimals, multiply first without the point and put the point in at the end: the result has as many decimal places as both factors together.

Watch negative numbers. (−3)² means (−3) · (−3) = 9. Minus times minus is plus, so the square of a negative number comes out positive. But −3² without brackets means −(3 · 3) = −9. The brackets decide what gets squared.

And two specials: 0² = 0 and 1² = 1. Zero and one do not change when you square them. On the other hand, no square ever comes out negative — you will need that right away in the next lesson, with square roots.

Example 1: 13²

I want 13². That is 13 · 13.

I split it: 13 · 13 = 13 · 10 + 13 · 3 = 130 + 39 = 169.

You do not have to multiply in columns. Splitting the number into tens and ones is faster and you mix it up less. Check with an estimate: 13 is near 10 and 10² = 100, but also near 15 and 15² = 225. The result 169 sits between them. It fits.

Example 2: (−6)² and −6²

They look almost the same, but they are not.

(−6)² = (−6) · (−6) = 36. The brackets say: square the whole number −6. Minus times minus gives plus.

−6² = −(6 · 6) = −36. Without brackets only the six is squared and the minus stays in front of the result.

One pair of brackets, a difference of 72. That is why brackets in maths are never pointless.

Example 3: 0.4²

0.4² = 0.4 · 0.4.

I calculate without the point: 4 · 4 = 16. Now the point: 0.4 has one decimal place, 0.4 too — the result has two. So 0.16.

Notice: 0.16 is smaller than 0.4. When you square a number between zero and one, the result shrinks. For numbers bigger than 1 it grows. That helps when you check with an estimate.

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The most common mistake: 5² = 10. No! 5² means 5 · 5 = 25, not 5 · 2. The exponent does not say “multiply by two”. It says “write the number twice in a multiplication”.

Practice: now you do it 🎲

Now try it for real. You will see all the questions at once, like on paper. スタート with ten. When it goes well, add more.

Calculate the square. Write only the result.

Practise the questions

On paper

Pencil in hand. Powers are learned by hand, not by eyes.

Write a table from 1² to 15². On each row write the multiplication: 6² = 6 · 6 = 36. Keep the table. You will need it all year.
Calculate and compare: (−5)² and −5². Write one sentence next to each result explaining why they came out different.
Calculate 0.3², 0.7² and 1.2². For each one write whether the result came out bigger or smaller than the base.
A square has side 9 cm. Calculate its area and write it with the unit (cm²).

Now you 💪

I can read 8² as “eight squared” and I know it is 8 · 8, not 8 · 2.
I know why (−4)² = 16, but −4² = −16.
I can calculate squares up to 15 without a calculator.

✅

Done when: You have a table from 1² to 15² on paper, you can explain the difference between (−5)² and −5², and you have a run of five correct in practice.

What to take from this lesson

n² = n · n. The exponent says how many times the number stands in the multiplication.
The second power is the area of a square with side n.
(−3)² = 9, but −3² = −9. The brackets decide.
The square of a number between 0 and 1 comes out smaller than the base.
Squares up to 20 are worth knowing by heart.