Square numbers and square roots

Equal rows

\(\displaystyle 2\)

Counters in each row

\(\displaystyle 5\)

A quick multiplication check: how many counters are there altogether?

Count five counters twice: five plus five

Each small box is one square cell. Predict the total number of cells in this larger square

Predict the total when there are four small squares along each side, then set the control to four and check. Next, work backwards: find the side count that makes a total of twenty-five cells. Count rows if you need to

Which relationship worked in both directions?

Squaring uses two equal factors

To square a number, multiply it by itself. The small raised 2 tells you to use two equal factors. It does not mean multiply by two

Read squared notation

Read this as four squared. It counts four rows of four cells

Zero works too

The same multiplication rule applies, even though zero cells do not make a visible array

One stays one

One row of one cell contains one cell

Evaluate this square without using the array

Multiply six by itself. Enter the resulting number, not the expression to be evaluated

A square root works backwards

A perfect square is the square of a whole number. Its square root is the non-negative number that multiplies by itself to make it. Non-negative means zero or greater

Read the root symbol

Eight squared is sixty-four, so the square root of sixty-four is eight

Check by multiplying

Do not halve the number. Check the proposed root by multiplying it by itself

Boundary cases

Zero and one are their own square roots

Find this square root. Use an equal-factor multiplication to check your answer

Which non-negative whole number times itself gives eighty-one? Enter that number, not a square-root expression

A student doubles the base when trying to square it. Which instruction fixes the method?

A student halves the number under a square-root sign. Which multiplication correctly checks the root instead?

Recover a side from an area

For a square, area is side length times side length. If you know the area, a square root gives the side length

Square centimetres count area

A square with area sixteen square centimetres has side length four centimetres, because four times four is sixteen

Length and area use different units

Give the side in centimetres, not square centimetres. In these questions the field supplies the length unit

Area

\(\displaystyle 121\text{ cm}^2\)

A square card has this area. Find one side length. The area is the side length multiplied by itself

Find two equal factors whose product is one hundred and twenty-one. Give the length in centimetres, not square centimetres or metres

Some roots lie between whole numbers

When the number is not a perfect square, compare it with the nearest squares below and above. Their roots give consecutive whole-number bounds. Consecutive means one apart

Compare squares first

Twenty is between sixteen and twenty-five

Then compare their roots

So the square root of twenty is greater than four and less than five. The less-than signs point towards the smaller number. No decimal approximation is needed

Root

\(\displaystyle \sqrt{59}\)

Choose the consecutive whole-number bounds. Compare neighbouring square numbers, not rounded decimal guesses

Bounds check