Finding square roots of numbers…
1. What is a square number? Choose which of the following are square numbers and
explain why or why not. Does a square number have to be a whole number? Does the
number that was “squared” need to be a whole number?
9, 25, 64, 100, 169, 56, 1.44
add stuff on perfect squares and square roots
Square numbers are usually whole numbers that are the square of whole numbers
(integers) eg 62 = 6 x 6 = 36 so 36 is called a perfect square.
The number that has been squared is called the “square root” of the “squared number”.
Most calculators have a special button that indicates a function for finding the “square
root” of a given number. Later?
2. Suppose that this square root function is not working or not available on the
calculator. We will find the answer by some different methods that use some
estimation and simple calculations.
Method 1
a. Suppose we want to know the square root of 75. ie 75 . We want to find a
number so that when that number is multiplied by itself, or “squared”, the answer is
75.
 Taking some sensible guesses is appropriate. We know that 82 = 64 and 92 is
81 so we can see that 75 will be somewhere between 8 and 9.
 Try, say, 8.5. Now 8.52 = 72.25 which is less than 75 so 8.8 is too small, so
we try another guess between 8.5 and 9, say 8.8.
o Calculator instructions  or 
which is quicker.
2
 8.8 = 77.44, which is greater than 75 so 8.8 is too big.
 Try 8.7
8.72 = 75.69 which is still too big.
 Try 8.6
8.62 = 73.96 which is too small.
 Try 8.65 which is between 8.6 and 8.7 8.652 = 74.8225
o On the calculator we can check if one number is less than another
using the Problem solving Manual mode:
o Press and underline Man and press to save it.
o Type in (press this button once for < and twice for >)
 and the calculator says either YES or NO to confirm the
statement.
o Students can use this message to check whether a guess is suitable or
not, or of course just try it out!
2
 8.67 = 75.1689
 8.662 = 74.9956, so now we need to try a number that is between 8.66 and
8.67
 8.6652 = 75.082225 which is getting closer….
A useful technique to reinforce the place value and sizes would be for the teacher to
record the guesses on a number line on the board, perhaps with another line below for
the squares….
Depending on how accurate we want the answer to be we can keep on “refining” our
guess until we are happy with the result. Using the square root button on the
calculator gives us 75 = 8.660254038 which would have taken a long to get to by
guessing! Perhaps 8.66 would have been accurate enough.
This is good practice for understanding decimal places etc.
b. Try these square root questions and record all your guesses and their squares as
you go, until you have an accuracy you are happy with.
i.
ii.
iii.
iv.
Find two whole numbers (integers) that the square root is between for:
95, 18, 120, 500, 2350.
Find the square root of each of these numbers correct to two decimal
places by using and checking suitable guesses.
Now check your answers on the calculator.
Can you find a number that the calculator cannot find a square root for?
c. Suppose the number is smaller than 1? eg 0.9 Would you expect the answer to
be bigger or smaller than 0.9? Use the calculator to find out. Investigate.
Method 2.
a. Another way of illustrating what a square number can represent is to draw a square
and find its area. The area is the “square” of the side length.
 What is the area if one side has length 5 units?
 What is the area if one side has length 2 units?
 What would the side length be if a square has area 9 square units (or units2) ?
 What would the side length be if a square has area 100 square units (or units 2) ?
 What would the side length be if a square has area 56 square units (or units2) ?
 What would the side length be if a square has area 1.44 square units (or
units2) ?
b. We can use this area concept idea to estimate the square root of any number by
following these steps…
i.
Given that we want to find the square root of eg 169, we can draw a
“square” with area 169 units2 and estimate its side length.
ii.
Suppose we guess 10 as the side length. The other side would be
169/10 = 16.9 units. This is clearly not a square! Use another guess eg
the average of these two sides.
iii.
Roughly average these two side lengths for another estimate eg
(10+17)/2 =13.5. It is not important whether 16 or 17 is used here. As
it happens in this example if 10 and 16 are used, 13 is the exact answer!
iv.
Now if one side is 13.5, the other is 169/13.5 = 12.5. The shape is still
not a square!
v.
Average 13.5 and 12.5 for the next guess = 13. Now 13 x 13 = 169!
We have a square, so the square root of 169 is 13.
c. Repeating this exercise to find the square root of 60.
i.
First guess eg 10, the other side is 60/10 = 6. Not a square.
ii.
Next guess (10+6)/2 = 8. Other side 60/8 = 7.5. Not a square.
iii.
Next guess (8+7.5)/2 = 7.75. Other side is 60/7.75 = 7.74. Fairly close!
iv.
Next guess (7.74+7.75)/2 = 7.745. Other side 60/7.745 = 7.746.
Closer!
v.
We could say the square root of 60 is fairly close to 7.74. Check 7.74 x
7.74 = 7.742 = 59.9. And 7.752 = 60.06.
d. Use this method to find the square root of ……
e. Check your answers on your calculator using the square root function..
3. Extension question
THE UNKNOWN SQUARE
a. What is the total area of this large
square in which A and B are squares?
A
36 m2
b. What is the area of the shaded
rectangles?
B
16 m2
Extension to perfect cubes and cube roots?
To find the “cube” of a number the number is multiplied by itself three times ( or is it
twice??) eg 53 = 5 x 5 x 5 = 125
Find 43, 63, 103, 2.63
To find the number which has been “cubed” we need to find the “cube root”. There is
no special button or function on the calculator to do this.
Using trial and error as in the square root examples, find the cube root of 27; 343;
1000; 54; 278.6
Playing with the  button:
Does the  button always make the number bigger?
Try  which = 1 000 000
Try  which is smaller than 100!
Try . This gives a much smaller answer…. 10. But this is the square
root of 100!
We know that 82 = 64. Check on your calculator.
We also know that 64 = 8. Check on your calculator.
This means that 640.5 = 8. Check  = 8.
But 43 = 4 x 4 x 4 = 64.
This means the cube root of 64 is 4.
So 64 ^??? = 4…….? Experiment with different decimal numbers in the ??? place
until you get exactly 4 as the answer.
Find a fraction that will get exactly 4 as the answer.
Find the cube root of 729.
Investigation
To find the square root of a number, the square root button can be used. It is also
possible to use the ^ button too.
Try  and  Each should give you the answer 4. You can
check that 42 =16 by pressing .
Make up some more questions like this so you are sure what is happening. Start with
some perfect squares that you know.
You can set these two operations automatically using the and  functions.
To set do this:
 Press . ( Nothing shows on the screen!)
 Put in the operation you want: in this case  which will find the
square root of any positive number.
 Press  again to lock it in, and you will see a very small Op1 symbol at the
top of the calculator screen.
Set  the same way but this time lock in .
Press  and you will see 25^Op1 and 5 will appear as the answer. Now do 5
 and you are back to 25.
(You can hide the operation that has been set if you wish to by following these steps:
 Press  and once.
 The selection +1 ? is displayed. This lets you select to either hide or show
the operation.
 Press to underline the ? and press  to confirm.
 Press to exit the menu. )
Start with any number greater than 1.
Press  repeatedly and see what happens. The answer is getting smaller and smaller.
Does it eventually reach 0? Investigate.