Squares & Square Roots Years 3 - 8

Summary Anyone can press a button on a calculator screen and be presented with a string of decimal places as the square root of a number, but what does it mean? This activity builds on the link between a square and its square root, and the array concept of multiplication. The activity has four parts which are developed over time. The whole activity need not be completed in one session. Parts 2 and 4 are activities which can be threaded into the curriculum for a few minutes a day, two or three times a week, over several weeks. Materials Calculator and Poly Plug for each child One full size Plug Frame for each child

Procedure Part 1: Making Squares Ask students to push all the plugs out of their red board and store them in the bag. Plug in 2 or 3 or 4 or 5 yellow plugs along the bottom row - your choice. Make rows the same length above this until you make a square. Now tell me the size of your square and I will write it on the board. There will be various responses to this request. Accept them all, but also try to represent each one on the board, under the heading Squares, as x rows of x. Now it's your choice again. You can add on one more row, or remove one row. Tell me the size you have now. Record these under the heading Not Squares. Now the records will be of the form x rows of y. What do you notice about the squares? The children will see that only one number is needed to say the size of a square, but you need two numbers for a 'not square'. Introduce this number as the Square Root of the square. A tree grows from its root and a square grows from this number, that's why it's called the Square Root. Let's imagine some plug squares. I will tell you the square root. You tell me how many plugs in the square. Also let the children try out some examples of their own with their partner. Now we will go backwards. I will tell you the number of plugs in the square. You tell me the square root. You can use your calculator to help if you like. Also let the children try out some examples of their own with their partner.

Content decimal calculations decimal interpretation decimal representation of a fraction division equations: creating/solving estimating number exploring large numbers multiples, factors & primes multiplication - array model multiplication problem solving properties of number recording - calculator recording - written square numbers/square roots times tables visual and kinaesthetic representation of number

Part 2: Calculator Squares

This game develops students' number sense related to those numbers which are perfect squares and prepares them for seeking square roots of numbers between perfect squares. To begin, children must pretend that their square root key doesn't work. They will now have to use their number sense to find square roots. Aim: To improve the ability to estimate products and develop visualisation of squares and square roots Materials: One calculator per child Rules Player A secretly enters a square number, for example 67 x 67 = and shows the result to Player B. It is good to restrict the game to 2-digit numbers until the players choose to use other numbers. Player B knows that this screen is the result of a number multiplied by itself. Player B has to discover the starting number (the square root) and reproduce Player A's screen, in as few trials as possible, using a trial and improve (guess, check & record) strategy. One point for each trial.

Example

• Player B will be shown 4489 in the example above.
• They might reason that 60 rows of 60 would be 3600 and 70 rows of 70 would be 4900 and guess 66 x 66 because it looks a bit past the middle.
• The result would be 4356.
• This is not quite big enough so the second guess might be 67 x 67, which would be correct.

4. When successful, expect players to visualise the plugs in 67 rows of 67 and do a quick sketch of this in their journal using arrows to show the number in a row and the number of rows, as well as the total of plugs.
5. Players then swap roles and the game is repeated to make one round.
6. After six rounds the player with the lower score is the winner.

Part 3: Non-Perfect Squares

You know a lot about squares made with whole numbers, like 11 rows of 11, or 20 rows of 20. These are called Perfect Squares. Today we will look at other numbers which make squares. Hand out the Plug Frame. Along the bottom row, colour in three and half plugs. Now let's see if three and a half rows of three and a half makes a square. Colouring three rows of three and a half is easy enough. When it comes to the half row, encourage discussion about how this could be done in a way that 'completes the square'. Ask the students to work out the total (equivalent) number of plugs coloured in. How could we use the calculator to check our total? Carry out the check. You may need to discuss the calculator's representation of one quarter as 0.25. Ask the students to try some other 'half' problems. They should sketch (or visualise) the square first then check with their calculator.

1. I am going to write a number on the board. It can be made into a square, but I don't think you will be able to picture it exactly at the moment. However, with the help of your calculator, I do think you will be able to work out its square root.
2. Write up 6.2 and, as you make a sketch like the one shown, say:

   Square Number = 6.2

   So, 6.2 is the square number. I want you to find
   the square root - the side length of the square.

3. Discuss how this could be done and perhaps model the beginning of the process on the board. Emphasise trial, record and improve.

   TRY ANSWER COMMENT Must be > 2 because 2x2 = 4 and 3x3 = 9 2.5 6.25 Too big - a bit 2.45 6.0025 Too small 2.49 6.2001 Wow! Almost .....

4. Allow time! This activity is almost magically engaging for most children (and adults). As they struggle to get closer and closer to 6.2, there is great deal of informal learning about decimals, including the fact that the calculator 'runs out' of places before an 'exact' answer is reached.

• Use the problem often as a component of a lesson, perhaps under the heading of Square Root of the Day.
• When appropriate, emphasise connections between the square number, the area of the square, the square root and the side length of the square.
• There is also the possibility of drawing a graph to link the number of plugs in one row of a square and the total number of plugs in the square. The points of this graph will appear to lie on a curve. Sketching in the curve will show all the square numbers in between and link them to their square roots... But perhaps that is another story!
• Consider the possibility of investigating cubes and cube roots in a similar way by building bigger cubes with unit cubes.

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