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Teaching for conceptual understanding requires children to understand how and why a concept works, rather than passively accepting abstract rules a teacher presents to them.
Teaching children that a square number is a number multiplied by itself is the final generalisation. Lots of teaching, learning and construction of a concept must come before. Deepening concepts and starting with ‘why’ is a powerful way to develop and strengthen understanding.
The following activities are designed to help you start with the concept of square numbers and take small steps to develop, strengthen and deepen understanding. They use a concrete, pictorial and abstract approach to allow pupils to experience the concept, thus making it more secure and memorable.
As this concept focuses on square numbers which are linked to square shapes and properties of shapes, it may be worth starting with a discussion on the properties of a square. Hand out this 2x2 table and ask pupils to fill it in as a starting point, thinking about why a square is a square:
Using squared paper, invite children to draw squares ranging from 1x1 to 12x12 in size. Ask pupils to calculate the area of the shapes. Explain that this is a square number. This visual representation deepens pupils’ understanding, while exposing the structure of the concept as a calculating area.
Alternatively, provide children with this multiplication grid and invite them to draw squares, starting from one in the top left-hand corner and increasing in size each time. Look at the number in the bottom right-hand corner to identify the square number.
Next, play ‘square, rectangle or stick’. Ask children to draw a 1x1 square on squared paper. Now, add one extra 1x1 square and see if you can create a square, rectangle or stick. Keep adding on extra 1x1 squares.
Make a note on this worksheet of how many 1x1 squares you need to create squares, rectangles and sticks.
Pupils should notice that squares can only be made from 1, 4, 9, 16, 25 (etc) squares. Alternatively, ask children to cut out squares and manipulate them on their work area to investigate this activity in a more concrete way.
Explore the structure of square numbers and allow children to look for patterns by giving pupils three grids: 0-100 in rows of 10; 0-99 in rows of nine; 0-96 in rows of eight.
Ask children to shade in the square numbers. What do they notice? The numbers go from no real pattern in the 10x10 grid to being lined up in columns in the 8x8 grid and subsequent grids after that. Extend this by asking children to draw more grids, getting shorter by one each time.
Making links and connections in mathematics is important for children to be able to reason and explain. Linking prime numbers and square numbers allows for rich discussions and the strengthening of both concepts.
After teaching or revising what prime numbers are, pose this question: can you make a square number by adding two prime numbers together?
Give children visual representations of square numbers with the number of squares visible inside each square to 12x12. Ask them to cut them out.
Show children how two prime numbers add together to make a square number, such as 11 + 5 = 16. Show this by colouring in the squares, like below:
Now children have had time to explore the concept in depth and secure their understanding of square numbers, consider an investigation to apply understanding. Try posing this question: which numbers can be made by adding two square numbers together? Cut out visual representations of squares to 12x12 to help.
Children who are confident can do it abstractly and systematically. For example, 1 + 4 = 5; 1 + 9 = 10; 1 + 16 = 17.
This activity can be done with counters, cubes, pencil and paper or abstract notations: think of a number, square it, subtract your starting number. Is the number you’re left with odd or even? What do pupils notice? The number is always even. For example:
Think of a number: 3
Square it: 9
Subtract the starting number: 9 - 3 = 6
In the example below, the blue squares show the number of parts subtracted from the square.
So, why is a square number a square number? Using the approaches in the activities above allows children to reason, make links, look for patterns, hypothesise and arrive at the generalisation that a square number is a number which is multiplied by itself. This is the end point; the generalisation; the rule.
Children will have been on a conceptual journey of discovery to arrive at this end point, rather than it being one of many abstract mathematical rules.
John Bee is head of KS2 and maths leader at a primary school. Follow him on Twitter at @mrbeeteach and visit his website at mrbeeteach.com.
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