These activities introduce working with the squares and cubes of numbers at Elementary.
Felt mat, short chains
NEW CASA EMAIL COURSE
Sign up for our email course that will guide you through the Montessori approach to the Casa.
All proceeds will go towards maintining Montessori Commons and improving the quality of the content.
Game 1 – shapes
Form chains with each short chains, beginning with the chain of three, after forming each shape ask the child to identify each shape, displaying them in a row, later you can ask the child to predict the next shape
Game 2 – concentric shapes
Form chains with each short chains this time forming then concentrically, beginning with the chain of three, putting the square around it and then the pentagon. Say, ‘The triangle is inscribed in the square, the square circumscribes the triangle and it is inscribed in the pentagon with circumscribes both the square and the triangle.’
The children realise something new about the chains, that number can be connected to geometry.
These form a series of short games
When to give the lesson:
This is a very early piece of work, to be done at the end of Casa or the beginning of Elementary
Concept and Notation
Squares, an exploration
Prepared tickets with notation of squares, blank tickets, pencil, short chains, the square of each chain.
‘Here we have a short chain of five, would you help me fold it? What have we made? (a square)
‘Look, here we have five on the base’ . Remove a ticket and write ‘5’ . ‘Look here, we have five on the side, I am going to write a ticket for that as well,’ Remove a ticket and write ‘5’.
‘Do you remember what we said?, This looks like a square, let’s check with a real square, when we want to talk about the real square we say squared’. Superimpose a square of five and then place it adjacent to the chain. ‘Yes, they are the same’.
Now when we talk about the ‘real square’ we call it five squared, and this is how you write it mathematically, you write ‘5’ first and a little ‘2’ in the right hand corner.’ Remove a ticket and write ’52’ .
‘Now we can make a mathematical statement, five taken how many times (five times) gives me 52’.
Arrange the tickets to read 5 x 5 = 52
Continue using other examples.
Squares, systematic exploration
Prepared tickets with notation of squares, blank tickets, all the short chains and their squares.
- ‘What do I have here?’ indicate the red bead, place it, ‘I have taken one once’, place a red bead. ‘I have one taken once, write a ticket ‘1×1’.
Which gives me 12, lets write a ticket, Write and place the ticket ‘12’. ‘What is it’s value?’ (1) Write ‘1’ on a ticket and place it.
- ‘I have two taken twice’, fold the two chain,‘Which is gives me 22’, place the bead square. Say, ‘Lets write a ticket’, write the ticket ‘2×2’.
‘This is two-squared’. Write and place the ticket ‘22’, and, ‘What is the value of 22? (4), Write the ticket for ‘4’ and place it.
- ‘I have three taken three times’, fold the three chain, ‘Which gives me 32’, place the bead square. Say, ‘Lets write a ticket’, write the ticket ‘3×3’.‘
This is three-squared’ write and place the ticket ‘32’, and, ‘What is the value of 32?’ (9), write the ticket ‘9‘ and place it.
- ‘I have four taken four times’, fold the four chain, ‘Which gives me 42’, place the bead square. Say, ‘Lets write a ticket’, write the ticket ‘4×4’.
‘This is four-squared’ write and place the ticket ’42’, and, ‘What is the value of 42?’ (16), write the ticket ’16‘ and place it.
Continue with the bead chains until 102 is 100.
Cubes, an exploration
Prepared tickets with notation of cubes, blank tickets, pencil, long chains and corresponding number of squares of beads and bead cubes.
‘Here we have a long chain of five, would you help me fold it? Lie the chain in a straight line, bending it after the fifth bar of beads to form five squares.
What have we made?…it looks like we have a square of five, taken five times. Super impose a square of five once over each square and build the five squares up in a tower. Write two tickets, firstly ‘5 x 5‘, place this beneath the first square and secondly, ‘5’ place this beneath the third square.
Ask, ‘What does it look like I’ve made?’ (a cube) ‘Let’s check with a real cube, when we want to talk about the real cube we say cubed’. Compare them with a cube of five. Indicating one face of the stack of squares say, ‘Look, what have we got here?,‘iI’s five squared’ (write a slip saying ’52 and place it standing up balanced on one face of the cube). Then look along another side of the stack of squares, say, ‘We have five lots’, write another ticket ‘5’ and place it against another side of the stack.
Refer at the cube, say, ‘I took five squared five times and this is how we write it, 5 and a tiny little 3 in the right corner’. Write a ticket with ’53’ and place it standing up along the side of the cube. ‘And what do we have here?’. Indicating the long chain say, ‘5×5 taken 5 times’ .
Now make the mathematical statement, forming the square first as the child knows this and then the problem and product.
‘Now we can make a mathematical statement, we have taken five cubed which is the same as five by five taken five times, this is the same as 53’. Move the tickets to write
52 x 5 = 5×5 x 5= 53
Continue doing other examples.
Cubes, systematic exploration
Prepared tickets with notation of squares, blank tickets, all the short chains and their squares.
- ‘What do I have here?’ indicate the red bead, place it, ‘I have taken one square taken once which gives me one cubed’, place another red bead.
Write and place a ticket ’12 x 1’. Say, ‘Which I know is the same as one taken once’, write a ticket ‘1x1x1’, “Which gives me 13, lets write a ticket, ‘13’ Write and place the ticket, ’13’, Ask,‘What is it’s value?’, (1) Write ‘1’ on a ticket and place it.
- Place two squares of two on top of each other, say, ‘I have two squares taken twice’, which is gives me 2cubed’, place the bead cube, say, ‘See, they are the same’.
Say, ‘Lets write a ticket’, write the ticket ’22 x 2’. Say, ‘Which I know is the same as two squares, taken twice, write a ticket ‘2x2x2’, ‘Which gives me 23, lets write a ticket. Write, ’23’ and place the ticket ’23’. ‘What is it’s value?’ (8) Write ‘8’ on a ticket and place it.
- Place three squares of three on top of each other, say, ‘I have three squares of three, which gives me 3 cubed’, place the bead cube, say, ‘See, they are the same’.
Say, ‘Lets write a ticket’, write the ticket ’32 x 3’. Say, ‘Which I know is the same as three squares, taken three times, write a ticket ‘3x3x3’, ‘Which gives me 33, lets write a ticket, Write, ’33’ place the ticket ’33’. ‘What is it’s value?’ (27) Write ’27’ on a ticket and place it.
- Place four squares of four on top of each other, say, ‘I have four squares of four, which gives me 4 cubed’, place the bead cube, say, ‘See, they are the same’.
Say, ‘Lets write a ticket’, write the ticket ’42 x 4’. Say, ‘Which I know is the same as four squares, taken four times, write a ticket ‘4x4x4’, ‘Which gives me 43, lets write a ticket, Write, ’43’ place the ticket ’43’. ‘What is it’s value?’ (64) Write ’64’ on a ticket and place it.
Continue with the bead chains until 103 is 1000.
This work reinforces the multiplication tables and the squares and cubes of the first ten natural numbers.
These are games discovered by children themselves
Game 1 – Making Shapes with Long Chains
Any set of long chain, squares and cubes
Take any long chain and join the sides to form a large version of the shape made in the introduction. Then take place the squares of the beads at each corner and the cube in the centre.
The children can continue to work with other sets
Game 2 – The Pyramid of Jewels
All of the squares
Starting with the 10 square place all of the squares on top of each other to form a pyramid.
‘Lets find the value of the pyramid’ , On paper record the value of each level.
102 = 100
82 = 64
Add them together to find the sum of the squares of the ten natural numbers
Game 3 – The Tower of Jewels
All of the cubes
Starting with the 100 square place all of the squares on top of each other to form a tower.
‘Lets find the value of the tower’ , On paper record the value of each level.
103 = 1000
83 = 512
Add them together to find the sum of the cubes of the ten natural numbers
The volume of the cubes of the pink tower is 3,025 cm3.
Game 4 – Saying the Multiplication Tables in order with beads
Bead box 1-9, one square for each number on a tray, paper for recording
Begin saying, ‘I will take one once is one’ take 1 unit, ‘one taken twice is two’, take another unit and place it below, ‘one taken three times is three’, take another unit then continue till ten.
‘Now I will take two once, which is two’, place a bar of 2 adjacent to the first unit bead. ‘Now I take two taken which is four’, place another bar of two, ‘two taken three times is six’, place another, ‘two taken four times is eight’, taken another bar of two continue till ten.
The Decanomial Games
Initial exploration with the beads
55 bars of bead bars 1-10, extra box to make changes for the adjusted decanomial
Say to the children, ‘I want to lay out all of the tables from 1 to 10, I am going to put one beed bar at the top of my mat to be the guide.’ Place one bard bar from 1 -10 horizontally across the top of the mat.
‘One taken once is one, and we have that here,’, indicate the bead bar of one.
‘One taken twice, one, two’ , lay out two bars of one beneath it, going vertically, ‘one taken three times’, one, two, three (3) Lay out three beads, say their names to ensure you have the right amount, this acts as a guide while building the rest. Verbalise the tables, continuing until 10, involving the children, and checking regularly.
‘We are going to each table systematically’ leaving a clear space between each table, complete each product. Continue verbalising the tables, ‘Two taken once is here, two taken twice is four’ Place two bars of two, ‘Two taken three times’, place three bars of two.
When complete say to the children, ‘When I look at the carpet of jewels I can see many squares. This is a square of one, I can exchange it for the square of one ‘, Exchange it, put the new bead at an angle. ‘This is two taken twice, it is two squared, I can change it for the square of two, three taken three times is the square of three’. Continue systematically, unless the children make other suggestions of true squares. When all have them have been exchanged,ask the children, Do you see where the squares are? They are along the diagonal, it’s as if the diagonal forms a backbone’.
This can begin in late Casa and is continued in early Elementary.
It is a small group activity, 2-3 children.
We form a sensorial layout of the multiplication tables, which we call the carpet of jewels.
Multiplicand on the horizontal and multiplier on the vertical
The Adjusted Decanomial
The carpet of jewels with the true squares along the diagonal
Join the children who have set this up on another day, changing parts using the commutative law so it becomes adjusted.Say, ‘Lets see if we can adjust it to make all of the beads green, beginning in the right angle. Let’s look at the twos, if we take the vertical bar of two beads and exchange it for green the lines are all green’. Exchange the bar of two. ‘Looking at the threes, if we taken the vertical three bar and exchange it for a pink bar of three’ Exchange the bar of three. ‘Then the two bars of three and exchange them for two three pink bards the line is all pink’. Exchange both bars of three.
Continue replacing the vertical beads, working through systematically, one level at a time. Continue verbalising, for example say, ’We are using the commutative law which says that it doesn’t matter which you take first, the multiplicand or the multiplier, if you take two six times it is the same as if you take six twice…and here we have the square, so six taken six times, it doesn’t matter which order you do it in’.
This can begin in late Casa and is continued in early Elementary. Suggest, ‘Let’s look at the top left corner and make all of the beads into green ones’.
The Numerical Decanomial
This is a series of envelopes containing pieces used to build the decanomial square.
- One envelope labelled, ‘squares’ containing pieces of yellow for all of the squares, with the product in the centre, and the notation 12, 22 to 102 in the top left corner.
Envelopes ‘1’ to ‘9’, (for the products of 2 to 10) these have a graduated number of pieces in each envelope, they have the product written in the centre but not the notation of the squares;
- Envelope 1 has 18 rectangular pieces, graded in size for the numbers 2 – 10, with two copies of each (one for the horizontal and one set for the vertical)
- Envelopes 2 to 9 rectangular pieces, there are less and less as their is no repetition of the commutative law.
With a small group, at a table.
Ask the children to remove the contents of the envelope labeled ‘squares’. Arrange them in a vertical line to the left of the table, saying, ‘One squared, that is one multiplied by one, which is one, two squared, that is two multiplied by two, which is four’ and so on, indicate the product written in the centre of the card.
Say, ‘Do you remember when we did the carpet of jewels?, Where were the squares? (in a diagonal) so lets arrange these squares along the diagonal’. Lay the squares corner to corner forming the diagonal.
Remove the contents of envelope 1, lay the cards to the left or where you are working, from the smallest to the largest, in the vertical and horizontal arrangements you will use later. Say, ‘The contents of this envelope will form the arms of our square’.
Begin with the horizontal saying, ‘two taken once is once, two taken once is two, three taken once is three’, laying out the cards along the top. Next place the cards along the vertical, saying, ‘Here we have two again, but this time it is one taken twice, which is two, one taken three times is three’, place the cards as you say the problem.
‘So now we have built our arms, this will help us with our multiplications’, take out the contents of envelope 2 and group them according to their orientation, this time simply group them, do not place them systematically. Take one card and say, ‘I have the product 16, what is it, it is two multiplied by eight, so where do you think it should go? Place it, paying attention to the order of the multiplicand and multiplier and using the arms to assist. Then take it’s pair according to the Commutative Law, saying, ’Look these is another 16 here, it means 8 multiplied by 2, where shall that go?’ Give a few more examples, giving the same verbalisation.
Take cards from Envelope 3,, the contents of the envelope begin after 32, so 6. Let the children taken over, they may not choose to put the Commutative Law cards together. Stay with the children for the first few envelopes, encouraging them to verbalise and not fit them into the pieces mechanically.
Putting away, begin with the outer arms, saying the tables as you collect them. Alternatively, begin with the contents of Envelope 1 and collect the multiples with the same product.
Reinforce the squares of the decanomial, further work with the Commutative and Distributive Laws, memorisation of multiplication tables and the squared products
If the material is made with squared card only write the product. If the children are making this they might write the problem on one side and the product on the number. The material must show the product.
The children’s work with the multiplication charts is a preparation for this, this is now a random exploration of the multiplications, showing the formation of the Commutative Law.
Building the backbone first and the fact that each envelope begins after the backbone highlights the squares of the numbers 1-9.
When to give the lesson:
After the sensorial layouts have been used, to reignite interest.
They need to know the multiplication tables and square notation, after the age of 6 years.
Follow up work:
After building the squares and the arms, the contents of the other envelopes can be mixed and the children build form this.
The children can draw the Numerical Decanomial on paper
The children can build the square backwards, beginning at the square of ten.
The children can build their own material with graph paper.
Transformations (From Bars to Squares to Cubes to the Tower)
Layout the decanomial with bead bars, with the colour pattern of the adjusted decanomial, but no true squares. An extra box to make changes for the adjusted decanomial, all of the squares and cubes from the bead cabinet, large mat, large sheet of squared paper
After the children have been working with the Numerical Decanomial, join children working with the sensorial decanomial or lay the beads out with them for this purpose. Say to the children, ‘We are going to change the carpet of jewels into a tower of jewels, we are going to see if we can change all of these bead bars into squares.’
Pass over the square of one and begin indicating the vertical bead bars of two, say, ‘I have one bead bar of two and another bead bar of two, that makes a square of two so I can exchange them for a square of two,’ Do this, then indicating the horizontal bars say, ‘I have two other bead bars of two, so I can exchange them for a square of two’ Do this.
Then indicate the vertical bars of three say, ‘I have three bars of three, I can exchange them for a square of three,’ Do this. Indicating the vertical say, ‘I have another three bars of three’, Indicate the horizontal bars, saying, ‘I can change these for a square of three’. Do so, ‘I have three other bars of three, I can change these for a square of three too’. Do this.
Indicate the vertical bars of four, say, ‘I have four bars of four I can make a square. I can make another exchange bars along the horizontal for a square of four.’ Indicate the horizontal bars and say, ‘I have four bars of four, I can change this for a square,’ Do this, then say, then count the other four bars and exchange them. For even numbers a square will be left missing.
Continue for the bars of fives, you can begin to count the bars, ‘one, two, three, four, five’, before replacing them with a square. After the first few examples you can work in whatever way you or the children wish.
Say, ‘We now have a decanomial made up entirely of squares, let’s collect all of the squares into the diagonal’, skip over one squared which is also one cubed, without saying anything, Then indicate the squares of two saying, ‘We now have two squares of two’, stacking them together as you count, ‘one, two’ and ask the children, ‘What are they the same as?’ (23). Replace both squares with a cube on the diagonal. Then indicate the squares of three, count them in the same direction as for the bars, stacking them, saying, ‘We have three squares of three, what is this the same as?’ (33) now exchange the squares for a cube on the diagonal.
Continue until the squares of ten saying, ’We have one, two, three, four, five’ stacking them along the base and then, ‘six, seven eight, nine, ten’ along the side, stacking them at the diagonal to make a cube. Exchange them for a true cube.
Say, ‘Let’s build the tower’, place then in a cube on the tens and so on, ‘So the carpet of jewels has now become a tower’.
After the lesson:
Encourage the children to reverse the process, immediately after exchanging the cubes for squares and to bars and individual beads.
Recording the work with the Decanomial
These recordings can be made during any point working with the sensorial explorations into the decanomial, later Casa, early elementary, particularly for children who need to work on their tables.
Good variety of squared paper, of different sizes and with different sizes of squares.
Part 1- with the carpet
Join the child with the decanomial carpet of jewels complete. Say, ‘Here we have one taken one time and write (1×1 = ) and there one taken twice, write (1×2 = ) and one taken three times, write (1×3 = ). Then do the two times table and continue up to ten.
The child can record the products in order, in whatever direction suits her needs from 1×1 to 10 x 10. It should fit onto one piece of paper.
Use only symbols the children are familiar with, using brackets, equals symbols depend on the child’s level.
The child may record over one or many days.
Part 2 – with the carpet
Invite the child to record the value of the carpet of jewels, ask, ‘What do we really have on the mat?’ (a square)
We have the square of one plus two plus three plus four’,
While writing (1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10).
When we have the square of something it means that we multiply it by itself, put the exponent 2 outside the brackets, so it reads, (1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10)2
Say, ‘We can expand this’, write;
(1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10) x (1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10)
Multiply it out with the children, ‘I will multiply everything in the first bracket by one’, make a tick above the ‘1’ of the multiplicand and write,
(1 x 1) + (2 x 1) + (3 x 1) + (4 x 1) + (5 x 1) + (6 x 1) + (7 x 1) + (8 x 1) + (9 x 1) + (10 x 1)
Leave a line for later calculations. Continue multiplying by 2, marking off the two of the multiplicand with a tick, before letting the children take over until the tens multiplicand.
(1 x 2) + (2 x 2) + (3 x 2) + (4 x 2) + (5 x 2) + (6 x 2) + (7 x 2) + (8 x 2) + (9 x 2) + (10 x 2)
Go back and fill in each partial product in the line below which has been left blank.
Add the partial products at the end of each line to find the totals of each multiplicand.
Then add the total value in a column addition at the end of each line, which is 3,025
Depending on the children, it may be better to do all of the expansions first and later the calculations or calculate as you go.
Older children may not want to record each multiplication and may record the products directly.
Part 3 – with the tower
Join the child with the decanomial tower complete. Say, ‘Here we have one cubed plus two cubed, plus three cubed, plus four cubed’ and write 13 + 23 + 33 + 43 + 53 +63 =73 + 83 =93 =103
Below it write the products 1 + 8 + 27 + 64 + 125 + 216 + 343 + 512 + 729 + 100
Add these, giving an answer of 3, 025.
Help the child to make the connection between the value of the product of the carpet and the jewel.
They are now aware of the square of Pythagorus, it is based on his theorem which states that ‘the sum of the first ten natural numbers squared is equal to the sum of the cubes of the first ten natural numbers’.
When to give the lesson:
For children between the ages of 6 and 8 years.
Part 4 – Extension
Join the child with the adjusted decanomial layout, draw their attention to the fact that we have
(1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10)2
Select a part of this, like
(1 + 2 + 3 + 4)2
Ask, ‘What does this actually mean, it means,’
(1 + 2 + 3 + 4) x (1 + 2 + 3 + 4 )
Say, ‘Sometimes we can use letters of the alphabet to help us do our maths so this morning we are going to use letters, let’s write it again using letters’, write
(a + b + c + d) x (a + b + c + d)2
We begin by multiplying everything in the first bracket by a then we put a tick over it to show it is done.
Say, ‘When we are using letters we always stay with their order so for the second half we change it around’, i.e. ba = ab
a2 + ab + ac + ad
ab + b2 + bc + bd
ac + bc + c2 + cd
ad + bd + cd + d2
‘Now we collect the terms, we put the ones that are the same together’
a2 = 2ab = 2ac + 2ad + b2+ 2bc + 2bd + c2 + 2cd + d2
‘And thats finished because when we are doing algebra we don’t have to write the terms’
This is another way to explore with the theorems of Pythagorus
The decanomial square is an exploration with the first ten natural numbers. There is a formula to find the sum of any set of numbers n2 + n
For example with the set (1, 2, 3, 4) take the highest number, 4 and square it, 16, plus that number 16 + 4, gives 20, and then divide the whole thing by two, gives 10. 1 +2 +3+ 4 + 5 = 10.
With the set (1, 2, 3, 4, 5), take the highest number e.g. 5 and square it, gives 25, plus that number 25 +5, gives 30, and then divide the whole thing by two, gives 15. 1 + 2 + 3 + 4 + 5 = 15.
Continue to say the multiplications for the bars of three and continue till nine, the children help.
‘Now we are going to see if we can exchange a row of identical beads for a square of each number, ‘One taken once is the same as the square of one so I can exchange it’, count one remove the bead and exchange it for one from they tray, ‘two taken twice will give me two squared, so I can change them, count two bars and remove them and exchange it for two squared from they tray, ‘Three taken three is the same as the square of three so I can exchange it’, count three bars remove the bead and exchange it for a square of three from they tray, ‘four taken four will give me four squared, so I can change it, count four bead bars, remove the bead and exchange it for a square of four from they tray.
I have one square and one taken nine times, look at the nine square, I have nine squared and nine taken once. I have my two squared and two taken eight times, with my eight I have eight squared and my eight taken twice. I have my three squared and three taken seven times, with my seven I have seven squared and my seven taken three times. I have my four squared and four taken six times, with my six I have six squared and my six taken three four. I have my five squared and my five five times.
10 =12 + 9
20 = 22 +16
30 = 32 + 21
40 = 42 + 24
50 = 52 + 25
60 = 62 + 24
70 = 72 + 21
80 = 82 +16
90 = 92 + 9
While recording say, ‘for one, what have I got? One squared plus nine’ then, ‘for two, what have I got? Two squared plus sixteen’.
Subtract each addition for the one above e.g. 16 -7 = 2 and 21 – 16 =5 and 24-21 = 3, continue.
When you subtract each difference the result is ‘0’
Game 5 – Squares of Numbers
Bead box 1-9, paper for recording
Record the squares of numbers (e.g. 12) and then the values of each square 12 is 1, 22 is 4)
Then find the difference between each value, continue subtracting.
Say, ‘The difference when I keep subtracting is 2, so what is the exponent?’ (2)
See sheet ‘squares of numbers’
‘Exponent’ refers to the number of the power, the exponent of squared is 2, cubed is 3
Game 6 – Cubes of Numbers
Bead box 1-9, paper for recording
Record the cubes of numbers (e.g. 13) and then the values of each cube 13 is 1, 23 is 8)
Then find the difference between each value, continue subtracting until the differences are the same
Say, ‘The difference when I keep subtracting is 2, so what is the exponent?’ (6)
The children may discover that the difference multiplied by the exponent gives the exponent of the next power, e.g. the exponent of 2 multiplied by the exponent 3 gives 6, the difference between the cubes. Therefore the difference of 6 multiplied by the exponent of four gives 24, the common difference when using the power of four.
Squared paper and bead squares
Say to the children, ‘We can do addition with squares and cubes of numbers, I want to add four squared and three squared. Do you know the value of four squared (16) and three squared (9) So four squared plus three squared equals’ (25).
While saying this write
42 +32 =25
16 + 9
Do a few examples.
42 +42 = 32
With the material take the eight squared, (do not represent the subtrahend) and cover one corner, the equivalent of 52 with paper, counting five horizontal and vertical beads outloud. Count what remains.
While saying this write
64 – 25 =
Take three lots of five bead squares
While saying this write
52 x 3 = 75
25 x 3
Note: ask the children the questions in the abstract and then check it with the squares.
Take out five skittles and a cube of five. Ask the children how I can share the cube and exchange it for five squares of five and share them.
The answer is the share one skittle has
While saying this write
53÷ 5 = 25
Operations with mixed numbers
Working without materials continue to explore,
(53 + 22) – 82
(125 +4) – 64
129 – 64
Repetition through variety, for intellectual development