Arithmetic

# Powers of Numbers

## Powers, using a base of 2

Material Description:

Box for Powers with a base of two:

• cubes and prisms
• coloured yellow, white, green

(small 1cm cubes for the follow up work)

blank tickets

pencil, paper

prepared tickets for follow-up work

Method:

### Making powers

Take a unit, ‘Here we have a unit, it is all on its own, it has no power.’, write a ticket – 1

‘Now I am going to take my unit and put another unit with it

Move the unit to the right and place another unit to it’s side a unit – ‘Now it has a power’, write 2

‘I am going to replace it’, Take the white cuboid and after checking it on all sides replace the two unit points with it, ‘So what did I have? I had one taken twice, I have two, we call this two to the first power’, flip the ticket and write 21.

‘To go to the next power I take my two twice’. Move the built blocks to the right and add the two unit cubes to the front. ‘I have a new power, two taken twice. Write 21 x 2 on the ticket. What did I do?, I have two taken twice, I have 2 to the second power’.  Flip the note and write 22.  ‘Do I have a piece to replace it with? , Yes I do’  Exchange it for the green surface.

‘To go to the next power I take my two to the second power once, then twice’. While saying this move the pieces to the right, Take the yellow point, check it on all sides and place it in front of the pieces. Write 22 x 2 on the ticket, ‘I have a new power, two to the second power, taken twice, I have two to the third power’.  Flip the note and write 23.  ‘Do I have a piece to replace it with? , Yes I do’. Make the exchange,

‘To go to the next power I take my two to the third power twice’. Move the piece to the right first and then adding the other block to the side. Write 23 x 2 on the ticket, ‘I have a new power, two to the third power, taken twice,  I have two to the fourth power’.  Flip the note and write 24.  ‘Do I have a piece to replace it with?, Yes I do’  Make the exchange.

Continue till you have a stack of two to the sixth power and have written the tickets.

### Deconstructing powers

How did I make that? What I did is I took two to the fifth power twice and move the original part to the ticket 25.  Leave the other blocks and say, what did I do to get two to the fifth power? We took two to the fourth power twice, move the block of 24 over the ticket. Keep going till they are all replaced, put the extra unit on one side, the 26 ticket is left empty. Say, ‘I took my unit two twice to get to the first power. I can talk about my unit in terms of the power of 2, it is a unit to the zero power’. Take the ticket with ‘1’ written on it, reverse it and write 20.

Aim:

To show the powers of numbers with a base of two

Notes:

Another name for a base is a factor.  Another name for a power is an exponent.

This is a basic lesson with abstract ideas, therefore the lesson can be repeated many times.  Children who grasp the concept can present to the others, it is given to children around the age of eight.  It is sensorial, the ideas are abstract but the children are not being asked to abstract.

We make two to the power of six and then undo it to reinforce the work, it is how the power of zero is introduced.

The material follows the point. line, surface pattern established in the Wooden Hierarchical Material.

On another day, say to the children, ‘We call two the base and the little number is called the ‘exponent’, this comes from the Latin, ‘exponare’, which means to expand.  We repeat the base the exponent number of times.’

The unit is always the building block, the first unit of each base is always the power of ‘0’, it has no group so it has no power.

To find the value you take the base the number of times of the value, e.g. 34 is 3x3x3x3

Discoveries

• The children discover that the unit is the base to the zero power.
• Children see a repetition of the geometric design (point, line and surface)
• Using the 1cm cubes shows that the size of the unit determines how far they can explore with the material.
• When moving from one power to the next power (e..g 21 x2 =22) you taken the power the base (21) number of times (x2)

When to give the lesson:

After the Wooden Hierarchical material with gives groups of ten, this work is presented soon after the work with squares and cubes of numbers and the notation.

Make the movements before writing the notation

After the lesson:

• The children repeat the presentation with the set of prepared tickets

• Encourage the children to record the values written on each side of the ticket, concentrating on the exponent first, e.g.

20 =1 =1

21 = 2 = 2

22 = 2×2 = 4

23 =2x2x2 = 8

24 =2x2x2x2 = 16

25 = 2x2x2x2x2 = 32

26 = 2x2x2x2x2x2 = 64

• Children can work with the 1cm cube, take a unit and say, ‘This has no power, If I put another unit with it it is two to the first power’, write the ticket, 21.

To go to the next power I must take this whole thing twice’, move the two units to the right, add the other piece. ‘Now I have my new unit which is two to the second power, it is two taken twice. Write a new ticket 22 and make the exchange.

Starting with the little box of cubes and the box for this presentation, you can reach the power of nine. Then deconstruct as before, finishing with the zero power as before.  Now the power of nine is empty and the tickets are all shown.

The power of three

## Powers, using a base of 3

Material Description:

Box for Powers of Three:

• cubes and prisms
• coloured yellow, white, green

small 1cm cubes

blank tickets

pencil, paper

prepared tickets for follow-up work

Method:

Making powers

This is a similar presentation to the powers using a base of 2. This time begin introducing the zero power, as the children are now familiar with it.

Put out the three yellow unit cubes and write a ticket 3 and then flip the ticket and write three to the zero power 30.  Say, ‘Can I exchange the blocks?’ (yes), exchange the cubes for a line.

To get to the next power, move the line to the right and add two cuboids, say,’I have taken three to the zero power three times’ write 30 x 3, say, I have three to the first power’ flip the ticket and write 31 Exchange the lines for a surface.

‘I want to taken this to the next power’,Move the surface to the right, and add another two surfaces, say ‘I have taken three to the power of one three times’, Write 31 x 3, ‘So what have I now?, I have three to the second power’, flip the ticket and write 32. Exchange the surfaces for a point.

‘Now I want to go to the next power, so I must taken my three to the second power three times’, move the point and add two additional cubes, ‘What did I do? I took two to the power of three three times’, Write 32 x3, ‘So what did I have? Three to the power of three’, flip the ticket and write 33.

Continue to three to the power of six

Deconstructing powers

Do this as for the power of two, only now the ticket 30 has already been written.

## Powers, using a base of 10: Reviewing the Wooden Hierarchical Material

Material Description:

Wooden Hierarchical Material

2 sets of prepared labels

• 1, 10, 100…1,000,000
• 10 to the powers up to 6 with multiplication on the back:

Front Back

100 1

101 10

102 100

103 1,000

104 10,000

105 100,000

106 1,000,000

Method:

Say, ‘We have been talking about the powers of numbers. Can you lay out the wooden hierarchical material?’. Then layout the cards saying their names, ‘This is the decimal system and ‘deci’ means ten, our numbers use the base of ten, so what do we have here? Pick up the card with ‘1’, ‘this is a unit’, turn over the card and change it’s position by ninety degrees, this is ten to the zero power, the unit is the first group and we know the first group has no power, zero power.’

Move to the card with ten. ‘Now we have ten, this is our first group’, turn over the card, ‘we have one unit to the first power, that is a unit to the power one. ’How do we get to the next group?’ (we times by ten), so what have we got?’ Turn the card of 100, ‘I have ten taken ten times, ten to the second power’

Ask the children if they notice anything about the number of zeros.

Discoveries:

The children may notice that there are the same amount of zeros as the number in the exponent.

They notice that the geometrical pattern point, line, surface is repeated

Aim:

To give the child access to keys, emphasising the relationship of the powers to their own number system.

After the lesson:

Play games with the numbers asking which is bigger, e.g. Which would you prefer 102 or 83,  103 or 310, showing the child how to make them

Exponential Notation – Operations

Multiplication

Material Description:

Material for the squares and cubes

Method:

‘Today we will do multiplication with the powers using the same base’,

Example 1

Write the example 33 x 32 =

Say, ‘I will take three to the third power, three to the second power number of times.  Indicating the multiplier, ask, ‘What is three to the second power?’ (9)

Say, ‘I have to build in the base of three’, (indicate the pink row). Then take the cube of three and sufficient pieces to form eight more cubes. Place the cube, (this is the point and represents 33).

Place two cubes with the original one in a line giving three to the power of four (this is the line)

Place another six cubes, three on either side to show three to the fifth power, making a square  (this is the surface)

Example 2

Write the example 22 x 24 =

Say, ‘I will take two to the second power, two to the fourth power number of times’  Indicating the multiplier ask, ‘How many is two to the fourth power?’ (16) Then, indicating the multiplicands base, ask, ‘What does this mean?’ (we are using the base of two, to the power of two) ‘The answer is two squared taken sixteen times’.

Remove a sixteen pieces of the square of two to form nine cubes, ‘I have to build in the base of two’ and place a square, now say, ‘I will go to the next power, which is three’, build a cube (this is the point and 23).

‘We want to take our answer in terms of the base of two to the next power, the power of four’ Arrange the pieces in two to the fourth power, by placing two squares in a cube beside it, making a line of two cubes .

And then continue to two to the fifth power by taking taking another two cubes (four squares), making a surface. And then two to the sixth power by taking another four cubes (eight squares), forming a point.  Doing this follows the established pattern of point, line, surface.

Write the answer 22 x 24 = 26

Give more examples till the children make the discovery. (e.g. 53 x 51 = 54, 43 x 40 =43)

Check the working and continue to build more questions

Discovery, when multiplying:

Children notice that when the base is the same all you have to do is add the exponents.

Division

Example 1

35 ÷  32 =

Build the dividend, three to the fifth power, (following the pattern point, line, surface, begin with a cube, three to the third power and then make a line of three, three to the power of a fourth, to make it three to the fifth power make it into a squared surface, so nine cubes)

As we are dividing by the second power we are using squares, so lay each piece out alone counting, ‘one times, two times, three times…twenty seven times. Three to the second power is contained within the dividend, which is three to the fifth power, twenty seven times. Twenty seven is also three to the power of three’

The answer is twenty-seven, which can be made into three to the third power.

35 ÷ 32 = 27 or 33

Example 2

54 ÷  53 =

Build the dividend, five to the fourth power, (following the pattern point, line, surface, take a cube, which is to the third power and then making five cubes in a line)

As we are dividing to the power of a cube we leave the pieces in cubes and separate them, saying, ‘I want to know how many times five to the third power is contained in five to the fourth power (five) and what do we know about five? (it is the first power) Lay each cube out alone counting, ‘one times, two times, three times…five times’.

The answer is five to the first power.

54 ÷  53 = 51

Give further examples, e.g. 42 ÷ 42 =40

Help the children to come to the discovery.

Discovery, when dividing:

If the base is the same all you have to do is subtract the exponent.

Notes:

We do operations with the same base.

Children can be told that their is a convention in Mathematics that for numbers to the power of zero the exponent does not need to be written, 3 plus 0 is 3.

When to give the lesson:

When the children have worked with the powers of 2, 3 and 10

When the children are familiar with the geometrical pattern (point, line, surface)

They can work with the notation of squares and cubes and have worked with multiplication of squares and cubes.

## Exponential notation with the decimal system

Material Description:

Pen and paper

Method:

### Introduction

‘Today we are going to express decimal system numbers using exponential notation’

‘if I write the number 125 in terms of the categories I have 100 + 20 + 5’. Write this

‘If I took 6, 346 I have 6,000 + 300 + 40 + 6’ Write this

Single categories

Now I will take the number 200, this time I will write it in terms of exponential notation,

200 = 2 x 100

= 2  x 102

6000 = 6 x 1000

= 6 x103

3 = 3 x 1

3 x 100

30 = 3 x 10

3 x 101

3, 642, ‘What is this really? Let’s decompose it.

3,642 = (3 x 1000) + (6 x 100) +(4 x 10) + (2 + 1)

‘Let’s express it in terms of exponential notation’

3,642 = (3 x 103) + (6 x 102) + (4 x 10) + (2 + 10)

‘There is another way of writing this, we can taken out the multiplication symbol because we know that when you put brackets, you multiply what is outside the bracket by what is inside the bracket.

We can express it like this, the numbers outside the brackets are called the co-efficient’

3,642 = 3(103) + 6(102) + 4 (10) + 2(10)

‘Now we are going to do some addition, we will taken 4,235 and add 1, 509’

4,235 + 1, 509

4, 235 = (4 x 103) + (2 x102) + (3 x10) + (5 x100)

• 1, 509 = (1 x103) + (5 x 102) + (0 x10) + (9 x 100)

5, 744 = (5 x103) + (7 x102) + (3 x 10) + (14 x 100)

‘Beginning with the units,first, expand the addends using exponential notation, then we translate them back into decimal system numbers’

500 + 700 + 30 +14

5, 744

### Subtraction

216 – 124

216 = (2 x 102) + (1 x 10) + (6 x 100)

• 124 = (1x 102) + (2 x 10) + (4 x 100)

92  = (0 x 102) + (9 x 10) + (2 x 100)

‘It is necessary to exchange, do not show this on paper’

Check with ordinary subtraction

### Multiplication

Example 1

To do this the children must have discovered that to multiply exponential notation with similar bases by adding the exponents.

42 x 3

42 = (4 x 10) + (2 x100)

x 3                     (3 x100)

(12 x 10) + (6 x 100)

126

First multiply the multiplicand, beginning with the units and then add the exponents.

Check using regular multiplication.

Example 2

1, 239 x 47

1, 239 =         (1 x 103) + (2 x 102) + (3 x 10) + (9 x 100)

x    47 =                                             (4 x10) + (7 x 100)

(7 x 103) + (14 x 102) + (12 x 10) + (63 x 100)

(4  x 104) +  (8 x 103)  +  (12 x 102) + (36 x 10)

(4 x 104) + (15 x103) + (26 x 102) + (57 x 10) + (63 x100)

40,000   +   15,000    +    2, 600    +    570      +      63

= 58, 233

First expand the decimal notation into exponential notation, then multiply the co-efficient of the units, tens, hundred and thousands of the multiplicand by the units of the multiplier.  Then do the same with the tens of the multiplier.

Then add the multiplications with both the units and tens multiplier.  Finally transforms the decimal numbers and add them.

### Division

Do this when the children have worked with the materials and have realised that to divide the exponents of numbers with the same base must be subtracted.

Example 1

6000 ÷ 30

6,000   =  6 x 103  =  2 x 102  = 200

30          3 x 10

First we expand using the powers. Then divide the dividend by the divisor, 6 – 3 is 2, then subtract the exponents, so here the power of three minus the power of 1 makes the power of 2, Finally translate the exponential numbers back to decimal ones.

Example 2

30,000  ÷ 5,000

30,000  =   3 x 104     =   3 x 10     =  30   = 6

5,000       5 x 103             5               5

Firstly, expand the decimal numbers into exponential ones. As you can’t divide the dividend by the divisor, three by five, you multiply what you subtract the exponents, here the power of four minus the power of three, to give the power of one and then work with what you have left (3 x10) and divide that by 5.

Example 3

75, 000 ÷ 15

75,000   = 75 x 103     5 x 103  = 5,000

15           15 x 100

First expand,divide the dividend by the divisor, 75 by 15 to give 5 and subtract the exponent.

Aim:

This helps the children to understand the powers of numbers.

It gives a new form of notation and prepares for scientific notation.

Notes:

These are simple exercises, based on the work in Casa when they used the decimal system cards, with the long multiplication whey are used to analysing the numbers on the bead frame notation paper.

With the keys which have been given with the addition and addicting in what they already know about multiplying and dividing with numbers with the same base, the children can do this work.

• To multiply add exponents with the same base.
• To divide subtract the exponents.

Arithmetic

Arithmetic