Arithmetic

# Laws of Multiplication

We begin with a single digit and then work into using more categories, when the children are familiar give laws.  The first part to the Passage to abstraction is putting in the signs, the second part involves expanding with the cards and the third using mental arithmetic, without the geometric presentation of the operation or products and finally working on paper.

## Single digits – to show the Commutative Law

Material Description:

Box of coloured bead bars 1 to 10 (for the multiplicand and to geometrically represent the operation)

Mat to work on

Sets of small decimal cards 1 – 3,000 (for second section)

One envelope

Box of printed digits 0-9 on grey cards (the grey represent the multiplier)

Method:

Say, I am going to do multiplication a different way’, I will take four five times’.  Take the bead bar four and the grey multiplier 5 and place them on one half of the black mat. Create the operation beneath by placing five bead bars four times.  The find the product and represent it beneath in vertical bead bars, (2 bead bars of ten).

Then say, ‘I will try five times four’, represent the five with the beed bars on the other half of the black mat, opposite to the original problem, represent the problem with four beed bars of five beneath and find the product, representing it in an identical fashion to the adjacent product.

Say, ‘This is a special law in Mathematics which says that the order in which you multiply does not effect the product.  This is called the commutative law.’

Aim:

To help the children become consciously aware of the commutative law of multiplication

Notes:

• The children have worked with the Commutative laws since beginning multiplication in the Casa, and the Distributive laws with the Decanomial (here the multiplier is distributive over the multiplicand and the multiplicand is distributive over the addition.  Now the laws are given consciously.
• We only explore the Commutative laws in relation to multiplication
• The laws are given very early in the Elementary class, serving as a transition exercises for children just from Casa, as the children only need multiplication, the exercises help with the consolidation and memorisation of multiplication tables and they help children to  become aware of the characteristic qualities of numbers so that they can express their discoveries and be given the terminology
• The work itself follows the pattern of sensorial, concrete experience before terminology and abstraction.

When to give the lesson:

Concept of multiplication, including knowledge of the tables, parallel work with the chequer board and flat beed frame.  This can be a first lesson at elementary, children can skip count to find the product if necessary. Give it early when the sensorial aspect will have a strong impact as it is tremendously important in imparting the intellectual idea

After the lesson:

Single digits

## Sum by single digit – to show the Commutative Law

Material Description:

As for the units, with many sets of brackets

Method:

Say, ‘Today we are going to do something new, some multiplication. I am going to take four and three as my multiplicand and to remind myself to keep them together I am going to put them in this envelope and I will take them five times’ (put a four and beed bar in an envelope and place a grey envelope and place a grey multiplicand card of five) Then take the beads out of the envelope and place them adjacent and say, ‘There is a special way to remind myself I will put them in brackets’ and place the brackets’.  Pointing at the multiplicand say, ‘What does it mean?,  I’m going to take my four five times and my three five times’

Lay out five sets of four bead bars and five sets of three bars, ‘Now I have done my multiplication’, turn over the five multiplicand card. Beneath each operation place the partial product in beads beneath them

Place other beads below this to show the addition of the partial products.  Read out the whole problem

Reverse the problem on the other side of the mat, the multiplicand becomes the multiplier and vice versa, five is represented in beads on the mat and the cards for four and three are placed in the envelope, then in the bracket.  Turn over the multiplicand cards once each operation has been completed.

Aim:

To help the children become consciously aware of the commutative and distributive laws of multiplication

Notes:

• The distributive law is that each term in the bracket is multiplied by the terms out of it, preparing the way for algebra.
• This is the first time the children have used brackets, the children do many examples, sensorially they discover the distributive law.
• The children can cease working with the envelope when they are ready, the adult continue to use it.

When to give the lesson:

Concept of multiplication, including knowledge of the tables, parallel work with the chequer board and flat beed frame.  This can be a first lesson at elementary, children can skip count to find the product if necessary. Give it early when the sensorial aspect will have a strong impact as it is tremendously important in imparting the intellectual idea

After the lesson:

After many examples, you introduce the distributive law, saying, ‘Look at the final products, we had two terms in our multiplicand and what did we do, we multiplied each term of the multiplicand by the multiplier’, indicating the first problem. ‘Then we had two terms in the multiplier and we multiplied the multiplicand by each of them’ indicating the second problem.

Later, if the children don’t realise you can suggest they add the grey cards together and multiply them by the other figure.

Sum by Sum – to show the Commutative Law

Material Description:

As for the units, with many sets of brackets and two envelopes

Method:

Say, ‘Today we are going to do something new, some multiplication, I am going to take five and four as my multiplicand’. First put the cards in an envelope, then say, ‘I will take then two and three times’ put the multiplier cards 2 and 3 in a separate envelope. Then open the brackets and lay out the beads of the multiplicand, and close them, then form the multiplier with brackets and the two cards in the envelope.

Lay out all four operations below, as before.  The add each partial product and find the whole product. Say, ‘When I took five and four, two and three times I got forty five.’

Reverse the problem on the other side of the mat, taking two and three in beads and five and four in cards. Say, ‘I took 2 and 3 four and five times and the product is 45 and I took 4 and 5  two and three times and had 45’.  Ask the children if it matters which you do first (no)

Then say, ‘Everything in the first bracket, must be multiplied by everything in the second bracket and each term in the multiplicand must be multiplied by each term in the multiplier’ (each term in the multiplicand everything in the second bracket must be multiplied by the multiplicand).

Aim:

To help the children become consciously aware of the distributive laws of multiplication.

Indirect preparation for binomial multiplication.

Notes:

• This exercise is very sensorial and must be done early enough to let the children work with many examples
• The work can be presented to younger children on an individual basis, older children can use this as a group to explore the properties of numbers.

When to give the lesson:

Concept of multiplication, including knowledge of the tables, parallel work with the chequer board and flat beed frame.  This can be a first lesson at elementary, children can skip count to find the product if necessary. Give it early when the sensorial aspect will have a strong impact as it is tremendously important in imparting the intellectual idea.

After the lesson:

After many examples, you introduce the distributive law, saying, ‘Look at the final products, we had two terms in our multiplicand and what did we do, we multiplied each term of the multiplicand by the multiplier’.  Indicate the first problem. ‘Then we had two terms in the multiplicand and we multiplied it by each term in the multiplier.’ Indicate the second problem.

Later, if the children don’t realise you can suggest they add the grey cards together and multiply them by the other figure.

## Passage to Abstraction,

### Stage 1 – using the symbols

Material Description:

As for the units and

Three sets of white cards (Decimal system cards for the product)

Two sets of the grey cards (the problem)

Box of printed digits 0-9 on grey and white card

Operation signs (+, -, X, /, =)

Sets of brackets (lots)

2 small envelopes that fit the 10 bead bar and the printed digit cards

Method:

– with the beads for the multiplicand and the grey cards for the multiplier and the operation signs

Place the multiplicand (6 and 3) in beads and the multiplier (2 and 4) in cards in separate envelopes.  Say, ‘We know that if we have it in the envelope we want to keep it together and we must use brackets, we are going to do something new today, we have six and three, which means six plus three so we are going to add a plus sign today’, after placing the brackets put a ‘+’ sign.  Do the same for the multiplier. Say, ‘Then we are going to multiply them by our multiplier so we are going to put in a ‘X’ sign.  Then we are going to find out what they equal so we will add a ‘=’ sign.  What must we do, we must take our six and our three twice and our six times our three four times’, lay out the operation beneath as before.  Continue finding the partial products and the product.  Represent the product in white cards at the end as you read it out and read out the whole multiplication.  Work out the partial products and the product with beads and represent the product with the decimal system cards.

Aim:

To help the children become consciously aware of the distributive laws of multiplication.

Indirect preparation for binomial multiplication.

Stage 1 – using the signs

### Passage to Abstraction, Stage 2 – Expansion using white cards

Material Description:

As for the units, two envelopes, with

One set of white cards (Decimal system cards for the product)

Two sets of the grey cards (the multiplier)

Box of printed digits 0-9 on grey and white card

Operation signs (+, -, X, /, =)

Sets of brackets (lots)

Method:

– with the beads for the multiplicand, grey for the multiplier, white cards to expand and operation signs

Place the multiplicand (4 and 5) in beads and the multiplier (6 and 2) in cards in separate envelopes.  Say, ‘We know that if we have it in the envelope we want to keep it together and we must use brackets, we are going to do something new today, we have four and five’, place the brackets put a ‘+’ sign, ‘six and two times’. Do the same for the multiplier. Say, Then we are going to find out what they equal so we will add a ‘=’ sign. Today we are going to do something different ,we are going to use the while cards to show all the multiplications we need to do, so what are we going to do? (take four six times)  Take white cards for four and six and then because we need to keep them together we are going  to put brackets around them, and then we are going to take five six times’.  Then turn the ‘6’ card over and do the same with the two multiplier.  Take the white cards and place the brackets and ‘x’ cards to show the four multiplication problems which must be solved.  Put ‘+’ cards between each set of brackets to show that later the partial products will be added and an ‘=’ sign at the end. Use beads to show the four operations beneath.   Represent the product in large cards at the end as you read it out and read out the whole multiplication.

Aim:

To help the children become consciously aware of the distributive laws of multiplication.

Indirect preparation for binomial multiplication.

Stage 2 – using the signs, showing the expansion in cards

### Passage to Abstraction, Stage 3 – Working without representing the operation

Material Description:

As for the units and two envelopes, with

Three sets of white cards (Decimal system cards for the product)

Two sets of the grey cards (the multiplier)

Box of printed digits 0-9 on grey and white card

Operation signs (+, -, X, /, =)

Sets of brackets (lots)

2 small envelopes that fit the 10 bead bar and the printed digit cards

Method:

• with the beads for the multiplicand, grey for the multiplier, white cards to expand and operation signs this time without showing the operations with beads

Place the multiplicand (6 and 3) in beads and the multiplier (4 and 7) in cards in separate envelopes.  Say, ‘We know that if we have it in the envelope we want to keep it together and we must use brackets, we are going to do something new today, we have six and three’, place the brackets put a ‘+’ sign, ‘four and seven times’. Do the same for the multiplier. Say, ‘Then we are going to find out what they equal so we will add a ‘=’ sign. Today we are going to do something different, we are going to use the while cards to show all the multiplications we need to do, so what are we going to do? (take six four times).  Take the white cards and place the brackets and ‘x’ cards to show the four multiplication problems which must be solved, turn over the grey cards when complete and add ‘+’ signs between the brackets and ‘=’ at the end. Say, ‘This time we are going to do the multiplication in our heads’. Ask the child what is six times four, place the white cards for 24 beneath the problem and continue.  Ask the child what the sum of the units for partial products are and represent it, carrying in the head, find the sum of the tens, represent it in decimal cards by the equals sign.  Read out the summary of the operations.

Aim:

To help the children become consciously aware of the distributive laws of multiplication.

Indirect preparation for binomial multiplication.

Stage 3 – the operations are done mentally

Notes:

• We introduce one stage at a time.

When to give the lesson:

After plenty of experience with the white cards

Further work:

Show the children to do it on paper, write the problem on paper and show the expansion beneath.

Work as before, ticking each digit of the multiplier when complete. With plenty of verbalisation find the partial products, the whole product and write the answer by the original problem.

## With terms Greater than the Units – beads and grey cards, working with beads

We show all the beads in tens so that the children get the pattern first, as we did with the sum by sum, collating the knowledge, bringing it together and then extending it.

Material Description:

Many golden beads in unit beads, ten bars and hundred squares

Mat to work on

Sets of small decimal cards 1 – 3,000 (for second section)

One envelope

Box of printed digits 0-9 on grey cards (the grey represent the multiplier)

Many sets of brackets and two envelopes

Grey slips and a black thick pen for the multiplier

Method:

Say, ‘I am going to do multiplication a different way’, I will take thirty two, twenty four times’.  Place the beads of the multiplicand in an envelope and take grey slips and a black pen and write the multiplier.  Lay the multiplicand and multiplier out on the mat in brackets, do not use signs. The find the product and represent it beneath in vertical golden bead bars, (2 bead bars of ten).  Say, ‘We take thirty twenty times’ (600) and lay out thirty golden bead bars of ten horizontally, under the ten bars of the multiplicand, then say, ‘I take two twenty times’ (40) and lay out these below the multiplicand, while laying out both sets of beads frequently show that you are checking how many you have.  Turn over the tens card of the multiplier.

Say, ‘I am going to do my multiplication by my four, three taken four times’, lay the bead bars of ten under the ones already placed, with a small gap, saying , this time I will put them here’Do the same for the units multiplicand.  Turn over the units multiplier.

To calculate the product take ten tens (top – left corner) and exchange them for a square of ten.  Do the same with ten tens from the column on the right, then back to the hundreds and then to the tens (following the pattern of building the square of the decanomial)

order of working

 1 2 5 3 4 6

Then exchange the units of the thirty and the units of the two, filling in going down the side.  (at this point only treat the beads times by the tens of the multiplier.

Exchange the bead bars for decimal system cards in their groups

 600 40 120 8

Exchange the cards till you have one card for each category, slide the decimal cards over each other to reveal the product.

Show the children how to put the symbols for the operations to write what has happened in a line at the top of the working and read through the summary.

‘Ask the children if anyone can see the multiplicand?’ (horizontal line) and, ‘Can anyone see the multiplier?’ (vertical line)

Aim:

To help the children become consciously aware of the binomial formation

Notes:

• Do not present with a number which can result in a square
• The material is laid out as for sum by sum
• To show the product we follow the pattern of the square of the decanomial and the pattern of the chequer board, to follow this sensorial impression

When to give the lesson:

After the work at the beginning of the chequer board has been complete and earlier lessons with the laws of multiplication and after a working knowledge of multiplication by tens and units (binomial multiplication) has been built.

After the lesson:

The children continue to work with many examples

### With terms Greater than the Units – Passage to Abstraction

Writing the problem cards, working without representing the operation

Material Description:

Many golden beads in unit beads, ten bars and hundred squares

Mat to work on

Sets of small decimal cards 1 – 3,000 (for second section)

One envelope

Box of printed digits 0-9 on grey cards (the grey represent the multiplier)

Many sets of brackets

White slips to write the whole problem and a black thick pen

Blank white cards instead of beads to represent the multiplicand and the numbered grey cards for the multiplier

Method:

Say, ‘Today we are going to do some multiplications using cards’.  As you write on cards say, ‘I am going to take ’(30+2) x (20+4)’.  Without using the envelope or beads lay out the problem in white and grey cards, the white cards are written as you go, using the addition, multiplication and equals signs.

Say, ‘I am going to do my multiplication thirty twenty times’,  write the small white cards and place the expansion in brackets ‘(30×20) (2×20) (30×4) (2×4), when it is complete add the addition signs. Turning over the multiplier cards as you go.

Ask the children, what is 30 x 20, place six hundred squares beneath in the same arrangement as above, then what is two times twenty and place the four bars of then to the right, then what is thirty-four times, place the twelve bards of the under the hundred squares and then two taken four times, put the unit beads in the bottom right corner.

order of working

 1 2 3 4

To calculate the product place decimal cards over the beads, starting with the units.  collect the cards, putting similar numbers together and exchanging them before overlaying them to get the product.  Lie the product in the top line and read through the summary.

Return to the slip of paper with the problem and write the product at the end.

Aim:

Direct preparation for squaring and square root

This brings together many parts of the laws that the children have worked with separately

The children learn something fundamental about the behaviour of the categories

Notes:

• Do not present with a number which can result in a square
• The material is laid out as for sum by sum
• To show the product we follow the pattern of the square of the decanomial and the pattern of the chequer board, to follow this sensorial impression

When to give the lesson:

After the work at the beginning of the chequer board has been complete and earlier lessons with the laws of multiplication and after a working knowledge of multiplication by tens and units (binomial multiplication) has been built.

After the lesson:

Early in elementary, when children continue to work with many examples, thousands do not appear in the binomial formation.  At some stage of their working they can be told, This is a binomial formation’

Later you can join the children and suggest doing it on paper.

## Working on Paper

(30 + 4 ) (20 +3) =

(30×20) + (4×20) + (30×3) + (4×3)

600 + 80 + 90 + 12

782

Explain the workings as you go and tick of the multipliers when they have been worked.

Check 34 x 23 with long multiplication

Encourage the children to explore using a variety of examples

Aim:

It encourages the children to work with the knowledge they already have

Direct preparation for squaring and square root

Arithmetic

Arithmetic

Arithmetic