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Classical Liberal Arts Academy

Classical Arithmetic I

Lesson 14.  Rules of Invention:  Multiplication

Before you begin:  Each lesson in the Classical Arithmetic program has three parts:  a Lesson, Memory Work and an Examination.  Some lessons will have exercises that must be completed before you take your examination.  When you have studied your lesson carefully and completed your memory work and exercises (if any), you should complete your lesson examination.  If you don't pass your examination, go back, review and try it again.  This page is designed to be printed for use away from the computer.

1.  Lesson

In our last lesson, we learned that the Latin word invenire means "to come upon" or "to find" something.  We have already learned about proofs and how reason leads us to connect arguments to make new conclusions.  For example, we learned that if 36 in. is equal to 3 ft. and that 1 yd. was equal to 3 ft., then 36 in. must be equal to 1 yd..  In this, what we know leads us--by reason--to the invention or discovery of something we wished to know or something that we simply didn't know before.  This is invention as it applies to Arithmetic.

There are four basic rules that help us to discover new knowledge in Arithmetic using what is already known to us.  They are Addition, Subtraction, Multiplication and Division.  In our last lesson, we learned about Addition and Subtraction.  Here, we will study the third rule of invention:  Multiplication.

WHAT IS MULTIPLICATION?

We learned that to add quantities quickly, we use addition.  Our goal was to discover a sum when two quantities are added together--and to discover it quickly.  We can memorize addition facts and then use our memory to quickly and accurately discover sums.  However, for many addition problems, there is an even faster rule for discovering sums:  multiplication.

In your study of addition, you memorized the addition facts in a table.  You memorized facts like "2 and 4 are 6", "3 and 7 are 10" and so on.  Addition as you have learned it works very well when we are adding two addends at a time. For example, if Mary had 3 lambs and Brian had 5 lambs, addition helps us to discover the sum of their lambs:  8.  What would we do, though if there were seven children and every one of them has 4 lambs?  How would we add 4 seven times?  We could count all the lambs, one-by-one and find that there are 28 in all, but addition would be quicker.  With addition, we could write the problem like this:

4 + 4 + 4 + 4 + 4 + 4 + 4 =    ?   

The use of addition would be faster than counting, but you can see that it is getting pretty complicated .  What if there were 12 children with 4 lambs each? We would have to write:

4 + 4 + 4 + 4 + 4 + 4 + 4 + 4 + 4 + 4 + 4 + 4 =   ?   

Adding like this gets more and more difficult the more times we must add a number, but this is exactly what multiplication is for. We can write these problems in a much simpler way and then, with some time and practice, memorize the answers.  Then, we can quickly solve problems like this in the future. 

Instead of writing out a long addition problem, we will state the number to be added and how many times it is to be multiplied (added to itself).  We use the symbol  × to signify multiplication. 

Rather than writing:

4 + 4 + 4 + 4 + 4 + 4 + 4 =   ?  , or

4 + 4 + 4 + 4 + 4 + 4 + 4 +4 + 4 + 4 + 4 +4 =   ?   

We can write:

4 × 7 =   ?  , or

4 × 12 =   ?   

We can read this as:

What is 4 added 7 times ?

What are 7 sets of 4?

What is 4 multiplied by 7?

What is 4 times 7?

What is the product of 4 and 7 ?

In each of these statements, 4 and 7 are called factors or efficients.  The first quantity is called the multiplicand (to be multiplied) and the second quantity is called the multiplier.  The product is the quantity to be discovered, which derives its name from the Latin word productus, which means the quantity brought forth. 

Remember:  Multiplication is simply a fast way to add a number to itself many times.

MULTIPLICATION MEMORY FACTS

By memorizing basic multiplication facts, we can quickly discover products from memory--and not need to add a list of numbers one-by-one.  The table below provides us with the multiplication facts that should be memorized.  The numbers zero and one are never used as factors because it makes no sense to take a number zero times or to take a number one time.  Thus, 2 is the least integer that we use for multiplying and is the first factor you should memorize. 

To use this chart, simply begin with the multiplicand 2 on the top row.  Moving down to the first product, read "2 taken 2 times is 4".  Then move to the right to the multiplicand 3.  Moving down to the first product read, "3 taken 2 times is 6", then, "3 taken 3 times is 9.".  Continue through the rest of the table, memorizing every product listed.  This table is very ancient and is called the Pythagorean Abacus, being named after the ancient philosopher Pythagoras who is believed to have invented it.

MULTIPLICATION FLASH CARDS

If you would like to create printed or hand-written flash cards to help you practice your memory work, you should make them to look like these examples.  The factors are on the bottom and the product at the top.  You should have one card for every product on the table above, for a total of 36 cards.  

You read these flash cards in two directions.  First, start with the factor on the bottom left as the multiplicand and read, "2 taken 5 times is 10".  Then, remembering that multiplication is commutative, start with the factor on the bottom right as the multiplicand and read "5  taken 2 times is 10".  When you have the facts learned well, have someone quiz you by covering the product and showing you the factors.  Then, have someone show you the product and one factor and read, "3 taken x times is 12.  What is x?"  You provide the missing factor in place of x, saying, "3 taken 4 times is 12, therefore x is 4."

MASTERING MULTIPLICATION

Students in most modern schools are allowed to use electronic calculators to solve their Arithmetic problems for them.   Students are able to solve problems in their books quickly, but they never learn the principles of Arithmetic and never become masters of it.  Our goals are much higher!   We need to understand the rules and reasons of Arithmetic as a way of growing in Wisdom--not just to pass a test with a calculator and be done with it.

Your multiplication table gives you the basic information your reason needs to solve many problems in Arithmetic.  To use this information like a master, you must know three basic rules for multiplication.  When you master these rules and use them with your multiplication memory facts, you will be able to solve many problems.  The question will ultimately be:  how well can you use your memory facts?

The three rules are these:

1.  Multiplication is Commutative.

2.  Multiplication is Associative.

3.  Multiplication is Distributive.

Let's learn what these rules mean and how to use them.

I.  MULTIPLICATION IS COMMUTATIVE

Remember that Multiplication is simply a fast way to add a number to itself many times.  Ultimately, any multiplication problem can be written as an addition problem--that's really all it is.  Let's look at an example that will help us learn a bit more about multiplication.

Above we said that if 7 children each had 4 lambs, we could discover how many lambs there were in all by addition:

4 + 4 + 4 + 4 + 4 + 4 + 4  = 28 

Here we have set the 4 lambs into seven groups.  If we use our multiplication table, we can find that the product of 4 taken 7 times is 28.  However, if these seven children wanted to set all of their lambs into 4 carts, there would be 7 lambs in each cart, like this:

7 + 7 + 7 + 7  = 28

What we see then is that if the multiplicand and the multiplier are exchanged, the product remains unchanged.   The Latin word for "exchanged" is commutativus, so we can say that multiplication is "commutative"--the factors can be exchanged.   We can write a formula using the indefinite quantities a and b to remember this:

a × b = b × a     or     ab = ba

For example, if a = 7 and b = 4, then once you learn that 7 × 4 = 28, you also know that 4 × 7 = 28.  Once you learn that 6 × 3 = 18, you also know that 3 × 6 = 18.  This is because multiplication is commutative.

HOW THIS HELPS US

When we know that multiplication is commutative, we then are able to know the answers to all basic multiplication problems by memorizing only half of them!  Therefore, it makes our memory work much easier.  If we memorize that "7 taken 2 times is 14", we can switch the factors to say, "2 taken 7 times is 14" also.  We only have to memorize those problems listed on Pythagoras' Abacus.  The rest we can figure out because we know that multiplication is commutative.

MULTIPLYING BY TENS, HUNDREDS, ETC...

When we multiply a quantity by 10 we take that quantity 10 times.  Therefore if we multiply 3 × 10, we are taking 3 ten times, which would look like this:

3 × 10 = 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 =   ?   

Knowing that multiplication is commutative helps us to solve this in a faster way.  Rather than adding 3 to itself 10 times, we can simply re-write the question to ask, "What is 10 taken 3 times?

3 × 10  =  10 × 3  =  10 + 10 + 10 = 30 

Therefore, the commutative property helps to see that anytime we multiply a number by ten, we simply add a zero after it to get the product.  3 taken 10 times is 30.  Just add a zero to 3!  If we multiply a number by 100, we make that many sets of 100, so we simply add 2 zeroes after the quantity we started with.  3 taken 100 times is 300.  Just add 2 zeroes to 3!  For thousands, we add three zeroes, and so on.  That is much faster than add 3 to itself 1000 times, isn't it...and we never need to use calculators.

II.  MULTIPLICATION IS ASSOCIATIVE

Our multiplication table has still more powers to help us once we learn more about multiplication!  Remember that Multiplication is simply a fast way to add a number to itself many times.  Ultimately, any multiplication problem can be written as an addition problem.  Let's look at an example that will help us see another way to use multiplication.

Above we said that if 12 children each had 4 lambs, we could discover how many lambs there were by addition:

4 + 4 + 4 + 4 + 4 + 4 + 4 + 4 + 4 + 4 + 4 + 4 =   ?   

Multiplication can help us find the answer to the same problem much faster.  We can re-write this problem using multiplication, like this:

4 × 12 =   ?     

This, however, leads us to a problem.  The multiplier 12 is not in our multiplication table!  How in the world can we know the answer to this problem if our multiplication table only goes up to 9?  Let's look back at the addition problem to think about this differently.  It should be pretty obvious that this addition problem can be broken up into pieces like this:

( 4 + 4 + 4 + 4 + 4 + 4 ) + ( 4 + 4 +  4 +  4 + 4 + 4 )  =   ?   

If we write this as a multiplication problem, we get this:

( 4 × 6 ) + ( 4 × 6 )  =   ?   

Remember that multiplication is simply a fast way to add a number to itself.  If we look at this problem as written above we are adding ( 4 × 6 ) to itself two times!  Therefore we can write this again as a multiplication problem:

( 4 × 6 ) × 2  =   ?   

Now, when we look back at our original problem, we can see that:

Since 12 = 6 × 2

Then 4 ×12  is equal to 4 × (6 × 2 )

When we compare this to the problem written just above, we can see that:

( 4 × 6 ) × 2  =   4 × (6 × 2 )

This is a long way of saying that anytime we write a multiplication problem, we can break each of the factors down into their own factors and multiply them in any order we please.  Sometimes, that will make multiplying large numbers much easier.

We can write a formula using the indefinite quantities a, b, and c to remember this.

a × ( b × c )  =  ( a ×  b ) × c   =  a × ( c × b )

For example, if a = 2,  b = 3 and c = 4, then:

2 × ( 3 × 4 )  = 2 × 12 = 24

( 2 × 3 ) × 4  = 6 ×   4 = 24

3 × ( 2 × 4 )  = 3 ×   8 = 24

HOW THIS HELPS US

When we know that multiplication is associative, we then are able to know the answers to more difficult multiplication problems by memorizing only those on Pythagoras' Abacus! 

For example, we do not memorize what "7 taken 30 times" is.  Most modern students would use a calculator to figure it out--but we don't need to.  We know that 30 is equal to 3 taken 10 times, so we can solve this problem by asking, "What is 7 × ( 3 × 10 ) ?" and then using our memory work to first solve 7 × 3 = 21  and finally multiplying that product by 10 to get 210.

WRITING OUT A MULTIPLICATION PROBLEM

Some multiplication problems would be too difficult for us to solve in our minds (at least now) and will need to be written down.  To write out a multiplication problem we write the multiplicand with the multiplier underneath.  Then we draw a line under the multiplier so that units are under units, tens under tens, etc..  If we were asked, "What is 7 taken 12 times?", we would write it like this:

The associative property teaches us that we can solve this in a different way.  Because 12 = 4 × 3, we can solve the problem like this: 7 × 4 × 3.  This can be solved in writing as follows--first finding the answer to 7 × 4 and then multiplying that product times 3.

But how can we find the answer to 28 × 3?  It's no easier than 7 × 12!  We will see that the next property is the one that helps us to multiply large numbers. 

III.  MULTIPLICATION IS DISTRIBUTIVE

Our multiplication table above appears to only give us the products for factors from 2 through 9, but it is much more powerful than that so long as we remember what multiplication is   Remember that Multiplication is simply a fast way to add a number to itself many times.  Ultimately, any multiplication problem can be written as an addition problem.  Let's look at an example that will help us see a different way to use multiplication.

Above we said that if 12 children each had 4 lambs, we could discover how many lambs there were in all by addition:

4 + 4 + 4 + 4 + 4 + 4 + 4 + 4 + 4 + 4 + 4 + 4 =   ?   

Multiplication can help us find the answer to the same problem much faster.  We can re-write this problem using multiplication, like this:

4 × 12 =   ?   

This, however, leads us to a problem.  The multiplier 12 is not in our multiplication table!  How in the world can we know the answer to this problem if our multiplication table only goes up to 9?  Let's look back at the addition problem to think about this differently.

4 + 4 + 4 + 4 + 4 + 4 + 4 +4 + 4 + 4 + 4 + 4 =   ?   

It should be pretty obvious that this addition problem can be broken up into pieces like this:

( 4 + 4 + 4 + 4 + 4 + 4 ) + ( 4 + 4 + 4 + 4 + 4 + 4 ) =   ?   

Now, if we write this as a multiplication problem, we get this:

( 4 × 6 ) + ( 4 × 6 ) =   ?   

Once we have re-written the problem, we can easily solve this, using our multiplication table.  While we could not find what 4 × 12 would produce, we can find that 4 × 6 equals 24.  Therefore, by adding each part, we can find the product we originally needed:

24 + 24 =  48, which is the product of 4 × 12.

What we see then is that multiplication works on each part of a factor just as it works on the whole factor.  The Latin word for "working on each part" is distributivus, so we can say that multiplication is not only commutative (as above) but also distributive.

We can write a formula using the indefinite quantities to remember this.

If b = y + z, then

a × b =  ( a × y ) + ( a × z )

For example, if a = 7 and b = 12, then:

7 × 12 = ( 7 × 4 ) + ( 7 × 8 ), or

7 × 12 = ( 7 × 3 ) + ( 7 × 9 ), or

7 × 12 = ( 7 × 6 ) + ( 7 × 6 ), or

HOW THIS HELPS US

When we know that multiplication is distributive, a whole new world of solutions opens up to us--and we only need to memorize the facts on Pythagoras' Abacus! 

Above we saw that the problem 7 × 12 is too difficult for us to figure out with our memory facts.  The commutative and associative properties didn't help us either.  However, the distributive property allows us to break difficult multiplications problems like this one in to easier parts, like this:

7 × 12 = ( 7 × 4 ) + ( 7 × 8 )

You can see that both of these simpler problems can be figured out from our memory work and then added together.  How easy is that!

7 × 12 = ( 7 × 4 ) + ( 7 × 8 )
7 × 4 = 28
7 × 8 = 56
28 + 56 = 84

Thus, 7 × 12 = 84

WRITING OUT A MULTIPLICATION PROBLEM

As we learned above, multiplication problems are often written down when they are too difficult to solve in our minds.  As you gain mastery in multiplication, you'll be able to solve more difficult problems without writing them down.  To write out a multiplication problem we write the multiplicand with the multiplier underneath.  Then we draw a line under the multiplier so that units are under units, tens under tens, etc..  If we were asked, "What is 7 taken 12 times?", we would write it like this:

Above, we learned that the associative property could not help us solve this problem--but the distributive property can!  Instead of multiplying 7 × 3 × 4, we will add the products of ( 7 × 10 ) and ( 7 × 2 ).  This will prove to be the easiest way to solve this problem.
 

Then, when we add 70 + 14, we have our product:  84. 

Most modern students are taught to write this down as follows, first multiply by units and then by tens:

This second form may be easier for writing, but the first is usually easier for multiplying in your mind.  Remember:  the goal is to solve problems with perfect accuracy at all times and increasing quickness throughout your life.  Your mastery will depend on how diligent you are in working with Arithmetic.

SUMMARY

In this lesson, we have learned the third rule of invention in Arithmetic:  Multiplication.  We have learned most importantly that "Multiplication is simply a fast way to add a number to itself many times."  We've learned that when we combine our knowledge of the Pythagorean Abacus and the rules of multiplication, we can solve many problems in Arithmetic.  Now it's time to put it all to work.

Directions:  The following questions help you to memorize the most important points of this lesson.  Commit them perfectly to memory and have a parent or praeceptor quiz you to test your mastery before taking your lesson exam.

101.  REVIEW:  What are the first four rules of Arithmetical Invention?

The first four rules of Arithmetical Invention are: Addition, Subtraction, Multiplication and Division.

111.  What is Multiplication?

Multiplication is the invention of a number or quantity called the Product, by taking or adding a given number called the Multiplicand as often as there are Units or Parts of a Unity in another number called the Multiplier.   

112.  By what other names are the multiplicand and multiplier called?

In multiplication, the multiplicand and multiplier are also called Efficients, or Factors.

113.  What is the Sign of Multiplication?

The Sign of Multiplication is ×, which is read "multiplied by" or "times", as 6 × 2 denotes the Product of 6 multiplied by 2, or 6 taken 2 times.  Indefinitely, b × d (or simply bd ) denotes the product of two given quantities signified by the species b and d.

114.  How do we multiply factors consisting of more than one term?

If one or both of the factors consists of more than one Term, connected by the signs  + and – , each factor is set in parentheses.  Thus, the product of a + b – c multiplied by x + z is written:  (a + b – c) × (x + z).  

115.  What are Coefficients?

Coefficients are numbers multiplied into Species, and when written next to Species signify how many times the Species are taken.  Thus, 5a means that the species a is taken 5 times, 14b means that b is taken 14 times.  Every Species without a Coefficient has Unity understood to be set before it, as a is 1a, b is 1b.

116.  What are the Dimensions of a Product?

The Dimensions of a Product refers to the number of literal (letter) factors in it, as aa is a product of two dimensions, 7abc of three, xxzz of four dimensions, and so on.

117.  AXIOM XVI:  If equal Quantities are multiplied by equal Quantities, the Products will be equal.

Directions:  When you have completed all of your assignments above, complete your lesson examination. 

Multiplication Memory Facts - Untimed |  2 Minutes |  1 Minute

Commutative Property Exercise - Untimed |  2 Minutes |  1 Minute

Multiplying by 10s, 100s, 1000s - Untimed |  2 Minutes |  1 Minute

Associative Property Exercise - Untimed |  2 Minutes

Distributive Property Exercise - Untimed |  2 Minutes

Directions:  When you have completed all of your assignments above, complete your lesson examination.