Definition/Summary/Review: Multiplication

Definition/Discussion: Multiplication

Definition Whole Number Multiplication:

Repeated Addition Definition: For any two whole numbers, a and b, we define a*b to be the sum of b added to itself a-many times. We write, a*b=(b+b+… +b). The numbers a and b are each called factors of the multiplication operation and the result is called the product.

When we multiply two whole numbers, a and b, we often say that we are finding "a times b" in reference to the number of times, a that we add the number b to itself. For this reason, multiplication is also called "fast or shortcut addition." We first teach that the operation 2*3 says to add three to itself two times, 2*3=3+3. Then after memorizing our single digit multiplication facts, we combine them with the associative, commutative and distributive properties of whole numbers as discussed below and mix them up with our place value system according to a recipe we call an algorithm to find the product of multiple digit numbers such as twenty-three by twenty-three (23*23) without having to add twenty-three to itself twenty-three times. When students are taught that all they need to know are the basic multiplication facts, that is the products of all the single digit numbers zero through 9 and some rules for putting them together, then they can find the product of any two whole numbers quickly no matter how many digits are in the numbers. And when students realize that there are many ways (algorithms) they can use to avoid the drudgery of adding up a column of multiple digit (e.g. adding 23 to itself 23 times), they can pick and choose among the different multiplication recipes (algorithms) and use the ones they like the best.

It also helps student learning and motivation if the teacher keeps the pencil and paper practice exercises no larger than the product of three digits by two digit numbers. It also can save trees if students are encouraged to use a calculator as the practical way to multiply such numbers today. When I was in high school, we were not allowed to use a slide rule on tests. My goodness! What would we do if we were ever alone on a desert island, needed to multiply a set of really large numbers and didn’t have our slide rule? If students have mastered their zero through nine products and have mastered at least one algorithm for multiplying three digits by two digit numbers, they know they can multiply larger digit numbers the same way if they ever had to. So let’s accept progress and move on.

The following write-up details explicitly every step we take in mathematical reasoning implicitly whenever we apply a multiplication algorithm. Because everyone knows the product of two times three, even many preschoolers, so we choose the example 23*23 in order to show focus on the process and not the result.

For example:

23*23 =(20 3)*23 by writing the left factor, 23 in expanded form 23=20+3.

=20*23 3*23 by distributing the right factor of twenty-three over the sum of twenty plus three.

We are all familiar with the algebraic way of writing out the right distributive rule, (a+b)c=(ac)+(bc) where we use juxtaposition of the letters to represent a product of two variables. Clearly, ac takes up less space then a*c. But numbers do not lend themselves to this notation. The product of two numbers requires that one use either the multiplication symbol or parentheses, for example, (2)(2)=4=2*2 ≠22. Continuing, …

=20*(20+3) 3*(20+3)

by rewriting the right factor of each term in expanded form, i.e. we expand 23=20+3 again.

=20*20 20*3 3*20 3*3 by the (left) distributive rule, a(b c)=ab bc.

NOTE: Our object in writing this product out in such detail is that we want you to appreciate how the FOIL algorithm enables you to get to this expression as the second not the fourth step as shown here. So to get to the next step we just acknowledge that 20*3=3*20, multiply and omit breaking down any other term than 3*20=3*(2*10)=(3*2)*10=6*10=60. Besides we do this to ad nauseam in the base five examples that follow this discussion.

=400+60+60+9 by multiplying out each term.

=400 + 120 + 9 by adding 60+60.

= 520+9 = 400 + 129 = 400+120 or any other way you want to find the sum by associating and commuting and otherwise using the basic properties of addition that out of habit, we take for granted.

Still on track but making an aside comment, it used to be fun years ago wowing students, engineering majors as well as mathematics education students, anytime you showed them the FOIL algorithm to find the product of two binomials. Now, it is rare that any student past middle school doesn’t already know it. Any way, back then the multiplication of variable binomials was most often taught by the same vertical algorithm we were taught to find the product of multiple digit numbers.

For example, . And few people knew the final sum spelled the anachronism FOIL that stood for a process that could be used to bypass having to write out any more steps than necessary to find the product of two binomial expressions, especially when the product was written horizontally, (a+c)*(x+y)=ax+ay+cx+cy. "FOIL" is the acronym for the process of finding the product of the first terms plus the product of the outside terms plus the product of the inside terms plus the product of the last terms to get the product of two binomial terms, whether they be algebraic or numerical.

Then they might apply the distributive property twice, first by a and c over (x+y), then by (x+y) over (a+c) getting a*(x+y)+c*(x+y)=(a+c)*(x+y) to check the answer. It seemed that the only ones who knew about, let alone used the FOIL Algorithm were senior level mathematics majors who had taken at least one upper division algebra course and not just understood but really comprehended the significance of what all those associative, commutative, distributive, etc. properties of a group were good for. Many students would even get A's in the theoretical classes being able to evoke out all the correct terminology and procedures and yet not recognize its usefulness in multiplying multiple digit numbers. My frustration was always that while I did recognize the advantages of the group properties when used in everyday arithmetic, I was too conditioned to the "put down your seven and carry the two" algorithms to be able to apply them in real life. If I wanted to buy three loaves of bread that were only about 49¢ a loaf for the big named brands, I had to stop in the middle of the aisle and try to picture how to bring down the seven and carry the two to add to the product of three times four to get the cost. Perhaps, it was partly because of my frustration that I was never able to pull-up these useful algorithms in practical real life situations that I always felt compelled to accept teaching schedules that included the mathematics education classes for pre-credential multiple subjects majors. I always taught one or more of these classes even as I was advising master degree students in algebra. The rest of my professional history is of no importance here, but my part of my main reason for writing this up is to deal with my frustration today that while the students immediately yell out to "FOIL it" to find such a product today, they do so with the same feeling they have about how a calculator gets its the answer so quickly even when multiplying enormously large multiple digit numbers together. The idea behind the "New Math" phase of the sixties was good in that one of its aims was to teach students how to use these easily applied theoretical algorithms mentally using the properties of the real numbers that made them valid. Many mathematicians then, just as now believe that if a student could become a better at solving pencil and paper word problems then they would transfer the strategies they learned there to their every day activities (and hence become better citizens, etc.)

If you want a challenge, try to find a1950’s algebra text that dared to mention the word “FOIL” even if they knew it was an algorithmic anachronism for a valid arithmetic process. While I enjoyed sharing the FOIL algorithm with the students then, it is frustrating how many students today blurt it out without any comprehension of what the product it gives means.

In the following we explain how the FOIL Algorithm works and why it always results in a correct answer (provided the single digit products and sums are correct) when applied to the product of any pair of binomials including two digit numbers.

We know that the area of a rectangle is the product of its dimensions that is the product of the length times the width. And even pre-school children putting puzzle pieces together know that the whole is equal to the sum of its parts. So with the advent of the NCTM Standards in the nineties, it became the rule for mathematics educators to use geometry to model many arithmetic concepts. In particular, multiplication was modeled using the additive properties of the area of rectangles. This model gives students a sense of applicability of what multiplication is good for as well a visual means to get a feeling for the relative differences in size among several sundry products of whole numbers.

We started using the geometric properties of areas of each of the sub-squares and in exactly the same order as we did before, namely, by the FOIL algorithm.

We know that the area of a rectangle is the product of its dimensions that is the length times the width. we get the same values as the areas of each of the sub-squares and in exactly the same order as we did before, namely, by the FOIL algorithm.					F O I L	=The product of the First digits, (basesbases) + =The product of the Outside digits (basesunits) + =The product of the Inside digits (unitsbases) + =The product of the Last digits (unitsunits).
		4*10		3			4*10	3
	4*10		(410)(410)= (44)*10	(410)3= (43)10		4*10	31* 10 3100 F	22*10 220 O
	4		4(410 )= (44)10	4*3		4	I (44)10 310	L 4*3= 22
And the sum of the areas of the sub-squares, 3100+220+310+22=

There are several models we can use to find the answer to a multiplication problem. These models enable the student to experience both a tactile and visual experience of the operation. As discussed below, an array is perhaps best and most used model to give the student a visual understanding of the commutative property of whole number multiplication.

the (fast addition) model (everyday definition): ab=(b+b+. . . +b), the value of the whole number b added to itself a many times

For example: 2*3=3+3 (two threes added), and because multiplication is commutative, 2*3=3*2 or three twos added, 3*.2=2+2+2.

Set (Cartesian Product) Definition:

The most common mathematical definition of multiplication found in textbooks today says that for whole numbers a and b, the product, ab, of a and b is the number of elements in the set Cartesian product of the sets A and B. That is, ab=n(AxB), where a=n(A) and b=(B) and AxB={(x,y) | x A and y B}. Therefore, the product, ab, of any two whole numbers a and b is the number of ordered pairs (x,y) in the Cartesian Product of sets A and B where the first element in each ordered pair is a member of the set A and the second element is a member of set B. The numbers a and b are called the factors of the product ab.

Models for Multiplication:

Number line model: The number line is as easily used as the array to model the product of two whole numbers. If we have a number line, then we can say that a*b is the total distance that a b-cricket covers in a-many jumps. It is not hard for students to accept that a b-cricket covers b-many units per jump, etc. In the past, it was most often the case that number lines were shown horizontally, across or as like the path one walks on a level field or vertically, up or as for many thermometers. With increasingly many teachers implementing the Standards into their classrooms, it is nice to see that future students will not be conditioned that horizontal or vertical are the only "correct" orientations that holds for number lines, rectangles or triangles all having a horizontal base, etc. A student can twist a ruler all around in space. Therefore, the association of the ruler with the number line makes it as sensible that a number line can be oriented diagonally, / or \ as well as up and across.

Array or rectangle model: An array is a rectangular grid having "a-many rows with b-many entries per rows" or "a-many rows by b-many columns." The array model derives from the formal Set or Cartesian product definition of multiplication as defined above and discussed below. This model is also aptly referred to as the rectangle model because a border around any array generates an a-width by b-length rectangle. And just as a paper cut-out rectangle can be held vertically as easily as horizontally, the area or number of square unit spaces that will cover an a by b rectangle is the same no matter how the paper is held, even / or \. But the horizontal vs. vertical orientation visually models the equivalence of a*b and b*a. It helps students to understand what we mean when we say that multiplication is commutative. And because a*b=b*a so clearly for all whole numbers, it obviates the hesitation of some students to accept the same equivalency for all real numbers.

Example: The matrix is a (2row)*(3col) array that would fit into any similar size 2x3 rectangular space.

Cartesian product model: The set definition given above tells us that a Cartesian product, AxB of two sets A and B model as defined above. Example: Let A be the set containing two pieced of fruit an apple~a and a banana~b., A={a,b}. Let B be a set with the three numbers 1, 2 and 3 as elements, B={1,2,3}. Then

AxB={(x,y)| x I a or b and y is 1 , 2 or 3} ={(a,1), (a,2), (a,3), (b,1), (b,2), (b,3)}

is the set of all possible ordered pairs of elements from the sets A and B. Clearly, 2=n(A) and 3=n(B) and 2*3=n(AxB)=6 . If the ordered pair notation (x,y) replaced the subscript x notation elements in the 2x3 array shown above, it would look like the following:

Tree* model: a diagram beginning with a-many objects shown vertically with b-many "branches" coming off of each of the a-many objects.

*Note: Billstein refers to the tree model of multiplication we give here as the Cartesian model.

NOTE: Models are like analogies, namely ways to clarify the understanding of a concept or operation. Most models enable you to physically visualize the component parts, whereas an analogy is more likely to give you a mental image to aid you in understanding whatever is intended. Notice the relationships that hold among the array and Cartesian product and the tree model. In an array, the elements in set A denote the row elements and the elements in set B label the columns elements. In the rectangle model, a is the width and b is the length. In the tree model, each element of set A is shown in the first column and is paired with each element b in set B one-by-on in the second column. A tree model could as well be shown in the shape of a T with the top row of the tree containing the elements in set A and the elements in set B shown as descending branches.

Properties of whole number multiplication:

For all whole numbers a, b, or c, the multiplication satisfies the following real numbers properties. Just as I use the anachronism C ica to help me remember the four real number properties of addition, I add D and Z at the end to get C icadz to help me remember the six properties of multiplication.

CL Closed ab is a unique whole number that belongs to the same set as do the elements a and b.

I Identity (R^{emember, must be unique)} a*1 = 1*a = a, the number 1 uniquely identifies all numbers with themselves under multiplication.

C Commutative ab = ba, says that the order of taking products does not numerically affect the product.

A Associative a(bc) = (ab)c says that the way in which any two numbers in a product are associated and then multiplied does not affect the numerical value of the product.

D Distributive over addition a(b+c)=ab+ac . (I call this the lunch bag rule because you need to buy 2 of each apples and bananas to make two lunch bags each one of which contains one each of an apple and a banana, i.e. 2a+2b=2(a+b)).

Z Zero property of multiplication 0*a=a*0=0, for all real numbers a.