Mathematics as a Second Language

Transcript Mathematics as a Second Language

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Arithmetic Revisited

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Lesson 2 Part 3.1

Whole Number Arithmetic

Multiplication

Multiplication of Whole Numbers Understanding the Algorithm When dealing with whole numbers, we have already seen that subtraction is a form of addition that we can view as “unadding”. In a similar vein, we may also view multiplication of whole numbers as a form of addition. More specifically, using place value and our “ adjective / noun ” theme, we can show that in place value format the multiplication algorithm is rapid, repeated addition. © 2010 Herb I. Gross

We will assume that most students know the tables and the traditional algorithm ( s ) for multiplication. However, if students memorize the algorithms without properly understanding them some very serious errors involving critical thinking can occur.

Note There is a raging debate about whether calculators hamper students’ efforts to understand mathematics. We should keep two points in mind. For one thing, if the traditional methods had been as successful as its advocates would have us believe, there might not have been a need for new standards and educational “reform”. © 2010 Herb I. Gross

Secondly, it is possible that students, if they do not fully understand what multiplication means, will not recognize when the answers they obtain are unreasonable. Moreover, this applies in equal measure to those students who use calculators as well as to those who learn algorithms by rote.

1 note 1 Prior to the advent of calculators, it was crucial for students to learn the arithmetic algorithms in order to compute. It was not as great a priority for students to understand the logic behind the algorithm . In other words, accuracy was more important than understanding. Nowadays, however, calculators can do the required arithmetic and as a result more emphasis is now being placed on understanding and applications.

Thus, we will accept the fact that most students can perceive that they have correctly assimilated the various multiplication algorithms by rote; yet because of subtleties that they overlook, they often make serious errors by not understanding each step in the process.

In this context, they might write… 4 1 5 × 1 0 1 4 1 5 + 4 1 5 4 5 6 5 2 4 1 5 × 1 1 4 1 5 + 4 1 5 4 5 6 5 note 2 Notice that even though the problem is written as 415 × 101, the computation was done as if the problem had been 415 × 11 (which is 4,565). Namely, in terms of our adjective / noun theme by placing the 5 under the 1, we multiplied 415 by 10 rather than by 100.

By understanding the multiplication algorithm students should realize that 415 × 101 must be greater than 415 × 100; and since 415 × 100 = 41,500 it is clear that 415 × 101 > 41

500.

There are many numbers that are greater than 41

500 but 4,565 isn’t one of them.

3 note 3 It is an important learning device, especially when we don’t know the correct answer, to realize that some incorrect answers are less plausible than others. Thus, recognizing what can’t be the correct answer can often serve as a clue for finding the correct answer.

Notes on Multiplying by a Power of 10 In Roman numerals, we multiply by ten by changing each I to an X; each X to a C; and each C to an M etc. Thus, to multiply XXXII by ten, we would change each X to a C and each I to an X, thus obtaining CCCXX… C C X X (32) (320) …which corresponds to 10 × 32 = 320.

Notes on Multiplying by a Power of 10 We obtain the same result in place value by placing a 0 after the 2 to obtain the fact that 10 × 32 ( = 32 × 10) = 320. More specifically, the 0 moves the 2 from the ones place to the tens place and the 3 from the tens place to the hundreds place.

4 note 4 Notice in this situation that 0 is not “nothing”. Rather it is used as a place holder that changes the nouns that 3 and 2 modify. That is by annexing the zero, the 2 now modifies tens instead of ones, and 3 now modifies hundreds instead of tens.

Notes on Multiplying by a Power of 10 In terms of our adjective / noun theme… 1 0 × 32 = 32 × 1 0 = 32 tens = 32 0 1 00 × 32 = 32 × 1 00

32 hundreds = 3,2 00 1, 000 × 32 = 32 × 1, 000

The Origin of the “>” and “<” Signs Because the equal sign (=) consists of two parallel lines, the spaces at each end are the same. Thus the symbolism was that since the spaces are equal the numbers at either end are also equal.

To indicate that 3 is less than 4, the two lines in the equal sign were pinched closer together at one end, with the understanding that the smaller number was placed next to the smaller space.

Because the equal sign was often written quickly, it was difficult to distinguish between a “sloppy” equal sign and the “less than” sign. To avoid confusion, the two lines at the smaller end were pinched together so that there was no space between them. Thus, 3 < 4 is an abbreviation for “3 is less than 4”. The thing to remember is that the smaller space (the closed end) is next to the © 2010 Herb I. Gross lesser number.

The Origin of the “>” and “<” Signs In this way we have two choices for writing that 3 is less than 4; namely, either 3 < 4 or 4 > 3. In either case the closed end is next to the lesser number; and when written in the form 4 > 3, we usually we usually read it as “4 is greater than 3”.

A common memory device (but hardly a logical device) is to memorize that the “arrow head” points to the lesser number.

Note Using a calculator does not make us immune from making the error described previously. More specifically, even with a calculator we can strike a key too lightly to have it register, and we can also type a number incorrectly. So even when we use a calculator to compute 415 × 101, we should still be aware of such “advance information” as 415 × 101 > 41

500.

There are more “concrete” explanations that can be helpful to the more visually oriented students. For example , in terms of money (and this is something all students can relate to), the person who has 101 checks each worth $415 has $415 more than the person who has only 100 checks that are worth $415 each. 5 note 5 Don't make the mistake of saying that since 101 is 1 more than 100; 415 ×101 is 1 more than 415 × 100. Remember that in terms of the adjective / noun theme, he only has 1 more check but 415 more dollars .

As a way to utilize a little geometry, we can think in terms of the area of a rectangle. For example , the rectangle whose dimensions are 415 feet by 101 feet has a greater area than the rectangle whose dimensions are 415 feet by 100 feet.

next Multiplication of Whole Numbers or Rapid, Repeated Addition In order to understand the traditional whole number multiplication algorithm , students should be nurtured to understand the concept of

rapid, repeated addition

Suppose you are buying 4 boxes of candy that cost $7 each. We could think of asking two questions based on this information… ( 1 ) “How many boxes of candy did you buy?” In this case we can see directly that the adjective is modifying the noun 4 phrase “boxes of candy”.

( 2 ) “How much did the 4 boxes of candy cost?” Explicitly, the 4 is still modifying “boxes of candy”, but to answer the question we see that it is being used to tell us how many times we are spending $7 .

The mathematical way of writing “ $7 times is to write 4 × $7 . 4

$7 four is called the 4th multiple of 7 ( dollars ).

6 More generally, no matter what number 7 modifies, 4 × 7 is called the 4th multiple of 7.

note 6 It is important to have students internalize the concept of multiple. As a preliminary step it might help to define the 4th multiple of 7 as the fourth number we come to if we are “skip counting” by 7’s. In this context, even if we do not know the answer in place value notation , 127 × 378 is the 127th number we come to if we are skip counting by 378's while 378 × 127 is 378th number we come to if we are skip counting by 127’s. More generally, the product of any two numbers is a common multiple of the two numbers.

In the expression 7 × 4… 4 and 7 are called the

factors

; and 4 × 7 is called the

product

of 4 and 7.

Note For example , we are viewing 4 × 7 as the 4th multiple of 7; that is, 7 + 7 + 7 + 7; and we are viewing 7 × 4 as the 7th multiple of 4; that is, 4 + 4 + 4 + 4 + 4 + 4 + 4. Clearly these two sums look different! In fact, it might be an interesting experiment to ask students questions such as the following one…

Which names the greater sum…

4 + 4 + 4 + 4 + 4 + 4 or 7 + 7 + 7 + 7?

For “self evident” reasons, students are willing to accept the fact that 4 × 7

7 × 4. Note However, there is a conceptual difference between buying 7 pencils at $4 each and buying 4 pencils at $7 each (even though the cost is the same in both cases).

$4 each $7 each 4

4 + 4 + 4 + 4 + 4 + 4 7 + 7 + 7 + 7 In fact, the previous question addresses this issue. Notice that the first group is the answer to the cost of buying seven pencils at $4 each; and that the second group is the answer to the cost of buying four pencils at $7 each. Certainly, the two events are quite different even though the total cost is the same in both cases.

In terms of our “ adjective / noun ” format, when we write 4 + 4 + 4 + 4 + 4 + 4 + 4, we may view 4 as the noun (because that’s the digit we see) and 7 as the adjective (because that’s the number of times 4 appears). However, when we write 7 + 7 + 7 + 7, it is 7 that plays the role of the noun (because that’s the digit we see), and 4 plays the role of the adjective (because that’s the number of times 7 appears).

We shall adopt the notation 4 × 7 to mean the 4th multiple of 7; that is, 7 + 7 + 7 + 7. If we mean 4 + 4 + 4 + 4 + 4 + 4 + 4, we will write 7 × 4.

7 The reason is somewhat related to the fact

‚

that we usually say such things as “four apples” rather than “apples four”.

8 note 7 This agreement is used in algebra note as well. For example , if we want to solve the equation 7x = 21, we divide both sides by 7 to obtain x = 3. In essence we were treating 7 as the adjective and x as the noun . That is we were saying that if seven x's are worth 21, each x is worth 3.

8 While it's important to distinguish the difference in meaning between 7 × 4 and 4 × 7; as far as adjectives are concerned, there is no harm done in confusing one notation with the other because as adjectives , 4 × 7 = 7 × 4 = 28.

next The Commutative Property of Multiplication Under the heading of “a picture is worth a thousand words”, notice how easy it is by rearranging tally marks (which we’ve written as “ dots ” for aesthetic reasons) to see why 7 × 4 = 4 × 7. © 2010 Herb I. Gross

Namely, we may view 4

7 as a rectangular array consisting of 4 rows, each of which contains 7 “ dots ”. That is… 1 8 15 22 2 9 16 23 3 10 17 24 4 11 18 25 5 12 19 26 6 13 20 27 7 14 21 28 However, if we look at the columns rather than at the rows, we see 7 columns, each of which contains 4 “ dots ”. In other words 4 rows of 7 “ dots ” is the same number of “ dots ” as 7 columns of 4 “ dots ” (28). © 2010 Herb I. Gross

Youngsters sometimes visualize square tiles more readily than they do dots or tally marks . Hence, by using a rectangular array of tiles we could indicate why 4 × 7 = 7 × 4.

The Area Model height 4 height 7 base 7 base 4 Here we have another application to geometry ( area ). Namely, the two rectangles above have the same area . To find the area we multiply the base by the height. In the first rectangle, the base is 7 and the height is 4, and in the second rectangle the base is 4, and the height is 7. Hence, 7 © 2010 Herb I. Gross × 4 = 4 × 7.

More generally, the

product

of two numbers does not depend on the order in which the two numbers are written. This is known as…

The Commutative Property for Multiplication.

Classroom Activity Ask the students to compute, the sum of ten 2’s, written as… 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 and see how long it takes them to discover that all they had to do was to “annex” a 0 after the 2; that is, 2 × 10 = 10 × 2 = 20.

9 note 9 We have to make sure that students do not just “blindly” annex a 0. Otherwise in the study of decimals they may make such errors as saying that 0.832 × 10 = 0.8320 rather than realizing that 0.832 × 10 = 8.32. © 2010 Herb I. Gross

Classroom Activity Then ask them to compute a sum such as… 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2…+ 2 100 terms …and see how long it takes them to discover that the sum is equal to 2 ×100 or 200. Such exercises should help students see why when we multiply a whole number by 100, we simply have to annex two 0’s.

Continuing with Multiplication It is not a particularly noteworthy saving of time to write, for example , 4 × 7 in place of 7 + 7 + 7 + 7. However, with respect to our “boxes of candy” situation, suppose we wanted to buy 400 boxes at a cost of $7 per box. It would indeed be very tedious to write explicitly the sum of four hundred 7’s; that is.... 7 + 7 + 7... + 7 © 2010 Herb I. Gross 400 terms

Note The above problem, when stated as a multiplication problem, should be written as 400 × 7. Writing the problem as 7 × 400 gives us the equivalent but simpler addition problem… 400 + 400 + 400 + 400 + 400 + 400 + 400

7 × 400

Note However, this obscures the fact that we want the sum of four hundred 7’s; not the sum of seven 400’s.

That is, while the answer is the same, the “mental image” is quite different.

10 note 10 Again notice the difference between buying 400 items at $7 each and buying 7 items at $400 each. The total cost is the same in both cases, but the event is different.

Using the adjective / noun theme there is another way to visualize the “quick way” of multiplying by 400. Namely, once we know the “number fact” that 4 × 7 = 28, we also know such facts as… 4 × 7 apples = 28 apples, 4 × 7 lawyers = 28 lawyers, 4 × 7 hundred = 28 hundred. © 2010 Herb I. Gross

The latter result, stated in the language of place value (namely we replace the noun “hundred” by annexing two 0’s) says that 4 × 700 = 2,800.

In other words, once we know that 4 × $7 is $28, we also know that 4 00 × $7 is $2,8 00 .

This observation gives us an insight to rapid addition. As an example , let’s make a multiplication table for a number such as 13 which isn’t usually included as part of the traditional multiplication tables. The idea is that we can think of 13 as being an abbreviation for the sum of 1 ten and 3 ones.

Thus, a “quick” way to add thirteen is to add 1 in the tens place and then 3 in the ones place.

For example , starting with 13 , we add 10 to get 23 and then 3 ones to get 26 . Starting with 26 we add 10 to get 36 and then 3 to get 39 . Adding 10 to 39 gives us 49 and then adding 3 gives us 52 .

Continuing in this way, it is easy to see that… 1 × 13 = 13 2 × 13 = 13 + 10 + 3 = 26 3 × 13 = 26 + 10 + 3 = 39 4 × 13 = 39 + 10 + 3 = 52 5 × 13 = 52 + 10 + 3 = 65 6 × 13 = 65 + 10 + 3 = 78 7 × 13 = 78 + 10 + 3 = 91 8 × 13 = 91 + 10 + 3 = 104 9 × 13 = 104 + 10 + 3 = 117 10 × 13 = 117 + 10 + 3 = 130 © 2010 Herb I. Gross

From the chart we see that at $13 each, 9 items would cost $117 . The problem with this approach is that it would be cumbersome, to say the least, to continue to go row-by-row to find the cost of, say, 234 items that cost $13 each .

In the next part of this lesson, we will demonstrate how our adjective / noun theme allows us to compute such products as 234 × 13 just by knowing the first nine multiples of 13.

Classroom Application Students often have trouble learning the multiplication tables. One reason for this is that the process is not much fun for them. However, there is a bit of mathematical humor that often shows students a “fun” way to learn the “9’s table”.

This method appears on the next slide.

Johnny is being tested on the 9’s table… He knows that 9 × 1 = 9 1 × 9 = Not knowing the other 2 × 9 = answers, he feels he should help the teacher 3 × 9 = 4 × 9 = correct his paper. So 5 × 9 = he counts the number he has wrong.

6 × 9 = 7 × 9 = To check his work he now counts starting 8 × 9 = 9 × 9 = 1 2 3 4 5 6 7 8 9 8 7 6 5 4 3 2 1 from the bottom of the list.

Obviously, Johnny is amazed when he gets his paper back with a grade of 100.

A Possible Teaching Moment Our experience shows that students enjoy seeing this “trick”. It might be nice to have them explain why this trick works, and why it doesn’t work for the other digits in the multiplication tables.