Multiplication and Division

Multiplication and Division 

Multiplication Multiplication is an aritmetic operation of the second kind,   where addition is the operation of the first kind.   Multiplication is defined using addition so it is an operation that has a level higher of complexity than addition, and so we say it is of the second kind. The signs used for multiplication are x and · and are spoken "times." 2 x 3  means  to add 2 to itself 3 times: 2 + 2 + 2 = 6 3  · 2 means to add 3 to itself 2 times: 3 + 3    =6 4 x 2 means to add 4 itself 2 times: 4 + 4 = 8 The result of multiplication is called the product. The word product is used two ways: The product of 3 x 4 is 12. The expression 3 x 4 is a product. On a side note, the original terms used for multiplication were "multiplicator times multiplicand equals product". Division Division is the opposite operation for multiplication. Instead of adding a number to itself we subtract a number from another repeatedly until we reach a difference that is less than the number we are subtracting.  It is an operation of the second kind since it is defined using subtraction, which like addition, is an operation of the first kind. The operation sign we use is ÷. For example: 12 ÷ 3 12 −  3  =  9    (once) 9   −  3  =  6    (twice) 6   −  3  =  3    (3 times) 3   −  3  =  0    (4 times) We subtracted 3 starting from 12 four times until we reached 0 so 12 ÷ 3 = 4. For example: 8 ÷ 4 8  −  4  =  4    (once) 4  −  4  =  0    (twice) We subtracted 4 starting from 8 two times until we reached 0 so 8 ÷ 4 = 2. We say "12 divided by 3 is 4."   And we say "3 divides into 12, 4 times." When we say "3 divides 12,"  we mean that "3 divides 12 evenly,"   which means there is no remainder in this division.  (Remainders will be defined shortly.) The number being divided is called the dividend.  The number that is doing the dividing is called the divisor  and the result is called the quotient. quotient = dividend ÷ divisor When teaching division start with divisions that do not create remainders.   Reinforce the multiplication after each division. After finding the result of the division, do the multiplication as a check written and spaced as the following two examples.   Doing this will reinforce both operations and how each number is related to the other.    Be careful to maintain the order of the numbers in the multiplication to make explicit the meanings of a dividend, a divisor and a quotient. 12 ÷ 3 = 4               4 · 3  = 12 8 ÷ 4 = 2             2 · 4  = 8 Now, we've seen the basis for multiplication and division.  To be successful requires us to change gears.  Every time we need to know what 6 · 5 is we cannot be adding 6 to itself 5 times.   And we don't need to be keying it into our calculator each and every time. The multiplication facts from the multiplication table for the integers from 1 to 10 must be memorized.  Yes, I said memorized. You can expect your child to have confidence in later mathematics if your child can handle routine multiplications and divisions mentally without having to resort to the calculator.  Numbers are to Mathematics as letters are to English.  Ignorance of number relationships leads to number illiteracy.   The fundamental difference between one thousand, one million, and one billion will otherwise be lost.   In other words, not spending $13 million has negligible effect on a national debt of $13 trillion even though it may sound good. Here's the table. This table works the same as the Addition table. The top left corner square has the letter 'x' which indicates the operation of this table, multiplication.   The leftmost column and the top row (with the gray backgrounds) are the numbers being multiplied.  The square at the intersection of the respective row and column is their product.  For example, the intersection of the row whose value is 6 and the column whose value is 7 contains the number 42, the product of 6 x 7. It is well worth the effort working with this table. First have your child create the table from scratch by filling in one row at a time as the facts are learned for that row, then move to the next row.  This is not an activity that is done all at once; rather, row by row until each row has been memorized.  Throughout this process your child will see connections/patterns between numbers in the table (which we will discuss next.) Do not focus on these patterns yet, but rather on memorizing the facts themselves. You can buy decks of flash cards to help memorize each multiplication and to speed up recalling each fact.  Index cards work just as well and are better if you have your child write each fact on the cards.  On each card you write in large letters on one side the multiplication fact, for example, 3 x 5 = 15. One the other side you'd write 3 x 5.  Now start with the side containing the complete fact, for example, speaking "3 times 5 is 15," etc.    Then turn the cards around and do the same, for example, "3 times 5 is 15."    In this case, however, the number 15 must be supplied by your child.  As facts are memorized remove them from the deck.  When the deck is empty the facts have been memorized.  Work with your child.  These cards can be used later to refresh these facts. There are three objectives you are to achieve with this table: to improve organizational skills to develop rapid recall from memory for the product of two numbers to see patterns between the numbers in the table And here's why. As problems increase in complexity, solving them requires better organization. Numbers and their relationships form the basis of Mathematics.  Being able to recall basic facts of numbers rapidly allows more focus on the problem at hand.  The process of memorization improves concentration as well. Finally, by studying the table, your child will see patterns that I discuss next. Patterns in the Multiplication Table Focus on the first row shown in the blue box below.  In this row we have the products of 1 with the numbers from 1 to 10.  We will call 1 the identity for multiplication since 1 multiplied by any number is that number. Now focus on the first column shown in red.  This column represents each number from 1 to 10 multiplied by 1.   We've reversed the order of multiplication. The result is the same, that is every number multiplied by 1 is that number.  (Changing this order is called the commutative property of multiplication.) Once it is understood that the product of 1 any number is that number, then the first row and first column can be dispensed with. Each row said aloud represents counting by 2s, 3s, 4s, etc.   For example moving across row 5 is 5, 10, 15, 20, 25, 30, 35, 40, 45, 50.  Row 7 is 7, 14, 21, 28, 35, 42, 49, 56. 63, 70.   Your child should learn how to count by each row as well. Now find the squares that contain the same number.  The following table shows a few examples. Focus on the squares containing the number 8.  The dashed red and green arrows shows the two products 4 x 2 and 2 x 4. This is the commutative property again.  Two other examples are shown in blue and black.  In the first we have the products 9 x 3 and 3 x 9 and in the second we have the products 10 x 7 and 7 x 10.  (Two other squares gave us the product 8, 1 x 8 and 8 x 1 but we covered these two squares when we covered the first row and the first column.) The point of this is we can eliminate half the table if we realize that it doesn't matter which order we multiply the two numbers.  We just reverse the order and we have the answer.   The next table illustrates this point. Finally the diagonal contains special numbers; the squares of the numbers from 1 to 10.   1, 4, 9, 16, 25, 36, 49, 64, 81, and 100.  The square of a number is that number multiplied by itself. Five more squares should be memorized as well, they are: 11 x 11 = 121 12 x 12 = 144 13 x 13 = 169 14 x 14 = 196 15 x 15 = 225 Now that we have the basic multiplication facts memorized, we can proceed. Let's multiply numbers with more than one digit.  Follow the arrows. Multiply 15 by 4 Write the numbers one above the other lining up the rightmost digits. (A) Multiply the rightmost digits.​​ In this case we have 5 x 4 = 20. 20 gets split like so: the 2 is written on top of the 1 (in 15) and the 0 is written below the 4. This 2 is called the “carry” since it carries over to the top of the next column to the left. (B) Now multiply 1 by 4 and add the 2 and get 6 . Write this 6 next to 0. The answer is 60. Multiply 6 by 12. Write the 12 first then the 6.  Always write the number with the most digits first. (A) 2 x 6 = 12. Write the 2 below the 6 and the 1 above the 1 in 12. (B) Multiply 1 by 6 and add 1 to get 7. The answer is 72. Multiply 324 by 7. Write the 324 first then the 7. (A) Multiply 4 x 7 to get 28. Write the 8 below the 7 and the 2 above the 2 in 324. (B) Now multiply 2 by 7 and add the 2 to get 14 + 2 = 16. Write this 6 next to the 8 and carry the 1 and write it above the 3 in 324. (C) Now multiply the 3 by 7 and add 1 to get 21 + 1 = 22. Write 2 next to the 6 and carry the 2 to the next column above the 0 that is next to the 3.   This 0 is used as a place holder.   Any number can be written with any number of zeroes preceding it.   For example, 5 = 05 = 0005, etc. (D) Multiply this 0 by 7 and add 2 to get 0 + 2 = 2.   Write this 2 in front of the 2 in 268. The answer is 2268. Multiply 24 by 71        When the multiplier has more than 1 digit, then we must do each one at a time and then add the results. (A) First multiply 4 by 1 to get 4 and write that below the 1. Now multiply the 2 by 1 to get 2 and write that below the 7. (method 1) (B) For the next number, 7, we push the top number 24 one digit left and write a zero as a placeholder where the 4 was.    Doing this accounts for the 7 being in the 10's place of 71 (pushing 24 one digit left is multiplying it by 10.)    Now we multiply 0 by 7 and write that 0 in the second row below the 24 created by the previous multiplication in (A). (C) Multiply the 4 by 7 and get 28. Write the 8 below the 2 in 24 and carry the 2 above the 2 in 240. (D) Multiply the 2 by the 7 and add the 2 to get 16.   Write the 6 in front of the 8 in 80 and carry the 1 above the implied preceding 0 of 240. (E) Multiply 0 by 7 and add the 1 to get 1 and write this 1 in front of 680.   Now add 24 to 1680. The answer is 1704. The previous example requires some explaining. Multiplying by the 1 in 71 is no problem. The 7 is the issue. Well first of all, there will be a row for each digit in the multiplier.   Each row is called an intermediate product. This intermediate product will be shifted left so that its rightmost digit lies directly below the digit being multiplied in the multiplier. Here's why: consider the previous example 24 x 71.   71 is the multiplier.   71 has two digits, 1 in the ones place and 7 in the tens place.   Remember that by definition 24 x 71 means to add 24 to itself 71 times.   We get the same result by adding 24 to itself 70 times and then to this sum add one more 24; we've still added 71 24s together.   Well 24 x 1 is 24.  24 x 70 is 24 x 7 x 10 which is 1680.  Now focus on 1680.  1680 is 168 shifted left by 1 with a trailing zero added.  Take another look at the previous example.  The second partial product with its leftmost digit shifted one left is exactly what 24 x 70 is. Multiply 307 x 624 We have 3 digits in the multiplier so we will have 3 intermediate rows to sum.  This is the second method (method 2) that avoids shifting the top multiplier left by one for each intermediate product. (A) Multiply 7 by 4 to get 28. Write the 8 below the 4 and the carry, 2, above the 0 in 307. Now multiply 0 by 4 and add the previous carry 2 to get 2 and write this 2 next to the 8. Multiply 3 by 4 to get 12. Write this 2 next to the 2 that is next to the 8 and the carry 1 above the leading 0 in 0307. Finally multiply the leading zero of 307 by 4 to get 0 and add the previous carry 1 to get 1. Write this 1 next to the 228 to get 1228 which is the first partial product. (B) We proceed as we did in (A) but multiply 0307 by 2. You will notice that I crossed out the previous carries since I've finished with them. Doing so avoids confusion. We know to start this partial product one digit to the left. To remember this I first write a zero below the 8. Now start multiplying by 2. There is only one carry in this case caused by multiplying 7 by 2 to get 14. (C) Notice the third partial product has two zeroes. This places the digit 2 of this product, 42, directly below the 6 in the number 624. I crossed out the carry from step (B).  Now add the partial products to get the answer 191,568. Remainders Not all numbers divide evenly.  For example, suppose we want to find the number required when multiplied by 3 equals 7?   We know that 3 x 2 is 6 and we know that 3 x 3 = 9.  There is no number between 2 and 3 when multiplied by 3 equals 7.   If we multiply 3 by 2 then add 1 we will get 7, i.e., 3 x 2 + 1 = 7.  This 1 is what we call the remainder after dividing 7 by 3.  We write this quotient as 7 ÷ 3 = 2 R1.  (For this topic the set of numbers is the whole numbers, {0,1,2,3,4,...}, fractional or real number results are not allowed.  This means that when a number is divided by a larger number the quotient is always 0 with some remainder.  In Computer Science this division is called integer division and results in zero anytime the dividend is less than the divisior.) Divide 60 by 4. Long division requires organization.  Columns must be kept straight.  Every step must be written out. With division we work left to right through the dividend.  we generate partial products again but subtract them as we go. 4 "goes into" 6   1 time.  Write this 1 above the 6. Take this 1 and multiply it by the 4 to get 4 and write this 4 below the 6. Draw a line under this partial product 4 and subtract it from the 6 to get 2.   This difference should always be less than the divisor, 4.  If not, then you need a larger number above the division bar (above the 6.) 4 does not go into 2, so drop down the next digit in the dividend, that is the 0 to the right of 6.   20 becomes your next dividend.   4 goes into 20,  5 times.  Write this 5 to the right of 1 above the 0 in 60. Now and multiply 5 by 4 to get 20.  Write this 20 below the 20 at the bottom and subtract it.   The result is 0.  Since we used the last digit in the dividend 60, we are done with the division.  This zero means there is no remainder.  The answer is 15. Divide 185 by 6. In this example the divisor 6 is larger than 1 in 185.   But we proceed as usual noting that 0 x 6 is 0.   1 ÷ 6 is 0. Write this 0 above the 1. Multiply this 0 by 6 to get 0 and write this below the 1.   Subtract 0 from 1 to get 0 and drop down the 8. 18 is the new dividend and 6 goes into 18,   3 times.  Write this 3 above the 8 in 185. Multiply this 3 by 6 to get 18 and write this 18 below the bottom 18. Subtract 18 from 18 to get 0. We have one more digit in 185, drop the 5 down next to the 0. 6 goes into 5,  0 times.  Write this 0 next to the 3 at the top. Multiply this 0 by 6 to get 0 and write it below the 5 at the bottom. Subtract this 0 from 5 to get 5.   There are no further digits in the dividend, 185, so we are done.  The answer is 30 R 6. Divide 1475 by 12. With some practice you should be able to use your multiplication facts to save a few steps. 12 goes into 1 zero times, 14 one time, but how about 147?  Immdediately, because you memorize the basic multiplication facts including the squares of 12, 13, 14, and 15, you recognize that 144 is the square of 12 and 144 is just slightly less than 147.  So, start with 12 as shown. IMPORTANT NOTES ABOUT THE NUMBER 0! We need to address the number zero. We've used the property that any number times zero is zero, .ie., k x 0 = 0.   From this equation we can say that 0 ÷ k = 0.  In other words zero divided by any number is zero,   well almost any number.   Consider 0 ÷ 0.   This quotient represents the number that when multiplied by zero equals zero.   Let this number be k, then k x 0 = 0.  The question is, what is this number k?  K can be any number; we cannot determine which number, so we say that k is indeterminate.  So, the quotient 0 ÷ 0 is indeterminate. We have one more case.  How about k ÷ 0, with k ≠ 0?   This quotient represents that number n which multiplied by zero equals k, i.e., n x 0 = k.   There is no number n multiplied by zero that results in another nonzero number, i.e., no such number has been defined.  So in this case we say that this quotient k ÷ 0 is undefined. To summarize: 0 ÷ k = 0 0 ÷ 0  is indeterminate k ÷ 0, with k ≠ 0,  is undefined So, in general we do not allow division by zero. We will see these special zero quotients later when we study the idea of a limit in Calculus. This is another FREE PRINTABLE presented to you from the K12math.com

Download our free math lesson plan template...and print!!