Adding Whole Numbers © Math As A Second Language All Rights Reserved next #5 Taking the Fear out of Math

next © Math As A Second Language All Rights Reserved Addition Through the Eyes of Place Value next The idea of numbers being viewed as adjectives not only provides a clear conceptual foundation for addition, but when combined with the ideas of place value yields a powerful computational technique. In fact, with only a knowledge of the 0 through 9 addition tables (i.e. addition of single digit numbers), our “adjective/noun” theme and our other rules allow us to easily add any collection of whole numbers.

© Math As A Second Language All Rights Reserved The main idea is that in our place value system, numerals in the same column modify the same noun; therefore, we just add the adjectives and “keep” the noun that specifies the place value column. Addition Through the Eyes of Place Value

next © Math As A Second Language All Rights Reserved To illustrate the idea, let’s carefully analyze the “traditional” way for how we add the two numbers 342 and 517. According to our knowledge of the place value representation of numbers, we set up the problem as follows… next hundredstensones

© Math As A Second Language All Rights Reserved In each column, we use the addition table for single digits. We then solve the above problem by treating it as if it were three single digit addition problems. hundreds tens ones adjectivenoun 3hundreds5 8 adjectivenoun 4tens1 5 adjectivenoun 2ones7 9 next

© Math As A Second Language All Rights Reserved Of course, in everyday usage we do not have to write out the names of the nouns explicitly since the digits themselves hold the place of the nouns. The numbers in the same column modify the same noun. Thus, we usually write the solution in the following succinct form… next 8 5 9

next © Math As A Second Language All Rights Reserved Since the nouns are not visible in the customary format for doing place value addition, it is important for a student to keep the nouns for each column in mind. For example, in reading the leftmost column of our solution out loud a student should be saying… “3 hundred + 5 hundred + 8 hundred” …rather than just using the adjectives, as in “ ”

next © Math As A Second Language All Rights Reserved In that way, one reads the answer as… “8 hundreds, 5 tens, and 9 ones.” Of course, in more common terminology (since we usually say “fifty” rather than “5 tens”), we read the solution as… “eight hundred fifty-nine.”

next © Math As A Second Language All Rights Reserved Using the Properties of Whole Number Addition In using the traditional format to perform the above addition problem, you may not have noticed our subtle use of the associative and commutative properties of addition

next © Math As A Second Language All Rights Reserved If we use the words “hundreds”, “tens”, and “ones”, is an abbreviation for writing… (3 hundreds + 4 tens + 2 ones) + (5 hundreds + 1 ten + 7 ones) However, in using the vertical form of addition, we actually used the rearrangement… (3 hundreds + 5 hundreds) + (4 tens + 1 ten) + (2 ones + 7 ones)

next © Math As A Second Language All Rights Reserved (3 hundreds + 4 tens + 2 ones) + (5 hundreds + 1 ten + 7 ones) (3 hundreds + 5 hundreds) + (4 tens + 1 ten) + (2 ones + 7 ones) So whether we did it consciously or not, the fact is that the vertical format for doing addition of whole numbers is justified by the associative and commutative properties of addition.

next © Math As A Second Language All Rights Reserved For example, suppose you have 3 hundred dollar bills, 4 ten dollar bills, and 2 one dollar bills… …and add 5 more hundred dollar bills, 1 more ten dollar bill and 7 more one dollar bills… $100 $10 $1 $10 $100 $1

next © Math As A Second Language All Rights Reserved Most likely you would compute the total sum in the following way… $100 $10 $1 $10$100 $1 next If you did this, you are using the commutative and associative properties of addition.

© Math As A Second Language All Rights Reserved Notice the difference between a job being “difficult” or just “tedious.” For example, we see from the computation that follows, it is no more difficult to add, say, 12-digit numbers than 3-digit numbers, it is just more tedious (actually, more repetitious). next Instead of carrying out three simple single-digit addition procedures, we have to carry out twelve. Note

next © Math As A Second Language All Rights Reserved In general, no matter how many digits are in the numbers that are being added, the process remains the same, but as the number of digits increases, the process becomes more and more tedious. For example, the numbers 234,267,580,294 and 352,312,219,602 are added as follows… next 2 3 4, 2 6 7, 5 8 0, , 3 1 2, 2 1 9, ,97 756,85

next © Math As A Second Language All Rights Reserved What Ever Happened to “Carrying”? Earlier generations used a technique for adding that was referred to as “carrying”. Nowadays the technique is more visually referred to as “regrouping”. Whichever way we refer to it, the idea behind it is best explained by our adjective/noun theme. next

© Math As A Second Language All Rights Reserved Notice that given a problem such as finding the sum of 35 and 29, a young student who just learned how to add two single digit numbers will often write the problem in vertical form and treat it as if it involved two separate single digit addition problems. For example, they would add 3 and 2 to obtain 5 next and then add 5 and 9 to obtain 14; and thus write…

next © Math As A Second Language All Rights Reserved This gives the appearance of obtaining the incorrect answer, 514. Yet if the adjective/noun theme is understood, it is not difficult to see that could also be the correct answer. next note 1 If we wanted to use grouping symbols, we could write 5(14) to indicate that there are 14 ones and 5 tens but this would quickly become very cumbersome as the number of digits increases

next © Math As A Second Language All Rights Reserved Line 1 and Line 2 in the chart below provide two different ways to represent the same amount of money. next $10 bills$1 billsLine 1514Line 264 However, if the nouns are now omitted, and all we see is Line 1 in the form 514, there is no way of telling whether we are naming 5 hundreds 1 ten and 4 ones or 5 tens and 14 ones.

next © Math As A Second Language All Rights Reserved The point is that as long as the nouns are visible it is okay to have more than 9 of any denomination. However, if we wish, we may exchange 10 $1 bills for 1 $10 bill. Thus, when we said such things as… “5 + 9 = 14, so we bring down the 4 and carry the 1”… next …we were merely saying that the statement “5 ones + 9 ones = 14 ones” means the same thing as the statement “5 ones + 9 ones = 1 ten and 4 ones”.

© Math As A Second Language All Rights Reserved To avoid such ambiguities in which 5 tens and 14 ones can be confused with 5 hundreds, 1 ten and 4 ones, we adopt the following convention (or agreement) for writing a number in place value notation… We never use more than one digit per place value column.

next Counting on Your Fingers Myth As teachers, we often tend to discourage students from “counting on their fingers”. We often say such things as “What would you do if you didn’t have enough fingers?” The point is that in place value, we always have enough fingers! © Math As A Second Language All Rights Reserved next

Consider, for example, the following addition problem… Notice that this result could be obtained even if we had forgotten the addition tables, provided that we understood place value and knew how to count. © Math As A Second Language All Rights Reserved next

next Remembering that numbers in the same column modify the same noun and using the associative property of addition, 3 …we could start with the 6 in the ones place and on our fingers add on nine more to obtain 15. © Math As A Second Language All Rights Reserved next note 3 Up to now we have talked about the sum of two numbers. However, no matter how many numbers we are adding, we never add more than two numbers at a time. For example, to form the sum , we can first add 2 and 3 to obtain 5 and then add 5 and 4 to obtain 9. We would obtain the same result if we had first added 3 and 4 to obtain 7 and then add 2 to obtain 9. next

next Then starting with 15 we could count three more to get 18; after which we would exchange ten 1’s for one 10 by saying “bring down the 8 and carry the 1”. We may then continue in this way, column by column, until the final sum is obtained. © Math As A Second Language All Rights Reserved next

next More explicitly… © Math As A Second Language All Rights Reserved next ones 5, , , = 18 ones tens= 20 tens hundreds = 17 hundreds thousands = 8 thousands 9, next

However, the point we wanted to illustrate in the previous example is the following… Even though there is a tendency to tell youngsters that “grown ups don’t count on their fingers”, the fact remains that with a proper understanding of place value and knowing only how to count on our fingers, we can solve any whole number addition problem. © Math As A Second Language All Rights Reserved next

In particular at any stage of the addition process, we are always adding two numbers, one of which is a single digit. One goal of critical thinking is to reduce complicated problems to a sequence of equivalent but simpler ones. Here we have a perfect example of the genius that goes into making things simple! © Math As A Second Language All Rights Reserved next

Same Sum Technique For students who find it difficult to regroup (as well as for students who like to see alternative approaches to problem solving) the “same sum” technique might pique students’ interest. It is based on the fact that the sum of two numbers remains unaltered if we add a certain amount to one of the numbers and subtract the same amount from the other number. © Math As A Second Language All Rights Reserved

next © Math As A Second Language All Rights Reserved More concretely, suppose that John and Mary have a total of 100 marbles and that John gives Mary 3 of his marbles. Even though Mary now has 3 more and John has 3 less, they still have a total of 100 marbles.

next © Math As A Second Language All Rights Reserved Suppose we want to compute the sum The problem would have been much less difficult if it had been So what we can do is add 2 to 298 and subtract 2 from note 4 You might want to postpone this method until after the students have studied subtraction. This will not change the sum =(679 – 2) + ( ) = 997 next

© Math As A Second Language All Rights Reserved By using the “same sum” technique and then adding the numbers in their original form, students get extra practice with addition as well as a good opportunity to internalize the structure of addition. For example, they could perform the computation below in the traditional way… Then they could add 4 to 296 and subtract 4 from 457, rewriting it in the form… = = 753 next

In the next presentation we will talk about various ways to estimate sums, especially when many large numbers are involved. © Math As A Second Language All Rights Reserved addition 1240