Whole Number Arithmetic through the Eyes of Place Value © 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 Preamble and Review next Until now we have been using tally marks and/or tiles as our “manipulative” for explaining the operations that are used in the arithmetic of whole numbers. However, as the numbers we use become greater and greater, the use of tally marks becomes, at best, cumbersome and, at worst, “impossible”.

© Math As A Second Language All Rights Reserved For example, by today’s standards it is not uncommon to talk about millions, billions, and even trillions, and yet have no idea of the magnitude of these numbers. ► A million seconds is about 12 days, but a billion seconds is about 31 years. People will often confuse a million and a billion, but they will never confuse 12 days with 31 years.

next © Math As A Second Language All Rights Reserved In a similar way the word “trillion” gives us no sense of how large a number it represents. Note that a trillion is 1,000 billion. ► If a billion seconds is around 31 years, a trillion seconds is about 31,000 years! Hence, if we were restricted to using tally marks and we could write 1 tally mark per second, it would take around 31,000 years just to write this number, and around 155,000 years to write 5 trillion tally marks. next

© Math As A Second Language All Rights Reserved So even though when we express a trillion in place value notation we sense that 1,000,000,000,000 is a very large number, we often have no idea of just how large. However, if our goal was to use tally marks to compute the sum… 3,000,000,000, ,000,000,000,000, and we were restricted to being able to write 1 tally mark per second continuously, it would take over 155,000 years to find the answer to this problem!

next © Math As A Second Language All Rights Reserved Even if we replaced the tally mark method by the abacus, the abacus would have to have 13 columns in order for us to be able to represent these numbers in place value notation.

next © Math As A Second Language All Rights Reserved However, our adjective/noun theme, together with the properties of arithmetic that we have discussed previously, used in conjunction with place value, allow us to deal with large numbers quite conveniently. Our “adjective/noun” theme is based on the following three “rules”.

next © Math As A Second Language All Rights Reserved Rule #1 When we say that two numbers are equal, we assume that as adjectives 1 they are modifying the same noun. note 1 We must be careful here to emphasize that we don’t have to view numbers as adjectives. In fact, depending on our point of view a number can be both an adjective and a noun. For example, on the number line 3 names the point on the line that is 3 units to the right of 0. In this context, 3 is a noun because it is the name of a point on the number line. On the other hand, the point 3 is located 3 units to the right of 0. In that context, 3 is an adjective modifying the noun “units”.

next © Math As A Second Language All Rights Reserved Rule #1 For example, 12 inches = 1 foot but 12 ≠ 1. In a similar way 1 = 1 but 1 inch ≠ 1 foot. This rather simple sounding rule will play a major role later in such topics as ratios and proportions.

next © Math As A Second Language All Rights Reserved Rule #2 The “usual” addition facts apply only when the numbers modify the same noun. In other words, when we say that = 5 we assume that 3 and 2 are modifying the same noun. For example… 3 dimes + 2 dimes = 5 dimes 3 nickels + 2 nickels = 5 nickels 3 dimes + 2 nickels = 40 cents

next © Math As A Second Language All Rights Reserved Rule #3 When we multiply numbers, we multiply the two adjectives to obtain the adjective part of the product, and we multiply the two nouns to obtain the noun part of the product. In other words, 3 × 2 = 6, regardless of what nouns 3 and 2 are modifying. However, what the 6 modifies depends on what the 3 and the 2 modify. next

© Math As A Second Language All Rights Reserved Rule #3 For example… 3 miles per hour 2 × 2 hours = 6 miles note 2 Actually in this case 3 is modifying the noun phrase miles per hour”. When we talk about the adjective/noun theme, the noun can be a noun phrase. For example, when we write 600,000 we may think of 6 as modifying hundred (in which case, 600 is modifying thousands) or we may think of 6 as modifying the noun phrase “hundred thousand”. 3 hundred × 2 thousand = 6 hundred thousand 3 note 3 When the nouns occur as words, we multiply them by writing them side by side. Thus, for example, hundred × thousand = hundred thousand. And if we multiply 2 inches by 5 feet, the answer is 10 inch feet. next

© Math As A Second Language All Rights Reserved Rule #1 The Closure Property And the rules (properties) of arithmetic that we have discussed so far are… If b and c are whole numbers, then b + c and b × c are also whole numbers. next Rule #2 The Commutative Property If b and c are whole numbers, then b + c = c + b and b × c = c × b.

next © Math As A Second Language All Rights Reserved Rule #3 The Associative Property If b, c, and d are whole numbers, then (b + c) + d = b + (c + d), and (b × c) × d and b × (c × d). Rule #4 The Distributive Property If b, c, and d are whole numbers, then b × (c + d) = (b × c) + (b × d).

next With the above rules as our prerequisite information, we will now look at whole number arithmetic through the eyes of place value. © Math As A Second Language All Rights Reserved addition