DEL OTORGAMIENTO DE AUTORIZACIONES DE EXPLORACIÓN Y

We give the formal argument that the identities (2.2) and (2.3) for all num- bers , m, n, lead logically to equation (2.1). We will in fact prove a little bit more, namely, that for any four numbers k, , m, n, the following holds: (2.8) (k + ) + m+ n = ( + n) + m+ k = ( + n) + (m + k). If we let k = 32,  = 25, m = 28, and n = 55, then equation (2.8) becomes equation (2.1).

To prove equation (2.8), recall the statements of equations (2.2) and (2.3):

( + m) + n =  + (m + n), (2.2)

m + n = n + m. (2.3)

Applying equation (2.2) to the three numbers  + n, m, and k, we get

which is exactly the second equality in equation (2.8). To prove the first equality, we do as follows: (k + ) + m+ n = (k + ) + (m + n) (apply (2.2) to k + , m, n) = (k + ) + (n + m) (apply (2.3) to n, m) = k +  + (n + m) (apply (2.2) to k, , n + m) = k + ( + n) + m (apply (2.2) to , n, m) = ( + n) + m+ k (apply (2.3) to k, ( + n) + m). This proves the first equality in equation (2.8), and therefore equation (2.8) itself.

This kind of formal argument is offered primarily for your benefit as a teacher or prospective teacher, and is not meant to be indiscriminately imposed on your elementary students. Hamlet may be great literature, but it is not the best thing to read to a toddler at bedtime. In the fifth or sixth grade, there may be occasion for you to introduce this kind of reason- ing, gently. But a teacher must exercise good judgment in not overdoing anything.

Exercises 53

Exercises

There are two rules about doing the exercises in this book: (i) Unless stated to the contrary, use only what you have learned so far in the book. (ii) Every answer must come with an explanation. The expla- nation may be in the form of the details in a calculation, or it may be a verbal citation of particular facts used in the text. Sometimes, for emphasis, an explanation is even explicitly demanded. But whatever it is, you will have to get used to never claiming anything without giving a reason.

1. Elaine has 11 jars, in each of which she put 16 ping pong balls. One day she decided to redistribute all her ping pong balls equally among 16 jars instead. How many balls are in each jar? Explain.

2. Before you get too comfortable with the idea that everything in this world has to be commutative, consider the following: (i) Let A1 stand for “put socks on” and A2for “put shoes on”, and let A1◦ A2 be “do A2 first, and then A1”, and similarly let A2◦ A1 be “do A1 first, and then A2”.3 Convince yourself that A1◦ A2 does not have the same effect as A2◦ A1. (ii) For any whole number k, introduce the notation, B1(k) is the number obtained from k by adding 2 to k. Thus B1(7) = 7 + 2, and B1(726) = 726 + 2. Similarly, let B2(k) be the number obtained from k by multiplying k by 5. For example, B2(3) = 3×5 while B2(79) = 79×5. Now, show that no matter what the number n may be, B1(B2(n)) = B2(B1(n)).

3. Find shortcuts to do each of the following computations and give reasons (associative law of addition, commutative law of multiplication, etc.) for each step: (i) 833 + (5167 + 8499), (ii) (54 + 69978) + 46, (iii) (25× 7687)× 80, (iv) (58679 × 762) + (58679 × 238), (v) (4 × 4 × 4 × 4 × 4)× (5 × 5 × 5 × 5 × 5), (vi) 64 × 125, (vii) (69 × 127) + (873 × 69), (viii) (125× 24) × 674+ (24× 125) × 326.

The purpose of the last exercise is not to get you obsessed with tricks in computations. Tricks are nice to have, but they are not the main goal of a mathematics education, contrary to what some people would have you believe. What this exercise tries to do is, rather, to make you realize that the basic laws of operations discussed in this section are more than empty, abstract gestures. They have practical applications too.

4. Prove the remaining three assertions in (2.7).

3You would think that A1◦A2means “doA1first, and thenA2”, but there are powerful rea- sons in mathematics, related to the notation of composing functions, that make the interpretation “doA2first, and thenA1” more natural.

5. Introduce the symbol ≤ between numbers as follows: for any two num- bers a and b, we say a ≤ b if a < b or a = b. (This symbol is sometimes referred to as a weak inequality.) Prove that all the assertions in (2.7) remain true if the strict inequality symbol “<” is replaced by the weak inequality symbol “≤”.

6. True or false: “For whole numbers a and b, the fact that a ≤ b is equivalent to the fact that b = a +  for some whole number .” Explain. 7. What is the smallest possible area of a rectangle whose length and width

are whole numbers which add up to 24? How big can the area be? 8. Let x and y be two whole numbers. (i) Explain why (x + y)(x + y) =

x(x + y) + y(x + y). (ii) Explain why (x + y)(x + y) = xx + xy + yx + yy. (iii) Explain why (x + y)2_{= x}2_{+ 2xy + y}2_{. (iv) Let x and y be nonzero;} compare (x + y)2 with x2 + y2 and explain in two ways why one is always bigger than the other—by a direct computation and by drawing a picture (cf. the Activity on page 51).

9. The following is how a fourth grade textbook introduces the associative law of multiplication.

Ramon buys yo-yos from two companies. He buys six different styles from each company and gets each style in 4 different colors. How many yo-yos does he buy in all?

Find 2×6×4 to solve. You can use the associative property to multiply three factors. The grouping of the numbers does not affect the answer.

Step 1: Use parentheses to show grouping. 2× 6 × 4 = (2 × 6) × 4

Step 2: Look for a known fact to multiply. 2× 4 is a known fact.

Step 3: Use the commutative property to change the order, if necessary.

(2× 6) × 4 = (6 × 2) × 4 = 6× (2 × 4) = 6× 8 = 48

Write down your reaction to such an introduction, and compare with those of others in your class.

10. Using only the laws for multiplication (equations (2.4) and (2.5) on page 43), write down an explanation of why, for four numbers , m, n, p, it is true that  m(np)= (p)(nm).

11. Using only the laws for addition (equations (2.2) and (2.3) on page 42), directly show that for four numbers , m, n, p, ( + m) + (n + p) =

Exercises 55

12. Let m and n be a 3-digit number and a 2-digit number, respectively. Can mn be a 4-digit number? 5-digit number? 6-digit number? 7-digit number? Explain your answer.

13. Let m and n be a k-digit number and an -digit number, respectively, where k and  are positive whole numbers. How many digits can the number mn have? List all the possibilities and explain.

14. Use mental math to decide which of the following is bigger: (a) 648×427 or 649× 426? (b) 207 × 816 or 206 × 819?

15. Suppose you have a calculator which displays only 8 digits (and if you have a fancy calculator, you will be allowed to use only 8 digits!), but you have to calculate 856164298× 65. Discuss an efficient method to make use of the calculator to help with the computation. Explain. Do the same for 376241048× 872.

16. How would you compute the square of 9,458,647,683 on a calculator with a 12-digit display?

Chapter 3