Mathematics 96 (3581) CA (Class Addendum) 2: Associativity Mt. San Jacinto College Menifee Valley Campus Spring 2013

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1 Mathematics 96 (3581) CA (Class Addendum) 2: Associativity Mt. San Jacinto College Menifee Valley Campus Spring 2013 Name This class addendum is worth a maximum of five (5) points. It is due no later than the end of class on Friday, 22 February. NOTE: You may need to study this entire handout carefully several times before you begin the exercises it contains. You may need to study example exercises carefully several times before you attempt the exercise sets that follow them. Also, the order in which the exercises occur may not necessarily be the order in which you complete them. If you find that the solutions to a particular exercise set elude you, skip to another one. You are being given (about) two weeks to complete this handout because you ll probably need to study it, attempt some of the exercises and then take a break, continuing with it a day or two later. The operations of addition and multiplication are associative. That is, the grouping when three constants or variables are combined utilizing either operation does not change the result. For example, the constants 2,3 and 5 add to 10 regardless of whether the 2 and 3 are added first [ (2 + 3) + 5 ] or the 3 and 5 are combined first [ 2 + (3 + 5) ]. The equality of these two groupings expresses the associativity of addition: (2 + 3) + 5 = 2 + (3 + 5). Consider the term x, the term 3y and the term 5z. Because addition is associative, adding these three unlike terms yields the same sum regardless of the grouping: (x + 3y) + 5z = x + (3y + 5z). Since multiplication is associative, utilizing x, 3y and 5z as factors, we have: [x(3y)](5z) = x[(3y)(5z)]. The following three equations express the associativity of addition: [-2 + 5] + 8 = -2 + [5 + 8] (a + b) + c = a + (b + c) ( + ) + Ө = + ( + Ө) 1

2 The following three equations express the associativity of multiplication: [-2 (5)] 7 = -2 [5 (7)] (ab)c = a(bc) ( ) Ө = ( Ө ) Utilizing variables, we can express the associativity of addition and multiplication as follows: The Associative Property of Addition If x, y and z are real numbers, then The Associative Property of Multiplication If x, y and z are real numbers, then (x + y) + z = x + (y + z) (1) (xy)z = x(yz) (2) Example 1. Express the associativity of addition utilizing the terms 4, 5a and 7b. Solution: We are asked to display that no matter how the terms are grouped, the resulting sum is the same. That is, adding the first two and then adding the third yields that same result as adding the last two and then adding the first. We can complete the exercise by following the pattern displayed in formula (1). That is, we ll replace (substitute) the x in formula (1) with 4, the y with 5a and the z with 7b. Then, formula (1) (x + y) + z = x + (y + z) becomes (4 + 5a) + 7b = 4 + (5a + 7b). One answer is: (4 + 5a) + 7b = 4 + (5a + 7b). The other answer is: 4 + (5a + 7b) = (4 + 5a) + 7b. 2

3 Example 2. Express the associativity of multiplication utilizing the factors 2, 3u and 5v. Solution: We are asked to display that no matter how the factors are grouped, the product is the same. Since the operation is multiplication rather than addition, we will utilize formula (2), substituting 2 for the x, 3u for the y and 5v for the z. That is, formula (2) (xy)z = x(yz) becomes One answer is: [2(3u)](5v) = 2[(3u)(5v)]. [2(3u)](5v) = 2[(3u)(5v)]. The other answer is: 2[(3u)(5v)] = [2(3u)](5v). (NOTE: As was true for exercises utilizing the commutative properties, you must use grouping symbols (e.g. parentheses) whenever following the order of operations would combine numbers in an order different from that intended by an associative law. For instance, the factor 3u should be surrounded by parentheses so that it is considered a factor in its own right. Otherwise, following the pattern of the order of operations agreement, the 3 and u would be interpreted as individual factors themselves and would be multiplied to surrounding expression separately, disrupting the expression of associativity. Also, it is important not to change the order of expressions as they appear from left to right as one passes through the equal sign. For example, while [2(3u)](5v) = 2[(5v)(3u)] is true, the order of the 3u and the 5v are different on the left side of the equation than on the right side. The instructions for Example 2 imply use of an associate law only; Here, however, the Commutative Law of Multiplication has also been employed. Therefore, this equation would not be a correct answer to Example 2.) Exercise 1. (To receive full credit (one point), you must complete at least three of the following four parts correctly). a. Express the associativity of addition utilizing the constants -8, 5 and 72. b. Express the associativity of multiplication utilizing the constants 7, -4 and 23. c. Express the associativity of addition utilizing the terms 6r, -9 and 3f. 3

4 d. Express the associativity of multiplication utilizing the factors 2 7v, 6 and 9s. There is nothing special about the letters x and y utilized in formulas (1) and (2) above. In Exercise 2 you will be asked to express the two associative laws utilizing a variety of symbols. Here are two examples. Example 3. Express the associativity of addition using the variables a, b and c. Solution: One way to interpret this request is that we are being asked to express the Associative Property of Addition using a, b and c rather than x, y and z used in formula (1). Therefore, replacing x with a, y with b and z with c in (x + y) + z = x + (y + z), we have One solution is: (a + b) + c = a + (b + c). (a + b) + c = a + (b + c). The other solution is: a + (b + c) = (a + b) + c. Example 4. Express the associativity of multiplication utilizing the symbols, π and. Solution: As in Example 3, we can interpret this request as a restatement of formula (2) using, π and rather than x, y and z. That is, with substituted for x, π substituted for y and replacing z, (xy)z = x(yz) becomes One solution is: ( π) = ( π ). ( π) = ( π ) The other solution is: ( π ) = ( π). Exercise 2. (To receive full credit (one point), you must complete at least three of the following four parts correctly). a. Express the associativity of addition utilizing the variables g, x and w. 4

5 b. Express the associativity of multiplication utilizing the variables t, a and k. c. Express the associativity of addition utilizing the symbols ε, π and ψ. d. Express the associativity of multiplication utilizing the symbols %, $ and &. Notice that Exercise 2 provides four additional ways to express associativity! That is, (a) and (c) are restatements of formula (1) and (b) and (d) are just restatements of formula (2). In Exercise 3, you will be asked to finish applications of one of the two associative laws. In other words, as written, each equation will be missing one or more symbols that you must insert to create an expression of associativity. Here are two examples. Example 5. Insert the missing symbol(s) (e.g. some combination of parentheses, an operation symbol, constants or variable expressions) to create an expression of associativity. (3 + 4) + = 3 + (4 + 5) Solution: Since we see addition and the equal sign, it appears we are to complete an application of the Associative Property of Addition. The right side of the equation, 3 + (4 + 5), looks complete, in that it could already be one side of an equation that expresses associativity (of addition). The left hand side, (3 + 4) +, appears to be missing something!! That is, if we inserted the constant 5 between the plus sign the equal sign, we d have the expression (3 + 4) + 5. The entire equation would then become: This equation takes the form (3 + 4) + 5 = 3 + (4 + 5). (a + b) + c = a + (b + c) where a corresponds to the constant 3, b corresponds to the constant 4 and c corresponds to the constant 5. But the equation (a + b) + c = a + (b + c) states that addition is associative (i.e. it s really The Associative Law of Addition, substituting the variables a, b and c in formula (1) for x, y and z). Therefore, inserting a 5 between the plus sign and 5

6 the equal sign completes an application of the Associative Property of Addition and thus completes the exercise. Example 6. Insert the missing symbol(s) (e.g. a combination of parentheses, an operation symbol, constants or variable expressions) to create an expression of associativity. 3yz = (3y) Solution: Since the lack of any other operation symbol (e.g. a plus, minus or division sign) implies the operation of multiplication, it appears we are to complete an application of the Associative Property of Multiplication. Currently, we have gibberish! This equation will need a lot of help if it s going to become an expression of associativity. Examining the left hand side, we can interpret 3yz as a product of 3, y and z. To create an expression of associativity, we must have the same factors in the same order on the right hand side as well. Therefore, we must insert a z at the end of the right hand side. We now have 3yz = (3y)z Unlike the equation we started with, this equation seems true (for all real values of y and z)! However, it s still not an expression of associativity. That s because the groupings on each side of the equation are exactly the same! If we followed the order of operations agreement on the left hand side, we d multiply 3 to y first and then multiply z. Unfortunately, we d do the same thing on the right hand side as well. The Associative Property of Multiplication groups factors on one side of the equation differently that it does on the other side. Therefore, we need to change the grouping on one side of the equation. Since the right hand side already has parentheses, let s insert parentheses on the left hand side. If we surround the y and z with parentheses, we ll be done. That is, we ll have 3(yz) = (3y)z. Notice that this equation now has the form a(bc) = (ab)c where a corresponds to the constant 3, b corresponds to the variable y and c corresponds to the variable z. But the equation a(bc) = (ab)c states that multiplication is associative (i.e. it s really formula (2) in The Associative Law of Multiplication utilizing the variables a, b and c instead of x, y and z). Therefore, inserting parentheses as we ve done completes an application of the Associative Property of Multiplication and thus completes the exercise. 6

7 Exercise 3. Insert the missing symbol(s) (e.g. a combination of parentheses, an operation symbol, constants or variable expressions) to create an expression of associativity. (To receive full credit (one point), you must complete at least three of the following four parts correctly). PLEASE USE A PENCIL OR INK OTHER THAN BLACK! a. m + (s + d) = (m + ) + d b. 34 j = ( 5 [34] ) j c [ ( 7-3s ) + n ] = 56 + ( 7-3s ) d. ( 8q ) 7 = q In order to recognize an application of a real number property, it is necessary to determine how a mathematical expression, say in an exercise, corresponds to a variable in the formula for the property. The following examples utilize the associative properties to illustrate this correspondence. Example 7. The following equation is an expression of the associativity of addition. (u + 4) + (5d g) = u + [ 4 + (5d g)] Comparing it to formula (1), which expression in the equation corresponds to the variable z in formula (1)? Solution: The equation above and formula (1), (x + y) + z = x + (y + z), are both expressions of the associativity of addition. One way to determine the correspondence requested in Example 7 is write down both equations in a vertical format as follows: (u + 4) + (5d g) = u + [ 4 + (5d g)] (x + y) + z = x + ( y + z ) Notice that as we read both equations simultaneously as we would read a book (from left to right), we see that the expression 5d - g in the upper equation always corresponds to the variable z in the lower equation. That is, both are always aligned vertically. This suggests that the equation takes the form of formula (1) (u + 4) + (5d g) = u + [ 4 + (5d g)] (x + y) + z = x + (y + z) where the 5d - g corresponds to the z (and u corresponds to x and 4 corresponds to the y). Therefore, the answer is: 5d - g corresponds to z (or z corresponds to 5d - g). 7

8 Example 8. The following equation is an expression of the associativity of multiplication. [9(3 x)]v = 9[(3 x)v] Comparing it to formula (2), which expression in the equation corresponds to the variable y in formula (2)? Solution: The equation above and formula (2), (xy)z = x(yz), are both expressions of the associativity of multiplication. One way to determine the correspondence requested in Example 8 is write down both equations in a vertical format as follows: [9(3 x)]v = 9[(3 x)v] (x y ) z = x( y z ) Notice that as we read both equations simultaneously as we would read a book (from left to right), we see that the expression 3 x in the upper equation always corresponds to the variable y in the lower equation. That is, both are always aligned vertically. This suggests that the equation [9(3 x)]v = 9[(3 x)v] takes the form of formula (2) (xy)z = x(yz) where the expression 3 - x corresponds to the y (and 9 corresponds to x and v to z). Therefore, the answer is: 3 - x corresponds to y (or y corresponds to 3 - x). Exercise 4. To receive full credit (two points), you must complete all four parts correctly. If you complete two or three parts correctly, you ll earn one point. If you complete less than two parts correctly, you ll receive zero points. a. The following equation is an expression of the associativity of addition. (96 + c ) + 3k = 96 + ( c + 3k) Comparing it to formula (1), which expression in the equation corresponds to the variable y in formula (1)? 8

9 b. The following equation is an expression of the associativity of multiplication. v [s (u 4)] = (v s)(u 4) Comparing it to formula (2), which expression in the equation corresponds to the variable y in formula (2)? c. The following equation is an expression of the associativity of addition. (8 + d) + (6 a) = 8 + [d + (6 a)] Comparing it to formula (1), which expression in the equation corresponds to the variable x in formula (1)? 9

10 d. The following equation is an expression of the associativity of multiplication. 72 [g(e + f)] = (72g)(e + f) Comparing it to formula (2), which expression in the equation corresponds to the variable z in formula (2)? 10

An Interesting Way to Combine Numbers

An Interesting Way to Combine Numbers Joshua Zucker and Tom Davis October 12, 2016 Abstract This exercise can be used for middle school students and older. The original problem seems almost impossibly