Hart Interactive Algebra 1 Algebraic Expressions The Commutative, Associative and Distributive Properties Exploratory Activity In elementary school you learned that 3 + 4 gives the same answer as 4 + 3 and that 3 4 gives the same answer as 4 3. These are both examples of the commutative property one for addition and one for multiplication. 1. In the chart below, circle all the statements that are showing commutative property for multiplication and underline all the statements showing the commutative property for addition. x 1 = x ( a+ y)( b+ x) = ( b+ x)( a+ y) ( xy) z = x( yz) ( ab) + c = c + ( ab) ( a+ b) + c= a+ ( b+ c) 7 8 = 8 7 (7 8) 9 = 7 (8 9) xy = yx a( b + c) = ab + ac (7 + 8) + 9 = 7 + (8 + 9) x+ 0 = x x+ y = y+ x ( ab) c = c ( ab) 2. In your own words, explain what the commutative property does. 1 x 1, x 0 x = a( b c) = ab ac 8 + 7 = 7 + 8 The commutative property allows you to switch the places of two variables or numbers when using addition or multiplication. Another important rule from elementary school was the associative property. This one deals with how numbers or variables are grouped. You learned that (3 + 4) + 5 will give the same answer as 3 + (4 + 5). This is the associative property with respect to addition. The associate property with respect to multiplication is similar (3 4) 5 = 3 (4 5). 3. Are all the rest of the statements, using the associative property? Put a star (*) next to any that are not showing either the associative or the commutative property. Properties S.59
Hart Interactive Algebra 1 4. The statements a( b + c) = ab + ac and a( b c) = ab ac are examples of the distributive property. Choose any three numbers for a, b and c to see that both statements are true. a =, b =, c = a( b + c) = ab + ac = and a( b c) = ab ac = We saw in Exercise 4, that no matter what three numbers we choose, the two sides of each statement would have to be equal. Mathematicians must prove that a statement is true for all values of a, b, and c. To do this they write a proof. Although we aren t ready to prove that the commutative, associative and distributive properties are true for all values, we can write proof to show how rewritten expressions are equivalent. 5. The statement we are trying to prove true is (xy)z = (zy)x. You can see that this is a combination of several properties used together. The steps for the proof are given below. Write in the property that was used at each step. Prove: (xy)z = (zy)x Work Reasoning 1. (xy)z 1. Given 2. z(xy) 2. 3. z(yx) 3. Commutative Property of Multiplication Commutative Property of Multiplication 4. (zy)x 4. Associative Property of Multiplication Properties S.60
6. In the next proof, the reasoning is given and you supply the work at each step. Prove: (x + y) + z = (z + y) + x Work Reasoning 1. (x + y) + z 1. Given 2. 2. Associative Property of Addition 3. 3. Commutative Property of Addition 4. 4. Commutative Property of Addition 7. Use these abbreviations for the properties of real numbers, and complete the flow diagram. CC + for the commutative property of addition CC for the commutative property of multiplication AA + for the associative property of addition AA for the associative property of multiplication Properties S.61
Lesson Summary Properties of Arithmetic THE COMMUTATIVE PROPERTY OF ADDITION: If aa and bb are real numbers, then aa + bb = bb + aa. THE ASSOCIATIVE PROPERTY OF ADDITION: If aa, bb, and cc are real numbers, then (aa + bb) + cc = aa + (bb + cc). THE COMMUTATIVE PROPERTY OF MULTIPLICATION: If aa and bb are real numbers, then aa bb = bb aa. THE ASSOCIATIVE PROPERTY OF MULTIPLICATION: If aa, bb, and cc are real numbers, then (aabb)cc = aa(bbcc). THE DISTRIBUTIVE PROPERTY OF MULTIPLICATION: If aa, bb, and cc are real numbers, then a(b + cc) = aabb + aacc. Properties S.62
Homework Problem Set 1. The following portion of a flow diagram shows that the expression aabb + cccc is equivalent to the expression cccc + bbaa. Fill in each circle with the appropriate symbol: Either CC + (for the commutative property of addition) or CC (for the commutative property of multiplication). 2. Fill in the blanks of this proof showing that (ww + 5)(ww + 2) is equivalent to ww 2 + 7ww + 10. Write either commutative property, associative property, or distributive property in each blank. (ww + 5)(ww + 2) = (ww + 5)ww + (ww + 5) 2 = ww(ww + 5) + (ww + 5) 2 = ww(ww + 5) + 2(ww + 5) = ww 2 + ww 5 + 2(ww + 5) = ww 2 + 5ww + 2(ww + 5) = ww 2 + 5ww + 2ww + 10 = ww 2 + (5ww + 2ww) + 10 = ww 2 + 7ww + 10 Properties S.63
CHALLENGE PROBLEM 3. Fill in each circle of the following flow diagram with one of the letters: C for commutative property (for either addition or multiplication), A for associative property (for either addition or multiplication), or D for distributive property. 4. What is a quick way to see that the value of the sum 53 + 18 + 47 + 82 is 200? 5. If aabb = 37 and xxxx = 1, what is the value of the product xx bb xx aa? Give some indication as to 37 how you used the commutative and associative properties of multiplication. Properties S.64
6. The following is a proof of the algebraic equivalency of (2xx) 3 and 8xx 3. Fill in each of the blanks with either the statement commutative property or associative property. (2xx) 3 = 2xx 2xx 2xx = 2(xx 2)(xx 2)xx = 2(2xx)(2xx)xx = 2 2(xx 2)xx xx = 2 2(2xx)xx xx = (2 2 2)(xx xx xx) = 8xx 3 7. Write a mathematical proof of the algebraic equivalency of (aabb) 2 and aa 2 bb 2. 8. Write a mathematical proof to show that (xx + aa)(xx + bb) is equivalent to xx 2 + aaxx + bbxx + aabb. Properties S.65
Spiral Review 9. Recall the following rules of exponents: xx aa xx bb = xx aa+bb (xxxx) aa = xx aa xx aa xx aa = xxaa bb (xx xxbb xx aa xx = xxaa xx aa aa ) bb = xx aabb Here xx, xx, aa, and bb are real numbers with xx and xx nonzero. Replace each of the following expressions with an equivalent expression in which the variable of the expression appears only once with a positive number for its exponent. (For example, 7 bb 2 bb 4 is equivalent to 7 bb 6.) A. (16xx 2 ) (16xx 5 ) B. (2xx) 4 (2xx) 3 C. (9zz 2 )(3zz 1 ) 3 D. (25ww 4 ) (5ww 3 ) (5ww 7 ) E. (25ww 4 ) (5ww 3 ) (5ww 7 ) Challenge Problem 10. Grizelda has invented a new operation that she calls the average operator. For any two real numbers aa and bb, she declares aa bb to be the average of aa and bb: aa bb = aa + bb 2 a. Does the average operator satisfy a commutative property? That is, does aa bb = bb aa for all real numbers aa and bb? Explain your reasoning. b. Does the average operator distribute over addition? That is, does aa (bb + cc) = (aa bb) + (aa cc) for all real numbers aa, bb, and cc? Explain your reasoning. Properties S.66