Module 1: Relationships Between Quantitates and Reasoning with Equations and Their Graphs Lesson 7: The Commutative and Associative Properties Last weekend, you and your friends were at Starbucks drinking some coffee (yuk). You then decided to walk to the theater to watch a movie. Along the way you pass Target. You want to know the total distance you walked. How do you find it? What if you want to find the distance from the theatre back to Starbucks? Is it the same? Starbucks → Target → Theatre = ______+______=_______ Theatre → Target → Starbucks = ______+______=_______ Words Ex amples
Words Examples Commutative
Pro perty of Addition + Symbols How to Remember It Symbols Commutative Pr operty o f Multiplicatio n How to Remember It 24 | P a g e L e s s o n 7 Module 1: Relationships Between Quantitates and Reasoning with Equations and Their Graphs Bag
A Bag
B 1st Count: ___ + ___ + ___ = ___ + ___ = ___
Bag
C ___ + ___ + ___ = ____________
2nd Count: ___ + ___ + ___ = ___ + ___ = ___
___ + ___ + ___ = ____________ I went grocery shopping at Fry’s and wanted to be sure I had all of the items on my list, so I decided
to count the number of items in my three bags. I counted the number of items in Bags A and B and
got a total. Then I added the number of items in the Bag C to that number. However, I wanted to
double check and be sure! This time I counted the number of items in Bags B and C and got a total,
and then added to that the number of items in Bag A. Will I still get the same answer? __________ Words Examples Symbols Associative Pro perty of Addition + How to Remember It Words Examples 25 | P a g e L e s s o n 7 Symbols Associative Pro perty of Multiplicat io n
× How to Remember It Module 1: Relationships Between Quantitates and Reasoning with Equations and Their Graphs Is it true for all real numbers !, !, and ! that (!+!)+! should equal (!+!)+!? (Note: The direct application of the associative property of addition only gives (!+!)+!=!+(!+!).) Draw a flow diagram and use it to prove that (!!)z= (!!)! for all real numbers !, !, and !. Use these abbreviations for the properties of real numbers and complete the flow diagram. !+ for the commutative property of addition !× for the commutative property of multiplication !+ for the associative property of addition !× for the associative property of multiplication 26 | P a g e L e s s o n 7 Module 1: Relationships Between Quantitates and Reasoning with Equations and Their Graphs Exercise 8 Let !, !, !, and !! be real numbers. Fill in the missing term of the following diagram to show that (! + !) + ! + ! is sure to equal ! + ! + (! + !). Numerical Symbol: A numerical symbol is a symbol that represents a specific number. For example, 0, 1, 2, 3, , −3, −124.122, !!, !! are numerical symbols used to represent specific points on the real number line. Variable Symbol: A variable symbol is a symbol that is a placeholder for a number. It is possible that a question may restrict the type of number that a placeholder might permit; e.g., integers only or positive real numbers. Algebraic Expression: An algebraic expression is either 1.
A numerical symbol or a variable symbol, or 2.
The result of placing previously generated algebraic expressions into the two blanks of one of the four operators ((__) + (__), (__) − (__), (__) × (__), (__) ÷ (__)) or into the base blank of an exponentiation with an exponent that is a rational number. Two algebraic expressions are equivalent if we can convert one expression into the other by repeatedly applying the commutative, associative, and distributive properties and the properties of rational exponents to components of the first expression. Numerical Expression: A numerical expression is an algebraic expression that contains only numerical symbols (no variable symbols), which evaluate to a single number. The expression, 3 ÷ 0, is not a numerical expression. Equivalent Numerical Expressions: Two numerical expressions are equivalent if they evaluate to the same number. Note that 1 + 2 + 3 and 1 × 2 × 3, for example, are equivalent numerical expressions (they are both 6) but ! + b +c and ! × ! × ! are not equivalent expressions. Lesson Summary The commutative and associative properties represent key beliefs about the arithmetic of real numbers. These properties can be applied to algebraic expressions using variables that represent real numbers. Two algebraic expressions are equivalent if we can convert one expression into the other by repeatedly applying the commutative, associative, and distributive properties and the properties of rational exponents to components of the first expression. 27 | P a g e L e s s o n 7