Skills you Need p. 76 1)5(b + 4) = 2)-3(2x + 5) = 3)4(-8 – 3q) = 4)-6(2b – 7) = -6x + (-15) or -6x b – 12q Preferred – -12q b – (-42) = -12b + 42

Algebraic Expressions 2.3 Parts I and II Combining Properties, Integer Rules, and Reviewing the Simplified Approach…..

Vocabulary Term: Constant: Like terms: Coefficient: Term: Constant: Like terms: Coefficient: a number or the product of a number and variable(s) * a term that has no variables have identical variables; terms that can be combined a number that multiplies a variable 7a + 4a + 3b - 6 The sign in front of the term IS THE SIGN OF THE TERM. When you think about “terms,” mentally separate the expression into “chunks.” We have four terms in this expression. They are 7a, 4a, 3b, and -6. A constant is a term that has NO variable. We have only one constant…… -6 Two of our terms (7a and 4a) have the same variable, making them like terms. Each variable has a coefficient, so we have three of them: 7, 4, and 3. Let’s use a chart to summarize….. We’ll fill in your chart in a minute.

7a + 4a + 3b - 6 Number of Terms CoefficientsLike TermsConstants 47, 4, 37a, 4a-6 Streamlining.... There are two approaches to teaching simplifying expressions. One involves rewriting expressions into addition expressions. 7a + 4a + 3b + (- 6)

7a + 4a + 3b - 6 Number of Terms CoefficientsLike TermsConstants 47, 4, 37a, 4a-6 The second approach is to “see” the invisible + sign between each term. 7a + 4a + 3b This allows you to use the sign in front of the term, as the term’s sign without a rewrite……most of the time.

The first approach is explained on p. 76 of your text. I will use both approaches on various problems. You choose what works for you. At some point, however, you will have to use single signs. Identifying Parts of an Expression ExpressionTermsCoefficientsLike TermsConstants 3m – 2n + n – 4 3m + (-2n) + n + (-4) 6+ 2s + 4s -4x 9m + 2r – 2m + r 9m + 2r + (-2m) + r 4 3, -2, 1-2n, 1n-4 3 2, 4 2s, 4s none 49, 2, -2, 1 9m, -2m 2r, 1r none

Your Turn…………work in your groups. Discuss your results. Identifying Parts of an Expression ExpressionTermsCoefficientsLike TermsConstants 7x + y – 2x – f f 2b + b – 3x x 4 7, 1, -27x, -2x-7 4 2, 4 2f, 4f 9, 3 3 2, 1 2b, 1b , 20 -3x, 20x 2, -5 7x + y + (-2x) + (-7) 9, (-3x) + (-5) + 20x

Today we…….. Identified the elements of algebraic expressions…… Terms, coefficients, like terms, and constants were identified and organized We tried to “streamline” our expressions as we work towards the algebraic process of simplifying these expressions

Today we will…….. Apply what we discussed about parts of an expression to simplify the expressions We will simplify these expressions by combining like terms We will ASK QUESTIONS as we go through the lesson!!

Simplifying Variable Expressions Using fewer terms Part II 2a a Think of the a’s as You can see that we can combine our because they are “like terms.” (2 + 3) + 4 = So….2a a = 2a + 3a + 4 = 5a + 4 We used the commutative property, and the distributive property to simplify this expression. We added the coefficients of our like terms to make fewer terms. Applying a property (not necessary to show on your work!!)

Practice Together…… 5y + y 3b – b -4m – 9m p + 6p – 4p Expressions are just stories waiting to happen! 5y +y y 3b - b b -4m – 9m -4 +(– 9) -13m p + 6p – 4p p + 6p + (-4)p – 4 Combine the coefficients of the like terms….. Deductive Reasoning: The process of reasoning logically from given facts to a conclusion. As you use properties, rules, and definitions to justify the steps in a problem, you are using deductive reasoning. p. 77 3

Example of an Algebraic Proof Simplify 4g + 3(3 + g) Given expression 4g g Distributive Property 4g + 3g + 9 Commutative Property g(4 + 3) + 9 Distributive Property 7g + 9 Simplified What do I want to see on your work??????????? 4g + 3(3 + g) Write the problem 4g g Distribute 7g + 9 Simplify The King likes for the constant to be written last!

Your Time to Try..... Simplify 6y + 4m -7y + m 4x + 3 – 2(5 + x) (7 - 3x)5 + 20x 6y + -7y + 4m + m ( I used a rewrite here) -y + 5m or -1y + 5m You could easily have combined 6 and -7 for the -1y 4x (-2x) w/o rewrite 4x x 2x – 15x + 20x Combine (-15)x and 20x 5x + 35

Try to “see” how the signs work. 5(n + 4n) - 3 Distribute the 5. Now combine your like terms 25n – 3 5n + 20n - 3

5 (m + 2) - 2m Use the distributive property to get out of the parentheses. 5m + 5∙2 - 2m 5m m 5m (-2m) Use the commutative property to change the order and get all the “monkeys” Together. Think of the “m” as monkeys!! 5m - 2m + 10 = 3m + 10

Simplifying Expressions First, with pictures……add the (4 - 2 ) (2 ) Now with variables…. 2 3 (3n + 2n) (5n) - 9 = 40n - 9

There is one area you can NEVER streamline….Please add this to your notes 2 ( 4 – 3t ) - ( -3 ) + 2t The section of – (-3 ) must be rewritten. This can be done in the same step as the distribution. 8 – 6t t 8 + (-6t) t Combine like terms and constants. -4t + 11

What have we done?? We have taken the concept of the distributive property and used it to simplify our expressions as we combine like terms. We are learning what we can do ……and what we cannot do!

Tomorrow we will apply these concepts to word problems. Area, Total Cost, and Error Analysis

Applications Finding Area Applying the distributive property and simplifying expressions: Remember that A = lw A = (w + 7) 4 4 w + 3 A = 4w a b A = lw A = 3 ( a + b ) A = 3a + 3b

7 y 5 A = lw A = 7 ( x + y + 5 ) A = 7x + 7y – 14 = 56 x

Word Problem – Pet Supplies p. 78 Gecko supplies: 4 plants for p dollars each A 10-gallon tank for $10 A water dish for $3 Write a variable expression and simplify: 4p = 4p + 13

Two more p – 3 (5p + 2 ) + 6 9p – 15p – p Simplify and evaluate. a = (-2) b = 5 c = (-3) c ( b – c ) + ba Watch the signs! The number you are distributing is (-). (-3)(5p) will be (-). (-3) (2) will also be (-). Substitute. (-3) [ 5 – (-3) ] + (5)(-2) Rewrite. (-3) [ ] + (-10) Solve. There is more than one way! (-3) [8] + (-10) = = -34

We have successfully simplified expressions and applied them to word problems..