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# 1.2 simplifying expressions and order of operations

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### 1.2 simplifying expressions and order of operations

1. 1. 1-2: Simplifying Algebraic Expressions
2. 2. IDing Parts of an Algebraic Expression Any thing separated by addition or subtraction is a TERM . A term that has NO variable is a CONSTANT . Any number that multiplies a variable is a COEFFICIENT . 7a 4a 3b 6 + + - 7a 4a 3b 6 + + - 7 a 4 a 3 b 6 + + -
3. 3. Parts of an Algebraic Expression/ Combining Like Terms Terms that have identical variables and exponents are LIKE TERMS . To SIMPLIFY Expressions with Like Terms, add or subtract them. If there are NO Like Terms, then you cannot simplify any more. 7a 4a 3b 6 + + - 8c² 4a -2c³ 9 + + +
4. 4. Name the coefficients, like terms and constants. <ul><li>There are 3 things they want from you, so get all 3!!! </li></ul><ul><li>6 + 2s + 4s </li></ul><ul><li>Coefficients: 2 and 4 </li></ul><ul><li>Like Terms: 2s and 4s </li></ul><ul><li>Constants: 6 </li></ul><ul><li>9m + 2r – 2m + r </li></ul><ul><li>Coefficients: 9, 2, and –2 </li></ul><ul><li>Like Terms: 9m and –2m </li></ul><ul><li>Constants: none </li></ul>
5. 5. Like Terms and the Distributive Property <ul><li>Simplify 5y + y = </li></ul><ul><li> = 5y + 1y  Identity Property of Multiplication </li></ul><ul><li> = (5 + 1)y  Distributive Property: Pull out Pikachu. </li></ul><ul><li> = (6)y  Simplify: PEMDAS </li></ul><ul><li> = 6y  Can’t simplify more because x is unknown. </li></ul><ul><li>Simplify: </li></ul><ul><li>-4m – 9m </li></ul><ul><li>p + 6p – 4p </li></ul>
6. 6. Evaluate. Justify each step . <ul><li>6y + 4m – 7y + m = </li></ul><ul><li> = 6y + 4m + (-7y) + m  Turn Subt. Into Addition. </li></ul><ul><li> = 6y + (-7y) + 4m + m  Comm. Prop of Addition. </li></ul><ul><li> = -1y + 5m  Simplify, Combine Like Terms </li></ul><ul><li> = -y + 5m </li></ul>These are the steps for Justifying. 7r + 6t – 3r – 13t =
7. 7. Order of Operations <ul><li>What is the Order of Operations? </li></ul><ul><li>It is a set of rules to find the exact value of a numerical expression. </li></ul><ul><li>Why do we use the Order of Operations? </li></ul><ul><li>A long time ago, people just decided on an order in which operations should be performed. It has nothing to do with magic or logic. It makes communication easier, and everyone comes up with the same answer. (MathForum.org/Dr.Math ) </li></ul>
8. 8. Order of Operations <ul><li>Use the phrase . . . </li></ul><ul><li>“ P l ease E xcuse M y D ear A unt S ally” to help remember the order in which to evaluate the expression. </li></ul><ul><li>PEMDAS </li></ul>
9. 9. P <ul><li>The P stands for parentheses and represents all grouping symbols. </li></ul><ul><li>( ), [ ], { } </li></ul><ul><li>Simplify within the grouping symbols first. </li></ul><ul><li>If there is more than one grouping symbol, simplify within the innermost symbol first. </li></ul>
10. 10. E <ul><li>The E stands for exponents. </li></ul><ul><li>Evaluate all powers. </li></ul>
11. 11. M & D <ul><li>M & D stand for multiplication and division. </li></ul><ul><li>You must simplify whichever comes first in the expression from left to right. </li></ul>
12. 12. A & S <ul><li>A & S stand for Addition and Subtraction </li></ul><ul><li>You must simplify whichever comes first in the expression from left to right . </li></ul>
13. 13. Simplify <ul><li>P First, work inside the brackets. </li></ul><ul><li>Evaluate inside the parentheses first: 4 + 2 = 6. </li></ul><ul><li>Then raise 6 to the second power: 36. </li></ul><ul><li>Now perform addition and subtraction from left to right: 36 - 8 = 28, now 28 +3 = 31. </li></ul><ul><li>Brackets are done. </li></ul><ul><li>M Multiply 31 by 4. </li></ul><ul><li>Final answer is 124. </li></ul>
14. 14. Algebraic Expressions <ul><li>An algebraic expression is an expression that contains at least one variable. </li></ul><ul><li>You can evaluate an algebraic expression by replacing each variable with a value and then applying the Order of Operations . </li></ul>
15. 15. Example: Evaluate a (5 a + 2 b ) if a =3 and b =-2 <ul><li>Substitute the values into the expression. </li></ul><ul><li>3[5(3) + 2(-2)] </li></ul><ul><li>Now apply the Order of Operations: </li></ul><ul><ul><li>Inside the brackets, perform multiplication and division before addition and subtraction </li></ul></ul><ul><li>5(3) = 15 and 2(-2)= -4 </li></ul><ul><li>3[15 + -4] then 15 + -4 = 11 </li></ul><ul><li>3[11] = 33 </li></ul>
16. 16. Formulas <ul><li>Formula is a mathematical sentence that expresses the relationship between certain quantities. </li></ul><ul><li>If you know a value for every variable in the formula except one, you can find the value of the remaining variable. </li></ul><ul><li>Examples of common formulas: </li></ul><ul><ul><li>A = lw </li></ul></ul><ul><ul><li>V = lwh </li></ul></ul>
17. 17. Example: Find the area of a rectangle if the length is 5 cm and the width is 9 cm. <ul><li>Apply the formula A = lw </li></ul><ul><li>Substitute the values of the variables </li></ul><ul><li>A = (5 cm)(9 cm) </li></ul><ul><li>A = 45 cm squared </li></ul>
18. 18. Example 1-4a Find the area of a trapezoid with base lengths of 13 meters and 25 meters and a height of 8 meters. Answer: The area of the trapezoid is 152 square meters. Add 13 and 25 . Area of a trapezoid Replace h with 8 , b 1 with 13 , and b 2 with 25 . Multiply 4 and 38 . Multiply 8 by .
19. 19. Example 1-4b The formula for the volume V of a pyramid is , where B represents the area of the base and h is the height of the pyramid. Find the volume of the pyramid shown below.
20. 20. Warm-Up Write an algebraic expression for each of the following. 5 minutes 1) twice the number n 2) half of the number n 3) 5 more than a number 4) Arthur is two years younger than Chan. Arthur is 21. How old is Chan? Translate to an equation and solve.
21. 21. 3.4 Expressions and Equations <ul><ul><ul><li>Objectives: </li></ul></ul></ul><ul><ul><ul><li>To translate phrases to algebraic expressions </li></ul></ul></ul><ul><ul><ul><li>To solve problems by writing and solving equations </li></ul></ul></ul>
22. 22. Example 1 Write as an algebraic expression. a) 3 times a number, plus 5 3n + 5 b) 12 less than the quantity 4 times a number 4n - 12 c) 8 less than half a number - 8
23. 23. Practice 1) 3 less than twice a number Write as an algebraic expression. 2) half the difference of a number and 1 3) 4 times the quantity 3 greater than a number 4) 2 fewer than the product of 10 and a number
24. 24. Example 2 This week Belinda worked 3 more than twice as many hours as last week. Let h be the hours worked last week. Write an expression for the hours worked this week. 2h + 3
25. 25. Example 3 The depth of the new well is 4ft less than three times the depth of the old well. Let w be the depth of the old well. Write an expression for the depth of the new well. 3w - 4
26. 26. Practice This year Todd sold five fewer houses than twice as many as he sold last year. Let n represent the number he sold last year. Write an expression of the number of houses that Todd sold this year. Translate to an equation.
27. 27. Example 4 A rectangular garden is 40ft longer than it is wide. The total length of the fence that surrounds the garden is 1000ft. How wide is the garden? Let w be the width of the garden. 4w + 80 = 1000 -80 -80 4w = 920 4 4 w = 230 Then w + 40 is the length of the garden.
28. 28. Example 5 On a committee of 18 persons, there were four more women than men. How many men were on the committee? Let m be the # of men on the committee m + (m + 4) = 18 2m + 4 = 18 -4 -4 2m = 14 2 2 m = 7 There were 7 men on the committee. Then m + 4 is the # of women on the committee