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Pre-Algebra Tools for Algebra and Geometry Order of Operations When a numerical expression involves 2 or more operations, there is a specific order in which these operations must be performed. In earlier grades you learned to solve numerical expressions using “order of operations” or as most of us say PEMDAS which stands for Parenthesis, Exponents, (Multiplication/Division) then (Add/Subtract) PEMDAS Please Excuse (My Dear) (Aunt Sally) Please Excuse (My Dear)(Aunt Sally) The reason (multiplication & division) and (add & subtract) are grouped is when those operations are next to each other you do the math from left to right. You do not necessarily do addition first if it is written next to subtraction. Variables and Expressions Variable – a symbol used to represent a quantity that can change. Coefficient – the number that is multiplied by the variable in an algebraic expression. Numerical expression – an expression that contains only numbers and operations. Algebraic expression – an expression that contains numbers, operations and at least one variable. Constant – a value that does not change. Evaluate – To find the value of a numerical or algebraic expression. Simplify – perform all possible operations including combining like terms. Numerical expression 8–6+2 Subtraction is first Remember when those operations, (multiplication & division) and (add & subtract) are grouped next to each other you do the math from left to right. When there are 2 or more operations, and we use grouping symbols such as parenthesis or brackets, you perform the inner most grouping symbol first. 2 + 3[5 +(4-1)²] 2 +3[5 + (3)²] 2 + 3[5 + 9] 2 + 3[14] 2 + 42 44 Summary of Basic Steps • Copy the problem. • Simplify any grouping symbols (such as parenthesis) first, starting with the inner most group. • Simplify any powers (exponents). • Perform the multiplication & division in order from left to right. • Do the addition & subtraction last, from left to right. • Remember, if operations are written next to each other, work from left to right. It is very important to understand that it does make a difference if the order is not performed correctly!!!!!!!!! 70 – 2(5 + 3) 70 – 2(8) 68(8) 544 incorrect (subtraction was done before multiplication) 70 – 2(5 + 3) 70 – 2(8) 70 – 16 54 correct Order of Operations is not an isolated skill. This skill applies to almost every topic in Math. Remember that “Aunt Sally” is used with evaluating formulas, solving equations, evaluating algebraic expressions, simplifying monomials & polynomials, etc… Order of Operations Get your pencil and calculator ready and try these problems. 1) 20 + 3(5 – 1) = 6) 48 / 3 + 5 = 2) 3 + 2²(1 + 8) = 7) 3(6 + 4)(5 – 3) = 3) (5 · 4)² = 8) 100 – 4(7 – 4)³ = 4) 2(3 + 5) – 9 = 9) 1² + 2³ + 3³ = 5) 2[13 – (1 + 6)] = 10) (24 – 6)/2 = Here is an example of an algebraic expression. 4 is the coefficient. X is the variable. 7 is the constant. Algebraic Expression 4x + 7 4 = coefficient x = variable 7 = constant Evaluating Algebraic Expressions To evaluate an algebraic expression, substitute a given number for the variable, and find the value of the resulting numerical expression. X – 5 for x = 12 (12) – 5 = 7 2y + 1 for y = 4 2(4) + 1 = 9 6(n + 2) – 4 for n = 5, 6, 7 38, 44, 50 Evaluate each expression for t = 0, x = 1.5, y = 6, z = 23 1) y + 5 6) 3(4 + y) 2) 3z – 3y 7) 3(6 + t) – 1 3) z – 2x 8) 2(y – 6) + 3 4) xy 9) y(4 + t) - 5 5) 4(y – x) 10) x + y + z Properties of Addition and Multiplication Problem Solving Commutative Property of Addition/Multiplication • The order in which numbers are added does not change the sum. 5+3=3+5 For any numbers a and b a+b=b+a • The order in which numbers are multiplied does not change the product 2·4=4·2 For any numbers a and b a·b=b·a Commutative – switching places or interchanging • Think of the commutative property as physically changing places, they commute or substitute one for the other. Associative Properties of Addition/Multiplication • The way in which addends are grouped does not change the sum. (2 + 4) + 6 = 2 + (4 + 6) For any numbers a, b, and c. (a + b) + c = a + (b + c) • The way in which numbers are grouped does not change the product. (6 · 3) · 7 = 6 · (3 · 7) For an numbers a, b, and c, (a · b) · c = a · (b · c) Associative – regroup the parts • The associative property can be thought of as “friendships” (associations). The parentheses show the grouping of two friends. They don’t physically move, they simply change the one with whom they are associating. Identity Properties of Addition/Multiplication • The sum of a number and zero is the number. 6+0=6 For any number a, a+0=a • The product of a number and one is the number. 6·1=6 For any number a, a·1=a Identity – “I” remain the same or “I” keep my identity. • The identity element here stays the same, so if “I” add zero “I” remain the same. If “I” multiply by one, “I” remain the same. Multiplicative Property of Zero • The product of a number and zero is zero. 5·0=0 For any number a, a·0=0 Inverse property of addition/multiplication. • The sum of a number and its opposite are equal to zero. 5 + (-5) = 0 For any number a, a + (-a) = 0 • The product of a number and its multiplicative inverse equals one. 2·½=1 For any number a, a · 1/a = 1 Inverse – what brings you back to the identity element using that operation? • Think of the inverse property as what would you need to add (multiply) to this number to turn it into an identity element? The additive inverse is the negative of the number, and the multiplicative inverse is one divided by the number. Distributive Property The sum of 2 addends (b + c) multiplied by a number (a) is the sum of the product of each addend and the number. 3(4 + 5) = 3(4) + 3(5) For any number a, b, and c, a(b + c) = ab + ac or (b + c)a = ab + bc The expression a(b + c) is read “a times the quantity b plus c” or “a times the sum of b and c” Distributive – multiply across the parentheses. • Using the distributive property lets you multiply each element inside the parentheses by the element outside the parentheses. Consider the problem to the left. The number in front of the parentheses is “looking” to distribute (multiply) its value with all of the terms inside the parentheses. Properties of Real Numbers Properties of Real Numbers Property Example 1 Commutative Property of Addition a+b=b+a 2+3=3+2 2 Commutative Property of Multiplication a·b=b·a 2 · (3) = 3 · (2) 3 Associative Property of Addition a + (b + c) = (a + b) + c 2 + (3 + 4) = 2 + (3 + 4) 4 Associative Property of Multiplication a · (b · c) = (a · b) · c 2 · (3 · 4) = (2 · 3) · 4 5 Distributive Property a · (b · c) = a · b + a · c 2 · (3 + 4) = 2 · 3 + 2 · 4 6 Identity Property of Addition a+0=a 3+0=3 7 Identity Property of Multiplication a·1=a 3·1=3 8 Additive Inverse Property a + (-a) = 0 3 + (-3) = 0 9 Multiplicative Inverse Property a · (1/a) = 1 3 · (1/3) = 1 10 Property of Zero a·0=0 5·0=0 Variables and Expressions As people age, their blood pressure rises. You can approximate a person’s normal systolic blood pressure by dividing his/her age by 2 and then adding 110. How would you write this problem in expression form to solve mathematically. The Language of Algebra Algebra, like any language, is a language of symbols. It is the language of math and must be learned as any other language. You know the symbols of division and addition, so you can write the bloodpressure relationship as: age ÷ 2 + 110 In arithmetic, you could write: □ ÷ 2 + 110 In algebra, we use variables, letters that represent unknown values. In this case the letter x: X ÷ 2 + 110 This is known as a algebraic expression. Expressions like a ÷ 2 + 110 can be evaluated by replacing the variables with numbers and then finding the numerical value of the expression. If Samantha is 18 years old, she could estimate her blood pressure by evaluating the expression, 18 ÷ 2 + 110 a ÷ 2 + 110 = (18) ÷ 2 + 110 = substitute 18 for a 9 + 110 = order of operations, division first 119 When reading a verbal sentence and writing an algebraic expression to represent it, there are words and phrases that suggest the operations to use. Addition Plus Sum More than Increased by Total In all Subtraction Minus Difference Less than Subtract Decreased by Multiplication Times Product Multiplied Each Of Division Divided quotent Translating Word Phrases into Math Expressions While the table on the previous slide gives you an idea about phrases that translate to math operations, being able to identify the key words that determine the operations (+, -, ·, ÷) that will be used to solve problems takes practice. Write an expression for each phrase. 1) 2) 3) 4) 5) 6) 7) 8) a number n divided by 5 the sum of 4 and a number y 3 times the sum of a number b and 5 the product of a number n and 9 the sum of 11 times a number s and 3 7 minus the product of 2 and a number x 6 less than a number x 7 times the sum of x and 6 Write an algebraic expression to evaluate the word problem: 1) 2) Samantha purchased a 200-minute calling card and called her father from college. After talking with him for t minutes, how many minutes did she have left on her card? Write and solve an expression to represent the number of minutes remaining on the calling card. Jared worked for h hours at $5 per hour. Write an expression to determine how much money Jared earned. How much money will Jared earn if he works a total of 18 hours? Evaluating Expressions Get your pencil and calculator ready and try these problems. Evaluate each expression if m = 4, n = 11, p = 2, q = 5. 1) 8 + n 2) 10 – q 3) 3m 4) 24 – 4q 5) 5m ÷ 2 6) 7) 8) 9) 10) n + 110 – 2p mn + p 10m + m 2m – p m+6÷p Combining Like Terms Using the language of algebra Combining Like Terms Have you ever heard the phrase “You are trying to compare apples to oranges”. Explain what you think this phrase means. Apples and oranges cannot be compared because they are unlike objects. Term – the parts of an expression that are added or subtracted. (x + 2) (2x – 4) Like terms – 2 or more terms that have the same variable raised to the same power. (in the expression 3a + 5b + 12a, 3a and 12a are like terms.) To simplify an expression – perform all possible operations, including combining like terms. Add or Multiply? x + x x x 1x + 1x = 2x x x Add or Multiply? x + y x 1x + 1y = x + y y x y A procedure frequently used in algebra is the process of combining like terms. This is a way to “clean-up” an equation and make it easier to solve. For example, in the algebraic expression 4x + 3 + 7y, there are three terms: 4x, 3, and 7y. Remember the 4 and 7 are coefficients. Let’s say we are given the equation below. It looks very complicated, but if we look carefully, everything is either a constant (number), or the variable x with a coefficient (4x). Remember, a coefficient is the number by which a variable is being multiplied (the 4 in 4x is the coefficient) The “like terms” in the equation are ones that have the same variable. All constants are like terms as well. This means 15, 10, 6, and -2 are all like terms, and the other is 4x, -3x, 5x, and 3x. To combine them is pretty easy, you just add them together and make sure they are all on the same side of the equation. Since the 15 and 10 are both constants we combine them to get 25. The 4x and -3x each have the same variable (x), so we can add them to get 1x. Doing the same on the other side we arrive at 25 + 1x = 4 + 8x. The process is still not finished. There are still some like terms, but they are on opposite sides of the equal sign. Since we can do the same thing to both sides we just subtract 4 from each side and subtract 1x from each side. What remains is 21 = 7x. Now it’s just a simple process of dividing by seven on each side and we arrive at our answer of x = 3. Combining like terms enables you to take that huge mess of an equation and make it something much more obvious to solve. Combine the following: 1) 14a – 5a 2) 7x – 3x 3) 12g + 7g 4) 7y + 8 – 3y – 1 + y 5) 5t + 7p – 3p -2t Simplify Algebraic Expressions by combining like terms. Simplify: 6(n + 5) – 2n = 6 (n) + 6(5) – 2n = Distributive Property 6n + 30 – 2n = 6n and 2n are like terms 4n + 30 Combine coefficients 6 – 2 = 4 Remember that a term like “x” has a coefficient of 1, so terms such as x, n, or y can be written as 1x, 1n, or 1y. Example: 2a + 5b + 5 – a + 3 How many terms are in this expression? What are the like terms? Simplify by combining like terms. a + 5b + 8 Example: 2 + 8(3y + 5) - y What would be the first step in simplifying the expression? Use the Distributive Property to simplify 8(3y + 5), 8(3y) + 8(5), 24y + 40 Combine like terms. 2 + 24y + 40 –y 42 + 23y Math Humor Teacher: Sam, please evaluate x + 4 if x = 7. Sam: Is this a trick question? Yesterday you told us that x = 2!