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Start thinking of math as a language, not a pile of numbers Just like any other language, math can help us communicate thoughts and ideas with each other An expression is a thought or idea communicated by language In the same way, a mathematical expression can be considered a mathematical thought or idea communicated by the language of mathematics. Mathematics is a language, and the best way to learn a new language is to immerse yourself in it. A SSE 1 Just like English has nouns, verbs, and adjectives, mathematics has terms, factors, and coefficients. Well, sort of. TERMS are the pieces of the expression that are separated by plus or minus signs, except when those signs are within grouping symbols like parentheses, brackets, curly braces, or absolute value bars. Every mathematical expression has at least one term. 3x 2 Has two terms. 3x and 2 5 A term that has no variables is often called a constant because it never changes. Within each term, there can be two or more factors. The numbers and/or variables multiplied together. 3x Has two factors: 3 and x. There are always at least two factors, though one of them may be the number 1, which isn't usually written. Finally, a coefficient is a factor (usually numeric) that is multiplying a variable. Using the example, the 3 in the first term is the coefficient of the variable x. The order or degree of a mathematical expression is the largest sum of the exponents of the variables when the expression is written as a sum of terms. 3x 2 Order is 1 Since the variable x in the first term has an exponent of 1 and there are no other terms with variables. Order is 2 5 x 3x 2 2 3xy 5x y 7 x 32 y 2 3 4 Order is 5 Now that we have our words, we can start putting them together and make expressions 3x 2 Translate mathematical expressions into English "the sum of 3 times a number and 2," "2 more than three times a number" It's much easier to write the mathematical expression than to write it in English Practice 1.1 Variables and Expressions 1. 10 less than x _______________ x 10 2. 5 more than d _______________ d 5 3. the sum of 11 and d _______________ 11 d d 11 t t 3 3 4. a number t divided by 3 _______________ A-SSE.A.1 Practice 1.1 Variables and Expressions A-SSE.A.1 5. 20 3 x 3 less than the quotient of 20 and x _______________ 6. 5 d 12 w the quotient of 5 plus d and 12 minus w _______________ 7. Write a rule in words and as an algebraic expression to model the relationship in each table. The local video store charges a monthly membership fee of $5 and $2.25 per video. $5 plus $2.25 times the number of videos; 5 2.25v Just the facts: Order of Operations and Properties of real numbers A GEMS/ALEX Submission Submitted by: Elizabeth Thompson, PhD Summer, 2008 Important things to remember • Parenthesis – anything grouped… including information above or below a fraction bar. • Exponents – anything in the same family as a ‘power’… this includes radicals (square roots). • Multiplication- this includes distributive property (discussed in detail later). Some items are grouped!!! • Multiplication and Division are GROUPED from left to right (like reading a book- do whichever comes first. • Addition and Subtraction are also grouped from left to right, do whichever comes first in the problem. So really it looks like this….. • • • • Parenthesis Exponents Multiplication and Division In order from left to right Addition and Subtraction In order from left to right SAMPLE PROBLEM #1 16 4(3 1) 22 11 3 16 4(2) 22 11 3 Parenthesis Exponents 16 4(8) 22 11 4(8) 22 11 This one is tricky! Remember: Multiplication/Division are grouped from left to right…what comes 1st? Division did…now do the multiplication (indicated by parenthesis) 32 22 11 32 2 More division Subtraction 30 SAMPLE PROBLEM 3(5) 65 3(2 3) 65 2 2 2 2 Exponents Parenthesis 75 65 10 3(25) 65 2 2 2 Remember the division symbol here is grouping everything on top, so work everything up there first….multiplication Subtraction Division – because all the work is done above and below the line 5 Order of Operations-BASICS Think: PEMDAS Please Excuse My Dear Aunt Sally • • • • • • Parenthesis Exponents Multiplication Division Addition Subtraction Practice 1.2 Order of Operations and Evaluating Expression Simplify 1. 42 __________ 4 4 16 2. 5 5 5 125 53 __________ 16 4. 4(5) __________ 8 20 12 2 3 15 33 27 27 12 6. __________ 5 3 83 A-CED.1 5 5 5 125 5 3. __________ 6 6 6 216 3 6 64(5) 33 5. 43 (5) 3(11) _________ 320 33 353 Practice 1.2 Order of Operations and Evaluating Expression Using the PARCC High School Assessment Reference Sheet. Evaluate each expression for the given values of the variables. FSA 7. Area of a triangle: b 6in and h 14in. F: 1 A bh 2 S: 1 A (6)(14) 2 A: A 42 in2 Practice 1.2 Order of Operations and Evaluating Expression Using the PARCC High School Assessment Reference Sheet. Evaluate each expression for the given values of the variables. FSA 8. Volume of a pyramid: B 18m and h 8m. F: 1 V Bh 3 S: 1 V (18)(8) 3 A: V 48 m3 Practice 1.2 Order of Operations and Evaluating Expression Using the PARCC High School Assessment Reference Sheet. Evaluate each expression for the given values of the variables. FSA 9. Find the value of x using the quadratic formula with a 1, b 2 and c 3 F: S: b b2 4ac x 2a (2) (2)2 4(1)(3) x 2(1) 2 4 12 x 2 24 x 3 2 24 x 1 2 10. The cost to rent a hall for school functions is $60 per hour. Write an expression for the cost of renting the hall for h hours. Make a table to find how much it will cost to rent the hall for 2, 6, 8, and 10 hours. 60h hours $ 2 120 6 360 8 480 10 600 Lesson Extension • Can you fill in the missing operations? 1. 2 - (3+5) + 4 = -2 2. 4 + 7 * 3 ÷ 3 = 11 3. 5 * 3 + 5 ÷ 2 = 10 Practice 1.3 Real Number and the Number Line Name the radicand of each of the following, then write in simplified form. 1. 64 radicand 3. 8 64 ___________, 64 ________ 1 1 ___________, radicand 36 36 36 1 ____ 6____ 25 3 5 15 2. 3 25 ___________,3 25 ________ radicand 4. 81 81 81, 100 ___________, radicand 100 100 9 ________ 10 Practice 1.3 Real Number and the Number Line Estimate the square root by finding the two closest perfect squares. 5. 51 49 perfect square < 51 < 64 perfect square 7 51 _______ set 6. A ___________ is a well-defined collection of objects. element 7. Each objects is call an ________________ of a set. 8. A ____________ subset of a set consists of elements from the given set. 9. U 2, 4,6,8 and A 2,8 , is A a subset of U? yes/no_________ yes no 10. U 2, 4,6,8 and A 2,3 , is A a subset of U? yes/no_________ Practice 1.3 Real Number and the Number Line Circle all the statements that are true. 11. 9 rational 15. rational 19. 100 rational 49 12. 5 irrational 13. 1 integer 3 16. 25 irrational 17. 9 whole 3 20. 4 2 irrational 21. 2.56 rational 14. 0 whole 18. 0 natural 22. 2 irrational An inequality is a mathematical sentence that compares the values of two expressions using an inequality symbol. The symbols are: ( >, <, , ) ______, less than _______,less than or equal to ______, greater than _______,greater than or equal to 3 3.5 7 23. What is the order of 3.51, 2.1, 9, , and 5 from least to greatest? 2 2 2, 7 5, 9, , 3.51 2 Properties of Real Numbers (A listing) • • • • • Associative Properties Commutative Properties Inverse Properties Identity Properties Distributive Property All of these rules apply to Addition and Multiplication Associative Properties Associate = group It doesn’t matter how you group (associate) addition or multiplication…the answer will be the same! Rules: Samples: Associative Property of Addition Associative Property of Addition (a+b)+c = a+(b+c) (1+2)+3 = 1+(2+3) Associative Property of Multiplication Associative Property of Multiplication (ab)c = a(bc) (2x3)4 = 2(3x4) Commutative Properties Commute = travel (move) It doesn’t matter how you swap addition or multiplication around…the answer will be the same! Rules: Samples: Commutative Property of Addition Commutative Property of Addition a+b = b+a 1+2 = 2+1 Commutative Property of Multiplication Commutative Property of Multiplication ab = ba (2x3) = (3x2) Stop and think! • Does the Associative Property hold true for Subtraction and Division? Is (5-2)-3 = 5-(2-3)? Is (6/3)-2 the same as 6/(3-2)? • Does the Commutative Property hold true for Subtraction and Division? Is 5-2 = 2-5? Is 6/3 the same as 3/6? Properties of real numbers are only for Addition and Multiplication Inverse Properties Think: Opposite What is the opposite (inverse) of addition? What is the opposite of multiplication? Rules: Inverse Property of Addition a+(-a) = 0 Subtraction (add the negative) Division (multiply by reciprocal) Samples: Inverse Property of Addition 3+(-3)=0 Inverse Property of Multiplication Inverse Property of Multiplication a(1/a) = 1 2(1/2)=1 Identity Properties What can you add to a number & get the same number back? 0 (zero) What can you multiply a number by and get the number back? 1 (one) Rules: Identity Property of Addition a+0 = a Samples: Identity Property of Addition 3+0=3 Identity Property of Multiplication Identity Property of Multiplication a(1) = a 2(1)=2 Distributive Property If something is sitting just outside a set of parenthesis, you can distribute it through the parenthesis with multiplication and remove the parenthesis. Rule: a(b+c) = ab+bc Samples: 4(3+2)=4(3)+4(2)=12+8=20 • 2(x+3) = 2x + 6 • -(3+x) = -3 - x Practice 1.4 Properties of Real Numbers A. B. C. D. E. Associative Property of Addition/Multiplication Commutative Property of Addition/Multiplication Identity Property of Addition/Multiplication Zero Property of Multiplication Multiplica tion Property of -1 What property is illustrated by each statement? C 4 x 1 4 x _____1. _____2. 3 (1 p) 3 ( p) E _____4. 4( x 1) ( x 1)4 _____5. B A 5 ( x y ) (5 x) y D : Give an example _____3. C m0 m _____6. xyz yxz B Practice 1.5 Adding and Subtracting Real Numbers Find each sum. 1. 8 5 3 7. 10 6 4 13. 10 1 11 2. 7 3 3. 6 4 10 4. 1 7 8. 15 6 9. 8 10 21 18 5 6. 5 9 11 4 6 2 14. 11 6 5. 2 9 15. 10. 7 16 11. 2 9 7 9 8 5 13 16. 7 12 5 12. 5 25 30 17. 12 10 2 Absolute Value. Simplify each expression. 18. 8 5 85 13 19. 7 4 11 11 20. 6 4 64 10 21. 1 7 22. 2 9 1 7 11 6 11 Opposites: additive inverse A number and its opposites are called _________________________________. State the opposite of result of each statement. 23. 3 5 24. 5 9 25. 6 (9) 26. 5 2 27. 2 8 2 2 4 4 3 3 7 7 10 10 Practice 1.6 Multiplying and Dividing Real Numbers Find each product/quotient. 1. 8 5 2. 7 3 40 7. 10 6 8. 15 6 10 2 5 18. 10 1 2 10(2) 20 9. 90 14. 15 5 4. 24 21 60 13. 3. 6 4 8 10 10. 3 1 19. 3 6 7 1 7 7 3 6 18 5. 2 9 7 18 45 7 16 11. 2 9 12. 5 25 8 8 16. 6 12 6 17. 3 1 1 4 5 20. 15 5 41 4 21. 6. 5 9 18 112 80 15. 1 7 12 10 8 125 2 8 1 4 12 2 5 5 12 30 2 Practice 1.7 Distributive Property What is the simplified form of each expression? 1 1. 5( x 7) 2. 12(3 x) 6 5x 35 4. (2 y 1)( y) 2y 2 y 7. 4(2 x 2 3 x 1) 8 x 2 12 x 4 36 2x 5. 4(2 x 5) 8x 20 8. 5 x(2 x 5) 10 x 2 25 x 3. (0.4 1.1c)3 1.2 3.3c 6. ( x 6) x 6 9. x( x 3) x 2 3x Practice 1.7 Distributive Property 10. Using the following expression: 3x 2 4 x 2 a. How many terms? _________ 3 3, 4 b. List the coefficients: _________ 2 c. List the constants: _________ What is the simplified form of each expression? 11. 3 y y 2y 14. a 3b a 4b 7b 12. 7 mn 4 5mn 4 12mn 4 15. x 5 y 3x 8 y 2x 3y 13. 3 y 2 x y 2 x y 2 2y 2 x y 2 16. 5 y 3 y 10 x 3x 2 y 4 y 7x Practice 1.8 An Introduction to Equations Tell whether each equation is true, false or open. Explain. 45 1. 14 22 2. 42 10 52 x Open True 3. 7 8 15 False Tell whether the given number is a solution of each equation. 4. 3b 8 13; 7 5. 4 x 7 15; 2 ? 3(7) 8 13 ? 21 8 13 Not ? 4(2) 7 15 ? 8 7 15 ? 15 15 Yes 6. 12 14 2 f ; 1 ? 12 14 2(1) ? 12 14 2 NO Practice 1.8 An Introduction to Equations Write an equation for each sentence. n7 8 6(n 5) 16 8. 6 times the sum of a number and 5 is 16. ________________________________________ 7. The difference of a number and 7 is 8. _________________________________________