Unit 1 Packet PDF

Unit 1 The Foundations of Algebra Name: ______________________________________  1.1: Real Numbers and the Number Line Square Root Expressions Perfect Squares ____________________ ____________________: the product of an integer and itself...

Unit 1 The Foundations of Algebra Name: ______________________________________  1.1: Real Numbers and the Number Line Square Root Expressions Perfect Squares ____________________ ____________________: the product of an integer and itself Perfect Squares 1 4 9 Simplifying Square Root Expressions 1. 2. 3. 4. Classifying & Comparing Real Numbers For each of the following, determine which subset(s) of real numbers the number belongs to. Place a check mark in each column that applies. Number Irrational Rational Integers Whole Natural Numbers Numbers Numbers Numbers  Number Irrational Rational Integers Whole Natural Numbers Numbers Numbers Numbers Comparing Real Numbers ____________________: a mathematical sentence that compares the values of two expressions using an inequality symbol To compare real numbers, we use inequality symbols: : ____________________________ : ____________________________ : ____________________________ : ____________________________ For each of the following, compare the numbers using an inequality. 5. 6. 7. 9. 8. 10. For each of the following, graph the given numbers on the number line. Then, list the numbers from least to greatest. 11. 12. 13. 14. 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 Name _____________________________________________________________ Directions: In each of the columns, you must find ALL of the values that are either Rational Numbers or Irrational Numbers. Choose the values which are Choose the values which are Rational Numbers: Irrational Numbers: Check ALL that apply. Check ALL that apply. 1. -13.3 a. 6.8282 2. 3 b. 25 4 1 3. c. 7 18 4. 3.63 d. 2 5 5. 6 e. 3 3 6. f. 17 81 5 7. g. 3 2 All Rights Reserved © MathBits.com  1.2: Properties of Real Numbers In math, properties are statements that are true for any real numbers. They justify steps when simplifying expressions and solving equations. Properties of Real Numbers Property Main Idea Algebraic Example Numeric Example Commutative _____________ Property of Addition does not matter! Commutative _____________ Property of Multiplication does not matter Associative _____________ Property of Addition does not matter! Associative _____________ Property of Multiplication does not matter! Identity _____________ Property of Addition _____________ (Additive Identity) _____________ Identity _____________ Property of Multiplication _____________ (Multiplicative _____________ Identity) _____________ Property of Zero by zero always equals zero.  Property Main Idea Algebraic Example Numeric Example Inverse Using the Property of ________________ Multiplication to cancel a value. Inverse Using the Property of ________________ Addition to cancel a value. ________________ Distributive Property a value to an expression inside parenthesis.  Name_____________________________________ Match the property in Column 1 with its example in Column 2. Place the letter in front of the property. Distributive Property A. (x + 2) + 1 = 1 + (x + 2) Additive Identity B. (7 + x) + 1 = 7 + (x + 1) Property Commutative Property C. x (1/x) = 1 of Addition Associative Property D. (ab) c = c (ab) of Multiplication Multiplicative Inverse E. 2(6 + x) = 12 + 2x Property Commutative Property F. (3x - 7) 0 = 0 of Multiplication Associative Property G. (-4 + (-a)) + (4 + a) = 0 of Addition Multiplicative Identity H. (x + 8) + 0 = (x + 8) Property Zero Property I. (2ab) 1 = (2ab) of Multiplication Additive Inverse J. (3 x) 4 = 3 (x 4) Property All Rights Reserved © MathBits.com  1.3 - Variables and Expressions Algebra uses symbols to represent quantities that are unknown or that vary. You can represent mathematical phrases and real-world relationships using symbols and operations. ____________________: a symbol, usually a letter, that represents the value(s) of a variable quantity ____________________ ____________________: a mathematical phrase that includes one or more variables ____________________ ____________________: a mathematical phrase involving numbers and operation symbols, but no variables Phrases for Translating Verbal Algebraic Expressions Operation Key Words sum, plus, and, total of, altogether, make, increased by, combined, add, together, more than, added to, in all difference, subtract, less than, decrease by, take away, gave, fewer than, minus product, times, double, triple, multiplied by, twice, multiple, increased by a factor of quotient, per, half, a third, a fourth, ratio of, divided by, divisor, percent, divided into, split is, are, were, was, sold for, will be, altogether, totals, yields times the sum of, the quantity of, twice the sum, plus the difference of, times the difference of Flipping Words: Addition & Subtraction Examples Translate the following phrases to algebraic expressions: Phrase Algebraic Expression 1. The sum of 5 and a number 2. 32 more than a number. 3. 58 less than a number. 4. 8 minus a number. 5. The difference of a number and 5. 6. 9 plus a number. 7. Subtract 15 from a number. 8. 23 added to a number. Multiplication & Division Examples Translate the following phrases to algebraic expressions: Phrase Algebraic Expression 9. The product of 4 and a number 10. 32 times a number. 11. The quotient of a number and 6. 12. 8 divided by a number. 13. Twice a number 14. One-third of a number. Writing Expressions with Two Operations Examples Translate the following phrases to algebraic expressions: Phrase Algebraic Expression 15. Twice a number plus 5. 16. 3 more than twice a number 17. 9 less than the quotient of 6 and a number 18. The product of 4 and the sum of a number and 7. 19. 8 less than the product of a number and 4. 20. Twice the sum of a number and 8. Using Words for an Algebraic Expression In Examples #1-20, you were given the phrase and asked to translate it to an algebraic expression. You can also translate algebraic expressions into word phrases. For examples #21-25, translate the algebraic expression into a word phrase. Phrase Algebraic Expression 21. 22. 23. 24. 25. Create Your Own Now that we have done some examples together, try to create some phrases on your own. Be prepared to share at least one with the class. Phrase Algebraic Expression Algebraic Expressions Name_______________________________ Directions: Match the algebraic expressions with their mathematical counterparts to find the answer to the question below. When you find a match, write the appropriate letter (or number) in the blank beside each problem. Use your answers to decode the message. What is the startling math fact about the letter “A” ? ___ ___ ___ ___ ___ ___ ___ ___ ___ 15 11 5 2 5 6 1 4 14 ___ ___ ___ ___ ___ ___ “ ___” ___ ___ ___ ___ ___ 13 5 15 15 5 2 9 6 4 9 4 3 ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___. 4 7 10 19 5 2 8 2 14 10 16 15 14 12 16 16 ____ 1. 6 more than a number Answer Bank A 3/x ____ 2. 8 decreased by twice a number P x-5 B 10 + x ____ 3. the difference of 7 and a number R 8 - 2x D 9-x ____ 4. 1 less than 6 times a number S 6+x E 2x – 4 ____ 5. twice a number diminished by 4 T 3 / (x – 5) F 4x - 2 ____ 6. 8 less than twice a number U 2(x + 4) H 4/x-3 ____ 7. the product of 2, and a number increased by 4 V 1 - 6x I 2x - 8 ____ 8. 2 less than four times a number W 5-x L 3 + 3x ____ 9. the quotient of 3 and a number Y 7-x M 4x - 3 ____ 10. 3 less than the product of 4 and a number Number Answers 1 4 / (x – 3) N 6x - 1 ____ 11. 3 less than the quotient of 4 and a number 0 x–9 O 3/x-5 ____ 12. 4 divided by the difference of a number and 3 ____ 13. 3 more than triple a number _____17. 9 minus a number ____ 14. the quotient of 3 and a number, decreased by 5 _____18. 5 decreased by a number _____15. the quotient of 3, and a number decreased by 5 _____19. 10 plus a number _____16. 9 less than a number _____20. 1 minus 6 times a number All Rights Reserved © MathBits.com “Ah-Bach” Series  1.4: Exponents, Absolute Value, & Order of Operations Exponents ____________________: used to shorten how you represent repeated multiplication such as ____________________: the number that is being multiplied repeatedly by itself ____________________: tells you how many times to multiply the base by itself For each of the following, simplify. 1. 3. 2. 4. Absolute Value ____________________ ____________________: distance away from zero on a number line For each of the following, simplify. 5. 8. 6. 9. 7. 10. Order of Operations P E MD AS For each of the following, simplify. 11. 16. 12. 17. 13. 18. 14. 19. 15. 20.  1.5: Evaluating Expressions ____________________: a mathematical phrase made up of numbers and/or variables that does NOT contain an equal sign ____________________: replacing a variable with another variable or value ____________________: determine the numerical value of an expression Steps to Evaluating Algebraic Expressions 1. Substitute/plug in the given values for each variable. Make sure to use parenthesis. 2. Follow the correct order of operations (PEMDAS) to simplify. 3. Go back and check over your work to make sure you did not make a silly mistake. For each of the following, evaluate. 1. Evaluate if. 2. Evaluate if. 3. Evaluate if 4. Evaluate if. 5. Evaluate if. 6. Evaluate if 7. Evaluate if. 8. Evaluate if.  Kuta Software - Infinite Algebra 1 Name___________________________________ Evaluating Expressions Date________________ Period____ Evaluate each using the values given. 1) y ÷ 2 + x; use x = 1, and y = 2 2) a − 5 − b; use a = 10, and b = 4 3) p 2 + m; use m = 1, and p = 5 4) y + 9 − x; use x = 1, and y = 3 5) m + p ÷ 5; use m = 1, and p = 5 6) y 2 − x; use x = 7, and y = 7 7) z( x + y); use x = 6, y = 8, and z = 6 8) x + y + y; use x = 9, and y = 10 3 10) 6q + m − m; use m = 8, and q = 3 9) p + 10 + m; use m = 9, and p = 3 11) p 2 m ÷ 4; use m = 4, and p = 7 12) y − (z + z ); use y = 10, and z = 2 2 13) z − ( y ÷ 3 − 1); use y = 3, and z = 7 14) ( y + x) ÷ 2 + x; use x = 1, and y = 1 ©O h280x1a2W OKBuIt1aK ySSoMfbt0w0a7rmes ILDL8CV.k d BArlolN Qrli3gAhZtEsN Yrwe7sPeVrSv3eFdV.x h 0M8a7d3ee mwEi8tnhc vIrnzfLiLnPiHtUeA vANlkgeeXb1rzaj d1y.v -1- Worksheet by Kuta Software LLC  1.6 - Simplifying Expressions ____________________: rewrite an algebraic expression in its simplest form (distribute, combine like terms; no like terms or parenthesis) ____________________: a number, a variable, or the product of a number and a variable (i.e., ) ____________________: the number in front of the variable ____________________: just a number; no variable ____________________ ____________________: have the same variable and exponent Identifying Parts of an Expression Expression Term(s) Coefficient(s) Constant(s) 1. 2. 3. 4. Combining Like Terms For each of the following, simplify by combining like terms. 1. 4. 5. 2. 6. 3. Examples with Exponents Terms with the same exponents are considered like terms. When adding like terms with exponents, do not change their exponents. 7. 8.  9. 11. 10. 12. Geometric Applications - Perimeter Write the perimeter of the following shapes as a simplified expression. (Perimeter = plus the rim) 1. 2. 3. 4. Combining Like Terms Name________________________________ Directions: Find the solutions to the following problems in the Answer Table at the end. When you find a match, write the appropriate letter in the hidden message below above the question number. Be on guard as there are more answers than are needed. __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ 19 9 19 6 6 13 19 16 9 12 13 9 13 5 1 15 13 11 __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ 10 3 13 11 7 19 15 15 17 13 2 1 21 8 15 13 4 3 19 9 6 __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ 10 15 13 17 20 14 1 6 3 7 19 9 12 19 9 6 5 13 7 6 __ __ __ __ __ __ __ __ __ __ __ __ __ __ __. 7 12 3 14 12 21 14 9 19 14 4 5 11 3 18 Combine like terms: Find the missing value if each statement is true. 1. 7 x 3 y 5 x 9. 6 x ( 3 ) 9x 2. 4 x ( 8 x) 10. 5 x xy 2 x 7 xy 3x 10 xy 3. 6 x 7 y 9 x 11 y 11. 4x x 6x 4. 2 x 8 3 x 2 7 x 8 12. 8 x xy 4 x 7 xy 4 x 11xy 5. 2( x y ) 5( x y ) 13. 6( x 2) 6x 6. 1.5 x 7.6 y 2.5 x 4.6 y 14. 3 8 4 7 7. 5( x 4) 3( x 2) 15. 5(8 13) 1 6x 3 4x 4 8. 7 x 8 3 x 15 16. 42 (3 7) 23 All Rights Reserved © MathBits.com “Ah-Bach” Series Express the perimeter for each of these figures: 17. 18. 19. 20. 21. Hint. Find the missing side first! ANSWER TABLE: A 12 B3 C 12x D 4 E 3x 4 y F1 G 12 x 2 H -18 I 5 x 10 J 2 x 14 K 6x 4 y L -24 M 12x N -8 O 2x 3 y P 7x 7 y Q 7x 8y R -2 S 4x 3y T x U 10 x 8y V 10 x 7 W 2x 26 Y 8x 12 Z 4x All Rights Reserved © MathBits.com “Ah-Bach” Series  1.7 - The Distributive Property The Distributive Property Algebraic Examples Numeric Examples Simplify each expression using the distributive property. 1. 4. 2. 5. 3. 6. Distribute and Combine! Remember, to simplify an expression means to ensure there are no parentheses and no like terms. In order to do this, distribute first (if there are parentheses), then combine like terms. 9. 12. 10. 13. 11. 14. Geometric Applications Area Write the area of the following shapes as a simplified expression. 15. Hint:. 16. 17. Hint:. 18.  1.8 - Adding & Subtracting Polynomials ____________________: a mathematical expression with two or more terms ____________________ ____________________: writing an expression with exponents in order from least to greatest and the variables in alphabetical order Standard Form For each of the following, write the polynomial in standard form. 1. 4. 2. 5. 3. 6. Adding Polynomials Steps to Add Polynomials: 1. Drop the parenthesis. 2. Combine like terms. 3. Write the simplified expression in standard form. For each of the following, simplify. 7. 10. 8. 11. 9. 12. Subtracting Polynomials Steps to Subtract Polynomials 1. Change the subtraction symbol to a. 2. Distribute the. 3. Combine like terms. 4. Write the simplified expression in standard form. For each of the following, simplify. 13. 16. 14. 17. 15. 18. Adding & Subtracting Polynomials “MATH LIB”! Directions: Simplify the expression at each station. Identify your answer and fill in the blanks at the bottom to complete the story. 1 2 3 4 5 6 7 8 9 10 © Gina Wilson (All Things Algebra), 2014  WRITE YOUR MATH LIB BELOW! (1) __________________________________ was (2) ____________________________ to be (3) _____________________________________________ with (4) (4) _______________________________ on (5) (5) _________________________________ at (6) (6) ____________________________ in (7) (7) _____________________________ wearing (8) (8) _____________________________ while (9) (9) ____________________________ because they wanted (10) (10) _______________________________________________! © Gina Wilson (All Things Algebra), 2014 Extra Notes Extra Notes Extra Notes Extra Notes