Clifton High School Mathematics Summer Workbook. Algebra 1

Transcription

1 1 Clifton High School Mathematics Summer Workbook Algebra 1 Completion of this summer work is required on the first day of the school year. Date Received: Date Completed: Student Signature: Parent Signature:

2 Dear Parents and Guardians, Attached is the mathematics workbook that your child is required to work on over the summer. Our goal is that your child will continue to work on appropriate math skills and concepts to maintain the progress made during the previous grade. This work will also help prepare your child for the next level. We have included a list of vocabulary words to define and several review sheets and problems that require written explanations. Please note that directions and sample problems are offered in each section for reference and review. Summer workbooks can be accessed online through the Clifton web site: click on: mathematics summer workbooks Please sign to indicate the date the packet was received and the date it was completed. Encourage your child to work through the booklet a section at a time during July and August. Your child s math teacher will collect the workbook during the first week of school. Giving time and thought to this work will help to maximize your child s grade on the test given in September. The test will be based on the work shown and will count as the first test of the school year. The grade will be determined as follows: Completion of the workbook on time will count as 0% of the grade. Performance on the test will count as 80% of the grade. Thank you for your cooperation in this matter. Sincerely, Jimmie Warren Mary Campbell Principal Supervisor of Mathematics 9-1

3 CLIFTON HIGH SCHOOL MATHEMATICS DEPARTMENT ALGEBRA I SUMMER WORKBOOK TOPICS COVERED Vocabulary Divisibility Rules, Prime & Composite Numbers GCF, LCM & Prime Factorization Fractions, Decimals and Percents Fraction Review (+, -, /) Variables and Expressions (includes Order of Operations) The Real Number System & Square Roots Real Numbers: Adding, Subtracting, Multiplying, Dividing Powers and Exponents Identity and Equality Properties Geometry and the Coordinate Plane HSPA: Patterns and Sequences Topics that you should also be familiar with are geometry formulas, tree diagrams, rounding and estimating, and the four basic operations with decimals. All pages MUST show the work in order for the answers to be accepted. All work should be written neatly on a separate page. This booklet must be kept neat and in order and is to remain in your notebook as a reference guide. Completion of this booklet is required by the first day of the school year.

4 4 VOCABULARY Match the given words to the correct definition. Write the answers on this page. absolute value equation GCF prime number sum base exponent integers product variable composite numbers expression LCM quotient < difference factors ordered pair rational number > 1) a mathematical sentence that contains an equal sign ) made up of quantities and the operations performed on them does not contain,, <,, >, ) a symbol that is used to represent a number 4) used to locate points (x, y) in the coordinate plane ) the solution to an addition problem 6) the solution to a subtraction problem 7) the solution to a multiplication problem 8) the solution to a division problem 9) whole numbers and their opposites { -,-,-1,0,1,,, } (symbol is Z) 10) a number that can be expressed in the form a/b, in which a and b are integers and b 0 (symbol is Q) 11) a number s distance from zero on the number line 1) the quantities that are multiplied in a multiplication expression 1) a whole number greater than one, with exactly factors, 1 and itself 14) a whole number greater than 1 that has more than factors 1) the greatest number that is a factor of two or more integers 16) the least positive integer that is divisible by each of or more integers 17) the x in an expression of the form x n 18) the n in an expression of the form x n 19) the symbol for less than ( ) 0) the symbol for greater than ( )

5 DIVISIBILITY RULES/PRIME & COMPOSITE NUMBERS PRIME NUMBER: a whole number with exactly two factors, one and itself COMPOSITE NUMBER: a whole number > 1 that has more than two factors The number 1 is neither prime nor composite. The following rules will help determine if a number is divisible by,,4,,6,8,9 or 10. A number is divisible by: if the ones digit is divisible by if the sum of the digits is divisible by 4 if the number formed by the last two digits is divisible by 4 if the ones digit is 0 or 6 if the number is divisible by and 8 if the number formed by the last three digits is divisible by 8 9 if the sum of the digits is divisible by 9 10 if the ones digit is 0 EXAMPLE: Determine whether 10 is divisible by,,4,,6,8,9,10 10 is divisible by since 0 is divisible by 10 is not divisible by since +1++0, which is not divisible by 10 is divisible by 4 since 0 is divisible by 4 10 is divisible by since it ends in 0 10 is not divisible by 6 since it is not divisible by 10 is divisible by 8 since 10 is divisible by 8 10 is not divisible by 9 since +1++0, which is not evenly divisible by 9 10 is divisible by 10 since it ends in 0 Therefore, 10 is divisible by, 4,, 8, 10 PRACTICE: Determine whether the first number is divisible by the second number. Write yes or no. 1) 489; 9 ) 714; 8 ) 74; 4 4) 000; 4 ) 6; 6) 616; 8 7) 1; 10 8) 0,44; 6 Divisibility rules will help tremendously when trying to simplify fractions.

6 6 GCF, LCM AND PRIME FACTORIZATION Prime Factorization: A whole number expressed as a product of factors that are all prime numbers A FACTOR TREE can be used to find the prime factorization of composite numbers. EXAMPLES: 7 0y x 1 y x y y x Since these factors are prime, the prime factorization of: 7 and the prime factorization of: 0y x y y x If you are factoring a negative #, factor out -1 first. PRACTICE: Factor completely. 1) 6 ) -8 ) 10x y 4) -7r 4 w GCF: the greatest number that is a factor of two or more integers STEP 1: factor each # completely STEP : circle all pairs of factors the # s have in common STEP : find the product of the common factors EX: Find the GCF of 4x and 6x 4 x 6 7 x x x 8x GCF is 8x PRACTICE: Find the GCF of each pair of numbers. ), GCF 6) 4, 4 GCF 7) 8n, 18n GCF 8) 1xy, 18x y GCF 9) 48ab, 7a b GCF LCM: the least positive integer that is divisible by each of or more integers STEP 1: Factor each # or monomial completely STEP : Write the prime factorization as powers STEP : Multiply the greatest power of each # or variable to find the LCM EX: Find the LCM of 0a and 4b 0a a 4b b b LCM 1 1 a 1 b 90ab PRACTICE: Find the LCM of each pair of numbers. 10) 1, 0 LCM 11) 4x, x LCM 1) w, 7w LCM 1) 4f, 10f, 1f LCM 14) 8, 16k, 7 LCM

7 7 FRACTIONS, DECIMALS AND PERCENTS FRACTIONS TO DECIMALS DECIMALS TO FRACTIONS 1 Use place value or 4 DECIMALS TO PERCENTS PERCENTS TO DECIMALS move the decimal places to the right move the decimal places to the left 0.4 4% 0.00.% 76% % % 0.% 0.00 FRACTIONS TO PERCENTS PERCENTS TO FRACTIONS A) use a proportion & cross multiply change the percent to a decimal and the n 100 so n 40 40% decimal to a fraction B) change the fraction to a decimal & 66.6% 0.6 move the decimal to the right places % PERCENT MEANS PER HUNDRED FRACTIONS MUST ALWAYS BE IN LOWEST TERMS PRACTICE: For #1-, change the fraction to a decimal, then to a percent 1) ¼ ) / 8 ) ½ 4) 9 / ) / 0 For #6-10, change the percent to a decimal, then to a fraction 6) % 7).% 8) 4.% 9) 7.% 10) 10% What if the decimal was a repeating decimal? Write 0. as a fraction.... to be done in class

8 8 PERCENT PROBLEMS TO FIND THE PERCENT OF A NUMBER: change the percent to decimal and multiply EX. 0% of % of AN EQUATION CAN BE USED IN EVERY PERCENT PROBLEM: REMEMBER: In math, is means equal, of generally means multiply by the number following the of ; if n is multiplied by a number, always divide by that number n6 means n 6 since 6 EX. 1: Find 0% of x x 4 EX. : 4% of is what number?.4 n n 14.4 EX. : 40 is 0% of what number? 40.0 y y EX. 4: 0 is what percent of 0? 0 n 0 n % When you need to find the percent, find the decimal and change it to a percent EX. : What percent of is 7? n 7 n 7. 0% EX. 6: What percent of 0 is? n 0 n % EX. 7: What percent of is 0? n 0 n % PRACTICE: (show work) 1) What number is 90% of 0? ) Find 1% of 600. ) 00% of 67 is what number? 4) What number is 0% of 10? ) is what percent of 60? ) What number is 98% of 0? 7) Joan s income is $190 per week. She saves 0% of her weekly salary. How much does she save each week? 8) Ninety percent of the seats of a flight are filled. There are 40 seats, how many seats are filled?

9 9 FRACTIONS To write a fraction in simplest form, divide both the numerator and the denominator by their GCF. A fraction is in simplest form when the GCF of the numerator and the denominator is 1. EX. 1) 6 9 EX. ) 1a c a a c c c a ac 4 a c c c c c PRACTICE: Write each fraction in simplest form x 1) ) ) 4) y r s ) 6 1r s 6a b 6) 4 18a b 7) Tara takes 1 vacation days in June. What fraction of the month is she on vacation? Express this answer in simplest form. 8) During a one-hour practice, Calvin shot free throws for 1 minutes. What fraction of an hour did he shoot free throws? Express this answer in simplest form. TO ADD AND SUBTRACT FRACTIONS: Find the common denominator and then add the numerators TO MULTIPLY FRACTIONS: multiply the numerators, then multiply the denominators TO DIVIDE FRACTIONS: keep the first fraction the same, change the sign to multiplication, then flip the second fraction (multiply by the reciprocal) IF MIXED NUMBERS ARE INVOLVED: always change the mixed number to an improper fraction FRACTIONS MUST ALWAYS BE WRITTEN IN SIMPLEST FORM! PRACTICE: 7 9) ) + 11) 4 1) ) ) ) ) ) ) 9 19) 7 0) ) A recipe calls for ½ tsp. sugar. If Sam wants to make ⅔ of this recipe, how much sugar should he use? ) If about ⅓ of the earth is able to be farmed and / of this land is planted in grain crops, what part of the earth is planted in grain crops? ) About 1 / 0 of the world s population lives in South America. If about 1 / of the world s population lives in Brazil, what fraction of the population of South America lives in Brazil? 4) The area of a rectangle is ⅔ yds. If one side is ⅓ yd, what is measure of the other side? show work on separate paper

10 10 VARIABLES AND EXPRESSIONS A variable is used to represent a number. The value of an expression may change as different numbers replace a variable. An expression may contain more than one variable. Use the correct order of operations when evaluating expressions. COEFFICIENT: the # multiplied by the variable (the coefficient of x is ) The correct order of operations is as follows: Parentheses and Exponents: left to right as they appear Multiplication ( ) and Division (/): left to right as they appear Addition (+) and Subtraction (-): left to right as they appear PE MD AS EX. 1: Evaluate a - b if a & b 4 EX. : Evaluate a (a b) if a 6 & b 4 a - b () (4) (6) (6 4) 0 () #7-9 are tricky! PRACTICE A: write an algebraic expression for each verbal expression 1) the sum of a number and 6: ) twice a number and 6: ) twice the sum of a number and 6: 4) the product of x and y: ) twice the product of x and y: 6) the square of the product of x and y: 7) the product of x and y-squared: 8) twice the difference of a and b: 9) the difference of a and b-squared: 10) less than times the difference of a and b: PRACTICE B: Evaluate each expression if a 1, b, x, and y 10 1) bx y ) 10b a ) 10(b a) 4) x + y 7 ) b + x a 6) a x + b y 7) y x b a 8) (y x) (b a)

11 11 THE REAL NUMBER SYSTEM & SQUARE ROOTS Types Symbol Definition Real Numbers R rational and irrational numbers Rational Numbers Q all numbers that can be expressed in the form a/b where a & b are integers and b 0 Irrational Numbers I numbers that never end and never repeat (ex: pi,,, ) Counting or Natural N {1,,,...} Numbers Whole Numbers W {0, 1,,,...} Integers Z {..., -, -1, 0, 1,,,...} RADICAL SIGN: A number is multiplied by itself to form a product called the SQUARE ROOT. 4 * 4 16, so the square root of 16 is is the principal square root of 16. In actuality, 16 has two square roots since -4 * -4 is also 16. Therefore, we can say that 16 ±4 (this is read as positive or negative 4 ). The negative square root is generally designated as Write the PERFECT SQUARES from 1 to 0: Since x * x x. x x Therefore: ** Find each of the following: A) 8 B) 4 C) OR 16 4 SO PRACTICE A: Name the set(s) of numbers to which each real number belongs (use the symbols above to represent the group) 1).14 ) 11 ) 6 4) 1/ ) - 4/9 6) -4 PRACTICE B: Evaluate each expression if c and d 1 if x 0 and y 19 7) a if a 8) 400 9) ± cd 10) ± x + y

12 1 REAL NUMBERS: ADDING AND SUBTRACTING ABSOLUTE VALUE: the distance from zero on the number line. ADDING If all numbers being added are positive, the answer is positive. EX: If all numbers being added are negative, the answer is negative. EX: If the signs are different, subtract and take the sign of the number with the greater absolute value. EX. 1: since 4 1 and the sign of 4 is positive EX. : since 4 1 and the sign of 4 is negative If there are more than numbers, add all the positives, then add all the negatives, then subtract the results and take the sign of the greater absolute value. EX: + (-) (-17) (+7+1) + ( ) () + (-) OR just perform the operations left to right: + - -; ; + 1 0; SUBTRACTING Add the opposite EX. 1: of the number that EX. : is being subtracted. EX. : EX. 4: 1 (-1) EX. : -1 (-1) REMEMBER: IF VARIABLES ARE INVOLVED, THE VARIABLES STAY THE SAME. Only like terms can be added or subtracted. EX. 1: n n n + -n -n EX. : 1xy 8xy 7xy Ex. : x y + x + 6y x + 4y ADDITION PRACTICE: 1) -n + (-1n) + 6p ) -ab + (-8ab) + 11ab ) k 7p 1k 4) -7e + (-10e) + 7e ) -8x + 9y + (-x) 6) 1 / + ( - / 4 )+( - / 4 ) SUBTRACTION PRACTICE: 7) -1x 0x 8) 1x 0x 9) 0x 1x 10) 1x (-0x) 11) -1x (-0x) 1) 1 / - / 4 - / 4

13 1 REAL NUMBERS: MULTIPLYING ( ) AND DIVIDING (/) AN EVEN NUMBER OF NEGATIVE SIGNS RESULTS IN A POSITIVE ANSWER. AN ODD NUMBER OF NEGATIVE SIGNS RESULTS IN A NEGATIVE ANSWER. EXAMPLES: 1) 4 1 and since there are two negative signs ) -4-1 and 4-1 since there is only one negative sign in each example ) since there are two negative signs (-x * x * -4x +4x ) 4) since there is one negative sign (-x * x * 4x -4x ) ) 6/ and 6/- since there are two negative signs 6) 6/- - and 6/ - since there is one negative sign in each example REMEMBER: MULTIPLICATION PRACTICE: DIVISION PRACTICE: 1) 6-6) 6 / ( -1 / ) ) -7y -9y 7) -6n / -7n ) -4n (-1) 8) -80x / 16x 4) -4a a b 9) -49n -7 ) 1 / -9 / - / 10) 0-6 STATE WHETHER EACH STATEMENT IS TRUE OR FALSE. 11) The product of two positive integers is positive. 1) The product of two negative integers is negative. 1) The product of one negative and two positive integers is positive. 14) The quotient of two negative numbers is positive. 1) The quotient of one positive and one negative number is negative.

14 14 POWERS AND EXPONENTS An exponent tells how many times to use the base as a factor. In the expression, the base is and the exponent is. EX. 1 8 EX. (x + 1) (x + 1)(x + 1) PRACTICE: Write each product using exponents. 1) ) ) n n y y y Write each power as a product of the same factor. 4) f ) (-k) 6) (y ) IMPORTANT NOTE: (-x) -x since (-x) (-x)(-x) x x is a negative number On a calculator, you MUST use parentheses to raise a negative number to a power. (-) MULTIPLYING BY POWERS OF since since since since since since ,000 since 10 1, ,000 since ,000 DIVIDING BY POWERS OF ,000 since since since since since since since 10 1, since ,000 MULTIPLICATION: if the exponent is positive, move the decimal to the right DIVISION: if the exponent is positive, move the decimal to the left PRACTICE: 1) ) ) ) ) ) ) ) ).7 10 ANY NUMBER RAISED TO THE ZERO POWER IS (The only exception is is undefined because division by zero is not allowed)

15 1 IDENTITY AND EQUALITY PROPERTIES Additive Identity Property Multiplicative Identity Property For any number a, a a a For any number a, a 1 1 a a Multiplicative Property of Zero For any number a, a 0 0 a 0 Substitution Property For any numbers a & b, If a b, then a may be replaced by b Reflexive Property Symmetric Property Transitive Property Distributive Property Commutative Property Associative property a a If a b, then b a If a b and b c, then a c For any numbers a, b, c: a(b+c) ab + ac and a(b-c) ab - ac For any numbers a & b, a + b b + a and a b b a For any numbers a, b, c: (a+b)+c a+(b+c) and (ab)c a(bc) PRACTICE: Name the property illustrated by each statement 1) ) (0)1 0 ) x 1 x 4) ) 6x + y y + 6x 6) (14 6) ) If x + y 9 then 9 x + y 8) 9r + 9s 9(r + s ) 9) If + 6 and 6, then + 10) (c + 6) + 10 c + (6 + 10)

16 INTEGRATION: GEOMETRY AND THE COORDINATE SYSTEM 16 According to legend, Rene Descartes got the idea for coordinate geometry while watching a fly walk on a tiled ceiling. The coordinate plane is a special kind of graph. The terms you should already know are: axes coordinate origin ordered pairs quadrants Additional terms you will be required to know at the completion of Algebra 1 are: Relation: a set of ordered pairs (x,y) EX: {(1,), (,4), (8,7)} Element: a member of a set; (1,) is an element of the set in the example of a relation Domain: the set of all first coordinates from the ordered pairs of a relation; in the EX: {1,,8} Range: the set of all second coordinates from the ordered pairs in a relation; in the EX: {,4,7} Mapping: pairs one element in the domain with one element in the range In the coordinate plane below, identify the quadrants using Roman Numerals (I, II, III, IV). Identify the x and y axes and the origin. Remember: these are ordered pairs (x,y) Order counts! x is always before y Quadrant I: (+,+) Quadrant II: (-, +) Quadrant III: (-, -) Quadrant IV: (+, -) PRACTICE: 1) The ordered pair (0,) is graphed on the -axis. ) The ordered pair (-,0) is graphed on the -axis. REMEMBER: Exactly one point in the plane is named by a given ordered pair of numbers. Exactly one ordered pair of numbers names a given point in the plane. An ordered pair consists of an x-coordinate and a y-coordinate and is written as (x, y). The origin is the ordered pair (0, 0).

17 17 HSPA: PATTERNS AND SEQUENCES Pattern: A repeated design or arrangement Sequence: a set of numbers in a specific order The numbers in a sequence are called terms. Famous patterns: 1) Fibonacci: 1, 1,,,, 8,,, Explanation: ) Perfect squares: 1, 4, 9, 16,,,, What is the 1 th term? What is the n th term? ) Triangular Numbers: 1,, 6, 10, 1, 1,,, Explanation: 4) consecutive numbers: x, x + 1, x +, x +,, consecutive even integers: y, y +, y + 4, y + 6,, consecutive odd integers: z, z +, z + 4, z + 6,, ) What is the next picture? What color is the 0 th picture? Why? 6) Arithmetic sequence: 1, 4, 7, 10,,, FORMULA: 7) Geometric sequence:, 4, 8, 16,,,,