Algebra 1 Summer Packet - Use This One

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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: _________________________________________

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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: • http://www.clifton.k12.nj.us/cliftonhs/index.html • 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 20% 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 Principal

Mary Campbell Supervisor of Mathematics 9-12

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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.

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VOCABULARY

Match the given words to the correct definition. Write the answers on this page. absolute value base composite numbers difference

equation exponent expression factors

GCF integers LCM ordered pair

prime number product quotient rational number

sum variable < >

1) ____________

a mathematical sentence that contains an equal sign

2) ____________

made up of quantities and the operations performed on them

3) ____________

a symbol that is used to represent a number

4) ____________

used to locate points (x, y) in the coordinate plane

5) ____________

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 {…-3,-2,-1,0,1,2,3,…}

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

12) ____________

the quantities that are multiplied in a multiplication expression

13) ____________

a whole number greater than one, with exactly 2 factors, 1 and itself

14) ____________

a whole number greater than 1 that has more than 2 factors

15) ____________

the greatest number that is a factor of two or more integers

16) ____________

the least positive integer that is divisible by each of 2 or more integers

17) ____________

the “x” in an expression of the form xn

18) ____________

the “n” in an expression of the form xn

19) ____________

the symbol for “less than” (2 __ 3)

20) ____________

the symbol for “greater than” (3 __ 2)

does not contain

=, ≠, <, ≤, >, ≥

(symbol is Z)

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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 2,3,4,5,6,8,9 or 10. A • • • • • • • •

number is divisible by: 2 if the ones digit is divisible by 2 3 if the sum of the digits is divisible by 3 4 if the number formed by the last two digits is divisible by 4 5 if the ones digit is 0 or 5 6 if the number is divisible by 2 and 3 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 2120 is divisible by 2,3,4,5,6,8,9,10 • 2120 is divisible by 2 since 0 is divisible by 2 • 2120 is not divisible by 3 since 2+1+2+0 = 5, which is not divisible by 3 • 2120 is divisible by 4 since 20 is divisible by 4 • 2120 is divisible by 5 since it ends in 0 • 2120 is not divisible by 6 since it is not divisible by 3 • 2120 is divisible by 8 since 120 is divisible by 8 • 2120 is not divisible by 9 since 2+1+2+0 = 5, which is not evenly divisible by 9 • 2120 is divisible by 10 since it ends in 0 Therefore, 2120 is divisible by 2, 4, 5, 8, 10

PRACTICE: Determine whether the first number is divisible by the second number. Write yes or no. 1) 4829; 9 ___ 2) 3714; 8 ___ 3) 724; 4 ___ 4) 3000; 4

___

7) 1355; 10 ___

5) 633; 3 ___

6) 616; 8 ___

8) 20,454; 6 ___

Divisibility rules will help tremendously when trying to simplify fractions.

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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: 75 30y2x = 2 ∗ 15 ∗ y2 ∗ x 3 ∗ 25

2∗3∗ 5∗y∗y∗x

3∗5∗5

Since these factors are prime, the prime factorization of: 75 = 3 ∗ 5 ∗ 5 and the prime factorization of: 30y2x = 2 ∗ 3 ∗ 5 ∗ y ∗ y ∗ x

If you are factoring a negative #, factor out -1 first.

PRACTICE: Factor completely. 1) 36 2) -28

3) 102x3y2

4) -72r4w

GCF: the greatest number that is a factor of two or more integers EX: Find the GCF of 24x and 56x2

STEP 1: factor each # completely STEP 2: circle all pairs of factors the #’s have in common STEP 3: find the product of the common factors

24 = 2 ∗ 2 ∗ 2 ∗ 3 ∗ x 56 = 2 ∗ 2 ∗ 2 ∗ 7 ∗ x ∗ x 2 ∗ 2 ∗ 2 ∗ x = 8x → GCF is 8x

PRACTICE: Find the GCF of each pair of numbers. 5) 22, 55 GCF = ___ 6) 42, 54 GCF = ___

8) 15xy2, 18x2y

GCF = ___

9) 48ab, 72a3b3

7) 8n, 18n GCF = ___

GCF = ___

LCM: the least positive integer that is divisible by each of 2 or more integers EX: Find the LCM of 30a and 45b2 30a = 2 ∗ 3 ∗ 5 ∗ a 45b2 = 3 ∗ 3 ∗ 5 ∗ b ∗ b

STEP 1: Factor each # or monomial completely STEP 2: Write the prime factorization as powers STEP 3: Multiply the greatest power of each # or variable to find the LCM

LCM = 21∗32∗51∗a1∗b2 = 90ab2

PRACTICE: Find the LCM of each pair of numbers. 10) 15, 20 LCM = ___ 2

13) 4f, 10f , 12f

2

11) 4x, 3x LCM = ___

LCM = ___

2

14) 8, 16k , 72

2

12) 3w , 7w LCM = ___ LCM = ___

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FRACTIONS, DECIMALS AND PERCENTS FRACTIONS TO DECIMALS 3

5 2

DECIMALS TO FRACTIONS

= 3 ÷ 5 = 0.6 3

Use place value 0.123 =

1000

25 1 9 = 2 or 2.25 = 2 100 4 4

= 2 + 3 ÷ 5 = 2.6

5

123

DECIMALS TO PERCENTS

PERCENTS TO DECIMALS

move the decimal 2 places to the right 0.45 = 45% 0.0032 = .32% 2.9 = 290%

move the decimal 2 places to the left 76% = 0.76 13.25% = 0.1325 0.25% = 0.0025

FRACTIONS TO PERCENTS

PERCENTS TO FRACTIONS

A) use a proportion & cross multiply 2 n = so n = 40 → 40% 5 100

change the percent to a decimal and the

B) change the fraction to a decimal &

66.6% = 0.6 =

decimal to a fraction

move the decimal to the right 2 places 2 5

2 3

PERCENT MEANS “PER HUNDRED”

= 0.4 = 40%

FRACTIONS MUST ALWAYS BE IN LOWEST TERMS

PRACTICE: For #1-5, change the fraction to a decimal, then to a percent 1) ¼ = ____ = _____ 4)

9

/5 = _____ = _____

2)

5

5)

3

/8 = _____ = _____ /50 = _____ = _____

For #6-10, change the percent to a decimal, then to a fraction 6) 52% = _____ = _____ 7) 33.3% = _____ = _____ 8) 4.55% = _____ = _____

9) 37.5% = _____ = _____

10) 10% = _____ = _____ What if the decimal was a repeating decimal? Write 0.5555555 as a fraction ..….. to be done in class

3) 2½ = ____ = _____

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PERCENT PROBLEMS TO FIND THE PERCENT OF A NUMBER: change the percent to decimal and multiply EX. 50% of 80 = .50 ∗ 80 = 40 22% of 90 = .22 ∗ 90 = 19.8 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 2n=6 means n = 6 ÷ 2 = 3 since 2 ∗ 3 = 6 EX. 1: Find 30% of 80. .30 ∗ 80 = x → x = 24 EX. 2: 45% of 32 is what number? .45 ∗ 32 = n → n = 14.4 EX. 3: 40 is 20% of what number? 40 = .20∗y → y = 40÷.20 = 200 EX. 4: 30 is what percent of 50? 30 = n ∗ 50 → n = 30÷50 = 0.6 = 60% When you need to find the percent, find the decimal and change it to a percent EX. 5: What percent of 35 is 7? n ∗ 35 = 7 → n = 7÷35 = .2 = 20% EX. 6: What percent of 250 is 25? n ∗ 250 = 25 → n = 25 ÷ 250 = .1 = 10% EX. 7: What percent of 25 is 250? n ∗ 25 = 250 → n = 250 ÷25 =10 = 1000% PRACTICE: (show work) 1) What number is 90% of 50?

2) Find 15% of 600.

3) 200% of 67 is what number?

4) What number is 30% of 120?

5) 2 is what percent of 60?

5) What number is 98% of 230?

7) Joan’s income is $190 per week. She saves 20% of her weekly salary. How much does she save each week?

8) Ninety percent of the seats of a flight are filled. There are 240 seats, how many seats are filled?

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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. 2) 15a2c3 = 3∗5∗a∗a∗c∗c∗c = 5a EX. 1) 36 = 2∗2∗3∗3 = 9 3∗a∗c∗c∗c∗c c 40 2∗2∗2∗5 10 3ac4

PRACTICE: Write each fraction in simplest form. 1)

16 36

5)

2)

=

45 48

3)

=

2 5 − 3r s = 3 6 − 12r s

49 56

6)

4)

=

9x 16y

2 3

=

3 3 6a b = 2 4 18a b

7) Tara takes 12 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 15 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: 9)

7 8

+

8

=

10)

1 9

+

2 8 + 12 = 3 9

14) 24

1 1 ∗ = 3 2

18) 2

13) 18

17 ) 4

5

5 6

11) 4

=

1 3 − 12 = 2 4

1 3 ∗3 = 9 5

15)

19)

3 4

−2

1 4

5 1 ∗ = 12 5

2 5 ÷ = 3 7

=

12) 10

6 7

−5

2 5

16)

4 5 ∗ = 5 14

20) 1

3 9 ÷ = 4 10

=

21) A recipe calls for ½ tsp. sugar. If Sam wants to make ⅔ of this recipe, how much sugar should he use? 2 22) If about ⅓ of the earth is able to be farmed and /5 of this land is planted in grain crops, what part of the earth is planted in grain crops? 23) About 1/20 of the world’s population lives in South America. If about 1/35 of the world’s population lives in Brazil, what fraction of the population of South America lives in Brazil? 24) The area of a rectangle is ⅔ yds2. If one side is ⅓ yd, what is measure of the other side? show work on separate paper

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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 “5x” is 5) 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 EX. 1: Evaluate 5a - 2b if a = 3 & b = 4 5a - 2b = 5(3) – 2(4) = 15 – 8 = 7

PE MD AS

EX. 2: Evaluate 5a – 2(a – b) if a = 6 & b= 4 5(6) – 2(6 – 4) = 30 – 2(2) = 30 – 4 = 26

PRACTICE A: write an algebraic expression for each verbal expression 1) the sum of a number and 6: __________ 2) twice a number and 6: _________ 3) twice the sum of a number and 6: __________ 4) the product of x and y: __________ 5) 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: __________ #7-9 8) twice the difference of a and b: __________ are 9) the difference of a and b-squared: __________ tricky! 10) 3 less than 5 times the difference of a and b: __________ PRACTICE B: Evaluate each expression if a = 1, b = 2, x = 5, and y = 10 1) bx – y

5) b + x ∗ a

2) 10b – a

6) a ∗ x + b ∗ y

3) 10(b – a)

7) y ÷ x ∗ b ÷ a

4) x + y ∗ 7 ÷ 2

8) (y ÷ x) ∗ (b ÷ a)

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THE REAL NUMBER SYSTEM & SQUARE ROOTS Types Real Numbers Rational Numbers

Symbol R Q

Irrational Numbers Counting or Natural Numbers Whole Numbers Integers

I N

Definition rational and irrational numbers all numbers that can be expressed in the form a/b where a & b are integers and b≠0 numbers that never end and never repeat (ex: pi, √2, √3, √5) {1, 2, 3, ...}

W Z

{0, 1, 2, 3, ...} { ... –3, -2, -1, 0, 1, 2, 3, ...}

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 4 Î √16 = 4 4 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 -√16 = -4.

Write the PERFECT SQUARES from 1 to 20: 12 = 52 = 92 = 22 = 62 = 102 = 32 = 72 = 112 = 82 = 122 = 42 = Since x * x = x2 …. √x2 = x

OR

172 = 182 = 192 = 202 =

Therefore: √52 = 5

** Find each of the following: B) √452 = A) √82 = √16 = 4 = 2 √4 2

132 = 142 = 152 = 162 =

√16 = √4 = 2 √4

C) √2382 = SO

√125 = √25 = 5 √5

PRACTICE A: Name the set(s) of numbers to which each real number belongs (use the symbols above to represent the group) 1) 3.145 ____________ 2) √11 ____________ 3) √36 ____________ 4) 1/3 ____________ 5) -√4/9 ____________ 6) -4 ____________ PRACTICE B: Evaluate each expression 7) √a if a = 225 8) √400

if c = 3 and d = 12 9) ±√cd

if x = 30 and y = 19 10) ±√x + y

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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: 3 + 4 + 5 = +12 If all numbers being added are negative, the answer is negative. EX: -3 + -4 + -5 = -12 If the signs are different, subtract and take the sign of the number with the greater absolute value. EX. 1: -3 + 4 = +1 since 4 – 3 = 1 and the sign of 4 is positive EX. 2: 3 + -4 = -1 since 4 – 3 = 1 and the sign of 4 is negative If there are more than 2 numbers, add all the positives, then add all the negatives, then subtract the results and take the sign of the greater absolute value. EX: 3 + (-5) + 7 + 15 + (-17) = (3+7+15) + (-5 + -17) = (25) + (-22) = 3 OR just perform the operations left to right: 3 + -5 = -2; -2 + 7 = 5; 5 + 15 = 20; 20 + -17 = 3 SUBTRACTING Add the opposite of the number that is being subtracted.

EX. 1: EX. 2: EX. 3: EX. 4: EX. 5:

3 – 2 = 3 + -2 = 1 4 – 10 = 4 + -10 = -6 5 – 6 – 12 + 2 = 5 + -6 + -12 + 2 = -11 15 – (-15) = 15 + 15 = 30 -15 – (-15) = -15 + 15 = 0

REMEMBER: IF VARIABLES ARE INVOLVED, THE VARIABLES STAY THE SAME. Only like terms can be added or subtracted. EX. 1: 3n – 5n = 3n + -5n = -2n EX. 2: 15xy – 8xy = 7xy Ex. 3: 2x – 2y + 3x + 6y = 5x + 4y

ADDITION PRACTICE: 1) -2n + (-1n) + 6p = ___

2) -3ab + (-8ab) + 11ab = ___

3) 2k – 7p – 1k = ___

4) -7e + (-10e) + 7e = ___

5) -8x + 9y + (-3x) = ___

6) 1/2 + (-3/4)+(-5/4) = ___

SUBTRACTION PRACTICE: 7) -15x – 20x = ___

8) 15x – 20x = ___

9) 20x – 15x = ___

11) -15x – (-20x) = ___

12) 1/2 - 3/4 - 5/4 = ___

10) 15x – (-20x) = ___

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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) 3 ∗ 4 = 12 and -3 ∗ -4 = 12 since there are two negative signs 2) 3 ∗ -4 = -12 and –3 ∗ 4 = -12 since there is only one negative sign in each example 3) -2 ∗ 3 ∗ -4 = +24 since there are two negative signs (-2x * 3x * -4x = +24x3) 4) -2 ∗ 3 ∗ 4 = -24 since there is one negative sign (-2x * 3x * 4x = -24x3) 5) 6/3 = 2 and –6/-3 = 2 since there are two negative signs 6) 6/-3 = -2 and –6/3 = -2 since there is one negative sign in each example REMEMBER:

−

2 −2 2 = = 3 3 −3

MULTIPLICATION PRACTICE: 1) 6 ∗ -2 = ___

DIVISION PRACTICE: 6) 6 / (-1/2) = ___

2) -7y ∗ -9y = ___

7) -63n / -7n = ___

3) -4n ∗ (-15) = ___

8) -80x5 / 16x3 = ___

4) -4a ∗ 5a ∗ 2b = ___

9) -49n ÷ -7 = ___

5) 1/3 ∗ -9/2 ∗ -2/5 = ___

10) 0 ÷ -6 = ___

STATE WHETHER EACH STATEMENT IS TRUE OR FALSE.

11) The product of two positive integers is positive. __________ 12) The product of two negative integers is negative. ____________ 13) The product of one negative and two positive integers is positive. __________ 14) The quotient of two negative numbers is positive. ____________ 15) The quotient of one positive and one negative number is negative. ____________

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POWERS AND EXPONENTS An exponent tells how many times to use the base as a factor. In the expression 23, the base is 2 and the exponent is 3. EX. 1 23 = 2 ∗ 2 ∗ 2 = 8 EX. 2 (x + 1) 2 = (x + 1)(x + 1) PRACTICE: Write each product using exponents. 1) 5 ∗ 5 ∗ 5 2) 2 ∗ 2 ∗ 2 ∗ 3 ∗ 3

3) 4 ∗ 4 ∗ 4 ∗ n ∗ n ∗ y ∗ y ∗ y

Write each power as a product of the same factor. 4) f5 5) (-k)2 6) (y – 2)3

IMPORTANT NOTE:

(-x)2 ≠ -x2 since (-x) 2 = (-x)(-x) = x2

–x2 is a negative number

On a calculator, you MUST use parentheses to raise a negative number to a power.

MULTIPLYING BY POWERS OF 10 36 ∗ 10‐3 = .036  since 10‐3 = .001  ‐2  36 ∗ 10 = .36   since 10‐2 = .01  36 ∗ 10‐1 = 3.6  since 10‐1 = .1  36 ∗ 100 =  36   since 100 = 1  1 36 ∗ 10  =  360  since 101 = 10  36 ∗ 102 =  3600  since 102 = 100  36 ∗ 103 =  36,000  since 103 = 1,000  36 ∗ 104 =  360,000  since 104 = 10,000

(-2)4 = +16

DIVIDING BY POWERS OF 10 36 ÷ 10‐3 = 36,000  since 10‐3 = .001  36 ÷ 10‐2 = 3600  since 10‐2 = .01  since 10‐1 = .1  36 ÷ 10‐1 = 360  0 36 ÷ 10  =  36  since 100 = 1  since 101 = 10  36 ÷ 101 =  3.6  36 ÷ 102 =  .36  since 102 = 100  36 ÷ 103 =  .036  since 103 = 1,000  36 ÷ 104 =  .0036  since 104 = 10,000

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) 15 ∗ 104 = _____

2) .46 ∗ 104 = _____

3) 1.78 ∗ 10-4 = _____

4) 15 ÷ 104 = _____

5) .46 ÷ 104 = _____

6) 1.78 ÷ 10-4 = _____

7) 61 ÷ 10-3 = _____

8) .132 ∗ 103 = _____

9) 35.7 ÷ 102 = _____

ANY NUMBER RAISED TO THE ZERO POWER IS 1. (The only exception is 00 …

00 is undefined because division by zero is not allowed)

60 = 1

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IDENTITY AND EQUALITY PROPERTIES Additive Identity Property Multiplicative Identity Property Multiplicative Property of Zero Substitution Property Reflexive Property Symmetric Property Transitive Property Distributive Property Commutative Property Associative property

For any number a, a + 0 = 0 + a = a For any number a, a ∗ 1 = 1 ∗ a = a For any number a, a∗0 = 0∗a = 0 For any numbers a & b, If a = b, then a may be replaced by b

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) 2)

21 + 0 = 21 (0)15 = 0

__________ __________

3) 4)

x 3 ∗ 1 = x3 4+3=4+3

__________ __________

5) 6)

6x + 2y = 2y + 6x (14 – 6) + 3 = 8 + 3

7) 8)

If x + y = 9 then 9 = x + y __________ 9r2 + 9s2 = 9(r2 + s2) __________

9) 10)

If 3 + 3 = 6 and 6 = 3 ∗ 2, then 3 + 3 = 3 ∗ 2 (2c + 6) + 10 = 2c + (6 + 10)

__________ __________

__________ __________

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INTEGRATION: GEOMETRY AND THE COORDINATE SYSTEM 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: {(1,2), (3,4), (8,7)} Element: a member of a set; (1,2) 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: {1,3,8} Range: the set of all second coordinates from the ordered pairs in a relation; in the EX: {2,4,7} 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 (0,3) is graphed on the ___-axis. 2) The ordered pair (-5,0) 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).

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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, 2, 3, 5, 8, ____, ____, ____ Explanation:

2)

Perfect squares: 1, 4, 9, 16, 25, ____, ____, ____ What is the 15th term? ____ What is the nth term?

3)

Triangular Numbers: Explanation:

4)

consecutive numbers: consecutive even integers: consecutive odd integers:

1, 3, 6, 10, 15, 21, ____, ____, ____

x, x + 1, x + 2, x + 3, _____, _____ y, y + 2, y + 4, y + 6, _____, _____ z, z + 2, z + 4, z + 6, _____, _____

5)

What is the next picture? What color is the 50th picture? __________

Why?

6)

Arithmetic sequence: FORMULA:

1, 4, 7, 10, ____, ____, ____

7)

Geometric sequence:

2, 4, 8, 16, 32, ____, ____, ____