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Pre Algebra Contents

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Order of Operations 1-1 Patterns and Variables 1-2 Translating Phrases to Algebraic Expressions 1-3 Properties 1-4/2-4 Simplifying Algebraic Expressions 2-5 Integers 3-1 Adding and Subtracting Integers 3-2/3-3 Multiplying and Dividing Integers 3-8/3-9 Factors and Multiples 5-1 Adding and Subtracting Fractions 6-6/6-7 Adding and Subtracting Mixed Numbers 6-8 Multiplying Rational Numbers 7-1 Reciprocals and Dividing Rational Numbers 7-2 More About Exponents 7-6 Scientific Notation 7-7 Solving Two Step Equations 8-1 Functions 8-4 More Simplifying to Solve Equations 8-6 Solving Equations with variables on Both Sides 8-7 Coordinate Plane 9-1 Graphing Linear Equations 9-2 Graphing Systems of Equations 9-9 Scatterplots 9-10 Graphing on the Number Line 9-11 Graphing Inequalities on the Coordinate Plane 9-12 Ratio, Rate, Proportion 10-1/10-3 Percent, Decimals, and Fractions 10-4/10-7 Finding a Percent of a Number 11-1 Finding the Percent One 11-2 Percent of Increase or Decrease 11-5 Calculating Simple Interest 11-8 Circles 12-8 Congruent Figures 12-10 Basic Geometric Figures 12-1 Angle Measure 12-3 Parallel and Perpendicular Lines12-4 Triangles 12-5 Polygons 12-6 Area of Rectangles and Parallelograms 13-1 Area of Triangles and Trapezoids 13-2 Area of Circles 13-3 Volume 13-6/13-8 Surface Area 13-9 The Basic Counting Principle 14-1 Permutations and Combinations 14-2 Probability 14-3 Frequency Tables 14-6 Square Roots 15-1 Aproximating Square Roots 15-2 Solving Equations: Using Square Roots Special Triangles 15-7 Special Triangles Trigometric Ratios 15-9

Order of Operations 1-1

1) Parenthesis 2) Powers 3) Divisions / Multiplication > (left to right) 4) Subtraction / Addition > (left to right) Algebraic expression involving multiplication and division 9xm= 9*m= 9(m)= 9m 7xaxb= 7*a*b= 7(a)(b)= 7ab y/3= y/3 ex.1) Evaluate m(7+w); m=9 9(7+3); w=3 9(10) 90 ex.2) Evaluate 8+ab/40; a=8 8+8*10 40 8+80 40 8+2 10 ex.3) Evaluate 3y+9 ; y=9 2y 27+9 18 36 18 2

Patterns and Variables 1-2


Translating Phrases to Algebraic Expressions 1-3

sum + difference - product * quotient / Translate to a numerical expression ex.1) the difference of 10 and 5 = 10-5 ex.1) 4 less than 10 = 10-4 ex.3) the product of 8 and 7 = 8*7 ex.4) 45 divided by 3 = 45/3 ex.5) a number increased by 7 = h+7 ex.6) 5 less than a number = n-5 ex.7) 35 increased by twice a number = 35+2*d ex.8) 1 less than a number divided by 10 = d/10-1

Properties 1-4 / 2-4

Commutative: a+b = b+a a*b = b*a ex.1) 5*7 = 7*5 ex.2) m+7 = 7+m Associative: ex.1) (98+47)+53 = 98+(47+53) 98+100 198 ex.2) 7*(5x) = (7*5)x 35x Identity: a+0 = a a*1 = a Distributive: a*(b+c) = a*b+a*c ex.1) a*(3+5) = 9*3+9*5 27+45 72 ex.2) 6(r+s) = 6r+6*5 6r+30 ex.3) (m+7)4 = 4m+28 ex.4) 86*48+86*52

Simplifying Algebraic Expressions 2-5

Simplify - replace with a simpler equivalent expression ex.1) Simplify 3(2n) (3*2)n 6n ex.2) (x+7)+6 x+(7+6) x+13 Term - parts of an expression separated by + or - 4m+2n-3 terms Like terms - terms with the same variable 2n and 3n are like terms, 4m and 2m are unlike terms ex.3) 2x+3x (2+3)x 5x ex.4) 25g+15g (25+15)g 40g ex.5) 2m+3m+m 6m ex.6) 4e+3f+e 5e+3f

Integers 3-1

Integers {.-3,-2,-1,0,1,2,3,.} Numbers such as 4 and -4 are opposites. The opposite of -4 is written -(-4) = 4. Absolute Value - distance from zero ex.1) | 7 | = 7 ex.2) |-4 | = 4 ex.3) | 0 | = 0 ex.4) Find m if | m | = 8. m = -8,8 Order from least to greatest ex.5) -1,-6,4 = -6,-1,4 ex.6) -2,|-3 |, -1,| 4 | 3 4 -2,-1,|-3 |,| 4 |

Adding and Subtraction Integers 3-2/3-3

Think about money (+ : $ you have) (- : $ you spent) ex.1) 3+-2 = 1 ex.2) -3+2 = -1 ex.3) -2+-3 = -5 To subtract, we add the opposite of the second number. |------| |------| |------| | 9-7 | = | 5-1 | = | 2-3 | = | 9+-7 | = | 5+-1 | = | 2+-3 | = |------| |------| |------| NOTE: -5 = -5 ex.4) 3-2 = 1 ex.5) -3-2 = -5 ex.6) 3- -2 = 5 ex.7) -3- -2 = -1

Multiplying and Dividing Integers 3-8/3-9

If the signs are the same the answer is positive. If the signs are different the answer is negative. ex.1) 4*-5 = -20 ex.2) -4*5 = -20 ex.3) -4*-5 = 20 ex.4) (-5)3 = -125 ex.5) 42/-7 = -6 ex.6) -42/7 = -6 ex.7) -42/-7 = 6 If n = + then n5 = + If n = - then n5 = - If n = + then n8 = + If n = - then n8 = -

Factors and Multiples 5-1

factors product 7*4 = 28 factors of 8 = 1,2,4,8 factors of 84 = 1,2,3,4,6,7,12,14,21,28,42,84 Divisibility - one number is divisible by another if their quotient is a whole number (no remainder) ex.1) Is 11 a factor of 143? ex.2) Is 9 a factor of 326? Divisibility rules 2 - even 3 - add digits, test 3 4 - test last 2 digits 5 - end in 0 or 5 Multiples - counting by a number ex3.) List the multiples of 5 0,5,10,15. ex.4) List the first 3 nonzero multiples of 11\ 11,22,33.

Adding and Subtracting Fractions 6-6/6-7

ex.1) 2/5 + 1/3 = 6/15 + 5/15 = 11/15 ex.2) 3/10 + 6/15 = 9/30 + 12/30 = 21/30 = 7/10 ex.3) -2/5 - 3/6 = -12/30 + -15/30 = -27/30 = -9/10 ex.4) 5/6 - -3/4 = 10/12 + 9/12 = 19/12 Reducing Fractional Expressions ex.1) 3mb/7mk = 3b/7/k ex.2) 4ef/12f = 1e/3 ex.3) 15ac2/25a2c 15acc/25aac 15c/25a 3c/5a Adding and Subtracting Mixed Numbers ex.1) The west wall measures 60 9/16" and 217 7/8". Find the total length? ex.2) 3M stock rose from 50 _ to 51 5/8. What was the increase? ex.3) GM stock fell from 52 1/8 to 50 3/4

Multiplying Rational Numbers 7-1

Multiply / Divide rules + + = + + - = - - + = - - - = + ex.1) 2/5 * 5/12 = 10/60 ex.2) -6/4 * -3/5 = 9/10 ex.3) 1/2 of 3/4 = 3/8 ex.4) 2 2/3 (1 1/5) = 16/5 ex.5) 3 1/12 (-2) = 37/6 ex.6) (5/8)2 = 5/8 * 5/8 = 25/64

Reciprocals and Dividing Rational Numbers7-2

- two numbers are reciprocals if their product is 1 ex.1) 2/3 in the reciprocal of 3/2 because 3/2 * 2/3 = 1 ex.2) what is the reciprocal of -2000m/3, _______ how many halves are in 3? 3/ 1/2 = 6 2 1/2 / 1/4 = 10 ex.3) 2/3 / 1/6 = -4 ex.4) -5/24 / 5 = -32/15 ex.5) 8/ (-3 _) = -1/24 ex.6) -3 1/8 / -2 1/12 = 3/2 (1 «)

More About Exponents 7-6

Review: Exponents are used to show how many times the some factor is repeated. exponent 43 = 4 x 4 x 4 = 64 base ex.1) 82 = 8*8 = 64 ex.2) 24 = 2*2*2*2 = 16 ex.3) (-.03)2 = -.03 x -.03 = .0009 Rule: To multiply two powers with the same base, we add the exponents. ex.1) 32 * 34 = 36 ex.2) 103 * 102 = 105 ex.3) 24 * 2 = 25 Try the following: 45/42 = 43 This suggests a rule for simplifying expressions in this form. Rule: To divide two powers with the same base, we subtract the exponents. ex.1) 25/23 = 22 ex.4) x6/x2 = x4 ex.2) (-3)7/ (-3)4 ex.5) m5/m ex.3) 104/10 ex.6) 34/34 you can also use the rule given above to simplify the expression 52/54 52/54 = 1/52 52/54 = 5-2 This shows that 1/52 is the same as 5-2 ex.1) 3-2 = 1/32 = 1/9 ex.3) (-2)-4 = 1/-2-4 = 1/16 ex.2) 5-3 = 1/53 = 1/125 ex.4) 4-1 = 1/-4

Scientific Notation 7-7

Review 103 = 1000 10-3 = 1/1000 = .001 105 = 100000 10-5 = 1/1000000 = .00001 106 = 1000000 10-6 = 1/10000000 = .000001 Scientific notation is used to simplify work with very small or very large numbers. 3.45 X 103 A decimal between 1 and 10 a power of 10 Write in standard form. ex.1) 5.8 x 103 = 5800 ex.2) 6.556 x 102 = 655.6 ex.3) 1.8 x 10-4 = .00018 ex.4) 4 x 10-2 = .04 Write in scientific notation. ex.1) 4567 = 4.567 x 103 ex.2) 1,234,000 = 1.234 x 106 ex.3) 234,000 = 2.34 x 105 ex.4) 50,000,000 = 5.0 x 107 ex.5) 0.000345 = 3.45 x 10-4 ex.6) 0.0206 = 2.06 x 10-2 ex.7) 0.000008 = 8.0 x 10-6 ex.8) 0.2004 = 2.004 x -10 Why are these not in scientific notation? ex.1)(10 is less than 12.5) 12.5 x 104 ex.2) 2 x 4-5 (the base must be 10) Give two reasons for using scientific notation. 1.) easier for large numbers 2.) nicer for comparing

Solving Two Step Equations 8-1

SOLVE review: m-17 = 24 3m = 5 m+5 = 11 m/4 = 9 Rule: Solving equations with combined operations A: Identify the order the operations were applied to the variable. B: Undo the operations in reverse order.

Functions 8-4

Definition: A function is a special relationship between two variables. Domain- number put into the function (input). Range- answer (output) Function notation: f(x) ex.) f(x) = 3 x +4 then f(5) = 3(5)+4 = 15+4 = 19

More Simplifying to Solve Equations 8-6

Like terms- have the same variable and exponent ex 3m+2m = 5m Unlike terms- cannot be simplified ex 3m+2k 3m2+2m ex.1) 9k+5 (k+7) = -49 ex.2) 2m+3(m-7) = 44 ex.3) 3(5n)+14+6n = 21 ex.4) -2(7c)-12+5c = 51

Solving Equations with Variables on Both Sides 8-7

1: Get the variable on one side 2: Solve as always ex.1) -4(2m-5) = -4m+10 -8m+20 = -4m+10 +8m +8m 20 = 4m+10 -10 -10 10 = 4m 4 4 5/2 = m ex.2) 2(m+2) = 2m+4 ex.3) 2(m-2) = 2(m+1)

Coordinate Plane 9-1
Graphing Linear Equations 9-2
Graphing Systems of Equations 9-9
Scatterplots 9-10
Graphing on the Number Line 9-11
Graphing Inequalities on the Coordinate Plane 9-12
Ratio, Rate, Proportion 10-1/10-3

Ratio- a comparison of one number to another. ex.) 3/5 or 3.5 or 3 to 5 Rate- a ratio that involves two different units. ex.) 216miles/4hours = 54m/h Proportion- An equation starting that two ratios are equal. We can use cross product to solve proportions. ex.1) 2/3 = 24/n ex.2) 9/12 = 3/n ex.3) 9/10 = n/22 ex.4) The lions in the zoo eat 40kg. Of food every 7 days. How many kg. do they eat in 30 days? Kg./days 40/7 = k/30 1200 = 7k 7 7 171 = k

Percent, Decimals, and Fractions 10-4/10-7

Percent means hundredths or "out of 100" Change to decimal. 1.) 43% = 43/100 = .43 2.) 90% = .9 3.) 2% = 4.) 34.7% = .347 5.) 10 «% = Change to percent. 1.) .27 = 27% 2.) .06 3.) .7 4.) .065 5.) .7/8 = 87.5%

Finding a Percent of a Number 11-1

Estimating Percents ex.1) 63%$ of 61 « of 60 = 30 ex.2) 27% of 79 ¬ of 80 = 20 Using Proportions ex.3) 90% of 25 90/100 = n/25 2250 = 100n 100 100 22.5 = n Using Decimals ex.4) 6.5% of $80 .065 x 80 = 5.20 ex.5) 15% of $25 .15 x 25 = 3.75 Using a calculator ex.6) 22% of 150 33

Finding the Percent One 11-2

Number is of Another "what percent" … n/100 of _ … ex.1) 17 is what percent of 25? n = 17 100 25 25n = 1700 25 25 n = 68% ex.2) What percent is 8 out of 13? n = 8 100 13 ex.3) What percent of 72 is 12? n = 12 100 72 ex.4) What % of 72 is 90? n = 90 100 72

Percent of Increase of Decrease 11-5

change original = _ % ex.1) My weight on January 1st = 140 My weight on February 1st = 144 change original = 4/140 = .0285 = 3% ex.2) Jenny runs the mile in 9 minutes in June. In July she can run it in 8 minutes. change original = 1/9 = .111 = 11%

Calculating Simple Interest 11-8

When you borrow money from a bank, credit union, or loan company you pay for the use of it. The amount you pay for the use of money is called interest. The amount of interest you pay depends upon the principle (amount borrowed), the rate (percent of interest) charged, and the length of time the money is kept (time). Interest = Principle * Rate * Time I = P * R * T ex.1) Calculate the interest ex.2) Principle = $80,000 on a $800 loan at 8% interest Rate = 8% per year per year if you paid it back in Time = 30 years 2 years. ex.3) You use your credit card ex.4) Principle = $5000 to buy $500 worth of clothes. Rate = 1.5% per month You have to pay 15% interest Time = 2 years per year. How much interest would you pay after 1 year? What did the clothes really cost? ex.5) You put $1000 in the bank ex.6) Principle = $1000 and leave it in for 3 months at Rate = 8% per year 2.5% interest per year. How much Time = 3 months interest do you make? How much money do you have now? ex.7) Principle = $1,000,000 Rate = 8% per year Time = 3 months

Circles 12-8
Congruent Figures 12-10
Basic Geometric Figures 12-1
Angle Measure 12-3
Parallel and Perpendicular Lines 12-4
Triangles 12-5
Polygons 12-6
Area of Rectangles and Parallelograms 13-1
Area of Triangles and Trapezoids 13-2
Area of Circles 13-3
Volume 13-6/13-8
Surface Area 13-9
The Basic Counting Principle 14-1

To find the total number of choices for an event, multiply the number of choices for each individual part. ex.1) How many outcomes are possible if you first toss a coin, then roll a die? 2 * 6 = 12 outcomes ex.2) How many different pizza combinations can you make with 2 crusts, 2 sauces, and 4 toppings? (only a one topping pizza) 2 * 2 * 4 = 16 ex.3) How many license plates can be made in Minnesota? 10 * 10 * 10 * 26 * 26 * 26 = 17576000

Permutations and Combinations 14-2

An arrangement of a group of objects in a certain order is called a permutation. ex.1) Andy, Bob and Chris are to be seated at 3 desks arranged in a row. How many ways can the 3 students arrange themselves? 3 * 2 * 1 = 6 ex.2) Steve, Rachel, Teddy and Kristin are on our 4x100 relay. How many running orders are there for this relay? 4 * 3 * 2 * 1 = 24 ex.3) 20 people are running for student council. How many different permutations are there? 20 * 19 = 380 (president & vice president) A selection of objects without regard to order is called a combination. ex.4) choose 2 of these 4: Art, Band, computer, Drafting. ex.5) Almond Joy, Butterfinger, Crunch, Dollar Bar; choose 3 of 4.

Probability 14-3

Probability = ways for an outcome to occur total outcome Flipping a coin: ex.1) P(getting tails) = « ex.2) P(getting heads) = « Rolling a Die: ex.3) P(rolling a 5) = 1/6 ex.4) P(not rolling a 5) = ex.5) P(rolling a 1 or 5) = 2/6 = 1/3 ex.6) P(rolling a 7) = ex.7) P(rolling a number <7) = 6/6 = 1 Note: If any outcome is certain to happen then the probability of that outcome is 1. If any outcome is impossible then the probability of that outcome is 0. ex.8) Find P(winning the daily 3) 1/1000 1/10 * 1/10 * 1/10 = 1/1000

Frequency Tables 14-6

Data is often recorded using a frequency table. This table shows how many students received each score on a math quiz. _ Mean = total points/number of scores = 468/27 = 17.3 Median = middle score = 17 Made = Score that occurs most often = 17 Range = Difference between the largest and smallest score = 20-13 = 7

Square Roots 15-1
Approximating Square Roots 15-2
Solving Equations: Using Square Roots 15-3
Special Triangles 15-7
Trigonometric Ratios 15-9


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