Abraham Clark High School ALGEBRA 2 SUMMER PACKET

Size: px

Start display at page:

Download "Abraham Clark High School ALGEBRA 2 SUMMER PACKET"

Imogen Davidson
5 years ago
Views:

1 Name: Abraham Clark High School ALGEBRA SUMMER PACKET INFORMATION/DIRECTIONS: All questions marked Practice must be complete and correct. You must return in September knowing how to do all the material in this packet. If there is any difficulty with the material within this packet, please use the below online resources for further eplanation. The main objective of the Algebra packet is that you have mastered these skills before entering Algebra. You will be given a series of quizzes in September to verify your mastery of these skills. This summer packet will count as the first test grade for marking period. This is a mandatory assignment by the Roselle Board of Education. WEBSITES: ORDERING AND COMPARING NUMBERS OPERATIONS WITH SIGNED NUMBERS OPERATIONS WITH FRACTIONS RATIO PROPORTION PERCENT ORDER OF OPERATIONS

2 LAWS OF EXPONENTS Go to eponents and eponential functions LITERAL EQUATIONS SLOPE OF A LINE WRITE THE EQUATION OF A LINE IN SLOPE-INTERCEPT FORM GIVEN TWO POINTS GRAPH THE EQUATION OF A LINE PARCC PRACTICE ACHS ENTERING ALGEBRA SUMMER 07 PACKET Page of 8

3 ORDERING AND COMPARING NUMBERS Eample: Graph the numbers on a number line, then write the numbers in increasing order. ORDERING AND COMPARING NUMBERS Practice: Graph the numbers on a number line, then write the numbers in increasing order. Given:,,,, 0 Step : Rewrite the numbers as decimals. When necessary, round the numbers to two decimal places.,.,, 0.66,.6 Step : Plot the numbers on the number line..,,,, 0 Step : Write the numbers in increasing order.,,,,,,, 0,., 0 8,., 7,,,,, ACHS ENTERING ALGEBRA SUMMER 07 PACKET Page of 8

4 OPERATIONS WITH SIGNED NUMBERS EXAMPLE ADDITION OF SIGNED NUMBERS: + + ( ) + ( ) + Step : Check if the numbers have the same sign. Step : Depends on the signs. If yes, find the sum and keep the sign ( ) 9 If no, find the difference and keep the sign of the larger digit. + ( ) + EXAMPLE SUBTRACTION OF SIGNED NUMBERS: ( ) ( ) Step : Rewrite the subtraction problem as an addition problem. + ( ) ( ) Step : Add the numbers as above. + ( ) ( ) 9 These rules work with real numbers (.) 8. OPERATIONS WITH SIGNED NUMBERS Practice: Add or subtract as indicated (.8).. (.8) (.) (.8) (.) ACHS ENTERING ALGEBRA SUMMER 07 PACKET Page of 8

5 OPERATIONS WITH SIGNED NUMBERS MULTIPLICATION OF SIGNED NUMBERS: 8() 8( ) 8( ) 8() Step : Check if the numbers have the same sign. Step : Depends on the signs. If yes, multiply the numbers and make the sign positive. 8() 8( ) If no, multiply the numbers and make the sign negative. 8( ) 8() DIVISION OF SIGNED NUMBERS: 8 8 ( ) 8 ( ) 8 () Step : Check if the numbers have the same sign. Step : Divide by following the same rules as multiplication. 8 8 ( ) 8 ( ) 8 () These rules work with real numbers..(.). OPERATIONS WITH SIGNED NUMBERS Practice: Multiply or divide as indicated...8(.6) (.6)..8 (.6)..8(.6) (.6).(.) (.6). (.) ACHS ENTERING ALGEBRA SUMMER 07 PACKET Page of 8

6 OPERATIONS WITH FRACTIONS Equivalent Fractions are fractions that are equal in value. Eample: A prime number is a number whose only factors are one and itself. The smallest and only even prime number is. is not a prime number. Eamples: () 7 7() A composite number is a number which has additional factors besides one and itself. OPERATIONS WITH FRACTIONS Hint: Use factor trees to prime factor. Eample: Prime factor 6. 6 ( ) Practice: Prime factor each number... Eample: (6) () () A factor is one of two or more epressions that are multiplied together to get a product. A product is the result of multiplication Prime Factorization means finding all the prime factors of a number. You do not need the factor when doing prime factorization. Eamples: 6 () ()() () 8 ()(7) (7) 98 (7)(7) (7 ) Simplified Fraction is a fraction in which the numerators and denominators do not have any common factors ecept. Eamples: Practice: Simplify each fraction. Show the prime factorization and canceling. Simplify a Fraction Using Prime Factorization Eample: Simplify Step : Write the numerator and denominator as a product of prime factors. 90 ( 7) ()()() Step : Cancel out all the common factors. 90 ( 7) ()()() ACHS ENTERING ALGEBRA SUMMER 07 PACKET Page 6 of 8

7 OPERATIONS WITH FRACTIONS ADDING AND SUBTRACTING FRACTIONS Adding Eample: + 0 OPERATIONS WITH FRACTIONS ADDING AND SUBTRACTING FRACTIONS Practice: Add or subtract the fractions as indicated. Step : Make sure that both fractions have the same denominator. The lowest common denominator is Multiply: 0 Step : Add the numerators together. DO NOT add the denominators. Add: Step : Reduce the answer if possible. 0 Subtracting Eample:. This is the same instructions as above ecept subtract in step instead of adding Step : Make sure that both fractions have the same denominator. The lowest common denominator is. Multiply: 6 Step : Subtract the numerators. DO NOT subtract the denominators Subtract: 6 0 Step : Reduce the answer if possible. 0 is in simplest form. ACHS ENTERING ALGEBRA SUMMER 07 PACKET Page 7 of 8

8 OPERATIONS WITH FRACTIONS MULTIPLYING AND DIVIDING FRACTIONS Multiplying Eample: Step : Multiply the numerators together and the denominators together. () () Step : Write the fraction. 6 Step : Reduce the answer if possible. 6 NOTE: Multiplication and division do NOT need to have common denominators. You can cancel before you multiply so you do not have to reduce at the end. OPERATIONS WITH FRACTIONS MULTIPLYING AND DIVIDING FRACTIONS Practice: Multiply or divide the fractions as indicated () () () () Dividing Eample:. 7 Step : Convert to multiplication by KCF (Keep-Change-Flip). 6. Step : Multiply as per previous section. () () Step : Write the fraction and reduce if possible. 7. This is in simplest form. 8. ACHS ENTERING ALGEBRA SUMMER 07 PACKET Page 8 of 8

9 RATIO RATIO PROPORTION PERCENT A ratio is a direct comparison of two different quantities. When writing a ratio, unless told otherwise in the problem, the quantity listed first in the problem is the first one in the ratio and the quantity listed second is the second one in the ratio. Eample: If there are boys and 6 girls in the room, what is the ratio of boys to the total number of students? Ratios can be written in several forms. The chart shows the different ways to answer the eample. RATIO SIMPLIFIED to 0 to : 0 : Eample: What is the ratio of a nickel to a quarter? or : is : Eample: On a triangle, each side measures cm, 0 cm, and 0 cm, respectively. In lowest terms, find the ratios of the lengths of the sides. The ratio is : 0 : 0. Each number can be divided by. Simplified: : : 6. RATIO PROPORTION PERCENT Ratio Practice: Find the ratio for each of the questions. Reduce when possible.. Find the ratio of two meshed gears if one gear has 8 teeth and the other has 6 teeth.. An auto dealership has the following annual break-down of vehicles sold: Sedans 60 Sports Cars 0 Minivans Trucks Total Vehicles 80 Find the following ratios. a) Sedans to sports cars. b) Minivans to trucks. c) Trucks to total vehicles.. Adam, Bob, and Carl share a job. Adam works hours, Bob works 9 hours, and Carl works hours. What is the ratio of the hours worked in simplest form? ACHS ENTERING ALGEBRA SUMMER 07 PACKET Page 9 of 8

10 RATIO PROPORTION PERCENT PROPORTION A proportion is an equation showing two ratios are equal. Etreme is another way of saying at the ends. RATIO PROPORTION PERCENT Proportion Practice: Find the proportion for each of the questions.. 8 Mean is another way of saying in the middle. etreme a b c d mean. 6 mean etreme Note: This is not the same mean as used in taking an average. Cross product: The product of the etremes is equal to the product of the means. Eample: Check to see if the proportion is a real proportion. Cross multiply to find the products. 6 () (6) 8 8 Since both sides are equal, it is a proportion. A proportion can be used to find missing information. Set up the proportion using a variable for the missing information. Make sure the set up is correct. If the proportion is set up incorrectly, the result will be incorrect. Cross multiply. Eamples: On a map, inch represents miles. Find the distance two cities are from each other if they are inches apart on the map. inches miles () () 0 miles. A certain factory produces 80 good motors for every defective motors. At this defect rate, how many defective motors will result with a production of 80 good motors?. The scale on a map of Europe states: cm 6 miles. If Rome is. cm from Paris on this map, find the distance in miles between these two cities.. A community college has a student to faculty ratio of to. If this college has faculty members, how many students are enrolled there? ACHS ENTERING ALGEBRA SUMMER 07 PACKET Page 0 of 8

11 RATIO PROPORTION PERCENT PERCENT A percent is out of 00. It is part of 00. The symbol used is %. Percent is a ratio or fraction where the denominator is 00. General percent equation: y is % of z, where: y the part of the whole the percent z the base. This can also be set up as a proportion. percent whole Eample (find the percent): What percent of 0 is 7? 7 is % of (7) 00 y z Using proportion: 00 part base 70% 7 0 Eample (find the base): 7 is 70% of what number? 7 is 70% of z 70z 00(7) 70z 70 Using proportion: 70z 00(7) 70z z z 0 z 0 RATIO PROPORTION PERCENT Percent Practice: Find the missing information using either method. Percent Practice: Find the missing information using either method.. What percent of 9 is 86?. 8 is what percent of 80? 0 00(7) 0 0 Eample (find the part): %. What is 6% of 7? What is 70% of 0? y is 70% of 0 0(70) 00y y 00 y 7. Find.% of 8. Using proportion: 0(70) 00y y y 00 y 7. 8 is % of what number? 6. 9 is 06% of what number? ACHS ENTERING ALGEBRA SUMMER 07 PACKET Page of 8

12 PERCENT DECIMAL FRACTION CONVERSIONS FRACTION TO Convert from fraction to decimal: Divide the denominator into the numerator. Eample: Convert Convert from fraction to percent: to a decimal. Divide the denominator into the numerator. Then multiply by 00. Eample: Convert 8 to a percent. ( 8)00 0.(00).% DECIMAL TO Convert from decimal to fraction: Determine the place value of the last number on the right that is not a zero. Remove the decimal point. Use the numbers from the decimal as the numerator. Use the place value found as the denominator. Simplify if possible. Eample: Convert 0. to a fraction. The place value of the is the thousandths place. 00 Convert from decimal to percent: Move the decimal place units to the right and put a percent sign after it. Eample: Convert 0. to a percent. 0..% PERCENT TO Convert from percent to decimal: Remove the percent sign and move the decimal place units to the left. Eample: Convert.% to a decimal..% 0. Convert from percent to a fraction: Percent is over 00. Remove the percent sign and place over 00. Do not remove the decimal yet. Look at how many places there are after the decimal. Remove the decimal and put add a zero to the denominator for each decimal place. Simplify if possible. Eample: Convert.% to a fraction RATIO PROPORTION PERCENT Conversion Practice: Fill in the table. Fraction Decimal Percent % 80% ACHS ENTERING ALGEBRA SUMMER 07 PACKET Page of 8

13 ORDER OF OPERATIONS When solving math problems with more than one operation, you must follow the correct order of operations. Go through the operations in the following order. Parenthesis (in order from inner to outer) Eponents (including radicals) Multiplication/Division (in order from left to right) Addition/Subtraction (in order from left to right) ORDER OF OPERATIONS Practice: Use order of operations to simplify the following problems. Show each step.. (7 ) 8 + Ignoring these rules will give different answers. Notice the difference the parenthesis makes in the two eamples below (7 + ) 8 0 Notice the difference the order of operations makes in the two eamples below Eample: Simplify ( + ) Step : Parenthesis + 7 (7) Step : Eponents (radicals) 7 9 Step : Multiplication/Division Step 6: Addition 9 (9) 7 + (9) NOTE: When you have a fraction, simplify the numerator and denominator first, then simplify the fraction. ACHS ENTERING ALGEBRA SUMMER 07 PACKET Page of 8

14 LAWS OF EXPONENTS PRODUCT RULE: a m a n a m + n (when multiplying like bases, add the powers) Eamples: a 7 a a a a 0. ( 6 )( ) ()() NEGATIVE POWER RULE: a n n a and n a a n (take the reciprocal of the variable to the negative power) NOTE: Apply the negative power rule to only negative POWERS. Eamples:. POWER RULE: (a m b m ) n a mn b mn (when taking a monomial to a power, multiply the powers including the coefficient). m 8 0 y n ny m 8 0 Eamples:. (a b ) a 8 b 6. (m n ) m 8 n 0 8m 8 n (6a b ) ( c d ) 8 6 a b 0 0 ( ) c d 8 6a b 0 0 c d LAWS OF EXPONENTS Practice: Simplify each epression using the laws of eponents. Write the answers as positive eponents.. ( 8 ) QUOTIENT RULE: a m a n a m - n (when dividing with like bases, subtract the powers) Note: it is always the numerator's power minus the denominator's power. Eamples: a 7 (a 8 )(a). (m n 6 )(m n). 7 mn mn m n 7 mn. 6 8a b c 7a b c ZERO POWER RULE: a 0 (any term to the zero power is one) Eamples:. (m n 7 ) 0. ( y 6 ) 0. (m 8 n )( mn ) 0 (m 8 n )() m 8 n ACHS ENTERING ALGEBRA SUMMER 07 PACKET Page of 8

15 LITERAL EQUATIONS Formulas are equations that state a fact or rule relating two or more variables. You can solve a formula for any of its variables using the rules for solving equations. Some common formulas: Temperature F 9 C + LITERAL EQUATIONS Practice: Solve each equation for the variable required. Show all work.. Simple interest formula: I Evaluate for p. 0 pt p Distance d rt Circumference of a circle C d Area of a triangle A bh Density d m v At times it is necessary to move the variables around in the formula without solving the formula. Use the rules of algebra to do this. Eample: Formula: A bh Solve for b. Perimeter of a rectangle formula: P L + W Evaluate for L. L Multiply both sides by A bh Divide both sides by h A h b Note: When solving literal equations, the result will be an algebraic epression, not a number. Eample: Formula: a + b c Solve for a. Volume of a cylinder: V r h Evaluate for r. Subtract b from both sides a c b Take the square root of both sides a c b ACHS ENTERING ALGEBRA SUMMER 07 PACKET Page of 8

16 SLOPE OF A LINE The slope of a line measures the steepness of the line. This is usually referred to as rise over run. Run means how far left or right you move from point to point. On the graph, that would mean a change of values. SLOPE OF A LINE Practice: Find the slope of the line through the given points.. ( 7, 0 ) and (, 0) m Rise means how many units you move up or down from point to point. On the graph that would be a change in the y values. The direction of the line will refer to the type of slope it has. Look at the diagram below.. (, 7) and (, 8) m Positive Slope Negative Slope. (00, 0) and (, 0) m Zero Slope Numerator 0 Undefined Slope Denominator 0 Slope Formula Given Two Points Given two points (, y) and (, y). (, 6) and (7, 7) m m rise run change in y changein y y Eample: Find the slope of the line that passes through the points (, ) and (, 7). m 7 ( ) ACHS ENTERING ALGEBRA SUMMER 07 PACKET Page 6 of 8

17 WRITE THE EQUATION OF A LINE IN SLOPE-INTERCEPT FORM GIVEN TWO POINTS The slope-intercept form of the equation of a line is y m + b, where (, y) is any point on the line, m is the slope, and b is the y-intercept. Eample: Write the equation of the line that passes through the points (, ) and (, 6). WRITE THE EQUATION OF A LINE IN SLOPE-INTERCEPT FORM GIVEN TWO POINTS Practice: Write the equation of the line that passes through the given points. Show all work.. (, 7) and (0, ) Step : Find the slope of the line. m 6 ( ) Step : Plug the slope and either on of the points into the formula. y m + b. (, 0) and (0, ) ( ) + b Step : Solve for b. ( ) + b 6 + b b. (, 6) and (, ) Step : Write the equation of the line by plugging in the values of m and b into the slope-intercept form. y m + b y. (, ) and (, 8) ACHS ENTERING ALGEBRA SUMMER 07 PACKET Page 7 of 8

18 GRAPH THE EQUATION OF A LINE The equation of a line can be graphed in several ways. One method is to use the slope and y-intercept of the line to draw the graph. GRAPH THE EQUATION OF A LINE Practice: The equation of a line can be graphed in several ways. One method is to use the Eample: Graph the equation y. Step : Find the values of m and b. m and b Step : Plot b on the graph. This is the y- intercept which means (0, ). Step : Use the slope to find a second point. m This means move up units and left unit from the slope intercept.. Graph y Step : Draw a line using the two points.. Graph y + Left Up ACHS ENTERING ALGEBRA SUMMER 07 PACKET Page 8 of 8

Connecticut Alternate Assessment: Individual Student Report Performance Literals Mathematics

Connecticut Alternate Assessment: Individual Student Report Performance Literals Mathematics Published November 9, 2016 Copyright 2016 by the Connecticut State Board of Education in the name of the Secretary