SUMMER REVIEW PACKET FOR STUDENTS ENTERING ALGEBRA Dear Students, Welcome to Ma ayanot. We are very happy that you will be with us in the Fall. The Math department is looking forward to working with you and watching you learn. Your teachers are very supportive and want you to understand and enjoy learning math. This packet is designed to help you to review the skills that are essential as a foundation for your algebra class. There are model problems which will help to guide you. INSTRUCTIONS: ) PRINT OUT PACKET ) SHOW YOUR WORK IN THE SPACE PROVIDED IN THE PACKET. ) NO CALCULATOR 4) FOR THE FRACTION PROBLEMS STARTING (p.-6) do the ODD problems only. ) HAND YOUR PACKET TO YOUR TEACHER ON THE FIRST DAY OF SCHOOL! GOOD LUCK AND HAVE A GREAT SUMMER! THE MATH DEPARTMENT Page of 9
Objective: Write an algebraic expression to represent unknown quantities. A variable is a symbol, usually a letter, used to represent a number. Algebraic expressions are combinations of variables, numbers, and at least one operation. Sum: addition Ex: x+ is the sum of a number and or added to a number Difference : subtraction Ex: x- is the difference between a number and or less than a number but x is the difference between and a number or decreased by a number Product: multiplication Quotient: Division Ex: 0x is the product of 0 and a number Ex: 0 x is the quotient of 0 and a number or 0 divided by a number But 0 x is quotient of a number and 0 or a number divided by 0 In each case, let x= the number. Be careful with difference and quotient!.) a number plus.) the sum of a number and 7.) the difference of and a number 4.) the difference of a number and ) a) Five less than a number 6.) Five decreased by a number 7) The quotient of 0 and a number ) Twenty divided by a number 9) The quotient of a number and 0 0) A number divided by 0 0. John is years younger than Bill; If john s age is x, what is Bill s age in terms of x 0. Bill is years younger than John. If John s age is x, what is bill s age in terms of x? Page of 9
Objective: Evaluate an algebraic expression. A variable is a symbol, usually a letter, used to represent a number. Algebraic expressions are combinations of variables, numbers, and at least one operation. Multiplication in algebra can be shown as 4n or 4 x n The variables in an algebraic expression can be replaced with any number. Once the variables have been replaced, you can evaluate, or find the value of, the algebraic expression. Examples: Evaluate 44 + n if n= 9 44 + n original expression 44 + 9 replace the variable with it s value solution.) Evaluate 0 + n if n =.) Evaluate n if n = 9.) Evaluate n + 9 if n = 4.) Evaluate 0n if n =..)Evaluate 4n k if n = 6 and k = 6.)Evaluate nk b + if b =., k =, and n = 7 Page of 9
Unit: KNOWLEDGE of ALGEBRA, PATTERNS, and FUNCTIONS Objective: Evaluate numeric expressions using order of operations. A numerical expression is a combination of numbers and operations. The Order of Operations tells you which operation to perform first so that everyone gets the same final answer. The Order of Operations is: Parentheses, Exponents, Multiplication or Division (left to right), and Addition or Subtraction (left to right.) Examples: 4 ( + ) original expression 4 6 - P simplify the expression inside the parentheses 4 6 4 E exponents 4 M,D multiplication and division from left to right 4 A, S addition and subtraction from left to right Show your steps. No calculator!.) ( + ) x.) x 4 7.) 6 x + 4.) 00 x 4.) 4 9 + x 6.) ( + ) x ( + 9) 0 Page 4 of 9
Unit: solve step equation with addition and subtraction Objective: Determine the unknown in a linear equation (addition & subtraction). Addition equations: Subtract the same number from each side of the equation so that the two sides remain equal. Subtraction equations: Add the same number to each side of the equation so that the two sides remain equal. Examples: b + = 6 original equation b = 4 original equation - - subtract from each side + + add 4 to each side b + 0 = solution b + 0 = solution b = simplify b = simplify.).) g + = s = 9.) 4.) m +. = 0. k. =..) 6.) w + 6. = g.7 = 49.7 Page of 9
Unit: Solve step equation with multiplication and division Objective: Determine the unknown in a linear equation (multiplication & division). In a multiplication equation, the number by which a variable is multiplied is called the coefficient. In the multiplication equation x =, the coefficient is. Multiplication equations: Divide both sides by the coefficient so that the two sides remain equal. In a division equation, the number by which the variable is divided is called the divisor. In the division equation 4 x, 4 is the divisor. Division equations: Multiply both sides of the equation by the divisor so that the two sides remain equal. Examples: 4b = 6 original equation m = 6 original equation 4 4 divide both sides by 4 6 x 6 m = x 6 multiply each side by 6 b = 4 solution m = 66 solution b = 4 simplify m = 66 simplify.) 7x = 6.) k = 9.) 4.) b =. n =. 7.) 6.) m = 4.7 p =.67 Page 6 of 9
Unit: NUMBER RELATIONSHIPS and COMPUTATION Objective: Read, write, and represent integers. Examples: Integer: Any number from the set { -,-,-,0,,, } Integers less than zero are negative integers Integers greater than zero are positive integers -6 - -4 - - - 0 4 6 Negative integers are written with a - sign Zero is neither nor positive Positive integers can be written with or without a + sign Write an integer to describe each situation EX: EX: a height increase of inches The word increase represents positive. Answer: or +. 0 feet below sea level The word below represents negative. Answer: - 0.) Write an integer to describe: The stock market increased 7 points.) Write an integer to describe: A loss of yards.) Write an integer to describe the situation: Nancy owes her friend $0 4.) Write an integer to describe: Frederick is located 90 feet above sea level..) Write an integer to describe: The temperature was below zero 6.) Write an integer to describe: The 6 th grade has fewer students than last year Page 7 of 9
FRACTIONS- Methods A. Changing a Mixed Number to an Improper Fraction Mixed number 4 4 Improper fraction - (contains a whole number and a fraction) (numerator is larger than denominator) Step Multiply the denominator and the whole number Step Add this answer to the numerator; this becomes the new numerator Step Carry the original denominator over Example #: = + = Example #: 4 9 4 = 4 9 + 4 = 40 40 9 B. Changing an Improper Fraction to a Mixed Number Step Divide the numerator by the denominator Step The answer from step becomes the whole number Step The remainder becomes the new numerator Step 4 The original denominator carries over Example 47 = 47 or 47 = 9, remainder Ans: 9 Page of 9
C. Reducing Fractions Step Find a number that will divide into both the numerator and the denominator Step Divide numerator and denominator by this number Example #: 0 = (because both 0 and are divisible by ) Example #: 4 = (because both 4 and are divisible by 4) D. Raising Fractions to Higher Terms When a New Denominator is Known Step Divide the new denominator by the old denominator Step Multiply the numerator by the answer from step to find the new numerator *Note: If the original number is a mixed number, convert it to an improper fraction before raising to higher terms (see Example #) Example #: = becomes = because = 4 and 4 = Example #: = 0 becomes = becomes 0 44 = 0 because 0 = 4 and 4 = 44 E. () Multiplying Simple Fractions Step Multiply the numerators Step Multiply the denominators Step Reduce the answer to lowest terms 4 4 Example: = 7 6 4 which reduces to Page 9 of 9
E. () Multiplying Mixed Numbers Step Convert the mixed numbers to improper fractions first Step Multiply the numerators Step Multiply the denominators Step 4 Reduce the answer to lowest terms 7 Example: = = 6 which then reduces to F () Dividing Simple Fractions Step Change division sign to multiplication Step Change the fraction following the multiplication sign to its reciprocal (flip the fraction around so the old denominator is the new numerator and the old numerator is the new denominator) Step - Multiply the numerators Step 4 Multiply the denominators Step Change the answer to lowest terms Example: = becomes which when solved is 6 F(). Dividing Mixed Numbers Step Convert the mixed number or numbers to improper fraction Step Change the division sign to multiplication Step Change the fraction following the multiplication sign to its reciprocal (flip the fraction around so the old denominator is the new numerator and the old numerator is the new denominator) Step 4 - Multiply the numerators Step Multiply the denominators Step 6 Change the answer to lowest terms 7 6 Example: = becomes becomes = 4 6 4 6 4 7 which when solved is 6 4 = which simplifies to 4 7 4 4 Page 0 of 9
G. Adding and Subtracting Fractions Step Find a common denominator (a number that both denominators will go into) Step Raise each fraction to higher terms as needed Step Add or subtract the numerators only as shown Step 4 Carry denominator over Step Change the answer to lowest terms Example #: 7 + = = 7 + = Common denominator is because both and will go into 4 7 which simplifies to Example #: 4 4 = Common denominator is 0 because both 4 and will go into 0 4 = 4 0 4 = 0 7 4 0 Example #: = = + = 9 4 = = = 7 ** Page of 9
INSTRUCTIONS: YOU SHOULD DO THE ODD PROBLEMS ONLY IN THIS SECTION ( See previous pages for instructions on how to do each type of problem) A. Write as an improper fraction... 4. 4. 6. 7 6. 6 7.. 4 9. 7 4 0.. 6. 6 B. Write as a mixed number.. 0 4. 9. 4. 9. 6 6. 4 7. 7. Page of 9
C. Write in lowest terms.. 6.. 4 4.. 0 6. 9 7 7. 4 49. 9.. 6 0. 0. 4 0. Page of 9
D. Find the missing numerator by raising the fraction to higher terms..? = 4. 7? = 6 64.? = 4 4.? = 9 7. =? 6. 4 = 0? 7. 4 =?. = 0? Page 4 of 9
E. Multiply.. 7 =. =. = 4. = 9 0 7 6. 7 4 = 6. = 7. =. = 4 6 64 9 9 9. 0 = 0. =. =. 4 = 4 6 6. = 4. 6 =. = 6. 6 = Page of 9
7. 6 =. 4 = 9. 4 9 4 4 4 = 0. = F. Divide as shown.. =. =. = 4. = 4 9 4 6 4. 4 = 6. = 7. 9 =. = 4 9. 4 7 = 0. =. 4 =. 4 = 7 9 Page 6 of 9
. 6 = 4. 4 =. 4 = 6. = 7. 6 4. = 9. 4 = 0. 7 = G. Add or subtract as shown.. 7 + =. + =. + = 4. + = 4 6. + = 6. + = 7. + =. + = 0 4 4 4 Page 7 of 9
4 9 7 9. + = 0. + =. =. = 6 9 0 6. 7 = 4. =. = 6. = 6 4 6 0 7. 6 9 =. 7 = 9. 4 6 = 0. 4 6 = Page of 9
Converting between decimals and fractions and percents: please read this carefully There are many different methods to convert between decimals, fractions and percents, so we must understand the meaning. Percent (%) means per hundred so % = /00 which means as a fraction it is 00 and as a decimal it is 00. which reduces to. A) To multiply by 00, move decimal point places to the right To divide by 00, we can move the decimal point over places to the left So. x 00 =.7 x 00 = 7.. x 00 = / 00 =. 7./ 00 =.7 /00 =. B) To multiply by, decimal point right digit To divide by 0, move left by digit. x 0 =. x 0 =. 7. x 0 = 7 /0 =../0=. 7/0=.7 See next page! Page 9 of 9
. Conversion between fractions and decimals A. From fraction to decimal : divide numerator by denominator Ex:.6 B. From decimal to fraction: write as a fraction using the place value of the last digit, then simplify if possible Ex: 4.4 00 4.4 0 79.79 000.0 00 4. Conversion between decimals and percents A. From percent to decimal Since percent means per hundred, it means divided by 00 So %. 00 (remember there s an easy way to divide by 00)..%.0 00 B. From decimal to percent: You can think of this as the reverse of the above, multiply by 00 So.. %..% There's more on next page- read it all carefully! Page 0 of 9
) Conversion between fraction and percents A. From percent to fraction, since percent means per hundred, write the number over 00 and simplify the fraction: % 00 % 00 0 4. 4.% but we don t want a decimal in the fraction so change to 00 4. 0 4 4.% which can be simplified to 7 00 0 000 0 B) From fraction to percent, we can either set up a proportion x 00 and cross multiply 00 x, and x=40, so it s 40% Or we can convert to a decimal first and change to a fraction.4 and multiply by 00 to get 40% Page of 9
Do the following and show work: Please do all problems on this page Write these decimals as fractions: 0. =. 0. =. 0.6 =. 0.0 =. 0.0 =. 0. =. 0.6 =. 0. =. Write these fractions as decimals: 0 7 =. =. =. 4 = 7 =. =. 0 9 =. 7 = Write these percentages as decimals: % =. 0% =. % =. 0% =. % =. % =. 67% =. 7.% =. Page of 9
4 Write these percentages as fractions: 0% =. 7% =. % =. 0% =. 40% =. % =. 4% =. % =. Write these decimals as percentages: 0. =. 0. =. 0.7 =. 0.07 =. 0.4 =. 0.09 =. 0.4 =. 0.7 =. 6 Write these fractions as percentages: 0 =. =. 0 9 =. 4 =. 4 =. 7 0 =. =. =. Page of 9
Complete this table. Fraction Decimal Percentage 0 0 0 7 4 0 9 4 4 Page 4 of 9
Fill the gaps in the table. YOU ARE NOT REQUIRED TO COMPLETE THIS PAGE, BUT PLEASE DO IF YOU HAVE TIME. Percentage Fraction Decimal 0% 0. 0 40% 0. 70% 0. 0 9 % Page of 9
Unit: NUMBER RELATIONSHIPS and COMPUTATION Objective: Identify and determine equivalent forms of proper fractions as decimals, percents, and ratios. Key Concept: Ratio: a comparison of two numbers A ratio can be written in ways: a:b a to b or a b EX: Write the ratio as a fraction simplest form: 4 wins to 6 losses 4 Since the ratio can be written as: we can the simplify to or : or to 6.) Write the ratio as a fraction simplest form: boys to girls.) Write the ratio as a fraction simplest form: 0 books to 4 magazines.) Write the ratio as a fraction simplest form: 0 circles to triangles 4.) Write the ratio as a fraction simplest form: cups to servings.) Write the ratio as a fraction simplest form: 0 cars to 00 trucks 6.) Write the ratio as a fraction simplest form: 9 pencils to pens Page 6 of 9
Objective: Compare and order fractions and decimals A. Ordering fractions only: ) rewrite each fraction as an equivalent fraction using the LCD ) Compare the numerators EX: order the fractions 7 ; ; from least to greatest ) LCD of,, and is 4 ) 4 9 4 7 4 4 ) Comparing the numerators: 7 B. Ordering fractions and decimals: ) Change the fractions to decimals ) Compare the decimals EX: order the numbers 0.; ; and 0. from least to greatest 0.7.000 ) 0. 7 4 60 ) Compare the decimals: 6 0. < 0.7 < 0. 40 Therefore: 0. 0. - 40.) SHOW WORK Order the fractions ; ; 6 4 from least to greatest.) Order the numbers 0.7; ; and 0. from least to greatest 4.) Order the fractions ; 7 0 ; 6 from least to greatest 4.) Order the numbers ; 0 ; and 0. from least to greatest.) Order the fractions ; ; 9 6 from least to greatest 6.) Which number has the greatest value? 0.94; 9 ; or 4 0 Page 7 of 9
Unit: NUMBER RELATIONSHIPS and COMPUTATION Objective: Determine 0, 0,, or 0 percent of a whole number. Example: Determine % of 40 Method : Write a proportion and solve x 00 40 x 4 40 0 x 4 0 40 0 x 40 40 so x 0 Method : Change the percent to a decimal and multiply %= 0. 0. X 40 = 0.00 Therefore % of 40 is 0. 40 X 0. 00 +00 0.00 Therefore % of 40 is 0..) Determine 0% of 6. Show using both methods Method :.) Determine 0% of 0. Show using both methods Method : Method : Method :.) Determine % of 0. 4.) Determine 0% of. Page of 9
.) 0% of the 0 students ate pizza for lunch. How 6.) Nadia saved 0% on her CD purchase. If the CD many students ate pizza? originally cost $4.90, how much did she save? 4 7) Juan answered questions correctly on his quiz. What percent of the questions did he get correct? ) 7% of the class completed their homework last night. What fraction of the class completed their homework? YOU'RE DONE! HAND IT IN TO YOUR TEACHER ON THE FIRST DAY OF CLASS Page 9 of 9