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Number Theory Important Concepts factor pair • • a pair of numbers whose product equals a given number dimensions of a rectangle factors A number that “fits evenly” into a given number. Examples The factor pairs of 18 are… 1 x 18 2x9 3x6 All the factors of 18 are 1, 2, 3, 6, 9, and 18 multiple • • what you say when you skip count by a given number the product of a given whole number and another whole number Some multiples of 4 are 4, 8, 12, 16, . . . . Any number has an INFINITE number of multiples. prime number • • a number with exactly one Examples of prime numbers are 2, 3, 5, 7, and 11. factor pair 1 is not a prime number, because it only has 1 factor, itself. has two different factors, 1 and the number itself. composite number A number that has three or more factors and two or more factor pairs Examples of composite numbers are 4, 6, 8, and 9. square number A number you get when you multiply two of the same numbers 16 = 4 x 4…..16 is the square number 36 Prime factorization Process of writing a number a s a product of prime numbers 2 x 18 9 x 2 3 x 3 Therefore, the prime factorization of 36 is….. 2 x 2 x 3 x 3 List all the factor pairs for 32._____________________________ List all the factors for 32. _________________________________ List the first four multiples for 32. ____, ____, ____, ____ Label each number prime or composite: 27_________ 19_________ 2_________ 36_________ Circle the square numbers… 30, 25, 100, 40, 64, 21 Write the prime factorization for … 45 84 PreAlgebra Concepts Important Concepts exponent In a power, the number of times a base number is used as a factor order of operations The rules which tell which operation to perform first when more than one operation is used. PEMDAS Parenthesis, Exponents, Multiplication, Division, Addition, Subtraction • • Examples In 53, the exponent is 3. So, 53 is equal to 5 x 5 x 5 or 125. Find the value of the expressions. 3+7x6÷3–4 = 3 + 42 ÷ 3 – 4 = 3 + 17 - 4 = Multiply OR divide in order from left to right. Add OR subtract in order from left to right. evaluating algebraic expressions Substitute values for the variables and evaluate using the order of operations above. 14 - 4 = = 13 Find the value of the expression if x = 10, n = 5. 15 + n = or 18x = 15 + 5 18 ( 10 ) = 20 = 180 Evaluate each algebraic expression for x= 1 5 and n = 2. Write your work on the lines provided. 1. 8.6 + n 2. 3 + 5x _________________ _________________ _________________ _________________ 3. 4x - 2n 4. x n _________________ _________________ _________________ _________________ Fractions, Decimals, and Percents Important Concepts fraction Examples Describes one or more parts of a whole that is divided into equal parts. 1 means 1 part out of a total of two equal parts 2 A number that is less than 1 but greater than zero. 1 5 = = 0.5 2 10 decimal percent 1 50 = = 50% 2 100 Percent means “out of 100”. improper fraction An improper fraction has a numerator that is greater than, or equal to, one. 8 7 1 means (1 whole) and more 7 7 7 mixed number 8 1 =1 7 7 A number that is both a whole number and a fraction. equivalent fractions • • fractions that are equal in value but have different numerators and denominators fractions that have the same amount 1 2 3 4 = = = =... 2 4 6 8 Complete the missing values based on the given fraction, decimal, or percent. fraction 1. decimal percent 1 83 % 3 2. 4 5 3. 5. 4. .13 7. 6. 8. 3 4 9. 20% 10. Operations with Fractions and Mixed Numbers Addition Algorithm 1. 2. 3. 4. 5. Find a common denominator. Write equivalent fractions. Add numerators and keep denominator. Add whole numbers, if necessary. Simplify your answer. Subtraction Algorithm 1. 2. 3. 4. Find a common denominator. Write equivalent fractions. Borrow if necessary. Subtract numerators and keep denominator. 5. Subtract whole numbers, if necessary. 6. Simplify your answer. Solve each problem. Use separate paper if necessary. NO CALCULATORS! 1. 4. 2 5 2 + 1 6 5 10 2 1 - 1 8 5 2. 4- 5. 6 5 8 3 4 + 2 8 7 Multiplication Algorithm 1. Change all whole or mixed numbers to improper fractions. 2. Multiply numerators. 3. Multiply denominators. 4. Simplify your answer. Look to simplify before you multiply by canceling a numerator with a denominator that has a common factor. 3. 6. 3 2 9 3 - 1 2 9 2 3 + 1 2 3 Division Algorithm 1. Change all whole or mixed numbers to improper fractions. 2. Keep the dividend and change the divisor to its reciprocal. 3. Then follow the rules for multiplication. OR 1. Change all whole or mixed numbers to improper fractions. 2. Find a common denominator for both fractions. 3. Write equivalent fractions. 4. Divide the numerators only. Simplify your answer. 1. 2 2 5 x 1 5 6 2. 5 ÷ 3 7 3. 7 9 x 1 1 2 3. 3 5. 2 1 ÷ 1 6 5 6. 9 13 ÷ 14 24 Decimals 2 9 x 1 2 3 Place Value: The position of a digit in a number that is used to determine the value of the digit. Addition Algorithm Subtraction Algorithm 1. Line up equal place values so you are adding equal sized pieces. 2. Put in zeros as place holders, if necessary. 3. Add beginning with the smallest place value. 4. Bring down the decimal point into the sum. Multiplication Algorithm 1. Multiply as you would with whole numbers. 2. Count the number of decimal places. 3. The total number of decimal places is where you put the decimal in the product example: 2 1 3.42 x 5 17.10 (The total number of decimal places was 2) 1. Line up equal place values so you are adding equal sized pieces. 2. Put in zeros as place holders, if necessary. 3. Borrow and rename when necessary. 4. Subtract beginning with the smallest place value. 5. Bring down the decimal point into the difference. Division Algorithm 1. If the divisor is a whole number, bring the decimal straight up into the quotient. Follow your division algorithm for whole numbers, adding zeros to the dividend as necessary. 2. If the divisor is a decimal number, multiply divisor and dividend by a power of ten that will make the divisor a whole number. Then follow your division algorithm for whole numbers, adding zeros to the dividend as necessary. Solve each problem. Use separate paper if necessary. NO CALCULATORS! 1. 123.5 x 0.25 2. 0.3 + 8.9 4. 264.051 – 2.3 5. 4.002 + 22 + 0.75 3. 5 – 0.671 6. 9.36 ÷ 12 Geometry Important Concepts/Definitions Polygon: a two-dimensional geometric figure formed of three or more straight sides Triangle: a polygon with three sides Equilateral triangles : all sides are the same length Isosceles triangles: at least two sides are the same length Scalene triangles: no sides are the same length Equilateral triangles : all sides are the same length Isosceles triangles: at least two sides are the same length Scalene triangles: no sides are the same length Right triangles: one angle measures 90° Acute triangles: each angle measures less than 90° Obtuse triangles: one angle measures more than 90° but less than 180° Quadrilateral: a polygon with four sides Parallelogram: both pairs of opposite sides parallel and equal in length Trapezoid: only one pair of parallel sides Rectangle: a parallelogram with four right angles Rhombus: a parallelogram with all sides the same length Square: a rectangle with all sides the same length Classify each polygon. Be as specific as possible. 1. 2. 3. _____________________ _____________________ _____________________ 4. 5. 6. _____________________ _____________________ _____________________ Area, Perimeter, and Volume Important Concepts/Definitions Perimeter: the distance around the outside of any polygon Formula: add the lengths of all the sides = units Area: the amount of surface a figure covers Formula: length x width = square units Volume : the number of cubic units needed to fill a solid figure Formula: length x width x height = cubic units Solve each problem. Use separate paper if necessary. NO CALCULATORS! 1. 2. 4.32 mm 3. 7 ft 12 3 2 in 7 5 ft 8 7.71 mm ` 5 4 in 6 Area_________________ Area _________________ Area_________________ Perimeter_____________ Perimeter _____________ Perimeter_____________ 5. 6. 4. 1 1 ft 8 1 2 ft 4 3 2 ft 8 Volume________________ Length: 7.31 cm 3 4 ft Length: 5 Width: 3.2 cm 3 Width: 8 ft Height: 4.28 cm 5 Height: 2 ft 7 Volume________________ Volume______________ Problem Solving • • • • • • Tips for Problem Solving Read the question first so you know what you need to find. Underline the important facts as you read the entire problem. Look for key words as you read. Plan how to solve the problem. Find the solution and check your solution for reasonableness/accuracy. Remember to label your solution. Answers should include… • Words that explain your strategy • Work that shows your calculations and steps • Picture(s) that visually show your understanding • Answer(s) • Label(s) Solve each problem. 1. During the month of April, rain gauges were set up in various locations to measure the amount of rainfall in inches. The line plot below shows the findings. April Rain X X X X X X X X X X X X X X X X X X X X X X X X 0 Inches • What was the total amount of rain collected by all of the rain gauges? • If all of the rain collected was poured equally among each of the rain gauges, how many inches of rain would be in each gauge? 2. Consider the following pattern: 1, 5, 25, 125. If this pattern of numbers continues, what are the next two numbers in the pattern? Explain. 3. Jim bought 5 pieces of wood. They measured 13.25 inches, 13.3 inches, 13.008 inches, 12.999 inches and 13.03 inches in length. List the pieces of wood in order from shortest to longest. 4. Joe wants to pack a box with books. He needs a box that has a volume of exactly 1,344 cubic inches. If the length of the box is 14 inches and the height of the box is 8 inches, what must be the measure, in inches, of its width? Find the ordered pair for each point. 6. H ( , ) 7. J ( , ) 8. L ( , ) 9. K ( , ) Measurement Conversion 52 weeks = 1 year 365 days = 1 year 8 fluid ounces = 1 cup 2 cups = 1 pint 2 pints = 1 quart 4 quarts = 1 gallon 1000 milliliters (mL) = 1 liter (L) 12 inches = 1 foot 10 millimeters = 1 centimeter 3 feet = 1 yard 100 centimeters = 1 meter 5,280 feet = 1 mile 1,000 meters = 1 kilometer 16 ounces = 1 pound 2,000 pounds = ton 1,000 milligrams = 1 gram 1,000 grams = 1 kilogram Solve each problem. Use separate paper if necessary. 1. A gallon contains 128 ounces. Paul wants to divide 3 gallons of apple cider equally among the 2 dozen friends at his party. How many ounces of apple cider will each friend receive? 2. If you only needed 1 cup of milk, what is your best choice at the grocery store – a quart container, a pint container, or a ½ gallon container? Explain.