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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. 1. Simplify expressions inside parentheses. 2. Find the value of all numbers with exponents. 3. Multiply OR divide in order from left to right. 4. Add OR subtract in order from left to right. 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 x 62 + 4 = 3 x 36 + 4 = 108 + 4 = 112 60 ÷ (12 + 23) ∙ 9 = = = = 60 ÷ (12 + 8) ∙ 9 60 ÷ 20 ∙ 9 3∙ 9 27 evaluating algebraic expressions Substitute values for the variables Find the value of the expression if x = 3, y = 5, and z = 20. and evaluate using the order of operations above. 2x + z ÷ y = 2 ∙ 3 + 20 ÷ 5 = 6 + 20 ÷ 5 = 6 + 4 = 10 Evaluate each algebraic expression for x= 12, y = 8, and z = 2. Write your work on the lines provided. 1. 40 – y2 ÷ z 2. ( x + y ) ∙ 32 ______________________________ ______________________________ ______________________________ ______________________________ ______________________________ ______________________________ 3. ( y + 1 )2 – x 4. 5x – z3 ______________________________ ______________________________ ______________________________ ______________________________ ______________________________ ______________________________ Data Analysis Important Concepts mean The sum of the values divided by the number of values in the set. median The number that marks the middle of an ordered set of data. mode The mode is the value that occurs with greatest frequency in a set of data. range The difference between the greatest and least values in a data set. Examples {1, 2, 3, 4} 1 + 2 + 3 + 6 = 12 and 12 ÷ 4 = 3 so the mean is 3 {4, 5, 6, 7, 8} The median is 6 because it marks the middle value when the numbers are ordered from least to greatest. {1, 2, 2, 2, 3, 4} The mode is 2 because it is the value that occurs most often. {1, 2, 3, 4, 5} The range is 4 because 5 – 1 = 4. Use the data table to answer each question. Calories of Selected Vegetables (per serving) 15 35 50 31 5 25 85 25 20 1. What is the mean of the calories? Round to the nearest tenth, if necessary. 2. Order the set of data from least to greatest. What is the median of the calories? 3. What is the mode of the calories? How do you know? 4. What is the range of the calories? Basic Integer Concepts Important Concepts integer The set of numbers that includes all the whole numbers and their opposites, including 0. negative integers All integers less than zero and found to the left of zero on the number line. positive integers All integers greater than zero and found to the right of zero on the number line. opposites Two numbers the same distance from zero but in opposite directions on the number line. *The sum of two opposites is 0. absolute value The distance a number is from 0 on the number line. Absolute value is never negative. Examples {. . .-4, -3, -2, -1, 0, 1, 2, 3, 4. . .} -8 -7 -6 -5 -4 -3 -2 -1 0 0 1 2 3 4 5 6 7 8 -4 and 4 -9 and 9 |-9| = 9 -36 and 36 |8| = 8 |-13| = 13 -17 and 17 |100| = 100 Identify numbers that meet the given requirements. 1. 2 positive numbers that are not integers ___________ __________ 2. 2 negative numbers that are not integers ___________ __________ Graph the given numbers on the number line. 3. 6, 0, -5, 1, -3 Insert <, >, or = to make a true statement. 4. -9 -8 5. -3 -6 6. -2 -4 Name the opposite of each integer given. 7. -12 __________ 8. 9 __________ __________ 10. 8 + 7 x 6 14 __________ Identify the absolute value. 9. -10 Ratio and Proportion Important Concepts Examples ratio A comparison of two numbers by division. Does NOT have to be a PART to WHOLE comparison. Can be written in 3 forms a , a to b, a:b b 3 , 3 to 7, 3:7 7 7 , 7 to 6, 7:6 6 ratio of triangles to circles ratio of nonpolygons to polygons Corresponding sides of similar shapes are in proportion. All measures shown are in units. proportion An equation that shows that two ratios are equivalent. 3 3 5 = 9 15 9 5 15 Use the numbers to write ratios for #1-3. Write each ratio in 3 forms. -14 35 1 2 24 100 -4 -8 9 16 1. ratio of square numbers to nonsquare numbers _____ _____ _____ 2. ratio of negative integers to positive integers _____ _____ _____ 3. ratio of abundant numbers to total numbers _____ _____ _____ Identify the missing value in the proportion. 4. 5 20 = 8 x 5. y 7 = 60 10 6. 72 9 = a 20 Probability Important Concept probability Probability is the likelihood that an event will occur. Probability can be expressed as a fraction, decimal, or percent. P(event) = number of favorable outcomes number of possible outcomes The probability of 2 events occurring can be found by multiplying the probability of each independent event. Examples You are rolling a standard number cube {1, 2, 3, 4, 5, 6}. Find the probability of each event. 1 P(5) = 6 3 1 P(even number) = = 6 2 P(composite number) = 2 1 = 6 3 P(4, 6) = 1 1 1 x = 6 6 36 P(1, prime number) = 1 1 1 x = 6 2 12 The letters T, H, E, O, R, E, T, I, C, A, and L are put in a bag. Find each probability. 1. P (A, I, or O) = ________ 2. P (not O or T) = ________ 3. P (U) = ________ You choose 1 letter from the bag and then put in back before choosing a second letter. Find each probability. 4. P (T, vowel) = ________ 5. P (E, T) = ________ Operations with Fractions and Mixed Numbers Addition Algorithm 1. 2. 3. 4. 1. Find a common denominator. 2. Write equivalent fractions. 3. Add numerators and keep denominator. 4. Add whole numbers, if necessary. 5. Simplify your answer. 5. 6. 1. 2. 3. 4. 5. 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. Subtraction Algorithm Find a common denominator. Write equivalent fractions. Borrow if necessary. Subtract numerators and keep denominator. Subtract whole numbers, if necessary. Simplify your answer. OR Find a common denominator. Write equivalent fractions. Change to improper fractions so no borrowing is necessary. Subtract numerators. Simplify your answer. 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. 5. Simplify your answer. Solve each problem. Use separate paper if necessary. NO CALCULATORS! 2 5 + 1 5 6 2. 5 ÷ 3 7 2 1 - 1 6 5 6. 2 1. 2 5. 10 1 x 3 2 3. 7 9 7. 10 x 1 1 2 ÷ 1 2 7 22 3. 3 2 9 - 1 2 3 8. 3 2 9 + 1 2 3 Operations with Decimals 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. 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. Multiplication Algorithm Division Algorithm 1. Multiply as you would with whole numbers. 2. Find the sum of the decimal places in both factors. 3. Count the same number of decimal places going right to left in the product and place the decimal. example: tenths x tenths = hundredths 1 place + 1 place = 2 places 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. 5.5( 9.9) 2. 0.6 + 5.8 3. 3.4 – 0.972 4. 25.041 – 8.3 5. 1.009 + 12 + 0.87 6. 936 ÷ 0.12 7. 0.492 ÷ 4 8. 12.2 ∙ 9. 43.59 x 0.1 1 2 Solve for n Making a fact family may help you. 2 1 = 3 4 5 2 5 + n = 5 3 6 2. n–4 2 n = 20 1 2 4. n 1 = 1 5 3 5. 4.3 + n = 13.42 6. n – 12.03 = 8.2 7. 0.7 ∙ n = 0.084 8. n ÷ 0.5 = 6.4 1. 1 3. Mixed 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. Solve each problem. Show work to support your answer. You may use a calculator on this section. 1. Laura has 121 carpet tiles. Each tile is a 1-foot by 1-foot square. Can she use all of the carpet tiles to make a square pattern without cutting any of the tiles? 2. A florist has 28 white roses, 35 red roses, 14 pink roses, and 70 yellow roses to use in making floral arrangements. He wants to use all the flowers and place an equal number of each color rose in each arrangement. Find the greatest number of floral arrangements he can make. 3. According to a survey, 9 in 10 teens volunteer at least once a year. Of these, onethird help clean up their communities. What fraction of teen volunteers help by cleaning up their communities? 4. Andrew is going to plant three new types of vegetables in his garden. The garden store sells packages of tomatillo seeds for $1.67, chili pepper seeds for $0.89, and pumpkin seeds for $2.32. Andrew buys one of each package. How much change will he receive from a ten-dollar bill? 5. Find the perimeter and area of the figure below. 7 cm 5 cm 6. Claire and Chad want to design a rectangular pen for their new puppy. They want the pen to have an area of 48 square feet. Fencing costs $0.85 per foot. What would be the dimensions and the cost of the least expensive pen Claire and Chad could build, assuming the side lengths are whole numbers? 7. A triangle has side lengths measuring 3 and 7. The measurement of the longest side is missing. Ted says that one possibility for the unknown side length is 11. Is Ted correct? Why or why not? 8. The figure below shows the dimensions of Adam’s bedroom. Adam wants to put new carpet in the room. How much carpet will Adam need? 6 3 2 3 4 9. The Make A Difference Club raised 30% of their fundraising goal of $200. What percent do they have left to earn? How much money do they have left to earn? 10. Alexa wants to use ready-made 6-foot long fence sections for her yard. The yard is a rectangle with dimensions 30 feet by 36 feet. How many fence sections will she need to enclose her entire yard? 11. The number of registered voters in the town of Cedarville is 8,900. In the last election, Mayor Burgis won reelection with approximately 72% of the vote. a. If approximately 52% of registered voters voted in the election, how many people voted? b. Based on your answer to part a, how many voters voted for Mayor Burgis? 12. The Midtown Middle School cheerleaders earned $175 at a car wash. If this amount is 25% of the cost of a new set of uniforms, what is the total cost for a set of uniforms? 13. Skateboards are on sale at Susan’s Skateshop for 30% off. If the regular price of a skateboard is $89, what is the discounted price with a 7% sales tax?