Algebra 1A and 1B Summer Packet

Transcription

1 Algebra 1A and 1B Summer Packet Name: Calculators are not allowed on the summer math packet. This packet is due the first week of school and will be counted as a grade. You will also be tested over the material in this packet. Come to school prepared!!! Solve each problem and place the answer on the line to the left of the problem. Adding Integers A. Steps if both numbers are positive. Example: Step 1: Add the two numbers = 7 The answer is 7. B. Steps if both numbers are negative. Example: (-5) + (-2) Step 1: Ignore the negative signs and add the two numbers = 7 Step 2: Put a negative sign on the answer. The answer is -7. C. Steps if one number is positive and one number is negative. Example: (-2) + 9 Step 1: Ignore the negative sign and determine which number is greater. 9 is greater than 2 The answer will have the sign of the greater number so the answer is positive because 9 is positive. Step 2: Subtract the smaller number from the larger number. 2 is smaller and 9 is larger 9 2 = 7 Step 3: Combine the sign of answer with the numeric answer. The answer is (-4) (-17) (-3) + (-6) (-21) (-45) (-34) 8. (-23) + (-12)

2 Subtracting Integers A. Two negative signs in a row = a positive sign. Example: 6 (-11) Step 1: combine the negative signs 6 (-11) = The answer is 17. B. Turn subtraction into adding a negative number. Example: 4 10 = 4 + (-10) The answer is -6. C. Use process for adding integers from the previous page. 9. (-4) (-17) (-3) (-6) (-21) (-45) (-34) 16. (-23) (-12)

3 Multiplying Integers A. The following signs are equivalent multiplication signs: * x ( ) B. If multiplying two positive numbers, answer will be positive. Example: (5) (8) Step 1: Determine the numeric answer by ignoring any negative signs. (5) (8) = 40 Step 2: Determine sign of the answer. Since both numbers are positive, the answer will be positive. The answer is 40. C. If multiplying two negative numbers, answer will be positive. Example: (-5) (-8) Step 1: Determine the numeric answer by ignoring any negative signs. (-5) (-8) = 40 Step 2: Determine sign of the answer. Since both numbers are negative, the answer will be positive. The answer is 40. D. If multiplying one positive number and one negative number, answer will be negative. Example: (-5) (8) Step 1: Determine the numeric answer by ignoring any negative signs. (-5) (8) = 40 Step 2: Determine the sign of the answer. Since one number is positive and one number is negative, the answer will be negative. The answer is (-4) (5) 21. (-17) (34) 18. (-3) (-6) (-21) 19. (13) (9) 23. (23) (-45) 20. (15) (-34) 24. (-23) (-12)

4 Dividing Integers A. The following signs are equivalent division signs: / B. If both numbers are positive, the answer will be positive. C. If both numbers are negative, the answer will be positive. D. If one number is positive and one number is negative, the answer will be negative. E. Steps: Example: 15 / 5 Step 1: Determine the numeric answer by dividing the two numbers and ignoring any negative signs. 15 / 5 = 3 Step 2: Determine the sign of the answer. Since both numbers are positive, the answer will be positive. The answer is 3. Example: (-25) / (-5) Step 1: Determine the numeric answer by dividing the two numbers and ignoring any negative signs. (-25) / (-5) = 5 Step 2: Determine the sign of the answer. Since both numbers are negative, the answer will be positive. The answer is 5. Example: 40 / (-8) Step 1: Determine the numeric answer by dividing the two numbers and ignoring any negative signs. 40 / (-8) = 5 Step 2: Determine the sign of the answer. Since one number is positive and one number is negative, the answer will be negative. The answer is (-12) / (-56) / (-7) 26. (-24) / (-8) 30. (-98) / / (-2) 31. (-28) / / (-16) 32. (-108) / 4

5 Adding Fractions A. The denominators are identical and both fractions are positive. Example: Step 1: Add the numerators = 5 Step 2: Keep the denominator the same, since the denominators for each of the fractions is the same. Answer is 5 9. B. The denominators are identical and both fractions are negative. 1 2 Example: Step 1: Adding the numerators. (-1) + (-2) = -3 Step 2: Keep the denominator the same, since the denominator for each of the fractions is the same. 3 Answer is. 4 C. The denominators are identical and one fraction is positive and one fraction is negative. Example: Step 1: Adding the numerators. 4 + (-5) = -1 Step 2: Keep the denominator the same, since the denominator for each of the fractions is the same. 1 Answer is

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7 Adding fractions continued D. Denominators are different. You may not add fractions unless they have identical denominators. You must modify the fractions to have identical denominators by finding the Least Common Denominator (LCD). Example: Step 1: Find the LCD. a) Find the prime factors of the denominator. The prime factors of the first denominator (4) are (2) (2) = 2². The prime factors of the second denominator (6) are (2) (3). b) Write the prime factors in exponential format. 4 = = 2 * 3 c) Find the greatest power of each unique prime factor. The unique prime factors are 2 and 3. The greatest power of 2 is 2 2. The greatest power of 3 is 3. d) Multiply these together = LCD. (2²) (3) = (4) (3) = 12 = LCD Step 2: Rewrite the fractions so that each has the LCD as its denominator. 3 5?? + = a) To compute the new numerators, look at each fraction individually. In the first fraction, multiply the old denominator (4) by 3 for it to be equal to the new denominator. Multiply the old numerator by 3 to compute the new numerator: 3 * 3 = 9. In the second fraction, multiply the old denominator (6) by 2 for it to be equal to the new denominator. Multiply the old numerator by 2 to compute the new numerator: (-5) * 2 = Step 3: Now denominators are identical. See previous instructions to add fractions. 1 Answer is

8 Subtracting Fractions Just like subtracting integers, change subtraction of a fraction to adding a negative fraction, remembering that two negative signs in a row equal a positive sign. You may not subtract two fractions unless they have identical denominators. Example: Step 1: Change it to the following: Step 2: Then follow the steps for adding fractions. a) The LCD is b) Add the numerators. 8 Answer is =

9 Multiplying Fractions A. If both numbers are positive, the answer will be positive. B. If both numbers are negative, the answer will be positive. C. If one number is positive and one number is negative, then answer will be negative. D. Fractions should be simplified/reduced/canceled before multiplying. Any numerator may be canceled only with any denominator. Look for a common factor. Example: 3 * Step 1: Reduce. a) The Numerator 3 and denominator 9 have a common factor of 3. Divide each by * 10 3 b) The Numerator 4 and denominator 10 have a common factor of 2. Divide each by * 5 3 Step 2: After simplifying the fractions, multiply numerator by numerator and then multiply denominator by denominator. a) Numerator: 1 * 2 = 2 b) Denominator: 5 * 3 = 15 Answer is * 63. * * 64. * * 65. * * 66. *

10 Dividing Fractions A. If both numbers are positive, the answer will be positive. B. If both numbers are negative, the answer will be positive. C. If one number is positive and one number is negative, then answer will be negative. Example: Step 1: Dividing by a fraction is the same as multiplying by its reciprocal. Change the division operation to multiplication and flip the second fraction. 4 3 * 7 2 Step 2: Determine if the expression can be simplified. a) Numerator 4 and denominator 2 have a common factor of 2. Divide each by * 7 1 Step 3: Multiply the two fractions using the steps in the previous section. Answer is:

11 Properties In Algebra 1, you will be asked to identify certain properties. These are properties that are the rules of Algebra that allow us to work the problems in certain ways. Here is a list of properties that you should be able to recognize: Associative Property: Addition: 1 + (2 + 3) = (1 + 2) + 3 Multiplication: 1 * (2 * 3) = (1 *2* 3) Commutative Property: Addition: = Multiplication: 1 * 2 = 2 * 1 Distributive: 2 (x + 3) = 2x + 6 Additive Inverses: 4 + (-4) = 0 Multiplicative Inverses: Additive Identity: = 1 1 4* 1 4 = Multiplicative Identity: 5 * 1 = 5 Determine if the expressions are True or False. If True, state what property is shown = (2 + 3) + 5 = 2 + (3 + 5) 77. 4(2 + 3) = 4(2) + 4(3) (-5) = (4 2) = (9 4) * (9 2) * 4 = 4 * = = 2

12 Order of Operations Operations are addition, subtraction, multiplication, division, square, etc. There is an order that operations are performed designated by the acronym PEMDAS. Operations are always performed from left to right. P = parentheses E = exponent M = multiplication D = division A = addition S = subtraction Practice: Example: (3 + 6) X 2 Step 1: Order of operations says parentheses is first. (3 + 6) = 9 Equation is now 9 X 2 Step 2: Perform the multiplication 9 X 2 = 18 Answer is ( ) (5 3)² (3 (6 5)) 89. 5² - (12 2) (6 2) (5 1)

13 Evaluating Expressions To evaluate an expression, substitute a number for each variable and perform the operation(s) paying particular attention to the order of operations. Example: 6x + 4y =? when x = 2 and y = 3 Step 1: Substitute 2 for x and 3 for y in the expression Step 2: Order of operations says do multiplication first. 6 2 = 12 and 4 3 = 12 Expression now is Step 3: Perform addition = 24 Answer is j = 5 and k = p = 4 and q = 6 k j 5 (6 + p) (q p) 92. m = 6 and n = x = 3, y = 5 and z = 5 m (n + 1) 2 + y x + z 93. x = 5, y = 5 and z = x = 2 and y = 4 x + y + 2z 5 + yx x

14 Rounding Decimals Rounding means reducing the number of digits while keeping the value similar. Rounding can be to any number of digits. Step 1: Determine the digit to which you are rounding (i.e. whole number, how many decimal places, etc. Step 2: Look at the digit immediately to the right and determine whether to round up or stay the same. a. If the digit to the right is 5-9, round up to the next value. b. If the digit to the right is 0-4, keep the same. Step 3: Eliminate all digits to the right of the rounded digit. Example: Round to the nearest whole number Step 1: Rounding to a whole number means no decimals. Step 2: The number to the right of the whole number is 4 therefore whole number stays the same. Answer is 12. Example: Round to 1 decimal place Step 1: Rounding to 1 decimal place means only 1 digit to the right of the decimal point. Step 2: The number to the right of 1 decimal place is 7 therefore round up the 1 st decimal place to 1. Answer is -2.1 Example: Round to 3 decimal places Step 1: Rounding to 3 decimal places means only 3 digits to the right of the decimal point. Step 2: The number to the right of the 3 rd decimal place is 4 therefore no rounding up. Answer is Round the following to the nearest whole number Round the following to 1 decimal place Round the following to 2 decimal places Round the following to 3 decimal places

15 Answer Key Adding Integers Subtracting Integers Multiplying Integers Dividing Integers Adding Fractions (Identical Denominators) Adding Fractions (Different Denominators)

16 Subtracting Fractions Multiplying Fractions Dividing Fractions Properties True, Commutative Property True, Associative Property True, Distributive Property False False True, Commutative Property False 82. True, Additive Identity Order of Operations

17 Evaluating Expressions Rounding of Decimals