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Grade 7 Math: Mr. McAuley Midterm Review Sheet: January, 2010 1. Write 136 402 in expanded form: __________________________________________ 2. Write 44 820 in word form: _______________________________________________________________________ 3. Write the following number in standard form: (5x10 000) + (8 x 1000) + (9x100) + (2x1) ______________________________ 4. Graph the following on a number line: the set of even numbers greater than 11: the set of integers greater than -4 5. Add the following. Show your work in each of the boxes. 9253 + 26 + 547 = 87 229 + 20 179 + 31 068 = 6. Subtract the following. Show your work in each of the boxes. 9774 – 2521 = 5 988 451 – 1 637 228 = 1 Grade 7 Math: Mr. McAuley Midterm Review Sheet: January, 2010 7. Multiply the following. Show your work in each of the boxes. 69 x 40 = 9735 x 8 = 532 x 607 = 8. Divide the following. Show your work in each of the boxes. 53 624 ÷ 8 = 752 ÷ 13 = 9. Solve the following word problem. Show your work and state the final answer in a word sentence. A 4-sided building had 112 windows on each side of the building. Vandals broke 31 windows on one side of the building and 89 on another side. How many windows were left? 2 Grade 7 Math: Mr. McAuley Midterm Review Sheet: January, 2010 10. Find the value of each of the following using the rules for Order of Operations. Remember BEDMAS. (4 + 7) x (14 - 9) = 26 – 18 ÷ 3 + 58 – 3 x 3 = 18-40÷5 15-4x2 9 – 21÷ 3 13 – 4 x 3 11. Evaluate each by substituting the given numbers for the variables. xx + 14, + 5y if x = 23 (n + 5y) – (3n + y), if n = 6 and y = 8 12. Fill in the missing numbers to complete the following sequences. 7, 11, 17, 25, _____, _____, _____ 4, 9, 7, 12, 10, _____, _____, _____ -10, -7, ____, -1, ____, ____ -94, -83, ____, ____, ____, -39 13. Complete the function chart below and draw a graph of y = 2x + 3. x y = 2x + 3 0 1 2 3 4 3 Grade 7 Math: Mr. McAuley Midterm Review Sheet: January, 2010 14. Circle the largest integer in each group. +3, -4, +5, -6, -7 -5, 0, +2, -1, +1 15. Write the set of integers described below. the integers greater than -4 but less than +3 16. Calculate the following: (-6) + (-9) = (+3) + (-9) = (-4) + (+7) = (+5) + (+38) = 0 + (-22) = (+89) + (-37) + (+28) = 17. Translate the following into an expression involving integers and find the integer that represents the final altitude of the airplane. An airplane was flying at 12 000 feet. It then went down 250 feet, went down another 415 feet and then rose 125 feet. What was its altitude then? 18. Subtract the following integers. (+13) – (+5) = (-12) – (+4) = (+7) – (+14) = (-11) – (-6) = (+5) – (-22) = (+30) – (+14) – (-3) = 19. Calculate. Watch the signs! (+8) x (+4) = (-5) x (-3) = (-8) x (+7) = (+66) ÷ (-3) = (+36) ÷ (+9) = (-270) ÷ (-30) = (+11) x (-7) = (-4) x (+43) = (-48) = (+6) 4 Grade 7 Math: Mr. McAuley Midterm Review Sheet: January, 2010 20. Solve each of the following. Show your work. y - 7 = 23 8x + 11 = 59 20m – 4 = 15m+ 11 7 + k = k + 14 2 3 21. Translate each of the following into an English statement. 4y - 17 _______________________________________________________________________ x + 10 _______________________________________________________________________ 22. Translate each English statement into a mathematical phrase or sentence. a number decreased by twelve ________________________________________________ The sum of fifteen and seven is a number. ___________________________________ 23. Solve the following using the proper procedure as follows: 1. Identify what you are looking for and give it a name (such as „x‟). 2. Translate the English sentence into a mathematical sentence (an equation). 3. Solve the equation that you created. 4. Answer the question with a final statement. If half of a number is decreased by 20, the result is 41. What is the number? 5 Grade 7 Math: Mr. McAuley Midterm Review Sheet: January, 2010 24. Is 21 885 divisible by each of the following numbers? Answer “Yes” or “No” by 2: _____________________ by 3: ______________________ by 5: _____________________ 25. Factor each of the following into prime numbers using either the Factor Tree Method or the Repeated Division Method: 630 168 26. List all the factors of 60. 40: 27. List only five multiples of 13: 15: 28. Circle the prime numbers in the following list. 1 2 3 9 11 51 29. Find both the GCF (Greatest Common Factor) and LCM (Lowest Common Multiple) of the following set of numbers. Show your work. (You may use any method you want.) (18, 45) GCF = LCM = 6 Grade 7 Math: Mr. McAuley Midterm Review Sheet: January, 2010 4 30. Write two equivalent fractions for . 8 36 31. Write the fraction in lowest terms. 60 32. Add or subtract the following and leave your answers in lowest terms (basic mixed fractions). 16 4 6 2 - = - = 20 20 7 3 1 4 1 1 + = 8 +2 = 5 9 3 4 33. Multiply the following fractions and leave your answers in lowest terms. 4x2= 15 x ½= 5 3 1 1 1/2 x 2/3 x 3/4 x 6/8 = 4 x2 = 2 6 7 Grade 7 Math: Mr. McAuley Midterm Review Sheet: January, 2010 34. Divide the following fractions and leave your answers in lowest terms. 4/9 ÷ 2/3 = 10 ÷ 4/7 = 3 7 2 5 2 ÷ = 4 ÷3 = 4 10 3 6 35. On your test you will also be given one problem to solve. In order to get full marks you will need to get the correct answer and demonstrate and state how you used at least three different strategies to solve the problem. Strategies that you may use to solve the problem are: Act it Out, Break into Parts, Calculator, Chart or Organized Listing, Computation, Deductive Logic, Diagram, Formula, Graphing, Grouping, Guess and Check, Check for Hidden Assumptions, Model, Narrowing In, Search for a Pattern, Simplify or Work Backwards. 8 Grade 7 Math: Mr. McAuley Midterm Review Sheet: January, 2010 Notes and Study Helps Some notes and special study helps are given to help with individual questions on the review sheet. These correspond with the question numbers on the review sheet. 1 and 3. When a number is written in the form of 67 502, we call this standard form. That same number can also be written in expanded form as follows to show the place value of each digit: (6 x 10 000) + (7 x 1000) + (5 x 100) + (2 x 1) You may leave out the zero digits in the expanded form, but you must include the zeros as place fillers in the standard form. (See assignment p. 3 for additional practice) 2. When writing a number in word form, think of the larger number in periods, such as the ones period, the thousands period, and the millions period and separate each period with a comma. For example, in the following number there are 326 in the millions period so we would write it as three hundred twenty-six million. The complete number would be written as follows: three hundred twenty-six million, five hundred fifty-one thousand, seven hundred eighty. The number 326 551 780 4. When you draw your number line, be careful to use a ruler and to space your points out evenly. Use circles on the number line to show the points you have chosen. For points that continue on infinitely, simply put three dots at the end of the number line. (See assignment p. 4 for additional practice) 5. When adding whole numbers, remember to line up the digits according to place value. Also, remember that you can use the inverse operation (subtraction) to check addition. You can also use estimation to check if your answer is reasonable. (See assignment p. 7-8 for additional practice) 6. When subtracting whole numbers, remember to line up the digits according to place value. Also, remember that you can use the inverse operation (addition) to check subtraction. You can also use estimation to check if your answer is reasonable. (See assignment p. 8-9 for additional practice) 9 Grade 7 Math: Mr. McAuley Midterm Review Sheet: January, 2010 7. When multiplying whole numbers, line up the digits according to place value. Remember to add a zero when multiplying by a number in the tens place; add two zeroes when multiplying by a number in the hundreds‟ place and so on. Also remember that you can use the inverse operation (division) to check multiplication. You can also use estimation to check if your answer is reasonable. (See assignment p. 9-10 for additional practice) 8. To divide questions given in the form of 116÷4, you need to rewrite them as follows: ___ 4)116 Line up your numbers in your quotient carefully. Remember the process of: estimate – multiply- subtract – check – bring down. (Repeat if not finished) (See assignment p. 10-11 for additional practice) 9. Be careful to choose the correct operation to reflect the information in the question. A sample question and answer is given below: Question: A parking lot had 76 rows for parking and spots for 30 cars in each row. The parking lot was full except for 8 spots in the last row. How many vehicles were parked there? Answer: Let v = vehicles. v = 76 x 30 – 8 v = 2280 – 8 = 2272 There were 2272 vehicles parked. (See assignment p. 11 for additional practice) 10. BEDMAS stands for brackets, exponents, division, multiplication, addition and subtraction. Division and multiplication are at the same level. Addition and subtraction are at the same level. When solving parts of the question at the same level, simply work left to right. (See assignment p. 15-17 for additional practice) 11. If x = 2, then 3x means 3 times x not 32. In questions involving substitution, the other rules for “Order of Operations” must also be followed. (See assignment p. 18-22 for additional practice) 12. To complete the missing numbers in sequences, you need to find the pattern of the numbers that are given and then extend the pattern. (See assignment p. 23 for additional practice) 10 Grade 7 Math: Mr. McAuley Midterm Review Sheet: January, 2010 13. To complete the chart simply substitute the number into the function rule. A sample is given below. If x = 0, then y = 0 + 5 or 5. If x = 1, then y = 1 + 5 = 6. x y=x+5 0 5 (2) 1 6 2 7 3 8 4 9 The corresponding x and y values are then put together to make an ordered pair that can be plotted on a graph. The ordered pairs for the chart above are (0,5), (1,6), (2,7), (3,8) and (4,9). The x value is always listed 1st in the ordered pair and the y value is listed 2nd. Once you have your ordered pairs, these can be plotted on the graph. To set up your graph, be sure to put the x-axis along the bottom going across, and the y-axis along the side going up and down. Plot the points on your graph on the lines, not in the spaces. When you have graphed all the points join them with a line. The above ordered pairs, when graphed should like the following. y 12 11 10 9 8 7 6 5 4 3 2 1 x 0 1 2 3 4 5 6 7 8 9 10 11 12 (See assignment p. 24-26 for additional practice) 11 Grade 7 Math: Mr. McAuley Midterm Review Sheet: January, 2010 14 and 15. The set of integers is as follows: I = {…, -4, -3, -2, -1, 0, +1, +2, +3, +4 …} The numbers farther to the right are the larger numbers. (See assignment p. 29-31 for additional practice) 16. Number lines can be used to help in the addition of integers. (See p. 31). You can also add integers by using the following two rules: 1. If the signs are the same, add the numbers and keep the same sign. Examples: (+3) + (+5) = +8 (-7) + (-5) = -12 2. If the signs of the integers are different, subtract the numbers and take the sign of the numerically larger number. Examples: (-9) + (+4) = -5 (+8) + (-3) = +5 (See assignment p. 31-35 for additional practice) 17. (See assignment p. 32 for additional practice) 18. To subtract integers, you can use the following rules. 1. To subtract a positive integer is the same as adding its opposite. Example: (-6) – (+7) = (-6) + (-7) = (-13) 2. To subtract a negative number is the same as adding its opposite. Example: (-8) – (-5) = (-8) + (+5) = (-3) (See assignment sheet, “Subtracting Integers” for additional practice) 12 Grade 7 Math: Mr. McAuley Midterm Review Sheet: January, 2010 19. To multiply integers remember the following rules: 1. If the signs are the same, the product is positive. Example: (+3) x (+4) = +12 (-3) x (-4) = +12 2. If the signs are different, the product is negative. Example: (+3) x (-4) = -12 (-3) x (+4) = -12 (See assignment sheets, “Multiplying Integers”and “Review: Multiplying and Dividing Integers” for additional practice.) To divide integers remember the following rules: 1. If the signs are the same, the quotient is positive. Example: (+12) ÷ (+4) = +3 (-12) ÷ (-4) = +3 2. If the signs are different, the quotient is negative. Example: (+12) ÷ (-4) = -3 (-12) ÷ (+4) = -3 (See assignment sheet, “Dividing Integers” “Review: Multiplying and Dividing Integers” for additional practice.) 20. Use the following rule to solve for unknown variables in an equation: To solve for an unknown variable in an equation, you must isolate the variable by doing the same thing to both sides of the equation. See instructions on the following pages for explanations on solving various types of problems: Type I Equations: p.36 Type II Equations: p. 37 Type III Equations: p. 39 Type IV Equations: p. 40 Ratio Type Equations: p. 42 (See assignment p. 36-43 for additional practice) 13 Grade 7 Math: Mr. McAuley Midterm Review Sheet: January, 2010 21 and 22. A list of some symbols, phrases and sentences that you will use for this type of question is found on p. 44 of your assignments. (See assignment p. 44-45 for additional practice) 23. A sample question and answer is given below: Question: Twice a number increased by nine is equal to fifty-one. What is the number? Answer: Let x be the number. 2x + 9 = 51 2x = 42 x = 21 The number is 21. When you have finished solving for the variable, remember to check your answer by substituting the number back into the original question to see if it works. For example, in the above question, if x = 21, then in the original question, 2 x 21 + 9 = 51 which is true. (See assignment p. 46-49 for additional practice) 24. See the handout sheet, “Divisibility Rules” to review the divisibility rules for 2, 3, 4, 5, 6, 8, 9, and 10. (See assignment p. 51 for additional practice) 25. See assignment sheets p. 55-56 for samples of the Factor Tree Method and the Repeated Division Method. Remember that when you are finished, all of the numbers at the end must be prime numbers. Remember that 1 is not a prime number. (See assignment p. 55-56 for additional practice) 26 and 28. Factors of a number are all the numbers that divide evenly into that number. If a number has only two factors (1 and itself) then it is a prime number. If a number has more than two factors, then it is a composite number. Divisibility rules can be used to check for factors. The factors of 13 are as follows: (It is a prime number) 13:1, 13 The factors of 24 are as follows: (It is a composite number) 24: 1, 2, 3, 4, 6, 8, 12, 24 (See assignment p. 52 for additional practice) 14 Grade 7 Math: Mr. McAuley Midterm Review Sheet: January, 2010 27. When you multiply a whole number by other whole numbers, all of the products are multiples of the original number. For example, the multiples of 5 are 5, 10, 15, 20, 25 and so on because 5 x 1=5, 5 x 2 = 10, 5 x 3 = 15, and so on. (See assignment p.53 for additional practice) 29. The GCF of two or more numbers is the Greatest Common Factor or the highest number that is a factor of all of those numbers. To find the GCF of numbers, you can start by writing each of the numbers as the product of prime factors. Then multiply any of the factors that are common to all of the numbers. Example: 20 = 2 x 2 x 5 36 = 2 x 2 x 3 x 3 The factors that are common to both of the numbers are 2 x 2 so the GCF is 4. The LCM of two or more numbers is the Lowest Common Multiple or the lowest number that is a multiple of all of those numbers. To find the LCM of numbers, you start by writing each of the numbers as the product of prime factors. Then multiply any numbers that appear in any of the prime factorizations by the most time they appear. In the above example, the factor 2 appears twice at the most, the factor 3 appears twice at the most and the factor 5 appears once at the most. Therefore the LCM is equal to 2 x 2 x 3 x 3 x 5 = 180 (See assignment p. 57-61 for additional practice) 30. Equivalent means equal value. To make equivalent fractions, you multiply or divide the numerator and denominator by the same number. For example, if I want to find an 12 equivalent fraction for I can do the following. 15 Multiply both the numerator (12) and the denominator (15) by the same number. 12 2 24 12 24 x = so and are equivalent fractions. 15 2 30 15 30 15 Grade 7 Math: Mr. McAuley Midterm Review Sheet: January, 2010 Divide both the numerator and denominator by the same number. A common factor of 12 and 15 is 3. If I divide both the numerator and denominator of the 12 4 fraction by 3, I will end up with the equivalent fraction . 15 5 To check if fractions are equivalent, you can cross-multiply. 3 12 = 3 x 40 = 120 and 10 x 12 = 120 so these are equivalent fractions. 10 40 (See assignment p. 64-65 for additional practice) 31. When writing a fraction in simplest terms, you must divide the numerator and denominator by the same number. Keep doing this until there is no number left, other than one, that can divide into both the numerator and denominator. (See assignment p. 66 for additional practice) 32. Rules for Addition of Fractions: 1. Rewrite the fractions with a common denominator. 2. Add the numerators but keep the denominators the same. 3. Simplify if possible to lowest terms and rewrite as a mixed fraction if necessary. Rules for Subtraction of Fractions: 1. Rewrite the fractions with a common denominator. 2. Subtract the numerators but keep the denominators the same. 3. Simplify if possible to lowest terms and rewrite as a mixed fraction if necessary. Rules for Addition and Subtraction of Mixed Numbers: 1. Change all mixed fractions to improper fractions. 2. Rewrite the fractions with a common denominator. 3. Add or subtract the numerators while keeping the denominators the same. 3. Simplify if possible to lowest terms and rewrite as a mixed fraction if necessary. (See assignment p. 70-75 for additional practice) (See assignment p. 67 for instructions on converting back and forth between improper and mixed fractions and for additional practice) 16 Grade 7 Math: Mr. McAuley Midterm Review Sheet: January, 2010 33. Rules for Multiplication of Fractions (see p. 76) 1. Indicate that you intend to multiply the numerators by placing a times sign between them. 2. Indicate that you intend to multiply the numerators by placing a times sign between them. 3. Simplify the fractions by dividing common factors into both the numerator and denominator. 4. Multiply the numerators and then the denominators. 5. Convert your result to a mixed fraction if you can. 3 6 3x6 3 x3 9 x = = = 8 5 8 x5 4 x5 20 (See assignment p. 76 for additional practice) Rules for Multiplying Mixed Fractions 1. Convert all mixed fractions to improper fractions. 2. Simplify the fractions by dividing common factors into both the numerator and denominator. 3. Multiply the numerators and then the denominators. 4. Convert the result to a mixed fraction if you can. 2 1 8 5 2 5 10 1 2 x 1 = x = x = =3 3 4 3 4 3 1 3 3 (See assignment p.77-78 for additional practice) 34. Rules for Division of Fractions 1. Change the question to a multiplication question by writing the reciprocal (multiplicative inverse) of the divisor (second fraction). 2. Proceed as you would in a multiplication question. 2 4 2 9 1 3 3 1 ÷ = x = x = =1 3 9 3 4 1 2 2 2 (See assignment p.79 for additional practice) 17 Grade 7 Math: Mr. McAuley Midterm Review Sheet: January, 2010 Rules for Dividing Mixed Fractions 1. Convert all mixed fractions to improper fractions. 2. Change the question to a multiplication question by writing the reciprocal of the divisor. 4. Proceed as you would in a multiplication question. (see p.80) 3 7 11 7 11 10 11 5 55 13 2 ÷ = ÷ = x = x = =3 4 10 4 10 4 7 2 7 14 14 (See assignment p. 80-81 for additional practice) 35. Problem Solving. See problem solving sheets for additional practice. 18

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