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Review Sheet for Math Midterm Exam Review Sheet for Math Midterm Exam Multiple Choice Identify the choice that best completes the statement or answers the question. _____ 1. Which number is divisible by 2 but not by 4? 100, 125, 150, 200 a. 150 b. 200 c. 125 d. 100 _____ 2. Which number is divisible by 3 and by 10? 606, 110, 260, 420 a. 606 b. 110 c. 260 d. 420 _____ 3. Write an algebraic equation for n multiplied by 5. n a. n - 5 b. n + 5 c. d. 5n 5 _____ 4. Evaluate the expression by replacing a with 13. a – 10 a. 23 b. 3 c. 130 d. 2 _____ 5. If n represents any term number, write a relation for the term. Term Number 1 2 3 4 5 6 Term 7 14 21 28 35 42 a. 7n b. 2n + 7 c. 2n d. n+7 _____ 6. Write a relation for the perimeter of the rectangle with length (n + 4) and width n cm. n n+4 a. (4n + 4) cm b. (2n + 4) cm c. n(n + 4) cm d. (4n + 8) cm 1 The actual test will be very similar to this practice test. Please see the attached pages for study notes to help you do each type of question. Review Sheet for Math Midterm Exam _____ 7. The are n players on a sports team. Each player gets 3 pairs of sox and 4 pairs are kept in reserve. Write a relation for the number of pairs of sox needed. a. 4n + 3 b. 3n + 4 c. 7n d. 12n _____ 8. Use algebra. Write a relation for the Input/Output table. Input x 1 2 3 4 5 Output 39 38 37 36 35 a. x – 40 b. x + 35 c. 40 – x d. 39 – x _____ 9. Write an equation for “I subtract 13 from a number. The answer is 24.” n a. = 24 b. n + 13 = 24 c. n – 13 = 24 d. 13 – n = 24 13 _____ 10. Write an equation for the situation. Patricia has p posters. She sold 7 and has 19 left. a. p = 19 – 7 b. p + 7 = 19 c. p – 7 = 19 d. p + 19 = 7 _____ 11. Let one white tile represent +1 and one black tile represent -1. Write the integer modeled by this set of tiles. ■□□■□□ □■■□■ a. +6 b. +11 c. +1 d. -1 _____ 12. Let one white tile represent +1 and one black tile represent -1. What sum is modeled by 17 positive tiles and 19 negative tiles? a. -1 b. +2 c. -2 d. +36 _____ 13. Add. (-5) + (+10) a. -15 b. +15 c. +5 d. -5 _____ 14. David gets on an elevator at the 36th floor. The elevator goes down 26 floors then up 17 floors. At what floor did it finally stop? a. -7 b. 27 c. 45 d. 79 _____ 15. Copy and complete. (+9) - □ = -18 a. -9 b. -27 c. 9 d. +27 2 The actual test will be very similar to this practice test. Please see the attached pages for study notes to help you do each type of question. Review Sheet for Math Midterm Exam _____ 16. The graph shows the high and low temperatures in a week from Sunday to Saturday. High and Low Tem peratures 20 Degrees Celcius 15 10 5 0 Su M Tu W Th F Sa Days On which of the following days was the difference between the high and low temperature the least? a. Sunday b. Monday c. Wednesday d. Friday _____ 17. A submarine at sea level dives 7m and then another 4 m. Write the final depth of the submarine as an integer. a. +11m b. +12 m c. -11m d. -12m 28 _____ 18. Write as a decimal. Identify the decimal as terminating or repeating. 36 a. 0.8; terminating b. 1.8; terminating c. 1. 7 ; repeating d. 0. 7 ; repeating _____ 19. Subtract. 12.39 – 7.735 a. -64.96 b. 6.496 c. 4.655 d. 20.125 _____ 20. Calculate. 10.45 + 6.59 – 0.7 a. 17.74 b. 16.97 c. 16.34 d. 6.35 _____ 21. Add. 0.9 + 0.89 + 0.789 a. 0.887 b. 2.579 c. 17.69 d. 7.89 _____ 22. Multiply. 2.4 x 60 a. 84 b. 14.4 c. 144 d. 1440 3 The actual test will be very similar to this practice test. Please see the attached pages for study notes to help you do each type of question. Review Sheet for Math Midterm Exam _____ 23. Divide. 2.12 ÷ 0.4 a. 5.3 b. 0.053 c. 0.53 d. 53 _____ 24. Evaluate 17.5 – 5 x 1.4 a. 10.5 b. 24.5 c. 63 d. 17.5 _____ 25. Write 59% as a decimal. a. 5.9 b. 0.059 c. 0.59 d. 59 12 _____ 26. Write as a percent. 50 a. 38% b. 24% c. 27% d. 42% _____ 27. What fraction of the diagram is shaded? 3 1 ///// a. b. ///// 8 24 ///// 1 1 c. d. 12 8 _____ 28. What fraction of the diagram is shaded? Write the fraction as a percent. 2 6 a. ; 30% b. ; 6% 10 100 5 2 c. ; 25% d. ; 20% 20 100 _____ 29. In your last 20 basketball games, you attempted 72 free throws and made 18 of them. Express your success in making free throws as a percent. a. 25% b. 26% c. 28% d. 24% _____ 30. Calculate 50% of 70. a. 18 b. 35 c. 4 d. 9 _____ 31. Jason wants to buy a bicycle that costs $395.00. His parents ask Jason to raise 40% of the money. How much does Jason have to raise? a. $154.00 b. $158.00 c. $118.50 d. $40.00 _____ 32. A circle has a radius of 36.9 cm. What is its diameter? a. 18.45 cm b. 12.3 cm c. 110.7 cm d. 73.8 cm 4 The actual test will be very similar to this practice test. Please see the attached pages for study notes to help you do each type of question. Review Sheet for Math Midterm Exam _____ 33. Which is not a relationship between the radius r and the diameter d of a circle? a. d ÷ 2 = r b. d = 2r c. r + r = d d. 2d = r _____ 34. What is the least number in the set? 11 9 1 4, , ,2 6 4 6 1 9 11 a. 2 b. c. 4 d. 6 4 6 _____ 35. Find the next two terms in the pattern. 0.277, 0.388, 0.499, 0.61… a. 0.721, 0.943 b. 0.721, 0.832 c. 0.832, 1.054 d. 0.832, 0.943 _____ 36. Determine the cost of 3.54 kg of apples at $1.05/kg. a. $4.59 b. $3.72 c. $0.46 d. $37.17 _____ 37. A circle has a diameter of 54.2 m. What is the radius? a. 162.6 m b. 27.1 m c. 108.4 m d. 18.1 m _____ 38. Four identical circles of the largest possible size are drawn on a square sheet of paper. The side length of the paper is 6.4 cm. What is the radius of each circle? a. 1.6 cm b. 6.4 cm c. 0.8 cm d. 3.2 cm _____ 39. Estimate the circumference of this circle. a. 21 m b. 42 m c. 17 m d. 84 m _____ 40. Estimate the circumference of this circle. a. 81 cm b. 547 cm c. 30 cm d. 41 cm 5 The actual test will be very similar to this practice test. Please see the attached pages for study notes to help you do each type of question. Review Sheet for Math Midterm Exam _____ 41. The circumference of a circle is 27 cm. Calculate the radius. Round your answer to one decimal place. a. 5.9 cm b. 4.3 cm c. 13.5 cm d. 8.6 cm _____ 42. Find the diameter of a circle with a circumference of 35.1 cm. Round your answer to one decimal place. a. 5.6 cm b. 11.2 cm c. 22.3 cm d. 17.6 cm _____ 43. Find the area of this parallelogram. a. 30 m 2 b. 50 m 2 c. 25 m 2 d. 100 m 2 _____ 44. Find the area of the parallelogram with base 45 cm and height 7.4 cm. a. 27.38 cm 2 b. 333 cm 2 c. 202.5 cm 2 d. 52.4 cm 2 _____ 45. Find the area of this triangle. a. 80 m 2 b. 18 m 2 c. 40 m 2 d. 160 m 2 6 The actual test will be very similar to this practice test. Please see the attached pages for study notes to help you do each type of question. Review Sheet for Math Midterm Exam _____ 46. Which triangle has an area of 6 square units? Q P R S a. Q b. R c. P d. S _____ 47. Find the area of this circle. Round your answer to two decimal places. a. 2678.65 m 2 b. 91.73 m 2 c. 669.66 m 2 d. 45.87 m 2 7 The actual test will be very similar to this practice test. Please see the attached pages for study notes to help you do each type of question. Review Sheet for Math Midterm Exam The circle graph shows how a college student breaks down her study time in a typical week. Study Time per Week English French 11% 23% Math 21% Art 14% History 31% _____ 48. What fraction of the time is spent on English and Math? 1 1 3 2 a. about b. about c. about d. about 2 3 4 3 _____ 49. Grade 7 students were surveyed on how many hours per day they spend on various activities. About how many hours per day are spent on school and homework? How Students Spend Their Time Socializing 13% Sleeping 33% School 25% Homework 8% Watching TV Eating 13% 8% a. 7 h b. 10 h c. 9 h d. 8 h _____ 50. Find the central angle of a sector that represents 90% in a circle graph. Round to the nearest degree if necessary. a. 342° b. 306° c. 32° d. 324° 8 The actual test will be very similar to this practice test. Please see the attached pages for study notes to help you do each type of question. Review Sheet for Math Midterm Exam Short Answer 51. Divisible by 3 Divisible by 2 Place the numbers in the Venn Diagram. 27, 36, 14, 8, 21, 42, 20, 33 52. Write an algebraic expression for the sentence. Subtract 14 from a number, then multiply by 3. x 53. Evaluate + 9 by replacing x with 30. 6 54. The pattern is formed using 1 cm 2 tiles. Term 1 Term 2 Term 3 a) What is the perimeter for each term? b) Write a relation for the perimeter of the nth term. 55. Write an equation for the sentence. Nine less than a number is 14. 56. Which equation has the solution x = 12 A: x + 6 + 18 B: 6x = 18 C: 18 + x = 6 9 The actual test will be very similar to this practice test. Please see the attached pages for study notes to help you do each type of question. Review Sheet for Math Midterm Exam 57. Let one white tile represent +1 and one black tile represent -1. Write the integer modeled by this set of tiles. ■ □ ■ □ ■ □ 58. Let one white tile represent +1 and one black tile represent -1. Add. □ ■ ■■ ■ □□□ ■ ■■ ■ □□□ + ■ ■ ■■ □ □ ■ ■ 59. Add (-7) + (-4) 60. Copy and complete. (-3) + = (+5) 61. Write the addition equation modeled by the number line. 62. Write the subtraction equation modeled by the number line. 63. Write 0.48 as a fraction in simplest form. 7 64. Write as a decimal. 4 5 65. Write 1 as a decimal. 6 10 The actual test will be very similar to this practice test. Please see the attached pages for study notes to help you do each type of question. Review Sheet for Math Midterm Exam 66. Use any method. Order the following numbers from least to greatest. 8 5 3 27 , ,2 , 3 2 5 10 67. Multiply. 3.6 x 4.4 68. Evaluate. (14.4 – 2.5) x 4.2 – 2.16 ÷ 0.6 69. A strip of cardboard measures 3 cm by 36 cm. a) What is the diameter of the largest circle you can cut from the strip? b) How many circles can you cut from the strip? 70. A bicycle wheel has a radius of 36 cm. How far will the bicycle travel if the wheel makes 400 rotations? 71. Calculate the area of the semicircle. Round your answer to the nearest square centimeter. ● 21.2 cm Problems 72. Write the least 3-digit number that is divisible by 3 and by 4. 73. A number is divisible by 5 and by 6. List at least 5 factors of that number. 74. The diagram consists of a square and a parallelogram. Find the area of the figure. 11 The actual test will be very similar to this practice test. Please see the attached pages for study notes to help you do each type of question. Review Sheet for Math Midterm Exam 75. The outer circle has a diameter of 10 cm. The two smaller circles are identical. What is the total area of the shaded regions? Round your answer to two decimal places. Show your work. 12 The actual test will be very similar to this practice test. Please see the attached pages for study notes to help you do each type of question. Review Sheet for Math Midterm Exam 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, 2, 51. Divisibility Rules: A number is said to be divisible by another number if the second number divides evenly into it and there is no remainder. For example 15 is divisible by 5 because when you divide 5 into 15 the answer is a whole number (3) and there is no remainder. Divisibility rules are rules that we can use to quickly check if a number can divide evenly into another number without having to do the actual division. The divisibility rules that students should know are as follows: A whole number is divisible by: 2 if the number is even (ends in 0, 2, 4, 6 or 8). 3 if the sum of the digits is divisible by 3. 4 if the number represented by the last two digits is divisible by 4. 5 if the ones digit is 0 or 5. 6 if the number is divisible by both 2 and 3. 8 if the number represented by the last 3 digits is divisible by 8. 9 if the sum of the digits is divisible by 9 10 if the ones digit is 0. (See Math Makes Sense 7, p. 6-13 for additional information and practice) 3, 4, 52, 53, 55, 56. A variable is a letter, such as n, that represents a number that can vary. An algebraic expression is a mathematical expression containing a variable; for example, 6x – 4 is an algebraic expression. Some examples of algebraic expressions and their meanings are as follows. In each case, n represents the number. Three more than a number: 3 + n or n + 3 Seven times a number: 7n Eight less than a number: n – 8 n A number divided by 20: 20 When we replace a variable with a number in an algebraic expression, we evaluate the expression. That is, we find the value of the expression for a particular value of the variable. (See Math Makes Sense 7, p. 16-19 for additional information and practice) 13 The actual test will be very similar to this practice test. Please see the attached pages for study notes to help you do each type of question. Review Sheet for Math Midterm Exam 5, 7. When we compare or relate a variable to an expression that contains the variable, we have a relation. A relation can be used to describe a pattern and to solve problems. In a growing pattern of numbers, a relation tells how each term of the pattern relates to the term number, rather than how each term relates to the term that comes before it. From a table of values, a relationship can be found. This relationship must be true for all terms in the table. Term # 1 2 3 4 5 6 Term 7 14 21 28 35 42 In the above table, if you look at Term 1, you might conclude that Term# + 6 = Term. This works for Term #1 because 1+6 = 7 but it doesn’t work for Term #2 because 2 + 6 does not equal 14, so this is not the relationship. You might also conclude that the relationship is Term# x 7 = Term. This works for Term#1, because 1 x 7 = 7. When we check it with the other Term #’s we see that it also works. i.e.) 2 x7 = 14, 3 x 7 =21 … Therefore we can conclude that we have found the relationship between Term # and Term for the whole table. Once we know the relationship for the table, we can use it to find the value for any Term if give the Term #, without having to calculate all the terms in between. For example, if the above table were extended the value of the term for Term # 500 would be 500 x 7 = 3500. (See Math Makes Sense 7, p. 20-24 for additional information and practice) 6. Perimeter is the distance around an object. The formula for the perimeter is: Perimeter = length + width + length + width or Perimeter = (2 x length) + (2 x width) x x+3 In the rectangle above, Perimeter = x + (x+3) + x + (x+3) = 4x +6 (See Math Makes Sense 7, p. 20-24 for additional information and practice) 14 The actual test will be very similar to this practice test. Please see the attached pages for study notes to help you do each type of question. Review Sheet for Math Midterm Exam 8. When a relation is represented by a table with consecutive Input numbers, a pattern can be observed in the Output numbers. When a relation is represented as a table of values, we can write the relation using algebra. Make sure that the relation is true for all pairs of input/output numbers. (See Math Makes Sense 7, p. 25-28 for additional information and practice) 9, 10. An equation is a mathematical statement that two expressions are equal. An equation must contain an “equal” sign. The following are examples of equations: 3n – 8 = 55 x + 5 = 10 k + 5 = 2k – 7 (See Math Makes Sense 7, p. 35-37 for additional information and practice) 11, 12, 57. 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. Colored tiles can be used to represent integers. A tile representing + 1 and a tile representing -1 form a zero pair. These tiles combine to model zero. ■■■■■ □□□□□□ In the example above each white tile represents + 1 and each black tile represents -1. Each of the five black tiles can be combined with a white tile to form a zero pair, which equals zero. When this is done there is still one white tile left over, so the total value of all the tiles combined is +1. (See Math Makes Sense 7, p. 52-55 for additional information and practice) 13, 14, 58, 59, 60, 61. Colored tiles, as explained above can be used to add integers. Number lines may also be used. An arrow to the right represents a positive integer. An arrow to the left represents a negative integer. The following number line shows the equation (+5) + (-8) = (-3) 15 The actual test will be very similar to this practice test. Please see the attached pages for study notes to help you do each type of question. Review Sheet for Math Midterm Exam For additional information and practice see the following from Math Makes Sense 7 Adding Integers with Tiles: p.56-59 Adding Integers on a Number Line: p.60-64 Instead of using tiles or a number line, the following rules may be used to add integers: 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 15. Colored tiles can be used to subtract integers. The concept of zero pairs may be used to subtract integers. To add integers, we combine groups of tiles. To subtract integers, we do the reverse, we remove them from a group. Adding a zero pair to a set of tiles does not change its value. For example, (- 3) + 0 = -3. If we do not have enough positive or negative tiles to take away from a group, we can add zero pairs until we have enough to take away. For example, for the question (+3) – (-2), we can do the following. We can represent + 3 as follows: □□□ If we want to subtract (-2) from this, we can’t because there are no black tiles to take away. However, if we add two zero pairs to the model it will look different as shown below but the value will still be the same. □□□□□ ■■ Now we have two black tiles, representing (-2) to take away. When we do, we are left with 5 white tiles as shown below, which has a value of (+5). □□□□□ Therefore (+3) – (-2) = (+5). 16 The actual test will be very similar to this practice test. Please see the attached pages for study notes to help you do each type of question. Review Sheet for Math Midterm Exam In addition to using tiles to subtract, we can also use a number line as explained later, or we can use the following rule to subtract integers: Instead of subtracting an integer, we can add its opposite. Example: (-6) – (+7) = (-6) + (-7) = (-13) Example: (-8) – (-5) = (-8) + (+5) = (-3) (See Math Makes Sense 7, p. 66-70 for additional information and practice) 16, 17, 61, 62. A number line can be used to subtract an integer. Usually this is done on a horizontal number line, but it can also be done on a vertical number line as in this question or as can be found on a thermometer. To subtract a positive integer on a number line, you move to the left. To subtract a negative integer is the same as adding its opposite, so you move to the right. On a vertical number line, to subtract a positive integer, you move down. To subtract a negative integer is the same as adding its opposite, so you move up. (See Math Makes Sense 7, p. 71-75 for additional information and practice) 18, 64. To convert a fraction to a decimal, you can divide the numerator (top number in the fraction) by the denominator (bottom number in the fraction). For example, ¾ as a decimal is equal to 3 ÷ 4, which equals 0.75. Because this has a definite number of decimal places, it is said to be a terminating decimal. To change 2/3 to a decimal, we calculate 2 ÷ 3 and get a result of 0.6666666666….When we do this we can never finish the calculation because some of the digits repeat forever. This is called a repeating decimal and it is written as 0. 6 . Sometimes it is easier to convert a fraction to a decimal if you first write it in simplest terms. For example 20/36 can be rewritten as 5/9, which is easier to divide. (See Math Makes Sense 7, p. 86-90 for additional information and practice) For certain fractions another method can be used to convert it to a decimal if an equivalent fraction can be found that has a denominator that is a power of 10 (10, 100, 1000,…) See sheet Fractions to Decimals. 19, 20, 21. When adding decimals, remember to line up the decimal point. 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. When subtracting decimals, remember to line up the decimal points. Also remember that you can use the inverse operation (addition) to check addition. You can also use estimation to check if your answer is reasonable. (See Math Makes Sense 7, p. 96- 99 for additional information and practice) 17 The actual test will be very similar to this practice test. Please see the attached pages for study notes to help you do each type of question. Review Sheet for Math Midterm Exam 22, 67. When multiplying decimals the decimals don’t have to be lined up. Instead just line up the numbers and multiply normally. When you have a final answer, you then need to decide where to place the decimal point. To do this, you add the total number of decimal places in the factors (the numbers being multiplied). There will be that many decimal places in the answer. For example if you multiply a number that has one decimal place by a number that has three decimal places, the product would have a total of four decimal places. 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 Canadian Mathematics 7, p. 94 for additional information and practice) 23. When dividing decimals, you need to start by making sure the divisor is a whole number. If it isn’t you must multiply both the divisor and the dividend by a power of 10 to make it a whole number. In the given question, 836.4 ÷3.4, you would begin by multiplying both the dividend and the divisor by 10 to get a new question, 8364 ÷34. Then you would divide normally. When you divide, place the decimal in the quotient (the answer) directly above the decimal in the dividend. If the divisor had two decimal places, you would have to multiply both the divisor and dividend by 100. If it had three decimal places, you would have to multiply by 1000, etc. Also remember that you can use the inverse operation (multiplication) to check division. You can also use estimation to check if your answer is reasonable. (See assignment Canadian Mathematics 7, p. 95 for additional information and practice) 24, 68. Order of Operations is the rules that are followed when simplifying or evaluating an expression. A word that is helpful in remembering the order the steps are to be done in is BEDMAS. In BEDMAS, each letter stands for an operation: B: Brackets E: Exponents D: Division or M: Multiplication A: Addition or S: Subtraction Division and Multiplication are at the same spot in the order so are simply done left to right. Addition and Subtraction are the same and are simply worked from left to right. The order of operations for whole numbers can be applied to decimals. (See Math Makes Sense 7, p. 108-109 for additional information and practice). 18 The actual test will be very similar to this practice test. Please see the attached pages for study notes to help you do each type of question. Review Sheet for Math Midterm Exam 25. Any fraction or decimal can be written as a percent, and vice versa. Percent is another name for hundredths. To change a percent to a decimal, begin by dropping the % sign and then divide by 100; (move the decimal point 2 places to the left). Example #1: Write 82% as a decimal. Begin by dropping the % sign. 82 Then divide by 100. 82÷100 = 0.82 Example #2: Write 40% as a decimal. Begin by dropping the % sign. 40 Then divide by 100. 40 ÷ 100 = 0.40 or 0.4 (See Math Sheet: Decimals to Percents and Math Makes Sense 7, p.111-113 for additional information and practice). 26. Some fractions can easily be converted to percents by first finding equivalent fractions with a denominator of 100 and then rewriting it as a percent. Example: Write 15/20 as a percent. Begin by writing an equivalent fraction for 15/20. 15/20 = 75/100 (Both the numerator and denominator were multiplied by 5) Then rewrite as a percent. 75/100 = 75% If the fraction cannot be written as an equivalent fraction with a denominator of 100, then a different method is used to convert the fraction to a percent. In such cases, you would divide the numerator by the denominator to get a decimal. Then you would move the decimal two places to the right and add a % sign. Example: Write 3/16 as a percent. Begin by dividing the numerator (3) by the denominator (16) 3 ÷ 16 = 0.1875 Then move the decimal point two places to the right and add a % sign. 18.75% (See Math Sheet: Fractions to Percents and Math Makes Sense 7, p.111-113 for additional information and practice). 19 The actual test will be very similar to this practice test. Please see the attached pages for study notes to help you do each type of question. Review Sheet for Math Midterm Exam 27. To find out what fraction of a diagram is shaded, you would first count the number of shaded parts. This would be the numerator or top number in your fraction. Then you would count the total number of parts (shaded + unshaded). This would be your denominator in the fraction. This assumes that all of the parts are an equal size. /////////////////////// /////////////////////// /////////////////////// /////////////////////// In the diagram above 4 parts are shaded, so 4 is the numerator. Altogether there are 10 parts, so 10 is the denominator. The fraction that is shaded then is 4/10. Sometimes fractions can be simplified into lower terms if there is a number that divides evenly into both the numerator and denominator. In this case, both 4 and 10 can be divided by 2, to get 2 and 5 respectively. Therefore, this fraction in lowest terms is 2/5. 28. To find out what fraction of the diagram is shaded, the above method can be used. To extend this to show the fraction as a percent, the fraction would need to be converted to a percent as explained in question 26. (See Math Makes Sense 7, p.111-113 for additional information and practice). 29. To find your success in making free throws as a percent, you would first need to express it as a fraction. The numerator would be your successful free throws. The denominator would be your total attempts. Then the fraction would need to be converted to a percent as explained in question 26. (See Math Makes Sense 7, p.111-113 for additional information and practice). 30, 31. To find a percent of a number, you can convert the percent to a decimal and then multiply. To change a percent to a decimal, begin by dropping the % sign and then divide by 100; (move the decimal point 2 places to the left). Example: Write 82% as a decimal. Begin by dropping the % sign. 82 Then divide by 100. 82÷100 = 0.82 For a question such as this you would do the following: Calculate 35% of 50. 35% = 0.35 50 x 0.35 = 17.5 (See Math Sheet, Percents to Decimals and Math Makes Sense 7, p.114-116 for additional information and practice). 20 The actual test will be very similar to this practice test. Please see the attached pages for study notes to help you do each type of question. Review Sheet for Math Midterm Exam 32, 33, 37. Radius is the distance from the center of a circle to any point on the circle. Diameter is the distance across a circle, measured through its center. Circumference is the distance around a circle, also known as the perimeter of a circle. The diameter is twice as long as the radius. d = 2r The radius is half the length of the diameter. r = d/2 The circumference is equal to 2 times π (pi) times the radius or π times the diameter. To calculate use 3.14 in place of π. C=2πr or C=πd To calculate the diameter if you are given the radius, multiply the radius by 2. To calculate the radius if you are given the diameter, divide the diameter by 2. To calculate the circumference if you are given the radius, multiply the radius by 2 and then by 3.14. To calculate the circumference if you are given the diameter, multiply the diameter by 3.14 To calculate the radius if you are given the circumference, divide the circumference by 2 and then by 3.14. To calculate the diameter if you are given the circumference, divide the circumference by 3.14. (See Math Makes Sense 7, p.130-137 for additional information and practice). 34, 66. Before fractions and decimals can be put in order they must first be converted into similar units. A few methods can be used to put numbers in order. Benchmarks: If a number line is drawn that includes benchmarks such as 1 , 1, 1 1 , 2 … 2 2 then the numbers to be compared can be positioned on the number line. Then they can be put in order by simply reading from the lowest number on the left to the highest number on the right. Equivalent Fractions: Before fractions can be compared, they must have the same denominator. If all of the denominators are the same, then the numerators can be compared. The fraction with the lowest numerator is the least and the number with the highest numerator is the greatest. 21 The actual test will be very similar to this practice test. Please see the attached pages for study notes to help you do each type of question. Review Sheet for Math Midterm Exam Decimals: If all of the numbers are converted to decimals, they can then be compared. When comparing decimals, it is sometimes easier if you add zeroes to the end of each number to make sure they all have the same number of decimal places. 2 Before mixed fractions (such as 3 5 ) can be compared or converted to equivalent fractions or to decimals, they should be written as improper fractions ( 17 ). To convert a mixed fraction to an 5 improper fraction (where the numerator is larger than the denominator), you must first multiply the denominator by the whole number part of the mixed fraction. Then add this to the numerator to get a new sum. Then place this sum as the numerator over the original denominator. In the example above, 5 x 3=15. Then add the numerator: 15+ 2= 17. Then place this over the original denominator: 175 2 17 Therefore 3 5 = 5 (See Math Makes Sense 7, p.91-95 for additional information and practice). 35. To complete the missing numbers in sequences, you need to find the pattern of the numbers that are given and then extend the pattern. When adding decimals, remember to line up the decimal point. Also remember that you can use the inverse operation (subtraction) to check addition. You can also use front-end estimation to check if your answer is reasonable. When subtracting decimals, remember to line up the decimal points. Also remember that you can use the inverse operation (addition) to check addition. You can also use front-end estimation to check if your answer is reasonable. (Front –end estimation is simply ignoring the part of the number after the decimal point and calculating the rest to get an approximate number. This won’t give you an exact answer but it will help you to see if your answer is reasonable and to show you if your decimal point has been reasonably placed. Example) By using front-end estimation on the question 15.147 + 3.3694, I can simply add 15 + 3 to know that the answer should be somewhere around 18. Adding and subtracting decimals are extensions of adding and subtracting whole numbers. Therefore rules such as carrying and borrowing still apply. (See Math Makes Sense 7, p.96-99 for additional information and practice). 36. When you multiply decimals, you need to line up the numbers rather than the decimal points. Ignore the decimal points while calculating, but then insert the decimal into your final answer. To determine where to insert the decimal point in the final product you must count the total number of decimal places (places after the decimal point) in the each of the numbers being multiplied. The final product will have this many decimal places. In the following example, a number with 3 decimal places is multiplied by a number with 1 decimal place. The final product, then, has 3 + 1 or 4 decimal places. 22 The actual test will be very similar to this practice test. Please see the attached pages for study notes to help you do each type of question. Review Sheet for Math Midterm Exam 2.083 x 33.2 69.1556 Front-end estimation may also be used to help place the decimal point. In the above example, if we ignore the numbers after the decimal point, we would multiply 2 x 33 = 66. Therefore, our final product should be somewhere around 66. If we had misplaced our decimal point so our answer read 6.91556 or 691.556 we could quickly tell that our answer wasn’t reasonable. (See Math Assignment, Canadian Mathematics 7, p. 94 for additional information and practice). 37. See Question 32. 38. For some questions, it is easier to visualize what is being asked for if you draw a diagram. A diagram for this question may look like the following. 6.4 cm The distance across both circles would be equal to 2 times the diameter or 4 times the radius. Therefore the radius of one circle is 6.4 cm ÷ 4 or 1.6 cm. (See Math Makes Sense 7, p.130-132 for additional information and practice). 39, 40. See question 32. C=2πr or C=πd To calculate use 3.14 for π. (See Math Makes Sense 7, p.133-137 for additional information and practice). 41. See question 32. r=C÷2π To calculate use 3.14 for π. (See Math Makes Sense 7, p.133-137 for additional information and practice). 23 The actual test will be very similar to this practice test. Please see the attached pages for study notes to help you do each type of question. Review Sheet for Math Midterm Exam 42. See question 32. d = C ÷ π To calculate use 3.14 for π. (See Math Makes Sense 7, p.133-137 for additional information and practice). 43, 44. The area of a parallelogram is equal to the area of a rectangle with the same base and height. To find the area of a parallelogram, multiply its base by its height. The formula for the area of a parallelogram is A = b x h. (See Math Makes Sense 7, p.139-142 for additional information and practice). 45, 46. The area of a triangle is one-half the area of a parallelogram with the same base and height. (See Math Makes Sense 7, p.143-147 for additional information and practice). 47. The area of a circle is π (or 3.14) multiplied by the square of its radius r; that is A= π r 2 . For this circle the radius is 4.2 m. The area would be 3.14 x 4.2m x 4.2m = 55.3896 m 2 To round the area to two decimal places (hundredths), you would need to check what number was in the next decimal place (thousandths). If it were 0-4 you would simply round off. (Drop everything after the hundredths place). If it were 5-9 you would round the number in the hundredths place up to the next number. In the example above, there is a 9 in the thousandths place, so you would round the digit in the hundredths’ place up from an 8 to a nine and drop everything after that. The area then would be rounded from 55.3896 m 2 to 55.39 m 2 . C(See Math Makes Sense 7, p.148-152 for additional information and practice). 24 The actual test will be very similar to this practice test. Please see the attached pages for study notes to help you do each type of question. Review Sheet for Math Midterm Exam 48. A circle graph is a diagram that uses parts of a circle to display data. In a circle graph, data are shown as parts of one whole. Each sector of a circle represents a percent of the whole circle. The whole circle represents 100%. If one sector of the graph was to represent 25% of the data, then it would be represented by a piece that took up 25% of the total circle. To estimate how much of a circle graph is represented by different data, you need to estimate how much of the total circle is represented by that data. If the percents are given in each sector, this can be used to accurately determine how much data is represented by each sector. (See Math Makes Sense 7, p.156-160 for additional information and practice). 49. To calculate how many stamps Carol has from each country, you need to multiply the percent given in each sector by the total number of stamps she has. For example, 25% of her stamps are from Italy so this would be 25% of 500 which equals 125. See question 30 for instructions on finding the percent of a number). To find how many more stamps she has from one country as compared with another country you need to find the difference by subtracting. (See Math Makes Sense 7, p.156-160 for additional information and practice). 50. The central angle of a sector on a circle graph is the angle located at the center of the circle for that sector. In a complete circle, there are 360°. On this circle graph, 25% of the budget is for rent. Therefore, the sector for rent takes up 25% of the graph and the central angle for that sector is 25% of 360° or 90°. To calculate the central angle, you must multiply the percent that represents the data by 360°. (See Math Makes Sense 7, p.161-164 for additional information and practice). 25 The actual test will be very similar to this practice test. Please see the attached pages for study notes to help you do each type of question. Review Sheet for Math Midterm Exam 51. 1 2 4 6 3 17 5 8 10 12 9 19 11 14 16 18 15 13 20 Divisible by 2 Divisible by 3 A Venn Diagram is used to compare and contrast different items, in this case numbers. In this Venn Diagram, the circle on the left contains all of the numbers up to 20 that are divisible by 2. The circle on the left shows all of the numbers that are divisible by 3. The overlapping region shows numbers that are divisible by both 2 and 3. Those numbers that do not fit into either circle because they are not divisible by 2 or 3 are placed outside the circles. See question 1 for information about divisibility rules. (See Math Makes Sense 7, p.8-12 for additional information and practice. Also see question 1). 52, 53. See question 3. 54. Since all sides of a square are the same length, the perimeter or distance around the square is equal to 4 times the length of one side. For additional information on relationships in patterns, see question 5. 55, 56. See Question 3. 57. See Question 11. 58-60. See Question 13. (See Math Makes Sense 7, p. 20-24 for additional information and practice) 61-62. See Question 16. 63. To change a decimal to a fraction, begin by seeing how many decimal places are in the decimal. If the decimal has only one decimal place, then write the number as a fraction with a denominator of 10. If there are two decimal places, write the fraction with a denominator of 100. After you have written the number as a fraction, it can sometimes be simplified by writing an equivalent fraction in lower terms. To do this simply divide the numerator and denominator by the same number, a number that divides evenly into both numbers. Example #1: Write 0.62 as a fraction. 26 The actual test will be very similar to this practice test. Please see the attached pages for study notes to help you do each type of question. Review Sheet for Math Midterm Exam Begin by counting the number of decimal places. There are two decimal places (numbers after the decimal), which means that the decimal extends to the hundredths’ place, so the fraction will have a denominator of 100. 0.62 = 62/100 This can be simplified by dividing both the numerator and denominator by 2. 62/100 = 31/50 Example #2: Write 0.4 as a fraction. 0.4 = 4/10 = 2/5 (See Math sheet Decimals to Fractions and Math Makes Sense 7, p. 86-90 for additional information and practice). 64. See Question 18. 65. Before converting a mixed number to a decimal you should first change it to an improper fraction. See notes for questions 34 & 18. (See Math Makes Sense 7, p. 86-90 for additional information and practice) 66. See Question 34. 67. See Question 22. 68. See Question 24. 69. To solve this problem, it would be good to draw a diagram to help you visualize the problem. The diameter would be the same as the width of the cardboard. The number of circles that could be cut from the strip would the same as the number of times 42 cm could be divided by 3.5 cm. 3.5 cm 42 cm (See Math Makes Sense 7, p. 86-90 for additional information and practice) 27 The actual test will be very similar to this practice test. Please see the attached pages for study notes to help you do each type of question. Review Sheet for Math Midterm Exam 70. A bicycle wheel is a circle. The circumference of the wheel is the distance around the outside of the circle. The circumference is the distance the bike would travel in one rotation of the wheel. See question 32. C=2πr To calculate the distance the bike would travel you have to multiply the radius by 2 and then by π (3.14) to see how far it would travel in one rotation of the wheel. Then you would have to multiply that total by 500 because there were 500 rotations of the wheel. (See Math Makes Sense 7, p. 133-137 for additional information and practice) 71. See question 47 for instructions on calculating the area of a circle. Since a semicircle is simply half of a complete circle, you would need to divide the area of the circle by 2 to find the area of the semicircle. (See Math Makes Sense 7, p. 148-152 for additional information and practice). 72. Use divisibility rules to list some of the numbers, starting at 100 (the first 3-digit number) that are divisible by 5. (The number ends in a 5 or 0). Then check which is the first of these numbers that is also divisible by 9 (The sum of the digits is divisible by 9). For similar questions, use the same process, though you will have to use different divisibility rules if different numbers are used. (See Math Makes Sense 7, p. 10-13 for additional information and practice). 73. If a number is divisible by two different numbers, then it must also be divisible by any of the factors of either of those numbers and by any of the factors of one of those numbers multiplied by any of the factors of the other number. The factors of 8 are 1, 2, 4 and 8. The factors of 9 are 1, 3 and 9. Therefore, any number that is divisible by both 8 and 9 must also be divisible by 1, 2, 3 and 4 and by any of the factors of one of the numbers multiplied by any of the factors of the other number: by 6 (2 x 3), by 12 (3 x 4), by 18 (2 x 9), by 24 (3 x 8), by 36 (4 x 9) and by 72 (8x9). (See Math Makes Sense 7, p. 12 for additional information). 28 The actual test will be very similar to this practice test. Please see the attached pages for study notes to help you do each type of question. Review Sheet for Math Midterm Exam 74. To solve this problem, you should break the question into parts. Calculate the area of the square first. A = L x W or 4 x side (since all sides of a square are the same length.) Then calculate the area of the parallelogram. A = b x h. (See question 43). Then add the area of the square and the parallelogram together. Then subtract the area of the triangle since this part is overlapping and you don’t want to count it twice. A = 1 b x h (See question 45). 2 (See Math Makes Sense 7, p. 139-147 for additional information and practice). 75. To solve this problem, you should break the question into parts. First calculate the area of the large circle. (See question 47) A= π r 2 . The diameter of this circle is given. To find the radius, you need to divide the diameter by 2. Next find the area of one of the smaller circles. Each circle is half the diameter of the larger circle and each radius is half the diameter of a smaller circle. Therefore, the radius of each of the smaller circles is the diameter of the larger circle divided by 4. A= π r 2 . To find the total area of both smaller circles, you need to multiply the area of one of the circles by 2. Finally, subtract the total area of both smaller circles from the total area of the large circle. This will leave the total area of the shaded regions. (See Math Makes Sense 7, p. 148-152 for additional information and practice). 29 The actual test will be very similar to this practice test. Please see the attached pages for study notes to help you do each type of question.