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Math in the Real World Unit (Level IV Graduate Math) Draft (NSSAL) C. David Pilmer ©2009 (Last Updated: April 2015) This resource is the intellectual property of the Adult Education Division of the Nova Scotia Department of Labour and Advanced Education. The following are permitted to use and reproduce this resource for classroom purposes. Nova Scotia instructors delivering the Nova Scotia Adult Learning Program Canadian public school teachers delivering public school curriculum Canadian non-profit tuition-free adult basic education programs Nova Scotia Community College instructors The following are not permitted to use or reproduce this resource without the written authorization of the Adult Education Division of the Nova Scotia Department of Labour and Advanced Education. Upgrading programs at post-secondary institutions (exception: NSCC) Core programs at post-secondary institutions (exception: NSCC) Public or private schools outside of Canada Basic adult education programs outside of Canada Individuals, not including teachers or instructors, are permitted to use this resource for their own learning. They are not permitted to make multiple copies of the resource for distribution. Nor are they permitted to use this resource under the direction of a teacher or instructor at an unauthorized learning institution. Acknowledgments The Adult Education Division would also like to thank the following NSCC instructors for piloting this resource and offering suggestions during its development. Eileen Burchill (IT Campus) Elliott Churchill (Waterfront Campus) Barbara Leck (Pictou Campus) Suzette Lowe (Lunenburg Campus) Nancy Harvey (Akerley Campus) Floyd Porter (Strait Area Campus) Brian Rhodenizer (Kingstec Campus) Joan Ross (Annapolis Valley Campus) Tanya Tuttle-Comeau (Cumberland Campus) Jeff Vroom (Truro Campus) Table of Contents Introduction…………………………………………………………………………... Negotiated Completion Date………………………………………………………… The Big Picture……………………………………………………………………….. Course Timelines……………………………………………………………………... iii iii iv v The Basics Order of Operations…………………………………………………………………... Fractions………………………………………………………………………………. Adding Fractions………………………………………………………………….. Subtracting Fractions……………………………………………………………... Multiplying and Dividing Fractions……………………………………………… Decimals……………………………………………………………………………… Percents………………………………………………………………………………. Ratio and Proportion……………………………………………………..…………... Signed Numbers……………………………………………………………………… All Together Now, Part I……………………………………………………………... All Together Now, Part II…………………………………………………………….. All Together Now, Part III……………………………………………………………. Reflect Upon Your Learning…………………………………………………………. 1 3 3 7 10 12 18 22 25 30 31 32 33 Math Games and Puzzles Addition Pyramids …………………………………………………………………… Find the Center Puzzles ……………………………………………………………… KenKen Puzzles ……………………………………………………………………… RAD Puzzles …………………………………………………………………………. Fraction Fury Puzzles ………………………………………………………………... Putting It Together: Math Games and Puzzles ………………………………………. Reflect Upon Your Learning ………………………………………………………… 34 37 39 43 45 48 50 Applying Math to the Real World Applications - Fractions………………………………………………………………. Applications - Decimals………………………………………………………………. Applications - Percents……………………………………………………………….. Applications - Ratio and Proportion………………………………………………….. Applications - Signed Numbers………………………………………………………. Reflect Upon Your Learning…………………………………………………………. Putting It Together: Part I…………………………………………………………….. Putting It Together: Part II…………………………………………………………… Putting It Together: Part III…………………………………………………………... Putting It Together: Part IV………………………………………………………….. Is It Reasonable?........................................................................................................... Reflect Upon Your Learning…………………………………………………………. 51 55 59 64 70 74 75 80 85 90 95 99 Post-Unit Reflections…………………………………………………………………. 102 NSSAL ©2009 i Draft C. D. Pilmer Soft Skills Rubric ……………………………………………………………………. 103 Answers………………………………………………………………………………. 104 Online Support………………………………………………………………………... 127 NSSAL ©2009 ii Draft C. D. Pilmer Introduction This unit is designed to review fundamental mathematical concepts (e.g. operations with fractions, operations with signed numbers, order of operations,…), both in and out of context, that are necessary to be successful in this course and in post-secondary studies that do not require an academic math credit. There are some requirements for this unit. 1. The unit will not be cut into sections such that learners will be assessed on only one mathematical concept at a time. That means you will not have a separate test on fractions, a separate test on decimals, and separate tests for the other concepts covered in this unit; tests will cover multiple concepts. 2. Most of the questions will be completed using paper-and-pencil techniques. Calculators will only be used on questions identified with the calculator icon (). 3. Learners are required to successfully complete the outcome and elaborations associated with math games and puzzles. That means all learners are expected to successfully complete a test on this material without the use of a calculator. 4. Learners who have completed Level III math are not required to complete all of this resource. They are only required to complete the following sections and be tested on all of these sections. Putting It Together: Math Games and Puzzles All Together Now, Parts I through III Putting It Together, Parts I though IV Is It Reasonable? Negotiated Completion Date After working for a few days on this unit, sit down with your instructor and negotiate a completion date for this unit. Start Date: _________________ Completion Date: _________________ Instructor Signature: __________________________ Student Signature: NSSAL ©2009 __________________________ iii Draft C. D. Pilmer The Big Picture The following flow chart shows the six required units and the four optional units (choose two of the four) in Level IV Graduate Math. These have been presented in a suggested order. Instructors and students may choose to alter this order to best serve the needs of the learner. Not all of the units take the same amount of time to complete. For example, the Consumer Finance Unit is quite short. By contrast, the Graphs and Function Unit and the Measurement Unit (A and B) tend to take the greatest amount of time. Math in the Real World Unit (Required) Fractions, decimals, percentages, ratios, proportions, and signed numbers in real world applications Math Games and Puzzles Solving Equations Unit (Required) Solve and check equations of the form Ax B Cx D , A Bx 2 C , and A Bx 3 C . Consumer Finance Unit (Required) Simple Interest and Compound Interest TVM Solver (Loans and Investments) Graphs and Functions Unit (Required) Understanding Graphs Linear Functions and Line of Best Fit Measurement Unit (Required) Part A: Imperial and Metric Measures Part B: Perimeter, Area and Volume Angles and Triangles Unit (Required) Angle and Line Relationships Similar Triangles Pythagorean Theorem Choose two of the four. Linear Functions and Systems of Equations Unit NSSAL ©2009 Trigonometry Unit iv Descriptive Statistics Unit Numeracy Unit Draft C. D. Pilmer Course Timelines Graduate Level IV Math is a two credit course within the Adult Learning Program. As a two credit course, learners are expected to complete 200 hours of course material. Since most ALP math classes meet for 6 hours each week, the course should be completed within 35 weeks. The curriculum developers have worked diligently to ensure that the course can be completed within this time span. Below you will find a chart containing the unit names and suggested completion times. The hours listed are classroom hours. Unit Name Minimum Completion Time in Hours 24 20 15 25 22 14 18 18 Total: 156 hours Math in the Real World Unit Solving Equations Unit Consumer Finance Unit Graphs and Functions Unit Measurement Unit (A & B) Angles and Triangles Unit Selected Unit #1 Selected Unit #2 Maximum Completion Time in Hours 34 28 18 30 30 16 22 22 Total: 200 hours As one can see, this course covers numerous topics and for this reason may seem daunting. You can complete this course in a timely manner if you manage your time wisely, remain focused, and seek assistance from your instructor when needed. NSSAL ©2009 v Draft C. D. Pilmer Order of Operations There is a specific order one must follow when calculating the value of an expression. 1. 2. 3. 4. Perform all operations within grouping symbols like brackets “[ ]”, and parentheses “( )”. Evaluate expressions with exponents. Perform multiplications and divisions in order from left to right. Perform additions and subtractions in order from left to right. People often remember this order using the acronym, BEDMAS. B - brackets E - exponents DM - division and multiplication AS - addition and subtraction Example 1: Evaluate 3 58 2 . Example 2: Evaluate 52 7 4 2 3 . 3 58 2 3 56 parentheses 3 30 multiplication 33 addition 52 7 4 2 3 Example 3: 2 Evaluate 17 8 23 4 5 . parentheses exponents multiplication 22 addition t hen subtraction Example 4: Evaluate 17 82 23 4 5 7 23 9 . 23 3 Here the fraction bar is like a grouping symbol. We simplify above and below the fraction bar separately. 7 23 9 23 3 769 83 10 5 2 92 23 4 5 parentheses 81 8 4 5 exponents 81 2 5 84 52 3 2 3 25 3 2 3 25 3 6 division subtraction then addition Questions 1. Evaluate each expression. (a) 14 35 2 NSSAL ©2009 (b) 3 45 8 1 Draft C. David Pilmer (c) 100 32 2 (d) 10 7 2 4 (e) 7 2 3 3 1 2 (f) 7 3 20 5 1 (g) 2 33 1 4 (h) 38 2 1 1 (i) 17 36 4 (j) 23 5 6 5 3 (k) 82 8 32 4 6 (l) 30 6 4 1 2 (m) 216 13 4 1 (n) 45 2 32 6 19 17 (o) 2 5 42 12 2 3 (p) 1 7 2 2 2 (q) NSSAL ©2009 2 3 79 6 26 4 14 10 8 18 24 3 (r) 2 Draft C. David Pilmer Adding Fractions To add like fractions (fractions that have the same denominator), add the numerators and write the sum over the common denominator. Express your answer in simplest form. Example 1: 3 2 7 7 5 7 The two fractions have a common denominator of 7. Example 2: 3 5 16 16 8 16 1 2 The two fractions have a common denominator of 16. Example 3: 7 6 8 8 13 8 5 1 8 The two fractions have a common denominator of 8. You have 3 sevenths plus 2 sevenths. The answer will be 5 sevenths. You have 3 sixteenths plus 5 sixteenths. The answer will be 8 sixteenths. The fraction is changed to its simplest form by dividing both the numerator and denominator by 8. You have 7 eighths plus 6 eighths. The answer will be 13 eighths. 13 is called an improper fraction. Improper fractions occur 8 when the numerator is larger than the denominator. The 5 improper fraction can be changed to the mixed number 1 . 8 To add unlike fractions (fractions that have different denominators), identify the least common multiple (LCM) of the denominators, write each fraction as an equivalent fraction whose denominator is the LCM, add the like fractions, and express the sum in simplest form. Example 4: 1 1 3 2 1 2 1 3 3 2 2 3 2 3 6 6 5 6 The denominators (3 and 2) are different. List the multiples of 3 and 2 separately. Multiples of Three: 3, 6, 9, 12, 15, … Multiples of Two: 2, 4, 6, 8, 10, 12, 14, … Notice that the least common multiple of 3 and 2 is 6. This means that the common denominator will be 6. Now you write each fraction as an equivalent fraction whose denominator is 6. You have 2 sixths plus 3 sixths. The answer is 5 sixths. NSSAL ©2009 3 Draft C. David Pilmer Example 5: 1 3 8 4 1 3 2 8 4 2 1 6 8 8 7 8 The denominators (8 and 4) are different. List the multiples of 8 and 4 separately. Multiples of Eight: 8, 16, 24, 32, 40, 48, … Multiples of Four: 4, 8, 12, 16, 20, 24, … Notice that the least common multiple of 8 and 4 is 8. This means that the common denominator will be 8. Now you write each fraction as an equivalent fraction whose denominator is 8. You have 1 eighth plus 6 eighths. The answer is 7 eighths. Example 6: 4 2 5 3 43 25 5 3 3 5 12 10 15 15 22 15 7 1 15 The denominators (5 and 3) are different. List the multiples of 5 and 3 separately. Multiples of Five: 5, 10, 15, 20, 25, … Multiples of Three: 3, 6, 9, 12, 15, 18, … Notice that the least common multiple of 5 and 3 is 15. This means that the common denominator will be 15. Now you write each fraction as an equivalent fraction whose denominator is 15. The improper fraction 7 22 is changed to the mixed number 1 . 15 15 To add mixed numbers, create a common denominator. Add the whole numbers, then add the fractions. If the fractional component is an improper fraction, some additional work will have to be done. Example 7: 1 4 7 5 6 9 1 3 4 2 7 5 63 9 2 3 8 7 5 18 18 11 12 18 11 12 18 NSSAL ©2009 Find the LCM of 6 and 9. Multiples of Six: 6, 12, 18, 24, 30, … Multiples of Nine: 9, 18, 27, 36, 45, … Now you write each fraction as an equivalent fraction whose denominator is 18. Add the whole numbers 7 and 5. Add the fractions 4 3 8 and . 18 18 Draft C. David Pilmer Example 8: 2 11 1 7 3 12 2 4 11 1 7 3 4 12 8 11 1 7 12 12 19 8 12 7 8 1 12 7 9 12 Find the LCM of 3 and 12. Multiples of Three: 3, 6, 9, 12, 15, … Multiples of Twelve: 12, 24, 36, 48, 60, … Now you write each fraction as an equivalent fraction whose denominator is 12. Add the whole numbers 1 and 7. Add the fractions Change the improper fraction 8 11 and . 12 12 19 7 to the mixed number 1 . 12 12 Add the whole numbers 8 and 1. Questions: 1. (a) 4 6 11 11 (b) 2 1 5 5 (c) 7 3 12 12 (d) 5 1 9 9 (e) 5 6 7 7 (f) 8 5 10 10 (g) 5 3 6 6 (h) 7 5 8 8 2. (a) 1 2 4 5 (b) 3 1 5 6 (c) 2 3 3 4 (d) 1 3 6 4 NSSAL ©2009 5 Draft C. David Pilmer (e) 3 1 8 6 (f) 4 5 9 6 (g) 2 1 3 12 (h) 4 8 5 15 2 1 3. (a) 3 6 5 4 1 5 (b) 4 7 3 8 1 7 (c) 1 2 6 12 4 1 (d) 6 1 7 2 5 3 (e) 2 4 6 8 1 3 (f) 4 2 5 10 NSSAL ©2009 6 Draft C. David Pilmer Subtracting Fractions Like addition, you need a common denominator to subtract fractions. To subtract like fractions (fractions that have the same denominator), subtract the numerators and write the difference over the common denominator. Express your answer in simplest form. Example 1: 5 2 7 7 3 7 The two fractions have a common denominator of 7. Example 2: 9 5 16 16 4 16 1 4 The two fractions have a common denominator of 16. You have 5 sevenths subtract 2 sevenths. The answer will be 3 sevenths. You have 9 sixteenths subtract 5 sixteenths. The answer will be 4 sixteenths. The fraction is changed to its simplest form by dividing both the numerator and denominator by 4. To add unlike fractions (fractions that have different denominators), identify the least common multiple (LCM) of the denominators, write each fraction as an equivalent fraction whose denominator is the LCM, add the like fractions, and express the sum in simplest form. Example 3: 2 1 3 2 2 2 1 3 3 2 2 3 4 3 6 6 1 6 Find the LCM of 3 and 2. Example 4: 5 1 8 4 5 1 2 8 4 2 5 2 8 8 3 8 Find the LCM of 8 and 4. NSSAL ©2009 Multiples of Three: 3, 6, 9, 12, 15, … Multiples of Two: 2, 4, 6, 8, 10, 12, 14, … Now you write each fraction as an equivalent fraction whose denominator is 6. You have 4 sixths subtract 3 sixths. The answer is 1 sixth. Multiples of Eight: 8, 16, 24, 32, 40, 48, … Multiples of Four: 4, 8, 12, 16, 20, 24, … Now you write each fraction as an equivalent fraction whose denominator is 8. You have 5 eighths subtract 2 eighths. The answer is 3 eighths. 7 Draft C. David Pilmer To subtract mixed numbers, create a common denominator. Subtract the fractions and then subtract the whole numbers. If the larger of the two mixed numbers does not have the larger proper fraction component, then borrow 1 from the whole number. Then add the 1 to the proper fraction to create an improper fraction. This will have to be done before the subtraction can occur. Example 5: 5 2 7 5 6 9 53 2 2 7 5 63 9 2 15 4 7 5 18 18 11 2 18 Example 6: 1 2 5 2 4 3 1 3 2 4 5 2 43 3 4 3 8 5 2 12 12 3 8 4 1 2 12 12 12 3 8 4 2 12 12 12 15 8 4 2 12 12 7 2 12 Example 7: 7 3 9 2 20 4 7 3 5 9 2 20 45 7 15 9 2 20 20 7 15 8 1 2 20 20 20 7 15 8 2 20 20 20 27 15 8 2 20 20 12 6 20 3 6 5 Questions: 9 2 10 10 (b) 11 3 12 12 (c) 7 1 8 8 (d) 9 3 16 16 2. (a) 3 2 4 3 (b) 4 1 5 2 (c) 5 1 6 4 (d) 5 1 8 6 1. (a) NSSAL ©2009 8 Draft C. David Pilmer (e) 5 1 6 2 (f) 9 3 10 5 4 1 3. (a) 6 2 5 3 3 1 (b) 8 3 4 6 8 1 (c) 7 1 9 6 1 5 (d) 4 2 3 7 2 4 (e) 9 3 3 5 1 7 (f) 5 1 2 8 NSSAL ©2009 9 Draft C. David Pilmer Multiplying and Dividing Fractions Multiplying Fractions To multiply two fractions, multiply the numerators and multiply the denominators. a c ac Therefore: Note: b and d are not equal to zero. b d bd Example 1: 4 3 5 7 43 5 7 12 35 Example 2: 3 2 10 5 3 2 10 5 6 50 3 25 Example 3: 3 2 4 5 13 4 5 1 52 5 2 10 5 Example 4: 1 1 1 2 2 6 3 13 2 6 39 12 13 4 1 3 4 Dividing Fractions To divide two fractions, multiply the first fraction by the reciprocal of the second fraction. Therefore: reciprocals a c a d b d b c ad bc Example 5: 3 4 7 5 3 5 7 4 15 28 NSSAL ©2009 Example 6: 3 7 4 20 3 20 4 7 60 28 15 7 1 2 7 Note: b, c, and d are not equal to 0. Example 7: 3 1 5 4 7 5 4 1 7 1 4 5 7 20 10 Example 8: 3 2 2 1 4 5 11 7 4 5 11 5 4 7 55 28 27 1 28 Draft C. David Pilmer Questions 1. (a) 4 1 9 5 (b) (d) 5 6 9 7 2 (e) 1 4 3 1 (f) 2 3 4 1 3 (h) 2 1 2 4 1 2 (i) 2 1 3 7 2 1 (g) 1 1 5 3 5 5 6 7 (c) 2 3 7 4 2. (a) 1 2 4 5 (b) 3 5 10 7 (c) 1 5 6 12 (d) 3 5 4 8 (e) 4 3 5 10 (f) 3 6 7 4 (g) 1 2 9 NSSAL ©2009 1 1 (h) 3 2 3 2 1 1 (i) 1 2 4 7 11 Draft C. David Pilmer Decimals Like fractions, decimals are used to represent part of a whole. Area Model Fraction Decimal 3 4 0.75 When a number is written as a decimal, it is comprised on three parts: whole number part, decimal point, and decimal part. Whole Number Part Decimal Part 571.806 Decimal Point Place names and place values for the whole number and decimal parts of fractions are shown in the chart below. 1000 100 10 1 Ten Thousandths Thousandths Hundredths . Tenths Decimal Part Units Tens Hundreds Thousands Whole Number Part 1 1 1 1 10 100 1000 10000 Example 1: Express the following numbers in expanded form. (a) 165.32 (b) 67.891 Answers: 3 2 10 100 8 9 1 (b) 67.891 60 7 10 100 1000 (a) 165.32 100 60 5 NSSAL ©2009 12 Draft C. David Pilmer Example 2: Write each number in words. (a) 35.6 (b) 165.32 (c) 67.871 (d) 307.05 (e) 2019.083 Answers: (a) 35.6 is “thirty-five and six tenths.” (b) 165.32 is “one hundred sixty-five and thirty-two hundredths.” (c) 67.871 is “sixty-seven and eight hundred seventy-one thousandths.” (d) 307.05 is “three hundred seven and five hundredths.” (e) 2019.083 is “two thousand nineteen and eighty-three thousandths.” Adding and Subtracting Decimals Adding decimals is similar to adding whole numbers. We line up the decimal points so that we can add corresponding place value digits (e.g. tenths with tenths, hundredths with hundredths, and so on). As with whole numbers, we start from the right and carry when it is necessary. Example 3: Add: 42.08 + 208.95. Example 4: Add: 36.07 _ 9.065. Answer: 1 Answer: 1 1 1 3 6 . 0 7 9 . 0 6 5 4 2 . 0 8 2 0 8 . 9 5 2 5 1 . 0 3 1 4 5 . 1 3 5 Subtracting decimals is similar to subtracting whole numbers. We line up the decimal points so that we can subtract corresponding place value digits (e.g. tenths from tenths, hundredths from hundredths, and so on). As with whole numbers, we start from the right and borrow when it is necessary. Example 5: Subtract: 57.62 - 6.18 Example 6: Subtract: 98.04 - 32.801 Answer: Answer: 12 7 6 2 1 8 9 8 . 0 4 0 3 2 . 8 0 1 5 5 7 6 . . 5 1 . 4 4 NSSAL ©2009 10 3 10 6 5 . 2 3 9 13 Draft C. David Pilmer Multiplying Decimals Step 1: Initially ignore the decimal points and multiply as if both of the factors are whole numbers. Step 2: Now the decimal point must be positioned in the product. The number of decimal places in the product is the sum of the number of places in the factors (count places from the right). Example 7: Multiply: 6.32 2.4 Answer: 6. 3 2 2. 4 Example 8: Multiply: 0.832 9.31 (2 decimal places) (1 decimal place) Answer: 0. 8 3 2 9. 3 1 (3 decimal places) 8 3 2 2 4 9 6 0 4 8 8 0 0 2 5 2 8 1 2 6 4 0 1 5. 1 6 8 7 7. 7 4 5 9 2 (3 decimal places) ( 2 decimal places) (5 decimal places) Dividing Decimals Step 1: Move the decimal point to the right in the divisor until the devisor is quotient a whole number. divisor dividend Step 2: Move the decimal point to the right in the dividend the same number of places as was done in Step 1. Step 3: Divide through as if you were dividing with whole numbers. Place the decimal point in the quotient directly above the new decimal point in the dividend. Example 9: Divide: 1.792 0.32 Example 10: Divide: 3.612 4.3 Answer: 1.792 0.32 becomes 179.2 32 because we moved the decimal point in both the dividend and divisor two places to the right. 5 .6 32 179.2 Answer: 3.612 4.3 becomes 36.12 43 because we moved the decimal point in both the dividend and divisor one place to the right. 0.84 43 36.12 344 172 172 160 192 192 0 0 NSSAL ©2009 14 Draft C. David Pilmer Questions: 1. Express the following numbers in expanded form. (a) 2482.6 (b) 7.93 (c) 265.147 (d) 40.0562 2. Write each number in words. (a) 14.9 (b) 3002.15 (c) 459.736 (d) 480.07 (e) 67.025 (f) 23.0578 3. Add. (a) 42.13 + 30.65 (c) 6.93 + 34.68 NSSAL ©2009 (b) 107.63 + 41.029 (d) 78.073 +105.96 15 Draft C. David Pilmer (e) 9.8562 + 6.2153 (f) 0.793 + 8.6254 (g) 32.06 + 7.42 + 11.23 (h) 0.645 + 1.39 + 2.0431 4. Subtract. (a) 46.37 - 14.12 (b) 27.891 - 4.24 (c) 328.46 - 41.28 (d) 489.231 - 25.65 (e) 3.2935 - 0.326 (f) 8.03 - 5.56 (g) 15.064 - 9.38 (h) 2.050 - 0.462 NSSAL ©2009 16 Draft C. David Pilmer 5. Multiply. (a) 6.4 2.8 (c) 40.5 5.23 6. Divide (a) 8.84 2.6 (c) 0.279 0.45 NSSAL ©2009 (b) 3.52 4.6 (d) 0.453 6.21 (b) 1.674 0.31 (d) 10.793 4.3 17 Draft C. David Pilmer Percents Percent means per one hundred. The % sign is used to show the number of parts out of one hundred parts. For example, 37% means 37 parts out of 100 parts. This particular percent can 37 also be expressed as the fraction and the decimal 0.37. 100 Area Model Fraction Decimal Percent 37 100 0.37 37% Changing Percents to Decimals Drop the % symbol and divide by 100 (i.e. move the decimal point two places to the left). Examples: 45% = 0.45 87% = 0.87 16% = 0.16 9% = 0.09 2% = 0.02 1.4% = 0.014 0.5% = 0.005 120% = 1.20 113% = 1.13 Changing Percents to Fractions Drop the % sign from the percent, place the number over 100, and simplify the fraction if possible. 60 36 55 60% 36% 55% Examples: 100 100 100 60 20 36 4 55 5 100 5 100 20 100 4 11 3 9 20 5 25 Changing Decimals to Percents Multiply by 100 (i.e. move the decimal point two places to the right) and add the % sign. Examples: 0.65 = 65% 0.19 = 19% 0.82 = 82% NSSAL ©2009 0.04 = 4% 0.07 = 7% 0.029 = 2.9% 0.009=0.9% 1.06 = 106% 1.13 = 113% 18 Draft C. David Pilmer Changing Fractions to Percents Convert the fraction to a decimal using division (by hand or with a calculator), and then convert the decimal to a percent (i.e. move the decimal point two places to the right and add the % symbol). 7 13 Examples: Convert to a percent. Convert to a percent. 20 25 0.52 0.35 25 13.00 20 7.00 60 125 100 50 50 100 0 0 7 13 0.35 35% 0.52 52% 20 25 Taking the Percentage of a Number If you need to find a specific percentage of a number, convert the percentage to a decimal and multiply that decimal by the number (by hand or with a calculator). Examples: Find 65% of 180. 0.65 180 117 Find 32% 0f 2100. 0.32 2100 672 Find 7% of 342. 0.07 342 23.94 Find 113% of 46. 1.13 46 51.98 Some of these questions can be done quickly and without a calculator if you are dealing with "friendly" percentages (e.g. 10%, 20%, 30%,…). Examples: Find 30% of 150 We know that 10% of 150 is 15, therefore 30% of 150 must be 45 (3 15). Find 20% of 320 We know that 10% of 320 is 32, therefore 20% of 320 must be 64 (2 32). Find 70% of 90 We know that 10% of 90 is 9, therefore 70% of 90 must be 63 (7 9). Questions: 1. Convert the following percents to decimals. (a) 38% = (b) 21% = (c) 37% = (d) 4% = (e) 6% = (f) 24.5% = (g) 3.4% = (h) 0.8% = (i) 105% = (j) 14.2% = (k) 210% = (l) 3.6% = NSSAL ©2009 19 Draft C. David Pilmer 2. Convert the following percents to fractions. (a) 45% (b) 70% (c) 28% (d) 80% (e) 6% (f) 42% (g) 72% (h) 120% 3. Convert the following decimals to percents. (a) 0.92 = (b) 0.47 = (c) 0.32 = (d) 0.07 = (e) 0.7 = (f) 0.007 = (g) 0.042 = (h) 0.206 = (i) 1.51 = (j) 1.8 = (k) 1.06 = (l) 1.034 = 4. Convert the following fractions to percents. You may use a calculator to complete this question. () (a) 2 5 (b) 14 25 (c) 17 20 (d) 5 8 (e) 3 8 (f) 15 16 (g) 26 25 (h) 9 8 5. Complete each of the following. You may use a calculator to complete this question. () (a) Take 35% of 6280. NSSAL ©2009 (b) Take 65% of 580. 20 (c) Take 9% of 1600. Draft C. David Pilmer (d) Take 15% of 74. (e) Take 3.5% of 56. 6. Complete each of the following. Do not use a calculator. (a) Take 10% of 430. (b) Take 10% of 5800. (f) Take 113% of 57. (c) Take 20% of 60. (d) Take 20% of 140. (e) Take 30% of 70. (f) Take 40% of 110. (g) Take 70% of 2000. (h) Take 60% of 400. (i) Take 20% of 2500. (j) Take 40% of 300. (k) Take 60% of 7000. (l) Take 30% of 1200. NSSAL ©2009 21 Draft C. David Pilmer Ratio and Proportions Ratios A ratio is the quotient of two quantities. A ratio can be written in two different forms: fractional notation and colon notation. For example, the ratio of 2 to 3 can be written as: 2 (fractional notation) 3 2:3 (colon notation) Ratios are easier to understand when they are written in their lowest terms (also called simplest form). For example, we can create a ratio of Ted’s new monthly savings to his monthly earnings. Initially the ratio was expressed as 125:1500. When the ratio is changed to its lowest terms, it is expressed as 1:12. That means that for every $12 that Ted earns, $1 is set aside as savings. Example 1: Write the ratio of $35 to $15 in fractional notation and express in lowest terms. Answer: $35 35 35 5 7 $15 15 15 5 3 Example 2: Write the ratio of 3.1 to 4.2 as a fraction in simplest form. Answer: 3.1 3.1 10 31 4.2 4.2 10 42 Example 3: Write the ratio of 0.14 to 0.2 as a fraction in simplest form. Answer: 0.14 0.14 100 14 14 2 7 0.2 0.2 100 20 20 2 10 Proportions A proportion states that two ratios are equal. For example, 5:7 = 15:21 is a proportion. We can read it as “5 is to 7 as 15 is to 21.” 13 26 For example, is a proportion. We can read it as “13 is to 9 as 26 is to 18.” 9 18 NSSAL ©2009 22 Draft C. David Pilmer When one number of the proportion is unknown, we can use cross products to find the unknown number. Procedure: 1. Find the cross products. 2. Set the cross products equal to each other. 3. Divide the number not multiplied by x by the number multiplied by x. Example 4: Find the value of the unknown number x. x 7 12 3 Answer: x 12 7 3 The cross products will be 3 x and 12 7 . 3x 84 Set the cross product equal to each other. 84 3 x 28 Divide the number not multiplied by x by the number multiplied by x. x Example 5: Find the value of the unknown number x. 9 1.5 x 7 Answer: 9 x 1.5 7 The cross products will be 9 7 and x 1.5 . 63 1.5x Set the cross product equal to each other. 63 1 .5 x 42 Divide the number not multiplied by x by the number multiplied by x. x Questions: 1. Write each ratio as a ratio of whole numbers using fractional notation. Write the fraction in simplest form. (a) 12 to 16 (b) 45 to 25 (c) 18 to 30 NSSAL ©2009 23 Draft C. David Pilmer 2. (d) 27 to 21 (e) 130 to 160 (f) 16 to 18 (g) 32 to 24 (h) 21 to 35 (i) 72 to 45 (j) 5.1 to 3.7 (k) 4.6 to 6.5 (l) 6.12 to 2.53 (m) 6.7 to 9.47 (n) 0.11 to 5.2 (o) 6.2 to 4 (p) 0.5 to 3.5 (q) 0.27 to 0.15 (r) 0.3 to 0.42 For each proportion, find the unknown number, x. () x 2 8 24 (a) (b) 21 3 5 x (d) NSSAL ©2009 28 24 x 18 (e) 15 2.5 6 x 24 (c) 10 x 18 45 (f) x 24 7.5 30 Draft C. David Pilmer Signed Numbers Much of the mathematics we use in our daily lives requires that we understand positive numbers (e.g. 0.4, 2, 127, 3/5). Negative numbers (e.g. -7, -0.5, -4/5), however, are extremely important as one continues their study of mathematics. negative numbers -4 -3 -2 -1 positive numbers 0 1 2 3 4 All signed numbers have magnitude (i.e. size) and direction (i.e. sign: positive or negative). For example the number -4 has a negative direction (to the left of 0 on the number line) and a magnitude of 4. For example the number -3.2 has a negative direction (to the left of 0 on the number line) and a magnitude of 3.2. For example the number +5 or 5 has a positive direction (to the right of 0 on the number line) and a magnitude of 5. For example the number +7.85 or 7.85 has a positive direction (to the right of 0 on the number line) and a magnitude of 7.85. Understanding the terms, magnitude and direction, is important when trying to explain the rules for addition, subtraction, multiplication and division of signed numbers. Adding Signed Numbers To add two numbers with the same sign, add their magnitudes, and keep the common sign. ex. 4 6 10 ex. 5 3 8 ex. 2 7 9 ex. 7.2 1.3 8.5 4 1 3 ex. 5 5 5 5 2 3 ex. 7 7 7 To add two numbers with different signs, subtract the smaller magnitude from the larger magnitude, and use the sign of the number with the larger magnitude. ex. 4 7 3 ex. 5 3 2 NSSAL ©2009 ex. 3 9 6 ex. 8 1 7 ex. 4.8 1.2 3.6 ex. 3.1 7.8 4.7 3 2 5 ex. 8 8 8 5 2 7 ex. 9 9 9 25 Draft C. David Pilmer Subtracting Signed Numbers To subtract one signed number from another, change the question from a subtraction question to an addition question, and change the sign of the number that was originally being subtracted. Once these changes have been made, follow the rules for adding signed numbers. ex. 2 5 ex. 3 4 2 5 7 3 4 1 ex. 4 6 4 6 10 ex. 1.4 0.8 1.4 0.8 0.6 1 2 ex. 5 5 3 1 2 5 5 5 2 6 ex. 7 7 4 2 6 7 7 7 Multiplying Signed Numbers To multiply two numbers with the same sign, multiply the magnitudes, and the resulting sign will be positive. ex. 6 3 18 ex. 5 4 20 ex. 2 12 24 ex. 0.4 0.3 0.12 2 1 2 ex. 15 3 5 6 3 2 ex. 35 7 5 To multiply two numbers with the different signs, multiply the magnitudes, and the resulting sign will be negative. ex. 6 2 12 ex. 8 4 32 ex. 0.2 12 2.4 ex. 0.4 0.7 0.28 4 1 4 ex. 45 9 5 10 5 2 ex. 21 7 3 Dividing Signed Numbers To divide two numbers with the same sign, divide the magnitude of the first by the magnitude of the second, and the resulting sign will be positive. ex. 6 2 3 ex. 8 4 2 ex. 35 7 5 ex. 3.6 0.6 6 To divide two numbers with the different signs, divide the magnitude of the first by the magnitude of the second, and the resulting sign will be negative. ex. 16 4 4 ex. 45 9 5 ex. 63 7 9 NSSAL ©2009 ex. 12 0.6 20 26 Draft C. David Pilmer Exponents and Signed Numbers When one writes a 4 it means a a a a . The a is the base, the 4 is exponent, and the entire expression a 4 is called the power. ex. 23 2 2 2 ex. 32 3 3 8 9 The challenging aspect of signed numbers and exponents occurs when one is attempting to understand if the base of the power is positive or negative. For example, many people make 4 the mistake of thinking that 2 and 24 mean the same thing; this is not the case. has a base of -2 and means 2 2 2 2 . Therefore 2 equals 16. 4 2 has a base of 2 and means 1 2 2 2 2 . Therefore 24 equal -16. 24 4 ex. 5 5 5 25 ex. 4 4 4 4 64 ex. 7 7 7 49 ex. 3 3 3 3 3 81 ex. 62 1 6 6 36 ex. 34 1 3 3 3 3 81 3 2 4 2 Order of Operations and Signed Numbers Order of Operations (BEDMAS) applies to all numbers. B – brackets E – exponents DM – division and multiplication AS – addition and subtraction Example: 30 3 2 4 10 8 10 8 18 Example: 2 8 5 2 3 6 Example: 3 7 4 2 7 4 8 7 32 39 Example: 2 6 8 5 4 Example: 2 7 2 4 5 2 6 3 4 6 3 16 7 2 4 5 2 Example: 23 18 9 2 8 18 9 2 8 2 2 8 4 8 4 4 2 3 16 13 NSSAL ©2009 2 7 6 5 2 7 36 5 2 7 36 10 33 2 27 Draft C. David Pilmer Questions 1. (a) 3 2 (b) 7 6 (c) 3 8 (d) 5 9 (e) 8 2 (f) 12 5 (g) 2 15 (h) 7 11 (i) (k) 9 2 (l) 10 2 (m) 15 9 (n) 21 8 (o) 9 7 (p) 11 8 (q) 3 9 (r) 7 5 (s) 6 7 (t) 8 4 (u) 7 8 (v) 9 6 (w) 27 9 (y) 28 7 (z) 32 4 2. (a) 14 6 (b) 48 8 (c) 9 7 (d) 3 19 (e) 17 8 (f) 7 10 (g) 30 10 (h) 12 14 (i) 10 8 (k) 4.5 9.1 (l) 2.5 0.5 (m) 1.4 1.1 (n) 0.6 4 (o) 5.1 4.3 5 1 (p) 9 2 1 4 (q) 6 6 4 2 (r) 7 7 (b) 3 (c) 5 (e) 14 (f) 26 (j) (j) 14 7 9.7 3.5 3. (a) 6 2 (d) 82 NSSAL ©2009 3 3 10 (x) 36 6 3 28 Draft C. David Pilmer 4. (a) 7 2 8 (d) 3 7 4 5 (b) 1 2 3 5 (c) 28 7 15 3 (e) 5 2 6 (f) 27 9 2 2 3 (g) 10 2 5 3 (h) 3 8 3 2 NSSAL ©2009 29 (i) 2 1 23 5 Draft C. David Pilmer All Together Now, Part I Complete all of the calculations without using a calculator. Show your work. (a) 13 5 (b) 78.14 - 42.8 (c) Find x. x 5 12 3 (d) 32 11 6 2 4 (e) Take 40% of 120. (f) 2 1 2 1 3 2 (g) 5.4 7.2 (h) 23 (i) 19.5 5 (j) 6 5 (k) 1 2 5 4 7 3 (l) 80.42 + 7.9 (o) 35 2 3 64 (m) 3 1 5 1 4 7 NSSAL ©2009 (n) 13 7 30 2 Draft C. David Pilmer All Together Now, Part II Complete all of the calculations without using a calculator. Show your work. (a) Take 20% of 70. (b) 6.3 2.5 (c) 2 7 4 7 (d) 4 1 2 3 9 2 (e) 52 (f) 8 4 2 (g) 9 7 (h) Find x. 2 6 7 x (i) (j) 168.45 - 92.8 (k) 5 2 10 3 32 1 (l) 3 8 (m) 41.4 9 (n) 1 2 5 2 2 3 (o) 18.45 + 9.813 NSSAL ©2009 31 1 3 6 2 4 5 Draft C. David Pilmer All Together Now, Part III Complete all of the calculations without using a calculator. Show your work. (a) (d) 1 1 4 1 6 3 11 2 (b) 11 7 (c) 5 1 7 2 8 3 (e) 1 7 4 2 4 2 (f) 16.2 6 (g) Find x. 5 x 4 12 (h) Take 30% of 220. (i) 28.56 - 6.902 (j) 14 (k) 7.4 3.1 (l) 3 5 1 4 8 6 (m) 9 3 5 8 22 30 (n) 9 3 (o) 5.61 + 63.815 NSSAL ©2009 32 Draft C. David Pilmer Reflect Upon Your Learning Fill out this questionnaire after you have completed pages 1 to 32. Select your response to each statement. 1 - strongly disagree 2 - disagree 3 - neutral 4 - agree 5 - strongly agree (a) I understand all of the concepts covered in the section, Order of Operations. (b) I do not need any further assistance from the instructor on the material covered in the section, Order of Operations. (c) I understand all of the concepts covered in the section, Fractions. (d) I do not need any further assistance from the instructor on the material covered in the section, Fractions. (e) I understand all of the concepts covered in the section, Decimals. (f) I do not need any further assistance from the instructor on the material covered in the section, Decimals. (g) I understand all of the concepts covered in the section, Percents. (h) I do not need any further assistance from the instructor on the material covered in the section, Percents. (i) I understand all of the concepts covered in the section, Ratio and Proportions. (j) I do not need any further assistance from the instructor on the material covered in the section, Ratio and Proportions. (k) I understand all of the concepts covered in the section, Signed Numbers. (l) I do not need any further assistance from the instructor on the material covered in the section, Signed Numbers. (m) I was able to complete all or most of the questions successfully in the three sections titled All Together Now. NSSAL ©2009 33 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 Draft C. David Pilmer Addition Pyramids With addition pyramids, the two numbers in adjoining boxes add to give the number in the box immediately above. 7 -8 3 4 3 9.2 -11 5 -2 2.6 -9 6.6 0.5 0.4 2.8 0.1 3.8 2.7 1.1 Insert the missing numbers in each of the following addition pyramids. 1. 2. 4 6 4 4. 6. 6 -2 -3 9 7. 8. 0.4 7 5 3. -7 3. 13 -5 9. 0.9 0.7 2.1 0.1 10. 1.6 11. 12. 7 6 13. 5 12 7 2 8 14. 21 5 15. 9 4 10 4 NSSAL ©2009 -7 5 34 -8 3 -6 Draft C. David Pilmer 16. 17. 5 -6 -10 -7 19. -5 -6 3 20. -9 3 21. -7 0.4 22. 1.5 23. 0.7 18. 2 0.8 2.3 0.6 0.4 24. 3 1.5 -1 1.9 0.6 2.6 1.2 25. 0.4 26. 1.8 27. 29 20 11 9 4 1 8 3 28. 3 6 2 2 29. 15 30. 12 5 13 6 9 8 31. 1 32. 9 -4 3 -6 -7 6 -2 NSSAL ©2009 -2 33. -6 3 7 -9 -5 2 1 8 35 Draft C. David Pilmer 34. 35. 1 36. 7 2.8 1.3 -5 0.7 -3 37. 0.3 0.7 1.6 0.9 38. 39. 3.2 1.4 0.9 NSSAL ©2009 0.2 0.1 3 1.4 4 1.8 1.3 0.6 36 1.5 0.5 Draft C. David Pilmer Find the Center Puzzles With these puzzles, you must determine the missing number in the center of the cross. You are provided with two numbers in each branch of the cross. Those two numbers are used to generate the number in the center but you are not told what operation (addition, subtraction, multiplication, or division) applies to each branch. However, you do know that each operation is represented by a single branch. Example 1 Find the missing number in the center of the puzzle. 12 Answer: The center number must be 4 because: 95 4 12 3 4 1 3 4 2 2 4 3 9 5 1 3 2 2 Example 2 Find the missing number in the center of the puzzle. -10 Answer: The center number must be -12 because: 0.6 20 12 10 2 12 18 6 12 3.6 0.3 12 2 -0.6 20 -18 6 3.6 -0.3 Questions Find the missing number in the center of each puzzle. (a) (b) 7 2 12 NSSAL ©2009 3 (c) 12 44 4 3 3 6 2 4 24 3 7 4 14 45 4 11 5 2 1 37 3 Draft C. David Pilmer (d) (e) -5 -1 18 -3 (g) 3 -2 5 -2 -10 -5 -18 -3 -5 2 2 3 (h) (i) 2.4 2 2 2.8 7 3 0.4 0.7 3.6 2.4 14 10 0.9 0.5 4 0.8 3 0.1 0.4 1.6 (k) -8 20 0.6 (l) -3.2 20 1 0.4 16 4 12 20 4 -6.3 -1.7 1.5 -0.5 -6 0.5 -24 -40 -2 -2 0.2 -1 (n) 5 4.2 1.8 (o) -0.3 1.2 -8 0.8 3 0.5 2.4 1.9 10 2.5 0.2 -2 0.2 0.6 1 -2 -4 -0.1 -8 -8 -0.3 (q) -2.4 1.6 5.6 (r) 0.8 0.6 NSSAL ©2009 -8 -2 -20 0.6 0.2 (p) -2 -4 0.1 (m) 30 12 0.3 (j) (f) 2 -0.3 0.9 -2.1 -1.9 -1 10 -5 0.5 -0.2 0.9 0.6 3 -8 -1.1 1.9 0.5 1 0.4 38 2 Draft C. David Pilmer KenKen Puzzles Insert the appropriate numbers into the grid such that: no number is repeated in the same row or column, and the numbers in the cages produce the target number using the indicated operation (e.g. -8 : find the numbers when multiplied give you -8). (a) 4, 5, 6 Puzzle 24 (b) 2, 6, 9 Puzzle 30 5 54 10+ (d) -3, -2, 5 Puzzle -15 0+ 6+ 6 12 15+ 6 -28 3 0.6 0.2 0.4 2.3+ 24 56 (f) -7, -6, 8 Puzzle -6+ -7 -5+ 2+ -48 42 -4 (g) 0.2, 0.3, 2 Puzzle NSSAL ©2009 21 (e) -4, -1, 7 Puzzle 3+ 0.5+ 11+ 18 9+ (c) 3, 7, 8 Puzzle (h) 0.4, 0.6, 3 Puzzle 1.2 4+ (i) 0.7, 0.8, 5 Puzzle 4 1.8 3.5 6.5+ 0.24 39 0.56 Draft C. David Pilmer (j) 2, 3, 4, 5 Puzzle 12 (k) 3, 6, 7, 9 Puzzle 15 2 10+ 8 27 7 18 10+ 40 63 5 7+ 6 (l) 4, 5, 7, 8 Puzzle 12+ 35 28 1+ -5 20 12 -3 -10+ -3 -5 -1+ -3 4 (o) -8, -7, 2, 9 Puzzle -6 56 -20 -6+ 9 -16 -63 -1+ 4+ -15 NSSAL ©2009 -20 40 (n) -6, -4, -3, 5 Puzzle -7+ 12+ -2+ 9+ -30 42 (m) -5, -3, 1, 4 Puzzle 15+ 56 54 18 -15+ 40 Draft C. David Pilmer (p) -9, -6, -2, 8 Puzzle 54 -72 (q) 0.5, 0.2, 1, 3 Puzzle -12 -1+ 1.2+ -16 0.5 1.5 0.5 -8+ 4+ 6+ -15+ 0.2 (r) 0.6, 0.4, 0.1, 5 Puzzle 0.5 0.24 3.5+ 6.8+ 0.2 0.7 4.2 1.7+ 0.06 27 5+ NSSAL ©2009 32 18 11+ 10+ 18 6 28 8 13+ 15 4 11+ 9+ 48 12 41 21 24 11+ 42 5 6 10 (u) 3, 4, 6, 7, 8 Puzzle 17+ 12+ 0.14 4.8 (t) 2, 3, 5, 6, 9 Puzzle 30 0.16 1.2 0.7+ 5.4+ 0.2 (s) 0.8, 0.7, 0.2, 6 Puzzle 3 0.5+ 3.2+ Draft C. David Pilmer (v) -5, -4, -2, 3, 6 Puzzle -6 20 2+ -5 10 1 8 6+ NSSAL ©2009 14 -16 1.2 18 7 -6+ (y) 0.8, 0.7, 0.3, 6, 9 Puzzle 2.4 2.7 4.8 0.18 0.6 -42 -21 1.2 8 4.6+ -12 -6 -17+ (x) 0.6, 0.5, 0.3, 2, 4 Puzzle 0.15 24 9+ -5 0.8+ 7 -14+ 6 -12 18 1+ 8 -30 (w) -8, -6, -3, 2, 7 Puzzle 0.8+ 0.24 2.5+ 6.3 42 1.1+ 4.2 0.7 7.2 15+ 1.8 9.7+ 6.7+ 1.1+ Draft C. David Pilmer RAD Puzzles Use the numbers supplied in the table below to correctly complete the RAD puzzle. Example: Original Puzzle: + Completed Puzzle: = = -6 -8 + = = = = = -4 = = = = + 2 + -9 3 -9 4 = -8 8 -7 9 = - = 36 = -6 -6 + 4 22 -9 30 1 = = = = = = -6 -2 = = = = = = -1 11 28 3 9 - + - 15 2 28 44 22 = + + 2 -9 = = -4 -7 = = 24 8 - -6 15 -6 -2 -1 1 2 11 24 30 36 44 Questions: (a) (b) - = = 3 34 - = 10 = + - + + - - + = = = = = = = = = 4 = = = = 8 = = 8 + 1 5 1 6 NSSAL ©2009 = = 2 7 = 2 9 = = 9 = 6 = = = + - 3 + = 1 6 4 18 - 21 = + - 2 3 4 4 10 12 19 40 43 5 - 1 7 1 8 = = 11 - 2 3 3 3 4 5 7 15 24 31 42 47 54 65 Draft C. David Pilmer (c) (d) - = + -10 -9 = = = = + + 8 35 + + = = -9 - + + 6 = = = -6 = = = = = = 3 -5 = = = = = = -10 = = = = = = + + 12 + 9 + + + = 6 = - -56 -36 -18 -7 -2 -2 -2 0 -6 1 -5 4 -5 -3 -3 21 30 54 + = -60 -56 -51 -9 -3 -1 2 6 (e) = -8 6 -4 - -7 -7 -6 -4 27 36 37 45 (f) - = + = + 0.2 + - 0.1 = + = - - + 2 4.3 = = = = = = 1.2 = = = = = = 4 3 = = = 0.4 = = 1 - 0.8 0.1 0.1 0.1 0.3 0.3 0.5 0.5 0.7 1.1 1.3 1.5 1.5 1.6 2.8 3 3.6 8 9 NSSAL ©2009 1.5 = = = + = = + - 0.2 0.3 0.6 0.7 0.7 0.8 2.1 2.4 2.7 3 5 5.2 44 = 1.8 + = = 1 - = = 30 = 1 6 1.9 2 18 21 Draft C. David Pilmer Fraction Fury Puzzles With the 3 by 3 fraction fury puzzle, your mission is to use the six 4 6 1 indicated numbers to create the desired nine fractions. However, 5 3 2 you are not permitted to repeat a number in a row or column (They can be repeated in a diagonal.). Hints are provided so that 1 5 4 you can figure out which fraction (proper or improper) belongs in each of the nine squares. If we look at the completed puzzle on 3 2 6 the right, the hint for the first square (top left) could have been " 1 0.2 ." There is only one possible solution based on this hint 2 1 5 4 6 4 3 and it is . For the second square (top center), the hint could 5 6 4 2 4 have been "Simplifies to 2." The possibilities are , , or . The can be eliminated 3 2 1 2 because we already used the 4 in the previous square (Can't repeat the same number in a row or 6 2 column.) At this point, you are unable to determine whether or belongs in square two. 3 1 You will have to look at the hints provided for the other squares before determining which of the two possible answers is appropriate for square two. Please note that you must be comfortable with fractions, decimals and percentages to complete these puzzles. 1. Complete the following 3 by 3 fraction fury puzzles. (a) 1, 2, 3, 4, 5, 6 Puzzle 20% 1.5 (b) 2, 3, 4, 5, 8, 10 Puzzle Double 1 1.2 – 0.4 Equal to 4 Little < 25% 75% 30% 3 Equal to 3 33 1 % 8 3 NSSAL ©2009 Little < 2 3 50% of 1 1 3 2 12 3 Half of 4 5 4 1 Double 12 4 3 45 Equal to 2 1 of 2 4 Draft C. David Pilmer (c) 1, 2, 5, 7, 8, 9 Puzzle 1.3 + 1.2 900% Little > 1 Half of 1 2 (d) 1, 2, 3, 4, 7, 10 Puzzle 1 17 4 18 1 1 6 Equal to 3 1 5 1 8 8 7 1 5 4 4.5 Double 0.1 400% 1 10 Between 1 and 2 1.5 1 2 1 Between 0 and 0.5 3 1.75 10% 2. Complete the following 4 by 4 fraction fury puzzles. In this case the hints have been provided to the right of the grid; as opposed to, below the grid. (a) 1, 2, 3, 4, 5, 6, 7, 8 Puzzle 350% 0.8% Double 1 Equal to 6 0.8 0.1 Equal to 1 2.75 – 1.5 2 5 6 1 Little < 1 2 1.2 3 1 46 of 14 4 Equal to 1 6 30 3 10% of 20 Little > 1 1 4 NSSAL ©2009 4 1 2 Draft C. David Pilmer (b) 1, 2, 3, 4, 5, 6, 7, 8 Puzzle Equal to 4 60% Little < 1 1 of 2 8 3 1 1 Half of 7 1 10 1.7 – 1.3 Equal to 2 4 Between 2.5 and 3.5 50% 12 1 Triple 2 1 7 9 3 200% 2 7 1 4 Double 1.25 7 (c) 2, 3, 4, 5, 6, 7, 8, 9 Puzzle 3 1 Equal to 2 2 120% 1 3 3.1 + 1.4 7 1 2.25 3 133 1 % 1 Equal to 7 Little < 0.25 2 Equal to 3 2 6 Half of 1 3 7 1 8 8 4 2 3 10 125% 0.5 Little < 3 1 2 (d) 3, 4, 5, 6, 7, 8, 9, 10 Puzzle 1 2 40% Triple 3 Half of 8 160% Equal to 2 6 NSSAL ©2009 47 1 2 Equal to 1 3 11 2 14 0.1 5 14 Little > 8 7 1 – 0.25 7 Little > 1 5 3 2 1 3 1 2 100% - 40% 2.5 3 Draft C. David Pilmer Putting It Together: Math Games and Puzzles The test on math games and puzzles portion of this unit will look very similar to the questions provided in this section. That means, if you do these questions with easy, then the corresponding test will be just as straightforward. 1. Complete the following Addition Pyramids. (a) (b) (c) 17 3.7 5 7 1.7 3 3 -7 5 0.6 -5 0.6 2. Complete the following Find the Center Puzzles. (a) (b) 15 3 4 3 24 2 -6 -2 -21 NSSAL ©2009 -7 1 1.2 0.9 0.2 0.1 16 3 4 -2 0.1 -30 -7+ 4+ 3 (b) 0.9, 0.6, 0.2, 3, 5 Puzzle 1 1.5+ -4+ -5 6 8 84 -1+ 1.8 2 3. Complete the following KenKen Puzzles. (a) -7, -5, -2, 3, 6 Puzzle -15 (c) -4 18 -9+ 0.8+ 10 8+ 48 0.12 3.9+ 3 3 4.5 1.1+ 15 2.7 0.12 Draft C. David Pilmer 4. Complete the following RAD Puzzles. (a) (b) -10 = = - 11 - 0.7 = = 18 + - + + - + = = = = = = = = = = 8 2 = = -48 = = = -8 = = = = 1.8 = = = + - 5.3 - 0.6 = -1 = + = -40 -22 -16 -15 -9 -2 -1 4 5 6 = 9 - + -8 -7 -3 -2 14 15 24 40 = = + 3.9 0.1 0.1 0.2 0.3 0.4 0.5 0.8 0.8 1.2 1.6 1.7 2.4 3 4 5 6 6.9 9 5. Complete the following 1, 3, 4, 5, 7, 8, 9, 10 Fraction Fury Puzzle. Double 1 7 0.1 Equal to 8 2 2 Half of 8 2.1 0.3 Little < 1 1.7 – 1.3 Triple 5 1 of 1.5 4 Double 35% Close to 0 49 1 21 7 10 NSSAL ©2009 200% - 40% 3 3 180% 1 9 10 Between 0.6 and 0.9 Draft C. David Pilmer Reflect Upon Your Learning Fill out this questionnaire after you have completed pages 34 to 49. Select your response to each statement. 1 - strongly disagree 2 - disagree 3 - neutral 4 - agree 5 - strongly agree (a) (b) (c) (d) (e) NSSAL ©2009 I understand all of the concepts covered in the sections concerned with Math Games and Puzzles. I do not need any further assistance from the instructor on the material covered in the sections concerned with Math Games and Puzzles. I remember how to use the necessary mathematical concepts (operations with whole numbers, fractions, decimals, percents, and signed numbers) to complete the math games and puzzles. I can complete these types of math puzzles more quickly and with greater easy than when I started these sections of the resource. I do not need any more practice questions. 50 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 Draft C. David Pilmer Applications - Fractions Example 1: The cross-sectional view of a pipe is provided. Based on the information in the diagram, determine the outer diameter of the pipe. 3 4 2 3 4 " 5 8 Example 2: 3 inch counter top is comprised 4 1 of a particle board core with inch laminate 16 glued on top. How thick is the particle board core? A particular " " 3 4 Answer: 3 5 3 3 2 5 3 2 2 2 4 8 4 4 2 8 4 2 6 5 6 2 8 8 8 17 2 8 1 22 8 1 4 inches 8 Example 3: For each hour that an oil burner runs, it uses 3 1 litres of fuel. If the burner is only running 4 2 for hour, then how much fuel is used? 3 Answer: 3 2 7 2 1 4 3 4 3 14 12 7 6 1 1 litres 6 NSSAL ©2009 " core Answer: 3 1 3 4 1 4 16 4 4 16 12 1 16 16 11 inch 16 Example 4: You have 10 pounds of flour in a bag. You are removing it from the bag using a container that 2 can hold pounds of flour. Assuming that 5 you are filling the container each time, how many times will you use the container to completely empty the bag? Answer: 2 10 5 10 5 1 2 50 2 25 times 51 Draft C. D. Pilmer Questions: 1. Determine the missing measurement. 7 5 inches 8 ? 1 8 inches 2 3 1 3 inch thick plywood, inch thick felt paper, and inch 4 16 8 thick shingles. What is the total thickness of the sheathing? 2. Roof sheathing is comprised of 3. Figure out the number of sheets of 4. If the diameter of a hole is 2 5 inch plywood in a stack 25 inches high? 8 7 inch, what is the radius of 8 the hole? NSSAL ©2009 radius 52 diameter Draft C. D. Pilmer 5. A motor is attached to a support block. Using the diagram, determine how high the motor’s shaft is off the ground? Motor 6. How many pieces of pipe that are 6 5 8 2 3 4 Support Block 2 foot long can be laid end to end to create a pipe 8 feet 3 long? 7. Find the thickness of a 1 1 1 inch board after inch is planed off? 16 4 8. A dog groomer uses a 36 gallon container to wash dogs. If she only fills it 3 full of water, 4 how many gallons of water does she use? 9. A baker used 2 1 2 cups of flour for rolls and 1 cups for cookies. How much flour was 2 3 used? NSSAL ©2009 53 Draft C. D. Pilmer 10. Find the perimeter of this regular hexagon. With any regular polygon, all the sides are of equal length. 11. How much longer is a 2 12. How many 2 3 inches 4 1 3 inch nail compared to a 1 inch nail? 2 4 2 cup sugar bowls can be filled using 10 cups of sugar? 3 13. Three quarters of a pizza is left over from last night’s party. You want half of the remaining pizza. How much of the original pizza does your piece represent? 14. Figure out the length of the following shaft. 1 NSSAL ©2009 54 3 4 " 2 3 8 " 1 1 2 " Draft C. D. Pilmer Applications - Decimals Example 1: A patient was given injections of 3.2 ml, 2.15 ml, and 1.9 ml of a particular medication. How much medication did the patient receive in total of that particular medication? Example 2: A school bus driver making $12.50 per hour was given a raise to $14.05 per hour. How much is the raise? Answer: Subtract the two numbers. Answer: Add the three numbers. 3 1 3 . 2 10 1 4 . 0 5 1 2 . 5 0 2 . 1 5 1 . 9 1 . 5 5 The driver received a $1.55 raise. 7 . 2 5 The patient received 7.25 ml of medication. Example 3: Determine the area of this rectangle. 5.1 m 2.7 m Answer: The area of a rectangle is found by multiplying the length by the width. 5. 1 2. 7 1 Example 4: A phone company is charging $0.06 per minute for long distance calls within Canada. If your bill for a long distance call within Canada was $4.32, how many minutes was the call? Answer: Divide 4.32 by 0.06. Change 4.32 0.06 to 432 6 by moving the decimal point two places to the right on both the dividend and the divisor. 72 6 432 (1 decimal place) (1 decimal place) 3 5 7 0 2 0 1 3. 7 7 42 12 12 0 (2 decimal places) The area of the rectangle is 13.77 m2. The call lasted 72 minutes. NSSAL ©2009 55 Draft C. D. Pilmer Questions: 1. A pair of jeans sells for $39.99 is marked down $7.49. What is the new price of the jeans? 2. A particular cut of meat costs $6.70 per kilogram. How much does 0.8 kilograms cost? 3. If Churchill Publishing shares can be bought for $5.80 per share, how many shares can be purchased with $2030? 4. Determine the perimeter of this triangle. 0.69 m 1.15 m 0.92 m 5. A patient is given 0.075 mg tablets each day for 7 days. How much medication of this type did the patient receive in total over that 7 day period? NSSAL ©2009 56 Draft C. D. Pilmer 6. Four students decided to split the lunch bill equally among themselves. If the bill came to $50.52, how much would each person pay? 7. Gasoline costs $1.12 per litre. How much will you pay for 18.5 litres? 8. When a particular piston in a car engine is cold, its diameter measures 8.579 centimetres. After the engine has been running, the piston becomes hot and expands. At that time, its diameter measures 8.593 centimetres. How much did the piston expand? 9. Akira and her partner traveled 526.3 km on day one, 488.5 km on day two, and 417.6 km on day three. How far did they travel over those three days? NSSAL ©2009 57 Draft C. D. Pilmer 10. A bottle initially containing 30 ml of medication has 2.7 ml removed. How much medication remains in the bottle? 9.49 cm 11. Determine the missing dimension. 2.81 cm ? cm 4.67 cm 12. A metalworker is casting a particular part. He randomly selects five completed parts and weighs them. Their masses are shown below. 134.5 g, 133.9 g, 134.2 g, 133.7 g, 133.7g Determine the average mass of these five parts. NSSAL ©2009 58 Draft C. D. Pilmer Applications - Percents Example 1: Tanya wants to leave a 20% tip on a meal costing $36. How much should she leave? Answer: Take 20% of $36 0.20 36 $7.20 Example 2: Angela conducted a survey in her adult education class and discovered that 15 out of the 24 learners had at least one child between the ages of 6 years and 18 years. (a) What percentage of learners in this class had at least one child between the ages of 6 years and 18 years? (b) What percentage of learners in this class did not have at least one child between the ages of 6 years and 18 years? Answer: 15 (a) 0.625 62.5% 24 9 0.375 37.5% (b) 24 24 23 22 21 20 19 18 17 16 Number of Individuals Example 3: Montez conducted a survey where he asked 40 individuals to indicate the number of working televisions they had at their place of residence. The results of this survey are displayed on the bar graph. (a) What percentage of the individuals has one television? (b) What percentage of the individuals has two televisions? (c) What percentage of the individuals has three or more televisions? (d) If we only considered individuals who had 2 or more televisions, what percentage of those individuals has 4 televisions? 15 14 13 12 11 10 9 8 7 6 5 4 Answers: 22 0.55 55% (a) 40 10 0.25 25% (b) 40 8 0.2 20% (c) 40 3 0.1666... 16.7% (d) 18 NSSAL ©2009 3 2 1 0 1 2 3 4 Num ber of Televisions 59 Draft C. D. Pilmer Example 4: Of 3600 vehicle accidents, only 2% were caused by mechanical failure. How many accidents were caused by mechanical failure? Answer: Take 2% of 3600 0.02 3600 72 accidents Example 4: The circle graph shows the sales at fast-food Hamburger Sales (USA) hamburger chains as a percent of total fast-food sales. It should be noted that $42 billion represents the total Others, 19% amount of money generated in one year by all of the fast-food hamburger chains in the United States. McDonald's, 43% (a) How much money did McDonald’s generate from Hardee's, 6% hamburger sales in one year? (b) How much money did Wendy’s generate from Wendy's, 12% hamburger sales in one year? (c) If Wendy’s could increase its hamburger sales by Burger King, 1% at the expense of MacDonald’s sales, how 20% would that affect the money generated by the sale of hamburger sales for both Wendy’s and MacDonald’s? (d) Presently what percentage of hamburger sales does Burger King and McDonald’s share between them? (e) If the total sales of hamburgers increased to $45 billion the following year, but the market shares remained the same, how much money would Burger King generate from hamburger sales? Answers: (a) Take 43% of $42 billion 0.43 42 $18.06 billion (b) Take 12% of $42 billion 0.12 42 $5.04 billion (c) Wendy’s: Now take 13% of $42 billion. 0.13 42 $5.46 billion (d) 20% + 43% = 63% (e) Take 20% of $45 billion 0.20 45 $9 billion McDonald’s: Now take 42% of $42 billion. 0.42 42 $17.64 billion Example 6: Janice is purchasing a coat that costs $69.99. What is the total cost including sales tax (13%)? Answer: Janice must pay 100% for the coat plus 13% for the tax. That gives us 113%. Take 113% of $69.99 1.13 69.99 $79.09 NSSAL ©2009 60 Draft C. D. Pilmer Example 7: Nita usually spends $250 per month on entertainment (e.g. movies, concerts, clubs). With the downturn in the economy, she has decided to reduce her spending on entertainment by 30%. How much will she now spend on entertainment after she initiates this plan? Answer: If the spending is reduced by 30%, that means that 70% of the spending is retained. Take 70% of $250. 0.70 250 $175 Questions: You may use calculator to complete these questions. () 1. Of the 128 people attending a movie, 96 bought popcorn. What percentage bought popcorn? 2. Jack wants to leave a 15% tip for a meal that cost $26. How much should he leave? 3. Nasrin brings home $2000 per month and budgets $350 for food. What percentage of her earnings does she budget for food? 4. The new local contract states that employees will receive a 4.5% salary increase. If an employee was making $24 000 a year, how much will he/she make under the new contract? 5. Of 120 adults interviewed at the mall, only 6 stated that they had used public transit to get to the mall? (a) What percentage of the people interviewed at the mall used public transit to get to the mall? (b) What percentage of the people interviewed at the mall did not use public transit to get to the mall? NSSAL ©2009 61 Draft C. D. Pilmer 7. F. Porter Contractors created a bar graph to illustrate the age ranges of their employees. (a) How many employees does the company have? (b) What percentage of the staff is between 20 and 29 years of age? (c) What percentage of the staff is 40 years of age or older? (d) What percentage of the staff is between 30 years and 49 years? (e) If two new employees were hired who were both 34 years of age, what percentage of employees would be between 30 and 39 years of age. Emergency Room Admissions Auto Accidents, 27% Home and Work Injuries, 34% Heart Attacks, 17% Other, 9% Respiratory Problems, 13% 13 12 11 10 9 Number of Employees 6. The circle graph shows emergency room admissions at a particular hospital over a one month period. During that time, a total of 420 were admitted. (a) How many individuals were admitted for respiratory problems? (b) How many individuals were admitted for heart attacks? (c) How many people were admitted for injuries associated with work injuries, home injuries or automobile accidents? 8 7 6 5 4 3 2 1 0 20-29 30-39 40-49 50-59 Ages NSSAL ©2009 62 Draft C. D. Pilmer 8. Jacob is presently paying $260 per month on electricity. The electricity is used for appliances, lights, heating, and the hot water tank. He is told that he can reduce this cost by 40% if he installs a woodstove. If he installs the woodstove, what should he expect his monthly electric bill to be? 9. (a) Jun is buying a video game that costs $49.99. What is the total cost including sales tax (13%)? (b) If Jun’s $49.99 video game was marked down 10%, how much would he have to pay in total (including 13% sales tax)? 10. If a metal rod, initially measuring 37.45 cm in length, expands by 0.1% when it is heated, how long will the rod be when it is heated? 11. You have $1600 in Digaflex stock. (a) If the stock drops in value by 5%, how much is your stock worth? (b) If the stock increases in value by 5%, how much is your stock worth? 12. A carpenter determines that he needs 840 board feet of flooring to complete a job. If an additional 10% must be factored in for waste, how many board feet of flooring should the carpenter order? NSSAL ©2009 63 Draft C. D. Pilmer Applications - Ratio and Proportion Ratios Example 1: Given the following rectangle: (a) Determine the ratio of its length to its width in fractional notation. (b) Determine the ratio of its width to its perimeter in colon notation. Answers: length 10 cm 10 10 2 5 (a) width 6 cm 6 62 3 (b) 6 cm 10 cm width 6 cm 6 62 3 perimeter 32 cm 32 32 2 16 3:16 Example 2: The Lake Fletchers canoe club entered a regatta. They took 48 female paddlers and 56 male paddlers. (a) Find the ratio of the male paddlers to female paddlers from this particular canoe club that participated in the regatta. (b) Find the ratio of the female paddlers to the total number of paddlers from this particular canoe club that participated in the regatta. Answers: male paddlers 56 56 8 7 (a) female paddlers 48 48 8 6 (b) female paddlers 48 48 8 6 total number of paddlers 104 104 8 13 Example 3: What is the alternator to engine ratio in colon notation if the alternator turns at 1150 rpm (revolutions per minute) when the engine is idling at 500 rpm? Answer: alternator 1150 rpm 1150 1150 50 23 engine 500 rpm 500 500 50 10 23:10 Example 4: You need 15 ft3 of cement to make 80 ft3 of concrete. Determine the ratio of cement to concrete in simplest form. Answer: cement 15 ft 3 15 15 5 3 concrete 80 ft 3 80 80 5 16 NSSAL ©2009 64 Draft C. D. Pilmer Proportions Example 1: A farmer uses 70 kg of chemical on a 20 acre field. How many kilograms of the same chemical will the farmer need to use on a 65 acre field? Answer: kilograms acres 70 x 20 65 20 x 70 65 20 x 4550 4550 20 x 227.5 Set the cross products equal to each other. x The farmer needs 227.5 kg of the chemical. Example 2: If you need 826 bricks to construct a wall that is 14 feet long, how many bricks will you need to construct a wall that is 48 feet long? Assume that the walls are the same height? Answer: bricks feet 826 x 14 48 14 x 826 48 14 x 39648 39648 14 x 2832 Set the cross products equal to each other. x You need 2832 bricks. Example 3: For a particular medication, the label reads “1.6 g in 20 cm3.” How many cubic centimeters are required to obtain 7.2 g of the active ingredient? Answer: grams cm 3 1.6 7.2 20 x 1.6 x 20 7.2 1.6 x 144 144 1.6 x 90 Set the cross products equal to each other. x NSSAL ©2009 You need 90 cm3. 65 Draft C. D. Pilmer Example 4: To estimate the number of fish in a lake, a fisheries officer catches 160 fish from the lake, tags them, and throws them back. He returns the next day and catches 70 fish. He notes that of these 70 fish, 28 have the tags that he attached on the previous day. Estimate how many fish are in the lake. Answer: tagged fish number of fish 28 160 70 x 28 x 70 160 28 x 11200 11200 28 x 400 Set the cross products equal to each other. x We estimate that there are 400 fish in the lake. Example 5: Ethan is using a map of the city to determine the distance between his house and his girlfriend’s house. The scale on the map indicates that every 4 centimetres on the map is equivalent to an actual distance of 1 kilometre. If the distance on the map between Ethan’s house and that of his girlfriend is 18.4 cm, what is the actual distance between the houses? Answer: map distance actual distance 4 18.4 1 x 4 x 118.4 4 x 18.4 18.4 4 x 4.6 Set the cross products equal to each other. x The houses are 4.6 km apart. Questions: You are permitted to use a calculator on these questions. () For questions for the first three questions, express all ratios using whole numbers and in simplest form. 1. Use the following triangle to answer parts (a), (b), and (c). (a) Find the ratio of the longest side to the shortest side. (b) Find the ratio of the longest side to the perimeter. (c) Find the ratio of the shortest side to the perimeter. 20 m 12 m 16 m NSSAL ©2009 66 Draft C. D. Pilmer 2. A poll at a local college found that 3220 students out of 4800 students were single. (a) Find the ratio of single students to the total number of students. (b) Find the ratio of married students to single students. (c) Find the ratio of married students to the total number of students 3. For every 100 tons of the earth’s crust there is 28 tons of silicon. What is the ratio of the weight of silicon to the weight of the earth’s crust? 4. A particular fruit punch asks that you mix 3 parts grapefruit juice to 4 parts cranberry juice. (a) How much grapefruit juice should be mixed with 1200 ml of cranberry juice? (b) How much cranberry juice should be mixed with 1500 ml of grapefruit juice? 5. The ratio of a quarterback’s completed passes to attempted passes is 3 to 8. (a) If he attempted 32 passes, how many would he have completed? (b) If he completed 15 passes, how many would he have attempted? NSSAL ©2009 67 Draft C. D. Pilmer 6. Jane can waterproof 450 ft2 of decking with 4 litres of sealant. How many litres will she need to seal a 1575 ft2 deck? 7. A quality control inspector examined 100 circuit boards and found 3 to be defective. At this rate, how many defective circuit boards would one expect to find in a lot of 2400? 8. The recommended daily allowance of protein for adults is 0.8 grams for every 2.2 pounds of body weight. (a) If someone weights 160 pounds, how much protein should he/she eat each day? (b) If someone is consuming 44 grams of protein each day, how much should that individual weight if they are attempting to follow the recommended daily allowance? NSSAL ©2009 68 Draft C. D. Pilmer 9. A sump pump discharges 320 litres of water in 5 minutes. How much time would it take for the same sump pump to discharge 800 litres of water? 10. The paper needed for a printing job weighs 5.5 kg per 500 sheets. How many kilograms of paper will be needed for a job requiring 13 500 sheets? 11. A builder sells a 1500 ft2 house for $187 500. If a 2200 ft2 house of the same quality is built by the same builder, how much would you expect it to cost? NSSAL ©2009 69 Draft C. D. Pilmer Applications - Signed Numbers There are several applications where signed numbers are used. Temperature is the most familiar application. If the temperature is reported as -10oC, you know that the temperature is 10oC below OoC. Sea level is another application. For example, the Badwater Basin in Death Valley has a reported elevation of -282 feet (or -86 metres). That means Badwater Basin is 282 feet (86 metres) below sea level. By contrast, Regina has a reported elevation of 1894 feet (or 577 metres). That means that Regina is 1894 feet (577 metres) above sea level. Businesses report their financial health using signed numbers. If a company reports a gain of $300 000, it would be recorded as +$300 000. If a company reports losses of $200 000, it is recorded as -$200 000. Financial statements from banks or credit institutions use signed numbers. If you deposit $50, it is recorded as +$50. If you withdraw $40, it is recorded as -$40. In golf, scores are often stated in terms of par. Suppose two golfers, Bill and Annette, are playing the same par 4 hole. If Bill completes the hole in 6 shots, his score on that hole is +2 because he made two shots above par. If Annette completes the hole in 3 shots, her score on the hole is -1 because she made one shot below par. When dealing with application, there are some key words that help identify the sign of a number. Negative below loss withdrawal decrease past below Positive above gain deposit increase future above Example 1: A football team gained 7 yards on the first down, lost 3 yards on the second down, and lost 6 yards on the third down. What was the overall change in position after the third down? Answer: 7 3 6 4 6 2 NSSAL ©2009 The position is -2 yards. This means that over the three downs they lost 2 yards. 70 Draft C. D. Pilmer Example 2: The lowest elevation in Africa is Lake Assal at -512 feet. The highest elevation is Mount Kilimanjaro at 19 340 feet. What is the difference in elevation between these two locations? Answer: 19340 512 19340 512 19852 They differ in elevation by 19 852 feet. Example 3: Submarine A is at a depth of -62 metres relative to sea level. Submarine B is three times deeper than submarine A. What is the depth of submarine B? Answer: 3 62 186 Submarine B is at a depth of -186 metres. Example 4: Kelly has been paying off her car loan. Last year she made monthly payments totaling $4944. How much were her monthly payments? Express your answer as a signed number. Answer: 4944 12 412 The monthly payments can be expressed as -$412. Example 5: The high temperature values over one week in January are found in the chart below. Day of the Week Temperature (oC) S -5 M -1 T +3 W +2 T -6 F -3 S -4 Determine the average high temperature over that one week period. Answer: 5 1 3 2 6 3 4 7 14 7 The average high temperature over the week was -2oC. 2 Questions: Answer the following questions and express your answers as signed numbers. 1. On the game show Jeopardy, contestants are penalized the value of the question when it is answered incorrectly. Suppose a contestant has a score of $800 and proceeds to answer a $1000 question incorrectly. What will the contestant’s new score be? NSSAL ©2009 71 Draft C. D. Pilmer 2. On Saturday, the temperature in Halifax was 3oC and the temperature in Sydney was -5oC. What was the difference in temperature between these two locations? 3. Ajay lost an average of 1.2 kilograms per week for 8 weeks. What was his net change in weight? 4. On the first nine holes, Mike Wier received 1 eagle (i.e. two strokes below par), 3 birdies (i.e. one stroke below par), 4 pars, and 1 bogie (i.e. one stroke above par). How far is Mike over or under par after the first nine holes? 5. During the first quarter of 2009, a company reported a net income of -$30 million. If this continued, what would the company’s net income be after 4 quarters? 6. Leo initially has $300 in his account. He then deposits a cheque worth $220. Over the next few days he writes a cheque for $260, and makes two ATM withdrawals, each of $140. What is the balance in Leo’s account? 7. The temperature of a particular object was initially -6oC. The object’s temperature increased by 2oC every minute. What was the temperature of the object after five minutes? 8. A scuba diver is initially at a depth of 7 m below the surface. He then dives down 4 m more. Find the diver’s present depth. NSSAL ©2009 72 Draft C. D. Pilmer 9. The temperature in the morning is -7oC. By noon, the temperature has risen by 3oC. Between noon and the evening, the temperature drops 6oC. What is the temperature in the evening? 10. A company’s net income for each quarter of a fiscal year is reported below. Quarter 1 2 3 4 Income (in millions) $1.2 -$0.9 -$1.7 $0.2 (a) What is the company’s net income for the year? (b) What is the company’s average quarterly income? 11. Jorell has been supplied with the high temperature reading for his town on January 10 for the last six years. Year 2003 2004 2005 2006 2007 2008 Temperature -10oC -6oC 1oC -4oC 3oC -8oC He figures that if he finds the average temperature over the last six years, he can use this value to predict the temperature on January 10, 2009. Find the average temperature. NSSAL ©2009 73 Draft C. D. Pilmer Reflect Upon Your Learning Fill out this questionnaire after you have completed pages 51 to 75. Select your response to each statement. 1 - strongly disagree 2 - disagree 3 - neutral 4 - agree 5 - strongly agree (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) NSSAL ©2009 I understand all of the concepts covered in the section, Applications - Fractions. I do not need any further assistance from the instructor on the material covered in the section, Applications - Fractions. I understand all of the concepts covered in the section, Applications - Decimals. I do not need any further assistance from the instructor on the material covered in the section, Applications - Decimals. I understand all of the concepts covered in the section, Applications - Percents. I do not need any further assistance from the instructor on the material covered in the section, Applications - Percents. I understand all of the concepts covered in the section, Applications - Ratio and Proportions. I do not need any further assistance from the instructor on the material covered in the section, Applications - Ratio and Proportions. I understand all of the concepts covered in the section, Applications - Signed Numbers. I do not need any further assistance from the instructor on the material covered in the section, Applications - Signed Numbers. 74 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 Draft C. D. Pilmer Putting It Together, Part I In the previous sections, you have looked at a variety of application questions but they have always been grouped together based on a particular mathematic concept (fractions, decimals, percents, ratio, proportion, and signed numbers). In this activity and those that follow, you will be expected to solve these same types of questions but you will have to figure out which particular mathematical concept will be used to answer the question. For the first activity sheet, hints have been provided in brackets so that you can go back in this resource and look up previously completed questions. Questions: Calculators can only be used with questions involving percentages or proportions. 1. Of 72 people who applied to a particular college program, the college accepted 55. What percentage of applicants did the college accept into this program? (Percents) () 2. Determine the perimeter of this rectangle. (Decimals) 45.2 cm 23.4 cm 3. On the game show Jeopardy, contestants are penalized the value of the question when it is answered incorrectly. Suppose a contestant has a score of $600 and proceeds to answer a $1000 question incorrectly. What will the contestant’s new score be? (Signed Numbers) 4. A motor brush is 1 7 3 inches long. How long is it after inch wears away? (Fractions) 16 8 5. The surface temperature on Mars varies greatly between night and day, and from the poles to the equator. At the equator, the temperature can reach at high of 27oC at midday. At the poles, the temperature can reach a low of -128oC at night. Determine the difference between the highest and lowest temperature found on Mars. (Signed Numbers) NSSAL ©2009 75 Draft C. D. Pilmer 6. A car depreciates (drops in value) as soon as it leaves the dealer’s lot. In the first year it typically depreciates by 20%. If the car initially cost $18 000, how much is it worth after one year? (Percents) () 7. Barbara earns $9.52 per hour. How much will she make before deductions if she works 23 hours? (Decimals) 8. The scale on a particular map indicates that 1 centimeter corresponds to 30 kilometres. How far apart are two towns if they are 16.3 centimetres apart on the map? (Ratio and Proportion) () 3 cup of vinegar. If you wanted to 4 triple the recipe, how much vinegar would you need? (Fractions) 9. A recipe for a homemade cleaning solution requires 10. Kelly has been paying off a loan. Last year she made monthly payments totaling $1560. How much were her monthly payments? Express your answer as a signed number. (Signed Numbers) 11. How much commission does a salesman receive on sales totaling $6200 at a 9% rate of commission? (Percents) () NSSAL ©2009 76 Draft C. D. Pilmer 12. A 180 pound individual burns 10.9 calories per minute while playing tennis. If this individual plays for 90 minutes, how many calories will he/she burn? (Decimals) 13. A fuel pump delivers 35 ml of fuel in 420 strokes. How many strokes are needed to pump 100 ml of fuel? (Ratio and Proportion) () 14. Partition walls separate the rooms of a house. If the partition wall is separating the garage 1 from main living space of the house, it must be clad with inch drywall on the one side and 2 5 inch drywall on the other side (i.e. garage side). If the studs used to construct the partition 8 1 walls are 3 inches thick, what is the total thickness of the partition wall when it clad on 2 both sides with drywall? (Fractions) 15. If 100 grams of ice cream contains 14 grams of fat, how many grams of fat are found in a 350 gram serving of the same ice cream? (Ratio and Proportion) () NSSAL ©2009 77 Draft C. D. Pilmer 16. A contractor estimates that it will cost him $127 000 to build a house for a client. If he needs to make 8% profit on the build, what should he charge the client on the build? (Percents) () 17. Four new snow tires cost $509.96. What was the price per tire? (Decimals) 18. The space shuttle completes one orbit every 1 1 hours. How many orbits does it complete in 2 24 hours? (Fractions) 19. Kate has been supplied with the high temperature reading for her small town on January 15 for the last five years. Year 2004 2005 2006 2007 2008 Temperature -6oC 1oC -4oC 2oC -8oC She figures that if she finds the average temperature over the last five years, she can use this value to predict the temperature on January 15, 2009. Find the average temperature. (Signed Numbers) NSSAL ©2009 78 Draft C. D. Pilmer Test Results 11 10 9 8 Number of Students 20. A college instructor was looking at her students’ test results. (a) How many students wrote the test? (b) What percentage of the class received a mark of 90% or greater? (c) What percentage of the class received a mark between 70% and 79%? (d) What percentage of the class received a failing grade (less than 60%)? (e) What percentage of the class received a mark of 80% or greater? (Percents) () 7 6 5 4 3 2 1 0 40-49 50-59 60-69 70-79 80-89 90-99 Mark on Test (%) NSSAL ©2009 79 Draft C. D. Pilmer Putting It Together, Part II Questions: Calculators can only be used with questions involving percentages or proportions. 1. James buys a T-shirt that costs $15.80 (taxes already included). If he pays with a $20 bill, how much change will he receive? 2. One third of a pizza is left over from last night’s party. You want half of the remaining pizza. How much of the original pizza does your piece represent? 3. A basketball player made 37 out of 46 free throws. What percentage is this? () 4. The community college accepts 3 out of every 7 applicants for their carpentry program. () (a) In 2006 the school received 371 applicants for carpentry. How many did they accept? (b) In 2007 the school accepted 147 applicants for carpentry. How many applicants applied? 5. The temperature of a particular object was initially -10oC. The object’s temperature increased by 2oC every minute. What was the temperature of the object after three minutes? NSSAL ©2009 80 Draft C. D. Pilmer 6. Figure out the length of side A. 13 16 7 8 " 5 8 " 3 1 8 " " Side A 7. A phone company charges $0.07 cents per minutes for long distance calls to the United States. If you talk to a friend in the US for 46 minutes, how much will you have to pay for the call? 8. A student answered 90% of the questions on a test correctly. () (a) If the test was comprised of 60 questions, how many questions were answered correctly? (b) If the test was comprised of 60 questions, how many questions were answered incorrectly? 9. It takes 38 litres of maple sap to produce 2 litres of maple syrup. () (a) How much sap is required to produce 11 litres of syrup? (b) How much syrup is produced using 142.5 litres of sap? 10. The water level in Fletchers Lake changed significantly over four months. In May it went up by 15 cm. In June, it went up by 5 cm. In July it went down by 7 cm. In August it went down 16 cm. How much had the water level changed by in that 4 month period? Express your answer as a signed number. NSSAL ©2009 81 Draft C. D. Pilmer 11. The inside diameter of a particular piece of copper tubing is 3 inch. The outside diameter is 8 9 inch. What is the wall thickness of the tubing? 16 12. A copper wire that is 750 feet long has a resistance of 1.89 ohms. What is the resistance in a wire of the same type measuring 2000 feet in length? () 13. Determine the unknown length, x, in the figure. 2.4 cm x 2.4 cm 12.53 cm 14. In a town of 6500 people, approximately 1200 are under the age of 25 years. () (a) What percentage of the town is under 25 years of age? (b) What percentage of the town is 25 years of age or older? NSSAL ©2009 82 Draft C. D. Pilmer 15. For each hour that an oil burner runs, it uses for 2 3 gallon of fuel. If the burner is only running 4 2 hours, then how much fuel is used? 3 16. Kimi initially has $200 in her account. She then deposits a cheque worth $120. Over the next few days she writes a cheque for $60, and makes an ATM withdrawal of $340. What is the balance in Kimi’s account? Express your answer as a signed number. 17. Determine the length of the shaft? 5 1 4 9 13 16 18. A phone company is charging $0.13 per minute for overseas long distance calls. If your bill for an overseas long distance call was $4.68, how many minutes was the call? 19. The yearly tuition at one college is $2700. The following year the tuition is supposed to increase by 4%. What will the new tuition be? () NSSAL ©2009 83 Draft C. D. Pilmer 20. Roof pitch is a way to describe the steepness of a roof. If a roof has an 8:12 pitch, it means that the roof rises 8 feet over a horizontal run of 12 feet. If a roof had an 8:12 pitch, what would be the height for a horizontal run of 21 feet? () 3 inches are cut from the 8 corners of a rectangular sheet of metal measuring 12 inches by 10 inches. After the four squares are removed, the sheet metal can be folded to create a box that is open on the top. What are the dimensions of the box? 21. Four squares having side lengths of 2 22. Kiana was making $11.40 per hour and then given a raise such that her new hourly wage is $13.15. How much is the raise? NSSAL ©2009 84 Draft C. D. Pilmer Putting It Together, Part III Questions: Calculators can only be used with questions involving percentages or proportions. 1. If three sheets of 3 inch plywood are glued together, how thick will the new sheet be? 8 2. A metal casting weighted 35 kg out of the mold. It weighed 28 kg after finishing. What percentage of the weight was lost in finishing? () 3. Jun had a car loan which he paid off by making 36 equal monthly payments. If his monthly payments were $204.36, how much did he ultimately pay back to the lending institution? 4. During the first quarter of 2008, a company reported a net income of -$1.5 million. If this continued, what would the company’s net income be after 4 quarters? 5. A salesperson is paid a $75 commission for selling $800 worth of goods. What is the commission on $1360 of sales? () NSSAL ©2009 85 Draft C. D. Pilmer 6. What is the difference in thickness between a 5 7 inch steel plate and a inch steel plate? 16 8 7. Nickel silver is 55% copper (Cu), 15% nickel (Ni), and the rest is zinc (Zn). () (a) What percentage of nickel silver is Zn? (b) If you had 60 kg of nickel silver, find the individual weights of the Cu, Ni, and Zn used in the alloy. 8. A scuba diver is initially at a depth of 5 m below the surface. He then dives down 3 m more. Find the diver’s present depth. Express your answer as a signed number. 9. A metal component for an engine must have 0.05 cm ground off. If the part initially measured 3.71 cm, what will be the measurement after the metal is ground off. 10. If the diameter of a hole is 2 7 inch, what is the radius of the hole? 8 11. On average, an adult’s heart beats 8 times in 6 seconds. How many times should an adult’s heart beat in 2 minutes? () NSSAL ©2009 86 Draft C. D. Pilmer 12. Jorell has $2800 in PanCanada stock. () (a) If the stock drops in value by 6%, how much is his stock worth? (b) If the stock increases in value by 6%, how much is his stock worth? 13. A company’s net income for each quarter of a fiscal year is reported below. Quarter 1 2 3 4 Income (in millions) $11 -$5 -$2 $8 (a) What is the company’s net income for the year? (b) What is the company’s average quarterly income? 15 15 inch by 1 inch has to have a hole 16 16 1 drilled in the center. If there must be of an inch of metal between the hole 4 and the side of metal, what is the diameter of the hole? 14. A small piece of metal measuring 1 15. A paramedic notes that the temperature of a patient is 101.7oF. After fifteen minutes, the temperature drops to 99.5oF. How much did the temperature drop? NSSAL ©2009 87 Draft C. D. Pilmer 16. A machinist can produce 6 parts in 40 minutes. If she can keep up this pace, how many parts can she complete in 6 hours? () 17. Tanya’s tuition was $2400. A loan for 3 of the tuition was obtained. How much was the 5 loan? 18. Pat DeCoste is a professional golfer from Shubenacadie, Nova Scotia. On the first nine holes, Pat obtained 1 eagle (i.e. two strokes below par), 2 birdies (i.e. one stroke below par), 5 pars, and 1 double bogie (i.e. two strokes above par). How far is Pat over or under par after the first nine holes? Express your answer as a signed number. 19. Determine the area of this rectangle. 7.3 m 5.2 m NSSAL ©2009 88 Draft C. D. Pilmer 20. The circle graph shows the different expenses associated with a business trip. Assuming that the individual spent $1200 in total for the business trip, answer each of the following questions. () (a) How much money was spent on lodging? (b) How much money was spent on transportation? (c) How much money was spent on food and entertainment? Business Trip Expenses Other 10% Entertainment 15% Lodging 30% Food 20% Transportation 25% NSSAL ©2009 89 Draft C. D. Pilmer Putting It Together, Part IV Questions: Calculators can only be used with questions involving percentages or proportions. 1. Determine the perimeter of this figure. 5.5 m 5.1 m 1.8 m 4.4 m 2. A 1.9 kg bag of grass seed covers 350 square feet of ground. How many kilograms of grass seed are needed to cover 4340 square feet of ground? () 3. The temperature at noon is 3oC. By supper time, the temperature has dropped by 4oC. Between supper time and midnight, the temperature drops 5oC. What is the temperature at midnight? 4. Janice earned $72 for working 7.5 hours. How much is her hourly wage? 5. A recipe calls for 1 1 cup of sugar. How much sugar should be used if only of the recipe is 2 3 being made? NSSAL ©2009 90 Draft C. D. Pilmer 6. A football team gained 8 yards on the first down, lost 2 yards on the second down, and lost 7 yards on the third down. What was the overall change in position after the third down? Express your answer as a signed number. 7. The property taxes for a particular city are described as $1.25 tax for every $100 of house value. () (a) If the house’s value is $228 000, how much would the property taxes be? (b) If you paid $2562.50 in property tax, what is the value of the house? 8. Determine the missing measure. 1 11 4 2 7 16 4 5 8 9. Tanya has to buy 27 feet of rope. If the rope costs $0.67 per foot, how much will it cost before taxes? NSSAL ©2009 91 Draft C. D. Pilmer 10. The human body contains 208 bones. Of those, 56 bones are found in the hands and feet. What percentage of the bones is found in the hands and feet? () 11. On 14 January 2009, 1 Canadian dollar was worth 0.6130 European euros. () (a) On this particular date, how much was $850 Canadian worth in euros? (b) On this particular date, how much was 700 euros worth in Canadian dollars? 5 inch plywood. If he stacks 10 sheets of plywood on top of each other, 8 how high is the stack? 12. Brian is stacking 13. Out of 250 circuit boards, 12 were identified as defective. What percentage of circuit boards was defective? () 14. The specifications for a drainage pipe are that the pipe must drop 1 inch for every 4 feet of pipe. How many inches should the pipe drop if it is 34 feet long? () 15. A grocery store sells their own brand of pop. During a sale you can purchase 5 bottles (2 litres) of pop for $4.00. Normally an individual bottle of pop costs $1.19. How much will you save per bottle if you purchase 5 bottles at the sale price? NSSAL ©2009 92 Draft C. D. Pilmer 16. A video gaming store tracked the sales of video games to different age groups over a weekend. The data was plotted on a bar graph. () Number of Individuals Weekend Video Sales 11 10 9 8 7 6 5 4 3 2 1 0 5 to 9 10 to 14 15 to 19 20 to 24 25 to 29 30 to 34 35 to 39 40 to 44 45 and over Ages (a) How many individuals purchased video games? (b) What percentage of video games was sold to individuals who are less than 10 years of age? (c) What percentage of video games was sold to individuals between the ages of 20 and 29 years of age? (d) What percentage of video games was sold to individuals who are 30 years of age or older? 17. Kendrick was making $12.75 per hour and then given a raise such that his hourly wage is $14.20. How much is the raise? 18. Determine the perimeter of this figure. 3 5 NSSAL ©2009 93 7 8 1 2 2 1 4 Draft C. D. Pilmer 19. The following table illustrates how a particular stock fluctuated over a five day period. Day Monday Tuesday Wednesday Thursday Friday Change in Stock Price in Cents -52 +33 -18 -23 +15 (a) How much did the stock change in price over the five day period? (b) What was the average daily change in the price of the stock over that period of time? 20. The bill before taxes for a meal is $48.50. () (a) What is the cost of the meal after taxes (13%)? (b) How much would you leave as a tip (15% of the complete bill)? NSSAL ©2009 94 Draft C. D. Pilmer Is It Reasonable? In the last few sections you have been completing a variety of word problems using a variety of mathematical concepts. When you complete a problem, do you take the time to consider whether your answer is reasonable given the problem or situation? Checking the reasonableness of an answer is an important skill in mathematics. Example 1: Maureen is organizing a field trip for five classes of grade 5 and 6 students. Buses will need to be rented to transport the students, teachers and chaperones. They will be taking 129 students, 5 teachers and 10 chaperones. If a bus can carry a maximum of 45 individuals, how many buses are required? Maureen completes the calculations. 129 5 10 144 3.2 (round 3.2 down to 3) 45 45 Maureen concludes that 3 buses are needed. Is this answer reasonable? Why or why not? Answer: Although we would typically round any number below 3.5 down to 3, it doesn’t make sense in the context of this question. Since each bus can only carry 45 students, we would end up having to leave a few individuals behind. In this case we must round 3.2 up to 4. Maureen should have stated that 4 buses are needed. Example 2: 1 5 inch in thickness is being glued to sheet of plywood that is 16 8 inch in thickness. How thick is the new sheet? Two individuals, Ken and Paul, attempted to answer the question separately. A laminate surface measuring Ken’s Answer: 5 1 5 2 1 10 1 9 8 16 8 2 16 16 16 16 Ken concludes that the new sheet will be 9 inch in thickness. 16 Paul’s Answer: 5 1 6 66 1 8 16 24 24 6 4 Paul concludes that the new sheet will be 1 inch in thickness. 4 Is either of the answers reasonable? Why or why not? Answer: Neither of the answers is reasonable. When you glue the two pieces of material together, the thickness should increase because the thicknesses are added together (not subtracted). Although Ken probably understands how to add and subtract fractions, he didn’t realize that NSSAL ©2009 95 Draft C. D. Pilmer the fractions should be added in this case. Paul, by contrast, knew that the thicknesses needed to be added but he didn’t know how to do this with fractions. He ended up with a 1 thickness of inch, which is far less than just the thickness of the plywood. This should 4 have told him that he made a mistake. Example 3: A triangle has side lengths measuring 2.15 m, 4.9 m, and 5.06 m. Three students were asked to determine the perimeter of the triangle. Their answers are supplied below. Which is a reasonable answer and why? Jeff’s Answer: 12.14 Dave’s Answer: 5.2675 Shirley’s Answer: 25.6036 Answer: To determine the perimeter of a triangle, we must add the three side lengths. If we round the numbers to the nearest unit, we get 2 5 5 12 . Jeff’s answer is the only reasonable answer. Example 4: Jun and his girlfriend, Nasrin, are dining out. The bill for the meal comes to $41.78. They decide to leave approximately a 15% tip. Jun thinks that they should leave $8.50. Nasrin thinks that they should leave $6.25. Which answer is more reasonable? Why is this so? Answer: We will round $41.78 down to $40. Ten percent of $40 is $4. Twenty percent of $40 is $8. That means that 15% of $40 will be halfway between $4 and $8. That gives us $6. We can conclude that Nasrin’s tip of $6.25 is more reasonable. Based on examples 3 and 4, we can see that having strong estimation skills is very useful in judging the reasonableness of an answer. Example 5: The ratio of female students in the college to male students in the college is 6 to 5. If there are 810 male students in the college, how many females are attending? Hassan works out that there are 675 females attending. Is his answer reasonable? Why or why not? Answer: Hassan’s answer is unreasonable. The ratio states that there are 6 female students for every 5 male students. This means the college has more female students than male students yet Hassan concluded that for the 810 male students there were 675 female students (the exact opposite). NSSAL ©2009 96 Draft C. D. Pilmer Questions: It should be noted that the most important aspect of your responses to these questions is your ability to explain in full sentences why an answer is reasonable or unreasonable. Do not complete the actual calculations to determine the reasonable answer. 1. A rectangle measures 2.1 cm by 6.3 cm. Three students were asked to determine the area of the rectangle. Their answers are supplied below. Which is a reasonable answer and why? Hinto’s Answer: 132.3 Kimi’s Answer: 16.8 Taylor’s Answer: 13.23 2. Three students are asked to figure out 52% of $38. Their answers are supplied below. Which is a reasonable answer and why? Jake’s Answer: $19.76 Nita’s Answer: $57.76 Pam’s Answer: $11.76 3. A carton of eggs contains a dozen eggs. Angela is hosting a large family reunion and has agreed to supply quiche for the meal. She needs to purchase 74 eggs for the recipe. How many cartons should she purchase? Three students completed the same calculations (shown below) but supplied different answers. 74 6.17 12 Which is a reasonable answer and why? Mary’s Answer: 7 Jim’s Answer: 6.17 Deangelo’s Answer: 6 4. Kirk is running a piece of lumber measuring 3 inch in thickness through a planer. The 4 1 inch. How thick is the lumber after it has passed through the planer? 16 Three students have attempted to answer the question. Which is a reasonable answer and why? 7 11 1 Bart’s Answer: Lisa’s Answer: Homer’s Answer: 8 16 6 planer will remove NSSAL ©2009 97 Draft C. D. Pilmer 5. If asked to evaluate 3.01 1.99 2.13 0.975 , which one of these answers appears reasonable and why? Answer A: 7.1449 Answer B: 9.46645 Answer C: 11.4262 6. The temperature in the morning is -3oC. It rises 5 degrees by noon and then drops by 7 degrees by supper time. What is the temperature at supper time? Which one of these answers appears reasonable and why? Answer A: 9oC Answer B: -5oC Answer C: -15oC 7. The price of a vehicle depreciates by about 20% one year after it is sold. If the car was initially worth $22 000, how much is it worth in one years time? Four students have attempted to answer the question. Jake’s Answer: $20 000 Shelly’s Answer: $4400 Bashir’s Answer: $26 400 Tylena’s Answer: $17 600 (a) Which is a reasonable answer and why? (b) Explain why the other three answers are unreasonable. 8. The ratio of the sales of red liquorish to black liquorish is 9 to 2. If the company sold 6300 units of red liquorish, how many units of black liquorish were sold? Three students have attempted to answer the question. Monica’s Answer: 1400 Julie’s Answer: 28 350 Montez’s Answer: 5800 (a) Which is a reasonable answer and why? (b) Explain why the other two answers are unreasonable. NSSAL ©2009 98 Draft C. D. Pilmer Reflect Upon Your Learning Fill out this questionnaire after you have completed pages 77 to 100. Select your response to each statement. 1 - strongly disagree 2 - disagree 3 - neutral 4 - agree 5 - strongly agree (a) (b) (c) (d) (e) (f) NSSAL ©2009 I understand all of the concepts covered in the sections, Putting It Together. I do not need any further assistance from the instructor on the material covered in the sections, Putting It Together. I can recognize which operation or operations (addition, subtraction, multiplication, or division) should be used to solve the application problem. I remember how to use the necessary mathematical concept (fractions, decimals, percents, signed numbers, ratio and proportion) to complete the application problem. I do not need any more practice questions. I can judge the reasonableness of an answer. 99 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 1 2 2 3 3 4 4 5 5 Draft C. D. Pilmer Post-Unit Reflections What is the most valuable or important thing you learned in this unit? What part did you find most interesting or enjoyable? What was the most challenging part, and how did you respond to this challenge? How did you feel about this math topic when you started this unit? How do you feel about this math topic now? Of the skills you used in this unit, which is your strongest skill? What skill(s) do you feel you need to improve, and how will you improve them? How does what you learned in this unit fit with your personal goals? NSSAL ©2009 100 Draft C. D. Pilmer Soft Skills Rubric Look back over the module you have just completed and assess yourself using the following rubric. Use pencil or pen and put a checkmark in the column that you think best describes your competency for each description. I will look at how accurately you have done this and will discuss with you any areas for improvement. You will be better prepared for your next step, whether it is work or further education, if you are competent in these areas by the end of the course. Keep all of these rubrics in one place and check for improvement as you progress through the course. Date: Competent demonstrates the concept fully and consistently Throughout this module, I… Approaching Competency Developing Competency demonstrates the concept most of the time demonstrates the concept some of the time Attended every class Let my instructor know if not able to attend class Arrived on time for class Took necessary materials to class Used appropriate language for class Used class time effectively Sustained commitment throughout the module Persevered with tasks despite difficulties Asked for help when needed Offered support/help to others Helped to maintain a positive classroom environment Completed the module according to negotiated timeline Worked effectively without close supervision Comments: (Created by Alice Veenema, Kingstec Campus) NSSAL ©2009 101 Draft C. D. Pilmer Answers Order of Operations (pages 1 to 2) 1. (a) 5 (g) 9 (m) 13 (b) 43 (h) 20 (n) 36 (c) 82 (i) 5 (o) 4 (d) 6 (j) 4 (p) 6 (e) 81 (k) 34 (q) 3 (f) 16 (l) 37 (r) 11 Adding Fractions (pages 3 to 6) 10 11 4 (e) 1 7 1. (a) 13 20 13 (e) 24 2. (a) 3. (a) 9 13 20 (b) 3 5 (f) 1 3 10 23 30 5 (f) 1 18 (b) (b) 11 5 6 1 (g) 1 3 (c) 23 24 5 12 3 (g) 4 (c) 1 3 4 (c) 3 2 3 1 (h) 1 2 (d) 11 12 1 (h) 1 3 (d) (d) 8 1 14 (e) 7 5 24 (f) 6 1 2 Subtracting Fractions (pages 7 to 9) 1. (a) 7 10 (b) 2 3 (c) 3 4 (d) 3 8 2. (a) 1 12 (b) 3 10 (c) 7 12 (d) 11 24 3. (a) 4 7 15 (b) 5 7 12 (c) 6 13 18 (d) 1 13 21 (e) 1 3 (f) 3 10 (e) 5 13 15 (f) 3 5 8 (e) 6 2 3 (f) 6 3 4 Multiplying and Dividing Fractions (pages 10 to 11) 4 45 13 (g) 1 15 1. (a) NSSAL ©2009 25 42 3 (h) 4 8 (b) (c) 3 14 (d) 10 21 (i) 3 102 Draft C. D. Pilmer 5 8 13 (g) 18 2. (a) 2 5 7 (i) 12 21 50 1 (h) 1 3 (b) (c) (d) 1 1 5 (e) 2 2 3 (f) 1 14 Decimals (pages 12 to 17) 6 10 1 4 7 (c) 200 60 5 10 100 1000 9 3 10 100 5 6 2 (d) 40 100 1000 10000 1. (a) 2000 400 80 2 2. (a) (b) (c) (d) (e) (f) (b) 7 fourteen and nine tenths three thousand two and fifteen hundredths four hundred fifty-nine and seven hundred thirty-six thousandths four hundred eighty and seven hundredths sixty-seven and twenty-five thousandths twenty-three and five hundred seventy-eight ten thousandths. 3. (a) 72.78 (e) 16.0715 (b) 148.659 (f) 9.4184 (c) 41.61 (g) 50.71 (d) 184.033 (h) 4.0781 4. (a) 32.25 (e) 2.9675 (b) 23.651 (f) 2.47 (c) 287.18 (g) 5.684 (d) 463.581 (h) 1.588 5. (a) 17.92 (b) 16.192 (c) 211.815 (d) 2.81313 6. (a) 3.4 (b) 5.4 (c) 0.62 (d) 2.51 (c) 0.37 (g) 0.034 (k) 2.1 (d) 0.04 (h) 0.008 (l) 0.036 Percents (pages 18 to 21) 1. (a) 0.38 (e) 0.06 (i) 1.05 9 20 3 (e) 50 (b) 0.21 (f) 0.245 (j) 0.142 7 10 21 (f) 50 7 25 18 (g) 25 4 5 6 (h) 5 2. (a) (b) (c) (d) 3. (a) 92% (e) 70% (i) 151% (b) 47% (f) 0.7% (j) 180% (c) 32% (g) 4.2% (k) 106% (d) 7% (h) 20.6% (l) 103.4% NSSAL ©2009 103 Draft C. D. Pilmer 4. (a) 40% (e) 37.5% (b) 56% (f) 93.75% (c) 85% (g) 104% 5. (a) 2198 (d) 11.1 (b) 377 (e) 1.96 (c) 144 (f) 64.41 6. (a) (d) (g) (j) (b) (e) (h) (k) (c) (f) (i) (l) 43 28 1400 120 580 21 240 4200 (d) 62.5% (h) 112.5% 12 44 500 360 Ratios and Proportions (pages 22 to 24) 3 4 4 (g) 3 670 (m) 947 9 5 3 (h) 5 11 (n) 520 3 5 8 (i) 5 31 (o) 20 9 7 51 (j) 37 1 (p) 7 13 16 46 (k) 65 9 (q) 5 8 9 612 (l) 253 5 (r) 7 1. (a) (b) (c) (d) (e) (f) 2. (a) 14 (b) 15 (c) 25 (d) 21 (e) 1 (f) 6 (d) (j) (p) (v) (e) -6 (k) -11 (q) 6 (w) 3 (f) (l) (r) (x) Signed Numbers (pages 25 to 29) 1. (a) -5 (g) -13 (m) 6 (s) 42 (y) -4 (b) (h) (n) (t) (z) 2. (a) -8 (g) 3 (m) 0.3 (b) -6 (h) -26 (n) 2.4 (c) -63 (i) -18 (o) -9.4 (d) -16 (j) -6.2 (p) -5/18 (e) 25 (k) -4.6 (q) -5/6 (f) 70 (l) -5 (r) 6/7 3. (a) 36 (b) -27 (c) -125 (d) -64 (e) -1 (f) -64 4. (a) 23 (d) 41 (g) -1 (b) -12 (e) 37 (h) -28 (c) -9 (f) -11 (i) 59 NSSAL ©2009 -13 -18 29 -32 -8 (c) (i) (o) (u) 11 7 -16 -56 104 4 -21 -3 54 7 -8 35 6 Draft C. D. Pilmer All Together Now, Part I (page 30) (a) 18 (b) 35.34 (d) 6 (e) 48 (g) 38.88 (h) -8 17 (k) 9 21 (j) 30 (m) 4 17 28 (c) 20 16 7 (f) or 1 9 9 (i) 3.9 (l) 88.32 (n) -20 (o) 1 All Together Now, Part II (pages 31) (a) 14 17 (d) 5 18 (b) 15.75 (c) 25 (e) 25 (f) -32 (g) -2 (h) 21 (i) 3 (j) 75.65 (k) 4 (m) 4.6 (n) 2 13 20 (l) -11 1 16 (o) 28.263 All Together Now, Part III (pages 32) 25 1 or 3 8 8 (d) 9 (g) 15 (a) 7 24 2.7 21.658 5 6 24 69.425 (b) -4 (c) 5 (e) 7 (h) 66 (f) (i) (j) 1 (k) 22.94 (l) (m) 4 (n) 27 (o) Addition Pyramids (pages 34 to 36) 1. 10 4 NSSAL ©2009 2. 6 13 5 3. 8 105 7 3 4 Draft C. D. Pilmer 3. -9 4. -7 7. -2 10. 11 6 5 13. 21 11 7 5 -9 -5 -8 22. 2 1.9 -7 2.2 0.7 0.4 0.6 16 1.2 3 106 -4 1 0.8 0.6 0.9 3.8 1.2 0.4 -1 1.3 0.4 24. 1.1 0.7 -5 2.3 2.3 1.9 0.9 -1 3 1.5 13 -6 21. 1.5 23. -6 12 4.2 -4 3 7 18. -4 20. 3 -8 9 4 -3 5 -6 3 15. -10 1 12 5 -3 -7 -6 20 8 -2 1.6 8 -5 17. -7 19. 2 10 -1 12 5 6 12. 10 14. 4 16. NSSAL ©2009 7 7 0.5 17 12 2.1 0.8 11. 2 9. 0.1 23 -5 0.9 0.7 -3 9 8. 0.4 6. -3 1.1 0.1 6 0.4 2.6 0.8 1.8 Draft C. D. Pilmer 25. 34 14 5 4 26. 20 9 1 28. 14 11 8 5 3 27 15 8 2 13 4 1 -6 3 34. 1 -14 4 1 4 5.7 0.9 NSSAL ©2009 0.2 0.3 1.4 0.6 0.8 5 0.6 107 -19 0 0.6 0.8 0.1 0.7 4 1.8 1.3 1.1 0.2 21 1.5 0.7 39. 1.1 2 2.8 2.5 0.9 -6 -9 8 3 0.6 3 1.3 1.6 -3 -11 36. 1.7 -2 -2 13 -15 10 -7 -5 6.1 1.8 7 4.8 3.1 2 2 10 0.7 3.2 1.2 1 12 1 33. 2.3 38. 1.4 5 8.1 1 4 6 3.3 -3 2.5 8 1 7 9 1.1 -5 6 4 9 11 35. 6 37. 8 -4 1 18 6 13 9 5 4 26 7 8 -5 5 7 -7 5 30. 15 32. -2 11 7 28 1 11 54 5 29 20 2 12 6 -4 3 27. 9 29. 7 31. 34 0.6 2.2 0.7 0.5 1.5 0.2 1.3 Draft C. D. Pilmer Find the Center Puzzles (pages 37 and 38) (a) (b) 7 2 12 (d) 3 9 44 4 3 3 6 2 8 4 24 3 7 4 11 14 45 4 11 5 2 1 (e) -5 2 -1 18 -3 -6 (g) (c) 12 (f) 12 3 -2 5 30 -2 -2 -10 -8 -2 -10 -5 -15 -18 -3 -4 -20 -5 2 2 3 (h) 0.3 2.4 3 (i) 2 0.1 2 0.7 0.6 0.2 0.4 2.8 7 3 0.4 1.2 3.6 2.4 14 10 1.4 0.9 0.5 4 0.8 3 0.1 0.4 1.6 (j) (k) -8 20 1 4 12 20 -8 -6.3 -1.7 1.5 -0.5 -3 -6 0.5 -24 -40 -2 -2 0.2 -1 4 (n) 5 1.2 -0.3 (o) -8 0.8 10 2.4 1.9 0.5 2.5 0.2 -2 0.2 -0.4 0.6 1 -2 -4 -0.1 -8 -8 -0.3 4.2 1.8 6 NSSAL ©2009 (l) 0.4 20 0.6 12 16 (m) -3.2 3 0.5 108 Draft C. D. Pilmer (p) (q) -2.4 (r) 0.8 -0.3 0.6 0.9 -5 1.6 5.6 -4 -2.1 -1.9 -1 10 -0.1 0.5 -0.2 -8 -1.1 1.9 0.5 1 0.4 0.9 0.6 1.5 3 2 KenKen Puzzles (pages 39 to 42) (a) (b) (c) 4 6 5 6 2 9 7 3 8 6 5 4 9 6 2 8 7 3 5 4 6 2 9 6 3 8 7 (d) (e) (f) 5 -2 -3 -1 -4 7 -7 8 -6 -3 5 -2 7 -1 -4 -6 -7 8 -2 -3 5 -4 7 -1 8 -6 -7 (g) (h) (i) 2 0.3 0.2 0.4 0.6 3 0.8 5 0.7 0.3 0.2 2 0.6 3 0.4 5 0.7 0.8 0.2 2 0.3 3 0.4 0.6 0.7 0.8 5 NSSAL ©2009 109 Draft C. D. Pilmer (j) (k) 4 3 5 2 9 3 7 6 3 2 4 5 7 6 3 9 5 4 2 3 3 9 6 7 2 5 3 4 6 7 9 3 (l) (m) 8 5 4 7 1 -5 4 -3 4 7 8 5 -3 4 1 -5 5 4 7 8 -5 -3 4 1 7 8 5 4 4 1 -5 -3 (n) (o) 5 -4 -3 -6 -7 -8 9 2 -6 -3 -4 5 2 9 -7 -8 -3 5 -6 -4 -8 -7 2 9 -4 -6 5 -3 9 2 -8 -7 NSSAL ©2009 110 Draft C. D. Pilmer (p) (q) -9 8 -2 -6 1 0.2 0.5 3 -6 -9 8 -2 3 0.5 1 0.2 -2 -6 -9 8 0.5 3 0.2 1 8 -2 -6 -9 0.2 1 3 0.5 (r) (s) 0.1 0.4 5 0.6 0.8 6 0.7 0.2 5 0.6 0.1 0.4 0.7 0.2 6 0.8 0.6 0.1 0.4 5 6 0.8 0.2 0.7 0.4 5 0.6 0.1 0.2 0.7 0.8 6 (t) (u) 5 9 3 2 6 8 4 6 7 3 6 3 5 9 2 7 3 8 6 4 3 2 6 5 9 3 6 7 4 8 9 6 2 3 5 4 7 3 8 6 2 5 9 6 3 6 8 4 3 7 NSSAL ©2009 111 Draft C. D. Pilmer (v) (w) -2 6 -4 -5 3 7 -3 -8 2 -6 3 -5 -2 -4 6 -8 -6 2 -3 7 -4 -2 3 6 -5 2 -8 7 -6 -3 -5 -4 6 3 -2 -6 2 -3 7 -8 6 3 -5 -2 -4 -3 7 -6 -8 2 (x) (y) 0.3 2 0.5 4 0.6 9 0.3 0.8 0.7 6 0.5 0.3 0.6 2 4 6 0.7 0.3 9 0.8 4 0.5 2 0.6 0.3 0.8 6 9 0.3 0.7 2 0.6 4 0.3 0.5 0.3 0.8 0.7 6 9 0.6 4 0.3 0.5 2 0.7 9 6 0.8 0.3 NSSAL ©2009 112 Draft C. D. Pilmer RAD Puzzles (pages 43 and 44) (a) 10 (b) - 6 = 4 = 12 3 34 - 24 = 10 = 7 + 3 - + + - - + 4 1 4 6 1 31 8 1 47 2 = = = = = = = = = = 4 3 40 5 = 8 = 2 3 = 9 = 54 6 = = = = = = = = = = 19 8 1 18 2 15 3 5 65 42 + - + + + - 2 5 21 3 = 7 = 9 - (c) -3 - 1 = 4 = 11 - 7 + -8 = 27 = -9 + 36 (d) - 4 = -7 = -56 8 35 + + + + - + -10 -9 1 54 -5 -7 -4 37 -51 6 = = = = = = = = = = 3 -5 30 -5 = -6 = -2 2 = -10 = -60 6 = = = = = = = = = = 21 -2 -36 -2 -18 45 12 -7 -56 -6 + + + + + -6 -9 9 + NSSAL ©2009 -3 = 6 = 0 - 113 + 6 = -3 = -4 - -1 Draft C. D. Pilmer (e) (f) 1.6 - 1.5 = 0.1 = 0.2 0.5 21 0.1 = 2.1 = 3 0.7 + + + - + - - + 2 0.5 1.1 0.1 8 18 1.9 0.6 2.7 4.3 = = = = = = = = = = 4 3 3.6 3 = 1.2 = 0.3 2 = 1.5 = 0.3 5 = = = = = = = = = = 9 30 1.5 1.3 2.8 2.4 1 0.7 1.8 0.2 - - + + + 0.7 0.8 0.4 - 0.1 = 0.3 = 1 - 1 = 0.8 = 6 - 5.2 Fraction Fury Puzzles (pages 45 to 47) 1. (a) 1, 2, 3, 4, 5, 6 Puzzle 1 6 5 6 2 4 1 2 4 4 1 5 1 2 9 7 1 8 9 7 NSSAL ©2009 8 2 5 4 (d) 1, 2, 3, 4, 7, 10 Puzzle 1 7 4 8 1 2 3 9 5 4 10 10 2 10 5 10 6 8 2 3 8 7 3 5 2 (c) 1, 2, 5, 7, 8, 9 Puzzle 5 8 3 3 3 4 5 2 (b) 2, 3, 4, 5, 8, 10 Puzzle 2 3 1 3 5 4 2 10 7 2 114 3 7 1 4 10 Draft C. D. Pilmer 2. (a) 1, 2, 3, 4, 5, 6, 7, 8 Puzzle 7 4 2 8 1 5 3 1 4 5 7 6 3 6 3 7 2 2 6 4 8 5 8 2 1 6 8 1 5 3 7 4 (b) 1, 2, 3, 4, 5, 6, 7, 8 Puzzle 8 3 2 4 6 5 1 7 5 2 2 1 7 8 6 6 3 6 4 5 4 3 1 3 8 1 8 7 7 5 4 2 (c) 2, 3, 4, 5, 6, 7, 8, 9 Puzzle 5 4 2 3 8 6 7 9 9 7 8 2 3 8 2 6 5 9 9 4 8 2 6 6 3 5 4 NSSAL ©2009 7 5 3 4 7 115 Draft C. D. Pilmer (d) 3, 4, 5, 6, 7, 8, 9, 10 Puzzle 5 4 3 10 9 10 8 9 4 6 3 10 3 7 5 8 7 6 8 4 6 8 7 5 7 6 3 9 9 10 5 4 Putting It Together: Math Games and Puzzles (pages 48 and 49) 1. (a) 32 17 10 6 (b) 15 7 4 0 5 8 3 3 5 -4 (c) -5 2 7 3.7 2 -7 -5 0.9 -2 0.3 1.7 1.1 0.6 0.6 0.5 0.1 2. (a) (b) 15 3 4 NSSAL ©2009 3 12 24 (c) -4 2 2 -6 -2 -8 -7 1.8 6 1 1.2 0.9 0.3 0.2 0.1 8 16 3 4 -2 0.1 116 Draft C. D. Pilmer 3. (a) -7, -5, -2, 3, 6 Puzzle (b) 0.9, 0.6, 0.2, 3, 5 Puzzle 3 -2 -7 6 -5 5 0.2 0.6 3 0.9 -5 -7 -2 3 6 0.9 3 0.2 5 0.6 -2 6 -5 -7 3 0.6 0.9 5 0.2 3 6 -5 3 -2 -7 0.2 0.6 3 0.9 5 -7 3 6 -5 -2 3 5 0.9 0.6 0.2 4. (a) (b) -40 -10 = 4 = 15 - 11 1.2 - 0.7 = 0.5 = 9 18 + - + + - + -8 -16 -2 -15 -3 0.8 0.4 0.1 0.2 6 = = = = = = = = = = 8 2 -48 6 = -8 = -1 0.3 = 0.6 = 1.8 3 = = = = = = = = = = 24 -1 -22 -9 40 1.6 5.3 2.4 1.7 6.9 - + - + - 5 0.8 -2 NSSAL ©2009 -7 = 14 = 9 + 117 5 = 4 = 0.1 + 3.9 Draft C. D. Pilmer 5. 1, 3, 4, 5, 7, 8, 9, 10 Puzzle 1 7 4 8 3 10 4 3 9 7 NSSAL ©2009 5 9 5 1 1 8 4 10 8 3 1 10 9 7 3 5 8 9 5 10 7 4 118 Draft C. D. Pilmer Applications - Fractions (pages 51 to 54) 1. 2 5 8 2. 1 5. 9 3 8 6. 12 9. 4 1 6 10. 16 13. 3 8 3 16 14. 5 3. 40 7. 1 1 2 11. 3 16 3 4 4. 1 7 16 8. 27 12. 15 5 8 Applications - Decimals (pages 55 to 58) 1. $32.50 2. $5.36 3. 350 shares 4. 2.76 m 5. 0.525 mg 6. $12.63 7. $20.72 8. 0.014 cm 9. 1432.4 km 10. 27.3 ml 11. 2.01 cm 12. 134 g Applications - Percents (pages 59 to 63) 1. 75% 2. $3.90 3. 17.5% 4. $25 080 5. (a) 5% (b) 95% 6. (a) 55 individuals (b) 71 individuals (c) 256 individuals 7. (a) 30 employees (b) approximately 23.3% (c) approximately 36.7% (d) approximately 66.7% (e) 43.75% 8. $156 9. (a) $56.49 (b) $50.84 10. 37.49 cm (rounded off) NSSAL ©2009 119 Draft C. D. Pilmer 11. (a) $1520 (b) $1680 12. 924 board feet Applications - Ratio and Proportion (pages 64 to 69) 1. (a) 5 or 5:3 3 (b) 5 or 5:12 12 (c) 1 or 1:4 4 2. (a) 161 or 161:240 240 (b) 79 or 79:161 161 (c) 79 or 79:240 240 3. 7 or 7:25 25 4. (a) 900 ml (b) 2000 ml 5. (a) 12 completed passes (b) 40 attempted passes 6. 14 litres of paint 7. 72 defective circuit boards 8. (a) 58.2 grams (b) 121 pounds 9. 12.5 minutes 10. 148.5 kilograms 11. $275 000 Applications - Signed Numbers (pages 70 to 73) 1. -$200 2. 8oC 3. -9.6 kg 4. -4 5. -$120 million 6. -$20 7. 4oC 8. -11 m 9. -10oC 10. (a) -$1.2 million (b) -$0.3 million 11. -4oC NSSAL ©2009 120 Draft C. D. Pilmer Putting It Together, Part 1 (pages 75 to 79) 1. 76% 2. 137.2 cm 3. -$400 5. 155oC 6. $14 400 7. $218.96 8. 489 km 9. 2 10. -$130 11. $558 12. 981 calories 13. 1200 strokes 14. 4 16. $137 160 17. $127.49 18. 16 orbits (b) 12.5% (e) 37.5% (c) 31.25% 4. 1 11 inches 16 5 inches 8 1 cups 4 15. 49 grams 19. -3oC 20 (a) 32 (d) 12.5% Putting It Together, Part II (pages 80 to 84) 1 of a pizza 6 1. $4.20 2. 3. 80% 4. (a) 159 applicants (b) 343 applicants 5. -4oC 6. 1 8. (a) 54 questions (b) 6 questions 9. (a) 209 litres (b) 7.5 litres 10. -3 cm 11. 7 inches 16 7. $3.22 3 inch 32 12. 5.04 ohms 13. 7.73 cm 14. (a) 18% (b) 82% 15. 2 gallons 16. -$80 NSSAL ©2009 17. 15 121 1 inches 16 Draft C. D. Pilmer 18. 36 minutes 19. $2808 20. 14 feet 1 1 3 21. 7 inches by 5 inches by 2 inches 4 4 8 22. $1.75 Putting It Together, Part III (pages 85 to 89) 1. 1 1 inches 8 2. 20% 3. $7356.96 4. -$6 million 5. $127.50 6. 7. (a) 30% (b) Cu: 33 kg, Ni: 9 kg, Zn: 18 kg 8. -8 m 9. 3.66 cm 9 inch 16 10. 1 7 inches 16 11. 160 times 12. (a) $2632 (b) $2968 13. (a) $12 million (b) $3 million 7 inches 16 15. 2.2oF 16. 54 parts 17. $1440 18. -2 19. 37.96 m2 20. (a) $360 (b) $300 (c) $420 14. 1 Putting It Together, Part IV (pages 90 to 94) 1. 16.8 m 2. 23.56 kg 4. $9.60 5. 7. (a) $2850 (b) $205 000 NSSAL ©2009 3. -6oC 1 cup 6 6. -1 yards 122 Draft C. D. Pilmer 8. 4 3 inches 16 11. (a) 521.05 euro 12. 6 1 inches 4 9. $18.09 10. 27% (b) $1141.92 13. 4.8% 14. 8.5 inches (b) 4% (c) 38% 15. $0.39 16. (a) 50 (d) 32% 5 inches 8 17. $1.45 18. 11 19. (a) -45¢ (b) -9¢ 20. (a) $54.81 (b) $8.22 Is It Reasonable? (pages 95 to 98) 1. Taylor’s answer of 13.23 is reasonable because 6 2 12 . 2. Jake’s answer of $19.76 is reasonable because 50% of 40 is 20. 3. Mary’s answer of 7 is reasonable because you have to round up from 6.17. 11 3 12 is reasonable because it slightly less than or . One could also 16 4 16 argue that Lisa is much smarter than Homer and Bart so there is a greater likelihood that her answer is right. 4. Lisa’s answer of 5. Round to nearest whole number and use order of operations. 3 2 2 1 3 4 1 12 1 11 Answer C (11.4262) is the reasonable answer. 6. If temperature initially starts below zero, rises slightly and drops back a little more than it rose. This means that the suppertime temperature should be close to the morning temperature. Based on this, Answer B (-5oC) is the reasonable answer. NSSAL ©2009 123 Draft C. D. Pilmer 7. (a) If the vehicle depreciates (goes down in value) by 20%, that means that 80% of the original price is retained. Based on this Tylena’s answer ($17 600) seems reasonable because it looks to be 80% of $22 000. (b) Although Jake’s answer is showing a drop in the value of the car, the drop isn’t even 10%. Shelly’s answer is extremely low. No one would buy a $22 000 vehicle if would only be worth $4400 in one years time. It appears that she figured out how much the value will drop, but didn’t work out the actual value. Bashir’s answer indicates that the value of the car appreciated (went up in value) which is not the case. 8. (a) According to the ratio there are 9 red liquorish sales for every 2 black liquorish sales. If one sold 6300 units of red liquorish, then you would expect that black liquorish sales would be much smaller; possibly between 1000 and 2000 units. That means that Monica’s answer (1400) is the only one that seems reasonable. (b) Julie’s answer is unreasonable because she is showing that black liquorish sales are far greater than red liquorish sales. Montez’s answer is unreasonable because she is showing that red liquorish sales are only slightly larger than black liquorish sales. NSSAL ©2009 124 Draft C. D. Pilmer Online Supports Order of Operations Learnalberta Exploring Order of Operations Use It Math Goodies Order of Operations with Exponents YouTube: Order of Operations YouTube: Watch Video on Order of Operations PEMDAS Fractions Algebra Help Fraction Operations Calculator Hot math Fraction Operations Shodor Interactive Fraction Quiz YouTube: Operations with Fractions YouTube: Fractions & Proportions Subtracting Mixed Numbers YouTube: Watch Video on Adding Mixed Numbers Pre Algebra Help YouTube: Multiplying & Dividing Fractions and Mixed Numbers Decimals Math Goodies Solving Decimal Word Problems Teachnology Decimal Math Worksheets Sinclair Decimal Word Problems Super Teacher Worksheets Decimals YouTube: Operations with Decimals YouTube: Adding Decimals YouTube: Subtracting Decimals YouTube: Multiplying Decimals YouTube: Dividing Decimals Percents Math Goodies Percent Unit 4 Sweethaven Working with Percents YouTube: Watch Video on Percent of a Number Math Help NSSAL ©2009 125 Draft C. D. Pilmer Ratio and Proportion Purplemath Ratios Proportions BBC Skillswise Ratio and Proportion CSG Ratio Proportion Calculator Teachers TV Math 4 Real Set A ratio and Proportion Algebra Lab Word Problems Proportions Introductory Consumer Math Proportion Word Problem Examples Sinclair Proportion Word Problems Handouts YouTube: Fractions & Proportions Understanding Math Ratios Signed Numbers Regents Mathematics A Signed Numbers Sweethaven Operations with Signed Numbers Southern Maine Community College Signed Numbers Amby Integers Operations with Signed Numbers YouTube: Lesson 101 Adding Signed Numbers YouTube: Lesson 102a Subtracting Signed Numbers YouTube: Lesson 102b Multiplying and Dividing Signed Numbers NSSAL ©2009 126 Draft C. D. Pilmer