Principles of Mathematics MPM1D


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1 Principles of Mathematics MPM1D Grade 9 Academic Mathematics Version A
2 MPM1D Principles of Mathematics Introduction Grade 9 Mathematics (Academic) Welcome to the Grade 9 Principals of Mathematics, MPM 1D. This fullcredit course is part of the new Ontario Secondary School curriculum. In this course, you will develop generalizations of mathematical ideas and methods through the exploration of applications, the effective use of technology, and abstract reasoning. You will investigate relationships to develop equations of straight lines in analytic geometry, explore relationships between volume and surface area of objects in measurement, and apply extended algebraic skills in problem solving. You will engage in abstract extensions of core learning that will deepen your mathematical knowledge and enrich your understanding. Material This course is selfcontained and does not require a textbook. You will require lined paper, graph paper, a ruler, a scientific calculator and a writing utensil. Expectations The overall expectations you will cover in the lesson are listed on the first page of each lesson. Lesson Description Each lesson contains one or more concepts with each being followed by support questions. At the end of the lesson the key questions covering all concepts in the lesson are assigned and will be submitted for evaluation. Evaluation In each lesson, there are support questions and key questions. You will be evaluated on your answers to the key questions in each lesson, the midterm exam and the final exam. Support Questions These questions will help you understand the ideas and master the skills in each lesson. They will also help you improve the way you communicate your ideas. The support questions will prepare you for the key questions. Copyright 00, Durham Continuing Education Page of 6
3 MPM1D Principles of Mathematics Introduction Write your answers to the support questions in your notebook. Do not submit these answers for evaluation. You can check your answers against the suggested answers that are given at the end of each unit. Key Questions The key questions evaluate your achievement of the expectations for the lesson. Your answers will show how well you have understood the ideas and mastered the skills. They will also show how well you communicate your ideas. You must try all the key questions and complete most of them successfully in order to pass each unit. Write your answers to the key questions on your own paper and submit them for evaluation at the end of each unit. Make sure each lesson number and question is clearly labelled on your submitted work. Midterm and Final Examination The midterm and final exams are weighted exams. Knowledge and understanding 0%; Application 0%; Communication 10%; Thinking/Inquiry 0%. What You Must Do To Get a Credit In order to be granted a credit in this course, you must Successfully complete the Key Questions for each unit and submit them for evaluation within the required time frame. Complete the midterm exam after Unit. Complete and pass a final examination. Your Final Mark Each Unit has lessons each worth % (10% per Unit x Units) 0% Midterm Test 0% Term Final Examination 0% Copyright 00, Durham Continuing Education Page of 6
4 MPM1D Principles of Mathematics Introduction Unit One Lesson One Introduction to ratios Simplifying ratios Equivalent ratios Solving proportions using ratios Rate Unit rate Percent Percent of a number Lesson Two Introduction to integers Addition and subtraction of integers Multiplication and division of integers Lesson Three Introduction to rational numbers Ordering rational numbers in numerical order Addition, subtraction, multiplication and division of nonfraction rational numbers Addition, subtraction, multiplication and division of fractions Lesson Four Introduction to exponents Multiplying powers of the same base Dividing powers of the same base Powers of powers Zero exponents Negative exponents Converting into scientific notation Converting from scientific notation Multiplying numbers in scientific notation Lesson Five Introduction to square root Pythagorean Theorem Copyright 00, Durham Continuing Education Page of 6
5 MPM1D Principles of Mathematics Introduction Unit Two Lesson Six Introduction to polynomials Like terms Addition and subtraction of polynomials Distributive law Multiplication and division of polynomials Simplifying expressions then using substitution Lesson Seven Introduction to algebra Solving for unknowns Checking solutions to algebraic equations Lesson Eight Introduction to problem solving steps suggested to problem solving Lesson Nine Introduction to slope Cartesian plane x and y coordinates on the Cartesian plane Plotting order pairs Quadrants of the Cartesian plane Recognizing positive, negative, zero and undefined slopes Using the rise and the run of a given line to find its slope Using a pair of coordinates of a line to calculate slope Lesson Ten Introduction to relationships in data Graphing relationship in data Working with table of values Creating graphs for tables of values Using graphs to solve related questions Copyright 00, Durham Continuing Education Page of 6
6 MPM1D Principles of Mathematics Introduction Unit Three Lesson Eleven Introduction to the line Standard form of an equation Yintercept form of an equation Converting to and from standard to yintercept form Parallel lines Perpendicular lines Point of intersection of two lines x and y intercepts Lesson Twelve Introduction to direct and partial variation The origin on a Cartesian plane m in the equation y = mx graphing equation of the form y = mx Introduction to partial variation b in the equation y = mx + b graphing equation of the form y = mx + b Lesson Thirteen Introduction to scatter plots Creating scatter plots Positive, negative and no correlation Determining the equation of best fit Extrapolation Interpolation Lesson Fourteen Introduction to averages (measures of central tendency) Mean average Median average Mode average Lesson Fifteen Introduction to perimeter and circumference Radius and diameter Calculations using pi (π) Solving perimeter/circumference questions using formulas and substitution Copyright 00, Durham Continuing Education Page 6 of 6
7 MPM1D Principles of Mathematics Introduction Unit Four Lesson Sixteen Introduction to area and surface area Radius and diameter Calculations using pi (π) Solving area and surface area questions using formulas and substitution Lesson Seventeen Introduction to volume Radius and diameter Calculations using pi (π) Solving volume questions using formulas and substitution Lesson Eighteen Introduction to angle geometry Angle types Angle properties Angle properties involving parallel lines and transversals Finding unknown angles with justification Lesson Nineteen Introduction to Triangles and Quadrilaterals Triangle types Triangle properties Quadrilateral types Quadrilateral properties Finding unknown angles with justification Lesson Twenty Introduction to Triangle medians and altitudes Median properties Centroid Altitude properties Orthocentre Copyright 00, Durham Continuing Education Page 7 of 6
8 Ratios, Rate and Percent Lesson 1
9 MPM1D Principles of Mathematics Unit 1  Lesson 1 Lesson One Concepts Introduction to ratios Simplifying ratios Equivalent ratios Solving proportions using ratios Rate Unit rate Percent Percent of a number Ratios A ratio is a comparison of two or more numbers with the same units. What is written first in the description matches with the first number in the ratio and what is second matches with the second and so on. Example The ratio of to is to. Ratios can written in the following three ways: 1. as a fraction,. as a ratio, :. with words, to Ratios in Lowest Terms A ratio is in lowest terms (simplest form) when the greatest common factor of the terms is one. 6: in lowest terms is :. Both terms are divided by two to produce a ratio in lowest terms. Example Give the ratio of A:B in lowest terms. A A A A B B B B B B B B B B A B A B B A B A A A B Copyright 00, Durham Continuing Education Page 9 of 6
10 MPM1D Principles of Mathematics Unit 1  Lesson 1 Solution The ratio of A:B is 10:1 or In lowest terms both 10 and 1 are divisible by so 10 1 Equivalent Ratios = or :. Any two or more ratios that simplify to the same lowest terms are equivalent ratios. Example Show that the following ratios are equivalent: 1:16 and 60:80 Solution 1:16 has a greatest common factor of reducing the ratio to : and 60:80 has a greatest common factor of 0 also reducing the ration to : 1 = 16 = or : and 60 0 = 80 0 = or : Solving Proportions Equivalent ratios are used when trying to solve questions involving proportion. Example Solve each proportion. a) : = x : 1 = 0 x Solution a) To find x, cross multiplication is used then is followed by division. : = x : is the same as x = 10 = x ()()=10 ()(x)=x 8 = x Copyright 00, Durham Continuing Education Page 10 of 6
11 MPM1D Principles of Mathematics Unit 1  Lesson 1 1 = 0 x Rate 1x = 80 x = 0 Rates compare two numbers that are expressed in different units. Unit rate is a comparison where the second term is always one. Example Write the following ratios as a unit rate. a) soft drinks for 1 people. $7 earned in 0 hours worked. Solution a) :1 = so x = = 1x = x Therefore there are two soft drinks per person 7 7 : 0 = so 7 x = = 0x x Therefore the unit rate is $11.88/hr. Copyright 00, Durham Continuing Education Page 11 of 6
12 MPM1D Principles of Mathematics Unit 1  Lesson 1 Support Questions 1. Write the following ratios in simplest form. a) 0: 7:1 c) 1:9. Write as a ratio in the lowest terms. a) cups of water for every cups of flour $1 saved for every $7 spent c) 8 people passed for every 10 writing a test. Write each ratio. a) clubs to diamonds hearts to clubs c) spades to diamonds d) spades to hearts. Write each ratio in simplest terms. a) home runs to home runs $10 to $ c) days to 8 days. On a hockey team 6 of 19 players are defensemen. Write each ratio in simplest terms. a) defensemen to team members team members to defensemen c) defensemen to nondefensemen 6. A gas tank in a car holds litres. The cost of a fill up is $8.. a) What is the unit price per litre? (round to the nearest cent) Using the answer in a, how much will litres cost? c) How many litres can be purchased with $1.00? 7. Johnny had 1 hits in 0 at bats. How many at bats are needed to achieve 100 hits? Copyright 00, Durham Continuing Education Page 1 of 6
13 MPM1D Principles of Mathematics Unit 1  Lesson 1 Percent Percent means the number of parts per hundred. Expressing a fraction as a decimal then as a percent Example Express the following fraction as a percent. Solution = = = 80% 1 st convert to decimal Multiply the decimal by 100 Percent of a Number Example a) Find % of 00. 0% of a number is 1. c) Calculate the total cost of a shirt costing $.99 with 7% GST. Solution of means multiplication a) % of 00 = (0.)(00) = 70 Therefore % of 00 is 70 Copyright 00, Durham Continuing Education Page 1 of 6
14 MPM1D Principles of Mathematics Unit 1  Lesson 1 0% of a number is 1 (0.0)(x) = 1 0.0x = 1 x = a number is an unknown (x) is means = x = 7 Therefore 0% of 7 is 1 c) Calculate the total cost of a shirt costing $.99 with % GST. Total cost = cost of item + tax % of.99 = (0.0)(.99) = $1. Rounded to the nearest penny Total cost = = $6. Support Questions Write the following ratios in simplest form. 8. Express each fraction as a percent. a) 7 8 c) Express each percent as a decimal. a) % 11% c) 0.9% d).% 10. Express each percent as a fraction in lowest terms. a) % 78% c) 0.% d) 1% 11. Find a) % of 00 70% of 0 c) 180% of 10 d) 0.7% of 1000 Copyright 00, Durham Continuing Education Page 1 of 6
15 MPM1D Principles of Mathematics Unit 1  Lesson 1 Support Questions 1. Determine the number in each statement. a) 0% of a number is 90. 8% of a number is. c) 1% of a number is A pair of jeans that normally sells for $9.0 is on sale for % off. What is the sale price of the jeans before taxes? 1. A MP player costs $19.9. What is the total cost of the MP player including % GST and 8% PST. 1. The population in a town increased by 80 citizens. This represents an increase of 6.% over last year. What was the population of the town last year. Key Question #1 1. Write the following ratios in simplest form. ( marks) a) 1:18 :1 c) 8:. Write as a ratio. ( marks) a) teaspoons of sugar for every cups of flour tshirts for every $00 raised c) 6 buckets of popcorn for every tickets soldö. Write each ratio. ( marks) a) happy faces to airplanes stars to flowers c) flowers to airplanes d) stars to happy faces øéö éö ööéøøééî öø. Write each ratio in simplest terms. ( marks) a) goals to goals $1 to $ c) people to people Copyright 00, Durham Continuing Education Page 1 of 6
16 MPM1D Principles of Mathematics Unit 1  Lesson 1 Key Question #1 (continued). On a soccer team of 11 players are defensemen. Write each ratio in simplest terms. ( marks) a) defensemen to team members team members to defensemen c) defensemen to nondefensemen 6. Express each fraction as a percent. ( marks) a) 11 8 c) 7. Express each percent as a decimal. ( marks) a) 7% 16% c) 0.7% d) 7.% 8. Express each percent as a fraction in lowest terms. ( marks) a) % 6% c) 0.0% d) % 9. Find ( marks) a) 9% of 00 6% of 100 c) 180% of 00 d) 0.% of The regular price of a DVD player is $8.99. It is on sale for 0% off. What is the amount of the discount? What is the sale price? What is the total price after 1% sales tax is included? ( marks) 11. The number of people in a town in 00 was 00. The population for 00 increased by.%. What was the population increase? What was the town s population in 00? ( marks) Copyright 00, Durham Continuing Education Page 16 of 6
17 MPM1D Principles of Mathematics Unit 1  Lesson 1 Key Question #1 (continued) 1. An instructor used the following equation to calculate a students final mark. Final Mark = 7% of term + % of exam mark. Suppose your term mark is 77%. How will your mark change as your exam mark has been taken into account? Copy and complete the table below: ( marks) Term mark Exam mark Final mark 77% 100% 77% 90% 77% 80% 77% 70% 77% 60% 77% 0% 77% 0% 77% 0% 77% 0% 77% 10% 77% 0% 1. Write to explain how your final mark in question 1 changes as your examination mark goes from great to not so good. ( marks) Copyright 00, Durham Continuing Education Page 17 of 6
18 Integers Lesson
19 MPM1D Principles of Mathematics Unit 1  Lesson Lesson Two Concepts Introduction to integers Addition and subtraction of integers Multiplication and division of integers Integers Integers are the set of numbers,, , 1, 0, 1,,, Adding and Subtracting Integers Example a) ( ) + ( ) (+) ( 7) Solution If these two signs are opposite then the sign becomes negative a) ( ) + ( ) = = 8 (+) ( 7) = + +7 = +1 If these two signs are the same then the sign becomes positive Support Questions 1. Add. a) (+) + ( ) ( 7) + ( ) c) ( 8) + (+8) d) (0) + ( 6) e) () + (+8) f) ( 7) + (+) g) (+1) + ( 7) h) (+7) + (+1) i) (+1) + ( ). Subtract a) (+1) ( ) ( ) ( ) c) ( 9) (+8) d) (0) ( ) e) (+) (+6) f) ( 8) ( ) g) ( 7) ( 7) h) ( 7) (+) i) (+1) ( ) Copyright 00, Durham Continuing Education Page 19 of 6
20 MPM1D Principles of Mathematics Unit 1  Lesson Support Questions. For each statement, write an expression then simplify. a) moves forward steps and moves backwards steps goes up an elevator 9 floors then down 6 floors c) a gain of $1 followed by a loss of $8 d) a loss of $ followed by a gain of $ e) a rise in temperature of 18 C then a drop of 0 C. For each statement, write an expression then finish the question. a) A plane goes up into the air 000m then increases another 100m and then decreases its elevation by 1000 m. Find the height of the plane. An elevator is on the 16 th floor. The elevator descends 8 floors, then ascends 1 floors to the top of the building. What floor is the elevator on? c) The open value of a stock is $. Later that day the stock increase by $8 only to fall back by $ to end the day. What is the closing value of the stock? d) At 8:00 am the temperature outside is 6 C. At noon the temperature had increased by 9 C. By :00 pm the temperature had increased another C and by 9:00 pm the temperature had decreased 7 C. What was the temperature at 9:00 pm? e) The opening balance in a bank account is $000. $ was spent on living expense. $17 was deposited into the account from salary. What is the closing balance? Multiplying and Dividing Integers The product or quotient of an integer question with an even amount of negative values will always give a positive answer unless one of the values is zero. The product or quotient of an integer question with an odd amount of negative values will always give a negative answer unless one of the values is zero. Example a) (+1)( )( )( 1)(+)( ) ( )( )( 1)(+1) c) () () Copyright 00, Durham Continuing Education Page 0 of 6
21 MPM1D Principles of Mathematics Unit 1  Lesson d) Solution a) (+1)( )( )( 1)(+)( ) = 60 ( )( )( 1)(+1) = 6 c) () () = +8 d) + 8 = 7 or (+60) negative signs in question gives positive answer negative signs in question gives negative answer Support Questions. Multiply. a) (+)( )( )( ) ( 7)( )(+1) c) ( 8)(+9) d) (0)( 6)( )( ) e) ()(+8) f) ( 1)(+) g) ( 1)( 7) h) (+7)( 1)(+1) i) ( 1)( )( 1)( ) 6. Divide a) (+8) ( ) ( 9) ( ) c) ( 1) (+) d) ( 1) (+) e) f) 9 ( + 8) ( + ) g) h) i) (+10) ( 10) ( ) ( + 7) 7 ( + 81) j) k) ( 9) (+) l) ( + 9) ( ) 7. For each statement, write an expression then finish the question. a) A person paid $ a day for 6 days on lunch. How much did this person spend in total? A person made $0 a day for 8 days worked. How much did this person make in total? c) A car traveled 00 km in hours. How many km did the car travel per hour? d) An individual times collected 8 stamps for his collection. Copyright 00, Durham Continuing Education Page 1 of 6
22 MPM1D Principles of Mathematics Unit 1  Lesson Key Question # 1. Add, subtract, multiply or divide as required. (1 marks) a) (+) + ( ) ( 7) + ( ) c) ( 8) + (+8) d) (0) + ( 6) e) () + (+8) f) ( 7) + (+) g) (0) ( ) h) (+) (+6) i) ( 8) ( ) j) ( 7) ( 7) k) ( 7) (+) l) (+1) ( ) m) (+)( )( )( ) n) ( 7)( )(+1) o) ( 8)(+9) p) (0)( 6)( )( ) q) ()(+8) r) ( 1)(+) s) (+8) ( ) t) ( 9) ( ) 6 u) 9 v) w) (+10) ( 10) x) ( + 9) y) ( 9) (+8) z) ( + 81) ( ). For each statement, write an expression then simplify. ( marks) a) moves backwards steps and moves forward 7 steps goes up 1 stairs then down 8 stairs c) a gain of $11 followed by a loss of $18 d) a rise in temperature of 1 C then a drop of 9 C. A hockey player was given $ for each of 6 goals she scored. Find her total earnings. ( marks). An overall loss of $ occurred over an 8 day period. Find the mean loss per day. ( marks). For each statement, write an expression then write its quotient or a product. ( marks) a) A company twice suffers a loss of trucks. On occasions, a baseball player lost 6 baseballs. c) The mean daily gain when $8 000 is gained over 6 days. d) A chicken place sells buckets of 0 pieces of chicken. 6. What did it mean when it is said that two negatives make a positive? Is this always true? Prove with examples. ( marks) Copyright 00, Durham Continuing Education Page of 6
23 Rational Numbers Lesson
24 MPM1D Principles of Mathematics Unit 1  Lesson Lesson Three Concepts Introduction to rational numbers Ordering rational numbers in numerical order Addition, subtraction, multiplication and division of nonfraction rational numbers Addition, subtraction, multiplication and division of fractions Rational Numbers Rational numbers are the set of numbers that can be written in the form m n, where m and n are integers and n 0. Rational numbers can always be represented on a number line. Rational numbers can be written as a decimal by dividing the numerator by the denominator. Examples of rational numbers. Example ,.7, 1.1, 1.67,,, 1, 8 9 a) State the rational number represented by the letter on the number line. A B C D Which rational number is greater?.7,.69 c) Which rational number is greater? 16, Copyright 00, Durham Continuing Education Page of 6
25 MPM1D Principles of Mathematics Unit 1  Lesson Solution a) A = ; B = 1.; C = 0.; D = 1.7 Since.7 means 70 parts per one hundred and.69 mean 69 parts per one hundred then.7 >.69 > means greater than and < means less than c).60 = and =.6,therefore,.6 >.60 or > Support Questions 1. Which rational number is greater? a), 1, 7 d) 1, 18 c),0.1 e) 0., 0. f).9, g), h) 0.1, 0.11 i), j), 8 9. List in order from least to greatest a) 0.6,,,1.1,, , 0.7,, 1.7,, , c),0.67,, 0.71, 1, Represent all of the following rational numbers on the same number line. a)..8 c).6 d).08 e).1 Copyright 00, Durham Continuing Education Page of 6
26 MPM1D Principles of Mathematics Unit 1  Lesson Adding and Subtracting Rational Numbers (Fractions) To add and subtract fractions a common denominator must be present. A lowest common denominator is best. Example Add or subtract as indicated. a) c) Solution a) 1 + = Only the numerators are added. + 1 = The denominator is the same, as required x = = x = = 9 or 1 is the lowest common denominator. In other words, is the lowest number that both 8 and 6 divide evenly into. c) 1 = 1 x 1 + = 6 = 1 1 = or 1 1 x + = 1 Support Questions. Add or subtract as indicated. a) c) d) e) (+.) (.9) f) (1.8) (6.7) 6 8 Copyright 00, Durham Continuing Education Page 6 of 6
27 MPM1D Principles of Mathematics Unit 1  Lesson g).9 + (.) h) (1.1) (1.1) i) 7 j) k) + l) Multiplying and Dividing Rational Numbers (Fractions) You should not use common denominators when multiplying or dividing fractions. Example Multiply or divide as indicated. a) c) 8 11 Solution a) 8 = Both 1 and 6 can be divided by so it is usually best to simplify before multiplying / / 1 17 = = = / 8 c) / = = = or / Copyright 00, Durham Continuing Education Page 7 of 6
28 MPM1D Principles of Mathematics Unit 1  Lesson Support Questions. Multiply or divide as indicated. Simplify all fraction answers. a) c) 1 6 d) 6 8 e) (+.) (.9) f) (1.8) (6.7) g).9 (.) h) (1.1) (1.1) i) j) 8 6 k) l) 1 10 Key Question # 1. Which rational number is greater? ( marks) a) 1, 9 d), 18 g) 6, 11 9 j), e) 1, c), , 00 f).7,.71 h) 0., 0. i), List in order from least to greatest. ( marks) 1 1 a) 0.1,,,1.1,,0,0.9, 1, 1.7, 7,0.9 1, c) 1,0.76,, 0.17,,0.1 7 Copyright 00, Durham Continuing Education Page 8 of 6
29 MPM1D Principles of Mathematics Unit 1  Lesson Key Question # (continued). Represent all of the following rational numbers on the same number line. ( marks) a).7.81 c) 100 d).06 e). Add, subtract, multiply or divide as indicated and simplify if needed. ( marks) a) + c) d) 9 e) (+.1) +(.1) f) (1.6) (.) g) (.7) h) (.) (.) i) j) k) + 7 l) 6 m) 8 n) o) 1 8 p) 7 8 q) (+7.1) (.9) r) (0.8) (6.) s) 8.9 (9.) t) (.) (.) 1 u) + 6 v) w) 9 x) Cliff fenced his yard using lengths of fencing that were. m,.7 m and. m respectively. The fencing cost $0./m. How much did the fence cost? ( marks) 6. Add 1 to the numerator of the fraction 1. How does the size of the fraction change? ( marks) Copyright 00, Durham Continuing Education Page 9 of 6
30 MPM1D Principles of Mathematics Unit 1  Lesson Key Question # (continued) 7. Add 1 to the denominator of the fraction 1. How does the size of the fraction change? ( marks) 8. How does the size of the fraction 1 change when both the numerator and the denominator are increased by 1? ( marks) Copyright 00, Durham Continuing Education Page 0 of 6
31 Exponents Lesson
32 MPM1D Principles of Mathematics Unit 1  Lesson Lesson Four Concepts Introduction to exponents Multiplying powers of the same base Dividing powers of the same base Powers of powers Zero exponents Negative exponents Converting into scientific notation Converting from scientific notation Multiplying numbers in scientific notation Exponents The raised value is the exponent The bottom value is called the base Together the base and exponent are called a power. in expanded form is. Example Evaluate each power. a) 10 Solution a) = x x = 1 10 = 10 x 10 x 10 x 10 x 10 = Copyright 00, Durham Continuing Education Page of 6
33 MPM1D Principles of Mathematics Unit 1  Lesson Support Questions 1. Evaluate. a) d) g) e) 8 c) 0.7 h) ( 6) Multiplying Powers f) i) (0.) 6 1 Exponent Law for Multiplying Powers: To multiply powers with the same base, keep n m n+ m the base and add the exponents. x x = x, where x 0, n and m are natural numbers. Example Write each product as a single power. a) c) Solution a) c) = = = 10 = = 6 = 6 1+ Any number without a visible exponent has an 1 exponent of one. i.e. 6 = 6 Copyright 00, Durham Continuing Education Page of 6
34 MPM1D Principles of Mathematics Unit 1  Lesson Dividing Powers Exponent Law for Dividing Powers: To divide powers with the same base, keep the n m n m base and subtract the exponents. x x = x, where x 0, n and m are natural numbers and n > m. Example Write each quotient as a single power. a) Solution a) 6 6 = 6 = = 10 = 10 1 Support Questions. Write each quotient or power as a single power. a) 9 9 c) d) g) e) 6 6 h) ( ) ( ) ( ) f) (0.) (0.) 7 ( 6) 6 ( ) Zero and Negative Exponents Zero exponent: 0 x is defined to be equal to 1; that is, 0 x =1 ( x 0). n Negative Integer Exponent: x n is defined to be the reciprocal of x ; that n 1 is x = x n (x 0 and n is an integer). Copyright 00, Durham Continuing Education Page of 6
35 MPM1D Principles of Mathematics Unit 1  Lesson Example  Evaluate each power. a) ( ) 0 c) d) Solution a) ( ) 0 =1 c) 0 0. =1 = 1 = 1 1 d) = = 8 Any positive value or any negative value in brackets with an exponent of zero is always equal to zero When a negative exponent occurs we rewrite the question with one as the numerator and the restated question without the negative exponent sign in the denominator This time the negative exponent is in the denominator so the denominator is rewritten as the numerator without the negative exponent Support Questions. Write each power as a positive then evaluate. 1 a) ( ) c) d) Evaluate. a) d) ( ) c). Write each expression as a single power then evaluate. a) c) 1 1 d) Copyright 00, Durham Continuing Education Page of 6
36 MPM1D Principles of Mathematics Unit 1  Lesson Support Questions (continued) 6. Use the law of exponents to evaluate each expression. 1 a) 6 ( ) ( ) ( ) c) 8 8 Power of Powers m mn Exponent Law for a Power of Power is ( ) x Example Write as a power with a single exponent. n x =, where m and n are Integers. a) ( 6 ) (( ) ) c) ( ) ( ) Solution a) ( ) 6 6 = 6 = 6 6 ( ) ( ) = ( ) = ( ) 6 c) ( ) ( ) 1 = = 6 0 Multiply the exponents together Remember to add the exponents here Support Questions 7. Write as a power with a single exponent. a) ( 7 ) 1 ( ) 0 c) ( ) 6 d) ( ) e) ( ) ( ) 6 9 g) ( 6 ) ( 6 ) h) ( 10 ) ( 10 ) 6 i) ( ) 0 f) ( 16) ( 16) Copyright 00, Durham Continuing Education Page 6 of 6
37 MPM1D Principles of Mathematics Unit 1  Lesson Scientific Notation Scientific Notation: a number expressed as the product of a number greater than 10 and less than 1 or greater than 1 and less than 10, and a power of 10. Scientific Notation is used to express either very large number or very small numbers. Example Write the following numbers in scientific notation. a) 60 Solution a) 600. When the value is equal to or greater than 10 or equal to or less than 10, we want the decimal between the first and second digit working from left to right. Decimal point is here and we want it there. Between the and becomes The exponent become by moving the decimal over positions between the and The decimal will be moved between the first and second nonzero digit working from left to right. The decimal will be move to this position. Becomes For all values between 1 and 1 a negative exponent will be used The exponent moved right decimal places therefore the exponent becomes If the decimal is moved left then the exponent will be positive and if the decimal is moved right the exponent will be negative. Copyright 00, Durham Continuing Education Page 7 of 6
38 MPM1D Principles of Mathematics Unit 1  Lesson Support Questions 8. Write in scientific notation. a) c) d) e) 1 00 f) g) h) i) j) k) l) Write as a numeral. a) d) c) Simplify to scientific notation. 7 a) (. 10 )( ) c) Key Question # 1. Evaluate. (9 marks) a) d) g) e) 1 7 c) 0. h) ( ) f) i) (0.). Write each quotient or power as a single power. (8 marks) a) 7 7 c) (.) (.) 6 9 ( ) d) e) 8 8 f) g) h) ( ) ( ) ( ) Write each power as a positive then evaluate. ( marks) a) ( ) 1 c) d) 10 ( ) Copyright 00, Durham Continuing Education Page 8 of 6
39 MPM1D Principles of Mathematics Unit 1  Lesson Key Question # (continued). Evaluate. ( marks) a) c) d) ( ). Write each expression as a single power then evaluate. ( marks) a) c) d) 6. Use the law of exponents to evaluate each expression. ( marks) 7 a) ( ) ( ) ( ) c) 7 7. Write as a power with a single exponent. (9 marks) a) ( ) 1 d) ( 6) ( ) 0 c) ( 10 ) e) ( ) ( ) 1 g) ( 7 ) ( 7 ) h) ( 9 ) ( 9 ) 9 i) ( 8 ) 0 f) ( 10) ( 10) 8. Write in scientific notation. (1 marks) a) c) d) e) 8 00 f) 000 g) h) 91 6 i) 8 10 j) k) l) Write as a numeral. ( marks) a) c) d) Simplify to scientific notation. ( marks) a) (.7 10 )( ) c) The mass of the Earth is approximately The mass of 7 the Sun is about. 10 times greater than the mass of the Earth. What is the mass of the Sun? ( marks) Copyright 00, Durham Continuing Education Page 9 of 6
40 MPM1D Principles of Mathematics Unit 1  Lesson Key Question # (continued) 7 1. The mass of the Sun is about.7 10 times as great as the mass of the Moon. What is the mass of the Moon? ( marks) 1. Two spacecraft named Veger I and Veger II were launched to visit the outer planets of our solar system. These spacecraft have now traveled so far they have left our solar system. Veger I is traveling at km/h and in years it will reach the star of Sirius. How far away is Sirius? ( marks) 1. Brenda wrote a number in scientific notation. She made a mistake. Instead of writing. 10 she wrote. 10. How many times as large as the correct number was Brenda s number? ( marks) 1. If two numbers in scientific notation are divided and their quotient is also in scientific notation, how is the power of 10 in the quotient related to the power of 10 in the two original numbers? ( marks) Copyright 00, Durham Continuing Education Page 0 of 6
41 Square Root Lesson
42 MPM1D Principles of Mathematics Unit 1  Lesson Lesson Five Concepts Introduction to square root Pythagorean Theorem Square Root Square root is a number, when multiplied by itself, results in a given number. Example Find the square root. a) Solution a) = ± ±1.  and +, when multiplied to itself, result in square roots questions always give answers Support Questions 1. What are the square roots of each number? a) 9 6 c) 1 d) 11 e) f) 8 g) 1 h) 16 i) 0. j) 1. k) 1 16 l) 9. Determine the values that satisfy the equations. a) n = x = c) w = d) t = 81. Determine the square roots of each number. a) c) 900 d) e) f) Copyright 00, Durham Continuing Education Page of 6
43 MPM1D Principles of Mathematics Unit 1  Lesson Pythagorean Theorem The Pythagorean Theorem is one method used to calculate an unknown side of a right angle triangle if the other two side are known. Pythagorean Theorem states a + b = c. Where the area of square a plus the area of square b is equal to the area of square c. a c The longest side of the triangle is the hypotenuse b Since the formula for the area of a square is A = s then to find the length of the a side of a square we square root the value of the area. Example Find the length of a side of a square that has an area of 100 Solution cm. Example The area of square was 100 then the side would be = ± 10 We only use the positive answer since you cannot have negative length. a) Find the length of the hypotenuse in the given triangle using Pythagorean Theorem. m h m Find the length of the missing side in the given triangle using Pythagorean Theorem. x 18 cm 1 cm Copyright 00, Durham Continuing Education Page of 6
44 MPM1D Principles of Mathematics Unit 1  Lesson Solution a) Using Pythagorean Theorem a + b = c a + b = c + = c = c = c The area of square created by the hypotenuse is so the length of its side is the square root of. Therefore, the length of the hypotenuse is m. Using Pythagorean Theorem a + b = c x a + b = c + 1 = 18 x x = 18 1 = 180 x 1. cm Therefore, the length of the missing side is 1. cm. Support Questions. Calculate the length of the third side of each triangle. Round to one decimal place. a) 8m 11. cm 1 m 1.8 cm Copyright 00, Durham Continuing Education Page of 6
45 MPM1D Principles of Mathematics Unit 1  Lesson. Calculate the diagonal of each rectangle. Round to one decimal place. 1. m a) 6 cm.7 m 8cm Key Question # 1. What are the square roots of each number? ( marks) a) 1 81 c) 100 d) 1 e) 0.6 f) 1.7 g) 1 h) 6 6. Simplify each expression. ( marks) a) c) + 6. Determine the values that satisfy the equations. ( marks) a) n = 81 x = c) w = d) t = Calculate the length of the third side of each triangle. Round to one decimal place. ( marks) a) 6m 17 m 10. cm 19. cm. Televisions are sold by the size of their screen s diagonal. What is the diagonal size of a television that has a monitor screen height of inches and a monitor screen width of 8 inches? ( marks) 6. Plot each pair of ordered pairs and calculate the distance between them. (6 marks) a) A(1, ), B(,7) C(,0), D(7,) c) E(,), F(1,) Copyright 00, Durham Continuing Education Page of 6
46 MPM1D Principles of Mathematics Unit 1  Lesson Key Question # (continued) 7. For a sloping ladder to be safe, the distance from the wall to the base of the ladder must be 1 of the vertical distance from the ground to the top of the ladder. A 16 m ladder is placed against a wall. How far up the wall will the ladder reach? ( marks) 8. Does Pythagorean Theorem work for nonright angle triangles? Prove your answer with examples. ( marks) Copyright 00, Durham Continuing Education Page 6 of 6
47 MPM1D Principles of Mathematics Unit 1 Support Question Answers Support Question Answers Lesson 1 1. a) :7 : c) 1:. a) : : c) :. a) : 6: c) : d) :6. a) : :1 c).1. a) 6:19 19:6 c) 6:1 6. a) L.71 = $.8 c) $ L L 1L = 8. x x = 8. x 8. = x = $0.71/ L 7. a) = 1x = 000 x = 67 at bats 0 x 8. a) = = 0% 7 8 = = 87.% c) 9 8 = = 11.% 9. a) 100 = = 1.1 c) = d). 100 = a) 100 = = = = = = c) = = = d) = = = 100 Copyright 00, Durham Continuing Education Page 7 of 6
48 MPM1D Principles of Mathematics Unit 1 Support Question Answers 11. a) c) 100 = = = = 1. a) c) 0.0 n = n = 0.0n = n = n = n = n = 00 n = = = 7 1. n = 0 1.n = n = n = = = $.17 GST = GST = 7.0 PST = PST = 1.00 Sales Tax = Sales Tax = $19.0 Total Price = $ n = n = n = n = 181 Lesson 1. a) 1 10 c) 0 d) 6 e) 10 f) g) 6 h) 8 i) Copyright 00, Durham Continuing Education Page 8 of 6
49 MPM1D Principles of Mathematics Unit 1 Support Question Answers. a) 1 c) 17 d) e) 1 f) 6 g) 0 h) 10 i) 6. a) + = = c) +1 8 = 7 d) + = 7 e) =  a) = = c) + 8 = 6 d) = 11 e) = 68. a) 10 1 c) 7 d) 0 e) 16 f) g) 7 h) 7 i) 0 6. a) c) 6 d) e) f) g) h) 6 i) 1 j) 8 k) l) 7 7. a) 6 = = 0 c) 00 = 80 d) 8 = Lesson 1. a) 0. =, = > 0.7 therefore > 1 0. =, > 0. therefore 1 > 7 Copyright 00, Durham Continuing Education Page 9 of 6
50 MPM1D Principles of Mathematics Unit 1 Support Question Answers c) = > 0.0 therefore 0.1> 0.0 d) 1 0. =, > 0.7 therefore 1 > 18 e) f) g) h) i) j) 0. = > 0..9 = >.90 therefore.99 >.9 7 = 0.8, > 0.8 therefore 0.1= > =, > 0.66 therefore 66 > 90 = 0.7, > > 0. therefore > 8 9 Copyright 00, Durham Continuing Education Page 0 of 6
51 MPM1D Principles of Mathematics Unit 1 Support Question Answers. a) c) 6 1 1,, 0,.1,, 0.6, , 0.7,,,, ,, 0.1,, 0.67, a) = = = c) d) = x or 1 x + = + = + = = x x = = = x x e) 7.1 f).9 g) 0.6 h. 0 i) j) k) l) x6 x or 1 x6 x = + = + = + = = x x or 1 x x = + = + = + = x x 10 1 x x = + = + = + = x x or 1 x x = = = = Copyright 00, Durham Continuing Education Page 1 of 6
52 MPM1D Principles of Mathematics Unit 1 Support Question Answers. a) c) d) = = = = = or / = = or 6/ = = = or / 9 9 / e) 0.8 f) 1.06 g) 0.87 h) 1.1 g) h) 1 + or 6 = = / or = = = 8 8/ 0 0 i) 1 = 6 6 = = 10 j) = = or i) j) / 6 6 = = or 1 / / / = = = or / / Copyright 00, Durham Continuing Education Page of 6
53 MPM1D Principles of Mathematics Unit 1 Support Question Answers Lesson 1. a) 6 6 c) 0.1 d) 9 6 e) f) g) h) 6 i) 8. a) 1 + = = 9 9 = 9 = 9 7. (.) = (.) = (.) = (.) d) c) ( ) + + e) g) = = = f) ( ) ( 6) = ( 6) = ( 6) = 0. = 0. h) ( ) ( ) ( ) = ( ) = ( ). a) 1 1 = ( ) = ( ) = = 16 ( ) 1 9 c) d) 1 = = = 10. a) 1 1 = = 1 1 ( ) = = ( ) c) = d) 1 1 = =. a) c) d) = = = = = = = = = = = = = = = 1 Copyright 00, Durham Continuing Education Page of 6
54 MPM1D Principles of Mathematics Unit 1 Support Question Answers 6. a) c) = = = ( ) = = = = ( ) ( ) ( ) ( ) ( ) = = 8 = 8 = 8 7. a) ( ) 1 7 = (7) = (7) ( )( 1) ()() 1 ( ) = () = c) ( ) 6 d) 0 (0)(6) 0 = = = 1 ()( ) 6 ( ) = ( ) = ( ) + ( ) ( ) = () = () () = () = () e) ( ) ()() ()() ()() ()() f) ( 16) ( 16) = ( 16) ( 16) = ( 16) ( 16) = ( 16) = ( 16) ()() ()() g) (6 ) (6 ) = (6) (6) = (6) (6) = (6) = (6) 6 9 ()(6) ()(9) h) (10 ) (10 ) = (10) (10) = (10) (10) = (10) = (10) = (6)(0) 0 i) ( ) = () = () = 1 8. a) 100 = = c) = d) = e) 100 = f) 1000 = g) = h) =. 10 i) = j) =.9 10 k) = l) = a) =.6 10 = 0.06 c) = 7800 d) = a) 7 (. 10 )(7. 10 ) = = = = ( 7) 7 Copyright 00, Durham Continuing Education Page of 6
55 MPM1D Principles of Mathematics Unit 1 Support Question Answers = ( ) = = c) = = = = ( ) + ( ) Lesson 1. a) 9 =± 6 = ± 8 c) 1=± 1 d) 11 =± 11 e) ±.8 f) 8 ± 7.6 g) 1 ± 8.1 h) 16 = undefined i). =± 0. j) 1. ± 1.1 k) = =± l) = = ± 9 9. a) c) n n = = n =± x = ± 11 x x = 11 = 11 w w w w = = = 81 =± 9 Copyright 00, Durham Continuing Education Page of 6
56 MPM1D Principles of Mathematics Unit 1 Support Question Answers d) t t t t = = = 1 =± a) 0.09 =± 0. 9 = ± c) 900 =± 0 d) =± 00 e) =± 000 f) =± a) a + b = c (8) + (1) = c = c = c = c 1. c. a) a + b = c (6) + (8) = c = c 100 = c 100 = c 10 = c a + b = c (11.) + b = (1.8) b = b = a + b = c (1.) + (.7) = c = c 1.1 = c 1.1 = c.9 c b b = 1. = 1. b 11.1 Copyright 00, Durham Continuing Education Page 6 of 6
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