Polynomials. Lesson 6

Transcription

1 Polynomials Lesson 6

2 MFMP Foundations of Mathematics Unit Lesson 6 Lesson Six Concepts Overall Expectations Simplify numerical and polynomial expressions in one variable, and solve simple first-degree equations. Specific Expectations Add and subtract polynomials involving the same variable up to degree three; Multiply a polynomial by a monomial involving the same variable to give results up to degree three; Solve first-degree equations with non-fractional coefficients, using a variety of tools and strategies; Substitute into algebraic equations and solve for one variable in the first degree. Polynomials Polynomials are a mathematical expression with one or more terms, in which the exponents are whole numbers and the coefficients are real numbers. Monomial is a polynomial with one term; e.g. -8 and x. Binomial is a polynomial with two terms; e.g. -4x 7 and y. Trinomial is a polynomial with three terms; e.g. 4x x 9 and w 7w. Term -4x -4x is an example of a term. x is called the variable or unknown -4 is called the coefficient Adding and Subtracting Polynomials To add and/or subtract polynomials the terms must be like terms. Like terms are terms that have the same variables with the same value of exponent for each variable. Example Underline the terms that are like terms. Copyright 005, Durham Continuing Education Page of 5

3 MFMP Foundations of Mathematics Unit Lesson 6 a) x, x, 7x, y, x b) 7 xy, 5x y, y x, 5 xy Solution Example Simplify. a) ( x ) 5 (x 7) b) ( a 4) (a ) c) ( 4w 6w 8) ( 9w w 5) Copyright 005, Durham Continuing Education Page of 5

4 MFMP Foundations of Mathematics Unit Lesson 6 Solutions Support Questions. State the like terms in each group. a) x, 5y, 5z, x, x, w, v b) 4x, y, 4z, y, y, 4w. Simplify. a) x 4x b) 5n 6n ) c) (8a a ) ( 5a 4a 7) d) ( 6x 5x ) (4x e) ( m n ) (7 6m n ) f) ( 6x ) (7 x ) g) (5 6w ) ( w ) h) (5x x) ( 4x 5x 5 x). Simplify. Then determine the value of the polynomial when n and when n. a) ( n 4) ( n ) b) (n 7n ) ( n 6n ) ) Copyright 005, Durham Continuing Education Page 4 of 5

5 MFMP Foundations of Mathematics Unit Lesson 6 Multiplying and Dividing Polynomials To multiply and/or divide polynomials the terms do not have to be like terms. Example Simplify. a) (4x )( x) b) c) ( w y ) (5wy) d) 6x y 8xy 00 x yz 5xyz Solutions a) (4x )( x) 4 x x x x or ( 4)( )( x )( x ) x x Use the multiplication of exponents rule b) 6x y 8xy 6 x x x y y y 8 x y Divide common factors x x y y x y or 6 x 8 x y y 0w w w y y c) ( w y ) (5wy) 0w y Copyright 005, Durham Continuing Education Page 5 of 5

6 MFMP Foundations of Mathematics Unit Lesson 6 d) 00 x yz 5xyz 4x 4xz y z Any base to the zero exponent equals Support Questions 4. Determine the product or quotient. a) (m )( 7m) b) (n ) ( 6n) c) e) 7ab ba 8a 7a d) ( x)(4xy)( y) f) (8z )(7) g) ( 4a b) (ab) h) (5x)(4x) i) 0 ab a Simplify. Then determine the value of the polynomial when a and when b. a) ( ab) b) ( ab)( 5ab ) Multiplying Polynomials with a Monomial This process requires the use of the distributive law. Example Expand. a) (x 4) b) x( x 5) c) ( p )( 5p) Solutions a) (x 4) 6x 8 ()(x) 6x and ()(-4) -8 Copyright 005, Durham Continuing Education Page 6 of 5

7 MFMP Foundations of Mathematics Unit Lesson 6 b) x( x 5) x( x) x( 5) 6x 0x c) -p ( p )( 5p) () - p (-5p) 9 p 5p Support Questions 6. Expand. a) x(x 9) b) ( 4n)(n ) c) b(b b ) d) ( x)(x ) e) ( 4m)(m m) Key Question #6. State the like terms in each group. ( marks) a) 4w,5w,5z,x, x,w,v b) 4x, x,4z,y, y,4w. Simplify. (5 marks) a) t 7t 5 b) 6 4r 5r c) 4n 4n 7n n 6 d) 4x x x 7 x e) 5m n 5 m 4n. Simplify. ( marks) a) ( 7x ) (4 6x ) b) c) (6x 7x) ( x 9x ) ( 7w ) ( 4 8w ) 4. Simplify. Then determine the value of the polynomial when n - and when n. (4 marks) a) (5n n 4) ( n 4n ) b) (7n 5n ) ( n 6n 8) 5. Expand. (5 marks) a) w(w 4) b) 4n(5n 9) c) c(7c 5c 6) d) ( h)(h 6) e) ( 6x)( x x) Copyright 005, Durham Continuing Education Page 7 of 5

8 MFMP Foundations of Mathematics Unit Lesson 6 Key Question #6 (continued) 6. Determine the product or quotient. (9 marks) a) (x )( 4x) b) (6m ) ( 9m) c) a b 5b a d) (8x )(y)( y) 75w e) 5w f) (d )(9d) g) ( a b ) (ab ) h) ( 7k)( k) i) 4 a b ab Simplify. Then determine the value of the polynomial when a - and when b. (4 marks) a) (ab) b) (4ab) 8. When the terms of a polynomial in x are arranged from the highest to the lowest powers of x, the polynomial is in descending order. Simplify the following polynomial in descending order then evaluate for n. ( marks) 6 ( n n) ( 5n n 6) ( 4n n ) 9. When are the sum, difference, product and quotient of two monomials also a monomial? ( marks) Copyright 005, Durham Continuing Education Page 8 of 5

9 Algebra Lesson 7

10 MFMP Foundations of Mathematics Unit Lesson 7 Lesson Seven Concepts Overall Expectations Simplify numerical and polynomial expressions in one variable, and solve simple first-degree equations. Specific Expectations Add and subtract polynomials involving the same variable up to degree three; Multiply a polynomial by a monomial involving the same variable to give results up to degree three; Solve first-degree equations with non-fractional coefficients, using a variety of tools and strategies; Substitute into algebraic equations and solve for one variable in the first degree. Algebra Solving Equations When solving algebraic equations we must try to think of a scale always in equilibrium (balanced). It is important to keep the scale balanced at all times. What you do to one side of the equation must also be done to the other side of the equation. You need to get all the terms with the variable to one side and the constants (the ones with out any letters) to the other. It does not matter which side you choose for the isolating of each. Copyright 005, Durham Continuing Education Page 0 of 5

11 MFMP Foundations of Mathematics Unit Lesson 7 Example Solve each equation algebraically. Check your solution a) 4 7k k b) x 7 8 Solution a) -4-7k k This side chosen for the k s -4-7k k -4-7k - k k - k -4-8k - k from both sides to keep scale balanced 4 from both sides to keep scale balanced k 4 Checking the solution - 8k 6-8k k - Divide 8 from both sides to keep scale balanced Should now check the answer - 4 7k k - 4 7(-) (-) Substitute your answer into the original equation same If both sides equal the same amount then your answer is correct. K - Copyright 005, Durham Continuing Education Page of 5

12 MFMP Foundations of Mathematics Unit Lesson 7 b) x 7 8 x x 45 x 45 x 5 Checking the solution x 7 8 (5) both sides equal the same amount therefore the answer is correct. Support Questions. Solve each equation. Check your solution a) 5 w 0 b) x 5 6 c) 0 t 7 d) 4 w w e) r 5 4r 7 f) 5 6x x 5 g) 7c c h) (w ) (5 w) i) 4 ( j) 7(j 0) j) 7 ( x). The formula for the area of a parallelogram is A bh. The area of a parallelogram is 4 cm and its base is cm. Using algebra, what is the height of the parallelogram?. The cost of a hall rental for a wedding is $500. Each meal at the wedding will cost and additional $5.00. a) Write and algebraic equation for the total cost of the wedding. b) Calculate how many people attended the wedding if the wedding cost totalled $ Copyright 005, Durham Continuing Education Page of 5

13 MFMP Foundations of Mathematics Unit Lesson 7 Key Question #7. Solve each equation. Check your solution. ( marks) a) w 6 b) n 7 c) 5 w 45 d) 5c 7c e) 6 6h h 5 f). x 4.x.5 g). x.(.4 x) 4 h) ( x ) i) 5 ( x ) 5 j) ( t 6) 0 k) (p ) (p ). The formula for the perimeter of a rectangle with length l and width w is Plw. A rectangular field is 0 m long and requires 45 m of fencing to enclose it. Determine the width of the field. ( marks). Volcanoes prove that the Earth s center is hot. The formula T 0d 0 is used to estimate the temperature, T degrees Celsius, at a depth of d kilometres (km). ( marks) a) What does each term on the right side of the equation represent? b) Estimate the depth where the temperature is 60 C. c) What is the approximate temperature at a depth of 6 km? 4. The cost, C dollars, to produce a school yearbook is given by the equation C n. Where n is the number of yearbooks printed. (4 marks) a) What does each term on the right side of the equation represent? b) Suppose there is $ 500 to spend on yearbooks. How many yearbooks can be purchased? c) How many yearbooks can be produced for $0 000? d) How much would 700 yearbooks cost? 5. Suppose you were asked to explain how to solve the equation ( x ) over the phone to a friend. Explain in detail the steps that you would tell your friend to solve the equation. (4 marks) Copyright 005, Durham Continuing Education Page of 5

14 Problem Solving Algebraically Lesson 8

15 MFMP Foundations of Mathematics Unit Lesson 8 Lesson Eight Concepts Overall Expectations Simplify numerical and polynomial expressions in one variable, and solve simple first-degree equations. Specific Expectations Add and subtract polynomials involving the same variable up to degree three; Multiply a polynomial by a monomial involving the same variable to give results up to degree three; Solve first-degree equations with non-fractional coefficients, using a variety of tools and strategies; Substitute into algebraic equations and solve for one variable in the first degree. Solving Problems using Algebraic Modeling Problem Solving Steps Example. Use a variable to represent the unknown quantity.. Express any other unknown quantities in terms of this variable.. Write an equation and solve. 4. Check your answer to the question. 5. State the answer to the question with a therefore statement. Members of a Girl Guide troop sold boxes of cookies to raise money for their year end camp. Brianna sold 8 more boxes than her friend Nicola. They sold a total of 46 boxes. How many boxes did each sell? Solution. Use a variable to represent the unknown quantity. Let b represent the number of boxes sold by Brianna.. Express any other unknown quantities in terms of this variable. If boxes sold by Brianna b; then boxes sold by Nicola b 8.. Write an equation and solve. Copyright 005, Durham Continuing Education Page 5 of 5

16 MFMP Foundations of Mathematics Unit Lesson 8 b b 8 46 b 8 46 b b 54 b 54 b 7 4. Check your answer to the question. LHS RHS b b State the answer to the question with a therefore statement. Brianna sold 7 boxes of cookies and Nicola sold 9. Example An airplane travels 7 times faster than a train. The difference in their speeds is 40 km/hr. How fast is each vehicle traveling? Solution. Use a variable to represent the unknown quantity. Let v represent the speed the train travels.. Express any other unknown quantities in terms of this variable. If the train travels at v; then the airplane travels 7v. Copyright 005, Durham Continuing Education Page 6 of 5

17 MFMP Foundations of Mathematics Unit Lesson 8. Write an equation and solve. 7v v 40 6v 40 6v v Check your answer to the question. LHS RHS 7v - v 40 7(70) State the answer to the question with a therefore statement. the train travels at 70 km/hr and the airplane travels at 490 km/hr. Support Questions. Find two consecutive numbers with a sum of 4.. A set of golf clubs and bag cost $5. The clubs cost $60 more than the bag. How much do the clubs cost?. Brianna and Noah ran as far as they could in 60 min. Brianna ran.5 km less than Noah. Together they ran 7.5 km in total. How far did each run. 4. Don and his brother Dan went fishing. Don caught 6 times more mass of fish than his brother. Together they caught 4. kg of fish. What mass of fish did each brother catch? Copyright 005, Durham Continuing Education Page 7 of 5

18 MFMP Foundations of Mathematics Unit Lesson 8 Key Question #8. Convert the following phrases into algebraic equations. (5 marks) a) Three times a number is twenty four. b) Twice a number increased by eight is forty four. c) Five less than four times a number is eleven. d) One more than triple a number is five less than double a number. e) A number decreased by seven is six times a number.. Solve for each of the statements in question one. (5 marks). The combined mass of two children is 75 lbs. The first child is four times the mass of the second child. What are the masses of the two children? ( marks) 4. A rectangle has a width that is cm less than its length. The perimeter of the rectangle is cm. What is the length and width of the rectangle? ( marks) 5. The same number of each type of coin has a total of $6.00. There are nickels, dimes and quarters. How many of each type of coin are there? (4 marks) 6. Are there more ways to solve problems such as the ones given in questions 5? Explain. ( marks) Copyright 005, Durham Continuing Education Page 8 of 5

19 Slope Lesson 9

20 MFMP Foundations of Mathematics Unit Lesson 9 Lesson Nine Concepts Overall Expectations Apply data-management techniques to investigate relationships between two variables; Determine the characteristics of linear relations; Demonstrate an understanding of the constant rate of change and its connection to linear relations; Connect various representations of a linear relation, and solve problems using the representations. Specific Expectations Interpret the meanings of points on scatter plots or graphs that represent linear relations, including scatter plots or graphs in more than one quadrant; Determine through investigation, that the rate of change of a linear relation can be found by choosing any two points on the line that represents the relation, finding vertical change between the points and the horizontal change between the points; Coordinate Plane Cartesian Plane uses the x and y axes to plot a point identified by a pair of numbers. If it is known that a point has an x-coordinate of, then this point could be located anywhere along the vertical line passing through on the x-axis. If it is known that a point has a y-coordinate of 5, then the point could be located anywhere along the horizontal line passing through 5 on the y-axis. The point of intersection of the lines formed by x - and y 5 is the location of a point given by the ordered pair (-, 5). x-coordinate y-coordinate Copyright 005, Durham Continuing Education Page 0 of 5

21 MFMP Foundations of Mathematics Unit Lesson 9 This grid system is the Cartesian plane Vertical axis is the y-axes The Cartesian plane is divided into four quadrants A Horizontal axis is the x-axes Example Plot and label each point: A(, 5), B(0, -6), C(-7, -4) Solution Copyright 005, Durham Continuing Education Page of 5

22 MFMP Foundations of Mathematics Unit Lesson 9 Support Questions. Properly draw and label a Cartesian plane then plot and label the following ordered pairs. A(, ), B(, 5), C(, -4), D(7, 0), E(0, 8), F(-4, -8), G(-5, 9), H(-, 0). State the quadrant for each of the ordered pairs plotted in question one.. The points (-, 5) and (, 5) are two vertices of a square. State all other order pairs that could be the other two vertices of the square. Slope Slope describes the steepness of a line or line segment; the ratio of the rise of a line or line segment to its run. Slope is either one of the following four types: a) positive b) negative c) no slope d) undefined rise The formula for determining slope is: slope. run Example a) Draw a ramp that would lie on the staircase b) State the slope of the staircase Copyright 005, Durham Continuing Education Page of 5

23 MFMP Foundations of Mathematics Unit Lesson 9 Solution rise slope run Example slope c) State the slope of the hypotenuse of the triangle given below. 4 Solution 4 rise slope run 4 slope slope Reduced to simplest form. Copyright 005, Durham Continuing Education Page of 5

24 MFMP Foundations of Mathematics Unit Lesson 9 Support Questions 4. Find the slope of each staircase. a) b) 5. This drawing represents the side view of road passing through a mountain range. The road moves from left to right. Calculate the slope of each line segment. 6. The slope of a line segment is 6. What is a possible rise and run? 7. The slope of a line segment is 4. What is the rise if the run is? 8. State the slope of each line segment. Copyright 005, Durham Continuing Education Page 4 of 5

25 MFMP Foundations of Mathematics Unit Lesson 9 Slope between two points on a Cartesian Plane The slope of line segment can be determined if two ordered pairs of the line segment are known. The following formula is used: Example slope y x y x z a) Find the slope of the in line containing the coordinate pairs A(7,) and B(-9,7). b) Find the slope of the in line containing the coordinate pairs A(-,) and B(4,6). Solutions ( x,y ) ( x,y ) The s in the first ordered pair denotes the x and y values of that term and the s in the second ordered pair denotes the x and y values of that term. a) (7, ) and (-9, 7) slope y x y x (7) () ( 9) (7) Substitute the values into the equation for slope. 4 6 Fraction needs to be simplified. 4 Therefore the slope is b) (-,) and (4,6). 4 slope y x y (6) () x (4) ( ) Therefore the slope is 7 5. Copyright 005, Durham Continuing Education Page 5 of 5

26 MFMP Foundations of Mathematics Unit Lesson 9 Support Questions 9. Find the slope of the in line containing the ordered pairs. a) A(4, -5), B(-4, 6) b) C(, 8), D(7, ) c) E(0, ), F(, ) d) G(5, 4), H(, -) e) I(, ), J(-6, -) f) L(, 7), M(, -) g) N(4, -6),P(, -6) Key Question #9. Properly draw and label a Cartesian plane then plot and label the following ordered pairs. (5 marks) A(, ), B(, -5), C(, 4), D(-7, 0), E(0, -8), F(, -8), G(-4, -9), H(, 0). State the quadrant for each of the ordered pairs plotted in question one. ( marks). The points (-4, ) and (4, ) are two vertices of a square. State all other order pairs that could be the other two vertices of the square. ( marks) 4. This drawing represents the side view of road passing through a mountain range. The road moves from left to right. Calculate the slope of each line segment. ( marks) 5. The slope of a line segment is. What is a possible rise and run? ( marks) 6. The slope of a line segment is -5. What is the rise if the run is 4? ( marks) Copyright 005, Durham Continuing Education Page 6 of 5

27 MFMP Foundations of Mathematics Unit Lesson 9 Key Question #9 (continued) 7. State the slope of each line segment. ( marks) 8. Find the slope of the line containing the ordered pairs. (7 marks) a) A(, -5), B(-, 6) b) C(, 6), D(5, ) c) E(, ), F(, ) d) G(-, 4), H(, -) e) I(, ), J(-6, ) f) L(4, 5), M(4, -7) g) N(, -5),P(, -8) 9. Decide if the following statements are true, sometimes true or not true. Explain. ( marks) a) An ordered pair with positive coordinate and negative coordinate lies in the rd quadrant. b) An ordered pair with both coordinates positive coordinates lies in the st quadrant. c) An ordered pair where the x and y coordinate are the same lies in the st or rd quadrant. Copyright 005, Durham Continuing Education Page 7 of 5

28 Relationships In Data Lesson 0

29 MFMP Foundations of Mathematics Unit Lesson 0 Lesson Ten Concepts Overall Expectations Apply data-management techniques to investigate relationships between two variables; Determine the characteristics of linear relations; Demonstrate an understanding of the constant rate of change and its connection to linear relations; Connect various representations of a linear relation, and solve problems using the representations. Specific Expectations Interpret the meanings of points on scatter plots or graphs that represent linear relations, including scatter plots or graphs in more than one quadrant; Pose problems, identify variables, and formulate hypotheses associated with relationships between two variables; Describe trends and relationships observed in data, make inferences from data, compare the inferences with hypotheses about the data, and explain any differences between the inferences and the hypotheses; Construct tables of values and graphs, using a variety of tools; Construct tables of values, scatter plots, and lines or curves of best fit as appropriate using a variety of tools; Determine values of a linear relation by using a table of values, by using the equation of the relation, and by interpolating or extrapolating from the graph of the relation. Relationships in Data Tables and graphs of data help to show the relationships between quantities. In mathematics the relationship between a pair of quantities is called a relation. Example Use the graph following to answer each question. a) State the percentage of Canadians who enjoy professional wrestling for each of the following years. Show the popularity in a table b) When did the popularity of professional wrestling reach 6% of all Canadians? Copyright 005, Durham Continuing Education Page 9 of 5

30 MFMP Foundations of Mathematics Unit Lesson 0 Solution a) State the percentage of Canadians who enjoy professional wrestling for each of the following years? Show the popularity in a table Year Popularity (%) b) When did the popularity of professional wrestling reach 6% of Canadians? Approximately 985 Copyright 005, Durham Continuing Education Page 0 of 5

31 MFMP Foundations of Mathematics Unit Lesson 0 Example Which graph below best represents each situation? a) the height of a person over time b) the height of roller coaster over time c) the amount of hours of sunlight over a year d) the number of D.V.D. players sold compared to selling price Solution i) ii) iii) iv) Which graph below best represents each situation? a) the height of a person over time b) the height of roller coaster over time c) the amount of hours of sunlight over a year d) the number of D.V.D. players sold compared to selling price i) ii) iii) iv) a d c b Support Questions. Use the graph following to answer each question. a) What year was Canada s population in the following years? b) When did Canada s population reach 5 million? Copyright 005, Durham Continuing Education Page of 5

32 MFMP Foundations of Mathematics Unit Lesson 0 Canadians Population Since Population (millions) Year. Brianna walks to her grandparents. This graph shows her distance from home during one of her walks. Describe her walk.. Noah is riding his bike from his grandparents to his home. Describe Noah s possible ride home Copyright 005, Durham Continuing Education Page of 5

33 MFMP Foundations of Mathematics Unit Lesson 0 Support Questions 4. Refer to the graph given below: a) How many textbooks make the following heights? 45 cm, 60cm, 75cm b) Approximately how high is the following number of textbooks? 7 books, 9 books, 4 books, books 5. a) Construct a graph using this data. Number of stairs climbed Height (cm) b) Did you join the points? Explain. c) What is the height of 5 stairs? d) How many stair will reach a height of 444 cm.? e) If the number of stairs is doubled will the height double? Explain. Copyright 005, Durham Continuing Education Page of 5

34 MFMP Foundations of Mathematics Unit Lesson 0 Support Questions 6. a) Construct a graph using this data. Radius of a circle, (cm) Area of a circle. ( cm ) b) Did you join the points? Explain. c) What is the approximate radius of circle with an area of 00cm? d) What is the approximate area of circle with a radius of 4.5 cm? e) If the radius is doubled will the area double? Explain. Graphing Relations Relations can be either linear or non-linear. Linear means the relation forms a single straight line and non-linear produces anything that is not a single straight line. y y The following formula is used: slope x x z Example Draw a graph of the relation described by the equation. a) y x Copyright 005, Durham Continuing Education Page 4 of 5

35 MFMP Foundations of Mathematics Unit Lesson 0 Solution a) y x First step is to make a table of values and choose values to place in x column of table x y Second step is to substitute each value into the equation to determine the y value. For x - y x (-) 6 7 x y For x - y x (-) 4 x y For x 0 Copyright 005, Durham Continuing Education Page 5 of 5

36 MFMP Foundations of Mathematics Unit Lesson 0 y x (0) 0 x y For x y x () - - x For x y y x () -6-5 x y Next, plot the coordinates (x, y) on a grid and join the points with a straight edge and label the equation. Copyright 005, Durham Continuing Education Page 6 of 5

37 MFMP Foundations of Mathematics Unit Lesson 0 Support Questions 7. Complete the table of values. a) y x 5 b) y x 6 c) y x x y x y x y The cost of D dollars, to print and bind y copies of a yearbook is given by the equation D 60 0n. a) Make a table of values to show the costs for up to 400 Yearbooks. b) Use the table to draw a graph. c) Use the graph to estimate the cost of 5 copies. d) Use the graph to estimate how many copies can be made for $50. Copyright 005, Durham Continuing Education Page 7 of 5

38 MFMP Foundations of Mathematics Unit Lesson 0 Support Questions 9. Complete a table of values and graph each relation. a) y x 4 b) y x c) C 5.5w Which of the following ordered pairs satisfy the relation modelled by y x 7. Show by substituting the values into the relation. a) (-, 6) b) (4, 5) c) (7,.5) d) (-4, 9) e) (-4, 6.5) Key Question #0. Which graph below best represents each situation? (4 marks) a) the height of a person over time b) the height of roller coaster over time c) the amount of hours of sunlight over a year d) the number of D.V.D. players sold compared to selling price i) ii) iii) iv). Complete a table of values and graph each relation. (8 marks) a) C n 0 b) y x 4 c) W 4.75n. 50 d) P l 5. The time that passes between the time you see lightning and you hear the thunder depends on your distance from the lightning. With each km from the lightning seconds pass. (4 marks) a) Make a table of values for distances from 0 to 5 km. b) Graph this relation. State whether it is a linear or non-linear relation. c) Using your graphed relation, how much time passes before you hear lightning that occurs 4.5 km away? d) Using your graphed relation, how far away are you if you hear the thunder in.5 seconds? Copyright 005, Durham Continuing Education Page 8 of 5

39 MFMP Foundations of Mathematics Unit Lesson 0 Key Question #0 (continued) 4. The amount a taxi driver charges a customer is given by the equation A.5k 5.5, where A is the total amount charged and k is the kilometres driven. (5 marks) a) What do the numbers in the equation represent? b) Make a table of values for distances from 0 to 0 km. c) Graph this relation. State whether it is a linear or non-linear relation. d) Using your graphed relation, how much is charged if a person goes 7.5 km? e) Using your graphed relation, how far can a person go in a taxi for $5? 5. Ashlee repairs DVD players. She charges $5 to inspect the problem and $0/h to repair the device. (5 marks) a) Write an equation to model this relation. b) Make a table of values. c) Graph this relation. d) Using your graphed relation, how long did it take Ashlee to repair the DVD player if she charged $55? e) Using your graphed relation, how much should Ashlee charge if it takes her 4.5 hours to repair the DVD player? Copyright 005, Durham Continuing Education Page 9 of 5

40 MFMP Foundations of Mathematics Support Question Answers Support Question Answers Lesson 6. a. b. x x, y, y. a. b. 5 x 4x x 4x x ) (4x ) x ( n 6n 5n 6n 5n ) 6n ( 5n) ( c. d. 4 6a a 7 4a a 5a 8a 7 4a 5a a 8a 7) 4a 5a ( ) a (8a 7x 0x 5 x 5x 4x 6x x 5 4x 5x 6x x) 5 (4x ) 5x 6x ( e. f. 8m 0 n 6m 7 n m ) n 6m (7 ) n m ( 9x 5 x 6x 7 x 7 6x ) x (7 ) 6x ( g. h. 5w w 6w 5 w 6w 5 ) w ( ) 6w (5 7x 0x 4x x 5x 5x 5x 4x x 5x ) 5x 4x ( x) (5x Copyright 005, Durham Continuing Education Page 40 of 5

41 MFMP Foundations of Mathematics Support Question Answers. a. b. ( n 4) ( n ) n 4 n n n 4 5n for n 5n 5() 0 9 for n 5n 5( ) 5 6 (n n n n for n n for n n n n 5 n 5 () n 5 ( ) 5 7 7n ) ( n 7n n 6n 7n 6n () 5 6n ) ( ) 5 4. a. b. c. (m )( 7m) (n ) ( 6n) m n d. e. f. ( x)(4xy)( y) 4x y 8a 7a 4a g. h. i. ( 4a b) (ab) a (5x)(4x) 0x 7ab ba 7 b (8z 56z )(7) 0 ab a a b 45 a b Copyright 005, Durham Continuing Education Page 4 of 5

42 MFMP Foundations of Mathematics Support Question Answers 5. a. b. (ab) (ab)(ab) 4a for a, b 4a b b 4() ( ) 4(4)() 6 6. a. x(x 9) x 9x b. c. b(b b ) b b b d. ( ab)( 5ab 5a for a, b 5a b b (60)( ) 60 ) 5() ( ) ( 4n)(n ) 8n ( x)(x ) x x n e. ( 4m)(m m) 4m 4m Lesson 7 a. b. c. x 5 6 5w 0 x w 0 x 5 5 x w 6 x 7 0 t t 7 0 t 7 t 7 t 7 d. e. 4w w r 5 4r 7 4w w r 5 5 4r 7 5 4w w 5 r 4r 4w w w w 5 r 4r 4r 4r w 5 8r w 5 8r r 4 w Copyright 005, Durham Continuing Education Page 4 of 5

43 MFMP Foundations of Mathematics Support Question Answers f. g. 5 6x x 5 56x 6x x 6x 5 5 4x x x 0 x h. i. ( w ) (5 w) w 6 5w w w w w w w w w 5w 5w 5 5 w 5 7c c 7c c 7c c 6 7c c c c 6 5c 6 5c c 5 4( j) 7( j 0) 4 8 j 4 j j 8j 4j 8j 70 4 j j j 66 j j j. 7 ( x) 7 6 x x x x x Copyright 005, Durham Continuing Education Page 4 of 5

44 MFMP Foundations of Mathematics Support Question Answers. A bh 4 () h Therefore the height of the parallelogram is cm 4 h 4 h 4 h h. a. C 5.00m500 b m m m m m Therefore 60 people would attend. Lesson 8. Let n be the first consecutive number. If the first consecutive number is n then the next number is n n n 4 n 4 n 4 n 4 n 4 n 7 the consecutive numbers are 7 and 7 Copyright 005, Durham Continuing Education Page 44 of 5

45 MFMP Foundations of Mathematics Support Question Answers. Let b be the cost of the bag If b is the cost of the bag then the clubs are b60 b b 60 5 b 60 5 b b 65 b 65 b 8.5 the bag cost $8.50 and the golf clubs cost $ Let b be the distance ran by Brianna If b is the distance ran by Brianna then Noah ran b.5 b b b b b 5 b 5 b 7.5 Brianna ran 7.5 km and Noah ran 0 km. 4. Let d be the mass of the fish Dan caught. If d is the mass of the fish Dan caught then Don s fish s mass is 6d. 6d d 4. 7d 4. 7d d.6 Don caught.6 kg of fish and Dan caught.6 kg of fish. Copyright 005, Durham Continuing Education Page 45 of 5

46 MFMP Foundations of Mathematics Support Question Answers Lesson 9. A. I B. I C. IV D. I and IV E. I and II F. III G. II H. II and III. rise 4. a. rise run m run b. rise - run m rise run 5. rise AB slope m run rise 0 EF slope m 0 rise run BC slope m run rise FG slope m rise 0 run CD slope m 0 run 4 rise 0 GH slope m 0 rise run DE slope m run rise HI slope m run Copyright 005, Durham Continuing Education Page 46 of 5

47 MFMP Foundations of Mathematics Support Question Answers 8 6. Answers may vary. 6 so a possible rise is 8 and a possible run is. 7. x 4 Therefore the rise is. x (4) x 8. rise a slope m run rise 0 b slope m 0 run rise c slope m run 8 rise d slope m run rise e slope m undefined run 0 rise f slope m run 9. a b c d slope m slope m slope m slope m y x y x y x y x y x y x y x y x (6) ( 5) ( 4) (4) 8 8 () (8) 6 (7) () 4 () () () (0) ( ) (4) 7 () (5) e f g y slope m x slope m slope m y x y x y x y x y x ( ) () 5 ( 6) () 8 ( ) (7) 9 undefined () () 0 ( 6) ( 6) () (4) Copyright 005, Durham Continuing Education Page 47 of 5

48 MFMP Foundations of Mathematics Support Question Answers Lesson 0. a. 6 million; 9 million; million; 6 million; 9 million b. 98 Canadians Population Since Population (millions) Year. Walked at a steady rate, then took a brief rest then walked slightly slower but steadily then reached her grandparents and finally walked quickly back to home.. Noah was riding his bike up a hill then once he reached the top rode his bike quickly down the hill. 4. a. 6,8 and 0 textbooks b. approx 48, 68, 0 and 5 cm. 5. Walk Those Stairs Height (cm) Number of Stairs Copyright 005, Durham Continuing Education Page 48 of 5

49 MFMP Foundations of Mathematics Support Question Answers 6. b. Yes, points are joined to see the possible values between points. c. 90 cm d. stairs e. Yes because this is a linear relationship. Circles 00 Area of Circle Radius of Circle (cm) 6 b. Yes, points are joined to see the possible values between points. c. 5.4 cm d. 6 cm e. No, this is not a linear relationship. 7. y x 5 b. y x 6 c. y w X x -5 Y -4 (-4) (-) () -5-5 (5) -5 5 X -x 6 Y - -(-) (-) (0) 6 6 -() 6 5 X w^ - Y - (-).5 0 (0).5 () (4) X Y Copyright 005, Durham Continuing Education Page 49 of 5

50 MFMP Foundations of Mathematics Support Question Answers b. c. $00 d. 00 yearbooks 9. a. X Y - ( ) ( ) (0) 4-4 ( ) 4 - () 4 b. X Y -8 ( 8) - -4 ( 4) 0 0 (0) 4 (4) 4 8 (8) 6 Copyright 005, Durham Continuing Education Page 50 of 5

51 MFMP Foundations of Mathematics Support Question Answers c. X Y ( ) ( ) (0) () () (-4, 9) y x 7 9 ( 4) Copyright 005, Durham Continuing Education Page 5 of 5