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1 WORK PROGRAM Chapter 10 Coordinate geometry Strands: Patterns and algebra, Measurement Substrands and outcomes: Algebraic techniques PAS4.4 Uses algebraic techniques to solve linear equations and simple inequalities Linear relationships PAS4.5 Graphs and interprets linear relationships on the number plane Coordinate geometry PAS5.1.2 Determines the midpoint, length and gradient of an interval joining two points on the number plane and graphs linear and simple non-linear relationships from equations Coordinate geometry PAS5.2.3 Uses formulae to find midpoint, distance and gradient and applies the gradient/intercept form to interpret and graph straight lines Coordinate geometry PAS5.3.3 Uses various standard forms of the equation of a straight line and graphs regions on the number plane Perimeter and area MS4.1 Uses formulae and Pythagoras’ theorem in calculating perimeter and area of circles and figures composed of rectangles and triangles Section Are you ready? (page 362) GC tips, Investigations, History of mathematics, Maths Quest challenge, 10 Quick Questions, Code puzzles, Career profiles SkillSHEETs, WorkSHEETs, Interactive games, Test yourself, Topic tests (CD-ROM) SkillSHEETs (page 362) 10.1: Substitution into a rule 10.2: Completing a table of values for a given rule 10.3: Plotting coordinate points 10.4: Matching a graph with its table of values 10.5: Plotting a line using a table of values Technology applications (CD-ROM) Learning outcomes PAS4.4 solving equations arising from substitution into formulae PAS4.5 reading and plotting ordered pairs on the number plane including those with values that are not whole numbers forming a table of values for a linear relationship by substituting a set of 2 10.12: Using Pythagoras’ theorem Plotting linear graphs (page 363) WE 1 Ex 10A Plotting linear graphs (page 366) History of mathematics: René Descartes (page 363) appropriate values for either of the letters and graphing the number pairs on a number plane MS4.1 using Pythagoras’ theorem to find the length of sides in right-angled triangles SkillSHEET 10.1: Cabri geometry: Linear PAS4.5 Substitution into a rule patterns and equations forming a table of values for a (page 366) (page 365) linear relationship by SkillSHEET 10.2: Cabri geometry: substituting a set of Completing a table of Coordinates and graphs appropriate values for either values for a given rule (page 366) of the letters and graphing the (page 366) Excel: Plotting graphs number pairs on a number SkillSHEET 10.3: Plotting (tables) (page 366) plane coordinate points Excel: Plotting graphs PAS5.1.2 (page 366) (tables) (DIY) (page 366) constructing tables of values SkillSHEET 10.4: Excel: Plotting graphs and using coordinates to Matching a graph with its (rule) (page 366) graph vertical and horizontal table of values Mathcad: Plotting linear lines (page 366) graphs (page 366) identifying the x-axis as the SkillSHEET 10.5: Plotting line y = 0 a line using a table of identifying the y-axis as the values (page 366) line x = 0 Game time 001 (page 367) graphing a variety of linear relationships on the number plane by constructing a table of values and plotting coordinates using an appropriate scale describing horizontal and 3 Gradient and y-intercept (page 367) WE 2, 3a–b, 4 Ex 10B Gradient and y-intercept (page 370) Maths Quest challenge: Q1–3 (page 374) 10 Quick Questions 1 (page 374) Investigation: Gradient and y-intercept (page 375) SkillSHEET 10.6: Measuring the rise and run (page 371) SkillSHEET 10.7: Finding the gradient given to two points (page 372) Worksheet 10.1 (page 373) Cabri geometry: Gradient (pages 368, 370) Excel: Gradient and intercepts (page 370) Excel: Gradient and intercepts DIY (page 370) Mathcad: Gradient (page 372) Excel: Gradient (page 372) Cabri geometry: Linear graphs and y-intercept (page 375) Cabri geometry: Linear equations and gradient (page 375) vertical lines in general terms (Communicating) PAS5.1.2 using the right-angled triangle drawn between two points on the number plane and the rise relationship Gradient = run to find the gradient of the interval joining two points determining whether a line has a positive or negative slope by following the line from left to right finding the gradient of a straight line from the graph by drawing a right-angled triangle after joining two points on the line identifying the x- and y-intercepts of graphs graphing a variety of linear relationships on the number plane by constructing a table of values and plotting coordinates using an appropriate scale PAS5.2.3 using the relationship rise gradient = to establish run 4 Determining the equation of a straight line (page 376) WE 5, 6, 7 Ex 10C Determining the Code puzzle (page 379) Investigation: Relating equations and graphs to values of gradients and intercepts (page 380) Game time 002 (page 378) SkillSHEET 10.8: Finding the gradient of a line from its equation (page 381) Mathcad: Equation of a straight line (page 378) Excel: Equation of a straight line (page 378) the formula for the gradient, m, of an interval joining two points (x1, y1) and (x2, y2) on the number plane using the formula to find the gradient of an interval joining two points on the number plane constructing tables of values and using coordinates to graph straight lines of the form y = mx + b recognising equations of the form y = mx + b as representing straight lines and interpreting the x-coefficient (m) as the gradient and the constant (b) as the y-intercept determining the difference in equations of lines that have a negative gradient and those that have a positive gradient (Reasoning) comparing similarities and differences between sets of linear relationships (Reasoning) PAS5.2.3 recognising equations of the form y = mx + b as representing straight lines and interpreting the x-coefficient 5 equation of a straight line (page 378) (m) as the gradient and the constant (b) as the y-intercept rearranging an equation in general form to the gradient/intercept form graphing equations of the form y = mx + b using the y-intercept (b) and the gradient (m) determining that two lines are parallel if their gradients are equal finding the gradient and the yintercept of a straight line from the graph and using them to determine the equation of the line determining the difference in equations of lines that have a negative gradient and those that have a positive gradient (Reasoning) matching equations of straight lines to graphs of straight lines and justify choices (Communicating, Reasoning) PAS5.3.3 describing the equation of a line as the relationship between the x- and ycoordinates of any point on the line 6 Sketching straight line graphs (page 382) WE 8, 9 Ex 10D Sketching straight line graphs (page 384) Investigation: Predicting a person’s height (page 385) 10 Quick Questions 2 (page 385) SkillSHEET 10.9: Solving Excel: Linear graphs linear equations that arise (page 384) when finding x- and Mathcad: Linear graphs y-intercepts (page 384) (page 384) SkillSHEET 10.10: Mathcad: Linear graphs — Graphing linear intercept form (page 384) equations using the xand y-intercept method (page 384) WorkSHEET 10.2 (page 384) finding the equation of a line passing through two points PAS5.1.2 identifying the x- and yintercepts of graphs identifying the x-axis as the line y = 0 identifying the y-axis as the line x = 0 graphing a variety of linear relationships on the number plane by constructing a table of values and plotting coordinates using an appropriate scale PAS5.2.3 recognising equations of the form y = mx + b as representing straight lines and interpreting the x-coefficient (m) as the gradient and the constant (b) as the y-intercept graphing equations of the form y = mx + b using the yintercept (b) and the gradient (m) finding the gradient and yintercept of a straight line from the graph and using them to determine the equation of the line 7 Graphs of linear inequalities (page 386) WE 10, 11, 12, 13 Ex 10E Graphs of linear inequalities (page 389) SkillSHEET 10.11: Checking whether a given point makes the inequation a true statement (page 389) Mathcad: Horizontal and vertical graphs (page 389) GraphEq (page 390) Graphmatica (page 390) applying knowledge and skills of linear relationships to practical problems (Applying strategies) PAS5.3.3 sketching the graph of a line by finding the x- and yintercepts from its equation PAS5.3.3 rearranging equations from the general form to the gradient/intercept form and hence graphing the line sketching the graph of a line by finding the x- and y-intercepts from its equation graphing inequalities of the form y < a, y > a, y a, y a, x < a, x > a, x a and x a graphing inequalities such as y a on the number plane by considering the position of the boundary of the region as the limiting case checking whether a particular point lies in a given region specified by a linear inequality graphing regions such as that specified by 8 Parallel lines (page 391) WE 14, 15 Ex 10F Parallel lines (page 393) Investigation: Investigating linear equations (page 391) Perpendicular lines (page 394) WE 16 Ex 10G Perpendicular lines (page 395) Investigation: Pairs of perpendicular lines (page 394) Maths Quest Challenge: Q1–2 (page 396) WorkSHEET 10.3 (page 405) GC tip — Casio: (page 391) Excel: Plotting points (page 392) Excel: Gradient (page 393) Mathcad: Gradient (page 393) Cabri geometry: Gradient (page 393) Excel: Parallel checker (page 393) GC program — Casio: Parallel checker (page 393) GC program — TI: Parallel checker (page 393) Excel: Perpendicular checker (page 395) x y 7, 2 x 3 y 5 PAS5.1.2 determining whether a point lies on a line by substituting into the equation of the line PAS5.2.3 rearranging an equation in general form to the gradient/intercept form determining that two lines are parallel if their gradients are equal using a graphics calculator to graph a variety of equations of straight lines, and compare and describe the similarities and differences between the lines (Applying strategies, Communicating, Reasoning) PAS5.3.3 rearranging equations in the gradient/intercept form to the general form describe conditions for lines to be parallel (Reasoning, Communicating) PAS5.2.3 rearranging an equation in general form to the gradient/intercept form PAS5.3.3 9 Distance between two points (page 397) WE 17, 18, 19 Ex 10H Distance between two points (page 399) SkillSHEET 10.12: Using Pythagoras’ theorem (page 399) Mathcad: Distance between 2 points (page 399) Excel: Distance between 2 points (page 399) GC program — Casio: Distance between 2 points (page 399) GC program — TI: Distance between 2 points (page 399) demonstrating that two lines are perpendicular if the product of their gradients is –1 showing that if two lines are perpendicular then the product of their gradients is –1 (Applying strategies, Reasoning, Communicating) proving that a particular triangle drawn on the number plane is right-angled (Applying strategies, Reasoning) PAS5.1.2 graphing two points to form an interval on the number plane and forming a rightangled triangle by drawing a vertical side from the higher point and a horizontal side from the lower point using the right-angled triangle drawn between two points on the number plane and Pythagoras’ theorem to determine the length of the interval joining the two points describing how the length of an interval joining two points can be calculated using Pythagoras’ theorem 10 Midpoint of a line segment (page 400) WE 20, 21, 22 Ex 10I Midpoint of a line segment (page 402) Mathcad: Midpoint of a segment (page 402) Excel: Midpoint of a segment (page 402) GC program — Casio: Midpoint of a segment (page 402) GC program — TI: Midpoint of a segment (page 402) Cabri geometry: Midpoint of a segment (page 402) (Communicating, Reasoning) PAS5.2.3 using Pythagoras’ theorem to establish the formula for the distance, d, between two points (x1, y1) and (x2, y2) on the number plane using the formula to find the distance between two points on the number plane using the gradient and distance formulae to determine the type of triangle three points will form or the type of quadrilateral four points will form and justify the answer (Applying strategies, Reasoning) PAS5.1.2 determining the midpoint of an interval from a diagram describe the meaning of the midpoint of an interval and how it can be found (Communicating) PAS5.2.3 using the average concept to establish the formula for the midpoint, M, of the interval joining two points (x1, y1) and (x2, y2) on the number plane using the formula to find the 11 Further equations of a straight line (page 403) WE 23, 24, 25, 26a–c Ex 10J Further equations of a straight line (page 407) Investigation: A Roman aqueduct (page 408) WorkSHEET 10.3 (page 408) Excel: Equation of a straight line (page 406) Mathcad: Equation of a straight line (page 406) Mathcad: Equation of a straight line (page 407) Excel: Equation of a straight line (page 407) midpoint of the interval joining two points on the number plane using the formula to find the distance between two points on the number plane explaining the meaning of the pronumerals in the formulae for midpoint, distance and gradient (Communicating) using the appropriate formulae to solve problems on the number plane (Applying strategies) PAS5.1.2 graphing a variety of linear relationships on the number plane by constructing a table of values and plotting coordinates using an appropriate scale PAS5.2.3 using the formula to find the gradient of an interval joining two points on the number plane finding the gradient and y-intercept of a straight line from the graph and using them to determine the equation of the line using the appropriate 12 formulae to solve problems on the number plane (Applying strategies) applying the knowledge and skills of linear relationships to practical problems (Applying strategies) PAS5.3.3 finding the equation of a line passing through a point (x1, y1), with a given gradient m, using: y ─ y1 = m(x x1) and y = mx + b finding the equation of a line passing through two points finding the equation of a line that is parallel or perpendicular to a given line Summary (page 409) Chapter review (page 410) ‘Test yourself’ multiple choice questions (page 412) Topic tests (2)