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Mathematical Studies Standard Level for the IB Diploma Scheme of work – Topic 5: Geometry and trigonometry Coursebook chapters 14–16 Introduction This scheme of work offers an example route through the specification with suggestions of activities and discussion points that you could consider along the way. For each chapter there are references to the associated PowerPoint file and interactive GeoGebra files, as well as to relevant websites and video clips. The PowerPoint files contain a number of discussion slides that raise Theory of Knowledge questions; students should be encouraged to talk about these both in maths lessons and during specific Theory of Knowledge lessons. The scheme of work, while not intended to cover the syllabus in full, does aim to provide a framework that you can supplement and adapt with your own activities and ideas. Note: Italic text describes suggested uses of the materials referenced. The GeoGebra files referenced in this scheme were created using GeoGebra version 4.2.31.0. Some functions, such as check-boxes, were introduced in this version; so if you have an older version of the software, you will need to download the latest version from the GeoGebra website to be able to use these resources to their full potential. Key [ppt] [tt] [ggb] [V] [www] [TOK] [GDC] PowerPoint activity textbook exercise GeoGebra activity video link useful website Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. Theory of Knowledge activity GDC question and worked answer Copyright Cambridge University Press 2014. All rights reserved. Page 1 of 9 Mathematical Studies Standard Level for the IB Diploma Topic 5: Geometry and trigonometry 18 hours Aims: To learn to analyse lines in two-dimensional space and identify the relationships between different lines To be able to calculate lengths in two- and three-dimensional space using trigonometry and Pythagoras’ theorem To be able calculate measurements such as areas and volumes in two- and three-dimensional space Phase Chapter 14: Equation of a line in two dimensions Estimated time allocation 5 hours Focus from the text Learning objectives The gradient of Be able to determine a line the gradient of a line from coordinates of points on the line or from the graph Activities Links [ppt] [TOK] Chapter 14 slides 2–3: Descartes Providing some historical background as well as a TOK link, slide 3 raises the question of whether graphical analysis has the same level of rigour as algebraic analysis. It offers an opportunity for students to discuss algebra and for the formality of written mathematics to be reinforced. [www] http://plato.stanford.edu/entries [ggb] ch14 gradient two points In this simple but useful interactive file, students can move two points around, and it will show the changes in x- and y-coordinates between the points as well as the right-angled triangle thus formed, enabling students to quickly calculate the gradient. A ‘show gradient’ button toggles the gradient on and off so that students can check their calculations. This is most suitable for use as a discovery activity, in which students could explore the link between the coordinates, the change in coordinates and the gradient to derive a method for working out the gradient directly from the coordinates of two points. Copyright Cambridge University Press 2014. All rights reserved. Page 2 of 9 /descartes-works/ Site dedicated to René Descartes [www] http://www-history.mcs.st-and. ac.uk/Mathematicians/Descartes.html History of maths page for René Descartes Mathematical Studies Standard Level for the IB Diploma [tt] Page 415 Exercise 14.1 ‘Finding and understanding gradients of straight lines’ The y-intercept Finding the equation of a straight line Know how to find the y-intercept graphically Be able to find the equation of a line both manually and using the GDC [ggb] ch14 equation of a line This interactive file facilitates exploration of the meaning of gradient and y-intercept. The line can be adjusted by moving the point marked with a blue dot. As the line changes, students can attempt to find the gradient and y-intercept manually. Buttons can be used to reveal the gradient and y-intercept coordinates along with two equation forms of the line. This is a flexible resource that could be adapted to suit many investigations, especially student discovery tasks. If necessary, you could provide more structure to the activity, for example by asking students to look at particular equations and then find the gradient and intercept to discover the link between the equation and the values. [tt] Page 419 Exercise 14.2 ‘Finding the equation of a line’ The equation of a straight line Know the different forms of the equation of a straight line; be able to find the equation of a line when presented with information about the line Copyright Cambridge University Press 2014. All rights reserved. [tt] Page 423 Exercise 14.3 ‘Further questions on finding the equation of a line, including equations of parallel and perpendicular lines’ [ggb] ch14 parallel lines This is an extended version of the ‘equation of a line’ GeoGebra file which contains two parallel lines, so that the relationship between the equations, gradients and intercepts can be explored. Again, the blue dot is the point that can be moved around. The ‘Show values’ button can be used to toggle the information on the Page 3 of 9 [www] http://www.mathsisfun.com/al gebra/line-parallel-perpendicular.html Web page summarising how to find equations of lines, including parallel and perpendicular lines [V] http://youtu.be/0r2XxCwgzSA 10-minute video showing examples of finding the equation of a line parallel to another line Mathematical Studies Standard Level for the IB Diploma screen, so you could start without the equations and values. Again, students could be encouraged to discover the relationships for themselves, or you could use the file for a demonstration displayed on an interactive whiteboard or by a projector. Review of Chapter 13 1 hour Drawing a straight line graph from an equation Know how to draw a straight line from its equation, both manually and with a GDC Chi-squared hypothesis testing Secure the skills of using χ2 hypothesis tests by undertaking past paper-style questions Copyright Cambridge University Press 2014. All rights reserved. [ggb] ch14 perpendicular lines This interactive file enables exploration of perpendicular lines to deduce the relationship between their gradients. The lines can be rearranged by moving the blue dot, and clicking on the ‘Show values’ button displays the equations together with the gradients and y-intercepts. The link between the equations is less obvious in this case, but it can be seen more easily when one gradient is integer-valued. It would be sensible to allow sufficient time for students to discover the gradient relationship for themselves, and you could set up structured exploration tasks to guide them along. [tt] Page 431 Exercise 14.4 ‘Drawing a line given the [V] http://youtu.be/EUzl8_aQ3xc equation’ 6-minute video demonstration of how to draw the graph of a line from its equation [tt] Page 405 Mixed examination practice Page 4 of 9 Mathematical Studies Standard Level for the IB Diploma Phase Chapter 15: Trigonometry Estimated time allocation 6 hours Focus from the text Trigonometric ratios Learning objectives Activities Links Understand the relationships between the sides and angles in a rightangled triangle; be able to use these relationships to find out missing information [ppt] Chapter 15 slides 2–3: Origins This short discussion activity highlights the historical significance of right-angled triangles in mathematics. It could be used as a short research task. [www] http://www.mathsisfun.com [ggb] ch15 similarity This file shows an image of two similar overlapping triangles and the ratios between their side lengths; it is intentionally drawn with the same labels for the common angle and sides as the image on the first page of Chapter 15, so that students or the teacher can refer to it. A slider can be used to vary the angle, and by moving the blue points you can change the size and orientation of the triangles; this demonstrates that the three ratios stay equal throughout such changes. [ggb] ch15 ratios This interactive file allows you to select the position of the angle within the triangle in relation to the right angle; it then shows the trigonometric ratios sin, cos and tan. You can move the corners of the triangle to see how this affects the ratios. Tick both A and C to view both sets of ratios simultaneously. The key point to notice is that the sides O and A switch around and that this swaps the values of sin and cos and inverts tan. [tt] Page 438 Exercise 15.1 ‘Using right-angled trigonometry’ Copyright Cambridge University Press 2014. All rights reserved. Page 5 of 9 /pythagoras.html This is a summary page which could be used to discuss the history of how the relationships between the sides of a right-angled triangle developed, from the earliest examples in China and Egypt through to its naming as Pythagoras’ theorem. [www] http://www.mathsisfun.com /sine-cosine-tangent.html [www] http://www.mathsisfun.com /algebra/trig-inverse-sin-cos-tan.html Mathematical Studies Standard Level for the IB Diploma Angles of elevation and depression Understand what these terms mean and be able to solve problems involving angles given in this context Harder Be able to answer trigonometry any question problems involving rightangled triangles The sine rule Be able to use the sine rule to calculate missing angles and sides The cosine rule Be able to use the cosine rule to calculate missing angles and sides Area of a triangle Be able to find the area of a triangle using the sine formula Copyright Cambridge University Press 2014. All rights reserved. [tt] Page 442 Exercise 15.2 ‘Solving problems involving angles of elevation and depression’ [tt] Page 445 Exercise 15.3 ‘Further two-dimensional right-angled trigonometry problems’ [tt] Page 449 Exercise 15.4 ‘Finding sides and angles using the sine rule’ [www] http://www.mathsisfun.com [tt] Page 452 Exercise 15.5 ‘Finding sides and angles using the cosine rule’ [www] http://www.mathsisfun.com /algebra/trig-sine-law.html /algebra/trig-cosine-law.html [ppt] [TOK] Chapter 15 slides 4–5: Generalisation This highlights the similarity between Pythagoras’ theorem and the cosine rule and can be used together with the following GeoGebra file to explore the concept of mathematical generalisation. [ggb] ch15 cosine vs pythagoras This interactive file demonstrates the relationship between the cosine rule for a non-right-angled triangle and Pythagoras’ theorem for an associated rightangled triangle. The blue corners of the non-rightangled triangle can be moved, and the calculations will adjust accordingly. [tt] Page 456 Exercise 15.6 ‘Finding areas of triangles using the sine formula’ [GDC] Land management Page 6 of 9 [www] http://www.mathsisfun.com/algebra/trig -area-triangle-without-right-angle.html Mathematical Studies Standard Level for the IB Diploma Review of Chapter 14 1 hour Constructing labelled diagrams Be able to solve more complex word problems requiring the drawing of a diagram [tt] Page 460 Exercise 15.7 ‘Further trigonometry problems, some requiring interpretation and construction of diagrams’ Equation of a line in two dimensions Secure the skills of finding and using equations of straight lines by undertaking past paper-style questions [tt] Page 433 Mixed examination practice Copyright Cambridge University Press 2014. All rights reserved. Page 7 of 9 Mathematical Studies Standard Level for the IB Diploma Phase Chapter 16: Geometry of threedimensional solids Estimated time allocation 4 hours Focus from the text Finding the length of a line within a threedimensional solid Learning objectives Activities Links Be able to identify right-angled triangles within a three dimensional solid and use these to find missing lengths [ppt] [TOK] Chapter 16 slides 2–3: Axiomatic systems This discussion is nearly an investigation or research project and could easily be adapted for this purpose. Slide 3 defines the rules for three-point geometry to introduce students to axioms and to stimulate a general discussion about this aspect of mathematics. The idea could be taken further by reviewing the properties of four-point geometry and quadrilaterals and then moving on to other polygons. [www] http://www.mathsisfun.com /geometry/solid-geometry.html [www] http://www.beva.org/math323 /asgn5/nov5.htm Extension of the idea for the TOK activity ‘Axiomatic systems’, this could be used to support further investigation [ppt] Chapter 16 slides 6–11: Right-angled triangles in three-dimensional objects These slides provide a visual summary of ways of creating right-angled triangles in solids. They could be used to introduce or review the properties of these shapes, and the diagrams are the same as those on the revision sheet for this chapter. [tt] Page 473 Exercise 16.1 ‘Finding lengths using Pythagoras’ theorem in 3D solids’ Finding the size of an angle in a threedimensional solid Be able to find missing angles in three-dimensional solids Copyright Cambridge University Press 2014. All rights reserved. [ppt] [TOK] Chapter 16 slides 4–5: Mathematical fact? This presents a challenging concept for students, which may surprise them as it suggests that the internal angles of a triangle don’t necessarily add up to 180°. On slide 5 are two images, one of which shows a normal triangle on the plane and the other showing a triangle on the surface of a sphere. The triangles are drawn accurately, and the angles given are the actual angles measured electronically (students will suspect it is a trick, but you can reassure Page 8 of 9 [www] http://en.wikipedia.org/wiki /Spherical_geometry Wikipedia page on spherical geometry Mathematical Studies Standard Level for the IB Diploma them that it isn’t). This could lead into a research project about distances of plane journeys around the globe and the shortest distance between two points. [ppt] Chapter 16 slide 12–15: Angle between a line and a plane These slides provide a visual summary of two methods for constructing a right-angled triangle in order to find the angle between a line and a plane. They could be used to introduce or review this situation, and the diagrams are the same as those on the revision sheet for this chapter. Review of Chapter 15 1 hour Calculating volumes and surface areas of threedimensional solids Become familiar with the formulas for volume and surface area of solids and be able to use them as required Trigonometry Secure the skills of problem solving using trigonometry by undertaking past paper-style questions Copyright Cambridge University Press 2014. All rights reserved. [tt] Page 477 Exercise 16.2 ‘Finding lengths and angles in 3D solids using trigonometry and Pythagoras’ theorem’ [tt] Page 485 Exercise 16.3 ‘Finding volumes and surface areas of 3D solids, particularly cylinders and spheres’ [tt] Page 463 Mixed examination practice Page 9 of 9 [www] http://www.learner.org /interactives/geometry/area.html Website of interactives for finding volumes and surface areas