# Geometry & Trig

'A guide for for teaching! Ideas and resources and lessons to look forward to'

*This page is good for the unit with this name in both the new and previous syllabi!*

This is a big, key topic in the syllabus that covers a variety of function families, both in the abstract and in context. Its a fabulous opportunity to show where the abstract and contextual meet, using functions as a tool to model the world around us. Its key because of the presumed knowledge involved, the presence of which, or not as the case may be, has a big impact on the way its delivered! This topic is the one that involves the most use of the GDC and as such provides the opportunity to help students become fluent with that tool.

This page is a guide to the topic, ideas for classroom activity that encourage critical thinking, practise exercises and some of the best resources available on the Internet to help! The overview written below aims to help teachers think about the main objectives of the module and potential opportunities and issues there are with teaching it! Please follow links to the pages on Ideas and resources, exercises and the internet guide.

#### Ideas and resources

Classroom experiences are the most important thing that we as teachers are responsible for. This links to a page of activities and ideas for teaching this unit that are aimed to be both engaging and effective in encourage students to enjoy, discover and understand mathematics. The list is a brief outline of an activity that links to a page that gives more detail and the associated resources. This is an area that should develop regularly. Think about subscribing to the RSS feed to get notifications of updates.

#### Practice

This page has a variety of tasks designed for practise and revision. Practice questions are an important part consolidating students' understanding of a concept. They can be found from numerous sources and the following is not intended as a 'fix all' solution. Most teachers appreciate that questions gathered from numerous different sources make the best diet!

#### The internet guide

This page is intended to list and outline some of the best resources available online to support the teaching and learning of this module. These will mostly be videos, virtual manipulatives and self help sites. Where appropriate, there will be a short commentary of what they are and how they might best be employed! It is also intended that this selection grow over time and that users contribute.

## Overview

This topic helps bring variety to the course. There is a natural link between the coordinate geometry and functions and algebraic skills are needed to solve the problems, but the trigonometry and the geometry of 3D shapes branch students out into a different field. Often this shows students specific strengths or weaknesses for this reason. Its a very visual topic that allows students to make yet more links between Mathematics and the world around them.

If done in order, the opening section on coordinate geometry follows on nicely from functions work. and leading into geometry. Working in the cartesian plane students look at the midpoints and the distance between two points, by using triangles and pythagoras theorem. Using two general forms for a straight line, they look at gradients and parallel and perpendicular lines. There follows a thorough look at applied trigonometry in 2 and then 3 dimensions, which leads nicely into the geometry of 3D shapes. The topic provides some nice opportunities for deductive proof and exposure to real life problem solving!

### Key elements

- coordinates, points and lines
- lines, properties and relationships
- right-angled trigonometry
- sine and cosine rule
- geometry of solids

Please refer to the subject guide for full details

### Teaching time

The recommended time is 20 hours and this is probably reasonable. In preparing for IB most students have probably seen right angled trigonometry and whilst it would be wrong to assume it was understood it is definitely helpful. Likewise, if pythagoras theorem, gradients and parallel lines are also quickly recognised then time can be saved here.

### Sticking points

In time these sticking points will be illustrated with a video or screen cast

#### Points and lines

At the very heart of understanding a generalised relationship is understanding that for a given straight line, y=mx+c, each x value has a corresponding y value. Translating this pair of values to a coordinates grid and back again is a regular sticking point. For example a question gives a segment of a straight line that intersects both the x and y axis. The equation of the line is given and students are asked to find the coordinates of the two intersections. Students are expected to recognise that either x or y is zero for each point and that the equation can be used to solve for the corresponding value. Sounds simple, but in fact draws and a considerable depth of understanding and relations.

#### Two forms of the straight line

In most cases students have been taught to this point that the equation of a straight line is y=mx+c where 'm' is the gradient and 'c' is the y-intercept. To introduce a new form of the type ax + by + d = 0 (note use of d to avoid confusion with associated meaning of 'c'^{[1]}) can create confusion. This is particularly so because none of the constants, a, b or d necessarily correspond to features of the function. To tackle this its worth demonstrating that the latter form only ever needs integer values and the associated advantages. It is also worth spending time on the algebraic manipulation required to change between forms. A simple 'matching pairs' activity is included under ideas and resources.

#### Algebraic manipulation

As mentioned above, the ability to manipulate equations helps when changing between forms of a straight line and when generating equations from given information. Equally, when solving problems involving geometric formulae, students can quickly get stuck when the unknown is not the subject of the formula.This can get in the way of understanding concepts and so should be reviewed before its applied. Another option here is to make use of a Computer Algebra System (CAS) like derive or TI-nspire.^{[2]}These programs allow students to experiment by, for example, 'multiplying both sides by 3'. The program generates the new equation correctly so the students can focus on adopting the right strategy.

#### Relations

When considering algebraic manipulation I think it is very important to consider the 'Relations' aspect of equations. For example, ifa=bc, then it follows that b=a/c and c = a/b. This can be illustrated with numbers like 8, 4 and 2 to bring it into reality and then applied, for example to problems with right-angled trigonometry, eg, if sinA = b/c, then it follows thatb = csinA and c= b/sinA. Below is a video reminder that may be helpful for students.

If these and similar relations are well understood then the notion of rearranging equations to change the subject becomes more intuitive.

#### Recognising problems

Often with the application of trigonometry there is a problem with 'recognising problems'. Decisions about which trig ratio should be used or whether the Sine or Cosine rule is most appropriate. This often arises form the view there there is a unique, correct approach to any given problem. This can be tackled by encouraging a more speculative approach to problem solving, where two or three attempts may be unsuccessful before a correct one is adopted. The following videos aim to demonstrate what is meant by the 'Speculative approach'

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### Teaching Opportunities

#### Investigation

Time is always the barrier to investigation, but the benefits are often worth the input. If students explore the patterns in the mathematics and discover the rules and relationships themselves then this can be both more engaging and more powerful. In this topic, there are opportunities for students to investigate things like, gradients of perpendicular lines, midpoints, trig ratios, trig rules and mensuration formulae. Obviously there is not likely to be time to do all of that but some would be good.

#### First principles

Trigonometry from first principles is brilliant example of how something apparently complex and abstract can be pulled back to a simple, easy to understand geometric property. Starting with similar triangles the whole concept of sine, cosine and tangent ratios can be defined and generated giving the topic in general more 'roots'. Whilst SOHCAHTOA will prove perennially useful, it can be very rewarding when students realise they remember their trig ratios in order to remember how to spell the acronym! This also helps with the 'speculative approach'.

#### Proof

The concept of a mathemetical proof is a fascinating one and very relevant to the study of 'Theory of Knowledge'. Opportunities to explore this with Maths Studies students can be few, but the ones there are should be explored. For example, using right angled trigonometry to prove the sine rule is generally approachable. Another good one, if slightly trickier, is the curved surface of a cone. In both of these cases, the resulting formulae will remain abstract iof their origins are not explored. All formulae can, of course, be applied, without being derived, but to spend some time deriving these is to unravel some of the fascinating mathematical mystery and empower students.

#### ICT

All of the above and more can, of course, be very well facilitated by the use of ICT^{[3]}. 'Dynamic geometry' allows a level of investigation and demonstration not really possible on paper. 'CAS' can help with algebraic manipulation to change the general form or rearrange. I have also experimented with 'Google Sketchup' to help create 2D snapshots of 3D situations.

#### Geometry around us

When studying 3D geometry there is an opportunity to draw students attention to the geometry of the world around them. What are the applications of geometry? The Earth is an approximate sphere, how far away is the horizon? How did this geometry develop? Shadows and the movement of the sun. Moving on from natural geometry, what about the buildings around us? prisms, cones, hemispheres etc. What is the role of geometry in architecture? These thoughts and more can inject terrific interest for a wide range of students.

### Ideas and resources

This links to a page of activities and ideas for teaching this unit. The list is a brief outline of an idea; some are just ideas and others link to a page that gives more detail and some resources. This is an area that should develop regularly. Think about subscribing to the RSS feed to get notifications of updates.

#### Footnotes

## Selected Pages

### Making Cones 29 September 2016

'Explore cones by making one! A great puzzle that helps to understand the formulae for cones'It sounds easy, but is it?...

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### Which Rule? 29 September 2016

'Master the art of working out when best to apply which rule to trigonometry problems'From SOHCAHTOA to cosine rule to Pythagoras's...

more

### Impossible Triangles 29 September 2016

Can these triangles exist?The main thrust of this activity is easily explained - What reasoning can you use to decide if...

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### Trig Calculator 29 September 2016

'Make your very own trig ratio calculator!'When you input sin 30 on your calculator you should get 0.5Careful to make sure...

more

### G & T Teaching Ideas 27 September 2016

Here you should find a wealth of resources that can be used by you and your students for teaching this topic. The resources...

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### Making Cones - TN 11 December 2012

The great thing about this activity is how it seems as though it is going to be easy and then isn't at all. This...

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