Introducing Trig
In this unit students will explore the importance of triangles, particularly right angled triangles, in the real world. Students will use practical measuring skills and calculations to find a pattern linking the ratio of the sides of a triangle with the angles.
This unit may take longer than one week, but introducing trigonometry using scale diagrams and practical measuring tasks allows for improved understanding of the concepts and how they apply to real life situations.
measure the lengths of the sides of sets of similar right angled triangles and find the ratio of sides
investigate the relationship between these ratios and the angle size
use calculators or tables to find the sine, cosine and tangent of angles
This unit develops concepts of trigonometry through practical measuring tasks which are done in groups. Initially the students explore the importance of triangles in constructions in the real world, and how scale drawings can be used to solve problems.
After a discussion about the limitations of scale drawings, the students develop the concepts of trigonometry through practical measuring tasks. Eight sets of similar triangles with eight different angles are prepared. Sets of these triangles are issued to each group, and the students measure the lengths of each side and calculate the ratios. The students have an opportunity to check the reasonableness of their answers by discovering that the ratios appear to be the same for the same angle and any apparent errors can be discussed.
At this stage, class results are graphed and the students discover that there appears to be a pattern occurring which again gives them an opportunity to discuss any apparent errors.
The ratios are now given a name – sine cosine and tangent – and the students are shown that by using trigonometry, heights and distances can be found more accurately than by scale drawings.
Some historical background into the development and use of trigonometry is interesting at this point.
Getting started
Bridge building activity
Each group has:
- 6 drinking straws
- 1 pair of scissors
- 1 ruler
- 2 sticky address labels
- 2 textbooks per group for the bridge supports
- 10c coins for weights.
The students should be asked to think of bridges in the district and what they have in common. The text books should be placed so that the distance between them is further than the length of one straw. The books represent the banks of a river infested with crocodiles. The group has to build a bridge to cross the river that will carry people from one side to the other using the straws and the sticky labels.
At the end of the time – probably 30 minutes – a piece of paper should be placed on the bridge, and it should be tested for strength using the 10c coins
The first lesson is for the students to discover that probably the best bridge that they can build using the equipment provided will involve triangles in some way
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Students need to investigate the use of right angled triangles as a mathematical
model by looking at structures and buildings in the school environment, and in other examples such as:- tents with guy ropes
- rotating clothes line
- pile of sand
- ice-cream cone
- ladder against a wall
- a pyramid
- arm of a crane
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Scale drawings can be used to solve problems involving triangles and the students need to be given the opportunity to solve at least one problem this way to discover the limitations of this method.
Examples: You have a window with the following dimensions 1.2m wide base, 4.1m high in the right side. 2.7 m on the left side. Use a scale diagram to find the sloping length at the top and then the angles at the top of the window.
Exploring
- Using the sets of similar triangles the students, working in groups, should measure the lengths of the three sides, and write the ratios as fractions and then decimals on worksheet 1. By working in groups of 4, each student will get to do the measurements and ratios for 2 triangles. The students should learn that in each set of similar triangles, the ratios of the pairs of sides are constant. For any result which looks different from the rest, the students should be encouraged to check first of all if the measurements, and then if the calculation of the ratios are correct.
Each group should now plot its mean values on three class graphs.
In collecting together and discussing the results for the whole class, it should be determined that the value of the ratio depends on the angles in the triangle, not the size of the triangle, that triangles that have the same ratios also have the same angles. and that the ratio increases as the angle increases for opp / hyp and opp / adj and decreases as the angle increases for adj / hyp .
Reflection
- It is time now to introduce the special names for the ratios and the use of calculators for determining the values of the ratios.
opp / hyp is sine of the angle (sin)
adj / hyp is cosine of the angle (cos)
opp / adj is tangent of the angle (tan)
- It would be also useful and interesting for the students to look at the historical background of trigonometry and how trigonometry is used various occupations eg surveying.
| Attachment | Size |
|---|---|
| Recording sheet 1.pdf | 38.6 KB |
| similar triangles.pdf | 33.17 KB |
Similar Resources
Trigonometric Applications Outside the Classroom
This unit follows on from others that introduce students to sine, cosine and tangent. In this unit students will gain practical experience in using trigonometry in practical situations; mainly outside the classroom.
Using Trigonometry
In this unit, students will explore the use of trigonometry to find unknown sides and angles in right-angled triangles, taking the concepts that were developed in the previous unit Introducing Trig and looking at different ways of practising the new skills. Students will then develop and reinforce skills to solve practical problems in trigonometry using the problem solving model
reality 
model 
solution 
check
It is important that good algebraic skills are used in setting out their work, and students can be encouraged to practice these individually and using cooperative logic problems in a group situation.
Investigating the Idea of Sin
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Scaling Our School
In this unit we look at estimating the length and height of buildings at your school. We then compare these estimates with the actual heights or lengths. Students will then build scale models selecting appropriate scales using ratio to translate the "real" dimensions into the dimensions for the scale model.
Cosine Rule
This unit explores the magnitudes of sides and angles of a triangle and leads to the discovery and proof of the Cosine Rule. This Rule is then used to solve triangles, some of which arise in practical situations. We note that the Cosine Rule is a generalisation of Pythagoras’ Theorem.
Note that this unit may contain more material than you need for some classes. You may want to choose from parts of this unit if you have not allotted 5 sessions on the Cosine Rule to your class.
