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Scheme of work – Cambridge IGCSE® Mathematics (US) 0444
Unit 8: Trigonometry (Core)
Recommended prior knowledge
Units 1 and 4 must have been completed. Students need a good understanding of where right angles facts occur, (quadrilaterals, diagonals intersecting, lines of
symmetry and edges intersections and angle in a semi circle),the link been square and square root and to find both on a calculator, and similarity.
Context
This is the fifth or five geometry units. Units 1 and 4 must have been completed. Both Pythagoras and Trigonometry are topics that require practice and experience
for students to use effectively. Delivering the entire unit as a block and not returning to the topic would not be recommended. The final section on identification of
question type can be a revision topic leading to the examination. The overlaps between Units 7 and 8 are such that the order of planning for both units needs to be
thought about simultaneously with respect to the choices outlined in Unit 7 but especially link to the slope of graphs in unit 7 and the m of y=mx + c to Tangent.
Students who are following the extended syllabus will move through this faster but need to have all these skills in place before working on the extended units, or
applying them in other areas of mathematics.
Outline
This unit covers the development of Pythagoras as a pattern, from diagrams and suggestions for approaching problems. Trigonometry is introduced from a set of
similar triangles. Learning to identify where right angles occur and selecting the right area of mathematics to solve missing angles and sides is also covered. Within
the suggested teaching activities ideas are listed to identify and remediate misconceptions and to pull learning through to the required standard. The learning
resources give both teaching ideas, summaries of the skills and investigative problems to develop the problem solving skills and a depth of understanding of the
mathematics, through exploration and discussion.
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Syllabus ref
Learning objectives
Suggested teaching activities
8.1
Use trigonometric ratios and
the Pythagorean Theorem to
solve right-angled triangles in
applied problems
Notes and exemplars
Problems involving bearings may be included.
Know angle of elevation and depression.
CCSS:
G-SRT6
G-SRT8
Learning resources
General guidance
Know all the places where right angles occur in rectangles, squares, kites
and rhombi, equilateral and isosceles triangles and where lines of
symmetry bisect odd sided regular polygons, angles in semicircle and
tangents to radii. (review of aspects of Unit 4).
Teaching activities
Set up a two way grid, ‘right angle(s)’ ‘no right angles’ along the top and
‘at a vertex’, ‘where diagonals cross’, ‘where lines of symmetry cross an
exterior line’ down the side and ask students to put as many polygons as
they can in the spaces.
General guidance
Develop understanding of Pythagoras rule and its use in finding missing
sides in right angled triangles. Ensure time is given to checking that
students can distinguish between problems that require the hypotenuse as
the answer and those that require one of the other two sides.
Teaching activities
Set up a worksheet with half a dozen right angled triangles with the
squares drawn on their edges, ask students to find the areas of the
squares and record in a table, so that the largest square (on the
hypotenuse) is in third column, smallest in first column and middle one in
the second column, ask what they can deduce. Use an interactive
geometry model (‘gcsemathstutor’ has one) to show it works for many
cases. Ask how this would help to find a missing side and model both for
finding the hypotenuse and for finding one of the non-hypotenuse sides.
Use this as revision of square and square root and finding both on a
calculator.
The ‘teachfind’ resources are a lesson plan and two interactive
spreadsheets (view at 100%) to find the next button and enable the
macro.
www.teachfind.com/nationalstrategies/notes-exploring-two-proofspythagoras-therom
www.teachfind.com/nationalstrategies/exploring-geometric-proofpythagoras-therom
www.teachfind.com/nationalstrategies/exploring-algebraic-proofpythagoras-theorem
www.gcsemathstutor.com/gg-ss-pythag01.php
www.mathsnet.net/dynamic/pythagoras/i
ndex.html
http://nrich.maths.org/1309
http://nrich.maths.org/6553
Find a bank of problems which require Pythagoras to solve them. If the
right angled triangle is not shown i.e. ask students to find the area of an
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Learning objectives
Suggested teaching activities
Learning resources
isosceles triangle given the lengths of all three sides. Get them
Students should draw the diagrams, identify the right angle(s) and then to
sort into two piles – finding the hypotenuse, finding a non-hypotenuse then
solve them.
Past Paper 13 June 2011 Q11
(syllabus 0580)
General guidance
Key skills for Trigonometry.
Identifying the sides of the triangle correctly.
Knowing the ratios.
Identifying which ratio to use.
Knowing whether to use the trig function or the inverse of the trig. function
and how these are related to button presses on a calculator.
Teaching activities
To develop trigonometry draw a right angled triangle that fills a page of
squared paper. Drop verticals inside the larger triangle between the
hypotenuse and the base to form a nest of similar right angled triangles.
Create a table with the base, heights and hypotenuse measured for each
of the six triangles. In a further three columns ask them to divide both the
adjacent and the opposite by the hypotenuse and the opposite by the
adjacent (you could do all six ratios possible if you want and there is time).
Discuss the fact that the ratios are almost identical going down a column
for the six triangles – you can go around the room and suggest to some
that you know that various answers/lengths need checking without telling
them how you know. Finally show students how to do a sine-1, cos-1 and
tan-1 on their calculators for the rough average value of each column (just
give them the button presses without telling them why) to discover the
same answer (approximately) . Finally measure the angle. After realising
the angle and the results from button pressing were the same discuss
what has happened and why by linking to similar triangle work if students
haven’t realised that that is why it works.
http://nrich.maths.org/5615
http://projects.exeter.ac.uk/csmsurvey/files/CSM10_Intro_to_trigonomet
ry.pdf
Past Paper 33 June 2011 Q6d
(syllabus 0580)
Past Paper 31 June 2011 Q10
(syllabus 0580)
Finally give students the three ratios as fractions.
Ask students to invent a Mnemonic to help them to remember the ratios.
e.g.
Silly Old Harry Caught A Herring Trawling Off America
Sine, opposite Hypotenuse.....
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Learning objectives
Suggested teaching activities
Learning resources
Ensure students understand that the angle has to be known to identify the
adjacent and the opposite. Give them a set of triangles in different
orientations with the right angle and one other angle identified. And ask
them to label the side opp, hyp adj or O, A H etc.
Teach one method for solving all problems.
1. Label triangle (O,A,H).
2. Identify the three facts (two given, one to find) on the diagram.
3. Decide which trig ratio it is because two sides are identified on the
diagram even if one is the ?.
4. Write down the statement in fraction form using the two given
facts with one unknown.
5. Rearrange if necessary to get the unknown on one side of
equation and the two knows on the other.
6. Decide whether to use the trig key or the inverse trig key on the
calculator.
7. Solve and round to 3 s.f.
Give students plenty of practice of a mixed bank of problems rather than
sets of sine, then sets of cosine etc.
It might be a good idea to ask students to sort a pile of problems into,
ones to find the angle, ones to find the hypotenuse and ones to find one of
the other sides at some stage. However, steps 1-7 are identical for all
problems.
Draw a 10cm circle on a coordinate grid, (centre the origin), marking off
10° angles from the origin to intersect with the circumference and noting
their coordinates, Plotting the x-coordinate divided by 10 against angle,
the y-coordinate divided by 10 against the angle, and the x-coordinate
divided by the y-coordinate against the angle either for the first quadrant
or for all 360° to give a different view of trigonometry. This isn’t essential
but gives breadth.
General guidance
Choosing the tool to solve the problem.
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Learning objectives
Suggested teaching activities
Learning resources
Students can mix up four types of questions, finding a side from an area of
triangle, trigonometry and Pythagoras and missing angle questions that
can be solved by other angle properties so give students experience of
identifying the question type.
Teaching activities
Print a mixture of questions, and ask students to sort them into the four
types before they try solving them. They may have to do a little work on
each problem to sort them and the discussion afterwards could be to
identify how they decided the type.
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