TRIANGLES
Assessment Criteria
AT
UAM
Level
5
Shape
5
Shape
6
Shape
Shape
Shape
6
6
6
Shape
6
Shape
7
Shape
8
Algebra
6
Algebra
7
Numbers
Calculating
5
7
Assessment Criteria
Identify and obtain necessary information to carry through a task
and solve mathematical problems
Use language associated with angle and know and use the angle
sum of a triangle and that of angles at a point
Deduce and use formulae for the area of a triangle and
parallelogram
Use straight edge and compasses to do standard constructions
Devise instructions for a computer to generate shapes and paths
Understand a proof that the sum of the angles in a triangle is 180
and of a quadrilateral is 360
Solve geometrical problems using properties of angles, of triangles
and of other polygons
Understand and apply Pythagoras’ theorem when solving problems
in 2D
Understand and use trigonometrical relationships in right-angled
triangles and use these to solve problems
Construct and solve linear equations with integer coefficients, using
an appropriate method
Use formulae from mathematics and other subjects; substitute
numbers into expressions
Round decimals to the nearest decimal place
Use a calculator efficiently and appropriately to perform complex
calculations with numbers of any size, knowing not to round
numbers during intermediate steps of a calculation
Cross-Curricular Links
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ICT (LOGO, Powerpoint, pamphlet, internet research, VLE); Performing Arts (script, presenting); History
(life of Pythagoras); DT (structure of pylons); RS (Star of David. Trinity), etc
Functional Skills
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Short task in Lesson Six for incorporation in pamphlet
Process Skills
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Conjecture and generalise.
Identify the necessary information to understand or simplify a context or problem.
Communicate own findings effectively orally and in writing.
Take account of feedback and learn from mistakes.
Discuss and compare approaches and results with others; recognise equivalent approaches.
Lesson One
LO:
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1) Understand and use the fact that the angle sum of a triangle is 180.
2) Calculate accurately, selecting mental methods or calculating devices as appropriate.
3) Conjecture and generalise.
Starter: Brainstorm in groups prior knowledge about angles and triangles [three straight sides, acute,
obtuse, right-angles, isosceles, equilateral, scalene, area, two right-angles make a straight line]. Define
and note keywords where required. After, reveal Learning Objectives and Keywords.
Main: Look at proof of angle sum using a paper triangle.
Main: Group based problem solving including basic algebra and probing questions:
an isosceles triangle has an angle of 32, what could the others be?
angles are x, x + 30, 2x – 10, what kind of triangle is it?;
what’s wrong with this list?
always, sometimes, never?
make up a question (based on a diagram) for which the answer is 52?
a triangle can have three acute angles, what other possibilities are there?, etc.
Plenary: Some groups present solutions to probing questions.
Lesson Two
LO:
1) Explain how to find, calculate and use the sums of the interior angles of polygons and the exterior
angles of polygons
2) Programme LOGO to draw a regular polygon.
3) Conjecture and generalise.
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Starter: Brainstorm and research names of other polygons. What’s the name of a 20 sided or a 1000
sided polygon? Where do the prefixes pent- and hex- come from, for example. What real life examples
are pupils aware of (e.g. the Pentagon, bees, etc). In groups, pupils combine triangles to make larger
polygons. Build on angle sum of triangle, looking at angle sum in quadrilateral, pentagon, etc. Use
specialisations to make generalisations. How can you calculate the angle sum in a 100-sided polygon?
How can you calculate each angle in a regular 100-sided polygon? How can you calculate each exterior
angle in a regular 100-sided polygon? Extend to splitting up an n-sided polygon into n triangles with a
shared vertex inside the polygon and how that relates to the sum of angles around a point.
Main: Teach basics of LOGO programming (fd, rt, repeat). Pupils write LOGO programmes to draw
polygons and suggestions are tried out on interactive whiteboard.
Plenary: CHICKEN FEED worksheet on exterior angles.
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Lesson Three
LO:
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1) Use formulae for the area of a triangle.
2) Round decimals to one or two decimal places.
3) Substitute positive integers into formulae.
Starter: Rounding Starter (ppt) (1 decimal place).
Main: Explain convention of labelling triangles (e.g. A, B and C for vertices opposite sides a, b and c).
Main: Consider different formulae for the area of a triangle – A = ½bh, A = ½abSinC, A =
s(s  a)(s  b)(s  c) [not expecting understanding of last two, but, hey, they’ve got their calculators
with them and they can press the sin and square root buttons rather than just multiplying). Given
triangles, select and apply appropriate formula. Consolidate rounding. Maybe make some observations
such as sin 90 = 1 and that sin 60 = sin 120. Emphasise that the base and height must be at rightangles to use A = ½bh. Could use A = ½abSinC to show, if not explain, why. Compare area of triangle
with area of rectangle. How does area of triangle allow you to calculate the area of a parallelogram or
even a trapezium?
Plenary: Discuss VLE and researching on the internet, leading to homework.
Homework: All groups given same right-angled triangle with a base of 15 cm and a height of 8 cm. Oddnumbered groups to research via VLE how to find the length of longest side, even-numbered groups to
research via VLE how to find the angle between 15 cm and the longest side (tan ratio).
Lesson Four
LO:
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1) Investigate and apply Pythagoras’ theorem.
2) Use tangent ratio in right-angled triangles.
3) Round decimals to one or two decimal places.
Starter: Pair up groups, 1 with 2, 3 with 4, etc. Odd-numbered groups teach even-numbered groups how
to find the length of longest side and even-numbered groups teach odd-numbered groups how to find
the angle between 15 cm and the longest side.
Main: Apply concepts to problems in double-sized groups. Individuals could pair up within groups to help
each other and peer assess. Might note that tan 45 = 1.
Plenary: Look at dissection method for proving Pythagoras’ theorem.
Lesson Five
LO:
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1) Investigate and apply Pythagoras’ theorem.
2) Use tangent ratio in right-angled triangles.
3) Round decimals to one or two decimal places.
Starter: Using provided clues (or as pre-prepared research), double-sized groups have to determine how
to find a shorter side in a right-angled triangle or a side in a right-angled triangle using the tan ratio.
Main: Apply concepts to problem solving in double-sized groups.
Plenary: Discussion – probing questions relating to how values change when angle is larger or smaller,
why does the calculator not give you a value for tan 90, how might you find the distance from one
corner of the room to another (perhaps even in 3D once 2D diagonal established). Could include some
Pythagorean triples.
Lesson Six
LO:
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1) Apply Pythagoras’ theorem, tangent ratio and area formula to functional task.
2) Solve simple problems involving ratio and proportion using informal strategies.
3) Know rough metric equivalents of imperial measures in common use.
Starter: Small projects given out randomly, one per group, to be self and peer assessed. Presentations
to be given in lessons nine and ten as well as a four A4 page pamphlet and an A3 poster in lesson eleven.
Projects are: (A) why are pylons made from triangles? (B) using the sin and cos ratios (as well as tan);
(C) triangle numbers (D) how can we construct a triangle given three sides and a compass? (E) how can
we find the area of an isosceles triangle; (F) find out about the life of Pythagoras and his cult; (G) what
is the religious significance of triangles; (H) the cosine rule. There will be starting points available on
the VLE. The four page pamphlet is to contain one page on their own specialist project, one page on
Pythagoras’ theorem (proof, worked examples of finding longest and a shorter side), one page on the
tan ratio (finding an angle or a side) and one page on the solution of a functional skills task. The poster
is to be on their own specialist project.
Main: Functional Skills task on area of fields using angles, area, Pythagoras, tan ratio, metric and
imperial conversions, ratio, basic money comparisons. It includes irrelevant information that needs to be
ignored.
Plenary: Each group has 30 seconds to present their solutions.
Lesson Seven
LO:
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1) Consolidate triangle facts.
2) Identify the necessary information to understand or simplify a context or problem.
Starter: Hand out Triangle Treasure Hunt sheets.
Main: Triangles Treasure Hunt – pupils travel around the school in groups looking at and answering
questions on triangles and other shapes.
Plenary: Puzzle for fast finishers. Discuss answers with class when all (most) have returned.
Lesson Eight
LO:
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1) Substitute positive integers into formulae.
2) Change the subject of a simple formula.
3) Communicate own findings effectively in writing.
Starter: Reverse Calculations – two methods for finding, say, the base of a triangle given its
perpendicular height and area: substitute and solve or rearrange the formula. Questions for groups.
Main: Preparation time for poster, pamphlet and presentation.
Plenary: Reverse Calculation Bingo
Lesson Nine
LO:
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1) Use straight edge and compasses to construct a triangle, given three sides.
2) Communicate own findings effectively, orally and in writing.
3) Take account of feedback and learn from mistakes.
Starter: Introduce the four presentations.
Main: Presentations A, B, C and D with time for questions after each one.
Plenary: Construction consolidation with pupils from presentation D circulating to help other groups.
Lesson Ten
LO:
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1) Use the cosine rule to solve 2D problems.
2) Communicate own findings effectively, orally and in writing.
3) Take account of feedback and learn from mistakes.
Starter: Introduce the four presentations.
Main: Presentations E, F, G and H with time for questions after each one.
Plenary: Cosine Rule consolidation with pupils from presentation H circulating to help other groups.
Allows practice of BODMAS.
Lesson Eleven
LO:
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1) Use the cosine rule to solve 2D problems.
2) Carry out calculations with more than one step, including using brackets and the square root key.
3) Identify the necessary information to understand or simplify a context or problem.
Starter: Two minuses make a plus (ppt)
Main: Consolidation on Cosine Rule. Opportunities for practising rounding. Note that cosine of an obtuse
angle is negative and hence two minuses make a plus. How does cosine rule relate to Pythagoras’
theorem? In what circumstances are they the same? What does that tell you about cos 90? Opportunites
for probing questions on the nature of BODMAS, substitution, squaring/square-rooting and negative
numbers, if not the cosine rule itself.
Plenary: Discuss findings.
Lesson Twelve
LO:
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1) Use formula for the area of a triangle.
2) Use tangent ratio in right-angled triangles.
3) Discuss and compare approaches and results with others; recognise equivalent approaches.
Fencing Problem (Calculate the area of a regular pentagon with a perimeter of 1000 m) – in groups, can
be guided. Extend to other regular polygons.