Download Maths SoW - Thinking Skills @ Townley

TRIANGLES
Assessment Criteria
AT
UAM
Level
5
Shape
5
Shape
6
Shape
Shape
Shape
6
6
6
Shape
6
Shape
7
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
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
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.
Angles in Triangles
LO:
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.
4) Explain how to find, calculate and use the sums of the interior angles of polygons and the exterior
angles of polygons
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Brainstorm 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. What if a right angle was 100o?(The what if key) Have a picture of an orchestra,
Ask how it relates to the topic (triangles- the picture key).
Discuss notation (e.g. given three vertices A, B and C, how could we denote the sides (i.e. a or BC)).
Proof of angle sum using a paper triangle. Try to find similarities between a triangle and a circle
(The commonality key)
Geoboards – make isosceles triangle, right-angled triangle, scalene triangle (can make equilateral
triangle on reverse); how many different isosceles triangles can you make on a three by three dot grid?
Worksheet on Angle sum (ANGLE SUM CODE), answer TOWNLEY.
Probing questions such as ‘if one of the angles of an isosceles triangle is 28, what possible values could
one of the other angles take?’ Write an angle on the board. If this is the answer what is the
question? (The question key) All shapes have angles is this true (The brick wall key)
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Plotting coordinates of points in four quadrants (mymaths > Shape > Coordinates > Coordinates 2 (for
four quadrants and recap on one quadrant).
Measuring Angles (mymaths > Shape > Angles > Measuring Angles).
Worksheet on Plotting Coordinates, Measuring Angles and Identifying Triangles (COORDINATES): first
question is two triangles to measure angles, second question is on special types of triangle and
identifying them, third question draws a robot from triangles. Could extend by: asking pupils to state
the coordinates of a triangle with a 45 angle, getting pupils to design their own pattern with
instructions, looking at relationship between the slopes of the perpendicular sides of the right-angled
triangles, looking at why equilateral triangles can’t be drawn on a coordinate grid. A man is found dead
in a locked room with a protractor wrapped around his head. How did it happen? (The
interpretation Key)
Algebra Cards – find value of each card when x = 30, which ones are the same value when x = 20, what
does 2x mean, put in order from smallest to largest when x = 12 (or which one would be in the middle),
which three could be the angles in a triangle when x = 38.
Non-algebraic linking activity such as ‘Call each of the equal angles in an isosceles triangle A and B and
the other angle C. What are the sizes of the angles if A is twice the size of C?’ etc. How many ways can
you work out the third angle in a triangle? (The variations key)
Solving equations based upon angles in triangles. Last activity leads into triangles with angles of (2x, 2x,
x) and (x, x, x + 30). Develop to problems involving equation solving such as angles of (3x + 17), (7x – 5)
and (5x + 18).
Worksheet on Solving Equations involving Angles (ALGEBRAIC ANGLES): solve equations then identify
what type of triangle each one is (e.g. isosceles, equilateral, right-angled, scalene).
Brainstorm and research names of other polygons (Brainstroming key). 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).
Handout (POLYGON GUIDESHEET), 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 100sided 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.
Worksheet on Angles in Polygons (POLYGON QUESTIONS) Make a statement that pupils have to
comment on ie A circle has one side (The ridiculous key)
Worksheet (CHICKEN FEED) on exterior and interior angles.
LOGO
LO:
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1) Programme LOGO to draw a regular polygon.
Teach basics of LOGO programming (fd, rt, repeat). Pupils write LOGO programmes to draw polygons
and suggestions are tried out on interactive whiteboard. How can we make our polygons larger, more
sides, can we change the instructions around? (The BAR key)
Logo cannot make a perfect heptagon. Why? What other shapes can’t it draw? (The interpretation
key)
Worksheet available on repeating (mswlogo_polygons).
Area of a Triangle
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.
4) Change the subject of a simple formula.
Rounding Starter (ppt) (1 decimal place).
Geoboards (pinboard triangles ppt), allowing counting of squares or conjecturing to find formula. How
many different types of triangle can you make? How many triangles can you make with an area of
6cm2? (Variations key)
A = ½bh. Encourage to perform in simplest order (e.g. if base = 9 and height = 6, don’t halve the 9).
Consolidate rounding. Emphasise that the base and height must be at right-angles to use A = ½bh.
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? Give triangle with the base and sloping height can you
find the area? (the alternative key)
Worksheets (FIND THE BASE and ODD ONE OUT).
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.
Plenary: Reverse Calculation Bingo What if you have a rectangle with an area of 10cm2. What
different triangles could I have? (The what if key)
Preparation for Pythagoras: 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.
Research via VLE how to find the length of longest side (hypotenuse).
Pythagoras’ Theorem
LO:
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1) Investigate and apply Pythagoras’ theorem.
2) Round decimals to one or two decimal places.
You are given a triangle with shorter side lengths 8 and 15. The length of the longest side is 17.
How can you get this? (The combinations key)
Choose a couple of groups to explain on board how to find the length of longest side of the triangle with
a base of 15 cm and a height of 8 cm.
Look at dissection method for proving Pythagoras’ theorem.
Worksheet on finding hypotenuse (Beans).
Use method for finding hypotenuse to consider how to find a shorter side given the other shorter side
and the hypotenuse. Write a list of instructions on how to find the shorter side of a right angle
triangle (BAR key).
Activity on Pythagoras cards
Worksheets (Mixed Pythagoras Bathroom Challenge and Crossword).
Group problems on area and perimeter and length of diagonal of a rectangle.
I have a cylinder shaped pencil case that has a diameter of 9.5cm and a length of 12cm. When my
pencil case is empty will my 15cm ruler fit in? (The question key)
Group Projects
LO:
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1) Communicate own findings effectively, orally and in writing.
2) Take account of feedback and learn from mistakes.
Small projects given out randomly, one per group, to be self and peer assessed.
why pylons are made from triangles; Why don’t we have triangular mirrors? (The disadvantages
key);
triangle numbers (include why they are useful);
Pascal’s triangle (include why it is useful);
constructing a triangle with a ruler and a compass;
life of Pythagoras;
triangles in spirituality and religion (find two reasons why triangles are used);
area of an isosceles triangle;
Babylonians and their base system (360 and approx 360 days in a year, 60 minutes in an hour).
Pupils to deliver presentations and produce an A3 poster, both on their own specialist project.
There will be starting points available on the VLE.
Constructions
LO:
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1) Use straight edge and compasses to construct a triangle, given three sides.
2) Use straight edge and protractor to construct a triangle, given two sides and angle between.
Following presentation on using a compass, consolidation with pupils from presentation circulating to
help other groups.
Worksheet (construction with a compass)
Find 10 different uses for a protractor. (The different uses key)
Invent a piece of equipmet that can measure angles as well as draw circles. (The inventions key)
Construction with a protractor (ppt) You need to draw a triangle with a bar of soap, a cat and a
bowl of cereal (The forced relationships key)
Video on each type of construction available.
Investigating Triangles
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Triangles Test
What type of angle do you think we will use in 2000 years? (had angles then came radians, The
prediction key)
What is wrong with this list? (ppt)
Worksheet (INVESTIGATING TRIANGLES).
Always, Sometimes or Never activity (ppt) State an angle that can never be the exterior angle of a
regular polygon (The reverse listing key).
How many different triangle related words can you come up with? (The alphabet key)