Lesson Observation - Townley
TRIANGLES
Assessment Criteria
|AT |Level |Assessment Criteria |
|UAM |5 |Identify and obtain necessary information to carry through a task and solve mathematical |
| | |problems |
|Shape |5 |Use language associated with angle and know and use the angle sum of a triangle and that of |
| | |angles at a point |
|Shape |6 |Deduce and use formulae for the area of a triangle and parallelogram |
|Shape |6 |Use straight edge and compasses to do standard constructions |
|Shape |6 |Devise instructions for a computer to generate shapes and paths |
|Shape |6 |Understand a proof that the sum of the angles in a triangle is 180( and of a quadrilateral is |
| | |360( |
|Shape |6 |Solve geometrical problems using properties of angles, of triangles and of other polygons |
|Shape |7 |Understand and apply Pythagoras’ theorem when solving problems in 2D |
|Algebra |6 |Construct and solve linear equations with integer coefficients, using an appropriate method |
|Algebra |7 |Use formulae from mathematics and other subjects; substitute numbers into expressions |
|Numbers |5 |Round decimals to the nearest decimal place |
|Calculating |7 |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
• 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
• 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)
• 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 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.
• 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: 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: 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: 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: 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.
o why pylons are made from triangles; Why don’t we have triangular mirrors? (The disadvantages key);
o triangle numbers (include why they are useful);
o Pascal’s triangle (include why it is useful);
o constructing a triangle with a ruler and a compass;
o life of Pythagoras;
o triangles in spirituality and religion (find two reasons why triangles are used);
o area of an isosceles triangle;
o 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: 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
• 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)
Triangles Test
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