Angles, triangles and polygons
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1 6 Angles, triangles and polygons Sample lesson plans Chapter overview Background This is the first shape chapter and starts by re-visiting the work on angles and angle properties students will have met at KS3. It goes on to the angles associated with parallel lines and then three-figure bearings are introduced with students asked to measure and draw bearings. The next sections look at triangles and their angles. Again a revision of work covered at KS3 is done before a more indepth look at quadrilaterals and their properties. This is then developed into work on polygons, looking at the properties of regular polygons and extending to interior and exterior angles of any polygon. Line and rotational symmetry conclude the chapter. Assumed knowledge This chapter assumes very limited prior knowledge. Main teaching objectives Student Book section Teaching objectives 6.1 Angles F3.2a recall and use properties of angles at a point, angles on a straight line; perpendicular lines and opposite angles at a vertex F3.2b distinguish between acute, obtuse, reflex and right angle; estimate the size of an angle in degrees F3.2c...use parallel lines, alternate angles and corresponding angles F3.4b understand angle measure using the associated language F3.4d measure and draw lines to the nearest millimetre, and angles to the nearest degree; draw triangles and other 2-D shapes using a ruler and protractor, given information about their side lengths and angles
2 2 Teachers support material 6.2 Three-figure bearings F3.4d measure and draw angles to the nearest degree 6.3 Triangles F3.2c use parallel lines, alternate angles and corresponding angles understand a proof that the exterior angle of a triangle is equal to the sum of the interior angles at the other two vertices F3.2d use angle properties of equilateral, isosceles and right-angled triangles 6.4 Quadrilaterals and other polygons F3.2c understand the consequent properties of parallelograms and a proof that the angle sum of a triangle is 180 degrees F3.2d explain why the angle sum of a quadrilateral is 360 degrees F3.2f recall the essential properties and definitions of special types of quadrilateral, including square, rectangle, parallelogram, trapezium and rhombus; classify quadrilaterals by their geometric properties F3.2g calculate and use the sums of the interior and exterior angles of quadrilaterals, pentagons, hexagons; calculate and use the angles of regular polygons 6.5 Symmetry F3.3b recognise and visualise rotations, reflections and translations, including reflection symmetry of 2-D and 3-D shapes, and rotation symmetry of 2-D shapes Common difficulties Using a protractor correctly.
3 Sample lesson plans Angles in parallel lines Introduction The aim of this lesson is to introduce angles in parallel lines and the associated language. Key teaching points: Angle properties associated with parallel lines Key words: Vertically opposite Perpendicular Parallel Transversal Corresponding Alternate Co-interior Prior knowledge: Properties of angles on straight lines and round a point. Links Links to the specification: F3.2c use parallel lines, alternate angles and corresponding angles F3.4b understand angle measure using the associated language Links to the Student Book: Pages Links to the Teaching and Learning Software: Missing angles starter (parallel lines) Angles in parallel lines core activity Alternate and corresponding angles tool Links to the Practice Book: Exercise 6ii
4 4 Teachers support material Teaching activity Starter: Teaching and Learning software starter: Missing angles starter (parallel lines) Alternative starter activity: Revision of previous lesson by giving students several diagrams to find the missing angles making sure they have reasons for their answers. For example: a b d c y y 140 Main activity: Discuss what parallel lines are and what angles are produced when a transversal is drawn. Introduce corresponding and alternate angles (F and Z angles are no longer acceptable answers at GCSE). Co-interior angles completes the work on angles in parallel lines. It is important that students understand they need to give their reasons in the setting out of answers. Students will need to work through several examples of solving missing angles to understand how this is done. The Angles in parallel lines activity on the Teaching and Learning Software can be used to help explain the concept. It is worth asking students to work through a question where there are two sets of parallel lines and asking different students to show alternative ways of getting the answer. There is often not one single solution to geometry problems. The Angles in parallel lines activity on the Teaching and Learning Software can also be used here to generate practice examples.
5 Sample lesson plans 5 Variation/extension: A true or false game relating to the facts they have covered in the lesson is a good way to reinforce the teaching points. Photocopy the following grid and ask students to identify which statements are true and which are false. Alternate angles are equal Co-interior angles add up to 360 A transversal creates pairs of equal angles Vertically opposite angles are equal Co-interior angles are also known as allied angles Corresponding angles add up to 180 Co-interior angles add up to 180 Co-interior angles lie on the outside of a pair of parallel lines Corresponding angles are equal Angles on a straight line add up to 360 Plenary: Using student interactive whiteboards, angles can be shown and students write down their type, or the type can be given and students have to draw them. Alternatively, an example can be put on the board and students asked to find different ways of finding the solution, giving reasons for their answers. Homework and consolidation Homework: Practice Book exercise 6ii Follow-up work: All angle facts form the basis for more complex geometry and can be found in sections 6.2, 6.3 and 22.4.
6 6 Teachers support material 6.2 Three-figure bearings Introduction This lesson covers the introduction to bearings. Students learn how to measure and draw bearings and a simple example of a back bearing is shown. Students will need a protractor. A 360 protractor makes this much easier, but only use if they will be available for examinations. Key teaching points: How to use bearings to describe directions Key words: Bearing Prior knowledge: Students should already know how to measure angles. Links Links to the specification: F3.4d measure and draw angles to the nearest degree Links to the Student Book: Pages Links to the Teaching and Learning Software: Three-figure bearings core activity Links to the Practice Book: Exercise 6iii Links to other subjects: Geography
7 Sample lesson plans 7 Teaching activity Starter: Ask a volunteer to draw an equilateral triangle on the board and label its vertices A, B and C. Ask a second volunteer to draw a second equilateral triangle over the first to form a star, labelling its vertices D, E and F. This needs to produce 6 small equilateral triangles and a regular hexagon. C D E A B Students work in pairs or small groups to work out all the angles shown on the diagram. What is the exterior angle at C? Can they find the total for the angles inside the hexagon? etc. (You may need to remind students that as the triangles are equilateral, each angle is 60.) Main activity: F Discuss with students how they would direct a stranger to find a particular student in the playground. Students might use a clock, say at 2 o clock, but they need to know where 12 o clock is. Move on to how ships know where to steer across oceans and planes to fly from one country to another use of compass and North as the starting point. Move on to increasing accuracy from the points of the compass to bearings using 360 and explain that to avoid confusion they are always written with 3 digits e.g. 90 = 090 as a bearing. Get student(s) to stand, designate a North point in the room and turn 090. This should emphasise the need to have a direction bearings always go clockwise from North. Students benefit from estimating bearings for themselves and using a student to face North and then ask students to work out the bearing from him to another student/the door etc, gives quick and easy practice at this before moving on to the drawing and measuring of bearings on paper.
8 8 Teachers support material To introduce back bearings, ask a student to walk on a bearing of 070 and stop. Discuss what they need to do to go back to their original position. How much do they turn 180? They were on turn = 250. An alternative method using parallel lines could also be taught as shown in Example 5 on page 144 of the Foundation Student Book. The animations in the Three-figure bearings activity in the Teaching and Learning software can be used to reinforce the teaching points described above. Show students a diagram, perhaps of 3 towns, and find the bearings from one to the other to complete the work on bearings. It is important to explain to students that they must draw in a North line if one isn t there, as all bearings must be measured from a North line. Variation/extension: The Three-figure bearings activity in the Teaching and Learning Software can also be used to give students practice at working out bearings and back bearings from one point to another in different situations. To further engage students interest you can replace the map provided in the activity with a map of your local area and ask students to work out the bearings from the school to their house and back. Plenary: What three things must you remember about bearings? Identify North and then ask students to estimate the bearings to four or five things/people in the room, from the teacher. Homework and consolidation Homework: Practice Book exercise 6iii Follow-up work: Section 17.6
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