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6
Angles, triangles
and polygons
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…
Sample lesson
plans
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 F3.2c …understand the consequent
and other
properties of parallelograms and a
polygons
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.
Sample lesson plans 3
6.1 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 141–43
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
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
38°
78° 63°
b
127°
112°
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.
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
Corresponding angles
add up to 180˚
Co-interior angles
add up to 360˚
Co-interior angles add
up to 180˚
A transversal creates
pairs of equal angles
Co-interior angles lie on
the outside of a pair of
parallel lines
Vertically opposite
angles are equal
Corresponding angles
are equal
Co-interior angles are
also known as allied
angles
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
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 143–5
Links to the Teaching and Learning Software:
Three-figure bearings core activity
Links to the Practice Book:
Exercise 6iii
Links to other subjects:
Geography
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
F
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:
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
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 070° + 180° 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