AO elaboration and other teaching resources
identify and construct right, acute and obtuse angles
begin to appreciate the degree, unit of measurement of angle
know the degree value of angles that are simple fractions of a whole turn
know that the angle at a point is 360°
Angle can be seen as and thought of in at least three ways.These are as:
- the spread between two rays
- the corner of a 2-dimensional figure
- an amount of turning
The final one of these underpins the others and leads on naturally to the definition of degree and the ability to measure angles with a standard unit. This leads students on to being able to apply their knowledge of angle in a variety of situations.
We see angle as developing over the following progression:
Level 1: quarter and half turns as angles
Level 2: quarter and half turns in either a clockwise or anti-clockwise direction
angle as an amount of turning
Level 3: sharp (acute) angles and blunt (obtuse) angles
degrees applied to simple angles – 90°, 180°, 360°, 45°, 30°, 60°
Level 4: degrees applied to all acute angles
degrees applied to all angles
angles applied in simple practical situations
Level 5: angles applied in more complex practical situations
The concept of angle is something that we see students developing gradually over several years. As their concept matures, they will be able to apply it in a range of situations including giving instructions for directions and finding heights. In the secondary school angle is used extensively in trigonometry (sine, cosine, tangent, etc. ) to measure unknown or inaccessible distances. This deals with situations where only right-angled triangles are present in 2-dimensional situations through to more complicated triangles in 3-dimensional applications.
Surprisingly these trigonometric functions are used in abstract settings too. At Level 8 and above they are used extensively in the calculus as means to integrate certain functions.
Outside school and university, angle is something that is used regularly by surveyors and engineers both as an immediate practical tool and as a means to solve mathematics that arises from practical situations. So angle is important in many applications in the ‘real’ world as well as an ‘abstract’ tool. This all means that angles have a fundamental role to play in mathematics and its application.
- Take a rectangle and discuss the size of the corner angles as being equivalent to quarter turns. Some suggested questions are:
What properties does a rectangle have?
What can you tell me about the angles of a rectangle?
- Now look at the side of a rectangle.
If we were facing this way along the side of a rectangle, what would we have to do to be facing in the opposite direction? (Make a half turn. )
- What do you remember about units?
What can you tell me about centimetres, kilometres, kilograms?
Show me 180° on this rectangle.
Can you show me 180° anywhere else in the room?
Can you turn through 180° clockwise and anticlockwise?
Make sure that they can see that straight lines are 180° and that that is the same as a half turn in either direction.
Put a straight line has 180° and a half turn has 180°, on a chart.
- Show the class the triangle below.
How many angles does a triangle have?
How big do you think the angles are? (Bigger or smaller than 180°?)
- Tear off the angles and show that they can be put together to make a straight line or 180°.
- Does this work for all triangles or was my triangle special in some way
Do all three angles of every triangle add up to 180°?
- Set the students the task of testing out that idea. Let them
- go back to their groups;
- draw a number of triangles;
- cut them out;
- tear off the three angles; and
- stick them together to see if the angles make up 180°.
- When they have each looked at at least three triangles get them back together to see what results they have. They should all have found that the three angles of any triangle add up to 180°.
- Discuss the chart (from Copymaster 1). First concentrate on the fact that the angles add up 180°. Then talk about the angles being sharp (acute angles) and blunt (obtuse angles).
- Put the chart on the wall.
- Recall the important aspects from the last lesson:
- the angle size of a straight line is 180°;
- the sum of the sizes of the angles in a triangle is 180°;
- a triangle can have acute and obtuse angles.
- Take a rectangle and cut it in half to make two smaller rectangles (see diagram below – cut along the dotted line). Put the two pieces close to each other. Point out that you now have two angles (marked in the diagram) that add up to 180°.
Why? (They are both equal and add to 180°. A half of 180 is 90. )
- Can you show me any more angles here that are equal to 90°?
Get the students to point to the four original angles plus the four new angles.
Why do you think that these angles are equal to 90°? (You can put them on top of the ones above and they are the same. )
- We call angles that measure 90°, right angles.
Can you tell me anything that you know that has right angles? (Various geometric shapes plus objects such as doors and windows. )
- Let the students go into their groups to produce at least 10 things, other than those discussed as a class, that have right angles.
- Compile a list from the students’ suggestions and put it on a chart.
- Now we want you to think about acute angles.
Can someone draw me an acute angle? (Have several acute angles to think about. )
How many degrees does an acute angle have? (This can’t be done accurately. The best that can be said is that an acute angle is less than 90°. )
How many degrees does an obtuse angle have? (Again this can’t be done accurately. The best that can be said is that an acute angle is bigger than 90°. )
- Take a rectangle and cut it along a diagonal.
What can you tell me about the two triangles that we have produced? (They have a right angle; the other angles are acute angles. )
- Get the class to work in their groups to produce a variety of right-angled triangles. Encourage them to make the triangles as varied as possible (skinny and fat, and tall and short). Ask them to think about the following questions and have an answer ready:
Do all your right-angled triangles have right angles?
Do they all have acute angles?
Do they all have obtuse angles?
What can you say about the sum of the two angles in a right-angled triangle that are not right-angles?
What can you do with two right-angled triangles that are exactly the same?
- As a class look at the variety of right-angled triangles that have been produced.
Then discuss what can you say about all off these triangles?
Do they all have right angles? (Of course. )
Do they all have acute angles? (Yes. )
Do they all have obtuse angles? Why? (This may cause some discussion. Let them put their ideas on the table and be prepared to defend them. Actually none of them do. Why? Because if there was one angle bigger than 90°, the sum of that angle and the right angle would be bigger than 180°. But we know that the sum of the angles of a triangle can only add up to 180°. )
So what can you say about the sum of the two angles in a right-angled triangle that are not right-angles? (Their sum is 90°. )
What can you do with two right-angled triangles that are exactly the same? (Put them together to form a rectangle. )
- Remember how we got a right angle?
(Recall how we cut a 180° angle in half.)
What sized angle would we get if we cut a 90° angle in half? How many degrees does it have? Why? (45 is a half of 90.)
Roughly what would it look like do you think?
Show me using as many ways as you can.
(This might be done by an half of a quarter turn, by folding a right angle in half, by holding two pencils together and opening them out to 1/8 of a turn, and so on. Check that their suggestions are about the right size and that they could find the answer fairly accurately.
- Give them a rectangular piece of paper and, in their groups, ask them to produce a 45° angle as accurately as they can. You will almost certainly need to give them help here.
There is more than one way to proceed. First, they might make a right angle in the same way that was done above. This is the most likely method. They then might try to cut the right angle accurately in half. If they do it this way, they will probably not quite manage to get an accurate 45°. Get them to check their accuracy by putting the two supposed 45° angles they have produced, on top of each other. (The accuracy of this method might be improved by doing it several times.)
Second, they might start with one of the right angles of the rectangle and fold it in half. If they then cut down the fold they will get a good approximation to the 45° that they want.
Third, they might first try to make a square. This can be done reasonably easily but many of them will require help to do that. Take the rectangle and fold down the shorter edge AB onto the longer edge AD. Make a cut up from B’ parallel to CD gives a square (along the dotted line). Using this square a 45° can be constructed by cutting from one corner to the other. (It might be a good idea to fold the square in half first. )
A likely wrong way that many of them might use is to just cut a diagonal AC from one corner to the other. This can be shown not to work as the two angles produced at A are not the same size.
- Get the class together and discuss the various ways that they have produced their 45° angles. Compare what they have produced by putting different angles on top of the other.
- Now give them the task of making a 60° angle using their piece of paper.
One way to do this is to use two folds along the side of a rectangle. When the three pieces fit each other exactly a 60° angle has been formed. This method has a self-check as the three angles that sit on top of each must be equal and 180/3 = 60. The difficulty here is to get the folds as equal as possible.
Another second way is to produce an equilateral triangle. This is a triangle in which all angles are equal. So each angle must be 180°/3 = 60°. Draw a straight line AB (see the diagram) and use the length of AB as the radius of a circle by setting the compass point at A and the pencil at B. Then draw an arc from A and an arc form B so that they meet at C. ABC is an equilateral triangle and all of its angles are 60°.
First construct a circle and mark any point on its circumference. Then, using the same radius, put the point of the compass on that point and mark another point on the circumference. Repeat until six points have been marked. These six points are the vertices of a regular hexagon.
- Encourage the class to use the various different ways to make a 60° angle.
- Get the class together and discuss the ways that they have produced their 60° angles.
Which do they think is the best?
Which did they enjoy doing most?
Compare the accuracy of what they have produced by putting the different angles on top of the other.
- Summarise what has happened today. They now know about 45° and 60° angles and how to produce them. Use their comments to make a chart that outlines the key ideas of the lesson.
- By questioning, recall the angle sum of a triangle. Emphasise that the sum of the angles of every triangle is 180°.
What is the angle sum of a square?
What is the angle sum of a rectangle?
- Show the angle sum is 360° by (i) adding four lots of 90°; and (ii) tearing off the angles of a rectangle and putting them together. In (ii) they make two lines and so add up to 2 x 180° = 360°. The important thing to note here is that they all fit together round a ‘point’. So the angle at a point is 360°. This could be added to the wall chart.
- Think about the fact that the angle sum of any rectangle is 360°. Ask:
What might this suggest? (That every four-sided figure has an angle sum of 360°.)
How would we test this? (By drawing some quadrilaterals and checking to see that they add together at a point.)
- Let the groups draw whatever quadrilaterals they like and check their angle sums.
- Get the groups to report back on what they have found. Do all the angles fit together round a point? Do they all add up to 360°?
- There is actually an easy way to see that the angle sum of a quadrilateral is 360°.
Can you repeat the argument we have just used?
Does it matter whether we use the diagonal above or a different diagonal?
- OK, so what is the case for pentagons?
Get them to investigate the angle sum of pentagons by either of the methods that have been used so far. When they report back look to see if they can justify their answers. Again the triangle approach will work. By dividing the pentagon into three triangles we see that the angle sum of any pentagon is 3 x 180° = 540°. The ‘tearing off of angles’ approach may be harder as it is not easy to see what 540° means.
- What is the next question we should investigate?
Get them to work on the angle sums of hexagons and report back. This time their answer should be 4 x 180° = 720°.
- Put the results that we have so far together in a table. Can they see the general pattern. What is the angle sum of any polygon?
- The angle sum of a polygon is 180° x (2 less than the number of sides). Add this to a wall chart.
- Recall the major things that have been done in the unit so far. This can be done by referring to the wall charts that have been made as summaries of the various pieces of work.
- Set them the task of making triangles with different properties, see Copymaster 2. They will need to use the construction methods that they have used earlier to make angles of size 45° and 60°. (Note that they can make the 45°, 45°, 90° triangle by cutting a square in half; we have already shown how to make a 60°, 60°, 60° triangle; the quadrilaterals can be made by combining these triangles in some way; the pentagons and hexagons can be made using a squares or rectangles and one of the triangles above).
- Ask the class to report back with their findings.