In this unit we develop the concept of angle to see that an angle may be constructed in a clockwise or anticlockwise direction. We see the effect of clockwise and anticlockwise turns on objects. We also think about corners of objects that are equivalent to quarter turns. We also think about whether a corner can be a half turn. These ideas are explored physically.
- Understand clockwise and anticlockwise directions.
- Understand that quarter half turns may be begun from any direction and not just from lines parallel to the classroom walls or other fixed lines.
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 right 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.
- Copymaster 1.
- Art equipment.
- Recall the ideas of quarter and half turns. (This is a revision of the work in Turns, Level 1.) This can be shown by a student or a toy turning. Think about objects that do or do not have corners that are equivalent to quarter and half turns.
- Ask three volunteers to stand at the front of the class and point their arm in some direction. Ask them to make a quarter turn without looking at the other two students. (If appropriate, you might like to heighten the dramatic impact of this by blindfolding the students.) With any luck some students will make their turns clockwise and the others anticlockwise. (If this does not happen automatically you may need to ask them if they can make a quarter turn another way.)
- Discuss the difference between the two types of quarter turns. Ask the three students to make a quarter turn. Mark where the students are pointing.
Can they reach the same point if they used the other kind of quarter turn? (Yes.)
How many quarter turns of one type will be the same as a quarter turn of the other type? (Three.)
- What happens with half turns? Is a half turn of one type equal to three half turns of the other type?
Ask them to discuss this in pairs and report back to the whole class. Children should be able to explain why the two half turns are the same.
- Ask them to consider contexts when it might be helpful to have names for the direction of our turns. Can they imagine times when it would be really important for people to know whether they needed to turn this way or that way? If no one offers the labels of clockwise and anticlockwise, then explain to them that we call the turn that goes in the direction of that the clock usually goes, the clockwise direction. The other direction is the anticlockwise direction. It might be useful to write this on a chart on the wall.
- Get the students to investigate on their own or in pairs, the link between clockwise quarter turns and anticlockwise quarter turns. They may want to use toy clocks or a drawing of a clock face. Ask:
How many clockwise quarter turns is the same as one anticlockwise quarter turn?
How many anticlockwise quarter turns is the same as one clockwise quarter turn?
- Gather the students together to discuss their results. Write the results on the chart.
- Draw a rectangle in the playground (or use a small rectangle in class). Have four students stand on the corners of the rectangle (or put four toys on the small rectangle).
- Have Mike look at Nell. What turn would Mike need to make in order to be looking at Jorge?
Have Jorge look at Karen. What turn would Jorge need to make in order to be looking at Mike?
Have Karen look at Jorge. What turn would Karen need to make in order to be looking at Jorge?
- Have Mike look at Nell.
Who will Mike be looking at if he makes a quarter turn clockwise?
Who will Mike be looking at if he makes a quarter turn anticlockwise?
Have Jorge look at Karen.
Who will Jorge be looking at if he makes a quarter turn clockwise?
Who will Jorge be looking at if he makes a quarter turn anticlockwise?
Have Karen look at Jorge.
In what two ways can Karen turn to look at Nell?
Have Jorge look at Karen.
In what two ways can Jorge turn to look at Mike?
- Ask the students to go off in small groups of four. Ask them to stand in a square and make up their own questions about clockwise and anticlockwise turns.
- Have the students return to the mat. Allow different groups to ask the rest of the class the questions that they had made up.
- Summarise the results of the discussions.
- It’s not just people that we can turn in clockwise or anticlockwise directions.
Look at the long thin rectangle in this diagram.
- Ask the students to close their eyes. Then ask a series of questions like the ones below. After each question let them open their eyes and tell you what they think will happen.
What happens to the rectangle if we turn it through a clockwise quarter turn about A?
What happens to the rectangle if we turn it through a clockwise half turn about A?
What happens to the rectangle if we turn it through an anticlockwise quarter turn about B?
What happens to the rectangle if we turn the rectangle through an anticlockwise half turn about C?
- Give the students Copymaster 1 and ask them to draw the result of some turns on the rectangles on that sheet. To produce the turns, roll a dice twice. The first roll will give you clockwise, if the number is even and anticlockwise otherwise. The second roll will give you half turn, if the number is even and quarter turn otherwise. Do this activity for each of the six shapes in Copymaster 1.
- Ask the class to show their answers after every ‘turn’.
In this session get the class to make up a piece of art using different rotations of one rectangle. They should colour in the rectangles they produce so that:
- the ones that are clockwise quarter turn rotations should be coloured red;
- the ones that are anticlockwise quarter turn rotations should be coloured blue;
- the ones that are clockwise half turn rotations should be coloured green; and
- the ones that are anticlockwise half turn rotations should be coloured yellow.
- Get the class to talk about quarter and half turns. Use questions such as
What kinds of turns have we been talking about this week?
How would you describe a clockwise quarter turn?
An anti clockwise half turn? Etc.
What happens to a rectangle when you give it an anticlockwise quarter turn about a corner?
What happens to a rectangle when you give it a clockwise half turn about a corner?
What is an anticlockwise quarter turn equal to? (Three clockwise ones.)
- Play ‘Simon says’ using clockwise and anticlockwise quarter and half turns.