Turns
In this unit we look at the beginning of the concept of angle.Here we are interested in students understanding quarter and half turns and to begin to see that ‘angle’ is something involving ‘an amount of turn’. These ideas are explored by using students’ bodies, toys, games and art.
show a quarter turn and a half turn in a number of situations
see that to quarter turns equals on half turn
recognise the ‘corner’ of a shape that is equivalent to a quarter turn
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.
Getting started
- Talk with the class about ‘turning’. This can be motivated by asking them directions from their classroom to somewhere else in the school. Emphasise ‘turning’ by asking them what they do when they get to a corner. Talk with them about what happens when they get to a T-junction or a cross road near the school. Ask them what they have to do if they want to go left or right. (They make a turn.)
- Tell them what a quarter turn is. Have a student come to the front and get them to put an arm straight in front of them. Turn the student through a quarter turn. Talk about what has happened to the direction of the student's’s arm.
- Repeat the demonstration with a toy. Using a toy animal, for instance, a student could show how to move the animal through a quarter turn.
- Give the student time to go and draw several examples of quarter turns. This may be done by using animal pictures, car pictures, or any other object. Emphasise that their drawings are not to be done in any great detail. It’s the idea of a quarter turn that is important.
- As you go around the class observing their drawings, check that they have the right concept and correct any misconceptions.
- Either in the same session or a later one, repeat the above with the idea of ‘half turn’. This can be introduced by thinking about times when they may have forgotten something on the way to school. They would have had to turn round and go back. This means they would have had to do a half turn. Activities similar to the ones above can then be used.
- Investigate the relation between quarter and half turns. Get them to see that two quarter turns is the same as one half turn.
Exploring
In the sessions that follow, the student produce artwork that they can assemble in their own ‘turns’ book. The quarter and half turn drawings that they have already done can be the first pages of this book. Some of these things can be done in conjunction with their normal artwork.
Session 1
Provide each student with a piece of string attached to a paintbrush. Show them how to fix one end of the string by using a drawing pin or the finger of one hand. Then show how they can make a quarter turn paint arc by sweeping the paintbrush through a quarter turn. Let them make ‘quarter paint turns’ in various colours. Check that their turns are approximately correct.
Having done quarter turns they can move on to half turns. Get them to see the link between quarter and half turns.
Save their work for their ‘turns’ book.
Session 2
This session is similar to that of the last session except that here the quarter turns are made using ‘combs’ the students make for themselves. To produce a comb, give the students cardboard rectangles and get them to cut out ‘teeth’ to make ‘comb’ shapes similar to the diagram below.
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By holding one end fixed, they should be able to rotate their ‘combs’ through quarter and half turns. Get them to do after dipping their combs in different coloured paint.
Give them the opportunity to make up patterns with their ‘combs’ based on quarter and half turns. Get them to see the link between quarter and half turns.
They might enjoy this activity and produce a number of pages of patterns. Let them choose the one that they like best to go into their ‘turns’ book.
While they are involved in this activity check that their ‘comb’ shapes do represent quarter and half turns. There is no need to measure their work precisely but their turns should be close to the right magnitude.
Session 3
Corners of shapes can also be thought of as quarter and half turns. The object of this session is to find corners of shapes that are equivalent to quarter and half turns.
- 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? - Point out that we can think of the corners of a rectangle as being made up of quarter turns. What other shapes can you think of that have corners that are quarter turns?
- Explore right-angled and other triangles as a class.
- Now look at shapes in the classroom that have quarter turn corners. Get them to make a class list.
- Get them to draw two objects from the classroom (that may or may not be on the class list) that have quarter turn corners and two that don’t.
- Add the drawings to their ‘turn’ book.
Reflecting
- 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 quarter turn? A half turn?
What objects do you know that have quarter turns?
How many quarter turns make a half turn? - Play ‘Simon says’ using quarter and half turns.
- When the Homelink has been used, these objects could also be discussed and added to the ‘turns’ books.
This week we have been thinking about quarter and half turns. We have seen how they can be produced by turning our bodies and moving objects. We have also seen that certain objects have quarter turn corners. Could you please help your child look through old newspapers and magazines and cut out two objects that have quarter turn corners and two objects that don’t have quarter turn corners?
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