In this unit we look at the beginning of the concept of angle. As students come to understand quarter and half turns, they also 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 two quarter turns equal one 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:
- an amount of turning
- the spread between two rays
- the corner of a 2-dimensional figure
Angle in the New Zealand Curriculum develops 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
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.
The learning opportunities in this unit can be differentiated by providing or removing support to students and by varying the task requirements. Ways to differentiate include:
- In session 1, providing students more opportunities to explore the concept of turning using themselves or objects instead of recording turns on paper.
- In session 2, having students work in pairs, with one student holding the end of the string still while the other draws the arc.
- In session 5, modifying the complexity of the course students are asked to follow.
The contexts for this unit can be adapted to suit the interests and experiences of your students. For example, the contexts for identifying shapes with quarter turns could be a local playground or building of particular interest. This could involve a trip to visit it, or photos could be used.
- Various toys that are available in the classroom
- Crayons in a variety of colours
- Drawing pins
- 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. Ask them what they have to do if they want to go left or right. (They make a turn.) Start recording some of the vocabulary being generated by the discussion related to turns: corner, turn, spin, circle, left, right, around etc.
- Demonstrate full, half and quarter turns. Use a big circular piece of paper or fabric, or alternatively a chalk circle drawn on carpet or concrete. Have a student come to the centre of the circle and put their arm straight out in front. Get someone else to place a marker on the edge of the circle showing where the person is facing and their arm is pointing. Demonstrate the full turn as the person slowly turns all the way around and ends up back at their beginning point. Have everyone trace the full turn on the ground with their finger. Choose another person to come to the centre of the circle, face the same starting point and demonstrate a half turn. How far will they need to go? Where should they stop? Stress the idea of ending up facing the opposite direction. Have someone mark where on the edge of the circle the half turn stops and the person ends up pointing. Get them to do another half turn. Where do they end up? So 2 half turns make 1 full turn? Have everyone trace the half turn on the ground with their finger. Repeat for quarter turn if the group is ready otherwise wait until they have had some practice doing full and half turns. For each demonstration, document where the pointing arm ends up, which way the person is now facing and what part of the circle the person has covered. This can also be recorded on circles on the whiteboard or modelling book.
- Repeat the demonstration with a toy. Using a toy animal, for instance, a student could show how to move the animal through a full, half and quarter turn.
- Give the students time to go and draw several examples of 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 turn that is important.
- As you go around the class observing their drawings, check that they have the right concept and correct any misconceptions.
- Create stories involving turns such as: forgetting something on the way to school when you would have had to turn round and go back. This means you would have had to do a half turn. Model the turn with your toy car or stick figure on the paper.
In the three sessions that follow, students produce artwork that they can collect in their own ‘Turns Book'.
Prior to this session tie pieces of string to enough crayons for each child to have one, with a few spares to avoid arguments about colours.
- Provide each student with a crayon with string attached. 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 arc by sweeping the crayon through a quarter turn. You will need to draw their attention to the importance of keeping the end of the string still and maintaining tension on the string.
- Ask students to make several ‘quarter turns’ in the same colour. Check that their turns are approximately correct.
- Having done quarter turns , students choose a new colour and create half turn arcs. Draw their attention to the relationship between quarter and half turns.
- Have each child choose a third colour and create some full turns. Draw their attention to the relationship between full, quarter and half turns.
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. As an alternative, you may prefer to have students use crayons lying flat to create the same effect.
- To produce combs, give the students cardboard rectangles and get them to cut out ‘teeth’ to make ‘comb’ shapes similar to the diagram below.
- By holding one end fixed, students should be able to rotate their ‘combs’ through quarter and half turns after dipping their combs in different coloured paint.
- Give students the opportunity to make patterns with their ‘combs’ based on quarter and half turns.
- Students could be encouraged to produce several pages of patterns. Let them choose the one that they like best for 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.
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 one student face another one. What turn would Mike need to make in order to be looking at Jorge?
Repeat for other examples.
- 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.
Does this triangle (right angled) have any quarter turns? (yes)
Are all the corners quarter turns? (no)
Do all triangles have quarter turns (no, provide examples that don't)
- Now look at shapes in the classroom that do and don't 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 to add to their Turns Book.
- Have students work in pairs to guide each other around a course using instructions involving quarter turns to the left and to the right. The course could be outside, possibly following a line drawn on a netball court, or they could be in the classroom, moving around the furniture.
- Bring the class together to talk about full, 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? How many half turns make a full turn?
- Finish with a game of ‘Simon Says’ using quarter and half turns.