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Turns

Keywords:
Purpose: 

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.

Achievement Objectives:

Achievement Objective: GM1-1: Order and compare objects or events by length, area, volume and capacity, weight (mass), turn (angle), temperature, and time by direct comparison and/or counting whole numbers of units.
AO elaboration and other teaching resources
Achievement Objective: GM1-3: Give and follow instructions for movement that involve distances, directions, and half or quarter turns.
AO elaboration and other teaching resources

Specific Learning Outcomes: 
  • 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
Description of mathematics: 

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.

Required Resource Materials: 
various toys that are available in the classroom
paint brushes
paint
string
drawing pins
cardboard
scissors
Activity: 

Getting started

  1. 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.) Start recording some of the vocabulary being generated by the discussion related to turns: corner, turn, spin, circle, left, right, around etc.
  2. 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. 
  3. 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.
  4. 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.
  5. As you go around the class observing their drawings, check that they have the right concept and correct any misconceptions.
  6. 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. 

Exploring

In the sessions that follow, the student produce artwork that they can assemble in their own ‘turns’ book.  The full, half and quarter 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.  Ask them to make ‘quarter paint turns’ in one colour.  Check that their turns are approximately correct. 

Having done quarter turns they choose a new colour and create half turn arcs.  Draw their attention to the relationship between quarter and half turns.

Choose a new colour and create some full turns. Draw their attention to the relationship between full, 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.

 

 comb

By holding one end fixed, they should be able to rotate their ‘combs’ through quarter and half turns 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.  

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.

  1. 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).

 diagram.

  1. 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?
  2. 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?
  3. Explore right-angled and other triangles as a class.
  4. Now look at shapes in the classroom that have quarter turn corners.  Get them to make a class list. 
  5. 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.
  6. Add the drawings to their ‘turn’ book.

Reflecting

  1. Get the class 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?
  2. Play ‘Simon Says’ using quarter and half turns.