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In this unit we look at the beginning of the concept of angle. As ākonga come to understand quarter, half and full turns, they also begin to see that ‘angle’ is something involving ‘an amount of turn’.

Achievement Objectives
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.
GM1-3: Give and follow instructions for movement that involve distances, directions, and half or quarter turns.
Specific Learning Outcomes
  • Demonstrate a quarter turn, half turn and a full turn in a number of situations.
  • Understand 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 perceived in at least three ways. These are as:

  • an amount of turning
  • the spread between two rays
  • the corner of a 2-dimensional figure

The concept "angle" in the New Zealand Mathematics Curriculum develops over the following progressions:

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 kura, 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.  Ultimately, angles play a fundamental role in mathematics, as an abstract tool, and in their application to "real world" contexts.

Opportunities for Adaptation and Differentiation

The learning opportunities in this unit can be differentiated by providing or removing support to ākonga and by varying the task requirements. Ways to differentiate include:

  • providing ākonga more opportunities to explore the concept of turning using themselves or objects as well as recording turns on paper
  • having ākonga work in pairs to encourage tuakana/teina. In session 2, one ākonga can hold the end of the string still while the other draws the arc.
  • modifying the complexity of the course ākonga are asked to follow
  • providing tools such as compasses, protractors, set squares and rulers for ākonga to explore with, and explicit instruction on how to use these to see angles. 

The contexts for this unit can be adapted to suit the interests and experiences of your ākonga. For example, the contexts for identifying shapes with quarter turns could be a local playground, marae or community garden. This could involve a trip to visit it, or photos could be used. Contexts for exploring the application of angles could include exploration of made up treasure maps, and could follow on from learning about how Māori and Pasifika settlers travelled to New Zealand.

Te reo Māori vocabulary terms such as koki (angle), huri (turn), haurua (half), hauwha (quarter), and koki huripū (full turn) could be introduced in this unit and used throughout other mathematical learning.

Required Resource Materials
  • Various toys that are available in the classroom
  • Crayons in a variety of colours
  • Paper (various sizes)
  • String
  • Drawing pins
  • Paint
  • Cardboard
  • Scissors
  • Chalk

Session 1

  1. Talk with the class about ‘turning’.  This can be motivated by playing Simon Says, and asking them directions from their classroom to somewhere else in the kura.  Emphasise ‘turning’ by asking them what they do when they get to a corner. Ask ākonga 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. Encourage bilingual and multilingual ākonga to share words for these terms from their home languages.
  2. Demonstrate full, half and quarter turns. Draw a large circle on carpet or concrete.  Alternatively, your kura may already have some pre-painted circles in your playground. Have one ākonga come to the centre of the circle and put their arm straight out in front. Get another ākonga to place a marker on the edge of the circle showing where the ākonga is facing and their arm is pointing. Demonstrate the full turn as the ākonga 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 ākonga 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? Emphasise the idea of ending up facing the opposite direction. Have an ākonga mark where, on the edge of the circle the half turn stops, and the ākonga ends up pointing. Get them to do another half turn. Where do they end up? So 2 half turns make 1 full turn? Have all ākonga trace a half turn on the ground with their finger. Repeat for a quarter turn, if your ākonga are ready. Otherwise wait until they have had some practice doing full and half turns. This activity could be repeated in pairs to reinforce learning. For each demonstration, document where the pointing arm ends up, which way the ākonga is now facing, and what part of the circle the ākonga has covered. This can also be recorded on smaller circles on whiteboards or in modelling books. 
  3. Repeat the demonstration with a toy. Using a toy animal, for instance, each ākonga could show how to move the animal through a full, half and quarter turn.
  4. Give ākonga time to act out and draw several examples of full, half and quarter turns.  This may be done by using objects from around the classroom, for example, cars, animals or teddy bears. Consider integrating contexts represented in other curriculum areas (e.g. dance) or texts you have recently read as a class.
  5. Kaiako can support their ākonga as they complete these acts and drawings by checking they have the right concept of turns and correct any misconceptions.
  6. Create stories involving turns such as: forgetting something on the way to kura when you would have to turn around and go back.  This means you would have to do a half turn. Model the turn with your toy car or stick figure on the paper, with the use of finger puppets, or with ākonga acting out the stories. Another idea could be pretending that your class is in a marching band and they are in a parade navigating the streets.

In the three sessions that follow, ākonga produce artwork that they can collect in their own ‘Turns Book' or display in the classroom.

Session 2

Prior to this session, tie pieces of string to enough crayons for each ākonga to have one each. Do this with a variety of colours. The string should be quite short (about 5cm long - although they can vary in length) so that the angles can fit on an A4 or A3 size piece of paper.

  1. Provide each ākonga 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.
  2. Ask ākonga to make several ‘quarter turns’ in the same colour.  Check that their turns are approximately correct. 
  3. Having done quarter turns, ākonga choose a new colour and create half turn arcs. Draw their attention to the relationship between quarter and half turns.
  4. Have ākonga choose a third colour and create some full turns. Draw their attention to the relationship between full, quarter and half turns.
  5. Their crayon turn arc artwork could be displayed or created into a 'Turn Book' for each ākonga or the whole class. 

Session 3

This session is similar to session 2, except that the turns are made using ‘combs’ the ākonga make for themselves. As an alternative, you may prefer to have ākonga use crayons lying flat to create the same effect.

  1. To produce combs, give ākonga cardboard rectangles and get them to cut out ‘teeth’ to make ‘comb’ shapes similar to the diagram below.
     Diagram of a cardboard comb.
  2. By holding one end fixed, ākonga should be able to rotate their ‘combs’ through quarter and half turns after dipping their combs in different coloured paint. The cardboard 'combs' can be wiped clean with a paper towel before changing colour paint. Alternatively, you could have a certain number of 'combs' at each paint colour and ākonga could move to the colour of choice.
  3. Give ākonga the opportunity to make patterns with their ‘combs’ based on quarter and half turns.  
  4. Ākonga could be encouraged to produce several pages of patterns. Let them choose the one that they like best for their 'Turns Book'. Alternatively, these could be displayed around the classroom.
  5. While ākonga 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 4

Corners of shapes can also be thought of as quarter turns. The purpose of this session is to find corners of shapes that are equivalent to quarter turns.

  1. Draw (with chalk) a rectangle in the playground (or use a small rectangle in class). Have four ākonga stand on the corners of the rectangle (or put four toys on the small rectangle).
  2. Have one ākonga face another one. What turn would Taika need to make in order to be looking at Jorge?
    Repeat for other examples.
  3. 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?
  4. Explore right-angled and other triangles as a class (mahi tahi model)
    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)
  5. Now look at shapes in the classroom that do and don't have quarter turn corners. Ākonga can work in pairs (tuakana/teina model could work well here) to find objects for both these categories. They could record their thinking by writing a list, drawing pictures or taking photos.
  6. Ākonga can share their findings with the class. Ākonga can draw or stick in two printed photographs of objects from the classroom that have quarter turn corners and two that don’t, to add to their 'Turns Book'.

Session 5

  1. Have ākonga work in pairs to guide each other around a course using instructions involving quarter, half and full turns to the left and to the right. A tuakana/teina model could work well here. The course could be outside, possibly following a line drawn on a court, or they could be in the classroom, moving around the furniture.  
  2. Bring the class together to talk about full, quarter and half turns (mahi tahi model). 
    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?
  3. Finish with a game of Whakarongo Mai Tamariki Mā (Simon Says) using quarter, half and full turns. If the kaiako says "Whakarongo mai tamariki mā quarter turn to your left" the ākonga do it, if the kaiako just says "Tamariki mā quarter turn to your left" the ākonga should remain still, otherwise they e noho.
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Level One