Angles - Level 3

1. Angles - Level 3 By: Sarah Lovett-Abela Amy Cadman Ghiwa Koubar Christine Mahlis

2. Importance of Angles Length and angle are arguably the two most basic concepts which we use to analyse our spatial environment, and they are also fundamental to the study of geometry. Most students master length early in primary school. However, angle can cause difficulties into secondary school (Mitchelmore, 1983).

3. Wilson and Adams (1992) Found that; • 98% of 145 YR 8 students & 99% of 143 YR 6 students did not refer to angles in defining properties of rectangle. • This shows that there is a lack of attention to the concept of angle in the early years. • They believe that the concept of angle measure as a turn can be developed as early as prep.

4. Overview of Dimension Space Measurement, Chance and Data Working Mathematically

5. 1. Conceptual focus – Angles Level 3 Dimension: Space –PP 2.5: • Representation of angles formed dynamically; for example, between the hands of a clock or between their own limbs, and explanations of these angles in terms of simple fractions of a complete revolution Space – PP2.75: • Construction of a model for an angle using rotation of lines

6. Conceptual Focus Dimension – Measurement, chance and data PP2.75: • Estimation of angle in terms of quarter turns and half turns Dimension: Working mathematically – PP2.75: • Use of materials and models to illustrate and test generalisations.

7. Students understanding of angles • Concept of angle develops over a period of time • A static view of an angle is difficult for young students to perceive • Students need good activities designed to help them explore angles and their properties relationships (Wilson and Adams, 1992, p7)

8. 2. Major Concepts According to Booker et al(2010)... The three aspects of angle are; • Angle as a corner • Angle as the shape made when two lines radiate from a common point • Angles as a measure of turn

9. Prior conceptual knowledge Number – PP1.50: • Use of half and quarter as a descriptor; for example, a quarter of a cake Space – PP1.75: • Specification of instructions for movement, including relative position and quarter turns left and right

10. Prior Conceptual Knowledge Working Mathematically – PP1.5: • Formulation and testing of conjectures using models that involve, for example, objects, patterns, shapes and numbers Number – PP – 2.0: • They describe simple fractions such as one half, one third and one quarter in terms of equal sized parts of a whole object

11. Prior Conceptual Knowledge Space – PP2.0 • At Level 2, students recognise lines, surfaces and planes, corners and boundaries; familiar two-dimensional shapes including rectangles, rhombuses and hexagons

12. Misconception 1:Distraction by arm length Students are distracted by the arm length and think that the length of the arms determines the size of the angle (DEECD, 2011).

13. Misconception 2: Orientation of angle • Students have difficulty recognising angles in different orientations (Close, 1982;Noss, 1987;Outhred, 1987 cited in Mitchelmore & White, 1998)

14. Misconception 3: Amount of turn • Students do not readily incorporate turning into their angles concepts (Mitchelmore & White, 1998). • They found that; • Less than 10% of grade 4 students mentioned turning when asked to give examples of angles.

15. Conceptual Difficulty: Mitchelmore and White (2000), found that students’ major difficulty in learning about angles is to recognise that the same idea is present in different situations (p3).

16. Realistic Context For example; • Some angles are horizontal (e.g. corner of table • Some angles are vertical (e.g. corner of window frame) • Some are solid (furniture corners) • Some are fixed (picture frames) • Some are moveable (scissors)

17. Realistic context cont... Children have to recognise that all these objects are similar , that is • They consist of two linear parts which cross or meet at a point. • (Mitchelmore and White, 2000)

18. Activity: Robo Maze This activity is modified from Wilson and Adams, (1992)

19. Justification: • Students use their bodies to experience the angle as they rotate right and left. • If students engage in physical activities first, then they will be able to apply their knowledge of angle as a measurement of something ‘dynamic’; That is turn. • Then they will not focus on irrelevant details such as length of arm (Clements and Burn, 2000)

20. References: Booker, G., Bond,D., Sparrow,L., Swan, P .(2010). Teaching Primary Mathematics(4th ed). NSW; Pearson Education DEECD (2011). Mathematics Developmental Continuum 3.25 Retrieved 25th August, 2010, from http://vels.vcaa.vic.edu.au/support/progression/maths.html#lev3 Clements, D.H & Burns, B.(2000). Students Development of Strategies for Turn and Angle Measure. Educational Studies in Mathematics, 41, 31-45 Mitchelmore, M.(1983). Children’s learning of geometry: Report of a cooperative research project. Caribbean Journal of Education, 10, 179-228. Mitchelmore, M .(1998). Young Students’ Concepts of turning and angle). Cognition and Instruction 16(3),265-284 Mitchelmore, M., & White, P.(2000). Development of angle concepts by progressive abstraction and generalisation. Educational Studies in Mathematics 41:209-238. Wilson, P., & Adams, V(1992). A Dynamic Way to Teach Angle and Angle Measure. Arithmetic Teacher 39(5) Victorian Essential Learning Standards (2011). Progression Point Examples, Retrieved 25th August, 2010, from http://vels.vcaa.vic.edu.au/support/progression/maths.html#lev3