MiS May 2021

14 Mathematics in School, May 2021 The MA website www.m-a.org.uk way not only helps connect the curriculum but also gives pupils a helpful conceptual schema for solving problems in other areas of mathematics. I hope this helps you to think about transformations as a way to teach quadrilateral properties and encourages you to delve more deeply into Don Steward’s inspiring back catalogue. Some of Don’s tasks that would make a good follow-up to this can be found here : https://donsteward.blogspot.com/2017/08/ quadrilaterals-on-5-by-5-dotty-grid.html The tasks accompanying this article are available here : www.doi.org/10.17605/OSF.IO/Z5JV7 Notes 1. https://donsteward.blogspot.com/2012/04/creating- quadrilaterals.html and here: https://donsteward.blogspot. com/2017/07/grid-kites-and-rhombuses.html 2. Of course, this works without the grid, but the grid makes the properties discernible. A dynamic geometry environment is also helpful. 3. Hewitt (2001, p.50) has a more detailed example using isosceles triangles. 4. Although he often gets credit, many of Pierre Van Hiele’s ideas built on observations from his wife Dina Van Hiele-Geldof’s doctoral work and their collaborative study of how geometric thinking develops. 5. …that mediocrity can pay to greatness. 6. On Don’s website (donsteward.blogspot.com/ ) there are many tasks that develop th inking about the gradient co ncept. A collection curated by Anne W atson is available here: w ww.pmtheta.com/ dose-of-don.html but of course I would encourage you to spend some time exploring Don’s blog and show your appreciation here: www.justgiving.com/fundraising/jessesteward References Golomb, S. W. (1966). Polyominoes: puzzles, patterns, problems, and packings . Princeton University Press. Hewitt, D. (2001). Arbitrary and necessary: Part 2 assisting memory. For the Learning of Mathematics , 21 (1), 44–51. Van Hiele, P. (1986). Structure and Insight: a theory of mathematics education . Academic Press Inc. Enlarging a triangle So far, we have generated all the commonly used special quadrilaterals by reflecting and rotating with the exception of trapeziums. As Don notes, trapeziums are not as straightforward to create but we can “cheat a bit and use an enlargement”. If we enlarge a triangle from any corner and ignore the original triangle then what can we deduce about the properties of the resulting shape? Figure 7:  Enlarging a triangle to create a trapezium. Learners I have worked with sometimes think the trapezium in Figure 7 is isosceles. It is worth thinking about how the square grid helps you decide and also allows you to justify the parallel sides (Note 6). At https://osf.io/etju9/ I have posted some activities to follow these examples. However, you might prefer the following more open version of this task. How many triangles can you draw on a 25-pin geoboard so that reflecting in a side creates a kite [arrowhead, rhombus, square]? How many triangles can you draw on a 25-pin geoboard so that rotating about a midpoint of a side creates a parallelogram [rhombus, rectangle, square]? How many triangles can you draw on a 25-pin Geoboard so that enlarging from a vertex and removing the original triangle creates a trapezium ? [isosceles trapezium]? (Keep the resulting image contained within the Geoboard each time) I hope these tasks give a flavour of some benefits of meeting the special quadrilaterals via transformations rather than via definitions. The key difference is that the properties we want learners to grasp are true by construction rather than by definition and are available through awareness rather than having to be told. Angles and sides are equal by construction. Symmetries appear by construction and not by definition. By limiting ourselves to working on a square grid, we can develop learners’ intuitive ideas about gradient, parallel and perpendicular. This happens alongside working mathematically as they notice, conjecture, and generalize about properties and relationships. Thinking in this Keywords:  Quadrilateral; Transformation; Reflection;  Rotation. Author: Tom Francome, Senior Enterprise Fellow, Centre for Mathematical Cognition, Schofield Building, Loughborough University, LE11 3TU e-mail: T.J.Franco e@lboro.ac.uk