HEARTLANDS ACADEMY LESSON STUDY SERIES RESEARCH LESSON 2 GEOMETRICAL REASONING

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1 HEARTLANDS ACADEMY LESSON STUDY SERIES RESEARCH LESSON 2 GEOMETRICAL REASONING Spatial intuition or spatial perception is an enormously powerful tool and that is why geometry is actually such a powerful part of mathematics - not only for things that are obviously geometrical, but even for things that are not. We try to put them into geometrical form because that enables us to use our intuition. Our intuition is our most powerful tool...(atiyah, 2001, p50) Previously, geometry and its reasoning faded from the UK mathematics curriculum, largely because it seemed very difficult to teach learners to prove things, so learners were encouraged to memorise proofs instead. Recent curriculum development in England has recognised a place for geometry: there are geometrical facts that govern how the material world works. More recent research in mathematics education has revealed some of the reasons why learners find proof challenging. The van Hiele (1986) levels give teachers a perspective on adolescents experience of moving from truth and rules belonging to adult authority to discovering that mathematical truth comes from the mathematics itself and so is in the view of learners themselves. Thus reasoning and proof in geometry offer learners an opportunity to become their own authority, within the social conventions of how mathematical reasoning proceeds, so that they need not take the word of any authority other than reasoning and mathematical structure. To date there has been little comparative work specifically on how teachers structure mathematics lessons to develop geometrical reasoning. Most of the difficulties encountered with proof have to do with knowing what you are allowed to assume, and what you are expected to justify. To prove, you need to appreciate that properties can apply to a variety of objects, and that reasoning proceeds on the basis of established and accepted axioms and derived properties only. At Heartlands Academy maths department the planning team decided to use visualisations to help students to understand geometrical reasoning better and aimed to find the answer to the research question below; Can we help the transition from oral reasoning to written reasoning through visualising? 1

2 Research Lesson Proving Which Shapes Can or Cannot Tessellate Lesson Summary Firstly students start to explore mathematical proof and reasoning by using visual examples. Secondly, Students will try to form mathematical arguments to convince others and themselves of their Learning Objectives 1. To begin to understand the value of proving mathematically, rather than testing various cases 2. Students to use visualisations clearly for their reasoning 3. Students to be able to write what they express in discussions on their worksheet. 4. To produce a clear and concise proof which can be understood by others Key Processes Analysing: Make a conjecture about proofs of shapes. Interpreting and evaluating: the concept of proof and reasoning 2

Heartlands Academy Lesson Study (2) Geometrical Reasoning Friday 4th December 2015 Teaching Questions and Notes Anticipated Student Responses Introduction: To recall and reflect upon the pre- task

Some examples show the pentominoes which students could not tessellate and which they thought were impossible, contrasting with other groups who had managed to tessellate a shape which others had

3 Heartlands Academy Lesson Study (2) Geometrical Reasoning Friday 4th December 2015 Teaching Questions and Notes Anticipated Student Responses Introduction: To recall and reflect upon the pre- task Examples of the work from the previous lesson are shown on the board. Some examples show the pentominoes which students could not tessellate and which they thought were impossible, contrasting with other groups who had managed to tessellate a shape which others had deemed impossible. 0.1 Last lesson we tessellated shapes, can anybody give me a good mathematical definition of what tessellation is? 0.2 Do you remember the shapes you can and you can t tessellate? 0.3 Why do you think you were able to tessellate some? 0.4 If we spend long enough attempting to tessellate a shape, who thinks that we could tessellate any shape? Tessellate Do not tessellate Task 1. Attempting to tessellate regular shapes 0.1 Some students may refer to tessellation as just being a shape moved around. Students to say tessellation is just repeating a shape without leaving any gaps. 0.2 The shapes could be tessellated were T,L at the same time some students could mention the same shapes as the shapes which couldn t be tessellated. 0.3 Students to realise the fact that some of the shapes they thought they couldn t tessellate, could have been tessellated if they had play around with their shapes more. 0.4 Some students to ask why are all the shapes made from pentominoes could be tessellated? 3

4 Students are asked to attempt to tessellate regular polygons, ranging from a triangle to a nonagon. They are given stencils to help them draw the regular shapes quicker and allow their diagrams to be more accurate. They must then state which regular shapes tessellate and show examples of each shape that they have attempted to tessellate. 1.1 Students are mostly expected to be able to tessellate the regular shapes which will tessellate and to have attempted others which will not. It is possible that in an effort to make (e.g.) the pentagon tessellate that they will alter the shape, causing it to no longer be regular. If this happens then their work will be shown to the rest of the class to ask why they have managed to tessellate the shape and whether this means that the shape does tessellate. 1.1 Which shapes were you able to tessellate and which shapes were you not able to tessellate? 1.2 If you were given more time do you think that you would have been able to find a way to make the pentagon tessellate? 1.3 Therefore do you think it is impossible to make a regular pentagon to tessellate? How do you know that you have not tried the correct way to tessellate it? Some of us struggled to tessellate some of the pentominoes last lesson, whereas others found a way to make it tessellate. They were convinced that some of them would not tessellate, yet somebody else found a way. How can you convince me that it is impossible for a regular pentagon to tessellate? 1.4 Would this work if the sides of the regular shapes were bigger? 1.2 Students are expected to assert that the pentagon, heptagon, octagon and nonagon are impossible to tessellate, using their many attempts to try to convince the class. The teacher will attempt to make them think that it is possible they may have missed something and not be convinced by the students arguments. 1.3 Students might be able to ask, could the reason be the gap between the shapes are for the angles of the corner? 1.4 Some students may have begun to notice that the shapes which 4

Students to think of angles around a point adding up to 360. Task 2.

5 tessellate will all fit around a point exactly. They will be encouraged to think further about how they could find the individual angles and what they know if the shapes fit together exactly around a point and try to link the two. Students to think of angles around a point adding up to 360. Task 2. Proving that all rhombuses, parallelograms and trapezia tessellate Students are separated into groups, which are all given a different quadrilateral which they have to produce a convincing argument that the shape always tessellates. One group will be given a rhombus, another will be given a parallelogram, another trapezium and another kite. They will work on their arguments as a group and once their arguments are put together they will swap their sheets with other groups. Students are expected to find this task relatively easy for rhombuses and parallelograms; however it is slightly harder for trapezia as there will be two variables which they are using. Students may struggle with the concept of generalised angles and writing some angles in terms of another angle. To aid them the groups have been split to contain children of varying abilities. If problems persist then they will be encouraged to think about what the angles would be if they were numbers. They will assess the other group s argument and try to follow it, if possible trying to improve the other group s argument in a different coloured pen. They might make mistakes on labelling their angles and using their knowledge of angles in parallel lines. 5

Some of the students might be able to label the opposite and equal angles wrong in a shape. Challenge: Teacher to address that during the walk in the lesson.

6 Some of the students might be able to label the opposite and equal angles wrong in a shape. Challenge: Teacher to address that during the walk in the lesson. There may be some confliction when the pupils assess each other s proofs as there are various ways to prove the Task 3. Proving that all quadrilaterals tessellate 3.1 Some students are expected to hypothesise that all quadrilaterals 6

After all of the groups have shown their proofs for specific types of quadrilaterals, they are asked to say something about what they think would happen for all quadrilaterals.

7 After all of the groups have shown their proofs for specific types of quadrilaterals, they are asked to say something about what they think would happen for all quadrilaterals. Students are given diagrams of a general convex quadrilateral tessellating and asked to use the table to prove that all quadrilaterals tessellate. One group will be given a tessellating general concave quadrilateral and asked to prove that it will also tessellate. 3.1 The students will be asked what is different about the general quadrilateral to the specific types of quadrilateral. 3.2 What can I assume about the general quadrilateral? 3.3 Is there any way that we could have saved time instead of proving for every different type of quadrilateral? will tessellate, and recall angles around a point and others are expected to hypothesise that only special types of quadrilateral will tessellate. 3.1 When proving, it is possible that some will try to use angle facts which may not be used (depending on whether they place extra parallel lines themselves). They will be asked if they think that there is an easier way and reminded of the work they did at the start of the lesson. Some students my claim that the proof is the same as previous proofs for other shapes (which they would be partially right). They will then be asked what is different about the other shapes and the general quadrilateral, if necessary asked to think about the shapes properties. 3.2 Students to respond we can tessellate all quadrilaterals. 3.3 Students might say you have a quadrilateral, you can actually split it into two triangles with a diagonal, and if you do this consistently throughout your tessellation you actually end up with a tessellation formed by two parallelograms, once again two easily tessellate- able shapes. Reflection of tasks and learning 7

8 Students are asked to reflect upon what they have shown in the lesson during each task. They are given a table to ask what they had proved for each task and why the task was useful. At the very end they are asked if they can answer questions from what they have proved. 4.1 If we had proved that all quadrilaterals tessellate first would we have to prove each of the separate types of quadrilateral tessellate? 4.2 What were the limitations of each of the things that you had proved? 4.3 What have we learned about proving? what do you think is useful about proof? 4.1 Students respond We could have proved that all quadrilaterals tessellate first but they would try for some shapes 4.1 We could have saved time instead of doing the proof for each shapes 4.2 Each time to remember the angles facts 4.2 We could use visualised examples to make proof easier. 4.3 Proof could generalise an answer to a question. 4.4 Proofing takes less time instead of testing for each time. 4.4 Why is proving useful compared to just testing different cases many times? 8

9 LESSON 1 INTRODUCTION TO TESSELLATIONS This lesson was taught prior to the main Research Lesson, which you are observing today. Lesson Summary Students begin by exploring the properties of shapes and then being able to tessellate shapes. Learning Objectives 1. To form and test conjectures about shapes. 2. To learn the concept of tessellation. Key Processes Analysing: Make a conjecture about the properties of shapes. Interpreting and evaluating: the concept of tessellation 9

10 Teaching Questions and Notes Task1: Teacher to show some pictures from patterns which represent tessellations and ask the students whether they know what are these patterns called Teacher to explain tessellations are patterns when shapes repeat themselves without any gaps between them or overlapping each other Teacher to give examples of shapes which could be tessellated such as equilateral pentominoes Teacher to give out tracing papers to students to tessellate their pentominoes. Task2: Teacher to ask students to tessellate a different pentominoes Teacher to stop the group and ask them what they can notice about their shapes Teacher to confirm the students understanding about the shapes and address the fact that they can t tessellate Teacher to give out some shapes which could be tessellated and not tessellated and without mentioning to students which one can be tessellated and can t be tessellated Teacher to finish the lesson by asking the students what are their findings in that lesson. Anticipated Student Responses a) Students could recognise the patterns similar to what is shown to them on the board. b) They might have heard of tessellations. c) Students to start drawing their shapes on their tracing papers and try to tessellate 6 shapes. d) They need time to rotate the shapes and fit them with each other. e) Students not to realise the fact that they cannot fit pentagons easily and try to reduce the size of them in order for them to fit together. f) Students to say it s not easy to tessellate them. g) Students to do the task and try to think which shapes can and can t be tessellated. h) Students Sir this is not Maths and why are we doing this and what the point of doing this? Tessellate Do not tessellate 10

11 Details of School Heartlands Academy, 10 Great Francis Street, Birmingham B74QR Contact details Name Lead Teacher Grant Portlock academy.org.uk Planning Team Knowledgeable Other Elnaz Javaheri Ferida McQuillan Aneesa Ayuub Colin Foster academy.org.uk academy.org.uk academy.org.uk Date of research lesson: Friday 4 th December 2015 Class Year group: Year 8 Time of research lesson: 11:25 to 12:35 Class Set (if set): 8MA1 (mixed ability) Maximum no. of visitors: 20 Permission to video lesson: Yes: x No: Relevant information about this class / particular needs of students: 11

Properties of polygons Year level: 6 7

Properties of polygons Year level: 6 7 Unit of work contributed by Anne Pillman, Marryatville Primary School, SA L6558 Exploring s. Copyright Alberta Education About the unit Unit description This unit