All entries for September 2010

September 28, 2010

Tessellations Lesson Plan


Year 7 or 8



Number of pupils


Start Time


End time



    (including place in relation to previous and future lessons)

    Previous lesson: on properties of regular polygons including number of sides and size of angles. Including group work so pupils will not need organising into tables for this lesson, just use the same ones. 5 groups of 5 pupils.

    Future lessons:

    • The shopping centre problem/ create a section of a beautiful wall poster for my classroom to explain tilings (emphasises on presentation and good explanations to appeal to other year 7 or 8's and the older pupils too!)
    • Making polyhedra (quick search found this website: Looks inspiring!).

    National Curriculum References

    (including key concepts, key processes and curriculum opportunities)

    Pupil learning targets

    To be shared:

    Must: Find at least 3 tiling patterns.

    Explain to a friend why the patterns you've found work, using maths words.

    Should: Find 5 tiling patterns

    Find patterns to explain why they work.

    Could: Attempt the shopping centre problem.

    Other learning targets:

    Develop reasoning skills.

    Personal Targets:

    To ensure equal participation by each pupil in a group, and that no-one is being left behind as the group moves forward.

    Not to give information away - chose questions carefully to get them to tell me what is happening. Focus on helping them organise their thoughts and prompt to use the proper terminology to describe what they mean.


    Seating plan/group lists.

    Whiteboard, pens and pre-prepared files.

    Shapes precounted and divided into equal sized packs (one pack per table)

    15x Semi-regular tiling worksheets (one per pair)

    15x shopping centre sheets (2 pages) (one each, not all will get that far)

    Stack of lined and square paper (Can use either, offered to do lined for writing explanations, squared for drawing pictures)

      Lesson Outline


      Teacher Activity

      Pupil Activity /Actions/ Reactions (expected answers in brackets)


      Have tessellation (of a relatively simple shape) projected on board before class arrives. Let pupils into classroom

      & quot;We've got some fun p rops to play with today, so be ready to work, quick! Don't need your books out, just your pencilcase."

      Should know where to sit from last lesson so will be relatively quick. Should be sufficiently interested.

      If take too long to settle class then skip parts of starter activity. Hopefully will be getting questions about the picture on the board which will draw in other pupils not settling down.



      [can be shortened]

      Here is a tessellation. Think about how you would describe it (allow 30secs).

      If answer is good but not sufficient ask someone else to build on it.

      Attention on the board.

      Take answers from approx 5 pupils (same shape repeated, no gaps in it, can be continued forever in all directions).

      [keep unless desperate for time]

      On whiteboard, have the triangle and hexagon picture with spare shapes at the side.

      What is different about this one? (2 different shapes)

      What rules are the same? (no gaps, can continue forever)

      Ask for volunteer to come up to the board and add some of the triangles/hexagons to the diagram. Ask someone else to explain how he knew where to put them.

      These are called semi-regular tessellations

      Why do you think they were called that? (obvious!)

      Learning objectives screen. Leave up for rest of lesson, no need for pupils to copy down.

      Get pupils to read it (modify this: hard to stop them just ignoring it. Question the class on what they are?)


      Instructions on the sheet, shapes in the pack, get started!

      4 volunteers to hand out first sheet, paper x2 and packs of shapes.

      To work on task without teacher intervention.

      Intervene only when a group is having trouble getting started : prompt them to start by creating the one on the sheet using the shapes handed out, then try to make a different one after that.


      [approx, monitor pupil's frustration levels]

      Emphasis importance of creating ways to describe what they are noticing and to record their findings (good and bad).

      Intervene with groups where workload is uneven - get all pupils involved. Only ask/answer vague or open ended questions, don't give anything away yet!


      Monitor whole class

      Can give away some help now.

      For groups with several tiling patterns: how did they find them? Looking for signs of generalisation - expect pupils to explain a specific pattern. Can they give any rules for what will or won't tessellate? (angles sum to 360 at a point restricts choice --> idea of looking at combinations of shapes around a vertex) Can you explain mathematically what you have found?

      For groups struggling to find patterns: Can they quickly show me what patterns they've tried already. What went wrong for (specific one/generalise to all of them)? How can this be avoided? Aim for them to discover looking around a vertex - lead in from "no gaps" idea in the starter if they don't bring it up themselves.

      For groups with all the tessellations: describe your findings mathematically. Be carefully to put a convincing argument together, write it on paper for me to take in. The best one will be typed up word for word to be used on a display about tessellations! (Next lesson - will get pupils to create an accompanying picture)

      For groups still only with few, come up with a way to describe what they are doing, can you use this rule to find another? (make sure they're on the right track so they don't fruitlessly stab in the dark)

      Once groups are happy with their convincing argument, start on the shopping centre problem. (To be continued next lesson)



      Class finish what they're doing, draw together to discuss findings.

      Get findings from each group, I pick a spokesperson from each group to give me one finding. Less interested in the tilings found, more interested in the maths behind it. Question to other members of the group (take volunteers): How did you figure that out? Can you show/explain why it works? If struggling, open the question to the rest of the class.

      Next lesson: those who started the shopping centre question, I expect you to explain the problem to the rest of the class. We'll be working on this. Also, I'll be picking the best reasoning to type up for a display. Next lesson someone gets to chose/make some semi-regular tillings to accompany it. (A way to involve the weaker pupils?)

      End: No-one leaves until I have all the sheets and paper for each group stapled together with everyone's name on their group's work.

      Hands up when you've written everyone's names down, I'll come around with a stapler as you pack up. Dismissed as groups once work is stapled and in my hand.

      Prompts and Notes (possibly including whiteboard content)

      Remember to check individual progress, not just group progress. Watch groups from afar to see interaction and division of tasks.

      Language and Vocabulary

      Tessellation, Semi-regular tiling.

      Triangle, square, hexagon.

      Angle, vertex.

      Differentiation Provision

      (SEN, G&T, fast workers, slow workers)

      G&T: The "convincing argument" must be very convincing: can ask them to add to it by looking over argument, finding a hole and asking them to plug it.

      SEN: Early prompting to go in the correct direction. Put together on a table and stop there frequently as circulate classroom to help direct their thoughts. Depends on exact SEN: eg dyslexia provision is there is no need to write, only one copy needed per table.

      Fast Workers: The Shopping Centre question.

      Slow workers: Don't have to find all the tessellations.

      Assessment Strategies

      (how you will know if your learning targets have been made)

      During lesson:

      After lesson:


      EDIT: I just came across this more-awesome-than-average real life tesselation

      Geogebra and Circle Theorems

      Some work-in-progress resources for teaching the circle theorems. I've been working on two of them, there are others done by the rest of my group for some of the other theorems and we're planning to link them all together via the blog. 

      The resources were made in Geogebra. Not a word I was familiar with until a week ago but I'm already very fond of it.  My aim before starting the course was to get very good with ICT. I'm currently hopeless. I didn't use Microsoft Office through the entirely of my degree and my school was incredibly poor with ICT. I didn't use ICT much at school, I have vague memories of one non-lesson with LOGO, but the teacher got confused and we didn't get anything done. There were some rushed attempts at using ICT with the class for when Ofsted visited but were never very useful. (I took great delight in reading the Ofsted reports - these consistently stated ICT as a very weak point for the school.)

      My opinions of ICT have softened in the years that have passed, my experience at first was that computers were rubbish and every attempt at using them in lessons wasted loads of time to set up and I never learnt anything from the activities anyway. I really want to be good with it, firstly because other people (Ofsted, the people who I want a job from) will probably like it, but secondly because I've seen real value in it. We get so conditioned into how shapes usually look (eg trapezium with the longer side on the bottom, right angled triangle with right angle in the bottom left corner) and rarely deal with crazy objects, like a really long and thin kite. I've really learnt from playing with Geogebra, just the simple task of creating valid shapes lead to discussion of things I've never thought about. If we drag the crossbar of a kite up to the very top we get a triangle. Are those triangles valid as kites? What if you pull the top point of the kite down below the crossbar to get an arrow sort of shape. Is that a kite? Discovering and posing these questions gives them far greater value to me than if I'd been given them on a plate. 

      There's one other piece of software we've been introduced too, called GridAlgebra. We've had two demonstrations of it and I warmed to it considerably during the second time. So far we've seen algebra as a "journey", it's a grid where moving down is multiplying and moving right is adding (divide and subtract are the expected inverses). It doesn't appeal much to me as a learner, when we were set questions in it I could only do them by looking away and doing the sum. I simply couldn't do it by visualising it as a journey. And therein lies it's value. It will be dead easy to write activities to appeal to pupils who think like me. Therefore, I need to put effort into finding things for pupils who don't think like me.

      A wide variety of approaches is key to getting maximum understanding over the class as a whole. I discovered this by chance while tutoring over the summer. The girl wanted help preparing for a retake she needed a C in. I'd explain stuff, we'd do some practice, she'd be able to do it. I'd come back the week after and some stuff would have gotten lost whereas some would be done perfectly (and perfectly every time). I tried a different approach every time and after 2 or 3 attempts she'd have something down. Except finding the coordinates of the midpoint of a line. It came up in every single past exam. It gave two co-ordinates in the first quadrant and asked for the midpoint of the line segment joining them. After the 5th week of her not retaining a method from the previous lesson, I turned to my boyfriend for ideas. This girl is very good at the handling data and we hit upon the idea of relating it to the mean. The mean finds the average of the values it's given. If we give it only two values, it will find the midpoint. So we just need to find the mean of the x values and the mean of the y values. 3 weeks later, because the family went on holiday, and of course she didn't do any practice in that time, she could do it instantly. Hallelujah. :D

      I've got familiar with Geogebra far quicker than I expected. The resources I've made could be (and hopefully, when I've learnt a bit more, will be) much better. There are limitations of the software that I've hit a brick wall trying to go around, I know exactly what I want to do but I can't find a way to do it! There is another session on it soon so I'm hoping to pick up a few more tricks. 

      Lesson Plan

      Writing about web page /cbates/entry/quadratic_equations_practice/

      Writing about an entry you don't have permission to view

      An attempt to link my name to my group's lesson plan which is on Chris's blog. 

      September 27, 2010

      Pre–reading scandal!

      The pre-reading for tomorrow has changed since last night! Very unexpected. I have proof it changed because I printed the page in order to ask various people where on earth I find the non-internet ones. There were two journals, one I eventually tracked for a forgotten corner of the library, and one from 1989 which the library didn't have. My workload for tonight is pretty manageable so I'm not begrudging the extra reading, it was pretty interesting actually and I expect I'll look at those journals (Micromath) again. The reading has been replaced by a lovely little video I found pretty interesting (and light "reading"). 

      Reflections on Week 51

      The end of my first week of my PGCE. Quite a new experience. I spent the week before in schools (Mon/Tue in a Primary and Wed-Fri in a Secondary school) to study the transition pupils face between schools. It's something I can relate to since I am in the process of the toughest transition of my life so far! I want to record how hard it is for this time next year when I'm an NQT struggling to keep up with my basic duties: I feel like that right now! 

      This is tough. Really tough. I've felt overwhelmed at times. But it's ok. Firstly, I just have to look around and see the faces of people who look like I feel. I'm less scared of failure than I was during my maths degree, as here I actually understand what I'm being told. It all makes sense. But there is a lot of it. There are longer university hours, and I have to be active in those hours instead of passively writing. The fact that they all are compulsory makes me scared of being ill or burning out, which is something that happened every term as an undergraduate and was only fixed by taking time out. No longer an option which adds to the stress levels. 

      I'm absolutely positive I'll get through the course. I'm thinking of it as an endurance test which has really helped me keep going. I've discovered I can work well enough even when tired. I've got into doing each day's reading at 7am that day (all the better to have it fresh in my mind, last minute isn't my style at all!) and even though I'm sleepy, I take it in. Doing maths at degree level required a certain mind set which was hard to attain. I've written my first essay while at my most rested and relaxed, but day to day stuff can be done while I feel tired or stressed. This is good, it means I don't need to worry about getting in that state, just ride it through. I'll never quit in any case, it's just not in my nature.

      Advice to me for when I start going into schools: discover priorities. I'm having trouble distinguishing these at the moment and it's highlighted just how vital they are! At the moment, all work is given equal, and high, priority and so I've worked insanely hard on some things that I should have taken less time on. Also in this, and personal to me, is to avoid my perfectionist tendencies. These are great for one long term piece of work, such as the master's essays we have to write. But this approach holds me back when I have to create large volumes of work, I want to write the most perfect lesson plan and have the most perfect resources to use in it. I've read some wonderful books which contain practical advice rather than theoretical info. One says "don't reinvent the wheel", all the resources you need are out there somewhere, don't spend ages making your own!. I need to remember that. 

      The other thing that's going to make or break me is handling conflict. I hate conflict. I've often let someone mistreat me rather than confront them about what they are doing. This had issues in my personal life, but also now, my professional one. There is inevitable conflict between teachers and pupils and I have to get used to jumping into it! I worked on this during my initial placement. The teacher had left the room and a boy threw a paper aeroplane across the room to his friend. I got it off the friend, by saying "give that to me" in a firm voice rather than the pitiful "give that to me?" which threatened to come out. I then told them in the same voice to get back to work. An instruction! That's just asking for trouble, what if they don't? Then it's a big deal and I must react again. But this is not my classroom and I don't have a clue what to do next. But they did, and it was over. 

      I'm going to have real trouble on placements talking to the classes. It's insane really, I confidently circulate any classroom and dive into any question I'm given. I didn't realise this was unusual until teachers told me that people observing usually hide at the back and avoid interacting with pupils. I'm happy with individuals or pairs or groups. Even relatively large groups. I can stand at the front of a class, and when observing, I prefer to do this as it implies teamwork with the teacher, I can see everyone, and I can quickly help the teacher when they ask for it. But stand me in front of a class and get me to speakand suddenly I'm petrified. Once I was explaining some sheet work to a perfectly nice class and my hands were shaking holding the sheets. I'm sure my nervousness is reflected in my body language and voice and it's not good! I've been thinking about ways to overcome this and I've found some excellent books dealing with the problems I worry about. Things like hiding nervousness will hopefully come with practice. I've seen someone go from PGCE student (while I did the Student Associates Scheme at the same school) with a lot to learn, to NQT (some voluntary work at the same school the following September) struggling to keep up, to teacher (initial placement) who is effective, and most importantly, incredibly happy. She's quite a lot like me so I'm just going to assume she's been through all the times I'll be struggling with and that I'll get through them and end up just as good at my job and happy with my life as she is. 

      It's been interesting to consider how we must do lessons which appeal to all our pupils. I've been reflecting on my time at school as well as my learning experience on the PGCE and it's already changed how I was thinking. One thing is that I like time to think. I'm a thinker and a reflector and I like to think carefully for as long as I want before I do something. This is probably why I never leave anything to the last minute, I like to allow ample time just in case I need it. I've been finding it hard to work to tight schedules in the seminar sessions, it's been stressful at times to be give x minutes to complete a task, and I find myself rushing at the start where possible, to create spare time at the end. This is especially the case when we have to report back to the class. It's tapped into something I thought was long dead (probably because it was never tested throughout my degree), my hatred of answering questions in class and fear of saying something wrong. It's so bad in fact, that at times when we've been asked a series of questions I really don't want to answer, that I can't learn anything at all because I'm too worried about not being able to answer. This is made worse by sitting near the back of the room, I feel so disconnected with everything happening. 

      I've rediscovered sides of my personality that I'd forgotten were really there. I love to read, I've done all the required reading plus loads more! I can't stop! I'm also more sociable that I realised. I love being alone and quiet, which is easy on a maths degree. But I also love to meet new people and share ideas. That's been really great. I have lots more to add but it's late now and the priorities thing is kicking in. I've written plenty and I have a long week about to start! 

      September 23, 2010

      Cheesy Maths Joke

      Why did the chicken cross the Möbius Strip?

      To get to the...

      September 2010

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