tag:blogger.com,1999:blog-25004470909237569982017-07-15T06:30:43.261-07:00Educating MrMattockPeter Mattocknoreply@blogger.comBlogger89125tag:blogger.com,1999:blog-2500447090923756998.post-3846135915461220062017-05-17T14:28:00.001-07:002017-05-17T14:28:05.767-07:00Malcolm Swan Day<div dir="ltr" style="text-align: left;" trbidi="on">Recently mathematics education lost one of its leading thinkers, Professor Malcolm Swan. The impact that Professor Swan had on developing mathematics teaching and mathematics teachers cannot be overstated, and also cannot be adequately described in words. This post is not an obituary, I didn't ever have the pleasure of meeting Professor Swan, but despite that I have been massively influenced by his resources and the development materials he has published, primarily for me in the Standards Unit (or Improving Learning in Maths).<br /><br />The purpose of this post is to highlight an opportunity to celebrate the life and work of this great Maths educator. Professor Swan's funeral is on Tuesday 23rd May, and so we are calling on Maths teachers to use Malcolm's materials in as many lessons as possible, and tweet pictures and examples using the #malcolmswanday<br /><br />For those people who may not realise what we have to thank Malcolm Swan for, his materials include:<br /><br /><ul style="text-align: left;"><li>the aforementioned Standards Unit, which can be found on mrbartonmaths website <a href="http://mrbartonmaths.com/teachers/rich-tasks/standards-units.html" target="_blank">here</a>.</li><li>the Mathematics Assessment Project materials, which have their own website <a href="http://map.mathshell.org/tasks.php" target="_blank">here</a></li><li>The 'How risky is life?' Bowland Maths project, which can be found <a href="http://www.bowlandmaths.org.uk/projects/how_risky_is_life.html" target="_blank">here</a></li><li>The Language of Function and Graphs - a fantastic book, which the Shell centre have kindly provided photocopiable masters on their site <a href="http://www.mathshell.com/materials.php?item=lfg&series=tss" target="_blank">here</a></li></ul><div>The posts and images tweeted on the day will be collated and given to his family as a tribute from maths teachers across the country to this inspirational hero of maths education.</div></div>Peter Mattockhttps://plus.google.com/113661418069132691177noreply@blogger.com0tag:blogger.com,1999:blog-2500447090923756998.post-68880455106179890072017-05-16T12:28:00.000-07:002017-05-16T12:28:39.637-07:00Approaches to teaching simultaneous equations<div dir="ltr" style="text-align: left;" trbidi="on">My esteemed colleague Mark Horley (@mhorley) wrote an excellent blog recently about the balance between the need for understanding when teaching simultaneous equations balanced against ensuring procedures are straightforward enough to support pupils ability to follow (read it <a href="https://mhorley.wordpress.com/2017/05/10/simultaneous-equations-refining-the-procedure/" target="_blank">here</a>). Reading his reflections led me to reflect on my own approach to simultaneous equations, as well as others I have previously seen, and one that occurred to me literally as I was thinking about them. This blog is designed to act as a summary and chart my journey through the teaching of this topic.<br /><div><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-Cv6u5dy_i-g/WRS6zGZirHI/AAAAAAAABmI/5E7nr0E9fvMIDZlTpjuwnLtAD1FtRDeVACLcB/s1600/Elimination.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="320" src="https://2.bp.blogspot.com/-Cv6u5dy_i-g/WRS6zGZirHI/AAAAAAAABmI/5E7nr0E9fvMIDZlTpjuwnLtAD1FtRDeVACLcB/s320/Elimination.png" width="204" /></a></div><div><b>Elimination</b>: This is probably the first method I used, and is definitely the sort of approach I was taught at school. Very much a process driven method, I can't remember understanding much about the algebra beyond the idea that I was trying to get rid of one variable so that I could find the other. I find that the subtraction often causes problems (which is partly why Mark's idea of multiplying by -2 instead of 2 is very interesting) and of course the method doesn't generalise well to non-linear equations. I can see this being a popular approach for those people teaching simultaneous equations in Foundation tier.</div><div><br /></div><div><br /></div><div><br /></div><div><br /></div><div><br /></div><div><br /></div><div><br /></div><div><br /></div><div><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-QRyQ_XZ1BM8/WRTAiLht38I/AAAAAAAABmY/XX90JzWx8NInHyuc1oUoJqZVty3yidwfACLcB/s1600/Substitution.png" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" height="400" src="https://2.bp.blogspot.com/-QRyQ_XZ1BM8/WRTAiLht38I/AAAAAAAABmY/XX90JzWx8NInHyuc1oUoJqZVty3yidwfACLcB/s400/Substitution.png" width="285" /></a></div><div><b> Substitution</b>: Another one from school,</div><div> this was the alternative I was taught to </div><div> elimination, which was mainly because it</div><div> was necessary to solve non-linear </div><div> simultaneous equations. I can't remember</div><div> it being the method of choice for myself </div><div> or any of my classmates, and that is </div><div> certainly borne out with my experience of</div><div> using it with any other than the highest </div><div> attaining pupils.</div><div><br /></div><div><br /></div><div><br /></div><div><br /></div><div><br /></div><div><br /></div><div><br /></div><div><br /></div><div><br /></div><div><br /></div><div><br /></div><div><br /></div><div><br /></div><div><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-aI57bZyc_k8/WRTENboahhI/AAAAAAAABmo/hoAYrXfu12cg0KCd3eVn67MTXTEMhxKkACLcB/s1600/Comparison.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="400" src="https://3.bp.blogspot.com/-aI57bZyc_k8/WRTENboahhI/AAAAAAAABmo/hoAYrXfu12cg0KCd3eVn67MTXTEMhxKkACLcB/s400/Comparison.png" width="231" /></a></div><div> <b>Comparison</b>: Similar to elimination, but for me less </div><div> process driven and more focused on understanding the</div><div> relationship between the two different equations. This </div><div> removes the difficulty around dealing with subtracting </div><div> negatives, and allows for the exploration of which</div><div> comparisons are useful and which aren't, so it is a little</div><div> less 'all or nothing' than the process drive elimination</div><div> approach. It also copes nicely with having variables with</div><div> coefficients that are the additive inverse of each other, for </div><div> example in the pair of equations above if instead of the</div><div> approach outlined we multiply the second equation by 3 </div><div> and get:</div><div><br /></div><div> 4x - 3y = 9 and 6x + 3y = 21</div><div><br /></div><div> then the comparison would be "the left hand sides have a </div><div> total of 10x, and the right hand sides have a total of 30, so</div><div> 10x = 30."</div><div><br /></div><div> This is the approach I used when recapping simultaneous </div><div> equations with my pupils in Year 11 and they certainly </div><div> took to it a lot better than the elimination or substitution </div><div> that had used with them the previous year.</div><div><br /></div><div><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-SJY9Lx689vs/WRTKl9ze-_I/AAAAAAAABm4/pv_abnrrplMi-eS8l2u05ACvBXmnoiMlACLcB/s1600/Transformation.png" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" height="400" src="https://1.bp.blogspot.com/-SJY9Lx689vs/WRTKl9ze-_I/AAAAAAAABm4/pv_abnrrplMi-eS8l2u05ACvBXmnoiMlACLcB/s400/Transformation.png" width="171" /></a></div><div> </div><div> <b>Transformation</b>: This approach is the</div><div> one I have very recently considered, but</div><div> not yet tried. The general idea is that you </div><div> isolate one of the variables, and then look</div><div> at how you can transform that variable in</div><div> one of the equations into the other. The</div><div> same transformation applied to the other</div><div> side of the equation then gives a solvable </div><div> equation. Although the equation may be </div><div> slightly harder to solve at first, I do believe</div><div> this approach has merit. I would suggest </div><div> that this approach develops pupils'</div><div> appreciation of the algebra and the</div><div> relationships between the different </div><div> equations in a similar way to the</div><div> comparison approach above. I can also see </div><div> this approach working for non-linear</div><div> equations, like the one below:</div><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-z9lK5oA1Eck/WRTPPRqzGgI/AAAAAAAABnE/9TnCpQCl6uwu2Ek-yobYI2quid5z62GeACLcB/s1600/Transformation%2B2.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="400" src="https://3.bp.blogspot.com/-z9lK5oA1Eck/WRTPPRqzGgI/AAAAAAAABnE/9TnCpQCl6uwu2Ek-yobYI2quid5z62GeACLcB/s400/Transformation%2B2.png" width="257" /></a></div><div><br /></div><div> </div><div><br /></div><div><br /></div><div><br /></div><div><br /></div><div><br /></div><div><br /></div><div><br /></div><div><br /></div><div><br /></div><div><br /></div><div><br /></div><div><br /></div><div><br /></div><div><br /></div><div><br /></div><div><br /></div><div><br /></div><div><br /></div><div><br /></div><div><br /></div><div><br /></div><div><br /></div><div>etc...</div><div><br /></div><div>I will almost certainly give this approach a try when I next teach simultaneous equations - when I do I will try and blog the results! </div></div>Peter Mattockhttps://plus.google.com/113661418069132691177noreply@blogger.com2tag:blogger.com,1999:blog-2500447090923756998.post-18016390359200609662017-05-11T11:39:00.001-07:002017-05-15T08:41:56.235-07:00Methods of Last Resort 4 - Comparing/Adding/Subtracting Fractions<div dir="ltr" style="text-align: left;" trbidi="on">Working with fractions is notoriously something that teachers complain about when it comes to pupils' understanding and ability to manipulate. As a result it often seems to me that working with fractions is a place where even the best maths teachers can often fall back into what Skemp would call 'instrumental understanding'; pupils mechanically following procedures rather than applying any understanding of the relationships between the different parts of the process or between the question and the result.<br /><div><br /></div><div>This was brought to mind for me recently when I saw the question below mixed into a group of questions about comparing fractions:</div><div><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-tT1V0xasX8Y/WQpDfQBSvGI/AAAAAAAABlc/hebUdPLrzGwsccYHQYZPAb-L9xgrQGBOwCLcB/s1600/Which%2Bis%2Bbigger.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="128" src="https://3.bp.blogspot.com/-tT1V0xasX8Y/WQpDfQBSvGI/AAAAAAAABlc/hebUdPLrzGwsccYHQYZPAb-L9xgrQGBOwCLcB/s200/Which%2Bis%2Bbigger.png" width="200" /></a></div><div class="separator" style="clear: both; text-align: left;">From the rest of the questions listed it was quite clear that the intention would be that pupils write the second fraction as a fraction of 30 so that the comparison between the numerators would yield clearly that the first fractions is bigger than the second. Which of course is completely apparent because the first is more than <span style="font-family: "calibri" , sans-serif; font-size: 11pt;">½</span> and the second less than <span style="font-family: "calibri" , sans-serif; font-size: 11pt;">½</span>. Any halfway competent mathematician wouldn't even bother equating the denominators, and this is the sort of thing I would want to highlight to pupils in order to try and develop their relational understanding.</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">The process of finding common denominators for comparing, adding and subtracting fractions is one that can easily become automatic for pupils, and I would argue that if pupils are to really understand fractions then they need to be able to take a more discriminatory approach. The following are all examples of questions that pupils could tackle without finding common denominators:</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-tCSuh-VqU9k/WQpRUEv009I/AAAAAAAABls/j8o6Bv8i7KwkfpP3Sa1ilR2eiR2BwYYpQCLcB/s1600/Questions.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="222" src="https://4.bp.blogspot.com/-tCSuh-VqU9k/WQpRUEv009I/AAAAAAAABls/j8o6Bv8i7KwkfpP3Sa1ilR2eiR2BwYYpQCLcB/s320/Questions.png" width="320" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: left;">I would argue that the first and second points are more easily done by converting to decimals than fractions (which people may or may not agree with), and that the last one certainly doesn't require a common denominator; the first is greater than ½ whilst the second is equal to ½.</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">So if you are truly committed to developing pupils' relational understanding of fractions then the next time you look at the sorts of comparisons or calculations that often benefit from converting into equivalent fractions with common denominators, it might be worth throwing in some examples and questions of calculations where this is a method of last resort.</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;"><br /></div></div>Peter Mattockhttps://plus.google.com/113661418069132691177noreply@blogger.com0tag:blogger.com,1999:blog-2500447090923756998.post-32730823087395304142017-04-16T06:36:00.000-07:002017-04-16T06:37:20.052-07:00Gradient of lines - a new approach<div dir="ltr" style="text-align: left;" trbidi="on">Recently I have been teaching the idea of gradient to Year 8, and I decided to approach things quite differently. In the past I would move quite quickly through the ideas of gradient as a measure of slope, finding gradients of lines plotted on a coordinate axes, then linking gradient and intercept to the equation of a line. From my experience this is a fairly standard approach and one that a lot of teachers use. My problem is that typically not too many pupils actually get success from this approach. It occurred to me that I could do a lot more to secure the concept of gradient, and I decided to spend significantly more time than normal doing this, with some surprising results.<br /><br />The first thing I did was to talk about different ways of measuring slope. Normally I would only focus on the approach I was interested in, but this time I talked about angles to the horizontal and the tangent function. I talked about road signs using gradients as ratios or percentages. Then I talked about gradient measure on a square grid. I have used different ways of defining gradient throughout my career, starting with the standard "change in y over change in x" before I realised this definition was more about how to calculate gradient on a axes rather than what gradient actually is. I played around defining gradient using ratios and writing in the form 1:n, which had some success for a while, but became cumbersome as ideas became more complex. The definition I have settled on for now is "the vertical change for a positive unit horizontal change", or as I paraphrased for my pupils "how many squares up for one square right?" The reason I like this definition is that it incorporates the ratio idea, works for square grids that may not include a coordinate axes, and I can see how it will help highlight gradient as a rate of change later on.<br /><br />From here we spent quite a number of lessons learning and practising the act of drawing gradients. We started with positive whole number gradients, drawing one short line, and then one line longer, so that we got pictures looking a little like this:<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-SPqfze85E_s/WPKHUI0mYMI/AAAAAAAABkw/_wnof17frWY3MGIDEqA-0TIEH90fNtyQgCLcB/s1600/Drawing%2Bgradient%2Bpicture.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://1.bp.blogspot.com/-SPqfze85E_s/WPKHUI0mYMI/AAAAAAAABkw/_wnof17frWY3MGIDEqA-0TIEH90fNtyQgCLcB/s1600/Drawing%2Bgradient%2Bpicture.png" /></a></div>What was really interesting at this point was dealing with the early misconception that the gradient of the right hand line was larger than the left, even though pupils had watched me draw both in precisely the same way. There was an idea, hard to shake, that a longer line meant a steeper gradient; I suspect because the focus was very much on 'how many squares up' the line was going. This did give me the opportunity to reinforce the importance of the single square right; this is an idea we had to keep coming back to throughout the topic.<br /><br />Once drawing positive integer gradients was secure, we turned our attention to negative integer gradients. Pupils were quick to grasp the idea of negative gradients sloping down instead of up, and I was sensible enough to throw some positive gradient drawing in with the negative gradient drawing so that we didn't get too many problems creeping in at this stage.<br /><br />With integer gradients well embedded, attention was then turned to unit fractions. There was a great deal of discussion about drawing 'a third of a square up' for a single square right. The beauty of our definition of gradient here was that it allowed us to use a proportional argument to build up to the idea of drawing 3 squares right to go single square up; if one square right takes you a third of a square up, then 2 right will take you two-thirds up and 3 right will take you three-thirds (i.e. one whole). What was very quickly showed up here was a lack of security with the concept of fractions and counting in fractions (this was Year 8 low prior attainers) and so I am sure that some pupils then started adopting this as a procedure. We were then able to build up to non-unit fractions, both positive and negative, all the time drawing one line short, and then at least one line longer (in preparation for the time where we would draw lines that span a whole coordinate axis).<br /><br />It was only after we had really secured the drawing of gradients of all types that we moved onto finding gradients of pre-drawn lines, which was simply then the reverse process, i.e. how many squares up/down for one square right? Again a nice proportional argument was used when the gradient was fractional. By the end of this there were pupils in the bottom set of Year 8 able to find and draw gradients like one and three-fifths.<br /><br />The next part of the sequence wasn't nearly as effective. I went back to the idea of linking gradient and intercept to equations, and although pupils were identifying gradients with ease, and drawing gradients with ease, the extra bits of y-intercept and algebraic equations wasn't so thoroughly explored and the kids struggled. I almost feel like I would have liked to have left this and then come back to it as an application of the work we had done on gradient later in the year; when I design my own mastery scheme I will almost certainly separate these parts and deal with gradient as a concept on its own before looking at algebra applied to straight line geometry at a different point in the scheme.<br /><br />My advice to anyone dealing with gradient would be to spend time really exploring this properly and not just rushing to using it to define/draw lines.</div>Peter Mattockhttps://plus.google.com/113661418069132691177noreply@blogger.com1tag:blogger.com,1999:blog-2500447090923756998.post-53582932424890287502017-04-15T12:14:00.000-07:002017-04-15T12:14:20.318-07:00The importance of evidence informed practice<div dir="ltr" style="text-align: left;" trbidi="on">I wanted to title this post the importance of evidence <b>informed</b> practice, but I cannot put bold words in the title unfortunately. There has been much discussion about this idea on edu-twitter recently, some of which I have involved myself in, and so I thought I would take the time to flesh my points out more fully in a blog.<br /><br />One of the quotes that I have seen that created a bit of controversy around this issue was used in the Chartered College of Teaching conference in Sheffield. The session delivered by John Tomsett, Head teacher of Huntington school in York and author of the "This much I know..." blog and book series. The quote was taken from Sir Kevan Collins, CEO of the Education Endowment Foundation:<br /><br /><div style="text-align: center;">"If you're not using evidence, you must be using prejudice."</div><div style="text-align: center;"><br /></div><div style="text-align: left;">This quote caused quite a bit of disagreement, with some people very much in favour of the sentiment, and some taking great exception to the provocative language used.</div><div style="text-align: left;"><br /></div><div style="text-align: left;">I had an interesting discussion on twitter about this quote, with my interlocutor seeming to hold to the viewpoint that because all children are different that any attempt to quantify our work with them is best avoided. Their argument goes that the perfect evidence-based model for classroom practice is an unobtainable dream, and so the effort to create one is wasted. To me the point of evidence informed practice is not to try and create the perfect evidence-based model, but rather to ensure teachers can learn from the tried and tested approaches of their peers; to stop them falling into traps that people have fallen into before, and to allow teachers to judge the likelihood of success of different possible paths. To bring another famous quote into the mix, "If I have seen further it is by standing on the shoulders of Giants." (Isaac Newton). In the same vein, we don't every new teacher to have to reinvent the wheel, we want them to be able to learn from those who have faced similar challenges and found solutions (or at least eliminated possible solutions).</div><div style="text-align: left;"><br /></div><div style="text-align: left;">One of the accusations that has been levelled at educational researchers is that they are 'experimenting on kids'. This is one of my least favourite arguments against evidence informed practice as its proponents must either be ignorant of how researchers operate or be feigning ignorance in order to make a point that isn't worth making. At some level everything we try in the classroom has a risk of failure; even the best practitioners don't get 100% understanding from every child in every lesson. The big point here though is that no one goes into the classroom with anything other than an expectation that what they are going to do is going to work, and this goes for researchers as much an any other professional, and is true in fields other than education. It would seem that some of the critics of evidence-based practice see researchers as a bunch of whacked-out lunatics wanting to try their crazy, crackpot theories out on unsuspecting pupils. In fact most researchers are following up on promising research that has already been undertaken, and so in theory their ideas should have a greater chance of success than a teacher whose view of the classroom is not informed by evidence. Even when researchers are trying totally new approaches, they are tried from a strong background and with a reasonable expectation of success. It is precisely the opposite of the view that some seem to hold, and in fact it is those who don't engage with educational research that are more likely to have some crackpot idea and then not worry so much about its success. </div><div style="text-align: left;"><br /></div><div style="text-align: left;">One of the situations I posed on twitter was the situation of the teacher new into a school, and therefore taking on new classes. Let us further suppose that said teacher is teaching in a very different setting to that which they are used to; perhaps a change of phase, a change of school style (grammar to comprehensive may well become more prevalent), or even just a change of area (leafy suburb to inner-city say). Now this teacher has two choices in order to prepare for their first day in their new classroom. Their first choice is to read something relevant and useful about the situation they entering, They could talk to teachers in their network that have experience in their situation, including in the school they are going to be working. They could inform themselves about the likely challenges, the likely differences, and the ways that people have handled similar transitions successfully in the past and then use this to make judgements about how they are going to manage this change. Alternatively they could not, either sticking blindly to their old practice, or making up something completely random. I know which one I would call professional behaviour. </div><div style="text-align: left;"><br /></div><div style="text-align: left;">When faced with this situation, the person with whom I was having the conversation sidestepped this choice and suggested that all would be well because they have a teaching qualification. Of course this ignores what a teaching qualification aims to do; the whole point of a teaching qualification is to lay down patterns for this sort of professional practice. This is one of the big reasons I was very much against the removal of HEI from teacher training. The idea of teacher training is to try and provide this dual access to practical experience through school placement along with skills in selecting and accessing suitable research and evidence from outside of your experience to supplement the gaps in your own practice. A teaching qualification has to be the starting point of a journey into evidence-informed practice, not the end point. One doesn't emerge from the ITT year as anything approaching the effective teachers that they have the potential to become; and the only way they will do so is by engaging with the successful practice of other teachers and using this to develop and strengthen your own practice and experience.</div><div style="text-align: left;"><br /></div><div style="text-align: left;">One other criticism levelled at those engaging with research and using it as the backbone of their practice is that the outcomes measured in order to test the success of the research are very often the results of high-stakes tests, and that these may not be the most appropriate measures of success. I have some sympathy with this point of view; I can see for example why people would baulk at the idea that the impact of using Philosophy for Children can and should be measured by their combined KS2 maths and English scores, which is what is happening in the EEF funded trial. However if we bring it back a notch we should ask ourselves what we are trying to achieve from the intervention. Ultimately I could argue that the purpose of any intervention in school is to try and make pupils more effective at being pupils, i.e. being able to study and learn from their efforts. Whether the intervention is designed to address gaps in subject knowledge, problems with learning behaviours or improve development in a 'soft-skill', the eventual intent is the same; that these pupils will be able to take what they have learned and use it to be more successful pupils in the future. Now I am not going to stand up and say that the way we currently measure outcomes from education is an effective way of doing so, but what I will say that is that however we choose to measure outcomes from education, any intervention designed to improve access to education has to be measured in terms of those outcomes. I am also not going to necessarily stand here and say that every single thing that goes on in schools should be about securing measurable outcomes for education (and I know many educators who would make that argument) but then I would argue that these things should not be attracting their funding from education sources. If an intervention is expected to benefit another aspect of a pupil's life, but it is not reasonable to expect a knock-on effect on their education (and when you think about it like that, it becomes increasingly difficult to think up sensible examples of interventions that might fit that bill) then it needs to be funded through the Health budget, or the Work and Pensions budget, or through whichever area the intervention is expected to impact positively.</div><div style="text-align: left;"><br /></div><div style="text-align: left;">Schools are messy places, subject to a near-infinite number of variables, very few of which can be controlled. It is virtually impossible to ensure that any improvement in results is due to one specific intervention; often several factors are at play. Does this mean, however, that we shouldn't experiment in the classroom, provided we have a reasonable expectation of success? Does this mean that we shouldn't attempt to quantify any success that we have that could, at least in part,be attributed to the change we made? Does this mean that we shouldn't share the details of this process, so that others can adopt and adapt as necessary, and then in turn share their experiences? To me this is precisely how a professional body of knowledge is built up, and so if teachers are going to lay claim to the status of 'professionals' then engagement with this body of knowledge has to be a given (provided they are well supported to do so). If you have the support to access this evidence, and then simply refusing to do so, then I would argue you certainly are using prejudice; either prejudice against the idea of research impacting your practice at all, or prejudice against the teachers/pupils that formed the research from which you might develop. Prejudice has no place in a professional setting, and no teacher should ever allow their prejudices to stand in the way of the success of the pupils in their care.</div></div>Peter Mattockhttps://plus.google.com/113661418069132691177noreply@blogger.com2tag:blogger.com,1999:blog-2500447090923756998.post-24416641778895833202017-03-01T08:14:00.000-08:002017-03-01T08:26:21.047-08:00Methods of last resort 3 - Straight line graphs<div dir="ltr" style="text-align: left;" trbidi="on">The linear relationship is probably one of the most fundamental relationships in all of mathematics. Functions that have a constant rate of change are the basis of our most rudimentary geometrical transformations, conversions and correlations. It should be fair to say that ensuring pupils have a proper grasp of linear relationships should be an important part of any mathematics curriculum; and yet many pupils are only given a very narrow view of these key mathematical constructs.<br /><br />Most pupils first view of the graphs of linear relationships between two variables are through algebra in the form <i>y</i> = <i>mx </i>+ <i>c</i>. Pupils will be given equations of this form, and asked to substitute to find coordinates and then plot coordinates to draw lines. <b>Some </b>pupils may be given the opportunity to draw parallels between the equation and the relationship between the variables <i>x</i> and <i>y</i> but not all. Eventually concepts like gradients and intercepts will be taught, and here is where the narrowing will begin. Most pupils will be given an algebraic definition of gradient, such as "change in <i>y</i> over change in <i>x</i>" or similar. Can we first be very clear from the start please that this is not what gradient is, this is just one way to find the gradient if you happen to know the horizontal and vertical distance travelled (for those people who think I am being picky, another way to find the gradient is to take the tangent of the angle the line makes with the horizontal, which is seldom taught in this way).<br /><br />What gradient actually is is the vertical distance travelled for a unit increase in horizontal distance. Dividing a given vertical by a given horizontal will calculate the the value, as will applying the tangent function to the angle made with the horizontal, but neither tell you what it actually <b>is</b>. Pupils should have a proper understanding of what gradient is, before they begin calculating it (in my opinion). But this is not actually the point of this blog post so I will get back on track...<br /><br />Once gradient is 'taught' the link between its value and the value of <i>m</i> in the formula given above is very quickly highlighted, often either explicitly or through some form of 'discovery'. Here comes the second narrowing - from this point onward virtually every attempt to ascertain the value of the gradient of a particular line when given any form of linear algebraic relationship invariably leads back to writing the equation in the form <i>y</i> = <i>mx </i>+ <i>c</i>. Remember lines are very often defined in a different form; <i>x</i> + <i>y</i> = 5, 3<i>x</i> + 2<i>y</i> + 4 = 0 etc. Ask any competent school age pupil to find the value of the gradient of these lines, and I will guarantee that the vast majority of the time a rearrangement into the form <i>y</i> = <i>mx </i>+ <i>c </i>is attempted if the pupil is even able to attempt the problem at all. And while this approach is perfectly correct and if done well will reveal the value of the gradient, it isn't the only approach; many pupils labour in ignorance when better methods may be applied.<br /><br />Take the line <i>x</i> + <i>y</i> = 5 for example. Now for most mathematicians it would be straightforward to rearrange this to give <i>y</i> = -<i>x</i> + 5, and hence find the value of the gradient of -1, and the y-intercept of (0,5). However I would argue at least equally straightforward would be to say "the points (0,5) and (5,0) are on the line, and so the value of the gradient = -5/5 = -1 and the y-intercept is (0,5) [and, by the way, the <i>x</i> intercept is (5,0) - which is not nearly so often asked about]. To be fair, there is probably not a huge difference in the mechanics, but as Anne Watson highlights in her blog (see postscript below) there is perhaps a difference in pupils understanding of what this line actually looks like, as well as providing more of an opportunity to reinforce the idea of vertical distance travelled for unit horizontal distance.<br /><br />If we then take the line 3<i>x</i> + 2<i>y</i> + 4 = 0, the rearrangement is a bit messier - I know plenty of pupils that wouldn't be able to rearrange successfully. However it is still a rearrangement that you would want pupils to be able to do and expect that they could if they had the proper grounding in inverse operations etc. The other side of this though is that I can quite quickly see that the point (0,-2) is on this line, and that the point (-1⅓, 0) is on this line. So I can also calculate the gradient as -2/1⅓ = 1<span style="font-family: "calibri" , sans-serif; font-size: 11pt;">½, </span>as well as tell you about the <i>x-</i>intercept and <i>y</i>-intercept. Perhaps even more straightforwardly I could have told you that the point (1, -3<span style="font-family: "calibri" , sans-serif; font-size: 14.6667px;">½</span>) is on the line, and so arrived at the value of the gradient immediately, I have gone 1<span style="font-family: "calibri" , sans-serif; font-size: 14.6667px;">½</span> units down when <i>x</i> increased by 1 (from 0 to 1).<br /><br />Whether you want to consider rearrangement to the form <i>y</i> = <i>mx</i> + <i>c</i> as a 'method of last resort' or not is up to you; clearly it is an important mathematical idea that relationships can be expressed in different forms. However I would suggest that it is not the only idea that pupils should be able to draw upon when talking and thinking about finding gradient values, and that we should be aiming to give pupils a range of strategies linked to a deeper understanding of what gradients, and lines of constant gradient, are.<br /><br />Postscript: Emeritus Professor of Education at Oxford University Anne Watson recently released a blog about a similar topic (and actually using one of the same equations!) <a href="https://educationblog.oup.com/secondary/maths/subbing-zero" target="_blank">here</a>. I have actually been writing this blog post since late January and was just trying to find time to finish it off, so wanted to go ahead and publish it anyway!<br /><div class="MsoNormal"><o:p></o:p></div><div class="MsoNormal"><o:p></o:p></div></div>Peter Mattockhttps://plus.google.com/113661418069132691177noreply@blogger.com0tag:blogger.com,1999:blog-2500447090923756998.post-75440822377492052242017-01-28T13:03:00.000-08:002017-01-28T13:06:02.226-08:00Multiplicative Comparison and the Standards Unit diagram<div dir="ltr" style="text-align: left;" trbidi="on">Recently I have been doing quite a lot of work with proportion (one way or another) across a lot of my classes. My Year 11 classes are looking at rates of change (gradient is a proportional relationship between change in x and change in y) and probability (the proportion of outcomes that fit a criteria) respectively. My Year 8 classes are working on probability and unit conversion. My Year 10 are working on compound units. I have been realising how versatile this diagram is:<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-xnVqyZauvAg/WIzcvi-2wEI/AAAAAAAABh8/RZiVDrqg6xUl1QElKzcdNqISWOtTkgwJwCLcB/s1600/Standards%2BUnit%2BDiagram.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="262" src="https://2.bp.blogspot.com/-xnVqyZauvAg/WIzcvi-2wEI/AAAAAAAABh8/RZiVDrqg6xUl1QElKzcdNqISWOtTkgwJwCLcB/s320/Standards%2BUnit%2BDiagram.png" width="320" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: left;">For those that don't recognise this picture, it is from N6 of the Standards Unit, which is about developing proportional reasoning. I call it 'The Standards Unit Diagram' whilst a Twitter colleague (@ProfessorSmudge) calls it a ratio table. It is probably also the best diagram I have ever seen for multiplicative comparison, which is pretty much the basis of all division and proportionality.</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">Lets say I want to convert between cm and metres, in particular 350 cm into metres. The diagram might look something like this:</div><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-tyjEH_mImLo/WIzsViZPkTI/AAAAAAAABiM/6PsV3gtaFkASh_bMAdoPYk11b7J7wjS1QCLcB/s1600/Standards%2BUnit%2BDiagram%2Bcm%2Band%2Bm.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="293" src="https://1.bp.blogspot.com/-tyjEH_mImLo/WIzsViZPkTI/AAAAAAAABiM/6PsV3gtaFkASh_bMAdoPYk11b7J7wjS1QCLcB/s320/Standards%2BUnit%2BDiagram%2Bcm%2Band%2Bm.png" width="320" /></a></div><div class="separator" style="clear: both; text-align: left;">This diagram really nicely shows off the twin relationships that are present in all proportional relationships, i.e. that one variable is always a certain number of times bigger than another (the conversion factor or rate of change, in this case 100 cm/metre) and the fact that any multiple of one of the variables is matched by a corresponding scaling in the second variable (in this case, the fact that the number of cm has been multiplied by 3.5 implies that the same also happens to the number of metres). Notice that it is only strictly necessary to find one of the relationships to solve the problem, but nonetheless it is clear that two exist (in this case, depending on the level of the pupils, the focus may be on the use of 100 rather than the scaling here). </div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">There is some anecdotal and written evidence (I remember reading an article but honestly can't remember what it was called) that most people will naturally focus on the scaling in a proportional problem, particularly if the scaling is obvious (the variable gets doubled or trebled for example), but what is nice about the diagram above is that it gives equal focus to both relationships that exist.</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">Below is one of the diagrams I used to highlight the commonality behind representation that was possible using this approach in my recent talk to Heads of Maths at the LaSalle Education HOM conference, sponsored by Oxford University Press. This diagram was used to solve the percentage problem "A jacket costs £84 inclusive of VAT at 20%. Work out the price before VAT." which is a fairly classic GCSE reverse percentage question.</div><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/--3LEMDqXRXw/WIzwo65k5SI/AAAAAAAABiY/_cE4x7_Y34kyk6e6vQoB0K4zZkb9Svl6gCLcB/s1600/Standards%2BUnit%2BDiagram%2BVAT%2Bproblem.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="https://1.bp.blogspot.com/--3LEMDqXRXw/WIzwo65k5SI/AAAAAAAABiY/_cE4x7_Y34kyk6e6vQoB0K4zZkb9Svl6gCLcB/s320/Standards%2BUnit%2BDiagram%2BVAT%2Bproblem.png" width="218" /></a></div><div class="separator" style="clear: both; text-align: left;">Now, as was pointed out in the session, what is clear from this diagram is that the most 'efficient' way to solve this problem is simply to divide 84 by 1.2. However the diagram does highlight a possible alternative, and more importantly highlights the commonality in the relationship here which is the essence of all proportion and division, namely "<i>100 is to 120 as 5 is to 6 as <b>what</b> is to 84?</i>"</div><div class="separator" style="clear: both; text-align: left;"><i><br /></i></div><div class="separator" style="clear: both; text-align: left;">I would argue quite strongly that very few pupils actually understand division and proportion as they don't understand that this comparison is at the heart of all of these types of relationship. Every division, every proportion are basically saying "If a is to b as c is to d then a proportion exists". The one that definitely caught the eye at the aforementioned head of maths conference was this demonstration of using the diagram to highlight the commonality of relationship when dividing with fractions, in this case solving the fractional division ¾ ÷ ⅚</div><div class="MsoNormal"><o:p></o:p></div><!--[if gte msEquation 12]><m:oMathPara><m:oMath><m:f><m:fPr><span style='font-family:"Cambria Math",serif;mso-ascii-font-family:"Cambria Math"; mso-hansi-font-family:"Cambria Math";mso-bidi-font-family:Calibri; mso-bidi-theme-font:minor-latin;font-style:italic;mso-bidi-font-style:normal'><m:ctrlPr></m:ctrlPr></span></m:fPr><m:num><i style='mso-bidi-font-style:normal'><span style='font-size:11.0pt; line-height:107%;font-family:"Cambria Math",serif;mso-fareast-font-family: Calibri;mso-fareast-theme-font:minor-latin;mso-bidi-font-family:Calibri; mso-bidi-theme-font:minor-latin;mso-ansi-language:EN-GB;mso-fareast-language: EN-US;mso-bidi-language:AR-SA'><m:r>3</m:r></span></i></m:num><m:den><i style='mso-bidi-font-style:normal'><span style='font-size:11.0pt; line-height:107%;font-family:"Cambria Math",serif;mso-fareast-font-family: Calibri;mso-fareast-theme-font:minor-latin;mso-bidi-font-family:Calibri; mso-bidi-theme-font:minor-latin;mso-ansi-language:EN-GB;mso-fareast-language: EN-US;mso-bidi-language:AR-SA'><m:r>4</m:r></span></i></m:den></m:f><i style='mso-bidi-font-style:normal'><span style='font-size:11.0pt;line-height: 107%;font-family:"Cambria Math",serif;mso-fareast-font-family:Calibri; mso-fareast-theme-font:minor-latin;mso-bidi-font-family:Calibri;mso-bidi-theme-font: minor-latin;mso-ansi-language:EN-GB;mso-fareast-language:EN-US;mso-bidi-language: AR-SA'><m:r>÷</m:r></span></i><m:f><m:fPr><span style='font-family:"Cambria Math",serif; mso-ascii-font-family:"Cambria Math";mso-hansi-font-family:"Cambria Math"; mso-bidi-font-family:Calibri;mso-bidi-theme-font:minor-latin;font-style: italic;mso-bidi-font-style:normal'><m:ctrlPr></m:ctrlPr></span></m:fPr><m:num><i style='mso-bidi-font-style:normal'><span style='font-size:11.0pt; line-height:107%;font-family:"Cambria Math",serif;mso-fareast-font-family: Calibri;mso-fareast-theme-font:minor-latin;mso-bidi-font-family:Calibri; mso-bidi-theme-font:minor-latin;mso-ansi-language:EN-GB;mso-fareast-language: EN-US;mso-bidi-language:AR-SA'><m:r>5</m:r></span></i></m:num><m:den><i style='mso-bidi-font-style:normal'><span style='font-size:11.0pt; line-height:107%;font-family:"Cambria Math",serif;mso-fareast-font-family: Calibri;mso-fareast-theme-font:minor-latin;mso-bidi-font-family:Calibri; mso-bidi-theme-font:minor-latin;mso-ansi-language:EN-GB;mso-fareast-language: EN-US;mso-bidi-language:AR-SA'><m:r>6</m:r></span></i></m:den></m:f></m:oMath></m:oMathPara><![endif]--><!--[if !msEquation]--><!--[endif]--><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-MvzOvOpGXh0/WIz0uqAmw_I/AAAAAAAABik/NNeU07X_-cIsnGHumqmm8eVcB4JIIRwQQCLcB/s1600/Standards%2BUnit%2BDiagram%2BFraction%2Bdivision.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="https://1.bp.blogspot.com/-MvzOvOpGXh0/WIz0uqAmw_I/AAAAAAAABik/NNeU07X_-cIsnGHumqmm8eVcB4JIIRwQQCLcB/s320/Standards%2BUnit%2BDiagram%2BFraction%2Bdivision.png" width="162" /></a></div><div class="separator" style="clear: both; text-align: left;">This can be summarised as "I don't know how three-quarters relates to five-sixths, but I know that it is the same as how 3 relates to twenty-sixths, which is the same as how 18 relates to 20, which is the same as how 9 relates to 10." The conclusion is that ¾ ÷ ⅚ = 9/10.</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">This way of viewing division as a proportional relationship, that can be manipulated in the same way as other relationships (i.e. as a multiplicative comparison) is a powerful interpretation, and one that I would argue that no pupil should be without. Even regular division of two integers can be seen in this way, particularly given the understanding that regular division is a multiplicative comparison to 1:</div><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-4IjCO4hpKms/WIz7dwptAUI/AAAAAAAABi0/2M2fjhaXpdIRSoLGM53CZzCJtUY7_ajDQCLcB/s1600/Standards%2BUnit%2BDiagram%2Binteger%2Bdivision.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="474" src="https://2.bp.blogspot.com/-4IjCO4hpKms/WIz7dwptAUI/AAAAAAAABi0/2M2fjhaXpdIRSoLGM53CZzCJtUY7_ajDQCLcB/s640/Standards%2BUnit%2BDiagram%2Binteger%2Bdivision.png" width="640" /></a></div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="MsoNormal"><o:p></o:p></div><div class="separator" style="clear: both; text-align: left;">So this is literally "<i>75 is to 15 as <b>what</b> is to 1?</i>" with the 'what' being 5, and similarly with the second "<i>23 is to 5 as <b>what</b> is to 1?</i>", with the 'what' this time being twenty three-fifths or alternatively four and three-fifths. Indeed, the earlier fractional division could well benefit from a final line showing the equivalent relationship to 1 as '9/10 to 1'.</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">Even if you don't ultimately like the diagram or the approach, I would argue that no pupil's (or teacher's) view of proportionality or division is complete without understanding this idea of multiplicative comparison. However you choose to represent or structure it, giving your pupils an insight into this aspect of division is pretty much guaranteed to give them a deeper insight into what it it means to think multiplicatively.</div><div class="separator" style="clear: both; text-align: left;"><br /></div><br /></div>Peter Mattockhttps://plus.google.com/113661418069132691177noreply@blogger.com2tag:blogger.com,1999:blog-2500447090923756998.post-84967321480225497582017-01-20T14:22:00.002-08:002017-01-20T15:04:41.679-08:00Christmas Mock Grade Boundaries - our story<div dir="ltr" style="text-align: left;" trbidi="on">Ok, I was wrong last time, this is definitely the most dangerous blog I have posted; if by dangerous I mean fraught with the capacity to be wrong and inconvenience a lot of people. So I will preface by saying I am very sorry if you base anything off of this post and it turns out to be wrong; we are all just guessing here really and guesses can go wrong. Still if it helps people clarify their own thinking, or supports people that wouldn't otherwise have a way of meeting the demands of their senior teams or other stakeholders then I suppose it is worth a little egg on the face if it turns out wrong. So here is the story of our Christmas Year 11 mocks and grade boundaries:<br /><br />We sat mock exams just before the Christmas holidays,which meant that by about a week after we returned from Christmas I had pretty much all of the results from our 285 pupils (of which about 250 or so actually sat all 3 papers). By this time I also had the results of one other school with about 180 pupils in Year 11, with about 160 that had sat 3 papers, and another much smaller school that were only going to sit two papers. I used a similar process that I had at the end of Year 10; I apply a scaling formula to the Foundation paper to make it directly comparable with Higher which has worked well in the past, and then applied the proportions and other boundary setting details which have been well publicised by the exam boards and great people like Mel at @Just_Maths. This led to this set of boundaries, which we applied to our pupils:<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-mB2NmLpVb4w/WIJ8j_0Wo_I/AAAAAAAABhU/bk79_BEbNdkULuAaFEepNHlHuulk8rC3QCLcB/s1600/Brockington%2BGrade%2BBoudaries.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="94" src="https://3.bp.blogspot.com/-mB2NmLpVb4w/WIJ8j_0Wo_I/AAAAAAAABhU/bk79_BEbNdkULuAaFEepNHlHuulk8rC3QCLcB/s640/Brockington%2BGrade%2BBoudaries.png" width="640" /></a></div><br />I wasn't completely enamoured with these - I knew for example that the 9 and 8 were lower compared to where I expect them to be in the summer, and in general I thought that maybe all of the higher scores were a little low (although as you go down the grades I expect them to be closer to the real end values in the summer of 2017). I did like the Foundation ones, they seemed to sit well with what I was expecting. Given that pupils still have 5 months before they sit the real thing though, I thought these were acceptable for now. At the time I couldn't make them public, as our pupils were not given their grades back until their mock results day today.<br /><br />Literally two days after we had inputted mock grades, AQA released the population statistics for the cohort. I was pleased to see that our Higher pupils had scored above average compared to the population, and our Foundation had scored lower. I took this to mean that our tiering choices were about right, although as any good statistician knows making judgements based on averages alone is a dangerous thing to do and we did have to look at the pupils at the lowest end of the higher paper scores as we had a large range of values.<br /><br />Although we had already set boundaries I work with a group of 5 other schools, many of which were doing their mock exams after Christmas and so would be needing boundaries - originally the plan would be to collect all of their results and set the boundaries (which would have given us a cohort over 1000, and so had at least some hope of being reasonable). With the support of some excellent colleagues who will remain nameless I managed to get hold of some data about the population rankings that were attributed to certain scores for Higher and Foundation. This allowed for the setting of the grade 7 and 9 (and therefore also 8) at Higher, based from last year's proportions and the tailored approach as outlined in the Ofqual documentation as well as the grade 1 at Foundation. The grade 4 proved more problematic, as there was no detail about how the Higher and Foundation rankings compared to each other (I am reliably informed that it is impossible to accurately do this without the prior attainment from KS2, although my scaling formula does seem to produce quite similar results).<br /><br />I was able to get hold (from a source who will definitely remain nameless) of the proportions of C grades that were awarded to 16 year olds last year for the separate tiers and based on this I was able to map out the separate values for grade 4 on the Higher and Foundation tier. This also allowed the setting of the 5 and 6 on the Higher tier, and 3 and 2 on the Foundation tier. Although it is still up for consultation (I believe), I also awarded the 3 using the approach that has been used in previous years for setting the E grade boundary on Higher, namely halving the difference in the grade 4 and 5 boundary, and then subtracting this from the grade 4 boundary.The trickiest one was actually the 5 boundary on Foundation, as there is no real guidance over this one; in the live exam I believe this will be set based on comparison of pupils scripts and prior attainment (although if anyone knows more about this I would be happy to be corrected). In the end I did have to make a bit of educated guess work with comparison back between my own papers, and ended up with boundaries for the whole AQA cohort that look like this:<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-CFJYqe3WanM/WIKHlk1eHGI/AAAAAAAABhk/UJj6X8hjX5YhjfTjHByoVQCjomiRBjN6wCLcB/s1600/AQA%2Bcohort%2Bboundaries.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="58" src="https://1.bp.blogspot.com/-CFJYqe3WanM/WIKHlk1eHGI/AAAAAAAABhk/UJj6X8hjX5YhjfTjHByoVQCjomiRBjN6wCLcB/s640/AQA%2Bcohort%2Bboundaries.png" width="640" /></a></div><br />I was quite pleased with the similarity of these to our boundaries, although it would appear my scaling formula is a little harsh to the Foundation pupils for mock exams (it does work quite well for real exams though). At this point though I should pass on some major health warnings and notices:<br /><br /><ul style="text-align: left;"><li>These boundaries are NOT endorsed by AQA, and they will rightly maintain that it is impossible to set grades or boundaries for exams without prior KS2 pupil data. Although this does use data available on the portal from the AQA portal, it is only my interpretation of it.</li><li>There are two big assumptions used to make these boundaries, which are unlikely to completely bear out in reality. In particular, there is an assumption that the proportions highlighted in the Ofqual document are going to pretty much repeat from last year to this year; i.e. that the cohorts from Year 11 in 2016 and 2017 are pretty similar. In reality we are told that Year 11 2017 have slightly higher prior attainment than those in 2016 (although the published data does say that the two are not directly comparable). The other major assumption is that the proportions of grade 4 at Higher and Foundation will roughly match the proportions of grade Cs awarded at Higher and Foundation last year. This assumption is certainly unlikely to be true, we are already hearing that schools are entering significantly more pupils at Foundation tier (myself included compared to the proportion I used to enter in my previous schools), which is likely to raise the quality of candidate at both Foundation and Higher tier. If this is the case for the current mock data it would have the effect of lowering the Foundation boundaries (although they seem to fit too nicely for me to believe they will go lower - just a gut feeling though) and raising the Higher boundaries (which seems likely in reality).</li><li>We mustn't forget that a lot can happen in the next 5 months, and I would expect most of the cohort to improve their scores; I would still expect the 9, 8 and 7 to be noticeably higher than these values in the summer, although I don't think the 4 boundary will shift up by as much as some people might think. In reality these boundaries are useful in the very specific circumstance that a pupil has completed all 3 papers from AQA practice set 3, and that they have done so after about a year and a bit to a year and half of GCSE course study.</li></ul><div>So that is our story, up until about 2 or 3 hours ago. If it helps people then great; if you disagree then fine; if you use it and it turns out wrong, well you were warned...</div></div>Peter Mattockhttps://plus.google.com/113661418069132691177noreply@blogger.com5tag:blogger.com,1999:blog-2500447090923756998.post-34808185293266831312016-11-24T15:02:00.001-08:002016-11-24T15:02:34.646-08:00New GCSE Grade Boundaries - my thoughts<div dir="ltr" style="text-align: left;" trbidi="on">I am going to start this blog by making the point clear, it is impossible to accurately grade pupils on the new GCSE for Maths. Completely impossible. Anyone that tells a pupil that they have achieved a particular grade is at best making an educated guess and at worst is making something up. If there is any way you can avoid giving pupils grades, making predictions of pupils eventual grades or even talking about future grades with any stakeholder then you should take the opportunity and avoid it like it is a highly contagious illness.<br /><br />That said, many schools are not giving departments and heads of maths the opportunity to avoid it. There are plenty of schools out there requiring staff to predict grades for pupils (some as low as in Year 7!), or provide current working grades. Even when schools don't require this, Year 11 pupils looking at the next stage are being asked for predicted grades in English and Maths from colleges or other post-16 providers. I have been in touch with many new department heads that are struggling to answer the demands of schools, parents and pupils with regards the new GCSE grading and so this post is designed to give some support and guidance for anyone who finds themselves in this unenviable position.<br /><br />You will hear people say that you cannot grade at all for the new GCSE, and I can see where they are coming from (see paragraph 1!). I do believe that it is possible to make some educated guesses about what the landscape is going to look like - we do have a reasonable amount of information to work on and one thing mathematicians are good at is building models for situations with many variables. We just have to be clear about our modelling assumptions and how that affects the accuracy of the predictions from the model. Lets start with the information Ofqual have provided:<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-xF5JD7Z-xYM/WDdMSXH9QmI/AAAAAAAABfM/LoNZC6WVTZgSSzIoCJ_1wLQynDaykPidACLcB/s1600/Ofqual%2Bpostcard.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="430" src="https://2.bp.blogspot.com/-xF5JD7Z-xYM/WDdMSXH9QmI/AAAAAAAABfM/LoNZC6WVTZgSSzIoCJ_1wLQynDaykPidACLcB/s640/Ofqual%2Bpostcard.png" width="640" /></a></div><br />This is probably the most viewed guide that teachers and schools have with regards the new grading. The key line in this is actually 'Students will not lost out as a result of the changes'. That means that if you have a kid in front of you that is a nailed on C for the old GCSE, they are at least a 4 on the new. Similar for A and 7, and G and 1. Of course this doesn't help with the borderline kids, but it is somewhere to start. The most updated postcard also has this information:<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-EQuQTK-Ry-w/WDdNzqvmUTI/AAAAAAAABfY/N_zfCAVZr9ouypgAR4UFUGDxdZJ1l8peACLcB/s1600/Extra%2BInfo%2Bfrom%2BOfqual%2Bpostcard.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="193" src="https://1.bp.blogspot.com/-EQuQTK-Ry-w/WDdNzqvmUTI/AAAAAAAABfY/N_zfCAVZr9ouypgAR4UFUGDxdZJ1l8peACLcB/s400/Extra%2BInfo%2Bfrom%2BOfqual%2Bpostcard.png" width="400" /></a></div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">What this means that if you are assessing pupils (mock exams or similar), once you have set the 4 and 7 boundary, you can set the 5 and 6 boundaries arithmetically. Although it doesn't say it here, I am reliably informed (he says, waiting to be shot down!) that the same is true for grades 2 and 3; they should be set equally between 1 and 4. The upper grades can also be calculated, using the tailored approach for grade 9. The tailored approach can be summarised as:</div><div class="separator" style="clear: both; text-align: center;"><i><br /></i></div><div class="separator" style="clear: both; text-align: center;"><i>Percentage of those achieving at least grade 7 who should be awarded grade 9 = 7% + 0.5 * (percentage of candidates awarded grade 7 or above).</i></div><br />By my calculations on last years figures, this will mean nationally about 15% of the pupils awarded 7+ will be in the 9+ bracket, which will end up being about 2.4% of the total cohort (based on 15.9% A* and A in 2016 translating to a broadly similar proportion for 7+). Of course if your cohort is very different to national then it shouldn't be massively far out to apply the tailored approach to your A and A* figure from last year (if you have one - I don't as this is the first year for GCSE) once you have adjusted for differences in the starting points of the cohort. This means we can have a reasonable stab at a grade 9 boundary for any mock exam we set. The grade 8 boundary should then be set halfway between 7 and 9.<br /><br />Using this approach it should be reasonable to generate some grade boundaries for a mock exam by looking at kids that would definitely have secured a C, A and G on the old GCSE exams, using their scores to set grade 4, 7 and 1 boundaries respectively, and then calculating the 9 and the others using the calculations Ofqual provides.<br /><br />Another approach that we (and several other groups have employed) has been to combine papers with other schools all doing the same board. This has allowed us to use proportional awards to set the 1, 4 and 7 boundaries statistically rather than through moderation - although it is still a bit unclear as to precisely what proportion will be used for the 4. This is the approach that the PiXL club among others also used, although from some points of view with varying degrees of success.<br /><br />This is all well and good for individual schools and cohorts, and setting retrospective boundaries when cohorts have already done mock exams, but what can we predict about the final exams? The true answer is very little, but perhaps not absolutely nothing. Using what we know it is possible to make some predictions about the likely distribution of the grade boundaries going forward, but with a very large margin for error built in, primarily because of the very different style that the assessment has which is very hard to quantify. We do know though that the balance of difficulty will shift in both sets of papers so that 50% of the Higher tier paper will be aimed at grades 7 plus, and similarly 50% of the Foundation tier will be 4+, which is between 10 and 20% increased on the current top two grades in each paper.<br /><br />We also know about the shift of material so that the Foundation tier will assess some material that is currently only Higher, and some of the material currently on Higher will no longer be assessed on Higher. Factoring all of this in we can make adjustments on current boundaries to make educated predictions at new boundaries. I will start by looking at the AQA boundaries for last year:<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-3YyHQN1jWyA/WDdafNCXX1I/AAAAAAAABfo/YmEp-N3a5G4OoFnmhduq3jkozpUL0DmFQCLcB/s1600/AQA%2BMaths%2Bgrade%2Bboundaries.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="68" src="https://4.bp.blogspot.com/-3YyHQN1jWyA/WDdafNCXX1I/AAAAAAAABfo/YmEp-N3a5G4OoFnmhduq3jkozpUL0DmFQCLcB/s640/AQA%2BMaths%2Bgrade%2Bboundaries.png" width="640" /></a></div><br />These are the boundaries converted to percentages for last year, and from these we can make some sensible adjustments. Given that there is now no D grade material on the Higher paper, it makes sense that the award of 3 (there is still a discretionary award of 3 similar to the current E award on the Higher) will come down towards where the E is now - around the 8 to 10% mark. The grade 4 will then have to come down as well to reflect the fact that all the D grade material is gone. With the D currently at 17.7% it is reasonable to predict that the 4 value will fall somewhere in the range of 15% to 25%. The B grade at 53.1% will also come down to nearer the current C grade - this won't translate automatically into 5 or 6, but given that B falls between 5 and 6 then 5 is likely to come in in the high 20s or low to mid 30s, with 6 likely to fall in the mid to high 30s to low 40s.<br /><br />It is almost certain that the grade 7 boundary will have to come down from 71% that the current A grade sits at. When you consider the loss of the D grade material which nearly all A grade+ pupils will be scoring well on, along with the increase in the amount of material at A grade/grade 7+ then one can justify quite a dramatic drop in the 7 grade boundary - with 50% of the paper at grade 7+ it is not outside the realms of possibility that the boundary for 7 will actually be below 50%. In reality something in the early to mid 50s is probably the most likely area for the 7 boundary, and almost certainly less than 60%. The 8 and 9 are probably the hardest to predict, because of the 9 calculation, and that 8 will be based on 9 and 7 together. It would be hard to see the 9 grade boundary being less than the current A* as this would defeat the whole reason for adding the extra grade into the top of the system. Currently 5.7% gain A*, so if 9 is going to halve this figure or better, then the expectation of a boundary somewhere between 90% and 96% would seem a fair prediction. If this is the case then the 7, 8 and 9 are going to be quite widely spaced, which is expected if they are going to allow distinguishing of candidates at the top end. If we take all of this into account, and apply to a total of 240 marks, we get boundaries somewhere around the ones below for the Higher tier:<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-I2erkDJwDP0/WDdoXvGWZpI/AAAAAAAABf8/yhqZpsoshNwK18A7Aq9geoRIn_kSN1BSACLcB/s1600/AQA%2BMaths%2Bgrade%2Bboundaries%2Bpredicted%2BHigher.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="62" src="https://2.bp.blogspot.com/-I2erkDJwDP0/WDdoXvGWZpI/AAAAAAAABf8/yhqZpsoshNwK18A7Aq9geoRIn_kSN1BSACLcB/s640/AQA%2BMaths%2Bgrade%2Bboundaries%2Bpredicted%2BHigher.png" width="640" /></a></div><br />I can see these being accurate to within 10 to 15 marks at a maximum, and significantly closer in some cases (points for me if I get any of the spot on!).<br /><br />Turning our attention to Foundation, we can do a similar 'analysis'. There is no reason that the grade 1 boundary should have to change much from the current G grade (except of course pupils really struggling to access the paper!) and so pupils are still likely to need in excess of 20% to be awarded a grade on Foundation (or perhaps a short way below). The most interesting here is the grade 4 boundary, with similar arguments for the 7 on Higher. There is reason to believe that this will have to come down significantly with the addition of extra, more demanding content in Foundation and the balance of the paper shifting to include more material at grades 4 and 5. A figure close to the current D grade percentage of around 55% seems rational, and it could even dip below 50% (I suspect that it won't as the balance of pupils sitting the Foundation paper instead of the Higher is likely to change so that there are more pupils that would score higher marks than currently sit the Foundation tier). Given this the grade 3 boundary and grade 2 boundary are calculable as equally spaced between the two. The grade 5 boundary at Foundation is probably the hardest to predict with any certainty as it likely to rely heavily on comparable outcomes with the Higher tier to set - if the 5 boundary for Higher has to be calculated then pupils awarded 5 on Foundation will need to be checked to make sure they are demonstrating similar understanding to those awarded 5 on Higher. I suspect it is likely to be above the current 66% for a C on Foundation, and have gone in on the low 70s. Based on this, my best guess for Foundation, with similar accuracy at all except Grade 5, looks like this:<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-IXaQeQalRcI/WDdubwgVFcI/AAAAAAAABgM/ky3QzeCgqk8mP5bszQqcCw4DNlCVYo-2QCLcB/s1600/AQA%2BMaths%2Bgrade%2Bboundaries%2Bpredicted%2BHigher%2Band%2BFoundation.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="62" src="https://4.bp.blogspot.com/-IXaQeQalRcI/WDdubwgVFcI/AAAAAAAABgM/ky3QzeCgqk8mP5bszQqcCw4DNlCVYo-2QCLcB/s640/AQA%2BMaths%2Bgrade%2Bboundaries%2Bpredicted%2BHigher%2Band%2BFoundation.png" width="640" /></a></div><br />A similar 'analysis' of the Edexcel boundaries yielded these results:<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-8SR-wc8dM94/WDdwjdi_00I/AAAAAAAABgY/1BJlwsIovE4TvkzHZQT-n4spa1X5hM5HgCLcB/s1600/Edexcel%2BMaths%2Bgrade%2Bboundaries%2Bpredicted%2BHigher%2Band%2BFoundation.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="62" src="https://2.bp.blogspot.com/-8SR-wc8dM94/WDdwjdi_00I/AAAAAAAABgY/1BJlwsIovE4TvkzHZQT-n4spa1X5hM5HgCLcB/s640/Edexcel%2BMaths%2Bgrade%2Bboundaries%2Bpredicted%2BHigher%2Band%2BFoundation.png" width="640" /></a></div><br />A big assumption here is that pupils continue to score better on Edexcel than on AQA, which by all accounts is not a good assumption to make. The tests from Ofqual suggested that pupils answered the AQA papers better than then Edexcel ones, so this second set of boundaries may well be less accurate than the others. Ultimately though, if you have nothing else you can use, and you absolutely must talk about grades etc with SLT, parents etc then this is the absolute best guess I can come up with; of course it remains to be seen how good a guess they are, so use these are your own peril as they come with precisely zero guarantees!</div>Peter Mattockhttps://plus.google.com/113661418069132691177noreply@blogger.com10tag:blogger.com,1999:blog-2500447090923756998.post-50174497503691388382016-11-06T03:54:00.000-08:002017-01-20T15:07:36.484-08:00Methods of last resort 2 - Order of Operations<div dir="ltr" style="text-align: left;" trbidi="on">Teaching the correct order of operations is possibly one of the most debated topics for maths teachers. In my #mathsconf8 session I was asked 'what is my problem with BIDMAS' and proceeded to outline times when this acronym is redundant (e.g. 4 x 3 ÷ 2) or even downright wrong (4 - 5 + 6 would mistakenly be given as -7 rather than the correct answer as 5). Various diagrams have been mooted as the solution to this, and there are several examples below:<div><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-BgNa9UQYC-0/WB8N4AL1sEI/AAAAAAAABew/DvDH9eXC3csOPoHekteFUfKAy31LFsGewCLcB/s1600/BIDMAS%2Breplacement%2Bdiagrams.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="105" src="https://2.bp.blogspot.com/-BgNa9UQYC-0/WB8N4AL1sEI/AAAAAAAABew/DvDH9eXC3csOPoHekteFUfKAy31LFsGewCLcB/s640/BIDMAS%2Breplacement%2Bdiagrams.png" width="640" /></a></div><div> </div><div>I have several issues with these diagrams, which can be summarised as:</div><div><br /></div><div>(a) It isn't specific enough for all of the possible functions that can be applied to numbers (even those that include square roots don't involve higher roots, and no mention of sin, cos, tan, log etc)</div><div><br /></div><div>(b) BRACKETS ARE NOT AN OPERATION (please forgive the shouting). This may seem like semantics but for me it is an important distinction - brackets are used to either alter or clarify the order of operations intended, but are not an operation in themselves (just a note on clarify, an example of this is 12 ÷ (3 x 4) needed clarity as without these brackets the answer would be 16 and not 1). If we are going to teach pupils to understand the maths they are doing then we need to be communicating understanding like this, and not allowing pupils to mistakenly believe that brackets are an operation themselves.</div><div><br /></div><div>But this post is not about teaching correct order of operations (although that segue has outlined my thoughts on it quite nicely); this is about when you wouldn't want pupils teaching using the correct order of operations in the first place. The example I used in my #mathsconf8 session was:</div><div><br /></div><h2 style="text-align: center;">673 x 405 — 672 x 405</h2><div>Any mathematician is definitely not applying the correct order of operations in this situation; and is quickly writing down that this is just 405. With the advent of 'teaching for mastery' gaining ground in mathematics education pupils are being increasingly exposed to questions like this when looking at distributive laws, or factorisation but I am yet to see it, or anything like it, thrown into a lesson on Order of Operations as a <b>non-example</b>. There is good evidence out there now to back up the idea that non-examples are important in communicating a concept and so if we are trying to communicate the correct order of operations we should be highlighting cases like this as when applying the correct order of operations is not wrong, but is just wildly inefficient compared to use of the distributive laws (in this case the formal statement would be something like 673 x 405 - 672 x 405 = 405 x (673 - 672) = 405 x 1 = 405).</div><div><br /></div><div>Some other examples of times when correct order of operations are an inefficient way to solve problems (particularly without a calculator) are:</div><div><ul style="text-align: left;"><li>12 x 345 ÷ 6</li><li><div class="MsoNormal">18<sup>2</sup> ÷ 9<sup>2<o:p></o:p></sup></div></li><li><div class="MsoNormal"><sup></sup></div><div class="MsoNormal">√128 ÷ √32 (although this one does require some real mathematical understanding)<o:p></o:p></div></li><li><div class="MsoNormal">372 + 845 – 369</div><div class="MsoNormal"><o:p></o:p></div><div class="MsoNormal"><o:p></o:p></div></li></ul><div>I would be exploring all of these questions prior to teaching the correct order of operations, and then including questions like it in the deliberate practice on the correct order of operations to ensure that pupils are recognising when <b>not</b> to apply them alongside when they are absolutely necessary.</div></div><div class="MsoNormal"><o:p></o:p></div></div>Peter Mattockhttps://plus.google.com/113661418069132691177noreply@blogger.com4tag:blogger.com,1999:blog-2500447090923756998.post-19704622731594459772016-10-21T13:01:00.000-07:002016-10-21T13:01:30.343-07:00Love teaching, love maths, love twitter.<div dir="ltr" style="text-align: left;" trbidi="on">As anyone who has known me for the last year and half will know, I love Twitter. As a medium for connecting educators and sharing practice I have not seen anything like it. I have probably had more professional conversations, attended more real CPD meetings and moved my practice on more in the last year and a half than in the previous 8 and half that I was working - and a lot of that can be attributed to Twitter. It is easy to begin to take the impact for granted once you have been used to it for a while, but then something will come along that makes you fall in love with it all over again. For me this happened very recently following the Secret Teacher article about teaching maths.<br /><br />Perhaps the thing I love most of all, more then twitter (although less than my family) is teaching maths. The joy of developing real understanding in pupils, seeing pupils go from nervous incomprehension to confident understanding is a joy that I am not going to soon tire of. Which is why articles like the Secret Teacher article make me so sad, when practitioners talk about how useless maths is for all but a small minority and how teachers are wasting time trying to teach all but a narrow set of skills to the majority I really do begin to despair of the poor opinion that some teachers have of pupils and of their role.<br /><br />Which brings me back to what makes me fall in love with Twitter all over again - the response from some of the colleagues, and people I now class as friends, was just brilliant. Within minutes we had responses like <a href="https://solvemymaths.com/2016/10/15/a-response-to-secret-teacher/" target="_blank">this</a> from Ed Southall (@solvemymaths) which so eloquently rebuts some of the poorer arguments in the article and really brilliantly we had a movement starting on Twitter courtesy of two of our newer teachers @MissBLilley and @Arithmaticks called #loveteaching.<br /><br />With the media and politicians seemingly fighting to report all of the ineptitudes and 'tribulations' (as the Guardian advertises for in its Secret Teacher blog), these two dedicated and driven young teachers have tried to take it upon themselves to be a big part of the opposite voice - the voice that highlights all of the things that we love about teaching and what is bringing and keeping those special people like these two ladies into the classroom. For me this is a perfect example of the power of platforms like Twitter to unite like-minded educators and provide a voice for the profession, and it makes me appreciate Twitter and the people I meet through it all over again.<br /><br />So I love Twitter, the camaraderie and the connectedness (if that is a word!); I love maths, the wonder and beauty, the way it has of revealing deeper and deeper insights for those that are prepared to work hard at it, but above nearly all I LOVE TEACHING.</div>Peter Mattockhttps://plus.google.com/113661418069132691177noreply@blogger.com0tag:blogger.com,1999:blog-2500447090923756998.post-63729837242778981222016-10-06T11:57:00.000-07:002017-01-20T15:07:36.487-08:00Methods of Last Resort 1 - Percentages<div dir="ltr" style="text-align: left;" trbidi="on">Following on from my session in Kettering at #mathsconf8 I will be writing a series of blogs about the areas of maths I find or figure out that might be better looked at separate to any problems that might be solved using a standard approach or a 'method of last resort'. The first area I want to look at is percentages.<br /><br />Because of the multiplicative nature of percentages there are lots of questions that can be solved without having to resort to approaches such as "Find 10% first..." or "What multiplier calculates...." or other standard approaches. The point I made at mathsconf is that I would want pupils to understand why these questions can be solved quickly and straightforwardly, and that actually by exploring the special nature of some of these calculations we can deepen pupils understanding of the topics - in this case percentages.<br /><br /><b>Find 32% of 75</b><br /><b><br /></b>This is the example I used at mathsconf. There are still plenty of teachers that don't realise that 32% of 75 is the same as 75% of 32, but once they see it they understand why. What I like is that in explaining why this is true really does get at the heart of percentages and how they are calculated and so it is a perfect little 'explain why' to stretch pupils as well as then serving as reinforcement of concepts for others.<br /><br /><b>Find 32% of 100</b><br /><b><br /></b>Try it; you will be surprised how many pupils don's immediately link the % with the 100 or are unsure when they want to say 'isn't that just 32?' Again this sort of question gets at the heart of percentages as parts of 100.<br /><br /><b>Find 32% of 50</b><br /><b><br /></b>If you have built up to it these are actually now becoming quite straightforward, but encouraging pupils to talk and explain why is still powerful.<br /><br /><b>Find 32% of 200, 300, 400 etc</b><br /><b><br /></b>I probably don't need to say much more at this point.<br /><br />As well as calculating percentages, equally there are similar questions for writing one value as a percentage of another. Again there are standard approaches for this (writing and converting fractions or similar) but there are questions that anyone with a real understanding of percentages would look at and solve. This set of questions comes from a well known worksheet provider; see if you can spot the ones that could be done without requiring the use of a standard approach or 'method of last resort'.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-a9jWLmiqdN0/V_acwNEWhsI/AAAAAAAABeI/uwm0D9PU7SQmjs81Fttrcxo9PJEZx0NAgCLcB/s1600/1st%2Bto%2B20.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="257" src="https://3.bp.blogspot.com/-a9jWLmiqdN0/V_acwNEWhsI/AAAAAAAABeI/uwm0D9PU7SQmjs81Fttrcxo9PJEZx0NAgCLcB/s640/1st%2Bto%2B20.png" width="640" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: left;"><br /></div>Even if you don't really know your fractions, questions 3, 5, 6, 7, 11 and possibly 12 and/or 17 can be solved using some relatively straightforward multiplication and division. Do we always teach pupils though that if they can see an obvious way to write it as 'a percentage of 100' that this will be much quicker than a standard approach, and more importantly to support them in understanding why this works which would lead to a deeper understanding of percentages as a whole.</div>Peter Mattockhttps://plus.google.com/113661418069132691177noreply@blogger.com3tag:blogger.com,1999:blog-2500447090923756998.post-26112966301326025732016-09-26T14:03:00.004-07:002016-10-02T03:43:57.333-07:00Variation in Mathematics<div dir="ltr" style="text-align: left;" trbidi="on">I am determined not to let my blog frequency slip below once a month no matter how busy I am; I probably have enough stuff for a year's worth of blogging at this point but one topic that I did want to discuss was the idea of variation and its use in mathematics teaching.<br /><br />I was lucky enough to be asked to host the twitter chat for the NCETM around this topic on 20th September and I jumped at the chance. The idea of variation is one that has interested me since I was observed teaching ratio and used these problems as my examples:<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-tNZsc8rB5RY/V-l-pqOSs_I/AAAAAAAABck/bkRnpcfpUZQ2RIbcuJ-Z6i12zJFjIEFigCLcB/s1600/Ratio%2Bexample.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="200" src="https://1.bp.blogspot.com/-tNZsc8rB5RY/V-l-pqOSs_I/AAAAAAAABck/bkRnpcfpUZQ2RIbcuJ-Z6i12zJFjIEFigCLcB/s640/Ratio%2Bexample.png" width="640" /></a></div><br />The visiting professor suggested that this series of examples showed the hallmarks of variation theory. This peaked my interest in the topic; I had known working in Oxford that Anne Watson and John Mason had done work on the idea but hadn't really had the opportunity to read any detail. I decided to look into variation a little more to see what it was all about.<br /><br />The first article I read was from Anne and John written for the Open University, and to this day remains one of my favourites on the subject. Entitled 'Seeing an exercise as a single mathematical object: using variation to structure sense-making' it really does give an excellent introduction to the idea of really thinking about and structuring the variation between questions or examples to allow pupils to make sense of how different facets of the situations effect the outcomes. One of my favourite activities from this article is:<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-5HPrhPkI9Uo/V-mFuAjvduI/AAAAAAAABc0/AmeXWT66WHk3ViMvGLGWpNcXVBWnLwbewCLcB/s1600/Multiplying%2Bbrackets%2Bexample.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="152" src="https://2.bp.blogspot.com/-5HPrhPkI9Uo/V-mFuAjvduI/AAAAAAAABc0/AmeXWT66WHk3ViMvGLGWpNcXVBWnLwbewCLcB/s640/Multiplying%2Bbrackets%2Bexample.png" width="640" /></a></div>I wont repeat Anne and John with all of the discussion, but the full article can be found <a href="http://oro.open.ac.uk/9764/1/06_MTL_Watson_%26_Mason.pdf">here</a> and I strongly encourage reading it.<br /><br />My other favourite article about variation is <a href="http://www.cimt.org.uk/journal/lai.pdf">this</a> one from the Centre for Innovation in Mathematics Teaching. The focus is very much on drawing out the misconception of eastern mathematics as relying a lot on rote learning, and could even been seen as a fore-running article to much of the recent focus on mathematics teaching approaches in the highest performing eastern jurisdictions. It is this article that gives me my clearest idea of the purpose of variation theory, namely<br /><br /><div style="text-align: center;">"the central idea of teaching with variation is to highlight the essential features of the concepts through varying the non-essential features"</div><div style="text-align: center;"><br /></div><div style="text-align: left;">The article also outlines nicely the difference between procedural variation and conceptual variation. During the chat I shared this activity which links to my original ratio problems and quite succinctly shows the idea of procedural variation.</div><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-tq9_l7KU1XI/V-mKHihfspI/AAAAAAAABdA/g7HDxKqzuzkLVzqu1Yq7sAtbKbi91vcjQCLcB/s1600/Juice%2Bproblem.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="366" src="https://2.bp.blogspot.com/-tq9_l7KU1XI/V-mKHihfspI/AAAAAAAABdA/g7HDxKqzuzkLVzqu1Yq7sAtbKbi91vcjQCLcB/s640/Juice%2Bproblem.png" width="640" /></a></div><div style="text-align: left;"><br /></div><div style="text-align: left;">As a guide to implementing different types of variation in the classroom this article is about as good as it gets. It certainly influenced my design of a department activity which resulted in some of these excellent activities (which haven't been formatted for pupil use yet!)</div><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-OJ7CWR6P48Q/V-mMESqSc9I/AAAAAAAABdM/AmWdMqmtEBsP7Qgr1R55DbeAEhMwIkI-ACLcB/s1600/Algebra%2Bexercise%2B1.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="206" src="https://4.bp.blogspot.com/-OJ7CWR6P48Q/V-mMESqSc9I/AAAAAAAABdM/AmWdMqmtEBsP7Qgr1R55DbeAEhMwIkI-ACLcB/s640/Algebra%2Bexercise%2B1.png" width="640" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-OXgb2oEQOmM/V-mMQ6t4vsI/AAAAAAAABdQ/MuA8Ki9hJrA0VBmzjKHQZuz38H4avW7jwCLcB/s1600/Algebra%2Bexercise%2B2.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="100" src="https://4.bp.blogspot.com/-OXgb2oEQOmM/V-mMQ6t4vsI/AAAAAAAABdQ/MuA8Ki9hJrA0VBmzjKHQZuz38H4avW7jwCLcB/s640/Algebra%2Bexercise%2B2.png" width="640" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-YT8fOiCwPLA/V-mMlwgIb0I/AAAAAAAABdU/qR6fIERivAQI8BnpkStSc2lnWlDzINQkwCLcB/s1600/Algebra%2Bexercise%2B3.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="324" src="https://4.bp.blogspot.com/-YT8fOiCwPLA/V-mMlwgIb0I/AAAAAAAABdU/qR6fIERivAQI8BnpkStSc2lnWlDzINQkwCLcB/s640/Algebra%2Bexercise%2B3.png" width="640" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: left;">I will definitely be using both of these articles with the work group we will be forming as part of our work as the lead secondary mastery school for the East Midlands South hub and would strongly recommend that anyone looking to ensure that every part of an activity is deepening pupils' understanding.</div><div style="text-align: left;"><br /></div></div>Peter Mattockhttps://plus.google.com/113661418069132691177noreply@blogger.com1tag:blogger.com,1999:blog-2500447090923756998.post-9562013413469403922016-08-07T02:35:00.003-07:002016-10-02T03:43:57.371-07:00Iteration and the new GCSE<div dir="ltr" style="text-align: left;" trbidi="on">So my blog frequency has become significantly lower recently - believe it or not I have been even busier than normal writing and sourcing resources for our new Year 7 mixed ability course, putting together topic tests for Year 7 and Year 11 (thanks AQA for all of your work putting your own topic tests together - I have stolen most of them!) and then writing the homework booklets for all three of my Year 11 schemes for term 1. All in all today is actually the first day since we broke up (bear in mind that Leicestershire broke up on 15/07/16) that I haven't been doing school work of some description - as a reward for finishing the homework booklets a day early I gave myself the weekend off!<br /><br />One of the things that I have had to sort out as part of writing the tests and homework booklets is finding sources of questions on iteration and numerical methods for solving equations, so I thought I would share some of the better ones here, and also offer some tips on designing your own.<br /><br />1) Check out A-Level worksheets - I dug through some of my old Core 3 resources (unfortunately I haven't taught A-Level for the last two years since moving to my new school) and found an ample supply of iterative formulae that were used. Some of them weren't suitable (too many natural logarithms and exponential functions) but many were with just some small adaptations. In particular a lot of A-Level questions ask pupils to show there is a root in a given interval using a change of sign approach and also ask pupils to justify why a given formula will converge to a solution. As far as I have seen the GCSE will not ask pupils to use a change of sign to show there is a root in a given interval,although to be fair it wouldn't be a bad thing to do with pupils as a way of tying roots of equations, graphs and iteration together. In addition it will definitely not require pupils to justify why a given iterative formula will converge, as this requires knowledge of calculus - although again it might be nice for the best mathematicians to look at this as a way of linking rates of change to iterative formulae. For some examples questions made from A-Level worksheets check out my Year 11 Higher or Higher+ term 1 and 2 homework booklets - there are a few pages on Iterative methods with a few exam style questions all taken from A-Level worksheets or similar.<br /><br />2) Exam board website - we are using the AQA exam board and they have a multitude of resources available for use with iteration. If you don't know AQA's site <a href="http://allaboutmaths.aqa.org.uk/">http://allaboutmaths.aqa.org.uk/</a> it is well worth getting yourself signed up for it. Browse to the New GCSE (8300) and select the Numerical methods section under Higher GCSE Algebra resources and you will find worksheets with some decent enough questions, as well as their topic test with some more. The one I really like though is their 'bridging' material, which can again be found under the New GCSE (8300) page. They have a lovely document in there called Pocket 4, which is all about iterative formulae. Although billed as a KS3 bridging material I would definitely save some of the later activities and use them during the actual GCSE teaching.<br /><br />3) Linked Pair Pilot - Although trial and improvement is not mentioned specifically in the new GCSE specifications, it is still being used under the guise of a numerical method. The Linked Pair Pilot papers, in particular the Applications 2 paper, has some nice examples of trial and improvement used to solve practical problems in geometry and other areas, which is nicely in keeping with the aims of the new GCSE. Often they have the tables printed on a separate page as well, which means you can feel free to not use them for the more confident mathematicians, just giving them the page with the question setup on instead.<br /><br />4) Pixi Maths - If you haven't seen Pixi Maths TES shop yet, then I would definitely head over there (<a href="https://www.tes.com/teaching-resources/shop/pixi_17#">https://www.tes.com/teaching-resources/shop/pixi_17#</a>). Pixi has created some lovely resources for a variety of topics, including iteration - <a href="https://www.tes.com/teaching-resource/iterations-11064012">https://www.tes.com/teaching-resource/iterations-11064012</a> although don't be fooled by the line that says trial and improvement has gone. Still there is a nice PowerPoint and activities which includes a jigsaw for the rearranging part of iteration and then a worksheet with some iterations to perform.<br /><br />5) Design your own - It isn't actually that tricky to design iteration questions, although there are a couple of things to beware of to ensure the question will work. Start with a polynomial set equal to 0; cubics are good as they can't be solved using other GCSE techniques (except if it has an obvious factorisation) and are guaranteed to have at least one root. From here you can do one of two things:<br /><br />(a) Use the Newton-Raphson formula:<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-UKLAWmGbYEI/V6b0sWtO0JI/AAAAAAAABbQ/f5x_1XqwRIE1F_JjvTPb9h8pCtNZdhmaACLcB/s1600/N-R%2Bformula.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="77" src="https://3.bp.blogspot.com/-UKLAWmGbYEI/V6b0sWtO0JI/AAAAAAAABbQ/f5x_1XqwRIE1F_JjvTPb9h8pCtNZdhmaACLcB/s200/N-R%2Bformula.png" width="200" /></a></div><div class="separator" style="clear: both; text-align: left;">The examples of exam questions I have seen using this formula have had the subtraction simplified to give a single fraction as the iterative formula, however I cannot see any reason why pupils couldn't be given the formula with the basic substitution already done and told to do a 'show that', i.e.</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-12EPLDn209s/V6b5XuhM0HI/AAAAAAAABbc/pcrwYSX36lQ6fnTW28IwxbjjaTSD3I41QCLcB/s1600/Example%2Bquestion%2Bfor%2BN-R.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="https://4.bp.blogspot.com/-12EPLDn209s/V6b5XuhM0HI/AAAAAAAABbc/pcrwYSX36lQ6fnTW28IwxbjjaTSD3I41QCLcB/s640/Example%2Bquestion%2Bfor%2BN-R.png" width="640" /></a></div><div class="separator" style="clear: both; text-align: left;"> </div><div class="separator" style="clear: both; text-align: left;">(b) Rearrange - the classic method for generating iterative formula is to rearrange the equation </div><div class="separator" style="clear: both; text-align: left;"><i>f</i>(<i>x</i>) = 0 into the form <i>x</i> = <i>g</i>(<i>x</i>). This is being used a lot in the new GCSE practice and sample materials which include asking pupils to show how a given rearrangement can be arrived at:</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-CNZb6xUkKlc/V6b8W0JuDsI/AAAAAAAABbo/AAsm9tjLh44KuV1kJGmldffZYHi_627AgCLcB/s1600/Example%2Bquestion%2Bfor%2Brearrangement.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="211" src="https://3.bp.blogspot.com/-CNZb6xUkKlc/V6b8W0JuDsI/AAAAAAAABbo/AAsm9tjLh44KuV1kJGmldffZYHi_627AgCLcB/s640/Example%2Bquestion%2Bfor%2Brearrangement.png" width="640" /></a></div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">If you use this approach to design your own question then a word of caution - not all possible rearrangements will find all of the roots. The best things do here is to check the graph of the rearranged function for the gradient in the locale of the root. The rule goes that if the gradient of the rearranged function around the root you are looking for is in the range (-1,1) then the formula will converge to the root there - if not then it wont. For example for the problem above the graphs of the original function and the rearranged function look like this:</div><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-EMkWXj443y4/V6b-GFbDLKI/AAAAAAAABb0/XtLKZms0YWodB0F5DHZKM5AbllRyiF2nQCLcB/s1600/Graphs%2Bfor%2Brearrangement.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="https://4.bp.blogspot.com/-EMkWXj443y4/V6b-GFbDLKI/AAAAAAAABb0/XtLKZms0YWodB0F5DHZKM5AbllRyiF2nQCLcB/s320/Graphs%2Bfor%2Brearrangement.png" width="281" /></a></div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">where the red graph is the original cubic and the blue graph is the square root function. You can see that there are actually three roots to the cubic, corresponding to the three points that the root function intercepts the line <i>y</i> = <i>x</i>. However the given rearrangement wont find the root that is slightly bigger than 2, as the gradient of the root curve is greater than 1 around that point. The rearrangement will quite happily find the other roots in the intervals (0,1) and (-1,0) as the gradients are close to 0 around these points. It is definitely worth just checking this if you are going to design your own rearrangement questions as you wouldn't want to give your pupils rearrangement that doesn't work!</div></div>Peter Mattockhttps://plus.google.com/113661418069132691177noreply@blogger.com1tag:blogger.com,1999:blog-2500447090923756998.post-61725986972506187882016-06-25T13:56:00.001-07:002016-10-02T03:44:37.798-07:00#Mathsconf7 - a cracking day out<div dir="ltr" style="text-align: left;" trbidi="on">I don't normally write blogs about conferences and events; there are much more eloquent people out there who normally do a great job of highlighting the key parts of the sessions they visit. Unfortunately quite a few of them were unable to attend this weekend and so the task falls to me to sum up my experiences of this wonderful weekend.<div><br /></div><div><u style="font-weight: bold;">Friday night</u> - the pre-drinks were great fun. Myself and Andy (@ColonelPrice) had a lovely dinner at the Cattle Grid (heartily recommended for anyone visiting Leeds in the future) followed a great catch-up with Mark (@EMathsUK) and the LaSalle team, Ben and some of the AQA team, Graham Cummings and @deko_j from Pearson, @dannytbrown @KristopherBoulton and @Naveenfrizvi. I also got to meet Douglas Butler (@DouglasButler1) for the first time (having missed his apparently fantastic ATM/Ma session in Leicester recently) as well as the dangerous duo of @AnandaCatterall and @MissVaseyMaths. As things wound down at Azucar we set off to find @El_Timbre, @missradders and @jennypeek to continue the drinking - nights out in Leeds are fun, but no more will be said!</div><div><br /></div><div><u style="font-weight: bold;">Start of the conference</u> - Following the usual messages from Mark and Andrew Taylor we were treated to a fantastic key note from professor Mike Askew (@mikeaskew26 which really highlighted some of the ways of working with pupils that actually do have impact in terms of pupils solving problems. In particular the importance of asking deep, exploratory questions like the one below. <img height="640" src="https://pbs.twimg.com/media/ClyUfDdWYAA8tVk.jpg" width="480" /></div><div><br /></div><div>Another important point raised here was the relative effectiveness of front-loading the lesson with examples and then pupils practising on lots of examples with a much more interleaved approach mixing worked examples with independent practice. The research Mike quoted suggests this second approach created much better outcomes in pupils compared to what might be considered the more traditional approach. Although there is lots to mention from this session, the one other thing I really want to mention is the use of little low-stakes quizzes on <b>prior</b> topics done half way through a current topic in order to refresh previous knowledge and understanding.</div><div><br /></div><div>Unfortunately due to technical difficulties I missed the speed-date but I am sure it was as useful and exciting as always.</div><div><br /></div><div><u style="font-weight: bold;">Session 1</u> - Avoiding misleading assumptions</div><div><br /></div><div>This was my first delivery of the day looking at the sometimes rather limited diet of examples that maths teachers have seen in the past, and therefore pupils see now. We played a game that tested the delegates creativity around designing examples. The full presentation can be found <a href="https://1drv.ms/p/s!AiVD5E48l4Wsh_dniyqYL8S6Mz_Z1g">here</a> and the major points of the game can be seen below:</div><div><img height="480" src="https://pbs.twimg.com/media/ClyozdrWIAAPZxB.jpg" width="640" /></div><div><br /></div><div><u style="font-weight: bold;">Session 2</u> - Questioning and Culture.</div><div><br /></div><div>My second delivery, along with @ColonelPrice making his #mathsconf debut. The session again seemed to be well received as myself and Andrew explored different aspects of questioning, including some brilliant responses to the request to come up with some non-standard questions to this stimulus (the idea of finding the equations of the lines given that the vertex at the bottom left has coordinate (0,0) was particularly inspired!). </div><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-yU-8XRpEQw4/V27p3dhWudI/AAAAAAAABa0/ShUPHjvMXqEc7hSit_uIcCNT2SzbHgr9gCLcB/s1600/Triangle%2BStimulus.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="200" src="https://2.bp.blogspot.com/-yU-8XRpEQw4/V27p3dhWudI/AAAAAAAABa0/ShUPHjvMXqEc7hSit_uIcCNT2SzbHgr9gCLcB/s320/Triangle%2BStimulus.png" width="320" /></a></div><div><br /></div><div>Again the presentation can be found <a href="https://1drv.ms/p/s!AiVD5E48l4Wsh_duJGCNfxQPRIXl3A">here</a>.</div><div><br /></div><div><u style="font-weight: bold;">Lunch and the Tweet up</u> - After less than inspiring fair in Peterborough we were treated to a very nice hot and cold selection for lunch. Of course no tweet up was ever going to be the same without @tessmaths there but our team of @MrBenWard, @HR_Maths @MissBLilley, @ColonelPrice, @missradders, @El_Timbre, @EJMaths and @MissBsresources really mucked in, and a great time was had by all! The puzzles proved particularly popular (with MissBLilley in particular more so then the delegates!) and congratulations to @hexagon001 for winning the triangles competition (with thanks to @EJMaths for donating the book used as the prize) and also to @KerryDunton for winning the smallest unique positive integer competition with a great choice of 6.</div><div><br /></div><div><u style="font-weight: bold;">Session 3</u> - Golden Age</div><div><br /></div><div>I always love talking about practice with @dannytbrown, I am only sad that I don't get the opportunity to do so very often. Listening to Danny talk about being present, being aware of our own awareness and noticing what leads to our actions before they happen, drawing on the work of Mason, Tahta etc, was hypnotic and Danny's clear passion but very deliberate approach is the perfect vehicle for delegates to slow down and really think about themselves in the classroom. This is possibly the only session I have not tweeted from as it was impossible to truly listen to Danny and distract yourself with a device at the same time, and would have been the antithesis of the whole session. Danny is clearly one of the deeper thinkers of our profession and everybody should take the time to listen to his thoughts and engage with the material he puts out in his excellent blog.</div><div><br /></div><div><u style="font-weight: bold;">Session 4</u> - Teaching for Depth</div><div><br /></div><div>The lovely ladies of the White Rose Maths hub (@wrmathshub) led by Beth talked about some of the work they have been doing to try and really ensure that the pupils across their area develop a really deep understanding of maths. Drawing on inspiration from Shanghai around ensuring pupils access truly intelligent practice and work with multiple representations the team have put together some excellent resources and assessments linked into their scheme of work. I will definitely be paying regular visits to their <a href="https://www.dropbox.com/sh/m8cffya8voqqgg1/AAANai1CLUB0AKVs9wBVuElza?dl=0">dropbox</a> when I am doing my own KS3 re-write next year and stealing as many of their materials as I can get away with!</div><div><br /></div><div>Of course one of the best parts of any #mathsconf is the chance to catch up with old friends and puts faces to the names of new ones and this was no exception - most of the names I have already mentioned and if I try and create a list here I will guarantee to miss someone out so I will simply say if I spoke to you today it was great to meet you/see you again and if I didn't then make sure you say "Hi" next time (particularly you @MrBartonMaths as I have seen you twice at mathsconf and haven't talked maths properly with you yet!).</div><div><br /></div><div>I cannot lavish enough plaudits on @EmathsUK and the @LaSalleEd team for the fantastic work they do three times a year to bring these events together and having started at #mathsconf4 I hope I am still around when they are doing #mathsconf40!</div></div>Peter Mattockhttps://plus.google.com/113661418069132691177noreply@blogger.com0tag:blogger.com,1999:blog-2500447090923756998.post-27549326602388040462016-05-26T13:38:00.001-07:002016-10-02T03:43:57.335-07:00Angles on 'straight lines' - tackling a key misconception.<div dir="ltr" style="text-align: left;" trbidi="on">As mentioned in my blog post a couple of days ago (I know, two in a week!) I have been teaching angle properties to a couple of different year groups. One misconception I kept bumping into is surrounding pictures like this:<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-hXrXmFC8xVE/V0dVNhZ6L_I/AAAAAAAABaA/Zwv50__FkX0HPPw0rvhkK6nlwTEiUdJbgCLcB/s1600/Example%2Bimage.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="257" src="https://1.bp.blogspot.com/-hXrXmFC8xVE/V0dVNhZ6L_I/AAAAAAAABaA/Zwv50__FkX0HPPw0rvhkK6nlwTEiUdJbgCLcB/s400/Example%2Bimage.png" width="400" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: left;">The key misconception I am talking about is this, "123 + <i>c</i> + <i>a</i> = 180 because those angles lie on a straight line".</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">For me, it is easy to sympathise with this, as of course these angles do "lie on a straight line". I think there are a couple of issues here and have been trying to deal with both through the topic. The first is a language issue, and the second involves the diagram.</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">The first issue is the idea of angles that 'lie on' a straight line. To me talking about angles on straight lines actually helps reinforce this misconception, rather than preventing it. Instead I think it is better to talk about angles that "form a straight line", this allows you to demonstrate that the angle 123 and <i>c</i> form a straight line, but that <i>a</i> is not needed to form the line, it is further down the line.</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">The other approach I have used alongside this is to get pupils to actually mark the point where the angles come together to form a straight line. Of course with the pupils being encouraged to look for angle properties rather than chase after particular angles (see my previous post), the conversation goes something along the lines of, "where do you see angles forming a straight line", followed by "can you mark where they form the straight line", which leads to pictures like these:</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;"><a href="https://3.bp.blogspot.com/-pevKDJJC5UA/V0ddVuWui6I/AAAAAAAABaU/XLaLH9bJTa48mmSgl6XUzpDA2Xx1VtcaQCLcB/s1600/Example%2Bimage%2B1.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="126" src="https://3.bp.blogspot.com/-pevKDJJC5UA/V0ddVuWui6I/AAAAAAAABaU/XLaLH9bJTa48mmSgl6XUzpDA2Xx1VtcaQCLcB/s200/Example%2Bimage%2B1.png" width="200" /></a><a href="https://2.bp.blogspot.com/-qyVpTTgTcuQ/V0ddVi3XQvI/AAAAAAAABaQ/WUldJMi9xNQdYH6H4ObfN59ZBNXDSqNAwCLcB/s1600/Example%2Bimage%2B2.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" height="128" src="https://2.bp.blogspot.com/-qyVpTTgTcuQ/V0ddVi3XQvI/AAAAAAAABaQ/WUldJMi9xNQdYH6H4ObfN59ZBNXDSqNAwCLcB/s200/Example%2Bimage%2B2.png" width="200" /></a></div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">These sorts of pictures really help show why the two (or three in the case of the upper line) angles are the ones that form the straight line, and that <i>a</i> is not involved.</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">My advice would be that when teaching angle properties, consider how the language you use supports pupils in identifying angle pictures correctly, and ways in supporting pupils on securing the correct angles as part of the correct pictures.</div></div>Peter Mattockhttps://plus.google.com/113661418069132691177noreply@blogger.com0tag:blogger.com,1999:blog-2500447090923756998.post-27645801799521588472016-05-24T14:05:00.000-07:002016-10-02T03:43:57.338-07:00Angle properties - don't go chasing angles...<div dir="ltr" style="text-align: left;" trbidi="on">Recently I have been teaching angle properties and calculations to Year 7 and Year 9. Particularly in Year 9 we have been exploring problems that require multiple properties and steps to arrive at a solution such as the problem below:<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/--46Irw0qNeQ/V0SrMLrhviI/AAAAAAAABZk/LJSiNYG_Dt8tcGfTrp_eij6OoufnCa43QCLcB/s1600/Example%2B2.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://4.bp.blogspot.com/--46Irw0qNeQ/V0SrMLrhviI/AAAAAAAABZk/LJSiNYG_Dt8tcGfTrp_eij6OoufnCa43QCLcB/s1600/Example%2B2.png" /></a></div>The approach I am taking here is not to focus on finding a particular angle, but rather than trying to focus pupils on the sorts of pictures they see. This means that instead of asking questions like "can you tell me the size of this angle?", I am asking questions like "Can you see any straight lines in the picture?". I am also modelling this process in examples, for example when we went through this example:<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-jqr1BVAi4mA/V0Std-PLYNI/AAAAAAAABZw/KkLsemRRWUQ0XW3UpxMoAPScMBUoMaDHgCLcB/s1600/Green%2B2.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="172" src="https://4.bp.blogspot.com/-jqr1BVAi4mA/V0Std-PLYNI/AAAAAAAABZw/KkLsemRRWUQ0XW3UpxMoAPScMBUoMaDHgCLcB/s200/Green%2B2.png" width="200" /></a></div><div class="separator" style="clear: both; text-align: left;">rather than trying to find <i>h</i> and then trying to find <i>i</i>, we instead just went through the different angle properties we knew and found angles that fit, including completely useless facts like 46 +90 + 44 = 180. Altogether we wrote down:</div><div class="separator" style="clear: both; text-align: left;"><i>h</i> + 46 + 90 + 44 + 61 + <i>i</i> = 360 (full turn)</div><div class="separator" style="clear: both; text-align: left;"><i>h</i> + 46 + 90 = 180</div><div class="separator" style="clear: both; text-align: left;">46 + 90 + 44 = 180</div><div class="separator" style="clear: both; text-align: left;">44 + 61 + <i>i</i> = 180</div><div class="separator" style="clear: both; text-align: left;">61 + <i>i</i> + <i>h</i> = 180 (all straight lines)</div><div class="separator" style="clear: both; text-align: left;"><i>h</i> = 44</div><div class="separator" style="clear: both; text-align: left;"><i>i</i> + 61 = 46 + 90 (both vertically opposite).</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">Only when we had written all of this down did we talk about and look at which bits of information may be useful in helping find <i>h</i> and <i>i</i> (quickly identifying multiple ways of finding both <i>h</i> and <i>i</i>) and eventually writing down the values of both angles.</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">This approach is definitely having an impact in terms of pupils working through these sorts of problems as they are less hung up on the fact that they can't immediately find values of an angle and are correspondingly (nice use of terminology!) more ready to make an attempt at these problems. This, coupled with a visualisation of walking down the paths that the diagram shows (more on this in a blog to come) seems to be a real support to pupils in working with these sorts of diagrams.</div><div class="separator" style="clear: both; text-align: left;"><br /></div><br /></div>Peter Mattockhttps://plus.google.com/113661418069132691177noreply@blogger.com0tag:blogger.com,1999:blog-2500447090923756998.post-79476027001718412462016-04-30T13:01:00.000-07:002016-10-02T03:43:57.361-07:00Dividing Fractions - not just KFC!<div dir="ltr" style="text-align: left;" trbidi="on">Is there anything with more potential for pupils to go wrong with in the arena of fractions than division by a fraction? Whether it is turning over the wrong fraction, both fractions, or not even having a clue about it, division by a fraction does seem to be a real stumbling block for a huge number of pupils. So I thought I would share the best 3 approaches I know to dividing by fractions.<br /><br /><b>1) Multiplying by the reciprocal</b><br /><br />This is basically where KFC comes from - although it is really important that pupils do understand the language of reciprocal and can identify reciprocals for areas of maths like functions. I like to build this by looking at unit fractions first, and definitely mixing up dividing both integer and fractional values, i.e.<br /><div class="MsoNormal"> 6 ÷ ¼<o:p></o:p></div><div class="MsoNormal"><br /></div><div class="MsoNormal"> ½ ÷ ⅓<o:p></o:p></div><br /><div class="MsoNormal"><br /></div><div class="MsoNormal"> ⅚ ÷ ⅛<o:p></o:p></div><div class="MsoNormal"><br /></div><div class="MsoNormal">Showing that these are the same as 6 x 4, ½ x 3 and ⅚ x 8 respectively is an important first step. Once this is secure we would look at dividing by a non-unit fraction as dividing by something <i>x</i> times bigger than the unit fraction, and so needing to divide by the unit fraction and by <i>x</i> i.e.</div><div class="MsoNormal"><br /></div><div class="MsoNormal"> ⅚ ÷ ⅘ = ⅚ ÷ <span style="font-family: "calibri" , sans-serif; font-size: 11pt; line-height: 107%;">⅕ </span>÷ 4 = ⅚ x 5 x <span style="font-family: "calibri" , sans-serif; font-size: 11pt; line-height: 107%;">¼ = </span>⅚ x 5/4 = 25/24</div><div class="MsoNormal"><br /></div><div class="MsoNormal">Highlighting and reinforcing the fact that 4 is the reciprocal of ¼, 3 is the reciprocal of ⅓, etc makes this approach complete.</div><div class="MsoNormal"><b><br /></b></div><div class="MsoNormal"><b>2) Dividing term by term</b></div><div class="MsoNormal"><b><br /></b></div><div class="MsoNormal">Although not an approach used a lot, this can be a really nice link to multiplication provided pupils can work with the fractions within a fraction that result. The idea centres on being able to divide numerators and denominators independently i.e. </div><div class="MsoNormal"><br /></div><div class="MsoNormal"> ⅚ ÷ ⅘ =<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-vTWbKbqCEfg/VyRObz5e9bI/AAAAAAAABYo/NU8jKG1a-FkzZlZmK0508wS0tzuU2GhUACLcB/s1600/Division%2Bmethod%2B2%2Bfirst%2Bimage.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://3.bp.blogspot.com/-vTWbKbqCEfg/VyRObz5e9bI/AAAAAAAABYo/NU8jKG1a-FkzZlZmK0508wS0tzuU2GhUACLcB/s1600/Division%2Bmethod%2B2%2Bfirst%2Bimage.png" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: left;">We can then proceed to multiply by 4/4 and by 5/5 (or alternatively simply by 20/20 if pupils will understand the reason for this in one step)</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-l6pvUasIiFU/VyRROhTcgJI/AAAAAAAABY0/OApDe64EWuQma_xS4ZVs4Gqls5RcrCfXQCLcB/s1600/Division%2Bmethod%2B2%2Bsecond%2Bimage.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://3.bp.blogspot.com/-l6pvUasIiFU/VyRROhTcgJI/AAAAAAAABY0/OApDe64EWuQma_xS4ZVs4Gqls5RcrCfXQCLcB/s1600/Division%2Bmethod%2B2%2Bsecond%2Bimage.png" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: left;"><b>3) Using common denominators</b></div><div class="separator" style="clear: both; text-align: left;"><b><br /></b></div><div class="separator" style="clear: both; text-align: left;">Like addition and subtraction (and particularly if you have already worked out common denominators for addition or subtraction) if fractions are given with a common denominator then dividing them can be quite straightforward.</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: center;"><span style="text-align: left;">⅚ ÷ ⅘ = </span></div><div class="separator" style="clear: both; text-align: center;"><span style="text-align: left;"><br /></span></div><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-lBVZLqNqftc/VyRTGFN1GtI/AAAAAAAABZA/PO9DgMjAV_g78OlS3V1IIMKmwWPH_BNUQCLcB/s1600/Division%2Bmethod%2B3%2Bfirst%2Bimage.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://2.bp.blogspot.com/-lBVZLqNqftc/VyRTGFN1GtI/AAAAAAAABZA/PO9DgMjAV_g78OlS3V1IIMKmwWPH_BNUQCLcB/s1600/Division%2Bmethod%2B3%2Bfirst%2Bimage.png" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: left;">The idea here is if you have 25 lots of <b>something</b> and you divide by 24 of the same <b>something</b> then you have 25/24 independently of the <b>something</b>. So</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-Jhr8evirJDo/VyRUmvVBpcI/AAAAAAAABZM/9Gppek-quLkgennmJZwM-i5-wfUDfVruwCLcB/s1600/Division%2Bmethod%2B3%2Bsecond%2Bimage.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://3.bp.blogspot.com/-Jhr8evirJDo/VyRUmvVBpcI/AAAAAAAABZM/9Gppek-quLkgennmJZwM-i5-wfUDfVruwCLcB/s1600/Division%2Bmethod%2B3%2Bsecond%2Bimage.png" /></a></div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">i.e. if we have 25 <b>thirtieths</b> divided by 24 <b>thirtieths</b> you have 25/24 independent of the original thirtieths.</div><div class="separator" style="clear: both; text-align: center;"><span style="text-align: left;"><br /></span></div><div class="separator" style="clear: both; text-align: left;">It may be that pupils will take to one method of dividing fractions over others, and that the pupils who grasp the concept quickly can work with all three, showing they are equivalent, choosing the optimum approach for different situations and in general working with all three to achieve true mastery of division by a fraction.</div></div><div class="MsoNormal"><o:p></o:p></div></div>Peter Mattockhttps://plus.google.com/113661418069132691177noreply@blogger.com0tag:blogger.com,1999:blog-2500447090923756998.post-84242669289984628932016-04-05T14:21:00.001-07:002016-10-02T03:43:57.366-07:00Parallel lines are the same length and other such nonsense!<div dir="ltr" style="text-align: left;" trbidi="on">Recently we have been talking about the messages and misconceptions we convey without meaning to. A colleague of mine (not in my school) put me onto one - when we draw parallel lines we nearly always draw them the same length. A quick google image search suggests that this is not just the maths teachers I know:<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-uHkcJ7nGcvw/VudHTsP3ryI/AAAAAAAABXk/tE0Gx4ZfBq8lpt4DfrZNPFa-7V1oTnm-A/s1600/Parallel%2Bline%2Bsearch.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="358" src="https://4.bp.blogspot.com/-uHkcJ7nGcvw/VudHTsP3ryI/AAAAAAAABXk/tE0Gx4ZfBq8lpt4DfrZNPFa-7V1oTnm-A/s640/Parallel%2Bline%2Bsearch.png" width="640" /></a></div><br />We can see that whilst most of the pictures do show horizontal or vertical lines, all of the pictures show parallel lines the same length. Whilst some might say that this isn't really significant, I wonder if it is not something we should be aware of anyway in terms of forcing ourselves to think about the implicit messages that we give to pupils alongside the explicit content or skills we are trying to teach.<br /><div class="separator" style="clear: both; text-align: center;"><br /></div>Another example that we have come across recently is that the equation 3x = 4 has no solutions because "three doesn't go into four". There was some debate as to whether this shows a general lack of understanding of division, or is a function of the fact that most equations we being to show pupils in there initial introduction to equation solving have whole number solutions. On the subject have you ever noticed that pupils struggle a lot more with equations of the form <a href="https://2.bp.blogspot.com/-DZahmCIb4aA/VudNJT8ZNbI/AAAAAAAABX0/QgSwuIJqjag7usLOe-v7CXxDJpMoB6oEw/s1600/fractional%2Bequations.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" src="https://2.bp.blogspot.com/-DZahmCIb4aA/VudNJT8ZNbI/AAAAAAAABX0/QgSwuIJqjag7usLOe-v7CXxDJpMoB6oEw/s1600/fractional%2Bequations.png" /></a> compared to 4<i>x</i> - 3 = 5? Could it be that on balance they see many more equations of the second type than the first?<br /><br />Some other areas of discussion:<br /><ul style="text-align: left;"><li>Index laws using a base that is not a single term and powers that are not integers or simple fractions.</li><li>Area of triangles where perpendiculars are horizontal and vertical.</li><li>Fractions - only ever talking about simplification of fractions with a numerator and denominator that are positive integers.</li></ul><div>There are lots of other patterns you can find in textbooks and other materials that teachers naturally draw on for their own examples - so my suggestion is to really think about the breadth of examples that are possible with the maths pupils are learning; and not just the typical examples you may have seen before.</div><br /><div class="MsoNormal"><!--[endif]--><o:p></o:p></div></div>Peter Mattockhttps://plus.google.com/113661418069132691177noreply@blogger.com0tag:blogger.com,1999:blog-2500447090923756998.post-25665918085727640102016-02-21T09:57:00.003-08:002016-10-02T03:43:57.331-07:00Probability without numbers<div dir="ltr" style="text-align: left;" trbidi="on">"There are <i>n</i> sweets in a bag. 6 of the sweets are orange. The rest of the sweets are yellow. Hannah takes at random a sweet from the bag. She eats the sweet. Hannah then takes at random another sweet from the bag. She eats the sweet. The probability that Hannah eats two orange sweets is <span style="font-family: Calibri, sans-serif; font-size: 11pt; line-height: 107%;">⅓.</span><br /><br /><div style="text-align: left;">(a) Show that <i><span style="font-family: "Calibri",sans-serif; font-size: 11.0pt; line-height: 107%; mso-ansi-language: EN-GB; mso-ascii-theme-font: minor-latin; mso-bidi-font-family: "Times New Roman"; mso-bidi-language: AR-SA; mso-bidi-theme-font: minor-bidi; mso-fareast-font-family: Calibri; mso-fareast-language: EN-US; mso-fareast-theme-font: minor-latin; mso-hansi-theme-font: minor-latin;">n</span></i><sup><span style="font-family: "Calibri",sans-serif; font-size: 11.0pt; line-height: 107%; mso-ansi-language: EN-GB; mso-ascii-theme-font: minor-latin; mso-bidi-font-family: "Times New Roman"; mso-bidi-language: AR-SA; mso-bidi-theme-font: minor-bidi; mso-fareast-font-family: Calibri; mso-fareast-language: EN-US; mso-fareast-theme-font: minor-latin; mso-hansi-theme-font: minor-latin;">2</span></sup><span style="font-family: Calibri, sans-serif; font-size: 11pt; line-height: 107%;"> – <i>n</i> – 90 = 0</span></div><div style="text-align: left;"><br /></div>(b) Solve <i><span style="font-family: Calibri, sans-serif; font-size: 11pt; line-height: 15.6933px;">n</span></i><sup><span style="font-family: Calibri, sans-serif; font-size: 11pt; line-height: 15.6933px;">2</span></sup><span style="font-family: Calibri, sans-serif; font-size: 11pt; line-height: 15.6933px;"> – <i>n</i> – 90 = 0</span> to find the value of <i>n</i>."<br /><br />Look familiar? This question caused massive controversy when it was released in summer 2015 as it was seen as too much like things to come - many felt that it was more like the sort of question we might expect in 2017 when the new '1-9' GCSE is first examined and had no place in the current GCSE. Whether you believe this or not, the point is clear that pupils need to understand the ideas of probability and apply them outside the realms of numerical chance. With that in mind I thought I would share some ideas about developing probability without giving (too many) values.<br /><br /><b>Probability and Proportion</b><br /><b><br /></b>I am surprised we do not see more links between probability and proportion as ultimately probability is a proportional idea, the chance of something happening is measured as a proportion of the things that are possible or as a proportion of a number of trials in an experiment. In the past proportionality has generally be pretty limited to calculating an expected number of trials that would satisfy the given condition. I think it is clear though that with anything up to 25% of the new GCSE paper content being linked to ratio and proportion I think that we will see a lot more questions linking these two topics in the future. Questions like the ones below could become much more common:<br /><br />1) A packet of sweets has orange, blackcurrant, strawberry and lemon sweets in the ratio 4:3:2:1. James and Sarah both buy packets of the same number of sweets. James doesn't like strawberry and so gives all of his strawberry sweets to Sarah. Sarah gives James all of her lemon sweets in return. If James takes a sweet at random from his bag, work out the probability that James take a lemon sweet.<br /><br />2) A childs' shape sorter has red, green, blue and yellow shapes. The number of red shapes is twice the number of green shapes. The number of blue shapes is twice the number of yellow shapes. In total the number of red and green shapes is twice the number total number of blue and yellow shapes. Work out the probability of a child selecting a red shape if the shape is taken at random.<br /><br />To be fair it strikes me that a lot of ratio and proportion question can be adapted to give a probability question - question (1) above could just as easily be "write down the ratio" rather than "work out the probability" and there are lots of ratio and proportion questions out there that could be adapted to this vein.<br /><br /><b>Probability and Algebra</b><br /><b><br /></b>Hannah and her sweets have given us a pretty clear indication that this will be a rich source of links for examiners to mine and again it makes perfect sense: if you understand the ideas of probability and algebraic expressions/equations then there should be no reason why you cant apply the two ideas together. We have also seen in the SAMs at least one question that has purely algebraic expressions inside a Venn diagram linked to probability for pupils to work with and I am sure we will see more examples in the coming years.<br />1)<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-ZqrDo10KWkk/VsnzUKnyRCI/AAAAAAAABXE/AuVwrld5XBo/s1600/venn%2Bdiagram%2Band%2Balgebra.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="412" src="https://2.bp.blogspot.com/-ZqrDo10KWkk/VsnzUKnyRCI/AAAAAAAABXE/AuVwrld5XBo/s640/venn%2Bdiagram%2Band%2Balgebra.png" width="640" /></a></div><br />2) A bag of counters contains red, blue and green counters. There is one more red counter than green, and one more green counter than blue. Stefan takes a counter out of the bag and puts it on the table, followed by a second. The probability that Stefan takes a blue followed by a red is 1/9. Calculate the probability that Stefan takes two greens.<br /><br />It strikes me that replacing lots of the numbers in current probability questions with letters will generate questions of this type, and so would be well worth some time in faculty meetings designing.<br /><br />No doubt at this point people will be thinking "yes but probability and statistics will only be 15% or so of the content..." and of course they are right, but don't forget that 15% of 240 marks is a good 36 marks, so there is plenty of space for one of two questions of this type to creep in, particularly as they can also count towards the 20% to 30% Algebra content or 25% to 20% Ratio content so I would suggest it is well worth building questions like this into your GCSE schemes.<br /><div class="separator" style="clear: both; text-align: center;"><br /></div></div>Peter Mattockhttps://plus.google.com/113661418069132691177noreply@blogger.com0tag:blogger.com,1999:blog-2500447090923756998.post-58510461588487126342016-02-02T14:53:00.001-08:002016-10-02T03:43:57.343-07:00Dimension and Pythagoras<div dir="ltr" style="text-align: left;" trbidi="on">My Year 10 have recently been working with Pythagoras in 3-D objects, and quite typically in my experience they were having difficulty identifying suitable triangles to calculate some of the lengths; particularly those lengths that go through a shape requiring multiple applications of Pythagoras. Being ready for this I decided to try an approach that I had been considering that links the number of applications of Pythagoras' Theorem to the number of dimensions that the line moves through. The approach met with some success and I can see how it might have real potential in linking to dimensional analysis so I thought I would outline it here.<br /><br />The first and one of the key points was to ensure that pupils understood that Pythagoras' Theorem is a relationship concerning area. Although we often use Pythagoras' Theorem to solve for missing lengths, the actual essence of the relationship is between the areas of three squares where two of them meet to form a right angle. The image below is one that is typically used to illustrate this (and one I have used lots in the past).<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-KrzGbCJ1Rp0/VrES1Ba5CjI/AAAAAAAABWI/5Wy0IHeYOJY/s1600/Pythagoras%2527%2BTheorem%2Band%2BArea.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="400" src="http://4.bp.blogspot.com/-KrzGbCJ1Rp0/VrES1Ba5CjI/AAAAAAAABWI/5Wy0IHeYOJY/s400/Pythagoras%2527%2BTheorem%2Band%2BArea.png" width="372" /></a></div><div class="separator" style="clear: both; text-align: left;">Once my pupils understand that this is a relationship area, the discussion is then turned to dimensions. What I am hoping to show pupils is that the two shorter sides of a right triangle are lines that only move in one dimension, whereas the diagonal moves in two (as the two sides are at right angles they can be considered to be two independent dimensions). So Pythagoras' Theorem can be thought of as a relationship that starts with lines moving in a single dimension, and relates them to a line moving in two dimensions (linking to area being a two dimensional concept).</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">Now let us consider a cuboid like the one below:</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-bcHWCQSD5jc/VrEYm5ox3UI/AAAAAAAABWg/V4l2pOI_uqU/s1600/Cuboid.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="361" src="http://2.bp.blogspot.com/-bcHWCQSD5jc/VrEYm5ox3UI/AAAAAAAABWg/V4l2pOI_uqU/s640/Cuboid.png" width="640" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: left;">An early job of work to do with pupils here is to make sure they can identify lines that move in one, two or three dimensions. Generally for me this leads to lots of gesturing around the room and drawing imaginary lines along walls and floors, as well as between corners across the room. Once pupils understand how these lines are moving we can start looking at which distance can be solved with only one application of Pythagoras' Theorem and which cannot. For example in the cuboid above the distances AF, BD, FC etc can all be solved directly using a single application of Pythagoras' Theorem as they are all lines that move in two dimension. </div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">Now let us consider a line that moves in three dimensions, for example EC. A tip from me on this, before drawing the triangle in the shape, try drawing the rectangle first (as below). For some reason pupils see this more easily that just the triangle.</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/--nBjtVQcKN8/VrEqnWsOHfI/AAAAAAAABWo/Id_-CscMPn4/s1600/Cuboid%2Bwith%2B3D%2Bline.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="362" src="http://4.bp.blogspot.com/--nBjtVQcKN8/VrEqnWsOHfI/AAAAAAAABWo/Id_-CscMPn4/s640/Cuboid%2Bwith%2B3D%2Bline.png" width="640" /></a></div><div class="separator" style="clear: both; text-align: left;">The discussion we had here is that we cannot expect to solve for lengths like this with one application of Pythagoras, because this length moves in all three dimensions and Pythagoras' Theorem is a two dimensional relationship. In order to solve this problem we are going to need two applications of Pythagoras, one to relate two of the one-dimensional lines with a line that moves in two dimensions (in the above case either EG or AC will do the job) and then to use this line moving in two dimensions with the line moving in the third dimension to relate to the line moving in all three dimensions.</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">To some this might seem like overkill, but what it does is give pupils an objective test as to whether a line can be solved using given information - if it moves in more than two dimensions it cannot be solved by a single application of Pythagoras' Theorem using lines that only move in one dimension. Of course from here you can complicate things and look at other three dimensional shapes, begin to make judgements about whether line can be considered to move in only one dimension (right-angled to each other), two dimensions or three dimensions. I also think it reinforces area as a key concept and will provide a nice link to dimensional analysis of different formulae when it comes to looking at that concept in more depth.</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">So in the future I think I will definitely be talking about Pythagoras' Theorem as an area relationship, and definitely be talking about dimensions that lines move in more formally with pupils; if your pupils are having trouble applying Pythagoras' Theorem to three dimensions why not try it as well?</div></div>Peter Mattockhttps://plus.google.com/113661418069132691177noreply@blogger.com0tag:blogger.com,1999:blog-2500447090923756998.post-23702322016247346852016-01-25T14:36:00.000-08:002016-10-02T03:43:57.328-07:00Proportion and straight line graphs<div dir="ltr" style="text-align: left;" trbidi="on">By now pretty much everyone involved in delivering the new Maths GCSE course in England are aware of the increased emphasis on pupils having proportional reasoning skills. Ratio, Proportion and rates of change as a separate strand of the GCSE is worth up to 25% of the assessed content of the qualification, and will appear linked with lots of other areas of maths. Already we are used to the idea of ratio and proportion appearing in trigonometry, scale diagrams, recipes, value for money, many different contexts; I want to look specifically at a couple of ways proportion appears in straight line graphs.<div><br /></div><div>Most teachers of the new GCSE (and quite possibly old) will be familiar with the obvious relationship that proportion has with straight line graphs; namely the graph of two variables that vary directly with each other. Graphs of the form <i>y</i> = <i>kx</i> are a fairly straight-forward link between proportionality and straight line graphs.</div><div><br /></div><div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-i3b0k0u97Aw/VqaZJ6W6c9I/AAAAAAAABVc/K9nlspQyt-k/s1600/y%2B%253D%2Bkx.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="400" src="http://4.bp.blogspot.com/-i3b0k0u97Aw/VqaZJ6W6c9I/AAAAAAAABVc/K9nlspQyt-k/s400/y%2B%253D%2Bkx.png" width="400" /></a></div><div class="separator" style="clear: both; text-align: left;">In this graph, the <i>y</i> values and <i>x </i>values are proportional, with the ratio <i>x</i>:<i>y</i> being 3:5. This kind of proportion should be relatively straight forward for any pupil that really understands proportion as an idea, and for those more graphically minded may even help with being able to visualise proportion. So what about this graph:</div><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-aMrjl0lQykk/Vqab0AjLvVI/AAAAAAAABVo/u7nJmAVEaag/s1600/y%2B%253D%2Bmx%2B%252B%2Bc.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="400" src="http://4.bp.blogspot.com/-aMrjl0lQykk/Vqab0AjLvVI/AAAAAAAABVo/u7nJmAVEaag/s400/y%2B%253D%2Bmx%2B%252B%2Bc.png" width="400" /></a></div><div class="separator" style="clear: both; text-align: left;">Clearly this is not a 'proportion' graph in the sense that <i>y</i> and <i>x</i> are not in proportion to each other. However if we take a closer look...</div><div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-XJOqILcaTfw/Vqaeay3YyOI/AAAAAAAABV0/Al3ABRveODM/s1600/gradient%2Bzoom.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="412" src="http://3.bp.blogspot.com/-XJOqILcaTfw/Vqaeay3YyOI/AAAAAAAABV0/Al3ABRveODM/s640/gradient%2Bzoom.png" width="640" /></a></div><div class="separator" style="clear: both; text-align: left;"><br /></div><div>Clearly there is a proportion going on here, but what is it? Of course in this case it is not the variables that are in proportion; rather it is <b>the rates of changes in the variables</b> that are in proportion. Specifically in this case that the change in <i>y</i> is half of the change in <i>x</i> (leading of course to the gradient of <span style="font-family: Calibri, sans-serif; font-size: 11pt; line-height: 107%;">½</span>). </div><div><br /></div><div>This proportionality is often overlooked, or at least not made explicit, but given that rates of change is now part of the new GCSE I think it will be worth highlighting the idea of a straight line as a line where the rate of change of <i>y</i> is proportional to the rate of change of <i>x</i> and that this proportionality is where we get the concept of gradient. This may well help pupils when it comes to rates of changes of curves by applying tangents; if pupils are already familiar with the idea of gradient at rate of change because it has been made explicit when working with straight lines the the concept should come more readily when moving on to rates of change of curves.</div><div><br /></div><div>So in order to ensure your pupils are ready for rates of change at GCSE, consider introducing them not just to graphs where the variables are in proportion, but also where the rate of change of one variable with respect to another is proportion: for if pupils can gain a deep understanding of how gradient links to proportionality then the beginnings of calculus are well within their grasp.</div></div>Peter Mattockhttps://plus.google.com/113661418069132691177noreply@blogger.com0tag:blogger.com,1999:blog-2500447090923756998.post-71744531274001737112016-01-16T07:24:00.000-08:002016-10-02T03:43:57.353-07:00Introducing surds - cutting out squares...<div dir="ltr" style="text-align: left;" trbidi="on">The 'recent' changes to the KS3 curriculum suggest that we should be given our pupils a basic understanding of surds and surd calculation prior to GCSE. Some people may be wondering about how to introduce the idea of surds to KS3 pupils and so I thought I would share one of the things I do with pupils, and that is to look at drawing and possibly cutting out squares.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-U-nVeYR4gxI/VppgLS8Ao_I/AAAAAAAABVA/xeK36jWSLeI/s1600/coin-grid.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="http://4.bp.blogspot.com/-U-nVeYR4gxI/VppgLS8Ao_I/AAAAAAAABVA/xeK36jWSLeI/s320/coin-grid.gif" width="320" /></a></div><br /><div dir="ltr" style="text-align: left;" trbidi="on"><br /></div><div dir="ltr" style="text-align: left;" trbidi="on">Typically this will start with some nice big squares. I actually quite like using inch-long squares; not because I have a real hankering to return to pre-decimalisation (I am too young to remember anything other than pounds and pence anyway!) but because I find centimetre squares too small and fiddly for this sort of work, and anyway I think there is something nice about reinforcing the concept of area by looking beyond the normal cm or metre squares. You can't buy inch square paper these days (or if you can I don't know where you can) but you can download square grids from the internet and stretch the image so that each square is 2.5 cm long. I need plenty of this paper as kids will need lots of attempts to try and fail; kids get to fail a lot here, so if you are looking to examine mindset as well this is a great activity to try.</div><div dir="ltr" style="text-align: left;" trbidi="on"><br /></div><div dir="ltr" style="text-align: left;" trbidi="on">I will start kids off by getting the to draw a square with 25 inch-squares inside, which most will do quite quickly. Next I will tell them to draw a square with 16 inch-squares inside of their 25 inch square so that they have this smaller square inside the larger square. The next part will be a discussion about what we might be able to say about squares with areas between 16 and 25 inch-squares, with the aim that pupils will realise that any square with an area between these two will have to have sides between 4 and 5 and therefore will be (a) drawable between the two squares we have already drawn and (b) have area made up part squares. Then comes the challenge (which at first to some pupils doesn't appear as much of a challenge): draw me the square that has an area equivalent to precisely 20 inch squares. Depending on the group this will proceed in one of two ways; either they will draw an attempt between the two squares they previously drew, bring it to me to measure (I keep a ruler that measures inches for this and other purposes), and then become frustrated when I show them that their area cannot be quite 20 inch-squares or alternatively (which I prefer) I will get them to draw a 4 x 5 rectangle and then cut their square up and see whether it can completely cover the rectangle (prompting to leave the 4x4 square intact and just cut the excess from around it and try and make it fit if necessary). This second approach is definitely nicer provided the pupils have the resilience to keep re-drawing the squares every time they make a mistake: of course sometimes because of small gaps between their pieces they will think they have completely covered the rectangle and you will have to show them (either by measuring the hole they cut or by talking about very small gaps etc) that they haven't got to exactly 20.</div><div dir="ltr" style="text-align: left;" trbidi="on"><br /></div><div dir="ltr" style="text-align: left;" trbidi="on">The purpose of the activity of course is to plant the idea in pupils head that the task may be impossible to perform in reality. I will often talk about accuracy of measuring instruments here as well, and get pupils to imagine rulers that could measure down to a millionth of an inch or more. What this allows me to do is introduce the idea of an irrational number in a way that speaks to pupils experience; they have seen first hand that the square root of 20 cannot be found as a decimal or fraction of the length of a real square. I can then talk about the fact that mathematically the number needs representing exactly rather than as a rounded value, and so the surd form is required, which leads to the need to be able to calculate with numbers in this form and so on. For me this approach is much more powerful than a simple calculator investigation and not just because it is more engaging for kids than just mindlessly punching numbers into a machine, but also because it really highlights the reality of irrational numbers: these are numbers that cannot be measured and cannot be represented in ways that have been used before.</div><div dir="ltr" style="text-align: left;" trbidi="on"><br /></div><div dir="ltr" style="text-align: left;" trbidi="on">So if you are looking for a nice concrete way to introduce the idea of surds that gives (for me) real insight into the fundamental nature of these type of numbers, then try getting your kids just drawing and cutting out squares.<br /><br /><br /></div></div>Peter Mattockhttps://plus.google.com/113661418069132691177noreply@blogger.com0tag:blogger.com,1999:blog-2500447090923756998.post-9119740012561702922016-01-06T13:43:00.004-08:002016-10-02T03:43:57.369-07:00Mode - Most unappreciated?<div dir="ltr" style="text-align: left;" trbidi="on"><div dir="ltr">For me, the mode is one of the most unappreciated averages that we teach in maths education. Perhaps because of its simplicity, perhaps because sometimes we feel it can contradict the idea of 'centralcy' that we look for in a good average, or perhaps because it isn't often talked about formally in many everyday situations, but rarely does mode get significant lesson time beyond its introduction in primary school. Often it is paired with median in a unit which then devotes a whole lesson or more to mean calculations; occasionally it is lumped into a general 'averages' lesson and becomes almost a footnote when looking at lists of numbers to get pupils to look at the list ready to find median and mean. In my opinion this is a real shame as the mode can be one of the most versatile and available averages, so I thought I would share with you some things that I like to do with the mode...</div><div dir="ltr"><br /></div><div dir="ltr">1) Mode from different representations</div><div dir="ltr"><br /></div><div dir="ltr">I love introducing the idea of mode as the most frequent item of data, and then challenging pupils to identify it in lots of different representations, such as bar charts, pie charts, tally charts, bar line graphs etc - recently I gave pupils a sheet with these images on and challenged them to find the mode from each situation: </div><div dir="ltr"><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-FuraKcd16sg/Vo1tACRreSI/AAAAAAAABUI/MuWndnfsfJk/s1600/Bar%2BChart.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="328" src="http://4.bp.blogspot.com/-FuraKcd16sg/Vo1tACRreSI/AAAAAAAABUI/MuWndnfsfJk/s640/Bar%2BChart.png" width="640" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-rJjT86RtpDo/Vo1tAW5zHII/AAAAAAAABUU/Es6zAoeNiAA/s1600/Bar%2BLine%2BChart.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="281" src="http://2.bp.blogspot.com/-rJjT86RtpDo/Vo1tAW5zHII/AAAAAAAABUU/Es6zAoeNiAA/s400/Bar%2BLine%2BChart.png" width="400" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-2H9Qp2X2D9s/Vo1tAdkg7bI/AAAAAAAABUM/wtbKHR7ajCQ/s1600/Pie%2BChart.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="264" src="http://1.bp.blogspot.com/-2H9Qp2X2D9s/Vo1tAdkg7bI/AAAAAAAABUM/wtbKHR7ajCQ/s400/Pie%2BChart.png" width="400" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-2yhtQCPfdNk/Vo1tAtnjx0I/AAAAAAAABUQ/Ij1XGK8JEzs/s1600/Raw%2Bdata%2Blists.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="38" src="http://4.bp.blogspot.com/-2yhtQCPfdNk/Vo1tAtnjx0I/AAAAAAAABUQ/Ij1XGK8JEzs/s640/Raw%2Bdata%2Blists.png" width="640" /></a></div><br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-usmAFQqWqw8/Vo1tAsRWgRI/AAAAAAAABUY/3ZaTrz02zFI/s1600/Tally%2BChart.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="294" src="http://3.bp.blogspot.com/-usmAFQqWqw8/Vo1tAsRWgRI/AAAAAAAABUY/3ZaTrz02zFI/s400/Tally%2BChart.png" width="400" /></a></div><div dir="ltr"><br /></div><div dir="ltr">We got a load of misconceptions out of the way here; a mode of 4 or a mode of 7 from the frequency table, a mode of 6 from the bar line chart, what happens when 241 and 242 appear the same number of times; stimulated a lot of discussion and conflict and led to some real understanding.</div><div dir="ltr"><br /></div><div dir="ltr">2) Make up a list of data.</div><div dir="ltr"><br /></div><div dir="ltr">A nice pre-cursor to more complicated problem solving is to just give pupils a mode and to ask pupils to come up with different lists of numbers that satisfy the conditions. As a simple example, the question might well be something like "4 numbers have a mode of 3, Give a possible list of the 4 numbers." This can then be complicated in the following ways:</div><div dir="ltr"></div><ul style="text-align: left;"><li>4 numbers have a mode of 3. What is the maximum number of 3s in the list? What is the minimum number of 3s?</li><li>4 positive whole numbers have a mode of 3. All of the numbers are 3 or less. Write all of the possible lists of numbers.</li><li>4 positive whole numbers have a mode of 3. The numbers add up to 10. Write down the four numbers.</li><li>4 positive whole numbers have a mode of 3. What is the minimum total that the four numbers can have? What about the maximum total if all of the numbers are less than or equal to 3? Less than or equal to 5?</li></ul><div>amongst other similar examples.</div><div><br /></div><div>These sorts of questions are nice to get pupils thinking and reasoning with mode; it is lovely to see them reason that the third list cannot have two 3s or realising that the fourth list cannot sum to 8.</div><br /><div dir="ltr"><br /></div><div dir="ltr">3) Modal mystery</div><div dir="ltr"><br /></div><div dir="ltr">Similar to above, designed to promote reasoning around the mode, these sorts of questions are lots of fun to throw at kids:</div><div dir="ltr"><br /></div><div dir="ltr"> 2 ............. 1 .................. 3</div><div dir="ltr"><br /></div><div dir="ltr">The above list of 5 numbers has 2 values missing. What could the mode be? What could it be if 3 is the highest number? If 1 is the lowest number? What about if we change the 3 to a 1? </div><div dir="ltr"><br /></div><div dir="ltr">4) Real life modes</div><div dir="ltr"><br /></div><div dir="ltr">Although rarely referred to formally as mode, a lot of statistics encountered in real life boil down to a mode. Whether it is votes on a popular TV reality show (X Factor, Strictly come Dancing, I am a Celebrity et al...) or likes on Instagram, giving a couple of examples like this and asking for more from pupils personal experience is a lovely thing to do with mode, because once you start to think about it, you can come up with loads!</div><div dir="ltr"><br /></div><div dir="ltr">So please, when you are teaching averages, don't just skip over the mode; there is so much more to come from this most common of measures.</div></div>Peter Mattockhttps://plus.google.com/113661418069132691177noreply@blogger.com0tag:blogger.com,1999:blog-2500447090923756998.post-155176326286737122015-12-27T02:13:00.001-08:002016-10-02T03:43:57.321-07:00Diagrammatic views of sequences<div dir="ltr" style="text-align: left;" trbidi="on">Expanding on my 'recent' (haven't blogged in ages admittedly) post about different views in algebra, I have been looking at the idea of showing different expressions using diagrammatic views of sequences, and thought I would outline a few thoughts here:<br /><br />1) Linear or quadratic?<br /><br />A really nice thing to do with pupils is to look at patterns that arise from (or generate depending on your point of view) linear sequences, compared to quadratic. In particular what is the difference between the way a linear pattern grows, compared to a quadratic pattern? Have a look at these patterns and see if you can decide whether they show linear or quadratic sequences without writing down the numbers:<br /><br /><div class="separator" style="clear: both; text-align: center;"></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-KSA9qP6pCRk/Vn3B4idztwI/AAAAAAAABQQ/VJdmhj-63MQ/s1600/Sequence%2B1.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="118" src="http://4.bp.blogspot.com/-KSA9qP6pCRk/Vn3B4idztwI/AAAAAAAABQQ/VJdmhj-63MQ/s640/Sequence%2B1.png" width="640" /></a></div><br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-5DNR8CsL4PA/Vn8O3KnqOkI/AAAAAAAABQo/9rwF3CfyMIk/s1600/Sequence%2B2.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="112" src="http://4.bp.blogspot.com/-5DNR8CsL4PA/Vn8O3KnqOkI/AAAAAAAABQo/9rwF3CfyMIk/s640/Sequence%2B2.png" width="640" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-Ok9oGWEHYac/Vn3B5ESvudI/AAAAAAAABQc/PpuTxsTR6pM/s1600/Sequence%2B3.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="186" src="http://1.bp.blogspot.com/-Ok9oGWEHYac/Vn3B5ESvudI/AAAAAAAABQc/PpuTxsTR6pM/s640/Sequence%2B3.png" width="640" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-9c3vTzl5--o/Vn8O3LqRUYI/AAAAAAAABQ8/pSFIMLNlQHI/s1600/Sequence%2B4.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="116" src="http://4.bp.blogspot.com/-9c3vTzl5--o/Vn8O3LqRUYI/AAAAAAAABQ8/pSFIMLNlQHI/s640/Sequence%2B4.png" width="640" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/--WyL7y5ICG4/Vn8g1F3NWTI/AAAAAAAABRs/EE0Vpq0StCA/s1600/Sequence%2B5.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="178" src="http://3.bp.blogspot.com/--WyL7y5ICG4/Vn8g1F3NWTI/AAAAAAAABRs/EE0Vpq0StCA/s640/Sequence%2B5.png" width="640" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: left;">Most people that know about sequences will be able to identify that the 1st, 3rd and 4th sequences are linear, because the same number of squares are added each time (the colours make this quite easy to identify), whereas in the second and third there are more squares of each colour - what is interesting though is to explore these views of the different sequences:</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;"><u>Linear</u></div><div class="separator" style="clear: both; text-align: left;"><u><br /></u></div><div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-sBiS5_xdZiM/Vn8duW8wvcI/AAAAAAAABRY/ztJfskNFJfQ/s1600/Sequence%2B1%2Bline.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="14" src="http://3.bp.blogspot.com/-sBiS5_xdZiM/Vn8duW8wvcI/AAAAAAAABRY/ztJfskNFJfQ/s640/Sequence%2B1%2Bline.png" width="640" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-rThp4D42QC8/Vn8duWpNLOI/AAAAAAAABRU/f0cPT87ccvA/s1600/Sequence%2B3%2Bline.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="14" src="http://3.bp.blogspot.com/-rThp4D42QC8/Vn8duWpNLOI/AAAAAAAABRU/f0cPT87ccvA/s640/Sequence%2B3%2Bline.png" width="640" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-32aiWzh6sLM/Vn8duBvllJI/AAAAAAAABRQ/3lVVl0rHvbk/s1600/Sequence%2B4%2Bline.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="28" src="http://3.bp.blogspot.com/-32aiWzh6sLM/Vn8duBvllJI/AAAAAAAABRQ/3lVVl0rHvbk/s640/Sequence%2B4%2Bline.png" width="640" /></a></div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">Each of the sequences 1, 3, and 4 can be rearranged to give these lines, , showing that they only grow in a 'linear' fashion, which doesn't work with the quadratic sequences as the number of squares is different each time (although you can technically rearrange them to make lines, they don't grow in a linear way).</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">Another interesting way to look at the linear sequences is using a graph:</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: center;"></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-5QOQd341HVQ/Vn8vNuNukAI/AAAAAAAABSw/ie_i7meezik/s1600/All%2B3%2Blinear%2Bgraphs.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="426" src="http://2.bp.blogspot.com/-5QOQd341HVQ/Vn8vNuNukAI/AAAAAAAABSw/ie_i7meezik/s640/All%2B3%2Blinear%2Bgraphs.png" width="640" /></a></div><div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;"><u>Quadratic</u></div><div class="separator" style="clear: both; text-align: left;"><u><br /></u></div><div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-CJgu4ajPTyQ/Vn8ibz6vTwI/AAAAAAAABR4/7VNF2ksk9iA/s1600/Sequence%2B2%2BSquare.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="112" src="http://3.bp.blogspot.com/-CJgu4ajPTyQ/Vn8ibz6vTwI/AAAAAAAABR4/7VNF2ksk9iA/s640/Sequence%2B2%2BSquare.png" width="640" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-1DfUilf8cCc/Vn8ib70D4AI/AAAAAAAABR8/nvT0PevohI0/s1600/Sequence%2B5%2Bsquare.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="178" src="http://1.bp.blogspot.com/-1DfUilf8cCc/Vn8ib70D4AI/AAAAAAAABR8/nvT0PevohI0/s640/Sequence%2B5%2Bsquare.png" width="640" /></a></div><div class="separator" style="clear: both; text-align: left;">Or if you prefer:</div><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-vSFAwh-_ZEQ/Vn-xN_7sVcI/AAAAAAAABTU/jGpbsyH1uG4/s1600/Sequence%2B2%2Bin%2Btwo%2Bdimensions.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="228" src="http://4.bp.blogspot.com/-vSFAwh-_ZEQ/Vn-xN_7sVcI/AAAAAAAABTU/jGpbsyH1uG4/s640/Sequence%2B2%2Bin%2Btwo%2Bdimensions.png" width="640" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-ERTDj_P1vao/Vn-xN4s0MLI/AAAAAAAABTQ/ATKkE1KtEms/s1600/Sequence%2B5%2Bin%2Btwo%2Bdimensions.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="258" src="http://2.bp.blogspot.com/-ERTDj_P1vao/Vn-xN4s0MLI/AAAAAAAABTQ/ATKkE1KtEms/s640/Sequence%2B5%2Bin%2Btwo%2Bdimensions.png" width="640" /></a></div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">What is nice here is that these sequences illustrate that quadratic sequences are the two dimensional extension to linear sequences. The graphs can also be used to illustrate the difference to a quadratic and the quadratic shape:</div><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-TM19GMBuL94/Vn83bUKGHkI/AAAAAAAABTA/OyqPhtexEx0/s1600/Both%2Bquadratic%2Bgraphs.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="640" src="http://4.bp.blogspot.com/-TM19GMBuL94/Vn83bUKGHkI/AAAAAAAABTA/OyqPhtexEx0/s640/Both%2Bquadratic%2Bgraphs.png" width="356" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: left;">Showing the curved nature of the quadratic graph as opposed to the straight line nature of a linear graph.</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">2) Different forms of an expression</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">Another possible use of these pictures is to illustrate the different ways of writing identical expressions, for example if we take sequence 1 from above without the colours:</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-7pb8d-LoU_4/Vn2-JvUyI6I/AAAAAAAABQI/5_3PLGJNk9A/s1600/4%2528n%252B1%2529.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="118" src="http://2.bp.blogspot.com/-7pb8d-LoU_4/Vn2-JvUyI6I/AAAAAAAABQI/5_3PLGJNk9A/s640/4%2528n%252B1%2529.png" width="640" /></a></div><div class="separator" style="clear: both; text-align: left;"><br /></div>It shouldn't be too hard to show pupils that the calculations for the number of squares in each successive pattern is 4 x 2, then 4 x 3, then 4 x 4 then 4 x 5, so in general 4(n+1). Consider the same picture with some slightly different colouring:<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-zU6xTw7am4E/Vn-zdfKptSI/AAAAAAAABTk/k7jfuEmsq7I/s1600/4n%2B%252B%2B4.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="118" src="http://3.bp.blogspot.com/-zU6xTw7am4E/Vn-zdfKptSI/AAAAAAAABTk/k7jfuEmsq7I/s640/4n%2B%252B%2B4.png" width="640" /></a></div><br />and we should be able to demonstrate that this is also 4n + 4 (the yellow squares given by 4n, and then 4 green squares on the end of each pattern). This is also true in quadratic sequences, taking sequence 2 from above:<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-kKJgn9LUC5Q/Vn-0-NrKxJI/AAAAAAAABTw/cW0iih03lhk/s1600/Sequence%2B2%2Bone%2Bcolour.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="112" src="http://4.bp.blogspot.com/-kKJgn9LUC5Q/Vn-0-NrKxJI/AAAAAAAABTw/cW0iih03lhk/s640/Sequence%2B2%2Bone%2Bcolour.png" width="640" /></a></div><br />Similar to above, the calculations this time are 1 x 2, 2 x 3, 3 x 4, 4 x 5, or in general n(n+1), if we then compare to the image below:<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-0-KQ0tW9BZw/Vn-1bV1tc_I/AAAAAAAABT4/BXcKhx9m1nQ/s1600/Sequence%2B2%2Btwo%2Bcolours.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="112" src="http://2.bp.blogspot.com/-0-KQ0tW9BZw/Vn-1bV1tc_I/AAAAAAAABT4/BXcKhx9m1nQ/s640/Sequence%2B2%2Btwo%2Bcolours.png" width="640" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div>We can show quite clearly that this is also n<sup>2</sup> + n.<br /><br />I am sure there are other uses I haven't yet thought of (I think it may be applicable to geometric and Fibonacci sequences as well, and possibly series at A-Level). When I get chance to explore more I will try and remember to write about it!<br /><br />P.S. - of course if you have multi-link cubes or similar then pupils can actually build these sequences, graphs etc. as well as just seeing or drawing the pictures.<br /><br /><div class="MsoNormal"><o:p></o:p></div></div>Peter Mattockhttps://plus.google.com/113661418069132691177noreply@blogger.com0