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Source: http://freedomtoteach.collins.co.uk/category/secondary-maths/feed/

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<title>Getting GCSE maths revision right!</title>
<link>https://freedomtoteach.collins.co.uk/getting-gcse-maths-revision-right/</link>
<dc:creator><![CDATA[stefanlesik]]></dc:creator>
<pubDate>Fri, 18 Nov 2022 15:24:42 +0000</pubDate>
<category><![CDATA[GCSE]]></category>
<category><![CDATA[Secondary]]></category>
<category><![CDATA[Secondary Maths]]></category>
<category><![CDATA[Education]]></category>
<category><![CDATA[GCSE Maths]]></category>
<category><![CDATA[GCSE revision]]></category>
<category><![CDATA[KS4]]></category>
<category><![CDATA[maths]]></category>
<category><![CDATA[Revision]]></category>
<guid isPermaLink="false">https://freedomtoteach.collins.co.uk/2022/11/18/getting-gcse-maths-revision-right/</guid>
<description><![CDATA[The run up to GCSEs can be a very difficult time for Year 11 students. Often, they are still learning … <a href="https://freedomtoteach.collins.co.uk/getting-gcse-maths-revision-right/">Continued</a>
The post <a href="https://freedomtoteach.collins.co.uk/getting-gcse-maths-revision-right/">Getting GCSE maths revision right!</a> appeared first on <a href="https://freedomtoteach.collins.co.uk">Collins | Freedom to Teach</a>.
]]></description>
<content:encoded><![CDATA[The run up to GCSEs can be a very difficult time for Year 11 students. Often, they are still learning new material but are aware of the need to practice and consolidate earlier learning so they are ready for their examinations. This year’s Year 11 students may feel even more challenged, as their earlier learning in secondary schools was disrupted by school closures during the Covid-19 pandemic. Collins have teamed up with White Rose Maths to produce a new set of <a href="https://collins.co.uk/pages/white-rose-maths-revision">revision guides for Edexcel and AQA</a> to support them with their studies.
<a href="https://freedomtoteach.collins.co.uk/wp-content/uploads/sites/87/2023/03/WRM-Hero-Banner-Out-Now664-x-432-1-1.jpg"><img fetchpriority="high" decoding="async" class="wp-image-9224 size-full aligncenter" src="https://freedomtoteach.collins.co.uk/wp-content/uploads/sites/87/2023/03/WRM-Hero-Banner-Out-Now664-x-432-1-1.jpg" alt="White rose maths revision guides. Text reads 'Available for Edexcel and AQA'" width="661" height="377" /></a>
Here are some answers to some Frequently Asked Questions about the guides. 
<h5>Why are there three levels of guides? </h5>
Firstly, rather than being split into just Higher and Foundation tiers, the <a href="https://collins.co.uk/pages/white-rose-maths-revision">Collins White Rose Maths revision guides</a> are more targeted to enable students to focus exactly on what they need. 
Aiming for a 4 – this revision guide support students who are looking for a standard pass in maths, covering all they need to know. 
Aiming for a 5/6 – as well as brief recap of earlier material, this revision guide focuses on the topics learners need to know to get a Grade 5 at Foundation tier or a 5 or 6 at Higher. Students are not looking at material that they already know or on topics they’re not likely to access. 
Aiming for a 7/8/9 – this revision guide provides some revision of earlier material but with a major emphasis on the topics students need to know to get the highest grades. 
<h5>How are they different from any other revision guides? </h5>
Compared to most revision guides, there’s a lot more information and a lot more practice! The material for each is broken down into four sections: 
Facts – what it’s about and the rules you need to know. 
<a href="https://freedomtoteach.collins.co.uk/wp-content/uploads/sites/87/2023/03/Picture1-1-3.png"><img decoding="async" class="alignnone wp-image-9218 size-full" src="https://freedomtoteach.collins.co.uk/wp-content/uploads/sites/87/2023/03/Picture1-1-3.png" alt="Image from White Rose Maths revision guides" width="356" height="201" /></a> 
Focus – a series of worked examples (not just one!) with step-by-step explanation of how to get to the solution. 
 <a href="https://freedomtoteach.collins.co.uk/wp-content/uploads/sites/87/2023/03/Picture2-7.png"><img decoding="async" class="alignnone wp-image-9217 size-full" src="https://freedomtoteach.collins.co.uk/wp-content/uploads/sites/87/2023/03/Picture2-7.png" alt="Image from White Rose Maths revision guides" width="540" height="293" /></a>
Fluency – practice questions on the basic idea of a topic. This is especially useful when revisiting maths students may not have seen for some time.
 <a href="https://freedomtoteach.collins.co.uk/wp-content/uploads/sites/87/2023/03/Picture3-5.png"><img loading="lazy" decoding="async" class="alignnone wp-image-9219 size-full" src="https://freedomtoteach.collins.co.uk/wp-content/uploads/sites/87/2023/03/Picture3-5.png" alt="Image from White Rose Maths revision guides" width="315" height="251" /></a>
Further – more challenging exam-type questions aiming for the higher grades that each revision guide caters for. 
<a href="https://freedomtoteach.collins.co.uk/wp-content/uploads/sites/87/2023/03/Picture6-1.png"><img loading="lazy" decoding="async" class="alignnone wp-image-9220 size-full" src="https://freedomtoteach.collins.co.uk/wp-content/uploads/sites/87/2023/03/Picture6-1.png" alt="Image from White Rose Maths revision guides" width="604" height="212" /></a>
<h5>What makes them “White Rose”? </h5>
Even though they are revision guides, the focus is on understanding the maths rather than “tips and tricks” that are easily forgotten. The models and representations used throughout the guides match those use in the White Rose resources, but students don’t have to seen these before to be able to learn from them. 
<h5><a href="https://freedomtoteach.collins.co.uk/wp-content/uploads/sites/87/2023/03/Picture5-1.png"><img loading="lazy" decoding="async" class="wp-image-9221 alignnone" src="https://freedomtoteach.collins.co.uk/wp-content/uploads/sites/87/2023/03/Picture5-1.png" alt="Image from White Rose Maths revision guides" width="456" height="103" /></a></h5>
<h5>Do these guides prepare students for all three GCSE papers? </h5>
Another key feature of these revision guides is that they prepare students for both calculator and non-calculator papers, including guidance on how to use a calculator when needed. 
 <a href="https://freedomtoteach.collins.co.uk/wp-content/uploads/sites/87/2023/03/Picture7-1.png"><img loading="lazy" decoding="async" class="alignnone wp-image-9222 size-full" src="https://freedomtoteach.collins.co.uk/wp-content/uploads/sites/87/2023/03/Picture7-1.png" alt="Image from White Rose Maths revision guides" width="604" height="101" /></a>
<h5>Are they for use at home or at school? </h5>
These revision guides can be used flexibly. The detailed explanations and worked examples mean they can be used independently by students, but there’s plenty of practice questions to use in revision lessons in schools too if needed. 
We hope you enjoy using these revision guides with your students! 
 
<h5>Ian Davies </h5>
Series editor – <a href="https://collins.co.uk/pages/white-rose-maths-revision">Collins White Rose Maths GCSE Revision Guides</a> 
You might also be interested in: <a href="https://freedomtoteach.collins.co.uk/how-to-smooth-the-transition-from-year-6-to-7-maths/">How to smooth the transition from Year 6 to 7 maths</a>
The post <a href="https://freedomtoteach.collins.co.uk/getting-gcse-maths-revision-right/">Getting GCSE maths revision right!</a> appeared first on <a href="https://freedomtoteach.collins.co.uk">Collins | Freedom to Teach</a>.
]]></content:encoded>
</item>
<item>
<title>Understanding and applying metacognition in your teaching</title>
<link>https://freedomtoteach.collins.co.uk/understanding-and-applying-metacognition-in-your-teaching/</link>
<dc:creator><![CDATA[stefanlesik]]></dc:creator>
<pubDate>Wed, 29 Jun 2022 14:03:39 +0000</pubDate>
<category><![CDATA[Secondary]]></category>
<category><![CDATA[Secondary Maths]]></category>
<category><![CDATA[adaptive learning]]></category>
<category><![CDATA[Collins]]></category>
<category><![CDATA[collins adapt]]></category>
<category><![CDATA[Dr Jonathan Firth]]></category>
<category><![CDATA[Jonathan Firth]]></category>
<category><![CDATA[metacognition]]></category>
<guid isPermaLink="false">https://freedomtoteach.collins.co.uk/2022/06/29/understanding-and-applying-metacognition-in-your-teaching/</guid>
<description><![CDATA[You might have heard of metacognition. This aspect of learning has been described as one of the most effective educational interventions that we can use.
The post <a href="https://freedomtoteach.collins.co.uk/understanding-and-applying-metacognition-in-your-teaching/">Understanding and applying metacognition in your teaching</a> appeared first on <a href="https://freedomtoteach.collins.co.uk">Collins | Freedom to Teach</a>.
]]></description>
<content:encoded><![CDATA[<h6 style="text-align: left">by Dr Jonathan Firth, University of Strathclyde</h6>
You might have heard of metacognition. This aspect of learning has been described as one of the most effective educational interventions that we can use<a href="#_ftn1" name="_ftnref1">[1]</a>. However, do we actually know what metacognition is? Are we all talking about the same thing?
The aim of this post is to make metacognition clearer and less subjective. After all, if it’s going to be effective – something you can use in your own teaching or learning – then you need to know how it works.
<a href="https://youtu.be/XwCReZTmc08">What is Metacognition? Click here to watch Jonathan Firth discuss the concept.</a>
<h6>Metacognition – the basics</h6>
When explaining metacognition to students, I often find that it helps to define cognition first. Cognition includes a set of everyday mental processes, including:
<ul>
<li>beliefs</li>
<li>paying attention</li>
<li>thinking</li>
<li>problem solving</li>
<li>reflecting</li>
<li>learning</li>
<li>memory</li>
<li>language</li>
</ul>
In short, it covers everything that your mind does on a moment-to-moment basis!
Metacognition means cognition about cognition. It’s also commonly defined as thinking about thinking – though, as you can see from the list above, that makes it sound narrower than it actually is. In terms of the cognitive processes listed, things like beliefs about memory would also count as metacognition, as would reflecting on your own use of language. And even talking about learning is metacognitive, too.
These examples show that metacognition is not something strange or additional to normal teaching processes. It is already happening in classrooms everywhere. You’re already developing metacognition with your learners every day!
Another very common example of metacognition in the classroom is when students feel that they are ‘stuck’. If they feel stuck on a task, they must have thought about their own learning or problem solving, and decided (rightly or wrongly) that they can’t make any further progress. It’s therefore a metacognitive process.
Something like growth mindset could also be seen as a form of metacognition. Whether learners do or don’t accept that ability can change through practice, this demonstrates a belief about learning.
Hopefully these examples make it a little clearer to see what metacognition is.
However, it would be a mistake to think that all complex or deep thinking is metacognition. Higher-order skills such as analysis are not necessarily metacognitive. Rather, they are just cognition! Likewise, something like problem solving is not necessarily a form of metacognition. It only becomes metacognitive if the problem solver stops to reflect on their strategy, or thinks about how much progress they have made.
<h6>Using metacognition in learning</h6>
It’s helpful to think about metacognition in terms of what happens before, during, and after learning<a href="#_ftn2" name="_ftnref2">[2]</a>:
1 Before: this is where a learner thinks about their cognitive processes in advance. For example, they might decide that they are going to take a particular strategy when writing an essay, perhaps because they are aware that other strategies have not been successful in the past. Or they may plan how they will revise for a test.
2 During: this is where learners think about or reflect on their progress as they do a task, and perhaps change strategy if things are not going well.
3 After: this is where learners reflect on their learning and make decisions about what to do next.
There is some interplay between the three stages above. For example, when students reflect on their learning this informs their planning, therefore looping back to the ‘before’ stage. Also, when thinking about their progress during a task, they may realise that something isn’t working well, and revise their plan. They may opt to change to a different strategy.
Despite these slight complexities, understanding the three stages provides a useful mental tool, and helps you think about how broadly metacognition applies.
This very process is supported by the Adapt platform, where students are continuously required first to retrieve their knowledge and decide on a strategy, and then to reflect on their level of confidence. The consistency of this process means that the ‘metacognition muscles’ are strengthened by regular sessions within Adapt.
Not forgetting that when you are thinking about your students’ learning, you are engaging in metacognition, too!
<h6>In practice in the classroom</h6>
When it comes to planning their attempt at a classroom task, learners may be better able to strategize if we take time to demonstrate worked examples<a href="#_ftn3" name="_ftnref3">[3]</a>. We can also make an effort to improve their metacognitive knowledge about learning, for example by:
<ul>
<li>Modelling strategies and ‘thinking aloud’ as they do so</li>
<li>Making them aware of effective learning strategies</li>
<li>Guiding them that their gut feeling about learning is not always accurate</li>
<li>Using accurate technical language when talking about strategies and cognition</li>
</ul>
In combination with these strategies, a little more planning time may pay dividends.
During the task, learners may then engage in what is known as metacognitive monitoring. This is where they think about their performance and progress. Do they realise how well or badly they are doing, for example, or notice when they have made a mistake?
The evidence suggests that students’ metacognitive monitoring is not always accurate – but it is better than chance or guesswork!<a href="#_ftn4" name="_ftnref4">[4]</a> This means that we should encourage learners to do this, and can do so by encouraging them to pause and think from time to time. Questions and reminders can nudge them to engage in monitoring, and can be built into tasks and worksheets (a similar strategy can be used for later reflection, for example via homework tasks).
Teachers may wonder how these strategies fit with cognitive load. Working memory capacity is limited, and thinking metacognitively (for example, monitoring your progress) adds extra load.
For this reason, it’s important to set aside time for these processes, particularly for younger learners whose working memory capacity is still developing. Even as little as a minute can make a difference. The key thing as that they are not being asked to monitor or reflect at the same time as doing a task. Doing two things concurrently is very demanding on working memory, and it’s only really possible for learners who are already highly experienced.
<h6>A problem with self-directed learning</h6>
If you consider the processes discussed above, it’s clear that students also need to engage in a lot of metacognition when they are studying by themselves. They have to decide what to study, when to study it, whether to stop work on one topic and switch to another, and so on. For this reason, metacognition researchers are very interested in self-directed learning.
One major problem that has arisen from this field of research is that learners are not very good at self-directed learning. This is partly due to motivation (they often leave things to the last minute)<a href="#_ftn5" name="_ftnref5">[5]</a>, but they are pretty bad at metacognition, too.
For example, most learners make flawed decisions about what to practise. They tend to focus their study time where they feel like they are making the most progress<a href="#_ftn6" name="_ftnref6">[6]</a>. This can lead to their skipping harder topics, or failing to consolidate easy ones (where they mistakenly feel they have nothing more to learn). In short, they underestimate both the cost of forgetting and the benefit of practice.
Within the Adapt platform, the opportunity to make these flawed decisions is greatly reduced The system moves the learner beyond the ‘easy’ topics (having first tested them), and focuses instead on identifying, practising and consolidating the harder or lesser-known elements of learning.
People tend not to fully understand memory, either. In particular, students tend to avoid subjectively-difficult strategies that slow down practice. They prefer to engage instead in passive strategies such as highlighting and re-reading texts, as these lead to a sense of rapid progress. They feel like they have understood and learned – but any such learning is shallow, and won’t last! Linked to the same issue, many learners avoid active revision strategies such as self-testing, and they don’t space out their practice over time.
These flawed study strategies lead to students getting worse grades<a href="#_ftn7" name="_ftnref7">[7]</a> than they otherwise might. Improving and guiding their metacognition as they study will help them to attain better, potentially making a big difference to outcomes.
As teachers and lecturers, we can play a role in tackling these errors. It is important for teachers to explain the rationale behind techniques such as retrieval practice (actively recalling things from memory, e.g. self-testing) and interleaving of practice problems<a href="#_ftn8" name="_ftnref8">[8]</a>, and to practise using evidence-based study strategies in the classroom.
Remember – we are the experts in learning, and can try to put ourselves in the position to gradually correct our students’ errors, guiding them towards more effective study habits. I think that this should happen quite early in schooling – long before young people face their first major exams<a href="#_ftn9" name="_ftnref9">[9]</a>.
<h6>Final notes</h6>
The examples in this article very obviously apply not only to classroom work in areas like science, maths and social science, but to other areas of the curriculum, too. A music student who is playing a musical scale may engage in metacognitive monitoring of their own skill and progress, and an athlete needs to think about scheduling their practice and adjusting elements of their physical performance.
Understanding metacognition and memory techniques are now central parts of Department of Education policy, and at times it can feel like a buzzword or fad in education. Everyone seems to want to be seen as metacognitive in their practice! Hopefully this article has convinced you that metacognition is an everyday part of learning, and not one that is beyond you or your students.
It’s also not something that can be left to take care of itself, however. Learners often avoid engaging in metacognition, and tend to have flawed beliefs about learning. We can tackle these issues in our practice. Hopefully these pointers will help you to recognise and develop metacognition in your own classroom.
 
Jonathan is a university teacher and researcher, and taught in secondary schools for many years. His current research interests focus on the psychology of education – specifically in how memory works, metacognition, and teacher professional learning. 
Twitter: @JW_Firth
ResearchGate: <a href="https://www.researchgate.net/profile/Jonathan-Firth">https://www.researchgate.net/profile/Jonathan-Firth</a>
Memory and Metacognition Newsletter: <a href="https://urldefense.com/v3/__https:/firth.substack.com/__;!!PH0vZokp8wwQNw!wGG9RfqY6w7tAr5JLxtgpncZppouFNLgBvJKV5YbM78S6V1YN4-L28M5EvInH_voYy8Rw9n0j8qb9B_gcnJnY2OqVVxH$">https://firth.substack.com/</a>
 
<a href="#_ftnref1" name="_ftn1">[1]</a> <a href="https://visible-learning.org/hattie-ranking-influences-effect-sizes-learning-achievement/">https://visible-learning.org/hattie-ranking-influences-effect-sizes-learning-achievement/</a>
<a href="#_ftnref2" name="_ftn2">[2]</a> As explained by Nelson and Narens (1994); <a href="https://psycnet.apa.org/record/1994-97967-001">https://psycnet.apa.org/record/1994-97967-001</a>
<a href="#_ftnref3" name="_ftn3">[3]</a> Muijs & Bokhove (2020). <a href="https://d2tic4wvo1iusb.cloudfront.net/documents/guidance/Metacognition_and_self-regulation_review.pdf">https://d2tic4wvo1iusb.cloudfront.net/documents/guidance/Metacognition_and_self-regulation_review.pdf</a>
<a href="#_ftnref4" name="_ftn4">[4]</a> Son & Schwarz (2002). <a href="https://psycnet.apa.org/record/2003-02861-001">https://psycnet.apa.org/record/2003-02861-001</a>
<a href="#_ftnref5" name="_ftn5">[5]</a> Kornell & Bjork (2007). <a href="https://link.springer.com/article/10.3758/BF03194055">https://link.springer.com/article/10.3758/BF03194055</a>
<a href="#_ftnref6" name="_ftn6">[6]</a> As discussed by Metcalfe and Kornell (2005). <a href="https://psycnet.apa.org/record/2005-04919-002">https://psycnet.apa.org/record/2005-04919-002</a>
<a href="#_ftnref7" name="_ftn7">[7]</a> Hartwig & Dunlosky (2012). <a href="https://psycnet.apa.org/record/2012-02361-015">https://psycnet.apa.org/record/2012-02361-015</a>
<a href="#_ftnref8" name="_ftn8">[8]</a> See Firth (2019). <a href="https://www.jonathanfirth.co.uk/blog/interleaving-using-it-in-the-classroom">https://www.jonathanfirth.co.uk/blog/interleaving-using-it-in-the-classroom</a>
<a href="#_ftnref9" name="_ftn9">[9]</a> Firth (2022). <a href="https://my.chartered.college/impact_article/understanding-the-human-mind-a-foundation-for-self-regulated-study/">https://my.chartered.college/impact_article/understanding-the-human-mind-a-foundation-for-self-regulated-study/</a>
The post <a href="https://freedomtoteach.collins.co.uk/understanding-and-applying-metacognition-in-your-teaching/">Understanding and applying metacognition in your teaching</a> appeared first on <a href="https://freedomtoteach.collins.co.uk">Collins | Freedom to Teach</a>.
]]></content:encoded>
</item>
<item>
<title>How to smooth the transition from Year 6 to 7 maths</title>
<link>https://freedomtoteach.collins.co.uk/how-to-smooth-the-transition-from-year-6-to-7-maths/</link>
<dc:creator><![CDATA[stefanlesik]]></dc:creator>
<pubDate>Mon, 06 Sep 2021 11:21:34 +0000</pubDate>
<category><![CDATA[Secondary]]></category>
<category><![CDATA[Secondary Maths]]></category>
<category><![CDATA[Collins]]></category>
<category><![CDATA[Collins White Rose Maths]]></category>
<category><![CDATA[Education]]></category>
<category><![CDATA[Key Stage 3]]></category>
<category><![CDATA[KS3]]></category>
<category><![CDATA[maths]]></category>
<category><![CDATA[maths activities]]></category>
<category><![CDATA[primary to secondary]]></category>
<category><![CDATA[primary to secondary maths]]></category>
<category><![CDATA[secondary]]></category>
<category><![CDATA[White Rose Maths]]></category>
<category><![CDATA[White Rose Maths for Key stage 3]]></category>
<category><![CDATA[White Rose Maths for KS3]]></category>
<guid isPermaLink="false">https://freedomtoteach.collins.co.uk/2021/09/06/how-to-smooth-the-transition-from-year-6-to-7-maths/</guid>
<description><![CDATA[It’s never easy for students to move from Year 6 to Year 7. For the vast majority, they move from … <a href="https://freedomtoteach.collins.co.uk/how-to-smooth-the-transition-from-year-6-to-7-maths/">Continued</a>
The post <a href="https://freedomtoteach.collins.co.uk/how-to-smooth-the-transition-from-year-6-to-7-maths/">How to smooth the transition from Year 6 to 7 maths</a> appeared first on <a href="https://freedomtoteach.collins.co.uk">Collins | Freedom to Teach</a>.
]]></description>
<content:encoded><![CDATA[It’s never easy for students to move from Year 6 to Year 7. For the vast majority, they move from being the biggest fish in a little pond to small fry in what feels like an ocean! Add to that the fact that their last two years of primary school have been severely disrupted by the global pandemic and some new Year 7s may be feeling the pressure more than ever this year. So how can the <a href="https://collins.co.uk/pages/white-rose-maths">Collins White Rose Maths series for Key Stage 3</a> help to smooth the transition?
<h6>It starts with something new</h6>
Unlike many schemes of learning, the White Rose Maths secondary schemes start with Algebraic Thinking rather than number. Although students may have met algebra briefly in Year 6, they won’t have covered it in detail, so this provides a clean slate and something fresh and exciting – especially for those students who have been struggling with arithmetic, possibly for several years. It’s also about algebraic thinking rather than just complicated use of symbols and designed to allow students to spot patterns and think creatively alongside their new learning. Here’s an example from the first section of the book:
<img loading="lazy" decoding="async" class="alignleft size-medium wp-image-9036" src="https://freedomtoteach.collins.co.uk/wp-content/uploads/sites/87/2023/03/White-Rose-Maths-1-1-300x174.png" alt="" width="300" height="174" />
 
 
 
 
There are so many possible answers to this question and it helps students to see that maths isn’t just about being right or wrong, but also about being imaginative and resourceful.
<h6>The focus is on understanding the maths</h6>
You need to be fluent and accurate to succeed in maths, and it’s much easier to do so if you understand what you’re doing. The Collins White Rose Maths books support understanding using “concrete manipulatives” – objects that help students to make sense of the maths they’re doing. Ideas for these, and for pictorial representations too, are provided at the start of every chapter of the books for the whole of Key Stage 3. Here’s an example for when algebraic notation is introduced in the second block of the book.
<img loading="lazy" decoding="async" class="alignleft size-medium wp-image-9030" src="https://freedomtoteach.collins.co.uk/wp-content/uploads/sites/87/2023/03/White-Rose-Maths-2-1-300x123.png" alt="" width="300" height="123" />
 
 
 
The models and representations are then used throughout the book to help students to understand things like “collect like terms” as shown in this worked example:
<img loading="lazy" decoding="async" class="alignleft size-medium wp-image-9031" src="https://freedomtoteach.collins.co.uk/wp-content/uploads/sites/87/2023/03/White-Rose-Maths-3-1-300x164.png" alt="" width="300" height="164" />
 
 
 
A cube symbol is then used in the exercises to show when manipulatives might be useful for students to use to help answer the questions, like this:
<img loading="lazy" decoding="async" class="alignleft size-medium wp-image-9032" src="https://freedomtoteach.collins.co.uk/wp-content/uploads/sites/87/2023/03/White-Rose-Maths-4-1-300x108.png" alt="" width="300" height="108" />
 
 
As you can see there’s a lot of practice too! Questions have been chosen carefully to consider what to vary and what to keep the same so students’ attention is drawn to the key points of learning.
<h6>Calculators are allowed!</h6>
A key feature of primary school maths learning is to get a secure grounding in number work, and although this is continued in the White Rose Maths Key Stage 3 scheme, it’s important that students learn how to use a calculator too, so this is another key new feature for students to enjoy as they start at their new schools. Here are some examples from the Algebra section from <a href="https://collins.co.uk/collections/secondary-secondary-maths-white-rose-maths-student-books/products/9780008400880">Student Book 1</a>:
<img loading="lazy" decoding="async" class="alignleft size-medium wp-image-9033" src="https://freedomtoteach.collins.co.uk/wp-content/uploads/sites/87/2023/03/White-Rose-Maths-5-1-300x70.png" alt="" width="300" height="70" />
 
 
As well as removing the possible fear of getting the answer wrong because of number work, this approach again helps students’ understanding by making them think about the relationships shown in an equation. If they are only presented with equations of the type “x + 4 = 10” where they can “spot” the answer x = 6, they don’t appreciate the inverse relationship of addition/subtraction they need to solve more complex equations. You can also see there’s a bar model symbol next to these questions. Most students will have met bar models in primary schools – definitely if they’ve followed the White Rose Maths primary schemes – so this is another useful support for transition as well as pictorial support for students to understand the relationships. Again these are illustrated in the textbook to help students understand.
<img loading="lazy" decoding="async" class="alignleft size-medium wp-image-9034" src="https://freedomtoteach.collins.co.uk/wp-content/uploads/sites/87/2023/03/White-Rose-Maths-6-1-300x162.png" alt="" width="300" height="162" />
 
 
 
<img loading="lazy" decoding="async" class="alignleft size-medium wp-image-9035" src="https://freedomtoteach.collins.co.uk/wp-content/uploads/sites/87/2023/03/White-Rose-Maths-7-1-300x99.png" alt="" width="300" height="99" />
 
 
Opportunities for learning calculator skills are then built-in throughout the rest of the book, and the Key Stage 3 series.
<h6>Becoming confident mathematicians</h6>
The White Rose Maths ethos is to support students to become confident, independent mathematicians. By creating a scheme and series of textbooks that build on their learning from primary school, using familiar approaches to increasingly complicated maths in fresh and exciting ways, we believe that students’ transition to secondary school will be much smoother. They’ll also have a much more positive attitude to maths and be more resilient as they’ll have bank of strategies and approaches they rely on when things get hard. The White Rose Maths motto is #MathsEveryoneCan and the Collins White Rose books will definitely help students on their way!
 
By Ian Davies & Caroline Hamilton, series editors of White Rose Maths for Key Stage 3
<a href="https://collins.co.uk/pages/white-rose-maths">Find out more about White Rose Maths for Key Stage 3 resources and download free samples to try with your class</a>
<a href="https://whiterosemaths.com/">Find out more about White Rose Maths and view their free schemes of work</a>
 
Liked this? <a href="https://freedomtoteach.collins.co.uk/category/secondary-maths/">Read more Secondary Maths articles</a>
The post <a href="https://freedomtoteach.collins.co.uk/how-to-smooth-the-transition-from-year-6-to-7-maths/">How to smooth the transition from Year 6 to 7 maths</a> appeared first on <a href="https://freedomtoteach.collins.co.uk">Collins | Freedom to Teach</a>.
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<title>Lessons learnt from the GCSE 9-1 Maths reform</title>
<link>https://freedomtoteach.collins.co.uk/lessons-learnt-from-the-gcse-9-1-maths-reform/</link>
<dc:creator><![CDATA[stefanlesik]]></dc:creator>
<pubDate>Mon, 13 May 2019 15:56:36 +0000</pubDate>
<category><![CDATA[GCSE]]></category>
<category><![CDATA[Secondary Maths]]></category>
<guid isPermaLink="false">https://freedomtoteach.collins.co.uk/2019/05/13/lessons-learnt-from-the-gcse-9-1-maths-reform/</guid>
<description><![CDATA[Wow! Where has the time gone.  It only seems like 5 minutes ago that the GCSE reforms were announced, and … <a href="https://freedomtoteach.collins.co.uk/lessons-learnt-from-the-gcse-9-1-maths-reform/">Continued</a>
The post <a href="https://freedomtoteach.collins.co.uk/lessons-learnt-from-the-gcse-9-1-maths-reform/">Lessons learnt from the GCSE 9-1 Maths reform</a> appeared first on <a href="https://freedomtoteach.collins.co.uk">Collins | Freedom to Teach</a>.
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<content:encoded><![CDATA[
Wow! Where has the time gone.  It only seems like 5 minutes ago that the GCSE reforms were announced, and here we are – 4 sittings later (don’t forget November!) with the 5th just around the corner. Having been through many reforms/changes/tweaks/updates over my 15-year teaching career (that has also passed in the blink of an eye!), I can honestly say … this was the biggest.  Coursework going (I liked this change), 3 to 2 tier, modular coming and going, little stars appearing next to the question numbers (quality of written communication questions!) and of course the introduction of 9-1.  This change not only ripped apart the content, it also ripped apart the currency (I’m still saying now ‘you need to get your grade C…’).
The content to get through for this new GCSE is MUCH bigger than the previous GCSE and most departments I meet are starting the course in year 9 to compensate for this. To be fair, maths is maths and as long as they have a sound understanding of the basics (a great KS1,2 and 3), starting in year 9 is a sensible option. I like to think our KS3 course (which is year 7 and 8 for us) is robust, enjoyable and naturally supports the GCSE, it almost feels right to be starting the GCSE as they enter year 9.
My biggest concern, still after 4 sittings, is getting the tier of entry correct. Although this concern is now for less students than it was for the first sitting in 2017, it’s still not a straight forward decision for those 3/4/5 borderline students. It’s around March/Easter time that I still find myself making some last-minute changes. The foundation and higher papers are such ‘different beasts’ … with the huge ‘number’ content in the foundation papers and huge ‘algebra’ content in the higher papers it’s not a simple decision to switch between the two. To help with this tiering decision, our lower Higher group (hope that makes sense) and our Foundation group follow the same content (the crossover material) until around Christmas in Year 11 with both groups being given regular ‘foundation paper challenges’ (this means the lessons can concentrate on the crossover material). The opportunity to regularly work through the start of the foundation papers means the students who remain on or move to Foundation are practising the start the papers (accompanying the crossover content in lessons) and the ones who end up staying on Higher have been developing their basic numeracy skills. It’s great to run these with mixed ability groups – the higher students quickly realise that the foundation paper is NOT an easier option! After Christmas, the lower higher group then adds higher only topics to the crossover topics in lessons (you know the big ones – CF, Box Plots, Histograms etc. etc.). With the students we are still unsure about – they remain in the higher group but with constant foundation paper challenges to allow us to bring these to Foundation a little closer to the exam if necessary (we are now at this point only talking about a couple of students).

I know I am not on my own when I say this – but getting the tier correct is not for school league tables, it’s for the student. Thinking that a pupil could not achieve what they deserve based on my decision is scary. The exams are still settling down … we are still getting our heads around exactly what a 9-1 paper includes. I understand that under historic reforms, grade boundaries normally increase and then settle after around 3 years. But this new GCSE has had significant changes from one sitting to the next as the examiners really understand what a realistic expectation is of what a 16-year-old can achieve in 90 minutes … so it may take a little longer to steady this ship. I, along with everyone who reads this, will be eagerly awaiting the next batch of results (and grade boundaries/mark schemes!) … so, my final words are GOOD LUCK ALL!
By Christian Seager
Christian Seager is part of the leadership team at
Alcester Academy, Warwickshire.  Christian led the maths team at Alcester
Academy to the TES Maths Team of the Year 2016, directed the maths results to
the dizzy heights they are now accustomed to with the progress 8 measure
consistently above 0.5 and not to forget, his previous school was Most Improved
School in England when the maths department was under his wing.  In
addition to his day job, he co-runs JustMaths that helps support maths
departments across the country.
The post <a href="https://freedomtoteach.collins.co.uk/lessons-learnt-from-the-gcse-9-1-maths-reform/">Lessons learnt from the GCSE 9-1 Maths reform</a> appeared first on <a href="https://freedomtoteach.collins.co.uk">Collins | Freedom to Teach</a>.
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<title>Any two will do: developing students’ modelling and problem solving skills</title>
<link>https://freedomtoteach.collins.co.uk/developing-modelling-problem-solving/</link>
<dc:creator><![CDATA[stefanlesik]]></dc:creator>
<pubDate>Thu, 02 Nov 2017 11:15:28 +0000</pubDate>
<category><![CDATA[A Level]]></category>
<category><![CDATA[Secondary Maths]]></category>
<category><![CDATA[a level]]></category>
<category><![CDATA[GCSE]]></category>
<category><![CDATA[GCSE Maths]]></category>
<category><![CDATA[maths]]></category>
<category><![CDATA[secondary maths]]></category>
<guid isPermaLink="false">https://freedomtoteach.collins.co.uk/2017/11/02/developing-modelling-problem-solving/</guid>
<description><![CDATA[Here is an activity which aims to help develop students’ modelling and problem solving skills. It requires knowledge of Pythagoras’ … <a href="https://freedomtoteach.collins.co.uk/developing-modelling-problem-solving/">Continued</a>
The post <a href="https://freedomtoteach.collins.co.uk/developing-modelling-problem-solving/">Any two will do: developing students’ modelling and problem solving skills</a> appeared first on <a href="https://freedomtoteach.collins.co.uk">Collins | Freedom to Teach</a>.
]]></description>
<content:encoded><![CDATA[Here is an activity which aims to help develop students’ modelling and problem solving skills. It requires knowledge of Pythagoras’ theorem, solving simultaneous equations and all things quadratical. Solutions to activities are included in order to highlight possible class discussion points.
The activity is also available to <a href="http://s20211.p595.sites.pressdns.com/wp-content/uploads/2017/11/Any-two-will-do-002.pdf" target="_blank" rel="noopener">download as a PDF</a><a href="http://s20211.p595.sites.pressdns.com/wp-content/uploads/2017/11/Any-two-will-do-Collins-Mark-Rowland-2017-1.pdf" target="_blank" rel="noopener">.</a>
Coverage of Assessment Objective have been flagged up.
For reference, the Edexcel version of AO3 is:
AO3.1 – translate problems in mathematical and non-mathematical contexts into mathematical processes 
AO3.2 – interpret solutions to problems in their original context and, where appropriate evaluate their accuracy and limitations 
AO3.3 – translate situations in context into mathematical models 
AO3.4 – use mathematical models 
AO3.5 – evaluate the outcomes of modelling in context, recognise the limitations of models and, where appropriate, explain how to refine them.
<a href="https://collins.co.uk/product/9780008205010/Bridging+GCSE+and+A-level+Maths+Student+Book+%5bSecond+edition+Second+edition%5d?" target="_blank" rel="noopener"><img loading="lazy" decoding="async" class="alignleft wp-image-7985" src="https://freedomtoteach.collins.co.uk/wp-content/uploads/sites/87/2023/03/Bridging-cover-1-215x300.jpg" alt="" width="149" height="208" /></a>
 
Please refer to the <a href="https://collins.co.uk/product/9780008205010/Bridging+GCSE+and+A-level+Maths+Student+Book+%5bSecond+edition+Second+edition%5d?" target="_blank" rel="noopener">Bridging GCSE and A-level Maths Student Book, 2nd edition</a> for questions designed to improve problem solving and modelling skills as well as ease the transition between these two qualifications.
 
 
<img loading="lazy" decoding="async" class="alignleft wp-image-7986 " src="https://freedomtoteach.collins.co.uk/wp-content/uploads/sites/87/2023/03/Bridging1-1.jpg" alt="" width="720" height="787" />
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
<img loading="lazy" decoding="async" class="wp-image-7987 aligncenter" src="https://freedomtoteach.collins.co.uk/wp-content/uploads/sites/87/2023/03/Bridging2-1.jpg" alt="" width="718" height="842" />
<img loading="lazy" decoding="async" class="aligncenter size-full wp-image-7998" src="https://freedomtoteach.collins.co.uk/wp-content/uploads/sites/87/2023/03/Bridging13-1.jpg" alt="" width="596" height="100" /><img loading="lazy" decoding="async" class="wp-image-7989 aligncenter" src="https://freedomtoteach.collins.co.uk/wp-content/uploads/sites/87/2023/03/Bridging4-1.jpg" alt="" width="722" height="668" />
<img loading="lazy" decoding="async" class="size-full wp-image-7997 alignleft" src="https://freedomtoteach.collins.co.uk/wp-content/uploads/sites/87/2023/03/Bridging12-1.jpg" alt="" width="670" height="375" />
 
 
 
 
 
 
 
<img loading="lazy" decoding="async" class="wp-image-7991 aligncenter" src="https://freedomtoteach.collins.co.uk/wp-content/uploads/sites/87/2023/03/Bridging6-1.jpg" alt="" width="725" height="422" />
<img loading="lazy" decoding="async" class="aligncenter size-full wp-image-7996" src="https://freedomtoteach.collins.co.uk/wp-content/uploads/sites/87/2023/03/Bridging11-1.jpg" alt="" width="677" height="502" />
<img loading="lazy" decoding="async" class="wp-image-7993 aligncenter" src="https://freedomtoteach.collins.co.uk/wp-content/uploads/sites/87/2023/03/Bridging8-1.jpg" alt="" width="707" height="693" />
<img loading="lazy" decoding="async" class="wp-image-7994 aligncenter" src="https://freedomtoteach.collins.co.uk/wp-content/uploads/sites/87/2023/03/Bridging9-1.jpg" alt="" width="711" height="252" />
<hr />
By Mark Rowland
Mark has been teaching Maths for over 20 years, mostly at a large F.E College. His publications include the <a href="https://collins.co.uk/product/9780008205010/Bridging+GCSE+and+A-level+Maths+Student+Book+%5bSecond+edition+Second+edition%5d?" target="_blank" rel="noopener">Bridging GCSE and A-level Maths Student Book</a>, teacher’s resources for <a href="https://collins.co.uk/category/Secondary/Maths/AQA+Core+Maths/?" target="_blank" rel="noopener">Level 3 Core Maths</a>, Further Maths books and contributions to a series of books for the new A-level Maths specification.
 
The post <a href="https://freedomtoteach.collins.co.uk/developing-modelling-problem-solving/">Any two will do: developing students’ modelling and problem solving skills</a> appeared first on <a href="https://freedomtoteach.collins.co.uk">Collins | Freedom to Teach</a>.
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<title>Did you know? Notes from the history of Maths</title>
<link>https://freedomtoteach.collins.co.uk/did-you-know-notes-from-the-history-of-maths-7-2/</link>
<dc:creator><![CDATA[stefanlesik]]></dc:creator>
<pubDate>Thu, 09 Feb 2017 15:22:23 +0000</pubDate>
<category><![CDATA[GCSE]]></category>
<category><![CDATA[Secondary Maths]]></category>
<category><![CDATA[Uncategorized]]></category>
<category><![CDATA[Collins maths]]></category>
<category><![CDATA[History of Maths]]></category>
<category><![CDATA[maths education]]></category>
<guid isPermaLink="false">https://freedomtoteach.collins.co.uk/2013/09/24/did-you-know-notes-from-the-history-of-maths-4/</guid>
<description><![CDATA[By Don Hoyle “A man and his dad put a bomb in the sink” Abstract: This article outlines some of … <a href="https://freedomtoteach.collins.co.uk/did-you-know-notes-from-the-history-of-maths-7-2/">Continued</a>
The post <a href="https://freedomtoteach.collins.co.uk/did-you-know-notes-from-the-history-of-maths-7-2/">Did you know? Notes from the history of Maths</a> appeared first on <a href="https://freedomtoteach.collins.co.uk">Collins | Freedom to Teach</a>.
]]></description>
<content:encoded><![CDATA[By Don Hoyle
“A man and his dad put a bomb in the sink”
Abstract: This article outlines some of the work of Matthew Stewart who was born 300 years ago. In particular, Stewart’s theorem, for which the title is a mnemonic.
<img loading="lazy" decoding="async" class="wp-image-7579 aligncenter" src="http://freedomtoteach.collins.co.uk/wp-content/uploads/2017/02/Matthew_Stewart_portrait.jpg" alt="Matthew Stewart" width="245" height="289" />
 
Early in 2017,<img loading="lazy" decoding="async" class="alignleft wp-image-7597 size-full" src="http://freedomtoteach.collins.co.uk/wp-content/uploads/2017/02/17-01-30-Did-you-know...jpg" alt="Simson Line" width="328" height="430" /> on January 15th, it was the 300th anniversary of the birth of Matthew Stewart. You might not have heard of this Scottish mathematician hence this article. He was born on the Isle of Bute and at the age of 17 went to work under the mathematician Robert Simson at the University of Glasgow.
Simson was a geometer and the ‘Simson Line’ is named after him. He also showed how the ratio of adjacent terms of the Fibonacci sequence tended to the golden ratio. He was also fascinated by Euclid’s lost books: “The Porisms”. The meaning of this word is vague and has been lost over time. Pappus, who wrote about these books of Euclid, described a ‘porism’ as something between a problem and a theorem. It isn’t a statement of truth, like a theorem, nor a construction and even the mode of reasoning involved has been lost. Stewart went on to study under Colin Maclaurin (as in the series) and, aged 30, succeeded him at the University of Edinburgh as Chair of Mathematics. Stewart’s interest in geometry enabled him to prove Kepler’s second law of planetary motion (‘The line joining the Sun to a planet sweeps out equal areas in equal times’). Kepler had formulated this and the other two laws from astronomical data in 1609. He didn’t know why they were true or how they related to one another. One of Isaac Newton’s motivations in publishing his great work the Principia in 1687 was to show how Kepler’s laws were derived from the inverse square law of gravitational attraction. Stewart proved the second law geometrically. The proof is hard to follow as it uses no diagrams¹! One result that can be followed is Stewart’s theorem². In the diagram, let a, b and c be the lengths of the sides of a triangle. Let d be the length of a line from a vertex to the opposite side a (called a ‘cevian’). If the cevian divides a into segments m and n, with m adjacent to c and n adjacent to b, the Stewart’s theorem states that:
b² m + c²n = a(d² + mn).
This can be rearranged as man+dad=bmb+cnc, for which there is the mnemonic used as the title for this article).
 This theorem can be proved using the cosine rule:
c² = m² + d² – 2dmcosθ 
 b² = n² + d² – 2dncosθ’ as θ and θ’ are supplementary cosθ’= – cosθ 
 b² = n² + d² + 2dncosθ 
Multiplying the first equation by n and the second by m and add to eliminate cosθ 
 b²m + c²n = nm² + n²m + (m+n)d² 
 = (m + n)(mn + d²) 
 = a (mn + d²) as a = m + n
<a href="http://freedomtoteach.collins.co.uk/wp-content/uploads/2017/02/Stewarts_theorem_svg.png"><img loading="lazy" decoding="async" class="aligncenter wp-image-7581" src="http://freedomtoteach.collins.co.uk/wp-content/uploads/2017/02/Stewarts_theorem_svg.png" alt="Stewarts theorem" width="309" height="242" /></a>
Diagram of Stewart’s theorem
Not all of Stewart’s work was successful. In 1763 he used Newton’s Theories to calculate the distance to the Sun from Earth and was out by over 25%.
<hr />
¹ In the second volume of “Essays of the Philosophical Society of Edinburgh” available to download from google books 
² By Krishnavedala – Own work, CC0, https://commons.wikimedia.org/w/index.php?curid=18036511 
This can be rearranged as man+dad=bmb+cnc, for which there is the mnemonic used as the title for this article).
<hr />
Collins publishes a range of key Maths resources across Key Stage 3, GCSE, and A-level. <a href="https://collins.co.uk/category/Secondary/Maths">Find out more.</a>
The post <a href="https://freedomtoteach.collins.co.uk/did-you-know-notes-from-the-history-of-maths-7-2/">Did you know? Notes from the history of Maths</a> appeared first on <a href="https://freedomtoteach.collins.co.uk">Collins | Freedom to Teach</a>.
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<title>Did you know? Notes from the history of Maths</title>
<link>https://freedomtoteach.collins.co.uk/did-you-know-notes-from-the-history-of-maths-7-3/</link>
<dc:creator><![CDATA[stefanlesik]]></dc:creator>
<pubDate>Thu, 09 Feb 2017 15:22:23 +0000</pubDate>
<category><![CDATA[GCSE]]></category>
<category><![CDATA[Secondary Maths]]></category>
<category><![CDATA[Uncategorized]]></category>
<category><![CDATA[Collins maths]]></category>
<category><![CDATA[History of Maths]]></category>
<category><![CDATA[maths education]]></category>
<guid isPermaLink="false">https://freedomtoteach.collins.co.uk/2011/11/02/did-you-know-notes-from-the-history-of-maths-3/</guid>
<description><![CDATA[By Don Hoyle “A man and his dad put a bomb in the sink” Abstract: This article outlines some of … <a href="https://freedomtoteach.collins.co.uk/did-you-know-notes-from-the-history-of-maths-7-3/">Continued</a>
The post <a href="https://freedomtoteach.collins.co.uk/did-you-know-notes-from-the-history-of-maths-7-3/">Did you know? Notes from the history of Maths</a> appeared first on <a href="https://freedomtoteach.collins.co.uk">Collins | Freedom to Teach</a>.
]]></description>
<content:encoded><![CDATA[By Don Hoyle
“A man and his dad put a bomb in the sink”
Abstract: This article outlines some of the work of Matthew Stewart who was born 300 years ago. In particular, Stewart’s theorem, for which the title is a mnemonic.
<img loading="lazy" decoding="async" class="wp-image-7579 aligncenter" src="https://freedomtoteach.collins.co.uk/wp-content/uploads/sites/87/2023/03/Matthew_Stewart_portrait-1.jpg" alt="Matthew Stewart" width="245" height="289" />
 
Early in 2017,<img loading="lazy" decoding="async" class="alignleft wp-image-7597 size-full" src="https://freedomtoteach.collins.co.uk/wp-content/uploads/sites/87/2023/03/17-01-30-Did-you-know..-2.jpg" alt="Simson Line" width="328" height="430" /> on January 15th, it was the 300th anniversary of the birth of Matthew Stewart. You might not have heard of this Scottish mathematician hence this article. He was born on the Isle of Bute and at the age of 17 went to work under the mathematician Robert Simson at the University of Glasgow.
Simson was a geometer and the ‘Simson Line’ is named after him. He also showed how the ratio of adjacent terms of the Fibonacci sequence tended to the golden ratio. He was also fascinated by Euclid’s lost books: “The Porisms”. The meaning of this word is vague and has been lost over time. Pappus, who wrote about these books of Euclid, described a ‘porism’ as something between a problem and a theorem. It isn’t a statement of truth, like a theorem, nor a construction and even the mode of reasoning involved has been lost. Stewart went on to study under Colin Maclaurin (as in the series) and, aged 30, succeeded him at the University of Edinburgh as Chair of Mathematics. Stewart’s interest in geometry enabled him to prove Kepler’s second law of planetary motion (‘The line joining the Sun to a planet sweeps out equal areas in equal times’). Kepler had formulated this and the other two laws from astronomical data in 1609. He didn’t know why they were true or how they related to one another. One of Isaac Newton’s motivations in publishing his great work the Principia in 1687 was to show how Kepler’s laws were derived from the inverse square law of gravitational attraction. Stewart proved the second law geometrically. The proof is hard to follow as it uses no diagrams¹! One result that can be followed is Stewart’s theorem². In the diagram, let a, b and c be the lengths of the sides of a triangle. Let d be the length of a line from a vertex to the opposite side a (called a ‘cevian’). If the cevian divides a into segments m and n, with m adjacent to c and n adjacent to b, the Stewart’s theorem states that:
b² m + c²n = a(d² + mn).
This can be rearranged as man+dad=bmb+cnc, for which there is the mnemonic used as the title for this article).
 This theorem can be proved using the cosine rule:
c² = m² + d² – 2dmcosθ 
 b² = n² + d² – 2dncosθ’ as θ and θ’ are supplementary cosθ’= – cosθ 
 b² = n² + d² + 2dncosθ 
Multiplying the first equation by n and the second by m and add to eliminate cosθ 
 b²m + c²n = nm² + n²m + (m+n)d² 
 = (m + n)(mn + d²) 
 = a (mn + d²) as a = m + n
<a href="https://freedomtoteach.collins.co.uk/wp-content/uploads/sites/87/2023/03/Stewarts_theorem_svg-1.png"><img loading="lazy" decoding="async" class="aligncenter wp-image-7581" src="https://freedomtoteach.collins.co.uk/wp-content/uploads/sites/87/2023/03/Stewarts_theorem_svg-1.png" alt="Stewarts theorem" width="309" height="242" /></a>
Diagram of Stewart’s theorem
Not all of Stewart’s work was successful. In 1763 he used Newton’s Theories to calculate the distance to the Sun from Earth and was out by over 25%.
<hr />
¹ In the second volume of “Essays of the Philosophical Society of Edinburgh” available to download from google books 
² By Krishnavedala – Own work, CC0, https://commons.wikimedia.org/w/index.php?curid=18036511 
This can be rearranged as man+dad=bmb+cnc, for which there is the mnemonic used as the title for this article).
<hr />
Collins publishes a range of key Maths resources across Key Stage 3, GCSE, and A-level. <a href="https://collins.co.uk/category/Secondary/Maths">Find out more.</a>
The post <a href="https://freedomtoteach.collins.co.uk/did-you-know-notes-from-the-history-of-maths-7-3/">Did you know? Notes from the history of Maths</a> appeared first on <a href="https://freedomtoteach.collins.co.uk">Collins | Freedom to Teach</a>.
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<title>Did you know? Notes from the history of Maths</title>
<link>https://freedomtoteach.collins.co.uk/did-you-know-notes-from-the-history-of-maths-7/</link>
<dc:creator><![CDATA[stefanlesik]]></dc:creator>
<pubDate>Thu, 09 Feb 2017 00:00:00 +0000</pubDate>
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<description><![CDATA[By Don Hoyle “A man and his dad put a bomb in the sink” Abstract: This article outlines some of … <a href="https://freedomtoteach.collins.co.uk/did-you-know-notes-from-the-history-of-maths-7/">Continued</a>
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<content:encoded><![CDATA[By Don Hoyle
“A man and his dad put a bomb in the sink”
Abstract: This article outlines some of the work of Matthew Stewart who was born 300 years ago. In particular, Stewart’s theorem, for which the title is a mnemonic.
<img loading="lazy" decoding="async" class="wp-image-7579 aligncenter" src="http://freedomtoteach.collins.co.uk/wp-content/uploads/2017/02/Matthew_Stewart_portrait.jpg" alt="Matthew Stewart" width="245" height="289" />
 
Early in 2017,<img loading="lazy" decoding="async" class="alignleft wp-image-7597 size-full" src="http://freedomtoteach.collins.co.uk/wp-content/uploads/2017/02/17-01-30-Did-you-know...jpg" alt="Simson Line" width="328" height="430" /> on January 15th, it was the 300th anniversary of the birth of Matthew Stewart. You might not have heard of this Scottish mathematician hence this article. He was born on the Isle of Bute and at the age of 17 went to work under the mathematician Robert Simson at the University of Glasgow.
Simson was a geometer and the ‘Simson Line’ is named after him. He also showed how the ratio of adjacent terms of the Fibonacci sequence tended to the golden ratio. He was also fascinated by Euclid’s lost books: “The Porisms”. The meaning of this word is vague and has been lost over time. Pappus, who wrote about these books of Euclid, described a ‘porism’ as something between a problem and a theorem. It isn’t a statement of truth, like a theorem, nor a construction and even the mode of reasoning involved has been lost. Stewart went on to study under Colin Maclaurin (as in the series) and, aged 30, succeeded him at the University of Edinburgh as Chair of Mathematics. Stewart’s interest in geometry enabled him to prove Kepler’s second law of planetary motion (‘The line joining the Sun to a planet sweeps out equal areas in equal times’). Kepler had formulated this and the other two laws from astronomical data in 1609. He didn’t know why they were true or how they related to one another. One of Isaac Newton’s motivations in publishing his great work the Principia in 1687 was to show how Kepler’s laws were derived from the inverse square law of gravitational attraction. Stewart proved the second law geometrically. The proof is hard to follow as it uses no diagrams¹! One result that can be followed is Stewart’s theorem². In the diagram, let a, b and c be the lengths of the sides of a triangle. Let d be the length of a line from a vertex to the opposite side a (called a ‘cevian’). If the cevian divides a into segments m and n, with m adjacent to c and n adjacent to b, the Stewart’s theorem states that:
b² m + c²n = a(d² + mn).
This can be rearranged as man+dad=bmb+cnc, for which there is the mnemonic used as the title for this article).
 This theorem can be proved using the cosine rule:
c² = m² + d² – 2dmcosθ 
 b² = n² + d² – 2dncosθ’ as θ and θ’ are supplementary cosθ’= – cosθ 
 b² = n² + d² + 2dncosθ 
Multiplying the first equation by n and the second by m and add to eliminate cosθ 
 b²m + c²n = nm² + n²m + (m+n)d² 
 = (m + n)(mn + d²) 
 = a (mn + d²) as a = m + n
<a href="http://freedomtoteach.collins.co.uk/wp-content/uploads/2017/02/Stewarts_theorem_svg.png"><img loading="lazy" decoding="async" class="aligncenter wp-image-7581" src="http://freedomtoteach.collins.co.uk/wp-content/uploads/2017/02/Stewarts_theorem_svg.png" alt="Stewarts theorem" width="309" height="242" /></a>
Diagram of Stewart’s theorem
Not all of Stewart’s work was successful. In 1763 he used Newton’s Theories to calculate the distance to the Sun from Earth and was out by over 25%.
<hr />
¹ In the second volume of “Essays of the Philosophical Society of Edinburgh” available to download from google books 
² By Krishnavedala – Own work, CC0, https://commons.wikimedia.org/w/index.php?curid=18036511 
This can be rearranged as man+dad=bmb+cnc, for which there is the mnemonic used as the title for this article).
<hr />
Collins publishes a range of key Maths resources across Key Stage 3, GCSE, and A-level. <a href="https://collins.co.uk/category/Secondary/Maths">Find out more.</a>
The post <a href="https://freedomtoteach.collins.co.uk/did-you-know-notes-from-the-history-of-maths-7/">Did you know? Notes from the history of Maths</a> appeared first on <a href="https://freedomtoteach.collins.co.uk">Collins | Freedom to Teach</a>.
]]></content:encoded>
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<title>Did You Know? Notes from the History of Maths</title>
<link>https://freedomtoteach.collins.co.uk/did-you-know-notes-from-the-history-of-maths-6-3/</link>
<dc:creator><![CDATA[stefanlesik]]></dc:creator>
<pubDate>Mon, 18 Apr 2016 11:36:54 +0000</pubDate>
<category><![CDATA[A Level]]></category>
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<guid isPermaLink="false">https://freedomtoteach.collins.co.uk/2012/02/03/did-you-know-notes-from-the-history-of-maths-2/</guid>
<description><![CDATA[‘The Man Who Knew Infinity’ In January 1916, Srinivasa Ramanujan (1887-1920), a self-taught mathematician working as a clerk in Madras, … <a href="https://freedomtoteach.collins.co.uk/did-you-know-notes-from-the-history-of-maths-6-3/">Continued</a>
The post <a href="https://freedomtoteach.collins.co.uk/did-you-know-notes-from-the-history-of-maths-6-3/">Did You Know? Notes from the History of Maths</a> appeared first on <a href="https://freedomtoteach.collins.co.uk">Collins | Freedom to Teach</a>.
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<content:encoded><![CDATA[‘The Man Who Knew Infinity’
<a href="https://freedomtoteach.collins.co.uk/wp-content/uploads/sites/87/2023/03/Ramanujan-1.jpg"><img loading="lazy" decoding="async" class="alignnone wp-image-6864" src="https://freedomtoteach.collins.co.uk/wp-content/uploads/sites/87/2023/03/Ramanujan-1.jpg" alt="Ramanujan" width="224" height="255" /></a>
In January 1916, Srinivasa Ramanujan (1887-1920), a self-taught mathematician working as a clerk in Madras, sent a letter to G.H. Hardy (1877-1947), one of the most famous English mathematicians of the day. Ramanujan sent letters to two others but they dismissed them as the work of a crank. This was Hardy’s initial impression but, on closer inspection, he was convinced the mathematics contained in the letter was the work of an unknown but brilliant thinker.
Ramanujan (pronounced Ra-man-ujan) claimed to have discovered a formula to calculate the number of primes below any given number. This was the central problem of contemporary number theory. Unfortunately, Ramanujan didn’t include the formula but there was sufficient other material to convince Hardy of Ramanujan’s brilliance. He was determined to get Ramanujan to come to Cambridge University. Bertrand Russell wrote at the time “In the hall I found Hardy and Littlewood<a href="#_ftn1" name="_ftnref1">[1]</a> in a state of wild excitement because they believe they have discovered a second Newton, a Hindu clerk in Madras on £20 a year”.
Srinivasa Ramanujan came from a very poor background. He is supposed to have written his mathematics using chalk on flagstones as he couldn’t afford paper. By the age of 13 he was already proving theorems of his own and re-discovering others including eix = cosx + i sinx (which becomes ‘the most beautiful equation in mathematics’, eiπ + 1 = 0, when x = π). Ramuanujan was devastated to find out that Euler had discovered this over 200 years earlier.
He and his family were strict Hindus and Ramanujan believed that he received his ideas from the family’s goddess Namagiri, consort of Lord Narasimha, the lion-faced, fourth incantation of Vishnu. It was only when Ramanujan dreamt that Namagiri commanded him to cross the seas, that he was persuaded to leave India.
<a href="https://freedomtoteach.collins.co.uk/wp-content/uploads/sites/87/2023/03/Ghhardy@72-1.jpg"><img loading="lazy" decoding="async" class="alignnone wp-image-6865" src="https://freedomtoteach.collins.co.uk/wp-content/uploads/sites/87/2023/03/Ghhardy@72-1-254x300.jpg" alt="Ghhardy@72" width="225" height="266" /></a>
H. Hardy was completely different. He was an atheist and had a fear of mechanical tools, never wearing a watch and loathing the telephone. He was unemotional and committed to rigorous proof. For Ramanujan, all that mattered was intuition and evidence. Hardy was quite witty and once told Bertrand Russell: “If I could prove by logic that you would die in five minutes, I should be sorry you were going to die, but my sorrow would be very much mitigated by pleasure in the proof”.
Ramanujan was interested in the study of partitions: the number of ways you can write a number as the sum of other positive numbers. For example, the number 4 has five possibilities: 1+1+1+1, 1+1+2, 2+2, 3+1, 4 so p(4)=5. You would think this was a simple situation with an equally simple formula to calculate p(n). The sequence goes 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101… Mathematicians had searched for centuries to find a formula. The Hardy-Ramanujan formula gives an approximation:
<a href="https://freedomtoteach.collins.co.uk/wp-content/uploads/sites/87/2023/03/FFTMAths1-1.png"><img loading="lazy" decoding="async" class="wp-image-6863 aligncenter" src="https://freedomtoteach.collins.co.uk/wp-content/uploads/sites/87/2023/03/FFTMAths1-1.png" alt="FFTMAths1" width="186" height="128" /></a>This formula, they showed, gets increasingly accurate as it gets larger. They also produced a formula for the exact number but it is too large to present here! Like all mathematicians, Ramanujan was fascinated by prime numbers. He devised many series approximations for pi, including this one:
<a href="https://freedomtoteach.collins.co.uk/wp-content/uploads/sites/87/2023/03/FTTMaths-Eq-1.png"><img loading="lazy" decoding="async" class="wp-image-6862 aligncenter" src="https://freedomtoteach.collins.co.uk/wp-content/uploads/sites/87/2023/03/FTTMaths-Eq-1-300x78.png" alt="FTTMaths Eq" width="407" height="106" /></a>
Even with n=0, this gives pi accurate to 6 decimal places. For each increase in the value of n, roughly eight new digits of accuracy are created.
Ramanujan found life in England hard. He was used to the warmth of southern India, hated wearing shoes and struggled to follow his strict Hindu diet. He was a pacifist during the First World War and he may also have contracted tuberculosis. In 1917, he tried to commit suicide by throwing himself under a London Tube train. Luckily the guard managed to stop the train in time but attempting suicide was illegal then and he only escaped going to prison by being confined to a sanatorium. It was during this time, when Hardy came to visit him, that the famous story of the taxi cab number originated. Struggling to make conversation, Hardy commented that the cab number was rather dull: 1729. Even on his sickbed, Ramanujan quickly protested that this was not the case: 1729 was the smallest number that can be expressed as the sum of two cubes in two different ways<a href="#_ftn2" name="_ftnref1">[2]</a>.
Ramanujan returned to India but died shortly afterwards, aged only 32. He left many notebooks crammed with his idiosyncratic style of mathematics. In one there is a table detailing the number of primes below 100 million. They are very close to being correct and are more accurate than the formula Ramanujan wrote about to Hardy originally. There is speculation that he might have discovered a new formula. In 1976 a lost notebook of his was found and mathematicians continue to study this and his other books. Maybe somewhere in Madras or the archives of Cambridge further work of Ramanujan is yet to be found.
 
<a href="#_ftnref1" name="_ftn1">[1]</a> J E Littlewood who worked closely with Hardy for many years.
<a href="#_ftnref2" name="_ftn1">[2]</a> 1729 = 13 + 123 = 103 + 93
The post <a href="https://freedomtoteach.collins.co.uk/did-you-know-notes-from-the-history-of-maths-6-3/">Did You Know? Notes from the History of Maths</a> appeared first on <a href="https://freedomtoteach.collins.co.uk">Collins | Freedom to Teach</a>.
]]></content:encoded>
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<title>Did You Know? Notes from the History of Maths</title>
<link>https://freedomtoteach.collins.co.uk/did-you-know-notes-from-the-history-of-maths-6-2/</link>
<dc:creator><![CDATA[stefanlesik]]></dc:creator>
<pubDate>Mon, 18 Apr 2016 11:36:54 +0000</pubDate>
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<guid isPermaLink="false">https://freedomtoteach.collins.co.uk/?p=629</guid>
<description><![CDATA[‘The Man Who Knew Infinity’ In January 1916, Srinivasa Ramanujan (1887-1920), a self-taught mathematician working as a clerk in Madras, … <a href="https://freedomtoteach.collins.co.uk/did-you-know-notes-from-the-history-of-maths-6-2/">Continued</a>
The post <a href="https://freedomtoteach.collins.co.uk/did-you-know-notes-from-the-history-of-maths-6-2/">Did You Know? Notes from the History of Maths</a> appeared first on <a href="https://freedomtoteach.collins.co.uk">Collins | Freedom to Teach</a>.
]]></description>
<content:encoded><![CDATA[‘The Man Who Knew Infinity’
<a href="http://freedomtoteach.collins.co.uk/wp-content/uploads/2016/04/Ramanujan.jpg"><img loading="lazy" decoding="async" class="alignnone wp-image-6864" src="http://freedomtoteach.collins.co.uk/wp-content/uploads/2016/04/Ramanujan.jpg" alt="Ramanujan" width="224" height="255" /></a>
In January 1916, Srinivasa Ramanujan (1887-1920), a self-taught mathematician working as a clerk in Madras, sent a letter to G.H. Hardy (1877-1947), one of the most famous English mathematicians of the day. Ramanujan sent letters to two others but they dismissed them as the work of a crank. This was Hardy’s initial impression but, on closer inspection, he was convinced the mathematics contained in the letter was the work of an unknown but brilliant thinker.
Ramanujan (pronounced Ra-man-ujan) claimed to have discovered a formula to calculate the number of primes below any given number. This was the central problem of contemporary number theory. Unfortunately, Ramanujan didn’t include the formula but there was sufficient other material to convince Hardy of Ramanujan’s brilliance. He was determined to get Ramanujan to come to Cambridge University. Bertrand Russell wrote at the time “In the hall I found Hardy and Littlewood<a href="#_ftn1" name="_ftnref1">[1]</a> in a state of wild excitement because they believe they have discovered a second Newton, a Hindu clerk in Madras on £20 a year”.
Srinivasa Ramanujan came from a very poor background. He is supposed to have written his mathematics using chalk on flagstones as he couldn’t afford paper. By the age of 13 he was already proving theorems of his own and re-discovering others including eix = cosx + i sinx (which becomes ‘the most beautiful equation in mathematics’, eiπ + 1 = 0, when x = π). Ramuanujan was devastated to find out that Euler had discovered this over 200 years earlier.
He and his family were strict Hindus and Ramanujan believed that he received his ideas from the family’s goddess Namagiri, consort of Lord Narasimha, the lion-faced, fourth incantation of Vishnu. It was only when Ramanujan dreamt that Namagiri commanded him to cross the seas, that he was persuaded to leave India.
<a href="http://freedomtoteach.collins.co.uk/wp-content/uploads/2016/04/Ghhardy@72.jpg"><img loading="lazy" decoding="async" class="alignnone wp-image-6865" src="http://freedomtoteach.collins.co.uk/wp-content/uploads/2016/04/Ghhardy@72-254x300.jpg" alt="Ghhardy@72" width="225" height="266" /></a>
H. Hardy was completely different. He was an atheist and had a fear of mechanical tools, never wearing a watch and loathing the telephone. He was unemotional and committed to rigorous proof. For Ramanujan, all that mattered was intuition and evidence. Hardy was quite witty and once told Bertrand Russell: “If I could prove by logic that you would die in five minutes, I should be sorry you were going to die, but my sorrow would be very much mitigated by pleasure in the proof”.
Ramanujan was interested in the study of partitions: the number of ways you can write a number as the sum of other positive numbers. For example, the number 4 has five possibilities: 1+1+1+1, 1+1+2, 2+2, 3+1, 4 so p(4)=5. You would think this was a simple situation with an equally simple formula to calculate p(n). The sequence goes 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101… Mathematicians had searched for centuries to find a formula. The Hardy-Ramanujan formula gives an approximation:
<a href="http://freedomtoteach.collins.co.uk/wp-content/uploads/2016/04/FFTMAths1.png"><img loading="lazy" decoding="async" class="wp-image-6863 aligncenter" src="http://freedomtoteach.collins.co.uk/wp-content/uploads/2016/04/FFTMAths1-300x208.png" alt="FFTMAths1" width="186" height="128" /></a>This formula, they showed, gets increasingly accurate as it gets larger. They also produced a formula for the exact number but it is too large to present here! Like all mathematicians, Ramanujan was fascinated by prime numbers. He devised many series approximations for pi, including this one:
<a href="http://freedomtoteach.collins.co.uk/wp-content/uploads/2016/04/FTTMaths-Eq.png"><img loading="lazy" decoding="async" class="wp-image-6862 aligncenter" src="http://freedomtoteach.collins.co.uk/wp-content/uploads/2016/04/FTTMaths-Eq-300x78.png" alt="FTTMaths Eq" width="407" height="106" /></a>
Even with n=0, this gives pi accurate to 6 decimal places. For each increase in the value of n, roughly eight new digits of accuracy are created.
Ramanujan found life in England hard. He was used to the warmth of southern India, hated wearing shoes and struggled to follow his strict Hindu diet. He was a pacifist during the First World War and he may also have contracted tuberculosis. In 1917, he tried to commit suicide by throwing himself under a London Tube train. Luckily the guard managed to stop the train in time but attempting suicide was illegal then and he only escaped going to prison by being confined to a sanatorium. It was during this time, when Hardy came to visit him, that the famous story of the taxi cab number originated. Struggling to make conversation, Hardy commented that the cab number was rather dull: 1729. Even on his sickbed, Ramanujan quickly protested that this was not the case: 1729 was the smallest number that can be expressed as the sum of two cubes in two different ways<a href="#_ftn2" name="_ftnref1">[2]</a>.
Ramanujan returned to India but died shortly afterwards, aged only 32. He left many notebooks crammed with his idiosyncratic style of mathematics. In one there is a table detailing the number of primes below 100 million. They are very close to being correct and are more accurate than the formula Ramanujan wrote about to Hardy originally. There is speculation that he might have discovered a new formula. In 1976 a lost notebook of his was found and mathematicians continue to study this and his other books. Maybe somewhere in Madras or the archives of Cambridge further work of Ramanujan is yet to be found.
 
<a href="#_ftnref1" name="_ftn1">[1]</a> J E Littlewood who worked closely with Hardy for many years.
<a href="#_ftnref2" name="_ftn1">[2]</a> 1729 = 13 + 123 = 103 + 93
The post <a href="https://freedomtoteach.collins.co.uk/did-you-know-notes-from-the-history-of-maths-6-2/">Did You Know? Notes from the History of Maths</a> appeared first on <a href="https://freedomtoteach.collins.co.uk">Collins | Freedom to Teach</a>.
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