A collection of teacher presentations and supporting notes for students are provided on this site. A number (certainly not all) are designed with some of the 'best practice' principles of explicit instruction in mind. A lot of the activities (sometimes contained on a slide within the presentation, most often provided separately, and/or via applets) take an inquiry approach (we share the IBs ATTL philosophy).
The below three/four provide examples of the type of presentations & student notes discussed, in detail, in this post:

SL Integration + Trapezium rule 

HL Type I & II errors 4.18 

HL Integration by substitution 5.11 

HL Differential Equations: Separation of variables 5.14

The debate rages on between direct/explicit instruction and inquiry/discovery etc learning. We think, like most teachers (probably?), that both 'camps' have a lot to offer. It's fantastic to see how much easier it is today for teachers to get access to cutting edge, and tried and tested ideas, from cognitive & neuro scientists, educational researchers and memory experts etc.

Clarity of explanation and attention to the visual layout of teacher working

I have found that changing my handwriting is not/has not been to date, an easy task! Slowing down certainly helps, but then I start worrying, on a long, complex process, that I’m losing my students attention/they’re losing the thread of the reasoning so far developed together, offering, in those pauses, opportunities for distractions to arise (I appreciate there are a range of options for tackling this, only one of which is expanded on here).
I think the ideal board for mathematics would be, as is seen in universities, three, really big black/whiteboards lined up in a row, and then make them interactive boards so they can be saved for students to consult at any time during the lesson: great for differentiated classrooms, which all classrooms are, even given homogenous attainment level groupings, from one topic to another, and for later reference and revision.
The ”three board reasoning layout” offers enough space for students to :

(a) see the whole process / reasoning from start to finish and back track if they’re unsure of any step so that they can:
(b) re-read and try and make sense of a particular step, freeing their mind from a specific confusion, and allowing it to re-focus on the current explanations being offered
(b) raise a hand, and point to the exact section of the board that they don’t follow
(c) allow the teacher to “emphasise/underline” and “replay” an explanation on the toughest/’key’ parts of the lines of reasoning, with the supporting visuals already present on the board, and ready to refer to (and students to “visualise”/”take a mental picture of”).

Something like this ‘three board workspace’, I’ve found, can be achieved by buying cheap, undercoat wall-paper from a DIY/Home improvement store and sticking it around the walls of the classroom ready for writing on, or using “window pens” if large areas of the wall space are actually ‘window space’: students can take photos which, over time/a number of classes/lessons, build into a series of notes.

However, it’s frustrating not to be able to easily conserve these on an editable interface.
and . .
how I end up laying out the working out, in response to questions that arise/whole and individual class discussions etc, , doesn’t always result in the ‘optimal’, visual presentation of each item, nor of how one concept links in to another.

The teacher slides presented on this site are my, current ‘best attempt’ to synthesise, from experience, the difficulties that students face with a given topic, choose worked examples, and ones for students to attempt, that focus on these difficulties and provide a clear line of reasoning, and links, between all the information offered on one slide (and the next).

Cognitive Load and Working memory

(with the evolving, nuanced schema being offered to teachers about this concept of ‘working’ memory).

Before we start on this topic, I really recommend listening to all sides of the debate on this complex issue. Some thought provoking nuances are offered in Guy Claxton’s: ‘The Future of Teaching, and the myths that hold it back’ (to capture attention, titles (perhaps have to), exaggerate their position/have an element of controverse in their title!) (this podcast, with James Mannion, is a good intro, see also the strong recommendations for this book, from well-respected educationalists: John Hattie and Dylan Williams, at the end of this blogpost.

"I do, we do, you do"

I found this podcast between Anita Archer and Ollie Lovell insightful and interesting.
Many of the teacher slides are designed with exactly this model in mind:
> I do: slides to work through to demonstrate the content in action.
> We do: slides to do together with students, working pairs/groups, feeding in to the teacher what they've understood as the next step or 'how we know', from the information given in the question, which technique may be applicable etc., so that class and teacher can work the question through together. The steps are covered one at a time, providing plenty of opportunity to discuss questions as they arise.
> You do: slides with questions for students to try for themselves.

Anita Archer emphasises the importance of the “we” do part of teaching and learning together with our students, and that this is a key step, too often omitted in the classes she visits and works with.

Many thanks to Craig Barton, Ollie Lovell and James Mannion for their excellent podcasts! Their channels provide listeners with access to 2 to 3 hour long, in-depth conversations, with a wide range of passionate, devoted and able educationalists, across many different fields and pedagogical inclinations (if you can, sign-up as a Patreon supporter to help them continue this incredibly valuable work).

Working Memory – animated slides

Most slides are animated so as to allow the teacher to control the amount of information the students are having to take in/discuss at any one time.

My personal preference for animations, which equally well offers the option to “show everything on the slide at once”, is linked to, whatever the final definition of working memory might be, the experience in the classroom of some students getting “overwhelmed” with new information without the necessary space to ask questions as they try and digest/make sense/take ownership of it.

Examples and NON-examples

One of the hardest skills, I think, in more complex mathematical techniques is identifying the conditions under which a certain technique is valid/can be applied.

Examples and, in particular, non-examples, can be a really helpful way to develop student’s ability to discern i.e. develop a deeper understanding of the mathematics, when a technique is, and is not, applicable.

Moreover, at HL Applications and interpretations, there are many “unsolvable” questions, given the syllabus content, but which are solvable, using techniques, that lie outside of the course.
Differential equations is a very good example of this, with a huge range of available techniques that many engineers&mathematicians have found are, “mostly solvable only via numerical methods”. Identifying some of these for students helps develop a deeper understanding of the ‘problems that humans have encountered for which this technique/area of mathematics, is their current, best response’ to overcoming/understanding, and hence influencing, those processes’.

'Edges' of the syllabus - carefully selected examples

Example questions to work through together, and student questions, are thoughtfully selected to concretise, with examples, the range/spread of questions that could be asked of them (thereby deepending their understanding, also, of the variety of contexts to which it may apply).

Formative Feedback & Worked Solutions

Formative feedback:
Whilst the ‘We’ do (from the questions students ask and help each other with) phase provides important information for the teacher on which students are likely to ask for further help first, and which parts of the process the class/individuals are struggling with most, the ‘You’ do is where a teacher can really see if students have started to take ‘ownership’/’understand’ the material.

Online platforms:
Teams, Google Meet/classroom, Zoom etc are continually evolving. We are lucky to have tablet computers (so students can write, with a stylus, their mathematics directly into their Onenote or other notes application) and the Virtual Network Computing (VNC) software, Impero, so that I can see, in real time, all students working out as they do it on the tablets. Going round the classroom/getting students to work in groups etc. also provides an effective means for students to get feedback on students understanding, but the VPN allows me to see all students screens at once and quickly see who is struggling, and who gets it, to either, discretely help a student advance (via private messaging, sending them to my working out etc) , or pair/group them with a student who could help. Numerous students all sharing their screens at once is also possible in Google meet (but the number is limited, at time of writing, in Teams and Zoom).

Non-tech approaches I’ve found effective at achieving the same (though requires, sometimes, a little extra preparation) is getting students working “in big” on large flipchart whiteboards/cheap, white home improvement paper stuck up around the walls/window pens on the windows of the classroom, allowing the teacher to “see at a glance” who is most likely to find some support beneficial and allows students to quickly and easily look at each other’s work to get help! Building an important culture of “the class as a team”, not as in competition with each other (there’s plenty of competition offered in the final exams!). Mini-whiteboards is another good option.

Worked solutions

John Hattie consistently stresses the efficacy of teachers sharing with students “what success looks like”. He cites providing answers, or better, full worked solutions, as a good way to do this (in one interview Hattie cites one of the best teachers he remembers that met this outcome from Hattie’s famous “visible learning” meta-analysis research: “Mr Tomlinson . . gave us hundreds of worked examples to learn from")

“If you are learning surface level information, the content, as contrasted with the deep learning, which is the relationship between the content . . then problem-based learning, inquiry based learning, is pretty useless . . but if you don’t teach the surface and the content, you’ve got nothing to enquire about. . . .  inquiry based learning is (often) introduced too early, before the students have the ideas. It’s introduced when some of the students have sufficient content they can go and implement the whole process of inquiry and problem-based, but other kids are left behind. One of the arts is to know when to introduce inquiry and problem-based, and there is a when time. If you’re trying to get the kids to build up sufficient knowledge and understanding and vocabulary, that’s the wrong time, once they have it, it is the right time. So yes, it can work, and the reason it turns up relatively low on the effect sizes (0.31) is that teachers introduce it far too early.” John Hattie on inquiry based learning

This blog post provides a useful summary of things to consider to ensure explanations are clear, well laid out and optimise the chances that all students integrate the information, and understanding, into their existing conceptual models & knowledge:

Like most complex systems, the many factors that contribute to "learning", across a wide range of individuals, cultural contexts, current experiences and divergent goals etc. is unlikely to be captured by a "one size fits all" solution.
Personally, in my limited experience, the 'best', in most disciplines, tend to be able to do 'everything'.

Differentiation and Mixed SL/HL classes

I have found making these presentations available to students has been incredibly helpful in managing the wide ranging demands of a mixed SL/HL class, and, more generally, meeting the differing needs of my students i.e. a very helpful resource for supporting in-class (and out-of-class) differentation.

John Hattie and Dylan Williams’ recommomendations of Guy Claxton’s book: “The Future of Teaching”.

John Hattie (Melbourne, Australia, graduate school of education and, most famously, author of the meta-analysis “Visible Learning”: “the largest ever collection of evidence-based research into what actually works in schools to improve learning”, regularly updated!): “so much simplistic nonsense is being touted about direct instruction and the knowledge rich curriculum, it is great to see someone finally talking sense. As a practising  cognitive scientist, Guy Claxton is perfectly equipped to take us beyond the familiar slanging match between traditionalists and progressivists . . he illuminates complex issues such as the function of knowledge, the psychology of creative and critical thinking, the true nature of memory, the culture of the classroom . . and the many purposes of education”.

Dylan William (Emeritus Professor of Educational Assessment, UCL, speaker and author of many highly influential educational publications): “people are often surprised that I endorse Guy’s work so strongly . . . in my own work . . I’ve focused on helping teachers do what they want, and need to do . . but supporting teachers to teach more effectively within an impoverished view of the overall purposes of education has real dangers, now more than ever.
While teacher lead approaches may be appropriate where the goal is to ensure students develop certain, important, well-defined skills, such as grammatical writing, accurate mathematical calculation or scientific reasoning, claims that all teaching should be of this kind, are way too sweeping. Those who argue for the superiority of direct instruction, no matter what you want your students to learn, remind me of the drunk, looking for his keys beneath the streetlamp, not because that’s where he dropped the keys, but because that’s where the light is: how you teach students has to be subordinate to what it is that you want your students to learn. Students’ . . dispositions, how they react to challenges and frustrations, what they believe about people who hold different beliefs to their own, their determination to do good in the world, these are arguably” the most important. Many teachers could probably identify with Dylan Williams description of his aims, in his twitter profile: “mostly interested in the power of education to transform lives, and how to do it better”.