Aug 30 2020

Bell Work

At the beginning of a lesson, I like to get everybody busy straight away, making a calm start to the lesson and very much like the idea of so-called ‘bell’ work. Give students a task that is simple to understand and requires no more than a simple instruction, question/s and/or diagram on the board (no technology required – unless you are in the room ahead of your students which offers more possibilities). This is a particularly useful idea if students arrive at different times. Students are expected to get to work as soon as they enter the room.

In current times, the students may well be in the room before their teacher, so could be given instructions at the end of a lesson on what they are expected to be working on at the beginning of the next lesson.

Some ideas

A short question or questions on a topic studied recently.
Ask students to write down all they can remember on any topic. They could perhaps draw diagrams or just jot down examples or vocabulary – anything at all – a ‘Brain Dump’, see ‘Brain Dumps: A small strategy with a big impact’ on Retrieval Practice.
Ask for some specific facts, eg write down the names of all the quadrilaterals they can with a quick sketch for each.
Students make up some short questions to review a topic – they could then put their questions to the class.

Provide students with a diagram, they write a question, (See ‘Here’s the diagram ….’). Particularly useful for providing a diagram or a question to write up quickly is Peter Mattock’ wonderful Goal Free Problems, a site he set up, in his own words “to allow teachers to access and share goal free problems created by myself and others. Goal free problems have been proven to support pupils in improving their knowledge and understanding by removing the cognitive load of the goal and therefore not prompting means-end analysis of a problem.” Here you will find hundreds of questions categorised by topic; there are also mixed questions available.

Prime numbers can be used for an exploring numbers type starter. Find numbers with exactly two factors. Three factors? (A square of a prime number). Four? Five? Or generalise (perhaps rather too long for a starter!) This investigation, How Many Factors on nzmaths requires students to find ways to group numbers, which numbers have only two factors and which have only three factors? For further ideas see these possibilities from Nrich. Two Primes Make One Square or Penta Primes for example could make suitable starters.
Prime Number ideas and numerous other ideas can be found in Colin Foster’s KS3 Instant Maths Ideas (3 books) which are freely available online; these contain a wealth of activities to try in the classroom. Colin Foster is a Reader in Mathematics Education in the Mathematics Education Centre at Loughborough University.

Also from Colin Foster on Nrich we have Mathematical Etudes where he discusses lovely rich tasks and tedious exercises! Note his Mathematical Etudes Project; scroll down the page for examples of Mathematical Etudes on Different Topics, there are many activities here for which instructions can be given to students easily.

Problems from Open Middle can be very simply explained, but can really get your students thinking.
Countdown type problems or Make 24, see 4 Numbers Game. If you just want to make a note of a few problems and solutions then you could use this page from ResearchManiacs.com. Print out some Make a number puzzles with solutions from Brain Food. Note the other problems available on Brain Food, a Logi-Number puzzle could be written up quickly for example. Many such problems are available, see for example As Easy As 1234 from MathsChallenge.net.
UK Maths Challenge questions can make excellent starters and you don’t even have to provide the multiple choice answers!
Some of the problems on Transum Software – Maths Starter of the Day are simple enough to easily write up on the board. Note that there is a complete index of starters including the topic of the starter. Many of the Shine and Write activities would also make good lesson starters.
A+ Click Maths provides another source of possible starters.
A book I like very much is ‘Thinkers’ from ATM, many questions here would be very simple to put to students at the beginning of a lesson.
At the end of a lesson – tell them what you expect them to do the minute they walk into the next lesson, so they know what there bell work is before thy even get to the lesson!

Mental Tests
Many schools are providing students with booklets for use in lessons. Alternatively or in addition to, why not give a mental test where the teacher simply reads a short question which the students can write down and then answer can make an excellent start to a lesson, or in fact can be used at any point in a lesson. These should be very much low stakes activities. There are many sources of questions you can use, for example, see

Class Quizzes from Corbettmaths for a collection of questions designed to help students remember key facts. Looking at these will probably give you ideas for writing your own quizzes too.

A really useful source of questions which can be used this way are the mental tests from CIMT; these are included with their resources for Years 7, 8 and 9 and also for GCSE. For Key Stage 3 (ages 11-14) scroll down this page for the Year 7, Year 8 and Year 9 course material, the resources include mental tests as part of the teacher support material. On the GCSE page scroll down to the teacher support material and note the mental tests available for most units, see this on Formulae for example.

Talking of mental tests reminds me of a long forgotten resource I used to use with Year 7 – The Three Little Pigs as a mental test. I found this many years ago on MathsStories.com as a free sample.

Why not start a collection of such ideas? Many of these ideas could also be used for those odd moments in a lesson when you find you have some extra time.

Mathematical Miscellany #46

A job I have been meaning to start for a while, a revisit to some of the much older Miscellany posts to remove any broken links and a reminder of the many excellent resources still available. This will be an ongoing project. For today a small selection of some enduring excellent resources from older posts.

From Fawn Nguyen comes the brilliant Visual patterns, note the menu; the Gallery includes blog posts from teachers and students who’ve used visual patterns in their classrooms.

For Problem Solving – GCSE Problem Solving Questions of the Day – Compilation from White Rose Maths available on TES Resources. The booklet contains over 50 problem solving questions suitable for KS3 and GCSE classes, answers are also provided. This resource along with other Problem of the Day resources is available from the White Rose Maths site.

Particularly excellent resources come from Andy Lutwyche, an author I have featured regularly on this blog; in fact looking back I see I recommended him as far back as 2012! Look at his excellent Erica’s Errors series for example and if we check on TES, these free resources also include plenty of errors from Clumsy Clive!

Mudd Math Fun Facts has resources which can make great starter activities, perhaps try Squares Ending in 5 and Multiplication by 11 both made excellent starters. I have looked at at proofs for these with students as well as enjoying the mental Maths tricks! You will find more lightening arithmetic suggestions on the site.

From the Science Museum, Mathematics in our World looks at how mathematics connects to so many aspects of our lives. Seeing the Spirograph, a favourite childhood toy reminds me of the brilliant digital version, Inspirograph by Nathan Friend. Try altering the gears so that the fixed and rotating gear are the same size, or make one size a factor of the other, make the two sizes have a common factor, or not! Investigate. You can change the colours too and create a work of Art!
More Spirograph resources.

I’ll end this week with a rather more recent resource, Forming Equations from Andy Lutwyche where students form equations from diagrams, the problems become more challenging but are accessible to a wide range of students.

Aug 16 2020

Mathematical Miscellany #45

Another collection this week, beginning with the latest update to my Knowledge Organisers page. Sarah Hall is creating a GCSE (‘WJEC flavoured’) collection. Sarah’s Knowledge Organiser resources can all be found (all free) on TES Resources. These have many clear illustrations and like others in the Knowledge Organiser collection, are very attractively presented.

Sarah Hall – Knowledge Organisers

A theme in posts recently has been A level resources – both Mathematics and Further Mathematics. Another recent and very welcome addition for Further Mathematics comes from the excellent Westie’s Workshop collection. This very clearly structured PowerPoint resource is for the new A-level Further Mathematics syllabus core content (CP1 & CP2) and uses questions from past papers also from the sample assessment material. The resource can be used by pupils for self-study or worked through in class.
Westie's Workshop
Clear notes at the beginning of the presentation explain how to easily navigate the slides. Full mark schemes are included.

You can view all the author’s resources with the most recent first on TES. Recent resources include IGCSE (Edexcel new course) questions and for Further Maths, Mechanics a Work Energy and Power teaching resource; you will also find resources for Mathematics A Level.
See also:
Further Mechanics 1

and for Mathematics A Level:
Mathematics AS Level Mathematics A Level

Each year, I have added a results page where you can find the results statistics and links to relevant blogs and articles. Clearly, 2020 is an exceptional year.

Check the blog from FFT Education Datalab
for the latest information including
A-Level results 2020: The main trends in grades and entries

Also, note that all the latest figures are available to explore on Education Datalab’s results microsite.

If you have not explored the new NCETM website yet, then do have a look. This is very well organised and easy to navigate. Have a look at the In the Classroom section for example where you will find materials, guidance and National Curriculum resources.
Check for example the very helpful National Curriculum Resource Tool which clearly shows the progression for topics from Year 1 through to Key Stage 3.

I like the way each area of the website links to related areas. For example from the National Curriculum Resource Tool we could choose Algebra which shows the content for Year 6 and Key Stage 3 Algebra; from that page, if you select Key Stage 3 Algebra you will not only find the content but a wealth of links and resources, a very valuable collection.

NCETM National Curriculum

Looking at the Articles for Algebra, there are seven articles – an excellent mix of case studies on work with pupils and also, for example, Anne Watson’s (2009) Key Understandings in mathematics learning, Paper 6: Algebraic reasoning (PDF), London: Nuffield Foundation

Standards Unit-Nottingham University

The Activities and Exemplification sections both provide examples and questions you can use in class. The Activities include resources from the excellent Standards Unit, links are provided to the individual activities hosted on STEM Learning. Another source for the Standards Unit comes from Nottingham University; everything is here in one place including all the professional development materials.

NCETM – KS3 Algebra Activities

NCETM – KS3 Algebra Exemplification

In this case in the Video section, we have 6 examples; I do like the inclusion of the Factorising with Multilink from Nrich.

Aug 9 2020

Mathematical Miscellany #44

Some updates this week…

MATHSgrader Wk1 Assessment Tier C

From Ross Blair on MATHSgrader comes a complete set of Interactive assessments to complement the Maths Genie homeschool program. The resources are provided in three tiers, Tier A roughly accords with grades 1-2. Tier B with grades 3-5 and Tier C is 6-9. Check the menus for Teachers and Pupils for easy site navigation.

For A Level Owen (Owen134866 on TES) has a library of Mathematics teaching resources, these are really clearly structured with step by step examples. Recently added, is A-level Core Pure Mathematics Year 2/A2 for the second year of Further Mathematics. These resources have been added to the Teaching Resources page for Further Mathematics. This Calculus content is part of the core Pure Mathematics specification common to all examining boards. For more on Further Calculus in Further Mathematics, see my post here.

Also note for Further Mathematics, Jack Brown who has created thousands of videos covering the complete A Level specification has been very busy with his Further Maths collection of teaching videos and exam paper walkthroughs; the easiest way to navigate the videos is through his website, TLMaths.com. A recent addition to the exam paper walkthroughs is the specimen paper for OCR MEI Modelling with Algorithms.

My post on Graspable Math proved popular last week and included a canvas of the problems from one of Dave Taylor’s wonderful Increasingly Difficult Question sets. I decided I liked the idea of having a canvas ready for his Simplifying Expressions, starting with IDQ-Simplifying Expressions 1, I opened it on my own canvas and adapted it slightly. I do like to keep all steps of the working displayed, so I have put the exercises on the left, creating a good space on the right.

One Incorrect - Don Stward - first problem Taking a canvas and adapting can help you learn to use the interface if you are unfamiliar with it. Using this One Incorrect canvas by Eric Weitnauer which is based on Don Steward’s one incorrect simplification I made a copy of Eric’s canvas and created one for the first problem from Don Steward’s blog post, discovering along the way that if you enter text for a web link, then Graspable Math inserts that as a link.

Note formulas on the menu.
$Graspable Math Formulae$

For a clear example of this feature in action, have a look at this video.

Aug 2 2020

Graspable Math

A revisit to Graspable Math.

Graspable Math offers a highly innovative interface for mathematical notation. You can read the Graspable Math story here.

You can learn a great deal about Graspable Math simply by experimenting, you can also find plenty of help and tutorials on the Learn section of the site, note the Gesture Library as well as the video tutorial collection. There is a YouTube channel here.

To experiment with the interface select Explore algebra online.

Graspable Math is easy to use, I decided I would solve an equation and wanted to show all the steps. To start, go to a blank canvas and choose Insert / Math Expression, I have used the method of selecting and holding the = sign to start as you can see illustrated in the video above; I was then able to enter an operation to apply to both sides of the equation.

equation to solve

$Graspabale Math - Solve Equation$

An alternative to the above is to use the settings menu; choose Dragging to apply the inverse operation to both sides.

We can also illustrate a solution graphically by inserting a graph to open a GeoGebra window.
$Graspable Math - GeoGebra$

Each expression has a circle at the end – simply drag that to the GeoGebra window. You will sometimes see more than one circle at the end of an expression, select to separate expressions hence showing all steps clearly.

The scrubbing feature is very useful, drag up or down on a number to change its value; the change will propagate through the rest of the working.

A great example of the use of the scrubbing feature can be found on this canvas created by Graspable Math for this lovely problem, Create a System of Two Equations by Daniel Luevanos on Open Middle, accessible for students yet such a great task for mathematical thinking. We could discuss inequalities here as well as simultaneous equations.

Graspable Maths have provided a ‘cheat sheet‘ providing a handy summary of how to use the interface.

Graspable Math took one of Dave Taylor’s wonderful Increasingly Difficult Question sets and created a canvas of the problems.
IDQ Changing the Subject

$Graspable Math Gesture Tutorial$ We now have Graspable Math Activities, Graspable Math have a free version and optional subscription (watch for information from Graspable Math on this, but teachers can be reassured that they can continue to use Graspable Math with their students). You can use the Activity Bank to learn how to use Graspable Math. Select Graspable Math Gesture Tutorial and you can watch demonstrations and try the features for yourself. This seems a really good way to learn to use the interface.

It is possible to insert a video onto a canvas, so you could inset a video and then follow the instructions on the canvas. Try this on solving equations.

Graspable Math – One Incorrect

A lovely way to practice – use this One Incorrect canvas by Eric Weitnauer which is based on Don Steward’s one incorrect simplification.

Also a good way to learn to use Graspable Math – save a canvas and adapt it, I made a copy of Eric’s canvas and created one for the first problem from Don Steward’s blog post, discovering along the way that if you enter text for a web link, then Graspable Math inserts that as a link.