# Building Houses

To use demonstrations of those wonderful applets such as Building Houses that work on modern browsers, go to the Numworx Secondary Education Site. To use the resources without saving any work you can simply Login as Guest. Whilst a guest login has restrictions you will be able to explore and use resources, many of which make excellent demonstrations.

A site very well worth exploring.

To find the resource so many teachers know as “Building Houses” select Old, then Secondary Education and then Geometry.

The Cube buildings exercises include 4 sets of 10 exercises to keep everybody busy! The last set of 10 involves building structures given only the silhouette – quite a challenge. It takes a little bit of pratise to become familiar with the mouse controls,  a left click adds a cube. To delete a cube hold the left click button and press the right click button also.

Further examples of these applets are on the page, Wisweb Applets HTML5.

# Mathematical Miscellany #36

Access Maths – Starters
On Access Maths you can now find a whole library of Starters covering Algebra, Sequences, Probability, Geometry and Number. The Starters have been designed to be used in the final term with a top set year 9 in the author’s school. These starters could be very useful for GCSE in Years 10 and 11.

Also on Access Maths, you can find all the Form Time Numeracy activities you ever wanted!

Though if you do want some more then look no further than Jonathan Hall’s (@StudyMaths) Form Time Ideas.

Topic Tests for Mathematics and Further Mathematics A Level

A growing set of really useful Topic Tests are available for both Maths and Further Maths A Levels from AQA. I like the index provided by the mapping documents, one for Maths and one for Further Maths. For each test there is a clear statement of what is assessed in the test which comprises two sections. The questions in section A test basics of the topic and those in section B require a bit more thinking. Mark schemes are provided for all tests.

AQA Topic Test – Series and Sums

Pyramid Puzzles – AndyLutwyche

Indices Pyramid Puzzles – Andy Lutwyche

I frequently recommend Andy’s excellent resources, recent resources include his pyramid puzzles, currently available, he has puzzles on Indices, Expressions and Calculations (designed to be done without a calculator).

I recently recommended Andy’s Non Examples – Reasoning tasks, this collection has now grown.

Non-examples Andy Lutwyche

I’ll end this collection with a little listening – literally a little listen from the Learning Scientists who include Bite-size research in their podcast collection. Their latest episode, Bite-Size Research on Seductive Details, looks at the details we include in our teaching that are not necessarily relevant for understanding the topic but make the lesson more interesting  – a good idea? Some further research into how you present new information to students.

# Sketching Graphs

GCSE Subject Content – Graph Sketching

For GCSE, students are required to be able to sketch graphs, an essential skill too for Advanced Level. Students can be taught graph sketching skills in Key Stage 3 as they study any work on graphs.

AQA – Higher Tier Teaching Pack

For AQA’s Maths Digest I recently recommended their Higher Tier Teaching Pack which contains a library of classroom resources.

On the same page, the word document, GCSE 8300 Topics list, is a useful index of resources and if we look at Algebra, for example, we see Sketching Graphs.

Selecting Sketching Graphs takes you to a page where you will see a choice of tabs, Plan, Teach and Assess, under Teach we have several resources for the topic.

Several resources are available including Craig Barton’s Diagnostic Questions which are ideal when looking at Graph Skectching.

Diagnostic Questions

Under Activities, you will see a favourite of mine, I often recommend Transum resources, Graph Match has various levels of activities in which a number of linear, quadratic, reciprocal and other graphs must be matched up with the given equations. Note too ‘More Graph Activities‘ including this lovely Advanced Level starter.

Transum – Graph Match

A recommendation under worksheets comes from CIMT, Chapter 13 of their GCSE material, is on Graphs and section 13.11 is on graphs of common functions is a very useful summary.

CIMT GCSE – Graphs

Note too, the Topic Tests, such a valuable resource as we have tests providing excellent coverage of the specification. Included in the Higher Topic Tests is one on Further sketching graphs.

AQA Topic Test – Further sketching graphs

There are two recommendations for (free) TES resources which we can see from the reviews have been very well received.

Under Activities, we have Recognising graphs – Higher GCSE and a suggested resource under Worksheets, Quadratic, Cubic & Reciprocal Graphs.

When using graphing software to illustrate graphs when discussing sketches, we can turn off the axis numbering.

# Further Maths – Further Calculus

Included in the detailed content statements on the material which must be included in A level specifications in further mathematics which makes up approximately 50% of the total content of A level further mathematics, we see the section on Further calculus.

Given that this is a compulsory topic, teachers of Further Maths can usefully look at specimen and practice materials for all the examination boards. We can also use technology to demonstrate and check answers. Additionally there are many useful resources offering notes and examples which are freely available.

WolframAlpha can very easily be used to check any volumes of revolution and also gives a clear visualisation of the solid formed; note the option to show the surface or solid formed.

CIMT – Further Calculus

The A Level notes from CIMT include some useful notes, activities and exercises for Further Mathematics, if we look at Chapter 8, Further Calculus in the Further Pure Mathematics section the we see that the chapter includes Volumes of revolution. We also have reduction formulae which is included in the optinal content for AQA, Edexcel and OCR Specification A.

For more notes, examples and exercises, not just for Calculus but many parts of the Further Maths specification, as well as CIMT, try AJ Hobson’s ‘Just the Maths’ or the Helm Project.

Many useful notes and examples can be found in AJ Hobson’s Just the Maths (individual pdfs hosted by UEA) or a complete pdf from the Math Centre: AJ Hobson’s ‘Just the Maths.

If you have not come across the HELM Project before, the project was designed to support the mathematical education of engineering students and includes an extensive collection of notes which include very clear worked examples. The complete set is hosted by the Open University. To access the Open University resources you will need to create an account (easy and free), this will also give you access to the numerous free online courses.

(For easy reference these notes have been added to the Teaching Resources page for Further Mathematics.)

Improper Integrals – University of Pennsylvania

Looking for resources on Improper Integrals, I came across this from the University of Pennsylvania, a clear presentation with worked examples.

Edexcel have a SolutionBank and GeoGebra interactives for their various texts for Maths and Further Maths freely available online. If we look at Core Pure Mathematics Book 2 for example, you can see the various chapter links. These take you to the interactives for each chapter and to full worked solutions for each exercise. Looking at Chapter 4 for example on Volumes of Revolution, we can see two GeoGebra Interactives allowing exploration of volumes of revolution around the x and y axes.

Edexcel Geogebra Interactive – Volumes of Revolution

# Happy New Year 2020

It’s that time of year again and we can play the 2020 Year Game in our January lessons. You can preview the 2020 game now, full rules are here. Can your students use the digits in the year 2020 and the operations +, -, x, ÷, sqrt (square root),^ (raise to a power), ! (factorial), and !! (double factorial) along with grouping symbols, to write expressions for the counting numbers 1 through 100?
My students have always been curious about the double factorial function.

Excel has a function for computing double factorials, illustrated here.

I like to show my students a few examples and see if they can work out what is going on!

Have a look at this article from Wolfram Math World or have a look at this article on Ask Dr Math. Note the relationship between the double and single factorial functions.

A great idea for a starter on return to school from Alex Bellos’s Monday Puzzle, note the end of the solutions post here where Alex Bellos describes a new New Year challenge from Inder J Taneja, a retired maths professor from Brazil; can your students write 2020 using only single digits? Solutions are provided for the digits 1 to 9.

Once again, Manan Shah has provided us with some puzzles to keep us busy, 20 in fact to keep us all busy!

I always read Transum’s Newsletter with interest, the newsletter published today notes some ideas on the number 2020 including an unusual property, something that last happened in 1210!
From MathWorld–A Wolfram Web resource. http://mathworld.wolfram.com/Self-DescriptiveNumber.html

I see MEI has also noted this property in their January item of the month, Autobiographical numbers. A little further research led me to Tanya Khovanova’s 2008 paper on Autobiographical Numbers. Whilst mentioning Tanya Khovanova, we should of course check her Number Gossip site for properties of 2020 and from her paper we have a reminder of the Online Encyclopedia of Integer Sequences.

We can also look at WolframAlpha which provides further information on the number properties of 2020 including what 2020 looks like in historical numeral forms. We could use the various WolframAlpha queries to learn how Babylonian, for example, numerals work. I have successfully used this as an interesting starter for January lessons.

The Babylonian system was a positional base 60 system, though interestingly uses ‘units’ and ‘tens’ symbols to create the 59 symbols needed.

For more on the Babylonian system including how fractions were represented see History of Fractions from Nrich.

We could look back and use the excellent MacTutor History of Mathematics Archive from the University of St Andrews, Scotland. We could check today or any day for Mathematicians who were born or died on that day.

The site is searchable in several ways, including the comprehensive index of History Topics.

Teacher Resources on Line

On the subject of dates and the new year, from trol, Teacher Resources on Line, we can make a calendar for 2020. I do like the fold and tuck models – no glue required Scroll down the page.

Wishing educators and students everywhere a very Happy New Year.