# Mathematical Miscellany #55

Featuring:

In Mathematical Miscellany #54 I featured two excellent resources from Curriculum for Wales; a third is now available, “The Foundations of Algebra” is suitable for progression step 3 of the new #CurriculumForWales (age 11). The workbook contains chapters on patterns, commutativity, distributivity & associativity.

I do like the above exercise which as the Teacher’s Guide acknowledges is based on Don Steward’s work, directed number arithmetic speed up and Chris McGrane’s Alternative representation of integers. A further useful resource for such an exercise is Jonathan Hall’s Directed Number MCQ Generator on MathsBot with which you can generate all the addition and subtraction multiple-choice questions you want; choose between Counters on or off.

As with the other two resources, a very comprehensive teacher’s guide is also available. You can see the contents here, this resource with its carefully chosen and varied activities and exercises will help students with the foundations of Algebra.

On the subject of negative numbers, from PhET simulations we have another excellent resource in their latest addition to the Mathematics collection. To use the number line as a model for ordering real numbers and also to illustrate operations with negative numbers we can use the excellent, Number Line: Distance. Also available are Number Line: Integers, and Number Line: Operations. All are excellent for students to explore.

This resource has been added to my post on Negative Numbers which looks at some resources to develop understanding of operations with positive and negative integers and exercises for practice.

A popular post on this blog is on Venn Diagrams, first written in 2016 this has recently been checked and updated with some new resources including always excellent resources from Amanda Austin on Dr Austin Maths. Included in her Probability resources you will find an excellent section on Set Notation and Venns.

GeoGebra retweeted this from Javier Cayetano Rod

…the translation:

“Adding some leaves to the stem of a flower can be the perfect excuse to talk about translations and turns in space @debora_pereiro . Added to the flower generator in @geogebrahttps://geogebra.org/m/duqthjva

Which in turn led me to this amazing GeoGebra collection, Flowers 3D, Author: Deborah Pereiro Carbajo.

Have look at Flowers from Curves, simply brilliant!

Explore the collection – you could be a while!

Complete Maths has made Robert Smith’s session “Web Autograph, a First Look” freely available

I wrote earlier this year on the excellent Math Whiteboard. This is completely free to use; If you create a whiteboard you can then get a link for that whiteboard which you can share. When I have created a whiteboard I then save a second copy so I can always return to the original.

With an individual subscription ( currently \$15 for a year) it is possible to access all the features of Fluid Math including as an authoring tool for creating Math Whiteboard activities. It is also possible to save your activities.

A new feature is available – the ability to create answer boxes to check for correct answers. You can see examples using this feature here.

By Colleen Young

# Negative Numbers

Students need to be confident in applying the four operations to positive and negative numbers; the image above shows the subject content included in GCSE (age 14-16) Maths specifications. KS3 (age 11-14) students too are required to be fluent with negative numbers.

A look at some resources to develop understanding of operations with positive and negative integers and exercises for practice …

To use the number line as a model for ordering real numbers and also to illustrate operations with negative numbers we can use the excellent PhET simulations, Number Line: Distance, Number Line: Integers, and Number Line: Operations. All are excellent for students to explore.

Create a free teacher account to access further help and information on any PhET resource including a very comprehensive guide to using the resource and suggested lesson activities. See Number Line: Integers and Number Line: Operations.

I’ll mention the ‘sign rules’ here as they cause much confusion and many misconceptions, we need to word these so carefully. In fact, we need some more examples to stress we are talking about adjacent like signs.
Look at -4−-3 and -4−-6 for example.

From Jonathan Hall, on MathsBot, generate all the addition and subtraction multiple-choice questions you want with his brilliant Directed Number MCQ Generator. I really like the option to have questions using counters of varying relative sizes.

Also from Jonathan Hall, try his Directed Number.

On Chris McGrane’s Starting Points Maths, check the Negative Numbers category. Several tasks are available.

We can use Graspable Math to check calculations.

We have numerous activities from Nrich.

From Don Steward we have many wonderful resources, see

Check this excellent post of his suggestions.

On Craig Barton’s Variation theory, you will find several negative number examples. Also from Craig Barton, try these two Maths Venns and of course Diagnostic Questions where we could search in several ways, but have a look for example at the collections including The Eedi Ultimate Scheme of Work.

From the brilliant Standards Unit, N8, Using directed numbers in context and N9, Evaluating directed number statements.

Colin Foster’s KS3 Instant Maths Ideas (3 books) are freely available online; these contain a wealth of ideas to try in the classroom. 1.13 in Number and Algebra is on Negative Numbers.

Transum has several activities including basic practice in the four operations.

Try the interactive tutorials on CIMT for these quizzes where students can check answers, Addition and Subtraction and Multiplication and Division.

Corbett Maths provides us, as always, with videos and many practice examples. Scroll down the list to Negatives for a set of resources.

On Dr Frost Maths, see the KS3 Negative Number resources, note that the downloadable resource (slides which cover the four operations) and the video can be accessed without signing in.

On Mathisfun you will find information on number lines including questions to try, examples and definitions for adding and subtracting positive and negative numbers and examples for multiplying and dividing negative numbers.

I’ll mention the ‘sign rules’ here as they cause much confusion and many misconceptions, we need to word these so carefully. In fact, we need some more examples to stress we are talking about adjacent like signs.
Look at -4−-3 and -4−-6 for example.

I like the wording here, students are often confused with sign rules when adding and subtracting positive and negative numbers.

# Mathematical Miscellany #32

A nice little starter from Transum, Small Satisfaction. I do like the related activities, I know some of my students will do this first level quickly, the second and third levels will provide them with something to think about, or perhaps the alternative, Satisfaction.

Remember that Transum provides starters for students all the way through to Avanced level.

A divisibility test for 7…
Staying with Transum, Divisibility Tests 2-12 provides a handy summary of divisibility tests. What really caught my eye here is the divisibility test for 7. Scroll down the page for the comment from Transum which tells the story of the origins of the test – a 12-year-old student. And note the proof – something for our A Level students perhaps!

Note the exercises to practise using the divisibility tests. Delightfully Divisible will keep everybody busy! See also, from Plus magazine – Finding the nine. (There is a link to a very clear solution in the video).

Mathisfun

Mathisfun is an attractively presented site, the divisibility rules are clearly explained and if you scroll to the end of the page you will see multiple-choice questions to try.

Alternative tests for 7 exist, but I like Chika’s Test!

The page on the divisibility tests for 7 is a reminder of the site – Mudd Maths Fun Facts. This searchable collection of Mathematics fun facts from Harvey Mudd College Math Department make ideal lesson starters or perhaps useful for those odd moments. Note the search on the left, it is possible to search by topic, difficulty level and keywords.

For example, try:

A blast from the past for many teachers perhaps, from MathsFrame, the ITPs have been made available for modern browsers – no Flash needed. Developed in 2002 to help with the teaching of maths in UK Primary Schools, the National Numeracy Strategy Interactive Teaching Programs cover many areas of numeracy including number, shape, measure and statistics.

To finish this collection, some Number Puzzles from Mathisfun.

Mathisfun number puzzle

Note the complete Mathisfun Puzzle Index.

# Happy New Year – 2017

Wishing educators and students everywhere a very Happy New Year.

Visit the page Happy New Year for ideas for Mathematics classrooms.

# Numbers – Visualizations

I have several references in various places on this blog to some great visualizations.
Time to put them all together!

Jeffrey Ventrella’s Composite Number Tree

From Jeffrey Ventrella this wonderful Composite Number Tree – I have used this successfully with many students. It makes a great starter. Students can work out themselves how the tree is being formed and comment on any patterns they notice.

Stephen Von Worley

Brent Yorgey

Another excellent visualization, animated factorization diagrams comes from Data Pointed. And here is Stephen Von Worley’s blog post, Dance, Factors, Dance which tells the tale of the animation. Noting his reference to Brent Yorgey’s factorization diagrams led me to Brent’s own later post, More factorisation Diagrams. I love Brent’s use of colour here. If you want even more on these great diagrams he has more information and links on this page on his blog, The Math Less Traveled.

Fawn Nguyen – Visual patterns

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.

On the subject of Diagrams generally I have several posts on the subject.