# Multiplication and Division

A look at a selection of resources for multiplication and division – from 2×1 to multiplication and division of polynomials!

The tutorial video is very clear on how this Mathigon resource can be used with spaced repetition, there is also, very usefully a Shuffle mode for simple practice.

Choose the Settings and Instructions icon for a summary of the resource and access shuffle mode to practice with a random subset from all decks at once. You can also jump to a later deck with more difficult questions and different visualisations.

For more practice try Transum for several resources for mastering tables, including, for Mixed Practice, Beat the Clock. Level 6 offers mixed multiplication and Division problems, later levels use all 4 operations.

Also from Transum, try Expedite:

For a challenge, try this Open Middle problem from Steve Phelps, Multiplying Two-Digit Numbers.

For a very clear demonstration of the area model for multiplication, see PhET Sims Area Model – Multiplication. You can enter your own calculations and as always with the PhET Simulations choose what you want displayed. For a look at how multiplication tables help understand multiplication, factoring, and division, see from Phet Sims – Arithmetic.

Following on from the area model, for your older students, try Divide it up from Underground Maths, a resource designed to help students to make links between multiplication and division of polynomials using multiplication grids. The problem is presented in the image here, but also provided is a warm-up activity and further notes

The UK National Curriculum specifies that:
Pupils should be taught to:

Year 6 (UK KS2 age 10-11)

• multiply multi-digit numbers up to 4 digits by a two-digit whole number using the formal written method of long multiplication
• divide numbers up to 4 digits by a two-digit whole number using the formal written method of long division, and interpret remainders as whole number remainders, fractions, or by rounding, as appropriate for the context
• divide numbers up to 4 digits by a two-digit number using the formal written method of short division where appropriate, interpreting remainders according to the context

Appendix 1 of the KS2 document includes the examples below and states that “the examples of formal written methods for all four operations illustrate the range of methods that could be taught. It is not intended to be an exhaustive list, nor is it intended to show progression in formal written methods. For example, the exact position of intermediate calculations (superscript and subscript digits) will vary depending on the method and format used. For multiplication, some pupils may include an addition symbol when adding partial products. For division, some pupils may include a subtraction symbol when subtracting multiples of the divisor.

And at KS3 (UK age 11-14) we are reminded that students should be able to “use the four operations, including formal written methods, applied to integers, decimals, proper and improper fractions, and mixed numbers, all both positive and negative”.

From MathsOnline, comes a clear step by step demonstration of long multiplication, (uses html). Each step is clearly explained. Long Division is also very clearly illustrated and for your older students we can extend to Algebraic Long Division. Refreshing the page generates a new example each time.

Mathisfun explains long division very clearly and additionally offers questions and worksheets. Similar resources are available for Long Multiplication and for

For examples, exercises and problems try the ever-reliable CIMT’s GCSE chapter 6 on the Number System – see Section 6.4.

# Mathematical Miscellany #35

Resources

Non-Examples – Expressions and formulae, Andy Lutwyche

From Andy Lutwyche try Non-Examples – Expressions and Formulae – Reasoning Tasks, a resource with seven sets of five questions and solutions, some of which are correct and some of which are not. Students decide which are correct and explain how they have come to their decision. The topics covered are simplifying expressions, substitution, expanding and factorising expressions including quadratics, rearranging formulae and algebraic fractions. Resources like this can promote excellent class discussion.

Also available is Non-Examples – Shapes (and Angles) – Reasoning Tasks.
For more on Non-Examples – see this post on Frayer Models.

To log or not to log – Underground Maths

To log or not to log – Underground Maths

I used a favourite Underground Maths resource this week – To log or not to log? This has worked really well every time I have used it. The activity requires students to think about the methods which could be used to solve the various equations. I have always found that in addition to working on indices and logarithms this task has exposed some misconceptions, with students trying to invent some new and invalid laws of logarithms!

This problem is classified as a Problem Requiring Decisions.

Students are often used to problems being posed in such a way that they have all the information that they require in order to start, and no more. Problems (especially from the real world) are very often not like this, and so resources of this type will give students the opportunity to develop the skills needed to deal with this. Some problems might not contain enough information, so students may need to decide on classifications, make assumptions or approximations, or do some research in order to move forward. Some problems might contain too much data, so that part of the challenge is to identify the useful information.

Here’s another Underground Maths task, Powerful quadratics, which will certainly give your students food for thought, as the authors state:

When students are familiar with concepts and ideas they often benefit from exploring them further to improve their understanding. These problems aim to allow this further exploration, and for example, might bring different techniques together, highlight interesting or unusual cases, or probe the definition of mathematical terms.

Puzzles
From Matthew Scroggs have a look at mscroggs.co.uk for a wonderful collection of puzzles including Advent and Christmas themed puzzles. Try Matthew’s Christmas Card 2019 with its nine puzzles – a great card for your mathematical friends!

Year 13 will get that card this week, thank you Matthew!

On the subject of Christmas, there are still loads of doors to open on those Advent Calendars and if you are looking for activities for the last week of term, try the Christmas 2019 collection.

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AQA – Maths Digest
From AQA, have a look at their Maths Digest written to support Mathematics Teaching and learning. Useful whichever examination board you use, the digest offers tips and resources. I do like “Small things make a big difference” on avoiding common exam mistakes. This PowerPoint highlights where marks are often unnecessarily lost.

“Top tips for perfecting exam techniques” by Julia Smith provides her top tips to help students perfect their exam technique and to help gain crucial marks. In the article, Julia refers to AQA’s list of command words, so useful to show your students.

In the Recommended resources section, I have reminded readers about AQA’s brilliant GCSE Mathematics: 90 maths problem  solving questions.

There is a helpful introductory section for teachers and note also the helpful Classification Tables by Strategy and by Content Area.

Em,  has a brilliant PowerPoint with all the questions and answers – see it here.

As part of the Maths Digest, you can also find information on AQA’s GCSE papers.

Working on Bivariate Data this week with my Further Maths students we were certainly able to use technology …

This GeoGebra applet allows students to move points and watch the effect on the line of best.

This can be used in class by asking students to plot the points, draw their lines of best fit and then comparing with the computer. This worked really well on my phone, I simply sent myself an email with the link and was able to move points easily. This could also be used with younger classes when talking about lines of best fit.

We can also demonstrate correlation coefficients and lines of best fit with this PhET simulation on Least Squares Regression.

Choose from a range of examples or choose Custom to add your own points and guess then check the correlation coefficient. You can also draw your own line of best fit and compare it to the theoretical line of best fit. Note the option to include residuals for both your own attempt and the line of best fit.

For more on resources for Regression see this post; you could also use Desmos, GeoGebra, Excel or WolframAlpha.

By Colleen Young

# Mathematical Miscellany #31

Underground Mathematics Review Questions
A project started recently to look at the various review questions available on Underground Maths to support teaching of the A Level specification now has further review questions added.

Read about the use of Review questions in the classroom on this Teacher Support page. The fact that these questions are designed to test understanding of one or more topics, exercise problem-solving and proof skills and help students make connections make them ideal for use in the A Level classroom. They can be used to introduce a topic or are also ideal for review, something which should be a regular part of linear courses.

For further questions to challenge your older students, see Challenging Mathematics Questions.

Posters and Displays

Looking at my blog statistics, I see the poster collection is very popular, it’s that time of year I think! I realised there were some missing gems and have added Ed Southall’s wonderful Mr Men collection (Note the two posts with poster packs and his delightful colouring book!

Also added is Clarissa Grandi’s display suggestions on her wonderful Artful Maths. Linking these two together, I do like the “Today I’m feeling…” collection!

Whilst there is much debate about classroom displays, see Mark Esner’s thoughtful TES article on the subject, we still may have corridors we would like to add some displays too or events where posters can be helpful. Many of the resources in the poster collection such as the Nrich collection  or Joey Kelly and Xi Yu’s Play With Your Math display very well on the whiteboard.

Puzzles

Earlier in the holiday I noted some puzzles in the post Puzzle Time; to add to those puzzles, some further puzzles for consideration from two favourite websites, Transum Mathematics and Mathisfun. Many of these would work as lesson resources.

For those still on holiday (the numbers are diminishing I know!) remember the #SummerMathsPuzzles from the Isaac Newton Institute for Mathematical Sciences. Each weekday throughout August the Institute publishes a maths-based puzzle.  Whilst the puzzles do not require sophisticated maths to solve, they are not be easy. A new puzzle becomes live each day in August and answers will be provided at the end of the month.

Mirror Writing
Visiting the Leonardo Da Vinci A Life in Drawing exhibition one sees many examples of his mirror writing. Curious to learn a little more, I came across information and additionally a lesson plan from the Museum of Science, Boston. Something a little different when teaching reflections perhaps?

UK Examination Results
I am keeping my Results 2019 page updated. The page includes links to all the results statistics and grade boundaries as well as links to ofqual’s apps and other news items.

By Colleen Young

# Challenging Mathematics Questions

…for age 16+

The following questions come from various university Admissions Tests but will be useful for any students really wishing to challenge themselves in Mathematics, particularly those students going on to study Mathematics beyond school.

Advanced Problems in Mathematics, Stephen Siklos. Step Paper I 1993

You can download a free copy of  Stephen Siklos’ Advanced Problems in Mathematics:Preparing for University. Whilst Advanced Problems in Mathematics has been written to support students taking STEP examinations, it is excellent preparation for any undergraduate Mathematics course. One aim of the book is to bridge the gap between school anduniversity.

Following each question, you will find a discussion and a full solution. The clear Contents page lists all 75 problems. Each problem has a title which is a direct link to the problem.

STEP (Sixth Term Examination Paper) Mathematics is a well-established mathematics examination designed to test candidates on questions that are similar in style to undergraduate mathematics. The specification is available on the Cambridge Assessment website.

You will find free STEP and AEA (Advanced Extension Award) solutions on MEI’s website. The STEP question papers are all available on the Cambridge Assessment website, note the STEP resources include a searchable database showing the topic of each question.

The AEA qualification from Pearson is based on the A Level specification and designed for the top 10% of students to help differentiate between the most able candidates. Note that AEA papers and mark schemes can be found here.

Underground Mathematics has STEP questions within their Review Questions. Each question comes with a fully worked solution.

See also, from Cambridge University, their STEP Support Programme. From the home page, access the resources, you will see STEP Support Programme Foundation modules, STEP 2 modules and STEP 3 modules. Any Mathematics students aiming for the highest grades, whether they are sitting university Admissions tests or not can benefit from studying for example the Foundation modules which provide a structured introduction to solving STEP problems.

From Nrich, Prepare for University.

To challenge your students, MAT,  STEP and AEA questions are all excellent. Dr Jamie Frost has created such a useful resource with his STEP, MAT and AEA questions all aligned to new A Level chapters. This document is 156 pages of categorised questions (brief answers are given). Also available is a pdf file of just the STEP questions.

For mark schemes see:

Further support for the Mathematics Admissions Test is available on Dr Frost’s site. The resources include topic based worksheets also a problem solving booklet.; the booklet has 151 problems which should certainly keep any students preparing for Mathematics university interviews busy!

Note the Underground Mathematics Review Questions include Oxford Mathematics Admissions Test questions and full solutions.

TMUA is a newer admissions test only one question is available on the Underground Maths site, however, papers from 2016 onwards and mark schemes here and there is much overlap between the specifications for the TMUA and other tests such as the Oxford MAT, so these questions should provide useful resources for students taking the TMUA examination. Interestingly, Durham University states that “Those students already registered for MAT may substitute those results in place of our own test, if they do not wish to take both.”

Warwick University advise taking one of MAT, TMUA or STEP. As Warwick state, these questions deepen students’ understanding of the A-Level material.

As already mentioned, TUMA papers and mark schemes are available from Cambridge Assessment and I would highly recommend the presentation introducing the test, from Julian Gilbey. As suggested – try the questions first (pdf file) before watching the presentation.

Talking to Julian Gilbey, he recommends for the TMUA, the importance of working through the Extended specification notes on the website, to learn about the logic side. (See Test Specifications for the specification and enhanced specification.) He also stresses that the more Maths you can do, the more you work on stretching problems and think hard about maths the better you will get at maths. Examples he mentions for resources are any questions on the Underground Maths website, (not just the review questions already mentioned here), UKMT and olympiad problems, STEP problems (probably just STEP I initially).

“And essentially your ability to ‘think mathematically’ and to solve mathematical problems is all that these tests are testing”

For further sources try UKMT Senior Maths Challenge Questions.

Dr Frost Maths.com

…and also see MadAsMaths with its many papers and solutions increasing in difficulty.

# Underground Mathematics – Review Questions

There are many Underground Mathematics Resource Types. I do like the Review Questions which in the words of the Underground Maths Team:

These are questions designed to test students’ understanding of one or more topics and to exercise their problem-solving skills. In many cases they can also be used as a classroom resource to help teach concepts and methods. They are mostly drawn from past examination questions and have been chosen as ones that are interesting in nature and require non-routine thinking. The hints and solutions are designed to explain the reasoning and highlight connections as well as giving the answer. In many cases, alternative methods or solutions are presented.

Read about the use of Review questions in the classroom on this Teacher Support page. The fact that these questions are designed to test understanding of one or more topics, exercise problem-solving and proof skills and help students make connections make them ideal for use in the A Level classroom. They can be used to introduce a topic or are also ideal for review, something which should be a regular part of linear courses.

You can browse all the Review questions or narrow your search by question type; note the O/AO-level questions which are questions from old papers. One can also search by line (Number, Geometry, Algebra, Functions or Calculus) and by Station.

The Underground Maths team put together a spreadsheet that offers resource suggestions for each of the content statements in the DfE specification.  Scroll down the page for the spreadsheet.

For each content statement, Underground Maths have suggested up to three rich resources and up to three Review questions. Each suggestion is hyperlinked to take you directly to the resource on the Underground Mathematics site. Resources that are particularly good at supporting the overarching theme of Mathematical modelling have been highlighted.

I have used many of the Review Questions successfully, something I will continue to do. My spreadsheet here gives the review questions by title rather than just number for each section of the subject contents. My starting point is the suggested review questions, most of which I have included; these are personal favourites. I can later add any additional suggestions.

The latest copy of the spreadsheet on this page on Underground Maths which is one of a series of pages on A Level Teaching Resources.