Mental Maths

mental maths cover image

Arithmetic is of great importance to most of us in our daily lives. Whether it is making calculations in connection with an occupation or working out the grocery bill most of us tend to use methods we learnt as children in school. But those methods are not the only ones available to us. The time and effort put to learning faster or easier to apply methods can be well worth while.

An over dependence on calculators means many students fail to develop their mental maths ability to the degree that they should. This can lead to problems of poor number sense, failing to see when an answer is of a totally wrong magnitude to be sensible,

Basic arithmetic is sometimes referred to as “social mathematics” because it is what most of us comes across in everyday life. When paying for groceries, reading reports on a survey in a newspaper or calculating how much you should be paid for two and a half hours work at a certain hourly rate we want to know how the figures work out. And when we are interested in the calculations concerned we usually have no great difficulty in working them out. A student beginning algebra may be confused when asked to do calculations involving variables like x and y, yet have no problem when asked to do the same calculation when the x is an ice-cream cone and the y a bar of chocolate. We can often put up mental road blocks based on some unfounded fear of maths for which there is no need. We can sometimes tell ourselves that we can not do sums in our heads. But we can. We do them all the time without being conscious of it. And like everything we can get better with a little practice. We can use little tricks or techniques for special cases.

Mental estimation techniques give us quick answers to everyday questions when we don’t need to know the answer to the last cent or decimal point. We estimate the answers to addition and subtraction problems by rounding, which can be useful when estimating the grocery bill. As each item is rung up, round it up or down to the nearest 50 cents.

Most people will be able to multiply or divide by powers of 10 by moving a decimal point or adding zeros. If asked what is 100 times 55 they will quickly reply 5500 without much thought or effort. By playing around or experimenting with numbers and spotting patterns we can get many more such tricks, shortcuts, techniques or whatever you want to call them. Playing such mind games will in turn help you to become better at seeing patterns in numbers, and this is an important part of studying mathematics. If studying arithmetic or geometric progressions for example, it is often necessary to be able to spot the pattern that determines the next number in the sequence. Spotting patterns can often help simplify mental calculations. You will not always have a calculator with you, or may not always want to be seen to use it.

Shortcuts are most useful when they help with something you use often. There is not much point in putting a lot of effort into learning shortcuts to do things that you never use in practice. But for things that you do often they are well worth while.

Mental calculations involve using specific techniques created for solving specific types of problems, rather than memorizing the answers to equations. There are many techniques for doing rapid mental calculations, particularly if you look beyond what is normally taught in schools. Unfortunately if you do not use them often you are likely to forget most of them.

 

What is mental maths used for?

Mental maths is the process of doing mathematical calculations in your head, without the use of a calculator, or pen and paper. We do this in everyday life. For example:

  • Working out the cost of sale goods when shopping. If there’s a 20% off sale, you’ll know exactly how much you expect to pay.
  • Calculating a tip. If you dine out and receive a good service, chances are you’ll leave a tip. Mental maths allows you to calculate how much a 10% or 20% tip would be.
  • Metric conversions. You don’t have to travel far to see measurement units change. Many of us still think in terms of miles for driving distances, but most road signs are now in kilometres. Similarly it allows you to easily work out the difference between inches and centimetres, pounds and kilos etc.
  • Working out exchange rates. If you holiday abroad, you may need to exchange currency to spend while you’re there. Mental maths makes it easy to work out how much value for money you’re getting, and how much currency you can expect to receive for your own currency.

There are many other places mental maths is used, probably without even thinking about it, in everyday life, such as cooking recipes, comparing values of products/services when shopping, working out a score/grade or calculating interest due on a loan.

Students preparing for certain exams, particularly where the use of a calculator is not allowed, will need a certain competency in mental maths.

For example, the Graduate Australian Medical School Admissions Test (more commonly known as the GAMSAT) is a test used to select candidates applying to study medicine, dentistry, optometry, physiotherapy, podiatry, pharmacy and veterinary science at Australian, British, and Irish universities for admission to their Graduate Entry Programmes (candidates must have a recognised bachelor’s degree, or equivalent, completed prior to commencement of the degree).

 

Benefits of Mental Maths?

Of course, many will argue that we now all have a calculator to hand in every life situation, thanks to ever-evolving smartphones. However, that’s not to say that mental maths teaching and skills are now redundant. There are plenty of benefits, stemming from good mental maths skills development.

At a basic level, things like concentration levels and listening skills are improved, and self-confidence is also improved as a result of practising mental arithmetic problem solving.

In addition, mental maths actually keeps our brains sharp, getting stronger and more efficient with use. That’s why it’s recommended that students continue practising and learning mental arithmetic throughout their education.

Mental maths also greatly improves a person’s number sense, which improves the ability to understand relationships between quantities, allowing logical thinking and plotting to develop.

Lets you work faster (less likely to run out of time in exam, more time to concentrate on more difficult parts, can follow the teacher in class sensibly).

By developing good mental maths skills from a young age, students are able to improve other skill-sets and easily work out answers to mathematical scenarios in everyday life.

People who are good at mental maths are generally more:

  • focused. “Scary” numbers do not faze them. They identify the logic required to solve the problem and then use skills already mastered to solve it.
  • Efficient. Being able to do simpler stuff quickly leaves them more time to deal with the more difficult elements.
  • Confident. Nothing succeeds like success. This carries over into other aspects of their lives.

Success in mental maths requires practice but not to the extent that you get tired of it and demotivated.   Have a look at Mental Maths.

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Happy Pi Day

Pi symbol

Pi Day is celebrated on March 14th (3/14) around the world. Pi (Greek letter “π”) is the symbol used in mathematics to represent a constant — the ratio of the circumference of a circle to its diameter — which is approximately 3.14159. Pi Day is an annual opportunity for maths enthusiasts to recite the infinite digits of Pi, talk to their friends about maths, and to eat Pie.

Pi has been calculated to over one trillion digits beyond its decimal point. As an irrational and transcendental number, it will continue infinitely without repetition or pattern. While only a handful of digits are needed for typical calculations, Pi’s infinite nature makes it a fun challenge to memorize, and to computationally calculate more and more digits.

Can anybody think of a greater waste of Time?

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Another handy way to produce Trigonometric Table

Another handy way to work out the table of trig values is to use your hand as an aid to recall as shown in the image below.

handy trig functions

For sin values count the number of fingers to the left of the one representing the choosen angle. Get its square root and divide by 2. For cosine values do the same with the number of fingers to the left of the choosen one.

For sin 60, you get 3 / 2.  For cos 60 you get √1 / 2 which is 1/2.

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Happy New Year

Janus

Janus the Roman god of beginnings, gates, transitions, time, duality, doorways, passages, and endings is usually depicted as having two faces, one looking to the future and the other to the past. Whether or not the month of January has actually been called after him, his image provides a good representation of this reflective time of the year.

From GMG’s perspective we can look back on the past year with satisfaction and forward to the new year with anticipation of further growth and development. Once again our exam students did us and themselves proud with the results they achieved.

We expanded our  offering with the launch of a the following online courses and resource packs:

Early in the new year we will add a resource for the new Junior Cert Business Studies course. June 2019 will be the first sitting of the exam with the common level paper in place of the separate ordinary and higher level papers up to this.

Wishing you a happy, peaceful and prosperous new year.

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Simple way to produce Trigonometric Table

I have been asked by a reader of this blog if I have any tricks to help remember values in the trigonometric table. Well, Mohamed, here is a way to produce a version of the table. You can easily add additional values to the simple table shown below by recalling the definitions of other functions.

Table of trig values

 

The steps to produce this are as follows:

  1. Draw a blank template for your table with six columns.
  2. Fill the cells of the first  row  with “angle” and the numbers 0,1,2,3, and 4 respectively.
  3. Divide across by 4.
  4. Get the square roots.
  5. Write each entry in simplest form.

You now have the sine values for angles of 0, 30, 45, 60, and 90 degrees.

Reverse the order of the results to get the corresponding cosine values.

Get the tangent values using tan = sin/cos.

If you want to extent the table to other values you can do so using:

cotan = 1/tan  = cos/sin          sec = 1/cos          cosec = 1/sin

 

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Work out rather than remember Trigonometric Formulae

If you can draw the picture, apply Pythagororas’ theorem and use similiar triangles you can work out the formulae.

Trigonometric Formulae

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GMG Launches Resource Pack for Leaving Cert Physics

LC Physics resource Pack

This resource includes:

A number of publications from Galway Maths Grinds:

Making the most of your Scientific Calculator

Time Management for Students

Google Search for Students

Making the most of the Formulae and Tables Booklet

Resources from the State Exams Commission website examinations.ie gathered here for convenient access:

Past exam papers

Marking Schemes

Advice from Chief Examinations Officer

Course Syllabus

Links to useful websites and video tutorials

Facility to download sheets of graph paper to print off.

Advice on how to prepare for and approach the examination

Some suggestions on how to remember things in physics

 

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A way to remember Physics Formulae related to Electricity

Distance Sppeed Time

It is not uncommon for students to make use of triangles like those shown on the left to help remember or quickly solve formulae based on equations of the form A=B.C or B=A/C.

While these can be quite useful as memory joggers, it would probably be too cumbersome for most people to try this method for remembering a lot of different formulae.

The use of the first letter of each word in a sentence or phrase to help recall an ordered list of items is another well used method of aiding memory. Ring Out Your Great Bells In Victory can be used to recall that the colours of the rainbow, the spectrum of white light, in increasing order of frequency, are Red Orange Yellow Green Blue Indigo and Violet.

The image below combines these two methods to help remember physics formulae relating to electricity.

Physics formulae - electricity

The top half is used to help explain the method implemented in the bottom half.

Step 1: draw five triangles laid out as shown.

Step 2: Add the letters

JWCJ

CVAMSW

AO

Step 3:  in three rows at the corners of the triangles. Make up a sentence to remember these letters in the given order. I have used

James Watt Can Justify

Clear Values And Serious Work

Acting Openly.

These stand for the units of measurement of different physical parameters as indicated in the bottom half of the image.

 

 

 

 

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New Course Launched – Resources for Junior and Leaving Cert Maths Students

Resources for Maths Students

GMG has today launched its online mini-course “Resources for Junior and Leaving Cert Maths Students“.

The purpose of this course to gather together in a cost-effective manner a number of publications of Galway Maths Grinds that are of value to students, particularly those studying for either the Junior Certificate or Leaving Certificate Maths exams, and to add further value by providing links to a number of other useful resources, such as a facility to download and print off graph paper.

The GMG documents included are:

  • Making the most of your Scientific Calculator
  • Time Management for Students
  • Google Search for Students
  • Junior Certificate Geometry Theorems
  • Making the most of the Formulae and Tables Booklet
  • How to Study Maths

The advantages for students of the “course” format over getting individual PDF documents to download include:

  • A lower price, by almost 30%, for the bundle compared to getting the individual documents separately.
  • Can access on-line any time to either download or view online compared to having one time access only to allow download to your own device.
  • Will automatically have access to any further updates or additions to the course.
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GMG launches Online Time Management Course

Personal Effectiveness is a measure of an individual’s overall ability to perform various tasks and activities in his or her educational and life endeavours. There are practical, straightforward skills and strategies needed to excel in the various realms in life. These personal effectiveness skills include amongst others; personal goal setting, time management, study skills, organization, problem solving, decision-making and efficient use of various tools and resources at ones’ disposal. The development of these skills allows individuals to become more capable, competent, and confident in themselves and their performance, and help them to be better able to reach the goals they set for themselves.

At Galway Maths Grinds we provide information and strategies to help individuals to make the most of their talents and abilities. Individuals can use these simple strategies and techniques to make the very most of their learning activities.

While you can not beat working one on one with an experienced mentor, we can only help a limited number of students in this manner. To cater for others we have a number of publications that can be purchased online and downloaded as PDF files.

Today we launch our Effective Time Management course which students can follow online at their own pace.

It launches at an introductory price of €23.37  until November 1st when it will revert to the normal price of €61.50.

If you are curious about the odd pricing, it is because we are using an American platform to host the course and it requires the course to be priced as a whole number of euro before VAT is added. VAT of 23% is then added because we are based in Ireland.  So the  €23.37  is made up of a pre VAT price of  €19 plus 23% VAT.

 

 

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