Happy Christmas

Christmas Greetings

Wishing all our clients and friends a very happy and peaceful Christmas and a prosperous new year.

We finish the year after Saturday 21st and restart on January 6th.

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Area of overlap of circles of equal radius where each touches the centre of the other

Problem: Given 2 circles that intersect each other in such a way that each circle passes through the other’s centre. What is the area of overlap if the radius of each is 4cm?


If we draw a line segment between the points of intersection of the circles it divides the required area in half. The portion on the left of this segment is a sector of the circle on the right and the area on the right is a sector of the circle on the left. Both sectors have the same area.

We can build Equilateral triangles with their vertices at the circle centres and the points of intersection, as shown below, whose side length is r:

The angle <AC1B is therefore 120°.

The red colour below outlines a sector of circle C1 with an angle of 120 degrees.

The portion of this on the right of [AB] is a segment of the circle on the left. This is half of the required area.

The area of the segment is equal to the area of the sector minus the area of triangle AC1B.

= π r2 (120/360°) – (r/2 x (r√3)/2)

= π r2/3 – (r2√3) /4

= r2( π /3 – √3 /4)

The total required area is twice this. = 2r2( π /3 – √3 /4)

Finally sub in the value for the radius of 4cm and let the calculator do the rest.

=17.197 cm2

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Junior Cert Results out Today

life signposts

Today is D-day for Junior Cert results. Students will be getting their results in school today or from 4pm they can get them online. In the scheme of things the results students get to day are of little real importance other than an indicator of how successfully they have worked on their subject matter over the last three years. The vast majority will do the Leaving Cert and then today’s results will not matter at all. That said, students who did well are entitled to mark their achievements while those who did not should take some time to figure out what they need to change to do better in the next examination.

Of 64,330 students from the Junior Cert class of 2019 who are receiving their results, 19 candidates will be celebrating 11 As or their near-equivalent under the new regime, Distinctions, followed by 88 with 10. In statistical terms these are outliers.

There are a lot of changes taking place in the Junior Cert exam and much debate as to the merits or otherwise of the changes. Some subjects, such as Science and Business, that were previously offered at higher and ordinary level are now only offered at common level. As the new Junior Cycle Framework continues to be phased in, June was the last time that Junior Cert students were examined in the old curriculums in Irish, French, German, Spanish, Italian, and CSPE (Civic, Social, and Political Education). They have all been replaced by new syllabi or subject ‘specifications’. We have a new “wellbeing” course. We have had the recent heated debate on whether history should be considered a core subject or not.

Today, students will receive the traditional alphabetical grades (A’s, B’s, C’s etc) in the majority of subjects. But, in the new subjects – English, Science, and Business Studies – grades are in a new format. – between 90% and 100% – now attracts a ‘Distinction’. The second highest mark – between 75% and 90% – is a ‘Higher Merit’. The new Junior Cycle Framework combines today’s results with the results of school-based assessments and projects into an overall ‘Junior Cycle Profile of Achievement’. This means home schooled students can only be marked for the written exams in June as the group projects must be done in school. There is an awful lot of change going on but I can find little evidence that improved levels of education will result. We will have to wait and see.

A sad reflection on our society is the fact that the Gardaí need to issue statements advising people to to take a few simple precautions to ensure their safety. Have a plan for the evening. How are you getting home? Are you taking a taxi, getting a lift or staying with a friend? Be mindful of your property. Keep your phone, wallet or purse on you at all times. If you find yourself in a scenario and sense there could be trouble, walk away. Call the Gardaí immediately if you feel the situation is getting out of hand. Think of the consequences for both yourself and others if you were to hit someone. Additional Gardaí will be patrolling from early this evening. We will be operating in a uniform and also plain clothes capacity and will be adopting a zero tolerance approach to anyone found behaving in an anti-social manner.

The message is celebrate sensibly and with respect for yourself and others. And remember good or bad, your results are just another little signpost on your journey of life. What you make of it is up to you.

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Success in Study as in Life is:

Success in Life

This morning I received an email typical of some I receive from parents a few times per year. It read as follows:

Hi Noel.
My son has just gone into junior cert and badly needs help to plan study and homework. He seems to waste lots of time with the book open and nothing sinking in. I feel if he could just be more motivated and organised he could be getting better grades. We also are to blame for not keeping a check on his work and its causing such worry. Any ideas where we could go from here? He has bags of books, folders and notes all over and I’m not qualified to help him. Help!

My reply would be something like this:

Hi (name),

You have hit the nail on the head. It comes down to motivation and organisation, and putting good habits or systems in place.

The natural tendency of many students is to spend too much time reading long sections of their textbooks without any real engagement, highlighting some of the text in the belief that this makes it stand out in their memory or copying text in their own handwriting. Unfortunately, these are the most inefficient ways to work.

Instead students should read shorter passages, then close the book and answer questions on what they have just read. Unfortunately this is the less likely thing most students do if left to their own devices. This is the reason homework is given, (or should be). School textbooks generally have questions at the end of each chapter or section of a chapter. Having a quick read of these questions before reading the actual text can be helpful because the student can then read with the purpose of finding the answers to those questions rather than just passively read without direction. If the textbook does not have questions, the student should close the book and spend a minute or two recalling in their own mind what they have just read, and make up and answer their own questions.

Should the student come across a question they can not answer after a genuine attempt, then ask for help at the first opportunity, be it from a teacher or classmate etc. Do not leave questions unanswered, because further lessons may be based on this information.

Several short bursts of activity will be more effective than a few marathon sessions.

Students also need to revise the work they did last week and a month ago to implant it successfully in their long term memory, so part of their study schedule should include quick revisions of work done earlier.

Eliminate distractions when studying. Watching TV, taking phone calls from friends or checking social media while studying does not work. You do one or the other.

Students need to accumulate facts, vocabulary and definitions before they make connections between them to develop understanding. Making flash cards can be helpful for learning short definitions. Drawing out mind maps can be helpful in finding connections between facts and seeing how one thing relates to another.

Before starting a study session have everything you will need readily available. (Pens, paper, textbooks, calculator etc.). Having to go search for something you need will break your concentration on the current activity.

These are some of the basics to take account of when studying. There is no magic bullet, so single big thing that will solve the problem in an instant. Instead, it is the repeated applications of the basics, over and over again that leads to success. This requires perserverance, a quality often in short supply in an age of instant gratification.


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Whether you use BOMDAS or BODMAS the answer is the same

Order of Operations

Apparently there is some discussion on the internet today as to the correct answer to this problem;  8 ÷ 2(2+2) = ?.

It is reported that “Even mathematics experts are wading in – why? Because people are coming out with two different answers: 1 and 16.”

Well, if your accountant or other mathematical expert is telling you the coorrect answer is 1, it may be time to review whether they are up to the job or not.

There is no conflict in using either BOMDAS or BODMAS as a mnemonic to help remember in which order to carry out the mathematical operations.

BODMAS stands for: Brackets, Orders, Division, Multiplication, Addition, Subtraction. It gives the order of priority in which to carry out operations. Anthing in brackets is done first. Next in priority is exponents (powers or indices are other names to signify the order of the number), followed by multiplication or division which have equal priority which in turn are followed by addition or subtraction which equally share the lowest priority.

The problem is not with the mnemonics but with some “experts” lack of  understanding of how to use them.  When operators of equal priority are encountered work from left to right across the equation.

Left to right across the equation to be solved is not necessarily the same as left to right across the way the mnemonic happens to be written. BOMDAS is the same as BODMAS. Or PEDMAS is the same as PEMDAS.

8 ÷ 2(2+2) =  8 ÷ 2 x (2+2) 

Brackets first:   8 ÷ 2 x 4

Division and multiplication have equal priority so work left to right on the equation:  in this case division comes first.   4 x 4 =16.


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Casio fx-83GT Plus Scientific Calculator upgraded to fx-83GTX

The fx-83GTX replaces the Casio Fx-GT scientific calculators  offering  additional features such as clearer display, clearer menus, faster processor and 14 additional functions. Allowed in every Irish exam where a calculator can be used. Recommended and approved for the  Junior and Leaving Certificate). The large Natural Textbook Display (Natural-V.P.A.M.) shows mathematical expressions like roots and fractions as they appear in your textbooks which increases comprehension because results are easier to understand.

Following nine years of service the fx-83/85GT Plus models are replaced by new improved models, the GTX CLASSWIZ range. Whilst retaining the familiarity and ease of use of the old GT Plus models, you will find the new features improve ease of use and aid comprehension. For example, all related statistical information is now shown on one screen.

New functions are:
• Equivalent ratios
• Improved statistics – Quartiles
• Table of values for 2 functions rather than just a single function

Key Benefits of Upgraded Casio FX83GTX / FX85GTX over Previous Model

  • New high resolution screen (easier to read)
  • New menu & display system (easier to access functions)
  • Keys, major function & operations in same place as old model
  • Results easier / quicker to view with scrolling screen
  • Improved statistics, tables & ratio functions
  • Increased number of functions: 276
  • Faster Processing of data
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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 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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