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## Visual Mathematics

The most powerful learning occurs when we use different areas of the brain together to solve problems. When students work with symbols, such as numbers, they are using a different area of the brain than when they work with visual and spatial information, such as an array of dots. Learning and performance is optimised when the two areas of the brain are communicating. Training students through visual representations may improve their maths performance significantly, even on numerical maths.

Students can be excited and inspired when they see mathematics as pictures, not just symbols. For example, consider how you might solve 18 x 5, and ask others how they would solve 18 x 5. Here are some different visual solutions of this problem.

Each of these visuals highlights the mathematics inside the problem and helps students develop understanding of multiplication. Pictures help students *see* mathematical ideas, which aids understanding. Visual mathematics also facilitates higher-level thinking, enables communication and helps people see the creativity in mathematics.

Mathematics is a subject that involves precise thinking. But it also involves creativity, openness to new ways of seeing things, visualisation, and flexibility in approach.

Nice neat formulae and procedures are used to solve familiar type questions posed in a familiar way. But understanding is required to solve them when they are presented in an unfamiliar manner. Students should be challenged to discover new ways in which to see and solve problems.

Take the following example.

A man is on a diet and goes into a shop to buy some ham slices. He is given 3 slices which together weigh ⅓ of a pound but his diet says that he is allowed to eat only ^{1}/_{4} of a pound. How much of the 3 slices he bought can he eat while keeping to his diet?

One approach would be to use ratios and algebra as follows:

3: ^{1}/_{3} = x: ¼

9:1 = 4x:1

4x = 9

x = 9/4 =2 ¼

Another would be to solve it visually as follows:

Both methods are equally valid. Both result in the same correct solution. Students who have difficulty with one approach may find success comes easily with the other. Don’t be afraid to experiment. For some doodling on scraps of paper may be an important part of learning and thinking their way through problems to the solutions.

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## Pascal’s Triangle and the Maths of Christmas

In the Christmas song, “The 12 Days of Christmas”, how many total gifts does my true love give to me? The gifts are:

A partridge in a pear tree,

Two turtle doves,

Three french hens,

Four calling birds,

Five gold rings,

Six geese a-laying

Seven swans a-swimming,

Eight maids a-milking,

Nine ladies dancing,

Ten lords a-leaping,

Eleven pipers piping,

Twelve drummers drumming.

There are two ways that people usually do this problem:

We can count the total number of each gift.

I get 1 partridge in a pear tree on each of the 12 days. 1 x 12 = 12

2 turtle doves on the last 11 days. 2 x 11 = 22

3 french hens on the last 10 days. 3 x 10 = 30

4 calling birds on the last 9 days. 4 x 9 = 36

5 gold rings on the last 8 days. 5 x 8 = 40

6 geese a-laying on the last 7 days. 6 x 7 = 42

7 swans a-swimming on the last 6 days. 7 x 6 = 42

8 maids a-milking on the last 5 days. 8 x 5 = 40

9 ladies dancing on the last 4 days. 9 x 4 = 36

10 lords a-leaping on the last 3 days. 10 x 3 = 30

11 pipers piping on the last 2 days. 11 x 2 = 22

12 drummers drumming on the last day. 12 x 1 = 12

That is a total of 364 gifts.

Or We could count the number of gifts I get on each day.

On the first day I get 1 gift

On the second day 1 + 2 = 3gifts.

On the third day 1 + 2 + 3 = 6 gifts.

On the fourth day 1 + 2 + 3+ 4 = 10 gifts.

On the fifth day 1 + 2 + 3+ 4 + 5 = 15 gifts.

On the sixth day 1 + 2 + 3+ 4 + 5 + 6 = 21 gifts.

On the seventh day 1 + 2 + 3+ 4 + 5 + 6 + 7 = 28 gifts.

On the eighth day 1 + 2 + 3+ 4 + 5 + 6 + 7 + 8 = 36 gifts.

On the ninth day 1 + 2 + 3+ 4 + 5 + 6 + 7 + 8 + 9 = 45 gifts.

On the tenth day 1 + 2 + 3+ 4 + 5 + 6 + 7 + 8 + 9 + 10 = 55 gifts.

On the eleventh day 1 + 2 + 3+ 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 = 66 gifts.

On the twelfth day 1 + 2 + 3+ 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11+ 12 = 78 gifts.

Adding: 1 + 3 + 6 + 10 + 15 + 21 + 28 + 36 + 45 + 55 + 66 + 78 = 364 gifts.

We are adding the triangular numbers. The triangular numbers can be found in Pascal’s triangle, so we can use Pascal’s triangle.

## Black Friday and Cyber Monday Deals from Galway Maths Grinds

**Thanksgiving**, or **Thanksgiving** Day, is an important public holiday, celebrated on the fourth Thursday in November in the **United States**. It originated as a harvest festival.

**Black Friday** is on the day after Thanksgiving. **Black Friday** 2016 will be on November 25. Based on data from previous **Black** Fridays, most **Black Friday** sales will start on Thanksgiving and extend through **Black Friday** weekend into Cyber Monday 2016.

“Black” refers to stores moving from the “red” to the “black,” back when accounting records were kept by hand, and red ink indicated a loss, and black a profit. Retailers hoped that good sales over the weekend would push any loss making enterprises into the black, that is profit.

**Cyber Monday** is a marketing term for the **Monday** after the Thanksgiving holiday. The term “**Cyber Monday**” was created by marketing companies to persuade people to shop online. Cyber Monday which occurs this year on November 28 is meant to be the online equivalent of “Black Friday”.

For Black Friday through to Cyber Monday Galway Maths Grinds publications can be purchased for just **99 cent** each instead of the usual **€2.99. **Just go through https://sellfy.com/GalwayMathsGrinds and use discount code **scout**.

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## Science Week 2016

Science Week will take place from 13th – 20th November and is coordinated by Science Foundation Ireland, as part of its education and public engagement programme. It is a collaboration of events involving industry, colleges, schools, businesses, communities, researchers and students throughout Ireland.

The Galway Science & Technology Festival is a two week festival taking place from November 14th – 27th in conjunction with Science Foundation Ireland’s National Science Week. The Festival involves both science and technology shows and workshops for primary and secondary schools throughout Galway City & County over a two week period and culminates with a large scale Science & Technology Exhibition in NUI Galway on Sunday 27^{th} November. This exhibition, in collaboration with NUIG, GMIT, Marine Institute and local companies, attracts approximately 20,000 people and is a very successful family day out, with families visiting from all around the country

## Maximise your grades with the Formulae and Tables Booklet

There is a lot of useful information in the Formulae and Tables booklet that will be available to students during their junior cert or leaving cert exams. But you need to put in some effort to be able to take best advantage of it. It is well worth the effort and does not have to be difficult if approached with a positive attitude and a curious mind.

My students quickly learn that I emphasise the importance of using and becoming familiar with it’s contents and its omissions. It is not only useful for the maths exams, but also for other science subjects with a significant maths content, such as physics, chemistry or applied maths. The formulae contained in the booklet are in effect concise summaries of the work of many great minds over the centuries and it would be foolish not to take advantage of such a valuable resource when preparing for and sitting your exams.

If you are serious about maximising your grades then make the most of the tools available to you including the Formulae and Tables Booklet.

I have revised and improved “**Making the most of the Formulae and Tables Booklet**“. The improved version gives several examples of its usefulness in other STEM subjects besides maths.

For the months of October and November you can use the discount code **scout** to purchase the updated “**Making the most of the Formulae and Tables Booklet**” from https://sellfy.com/p/qR3F/ for 99 cent instead of the usual €2.99.

Queries from any book publishers that might be interested in helping bring this to a wider audiance are welcome to contact me.

## Junior Cert Results out today

Over 60,000 Junior Cert students receive their results today. After 4pm they can collect their results from their school or check them online.

In order to check your results online you must have your Examination Number and the Personal Identification Number (PIN) supplied to you before the commencement of the examinations. If you require confirmation of your examination number and/or PIN number you should contact your school.

If you think you should have got a higher mark results can be appealed up to 5pm on Friday, September 30th, at a cost of €32 per subject.

If you did not get what you hoped for do not worry. Most students go on to do Leaving Cert and after that no-one will ever care about what results you got in the Junior Cert. Just use it as a learning experience for the future.

The current system’s replacement, the Junior Cycle Student Award, or JCSA, will be rolled out subject by subject between now and 2022. The idea is that if we shift focus from what students are taught to how and what they learn, and encourage independent thinking, learning and real engagement with education, we will end up with students who are more flexible and capable of adapting to ever-changing learning environments.

The changes have already begun, starting with English, and involve some continuous assessments and less emphasis on and final phasing out of the final Junior Cert exam as we know it. It remains to be seen how successful it will be at obtaining the hoped for results.

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## Big enough to compete; small enough to care

If you find it cheaper anywhere else let us know.

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## Mental Maths Game

You’ll need all your mental maths skills to escape with the bio-rods in this exciting Mission 2110 game. http://downloads.bbc.co.uk/bitesize/ks2/maths/activity13/activity.swf

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## Cool Interactive Periodic Table of Elements

TED-Ed Interactive Periodic Table With Video Lessons for Every Element.

This work is a fruit of a partnership between TED-Ed and Brady Haran, the creator of the popular YouTube channel Numberphile. The purpose behind this period table is to provide students with both a refresher and a study aid to help them better understand the different chemical elements and their properties.

This periodic videos contains facts on almost all elements, from Hydrogen to Ununennium and beyond. And for each element, a video lesson is accorded. So students will be able to watch the video and use the TED Ed platform to engage with its content.

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