## The Logic of Maths

May 26, 2014 § 1 Comment

Part of mathematical ability comes down to logic. People who are logical are able to follow mathematical processes and understand the mechanisms behind the question. This is especially helpful when it comes to solving word problems, a particular obstacle to lots of maths students.

Here’s one of the word problem questions from a GCSE Higher Maths Mock Paper:

Janice asks 100 students if they like biology or chemistry or physics best.
38 of the students are girls.
21 of these girls like biology best.
18 boys like physics best.
7 out of the 23 students who like chemistry best are girls.
Work out the number of students who like biology best.

This question can easily get you in a tangle! It needs to be tackled logically. Often, diagrams and charts are a good way to order your thought processes. For this particular question, a two-way table is helpful. We know that there are 100 students in total. 38 are girls, so 62 are boys. 21 girls like biology, 18 boys like physics, 7 girls like chemistry and 16 boys like chemistry. If we put those numbers into a table, we can then work out the rest.

So, the answer is 49.

## The Great British Heatwave!

With our little mini heat wave this week, all eyes are on the weather forecast to see how long the temperature will stay at shorts-wearing level! So, where did our temperature scales come from.

We mainly use two temperature scales: Fahrenheit and Celsius.

The Fahrenheit scale was devised by a German physicist called Daniel Gabriel Fahrenheit in 1724. It has two fixed points: the freezing point of water is 32 degrees and the boiling point of water is 212 degrees. The difference between those two points is 180 degrees.

The Celsius scale, meanwhile, originates with a Swedish astronomer called Anders Celsius. The Celsius scale is preferred in modern use because it has nice round numbers! The freezing point is 0 degrees and the boiling point is 100 degrees, so the difference is 100 degrees rather than 180. This fits in better with our metric system, so is now much more widely used. The Celsius scale is also aptly called centigrade because, of course, centi means 100.

The formulas for swapping between the two are:

Fahrenheit to Celsius: (Fahrenheit – 32) x 5 / 9
Celsius to Fahrenheit: (Celsius x 9 / 5) + 32

And those will continue to be our temperature scales until the British devise a new scale that goes between the two points at which we moan about it being too hot or too cold!

And for kids who’d like to practice their thermometer skills, try taking a journey to Penguin City.

Enjoy the sunshine everyone!

## Don’t Be Negative – A Common GCSE Maths Mistake!

Okay so this might be a bit of a boring blog post! But, with the GCSE exams round the corner, I thought I would highlight one of the most common mistakes seen when marking GCSE maths papers.

That problem is negative numbers. They just get students in a right old pickle! And that’s why they crop up every year in the exam papers! But they won’t be obvious . . . . they are often very sneaky and catch people off guard. For instance, you will see minus signs when expanding brackets or in substitution or indices questions.

So, here are the simple rules:

MINUS X MINUS = PLUS

MINUS X PLUS = MINUS

Take this past paper question as an example:

V = 3b + 2b2

Find the value of V when b = –4

First, substitute the number into the equation.

3b + 2b² =   3 x -4 + 2 x -4²

3 x -4 is PLUS x MINUS, so it becomes a MINUS: 3 x -4 = -12

2 x -4² is a bit more complicated because we need to do the squaring first:
-4² = -4 x -4 which is MINUS x MINUS, so it becomes a PLUS. -4 x -4 = 16

2 x 16 = 32

So, 3b + 2b² =   3 x -4 + 2 x -4² = -12 + 32 = 20

Just take it step by step and CHECK YOUR WORK! We wish everyone doing the GCSE exams all the best.

## Fractions for Five Year Olds

The new National Curriculum will soon be sailing in to schools this September. It is set to make quite a difference and it’s exciting to see whether it will be a success.

One of the biggest changes is the step-up in terms of introducing topics much earlier than before. Now, your 5-year olds will be learning simple fractions, such as halves and quarters. This may sound crazy! But, don’t forget that the younger they are, the more they are able to learn and absorb.

The Department for Education has had the new curriculum published on their website for quite some time . . . . and it looks like the kids (and the poor teachers!) are going to be packing a lot in! These links show the new primary curriculum for maths and English – it shows you the topics that are going to be covered in each year group:

Primary Maths Curriculum From 2014

Primary English Curriculum From 2014

And, if you want a heads-up on simple fraction work, we’ve got it covered: How To Do Fractions

## The Importance Of Angles

Say the word “angles” to a class of 10 year olds and everyone sighs! But without certain angles, life would be very different.

Think about this angle: 23.4 degrees. That is the size of our Earth’s axial tilt. Earth isn’t upright as it travels round the Sun; instead, it has a slight lean. And it’s quite a tilt – the Leaning Tower of Pisa only has a tilt of 4 degrees, so it’s pretty straight in comparison!

That angle of 23.4 degrees is more than just an interesting fact. Without that angle of tilt, we would not get our seasons. During summer in the Northern Hemisphere, the North Pole is tilted towards the Sun, so the Northern Hemisphere gets more sunlight and warmth than the Southern Hemisphere. Then, in winter the North Pole is tilted away from the Sun, so there is less sunlight and it becomes much colder in the Northern Hemisphere. And, of course, during autumn and spring, both hemispheres are receiving equal sunlight.

Yes, we enjoy the variation of the seasons, but more than that they are vital to our survival. If the Earth’s tilt was different, we would not get the same regulation of temperature that we have now. The extremes caused by more or less tilt would make life very difficult – impossible in certain parts of the Earth. Not just that, without seasons we would not be able to grow food in the same way. Just think about how important it is to have the variations that make seed germination and plant growth possible. Without the seasons, food production would be seriously affected.

So, yes, angles may be incredibly boring when you’re 10! But, maybe certain angles need a little respect every now and then!

## The 11 Plus Fear Factor!

The Eleven Plus – a term that strikes fear into the hearts of parents across the country! In the business I am in, every day is made up of phone calls from parents looking for guidance, advice and reassurance that they are doing the right thing.

At the end of the day, there is no single “right way” of doing things; every child is different and requires a different approach to learning. Some children pick it all up “just like that”, while others need a slow and gradual approach to make sure it all sinks in. So, having a parent call, reel off their entire 11 plus revision schedule for the past year and ask you if they are doing the right thing, having only spoken to them once and never having actually met their child, is easier said than done!

But, of course, having seen hundreds of parents go through this every year has given some insight into the “krypton factor” of 11 plus success. And the keys, really, are:

• Start revision early enough to avoid panic, but not so early that they get fed up with it!
• Practice makes perfect, so make sure they get plenty of practice in all the subject areas covered in the exams you are entering for. But, don’t overload them – they are only 9 or 10, after all.
• Don’t push your children to be something they are not. If they are not cut out for grammar school, it is not fair to push them into it. Even if they pass, this is no guarantee of success in the high-pressure world of grammar school.
• Don’t let other parents worry you! The phrase “I’ve just found out Mrs So-and-So’s little boy has had a tutor since he was 8” is an all-too-common one! That is their choice, but it doesn’t mean to say that would have been the best option for your child.

Despite the efforts of the education authorities to make 11 plus exams “tutor-proof”, I do believe that tutoring still has a place in the 11 plus preparation process. The 11 plus exams have become more and more difficult and often contain topics that students may not have even learnt at school yet. Tutoring has the benefit of being able to cover all the elements required in an environment where they can get the support that they need from people who know the current methods of teaching.

I recently spoke to a parent who said that they had been going through the work at home, but had now got to the stage where they could no longer help with the questions. So, they decided to get a bit of help from a tuition company. I thought this was a very balanced way of dealing with things. Just because you haven’t had a tutor the whole time doesn’t mean that your child is going to fail! If they are bright enough and get the right help (whether it be from you or from someone else), they are on the right track.

I guess the point I am trying to get across is that the key to success is to find the balance. You’ve got to put the work in, but don’t let it take over your life!

## Busy Bee Geometry

“When there is no honey, the Pooh takes his tummy wherever the honey will be,” sang Pooh.

But, whilst poking his paw into the nearest beehive, did he ever stop to think about the ingenuity of the humble honey bee? Let’s hope not, a mathematical analysis surely wouldn’t make for a good Winnie the Pooh episode!

We are all used to the hexagonal shape of the cells within a beehive. But, why do the bees choose hexagons?

It was way back in the 4th century that Pappus, an Ancient Greek geometer, started to think that maybe there was more to it than just being pretty . . . it’s actually a very efficient building technique. To make the most efficient use of space, they need to use shapes that tessellate – that means shapes that fit neatly together without leaving gaps.

But, you may say, why not use other shapes that tessellate, like squares or triangles? Well, for one thing, regular hexagons have a smaller overall perimeter than squares and triangles – once they have built one hexagon, they have already built one of the walls for 6 other hexagons, so it makes the workload much quicker and easier. Plus, the regular hexagon pattern has now been proved to be the most efficient pattern for curved walls. This means that the individual cells can bulge with honey and still make the most efficient use of the space.

So, next time you are using a protractor to measure an angle, spare a thought for the worker bees that are instinctively setting the walls in their beehives at exactly 120 degrees without so much as a spirit level!

## Maths Star’s brand new Level 5 Mathematics Pack is now available to download!

The level 5 maths pack covers 36 all-important topics aimed at students in Years 5 and 6 at school. The 36 topics have been chosen to relate to the 11 Plus maths topics, so they are a perfect revision aid for the exams. They also build important SATs skills and help to revise all the different areas of maths that are likely to appear in Year 6 SATs tests.

Each topic has a full and detailed explanation with exam reminders, followed by a series of practice questions. At the end of the pack are 7 revision sheets that students can use during their final week before the exam to brush up on their skills.

You can choose to have the pack emailed to you or you can pay extra to have it printed and posted to you. Emailed versions will be attached to your confirmation email. Posted versions will be with you within 5 working days. If you have any questions or have problems with your order, please contact us at mathsstaruk@gmail.com

Click here of list the topics covered in this pack

## Brightspark Education Review 2 – The Lessons!

February 5, 2011 § 1 Comment

Online lessons can cause some controversy – the idea of learning online makes people very skeptical. But . . . you don’t know unless you try! So, I gave it a go and had 2 lessons with Brightspark Education.

SOFTWARE and SETUP
I have worked with the idea of online teaching for a few years now and I know how difficult it can be getting it set up. Unfortunately, no matter how good your software is, it can’t account for parents that are not too good with a computer! But, I must admit that Brightspark’s set up was very easy to use. I did not have any problems with the sound (it was very easy to adjust the audio settings) and the writing tools were easy to use. I’m sure that most children would not have any problem using them. Attending a session was also very easy. I just made sure I had Flash installed, logged in to my account and clicked a button! The instruction manual I received when I booked made it all very hassle-free.
Overall: 9 out of 10!

THE TEACHERS
I had two different teachers in my two lessons. My first teacher was “Jack”. As you may know, all of Brightspark’s tutors are in India, so Jack had an Indian accent! He was, however, very easy to understand. He was fun to learn from and made sure I was confident in the topic. My lesson was mean, median and mode, but when I said I wasn’t too good at division, he opened up a new whiteboard and went over it with me, then we continued with the lesson. He was full of praise and was always putting up fun pictures of smiley faces and clapping hands when I got things right – just what children want and need. My second teacher was a little less inspiring (I can’t even remember her name!) but that could have been the topic (we’ll get on to that in a minute!). She was, however, clear and easy to understand.
Overall: 8 out of 10 (10 out of 10 for Jack!)

THE CONTENT
First Lesson: Mean, Median, Mode and Range. This was great. There were presentations to teach the different parts of the lesson, followed by lots of different example questions I could try.
Second Lesson: Congruent Figures. This was a bit boring! The problem here seemed to be a lack of material, as after about 10 minutes, we had exhausted the resources and the teacher spent the rest of the time making up questions . . . that were all the same and very boring! It certainly wouldn’t have captured a 10 year old’s imagination! This is an issue that can be easily fixed though and I am sure that if a child was having a real problem with the topic, it would take longer.
Overall: 5 out of 10

All in all, I was very impressed. Of course, I only saw a small amount of what they can do, as they cover a wide range of ages and topics. However, from what I did see, it would definitely be worthwhile for children that need a more interesting tuition method! Watch this space for my third and final review – coming soon!

## Fascinating Fibonacci Flowers!

January 29, 2011 § 1 Comment

Have you ever seen a girl in the playground picking petals off of a flower one by one – “he loves me, he loves me not”?!

At that age, you don’t think about how amazing nature is and the tiny details that make it perfectly designed. Even nature conforms to mathematics; in fact, mathematics came from nature first! Flowers, shells and, believe it or not, pineapples were using the Fibonacci sequence way before Fibonacci was even born!

Take a close look at a sunflower head. Have you noticed that the seeds are arranged in a spiral pattern. Notice, too, that they are REALLY tightly packed! Plants grow new bits from a central point called the meristem. Every new growth comes out of the centre at an angle with the previous growth. However, only with a VERY specific angle can the plant make the most economical use of the space. This angle is called the Golden Angle and is approximately 137.5 degrees. If they didn’t grow at this angle, the plant would form its growths (seeds, petals, etc) in rows rather than spirals and you would end up with gaps. Amazing stuff!

So, now that you are baffled with science, here comes the mathsy bit! The Fibonacci Sequence is made up of 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, etc (each number is equal to the sum of the previous two). The number of spirals that form from plants using the Golden Angle is a number from the Fibonacci Sequence. For example, buttercups have 5 petals, asters have 21, Michealmas daisies have 55 or 89.

The Golden Angle appears in shells, too, so the number of spirals in a shell will be part of the Fibonacci Sequence. Pine cones will have a Fibonacci number of spirals coming from the centre. And, yes, pineapples follow the sequence too! Look closely and you will see that there are three distinct sets of spirals, usually a set of 8, a set of 5 and a set of 13.

In fact, you will find the Fibonacci Sequence in most fruits and vegetables, even your cauliflower! So, remember, it’s not mathematicians that create the maths; they are just the ones that find it. Your pineapple is more than just a fruit – it is a marvel of mathematics.

It’s a shame it doesn’t make cauliflowers any more exciting!