For anyone who is into Strictly Come Dancing, you will know that one of the celebrities is Johnny Ball. Johnny Ball is a TV personality from the 70s and 80s, and is known for popularising maths and science for children.
A little while ago, I can't remember exactly when, my mum bought me a book written by Johnny Ball. It is called Ball of Confusion, which is a collection of puzzles to do with maths, inspired by when he presented a puzzle on Zoe Ball's (his daughter) Radio 2 show. As Strictly just started this last weekend, I thought I would give a few puzzles from the book for you to ponder over.
1. Fumbling in the Dark
The lights have gone out - I said we should never trust wind farms - and I have to search my wardrobe in the dark for a pair of shoes and socks.
I have 3 pairs of shoes, 12 pairs of black and 12 pairs of brown socks.
How many of each do I need to take, to be sure I have a matching pair of each to wear?
2. Animal Farm
A farmer has 20 goats, 30 cows and 50 horses. How many horses would he have if you called the cows 'horses'?
3. Sum for simpletons
Look at this: 12 = 3 x 4.
Can you think of four other consecutive digits that do that?
4. An up and down kind of existence
A monk lives on a river bank in a valley. Once a week he must travel to the monastery at the top of the Lonely Mountain. He sets off at dawn, as the road is steep and long, and he arrives as the Sun sets. Next morning he has a lie in and leaves for home at 10am. As the road is downhill all the way, he arrives home at around 3pm.
What are the chances of the monk being at exactly the same point on the road, at exactly the same time on both days?
Answers on Wednesday! (Just to annoy my sister :P)
Showing posts with label maths. Show all posts
Showing posts with label maths. Show all posts
Monday, 8 October 2012
Tuesday, 21 August 2012
Results Day
This Thursday was my A-level results day. The day that would decide my future. I was surprisingly calm the night before, and managed to sleep quite well, when normally I have trouble sleeping when I'm anxious about something. That may have been because I got a phone call from school earlier that day asking me to come in at 9am to talk to the newspaper reporters. It seemed like a good sign.
I was still calm when I woke up on Thursday morning and had a shower. More good signs came when I heard my mum sing "Celebration" really loudly from her room. I had left my UCAS login details with my step-dad so that he could check whether I got into University or not and find out if it was safe for him to go to the results day at the school he works at. From all of these clues, I wasn't too nervous to log into UCAS for myself. Although I wasn't entirely surprised to see confirmation of my place at Warwick (my first choice) I was enormously relieved. It was quite stressful not knowing if I had done well enough in my exams. I wasn't worried about getting an A* in my maths, and I was fairly confident I was going to get an A in further maths, but not as confident about whether I was going to get an A* in it. The exams I was most worried about were business, as I'm a lot more comfortable with numbers than words, and the STEP papers, as they were really hard, and I had no idea how the marking and scoring worked with them.
Knowing that I had already got my place at Warwick made going into college to collect my results a lot easier. My whole family came with me to school so that we could go to Wetherspoons for a celebratory breakfast afterwards. My results showed that I got an A* in maths, an A* in further maths and an A in business. I was so happy! I worked really hard to get those results, and it was nice to see that it paid off. My teachers were really pleased as well. They kept taking pictures of me and my friends, who also did very well. I was also interviewed by the reporter for the local newspaper.
The rest of the day was a lot of talking to friends and relatives about my results. My boyfriend got his AS results as well. He did a lot better than last year with two Cs, but he was disappointed with his History result, so had to go back to school to talk to the head of college. When he came back with the good news that he was allowed to go into his next year, but the slightly less good news of having to wait for them to talk to his History teacher about whether he can carry on with that or not, he also brought me a surprise.
Crazy colourful flowers! He always picks the best flowers for me :)
My sister made a mad celebration dough dinner.
My mum decorated a plainly iced madeira cake to incorporate my name and my results...
...which quickly turned into fraction cake.
A close family friend was visiting, ready for the V festival at the weekend, and made me a lucky mascot out of the champagne (which he very kindly bought for me) cork and a coin. Apparently it's a tradition to put a coin in the cork of a celebration bottle of bubbly to give to the person the celebrations are based around for luck. I've never heard of the tradition, but I like it! I named him Fermat, after a mathematician :)
It was a really good day, and I can't wait to go to Warwick now!
I was still calm when I woke up on Thursday morning and had a shower. More good signs came when I heard my mum sing "Celebration" really loudly from her room. I had left my UCAS login details with my step-dad so that he could check whether I got into University or not and find out if it was safe for him to go to the results day at the school he works at. From all of these clues, I wasn't too nervous to log into UCAS for myself. Although I wasn't entirely surprised to see confirmation of my place at Warwick (my first choice) I was enormously relieved. It was quite stressful not knowing if I had done well enough in my exams. I wasn't worried about getting an A* in my maths, and I was fairly confident I was going to get an A in further maths, but not as confident about whether I was going to get an A* in it. The exams I was most worried about were business, as I'm a lot more comfortable with numbers than words, and the STEP papers, as they were really hard, and I had no idea how the marking and scoring worked with them.
Knowing that I had already got my place at Warwick made going into college to collect my results a lot easier. My whole family came with me to school so that we could go to Wetherspoons for a celebratory breakfast afterwards. My results showed that I got an A* in maths, an A* in further maths and an A in business. I was so happy! I worked really hard to get those results, and it was nice to see that it paid off. My teachers were really pleased as well. They kept taking pictures of me and my friends, who also did very well. I was also interviewed by the reporter for the local newspaper.
The rest of the day was a lot of talking to friends and relatives about my results. My boyfriend got his AS results as well. He did a lot better than last year with two Cs, but he was disappointed with his History result, so had to go back to school to talk to the head of college. When he came back with the good news that he was allowed to go into his next year, but the slightly less good news of having to wait for them to talk to his History teacher about whether he can carry on with that or not, he also brought me a surprise.
Crazy colourful flowers! He always picks the best flowers for me :)
My sister made a mad celebration dough dinner.
My mum decorated a plainly iced madeira cake to incorporate my name and my results...
...which quickly turned into fraction cake.
A close family friend was visiting, ready for the V festival at the weekend, and made me a lucky mascot out of the champagne (which he very kindly bought for me) cork and a coin. Apparently it's a tradition to put a coin in the cork of a celebration bottle of bubbly to give to the person the celebrations are based around for luck. I've never heard of the tradition, but I like it! I named him Fermat, after a mathematician :)
It was a really good day, and I can't wait to go to Warwick now!
Monday, 13 August 2012
Monday Maths Madness 4
Is this pie, or is this pi? Or maybe a pi pie? Whatever it is, it looks tasty.
My mum bought me a book called "Mr. Messy's Guide to STUDENT LIFE". My boyfriend found this page particularly funny.
Maths is everywhere! Even in the bike oil on my step-dad's leg...
Tuesday, 7 August 2012
The Cake Ratio
My sister has been doing a lot of cooking recently to put into her new blog, The Disaster Kitchen. One of the things she cooked was a cake. The first cake she cooked, and she will admit this, wasn't brilliant. Apart from anything it tasted kind of herby... To be honest that was mostly to do with her spoons, but because she followed the recipe, rather than doing what we normally do, it meant she used the spoons rather than electric whisks.
Now what we normally do, is follow the Cake Ratio. This is the Cake Ratio:
Basically, different multiples of the basic 2:2:2:1 ratio for self raising flour to sugar to butter and to eggs makes different sized cakes. 2:2:2:1 makes a cake to share between two people. I've not tried this yet, but my sister says its good, so I may have to soon! 4:4:4:2 is for a cake that you make in a single bowl. We usually use this for quick cakes we cook in the microwave, like golden syrup sponge cake (or pongey as we call it for some reason I'm not entirely sure of). 6:6:6:3 is for the classic sandwich cake (I don't know what the actual name for them is) where you put jam or butter cream or chocolate sauce in the middle.
And that is the simplicity and magic of the cake ratio. Just throw everything into a bowl, add a bit of milk, and whisk it until it is smooth. Then cook in the oven or the microwave, and enjoy!
Now what we normally do, is follow the Cake Ratio. This is the Cake Ratio:
Basically, different multiples of the basic 2:2:2:1 ratio for self raising flour to sugar to butter and to eggs makes different sized cakes. 2:2:2:1 makes a cake to share between two people. I've not tried this yet, but my sister says its good, so I may have to soon! 4:4:4:2 is for a cake that you make in a single bowl. We usually use this for quick cakes we cook in the microwave, like golden syrup sponge cake (or pongey as we call it for some reason I'm not entirely sure of). 6:6:6:3 is for the classic sandwich cake (I don't know what the actual name for them is) where you put jam or butter cream or chocolate sauce in the middle.
And that is the simplicity and magic of the cake ratio. Just throw everything into a bowl, add a bit of milk, and whisk it until it is smooth. Then cook in the oven or the microwave, and enjoy!
Tuesday, 31 July 2012
University
When I was first looking at Universities, I didn't know where I wanted to go, or what I wanted to do. At the time I was enjoying ICT and thinking that I maybe wanted to do something creative to do with ICT. I couldn't find anything that grabbed my attention, so started to look at business courses. I was worried then about having a subject that would involve too much writing and essays, so I started to look for business courses that let me do other modules in maths.
In all this time I never thought of studying just maths. I always thought it would be too hard or too boring. But after looking at other courses, and not being particularly inspired by any of them, I settled on maths, simply because it's what I'm best at, and I didn't want to do anything else. But the more I thought about it, the more excited about the idea I got.
Maths has always been my favourite subject. It is logical, it doesn't involve many words, and it has a right answer. I hate it when you have to write an essay or something, and the teacher says "there is no right or wrong answer". How do you score points then? How do you know if what you have written is going to get you a good mark? I just don't, so that is why I like maths. I can usually tell if I've done well on a paper, just because there are methods to answering the questions.
My teachers were happy with the choice I'd made as well. They believed I was smart enough for it, and they knew that I enjoyed the lessons so they thought that I would be very interested in the subject at a degree level. With their support, I started to believe that I could do it as well.
Last summer, a close friend of our family kindly bought me a couple non-academic maths books (one of which is the book I talked about in a previous post). I read The Number Mysteries by Marcus du Sautoy first. I found it surprisingly interesting and easy to read. I never realised how much more there is to maths than what you learn in the classroom. Reading this book made me think more about what I could learn at University, and how maths applies to the world, and I started to think more and more that it was something I wanted to do. Since then I have listened to a podcast and watched a TV series about the history of mathematics (both by Marcus du Sautoy as well). I have been given and bought a few more books that I have read various chapters from, and started this blog!
The next challenge, after getting exciting maths, was to choose which Universities to apply for. I had no idea where I wanted to go. I knew that I didn't want to go to Cambridge or Oxford, they just didn't seem right for me. Other than that, I didn't really know. My Step-dad suggested that I look at the Universities in the Russell group because they are below Oxford and Cambridge, but they are still high standing Universities. I ordered prospectuses from all the ones that weren't too far away, and looked at the courses they offered. The maths courses were all pretty much the same, as I decided I just wanted to do the basic mix of pure and applied mathematics, so I could find out what I like and specialise later on.
After a long time of still not really knowing what to do, my mum suggested a road trip to fit in with teaching my sister how to drive on motorways and to let me have a quick look at a University for myself. I'm not sure how we ended up choosing Warwick, whether it was purely because it was one of the closer ones, because the prospectus was on top of the pile, or because we used to go to Warwick castle a lot, but that is where we went. Although we only got an outside look at the Uni (it wasn't actually an open day), we had a drive around the town, and found a little shopping centre to get some lunch and we bought a couple of books. Mum kept saying things like "this could be where you do your shopping for food to survive!" which was a little bit scary, but kind of cool. We also decided that the maths course there would be ideal, because there was a lot of room for flexibility if I wanted to take a module of something else to break up the maths.
After that, everything got a bit easier. I applied to Warwick, Bath, Nottingham, Liverpool and Leeds, mostly based on the grades, and the flexibility of the courses. I went to open days to all of them (except Bath, because no one could take me and I wasn't ready to go on my own after having just gone to Liverpool by myself). I got offers from all five, which was very exciting, and I decided that I wanted Warwick as my first choice, and Nottingham as my back up.
I found out, at some point, that I would have to take an additional paper alongside my A-levels to get into Warwick. The STEP papers use A-level syllabuses, but the questions are designed to be more like the kind of questions you would see at University, so it tests you on how you would cope with degree level work. I surprised myself by not being put off by this. Even when I tried some questions and they were really hard, and I didn't really get them, I kept going with it. The day I got an answer (or close to part of an answer maybe...) I was so ridiculously excited. It was a really good feeling. I think it helped that my teacher thought that I could do it, and my mum said, even if I don't do very well in the exam, just the process of doing it will help me so much, and she is so impressed that I even had a go. The exams were hard. One of them more so than the other. But I gave it my best shot, and I was really quite pleased with what I did for one of them, so fingers crossed!
Even if I don't get into Warwick (the grades are really rather high!) I will still be happy to go to Nottingham. So from starting out with no idea what I want to do, I've managed to get to a position where, no matter what happens, I will be sorted for the next 3 or 4 years. After that, I have no idea what I'm doing!
In all this time I never thought of studying just maths. I always thought it would be too hard or too boring. But after looking at other courses, and not being particularly inspired by any of them, I settled on maths, simply because it's what I'm best at, and I didn't want to do anything else. But the more I thought about it, the more excited about the idea I got.
Maths has always been my favourite subject. It is logical, it doesn't involve many words, and it has a right answer. I hate it when you have to write an essay or something, and the teacher says "there is no right or wrong answer". How do you score points then? How do you know if what you have written is going to get you a good mark? I just don't, so that is why I like maths. I can usually tell if I've done well on a paper, just because there are methods to answering the questions.
My teachers were happy with the choice I'd made as well. They believed I was smart enough for it, and they knew that I enjoyed the lessons so they thought that I would be very interested in the subject at a degree level. With their support, I started to believe that I could do it as well.
Last summer, a close friend of our family kindly bought me a couple non-academic maths books (one of which is the book I talked about in a previous post). I read The Number Mysteries by Marcus du Sautoy first. I found it surprisingly interesting and easy to read. I never realised how much more there is to maths than what you learn in the classroom. Reading this book made me think more about what I could learn at University, and how maths applies to the world, and I started to think more and more that it was something I wanted to do. Since then I have listened to a podcast and watched a TV series about the history of mathematics (both by Marcus du Sautoy as well). I have been given and bought a few more books that I have read various chapters from, and started this blog!
The next challenge, after getting exciting maths, was to choose which Universities to apply for. I had no idea where I wanted to go. I knew that I didn't want to go to Cambridge or Oxford, they just didn't seem right for me. Other than that, I didn't really know. My Step-dad suggested that I look at the Universities in the Russell group because they are below Oxford and Cambridge, but they are still high standing Universities. I ordered prospectuses from all the ones that weren't too far away, and looked at the courses they offered. The maths courses were all pretty much the same, as I decided I just wanted to do the basic mix of pure and applied mathematics, so I could find out what I like and specialise later on.
After a long time of still not really knowing what to do, my mum suggested a road trip to fit in with teaching my sister how to drive on motorways and to let me have a quick look at a University for myself. I'm not sure how we ended up choosing Warwick, whether it was purely because it was one of the closer ones, because the prospectus was on top of the pile, or because we used to go to Warwick castle a lot, but that is where we went. Although we only got an outside look at the Uni (it wasn't actually an open day), we had a drive around the town, and found a little shopping centre to get some lunch and we bought a couple of books. Mum kept saying things like "this could be where you do your shopping for food to survive!" which was a little bit scary, but kind of cool. We also decided that the maths course there would be ideal, because there was a lot of room for flexibility if I wanted to take a module of something else to break up the maths.
After that, everything got a bit easier. I applied to Warwick, Bath, Nottingham, Liverpool and Leeds, mostly based on the grades, and the flexibility of the courses. I went to open days to all of them (except Bath, because no one could take me and I wasn't ready to go on my own after having just gone to Liverpool by myself). I got offers from all five, which was very exciting, and I decided that I wanted Warwick as my first choice, and Nottingham as my back up.
I found out, at some point, that I would have to take an additional paper alongside my A-levels to get into Warwick. The STEP papers use A-level syllabuses, but the questions are designed to be more like the kind of questions you would see at University, so it tests you on how you would cope with degree level work. I surprised myself by not being put off by this. Even when I tried some questions and they were really hard, and I didn't really get them, I kept going with it. The day I got an answer (or close to part of an answer maybe...) I was so ridiculously excited. It was a really good feeling. I think it helped that my teacher thought that I could do it, and my mum said, even if I don't do very well in the exam, just the process of doing it will help me so much, and she is so impressed that I even had a go. The exams were hard. One of them more so than the other. But I gave it my best shot, and I was really quite pleased with what I did for one of them, so fingers crossed!
Even if I don't get into Warwick (the grades are really rather high!) I will still be happy to go to Nottingham. So from starting out with no idea what I want to do, I've managed to get to a position where, no matter what happens, I will be sorted for the next 3 or 4 years. After that, I have no idea what I'm doing!
Wednesday, 25 July 2012
Alex's Adventures in Numberland - Chapter 0
In the introduction to Alex's Adventures in Numberland, Alex Bellos explains that he has a degree in mathematics and philosophy. He entered into journalism to continue with philosophy and abandon his mathematics. While working as a journalist though, he was surprised by how innumerate most journalists are. After a few years, Alex was trying to find what he wanted to do, when he delved back into the world of mathematics. As an adult, he wasn't looking at exam maths, but anything that he found interesting and curious. This is where he got the inspiration to write this book.
The book starts at chapter 0, to emphasise that the content covered in the chapter is pre-mathematics, how numbers emerged. In the contents page, the chapter is described as follows:
A Head For Numbers - In which the author tries to find out where numbers come from, since they haven't been around that long. He meets a man who had lived in the jungle and a chimpanzee who has always lived in the city.
Basically, the chapter is talking about number instinct. Do people everywhere have the same instincts when it comes to numbers? What about babies? Animals? Reading the chapter, I have learned some interesting things.
Animals
There is an interesting case about a horse called Clever Hans, who was thought to be able to perform simple arithmetic by stamping out the correct answer with his hoof. A committee of scientists investigated the horse to see if the act was just a trick, but decided that the horse was in fact doing the maths. However, a psychologist also investigated the horse, and noticed that he reacted to slight changes in his trainer's face as he reached the correct answer. The trainer wasn't even aware that this was happening, but the horse was very sensitive to facial changes. So the horse wasn't doing arithmetic, but I still think it was incredibly clever for the horse to be able read these changes.
In Japan, they wanted to test the mathematical abilities of animals without the chances of human cues, intentional or not. This is where Ai came in. Ai is a chimpanzee from West Africa, brought over to Japan in the late 1970s. The aim was to teach Ai to count without human interaction, but using touch screens and apple cube treats. Ai became the first non-human to count with Arabic numerals (1, 2, 3, etc.) In order to count, you need to understand two concepts about numbers, which are quantity (cardinality) and order (ordinality). Ai grasped these concepts fairly quickly, and so they tested her (and her son) with flash memory tests. The numbers 1-5 were arranged randomly on screen for just over half a second, and then covered so that Ai had to tap the numbers in order from memory. Compared to a sample group of Japanese children, the chimpanzees continued to perform well when the time the numbers were visible dropped, whereas the children dropped below 50% success rate. This showed that the chimpanzees have an incredible photographic memory, which could be down to having to make snap decisions in the wild about things like the number of foe there are.
Babies
Other tests have been conducted to find out what instinctual maths abilities babies might have. To start with a puppet is placed on a stage. Then a screen is placed in front of the puppet. Another puppet is shown to be placed behind the screen. When the screen is removed, either 1, 2 or 3 puppets will be revealed. It was found that babies stared longer at the puppets when the maths didn't add up, suggesting they were expecting only 2, which in turn suggests they have some basic understanding of numbers. This is backed up by the fact that they didn't stare for longer if the puppets had changed completely, only if there were the wrong numbers. Obviously it is difficult to tell for sure if this is because of a basic grasp for numbers, because you can't just ask a baby, but the research has shown some interesting results.
There are loads more examples of comparisons between maths you learn and maths you are born knowing in the chapter. For example, it talks about tribes that only have numbers for 1, 2, 3 and many. It is really interesting to see how maths is thought about and used by other cultures and creatures and I recommend reading it. Being written by a journalist for people other than mathematicians, it is very easy to read and understand. It isn't so much about how maths works, but how people use it.
Here is the Amazon link for it.
The book starts at chapter 0, to emphasise that the content covered in the chapter is pre-mathematics, how numbers emerged. In the contents page, the chapter is described as follows:
A Head For Numbers - In which the author tries to find out where numbers come from, since they haven't been around that long. He meets a man who had lived in the jungle and a chimpanzee who has always lived in the city.
Basically, the chapter is talking about number instinct. Do people everywhere have the same instincts when it comes to numbers? What about babies? Animals? Reading the chapter, I have learned some interesting things.
Animals
There is an interesting case about a horse called Clever Hans, who was thought to be able to perform simple arithmetic by stamping out the correct answer with his hoof. A committee of scientists investigated the horse to see if the act was just a trick, but decided that the horse was in fact doing the maths. However, a psychologist also investigated the horse, and noticed that he reacted to slight changes in his trainer's face as he reached the correct answer. The trainer wasn't even aware that this was happening, but the horse was very sensitive to facial changes. So the horse wasn't doing arithmetic, but I still think it was incredibly clever for the horse to be able read these changes.
In Japan, they wanted to test the mathematical abilities of animals without the chances of human cues, intentional or not. This is where Ai came in. Ai is a chimpanzee from West Africa, brought over to Japan in the late 1970s. The aim was to teach Ai to count without human interaction, but using touch screens and apple cube treats. Ai became the first non-human to count with Arabic numerals (1, 2, 3, etc.) In order to count, you need to understand two concepts about numbers, which are quantity (cardinality) and order (ordinality). Ai grasped these concepts fairly quickly, and so they tested her (and her son) with flash memory tests. The numbers 1-5 were arranged randomly on screen for just over half a second, and then covered so that Ai had to tap the numbers in order from memory. Compared to a sample group of Japanese children, the chimpanzees continued to perform well when the time the numbers were visible dropped, whereas the children dropped below 50% success rate. This showed that the chimpanzees have an incredible photographic memory, which could be down to having to make snap decisions in the wild about things like the number of foe there are.
Babies
Other tests have been conducted to find out what instinctual maths abilities babies might have. To start with a puppet is placed on a stage. Then a screen is placed in front of the puppet. Another puppet is shown to be placed behind the screen. When the screen is removed, either 1, 2 or 3 puppets will be revealed. It was found that babies stared longer at the puppets when the maths didn't add up, suggesting they were expecting only 2, which in turn suggests they have some basic understanding of numbers. This is backed up by the fact that they didn't stare for longer if the puppets had changed completely, only if there were the wrong numbers. Obviously it is difficult to tell for sure if this is because of a basic grasp for numbers, because you can't just ask a baby, but the research has shown some interesting results.
There are loads more examples of comparisons between maths you learn and maths you are born knowing in the chapter. For example, it talks about tribes that only have numbers for 1, 2, 3 and many. It is really interesting to see how maths is thought about and used by other cultures and creatures and I recommend reading it. Being written by a journalist for people other than mathematicians, it is very easy to read and understand. It isn't so much about how maths works, but how people use it.
Here is the Amazon link for it.
Sunday, 3 June 2012
Mechanics
This was my mechanics revision. My last mechanics exam was on Friday. I don't need these notes any more :)
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