tag:blogger.com,1999:blog-44269037337349441802024-03-13T22:42:05.375-07:00Derek Haylock - AuthorDerek Haylockhttp://www.blogger.com/profile/10959138785915870717noreply@blogger.comBlogger192125tag:blogger.com,1999:blog-4426903733734944180.post-85932155053945637112016-11-20T08:41:00.000-08:002016-11-20T08:41:24.118-08:00Formula for escalator stepsHere's my solution to the escalator problem in my previous post.<br />
<br />
Per minute, I climb <i>n</i> steps and the escalator moves along by <i>s </i>steps. So relative to the bottom of the escalator I am travelling at the rate of <i>n</i> + <i>s</i> steps per minute.<br />
<br />
If there are <i>p</i> 'visible steps' on the escalator and I take <i>t</i> minutes to get to the top, then <i>t = p/(n + s)</i>.<br />
<br />
But in <i>t</i> minutes I actually climb <i>nt</i> of the moving steps. This is the variable I call <i>m,</i> the actual number of steps that I take in getting to the top.<br />
<br />
So <i>m </i>= <i>nt</i> = <i>np/(n + s).</i><br />
<i><br /></i>
So below is my formula!<br />
<br />
<i>m = </i><i>np/(n + s).</i><br />
<i><br /></i>
I must admit I never expected it to be as complicated as this.<br />
<br />
To understand this formula, think of it as saying that the proportion of visible steps I climb is the ratio of my speed (<i>n</i>) to the sum of my speed and the speed of the escalator (<i>n + s</i>).<br />
<br />
For example, if the escalator has 30 'visible steps' and is moving at 90 steps per minute, my speed is 120 steps per minute, then the proportion of the 30 steps I will tread on will be 120/210, or 4/7 of them, which is about 17.<br />
<br />
If you plot <i>m </i> against <i>n</i>, from <i>n</i> = 0, you get a curve that starts at the origin (<i>m</i> = 0) and tends to <i>p</i> as <i> n</i> tends to infinity.<br />
<br />
When <i>n</i> = <i>s</i>, notice that <i>m</i> = <i>s</i>/2, which makes sense intuitively: if I climb at the same rate as the escalator is moving I will cover half the steps before I get to the top.<br />
<br />
Note also that as <i>n</i> increases the gain in my step count decreases. For example, with a 30-step escalator, and a speed of 90 steps rising per minute ...<br />
<br />
When <i>n</i> = 0, <i>m </i>= 0<br />
<br />
When <i>n </i>= 60, <i>m </i>= 12<br />
<br />
When <i>n</i> = 120, <i>m </i>= 17<br />
<br />
When <i>n</i> = 180, <i>m </i>= 20<br />
<br />
When <i>n</i> = 240, <i>m </i>= 22<br />
<br />
When <i>n</i> = 300, <i>m </i>= 23<br />
<br />
It's intriguing to think about how the graph behaves for values of <i>n</i> less than zero down to <i>n</i> = –<i>s. </i>These values of <i>n </i>correspond to walking down the escalator when it is going up. Provided you walk down slower than the escalator is going up, then you will get to the top! The number of steps you take, <i>m, </i>will turn out to be negative because you are walking down the escalator. <i> </i>If you walk down as fast as the escalator is going up (<i>n</i> = <i>–s</i>) then the formula involves division by zero, which is impossible: you never get to the top. In fact, you never get anywhere.<br />
<br />
You can have even more fun now, by thinking about how the formula works for climbing up an escalator going down!<br />
<br />Derek Haylockhttp://www.blogger.com/profile/10959138785915870717noreply@blogger.com30tag:blogger.com,1999:blog-4426903733734944180.post-86161609396448606812016-11-15T03:31:00.000-08:002016-11-15T03:31:37.839-08:00How many steps on an escalator?One of the joys (or burdens?) of being fascinated by mathematics is that you see interesting mathematics problems all around you in your everyday life!<br />
<br />
I have recently acquired one of those little devices you attach to your belt that counts the number of steps you make in a day. The result of this is that you get obsessional about maximising your number of steps in every situation. There is even a temptation to mince along with very short steps rather than my usual manly stride, just to increase the step-count!<br />
<br />
Today, I had to go up an escalator in a shopping centre. I never stand still on an escalator, unless it is totally blocked with other people. So, I found myself thinking about how many steps might I manage going up this escalator. It seemed intuitive to me that the faster I went the more steps I would get in before I reached the top. After all, if my speed was zero (i.e. standing still on one step) then my step-count for the escalator ride would also be zero.<br />
<br />
Intuitively I imagined that there would be a linear relationship here. So, if I went twice as fast I would get in twice as many steps before I reached the top. But, wait a minute! That can't be true. Because I could never get more than the total number of visible steps on the escalator, however fast I went. In fact, to get in a number of steps equal to the number of visible steps on the escalator I would have to travel at an infinite speed!<br />
<br />
This was getting so interesting, I forgot to do my shopping and just walked home puzzling it out.<br />
<br />
It's a really nice mathematical problem. I managed to calculate, for example, that ...<br />
<ul>
<li>if there are 30 visible steps on the escalator (i.e. the number of steps you would see if the escalator were stationary), and</li>
<li>the steps are disappearing at the top at the rate of 90 steps per minute, and</li>
<li>I go up the escalator at the rate of 120 steps per minute (pretty fast),</li>
</ul>
then I will add 17 steps to my step-counter.<br />
<br />
If I went twice as fast up this escalator, doing 240 steps per minute, I calculated that I would increase my step-count to 21 or possibly 22 steps. And at half the speed, i.e. 60 steps per minute, I would do 12 steps before reaching the top of the escalator.<br />
<br />
I'll leave this as a challenge for readers ... can you produce a formula for the number of steps I add to my step-counter (<i>m</i>) if there are <i>p</i> visible steps; the steps are disappearing at the top at the rate of <i>s</i> steps per minute; and I go up at the rate of <i>n</i> steps per minute?<br />
<br />
<i>m </i> = ....?<br />
<br />
And what would the graph look like, if you plotted <i>m</i> against <i>n</i>?<br />
<br />
Solution in my next post.Derek Haylockhttp://www.blogger.com/profile/10959138785915870717noreply@blogger.com1tag:blogger.com,1999:blog-4426903733734944180.post-73447720138749579472016-11-04T05:20:00.000-07:002016-11-04T05:20:12.378-07:00Average ageAs I get older I find that I pay more attention to the ages of the people who appear each day in the obituary pages of <i>The Times</i>! Now I find that I have started to calculate the average (mean) age at which they died – and I am encouraged if it is greater than my age (it usually is!).<br />
<br />
But, I have an interesting mathematical observation, related to the issue of rounding errors, to offer you.<br />
<br />
Take this example. Four people are listed in the obituaries, dying at the ages of 69, 73, 76 and 80. What is the average (mean) age at which they died?<br />
<br />
Now, 69 + 73 + 76 + 80 = 298. Divide this by 4 and we get 74.5 years.<br />
<br />
But, this is NOT the average age of these four individuals.<br />
<br />
Remember that when we say that someone died at the age of 69 this means that they could have been as little as one day short of their 70th birthday. Different conventions for rounding up or rounding down are used in various contexts. We always round <i>down</i> to the year below when we give someone's age in years.<br />
<br />
So, the best estimate for the average age of the four individuals in this example would be the mean of 69.5, 73.5, 76.5 and 80.5 years. And that gives the mean age as 75 years.<br />
<br />
At my age, let me assure you, that extra half a year is quite significant!<br />
<br />
<br />Derek Haylockhttp://www.blogger.com/profile/10959138785915870717noreply@blogger.com0tag:blogger.com,1999:blog-4426903733734944180.post-50822408374022867342016-09-06T02:29:00.001-07:002016-09-06T02:33:39.903-07:00Calculations: foundations of mathematics?The myth that the mastery of the processes of written calculations, particularly long multiplication and division, is fundamental to doing mathematics continues to be perpetrated by people with political influence and control of our school curriculum. This myth was exposed for me recently in the experience of helping one of my grandsons prepare for his A-level mathematics examination.<br />
<br />
We worked together through loads of questions from past examination papers – pure maths, mechanics and statistics. I made the following observations.<br />
<br />
Not once in doing A-level mathematics was he required to do a written calculation, since he always had a calculator to hand. His calculator skills were stunning and showed real mathematical understanding, in terms of processing the steps of a complex calculation in the appropriate order, in selecting the correct function keys and handling brackets, and doing this with speed and accuracy.<br />
<br />
More important than written calculation skills were the ability to interpret the calculator answer and checking whether it looked reasonable. Additionally, with a little encouragement from me, he improved markedly in using mental strategies for calculations that could be done more efficiently that way than by resorting to the calculator.<br />
<br />
But, I repeat, <i>not once </i>did he use a formal written calculation procedure. Yet, there he was doing advanced level mathematics! If he had had to take his eye off the structure of the problem to do a written calculation it is very likely that he would have lost his grasp on where he was going.<br />
<br />
For centuries mathematicians have devised ways of avoiding or reducing the demand of written calculations, simply because they get in the way of the real mathematics and effective problem-solving, and take up too much of your precious time. So, we had Napier's bones, and logarithm tables and slide rules, and so on. Now we have modern technology, so please let's give younger children the chance to use it and start doing real mathematics.<br />
<br />Derek Haylockhttp://www.blogger.com/profile/10959138785915870717noreply@blogger.com4tag:blogger.com,1999:blog-4426903733734944180.post-50926671835160240472016-06-21T02:14:00.005-07:002016-11-04T05:21:44.553-07:00Haylock and CockburnMy apologies to any readers of this blog for a long period of silence. Anne Cockburn and I have been busy working on a fifth edition of our Book, Understanding Mathematics for Young Children. We finished this last week and have sent off the 'manuscript' to the Sage Publications.<br />
<br />
The word 'Manuscript', which is still used in publishing circles, means, of course, 'written by hand', which is strangely archaic, given that no pens and no paper were involved in either the writing process or the submission of the new edition to the publisher.<br />
<br />
We hope to see the new edition on the shelves (another anachronism, since most of the sales will be online!) in the first half of next year.<br />
<br />
We have had to rework the book to ensure that it is consistent with the language and content of the new primary mathematics curriculum in England. This has introduced some new material for Year 2 children, such as fractions. The new curriculum has meant a general shift of content down from Year 3 to Year 2 (and likewise from Year 4 to Year 3). So, to avoid our book getting even longer, we have had to make a decision to reduce the age range covered. So, it is now described as 'a guide for teachers of children aged 3–7 years' (rather than 3–8). This is a better fit for teacher training courses anyway, since it is now clearly aimed at Early Years Foundation Stage and Key Stage 1.<br />
<br />
So, with that done, I hope to get back to writing the occasional blog again! Watch this space.<br />
<br />Derek Haylockhttp://www.blogger.com/profile/10959138785915870717noreply@blogger.com2tag:blogger.com,1999:blog-4426903733734944180.post-32635377735984177582016-03-04T03:23:00.002-08:002016-11-04T05:22:38.985-07:00Roman numerals<span style="font-family: georgia, 'times new roman', serif;">Here are three questions about Roman numerals that genuinely puzzle me! </span><br />
<div class="B1TFBox1textfullout">
<span style="font-family: "georgia" , "times new roman" , serif;"><span style="color: black;"><span style="font-size: small;">First, why, in the twenty-first century, do we persist in using Roman numerals in particular contexts, such
as the hours on clock or watch faces, and dates on buildings or at the end of a movie? </span></span></span></div>
<div class="B1TFBox1textfullout">
<span style="font-family: "georgia" , "times new roman" , serif;"><span style="color: black;"><span style="font-size: small;">Second, in such a technological age, can anyone really
justify the inclusion of Roman numerals in the statutory English primary
mathematics curriculum and in the associated national assessment of
mathematics? </span></span></span></div>
<div class="B1TFBox1textfullout">
<span style="font-family: "georgia" , "times new roman" , serif;"><span style="color: black;"><span style="font-size: small;">Third, there’s something odd I’ve noticed recently about Roman numerals on clock and watch faces. It is important for teachers to be aware of this, because there are limited contexts for
assessment items in Key Stage 2 national mathematics tests in this country, so clock and watch faces with Roman numerals turn up often.</span></span><span style="color: black;"><span style="font-size: small;"> </span></span></span></div>
<div class="B1TFBox1textfullout">
<span style="font-family: "georgia" , "times new roman" , serif;"><span style="color: black;"><span style="font-size: small;">In an early form of Roman numeration, the numbers
we call ‘four’ and ‘nine’ would be represented by IIII and VIIII. A later
development was to represent these more concisely as IV and IX, the convention
being that when a letter representing a smaller value is written in front of
another letter, then the value is to be subtracted, not added. So, XC would
represent 90 (100 subtract 10). So, here’s what we have noticed: in most cases
where Roman numerals are used on a clock or watch face, the four is written
using the early system (IIII) and the nine is written using the later system
(IX). Check this out and see if we are right. The question that puzzles me is,
simply, why?</span></span></span><span style="font-size: small;"><span style="font-family: "trebuchet ms" , sans-serif;"></span></span>
</div>
Derek Haylockhttp://www.blogger.com/profile/10959138785915870717noreply@blogger.com4tag:blogger.com,1999:blog-4426903733734944180.post-42594270493744064232016-02-25T03:30:00.001-08:002016-02-25T03:30:26.866-08:00The Haylock BookmarkWhen I am in the middle of reading a book I need a bookmark that not only helps me to find the page I am on, but also exactly where on the page I should start to continue my reading! I offer readers my design for an effective bookmark for this purpose and a simple example of mathematics applied to another area of the curriculum!<br />
<br />
First measure the height of the page of the book (<i>h</i> mm) and the height of the smaller of the top and bottom margins for the text on the page (<i>x</i> mm). Cut a rectangular piece of firm card so that it has a length that is equal to <i>h</i>/2 + <i>x. </i>The width can be whatever you wish<i>,</i> say, about a third of the width of the book.<i><br /></i><br />
<br />
For example, for a standard paperback novel I find that <i>h </i>= 198 and <i>x</i> = 19. So we need the bookmark to be 99 + 19 = 118 mm long.<br />
<br />
Then on one side of the card draw a double-headed arrow <i>x</i> mm from the top (i.e. 19 mm for our standard paperback). The diagram shows a bookmark made for this standard paperback.<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://1.bp.blogspot.com/-xIhDa1-XLxw/VrzKEaA5BAI/AAAAAAAAALg/_KPXgxaBFNU/s1600/2016-02-11_17-50-00.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://1.bp.blogspot.com/-xIhDa1-XLxw/VrzKEaA5BAI/AAAAAAAAALg/_KPXgxaBFNU/s1600/2016-02-11_17-50-00.png" /></a></div>
<div class="separator" style="clear: both; text-align: center;">
</div>
That's it! Now, when you have finished your reading session place the bookmark in the fold of the book on the opposite page to the one you are on, with the arrow showing and pointing to your place on the page. This might involve rotating the bookmark, of course, depending whether you are on the top half or the bottom half of the page. Then just close the book. When you open it next time just make sure you have the side of the card with the arrow visible, and remember that the arrow tells you where on the <i>opposite</i> page you should start reading again!<br />
<br />
With this system any starting point on either the even-numbered page (verso) or the odd-numbered page (recto) can be marked, without the bookmark sticking out of the closed book (which I don't like).<br />
<br />
The diagrams below show the bookmark placed inside the book to mark (a) a point half-way down the left-hand page; and (b) a point near the bottom of the right-hand page.<br />
<br />
(a)<br />
<a href="http://4.bp.blogspot.com/-X4aHuvPeyUg/VroGCbwOzHI/AAAAAAAAALM/Jt7OREZvARQ/s1600/Example%2B1.jpg" imageanchor="1"><img border="0" height="239" src="https://4.bp.blogspot.com/-X4aHuvPeyUg/VroGCbwOzHI/AAAAAAAAALM/Jt7OREZvARQ/s320/Example%2B1.jpg" width="320" /></a><br />
(b)<br />
<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://3.bp.blogspot.com/-ZO8aaSM_DHI/VroGbMWHH4I/AAAAAAAAALQ/NQxaMEwjzqk/s1600/Example%2B2.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="216" src="https://3.bp.blogspot.com/-ZO8aaSM_DHI/VroGbMWHH4I/AAAAAAAAALQ/NQxaMEwjzqk/s320/Example%2B2.jpg" width="320" /></a></div>
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Well, it works for me. And it is a good application of simple measurement and spatial reasoning!Derek Haylockhttp://www.blogger.com/profile/10959138785915870717noreply@blogger.com1tag:blogger.com,1999:blog-4426903733734944180.post-49102956513255876492016-02-09T04:02:00.000-08:002016-02-09T04:02:57.635-08:00Mastery and understanding mathematics
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<span style="font-size: small;"><span lang="X-NONE">In the
context of the challenge to raise standards in mathematics in schools in
England the word ‘mastery’ has recently become prominent in the vocabulary of
the English mathematics curriculum (NCTEM, 2014, www.ncetm.org.uk/public/files/19990433).
It is reassuring to note that the way in which the word ‘mastery’ is being used
is entirely consistent with the approach to children’s learning of mathematics
that I have promoted in my own writing. </span></span></div>
<span style="font-size: small;">
</span><div class="BTFBodytextfullout" style="line-height: normal;">
<span style="font-size: small;"><span lang="X-NONE">Mastery
is seen as children developing fluency in mathematics alongside a deep
understanding of mathematical ideas and processes. So, for example, teaching
approaches for mastery should ‘foster deep conceptual and procedural knowledge’
and ‘exercises are structured with great care to build deep conceptual
knowledge alongside developing procedural fluency’ (op.cit.). This is a key
principle in teaching mathematics to young children: that mastery of the
subject is not achieved simply by repeated drill in various procedures.
Instead, the focus is on the development of understanding of mathematical
structures and on making connections.</span></span></div>
<span style="font-size: small;">
</span><div class="BTFBodytextfullout" style="line-height: normal;">
<span style="font-size: small;"><span lang="X-NONE">Making
connections in mathematics – a recurring theme in all my books – ensures that
‘what is learnt is sustained over time, and cuts down the time required to
assimilate and master later concepts and techniques’ (op.cit.). Nearly all
mathematical concepts and principles occur and can be applied in a wide range
of contexts and situations. Because of this, the deeper understanding central
to mastery in mathematics is facilitated by a wide variation in the experiences
that embody mathematical ideas. </span></span></div>
<span style="font-size: small;">
</span><div class="BTFBodytextfullout" style="line-height: normal;">
<span style="font-size: small;"><span lang="X-NONE">For
example, mastery of the 5-times multiplication table by Year 2 children is not
just a matter of memorizing a chant that begins ‘one five is five, two fives
are ten …’ – although that is part of it. It would also involve, for example:</span></span></div>
<span style="font-size: small;">
</span><div class="BTFBodytextfullout" style="line-height: normal; margin-left: 36pt; text-indent: -18pt;">
<span style="font-size: small;"><span lang="X-NONE" style="font-family: Symbol;"><span>·<span style="font-family: "Times New Roman"; font-feature-settings: normal; font-kerning: auto; font-language-override: normal; font-size-adjust: none; font-stretch: normal; font-style: normal; font-synthesis: weight style; font-variant: normal; font-weight: normal; line-height: normal;"> </span></span></span><span lang="X-NONE">connecting each result in the table with a collection of 5p coins
and the total value;</span></span></div>
<span style="font-size: small;">
</span><div class="BTFBodytextfullout" style="line-height: normal; margin-left: 36pt; text-indent: -18pt;">
<span style="font-size: small;"><span lang="X-NONE" style="font-family: Symbol;"><span>·<span style="font-family: "Times New Roman"; font-feature-settings: normal; font-kerning: auto; font-language-override: normal; font-size-adjust: none; font-stretch: normal; font-style: normal; font-synthesis: weight style; font-variant: normal; font-weight: normal; line-height: normal;"> </span></span></span><span lang="X-NONE">articulating the pattern of 5s and 0s in the units position in the
odd and even multiples of 5; </span></span></div>
<span style="font-size: small;">
</span><div class="BTFBodytextfullout" style="line-height: normal; margin-left: 36pt; text-indent: -18pt;">
<span style="font-size: small;"><span lang="X-NONE" style="font-family: Symbol;"><span>·<span style="font-family: "Times New Roman"; font-feature-settings: normal; font-kerning: auto; font-language-override: normal; font-size-adjust: none; font-stretch: normal; font-style: normal; font-synthesis: weight style; font-variant: normal; font-weight: normal; line-height: normal;"> </span></span></span><span lang="X-NONE">explaining how to get from 4 fives to 8 fives by doubling;</span></span></div>
<span style="font-size: small;">
</span><div class="BTFBodytextfullout" style="line-height: normal; margin-left: 36pt; text-indent: -18pt;">
<span style="font-size: small;"><span lang="X-NONE" style="font-family: Symbol;"><span>·<span style="font-family: "Times New Roman"; font-feature-settings: normal; font-kerning: auto; font-language-override: normal; font-size-adjust: none; font-stretch: normal; font-style: normal; font-synthesis: weight style; font-variant: normal; font-weight: normal; line-height: normal;"> </span></span></span><span lang="X-NONE">explaining how to get from 6 fives to 7 fives by adding 5;</span></span></div>
<span style="font-size: small;">
</span><div class="BTFBodytextfullout" style="line-height: normal; margin-left: 36pt; text-indent: -18pt;">
<span style="font-size: small;"><span lang="X-NONE" style="font-family: Symbol;"><span>·<span style="font-family: "Times New Roman"; font-feature-settings: normal; font-kerning: auto; font-language-override: normal; font-size-adjust: none; font-stretch: normal; font-style: normal; font-synthesis: weight style; font-variant: normal; font-weight: normal; line-height: normal;"> </span></span></span><span lang="X-NONE">counting in steps of five along a counting stick;</span></span></div>
<span style="font-size: small;">
</span><div class="BTFBodytextfullout" style="line-height: normal; margin-left: 36pt; text-indent: -18pt;">
<span style="font-size: small;"><span lang="X-NONE" style="font-family: Symbol;"><span>·<span style="font-family: "Times New Roman"; font-feature-settings: normal; font-kerning: auto; font-language-override: normal; font-size-adjust: none; font-stretch: normal; font-style: normal; font-synthesis: weight style; font-variant: normal; font-weight: normal; line-height: normal;"> </span></span></span><span lang="X-NONE">knowing that, say, ‘3 fives are fifteen’ is what you use for the
cost of 3 books at £5 each;</span></span></div>
<span style="font-size: small;">
</span><div class="BTFBodytextfullout" style="line-height: normal; margin-left: 36pt; text-indent: -18pt;">
<span style="font-size: small;"><span lang="X-NONE" style="font-family: Symbol;"><span>·<span style="font-family: "Times New Roman"; font-feature-settings: normal; font-kerning: auto; font-language-override: normal; font-size-adjust: none; font-stretch: normal; font-style: normal; font-synthesis: weight style; font-variant: normal; font-weight: normal; line-height: normal;"> </span></span></span><span lang="X-NONE">constructing patterns with linked cubes that show 1 set of five, 2
sets of five, and so on;</span></span></div>
<span style="font-size: small;">
</span><div class="BTFBodytextfullout" style="line-height: normal; margin-left: 36pt; text-indent: -18pt;">
<span style="font-size: small;"><span lang="X-NONE" style="font-family: Symbol;"><span>·<span style="font-family: "Times New Roman"; font-feature-settings: normal; font-kerning: auto; font-language-override: normal; font-size-adjust: none; font-stretch: normal; font-style: normal; font-synthesis: weight style; font-variant: normal; font-weight: normal; line-height: normal;"> </span></span></span><span lang="X-NONE">filling in the missing number in number sentences like ‘6 × </span><span lang="X-NONE" style="font-family: "Courier New";">□</span><span lang="X-NONE"> = 30’.</span></span></div>
<span style="font-size: small;">
</span><div class="BTFBodytextfullout" style="line-height: normal;">
<span style="font-size: small;"><span lang="X-NONE">To teach
for this kind of mastery teachers themselves need a deep structural
understanding of mathematics, an awareness of the range and variety of
situations in which a mathematical concept or principle can be experienced, and
confidence in exploring the connections that are always there to be made in
understanding mathematics. Any teachers looking for this? I can recommend one or
two books.</span></span></div>
Derek Haylockhttp://www.blogger.com/profile/10959138785915870717noreply@blogger.com2tag:blogger.com,1999:blog-4426903733734944180.post-28411674489418311692015-12-31T08:57:00.000-08:002015-12-31T08:59:05.126-08:00Three halves make a whole?Our Prime Minister, David Cameron, came in for some criticism this year for his use of the word 'half', when, in answer to a question about what were the Houses of Parliament like, he replied, 'half museum, half chapel, half school'. 'You can't have three halves!' was the outcry.<br />
<br />
Obviously, what he said is not intended to be an exact statement of proportions, but even so this looks as sloppy as the often-heard 'I'll have the bigger half'.<br />
<br />
However, on reflection, Cameron is OK in what he says here. You <i>can</i> have three halves of a set, for example, if the halves overlap. For example, a teacher could say, half of the children in my class are Asian, half of them are male, and half of them walk to school.<br />
<br />
So, if, we were to assume that there are 100 quintessential qualities or features that characterise the Houses of Parliament, it is quite possible that 50 of these are shared with those of a museum, 50 of them are shared with those of a chapel, and 50 of them are shared with those of a school. This is because the quintessential qualities of museums, chapels and schools are not unique to each of these kinds of places.<br />
<br />
So, it seems alright to say that Parliament feels like half museum, half chapel and half school!<br />
<br />
There! It's not often you get me supporting our Prime Minister; but fair's fair.<br />
<br />
<br />
<br />Derek Haylockhttp://www.blogger.com/profile/10959138785915870717noreply@blogger.com0tag:blogger.com,1999:blog-4426903733734944180.post-9633005934245124582015-11-05T07:23:00.000-08:002015-11-05T07:23:58.284-08:00Back to back?The BBC news announced recently that New Zealand had won two rugby world cups 'back to back'. Well done, New Zealand. But not so well done, BBC.<br />
<br />
How has it happened that 'back to back' has come to mean 'in succession'? Given that this question is about the language we use to describe spatial or temporal relationships, I think that, as a mathematician, I am entitled to a little light-hearted rant.<br />
<br />
I hear people talking about watching successive episodes of a TV programme they have recorded and saying that they watched them 'back to back'. That's just daft. The back of the first episode (i.e. the end) is not followed by the back of the second episode, but by the front of it (i.e. the beginning). So, if anything, they are watching the two programmes 'back to front'.<br />
<br />
The pastor in our church announces that we are going to sing two songs 'back to back'. It would clearly make more sense to say that we'll sing them 'back to front'! But 'in succession' or 'one straight after the other' would perhaps be less ambiguous.<br />
<br />
When I was a lad, growing up on a council estate in London, 'back to back' described houses, where the back of one house faced the back of another, often with very little space between them for our 'back yards'. Those were the days when the phrase 'back to back' actually meant something; when we got the geometry right.<br />
<br />
This phrase 'back to back' seems to be associated with the image of books or DVDs stacked on a shelf, cover to cover. But when they are stacked like this the back of one book is touching the <i>front</i> of the next book. This is related to a well-known little mathematical puzzle ...<br />
<br />
There are ten paperbacks, each width one centimetre, standing neatly and upright on a bookshelf, with no gaps between them. Working left to right, a bookworm starts at the beginning of the first book and chews horizontally through to the end of the tenth book. How far does the bookworm travel?<br />
<br />
No, it's not 10 cm!<br />
<br />
I enjoyed Tim Vine's one liner about this irritating phrase: 'my wife and I watched two episodes of Downton Abbey back to back; fortunately I was the one facing the television.'<br />
<br />
By the way, the bookworm travels about only 8 cm.Derek Haylockhttp://www.blogger.com/profile/10959138785915870717noreply@blogger.com0tag:blogger.com,1999:blog-4426903733734944180.post-69931897507633686122015-10-07T03:41:00.002-07:002015-10-07T03:41:34.977-07:00Less Than Zero<i>The Times</i> has been running a feature called 'Grammar for Grown-Ups', written by John Sutherland, Emeritus Professor at UCL. As we would expect it generates the usual battles between the traditional pedantics who believe passionately in the validity of randomly imposed and sometimes archaic rules (such as the one about split infinitives) and the more liberal modern linguists who are happy for 'rules' to be determined by usage, rather than the other way round.<br />
<br />
I think I know most of the so-called rules and, to avoid confrontation with my copy editor, I try to stick to them in formal writing – even though I know in my heart of hearts that many of them are unjustified and unnecessarily rigid. And, if I do ever pick people up about dangling participles or hypercorrection of 'you and me', then it is done (usually) tongue in cheek. You can now challenge me about starting a sentence with the word 'And'!<br />
<br />
But I must take issue with John Sutherland about his claim that if you follow the rule about the distinction between 'fewer' and 'less' then the title of Bert Easton Ellis's book, <i>Less Than Zero</i>, should be <i>Fewer Than Zero</i>.<br />
<br />
His argument is that zero is a 'counting number' and therefore 'fewer' is correct. He's wrong.<br />
<br />
Zero can be a counting number describing an empty set, but it can also be an ordinal number describing a position; it is an integer separating negative integers from positive integers on various measurement scales; and it is a real number representing a unique point on the number line. Only when comparing an empty set with another set would the word 'fewer' be associated with zero. [A has zero marbles, B has 7 marbles, A has fewer than B.]<br />
<br />
But if we are referring to a number as an abstract entity, not as an adjective attached to a set of things, then it is a singular noun. So we can form sentences about, say, 7, that begin '7 is ...'. Examples would be: 7 is greater than 5; 7 is a prime number; 7 is a factor of 21; and so on. So, we would correctly say, 7 is <i>less than</i> 9. Likewise, zero is <i>less than</i> 7. Or, indeed, 'negative three is <i>less than zero</i>'.<br />
<br />
We always talk, correctly, about negative numbers being numbers<i> less than zero</i>. They are not cardinal numbers that describe sets of things, so it is incorrect to use the word 'fewer' when making statements about negative numbers. 'Negative three is fewer than zero' sounds bizarre: it seems to imply that there is a set of 'negative three things' that is being compared with an empty set. In none of the contexts in which negative numbers describe actual things (temperatures, bank balances, heights above sea level, and so on) does it make any sense to use 'fewer'. Would anyone say 'my bank balance is fewer than zero'?<br />
<br />
Finally, we should note that 'fewer' can only be applied to whole numbers (because they can describe sets). It cannot be used with non-whole numbers. Is there a meaningful sentence that begins '0.2 is fewer than ...'. So, a number 'less than zero' could be the number –0.2. Once again, this number is never going to be described as 'fewer than zero'.<br />
<br />
<br />
<br />
<br />Derek Haylockhttp://www.blogger.com/profile/10959138785915870717noreply@blogger.com0tag:blogger.com,1999:blog-4426903733734944180.post-49618883753042294632015-09-26T10:35:00.002-07:002015-09-26T10:35:54.646-07:00Measuring scalesOne of the important principles to understand about measurement is that there are different kinds of measuring scales. And what you can do with a particular set of measurements will depend on what kind of measurement scale is involved.<br />
<br />
For example, you can compare two lengths (say, <i>a</i> = 6 metres and <i>b</i> = 2 metres) by their <u>ratio</u> (<i>a</i> is 3 times longer than <i>b</i>). So, 'length' is what is called a ratio scale. One of the features of a ratio scale is that 0 represents 'nothing'. So, '0 metres' is no length at all.<br />
<br />
Temperature measured in degrees celsius is not therefore a ratio scale, because '0 degrees' does not represent 'no temperature at all'. So, you could not compare, say, a temperature of 12 degrees with one of 4 degrees, by saying that the the first one is three times hotter! To compare these you could only use the <i>difference</i> in temperature: the first is 8 degrees hotter than the second. This is an example of an <u>interval scale</u>.<br />
<br />
A more primitive type of measurement, often met in everyday life, is an ordinal scale. Essentially all this does is to put various entities into order, according to some criteria.<br />
<br />
An interesting example of the misuse of an ordinal scale is provided by the Amazon customer reviews for a book. To give a book 3 stars means simply that it is judged better than 2 but not as good as 3, by whatever criteria the reviewer is using.<br />
<br />
The thing to note is that you cannot do arithmetic with ordinal scales. You cannot say that a 4-star rating is twice as good as a 2-star rating; or even that the difference between a 2-star rating and a 3-star rating is in some sense 'the same size' as the difference between a 3-star rating and a 4-star rating! So, you definitely cannot add up all the customer ratings and divide by the number of them to calculate an average rating (i.e. the arithmetic mean). That could only be done if the interval between various ratings represented the same amount of something! If there was something that could be measured that we could call, say, a 'unit of worth', so that 1 star meant 1 unit of worth, and 2 stars meant 2 units of worth, and so on, and 0 stars meant 'absolutely worthless', then we would have a ratio scale and we could merrily calculate averages and so on. But customer ratings are not like that!<br />
<br />
I mention this because I noticed that my book, <i>Mathematics Explained for Primary Teachers</i>, is given an average rating of what looks like four and a half stars. This is the result of doing some arithmetic with 21 five-star reviews, 3 four-star, 1 three-star and 1 one-star. [This 1-star reviewer wrote 'not fore parents' (sic).] If you were to treat these as measurements on a ratio scale, the arithmetic mean would be 4.65, which appears in an icon as 4.5 stars.<br />
<br />
But, as explained above, you cannot do arithmetic with an ordinal scale, so the use of the arithmetic mean is unacceptable to find an average. What you could use for an average ranking is either the median or the mode, since these require nothing more than putting items in order or putting them into a set number of categories.<br />
<br />
If you use the median as an average you line up all the reviews from the lowest to the highest and choose the one in the middle to represent the whole set. In this case the median value is 5 stars!<br />
<br />
If you use the mode then you just select the most commonly occurring grade: which again is 5 stars.<br />
<br />
So, clearly, if the maths is done appropriately, using either of the legitimate averages for ordinal measurements (median or mode), then my book should get an average ranking of 5 stars! I rest my case.<br />
<br />Derek Haylockhttp://www.blogger.com/profile/10959138785915870717noreply@blogger.com1tag:blogger.com,1999:blog-4426903733734944180.post-67207791300979457712015-08-25T11:51:00.001-07:002015-08-25T11:51:22.324-07:00Shortage of Mathematics TeachersRecent press reports have highlighted concern about insufficient mathematics graduates being recruited into the teaching profession. <i>The Times</i> headline invited us to be horrified that many maths classes in secondary schools in this country are being taught by PE teachers!<br />
<br />
My immediate reaction was: so, what's new? My second reaction was that this seemed to suggest that PE teachers must be quite incapable of teaching anything other than forward rolls and cartwheels. I am a mathematics graduate, but I am also a teacher. In my time I have on occasions taught music, RE, Games (cricket/hockey) and science at secondary school level, and just about everything at primary level. So, in principle I can't be horrified at the thought of PE teachers filling in some gaps in the mathematics lessons timetable!<br />
<br />
The key question, of course, is, why are there not more mathematics graduates going into teaching? Of course, with a good maths degree there are more lucrative professions than teaching available to a new graduate. Early on in my career I was very nearly recruited by a major computing company; but at the last stage I declined their offer of a very well-paid job. My reasons were simply that I loved teaching, that I looked forward to going to work every day, that helping youngsters learn with understanding and enjoyment gave me great satisfaction, and that I really felt that I was making a difference.<br />
<br />
Teaching was a respected profession and I felt privileged to be part of it. In my view this is what has changed. Our press and our politicians are constantly running down the profession and undermining the morale of teachers. Government ministers interfere in educational practice at a level of detail that is intolerable. For example who decides what proportion of marks in the end-of-KS2 mathematics tests should be awarded for the context-free written calculation papers? Not teachers! Not mathematics educators. Not any professional body with expertise in the field. No, a government minister decides and imposes their decision on the testing agency. And how much attention was given to the view of the profession in the consultation over the National Curriculum? (That's a rhetorical question!)<br />
<br />
If we want to recruit more able graduates in mathematics – and other subject areas where the financial rewards elsewhere are much more attractive than that of a career in education – then a major priority is to restore the professionalism of teachers; to make teaching a profession which is respected; to trust teachers and to recognise their professional judgement. Teachers need professional conditions of employment, with the expectation for properly-funded professional development opportunities throughout their career. Educational policy related to the practice of teaching and children's learning must be informed first and foremost by the profession itself. Teaching should be a profession that bright young people will be proud to be part of.<br />
<br />
---------------------------------------------------------------------------------------------------------------------------<br />
Now, back to the real-life mathematics problem I set in my previous posting. I was charged £10.99 for the lawn feed, which with my 10% membership discount (£1.10) came to £9.89. But the lawn feed should have been sold at 80% of £10.99, namely £8.79, which with my membership discount would have cost me £7.91. So, the refund should have been £9.89 – £7.91, which is £1.98. I was cheated. The garden centre should have given me a refund of £1.98, but they gave me a refund of only £1.10. They owe me 88p. Next time I go I might consider stealing a daffodil bulb.<br />
<br />
<br />
<br />Derek Haylockhttp://www.blogger.com/profile/10959138785915870717noreply@blogger.com0tag:blogger.com,1999:blog-4426903733734944180.post-43087939800063528262015-08-18T06:42:00.000-07:002015-08-18T06:42:22.806-07:00Discount problemSeeing a 20% reduction in the usual price of £10.99 for a pack of Scotts Lawn Builder in my local garden centre I was overtaken by a moment of untypical enthusiasm to do something about the state of the fading green patch in the middle of our small plot. With an additional 10% discount for having a membership card, I was well on track for a bargain here!<br />
<br />
Unfortunately the 20% reduction had not been entered into the system and my 10% membership reduction was applied to the full price of £10.99. Because I had purchased some other items as well, I did not immediately spot this and paid what they asked.<br />
<br />
When I looked at my receipt, I saw the error and drew it to the attention of the garden centre staff. They checked and agreed that an error had occurred.<br />
<br />
While the queue behind me grew longer and longer the cashier struggled to work out my refund. The problem proved to be beyond the smartness of their computerised system and the mathematical ability of the cashier. In the end they gave me a refund of £1.10, which I accepted meekly, knowing that this could not possibly be right!<br />
<br />
What refund should I have received? More than that? Less than that?<br />
<br />
Ironically, several weeks later the pack remains unopened in my garden shed and the lawn looks as bad as ever.<br />
<br />
The solution will appear on this blog in about a week or so.<br />
<br />
I can't promise when the lawn will be treated.<br />
<br />
<br />
<br />
<br />Derek Haylockhttp://www.blogger.com/profile/10959138785915870717noreply@blogger.com0tag:blogger.com,1999:blog-4426903733734944180.post-42510744764475300652015-07-16T09:17:00.000-07:002015-07-16T09:19:37.843-07:00Australian mathematics curriculum<span class="Apple-style-span" style="font-family: Times, 'Times New Roman', serif;">My apologies to anyone out there who might read this blog from time to time. I have sadly neglected you for the last three months or so, such is the busy nature of my life as a 'retired' academic! I am now back from a wonderful holiday in Provence and able to give some time to writing the occasional post.</span><br />
<span class="Apple-style-span" style="font-family: Times, 'Times New Roman', serif;"><br /></span>
<span class="Apple-style-span" style="font-family: Times, 'Times New Roman', serif;">At my publisher's request I have spent quite a bit of time mapping the Australian primary mathematics curriculum to the content of the 5th edition of <i>Mathematics Explained for Primary Teachers</i>, to help them with marketing the book in that country. I must say I found the curriculum considerably more intelligent in its construction and content than the shoddy and embarrassing curriculum that we now have in place in England. It has quite clearly been written by people who actually know something about mathematics and mathematical pedagogy.</span><br />
<span class="Apple-style-span" style="font-family: Times, 'Times New Roman', serif;"><br /></span>
<span class="Apple-style-span" style="font-family: Times, 'Times New Roman', serif;">I drew on this Australian material when talking about algebra at a conference for primary school teachers in Cambridgeshire a few weeks ago. We noted that the English curriculum does not recognise algebra until Year 6! By contrast the Australian curriculum has an algebra strand all the way through from the start of primary education.</span><br />
<span class="Apple-style-span" style="font-family: Times, 'Times New Roman', serif;"><br /></span>
<span class="Apple-style-span" style="font-family: Times, 'Times New Roman', serif;">For example, all these are given as examples of algebraic reasoning developing
in earlier primary years:</span><br />
<span class="Apple-style-span" style="font-family: Times, 'Times New Roman', serif;"><br />
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<div class="MsoListParagraphCxSpFirst" style="line-height: 150%; margin-bottom: 4.0pt; margin-left: 36.0pt; margin-right: 0cm; margin-top: 0cm; mso-add-space: auto; mso-list: l3 level1 lfo2; mso-outline-level: 1; text-indent: -18.0pt;">
<!--[if !supportLists]--><span class="Apple-style-span" style="font-family: Times, 'Times New Roman', serif;"><span style="font-family: Symbol; font-size: 12.0pt; line-height: 150%; mso-bidi-font-family: Symbol; mso-fareast-font-family: Symbol;"><span style="mso-list: Ignore;">·<span style="font: 7.0pt "Times New Roman";"> </span></span></span><!--[endif]--><span style="line-height: 150%;">Sort and classify familiar objects
and explain the basis for these classifications<o:p></o:p></span></span></div>
<div class="MsoListParagraphCxSpMiddle" style="line-height: 150%; margin-bottom: 4.0pt; margin-left: 36.0pt; margin-right: 0cm; margin-top: 0cm; mso-add-space: auto; mso-list: l3 level1 lfo2; mso-outline-level: 1; text-indent: -18.0pt;">
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with objects and drawings<o:p></o:p></span></span></div>
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missing elements<o:p></o:p></span></span></div>
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<!--[if !supportLists]--><span class="Apple-style-span" style="font-family: Times, 'Times New Roman', serif;"><span style="font-family: Symbol; font-size: 12.0pt; line-height: 150%; mso-bidi-font-family: Symbol; mso-fareast-font-family: Symbol;"><span style="mso-list: Ignore;">·<span style="font: 7.0pt "Times New Roman";"> </span></span></span><!--[endif]--><span style="line-height: 150%;">Solve problems by using number
sentences for addition or subtraction<o:p></o:p></span></span></div>
<div class="MsoListParagraphCxSpMiddle" style="line-height: 150%; margin-bottom: 4.0pt; margin-left: 36.0pt; margin-right: 0cm; margin-top: 0cm; mso-add-space: auto; mso-list: l4 level1 lfo4; mso-outline-level: 1; text-indent: -18.0pt;">
<!--[if !supportLists]--><span class="Apple-style-span" style="font-family: Times, 'Times New Roman', serif;"><span style="font-family: Symbol; font-size: 12.0pt; line-height: 150%; mso-bidi-font-family: Symbol; mso-fareast-font-family: Symbol;"><span style="mso-list: Ignore;">·<span style="font: 7.0pt "Times New Roman";"> </span></span></span><!--[endif]--><span style="line-height: 150%;">Describe, continue, and create number
patterns resulting from performing addition or subtraction<o:p></o:p></span></span></div>
<div class="MsoListParagraphCxSpMiddle" style="line-height: 150%; margin-bottom: 4.0pt; margin-left: 36.0pt; margin-right: 0cm; margin-top: 0cm; mso-add-space: auto; mso-list: l4 level1 lfo4; mso-outline-level: 1; text-indent: -18.0pt;">
<!--[if !supportLists]--><span class="Apple-style-span" style="font-family: Times, 'Times New Roman', serif;"><span style="font-family: Symbol; font-size: 12.0pt; line-height: 150%; mso-bidi-font-family: Symbol; mso-fareast-font-family: Symbol;"><span style="mso-list: Ignore;">·<span style="font: 7.0pt "Times New Roman";"> </span></span></span><!--[endif]--><span style="line-height: 150%;">Explore and describe number patterns
resulting from performing multiplication<o:p></o:p></span></span></div>
<div class="MsoListParagraphCxSpMiddle" style="line-height: 150%; margin-bottom: 4.0pt; margin-left: 36.0pt; margin-right: 0cm; margin-top: 0cm; mso-add-space: auto; mso-list: l4 level1 lfo4; mso-outline-level: 1; text-indent: -18.0pt;">
<!--[if !supportLists]--><span class="Apple-style-span" style="font-family: Times, 'Times New Roman', serif;"><span style="font-family: Symbol; font-size: 12.0pt; line-height: 150%; mso-bidi-font-family: Symbol; mso-fareast-font-family: Symbol;"><span style="mso-list: Ignore;">·<span style="font: 7.0pt "Times New Roman";"> </span></span></span><!--[endif]--><span style="line-height: 150%;">Solve word problems by using number
sentences involving multiplication or division <o:p></o:p></span></span></div>
<div class="MsoListParagraphCxSpMiddle" style="line-height: 150%; margin-bottom: 4.0pt; margin-left: 36.0pt; margin-right: 0cm; margin-top: 0cm; mso-add-space: auto; mso-list: l4 level1 lfo4; mso-outline-level: 1; text-indent: -18.0pt;">
<!--[if !supportLists]--><span class="Apple-style-span" style="font-family: Times, 'Times New Roman', serif;"><span style="font-family: Symbol; font-size: 12.0pt; line-height: 150%; mso-bidi-font-family: Symbol; mso-fareast-font-family: Symbol;"><span style="mso-list: Ignore;">·<span style="font: 7.0pt "Times New Roman";"> </span></span></span><!--[endif]--><span style="line-height: 150%;">Use equivalent number sentences
involving addition and subtraction to find unknown quantities<o:p></o:p></span></span></div>
<div class="MsoListParagraphCxSpMiddle" style="line-height: 150%; margin-bottom: 4.0pt; margin-left: 36.0pt; margin-right: 0cm; margin-top: 0cm; mso-add-space: auto; mso-list: l2 level1 lfo5; mso-outline-level: 1; text-indent: -18.0pt;">
<!--[if !supportLists]--><span class="Apple-style-span" style="font-family: Times, 'Times New Roman', serif;"><span style="font-family: Symbol; font-size: 12.0pt; line-height: 150%; mso-bidi-font-family: Symbol; mso-fareast-font-family: Symbol;"><span style="mso-list: Ignore;">·<span style="font: 7.0pt "Times New Roman";"> </span></span></span><!--[endif]--><span style="line-height: 150%;">Use equivalent number sentences
involving multiplication and division to find unknown quantities<o:p></o:p></span></span></div>
<div class="MsoListParagraphCxSpMiddle" style="line-height: 150%; margin-bottom: 4.0pt; margin-left: 36.0pt; margin-right: 0cm; margin-top: 0cm; mso-add-space: auto; mso-list: l5 level1 lfo6; mso-outline-level: 1; text-indent: -18.0pt;">
<!--[if !supportLists]--><span class="Apple-style-span" style="font-family: Times, 'Times New Roman', serif;"><span style="font-family: Symbol; font-size: 12.0pt; line-height: 150%; mso-bidi-font-family: Symbol; mso-fareast-font-family: Symbol;"><span style="mso-list: Ignore;">·<span style="font: 7.0pt "Times New Roman";"> </span></span></span><!--[endif]--><span style="line-height: 150%;">Continue and create sequences
involving whole numbers, fractions and decimals<o:p></o:p></span></span></div>
<div class="MsoListParagraphCxSpLast" style="line-height: 150%; margin-bottom: 4.0pt; margin-left: 36.0pt; margin-right: 0cm; margin-top: 0cm; mso-add-space: auto; mso-list: l5 level1 lfo6; mso-outline-level: 1; text-indent: -18.0pt;">
<!--[if !supportLists]--><span class="Apple-style-span" style="font-family: Times, 'Times New Roman', serif;"><span style="font-family: Symbol; font-size: 12.0pt; line-height: 150%; mso-bidi-font-family: Symbol; mso-fareast-font-family: Symbol;"><span style="mso-list: Ignore;">·<span style="font: 7.0pt "Times New Roman";"> </span></span></span><!--[endif]--><span style="line-height: 150%;">Describe the rule used to create the
sequence<o:p></o:p></span></span></div>
<!--EndFragment--><span class="Apple-style-span" style="font-family: Times, 'Times New Roman', serif;"><br /></span>
<span class="Apple-style-span" style="font-family: Times, 'Times New Roman', serif;">My contention has always been that the most fundamental component of mathematics education is not doing harder and harder calculations (the message in the English curriculum) but algebraic thinking and reasoning. This is where real mathematics emerges, powerful, widely applicable and creative. So, a proper mathematics curriculum must recognise and seek to develop algebraic reasoning right through the primary years; and not see it just as an add-on in Year 6. So, well done, Australia!</span><br />
<span class="Apple-style-span" style="font-family: Times, 'Times New Roman', serif;"><br /></span>
<span class="Apple-style-span" style="font-family: Times, 'Times New Roman', serif;">But I still hope you lose the Ashes.</span><br />
<span class="Apple-style-span" style="font-family: Times, 'Times New Roman', serif;"><br /></span>
<span class="Apple-style-span" style="font-family: Times, 'Times New Roman', serif;"><br /></span>
<br />Derek Haylockhttp://www.blogger.com/profile/10959138785915870717noreply@blogger.com2tag:blogger.com,1999:blog-4426903733734944180.post-59631935626278757992015-04-22T09:48:00.002-07:002015-04-22T09:48:28.219-07:00Solution for Cheryl's birthdayIt is 16 July! Below is the explanation of this very pleasing problem.<br />
<br />
Cheryl has told A and B that her birthday is one of these ten dates:<br />
<i><br /></i>
<i>May 15, 16, 19</i><br />
<br />
<div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;">
<i>June 17, 18</i></div>
<div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;">
<i>July 14, 16</i></div>
<div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;">
<i>August 14, 15, 17</i></div>
<div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;">
<i><br /></i></div>
<div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;">
She has told A the month and B the number of the day, and they both know this.</div>
<div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;">
<br /></div>
<div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;">
<i>A says, 'I do not know C's birthday.</i></div>
<div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;">
<i><br /></i></div>
<div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;">
This tells us nothing, because whatever month he had been told A would not be able yet to deduce the birthday.</div>
<div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;">
<i><br /></i></div>
<div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;">
But B <u>would</u> know C's birthday if he had been told that the day was, say, 19: because 19 occurs in the list of options only in May. Likewise B would know the birthday if had been told that the day was 18: because this occurs only in June.</div>
<div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;">
<br /></div>
<div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;">
So, then A says:</div>
<div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;">
<br /></div>
<div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;">
<i></i></div>
<div style="font-style: normal; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;">
<i><i>'... but I know for sure that B does not know either.'</i></i></div>
<div style="font-style: normal; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;">
<i><i><br /></i></i></div>
<br />
<div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;">
A would not be able to say this, if the month he had been told was May or June, because for each of these two months there is the possibility that B actually knows the birthday already.</div>
<div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;">
<br /></div>
<div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;">
From this B (and we) can deduce that the month of C's birthday is neither May nor June.</div>
<div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;">
<br /></div>
<div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;">
B replies: </div>
<div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;">
<i><br /></i></div>
<div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;">
<i>'At first I did not know C's birthday, but I do now.'</i></div>
<div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;">
<i><br /></i></div>
<div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;">
This means that the number of the day (which B knows) must occur in either July or August but NOT in both these months. For example, if B knew the day was 17 then he would know it had to be August 17. Likewise, B would know the birthday if the day was 16 (July) or 15 (July).</div>
<div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;">
<br /></div>
<div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;">
A has deduced all this as well! He knows that the day is 15, 16 or 17. But A also knows the month. And he then replies:</div>
<div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;">
<br /></div>
<div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;">
<i>'Now I know as well!</i></div>
<div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;">
<i><br /></i></div>
<div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;">
Now A could only know the birthday if the month (which he knows) has only one of these three options in the list: 15, 16 or 17. That month is July.</div>
<div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;">
<br /></div>
<div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;">
This is how we (and B presumably now) know that Cheryl's birthday must be 16 July.</div>
<div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;">
<i></i></div>
<div style="font-style: normal; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;">
<i><i><br /></i></i></div>
<div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;">
<i><br /></i></div>
<br />
Derek Haylockhttp://www.blogger.com/profile/10959138785915870717noreply@blogger.com0tag:blogger.com,1999:blog-4426903733734944180.post-19111381647631477652015-04-21T08:32:00.000-07:002015-04-21T08:32:03.436-07:00Solution to the hats logic problemHere is the solution to the easier logic problem in my previous post:<br />
<br />
<span class="Apple-style-span" style="color: #333333; font-family: Arial; font-size: 13px; line-height: 20px;"><i>A, B and C are told that in a bag there are 3 red hats and 2 blue hats. B is blindfold, so is unable to see anything. One hat is put on each person's head and they have to work out what colour hat they are wearing. A and C can see the hats on the other two, but none of them can see their own hat.</i></span><br />
<span class="Apple-style-span" style="color: #333333; font-family: Arial; font-size: 13px; line-height: 20px;"><i>A says: I do not know what colour my hat is.</i></span><span class="Apple-style-span" style="color: #333333; font-family: Arial; font-size: 13px; line-height: 20px;"><br /></span><span class="Apple-style-span" style="color: #333333; font-family: Arial; font-size: 13px; line-height: 20px;"><i>C says: Nor do I.</i></span><span class="Apple-style-span" style="color: #333333; font-family: Arial; font-size: 13px; line-height: 20px;"><br /></span><span class="Apple-style-span" style="color: #333333; font-family: Arial; font-size: 13px; line-height: 20px;"><i>B says: Then I am wearing a red hat.</i></span><span class="Apple-style-span" style="color: #333333; font-family: Arial; font-size: 13px; line-height: 20px;"><br /></span><span class="Apple-style-span" style="color: #333333; font-family: Arial; font-size: 13px; line-height: 20px;"><i>How did B work that out?</i></span><span class="Apple-style-span" style="color: #333333; font-family: Arial; font-size: 13px; line-height: 20px;"><br /></span><br />
<br />
B considers the possibility that the hat on her head is blue.<br />
Assuming this, when A says, I do not know what colour hat I am wearing, it follows that C must be wearing a red hat: because if C were wearing a blue hat, A would see two blue hats and know that her own hat would have to be red.<br />
But C would also work this out! So C would be able to deduce that she is wearing a red hat.<br />
But, even after A has spoken, C does not know what colour her hat is.<br />
So B deduces that she cannot be wearing a blue hat.<br />
Hence she knows that she is wearing a red hat.<br />
<br />
Cheryl's birthday to be disclosed in my next post ...<br />
<br />Derek Haylockhttp://www.blogger.com/profile/10959138785915870717noreply@blogger.com0tag:blogger.com,1999:blog-4426903733734944180.post-41833764230571077192015-04-18T12:01:00.001-07:002015-04-18T12:01:15.969-07:00Cheryl's birthdayI'll give my solution to this logical reasoning problem in my next post, so anyone reading this has the chance to solve it themselves first. This is the problem from a Singapore mathematics test that has apparently 'gone viral'. It was constructed by Dr Joseph Yeo Boon Wool, a mathematics professor at the Singapore National Institute of Education.<br />
<br />
So here is the actual problem.<br />
<br />
<i>Albert and Bernard want to know the birthday of their new friend Cheryl. She tells them that it is one of the following ten options:</i><br />
<i><br /></i>
<i>May 15, 16, 19</i><br />
<i>June 17, 18</i><br />
<i>July 14, 16</i><br />
<i>August 14, 15, 17</i><br />
<i><br /></i>
<i>She whispers in A's ear the month of her birthday and tells B that she has done this.</i><br />
<i>She then whispers in B's ear the day of her birthday and tells A that she has done this.</i><br />
<i><br /></i>
<i>Then A says, 'I do not know C's birthday; but I know for sure that B does not know either.'</i><br />
<i>B replies: 'At first I did not know C's birthday, but I do now.'</i><br />
<i>A replies: 'Now I know as well!'</i><br />
<br />
This is an excellent example of a logical reasoning problem that involves making deductions from what people say about what they know or do not know. These puzzles always assume that all the people involved have high powers of deductive reasoning, so you can assume that if something can be deduced they will deduce it!<br />
<br />
Here is another example, much easier than finding Cheryl's birthday!<br />
<br />
<i>A, B and C are told that in a bag there are 3 red hats and 2 blue hats. B is blindfold, so is unable to see anything. One hat is put on each person's head and they have to work out what colour hat they are wearing. A and C can see the hats on the other two, but none of them can see their own hat.</i><br />
<i>A says: I do not know what colour my hat is.</i><br />
<i>C says: Nor do I.</i><br />
<i>B says: Then I am wearing a red hat.</i><br />
<i>How did B work that out?</i><br />
<i><br /></i>
Solutions in my next post.<br />
<i><br /></i>Derek Haylockhttp://www.blogger.com/profile/10959138785915870717noreply@blogger.com0tag:blogger.com,1999:blog-4426903733734944180.post-33695268543132494472015-03-27T13:45:00.000-07:002015-03-27T13:45:20.500-07:00How old are you?In our church we have a spot in our Sunday morning services where we invite children who have had a birthday that week to come to the front to receive a small gift. 'When was your birthday' said the pastor to young William. 'On Friday' he replied. 'And how old are you?' 'Four and a half', replied William!<br />
<br />
Very interesting that reply, isn't it? On Friday he was four. But now he is more than four. Hence the 'four and a half'!<br />
<br />
When we teach children simple fractions, like halves and quarters, we tend to major on the idea of a fraction as one or more equal portions of a whole unit. So we use images like a pizza or a rectangle cut up into a number of equal parts. But William's response reminds us that children's early experience of fractions also includes the idea of a fraction describing a point on a number line, a point 'lying between' one integer and the next. So 'four and a half' means a point on a time line 'somewhere between four and five'.<br />
<br />
I suggest that we would do well to make much more of representing fractions as points on number lines, especially if we are teaching children about mixed numbers.<br />
<br />
Thanks, William. And happy four and halfth birthday!<br />
<br />
<br />Derek Haylockhttp://www.blogger.com/profile/10959138785915870717noreply@blogger.com0tag:blogger.com,1999:blog-4426903733734944180.post-81201299190364141362015-03-05T07:48:00.000-08:002015-03-07T04:46:15.371-08:00Yet another error in new Mathematics Primary Curriculum: 'irregular'Call me pedantic if you wish, but I do think that mathematical terms used in the Mathematics National Curriculum should be used correctly. The misuse of the word 'irregular' is my latest find. This occurs in the measurement section of the Year 5 Mathematics Curriculum programme of study:<br />
<br />
<i>'Calculate and compare the area of rectangles (including squares) ... and estimate the area of irregular shapes.'</i><br />
<i><br /></i>
This statement seems to imply that a non-square rectangle is a 'regular' shape. The word 'irregular' is used here for two-dimensional shapes where you cannot calculate the area exactly using a formula, such as the outline of an island or that of a fried egg. These are irregular, but so are all rectangles that are not squares, all triangles that are not equilateral, and so on.<br />
<br />
So, let's tidy this up. A regular two-dimensional shape is one where all the sides are equal in length and all the internal angles are equal. So, the only quadrilaterals that are regular are squares. All others are irregular, including non-square rectangles.<br />
<br />
All these shapes, for example, are <u>irregular</u> polygons:<br />
<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="http://1.bp.blogspot.com/-134AFGhUPls/VPrx75AHVKI/AAAAAAAAAKc/uwPWTWGsCa4/s1600/irregular%2Bshapes.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://1.bp.blogspot.com/-134AFGhUPls/VPrx75AHVKI/AAAAAAAAAKc/uwPWTWGsCa4/s1600/irregular%2Bshapes.jpg" height="154" width="320" /></a></div>
<br />
The National Curriculum needs another way of describing the other kinds of 2-D shapes that it has in mind for which Year 5 children should learn to estimate the area. I usually call them non-standard shapes or non-geometric shapes.Derek Haylockhttp://www.blogger.com/profile/10959138785915870717noreply@blogger.com0tag:blogger.com,1999:blog-4426903733734944180.post-40022989930377272862015-02-19T07:08:00.000-08:002015-02-19T07:08:11.473-08:00A further thought on composite numbers definitionGoing back to the erroneous 'definition' of composite numbers in the Year 5 Programme of Study for the Mathematics National Curriculum for England ... where we read that pupils should learn about 'composite (non-prime) numbers'.<br />
<br />
In my previous post I pointed out that you cannot use 'non-prime' as a synonym for 'composite' because the integer 1 is neither prime nor composite.<br />
<br />
On reflection, I realised that the error here is much more substantial than this. The concept of 'prime' applies only to positive integers. So, a prime number can be defined as an <i>integer</i> with precisely two factors (which will be 1 and itself).<br />
<br />
This means that 'non-prime numbers' would include all numbers that are not positive integers. So, the identification of composite numbers with non-prime numbers would imply that numbers such as –3, 2.4, ⅚ and √2, for example, are all composite!<br />
<br />
Of course, they are not! 'Composite' is also a concept that applies only to positive integers – those integers with two or more factors greater than 1.<br />
<br />
That was a sloppy bit of work by whoever wrote that programme of study.<br />
<br />
---------------------------------------------------------------------------------------------<br />
<span class="Apple-style-span" style="font-family: Arial, Helvetica, sans-serif; font-size: x-small;"><i><br /></i></span>
<span class="Apple-style-span" style="font-family: Arial, Helvetica, sans-serif; font-size: x-small;"><i>Any reader looking for professional help in the area of primary mathematics education might like to take a look at the recently-launched website of my colleague, Ralph Manning: <a href="http://www.manningeducation.co.uk/" style="color: #0563c1; text-decoration: underline;">www.manningeducation.co.uk</a></i></span><br />
<span class="Apple-style-span" style="font-family: Arial, Helvetica, sans-serif; font-size: x-small;"><i><br /></i></span>
<br />Derek Haylockhttp://www.blogger.com/profile/10959138785915870717noreply@blogger.com0tag:blogger.com,1999:blog-4426903733734944180.post-48126887102654460242015-02-13T03:21:00.000-08:002015-02-13T03:21:14.924-08:00Error in Primary Mathematics National Curriculum<span class="Apple-style-span" style="font-family: Times, 'Times New Roman', serif;">Come with me to the Year 5 Mathematics Programme of Study for England, where we read that pupils should be taught to ...</span><br />
<span class="Apple-style-span" style="font-family: Times, 'Times New Roman', serif;"><br /></span>
<span class="Apple-style-span" style="font-family: Times, 'Times New Roman', serif;">"know and use the vocabulary of prime numbers, prime factors and composite (non- prime) numbers"</span><br />
<span class="Apple-style-span" style="font-family: Times, 'Times New Roman', serif;"><br /></span>
<span class="Apple-style-span" style="font-family: Times, 'Times New Roman', serif;">This was clearly written by someone who doesn't!</span><br />
<span class="Apple-style-span" style="font-family: Times, 'Times New Roman', serif;"><br /></span>
<span class="Apple-style-span" style="font-family: Times, 'Times New Roman', serif;">'Composite number' is not synonymous with 'non-prime number'.</span><br />
<span class="Apple-style-span" style="font-family: Times, 'Times New Roman', serif;"><br /></span>
<span class="Apple-style-span" style="font-family: Times, 'Times New Roman', serif;">A composite number is a positive integer with two or more factors greater than 1.</span><br />
<span class="Apple-style-span" style="font-family: Times, 'Times New Roman', serif;"><br /></span>
<span class="Apple-style-span" style="font-family: Times, 'Times New Roman', serif;">So, composite numbers are: 4, 6, 8, 9, 10, 12, ...</span><br />
<span class="Apple-style-span" style="font-family: Times, 'Times New Roman', serif;"><br /></span>
<span class="Apple-style-span" style="font-family: Times, 'Times New Roman', serif;">The error in the National Curriculum is to overlook the fact that 1 is NOT a composite number. Of course, it is not a prime number either. So, 1 is the only integer that is neither prime nor composite.</span><br />
<span class="Apple-style-span" style="font-family: Times, 'Times New Roman', serif;"><br /></span>
<span class="Apple-style-span" style="font-family: Times, 'Times New Roman', serif;">OK, so it is only a small error. But it is still embarrassing to have a mathematical error in our National Curriculum: to add to all the pedagogical errors.</span><br />
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<span class="Apple-style-span" style="font-family: Times, 'Times New Roman', serif;"><br /></span>Derek Haylockhttp://www.blogger.com/profile/10959138785915870717noreply@blogger.com0tag:blogger.com,1999:blog-4426903733734944180.post-70289255987519526262014-11-18T02:38:00.003-08:002014-11-18T02:39:16.369-08:00Outrageous Performance Descriptor Maths KS1The Department for Education in England is engaged at present in a consultation about their proposed performance descriptors for children in Key Stages 1 and 2 in relation to the new primary curriculum.<br />
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I have just been looking at the mathematics descriptors for Key Stage 1.<br />
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The proposal includes the requirement that children who are to be judged as working 'at the national standard' at the end of Year 2 will be able to ...<br />
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<i> add and subtract numbers using ... the written columnar methods ...</i><br />
<i><br /></i>
They can't be allowed to get away with that! Columnar methods for addition and subtraction are NOT in the curriculum for Key Stage 1 mathematics. They are mentioned in the non-statutory guidance for Year 2, but they are quite clearly not required in the actual curriculum.<br />
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Non-statutory guidance is non-statutory!<br />
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But these proposed performance indicators will be statutory. Key Stage 1 teachers will be requried to assess pupils against these criteria. So, <i>de facto</i>, they will become the key focus of the curriculum.<br />
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This is an outrage. We must protest strongly at what looks like a deliberate attempt to give non-statutory advice about formal traditional written calculation methods the same status as what is in the actual curriculum.<br />
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<br />Derek Haylockhttp://www.blogger.com/profile/10959138785915870717noreply@blogger.com1tag:blogger.com,1999:blog-4426903733734944180.post-5722210404651662542014-11-16T07:13:00.000-08:002014-11-17T01:58:41.575-08:00It's a long way to 67P/Churyumov–Gerasimenko<span class="Apple-style-span" style="font-family: Times, 'Times New Roman', serif;">As readers of this blog will know, I enjoy trying to find ways of getting our heads around large numbers and huge quantities. In my experience children in primary schools also find large numbers fascinating, and, with the help of a calculator and some approximate mental calculations, we can play with big numbers and do some interesting mathematics.</span><br />
<span class="Apple-style-span" style="font-family: Times, 'Times New Roman', serif;"><br /></span>
<span class="Apple-style-span" style="font-family: Times, 'Times New Roman', serif;">This last week saw the landing of 'Philae' on Comet 67P/<span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px; line-height: 16px;">Churyumov–Gerasimenko. The newspaper reports told us that this comet was 300 million miles away. That's a long way, I'm sure, but how can we understand a distance like this? I always try to connect these big numbers with our own experience.</span></span><br />
<span class="Apple-style-span" style="font-family: Times, 'Times New Roman', serif;"><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px; line-height: 16px;"><br /></span>
</span><br />
<span class="Apple-style-span" style="font-family: Times, 'Times New Roman', serif;"><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px; line-height: 16px;">So, let's imagine I start driving my car at an average speed of 45 mile per hour. How long is it going to take me to clock up 300 million miles, assuming I have a co-driver and we manage to drive night and day non-stop? </span></span><br />
<span class="Apple-style-span" style="font-family: Times, 'Times New Roman', serif;"><br /><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px; line-height: 16px;">Well, let's see 45 mph is 45 × 24 miles per day, which is 1080 miles per day.</span></span><br />
<span class="Apple-style-span" style="font-family: Times, 'Times New Roman', serif;"><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px; line-height: 16px;"><br /></span></span>
<span class="Apple-style-span" style="font-family: Times, 'Times New Roman', serif;">That's about 1080 × 365.25 miles per year, which looks like approximately 400,000 miles per year.</span><br />
<span class="Apple-style-span" style="font-family: Times, 'Times New Roman', serif;"><br /><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px; line-height: 16px;">So to find how many years it will take me ... careful with all these zeros ... we need 300,000,000 divided by 400,000, which is 750 years!</span></span><br />
<span class="Apple-style-span" style="font-family: Times, 'Times New Roman', serif;"><br /><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px; line-height: 16px;">So, put it like this: imagine I had started this epic drive in 1264, back in the Middle Ages, during the reign of Henry III, in the middle of the Crusades, when Marco Polo, that intrepid traveller, was 10 years old; I would have continued on driving through all the subsequent Henrys, through the reigns of the Tudors and Stewarts, and the Civil War; I would have still been driving when Beethoven was born and when he died; and when Queen Victoria came to the throne and when she died; and still driving during the two World wars, mysteriously passing the day of my own birth, and right through to the present day, and I would be now just closing in on the 300 million miles target!!</span></span><br />
<span class="Apple-style-span" style="font-family: Times, 'Times New Roman', serif;"><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px; line-height: 16px;">Yes, it's a long way to </span>Comet 67P/<span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px; line-height: 16px;">Churyumov–Gerasimenko!</span></span><span class="Apple-style-span" style="font-family: Times, 'Times New Roman', serif;"><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px; line-height: 16px;"><br /></span></span><br />
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<span class="Apple-style-span" style="font-family: Times, 'Times New Roman', serif;"><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px; line-height: 16px;"><br /></span></span>
<span class="Apple-style-span" style="font-family: Times, 'Times New Roman', serif;"><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px; line-height: 16px;"><br /></span></span>
<span class="Apple-style-span" style="font-family: Times, 'Times New Roman', serif;"><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px; line-height: 16px;"><br /></span></span>
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<span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px; line-height: 16px;"><span class="Apple-style-span" style="font-family: Times, 'Times New Roman', serif;"><br /></span></span>Derek Haylockhttp://www.blogger.com/profile/10959138785915870717noreply@blogger.com0tag:blogger.com,1999:blog-4426903733734944180.post-20905378762307704302014-11-07T04:31:00.000-08:002015-02-13T03:27:34.774-08:00Rounding to the nearestI occasionally find myself having a disagreement with teachers and others involved in assessment who want to set children a test question in which to get the mark you have to assume that, for example, 148.5 rounded to the nearest whole number is 149. My argument is that in this situation there is no nearest whole number. The number 148.5 is as near to 148 as it is to 149. The only way you could decide which to round it to (if at all) would be to look at the context that gave rise to the number 148.5.<br />
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There is a myth that the convention is that if it ends in a 5 you round it up. I won't go along with this, because it is unnecessary and unjustified. Here's an example that illustrates my point.<br />
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I have a discount card from Waitrose that allows me 10% off any purchase. I buy a packet of biscuits that costs £1.65. For convenience, I'll write this in pence, as 165p. So ...<br />
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<i>original price = 165p</i><br />
<i>discount = 16.5p</i><br />
<i>reduced price = 148.5p.</i><br />
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So let's assume we are to round the results to the nearest penny, because we can only deal in whole numbers of pence in Waitrose. If we use the rule of 'rounding up when it ends in a 5', we get:<br />
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<i>original price = 165p</i><br />
<i>discount = 17p (to the nearest penny)</i><br />
<i>reduced price = 149p (to the nearest penny).</i><br />
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This is plainly impossible! A discount of 17p gives a reduced price of 148p.<br />
So if the discount is rounded up, the reduced price has to be rounded down; or vice versa. So, assuming the generosity of Waitrose, I would expect:<br />
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<div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;">
<i>original price = 165p</i></div>
<div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;">
<i>discount = 17p (rounded up)</i></div>
<div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;">
<i>reduced price = 148p (rounded down).</i></div>
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This is why I refuse to accept the so-called convention! Context is everything.<br />
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I suggest that we just do not set a context-free maths assessment question about rounding a number ending in a 5 to the nearest something, if in fact there is no nearest something.<br />
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Bad assessment questions (but good for class discussion):<br />
1. Round 6.05 to 1 decimal place.<br />
2. Round 125 to the nearest ten.<br />
3. Round 3500 g to the nearest kilogram.<br />
4. Round 3 minutes 48.65 seconds to the nearest tenth of a second.<br />
<br />Derek Haylockhttp://www.blogger.com/profile/10959138785915870717noreply@blogger.com0