Mathematical debates, and Bounds

It’s possible that what I am about to explain is already completely obvious to many a maths teacher.  But after 5 years of teaching, it’s only just dawned on me (a recurrent theme of this blog by the way!) so I thought I’d share.

I like teaching the concept of Bounds.  It highlights a fundamental difference between concepts that exist in Maths and the real world.  We measure things using numbers as continuous variables.  When we measure the length of a table, say, as being 120cm, we are rounding to the nearest centimetre.  We could be more accurate and say it is 120.4cm or even 120.4187234cm but this is still an approximation.   It reminds me of my engineering days, looking at milling machines that were capable of cutting to the nearest 0.1mm. I used to wonder about how those milling machines were made and the tolerances required in the dimensions of their components.  What machines were used to make the components? And what about the machines that were used to make those machines? Where does it stop?!

But one of the stumbling blocks with bounds is always with the Upper bound.

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“But you told us that 5 rounds up!” is the usual complaint.  And then we get into a heated arguement about whether:

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Which of course it is, for the same reason:

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(these images from Don Steward can be useful for this one).
Sometimes that argument can be fun, but I think that as teachers we need to beware of going down rabbit holes. The more inquisitive students might find this philosophical debate stimulating but it can be a turn-off for others. And even the ones that do actively engage in that discussion may not be convinced at the end of it.  It’s just another one of those decisions that we make in the moment as we read the mood in the class.  In making that decision we should ensure that we are taking into account the feelings and needs of all students, not just the more vociferous ones.
So, to the point of this post, how to move this debate on.

Earlier this week, I was marking this GCSE question, which most of my students, including those sitting Foundation were answering no problem.

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I’m pretty sure that I have previously suggested that students state the “Lower Bound” and the “Upper Bound” almost as though they are two separate answers.  But expressing the range of possible values in this way makes more sense. It is also entirely consistent with the well-understood rule that “5 rounds up”. In the question above, if the actual length was 13.5, it would be rounded up to 14, but a value below 13.5 would not.  Whereas a measured value of 12.5 would be recorded as 13.

All makes perfect sense now!

 

 

Comparing Fractions

There is something very simple about a task which presents two numbers and simply asks “which is bigger?”.  This should be done using mathematical notation, i.e. using the < > symbols. I have seen these being introduced successfully in Year 1 without any mention of crocodiles, or such similar unhelpful “stories”.  But my Year 7 class still insist on calling them crocodiles and drawing teeth on them.  But hey, I have bigger battles to fight…

As well as comparing 2 fractions we can put multiple fractions into order from smallest to largest. There is a significant range of difficulty in this apparently simple task.

  1. Comparing fractions of the same denominator
  2. Unitary fractions with different denominators
  3. Same numerator, different denominators
  4. Different denominators where one is a multiple of another.
  5. Different denominators where a common denominator needs to be found for both fractions.

Alongside all of these there may also be strategies where learners are using known facts or doing calculations to convert to decimals or percentages, e.g. 1/2=0.5, 2/5=0.4, therefore 1/2 > 2/5.  That is not the intention of this task (it is of a different task here) but in the end we want learners to be able to play with all these ideas and I can’t really control, nor would I want to control, the order in which they coalesce in students’ mind.

Here is a simple set of cards that I used recently.  I got the students to do the last bit of cutting to turn each strip into the 3 separate cards.  I also told them that there is deliberately some space alongside the fraction to enable them to write equivalent fractions if they needed to.

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I gave them out a strip at a time, the idea being that they were to “slot in” the subsequent fractions to maintain the order.  The fractions are carefully chosen, so that each time they get a new strip they are having to apply the next level of reasoning.  The first set are simple but this can end up quite challenging especially if they chose their own more “exotic” fractions.

It can be a bit of a hassle preparing and managing card sort exercises in the classroom.  Whenever I see a resource that is created as a card sorts, I always think, could students get the same benefit by just writing in their books. But for some tasks such as this, I think it is worth it as it enables a richer discussion and the possibility for learners to easily changing their mind as they are building understanding.

 

Zoomable Number Line

This little gadget on Mathisfun.com is so illuminating for teaching decimals. I always get a positive response from students when I show it.

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I think it can be used it in a number of ways:

  1. Predict what happens when I zoom in. As you zoom in, first the tick marks appear and then at the next level of zoom, numbers start appearing.  It’s a great way of “bridging” from the familiar to the new.  A number line is likely to be a familiar concept.  However, what is “in between” 0 and 1? An intentionally ambiguous question. Students are likely to say “half” or 0.5.  How else can we show half? What other numbers are in between 0 and 1?
  2. Why are there 10 ticks between 0 and 1? We have divided 1 into 10 equal parts, what is each part.  How else can we represent 1/10?

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From there we “zoom” into the next level of hundredths

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This table might be useful as some practice to relate fractions to decimals.  I would love to hear some comments on this.

capture

 

 

 

 

When is a Quadratic “factorisable”?

There are 3 standard ways of solving quadratic equations once they are in the form:

ax² + bx + c = 0

They are:

  1. Factorise
  2. Complete the square
  3. Use the formula

I think I generally teach them in that order probably without much thought as to why. I guess the formula needs to be derived by using completing the square and factorising seems to follow on from multiplying out double brackets, which comes before all of this.  The I question that I sometimes get from students is “what’s the point in learning factorising if the two other methods always work?”.  Well, it’s quicker and you can do it on a non-calc exam is probably a standard response.

But have you tried using the formula without a calculator to solve a quadratic that you know will factorise?  Have a go.  Plug this:

x² - 3x - 28 = 0

into this:

download

…and solve without a calculator.

It’s a surprisingly satisfying experience, one that I would not want to deny my students.

You’ll need to know your square numbers because b² – 4ac will always give you a square number for quadratics that factorise. But the arithmetic is perfectly reasonable and is likely to be so for most quadratics that can be factorised.

When I did this recently, I had a great question from one of my students, as I was aching my brain trying to make up a quadratic that I knew would factorise.   “If you just picked one randomly, what are the chances that you would be able to factorise it?”

I’ve since had chance to investigate this further.  It’s a great question and there is a lesson in here, or at least an extension question to explore once the fundamentals of using the formula are secure.

I started approaching it by using the formula and focussing on b² – 4ac and what values of a, b and c would yield square numbers.  To simplify the problem, I started with a=1, so I was looking for when b² – 4c = 1,4,9,16,25, etc.

I then looked at it from the other end, i.e. starting with e.g. (x+1)(x+n) what values of a, b, and c are yielded.  Then vary further to look at (x+m)(x+n).  I just started with a few values of n and m to see if I could spot patterns.  I won’t spoil the fun by revealing those patterns, but this is very open-ended and could provide some intrigue for the right learners.

 

Concentric Equilateral Triangles

The red equilateral triangle side length 4cm sits inside the larger pink equilateral triangle such that the “border” is 1cm wide.

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What is the ratio of the height of the red triangle to the height of the pink triangle?

Can you solve using trigonometry or only using Pythagoras?

The border is now 2cm, whilst the side length of the red triangle remains 4cm.  What is the ratio of heights now?

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Explore what happens for other border widths.  Can you generalise for any border width w?

Geogebra file here.

Spoiler here.

I got a few solutions posted on Twitter for this, but the most elegant (so far…) has to be this from @mathforge. As he said: no Trig, no Pythagoras, just ratio.

C9pI4kaWsAATi1m

 

 

Recurring Decimals

First up, this post is NOT going to be about this:


This is what came to mind when joining Derek Ball’s session at the ATM conference this week, entitled “Recurring Decimals”.

In fact, we were going the other way: converting fractions to recurring decimals, specifically looking at fractions where the denominator is prime.  It was a fascinating session, great for deepening subject knowledge.  This blog post is my attempt to reflect on what I learned and my thoughts about how I might use this in the classroom, probably Year 10 or 11, but really any group that is confident with bus stop division could investigate this.

I was already aware of some pretty cool things that happen with sevenths, mainly from Don Steward’s blog:

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Why should this happen? Why do we only see these six digits with sevenths? The process of bus stop division helps us see why and this is where I feel I would start with a class. This is good practice of a technique that should be secure but often isn’t in Year 9 / 10.  It helps learn the seven times table and I don’t think it is too tedious to ask students to perform these six calculations manually:

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Some learners might want to find 2/7 by doubling 1/7.  And then maybe find 3/7 by adding 1/7 to 2/7 and so on.  Even if they stick with the bus stop division, they will find that they are essentially doing the same six calculations:

10 ÷ 7 = 1 rem 3
20 ÷ 7 = 2 rem 6
30 ÷ 7 = 4 rem 2
40 ÷ 7 = 5 rem 5
50 ÷ 7 = 7 rem 1
60 ÷ 7 = 8 rem 4

The ones digit is always zero and the tens digit must be less than 7 so there are only these 6 options. Can we extend this rule to other fractions with denominator less than 10?  Of course, all others except 3, 6 and 9 will terminate – why is this?

There are also some interesting things to notice about the 1/7 “wheel” before moving onto higher primes.  I won’t spoil it for you, but suggest you include the fractions around the outside of the wheel to spot some of the patterns.

We then moved onto looking at 13ths.  11th are interesting too for different reasons, so I can see why we went onto 13th because there are some surprising relationships between 7th and 13ths. At this point, we started using calculators and I would do the same with a class. Or even better, open a spreadsheet, which I am always keen to do!

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I’m not sure of the value of kids typing 12 calculations into their calculators, so I might give this image above as a print out for them to write on. Hopefully they will soon spot that there are two sets of recurring digits, i.e. 076923 in 1/13 + 5 others and 153846 in 2/13 + 5 others and which, this time can be written as two wheels:

Ideally at this point you’ll have the class hooked and they would be asking all sorts of questions.  Well, maybe enough of them to get everyone else thinking.  I would try really hard to encourage the students to come up with these questions.  This is what I hope they will ask, but I hope they will also ask questions I hadn’t thought of, something which is a really special moment in any lesson.

  • Do we see the same patterns in the sevenths wheel as we do in the thirteenths wheel?
  • Is there something special about 6 points around the wheel?
  • Why are there 2 wheels? Are there any patterns across the two wheels?
  • What about other fractions, do they have multiple wheels?

On the image above, I have shown that as you move clockwise around the wheel, you are effectively multiplying by 10.  This is obvious going from 1/13 to 10/13 but actually occurs for all other hops if we ignore the whole number part.  i.e.

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So if go 3 hops we have multiplied by 1000.  I think this goes some way to explaining why the fractions that are opposite each other must sum to 1.

The next prime fraction: 17ths recur after its maximum of 16 digits, so effectively we have one, rather large wheel. Unfortunately, Excel gives up after 16 decimal places (please let me know if there is a way around this) but you can still see some patterns in here:

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Beyond that, we found that the following fractions recur after the maximum number of digits so have only one wheel: 3,7,17,19,23,43,59,61 – someone had a much better calculator than me that showed 32 digits!

Other fractions worth exploring are:

  • 31ths – contain 2 wheels of 15
  • 37ths – contain 12 wheels of 3
  • 41th – contain 8 wheels of 5

With the fractions that recur after a relatively few numbers of digits, we can find factors.

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Because 1/41 is a decimal that recurs after only 5 digits, if follows that 41 must be a factor of 99999.

And in the larger wheels there are all sorts of patterns – pentagons and triangles in the wheel of 31ths apparently.

I’m not sure how far I’d go with a class, there could be several lessons worth in here. It would depend on how they responded, of course.  Maybe 13ths would be sufficient unless…

This post captures only some of what we worked on in this session and highlights to me the depth of subject knowledge that can be gained by attending the ATM conference. Other sessions were more pedagogical in nature, but this one was pure fascinating mathematics and I was grateful to be surrounded by so many knowledgable and friendly people!

 

Ideas for better maths teaching