Category Archives: #mixedattainmentmaths

Generalising Surface area

What is the surface area of a cube of side length 1?
If we then cut this cube in half, and throw one of the halves away, what is surface area of the remaining cuboid?
Repeat the process, cut the shape in half along the same plane. What pattern can you see?

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What is the general formula for the surface of a cuboid of width 1, depth 1 and height h?

What is the general formula for a cuboid of width 1, depth d and height h?

What is the general formula for a cuboid of width w, depth d and height h?

What other 3D shapes can you find the general formula for the surface area? Try:

  • A tetrahedron, side length a
  • A square based pyramid, base length a, height, h
  • A cylinder radius r, length l

 

Trigonometry, another way

Trigonometry falls firmly into the camp of one of those areas which I don’t feel I have cracked yet.  If “cracked” means finding a bomb-proof way to introduce it to any class I encounter, then maybe I never will!  Either way, I am always interested in different approaches to teach this topic which many students seem to struggle with at first.

This approach builds on the presentation that Mike Ollerton gave at the recent Mixed Attainment Maths conference in Sheffield (keep an eye on the site for details of the next conference in November!)

I didn’t actually attend Mike’s presentation – I was too busy giving my own – but Mike has kindly shared his ideas and I have been thinking about how I might use Geogebra as a tool to aid learning.

As with most of my use of Geogebra, I am using it as an exposition tool to structure whole-class questioning and discussion around. In an ideal world, I might get students to do this themselves, but that is not practical in my classroom. I feel that starting Geogebra from a blank sheet can be nearly as powerful as them doing it themselves and is likely to be a much more efficient use of lesson time.

The basic principle used is that of rotating a fixed line segment, a “spinner” if you like, around a point.  We are aiming to explore the co-ordinates of the point at the end of the line as the angle increases from 0º to 90º (and beyond) in a table.

I must say that from this point onwards this post is not Mike’s recommended approach (which is here) – but my interpretation of it using Geogebra.

So, first step is to form the spinner by by plotting points at (0,0) and (1,0), zooming in and connecting the points with a line segment. There is something in observing what happens to the scale on the axes as we zoom in.

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Next, we create the angle by using the Angle with Given Size tool. As the prompt says when you hover, the tool: select leg point, then vertex.  Rather than fix the angle, I want to make a slider so I can easily change it. I set the angle “a” making sure to leave the degree symbol in place (otherwise you get radians).  The slider then needs tweaking by double-clicking to set the max, min and increment.

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Next, I need to make the line segment a bit bolder by right clicking on it…

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..and change the properties of the point so it shows the coordinates to 2 decimal places, also using right click.

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We now have a tool that can tell us the co-ordinates.  Before using this, however, I think I would want pupils to do some work on paper, using Mike’s handout to get a feel for the numbers and get their own results.  To fill in this table:

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I feel that it’s useful that they have the opportunity to correct any measurement inaccuracies before the next step and this is where the computer helps.

As per the worksheet, a series of questions can be posed before using the calculator’s Sin and Cos functions to complete the following:

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From there we can easily start exploring what happens when the side length is not 1 and use the ideas of scaling and similar triangles.

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And then, of course change the angle again.

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Once this is all set up, it’s easy to display / hide the coordinates and maybe so some miniwhiteboard work to assess how well the class has grasped the use of the Sin and Cos functions. And then keep the coordinates, but hide the angle to demonstrate inverse Sin and Cos.

With some practise and familiarity with Geogebra you are spending less than 3 minutes on the computer.  If you like the idea of “Geogebra from a blank sheet”, click on the Geogebra category link at the top to see posts on using the same idea for other topics.

Four operations of fractions by folding paper

Another idea from Mike Ollerton’s workshops.  This file gives a comprehensive explanation of the activity, which starts like this:

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I have used a similar task before but I realised that I had missed a key step which is to label each fraction after folding:

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The file then goes on to describe how to use this for demonstrating all four operations: add, subtract, multiply, divide. It’s a lovely way of reinforcing the concept of equivalent fractions at every stage.

My only reservation with this task is that doing the folding in the first place might be a barrier for some learners.  Especially folding something into thirds – it’s not straightforward.

I have added some Powerpoint printables that provide guidelines along which to fold. Note these are set up as A4, so print them 2 to a page and then cut.

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In a sense, I can see that this might detract from the notion of “folding in half” because it becomes “fold along that line”.  I haven’t had enough experience of which is the “better” way to do this – I’d be very happy if anyone wanted to share their thoughts!

 

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.

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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:

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…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.

 

Fraction problems

These problems are ones that are made much clearer by drawing a rectangle to represent the “whole” and then deciding how to divide it into equal parts.  The numbers are not too tricky but interpreting the question might be:

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These are not intended to be fraction of an amount questions.  An approach could be to decide upon an amount, but the intention is to direct students to drawing a representation of each question.

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