Is this a graph of a quadratic function?

Here’s an idea that came to me in Anne Watson’s plenary today at the MEI conference in Keele.  I think it’s quite challenging but I haven’t used it with any classes yet, so have no idea really!

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I expect students to think of translating the graphs along x and y so that the turning point ends up at the origin.  Then they can test whether y = x² .  What other methods might you use? What other prompts might you give  to learners that are stuck? What additional questions could you ask?

Here is the above image as a pdf.  I created it in Excel here as a scatter graph with a smoothed curve, so can tweak the values if you wish.

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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!

 

Just follow these steps and you’ll be OK at Enlargements

Here’s a statement which I don’t think will be too controversial – I would have thought maths teachers the world over would agree with this:

I want my students to gain a deep understanding of the mathematics, not just follow a procedure to get the right answer.

This is our aim. We don’t always get there. We have different ways of getting there. I have recently re-read this seminal paper by Skemp from 1976.

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In it he talks about Relational Understanding, which I have generally thought of as understanding of concepts, and Instrumental Understanding, which I think of as understanding of procedures.  In my teaching I have been inclined to build conceptual understanding first, and then see what methods make sense from there.  However, I’m starting to think that it’s not that simple.  There are some situations where some instrumental understanding might come first and act as a foundation on which to build relational understanding.  Ultimately we want both, but the order in which we achieve this is not always the same.  We should not dismiss a didactic approach that provides a clear sequence of steps and worked examples as a part of the journey to a deeper mathematical understanding.

I recently observed a colleague teaching the topic of Enlargements to a Year 11 revision class, who are entered for the Foundation paper.  He used a method which I don’t think I had thought of before.  It was heavy on the Instrumental Understanding, but it worked and the students were doing some more tricky fractional and negative enlargements with centres of enlargement not at the origin.  About as hard as it gets for these types of exam questions (thanks for Maths Genie for these examples)

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So, what was this wonderful method?  Well, it might be nothing new to you dear reader, but it used vectors.  It relied on students being secure with describing the translation between two points using vector notation.  Given that this is how Translations are described and that we typically teach Translations (along with Reflections and Rotations) before Enlargements in a topic called Transformations, this should be a build on / consolidating of what was learned a few lessons ago.

The steps go something like this.

  1. Label the vertices of the shape you’ve been given (say A,B,C,D, something like that)
  2. Circle the centre of enlargement, CoL (helpful to distinguish it from a vertex later)

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3. Find the vector that moves you from the CoL to each vertex.
4. Multiply each vector by the scale factor

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5. Now use those vectors to plot the new points, starting from the CoL again. Connect the vertices to form the shape.

6. Finally draw in some ray lines to convince yourself that you have not made any mistakes.

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I think that’s quite a neat method that enables us to go straight to what might be seen as the most difficult example. The only real difficulty here, however is multiplying a (pretty simple) negative fraction by an integer.  Something which should be secure by the time this topic is being taught. And if it’s not secure, well here is an opportunity to practise it without detracting too much from the main objective.

Another benefit, is that it might be easier for students to plot points by counting squares rather than draw accurate, extended ray lines, as pointed out by Mr Blachford on Twitter.

Once students have done a few examples, we can draw attention to some things, for example:

  • If the scale factor is >1, it gets bigger, if it’s <1 it gets smaller
  • The scale factor applies to each side length of the shape (but not the area…)
  • Negative scale factors always place the image the “other side of the CoL”
  • All ray lines must go through the CoL. This is how it is constructed.

I would use Geogebra (as in fact I did to create these images) to examine what happens when we move the CoL. (Note: Geogebra uses the US terminology Dilation rather than Enlargement – which actually is more descriptive of what we are doing, isn’t it?!)

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And from there we can go onto questions that show the enlargement and ask for a description.  I would use Geogebra for that bit as well as it is very easy to create images that can then be used to ask learners to describe what they see.

Just for fun, I had a go at a 3D version in Geogebra.  I’l leave you to decide whether or not it adds anything. You should be able to access it here. This is what it looks like:

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This is an example.  It’s one way of doing it. For other topics, for example fractions, I would prefer to spend a decent chunk of time on building conceptual understanding before focussing on algorithms to get right answers. But that’s for another post…

 

 

 

 

Interleaving Algebra and Perimeter

Over the last couple of years we have organised our Year 7 curriculum so we do some introductory algebra early on.  Forming expressions from words and collecting like terms, would be topics that I would put in the introductory bucket. The benefit of this is that it can be interleaved into various other topics to extend thinking and promote generalising.  Perimeter is an example of this, when we can give side lengths letters instead of numerical values.

An alternative to presenting a bunch of text-book type questions is to investigate a simple 4 piece Tangram, as described in this task from Mike Ollerton.

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(Click on the image to access the full Word document)

The task as presented is primarily an exercise in shape, but I might use a slightly modified version of the Tangram to focus purely on perimeter.

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Before asking students to cut the triangles out of the shape, we might agree on some labels for the side lengths.  If we focus just the shorter sides, we could call Triangle A’s short lengths a, Triangle B’s short lengths b, and Triangle C’s short lengths c, so we end up with something like this.

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Again, before getting to work with the scissors we could discuss how we might describe the hypotenuse lengths in terms of a, b and c.  And in fact, if we really need c at all…

Using this notation for the side lengths, we can then cut up the 4 shapes and generate many other shapes and find their perimeters.  Using 1 shape, 2 shapes, 3 shapes, all 4 shapes: what is the shape with the longest perimeter? The shortest perimeter? What is the difference between longest and shortest?  What shapes are different but have the same perimeter – can we prove this using algebra?

It hopefully presents a need for “collecting like terms” as well as some introductory practice in using the technique.

9-pin Geoboard

A second task that links algebra to perimeter uses a 9-pin geoboard.  This sounds fancier than it is. You don’t need the actual boards, students can create their own in their books or you can give them some dotty paper. First, we tell students that we are going to make triangles using only these lines a, b, c. This is a key image that we will need to refer to either on the board or on a handout.

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The next task is to construct triangles using various combinations of these lengths. Each triangle must fit within the 3×3 array.  Depending on the class and your objectives for the lesson, at some point you show them that there are only 8 “different” triangles. An opportunity here for a discussion on what we mean by different and what congruency is.

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Once we have these, we can go through the process of calculating the perimeter for each one using the a, b, c notation we introduced earlier.

There are some options from here.  Mike’s suggestion is that we order these from smallest to largest.  We could do that by just looking at the shapes and having a guess. It’s pretty obvious that C is smallest although some of the in-between ones are harder to see. We could establish that a<b<c in the first picture (again by looking).  Then we would also need to decide, for example which is bigger: 2a or c.  With an older group, you might even use Pythagoras and express c and b in terms of a using surd form.

By looking at the differences between each shape’s perimeter, we start dealing with negative quantities of a, b, and c.  If we then sum up all those differences, we should end up with an expression the same as the difference between the smallest and largest, with the c’s cancelling out. Which is quite satisfying and obvious if you think about the expressions lined up on a number line.

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There is a fair degree of flexibility within tasks like these and I believe that as teachers we need to select carefully what routes we expect to go down in a lesson. There is a danger that we try to encompass too many different topics in one go and if all of these topics are new to a class then they (and you!) are likely to lose track of what they are actually meant to be learning in this lesson. However, if you are confident that the learners in your class are secure with certain concepts (in this case collecting like terms with negative coefficients) then it is a good way to consolidate and practice this knowledge whilst pushing into new territory.

 

 

 

Birth Date Values

One of the great things about my role(s) this year is that I have had the opportunity to meet some fantastic Maths teachers and educationalists and last week I hosted Mike Ollerton for two separate events.  Mike has made many significant contributions to Mathematics education over the years and he has kindly permitted me to write about the ideas he shared with us last week.

This is a simple activity that feels quite fun and personal but could lead to some rich discussions. Mike’s description of it is here:

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After going round the class, asking several children for their BDVs, there are many questions which might present themselves. Can you ask children to work out someone else’s birthday given their BDV?  Mike suggests lots more questions:

  • Which BDVs only have one birth date?
  • What are the minimum and the maximum BDVs in a class?
  • Which BDVs have the most dates?
  • What is the smallest BDV which cannot be made?
  • What is the largest unique BDV?
  • Which dates are square BDVs?
  • Which dates are triangular BDVs?
  • In a group of people who has the average BDV?

What other problems can you devise based upon BDVs?

Simultaneous Equations, refining the procedure

Going over Simultaneous Equations today with Year 11, we all agreed that the thing that is most confusing about solving equations like these are the negative numbers:

4 - 2x = 9 + 6y
6 - 2x = 7 - 2y

We also agreed that we much prefer these types of questions when we heave a sigh of relief realizing that we can add instead of subtract.

4 + 2x = 9 + 6y
6 - 2x = 7 - 2y

Adding is easier. We have one less choice to make and we don’t need to keep track of which is the minuend and which is the subtrahend.

This is subtle.  I would always start teaching simultaneous equations using some real-life examples (the ideas in this post).

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I think real-life examples work in this case, because it is apparent what “extra” you are getting for the extra money.  These problems create the need for the algebra – it becomes a way to represent what is going on by writing less.  With these problems, everything is positive so it makes sense to think of the difference between the two situations or the two equations.

From there we can introduce the need to “multiply up” to get the same co-efficient for one of the terms.   In my experience, most students don’t struggle too much with this concept.

What gets tricky is subtracting one equation from the other when negative signs are involved.  We might identify the problem as being one of a lack of mastery of a fundamental concept – in this case negative numbers.  So before teaching these types of simultaneous equations we could do lots of practice and drills on negative numbers.  But I’m not sure that is always a helpful approach.  Some things are just more confusing (present a higher degree of cognitive load if you like).   There is a lot going on, and lots of it needs to be done mentally. So, maybe there is a case here for explicitly teaching an easier technique which is less prone to error.

A simple way to avoid subtraction is by always ending up with equations with opposite coefficients of one of the variables.

So, for example when solving the following:

4x - 3y + 1 = 0  (A)
3x - 7y + 15 = 0  (B)

multiply (A) by 3 and (B) by -4. A valid shortcut / rule to remember here is that multiplying by a negative simply “flips the sign” of any term it is being applied to. So we end up with:

12x - 9y + 1 = 0
-12x + 28y - 60 = 0

It’s easy to check that the sign of each term has been flipped before then proceeding to add the equations.

As any Maths teacher would, I believe strongly in teaching for deep understanding of concepts, not blind memorisation of rules (Nix the Tricks is a frequent reference point). But there is also a place for remembering certain procedures to reduce cognitive load (times tables being the most obvious example). Dani Quinn has written a great post on this in relation to “moving the decimal point” here.   Manipulating Simultaneous Equations like these is another example of when there is a case for a bit of explicit teaching of a method.

 

 

Ideas for better maths teaching