Lessons taught; Lessons learnt

I am at the Maths Association conference, being held in Robinson College, Cambridge. Over the next few days I’ll be writing reports and reflections on the sessions I attend.

Session title:

Embedding Rich Tasks into the Curriculum

Session speaker:

Charlie Gilderdale

What was meant to happen:

We have been working with a number of schools trialing how problems on the NRICH website can be integrated into the curriculum. Find out how this is being done and see how our mapping documents can help your school. Of course, we will do some problem solving along the way.

What actually happened:

I have been to sessions run by Charlie Gilderdale before, and they have all been great fun. He is one of the driving forces behind NRICH, a wonderful resource of over 5000 maths problems, investigations and activities (many with teacher notes and solutions). They have also recently started to build up documents which map their activities into the UK maths curriculum. This was meant to be the focus of the session, but happily we spent most of the time working on an excellent set of problems which would be a great way to lead into some circle theorems.

Charlie started by showing us the Virtual Geoboard available on the NRICH site. If you’ve never seen a Geoboard in real life, it’s just a board with pins sticking out. You can then use rubber bands to create various shapes. The virtual geoboard lets you do this… virtually.

He then handed out a few pages of grids with 9-pin circular geoboards to everyone, and asked a simple question: What angles can we find, if the only things we are allowed to draw are segments joining two points on the geoboard? In particular, can we find angles of 10 degrees, 20 degrees, all the way up to 360?

This triangle, for example, gives us 40 degree and 70 degree angles:

We were asked to assume that we had primary-level knowledge about angles. So, we weren’t allowed to use any circle theorems, but were assumed to know that the angles in a triangle add to 180, the angles around a point add to 360, that vertically opposite angles are equal, etc.

Over around 15 minutes, we managed to find all the required angles except 150 and 170 degrees — can you fill in these gaps?

We were then asked to draw as many quadrilaterals as possible using these 9-pin Geoboards (without using the central vertex). Here is one example:

This lead to a number of interesting discussions:

• When are two quadrilaterals different, and when are they they same? We settled on saying that two would be the same if we could cut one out and place it exactly over the other (in other words, rotations and reflections don’t make new shapes).
• How can we try to systematically find examples?
• How do we record the examples that we find?
• How we could convince ourselves, and others, that we had found all the quadrilaterals?

Having found these quadrilaterals we were then asked to calculate their angles. Again we only used late-primary maths. We found a number of ways to calculate the angles, and particularly noticed how symmetry helped to make the calculations easier.

Charlie then pointed out that the results obtained from this activity could be used to get students to notice that the opposite angles in a quadrilateral added to 180. Would this happen if we changed the number of dots around the circumference? He handed out some 12- 15- and 18- pin Geoboards, and asked us to check, and then to try and develop a proof.

We soon found a very nice visual proof that the opposite angles added to 180 when the centre of the circle was ‘inside’ the quadrilateral, and were asked to try to develop a similar visual proof for the other case. He noted that this could be a great way to introduce this circle theorem to children, without having to immediate descend into the murky world of algebra, but also without copping out and just getting students to measure angles with protractors.

(Incidentally, there are a variety of other Geoboard-based investigations on the NRICH site. To find them just search for geoboard.)

In the last couple of minutes of the session we moved onto the mapping documents, which can be accessed from this section of the NRICH site. They are mostly useful for teachers in English schools, but could also be useful for teachers outside the UK, to help navigate around the multitude of problems on the site.

In summary, then, a fun hour, working through an investigation that was accessible, enjoyable, and has extensions leading in a number of different directions.

Thoughts:

It is not hyperbole to say that NRICH has developed into one of the best free maths teaching resources on the internet. Every month they present a selection of their problems, arranged around a particular theme, and at a variety of difficulty levels. They encourage students to write in with their solutions, which may get added to the site — several of my students have submitted solutions, although none of them have been featured so far.

The addition of the mapping documents has made it even more useful for those of us in the UK, because we no longer have the danger of spending a very enjoyable couple of hours on the site without actually coming out with anything we can use in a lesson!

I really liked the particular activity that was introduced in this session, and am looking forward to trying it out with the Primary Maths Club that I’m running this year.

I was browsing through the Nrich website yesterday, and found this interesting problem:

If you are given the coordinates of the midpoints of the edges of a pentagon, can you find the coordinates of the vertices of the pentagon?

This was, apparently, originally given as a question in an Oxford extrance paper from 1926!

If you follow the link, you’ll see that the Nrich site gets you to investigate this problem with pentagons, triangles, and quadrilaterals, using the excellent free dynamic geometry app Geogebra. I use Geogebra regularly in my teaching, so I’m happy to see it appearing in well-known sites like Nrich.

I’ll let you think about this problem by yourself (Nrich gives the hint that you should think about simultaneous equations), and will focus on just one aspect of it in this post…

Midpoints of a Quadrilateral

It’s very easy to convince yourself by messing around with the dynamic geometry that it’s always possible to find a pentagon or a triangle with any given midpoints, but that quadrilaterals are more tricky. Indeed, the default configuration of midpoints presented to you seems to be impossible.

Why is this? Perhaps there is a special property which needs to be satisfied by the midpoints of quadrilaterals. A great way to look at this is to set up the situation in a dynamic geometry package, and see if anything interesting suggests itself for further investigation:

Geogebra applet will be here if you enable Java.

(Source: midpoints_of_a_quadrilateral)

It appears that the midpoints always form a parallelogram, and that this holds even when we don’t have a ‘proper’ quadrilateral, but a bow-tie shape.

How could we prove this? One route is to use vector arithmetic.

Using Vectors

Let’s call the four points of the quadrilateral P₁, P₂, P₃, and P₄, and the four midpoints M₁, M₂, M₃, and M₄.

A sample quadrilateral with midpoints

We have

M₁ = ½ ( P₁ + P₂ )
M₂ = ½ ( P₂ + P₃ )
M₃ = ½ ( P₃ + P₄ )
M₄ = ½ ( P₄ + P₁ )

Now, we want to show that the quadrilateral formed by the midpoints is a parallelogram. We can do this if we show that the vector that takes us from M₁ to M₂ (M₂ – M₁) is the same as the vector that takes us from M₄ to M₃ (M₃ – M₄), and that the vector that takes us from M₂ to M₃ is the same as the vector that takes us from M₁ to M₄. Both of these come from the observation that

M₁ + M₃ = ½ ( P₁ + P₂ + P₃ + P₄ ) = M₂ + M₄,

from which we can conclude that M₂ – M₁ = M₃ – M₄ and M₃ – M₂ = M₄ – M₁. M₁M₂M₃M₄ is therefore a parallelogram.

Alternative Proofs and Further Questions

This seems like the type of property which could be proved in many different ways. While I like using vectors, the traditional way to prove something like this would be to use synthetic (Euclidean) geometry. I’ve never been particularly great at this, but if someone has a simple proof in this style, I’d love to see it.

This also raises several further questions:

• When is the parallelogram a rectangle?
• When is the parallelogram a square?
• What happens for hexagons?

Next in this series

Midpoints (2): Pentagons