Often I am asked how we decide what goes on the NRICH website. In
the same way that authors find it difficult to answer the question,
"Where do you get your ideas", it's not a straightforward question
for me to answer. People sometimes send us ideas for problems,
and old books are a great source of rich starting points. Many
ideas that have been around for a long time are well worth dusting
off and revisiting with a fresh twist.
For me, what tends to happen is that I'll play with a piece of
mathematics either on my own or with another member of the NRICH
team, keeping a note as I go of what questions occur to me and what
strikes me as interesting. I work closely with Charlie Gilderdale
on the Stage 3 and 4 parts of the site, so once we have an idea
that we think can be developed into a problem or game for the site,
we think about where it fits into the curriculum and what age of
learner it might suit.
Once the problem has been written up with a student audience in
mind, we think about how we would use the task in the classroom,
and reflect on similar ideas that we have used when teaching that
topic in the past. Whenever we have the opportunity, we try out a
new idea in a pupil workshop so that we can draw on direct
experience of using the task when we write the Teachers' Notes.
This month's site has been developed with the new academic year in
mind, so apologies if it's not a new school year where you are,
though we hope there will be something here for everyone. We are
aware that there are challenges involved in using rich tasks for
the first time, so we wanted to publish a collection of engaging
resources with no theme other than offering ideas suited to the
start of the year. One example is the stage 3 problem
Diminishing
Returns. The problem begins with an image showing seven nested
squares, and we have suggested in the Teachers' Notes that learners
draw the image and calculate the proportions of the total coloured
in each colour. The task goes on to explore what would happen if
the pattern continued into the centre of the page.
I think this makes an excellent task for the start of a new term
for several reasons. Firstly, it is an example of a Low Threshold,
High Ceiling task - everyone can make a start on it, but there
are opportunities for high levels of mathematical thinking. The
first part of the problem offers practice on working with
fractions. This provides an excellent assessment opportunity
for teachers with new classes, to watch how learners tackle the
problem and see the what prior knowledge they bring to the task. We
suggest in the Notes that the class could discuss different methods
they used for working out the fractions - establishing a
classroom culture where methods are shared is a great thing to do
early on with a new class. Then in discussion on what might happen
if the pattern could continue forever into the centre of the page,
learners are asked to come up with convincing explanations of their
thinking. This is particularly useful in establishing the maths
lesson as a place where ideas are explored and proved, and
overcoming some learners' preconceived idea of maths as a subject
where a method must be learned in order to progress to a single
correct answer.
Another problem from our September Collection is the Stage 4
resource
Curvy Areas. As
with Diminishing Returns, all learners should be able to make a
start on this problem. As far as curriculum content is concerned,
there are a variety of topics met in the task. The problem begins
with constructing a diagram made from arcs, which could be used to
give much-needed practice in using a pair of compasses. It goes on
to look at calculating areas of the regions of the shape, a
possible introduction to finding the area of sectors of a circle,
and then leads to some sequences of areas which can be described
using algebra. It can seem hard in a busy curriculum to find time
for maths enrichment, but a task such as Curvy Areas covers diverse
topics from the curriculum in an engaging context, and offers the
chance for learners to be surprised by the results they find.
Rather than spending separate lessons on construction, area of
circles, and sequences, these skills can be taught as learners meet
them through the investigation.
Teachers using NRICH for the first time can be daunted by the sheer
volume of resources on the site, which is one reason why we have
our Curriculum Mapping Documents. Each problem we've identified as
having strong curriculum links appears on the document at the point
most appropriate to its curriculum content. However, that doesn't
mean it's the only place that task can be used - many of our
problems could be encountered for the first time in Year 7 and then
revisited in Year 10 or 11 to be tackled at a higher mathematical
level.
Finally, I'd like to share some of our thinking about the problem
Shady Symmetry.
This problem originally appeared on the site a few years ago with
the title 'Isometrically', and challenged learners to work
systematically to find all the examples of a symmetrical pattern on
a triangular grid by shading in four triangles. In rewriting the
problem, we've chosen to use a smaller grid, and also to offer the
choice between triangular and square grids.
Convincing someone else that your solution is complete is an
important skill to learn in mathematics. The problem is
written with the hope of encouraging discussion of the different
ways of working systematically on this problem. Of course another
important consideration is that these symmetrical patterns are very
pleasing to the eye and can be used to fill empty display boards
with students' work at the start of term!
In the Teachers' Notes, we have suggested that learners decide for
themselves a line of enquiry to explore. This is a key feature in
many recent NRICH tasks, as we want to model the way the maths
community works, and to encourage learners in classrooms to
consider themselves research mathematicians working to find
something out. However, for classes who are working in this way for
the first time, it can be very daunting to be asked to come up with
conjectures and lines of enquiry, so we have put some suggestions
in the problem for teachers to use to prompt their classes.
Our hope is that the problems we chose to publish this month will
whet both teachers' and learners' appetites for learning through
working on rich mathematical tasks. If you are new to NRICH, we
hope you will look back at our recent problems too - some
highlights have been
collaborative maths,
group-worthy tasks,
art, and
scientific exploration. And of course we look forward to
reading solutions submitted by students, so come back next month to
see if your classes get a mention!