Multiplying and Dividing by 10, 100 and 1000

I wanted to share an example of how ‘What if…’ can be used to open up new ways of exploring concepts that might not appear obviously open to investigation. In this case it is multiplying and dividing by 10, 100 and 1000. Obviously this is an important concept and one that we want children to be able to be fluent in.

I have a few strategies for developing starting points and one of them is to arrange them in some kind of diagram. In this case the idea was to build in multiplying and dividing by 10, 100 and 1000 in a ring. In its simplest form, we can think about just trying to perform operations so that we end back with 324. With some I might give them a range of numbers that they could put in the circles as support. 10, 100 and 1000 ring10, 100 and 1000 ring filled in.

Often I’ve used the POG format with this kind of thing (get them to make a peculiar and ordinary example and then think about what needed to be true in general). The general being something around that the amount multiplied needs to be the same as the amount divided.

But there is plenty of possibilities for the use of ‘What if…’, to make it more challenging perhaps or just to allow them to go down their own direction.

The obvious one is what if they had more circles in the ring? An odd number makes for an interesting case with the fact that you then can’t have the same number of divisions as multiplications.

We can though be far more creative and perhaps initially it might come from you but soon the children will come up with their own ideas and rules. What if no circle could have the same value? What if we had two rings and the circle in the same position on both rings couldn’t have the same value? A wealth of opportunities present themselves that could be pitched at different levels. I’ve done this idea with both Years 4 (with just integers) and 6 (where we focused a lot more on not allowing the same value more than once) but I see no reason why it wouldn’t be effective in lower Key Stage 3 as well.

This is the power of ‘What if…’, it really allows interesting new areas of exploration to open up.

Advertisements

The game of ‘what if’

Mathematics always amazes me in terms of how something simple can produce something seemingly magical. Andrew Blair over at inquirymaths.co.uk has been a real inspiration in this. He states that:

Inquiry Maths is a model of teaching that encourages students to regulate their own activity while exploring a mathematical statement (called a prompt). Inquiries can involve a class on diverse paths of exploration or in listening to a teacher’s explanation. In Inquiry Maths, students take responsibility for directing the lesson with the teacher acting as the arbiter of legitimate mathematical activity.

I think this idea of self-regulation is a really key aspect. Now this doesn’t mean the absence of teaching or the absence of curriculum content. I have found many times that teachers have a belief that mathematical investigations/inquiries are somehow separate from the day to day mathematics and the curriculum objectives that they need to cover.

But actually, let us take a step back and consider what maths is in the first place. I happen to really like this definition:

Mathematics is a collection of extended, collaborative
games of ‘what if’, played by mathematicians who make up sets of rules
(axioms) and then explore the consequences (theorems) of following
those rules.

It is this game of ‘what if’ that has been the source of many lesson ideas for me. To give an example of that, let me show you how one lesson came to be.

My Year 6 class were studying angles, I asked myself, what if I take a rectangle and draw a line at the same angle from each corner?

Capture

Immediately, some interesting things crop up. I have a rectangle in the middle and triangles and quadrilaterals surrounding it. I wonder if that is always the case. What if I wanted to know all the angles in the intersections between the lines and the lines and the edge of my bounding rectangle? Are they determinable?

This is essentially what my class set out to investigate. As mentioned before though, self-regulation is the key. Borrowed from Andrew’s excellent site is the use of regulatory cards. Working in a primary school, I use a set of cards that I have feel work well for me but do look at various different options here. Here are my set:

Capture

In keeping with the definition of mathematics above, I have a ‘What if…?’ card.

That’s because I want them to take this initial idea or prompt and go in their own direction. An obvious one would be, what if I used a different bounding shape? Or what if I used different angles – a, a + 10, a + 20, a + 30. This can be so empowering. The children are able to challenge themselves at an appropriate level, everyone capable of making different discoveries. A lot of the fear of maths evaporates because they became the expert on their line of inquiry.

But embedded into this lesson are some really useful curriculum objectives:

Year 5

  • draw given angles, and measure them in degrees (°)
  • identify:
    • angles at a point and 1 whole turn (total 360°)
    • angles at a point on a straight line and half a turn (total 180°)
    • other multiples of 90°

Year 6

  • compare and classify geometric shapes based on their properties and sizes and find unknown angles in any triangles, quadrilaterals, and regular polygons
  • recognise angles where they meet at a point, are on a straight line, or are vertically opposite, and find missing angles

And who knows where else it will lead us.

This is the power of playing what if. Just trying out things, seeing what the consequences are can produce astonishing results. Sometimes little experiments and doodles produce great mathematical discoveries. The Ulam spiral is a very well known example of this.

Doodle yourself, try a what if, and see what you discover.

P.S. It has been a long long time since my last post. Hopefully not as long until my next one!

Maths, Wondrous Maths – Feeding children’s curiosity

It is very sad, but even as many children reach Years 5 and 6, you can see that already they have a negative attitude towards maths. Even those who are very capable sometimes aren’t motivated or interested.

This can be down to a number of things, but ultimately the abstract nature of what we are teaching the children can make it difficult to identify why what they are doing is interesting or important. The ease of using formal methods of teaching with rigid paths can leave little room for exploration or generation of interest.

Often mathematics is made more interesting by using real life contexts. This can be brilliant, and produce fantastic maths and I don’t want to take anything away from using this approach but sometimes the links are there just for the sake of it. My biggest mistake back when I was an NQT was trying to put contexts into everything. Isn’t it possible to make mathematics on its own truly wondrous? Can’t we generate enthusiasm for mathematics itself without relying on tenuous links? In the long term, by approaching maths as such, children can become keen and engaged, desperate to learn more.

One of the more abstract and difficult areas that primary schools have to approach is the concept of decimals. Ask most children what a decimal is, you can almost be guaranteed to be greeted with a confused response. How are they supposed to enjoy something they have no concept of whatsoever?

guinea pig and bee
Using animals to reinforce relative size

My first decimals lesson with a class is always very hands-on. First we talk about why we need decimals, imagining a place where you could only be 1 or 2m tall, where everything has to be sold in whole number of pounds or all time is given in seconds because minutes would be too long. The ridiculousness of it is a great starter. Then given strips of paper with lengths illustrated in whole, tenths and hundredths (see left) we are off measuring different things, initially using either whole metres or whole tenths, then using hundredths as well.

Often money is used as a first approach to decimals, but I don’t think it is as effective as measurement. 10 lots of 1p doesn’t have the same visual reinforcement that measurement does and it is through these lengths that the children are able to establish the key ideas of exchanging and the significance of 10 in our number system.

The realisation that with each decimal place the value was 10 times smaller produced some really great pidiscussion. Having been asked to imagine how big 0.001m was one child responded with, “Imagine what the 1000th decimal place would look like!” The idea blew their minds. I mentioned that some decimal numbers go on forever and many of them had heard of pi. I quickly showed them a website showing one million digits of pi, suddenly my class had an obsession with it.

Another example I have from this year is with multiplication. Mathematics is brilliant for patterns and finding patterns. I set two challenges, the children chose which one to have a go at. One was to find a TU x TU that resulted in an answer as close to but below 1000. The other, I gave them the example of 32 x 46 = 1472 and if I reverse the digits of both my numbers I also get 23 x 64 = 1472 (taken from the excellent Inquiry Maths website). It took something that they felt fairly confident with (2 digit multiplication) and added an edge that really tested their abilities as mathematicians. When we compiled examples at the end and looked for patterns they really were keen to understand why it only worked sometimes.

Sometimes it is giving time to questions they might have. The other day a child asked me if decimal numbers can be odd or even. This is where maths can be so interesting. With the opportunities for these kinds of discussions the children look forward to maths. Sometimes you can throw these questions at them as starters;

It isn’t always easy, and it isn’t always going to be exciting. I know with my lessons I there have been plenty of times where I thought I was fostering curiosity and instead have created confusion. As well as this, sometimes the curriculum makes it difficult. But the long term benefits of creating an environment where children are curious about maths can be so rewarding. Having the opportunities to look for patterns, to understand maths, can make it a whole new subject for some. Ultimately, when they are enthused by the maths itself they are so much more willing to try and grasp those difficult concepts.