Thinking Outside the 'Box'

I'm a big fan of the array or 'box' method for multiplication (as I blogged earlier here.)  A twitter chat with Britnny Schjolin last week raised this troubling point however:

I know that many of my colleagues are also impressed by it even though they, like me, might not have see it when they were students. I have also worked with colleagues who have openly stated that the 'box' method is not the proper way to show your work or that they don't like it so they won't show this to their students. I'm not sure how widespread such attitudes are but I honestly feel that we are doing our students a huge disservice by not showing them such a powerful representation that allows for so many different connections to be made. If this means that we as teachers should learn something new, then so be it: as educators we must always be prepared to learn new things.

To show how useful I have found this, here is how I recently solved a problem that I came across by using arrays or the 'box' method. 

Prove that the product of four consecutive numbers is always one less than a perfect square.

I started pretty conventionally by trying to generalise the product of four consecutive numbers:

Well, I don't fancy working out that product, but I know if I rewrite the four consecutive numbers like so:

Now I can rearrange to make use of the difference of squares to make something a little more delightful:

 A quick array is drawn to help me work out this product:

Since I have to prove that the product is one less than a perfect square then I need to consider this:

I am good at factoring quadratics by inspection but not so good with quartics! However, I now decide to draw a square array to help me factor by working out the components of each side. The first part solves itself:

To get the 2a³ term, I need to split this symmetrically across the square and think what the next component must be. This makes things very clear:

 This also helps me get the middle product:

 Now to get the -a² term, I need to have a -a² in each of the top right and bottom left cells:

This immediately gives me the last component from which I can write the term as a perfect square:

Thus the product of four consecutive numbers is always one less than a perfect square.

Now, before I learned about the array or 'box' method, I would have chugged through with the algebra and probably would have eventually reached the same conclusion. However, now I can attack and solve such problems in a fraction of the time and with more clarity. This is why we must teach this method:

It is an incredible mathematical tool.

It is not a new idea either. Recently, for fun, I have decided to work my way through Silvanus P. Thompson's classic Calculus Made Easy and I came across this:

Array models were being used back in 1910!

Open Middle Fraction Problems

I have often shared the Open Middle website as a source of good thinking problems with the teachers that I work with. This week I created a couple of my own and used them with a grade 6 class who have been working on equivalence in fractions, decimals and percentages. The first problem (given using verbal instructions) I gave was this: 

Use any of the digits 0 to 6 once only to satisfy the statement below.

We used visibly random groups to get the students into threes and gave each group seven tiles to work with. As we began to walk around the room, a few students asked 'What do I have to do?' Taking the lead from Peter Liljedahl's work, I smiled and told them to find out from a friend.

We could see some students getting stuck or making mistakes but resisted the urge to jump in and show them how to get a solution. 

We let them think their way out of it.

Sure enough, we soon heard some 'A-ha's around the room as students realised that the first space had to be a zero, and as they figured out the decimal equivalent of a fraction:

Once they had found one solution, I asked them to find me another. Some used two-fourths as the fraction, others used one-fifth or two-fifths:

One group bent the rules a little bit:

Another group tried something similar but put 2.5 instead of 0.25. I simply pointed to this and asked 'Is this more than one or less than one?' and walked away. When I came back, they had corrected it.

Another group tried this:

When I asked about this, they knew that two-sixths was one-third but they also thought that its decimal equivalent was exactly 0.3. As they knew how to use a calculator to change a fraction into a decimal, I asked them to do this for one-third. This made it easier for me to convince them that 0.3 is different to 0.333333333... 

As we had fifteen minutes left, I gave a variation of a question that I tweeted last week that proved popular:

Any thoughts we might have had about this question being tricky for them were soon dispelled:

As we circulated, we could hear the students justify their solutions and, if we were unsure, simply asked a question like "Can you convince me that two-sixths is less than five-eighths?"

One student asked if it was OK to use a fraction whose numerator was larger than its denominator. This led into a nice discussion about improper fractions and how these are all larger than one:

We could see a couple of groups who found solutions quite quickly so we gave them an added challenge by removing the '1' tile. They relished the added level of difficulty:

All in all, it was a really pleasing lesson that allowed the students to show us their problem solving skills as well as allow us as teachers to assess their understanding of fractional equivalence.