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Five ideas for square numbers

25/2/2016

 
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Post 25/29 in the Staffrm #29daysofwriting challenge: Lesson ideas for exploring square numbers
As we've got two squares in today's date, I thought I'd go for a square themed post. Also I've just realised that this year contains 25/04/16, and 25/09/16, both even better excuses for a lesson on square numbers.

​These are all great investigations/rich problems with plenty of practice working with square numbers - a little more interesting than fifteen questions out of a textbook. If pupils have to learn facts, such as the squares to 15, I think it's better to embed them into something a little mathematically rich than just drill.


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I thought Christmas was over...?

4/2/2016

 
Post 4/29 in the Staffrm #29daysofwriting challenge: Fermat's Christmas theorem
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Pick of Twitter 20/09/15

20/9/2015

 
Practice for high attainers, a pile of vector resources and a bit of mathematical colouring 
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Mental squares

23/2/2015

 
Thank you, Internet, for wasting my evening. I don't know how I found this on YouTube - one of my random click-a-thons searching for goodies, and I'm a sucker for "magic maths" tricks. This one is how to square any (two-digit) number mentally.
If you don't want to watch the video, here's the basic idea with 33² as an example:
  1. Work out the difference between 33 and the nearest 10 (in this case, 30). Difference = 3.
  2. Add the difference to 33 to get 36. Subtract the difference from 33 to get 30.
  3. Multiply these two numbers together (30 x 36). This is (apparently) the advantage of this method - it's easier to do 30 x 36 in your head than a four-way decomposition. 
  4. Finally, square the difference (3² = 9) and add it on to 30 x 36 to get a final answer of 1080 + 9 = 1089.

And here's my workings:
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So I did this one...then tried another. 
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Cute? Convoluted, really, and not particularly workable in the curriculum (I'd much prefer a sensible approach with proper multiplication), but then I got one of those "oh now how does that work?" moments. I could see how it was kinda linked to quadratics and square areas, so I did some doodling. After many aborted attempts, I came up with this: 
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Take a square with side length 33(cm if you insist on units). You can then chop a rectangle off the edge of length 3 (making an easier multiplication of 30 x something) and whack that on the bottom. So then you've got a 30 x 36 rectangle, and the little square left over to add on.

Then, because I'm a mathematician and I like doing things properly, I tried to make it work with algebra:
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Typical quadratic expansion, but we think about a as the tens digit and b as the units, because we're trying to change the square into a rectangle with length a (so we have a multiple of 10 to calculate with). Then:
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So we've changed the problem from (a + b)² (difficult) to a(a + 2b) + b² (messy but easier mentally). I was quite pleased with this!

Just thought this might be a nice challenge problem to give to pupils when working with quadratics (or square numbers for that matter) - show them the video, then get them to explain why it works - numerical or algebraic approach. I do think this lends itself to an exploration of the link between square numbers, area and quadratics, which I think is often overlooked.

Growing squares

29/1/2015

 
This investigation is great for getting pupils to “spot” square number patterns – it’s particularly useful as a little starter a couple of weeks after completing the topic to remind them of the sequence of square numbers. Pupils could also make the patterns from multi-link cubes and then demonstrate how the top and bottom patterns can be turned into squares.

I’ve also used this as a time-filler activity with half a class of Year 10s and asked them to come up with the nth term rule for the top and bottom patterns (n² and (n – 1)² respectively), then combine this to get the nth term rule for the complete pattern (gives 2n² - 2n + 1 when expanded and simplified). They then used this to predict and check the next pattern in the sequence.
Growing Squares investigation | NorledgeMaths
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Square differences

28/1/2015

 
Pupils explore which numbers can be written as the difference of two square numbers; a nice way to practise calculating with square numbers, but also encourages them to pattern-spot at the same time.

All the odd numbers can be written as the sum of two consecutive square numbers. Some pupils notice this very quickly, but some will need to try numbers for themselves first. It helps to ask pupils to look for patterns in answers they already have, then ask them to think about whether this pattern could apply to other numbers. It’s a little trickier for pupils to work out the other group – the multiples of 4.

It’s highly unlikely that proofs of these will be applicable when first introducing square numbers to a group. However, the activity could be revisited in the context of lessons on proof or expanding quadratics.

Square Differences investigation | NorledgeMaths
Square Differences solutions | NorledgeMaths
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Happy numbers

28/1/2015

 
This is one of my absolute favourite investigations to use when teaching square numbers. It can fit quite nicely into 15/20 minutes as a simple activity, or extend to a whole lesson with the class working together.

To find out if a number is happy:

  1. Pick any two-digit number (e.g. 23).
  2. Square and add the digits (4 + 9 = 13).
  3. Repeat (1 + 9 = 10).
  4. And again (1 + 0 = 1).

A number is happy if it eventually gets to 1. There are 20 happy numbers between 1 and 100; asking pupils to find two or three happy numbers works well as a quick activity, possibly towards the end of a lesson introducing square numbers.

However, the most interesting part is what happens with numbers that aren’t happy. Every number that isn’t happy eventually “spirals” into a repeating loop:

4 → 16 → 37 → 58 → 89 → 145 → 42 → 20 → 4

By extending the time allowed, pupils quickly discover this loop for themselves. Some then realise that they can rule out reverse digits (e.g. if 42 isn’t happy, then 24 won’t be either). It’s quite powerful to ask pupils to work in pairs or groups, as this allows lots of opportunity for planning and discussion.

I’ve previously used the board to map this “spiral” out for pupils, asking them to come up and add any numbers that they can. I also spent one very rainy Sunday afternoon transferring this map to a PowerPoint slide with colour coding (cause I'm a loser)... I think it looks very satisfying!
Happy Numbers | NorledgeMaths
Happy Numbers spiral | NorledgeMaths
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