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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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Grid lines - a HCF and LCM investigation

22/2/2016

 
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Post 22/29 in the Staffrm #29daysofwriting challenge: An interesting way to investigate HCF and LCM
For my remaining full week, I'm going to try and post one rich resource and discussion. I found a nice prompt yesterday in a pack of really old investigations, and thought I'd look into it in a little more detail today. I'd got this filed under HCF and LCM, and have no idea of the origin, but it looked pretty simple so I sat down with pen and paper to try and figure it out.

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Discovering pi

11/3/2015

 
As Pi Day rolls round again, I've pinched one of my colleague's "Pi Kits" and have now done this lesson with Year 7, 8 and 9 this week. It's my favourite way of teaching circumference and area of a circle, both the first time pupils encounter it and as a reminder before teaching any new circle topics. I'm surprised that some pupils have often come across pi prior to this lesson, or can even quote the formulae for circumference and area, but have almost no understanding of what it is and why it's so fundamentally important in circles.

I changed things a bit this year; usually I just investigate circumference and diameter, then do a bit of bluffing about how pi miraculously "pops up" in the area formula too. However, I extended my investigation this year to include area for some groups, and it worked really well.
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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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