Great Fractions Resources from Solve My Maths

This blog post from Solve My Maths is a treasure trove of deep thinking fractions questions.

This image inspired me to plan a group activity with a class.

number-line

Give each group a set of coloured stickers (8 different colours – getting ones that exactly match the picture could be a challenge! I might need to make my own and then take a photo.  Maybe a grid in the background would be helpful if I do this…)

Show the left-most and the right-most column of numbers on this image and hide the stickers in the middle.  Their challenge is to fill in the ones in the middle.  They can use calculators if they like – seeing a fraction as the equivalent of the operation of dividing numerator by denominator is useful.  Work together, only place the sticker if everyone else in your group agrees.  I think this will really get them talking.

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Do try this at home!

I liked the look of this Nrich task thinking that my Year 7s, who have shown some appetite for investigative tasks, would enjoy it.

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It’s a great task, but like so many investigations, you really need to have a go at it first yourself.  I did it with my daughter (Yr7) and we took nearly an hour to find the 3×3. And I was trying pretty hard!  Interestingly the 4×4 was much easier.  Here are our combined efforts!

   It’s going to be critical how I introduce / explain the task, so I will create a notebook file to help with this.

I have also created a worksheet as I feel that my class will need a bit more scaffolding on this task.  I now know that I will encourage them to move onto the 4×4 if they get fed up with the 3×3.

I’ll update this post tomorrow once I have taught the lesson.

The beauty of the very very big and the very very small – Standard form

Scale of the Universe is one of the nicest websites I think I have ever seen. It’s fascinating in its own right, but is also a nice resource for looking at very small and very big numbers.

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I put together this sheet to use with students to give them some tasks to do whilst playing with the site. The idea is that it gets them used to using standard form numbers and demonstrates why it is actually useful and better than writing all those zeros!

Adding colour to Sequences

After browsing the ever excellent Don Steward’s Median site for sequences, I found this on Linear and Quadratic growths.

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It got me thinking about using shapes to represent sequences and in particular using different colours to represent different sequences laid on top of each other.  For example, the following patterns produce a linear sequence.

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However, there are a couple of ways you could look at this which combine sequences which are arguably simpler.

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Or:

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We can then extend this on to quadratic sequences.  This is nice: one way we can see the sequence of square numbers, the other way we can see how multiplying one dimension by the other leading to some brackets which can be multiplied out.

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There are a bunch of these in this file.

There are also over a hundred sequences at visualpatterns.org a website entirely dedicated to, well, visual patterns.

Get your students to create their own once they have got the idea.  Could make for some great wall displays!

On quadratic sequences specifically, a nice worksheet here from @solvemymaths.

Finally, back to Don Steward with this which is actually just a series of terms stacked on top of each other.

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Are you sure you can prove that?

A nice couple of demonstrations of what makes a proof, i.e. just because you’ve got lots of examples doesn’t mean you’ve proven it.

Mathematicians of the 18th century proved that numbers 31; 331; 3331; 33331; 333331; 3333331; 33333331 are all primes.  This was no mean feat without a calculator.  It was a big tempation to think that all numbers of such kind are primes.  But, the next number is not a prime:

333333331 = 17 x 19607843 

Another classic example is the question of how many areas you get when you cut a circle with chords formed by joining points on the circumference.

CircleCuttingCircumference_1000

You might think you’ve spotted a pattern of doubling each time (or 2^n).  And indeed the next one is 16.  But the one after that is 31.

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The formula is not quite so straightforward and involves combinations:

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In its expanded form it looks even more crazy!

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Wolfram has more details on this problem here.

And some nice discussion of the problem here.

Ideas for better maths teaching