Geometry Problems

Having now taught this lesson, I’ve edited a few things. My first idea was to do it on squared paper and count squares.  But there were too many squares and the totals we got were widely varying due to how students estimated partial squares.  Hopefully, it’ll work even better when I do it this way next time…

This problem was posted on Twitter last week by @solvemymaths. CHzJeeAWEAAISYR.png-large I must confess I was pretty slow to solve this! It’s a lovely problem – so easy when you know how.  I created this Geogebra file.  It didn’t help me solve it directly but it did answer the question of whether this is a fixed shape or not, which I was struggling to visualise from the diagram. It isn’t fixed and that gave me an idea for a lesson. Rather than just give my students the problem, we will draw it first to provide that hook, the hunger to answer the question, “Why does that happen?”. I also provides a bit of practice using a compass, a skill that is needed at GCSE and one which we don’t spend much time practising.

Here’s how I would now run the lesson (key maths words italicised):

1, On plain paper everyone draw a circle using a pair of compasses.  You chose the radius, anything between, say 1cm and 6cm. Make sure you write down your radius. Calculate the area of this circle.

2, Then, using a ruler, carefully draw a tangent, doesn’t matter where it is on the circle.  Make the line nice and long, using a 30 cm ruler if you have one.

3, Then, using compasses again, mark 5cm either side of where you tangent touches the circle. Label the points A and B.

4, Next draw a second concentric circle, going through A.  It should also cut B if you have drawn accurately. You should have a shape looking something like this:

Screen Shot 2015-06-21 at 16.24.16or this Screen Shot 2015-06-21 at 16.24.45

5, Now, calculate the area of the big circle.  You’ll obviously need to measure the radius of the big circle first.

6, Now find the area of the orange ring. Compare your area with that of others around you.

Where you go next will depend on the class. I’m not going to give you a full solution here, that would spoil the fun! But the answer is 25π…

Advertisements

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.

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.

Screen Shot 2015-06-07 at 17.59.45

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.

Screen Shot 2015-05-14 at 22.20.30

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.

Screen Shot 2015-05-09 at 20.51.43

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.

Screen Shot 2015-05-09 at 20.55.41

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

Screen Shot 2015-05-09 at 20.56.53

Or:

Screen Shot 2015-05-09 at 20.58.25

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.

Screen Shot 2015-05-09 at 21.00.34

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.

Screen Shot 2015-05-09 at 21.07.58

Ideas for better maths teaching