Introducing Pythagoras

Until recently, I'd always introduced Pythagoras's Theorem in the same way: by showing a picture of a cartoon man in a toga and writing a squared + b squared = c squared on the board.

Not very inspiring! (Although I did manage to write the squared symbol properly, something I haven't figured out on here yet)

THEN a colleague of mine told me that she never even mentioned the letters a, b and c, she just showed them this picture of Perigal's dissection, and by discussing it, her students arrived at their own method.

So I tried it with 2 of my classes recently and it went down a storm. They came up with the fact that the area of the two smaller sqaures added together to make the larger square, then I added numerical values for the lengths of the shorter sides and set them to work figuring out the length of the hypotenuse. Each group managed to figure it out for themselves and then apply the same system to simple questions involving both the hypotenuse and a shorter side.

It's a shame to deprive anyone of algebra, so the next lesson I asked both classes if they could see how the formula  a squared + b squared = c squared would apply to Pythagoras's Theorem and they picked it up pretty quickly.

I think starting from the visual basis for the theorem helps students to think it through logically. In the past, I've always seen furrowed brows and cross expressions when I tried to show weaker students how to rearrange the formula to find a shorter side. Introducing Pythagoras using the picture first helped students to understand it more intuitively. In fact, it was a student who usually complains at the merest whiff of algebra, who suggested that the formula for a shorter side would be c squared - b squared = a squared.


Once again - not rocket science. But pretty useful.

One Slide, Lots of Questions: Sequences

Following in the footsteps of the lazy teacher again, this lesson took no planning time whatsoever. I found the slide on the TES (thank you, wonderful contributors to the TES!) but rather than using the whole powerpoint, I just took this slide, and asked the students to come up with the questions.

The students drew out diagrams of 1 table, 2 tables, 3 tables etc, then worked out the nth term, then used the nth term to calculate how many people could sit at different numbers of tables.
I then asked the class to suggest questions. They wanted to know how many tables you would need for 100 people - so we formed and solved equations. Then they wanted to know if you could arrange the tables in a different way, so we did that and discussed which method was the most efficient.
Then they wanted to design their own tables, so we did that too and I got them to invent questions about their sequences and swap them between pairs.

I'm sure thousands of teachers have taught thousands of lessons like this before, but for me the unusual thing about this lesson was that it only required 1 resource and the students came up with the questions. I think a lot of us teachers feel under pressure to have the whole lesson mapped out to the millisecond, but if you have a rich enough resource and you're able to point students in the right direction when it comes to questions, you can have a open ended lesson that really deepens their understanding. AND it is much quicker to plan!!

I'm going to keep challenging myself to get as much out of 1 slide as possible.....

One Slide, Lots of Questions: Shape

This is an idea that I've cultivated since listening to the wonderful Jim Smith (AKA the lazy teacher)

He pointed out that a lot of time and effort is spent making resources that only last 5 minutes. The challenge is to create a resource that will generate lots of discussion / questions / reflections.

I'm going to try and build up a collection of these sorts of slides so here is one to get me started.

 Angles, Area and Shapes revision for y11

What did y11 have to say? Here is a summary:

• They're all different colours (there's always one)
• How to calculate the interior and exterior angles of each shape
• The formula for the sum of the interior angles of any regular polygon
• What was meant by the word regular
• How to calculate the area and perimeter (including breaking the hexagon into 2 trapeziums)
• The fact that they were all enlargements but there was no centre of enlargement
• How many lines of symmetry each shape had
• The order of rotational symmetry

Ok ok - so they had some hints from me! But it was an incredibly useful revision aid and took me about 2 minutes to make. It was certainly better than writing out loads of questions to get them to think of the same material.

Graphs of linear equations - a useful resource

Yay for straight line graphs!

This was a topic that I used to dread teaching, but I've had much more success recently so I thought I'd reflect on why that was.

Straight line graphs are an excellent topic to get students to investigate things for themselves. Give them some y = mx +c type functions, ask them to plot them and tell you what they notice.

Aha! The ones that end in a 5 all cross the y axis at 5. What a coincidence! And the ones that all start with a 2 go up in 2s! Amazing!  etc etc.

I used this approach right from the beginning of my teaching practice, but I still found that some students struggled to understand the generalised form of the equation. Some of them noticed the links straight away and the whole class moved on, but a significant minority were left with only a superficial understanding and they got very stressed out when they couldn't understand something that appeared to be so easy for the others.

I'm sure there are lots of things that I could do to improve my pedagogy for this topic, but I kept having the same response with different classes ...... until...... I made some laminated copies of a 10x10 coordainte grid to use for these lessons. I made 32 (1 for each student) and made sure they were big enough for me to see which coordinates students had plotted when they held them up from the back of the room.

Well the simple ideas are often the best and this turned out to make a big difference.

Firstly, using whiteboard pens on these grids encourages those students who are terrified of making a mistake to just experiment and stop worrying. And those kids who get angry easily - the ones who rip up their pages when they make one error - it calms them down completely. None of them mind rubbing out something written with a whiteboard marker.

But most importantly, it allowed me to use QUESTIONS much more effectively. I could get the whole class, or just a small group to show me this sort of thing:

the coordinate (0,3)
a line that intercepts the y axis at (0,7)
any line with a gradient of 2
another line parallel to your previous one that passes through the point (0,5)

and then:

the line y = 2x + 5
a line with a gradient of -3 or 4/5
a line that is perpendicular to y = 2x + 5


I could straight away see which students were getting muddled up and why. I could set different tasks for different students and I knew exctly who needed what level of challenge. I wasn't moving the whole class on anymore when only half of them were really ready.

 Our HoD has made laminated coordinate grids for eveyone in the department now.
Not rocket science. But pretty darn useful.

Hellooo!

Hello potential readers of my new blog.*

 I'm hoping to use this space as a place to collect and muse on new ideas for maths teaching. Akin to a magpie, I'm looking for shiny new ideas that will:

-> Make lessons more engaging
-> Help provide real life examples
-> Reduce planning and preparation time!
-> Help my students gain a deeper level of understanding and appreciation of mathematics

I've always found it helpful to write things down, so I'll use this blog to reflect on things that I've done in lessons, but I'm most interested in other people's resources and ideas. I'll collect and share links and examples of anything I think is useful.

Laters,

The MM


*Hmmm. Writing this post in the belief that someone might actually read it is starting to feel a bit naive. In fact, it feels a lot like shouting "hello" down a big hole in the ground on the off chance that someone from Australia might reply and start an interesting conversation. I'm hoping that only one of these events has a probability of 0.