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Squares vs rectangles

One of the persistent misconceptions I come across as a teacher is the question of what exactly is it that makes a shape a rectangle? This is something I mentioned briefly on episode 48 of Wrong but Useful (coming soon here) but I think it’s worth getting to a wider audience. 

Chances are that if I ask you to imagine a rectangle, something like this will come to mind:


I suppose it’s fair to say that if you’re reading this then maybe it might be more like this one:


I think the reason the first rectangle I showed is most likely is because that’s how they usually appear in ‘my first shapes’ books. Note that they almost always have their edges parallel to the sides of the book and certainly have “two long sides and two short sides”. However the necessary requirements for a shape to be a rectangle are:

  • It’s a closed quadrilateral (a 2d shape with four straight sides and no gaps)
  • It has four right angles

That’s it. 

Easy. 

However, in our Equable Shapes project, students are asked to explore shapes where the perimeter and area have the same numerical value. We start with rectangles and sooner or later, someone finds that 4 by 4 works (area and perimeter both 16). At this point all hell breaks loose and the two sides are formed with the “you can’t have that because sir said it had to be a rectangle” gang squaring* off against the “squares are special rectangles” crew. 

Inevitably this leads to a lot of unpicking and most students believe they have been told explicitly at some point in the past that reactangles have two long sides and two short sides. Obviously I delicately correct this by going through what I mentioned above but some students still struggle to get that all squares are rectangles but it doesn’t follow that all rectangles are squares. The most effective way I’ve found to help illustrate the point is with furniture.

“Do you know what the word furniture means?”

“Yes”

“Are you happy that all wardrobes count as furniture?”

“Yes”

“Are you also happy that not all furniture is a wardrobe?”

“Yes”

“Then you can understand about the squares and rectangles.”

“Oh. I think I get it now.”

So, there we have it. Squares are rectangles, rectangles might be squares and oblongs are something for another post.

Why not try asking people you work with whether a square is a rectangle? The 

*pun very much intended

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First lesson with year 7 and 8 – Caterpillars

It’s been ages since I’ve blogged. You know what it’s like! Anyway, I intend to do some more so, here goes!

*******************************************************************************

What do you do in your first lesson with year 7? Rules? Expectations? I certainly do mention those things but usually, I like to do some maths and the investigation I choose is Caterpillars. I’ll explain it first then discuss why I like it.

What is ‘Caterpillars’?

Caterpillars have three simple rules:

  • Stop when you get to 1
  • Even numbers – halve them
  • Odd numbers – add one

and that’s it. I demonstrate with 14 like this:

caterpillar14

and then just say things like “I wonder if you can get a caterpillar as long as mine”, “Can anyone get one longer? or shorter?”. Then I let them try some of their own recommending (strongly) that their starting number is under 100.

After a short while, I ask people to make comments on what they’ve noticed and it’s often:

  • They end 4,2,1
  • They all get to 1
  • Odds are better than evens

Amongst other things. This is a project that will comfortably take the first lesson.

So, why do I like it?

The actual operations are very straight forward as they are just halving and adding one. This investigation quickly show you who your weaker students are as they struggle to halve.

The project introduces a clear set of rules to follow. Which students bother to check whether the number is odd/even and which simply alternate the rules?

Who ignores the suggestion to start under 100? Why have they done that? It’s a great chance to tackle the misconception that picking higher numbers shows that you’re cleverer.

What numbers do they start with? Are they being remotely systematic in their choices?

Who can articulate a conjecture clearly? Who can do it concisely?

There’s frequently a chance when someone’s conjecture is contradicted by someone else and we get a chance to discuss how to handle this. The conjecture is wrong now, but it wasn’t at the time. Such a crucial way to show that someone has learnt something is that they’ve found something that doesn’t work!

Also in this project, you’ll eventually end up discussing whether decimals are odd or even. Is 5.2 even? What if it was? How do you know a number is even/odd? There’s a lot to discuss here and you will uncover misunderstanding about what odd and even means.

Someone might choose a negative number, presenting you with an early chance to think about adding one to a negative. A student may well choose zero and the infinity discussion to follow is always a nice moment!

Finally, I like that the longest caterpillar is fairly surprising. I won’t spoil it here but I honestly think this project starts the year with an air of curiosity and a chance to show determination while telling a lot about the students in front of you.

Let me know what your starting lessons are in the comments!

New podcast: Maths Snippets

There’s a new maths podcast on the block called Maths Snippets. It’s recorded by Anda and Katie from Brix learning (@BrixLearning) and the first episode focuses on Ada Lovelace. It’s a good first episode, short and to the point. While it’s obviously no Wrong, but Useful, I still thoroughly recommend you go and listen to it!

Looking forward to the next one already. 

How funny are these jokes? Results!

In February, I asked people on twitter to rate some jokes so I could compare their responses with the responses of my year 12 stats students.

Here are the jokes:

  1. I used to train racing snails. One day, I took the shells off to see if they’d go faster. It didn’t really work and, if anything, it made them more sluggish.
  2. I can’t stand Russian dolls. They’re so full of themselves.
  3. I’ve always wanted a job putting up mirrors. It’s something I can really see myself doing.
  4. An ex-student of mine said he’d been doing a building course and could build me a ‘wishing wall’ in my garden. I thought, bless him. He means well.
  5. Here’s a site for sore eyes: http://www.conjunctivitis.com
  6. Someone complimented me on my driving the other day. I got back to the car and there was a little sign saying “Parking: Fine”, which was nice of them.
  7. My cat’s a genius. I asked her what 2 minus 2 was and she said nothing.
  8. Somebody stole my mood ring. I’m not sure how I feel about it.
  9. Cheer leading exams are easy. You go in and shout “Give me an A”.
  10. I’m very good at maths. I understand 110% of it.

After being rated 324 on a scale of 0 (not funny) to 4 (hilarious) here are the average results:

jokes-averages-internet-2017

(The one on the end is the 110% joke.)

If you’d like to see the raw data and the jokes in a word document, then there’s a link to a dropbox file here.

I got my students to rate the jokes themselves (they seemed to conclude they weren’t very funny). Then, they compared their ratings with the average score for each joke from the internet. Students also compared their ratings with another student to see if they had similar senses of humour or not.

Hexagons in nature and a Turtle question

I’m sure you’re familiar with the good old honey bees and their hexagonal structures. After a trip to the living rainforest, I found out that it seems that bees aren’t the only creatures that like their hexagons!

This tortoise has irregular hexagons on its shell. (Fun fact – tortoises shells aren’t watertight. If they get in deep water, they fill up and will sink. Tortoises are not turtles.)

img 1666

Tortoise

This lovely carpet python and lizard both have hexagonal skin! (I was disappointed that the python wasn’t 3.14 metres long.)

As I was looking at the turtles, I was struck by these numbers and their potential for a maths class.

Somewhat frighteningly, 17.4 tonnes of live turtles are shipped from Vietnam to China every day. If the turtles are 2kg each, how many is that? How many of our classrooms would this many turtles fill up? Would it fill the school hall? How many of them is this in a year?

What assumptions are we making in modelling these?

Is 17.4 tonnes a day plausible? How many turtles are there in the world?

It can be (legitimately) that they are given very little space and ‘we’ aren’t concerned about their comfort levels.

Wrong, but Useful episode 41

On this month’s episode, myself and @icecolbeveridge are joined by @evelynjlamb, who is Evelyn Lamb in real life. She writes the Roots Of Unity column for Scientific American.

Nice numbers

We discussed the idea of some numbers being ‘nicer’ than others. This really is a bit a weird topic and hard to describe but I think we all have an inner feeling of which numbers seem nice. You may have heard someone say, “ooh, let’s make it a nice, round number” for example. Alex Bellos has written about how people have favourite numbers and that the most popular choice is 7. My current favourite number is Belphegor’s Prime:

1000000000000066600000000000001

3s, 6s, and 9s

I also talked about taking a string of 3s (or 6s or 9s) and squaring it. See what happens.

33333² = 1111088889

15* kievs

We talked about when 15 = 16 or when 15 is a square number:

Some questions that came from twitter:

  • What are bounds for the radius of a kiev?
  • What’s the minimum number of kievs you might get in a bag?
  • How can they be so precise about the amount of salt in a kiev that seem to vary in size?
  • If the * is a power, what does it equal?
  • What percentage extra did I get?
  • How many chickens were harmed to make this bag?

A game or two (Two. It’s definitely two.)

I also made a hash of describing a game called Areamaze. Evelyn did a better job of describing Euclid, the Game. Try them yourself!

Thanks for listening. If you’d like to be on an episode, let us know.

TES resources 5*

As a member of the TES maths panel, I review resources periodically. Here are the highlights from this batch, all of which I awarded 5 stars to: (for reference, I look through about 50 resources so you can tell these ones are good!)

Dividing by Decimals by amurra8827

This is a very good resource. It has a lot of questions to do with decimal division and explores it in a sensible, understandable way.

Year 7 Place Value by mathshub

An exceptionally well planned power point with a clear level of deep understanding and thinking involved.

C1 and C2 revision cards by mrsmorgan1*

Absolutely excellent for revision. I can’t really believe I haven’t seen something like this before!

*I had credited this resource to someone else but they appear to have simply taken Jo’s resource and uploaded it as their own. I’ve removed my review for them and reported the resource.

They also inspired me to write and upload two new resources of my own:

Quiz for GCSE Maths Formulae to memorise

Frequency trees – exploration and explanation

And here are the ones I rated as 4 stars:

4 Times Table Mastery Check

https://www.tes.com/teaching-resource/4-times-table-mastery-check-11418719

A formula for Cubic Equations

https://www.tes.com/teaching-resource/a-formula-for-cubic-equations-11482366

Angles Rules in Regular Polygons (Exterior, Interior, Central, parallel lines)

https://www.tes.com/teaching-resource/angles-rules-in-regular-polygons-exterior-interior-central-parallel-lines-11448318

AQA Further Maths – Unit 2 Algebra II – Rearranging Formulae.

https://www.tes.com/teaching-resource/aqa-further-maths-unit-2-algebra-ii-rearranging-formulae-11413527

Confidence Intervals demonstration on Excel

https://www.tes.com/teaching-resource/confidence-intervals-demonstration-on-excel-11464752

Exploring the equations of circles

https://www.tes.com/teaching-resource/exploring-the-equations-of-circles-11436039

Halloween Equations

https://www.tes.com/teaching-resource/halloween-equations-11397398

Maths Grid Game

https://www.tes.com/teaching-resource/maths-grid-game-11409413

OCR Maths: Foundation GCSE – Check In Test 8.04 Properties of polygons

https://www.tes.com/teaching-resource/ocr-maths-foundation-gcse-check-in-test-8-04-properties-of-polygons-11411699

Order of Operations

https://www.tes.com/teaching-resource/order-of-operations-11393898

Shadow Shapes

https://www.tes.com/teaching-resource/shadow-shapes-11460687