240. Cogged up

It’s amazing what maths you see when you go for a walk along a canal on a beautiful afternoon. After helping a canal boat through a lock, the following problem occurred to me: how many times must you turn the handle to raise the sluice gate?

Fact: The sluice is controlled by a series of cogs. The handle turns a ratcheted cog with eight teeth.

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Fact: The handle turns a small cog with thirteen teeth.

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Question: The next cog has ten teeth on a quarter of it’s circumference. How many is this in total?

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Fact: This large cog is attached to a small cog with ten teeth, which lifts the vertical post.
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Question: From the picture can you estimate how many teeth are on the vertical post?

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Question: Given all this information how many turns does the handle need?
Extension: Look at this picture. What is the angle between the foot supports?
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239. Introduction to Arithmetic sequences

Here’s a quick post for all those of you teaching Arithmetic Sequences. Whether you are teaching Level 3 Algebra or the C1 A-Level module, the jump from GCSE Nth term to the form ‘a + (n-1)d’ can be unnecessarily tricky. To help with this I’ve written a starter booklet for Arithmetic Sequences. You can download it here:

Introduction to arithmetic sequences

By the way, if your students confuse the vocabulary ‘sequence’ with ‘series’ get them to think about television. A normal television series ends, so an arithmetic series must end too!

238. Duck repairs

Duck Brand tape – it’s brilliant!

You only have to look at post 234 to see a great learning opportunity with duct tape … but the crazy patterns on Duck tape make it even cooler!

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The older designs are often discontinued and reduced which is a great chance to bulk buy. Apart from the obvious craft applications you can use it for quick book repairs. It’s much quicker and neater than tape.

Torn spine?

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Cover come off?

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You could even use different patterns to identify different groupings.

237. Quick Starter

Don’t you just hate it when students forget basic key skills? Especially those at the higher end of Year 11 or studying A-Level, who should have a better core knowledge. What if there was a magic tool which began to address this issue?

Skills required

  • Comparing fractions
  • Trigonometric ratios
  • Simplifying surds
  • Rationalising surds
  • Pythagoras

Equipment

You will not need:

  • Worksheet
  • Powerpoint
  • Printer
  • Laminator
  • Calculator

Magic Tool

  • One board, with pen

Activity

Quite simply draw the four diagrams below on the board and ask the following questions:

Triangle Problems

  1. Which has the largest sine ratio: A or B?
  2. Which has the largest cosine ratio: C or D?
  3. Which has the smallest tangent ratio: A, B, C or D?
  4. Extension: Calculate the missing angles and areas (Calculator allowed)

It takes moments to draw the questions on the board, but the discussion can take some time and addresses several basic skills. You can change the numbers to adjust the level of challenge.

236. An Emerald Adventure

There are assorted holidays coming up and the weather is getting grim. Time to put your feet up and exercise your brain.

Mathematics of Oz

If you like killer Sudokus, logic problems and applied Maths, you’ll love this book. It follows the adventures of Dorothy Gale as she battles her wits against Dr Oz in her journey through the alien world of Oz. The problems are graded so you can work your way up to the harder questions.

You can download a free sample here: Cambridge Press

Buy on Amazon (UK)

I have used problems from this book with all ages of senior school student from able Year 8 to Further Maths A-level students. A word of warning though – check the difficulty level before you let students loose on the problems!

235. Which witch is which?

Whether you are on half term holiday this week or next, I’m sure you’ll have time for this little number skills starter.

worst witch

Image Credit: Jill Murphy, ‘The Worst Witch’ – a children’s classic, which I highly recommend.

Can you help Wanda, the Grand High Witch, to find the local reporter hiding at her Halloween Girls Night Out? Solve the number problems and unveil the imposter.

Which witch is which? (pdf)

This starter or homework activity includes order of operations, factors, prime numbers, addition and multiplying (written method).

Happy Halloween!

(Updated: 1st Nov 2017)

234. (Students) cubed

Here is a quick fun starter to get your class thinking about dimensions and volume.

Question: How many students can you fit into a metre cube?

The discussion will probably include:

  • ‘How big is a metre?’ (find a metre stick)
  • ‘Did you mean standing in a square?’ (no)
  • ‘How big is the student?’ (average – that answer annoys students)
  • And finally ‘Huh?’ (ask the person next to you to explain)

Whilst this is going on make a metre square on the wall and the floor, using duct tape. The inner measurements of the cube are 1m, the rest is just tape border.

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The Predictions
Draw up a quick tally chart of how many students they think will fit. A bright child will usually ask how are you going to find out. Easy …

Put students in the cube
Let them put themselves into the confines of the cube. Cue bouncy boys squashing up. Then remind them it can’t be higher than a metre. You might find it useful to have two spare students hold metre sticks vertically at the non-wall end to define the end of the cube.

We managed nine boys, plus gaps at the top for bits of a tenth boy – it wasn’t ethical to chop one up and sprinkle the bits. So we imagined the tenth person balanced on the gaps around their shoulders.

Ten? That is a new record for this activity!

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The Point

  • Volume is the space inside a 3D shape.
  • One metre cube is bigger than you think.
  • It’s a memorable activity to refer back to.

Even better if …

I’d love to get sturdy board covered in birthday (or Christmas) paper to put under and around the cube to start a discussion about surface area. You could make a big show of unfolding the cube and laying the wrapping out on the floor to form a huge net.

 

Note
I used to do this by taping metre sticks into a cube, but they fell apart easily. In some schools three metre sticks is a challenge, twelve would be a miracle find. Duct tape works much better!