Folding Fractions

Skills Practiced:

• Spatial reasoning
• Fractions of shapes
• Multiplication of Fractions
• (Potentially) Pythagoras

Material:

• Paper cut in to squares
• Display Board

Instructions

• Each student gets a square piece of paper.
• Ask if anyone can fold the square into a shape that is 1/4 of the original size. Ask them to explain how they know that it is exactly 1/4.
• Project the fractions on the picture below on to the board.

• Students aim to make those fractions from the original square.
• Every time they get an answer, they must convince their partner that it is correct before they put their name on it and blue-tack it to the board. Multiple different answers are allowed for each fraction
• Below this (or on another board(, students put up any other fractions they can find.

Example work

Extensions and Variations:

• Find the same fractions as above, but now your final shape must be a square.
• https://www.youcubed.org/task/paper-folding-fun/ 

Interactive: Polyrhthmic fractions

I've been coding again! See my latest maths game below...

I was inspired by how some of favourite musicians (Joanna Newsom)
overlap simple rhythms containing quarter, sixth and eighth notes and make something that is subtly complex and interesting. The above version is a much more advanced version of these polyrhythms.

I thought that this was an interesting way to look at how fractions add up. I also wanted the game to be a little more informal and sand-boxy than the last, so it allows you to try things out and see what works. Have a play:

 

Direct students to a fullscreen version with this memorable address: thedrummer.rocks.it

Its not perfect. Some of the problems are solvable, others aren't (because my original file has corrupted!!!)

• there's no way to tell which achievements you've done (i was implementing this when the file corrupted)
• you can't save and share your beats, which would have given an incentive to make a good one.
• the tradeoff with it being informal and intuitive is that its gamable and you can get away with using little fraction knowledge - I would solve this by having a worksheet of questions to answer (but only once people have already got in to playing the game). It feels like a cheap cop-out though. What could I have done to make it organically more challenging/rigorous? I would love your thoughts on this.

Lesson Sketch: mathematics of a cake sale

Intro:

There is a cake sale coming up and you plan to sell these flapjacks:

How much should you charge for one? Why? Will that make me a profit? How do you know? How much profit would I make? How could I work it out?

Main:

Give pupils the recipe: http://allrecipes.co.uk/recipe/3846/apricot-oat-flapjacks.aspx and either a table of prices or access to a supermarket website, such as http://www.tesco.com/
Pupils work out how much it costs to make a batch of 8 flapjacks and the profit they could make on each one sold.

Extension:

What are the most expensive parts of these flapjacks? Try to replace some things in the recipe with something similar in order to make the total cost lower. How much profit would you make now? How cheap could you make them?
Can you find a different cake recipe that would be cheaper to make?

Further Discussions:

This cake is slightly cheaper to make than the flapjacks:

Should I make these instead of the flapjacks? Why? Why not? What other considerations should I be making apart from price? (Time, effort, equipment needed, etc.)

You may discuss how some people factor time as money using a basic wage.

Lesson Sketch: The Money Game - Adding and subtracting from negatives

This is an adaptation of the bank balance activity found here. Basically students pick up a card and either add the money on the card to their account or take it away. To make it a little more interesting and to force them in to negatives (what I wanted them to be practicing), I made it in to a board game. The board is in the final slides of the presentation below:


Link to document       Link to cards and balance sheet (the larger amounts are for the purple, 'paid in' and the red, 'paid out' cards)

The game is kinda rubbish; there's no choice and it just involves calculation after calculation. However, I thought I'd take a leaf out of Shawn Cornally's book and leave it that way:

The point of Laser Pandas was to be a bad game. My premise is, and always will be, that teachers take far too much of the cognitive load onto themselves. We get caught up in the minutia of lesson planning and quickly forget that this is where all the thinking is done.
My students took to Laser Pandas quite willingly, but for the wrong reasons. As you can imagine, teenagers in a school will flock to nearly anything that doesn’t smell like sitting in a seat and listening. This is embarrassing for what we call “education”, and only good for me because even my most paltry attempts at fun will be considered monumental by comparison.
Here’s the point: The students are to create new rules for the game. They are to make it more accurate.

My game worked pretty much exactly how he has described; the kids enjoyed it because it didn't look like a normal, boring lesson, and the second half of the lesson, when I asked them to improve the game, was where much of the reasoning skills and creativity was shown.

I was really impressed with the sort of improvements they made to the game, from the simple to implement, such as 'calculator tax' (you can use a calculator whenever you want but it costs £10 each time) and cards with interesting maths puzzles, to complete game overhauls, such as having wage and bill cards (which you get at the start but take effect each turn) and having the dual goals of ending with most money AND getting married/having children/obtaining the material goods to look after them.

I enjoyed teaching this lesson. I am embarrassed that this crappy game was more appealing than my regular lessons. One day I want this not to be the case.

Lesson Sketch: Land speed

Hopefully I'll get back into blogging soon, but for now here's a quick puzzle about converting units:

Have a guess: which of these animals is fastest/second fastest/slowest?
Using the measurements given, put the animals in order from fastest to slowest.

Cheetah - 70 miles per hour

Snake - 9 metres per second

Human - 751 metres per minute

Grizzly Bear - 1159 kilometres per day

Reindeer - 52 kilometres per hour

Elephant - Can travel 1 kilometre in 89 seconds

How much faster than Ussain Bolt could a cheetah run the 100m?

Answer:

Year 7 Maths Attack! follow-up

Follow-up to my post on taking year 7 pupils to teach in the primary school. It seemed to go great! All the year 4s and 7s enjoyed it (though the year 7s got a little bored of having to do the same activity four times). The twister and top trumps worked perfectly and the year 7s were giving some great little hints as the young 'uns got stuck. Everyone enjoyed the making shapes activity, but the year 7s couldn't think up enough shapes on the spot. When we do this next week I will make some cards with shape names on. The year 7s still have to remember what the shapes and their properties are.

The only activity that didn't really work was the estimation game. Turns out if you give year 4s beads, without a teacher around, they will throw them at each other. My year 7s, who didn't know how to stop them, decided to join in and I had to sit with that group most of the time. I think that I will exchange this game for something else, probably some arithmetic games with a 20 sided dice my class can make.

I was really happy with how it went (as you can probably tell). The year 4s had a good time, and the year 7s got alot of practice of their own skills without ever really realising it.

Year 7 Maths Attack!

The school I'm at has a primary school part, which, though attached, isn't very integrated with the larger school. I've been planning an event where a small group of my year 7 group will run workshops in a year 4 numeracy class. There will be four activities all going on at once, and the year 4s will rotate to each in small groups. I'm hoping that my year 7s will be up to this, so that I can spend some of my time taking photos and not spend all of it helping them out. Anyway, I thought it would be nice to share the activities I will be doing and the resources that go with them.

1. Numerical twister: Like normal twister, but each spot has a number on. The caller calls out a number and you have to make a sum that is equal to it. Can increase difficulty by saying that the sum must include 3 numbers or must include a multiplication.
2. Fractions and Decimals top trumps (google docs seems to mess it up a bit, but just click the link to download the original): Top trumps cards with a combination of decimals, fractions, large numbers and fractions of amounts. This is the one I see most potential problems with, and I will need to train my year 7s on what to do when the year 4s don't know how to work it out (hopefully they will be able to show them using a mini-whiteboard).
3. Estimation game: pupils have sock puppets on (making it a bit more appealing, but also to make it harder to feel out the answer). They each grab at some beads in the middle then try to estimate how many they have. Once pupils get good at this, add a question at the end of guessing how much everyone has (activity stolen from here - just under 10 minutes in).
4. Making shapes: Have a large elastic circle. Pupils step inside it and stretch it apart with their legs, trying to work out how to make different shapes (activity stolen from here).

I'm really hoping that I can pull this off. I'll let you know how it goes.

Except that it doesn't work!

So in my last post I got all excited about the Helter Skelter video from the BBC clips site. I feel like it is my first true WCYDWT.

How I would have done it

• Play the video of the man building the helter skelter. Either play the whole thing or pause it half way, depending on how good the data comes out.
• Ask "How long would it take to build it twice as high/complete it?" or possibly a better motivator for a formula, " How long would it take to add another 100 sections?"
• Someone will probably say "twice as long". A bright spark may or may not notice that it will take him longer to walk up it each time as it's getting longer.
• At this point we take guesses and discuss how to solve it. Pretty soon we will be collecting data and plotting it. If we just plot the time as he completes each section, we end up with a curve, which is hard to extrapolate from, but splitting it into amount of time taken for each individual section we get a nice straight line. Pretty soon we're finding the sum of an arithmetic sequence.

How exactly to work smoothly through this last point I had yet to work out. It's become moot though, as the data just doesn't work:

 This is a graph of the number of sections done against total time when completed. It seems like a fairly nice curve (after you've accounted for the times he goes off to do other things), if a little too un-curvy to convince anyone that a straight line wouldn't do. However, plotting the time taken for each piece to be put in place we get this:

My heart sank when it came out. Its just a bit rubbish really; a totally unconvincing line that just doesn't reward us for all that hard work. As painful as it was too see it go wrong, I feel that I've learned alot from trying to work out how to pitch this in the class and I'm still pleased with the initial idea.

I guess now I leave this idea to die, until the fair next comes to town.

Lesson Sketch: Scale

Inspired by this I took a couple pictures, such as a picture of me with their textbook to work out my height, for pupils to iterate this idea of scale. What I really wanted was to find a picture of a claw scratch on a door, but I think that what I did find is even better.

An online news article was talking about a big cat that was spotted in the rural village of Thrupp. It was sighted many times but there was no evidence of it till someone took this picture:

Its a great picture for scale and we had plenty of animated class discussion speculating on the size of the cat. What was great about this is that we could confirm the actual size as the cat was later caught on camera:
http://news.bbc.co.uk/local/gloucestershire/hi/people_and_places/nature/newsid_8491000/8491497.stm

Lesson Sketch: Compound Interest

 

I'm not entirely sure whether my pupils will have the same perplexity as I do with this. This is from the film 'Swordfish', but I seem to remember this happening in quite a few other films too (contact me if you can name any). The natural question I get is: "What, really? That interest that in my bank account amounts to such a piddling amount can take you from $400 million to $9.5 billion in just 15 years?"

The problem with this might be of course that a lot of my pupils may not have a bank account yet and they aren't maths nerds (yet).

What I'm going to do:

Introduce the session:

• Show the video.
• Ask what question comes to mind. Do we think their answer is too high/too low? How could we show their answer is too high (upper bound)?
• What information would we need to answer this question (Interest rates since 1986)? Where can we find this information (can lead them here if they can't find one themselves)?

Main section:

Pupils should be ok from here if they know how to find the multiplier. Pupils finding 5% first then adding it on will get frustrated at the number of repeated calculations they need to do and will be gagging for an easier way.

End:

No way I can see of verifying your answers unfortunately, so we will have to compare our answers with each other then share our disappointment with the accuracy of the movie.

Extension questions:

• What interest would we need in order to get $9.5 billion from $400 million in 15 years?
• (Super-extension) How many years would it take if we had a fixed rate of 7% interest?