Saturday, 21 April 2012

Cross Number Puzzle

Have a go at this Cross Number Puzzle created by a boy in Grade 6. I might give some hints and solutions in the "Comments" section, if I can work out the answers myself...

Across
1. When fractions aren't tasty enough
3. You won't find the answer that easily
4. A calculator has to pay the __ __ __ __
5. "Half a lucky number, half an unlucky number, then twice!" - Julius Caesar
8. It's square, prime AND prime x triangle AND ninth power - prime...
10. 5th show on Channel 7 x perfect? Only second grade...
11. Oranges? No, pears...I have 5 (NOT 10)
13. The anti-nine of 29296
14. 10[SHIFT6]2 - 2[SHIFT6]3 less than 10 across

Down
1. It's letters are in order, but its numbers are backwards
2. Cubed, it's 205 x 10[SHIFT6]3 rounded to the nearest 1000
4. A + B + C = 18; A x B x C = 196; ABC = __ __ __
6. Put a 9-sided square with what 4 x 5 chewed to make a perfect match
7.    10[SHIFT6]6 - (mailbax + 1 ion)
[LUCKY?]
8. [Dwarves] x [seas] x [sins]
9. Bury 18 and 13 in a square
11. 50% evil
12. The answer according to a colour?

All rights to this puzzle remain the property of the writer.

Tuesday, 17 April 2012

Tips for Successful Tiger Hunting

Introduction to Tiger Hunting

Many times I have seen enthusiastic tiger hunters come unstuck because they failed to follow a few simple guidelines. Please consider these tips before your next expedition.

1. Preparation is Everything

You don't want to be caught out in the middle of the jungle when you suddenly realise that you're not too sure which end of the twacker to hold. Practice the skills you are going to rely on until they are second nature. It's too late once the hunt is on. Also make sure you have all the kit you might need. You won't always be able to pop home for a jiffy to get another case of Pimms if you're caught short.

2. Carry More than One Tiger Catching Device

Don't rely on just the one solution to the tricky business of hunting a tiger. The successful tiger hunter carries a variety of useful devices and tools that he is able to employ in the many different contexts in which he will find himself. Be prepared to consider the challenge from multiple perspectives.

Remember, a big gun is not always the most effective tool.

3. Teamwork

Work together. Cooperate.

4. Some Days the Tiger Will Eat You
(Some Days You'll Eat the Tiger)

Be prepared to admit defeat. Sometimes it is better to fight and run away but always remember

5. In Conclusion

Inquiry in Mathematics should have many of the elements of a tiger hunt - the search for the mysterious, the excitement and adrenaline of venturing into unknown territory and the thrill of pursuing the elusive shadow - all without the profound calamity of driving to the brink of existence a noble and beautiful animal.

Footnote - There are no wild tigers in Australia.

They must have been hunted to extinction by enthusiastic mathematicians.

Permission to use these photos is pending

Monday, 16 April 2012

I Can't Do this! It's Impossible!

Is this a valid response to a maths question? Is this an appropriate way for any student of mine to speak in class?

Sure. Particularly when the question being asked was deliberately.....impossible.

So What Was The Problem?

The kids were asked to make a square. Then, using one single straight cut, they had to produce an equilateral triangle.

The results were many and varied.

What did they do?

Many kids rushed straight in and went "Snip! - Done it!", holding aloft a beautiful triangle. But was it equilateral? Well, maybe not but the problem was half solved - at least we had produced a triangle.

That Beautiful Moment of Confusion and Panic

Next ensued that period of chaos as kids rethought their initial ideas and tried a variety of different techniques. Not surprisingly, every triangle produced was a right-angled triangle.

And then, the Moment of Revelation

Having battled with the idea for 15 minutes, one student finally sat back and said, "This is impossible!" Whether he meant, "This is impossible for me." or "The is mathematically impossible." I'm still not sure but it doesn't really matter - he had come to the realisation that maybe there was no solution.

Almost simultaneously another, more mathematical, student observed that the hypotenuse (not her word) would always be longer than the other 2 sides, no matter how much longer you made the sides. Focusing solely on the sides, she realised that she could not get them all to be the same length.

And then the conversation was deftly redirected (by me) to consider the internal angles. To be an equilateral triangle, all angles needed to be 60 degrees. Suddenly, the kids saw that no matter where they cut a line in the square, there was always going to be a 90 degree angle.

Is it Fair to Ask Impossible Questions?

Fair? Unfair? Right? Wrong? Is there a moral dimension to this? - I doubt it.

What I do know is that this provocation produced some good thinking. Kids had to work from what they knew about shapes to think about what they were trying to do. When they exhausted all their possibilities, they were left with a solution - maybe I've proved that it can't be done.

Sunday, 15 April 2012

Can you change shape?

The Shape of Things in 4BF

We set out to see what shapes we could "cut" from other shapes using only a single straight cut.

Here's how we went.

1. Can you make a rhombus from a parallelogram?

Not bad for a start - hope those sides are even in length.

2. Can you make a parallelogram from a rhombus?

That bottom shape looks a bit uneven.

3. Can you make a rectangle from a square?

Yep - not too hard

4. Can you make a trapezium from a hexagon?

Nice work. Cutting from corner to opposite corner

5. Can you make a triangle form a rhombus? Can it be an equilateral triangle?

Hmm, I can do the triangle bit but I don't think they're equilateral are they? Might need to tweek the angles and cut on the other diagonal.

6. Can you make a triangle from a trapezium? Can it be an equilateral triangle?

Yes I can - as long as one of the existing corners on the trapezium is 60 degrees.

7. Can you make an irregular pentagon from a regular pentagon?

Careful to cut from one of the corners so that the remaining shape still has 5 corners and an extra one isn't added. Great move!

8. Can you make an equilateral triangle from a square?

Like a rabbit in the headlights, we hit a problem and froze - can't go forward, can't go backward. What do I do? Nothing - I just leave the square as it is.

Conclusion

Some interesting thinking revealed in this exercise. Students showed understanding of the properties of 2D shapes. Most found solutions without having to resort to cheating (cutting multiple lines, cutting lines that weren't straight etc).

The activity was designed to culminate in an impossible problem. You cannot cut a line through a square that will result in an equilateral triangle because wherever you cut you will include a right angle.

So how did the kids handle this? How did they respond?

Stay tuned for the next post, "Asking Impossible Questions"

Footnote

Thanks to Mr Richard Black (@CapitanoAmazing), colleague and author of the blog "When 4th Grade Kids Ask The Big Questions?" http://4thgradebigquestions.blogspot.com.au/, for his inspiration, creation and initiation of the activity described above.

Monday, 9 April 2012

Some great ideas for investigating "Time'

So you want to investigate "Time"?

Investigating how to measure time? Here's a few different ideas that you might like to use as a provocation to get the kids thinking.

One-Handed Clock

I got this idea from a colleague on Twitter (@turtletoms) and I think it sounds great. Fortunately I dropped my classroom clock last week when I was taking it down to change for the end of daylight savings. The clock ended up in pieces but I should be able to resurrect it for school tomorrow morning.

So, remove the second and the minute hands from the clock and leave only the hour hand. Use this as the timepiece for the day (or week) so that the kids get the idea of what the hour hand does. Where is it at the half hour? At quarter past? At ten minutes to? Hopefully they will also get to appreciate the need for the other hands. A great way to explore what the clock does and the relationship between the parts.

Digging a bit deeper, it seems that clocks originally only had one hand. People living in countries with very old clock towers may be familiar with this fact.

Timeline - Till Rolls

I acquired a box of till rolls from my wife's pharmacy when they upgraded their cash registers to a point of sale system a few years ago. The new machines use different sized paper so I got all the old ones. The long strips of paper are really useful in lots of Maths investigations.

I gave the kids a long strip of paper each. I then asked them to divide the paper into 24 equal sections - we used the width of their ruler for convenience. Then they drew in each box what they were doing in each of the hours and labelled the times appropriately. If you wanted to introduce the idea of a time-scale, you could make each section 6cm long so that 1mm = 1 minute. The strip would need to be 144cm long (24 x 6cm).

You can also use other time scales - weeks, months, years etc.

I have also seen a strip of paper wrapped around the circumference of a wall clock and the parts of the hour (half, quarters, 5 minute increments) were then marked on the paper. When the strip of paper is unrolled, the students can see the "hour" as a length and each of the time divisions as a fraction of the whole or as a number line.

School Bells

A favourite little Glasga school yard song goes....

The bell, the bell, the b-i-l

Tell my teacher I'm no' well
If I'm late, shut the gate
The bell, the bell, the b-i-l

Apparently poor spelling is a consequence of truancy.

Anyway, at our school we have a bell that rings to indicate break times, probably much like most schools around the place. I am a bit perplexed that everyone, children and adults alike, accept that the time the bell rings is the actual time it is meant to ring.

You can use one of many on-line clocks to check the accuracy of your school bells:

http://www.thetimenow.com/ - this is not very easy for people with visual impairment to see

www.thetimenow.com - this one is easier to see
http://www.timeanddate.com/
https://greenwichmeantime.com/
http://www.time.gov/ - time in USA

..and there are many more.

Put the digital clock of your choice up on a screen and keep a record of when your school bells rings. Compare this with when the bell is supposed to ring.

How many days old are you?

There are web-sites where you can plug in your birthdate and it will calculate your age such as:

http://www.capoferri.com/howold.htm
http://www.time-for-time.com/howold.htm
http://www.onlineconversion.com/howold.htm

...and the list goes on.

But, before showing these to the kids, I would be really interested to see how they would go about working it out. This is not an open-ended task but it has great potential to explore problem solving strategies. Allow significant time for kids to work through a series of mistakes and miscalculations (we call these "opportunities to learn" in 4BF). See if they can come up with a "formula" that can be applied to any birthdate.

Then I'd let them loose on the websites to see how accurate their own calculations are.

Stop the Clock

And just for fun, a favourite website of mine that is pure enjoyment - the kids and I all love the big bang noise when you finish and it says, "STOP.....THE.....CLOCK!"

Give it a go:

Maths Clock

And here is a picture of the classroom clock from my daughter's maths classroom:

Thanks Mr H for the inspiring idea!

Historical Note

I was reading a book last year called "The Case of the Hail Mary Celeste", a speculative fiction book by Malcolm Pryce. In the book, Pryce mentions that time zones were not formalised until the advent of railway lines. The railway timetable meant that some sort of standardisation was needed across countries where previously each town and city had had marginally different times based on their meridian of longitude.

"No way," I thought. "Interesting idea to make your book that bit quirkier than it already is..."

But he was right. From about 1840, the UK, USA, India and Europe all had to find a way to

Across England there were towns that "held out" against the imposition of "London time", such as Bristol and Exeter. In Bristol, for example, the town clock introduced an additional minute hand two, one to show "London" or "Railway" time and one to show true local time.

Bristol clock with and extra minute hand.

...and the book that prompted my personal inquiry into time.

Friday, 6 April 2012

How did they measure time back before time began?

Introduction

In 4BF we are currently investigating "Time". I wasn't really interested in printing off pages of clock faces from the free download sites. I wanted to get a bit deeper into the concept of measuring time itself. Yes, we did spend some time revising our skills and discussing the mechanics of telling the time but the focus was always going to be on something bigger and better.

Here's where we went...

Using Sand Timers

After a bit of a discussion on historical lines, the kids identified a few early time-measuring devices. We had a good collection of sand timers in the storeroom, so we took them outside and had a play.

Activity 1 - see how many times you can walk around the quad before your sand timer runs out. Repeat this three times.

A pretty basic measurement activity but highly enjoyable.

Using Water Timers

Activity 2 - see how long it takes to empty your water bottle into the garden. Repeat this three times.

By the time we got to the second activity, the kids were beginning to question why they had to do each activity multiple times.

Using Rice Timers

Activity 3 - see how long it takes to pour a container of rice through a funnel and into a second container. Repeat this three times.

We used stopwatches to time Activity 2 and 3. By timing each activity multiple times, the kids were able to see the variability of each method and question the accuracy of water timers and such devices.

SO FAR...

Well, so far it's been a lot of fun but it was still pretty much teacher-centred. I asked some questions and gave out some equipment and the kids went off and explored the way I wanted them too. It was time to loosen up a bit and see where the kids wanted to head.

Activity 4 - design and build a device to measure 10 seconds.

After some more discussion about measuring time and what it all meant, we agreed that it might be interesting to see if we could build out own timing device. Some suggestions were going to be a bit impractical (in a school context - like how to measure really long periods of time like seasons or years - but it showed that someone was thinking big picture ideas) we settled on measuring 10 seconds.

The children were given two days to work on their device, some class time but most kids worked at home on testing and producing the device. Results were all really exciting and the kids spent significant time trialling and modifying their designs. Here are a few examples of what was produced:

A simple "sand" timer using 2 plastic bottles and rice. Interestingly, the timing was faster when pouring from the bigger bottle to the smaller than when it was inverted. This provoked much discusion about why, what caused this, did the sizes of the opennings matter, etc

This was a rather complicated, but effective, arrangement of three plastic cups connected with string. The red-coloured water in the top cup (note the fill linein black) runs through a hole in bottom, down the string into cup #2 and then through a hole in the bottom of that into cup #3.

Following more conversation about alternative timing devices, three kids came back to school the next day with candles that they had scaled to measure time. One girl had bought 2 identical candles and set the timer on her iPod to beep each hour. She light one candle and saved the second one. Then she marked the scale on the unused candle each hour when the iPod went off. I loved the juxtaposition of the iPod and the candle clock!

Here's a pic of one of the candle clocks. We have a few more in the pipeline and will post them when they are ready - hopefully tomorrow.

All in all, lots of fun, some good inquiry, lots of questions and I think we all learnt something about measuring time!

Authentic Inquiry Maths

Did you ever read a book called “The Number Devil”  by Hans Magnus Enzensberger? If you ever see a copy of it, grab it and have a look. In chapter one, Robert, the hero of the story, meets the Number Devil. The Number Devil explains to Robert that knowing a little bit of arithmetic, such as addition and subtraction, is quite useful for when the batteries of your calculator run out but really it has little to do with mathematics.

How often do teachers fail to grasp the distinction? How often do we overstate the importance of the “skills” that we fail to recognize the importance of their application? It’s like a football team that focuses so much energy on their training sessions that they forget to turn up to play their game.

The place of skills training

Make no mistake – a good level of competence and fluency with operational skills is useful in producing fast and accurate calculations.

Make no mistake – the skills are not an end in themselves.

TRAIN for the GAME

Just like a sports coach, I believe that students need to train to develop skills.

By “TRAIN” I mean:

Tedious Repetitive Activities Involving Number

Just as in a sporting context, what we do in training is useful. A football coach will drill players over and over until they get mastery of the skill.

But this is not the end of the process. There are no trophies for the team that trains the best or has the best drilled skills session. The skills need to be employed and articulated in the game context.

By “GAME” I now refer to:

Genuine Application of Mathematical Experience

Children have a great capacity for learning and replicating very sophisticated processes and operations. Does this necessarily imply understanding? I’d be inclined to say no – I can’t really observe understanding until I see how students apply their knowledge or skills to a real-world problem.

Just as the coach wants to see his players apply their ball skills in the game, I want to see my students apply their maths skills in the real world.

So what is Authentic Inquiry Maths?

Mathematics is a language that helps people understand the world in which they live.

When we get students who are keen to apply their knowledge and skills to make sense of the world around them, then we see authentic inquiry learning happening.

To get to this point, there are a few things that I feel are central to authentic inquiry in general and maths in specific:

1.     Inquiry is in response to a provocation or problem drawn from real-life experience.
2.     The provocation may be chosen or decided by the teacher but the direction and focus of the inquiry is determined by the student.
3.     Inquiry is an open-ended process that follows a cycle – good inquiry leads to further provocations and problems to investigate.
4.     Collaboration is central to effective inquiry to ensure multiple perspectives and relevance to a diverse audience.
5.     An inquiry moves the student from the known into the unknown. It needs to challenge the student to question his or her own understanding.
6.     An inquiry is transdisciplinary. In maths, this means a shift in thinking away from “pure maths” to a focus on “real maths”.

On this framework I intend to develop my own exploration into the world of maths and maths education. As a teacher and learner, I aim to follow my own advice – to TRAIN for the GAME.