exzuberant

How to get your math class SCREAMING

2013-03-17T10:54:00.001+11:00

I know students are not supposed to be using mobile phones in class for private communications*, but I couldn't help but smile when one of my students showed me a text message she had just received from a friend in the class next door : "What are you guys doing in there? We can hear you screaming!"

If you don't know who these five boys are,
then you're definitely not teaching at a girls school.

So a little context first (context is everything!): This class doesn't really like maths that much - they tolerate it - I try my best to make it relevant and pleasant, trying to raise their confidence and skill levels. We've been studying a fairly dry topic for the last few weeks - they've done reasonably well in the topic test but need more practice. Looking for something engaging to make the second half of a long double period interesting, I turned to Stu Hasic's Quiz Boxes.

Download Stu's Quiz Boxes at http://quizboxes.com/

Quiz Boxes offers a Jeopardy! style game with questions of increasing complexity organised into categories, with a high stakes question at the end. Students love this game - and with careful planning and implementation (you will need to design the questions) it makes for a terrific fun period with high levels of engagement and gets students doing a lot more maths revision than they might have otherwise intended :-). There are many ways you can use Quiz Boxes so I would like to share an approach I have found that works well for classes of all levels of maths achievement.

Quiz Design

Choose categories that students are interested in. Current hot topics are "One Direction", "Justin Bieber", "Beyonce", "You Tube Hits" and "In the Movies". Find whatever your class is interested in. Once they play the game, they will suggest topics to you. Since I don't know that much about One Direction, I go to Wikipedia and collect the factual information I need. Find some obscure information for the harder questions. Your students will be amazed you know something so detailed about One Direction - and infuriated most of them don't know it. I like to use student interests for half the categories, and use more explicit math topic categories for the rest.
Work maths into the "non-maths" categories. For example, my third question on One Direction was "What percentage of One Direction are boys?". OK - it is a simple question - but it reinforces the idea that 100% means "all". One question I found generated interesting responses was "How many records has Beyonce sold?" - which gave a good opportunity to explore estimation. Another One Direction question: What is the name of the band member who is last in alphabetic order?" Again - it's easy, but it gets some mathematical thinking happening.
Make the maths category questions easy at the start You want students to engage with the maths categories. I always start with easy questions - if you make them too hard, students will turn off - it's not a game any more. I save the harder questions for the 800 and 1000 point questions. I make the end-game question a more challenging - but doable - math question on the current topic.

The Quiz editor in Stu's Quiz Boxes.
I find I can reuse the quizzes across many grade levels,
making this an efficient use of lesson preparation time.

Playing the game

This game is so much fun, and the students get so excited, it's essential to have a management strategy.

Every group gets a chance to answer the question. This is perhaps the biggest change I make to playing the game: I don't have a "first-answer-wins" approach. In a classroom of 30 students, it's impossible to work out who gave the first answer and the noise levels are impossible if you go this way. Instead every group has a mini whiteboard to write their answer (you could use just a sheet of paper). Once I see a group has a quality answer (doesn't have to be correct - just interesting), I yell "2 minutes" and give all the other groups time to complete. When I call "time up", we look at all the answers and every group that has a correct answer gets the points.
Encourage group checking of answers. As the questions get harder and are worth more points, I ask each group to ensure everyone agrees on the answer before presenting it. This gives the group a chance to teach the content to each other. It's wonderful to see students try to convince each other their answer to a maths question is correct.
Noise level management. This is hard because it's so exciting. Never have you seen a class so interested in knowing what 8% of $200 is! As the noise level rises you'll have to calm the class down.
Prizes. I confess to motivating with a very small chocolate prize. I give one to every student at the end and don't buy into "but we won...." discussions - as far as I'm concerned everyone is a winner if they participated :-) Waving the packet at the start of the game gets their attention - but it's amazing how quickly the students forget about the chocolate and become obsessed with winning game points.

Special thanks to Stu Hasic who so kindly donated Quiz Boxes to the education community. I highly recommend you try Quiz Boxes with your classes. And over time you will develop a bank of quizzes which you can share with other teachers in your faculty - or maybe even at Stu's website.

Practicalities

Here's what you need:

A data projector (or an Interactive White Board)
A copy of Quiz Boxes - free download from Stu's web site
A pre-prepared quiz. It can take a good hour to design a quiz, but you will find you can reuse quizzes across many year levels and they stay current for several years. You might like to challenge your class to design quiz questions for a category - although this will take some time and planning.
Students arranged in groups - maximum six groups for Quiz Boxes.
Mini-whiteboards OR a pad of paper per group.
Solid walls between you and the classroom next door. Close your windows :-)

Getting the most out of graphing software

2013-02-03T11:15:00.000+11:00

"GeoGebra is your friend!" - my students must have heard me say it a hundred times. If a student asks me about a homework question, they know my immediate response : "Did you check what it looked like in GeoGebra?". If they haven't, then I will usually ask them to sit with me while we explore it together using the software.

Some teachers worry using mathematics software will weaken student's skills, but here's a mantra I recite in class which I believe not only develops mathematical skills but also stimulates deeper learning:

I believe the essential ingredient in using graphing software to answer questions is to stop and think before using the software and then predict what you expect the software to display. If you are fortunate, you'll find the software doesn't match your prediction. I say fortunate because you have discovered a misconception, an error - or in some cases, managed to confuse the software. Prediction and the subsequent reveal of an incorrect prediction is a powerful learning tool. With a positive attitude to the error monster this revelation will stimulate questions and further exploration.

Another key learning idea I advocate is to take a few extra minutes once you have your answer to extend the problem with some "what if?" questions: "What if I changed that positive x to a negative x? What if that was to the power 3, not power 2? What if that parameter was 4 not 5? Can I reflect that curve?" Here the power of the software comes to the fore: we can ask many questions and rapidly get answers - something not possible in reasonable time without the software. Of course students won't have the time to do this for every question, but even just doing this once in a study session is rewarding.

One more powerful pedagogical factor is at work when students use a graphing tool to help with their homework: they are forced to translate their problem into a representation suitable for the tool. For example, an algebraic equation has to be split into two (or more) graphs and intersections found. This serves to build and reinforce understanding of the links between the different forms of mathematical representation. Often a student needs break down the problem into steps, introducing parameters and intermediate results or constructions, providing 'hooks' they can use to explore how the problem changes as parameters are changed.

A topic I recently taught was based totally on drawing graphs by hand - and students have to be able to do this in an exam situation, without software. For a course like this, I think the graphing software is an even more valuable learning tool. Why check your answers in the back of the book when you can do this:

This approach means students are still learning to work by hand - and maximising the benefits of having software during the learning of the topic - without becoming dependent on it - a bad thing at exam time!

So to my way of thinking, there's no question dynamic geometry software is a powerful learning tool: when coupled with a mindset that thinks and predicts prior to using the software, and then extends a problem through questioning and exploration with the software - it's like having a personal tutor. GeoGebra is indeed your friend!

Practicalities: There's lots of good quality dynamic geometry and algebra software available to students: I'm a big GeoGebra fan, and I also like the Desmos tool. I'm beginning to really appreciate AutoGraph - but sadly the cost factor rules it out for most of my students. For intensive algebraic work, I point my students at WolframAlpha - especially the WolframAlpha iPad app which is great value.

Images of Integration

2012-12-16T14:28:00.001+11:00

Slice and dice: that's how I think about the calculus topic of Integration - take something complicated, slice into increasingly fine slices, then put it all back together. In my quest to encourage my students to see this theme in the wonderful world around them, here is a selection of images I used this term to help show the idea, generously made available by people around the world through a Creative Commons License on Flickr. If you're taking great photographs - think about sharing them under Creative Commons - a wonderful resource for teachers to help inspire students.

Bay St, St Louis Bridge by Alaskan Dude on Flickr

La Agora, be el.manu on Flickr

L'Hemisferic by el.manu on Flickr

Tower by timtom, on Flickr

Untitled, by SymoO, on Flickr

The idea of looking for visual representations inspired one of my students to take a photo of the magnificent Neuroscience Research Australia under construction across the road from our school - which just screams at me "Area under the curve!" every time I walk past it.

Neuroscience Research Australia building 2012 - under construction.
Photo by J Yu - used with permission.

This is part 3 of a series of posts on teaching Integration.
Part 2: Exploring Inequality - an entry point to calculus

Exploring inequality : an entry point to calculus

2012-12-15T14:09:00.001+11:00

"Have you ever noticed .... ", I said to my senior maths class, as I walked in bearing a huge and very obvious glass bowl containing about 40 packets of Smarties, ".. how some people seem to have so much more than other people?"

"Taking it Back - Occupy Oakland" by Glenn Halog
http://www.flickr.com/photos/ghalog/6271929376/in/photostream/ CC-BY-NC-2.0

I then proceeded to "share" out the Smarties: first I gave 20 of the 40 packets to one student - making a huge pile on her desk. Her eyes popped out - while the other students looked with disbelief and some concern for their own anticipated share. I gave a wicked grin and 10 packets to the student next to her. To the rest of the class I handed out 2 or 1 packets - apart from a few students at the end of line who received nothing. Oh the looks they gave me!

And so we started a lesson exploring the question of how we could measure income distribution - a hook (although the class didn't know it yet) - to introduce our next calculus topic: integration. Here are some notes on my first attempts at a lesson design using an idea from economics as a motivation why we might want to find the area between two curves. But first a big thank-you to mathematics teacher Alastair Lupton who showed me how to bring the Gini Coefficient into the classroom and encouraged me to try it out in my classroom.

So here's the sequence I tried this year.

Step 1: Build interest in the problem. With strict instructions not to eat or worse yet - share - their Smarties, we looked at a short OECD video about the rising inequality in income distribution:

Depending on the time available, you might want to explore some other video material, perhaps some recent news footage of the Occupy movement protests, or look at some studies of global income distribution.

Step 2: Thinking how to organise the data: I lined up the students, holding their very unequal distribution of Smarties. We ordered the line by 'income' and partitioned into 5 groups - helping the students see the organisation of the data into quintiles. We returned to our desks and looked at some local and international data on income distribution, also organised into quintiles. Here is some recent Australian data:

Click on the image for a larger view.
Source: Australia Bureau of Statistics 6503.0Household Expenditure Survey and
Survey of Income and Housing User Guide 2009-10

Step 3: Ask the question: "How could we measure inequality?" This isn't easy or obvious. Give the class some time to explore ideas. Then it's time to look at how economists do it...

Step 4: Develop the idea of graphing cummulative quintiles. After trying some different ways to plot our quintiles, I showed the students how the economists do it: reorganising the data into cummulative quintiles. This allows us to make normalised curves which work for all situations, regardless of the size of the total income pool. We drew our first Lorenz Curves:

The Lorenz Curve is used to calculate the Gini Coefficient. The area A is the difference from total equality.
The larger the area A as a proportion of the total area A+B, the greater the inequality.
Source: Wikipedia Lorenz Curve Image by Reidpath,

To help explore the idea, we discussed what the Lorenz Curve would look like if one person had all the Smarties, and if all the Smarties were shared equally. We also considered if the curve would ever go above the "Line of Equality" (it won't!). We selected different data sets (see references below) and plotted them. Here is the 1993 World Bank data for Nigeria plotted in GeoGebra, with a polynomial fitted to the curve:

By modelling the curve with a polynomial, we can use integration
to calculate the area under the curve and hence the area between the curves.
Data is entered into the GeoGebra Spreadsheet window, then plotted and
a function calculated to fit the data using FitPoly[].

Step 5: Ask the question again: how could we measure the inequality? After looking at a few different data sets, students will quickly come to the conclusion that measuring the area between the line of equality and the Lorenz Curve will give us a nice single number. And now you have them hooked: here's a very interesting and practical reason we might want to be able to calculate the area between two curves.

Step 6: Declare a communist revolution. I then ordered a redistribution of the Smarties so everyone was equal. This was actually quite funny because several of my diet conscious students insisted they did not want any Smarties. Tongue-in-cheek I told them this was not an option - it was a revolution and everyone had to be equal whether they wanted it or not! A nice opportunity to open up the discussion to different views about income distribution. I gave my students a selection of recent articles from The Economist which seemed to provide a good balanced discussion on the topic.

Step 7: Begin the mathematical discussion on ways to calculate the area between the two curves. Your students will have many useful ideas! Try them out with the tools available. And now you're ready to start a calculus based exploration: What is the area under a curve?

Where could you go with this lesson idea?

Get students to make up a small poster using their data and stick them up on the wall. Then as you move through the Integration topic, you can refer to them in the context of each new idea.
Once students know how to integrate, get them to model their curves as a polynomial - I like to use the GeoGebra FitPoly[]function - and then do calculate the integral, comparing their result to given Gini Coefficient for the data set.
The student data makes for a great application of the Trapezoidal Rule : they can calculate the area without knowing the equation of the curve. A good example of why you might want to use the numerical approaches to calculating integrals.
Challenge activity: calculate the area under the curve using Simpson's Rule. If you only have the standard Simpson's Rule, you can't do it because there are an even number of data points! But there is more than one Simpson's Rule - challenge your students use the internet to find one that will work for this data. [Hint: Simpson's 3/8 rule will work].
Apply the concept of the Lorenz Curve to another field of study. An interesting application is to social networks - some people contribute significantly more than others, while others 'lurk' in silence. I use edmodo with my class and there is a high degree of inequality in the number of postings per student - counting postings per students could make for an interesting Lorenz Curve.

Thinking beyond the mathematics:

Talk to the economics teachers at your school. I discovered mine do teach the Gini Coefficient, but they don't go into how it is calculated. I think it could be a very powerful lesson to develop a sequence of combined economics/calculus lessons with an economics teacher at your school. The more I explored the subject, the more interesting I found it. Options to consider include: the effects of taxation policy on the Lorenz Curve; the differences in the Gini Coefficient between different types of economies; differences within one country over a time series; challenges to the validity of the measure; economic and social arguments on the topic of income distribution. All highly suitable for deeper mathematical and social science exploration.
Take some time out to look at the Gap Minder website which options to view the data through the Gini Coefficient.

Resources

The Wipedia pages on the Lorenz Curve and the Gini Coefficient are a good starting place, with good entry points to more nuanced discussion on the use of the Gini Coefficient.
Some excellent Australian data and good explanatory notes in the ABS publication 6503.0 Household Expenditure Survey and Survey of Income and Housing User Guide 2009-10. See Table A1 in Appendix 1, and a high quality discussion on the Gini Coefficient in Appendix 3.
International data from World Bank: World Development Report 2000/2001: Attacking Poverty - Table 5 Distribution of Income or Consumption
The October 23, 2012 issue of The Economist contains some excellent articles on the challenges of income inequality seen from a pro-Free Market view. I found these particularly interesting given one could hardly call The Economist left wing!

Some teaching reflections:

The students really loved the lesson - they were engaged and it was interesting.
I planned carefully for my 'inequitable Smarties distribution'. Our class was well established and we knew each other well enough that my students would know I was up to something and trust me when I played this game. I also made sure the students who didn't receive Smarties were the most resilient, confident students.
I did however make the mistake of trying to do this opening lesson in a single 50 minute period - it wasn't enough time and I rushed it, making it less student centred than I had hoped. This lesson needs a double period to do it justice.
Is it worth taking the time out from a busy course to do this activity? I think so. Once I realised I could leverage this work into my teaching of the Trapezoidal Rule, Simpson's Rule, the area between two curves and also do some polynomial modelling, I saw it was a lesson that just "keeps on giving".
Coming from a physics background, it was wonderful to find an interesting and practical application of calculus to a completely different field. Many of my students are planning a career in business and are interesting in economics - here was something to show them the calculus applied to money as much as to speeding particles!

This is part 2 of a sequence of posts on teaching integration.
Part 1: Slicing and Dicing. Part 3: Integration in the world around us

Slicing and Dicing

2012-12-09T10:35:00.002+11:00

To my way of thinking, the topic of Integration is all about 'slicing and dicing' - thinking about what happens when you take an object and chop it into increasingly thinner slices, then put those slices all back together again. Here's a fascinating and gruesome hook I used in my senior mathematics class this year to consolidate* the theme of "slicing and dicing": What would happen if you sliced up a human being?

Warning: This content is only suitable for a senior class, and you should warn students there are medical images coming up. There won't be any blood, but it might affect sensitive students and the dissection of human bodies may not be culturally appropriate in your classroom.

First we start in reverse, using a scene from one of my favourite science fiction films "The Fifth Element"

Then let your students know the images of the human body used aren't computer generated, but actually come from The Visible Human Project. Cue in this video clip:

My students were grossed out and fascinated - and then asked to see it several more times! It took them a while to come to terms with the fact the images weren't generated using a medical scanning device, but by actually slicing up a body. Lots of questions followed!

Depending on time and if you think this is a good idea or not, there are some websites where students can use an online Java application to dynamically explore the data by selecting their own slices in any orientation and see the resulting image created by reassembling the original slices to your specification.

Here are two websites I found worth exploring:

http://visiblehuman.epfl.ch/index.php

Where to next? Many options for discussion about:

the mathematics and computation required to reassemble the data so that different views can be constructed.
the ethics of using bodies from condemned prisoners for science.
the value of the data from The Visible Human Project - there were scientific as well as ethical criticisms of the project.
Recent advances in 3D printing technology to "print" biological components using layers of living cells. A long term goal is to print transplant organs using cells from the donor. A quality video from ABC Catalyst program at http://www.abc.net.au/catalyst/stories/3618385.htm (starting at 00:03:00).

One of my students later told me the data from The Visible Human Project is also used in a (rather violent) Japanese manga film Gantz.

(*) I used this lesson idea in the middle of the topic sequence. For my first Integration lesson, I went down a different path - but that's for the next post! Part 2: Exploring Inequality

Still going ...

2012-12-02T08:30:00.001+11:00

Photo: "Yellow" by darkmatter CC-BY-NC-ND
http://www.flickr.com/photos/cdm/84202849/

It's been a very long and tough final school term. I'm still running the "marathon" - albeit limping on some days. Ran headlong into some very steep hills (teaching Mathematics Extension 2 for the first time, in addition to teaching Mathematics Extension 1 for the first time... madness!). Combine this with the normal teaching load, writing over a hundred school reports and accumulated sleep deprivation - not good. Running too fast, too hard - feels like I've done a year's work in a term. In recovery mode now - still hundreds of end-of-year papers to mark but only a few weeks to go!

Like all marathons though, the experience is amazing - the views incredible. Lots of teaching ideas share in this blog once my energy levels are restored.

The blue shark of full mastery

2012-10-20T10:15:00.000+11:00

"Sir, does mastery count for more than the test mark?", asked one of my Year 7 students this week. I beamed back - "YES!" Slowly but surely, I'm weaning this class to look beyond their scores ("You got 95%! I got 98%!" ... yes - it's a high achieving class :-) ) and focusing on mastery. Recently I have been making little mini-report cards which I staple onto the end-of-topic test paper:

My classes now have a symbolic language for achievement levels : the red dolphin stamp is 'Not Demonstrated' and 'Starting Out', the orange seahorse is for 'Progressing', and the orange killer whale is for "Mastery". If you get Mastery for all the standards, you also get the blue shark. I find the visual imagery helps focus on achievement of the standards. And it doesn't just work for Year 7 - even my Year 12 students like the blue shark.

My goal with these mini report cards is to make the standards and the student's achievement of those standards prominent - the topic test score is there, but it doesn't dominate the feedback. Why? Because even in this high achieving class, a score of 90% means there is something students can improve on - and I want to focus on that specific item. I try to write a helpful comment, focusing on the standards that need work and some ideas how the student can advance that standard. While the students are looking at their test, I walk around the class and try to chat to every student about their achievement in terms of the standards and what we can do to raise them (that can be hard with 28 students in 30 minutes!).

These little report cards though reveal a deeper change in my approach to Standards Based Grading....

SBG: Where I'm at now

Time pressures have taken their toll on my loftier goals of high precision SBG implementation - and I have found I'm migrating closer to what Frank Noschese calls "Keep It Simple Standards Based Grading. Now that I have simplified the system, I find it also makes it clearer and more approachable to students.

Less standards per topic - especially for junior classes. My lists are still too long for senior classes - mainly because I am trying to cover all the syllabus points (there are a lot!).

Achievement levels: I'm happy with the language of my achievement levels "Not Demonstrated/Starting Out/Progressing/Mastery" - I believe they give clear and honest feedback without being discouraging - they don't say 'you failed' - they say 'you're not there yet'. I'm not comfortable with a simple Yes/No binary decision because I want the levels to support my goals for student motivation and engagement - to reinforce they are on a learning path - I want to recognise their 'progress so far'. A sheet full of 'No' results isn't going to encourage lower achieving students.

The role of quizzes: I have effectively stopped using quizzes for grades. Woah - that's a big departure from the SBG ethos! Why? Because I believe that meeting standards once in a quiz isn't enough : the student has to retain the standard. So for me, the end of topic test does matter. If a student could demonstrate the standard in quizzes during the topic, but can't demonstrate them at the end of the topic, I think there is a problem. But I haven't abandoned quizzes - on the contrary, they are a key part of my formative assessment strategy. I still give regular quizzes and use the dolphins, seahorses, killer-whales and blue sharks to give feedback during the topic. I do record the quiz results to help direct my teaching of the topic. But the difference is quizzes taken during the teaching of a topic don't count toward grades. I save that for the end of the topic. If a during-topic quiz shows me a few students need help on a specific standard, I give them specific support. If I see many students need help on a specific standard, then I alter the teaching the next day and put this standard in the next quiz for the whole class. So I don't do repeat attempts on quizzes, and I don't try to juggle grades based on quizzes and quiz retries.

The role of topic tests: I use the topic test to decide the level of achievement for each standard and report this to students with their topic test mark. I do this by grouping test questions against standards - either explicitly in the test design, or working backwards from a preexisting test. This does mean marking takes longer, but it gives much more useful feedback than a single test score. The results should not be a big surprise because the quizzes have been giving the student feedback along the way. Retry attempts happen after the topic test, I give students the chance to improve their topic grade by taking quizzes or alternate tests for specific standards. That's how they can change their topic grade. In my grade book I have the topic test result (which stays constant), and an array of standards achievements which can be updated by retries.

Topic test result is recorded, along with initial end-of-topic achievement of standards.
I use red-orange-green traffic light indicators to quickly spot areas of concern.
Students can improve their standards results after the topic test by taking quizzes.

The final grade: I blend the topic test (snapshot in time result), with the standards achievement levels (which students can change through post-test quizzes) - giving more weighting to the standards indicators than to the topic test results. Why? Because I want students to have the opportunity to raise their grade through further effort. This reduces test anxiety and redirects the learning focus to mastery.

And back to the blue shark .... stamps are fun - kids (and teachers!) love them. And when it comes to assessment, having a discussion about whether you got a seahorse, a killer whale or a blue shark - well it just takes some of the sting out of assessment and helps everyone realise the symbol or the grade isn't what's important : it's working towards mastery that counts.

Woot! A blue shark!
I get my animal stamps from www.allyoucanstamp.com

A note on my constraints: I am the only teacher in my faculty using SBG - so I have to maintain the topic test results to allow for comparison across classes. With middle level classes, my grading system has to be consistent with other teachers (since we rank across the cohort) so my grades have to come exclusively from the tests. Perhaps one day I might be able to convince my colleagues to allow retries for the grading in these classes. For the senior years there are statutory regulations on assessment policy which are sadly high-stakes, single-attempt only assessments. So for the higher level classes, I can only use SBG to guide my formative assessment. My hope is that this translates into the summative assessment results.

Your thoughts? Have I oversimplified SBG? How could I improve this approach?

Polynomial stories

2012-10-13T10:07:00.000+11:00

What's not to like to about polynomials? They look amazing - and they are just great fun to play with - especially if you have dynamic graphing software to explore their shapes. Here are a few teaching ideas I developed over the last few weeks.

First and foremost we need a character : meet Polly the Amazonian parrot. There's a reason she is from the Amazon... you'll see soon.

Amazona agilis by Jacques Barraband(1767-1809)
http://commons.wikimedia.org/wiki/File:Amazona_agilis_-_Barraband.jpg

Polly featured throughout my lessons - my favourite was the zero polynomial $P(x) = 0$

Reminds me of a Monty Python sketch ....

and later when we looked at taking the second derivative, then the third, fourth, and fifth derivatives - the disappearing Polly:

The Disappearing Polynomial

I'm a big believer in having 'characters' to help teach mathematics - I think they act as 'mental anchor points' to help link related concepts, and then make high level linkages across topics more visible - as in the example above of the disappearing polynomial.

Then we need to explore the properties of polynomials. If we have a quartic polynomial, how does its graph change if we change the roots? I got a good response to my students with this homework exploration tool - I used it for a 'flipped' lesson:

Locus and Parabola Make a Parabola GHT0301

Lesson 02 FLIPPED Graphs of Polynomials

Early on in the presentation of polynomials, I think it's a good idea to show some of the interesting applications:

Using polynomials for modelling. I used this example of a photograph of the Amazon river, loaded into GeoGebra, fitted to a polynomial using the FitPoly function. Why would we want to do this? I suggested in this case, having an equation for the Amazon River could help us model water flow - perhaps helped by working out the gradient function:

Amazon River – photo from NASA.
Curve fitting using GeoGebra FitPoly[] function.
http://commons.wikimedia.org/wiki/File:Amazon_57.53278W_2.71207S.jpg

Using polynomials to approximate other functions: A good time I think to introduce the Taylor Series:

No need to go into deep explanations - just show what is possible with polynomials. I returned to this idea in the next topic when showing higher derivatives, and have another visit planned when we do complex numbers to help demonstrate the famous Euler Formula.

And finally, after demonstrating the closure of polynomial operations for addition, subtraction and multiplication, students may find it interesting to learn about the role polynomials play in many encryption systems.

I looked hard for an online paper suitable for advanced high school students - the best I have found so far is Christopher Cooper's notes for his "Languages and Machines" course at Macquarie University.

More GeoGebra HowTo Sheets

Newton and Descartes channel Dan Meyer

2012-10-04T18:14:00.003+10:00

There's a definite pause the first time you show parametric equations to students well conditioned to Cartesian representations. I like to imagine Descartes himself staring at the equations pondering : "Why would you do that????"

We're not in Kansas any more!
Descartes: "Why would you do that? It's the same end result!"

Here are three teaching ideas I used this year with my senior mathematics class which may supplement the traditional approach of showing the different representations are functionally equivalent.

1. Extend the function machine idea to show a weird new parametric function machine. Now we have two outputs! Here are the two function machine images I use for my resources:

Based on a function machine diagram at http://raider.mountunion.edu/ma/MA125/Fall2011/Chapter7/IntroToFunctions.html
I removed text from the original image, then adjusted it to make the parametric machine.

2. Explore the reasons why we might want to use parametric expressions to describe a relationship.

The best I answer I came up was this (click on the image for a larger view):

Newton and Descartes ponder Dan Meyer's "Will it hit the hoop" lesson.
My students did this activity in a previous lesson, so they got the joke.

In other words, a parametric description of this scenario lends itself to a deeper understanding of the physics of the situation.

Another reason for using parametric equations is that the maths can be much more interesting - and possibly a lot easier to work with. Parametrics also give us another way to get a feel for the constraints at work in a locus. I love this wonderful "move the robot" explanation from James Tanton - and it speaks to my IT background where parameter go in, and things move accordingly!

3. Get a feel for parametrics by controlling the parameter using dynamic geometry software. I found it really helped my students to build a parametric representation, then adjust the parameter by moving sliders and then seeing points move under their control. Actually touching and moving and parameter reinforces the idea of a point travelling along a path under a constraint. Here is a resources for students to explore parametric representation of the parabola using GeoGebra:

HowTo Guide: Exploring the Parametric Representation of the Parabola

This guide is part of my collection at GeoGebra HowTo

Toys and tools for exploring the Parabola

2012-09-22T08:51:00.001+10:00

Following on from three ideas to introduce locus, here are three ideas I used to help make the locus of the parabola come alive for my students. Regular readers of this blog will know how much I believe in the benefits of hands on exploration of mathematical objects - and these are very hands on!

The three ideas are:

Use a MIRA mirror to construct a parabola. My senior students loved this activity- a chance to revert to back to childhood, while still being challenging.
Use GeoGebra to construct a parabola given any arbitrary focus point and directrix. Try this with non-standard orientations.
Be entranced by a wonderful 3D optical illusion toy that exploits the properties of the parabola.

1. Using a MIRA mirror (MIRA math tool) to construct a parabola
I'm extremely fortunate to have a box of these in my faculty storeroom:

Source: http://www.enasco.com/product/TB14953T

While they look like tools for the junior math room (and they are wonderful to use in this context!), there's no reason our senior students should be locked out from using them! Here is a worksheet that give instructions on constructing a parabola with the MIRA mirror. It's a really fun activity - a chance for senior students to play a little - and a great opportunity to ask the "why" question - reinforcing the idea of locus and the locus definition of a parabola.

Locus and Parabola MIRA Parabola GHT0501

2. Constructing a parabola using GeoGebra.
Why do I need special help to construct a parabola in GeoGebra you may say? Of course GeoGebra can construct a parabola with zero effort. But this guide explores how to construct a parabola using the locus approach.

Can you find the parabola given any arbitrary focus
point F and directrix AB?

Locus and Parabola Make a Parabola GHT0301

3. Discover something special about the parabola
A terrific toy worthy of being in your mathematics (and science) classroom is the Optigone Mirage®.

The Optigone Mirage® is a pair of twin parabolic mirrors, arranged to project
a 3D image floating above the top of the kit. In this image from a paper by Christian Ucke,
the pig is actually inside the mirrors.

As always, encourage students to play with the toy (not that they will need encouragement - my students had their smartphones out takings photographs within seconds!), then ask the Why? question. Even though I purchased* one of these for my senior mathematics class, students across all my classes were entranced by it - and it gave me great pleasure to say to the juniors "you will learn how it works in your maths and science classes in a few years".

* Sadly I could not find a convenient way to purchase one of these in Australia - so I ended up buying a clone from Australian Geographic.

How do you do that in GeoGebra?

2012-09-09T13:15:00.000+10:00

Looking back on my own high school mathematics education, I realise I never really knew what a parallelogram was. I never knew how it 'worked', how its angles and diagonals operated, how they changed when the slope of the parallel lines was changed. The rhombus? All I could really say - if I remembered it at all - was it was a kind of squashed up square. If only I had been given a dynamic geometry tool to play with! As a teacher now, I strive to have my students actually touch mathematical objects - to move them, push them, pull them, to watch what happens. I'm convinced that if students do that, so long as they are reflecting on what is happening to the objects (and why), they will remember them for life. And the ideal tool for hands-on interaction: GeoGebra. Free software, runs on Windows, Apple and Linux (anything that runs Java), backed by a community of hundreds of thousands of teachers using and sharing GeoGebra resources.

A resource I haven't found yet though is a set of simple, one page instructions I can give to students showing how to construct a certain mathematical objects in GeoGebra, so I have begun building some.

Here's the first installment:

For Junior and Middle School

How to construct a parallelogram.

How to construct a rhombus. This one appears simple but can be confusing - practice it first before giving to students.

For Senior School

How to construct a parabola using a focus point and a directrix.
How to build a polynomial factor exploration tool.

I have put these links on a new GeoGebra HowTo page in this blog. These files are also available at the Maths Faculty sharing repository.

Three teaching ideas for introducing locus

2012-09-01T08:45:00.001+10:00

Here's three ideas I use for introducing locus.

1. What makes these shapes? Have you ever looked down from an airplane window and seen a sight like this? Something you will often see above large scale farming areas:

Centre Pivot Irrigation - an example of locus of the circle.
Photo: Etienne Boucher CC-BY-BY-NC-SA

http://www.flickr.com/photos/djof/147222315/

Give the students some time to digest these images - you will hear lots of wild theories.

Then show this picture and see if they can work it out:

“Centre Pivot Irrigation” Matt Green CC-BY-NC-SA

http://www.flickr.com/photos/imjustwalkin/4875949345/

The Wikipedia page on Centre Pivot Irrigation has lots more interesting information.

2. Acting the Goat. Tie yourself with a rope to a chair or desk and pretend to be a goat. Explain how goats will eat absolutely everything in sight. Model the behaviour. A good laugh - and your class won't forget the locus of a circle or the concept of a constraint determining the locus. Then extend the idea to different situations of a goat on a leash (tied to a fence with a sliding leash, etc).

Built the idea of a locus by "acting the goat".

3. Who cares about locus? Show an image like this:

Astronaut Stephen K. Robinson, STS-114 mission specialist,

anchored to a foot restraint on the International Space Station’s Canadarm2.
Photo: NASA http://spaceflight.nasa.gov/gallery/images/shuttle/sts-114/html/s114e6647.html

This person certainly cares about locus! Also a good opportunity to raise awareness of the International Space Station. More information on the Canadarm2 at http://science.nasa.gov/science-news/science-at-nasa/2001/ast18apr_1/

I said three ideas? Sorry - I can't resist sharing three more ...

4. "Locus Pocus" : A high quality video about locus well worth showing in class is Erica Morabito's Locus Pocus.

I like this video because it creates some memorable visual representations of specific locus problems that students need to know while also being entertaining and succinct (lesson time is a precious resource!).

5. Introduce the ellipse : Most students have an idea what an ellipse is, but very few know how to make them or the locus idea behind them. This YouTube clip give a good demonstration:

and I wrap up this discussion with a picture of my favourite example of an ellipse in action:

Locus of the ellipse at work in the universe:
a stunning NASA image of Io orbiting Jupiter.
http://www.nasa.gov/mission_pages/juno/multimedia/pia02879.html

Yes - another segue into astronomy. Locus is manifest throughout our universe - it would be remiss of us not to show such stunning images of it at work.

6. Work the Geometry 'vs' Algebra dynamic: I find it helps students to be explicit about the two different approaches to curves (geometric versus algebraic). It seems to me many students prefer the algebraic - so we need to work extra hard to show the virtue and value of the locus view. I reintroduce our fellow travelers Euclid and Descartes and show the dynamic at work - that we want to be able to switch between them with ease.

Every time I have an idea or problem to develop with the class, I invoke the Euclid/Descartes duo. It's fun to have great mathematicians in the classroom with you - and it helps develop a broader of mathematics as a dynamic, developing intellectual inquiry - something students can participate in and perhaps even extend.

Next post in this sequence: Teaching ideas for introducing the locus of the parabola.

Moves in translation with Miss Anna Parabola

2012-08-25T16:09:00.000+10:00

Some concepts are so powerful in mathematics, they just keep popping in your course again and again - like old friends. Such an idea is the translation of a curve in the number plane. In my class, this old friend has a name: Miss Anna Parabola. Anna has been making an appearance throughout my course, starting off with an introduction to the quadratic function.

Miss Anna Parabola demonstrates $y = x^2, y = -x^2, y = x^2 + k$.
Ballerina: Alicia Alonso in 1944, photographed by Gjon Mili for Life magazine

I will admit to raising and lowering my arms in the different ballet positions in class, standing up on chairs (against OH&S regulations ....) - but I vehemently deny donning a tutu.

I knew I was onto a good thing when I started teaching the topic "Locus and the Parabola". One of our textbooks spends an arduous 35 pages (no kidding) going through all the iterations of the different orientations and translations of the parabola - but I realised with our class understanding of Miss Anna's dance moves, we could collapse the entire thing into two lessons: one lesson to cover the different orientations, one to cover the translation.

The Four Standard Orientations of the Parabola -
as interpreted by Miss Anna Parabola (aka Alicia Alonso)
(Click image for a larger view)

Translating the vertex.

And it worked : my students can now do this effectively and efficiently. They connected our previous work on shifting curves like $x^2 + y^2 = 25$ to $(x-2)^2 + (y+4)^2 = 25$ to this work on shifting the parabolas. We cracked what would otherwise be a very arduous (and boring) part of the topic by focusing on the key idea of 'moves in translation'. I'm a big fan of creating characters and story to build a narrative in the course, so I was thrilled to see the work from previous topics developing Anna Parabola pay off like this.

Oh - and in case you haven't heard of him, Anna has a new friend: Billy the goat. Billy helps develop the idea of locus : if you tied him to a fence, he would happily devour everything around him, following the locus constraint imposed on him. And yes - I do admit to tying myself to a desk and 'acting the goat'. How am I going to live this down....?

PS: I'm not sure this trick would work at a boys' school.... Might have to invent a rugby player in motion...

Art photography in the maths classroom - thanks to flickr

2012-07-22T10:08:00.000+10:00

So much of the mathematics content we teach appears to many students to be fairly dry on the surface - we hope we bring them on the journey to see its wonder, but there is nothing like a great visual to create interest and start a quality discussion. Recently I tapped into an amazing resource: flickr photos provided under Creative Commons licenses. There are thousands of high quality inspirational photographs available just waiting to go into a mathematics lesson.

For the topic "The Quadratic Function", water was my visual theme. I use just one of these images for each lesson - we start the lesson with a full screen, high definition version and then I use smaller versions to create a visual link for transitions between lesson activities.

All images under Creative Commons. Click on the photograph for details.

For the topic "Locus and the Parabola" I blended an astronomy theme (think: parabolic reflectors) and more abstract designs:

All images under Creative Commons. Click on the photograph for details.

These images look spectacular in high definition projected onto a screen in class - and give an opportunity to engage in some broader discussion about the application of mathematics, and the relationships between art, science and mathematics. Interesting to see students also noticing the use of Creative Commons - a chance to model good practice and spread the CC message.

Because the creators of these images generously put their work 'into the commons', I can use them in my own works and then in turn, share those works with other teachers without any copyright constraints - hopefully helping students in many other classes.

Finding Creative Commons licensed content on flickr
Use the Advanced Search feature:

scroll down and select these boxes:

I also select "Interesting" which tends to return richer images. When using the image, make a CC attribution and provide a link back to flickr page. I also like to leave a thank you comment to author. And thank you flickr for coming to the CC party. Now if only Google would make CC searching available on its main image search page! I believe it is there - just hidden - and life is too short to look for hidden options.

The monkey and the mathematician learn calculus

2012-07-05T21:52:00.003+10:00

"Even a monkey can differentiate" - that's how I described the rules based approach that seems to dominate so many students' (and teachers') interaction with calculus. Coming from the "teaching for understanding" camp, I made a very deliberate and careful attempt in my first teaching of calculus to emphasise understanding as opposed to a formulaic, mechanical approach to the subject. And yet - a few weeks later, I've come to embrace my inner monkey. There is a place for mechanical, automated rule based thinking in mathematics - and I'm now leaning to the view we need to make room for both the monkey and the mathematician.

Here's the monkey at work:

No disrespect - WolframAlpha is an amazingly powerful tool, but it reminds
us differentiation can be done without understanding.

As I worked through the basic rules of differentiation with my class, I found myself continually looking at the rules from the 'monkey' viewpoint as well as the 'understanding' viewpoint.

Differentiation from first principles
Monkey: "Substitute in the values correctly, expand, pray you can factorise out the bottom, then shrink the delta-x to zero."

Mathematician: Understanding the central principle. The meaning behind every element of the fundamental equation is pivotal - it's like a little prayer in our holy canon. If you have to memorise the formula, you haven't understood it. Visualise the image of the secant becoming a tangent and just write down the description of the process: $f'(x) = \lim_{\Delta x \to 0}\frac{f(x + \Delta x)-f(x)}{\Delta x}$. OK - now release your inner monkey and finish the work.

The Chain Rule
An exploration using Marc Renault's amazing Chain Rule analogy interactive gives our mathematician side a boost here. For our monkey side, we developed the language of 'inside' and 'outside' to describe composite functions - modelled on Russian dolls. Here's how I summarised the two approaches:

Click on the image for a larger view

The Product Rule
I'm a big believer in showing the geometric justification - and it's more credible than the limits sleight-of-hand pulled by high school text books. That's for the mathematician. For the monkey, we learn the rule - and I like a cross-product type visualisation:

Click on the image for a larger view

The Quotient Rule
Last but not least, the quotient rule. I think it's important for the mathematician to see the connection to the Chain Rule and the Product Rule ("so that's why there is squared in the denominator!") For the monkey - well it's another pattern to get into the habit of using:

Click on the image for a larger view

Who's more important: the monkey or the mathematician? As much as I initially laughed at my inner monkey, I've come to value him. I don't think we need to choose between the modes of working - there is value in both. I suspect it's about 'reducing cognitive load' - with a reliably functioning monkey, we can perform low-level functions without too much thought, saving our awareness to concentrate on the more complex ideas at hand. The only danger with that monkey is too many bananas and we can forget the meaning behind the operations....

The wisdom of Year 7 : thinking about groups

2012-06-23T10:56:00.000+10:00

It's been nearly a year now since I changed my classroom configuration from rows to groups:

Overall I'm very pleased with the results - it's working for almost every class. The one class where I have wondered if I should revert to rows have begged me not to - challenging me to think more deeply about my classroom management for this class - I'm working on it! But there is no looking back now. These insightful comments from my Year 7 students, given as anonymous student feedback, reveal the benefits and challenges of setting up group tables.

Group Table Configuration : The Good

"Helps with learning because always someone u can ask 4 help"

"You can ask for help when the teacher is busy"

The group configuration helps deal with the challenge of answering questions from thirty students at once. Sometimes the group may come up with the wrong answer - but I rarely see this happen and am much more likely to detect any misconceptions if four students share them. What I do see on many occasions is students debating the answer, and they will ask for help if they aren't sure of their answer.

"Interesting learning from different perspectives"

Students discover how their peers view and understand the content - enriching their own understanding, and providing opportunities to develop metacognition : becoming aware how knowledge is obtained and processed.

"Let's me compare my answers and help people"

A powerful gift to offer to students : creating an environment for helping each other - developing generosity. One of my four pillars from the Circle of Courage.

"It helps cos if u don't understand something and 2 shy 2 ask u can ask ur friends"
Wow! How many students are held back because of this? A powerful insight on how a group table structure can help overcome emotional, personal and social barriers to learning.

"No need to be a loner - there are people around to help and support"
"Being alone is lonely"
How can we know the emotional needs of all students - let alone be able to help thirty students? Sitting students together, supporting them during class time to be together may just turn out be very important to some students who might be alone at other times. I was moved to read these comments.

Group Table Configuration: The Bad
"I would like to sit with different people"
I'm still uncertain if it's best to organise students or let them self-select groups. I worry about bullying and social exclusion, allowing students to set up hierarchies "you are in my group, you aren't". For now my answer is I assign the groups for Year 7 and Year 8 - and consider any problems on a case by case basis. A part of me also thinks it's important I maintain control of the seating. Any ideas welcome!

"I can never see the board properly"
Ouch. This is the biggest issue - and I think it's serious - especially since perhaps 30% of my lesson time is whole class instruction. Research that argues for sitting in rows claims this is the major problem with the group configuration. There are four seats in my configuration where this is a problem. I wish my classroom was wider to optimise the layout, but some tables don't get a good view of the board. I'm now going to establish it as a norm that those four students to turn their chairs to face the board during whole class instruction.

"Sometimes people just give you the answer"
An insightful comment from one student! Group configuration discourages solitary work - which is essential at times - and allows for students to just give each other answers. I often use an A/B/C/D paper approach to ensure each student at the group table has a different set of problems to work on - getting help has to be real help, not answers. But time doesn't always permit this, and if we are using the textbook, they are working on the same questions. I think I will need to be more explicit with students about ways of helping to maximise the learning.

Something not mentioned in the feedback is groups can encourage off-task behaviour and conversation. Fortunately with this class, that's not a problem - when they do go off-task (they are students!) they respond quickly to my request to get back to mathematics. This isn't the case with all my classes - more on that in another post.

The Verdict?
Unquestionably (for me) : Yes. The learning and social benefits are so high, it's worth persisting to deal with, or minimise the down sides. I'm looking forward to seeing how the comments next week from the class where I am having some class management issues will compare to those quoted above.

Note: My student feedback forms have an Opt-In indicator "Tick if you are OK for these comments to be shared with others". The forms themselves are completely anonymous, allowing for students to give me frank feedback without concern for any consequences.

It's trigonometry Jim, but not as we know it!

2012-06-11T11:42:00.000+10:00

Do you remember high school trigonometry? Was it a blurred sequence of formulae with a recipe book of incantations for solving standard exam questions? I'm afraid to say that was my experience and it has taken many years (decades!) for me to see the full beauty and unity of the subject. Teaching it for the first time this year I was determined not to inflict the same on my students. My attempt at something a little different was based on a few key metaphors and heavy use of diagrams and graphs, supported by stories to help students see why trigonometry is indeed an interesting study for scientists, mathematicians and historians.

Forget triangles - let's cast a horoscope!
In senior school trigonometry, we're not just playing with triangles any more. Welcome to the weird and wonderful world of circles and periodic functions. I think a good way to introduce the unit circle is to place it in the historical context of our ancient star gazers watching the heavens. That's our unit circle: the night sky. I suspect our ideas of angles, triangles and projections onto a circle owe as much to calculating the pharaoh's horoscope as they do to measuring irregular sizes strips of land to work out the taxes owed him.

Trigonometry : was it really about casting the daily horoscope?
Celestial dome cartoon (inset) from http://www.herongyang.com/astrology_horoscope/Astronomy_The_Celestial_Meridian_and_Zenith.html

Now that we have the idea of the rotating angle, and students see (or better yet explore) the sine curve, I think it's well worth showing why this is such a relevant and powerful idea. Share with students some of the interesting physics - show wave motion, show how different light frequencies relate to different colours. Hint at the mathematical treasures that await them: I showed some pictorial sequences introducing Fourier analysis. Trigonometry is about so much more than triangles - and it's relevant, interesting and surprising.

Round about and round about we go : the trigonometry carousel
A powerful metaphor for explaining the rotating angle and the periodicity of trigonometric function is the carousel:

Think of the rotations around the unit circle like riding a carnival carousel.
Carousel photo CC-BY-NC- SA http://www.flickr.com/photos/jaremfan/3478916095/
Carousel horse photo CC-BY-NC-SA http://www.flickr.com/photos/birminghammag/6045458462/

This emphasises the periodicity of the rotation and allows extension to the idea of spinning faster, going from $\sin(x)$ to $\sin(3x)$, or slower, $\sin(\frac{x}{2})$, and then spinning wider to $5\sin(x)$.

Draw a diagram!

At this stage I believe it's worth spending quality time looking at the different graphs of $\sin(x), \sin(3x), \sin(\frac{x}{2}), 5\sin(x), 5\sin(3x)$. This helps visualise the functions and helps avoid the problems we seen when students start working with $\frac{\sin(3x)}{3}$. Time also to bring out your function machine analogies and emphasise that $\sin()$ is function operator - not a multiplication of $\sin \times x$.

When it comes to solving trig equations, so many text books are filled with pages of algebra and barely a single diagram. Want to really understand the equation $4\sin(3x) = 1$ in the range $0^\circ \leq x \leq 180^\circ$ ? Draw the graph (sketch or use your favourite graphing tool):

Now we have many stories to tell! Why are there so many solutions? Because your students understand we have periodic functions (riding the carousel) and have seen the difference between the graphs of $\sin{x}$ and $\sin{3x}$ the reason for all those solutions becomes less mysterious. And a quick sketch can help check the solutions. Drawing the graphs of trig equations also reinforces the difference between equations (sometimes true) and identities (always true) - another source of confusion when students start trigonometry at this level.

A helpful place to use graphs is when teaching the Auxiliary Angle transformation of $a\sin(x) + b\cos(x)$. Remarkably, very few textbooks show the graphs. I started with the graphs: looking at the unexpected result that when we add a $sin()$ curve to a $cos()$ curve, we get another periodic function - just with a different amplitude and a slight phase shift.

Yes - that's physics coming in - I share this language with the students - many of them are studying physics. Once you show this remarkable graph, actually working out the equations for the transform is straight forward and it makes sense - it's not just abstract symbolic manipulation.

Trigonometric Identities : Same person, different clothes
A helpful metaphor to distinguish identities from equations (why oh why do we do regularly use the same equals sign to mean different things?):

Same equation, different clothes.
Superman/Clark Kent graphic by Ian MsQuee
http://ianmsquee.deviantart.com/gallery/3370060?offset=24#/d1onlf2

Later, when it comes to teaching the transformations, I add in the telephone box:

The half-angle t-transform helps Superman transform back into Clark Kent.
And Clark Kent is easier to pin down (solve) than Superman!

Trigonometry at this level really is a lot of fun - it brings together many different ideas and skills, producing some beautiful and unexpected results. And so many more wonderful surprises to come for those students who will later visit the world of complex numbers! Hopefully we can share that wonder with our students, so they don't just view trigonometry as a set of definitions, formulae and algebraic manipulations, but instead develop a strong intuitive feel for working with the circular functions.

And don't forget the graphs - or as a colleague repeatedly tells her class "DRAW IT!"

Some Teaching Resources

The Maths 300 "Trigonometry Walk" lesson (subscription required - but an internet search will find you some worksheets) is an outdoor exercise that helps students get a strong sense for the idea of projecting onto the unit circle.
James Tanton's whimsical Squine and Cosquine presentation explores what would happen if we used a unit square instead of a unit circle. Perhaps save after students have consolidated the knowledge?
Vi Hart's 13 minute video "What is up with noises?" is a wonderful exploration of the physics of sound, music and hearing. A little long to fit into a busy schedule, but good for a rainy end-of-term day.

LaTex in Google Blogger

2012-06-11T08:47:00.002+10:00

Thanks to this post on A Rambling Soul I now have easy-to-use built in LaTex in this blog!

So now I can write $ax^2 + bx + c = 0$ and $\frac{ \sin{x}}{\cos{x}} = \tan{x}$ without pain!

For Google bloggers: you will need to go into the template and select the option to directly edit the HTML template.

Countdown to Transit of Venus

2012-05-26T09:30:00.000+10:00

A curious composite image from the TRACE solar observation satellite, watching the planet Mercury move across the sun in 2003.

Parallax shift as recorded by TRACE satellite (orbiting Earth)
recording the Transit of Mercury 2003.
Image source http://trace.lmsal.com/POD//

No - Mercury isn't wobbling ... it's the satellite taking the photos that is moving, orbiting Earth on a North-South path. Mercury thus appears to move up or down, depending whether the satellite is North or South of the equator when the image is taken. While man-made satellites able to photograph Mercury passing the sun are relatively new, people have been measuring the parallax shift during the transit of Venus since 1761 by sending observers to different points across the globe. Australians feel a particular affinity with the Transit of Venus: measuring it was a key motivation for Cook's voyage on the Endeavour. In two weeks, Venus does it again - the last chance to see it in your lifetime. Australian students and their teachers are particularly fortunate as the ToV event starts and ends with the school day on June 6th.

Looking for activities to do with students? http://transitofvenus.org/ has an impressive one-stop collection of resource links. With so many to choose from, here's a short list we are using in our mathematics faculty to prepare students for watching the event:

Two videos we found to be high both engaging and high quality, with mathematical content suitable for all ages:

For Years 7-10: A great hands on paper-plate activity designed by Kathryn Williamson, offering many opportunities to explore angles and the use of decimals.
For Years 10+ : An excellent mathematical exploration from NASA educators at http://sunearth.gsfc.nasa.gov/sunearthday/2004/2004images/VT_Activity3.pdf (apologies for the deep link - I can't locate a top level link at SunEarthDay). Requires knowledge of similar triangles and right angle trigonometry. Also http://spacemath.gsfc.nasa.gov/transits/TRACEmerc.html offers a modern approach using the TRACE satellite data.

If you like the idea of using satellite imagery to demonstrate the parallax shift, it's interesting to compare image from TRACE (which does orbit the Earth), to that taken by SOHO (which does not orbit the Earth):

No parallax shift from SOHO images!
Transit of Mercury 2003. Photo: NASA.

Still alive ...

2012-05-12T08:47:00.000+10:00

It's been a long time between posts .. but rest assured I'm still alive - just swamped by teaching right now. Hope to write some more soon on the following topics:

Useful metaphors for teaching senior trigonometry

Ideas our school is doing for the Transit of Venus 2012 - with some resources

but in the meantime ...

A notice to all new teachers: Get yourself to the doctor ASAP and get vaccinated for whooping cough. Yes - you probably got vaccinated as a child, but immunity only lasts 10 years. I really really wish someone had told me to do this. Yes - I caught it - most likely from someone at school - I have only just stopped coughing after three months of hacking. And even worse, I could have spread it around. Fortunately my doctor picked it up and gave me the treatment to stop it being infectious. Sadly the treatment doesn't cure the cough - making for a very unpleasant time the last few months. As a new teacher you are particularly vulnerable to catching and spreading the disease. In New South Wales, Australia, there are 20,000 cases diagnosed per year - and probably many many more go undetected - because adults just put up with it. The vaccine is normally combined with a Tetanus booster - just do it!

TeachMeet East At Kambala

2012-04-03T19:22:00.000+10:00

A great way to get inspired: catch up with other teachers at a Teach Meet - come join the movement! Organised by teachers for teachers, Teach Meet seems to be really taking off in Australia. Thanks to the teachers at Kambala school for hosting us tonight.

My contribution:

The Function Zoo a Group Exploration Lesson Design v5 Annotated

Reversing the Meat-a-Morphosis machine : inverse functions

2012-03-31T08:51:00.000+11:00

A short followup to the post Two Ideas for Introducing Functions.

Yesterday I discovered another payoff from the hilariously gruesome Meat-a-Morphosis video: a powerful and memorable analogy for Inverse Functions. Imagine if we ran the machine backwards - if we put chicken nuggets in and the original chickens came out? My students thought this was hilarious.

Inverse Functions: What if you put in nuggets and out came chickens?
Image adapted from original at Meat-a-Morphosis ( Patty Hill & Michael Word)

Even more fun: for students still coming to terms with f(x) notation, faced with this:

doesn't this convey the same idea even more clearly?

Put a chicken into the nuggetiser, then into the de-nuggetiser and you get back your chicken
(OK , maybe with few ruffled feathers and a lot of squawking!)

Who would have thought there was so much laughter in a lesson on functions? Thanks to the Meat-a-Morphosis team at Kealing Middle School, Austin Texas for a wondeful teaching resource.

Who's afraid of the error monster?

2012-03-24T10:34:00.000+11:00

Maybe it's built into our very survival instincts : if something is wrong, it's uncomfortable - so run away and hide from it. We see it in class every day, every hour - students (and teachers!) running away from errors. No-one wants to be wrong, or even worse, be seen to be wrong. And yet, when it comes to learning, "errors" are valuable tools. I see it as one my most important roles as a teacher to convince students not to be afraid of errors - on the contrary, to look for them, appreciate them and share them.

When I talk about errors with my students I introduce them to the Error Monster:

Image from
http://conservationbytes.com/2009/10/21/sleuthing-the-chinese-green-slime-monster/
(by CJA Bradshaw?)

We discuss ways in which this scary monster is in fact a good friend - how every error we make, or another person makes, is a valuable gift to our learning. I encourage students to face their monster head on - whenever they do an assessment and see their mistakes, to run towards their monster and embrace it:

Adapted from CJA Bradshaw's slime monster.

Now our monster becomes an object of fun and affection - helping overcome embarrassment and disappointment at making mistakes - and allowing us to instead focus on resolving those errors.

Teacher notes:

I like to use the analogy of a blind person using a walking cane. How could they see where to go, if they didn't make "mistakes"? It's only by having the cane bump into things the person can see where to go. Errors help guide us on our learning path.
You have to walk the walk : be happy - and not embarrassed - to face your own errors in class. I highlight to my students my specific weaknesses when doing algebra : I know (and they know) I make silly errors with signs and expansions - so I laugh at my monster and then keep a careful eye out for him. I make a show in front of the class of checking for my common errors. Hopefully over time I will get better at these!

A visit to the Function Zoo

2012-03-17T10:21:00.000+11:00

Do you remember your early encounters with the animal kingdom? So many wonderful different animals - it may even have been a bit overwhelming at first. But very quickly we learnt to group the animals into a scheme that made sense to us. In mathematics we have a similar extravaganza of different 'animals', which can be overwhelming for students to make sense of. Enter the idea of The Function Zoo - first introduced to me by Mary Barnes in her amazing Investigating Change books.

Here is how I worked the idea into a Year 11 class, several lessons into the Functions topic:

A look at the different species of animals ....

... and how we might organise them.

The challenge:

Students worked in groups of four, using large sheets of butcher paper to sketch their ideas. There were at least two laptops per group and the students had just enough GeoGebra skills to be able to turn algebraic expressions into graphs.

The results were incredible: great conversations between students about functions. With GeoGebra on hand, I was able to encourage students to explore their questions, rather than give them answers, and even ask them more questions if they were ready for it.

Twenty minutes later I quietly threw this slide on the screen but otherwise said nothing:

The groups noticed it soon enough - and went wild. Seeing a few more functions they knew but had forgotten gave them new energy to keep going. Others asked each other questions, trying to work out the graphs they didn't recognise. Most recognised the last graph from our "explore your calculator" game. We then debriefed as a class, and explored why the idea of the Function Zoo is helpful and interesting. Apart from the obvious benefit of being able to organise our thinking, the real benefit comes in being able to make connections - as I suggested in these slides:

As often happens in student exploration activities, the class produced something unexpected, a gift from them to extend the lesson idea. One group drew the absolute value of a quadratic function - a blend of two of our function families. We decided this new function was like the cross-species breeding you sometimes see on display at the zoo : the lion bred with a tiger to make a liger.

Absolute value of a quadratic function : a "liger" in our function zoo.
Liger drawing: St Hilare (1772-188). Function by GeoGebra.

A fun and powerful idea - allowing students to see that even quite unusual functions can be seen as blend of function attributes they already know how to work with.

The full lesson PowerPoint is available for download and customisation to your needs at http://www.scribd.com/doc/85668117/Lesson02-03-Function-Zoo-Public

Teaching Notes:

A graphing tool makes a huge difference to the success of this lesson. Without it, students would spend a very long time plotting to explore their ideas. There is time for careful plotting later - this lesson is about seeing the bigger picture.
I found the group structure allowed for a high degree of differentiation - I could customise leading questions for each group, depending where they were up to on the functions journey.
I can't stress enough the value of developing students' GeoGebra skills (or other computer graphing application) when doing mathematics at this level. I sneak some GeoGebra learning into every lesson - even if it's just the class watching me do a quick check of an equation or a graph. Show them one small GeoGebra idea per lesson and by the end of term they will know the product well - especially if they are using GeoGebra at home as part of their study.
Why am I such a GeoGebra fanboy? Most importantly because all my students can download a copy to use home. GeoGebra is free and runs on Windows and Macintosh and it doesn't need an internet connection to run.

Two ideas for introducing functions

2012-02-22T20:11:00.000+11:00

Here's two ideas for introducing functions to your class - none of them original, but I used them today and was pleased with just how well they worked.

1. Watch the Meat-a-Morphosis video

This amazing video is a winner with students and teachers. Powerful, simple and clear ideas about functions wrapped up in deliciously gruesome humour.

Before watching the video: We spent a few minutes exploring the key idea of a function as a 'machine' that maps values of input variables (in the domain) to output variables (to a range), looked at the function notation f(x), and tried a few practice examples using f(x) substitutions.

After the video: We discussed the 'function machine' analogy and reviewed some of the fun examples in the video with their corresponding mathematical analogy.

2. Explore an unknown function on the calculator : the ln() function.

This idea comes directly from Mary Barnes' wonderful "Investigating Change" books*. Let the students know they have their very own "function machine" : their calculator. Ask them if they ever wondered what the ln() button does?

Students have been carrying this function machine with them for years.
So what is that ln() button all about?

Let's investigate! I gave each group of students some sheets of butcher paper and pens, and asked the question: "What does this function do to numbers? What is its domain and range?" - then let them at it, encouraging them to write, sketch, draw on the paper to show their thinking.

The results were astounding. As the work progressed, I dropped some hints to different groups to try different types of values and commented loudly (so other groups could hear!) when I saw group making a nice table or beginning to construct a graph. Some groups discovered logarithmic properties - that ln(100) was double ln(10), one group noticed ln(2) + ln(5) = ln(10), while others had discussions about asymptotes or debated with each other if their calculators were doing the right thing - the numbers seemed so odd and error messages kept coming up. For groups running ahead, I sketched y=x on top of their graph and asked them to draw a reflection. They recognised the resulting graph as an exponential one.

After the activity: I fired up GeoGebra on my board, showing how to graph the function (they groaned, having spent a long time doing it by hand :-) ), then we zoomed in and out to explore the interesting parts, referring to conversations and discoveries made by the class. Then a good discussion on how to determine the domain and range. My not-so-secret agenda is to convince the students the value of GeoGebra for this course - coming soon to a lesson near you!

I highly recommend this activity. Don't rush it - it will take at least fifteen minutes. Many great opportunities to develop and practice mathematical investigation skills.

* See http://books.google.com.au/books/about/Investigating_change.html?id=BjOJBR54jkIC for a preview.

Some teaching thoughts:

I was surprised how much the activity of exploring ln() on a calculator allowed for differentation through asking different groups different questions. One group finished early, so I gave then the challenge to investigate the hyp() button (hyperbolic trig functions .. hehe!).
I don't think it's a problem to explore the ln() function a good six to twelve months before we might otherwise look at it. Not knowing about the function is the whole reason the exercise works.
I think it's a mistake to start with the formal definition of a function that distinguishes between a relation and function. This puts to much focus on the idea of one-to-one mapping, before the deeper idea of the mapping aspect of functions. Start with an interim definition of mapping of a domain to a range - the refinement can come next lesson. This is also Mary Barnes' approach.
Be ready to explain why we care about functions, as distinct from just working with our usual y = x + 2 expressions. To my thinking, the answer is that functions are themselves distinct mathematical objects - taking us to the next level of abstraction from number -> variable -> function. Equally importantly, functions are the powerful idea we use for mathematical modelling.