Streamlining nearly done

I’m hoping to have some new maths animations up this next week some time. It’s been a bit longer since the last update as I’ve been working on streamlining the website. That work is close to complete and if you have a slower connection you should hopefully notice a difference when the new animations go up.

After that I’ll focus for a while on developing new animations and I’ll start research towards writing an html5 version of the site.

At some stage I’ve also got to increase my social media presence so I can promote the site more….

Working towards an html5 version

I’m always working on new animations but these days I’m also working towards an html5 version of

The first step is to streamline the current flash version. Currently the animations are coded into the base swf but I’m working on having them load dynamically as needed.

I’m enjoying the learning curve and have been reading up on xml, html5 and css3. I’ll keep you posted on this process!

New features at search facility, instant feedback and more!

Welcome back for the new year. I hope you have had a great holiday season.

I’ve been working on some improvements to


There is now a search option available:

screenshot showing the new search facility

I will improve the search facility over the next few weeks by adding more metadata.

Instant feedback

It’s now even easier to give direct feedback:

You don’t even need to provide your name or email (unless you want a reply).


I’ve been working on a few other minor improvements:

  • I’ve improved the sets of related animations that appear when you click on the ‘links’ button.
  • I’ve added more ‘breakpoints’ to the animations so that the ‘step forwards’ and ‘step backwards’ buttons at the top of the remote control are more useful.
  •  I’ve also made a few minor changes to the remote control.

I wish you all the best for your teaching this year and I look forward to adding new animations and new features.

Seasons greetings

I would like to sincerely thank everyone that has supported this year including those people that have provided feedback and those people who have helped to spread the word about the website.

Next year I’ll be adding new animations and several new features.

The next update will be in the first week of Janary, 2012.

I wish you well over the holiday season.

Tal Greengard.

New feature: full screen

It’s really great to be getting some feedback and helpful suggestions from teachers. A few  teachers have suggested adding the ability to enlarge the animations. So I’ve added a full screen option. It is now available on the remote control:

screenshot of which can now play full screen!

Full-screen is available on the remote control









When checked, the animations will resize to fill the browser window.

If you have a small screen, pressing F11 will also help.

I’ve tested it in a few browsers but let me know if there are any problems and thanks again for the feedback!


Looking at ratios in different ways

I’ve been adding some new ratio animations and I thought that I would write some posts explaining the rationale behind them.

Let’s take the ratio 3:2. There are many ways of thinking about this ratio. A common way to think of it as ‘selfish’ sharing (“Three for me, two for you, three for me, two for you….”). More formally we write statements like “For every 3 bananas that Jack gets, Jill gets 2 bananas.” or “There are 3 Litres of blue paint for every 2 litres of red paint”.

I use these sorts of ideas to explain ratios in the first lesson and I often organise the students into groups and pass around counters. I ask my students to solve some problems such as:

“Jan and Dia share 25 oranges in the ratio 3:2. How many oranges do they each get?”

I’ve made an animation that shows how the students might solve this with their counters and I like to give them several problems of this type (decomposition in a given ratio) before we move on. It depends on the class, but at some stage they can move from using the counters to drawing diagrams instead.

There may be some students who will start to see faster ways to solve the problem, and sometimes I invite them to share their method with the class. It depends on the student and the class as there are advantages and disadvantages to this sharing. It’s a risk because it might confuse or intimidate some students who aren’t ready to hear it, but it could also help some students understand the concept and students can explain things in ‘kidspeak’ better than I can.

Once the students are feeling comfortable with the counters and diagrams (possibly in the next lesson) I move on to a second type of problem:

“Jan and Dia share oranges in the ratio 3:2. If Jan gets 12 oranges, how many oranges does Dia get?”

I’ve made an animation that shows how the students might solve this with counters, and again I give the students several problems of this type to solve in groups with counters and later with diagrams.

For each problem it’s a good idea to ask “What does each counter represent?”. The students should always be able to provide the answer “Each counter represents 1 ____”. If my students answer, for example,  ”Some counters represent red paint and some blue paint”, then I suggest to them that “each counter represents 1 Litre of paint”. They quickly see that in each problem we are dealing with objects of the same type. This pre-empts a discussion (in a later lesson) of the difference between ratios and rates and the formal definition of a ratio.

Solving these problems with concrete materials and diagrams helps students to develop an intuitive grasp of how ratios work.

The next step is to start to look at equivalence and simplification (in a very concrete and intuitive way of course) and I will describe this in my next post.


Chance and mandatory pre-commitment

A few days ago Dave Radcliffe (@daveinstpaul ) tweeted an interesting problem:

Consider this game. Start with $100 and toss a coin repeatedly. If heads, the pot is increased by 50%, otherwise it is decreased by 40%. Would you be willing to play this game and if so for how many rounds?

It’s a tempting proposition. We stand to gain more than we stand to lose and there are equal odds of winning and losing.

I thought at first that the answer is “Yes and for as many rounds as possible!”. Curiously it doesn’t quite work out that way.

Recently in Australia there has been a debate about mandatory pre-commitment. Should gamblers be forced to determine a limit on the amount of money they will gamble before they start gambling? Will this help problem gamblers?

Dave Radcliffe’s problem implies a type of pre-commitment. We pre-commit to a certain number of rounds of the game.

Suppose we pre-commit to two rounds. There are four equally possible outcomes: 

  • Two heads, in which case we finish with  $225
  • One head, then one tail, in which case we finish with $90
  • One tail then one head, in which case we also finish with $90
  • Two tails, in which case we finish with $36

The expected value is the average of these which is $110.25. So the expected return is positive ($10.25) but notice that the odds of a positive return are 1 in 4.

No matter how many rounds we pre-commit to the expected return will be positive. In fact, the more rounds we pre-commit to, the greater the expected value (100*1.05^n, where n is the number of rounds we pre-commit to).

But if we are only allowed one game it would be unwise to pre-commit to too many rounds.

Here’s another way of looking at it. If the coin shows a head, followed by a tail, we multiply the money we start with by 1.5*0.6. In other words we multiply by 0.9 .

If we play many rounds, then we might multiply the money we start with by:


depending on the order of heads and tails.

The more rounds we play of the game the more likely it is that there will be a roughly equal number of heads and tails. So when we rearrange the terms in this long product we are likely to end up with a large number of pairs like this:


There may be a few extra 1.5′s at the end or a few extra 0.6′s but in a long game with many rounds we would expect there to be a roughly equal number of heads and tails.

So now we have

0.9 * 0.9 * 0.9 ……..

… with a few extra factors at the end. As we play more and more rounds there are probably quite a lot of these 0.9′s which multiply to give quite a low factor so we very probably end up losing most of our money.

So the more rounds we play, the greater the expected return AND the lower the probability of a positive return. 



Points to bring out when we teach algebra with cups and counters

This the fourth post in a series about using cups and counters to introduce concepts in algebra. In my last post I started to describe a teaching sequence. We got to the point where the students were writing expressions for the number of counters in a diagram.

It’s a good idea to mention to the class that if two cups are marked with different variables they may have a different number of counters but they also could have the same number of counters. One misconception among students who are being introduced to algebra is that different variables always stand for different numbers.

As the lesson progresses students may offer different correct answers to a question. They may change the order of addition (3+x or x+3). If this happens I often refer them back to the first example and point out ’3+5′ and ’5+3′  are both correct sums. You might also have students who suggest multiplication as an alternative to repeated addition. For example:

 Repeated addition or multiplication?

Here some students may answer ’a+b+b’ and other might answer ’a+2 x b’ . It’s a good opportunity to remind students of the relationship between multiplication and repeated addition. I usually provide some more examples and ask for both answers. I always leave the multiplication symbol in the answers at in my early algebra lessons as it can mimise confusion. At this stage we’re not concered about simplification so there’s no need to omit the multiplication symbol.

With these points in mind it’s time to start handing out the cups and counters to the class. I’ll describe how I like to do this in my next post in this series.

Eliminating positive and negative signs with the magic elevator

I’m writing a series of posts about my magic elevator animations, and this is the fourth and final post in the series. So far I’ve explained the rationale for the animations, then started to describe a teaching sequence. In my last post we got to the point where students could see that adding a positive is equivalent to subtracting a negative (of the same magnitude) and that adding a negative is equivalent to subtracting a positive (of the same magnitude).

After some revision of the previous lesson, here’s what I do:

  1. I remind the class that we don’t need to write positive signs. If there is no negative sign in front of a number (and it is not zero), it is automatically positive. We look at some number sentences and rewrite them without positive signs. For example, we might rewrite: ‘+2++3=+5′, ‘+2-+1=+1′, “-3++1=-2″ and “-1++4=+3″.
  2. I then provide some sums and differences to evaluate using the magic elevator. I leave out the positive symbols. For example, I might write ‘-1+4′ on the board (but not ‘-1++4′ as I would have in the previous lesson). We notice that the ’4′ is positive so there are 4 happy people getting on the elevator. I might write several of these on the board and ask the students to evaluate them in groups. (e.g. ’2+1′, ‘-3+2′,’3-4′,’1-5′ etc.)
  3. I write ’3+-2′ on the board and remind the class that in our last lesson we learned that adding a negative has the same effect as subtracting the positive. I write  ’3-+2=1′ on the board. Usually they will point out that I don’t need to write the positive sign, otherwise I will ask “Is there a shorter way of writing this ? Can we leave anything out?”. We now have ’3-2′ on the board. We do this with several similar sums (e.g. ’5+-3′ becomes ’5-+3′ which becomes ’5-3′, ’4+-5′  becomes ’4-+5′ which becomes ’4-5′ etc.)
  4. The students quickly see that we can replace ‘+-’ (adding a negative) with ”-’ (subtraction). I give then several sums where a negative number is added and ask them to find the answer. Sometimes they need the elevator at this stage but often they don’t.
  5. I write ’3- -2′ on the board and remind the class that instead of subtracting a negative we can add a positive. When I write ’3++2′, .they remind me to leave out the positive sign so we now have ’3+2′ . We do this with several similar sums (e.g. ’4- -1′ becomes ’4++1′ which becomes ’4+1, ‘-2- -3′  becomes ‘-2++3′ which becomes ‘-2+3′ etc.
  6. The students quickly see that we can replace ’–’ (subtracting a negative) with ‘+’ (addition).  My cousin, who is also a teacher likes to remind her students that subtract means “add the opposite”. I give them several sums where a negative number is subtracted and ask them to find the answer, which they can now do without using the magic elevator.

Phew! What a lesson and what a post! You might like to do the last two steps in a seperate lesson. It depends on the class and how fast they are catching on.

This is only one of many, many ways to teach this topic. If you have any feedback or any suggestions I would love to hear from you. Why not add a comment below?




The magic elevator and equivalent expressions

This is the third post in a series about my magic elevator animations. In my last post I started to describe a teaching sequence. We had just reached the point where students were finding answers by translating number sentences such as ‘-3++4′ to magic elevator scenarios (‘starting on the third basement level, 4 happy people get on the elevator’). Here’s what I do in the next lesson (after revising those number sentences!).

  1. Draw the elevator diagram on the board again. Ask questions like: “If the elevator is on the second floor, what are two ways that it could move to the fifth floor?”. When students provide the answers I translate them into number sentences on the board (’2++3=5′, ’2- -3=5′). In this step I only ask questions where the elevator moves upwards. By the end of this step I have several pairs of equations on the board (e.g. ’2++3=+5′, ’2- -3=+5′  and ’1++2=+3′, ’1- -2=+3′ and ‘-2++4=+2′, ‘-2- -4=+2′.
  2. I ask the class to tell me what they can learn from these equations? What do the pairs have in common? We draw a conclusion together that subtracting a negative has the same effect a adding a positive (with the same magnitude).
  3. I repeat step 1 but ask questions that involve the elevator moving downwards. For example “If the elevator is on the second floor, what are two ways that it could move to the ground floor?”. Again I end up with several pairs of equations on the board (e.g. ’2+-2=0′, ’2-+2=0′ and ‘-1-+3=-4′,’-1+-3=-4′ and ’5-+2=+3′,’5+-2=+3′).
  4. I ask the class what we can learn from these new equations? What do the pairs have in common? We draw a new conclusion that adding a negative has the same effect as a subtracting a positive (with the same magnitude)
  5. At this point I write ’2       =-3′ on the board and ask the class to find two ways to complete the number sentence (in this case ’2+-5=-3′, ’2-+5=-3″). They will need to add an operation and a second number. I then organise the students into groups and ask them to create similar ‘fill in the blanks’ questions for each other.

I try to give the class an opportunity to sleep on this before I do more. I will continue this teaching sequence in a later post.