Wednesday, September 13, 2017

More Than One Way to Crack an Egg

Last week I posted this photo onto my Twitter account and it received a lot of interest:
I often am asked to help out with the initial Math assessment for students who have come from different countries into our board. These were two solutions from two different students, both from Vietnam. The first thing that struck me was that these methods were completely new to me:
Many folk agreed. Others still wondered why these methods were chosen. I agreed with Matt Dunbar that my preferred method would be this: 
My mantra has always been: 'First, isolate the variable'. Yet this is not what these students did. Other folk wondered, in the case of the student on the right, what the student was thinking:
So this week, I happened to see this student again and asked him if we would be happy to do some more questions for me. Fortunately, he was happy to help! Here's the first question I gave:
OK, so this tells me that his very first solution wasn't a one off: this 'first get-a-common-denominator' approach is a go-to strategy for him. I next gave him this question:
So now I see he is also paying attention to the numerator of the fraction in the first line. But still, he uses the 'first get-a-common-denominator' approach. I was now curious as to what he would do with a simpler equation:
So in this case, he does isolate the variable first: verrrry interrrresting! But now I want to see what he does when there is more than one denominator:
At this point, I am grinning like a Cheshire cat. What a neat way of solving this! When I asked him if everyone learns this method in his Vietnamese school, he replied that they did. So I showed him how I would have solved these (isolate first, then get rid of the denominators) and it was nice to see him smile and nod and say "Cool!"
I love it when I see something new like this. It reminds me that what we take for 'standard' here might not be standard everywhere. That doesn't mean to say that I will change the way I do these types of questions myself (I still like 'my' way!) but knowing that there is more than one way to crack an egg will help me as a teacher help students who might not get 'my' way.

Friday, June 23, 2017

Summer Math

With many students getting ready for the summer break, there are also many parents wondering how they can keep their sons' and daughters' entertained for six weeks and more. I am one of those parents! So here are some math-based ideas to try that are not only fun but will also help keep math skills honed until September.

Dice Games

Three words: Shut-the-Box. Farkle. Yahtzee.

OK, that might be five words but who's counting? These are great games for any age and are a wonderful way of getting students to practice their number sense. I particularly like Shut-the-Box: it is something I have played over a cup of tea with my children in a Tim Hortons.











Card Games
These can range from simple games like Snap, Pairs or Marilyn Burns's Oh No 99! game to more complicated games like Euchre and (my favourite) Cribbage. Playing these games necessarily involves using Math: from simple comparing of numbers (Snap) to more complicated decision making (in Cribbage, which of these two cards should I put in my opponent's box?)
You can also practice your math facts by playing a 'War' type game with cards. Split a pack of cards evenly between two people. Each player then simultaneously turns over the top card of their deck and places it on the table next to the other player's card. The first player to call out the total of the two cards gets to keep the cards. Keep playing until all the cards are used up. This game can be adapted so players have to get the product of the two cards, or to treat black cards as positive and red cards as negative and to get the total or product of these integers.



Spatial Reasoning Games
Math is more than number sense so it's important to work on our spatial reasoning. Kanoodle is probably one of the most engaging puzzles that I know of and has been a huge hit with any kid (and adult) that I've shared it with.






Other great games include Tantrix...

...and Pentago.

And if you want something for your tablet, I highly recommend the app Flow Free.


Puzzles
Try a yohaku puzzle each day!


For older students, try one of the many from the yohaku website or Twitter feed @yohakupuzzle. Or, if you have younger kids, create your own and leave them on the fridge!

I'd also recommend kakuro and KenKen puzzles. There is also the 100 Day Challenge at brilliant.org as well as the Math Before Bed site for younger kids.

Road Trip Math
If you are going on a road trip, then get your kids involved with this! Show them a map of your route. Better still, print off a copy of the map and mark on your location every hour and to note how many kilometres you have travelled: this might help kids answer their favourite question: "When are we going to get there?" Make a note of gas prices on your route: are they more expensive or less expensive than where you live? When you fill up with gas, mark the location on your map. How many litres did you pump in?  Using your map, how far do you think you will go before you need to fill up again.

Sporting Math
If your child  are following their favourite team, get them to collect data of how their team is doing. From simple bar charts to keep track of wins/losses to more detailed things such as number of runs/hits, batting averages etc. If you are watching a game or a sporting event, casually ask your children questions like:

  • By how many runs/goals/points are we winning?
  • How many runs/goals/points have been scored in total?
  • Have we had more running yards or passing yards?
  • How many minutes are left?

Or if you are going to watch the World Athletic championships in London, how about a question like "Who do you think will win: Bolt or De Grasse? By how much?"

BBQ Math
There are even opportunities to practice math when you are having a barbecue. From simple questions like "How many burgers are on the grill? How many will there be when I flip them over?" to using a meat thermometer to measure the temperature of the chicken breasts (we don't want to eat undercooked chicken, do we?!)

Hope that you have fun using these ideas!

Monday, May 29, 2017

Triangle Numbers: they aren't square!

I was sharing some web resources with some teachers recently and showed them the Who Am I puzzles from Solveme.edc.org . One such puzzle is shown below:
An honest question was asked: "I know what square numbers are but I don't know what triangle numbers are. Can you tell me what they are and why we need them?" As there were some blocks nearby, I decided to try a concrete-diagrammatic-symbolic approach with these adults.

Concrete
Triangle numbers are created in this way: start with a row of 1. This (1) is the first triangle number. (OK, some people say 0 is the first triangle number but I went for a natural number approach!). To get the second triangle number, add a row of 2 below the first (to get 3). Now to get the third triangle number, add a row of 3 (to get 6). To get the fourth, add a row of 4 (to get 10) and so on.

So the first few triangle numbers are 1, 3, 6, 10, 15, 21, 28, ...

So why do we need them? Well, they can pop up in certain problems, especially ones which involve finding the sum of the first n numbers. We know that the sum of the first four natural numbers, 1+2+3+4, is 10 which is also the 4th triangle number. I showed them this vine to help them visualise what is going on:



Diagrammatic
So how does this help us work out, say, the hundredth triangle number? Well, it's just the same as 1+2+3+...+100. This will take too long to build with blocks but a diagram will help (see right).
I asked them what size rectangle, these two 'triangles' would make and they quickly told me a 100 by 101 rectangle. This would have an area of 10100 units which means that one of the 'triangles' would be half of this. In other words:

1+2+3+...+100=(101×100)÷2=10100÷2=5050.
So far, so good, but can we now generalise this result?

Symbolic
I related one of my favourite math stories: of how a young Gauss was given a problem by a possibly hungover teacher in the hopes that it would keep Gauss and his classmates busy for the morning. The teacher had barely sat down when Gauss tossed his slate on his desk and in his peasant, Brunswickian dialect said, "Ligget se" or "'Tis there." I showed them his approach:

As I was doing so, the teacher who asked the initial question suddenly shouted out "Oh, shut up! He just added the two triangles like you showed earlier!"
Joyous math, for sure.
Now I could have explained triangle numbers by jumping straight to this formula, but I doubt very much if this would have led to the same sense of excitement and understanding that I saw. It also got me thinking that sometimes it doesn't matter whether or not triangle numbers have practical applications: triangle numbers even on their own are just incredibly cool.

The nrich site has a wealth of problems that involve triangle numbers. Triangle numbers (or near triangle numbers) can also be used when solving kakuros such as this one:

Friday, May 19, 2017

A Concrete-Diagrammatic-Symbolic Approach to Dividing Fractions

As quickly as you can, look at the question below and write down the answer:
This is a question I have shared with many adults (teachers and parents) and I often get the answer 1⅕. Likewise, if I ask students to do 6÷½, I often get the answer '3'. There is a major misconception here that needs to be addressed:
Dividing by a half ≠ Dividing in half
When I start students on the road to dividing fractions symbolically, I will avoid such sayings as:
'Ours is not to reason why, just invert and multiply.'
It can cause no end of misconceptions. Instead I want to ensure that they start by understanding that they know that dividing by a half is equivalent to doubling a number. This is something we can develop concretely. For example, for this question:
we can show using pattern blocks:
The trapezoid is one half of the hexagon, so how many of these are needed to make 5 hexagons?
By repeating this for similar questions and then recording our results, we are in a position to think about what dividing by a half is equivalent to:
By using the concrete, students are in a better position to see that dividing by a half is equivalent to multiplying by 2.

This can now be extended to dividing by a third:
Again, by using the concrete, students are in a better position to see that dividing by a third is equivalent to multiplying by 3. In fact, I often see students at this point not even 'complete' the division (as in the photo above) because they can see the solution. I can repeat this for dividing by a quarter, dividing by a fifth, dividing by a tenth etc. and record the results thus:
So what if we are dividing by fractions other than unit fractions? For example:
Here, I might use a diagrammatic approach with a number line:

The big idea I want to get at here, is not so much the answer (6) but the effect of dividing by two-thirds. Again, recording results might give something like:
When students notice that dividing by two-thirds is the same as multiplying by 1½, we can then show that this is equivalent to multiplying by three-halves. We can then extend this idea with other examples  and maybe get a set of results like this:
I find that with this progression, students are in a better position to see that dividing by a fraction is equivalent to multiplying by its reciprocal.
At this stage, I feel that students are ready to appreciate a symbolic approach to dividing by fractions. There are some prerequisites that students must be comfortable with though:
  • Dividing any number by 1 gives the same number.
  • Multiplying a fraction by its reciprocal gives 1.
  • Multiplying the numerator and denominator by the same number gives an equivalent fraction.

Now we can show:
I have yet to see a visual that shows that dividing by a generic fraction is the equivalent to multiplying by its reciprocal. If you know of one, please let me know!

I certainly want students to be able to divide fractions symbolically. However, I have found that I can't just jump to this stage. Students will greatly benefit from this concrete-diagrammatic-symbolic development.




Thursday, April 6, 2017

Joyous Maths: Dan's Favourite Pattern

The wonderful James Tanton speaks of the importance of getting our students (indeed everyone) to experience 'joyous maths'. One of the most satisfying sounds for any teacher is the squeal of delight and the "Oh, that is so neat!" when a problem has been solved. My colleague Dan Allen (past-president of OMCA) recently confessed that this is his favourite visual pattern.

A typical question to accompany a question like this might be 'How many squares are in the hundredth term?' or even 'what is the nth term?' In the past, I have dutifully created a table of values and used all sorts of algebraic techniques to come up with such an algebraic rule. This is fine, to an extent, but if this is all we do (or get our students to do) then we are missing out on a grand opportunity to do some beautiful maths.
So why does Dan get so excited about this pattern? Think about how many different ways it can be seen:
Perhaps you see it as an inner rectangle and two squares. Generalising this suggests an nth term of (n-1)(n+1)+2 


Maybe you see an inner square and two identical rectangles. Generalising, this suggests an nth term of (n-1)2+2n
Maybe you see an opportunity to complete an 'outer' square with two identical rectangles. Generalising, this suggests an nth term of (n+1)2– 2n
Or maybe you transformed the pattern by taking the top layer of each term, rotating it 90˚ to create a large square and a small square. 



Generalising, this suggests an nth term of n2+1.

In fact, much fun can be had showing that all these general terms do simplify to n2+1.

Visual patterns like these are so important as they allow us to be able to decompose shapes in different ways. This in turn helps us generalise by moving along the Next, Near, Far, Any continuum. Indeed, I am noticing how effective it is to get students (and adults) to generalise by asking them to 'draw' the hundredth term. When I did so recently in  Grade 9 class, the students moved on from drawing individual tiles to thinking about the dimensions of the inner rectangle and thence how to write this algebraically:

So I set myself a challenge: create a new visual pattern that will become Dan's new favourite puzzle. This in itself involved much problem solving and eventually I came up with this:

Using decomposing strategies like those shown above, how can we describe the hundredth term? the nth term? How many ways can we see this?

I showed it to Dan. He liked it (how much I'm not sure) and funnily enough, he saw it in a different way than I. He then set about creating a 'decomposable' visual pattern of his own:

So, be honest, which of these is the most joyous?


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These diagrams were created using the Colour Tiles from the  mathies.ca site.