Monday, November 20, 2017

Using Lego to Build Fractions

Having seen the occasional tweet about using Lego to teach fractions, I was curious as to how effective this might be. Today I tried this for the first time with some Grade 4 students. These are students who have not yet learned about equivalence but have learned about unit fractions and how we can use this knowledge to label fractions.
I worked in the school's Learning Commons and made use of the Lego board that was placed on the wall.
Each pair of students had there own board and a tub of Lego blocks to work with. I started by asking them to find a 2-by-4 block and to fix that to their board. Then I asked students to find a block, or blocks, that was one-half of the original block. Some chose a 2-by-2 block, others chose two 2-by-2 blocks. We also had students use a 1-by-4 block or four 1-by-1 blocks.
In a similar way, I asked students to show me what one-fourth (or one-quarter) might look like. As I listened to the students think there way through this, it became clear that even though they might know the fraction words 'half' or 'fourth', that they did not necessarily know how this connected to the number of parts needed to make a whole. Indeed, when we tried to do some skip counting by fourths as a class, I could tell that there was some uncertainty. To clarify this, I gave them the following challenge:

If a 2-by-4 block is one whole, build seven-halves and tell me another way to say this.

As they did this, I moved from group to group and modelled some skip counting aloud with them:
"One half, two halves, three halves, four halves, five halves, six halves, seven halves."
Now we could use a 2-by-4 block to show that this was equivalent to 3 and one-half:
The students seemed to like this visual 'proof'.
The next challenge was to build twelve-halves and to find what this was equivalent to. The students were able to do this more quickly now and tell me the correct answer. If some students put all their blocks together like this:
we suggested that that their work might be more clear if they leave a gap between each whole:
Finally, we asked them to build six-fourths and to find out what this is equivalent to. Again, they were able to build this quickly:
Some students argued that this was one-and-one-half. Others argued that it was the same as one-and-two-quarters. Then one student suggested that it this meant that one-half was the same as two-quarters.
BINGO!
I seized on this idea: "Who agrees with this suggestion?" 
Everyone did. Unfortunately, the bell rang for recess but it gave me plenty to think about what happened in the lesson and what the students are now ready for. I liked working with the Lego mainly because it was easy for the students to organise their thinking: once the blocks were placed, they stayed put and didn't get knocked all over the place. In fact, I could pick up one student's work and easily show the whole group. 
One drawback of using Lego is that the fractions that you can use are somewhat limited, so I would have to think more carefully about what models and blocks I could use to show thirds, fifths, sixths, tenths etc.

I actually did a similar activity at a Parent Council meeting for one our elementary schools last week but with pattern blocks. I showed the parents a hexagon and asked them to show me what one-third of this was. After they confirmed it was the blue rhombus, I challenged them to build eight-thirds and to then tell me another way of saying this. Once they had lined up eight of these rhombii, they carefully arranged six of these into two hexagons and were able to tell me that eight-thirds is equivalent to 2 and two-thirds. A number of parents actually said "That's why it works!" It was a perfect moment to show them the power of the Concrete-Diagrammatic-Symbolic continuum and how we can use this to develop students' understanding of fractions:

My overall learning of this is then as follows:
1) Equivalence is a key concept in fractional understanding. Without it, fractional computations are built on shaky foundations. Students need to develop this knowledge of equivalence initially through concrete activities before moving on to diagrammatic and thence symbolic activities.

2) Skip counting with fractions is an important prerequisite for developing an understanding of equivalence. Again, this should be developed concretely first before moving on to diagrammatically (number lines) and then symbolically.

Wednesday, October 25, 2017

Exploding Dots and Math Bumps

Last week saw the end of the first ever Global Math Week and what a successful week it was: over 2 million teachers and students took part! I am proud to have been an ambassador for this and to have brought the joy of exploding dots to many teachers and students. If you haven't heard about James Tanton's Exploding Dots then you need to check out this site. Basically, exploding dots are a way of visualising a journey of mathematical ideas from primary to senior grades.
In preparation for Global Math Week, my colleague, Dan Allen, and I held a number of sessions for interested teachers to introduce them to the idea of exploding dots and how they could incorporate this into one of their lessons that week:
For Global Math Week itself, I went into Grade 2, 3, 4, and 7 classes. The students really liked the idea that they were part of a worldwide event and that they were solving the same problems as students in Australia, China, India, Germany, Tanzania or wherever they had friends and relatives.
I started with simple 2 to 1 and 3 to 1 machines. Here, a grade 4 student is writing 7 using a 2 to 1 machine. Listen to all the kabooms happening in this clip:
We quickly learned that we can't use our normal number words to describe our results so instead of saying "One hundred eleven" we said "one, one, one". Then we used a 10 to 1 machine and found out that twenty-three could be written as...23! Here, I could explain to students that nearly all of the math they have learned so far has been in a 10 to 1 machine so in this case we could use our number words 'twenty three'. 
For the Grade 4 and 7 students, I then tried a 3 to 2 machine. 
Kabooms galore! It was neat to see the students taking care to make sure that they did the explosions correctly and checking with each other to see if they got the same result. Where they were discrepancies, they sought to convince each other of the correct answer. 
Finally for the Grade 7s, I tried an operation with them using a 3 to 1 machine:
This they did with no further instructions from me:
From a personal point of view, there is something about Exploding Dots that brings out a beauty I'd never considered in polynomial division. When Sunil Singh first introduced us to exploding dots last year, he challenged us to do 1÷(1–x) and 1÷(1-x^2) using this method. Even though I knew how to do these using more conventional methods, I was gobsmacked by the visuals produced:
I tweeted my excitement to James Tanton who then sent me another challenge. When I got stuck into this, something so surprising and wonderful happened, that I experienced what can only be described as 'Math Bumps':
Joyous maths indeed. 
My sincere thanks go to James Tanton and all at the Global Math Project for helping to spread joyous math everywhere.

Monday, September 25, 2017

Fun for All the Family

I'm always on the look out for math problems that can be used with people of any age. These puzzles are usually easy to explain but not so easy (or obvious) to solve. Here is one that I recently created and posted on our board's Math Twitter account (@DCDSBMath) that has proved popular with kids and adults alike.
To clarify: you can use any number of pieces to make a rectangle so right from the get go it has multiple entry points.
I gave this task to our principals and Math Lead teachers this week and as they worked through it, I was struck by the energy in the room. First two pieces were put together to make rectangles

then three:

and then the question was asked, "Is it possible to use all six pieces?" I put on my best enigmatic smile and said, "Perhaps!" With no immediate solution obvious, these adults had to persist at trying different arrangements.  Snippets of conversations from each table showed some common ground to the thinking going on:
"Are we allowed to flip the shapes?"
"How big must the rectangle be?"
"Either 6 by 4 or 3 by 8."
Then, around the room, shouts of delight went up and high fives abounded as different solutions were found.

When I made this puzzle, I did so knowing that this solution could yield a second solution.

As we moved from table to table airplaying the solutions, I was pleasantly surprised to see some I had not thought about. These two are a nice variation of each other:


As are these two:


But this one really toasted my crumpet:

Which of course begs the question: Can we find all the solutions? I'll leave that up to the reader to figure out.
A question arose during the session as to how to make this task more accessible to students who might have a visual-processing LD. One possible accommodation might be to colour-code the pieces and provide a template for them like this:


What I like about this problem is that not only is it great for developing spatial reasoning, it is one that can be attempted by adults and children alike: it is a problem that would be great for families to do together! And if you like problems such as this, then you will also love games such as Blokus and Kanoodle.
Or for those of you of a certain age, Tetris!

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.