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Just made a couple of videos explaining multiplication in a visual way, using the ‘Box Method’
I’m working on adding nicer text and math symbol support in GridMaths, so formulas will look like they should.
LaTex is the ‘gold standard’ but I’m thinking of using a simpler alternative which gives most of the benefit, namely ASCIIMathML. I think its quite a nice format, something that covers most of the things a highschool Maths student will encounter, with a reasonably predictable and short syntax.
I’m experimenting with the best way for GridMath UI to enter math expressions, so it may be that you don’t normally see the text representation, but it may be exposed when you need to edit. I’ll use MathJax to render this, so it may be that you can edit and enter LaTex directly if you are a power user and know it well.
In my googling, I happened upon this video ahowing how to do long-multiplication of polynomials.. and thought I’d try the same problem using the Grid or Box method :
For some students it might bridge the gap and give them a helpful intermediate step so they see it as the same kind of thing as multiplying integers. I would also try out the box approach to introduce multiplying mixed numbers with fraction parts, and numbers with decimals, and so on – as it relates back to the earlier understanding of multiplication as area of a rectangle.
So GridMaths.com is in open beta, now works reasonably well on iPad and recent desktop browsers. Android browser support coming sometime soon.
Heres a pic my 9yo created while testing things out on the iPad.. with the obligatory battle scene [ dad vs browser quirks ? ] :
Here’s a couple of sheets on how I like to present long multiplication…
Firstly a concrete rectangle where you can actually count the squares to satisfy yourself its right…
Then move to a more compact form, which still shows the rectangle grid, but not to scale..
This shows that lots of digits should not induce panic.. the same systematic approach works [ which is why computers can do multiplication so well ].
I think having the box grid is a nice way to remember where all the pieces come from [ single digit products ]. Also I think it really helps to use the blank grid spaces, rather than fill in every 0.
The lattice method is slightly more compact, but I think this box approach reminds students of whats really going on.
Thanks for all the emails and encouragement so far as I build this.
Enjoy, and let me know how your using GridMaths.
Just added a color palette to GridMaths, so it now has rectangles, lines, ellipses in few simple colors and weights. The idea is to help make diagrams clearer, and help with Venn diagrams and other cases where you want to group things together.
Heres a grid sheet comparing fractions : 2/3 and 4/5 … then we show the product and sum using the same visual representation.
If you look carefully, you may notice that I ‘abuse’ the grid in that each grid square is 1/5 high and 1/6 wide – so a 1 unit x 1 unit square is actually 6×5 grid squares. I think this is a legal abuse, in that we often graph things with different x and y axis scales.
The alternative is to have a resizeable non-square grid.. which I think creates too much complexity for not much gain. The philosophy of GridMaths is “keep it simple”, so you can do the basics quickly and easily. I used to think more features = more power = better, but after my son showed me the Minecraft game, and all the cool things people have built with it, I changed my mind about this. I kept thinking.. why is Minecraft so popular, when you can do all that and more in Blender ? Blender is a superb free 3D modelling package, which like all powerful modelling packages takes a while to become proficient at. So, it dawned on me that Minecraft is so brilliant because it brings down the barrier to entry, and makes the 80% of things you need to do to make a world, really easy and quick for everyone.
So I’m adding the most useful things to GridMaths in a way that keeps it really simple. Its not an algebra system, it most likely wont have handwriting recognition… but it should be a really fast way for math teachers to make Math diagrams and for students to do a wide range of worked Math problems, replacing grid paper for that 80% of tasks and adding some nice features. Its kind of like an infinite supply of grid paper, that weights nothing and can be erased and replayed, and saved for later use And.. you can step forward and back thru your edits and change stuff, and students can step thru a worked problem. And you have counting beans and … :-]
I did a couple of worksheets on Multiplication, to test out the GridMaths tools.
My feeling is that multiplication should be taught with Rectangles. So once students are familiar with counting in groups, they can play around with something like this interactive demo to get a basic feel for the distributive rule.
The distributive rule says we can divide up A and divide up B any way we like, and the product is just all the pieces of A times all the pieces of B, summed up together.
For humans with 10 fingers, its easy to break A and B up by 1s,10s,100s, which means we need to multiply each digit of A by each digit of B, then sum them up.
So long multiplication kind of falls into place, as a way of keeping the pieces straight :
Then the Lattice method can be seen as just another convenient system for keeping all the pieces organised and lining up the 1s,10s,100s etc :
The first time I saw this Lattice method on Hacker News it looked like Magic.. but after you do one you realize it really is the same thing, just a cool way of keeping track of the pieces.
Just wanted to share a small ‘teaser’ from a project I’m really excited about, GridMaths.com
I’ve just finished adding support for the buttons keyboard which lets you make cool counting lessons for younger kids, so here’s a screenshot teaser to enjoy before we go live with the beta site in a few days :
If you haven’t guessed already from the above pic, the idea of GridMaths.com is to have an interactive quad grid paper environment in your browser or tablet, where you can work out math problems like long division, multiplication and so on.
My 9yo son has been doing lots of KhanAcademy exercises, working his way up to multiplying decimals..[ which Im pretty chuffed with ]. You really need grid paper to do those as you have lots of adding up in columns.. so anyway we spent ages getting quad ruled grid paper in Bangkok, as this is less common, or at least not easily found in the otherwise excellent stationary shops in the main shopping malls.
Some of these problems take a fair bit of working, so its easy to make typo mistakes .. it seemed to me if you could type in a grid that would really help. When I thought about how much time Id spent hunting down stationary and that Id spent about $50 in the last year on grid paper… something kind of clicked and I decided to make a prototype and see if it would help. thus was born GridMaths.
Heres a worked example, of a ‘long multiplication with decimals’ problem done on a GridMaths sheet :
Counting with coffee Beans on grid paper
When I started teaching my son maths we did a lot of counting coffee beans. We would arrange 12 beans in groups of 3 and groups of 4… The idea was to start building up intuition and then segway into multiplication.
Don’t ask me why coffee beans.. I must be a bad parent! The beans just happened to be handy, being dark they were easily visible when put on 5mm math grid paper and fit nicely in the squares. You can use small buttons or raisins even. Its pretty tactile, so works okay for young kids [ as long as they are old enough not to swallow small things ]
Abstracting to Multiplication
Later we drew the ‘beans’ in the squares and then just drawing a circle in the box instead of placing a bean. This gradually led to tracing out the rectangle outlines of the groups and let the squares on paper take the place of actual beans.. so it abstracts really well in a fairly natural and unforced way.
There are some nice ‘tricks’ you can do on squared paper that grow out of this approach –
- every rectangle is a multiplication product, so you can use it to figure out any times table question
- you can work out all the times-tables and write them in the top right square before reciting them
- introduce distributive property (a+b)*c = a*c + b*c by showing the rectangles add up
- ask.. are there any numbers that cant be made by a rectangle product? [ yes..prime numbers]
- show square numbers – 1,4,9,16 …
- you can show how to add 2n+1 to n squared to get the next square number
- you can introduce series – 1+3+5+7+ … and show how they sum to make the square numbers
- make a stepped-triangle 1+2+3+4+5+6 and show how two of these can be put together to make a rectangle, which leads to sum 1 to n = n(n+1)/2
So this approach leads very naturally into some really nice mathematics. Along the way it reinforces the rote learning of times tables (auditory repetition) with visual intuition.
The web app – introducing Doctor X
I thought there was probably an App or web page to do this kind of thing interactively. I googled around and found lots of times table grids, math systems but nothing that seemed to take the grid-paper-rectangles-and-counting-beans approach into an interactive medium.
Anyway, here is the current version of the visual grid calculator for kids, which I’m calling :
When I get time Ill make some more in-depth tutorials and worksheets on some of the concepts I mentioned above. Let me know what you’d like to see,