Tuesday, 31 July 2012

5 Ways With Multiplication

There's more than one way to skin a cat

And there's more than one way to deal with Multiplication. So if your kids are struggling with multi-digit multiplication, try one of these alternatives:

1. The Vertical Algorithm

Okay - so here's old faithful. You get the top number and multiply it by the bottom number. Basically it's breaking the 2-digit algorithm into 4 single digit operations, using your knowledge of place value and the distributive law.

The example on the right simplifies this into two steps.

Many kids, and teachers, think that this is the only way to do multi-digit multiplication.

Let's hope they get a look at this page....

2. Draw it as a rectangle

We use multiplication to calculate the area of rectangles. This method can be used to represent the actual multiplication algorithm as well.

In the example, the rectangle drawn is 32 units high and 43 units wide.

Then the rectangles is divided up into simple boxes that are 10 x 10. These are easy to add up - they are equal to 100 each.

The left over bits at the top and sides are smaller rectangles that can be added on.

3. Using the diagonal grid - The Lattice Grid

Here's a great method to add to your repetoire.
Write the numbers as shown with a diagonal grid or lattice pattern as shown.
As you multiply each number, the result is written in the corresponding triangle.
When you finish, add along the diagonals. The lowest diagonal is units column, the next is the tens column etc.
The link below is a clip from Khan Academy working through an example - worth a look.


4. Visual Multiplication

Here's a really visual way to show multiplication.

Represent each multiplicand as a series of parallel diagonal lines.

In the example, 43 is shown as 4 lines and 3 lines sloping up from the left to the right.

The other multiplicand is 32, 3 lines and 2 lines crossing over the first line and going downwards from left ot right.

5. Russian Multiplication

My favourite - love the way this works.

1. Write the two multiplicands beside each other.

2. Halve the number on the left, double the number on the right. If the number on the right ends in a fraction, round it down to the next whole number.

3. Continue until the number on the left gets to 1.

4. Cross out the number on the right if the corresponding number on the left is even.

5. Add the remaining numbers in the right column.

Hope this gives you a few ideas to try out.

If you have any additional suggestions, please leave a comment - I'd love to hear of more ways to work with multiplciation!

Sunday, 29 July 2012

10% Off $748?

I was buying a new laptop for my daughter


It's her birthday and she had asked for a new laptop. I was very keen - it meant I could repossess mine that she had been using for school work.

So I'd been looking around for a few weeks and identified the one that we were happy with.

I went off to do the purchase and found the model we wanted reduced from $798 to $748. 


On finding the shop assistant. I asked him if I could purchase the said computer and he headed out to the storeroom to locate it. 
After a 10 minute delay, he came back and said the display model was the last one left but I could have it for a further 10% off.

Deep joy!

So we packed up the display model and headed for the checkout.

This is where the fun started.

So, what's 10% off $748?

Being unable to work this one out for himself, the young salesperson turned to the girl at the next register. They put their heads together and pulled out a calculator and a folder with instructions on how to work out a discounted price.

I was about to say, "It's $74.80 so the new price is $673.20." but I held my tongue to let them do the calculation.

"I'm not too good at this. I should have listened more in maths class," the young man said apologetically as he produced the final total.

Yep - the calculator agreed - 748 divided by 10 was indeed 74.8.

Relieved, the salesman rang up the new price on the till - $1496.

I pointed out that this was not quite correct and he made a few adjustments and said, "Oops - it should be $673.20."

As I was about to pay this amount, he stopped suddenly and said, "Hang on, now I need to take 10% off that, don't I?"

Probably could have got my daughter a $798 laptop for $605.88.

Saturday, 28 July 2012

Proving the Distributive Law

What is the "Distributive Law"?

You will no doubt remember your old maths teacher telling you this one:

57 x 32 = (50 x 30) + (50 x 2) + (7 x 30) + (7 x 2)

or even more simply

5 x 12 = (5 x 10) + (5 x 2)  
or   (5 x 6) + (5 x 6)   

or   (5 x 8) + (5 x 4) etc

I thought it might be good to get the kids to show this one visually and to have them prove the Distributive Law.

Let's see what they came up with...

Alright - this is 6 x 5 = 30...........................and this shows (6 x 3) + (6 x 2) = 30

And here we have 8 x 3 = 24............which can be shown to be (5 x 3) + (3 x 3) = 24

and finally 7 x 4 = 28.......................which can be seen to be the same as (5 x 4) + (2 x 4) = 28

Nice work class!

The conversation continued....

Having had a chance to model these facts and use the Distributive Law a bit, we then sat down to talk it through.

"So, the 12 times tables is the same as the 10 times tables plus the 2 times tables!"

"Yes because 12 x 3 is 36 and 10 x 3 plus 2 x 3 is 36 too!?

"So the 15 timnes tables is...?"

"Oh, that would be the 10 times tables plus the 5 times tables."

"Right! And the 17 times tables....?"

"It's the 10 times tables plus the 7 times tables."

"Good work. Then how about the 26 times tables?"

"Hmm, that would be the 2 times tables plus the 6 times tables."

WHAT?! Fantastic - an opportunity to learn!

And so we continued the conversation.

Why is multiplying by 26 NOT the same as multiplying by 2 and multiplying by 6? And why WOULD multiplying by 26 be the SAME as multiplying by 20 (not 2) and multiplying by 6?

Place value, Distributive Law, times tables facts, arrays, modelling - a lot of interesting stuff in what could have been a bit of a tedious, repetitive activity involving number.

Thursday, 26 July 2012

If You Don't Have a Word For "Zero"...

...can you have the concept?


And yes.

Sound confused? Let me explain.


There is a school of thought in Linguistics called Whorfianism, named after Benjamin Lee Whorf, who proposed that our understanding of the world around us is framed by the language we use to describe it.

"The 'real world' is, to a large extent, unconsciously built upon the language habits of the group."
Sapir, E. (1929). 'The Status of Linguistics as a Science". I E. Sapir (1958): Culture, Language and Personality (ed. D. G. Mandelbaum). Berkeley, CA: Univesity of Califronia Press as quoted in "The Sapir-Whorf Hypothesis" by Daniel Chandler at http://www.aber.ac.uk/media/Documents/short/whorf.html

Benjamin Whorf

If this is so, then our understanding of Mathematics is determined by the language we use to express it. And if we have no word for "zero" then we can have no idea of the concept of "zero".

Ancient Rome had no zero as a part of its numeric system. One of the benefits of using the Hindu-Arabic number system is the zero and having ten unique digits that we can use to make a base10 number system. Roman numerals had no such features and was completely awkward as a number system for calculations. 

Edward Sapir - the other half of the Sapir-Whorf Hypothesis

You've probably heard the case quoted of the Inuit language having more than 100 words for snow. Australian indigenous languages are often quoted as having the opposite issue - not a surplus of words but a dearth of words specifically in relation to numbers. Some of these languages are cited as having a numerical vocabularly of only two words - "one" and "many".

But does this mean that Australian indigenous people can't tell the difference between 3 objects and 100 objects, just because they don't have the words for "three" or "hundred"?


The problem often identified with Whorfianism is that it suggests that language is formulated before the actual concept it is going to describe. Is that possible?

Probably not. And then you could have lots of arguments about what came first, the chicken or the word for chicken.

SO, while it may be true that language has a part to play in understanding and concept development, not many linguists would subscribe to pure, hard-line Whorfianism.

And even if you don't reject the hypothesis outright, there may be some middle ground where you can actually argue that.....

.....without a word for "zero" you can still have a concept of what "zero" is - but it may not necessarily be the same concept as those who do have the word.

Hope that clears it up.

And never forget the significant role that language plays in maths.

Wednesday, 25 July 2012

5 Things to do with Triangles

Looking for something to do with triangles?

I was scratching around for some things to do with triangles to show the kids some interesting properties of these cool shapes. I didn't want to get into Pythagoras at this point but lots of the websites and Youtube clips I found were pretty pedestrian.

So, here’s my contributions about what you can do with triangles. Hopefully they are a bit more stimulating and provocative than “put a triangle on top of a square to make a house”…

1. How many degrees in the sum of the angles of a triangle?

Cut out a triangle. Put a dot in each corner. Cut off the corners. Reassemble them to make a straight line.
How many degrees are there in the sum of the angles of a triangle?
What would happen if you used the corners from a quadrilateral? pentagon? dodecagon?

2. Diagonals making new triangles

Divide polygons up into triangles using diagonals going from one corner to another. How many triangles do you end up with if you start with:
a) a quadrilateral and join up each corner with straight line diagonals?
b) a pentagon and join up each corner with straight line diagonals?
c) a hexagon and join up each corner with straight line diagonals?

What is the pattern you can see emerging here? How many triangles would you expect to make with an octagon? And is there something special about the middle shape you create when you start with an odd-sided polygon?

3. Cross sections of a tetrahedron

Make a playdough model of a tetrahedron. Can you make a shadow that is:
a) an equilateral triangle?
b) a square

Use a plastic knife to cut a cross section. Can you make a cross-section that is:
a) an equilateral triangle?
b) a square

4. Making 3D shapes from triangles

Using triangles made from toothpicks and blutack, make 3D shapes.
a) How many triangles do you need to make a tetrahedron?
b) Ditto for an octahedron?
c) And also an icosahedron?

Have a go at making some of these shapes from cardboard. 
For some useful nets try:

5. Dividing up a triangle

Using a single straight line, cut a triangle into:
a) 2 triangles
b) a triangle and a quadrilateral
c) 2 quadrilaterals (Hmm...not sure about this one - I had an idea when I first wrote it but now I can't remember what it was. Any suggestions?)
d) anything else? (Hint – get away from the literal, flat view and think lateral)

....and have some fun!

Tuesday, 24 July 2012

Students Reflecting on How They Feel About Maths

Times Tables Are Evil

Well, that's what the new boy in my class announced on Day 1 in our first Maths session together.

And it made me wonder.....

.....how can a 9 year old have such strong negative feelings about multiplication facts?

.....what can I do to change this negativity?

.....what are the other kids feeling about Maths in my class?

So what do the kids think about their performance in Maths?

I asked the kids to reflect on their feelings about Maths, specifically about their level of confidence with mathematical thinking. I wanted to try to avoid performance-based self assessment that was confined to "I get good marks, therefore I'm good at Maths" type of comments. This was part of a Semester 1 reflection devised by my good colleague Capitano Amazing (4thgradebigquestions.blogspot.com.au/)

There was also a sliding scale where the children were asked to tick where they felt they were along a continuum from extremely confident to totally confused.

The continuum showed that about half the class is extremely confident and about half are somewhere in the middle. Two students indicated that they were totally lacking in confidence. This is great - now I know what I can do and where I can start with them.

The comments also were interesting - some good, some bad, some very revealing.

Here is a summary of what they said:

The Good

I love Maths. I can safely say I'm good at it too. I try to find Maths in everything I do.

I love doing Maths because it is really fun when we learn something new.

I love Maths!

Maths is awesome!

I like learning my times tables.

I love solving problems.

The Bad

Because I'm horrible.

The Revealing

Sometimes it's really easy and sometimes I think, "Oh no, what am I going to do?"

I don't always understand Maths questions.

I would like to get a bit more confidence in Maths.

I have trouble concentrating.

Most things I am confident with but others things like "Time" and "Division" I need to work on.

Although everyone keeps insisting I'm good at Maths I don't think so because my Mentals is horrible and I don't have the habit to check my answers twice.

I have trouble with carry the one and where to put it. Let's chat please.

So what?

Students have a very significant insight into their own learning. Ask them, they will tell you all about themselves if they are given the opportunity.

Self evaluation gives me so much information to work through that I could not get from other sources. 

As a result, in class we will be spending more time looking at Division, Time and mental strategies.

And yes...we will chat. I will tell you where to put the one when you carry it.

Tuesday, 17 July 2012

Maths and Making a Quilt

My Youngest Daughter is Making a Quilt

We're on holidays and the kids were looking for something to do. My youngest daughter decided to have a go at making a quilt. It was not surpirsing really - sewing is a bit of a family passion - my mum, sister, wife and eldest daughter are all more than competent on a sewing machine. We have several great quilts around the house that are always in demand on winter evenings in Canberra.

So, after a trip with mum to the material shop, Daughter #2 sat down and started to plan out the masterpiece.

"Hey Dad, what's 13 divided by 8?"

And being a good teacher, I answered a question with another question.

"Why do you need to know that?'

"Well, I'm making a pattern that's 10 rows of 8 squares going across. I've got 13 different patterns and I need to know how many of each colour I'm going to need," she replied.

"Right," I said. "And you're thinking that will help you work it out?"

"Yep, because there's going to be 80 squares so if I divide by 8 then I can just move the decimal point over one jump."

Hmm, good strategy but based on faulty reasoning.
And being a good teacher, I let her follow this line of thinking to see where it took her.

It didn't take long.

"Hang on, I think I'm doing it the wrong way round! I need to divide 8, or 80, by 13. So, that's 6 remainder 2. That means I need 6 squares of each pattern but 7 squares of two of the patterns!"

I knew we'd get there in the end.

Here's a patchworking puzzle to try with your kids

Plan out a quilt that is made up of 8x10 squares.

You have 13 different patterns:

3 pink    3 blue   3 green   2 purple   1 brown   1 yellow  

Same colours cannot touch each other.

And if you want to make it harder (impossible?) no row or column can have all 3 pinks, all 3 blues or all 3 greens.


And my daughter's quilt?

It's looking great!

Wednesday, 4 July 2012

Beware the Guru

I was at a meeting a few weeks ago

...and three of the teachers there were introduced by their colleagues as "maths gurus" from their respective schools. Strangely, none of them were bald nor did they show any signs of having spent significant periods of time in their recent history on top of remote mountains.

But it made me think.

3 Reasons You Don't Want to be a Guru

1. You are being set up for failure. It might be fine for now but just wait for that moment when someone finds you using a worksheet you've downloaded from Superteacher.com. "I thought you were a maths guru! You're just like the rest of us!" they will exclaim, all illusions shattered.

2. You might actually believe what people are saying about you. If you think you have reached nirvana already there is no incentive to keep trying to do any better.

3. Your expertise becomes an excuse for other teachers to do nothing, know nothing, try nothing and learn nothing. "I'm not a maths guru like you. Can you do it for me?"

Ultimately We Are All Learners

I am a learner, on the same journey as the rest of my professional colleagues. I have the advantage of age which gives me some degree of perspective when I sit down and reflect on where I've been and where I'm going.

It certainly doesn't make me a keeper of the secret of life, the universe and everything.