## Thursday, 29 November 2012

### Mastering Math - The Musical

Sorry I haven't posted much about what's been happening in the class for the last week or so. We've been too busy to do much Maths - we're in rehearsal for our class assembly item.

On Friday we are presenting a play (with songs).

It is called "Mastering Math" by Ron Fink and John Heath.

It's going to be an epic performance - do the words "Ben Hur" suggest anything regarding the scale of the enterprise?

The first rehearsal we timed took 38 minutes and 6 seconds.

Yesterday's dress rehearsal went for 38 minutes and 6 seconds.

A spooky coincidence or perfect timing?

It's a great play (with 10 songs!) that teaches problem solving using 4 steps:

This is the model used in the play for Problem Solving

And it gives 6 problem solving strategies:

1. Guess and check

2. Work backwards

3. Draw a table or a chart

4. Act it out

5. Draw a picture

6. FInd a pattern

I strongly recommend this play. It's fun, it engages the kids and it has some great music.

We loved it.

"Mastering Math" is available through Bad Wolf Press at:

But I figure, when you find something that works, share it with the world. It is hard to find good plays and musicals that are at the right level for primary kids.

## Friday, 23 November 2012

### I''m moving my house to Equador

Year 6 students have been designing and building scale models of houses as part of their inquiry into housing and poverty. After producing a plan for their houses, they started to build them.

Much interesting maths ensued with the application of the scale to the materials.

This student decided to build a house with curved walls
- lots of fun working out how to do it with straight lines

As the houses neared completion, thoughts turned to power supply.

One enterprising group decided to use wind power and constructed their own scale model of a commercial wind turbine, based on the Capital Wind Farm across the border near Lake George which are 124m from the base to the tip of the top blade.

That's the house down there - and that's how big the wind turbine is!

Other groups considered solar power and constructed mini panels for the roofs of their houses.

"Hmm," said one boy, "The panels need to be angled towards the sun to achieve maximum efficiency in energy production."

"What angle would that be?"

"Well, the angle of the sun at its highest point from the horizon I suppose."

"But doesn't that change with the seasons?"

"So can we make the panels adjust and rotate to face the sun automatically?"

"Hang on a minute. If we face the panels towards the sun we can only fit three onto the roof. We can fit four onto the roof if they are square to the edge of the roof."

"But they will produce less energy because they don't face the sun."

"Yeah but we'll have an extra panel so that's 33% more."

This one had plenty of solar cells - but lost some room by placing them on an angle

Enter a third student.

"Are you using solar panels?" asks the first student.

"Yes but I've built a house with a flat roof. How am I going to angle them?"

"A flat roof would be perfect if the sun was straight above it all the time."

Hmm, the flat roof creates it's own problems

"Where would that be?"

"You know - somewhere where the latitude is 0 degrees."

"What? Like the equator?"

"Yeah."

Grabs an atlas and flicks through to a world map.

"I think I'm going to move my house to Ecuador."

## Thursday, 22 November 2012

### Race Around the World

You've probably seen that TV show where teams get to race around the world looking for clues and solving problems.

We today we did our Maths equivalent. It was a lot of fun.

Here's the activities we included in the programme:

1. Throw a bean bag 3 times. Measure it. Add up all the measurements.

2. Find angles in the environment. Find 3 of each type - acute, right, obtuse, straight, reflex.

3. Find shapes in the environment - spheres, prisms, cylinders

4. Make your own 20cm ruler, indicating cms and mms. Use it to measure an exercise book, your teacher's wrist and a pencil.

5. Recite your 2x and 8x tables. Time them with a stopwatch. What is the difference between them in time taken?

6. Find a clock in a classroom. Draw what time it is. Convert this to digital time. Calculate how long until home time.

7. Sort out a load of coloured pencils. How many of each colour? How many altogether?

8. Count how many crayons in the big tub. How many would each class get if they were divided between the four Year 4 classes? How many of each colour?

9. Find 2 items that have a perimeter less than a metre and 2 that are greater than a metre.

10. If 7 is the answer, what is the question? Write some algorithms and word problems to match this answer.

## ...or why my kids all "failed" at decimals.

We'd been doing lots of inquiry into fractions and it was time to explore the link with decimals.

"Easy," thinks I, "Just jump in and start with a few simple decimals that they already know like 0.5 and 0.25 and then we can escalate it from there!"

So why were they staring at me with blank faces when I started going on about tenths and decimal points and converting and notation?

Because I had made a big mistake - I had overlooked the importance of tuning in as the first step in inquiry.

And because of this omission, the learning that did eventuate was uneven, unsystematic and unreliable - the kids didn't really have a stable base on which to build their new thoughts and ideas.

## So who "failed" - the kids or the teacher?

If I had looked at the results of the pencil and paper test that followed this experience, I could have justifiably "failed" a large proportion of the class.

But I'm smarter than that.

Of course, alarm bells started ringing the minute someone tried to explain to me that ¾ was the same as 0.34.

And it was at that point I started to reflect on my own professional practice and I came to the conclusion:

"Tuning In" is the critical first step in inquiry.

To "tune in" is to:

Establish "the Known"
- for we are about to embark into "the Unknown"

Connect to students' lives
- so that it is going to make sense to them

Define the sense of purpose for the inquiry
- validates that what we are about to do is important

Identify first thinking
- what are our initial thoughts?

A first invitation for questions
- an opportunity to set a few directions

(This is paraphrased from the work of Kath Murdoch, Australian inquiry educator and well respected education consultant)

## FAIL - First Attempt In Learning

In our class, we have stolen this acronym for "FAIL" - it is quite common on Twitter as it is a pithy little quote that is easy to remember. And it's less than 140 characters.

So, as a life-long-learner, I am ready to put my hand up and say, yes I failed - but I'm still learning.

## It's that time of year....

For us in the southern hemisphere, we are rapidly approaching the end of our academic year. This means report writing and assessment, sometimes in that order.

And when he thinks of assessment, a young man's heart quickly turns to pencil and paper tests.

So I think it's time we sat down and had a talk.

## What are the good things about pencil and paper tests?

1. They are convenient for teachers to administer to a large group of students. If convenience is a priority in education, then pencil and paper tests are the way to go. And sometimes when you have to assess a really large group, like the entire population of the country, this seems the best way to do it.

2. They give you quick results. Yep - you certainly get a result - a number, a score, a grade. And it is these numbers and grades that are required of us by parents, administrations and government offices.

3. They can assess knowledge. SOMETIMES a pencil and paper test will also provide an insight into student thinking and how they are approaching a problem. RARELY will a pencil and paper test lead to an effective assessment of higher-order thinking. MOSTLY it will focus on knowledge, understanding and application.

4. They are what we are familiar with. We are comfortable with them as teachers. They've been around for years. We did them when we were kids and we turned out alright.

5. They are "fair". Everyone gets the same set of questions so everyone gets an equal chance to perform. They are designed by professionals (ie you and me) and are worded to avoid bias, cultural prejudice and advantage to any particular class or group.

1. Their convenience for teachers can indicate a lack of formative assessment and informed observation. I have done this myself when I have not been attending to formative assessment of students during maths inquiries. To quickly throw together a "pop quiz" or similar task is effectively an admission that I took my eye off the ball and I'm trying to get back on top of my assessment by taking short cuts. Not good pedagogy.

2. They reduce student understanding to a score or grade. While this may be a requirement (certainly in my country - if we don't give A-E grades we lose government funding), it is meaningless. How can one student's entire mathematical experience be summarised in one of 5 assessment grades?

3. They do not reveal the depth of thinking and understanding of the student. Pencil and paper tests provide little opportunity for students to actually explain what they mean. Our Australian national testing system (NAPLAN) requires students to choose from multiple choice answers and colour in a circle that is corrected by a computerised scanner. Even the four or five open questions at the very end of the testing require no more than a series of calculations or correctly worked algorithms.

4. The "old ways" were designed for industrial-revolution style education factories. 21st century education is heading in a different direction. It is no longer satisfactory to maintain ineffective pedagogy just because "it is the way we've always done it". We need to be thinking of new ways to assess which may involve collaboration, open communication between students, sharing of resources and ideas, presentations of multiple solutions to open-ended tasks...

5. They are not fair. Having the "same" test is not the same as having an "equal" test. All students bring their own personal, physical, cultural and social baggage with them into the classroom. This is why we differentiate. How can you explain on a pencil and paper test that your blood sugar is low (or high)? that your parents are separating? that you've never been on a train? that you've never seen a \$100 note? that you have a migraine? that your allergies are playing up? that the boy next to you has been bullying you for the last 4 weeks? that you can't think straight when the teacher puts a timer on and says you have 30 minutes to complete the 7 page test?

## So what?

Well, this rant may not have done much for you but it has certainly helped me clarify a few of my own ideas:

1. I think pencil and paper tests have (very) limited value.

2. I think I need to be more creative in my design of assessment tools.

3. I need to re-examine my pedagogy and ask the big questions again..... why am I doing this? is this the best way to do it?

## As we close in on 10 000 hits on this blog...

I thought it might be good to steal some ideas from a web page "What's Special About this Number?"

## http://www2.stetson.edu/~efriedma/numbers.html

Yes - I e-mailed the author of the site and got permission to do this. Thanks Erich!

And I edited the list a bit to pick out the numbers that I understood...

## So here's some interesting numbers between 9500 and 9999

9500 is a hexagonal pyramidal number.

9513 is the smallest number without increasing digits that is divisible by the number formed by writing its digits in increasing order.
9519 has a 4th power that is the sum of four 4th powers.
9538 is a value of n for which 4n and 5n together use each digit exactly once.
9541 is a value of n for which n and 8n together use each digit 1-9 exactly once.
9542 is the number of ways to place a non-attacking white and black pawn on a 11×11 chessboard.
9551 has the same digits as the 9551st prime.
9552 and the following 34 numbers are composite.
9563 = 9 + 5555 + 666 + 3333.
9568 = 9 + 5 + 666 + 8888.
9576 = 19!!!!!
9592 is the number of primes with 5 or fewer digits.
9602 has the property that if each digit is replaced by its square, the resulting number is a square.
9615 is the smallest number whose cube starts with 5 identical digits.
9627 is a value of n for which n and 5n together use each digit 1-9 exactly once.
9629 is a value of n for which 2n and 7n together use each digit exactly once.
9632 is the number of different arrangements of 4 non-attacking queens on a 4×14 chessboard.
9639 has a 4th power that is the sum of four 4th powers.
9643 is the smallest number that can not be formed using the numbers 20, 21, ... , 27, together with the symbols +, –, × and ÷.
9648 is a factor of the sum of the digits of 96489648.
9653 = 99 + 666 + 5555 + 3333.
9658 = 99 + 666 + 5 + 8888.
9677 is a prime that remains prime if any digit is deleted.
9701 has a square whose digits each occur twice.
9721 is the largest prime factor of 1234567.
9723 is a value of n for which n and 5n together use each digit 1-9 exactly once.
9724 = 1111 in base 21.
9728 can be written as the sum of 2, 3, 4, or 5 positive cubes.
9753 is a value of n for which 4n and 5n together use each digit exactly once.
9767 is the largest 4 digit prime composed of concatenating two 2 digit primes.
9768 = 2 × 22 × 222.
9779 has a square root that has four 8's immediately after the decimal point.
9786 has a square whose digits each occur twice.
9790 is the number of ways to place 2 non-attacking kings on a 12×12 chessboard.
9793 is the smallest number that can be written as the sum of 4 distinct positive cubes in 5 ways.
9796 has the property that dropping its first and last digits gives its largest prime factor.
9797 is the product of two consecutive primes.
9801 is 9 times its reverse.
9841 = 111111111 in base 3.
9856 is the number of ways to place 2 non-attacking knights on a 12×12 chessboard.
9862 is the number of knight's tours on a 6×6 chessboard.
9872 = 8 + 88 + 888 + 8888.
9876 is the largest 4-digit number with different digits.
9878 has a 10th power whose first few digits are 88448448....
9900 = 100110101011002 = 990010 = 188119 = 119921, each using two digits the same number of times.
9920 is the maximum number of regions a cube can be cut into with 39 cuts.
9933 = 441 + 442 + . . . + 462 = 463 + 464 + . . . + 483.
9945 = 17!!!!.
9973 is the largest 4-digit prime.
9999 is a Kaprekar number.

## Tuesday, 13 November 2012

### Improper Fractions and Mixed Numbers

Here's a really great idea - great because it's simple, visual and something I stole from someone else (many thanks, @CapitanoAmazing - brother in collaboration).

It is a great way to demonstrate the relationship between improper fractions and mixed numbers.

## Setting it up

Give each kid in the class a stack of 5 triangles. (Right-angled triangles are good because they can be re-assembled to make squares. Any shape will do as long as they have a stack of identical shapes. Circles are problematic - they don't tessellate.)

Tell the class that each triangle represents one half.

(Therefore you have 5 halves or 5/2)

Ask them to make as many "wholes" as they can.

Hey look! I can make 2 wholes and one half left over.

So then, 5 halves or 5/2 = 2½

Keep playing with this idea but change the number of "halves" you start with.

Try it with 7 halves. Or 13 halves. (You need to use odd numbers so that you end up with a mixed number at the end. )

## Stepping it up

Alright, now we can work with halves, lets call the single triangle a different fraction, such as a quarter.

Now, how many wholes can you make?

Yes - I can make 1 whole and 1 quarter left over.

So that means 5 quarters or 5/4 = 1¼

## Going Further

Now it's time for the kids to do their own thing.

Get them to decide what fraction each triangle will represent.

And see if they can identify a relationship between an improper fraction and a mixed number.

Then get them to explain it to their peers.

Look!   11 thirds = 3

Yes! 11 quarters =

And 10 thirds = 3

## So what?

This is great because it is so visual. This is important especially when dealing with abstract concepts such as improper fractions or mixed numbers. It helps if the kids can see something to show what they are exploring. Physical manipulation of objects reinforces what they are conceiving in their brains.

And it's always fun to get the blocks out and have a play.