Training For Maths
Years 4 - 10

Preparation

Fold a piece of paper in half like this.

Pinch along the crease with your thumb nail.
Then you can tear it into 2 pieces.
Or you can cut with scissors.
Use one piece.
Fold and tear (or cut) it into 4 pieces like this.

Do the same with the second piece.
Then tear each of these four (4) pieces in half.
You will finish with these pieces.

The short ones are 1 unit train carriages.
The long ones are 2 unit train carriages.

Now let's play trains. Toot! Toot!

 

3 Unit Trains

  • Both of these trains are 3 units long.
  • The engine is pulling the -21 train.
    The other train is the -111 train.
  • There is only one more way to make a 3 unit train. What is it?
  • In your journal sketch all the 3 unit trains.
    (The third one is in a photo in the Answers below.)
    Write each train in numbers too.

Shorter Trains

  • In your journal sketch all the trains shorter than 3 units.
    Write them in numbers too.
    There are four (4) trains shorter than 3 units, but one is really sneaky.
    (The sneaky one is in the Answers below.)

Challenge 1

  • Make and record all the six (6) unit trains.
Have fun exploring Training For Maths.

Challenge 2

  • A mathematician's work begins with an interesting problem.
  • A problem is a problem because it doesn't have an answer yet.
When a mathematician thinks they have found the answer, no one else can tell them they are correct, because no one else knows the answer.
That's when a mathematician asks, Can I check this another way?.
  • If they try the same problem two (2) different ways and get the same answer, they can be more sure that they are correct.
Look at the Strategy Toolbox on this Working Like A Mathematician page and choose one that helps you find all the 6 unit trains another way.
  • Try out your other way. Do you get the same answer for 6 unit trains?

Digging Deeper

Train
Length
No. of
Trains
0 1
1 1
2 2
3 3
4 .....
5 .....
6 13
This table shows the data we have so far.
  • Copy the table into your journal and work out the missing numbers.
    Your paper carriages will help you think.
If you find the right number for 4 and 5 length trains, there will be a rule that connects the numbers down the second column.
  • With that rule you can make a hypothesis (an intelligent guess) that the number of ways to make a 7 length train is 21.
You can check your hypothesis by writing out every possible case for 7 length trains.
  • There are at least 2 ways to plan that out.

Counting 7 Unit Trains
Your paper carriages will help you think.

Plan A
Start by thinking about Size 2 carriages:

Can I make a 7 unit train that has 4 Size 2 carriages?

  • No. It would be too long.
Can I make a 7 unit train that has 3 Size 2 carriages?
  • Yes.
    Now I need one Size 1 carriage and I have to find all the places I can put it.
Can I make a 7 unit train that has 2 Size 2 carriages?
  • Yes.
    Now I need three Size 1 carriages and I have to find all the place I can put them. They could be in ones, or twos, or threes.
Can I make a 7 unit train that has 1 Size 2 carriage?
  • Yes.
    Now I need five Size 1 carriages and ...
Can I make a 7 unit train that has 0 Size 2 carriages?
  • Yes.
    ...
Plan B
Start by thinking about the total number of carriages:

Can I make a 7 unit train that has 8 carriages?

  • No. It wouldn't be 7 units long.
Can I make a 7 unit train that has 7 carriages?
  • Yes. That would be seven Size 1 carriages.
Can I make a 7 unit train that has 6 carriages?
  • Yes. That would have five Size 1 carriages and one Size 2 carriage. Now I have to count all the ways to arrange those.
Can I make a 7 unit train that has 5 carriages?
  • Yes. That would have three Size 1 carriages and two Size 2 carriages. Now I have to count all the ways to arrange those.
Can I make a 7 unit train that has 4 carriages?
  • Yes. That would have one Size 1 carriage and ...
Can I make a 7 unit train that has 3 carriages?
  • No. The longest train with ...

Choose one of the plans and use it to check that there are 21 trains that are 7 units long.

Fibonacci Numbers

The sequence of numbers in Training For Maths is part of the Fibonacci Numbers which are:

  • 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ...
0 and 1 start the sequence. After that, each new number is the sum of the two that come before it. This sequence was first published in Europe by Leonardo Fibonacci. He probably learnt about it by studying mathematics from Arabian countries. He spent a lot of his early life studying mathematics in several countries.

This list shows the first eleven (11) Fibonacci numbers.

  • Guess a 'between' for the 25th Fibonacci number and write it in your journal.
    I think it will be between ... and ...
  • Copy and continue Fibonacci's list to see how close your guess was.
    (You can use a calculator when you need to.)
If you know how to use a spreadsheet (or you ask someone to show you) you can easily 'teach' it to make a list of the first 100 Fibonacci Numbers.
You will be amazed to see how many digits there are in the 100th Fibonacci Number.

You can learn more about Fibonacci Numbers by web searching Fibonacci in Nature.
Perhaps you will want to do a project about the man and his numbers.

Just Before You Finish

For this part you need your maths journal and your Working Like A Mathematician page.
  • Read again what it means to work like a mathematician.
  • In your journal, copy and finish this paragraph:
    Working with Training For Maths I was a successful mathematician because...

 

Answers & Discussion

Answer: 3 unit trains & shorter

  • The third 3 unit train is the -12 train.
  • The trains that are three units long are ... -111 ... -21 ... -12.
  • The trains that are two units long are ... -11 ... -2.
  • The trains that are one unit long are ... -1.
  • The tricky one is the zero train. There is one way to make it. Just the engine.

These notes were originally written for teachers. We have included them to support parents to help their child learn from Training For Maths.

Send any comments or photos about this activity and we can start a gallery here.

 

Maths At Home is a division of Mathematics Centre