Training For Maths
Years 4 - 10
|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.
- 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.)
- Make and record all the six (6) unit trains.
Have fun exploring Training For Maths.
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.
- A mathematician's work begins with an interesting problem.
- A problem is a problem because it doesn't have an answer yet.
That's when a mathematician asks, Can I check this another way?.
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.
- If they try the same problem two (2) different ways and get the same answer, they can be more sure that they are correct.
- Try out your other way. Do you get the same answer for 6 unit trains?
||This table shows the data we have so far.
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.
- Copy the table into your journal and work out the missing numbers.
Your paper carriages will help you think.
You can check your hypothesis by writing out every possible case for 7 length trains.
- With that rule you can make a hypothesis (an intelligent guess) that the number of ways to make a 7 length train is 21.
- There are at least 2 ways to plan that out.
Counting 7 Unit Trains
Your paper carriages will help you think.
Choose one of the plans and use it to check that there are 21 trains that are 7 units long.
Start by thinking about Size 2 carriages:
Can I make a 7 unit train that has 4 Size 2 carriages?
Can I make a 7 unit train that has 3 Size 2 carriages?
- No. It would be too long.
Can I make a 7 unit train that has 2 Size 2 carriages?
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 1 Size 2 carriage?
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 0 Size 2 carriages?
Now I need five Size 1 carriages and ...
Start by thinking about the total number of carriages:
Can I make a 7 unit train that has 8 carriages?
Can I make a 7 unit train that has 7 carriages?
- No. It wouldn't be 7 units long.
Can I make a 7 unit train that has 6 carriages?
- Yes. That would be seven Size 1 carriages.
Can I make a 7 unit train that has 5 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 4 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 3 carriages?
- Yes. That would have one Size 1 carriage and ...
- No. The longest train with ...
The sequence of numbers in Training For Maths is part of the Fibonacci Numbers which are:
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.
- 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ...
This list shows the first eleven (11) Fibonacci numbers.
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.
- 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.)
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