Training For Maths

Task 137 ... Years 4 - 10

Summary

Easily stated and easily started, but containing plenty of challenge. The story shell of an engine followed by a sequence of carriages, each of which is either one or two units long, offers the challenge of counting the number of different trains that can be created for a given train length.
 

Materials

  • One 'train engine'
  • Four 2-unit 'carriages' and eight 1-unit carriages

Content

  • pattern and order
  • combination theory
  • recording in a table
  • Fibonacci Numbers
  • Binet's Formula
  • Golden Ratio
  • Pascal's Triangle

Training for Maths

Iceberg

A task is the tip of a learning iceberg. There is always more to a task than is recorded on the card.
   

The problem is simplified if students realise that the 2 unit carriage 'controls' the count. If we categorise trains by the number of 2-unit carriages then the other carriages in the train, if they exist, must be 1-unit ones. Therefore, the 6 unit trains are:

  • 1 train with 0 x Size 2 carriages
    -111111
  • 5 trains with 1 x Size 2 carriage
    -21111, -12111, -11211, -11121, -11112
  • 6 trains with 2 x Size 2 carriages
    -2211, -2121, -2112, -1221, -1212, -1122
  • 1 train with 3 x Size 2 carriages
    -222
and since these are the only possibilities for trains with Size 2 carriages, the total number of 6 unit trains is 13.

A different (but related) way to investigate this problem is to look at the possible number of carriages in the train rather than the length of the carriages. The maximum number of carriages for a 6 unit train is six. So, we can count:

  • 1 train with 6 carriages
    -111111
  • 5 trains with 5 carriages
    -21111, -12111, -11211, -11121, -11112
  • 6 trains with 4 carriages
    -2211, -2121, -2112, -1221, -1212, -1122
  • 1 train with 3 carriages
    -222
  • 0 trains with less than 3 carriages because none of -22, -11, -21, -12, -1 or -2 make a 6-unit train.
which again is a total of 13.

To successfully investigate further, as encouraged by the card, requires finding careful counting strategies which ensure that all the trains have been found for various lengths. The card offers answers for the 3-unit train, but begs the question of whether the three given are the only answers. Are they?

Clearly the task is expecting students to apply the Working Mathematically Process. That is, to play with more train lengths, collect data about each, organise the data, build and check hypotheses based on the data and publish the results.

Representing organised data into a table showing the total number of trains for each length gives:

Train
Length
No. of
Trains
1 1
2 2
3 3
4 5
5 8
6 13

which almost offers a pattern in the second column. If we then allow for the 0 unit train - which can be made one way by not making it at all - the table becomes:

Train
Length
No. of
Trains
0 1
1 1
2 2
3 3
4 5
5 8
6 13

then the first two rows of the second column provide the seed for the other numbers of that column, each of which is found using Fibonacci's rule of adding the two previous numbers. Our train totals appear to be Fibonacci Numbers! and if that is true we could predict that the number of 7-unit trains would be 21. Is it?

The answer is yes, but it is a little tricky - bit worthwhile - to confirm that by writing them all out. Beyond 7-unit trains though it becomes increasingly tedious to write out all the trains, so it's a good thing that we have found a pattern that will allow us to calculate the total number of trains for an n-unit train.

Or have we replaced one tedium with another? Try calculating the number of trains for a length of 25 units using Fibonacci's rule.
...
Aha! at this point we might discover the value of a spreadsheet.

Although the spreadsheet approach is very powerful, it still doesn't allow us to calculate the number of trains given any length of train (n). All calculators are limited by their technology. It will only be possible to calculate in this way if it is possible to calculate the nth Fibonacci Number. There is a sophisticated formula, called Binet's Formula, which does this (again, almost because technically it only gives a very accurate approximation). There are several web links to this. One useful one is:

Intriguingly, this formula involves the Golden Ratio: (1 + Sq.Rt.[5]) / 2.

The intrigue is extended a little further if we return to our 'written out' trains and ask:

  • For a given train length, how many ways 1, 2, 3, 4, ...carriage trains can be made?
The organised data for this question is:

Carriages

Train Length

1 2 3 4 5 6 7 8 9 10
1 1                  
2 1 1                
3   2 1              
4   1 3 1            
5     3 4 1          
6     1 6 5 1        
7       4 10 6 1      
8                    
9                    
10                    

and various patterns have begun to appear that will assist in completing this table and any extension of it. Look carefully and you will even find the beginning of Pascal's Triangle.

Whole Class Investigation

Tasks are an invitation for two students to work like a mathematician. Tasks can also be modified to become whole class investigations which model how a mathematician works.
   

If you don't have access to 1- and 2-unit blocks like those in the task, or perhaps to appropriate Lego or Duplo, then perhaps the easiest way for pairs of students to make carriages is by paper folding. Each person first folds a piece of A4 paper to make a square. The extra rectangle is removed (and no doubt some will want to turn it into the engine). Each square piece is then folded into eight sections. One person cuts along the folds to make four 2-unit carriages. The other cuts along the folds to make eight 1-unit carriages.

It is also good practice to introduce the problem in a fish bowl situation on the floorboard (or a central table) using larger cardboard squares and rectangles which you have prepared earlier. Twenty centimetre squares work well. With this introduction and their own train making equipment, the way is open to explore the mathematics above.

At this stage, Training For Maths does not have a matching lesson on Maths300.

Is it in Maths With Attitude?

Maths With Attitude is a set of hands-on learning kits available from Years 3-10 which structure the use of tasks and whole class investigations into a week by week planner.
   

The Training For Maths task is an integral part of:

  • MWA Pattern & Algebra Years 3 & 4
  • MWA Number & Computation Years 9 & 10

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