Number Discs

Task 171 ... Years 2 - 10

Summary

This task is virtually content free. It only relies on reasoning. Students have seven numbered discs and to solve the puzzles in anything other than a random way, they need to identify the limits created by the way the discs are numbered. In other words they need to find information in the task that isn't actually stated - just as students need to do with some textbook and examination exercises and, in particular, just as a mathematician must do in their work.

Success can come at several levels. Simply finding one answer is sufficient, as long as students are encouraged to come back to the task at another time. A much deeper level is to examine the various puzzles with a view to finding all the solutions. Then there are the extensions suggested by the structure of the task.

This cameo includes two Investigation Guides in the form of templates, one of which supports making your own number discs from plastic screw caps.

 

Materials

  • Seven discs, each numbered in a particular way

Content

  • combination theory / ordered arrangements
  • concept of proof
  • reasoning
  • recording mathematics
Number Discs

Iceberg

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

In the first instance, students might offer a solution to the first puzzle with the letters on the discs oriented in any rotation. This can be accepted, but discussion will also be needed. If the orientation of the identifying letters didn't matter, then the discs could be put down in any order and the numbers made to touch. It would only be necessary to rotate each one around its centre.

That wouldn't really be much of a puzzle would it? So would you like to try the puzzle again but this time keep all the letters the right way up?
When this is the guideline for the task, the students need to identify that what really matters is the numbers to the left and right of the letter. The other numbers can't match without rotation.
Disc Left No. Right No.
A 1 4
B 5 1
C 1 6
D 1 3
E 5 1
F 4 5
G 3 1
Making a table like this removes the 'noise' from the other numbers and perhaps helps students identify that every number except 6 has a match in the other column. So, if the puzzle is going to work in a line, the 6 must be where it doesn't need a match. Hence, because it is on the right hand side of C, the implication is that C must be at the right hand end of the line.

Now the mathematician is in a position to use the strategy of working backwards. Place C, then look for a disc with one on its right to match the 1 on C. The table tells us there are three choices. Make a choice and explore the next possibility. Continue this process until a solution is found or the sequence proves impossible. Retrace your steps and make alternative choices where possible.

One solution is:

There are at least three more solutions. (Try not to look until you have attempted to find them.) The bigger problem is to prove that there are only four solutions in all.

The second question also has limits. As students play with the 'hexagon' shape (where is the hexagon and how do we know it must be a hexagon) they will start to realise that at the top and bottom it is left and right numbers that have to match, and in the other places it it numbers in the other positions. Now the problem can be broken into smaller parts based on all the possibilities for the top two discs to match.

The table above shows that:

  • If the 1s are to match, there are 3 possible discs that could be in the left top position of the hexagon.
  • There are also 3 possible discs for the right top position if the 1s are to match.
  • So between them there are 9 possible starting matches to explore.
The table also shows that there is no need to think about 2s or 6s matching at the top. However 3, 4 and 5 need to be checked using an elimination process similar to the straight line situation.

One solution is:

Let us know when you think you have found them all. One of them is very special. When you arrange the six the left over one fits exactly in the middle:

Extensions

  1. It happens that the seven letters of the first puzzle are the same as the letters used to identify musical notes.
    • What happens if the letters of each solution are played in sequence on a keyboard? Does the sequence sound 'good' to your ear?
  2. Each disc is a different way to write the numbers 1 - 6 equally spaced around the circumference of a circle. The discs show 7 ways.
    • Can you find another way?
    • How many ways are there?
    Note: In this extension labelling the discs A, B, C, ... isn't involved. It's just about the order in which the digits are written. To avoid any orientation question it might be best if the numerals are written with their 'heads' pointing into the middle. (Like children lying in a circle with their their feet touching the circumference and their heads pointing along a radius towards the middle.)

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.
   

Screw caps from milk or fruit juice containers is one way to explore this task as a whole class investigation. These caps are about the same size as the discs and are comfortable to pick up and move. Each pair of students can make their own set of discs using this template. You will also need fine point permanent markers or something else that will write well on the plastic caps. Ask the art teacher about these.

The three challenges on the task card can then be explored under the teacher's guidance as described above. The focus of the lesson is that we can work like a mathematician by applying the reasoning processes and strategy tools a mathematician uses.

If any of the students want to attempt the second extension above this template might help.

At this stage, Number Discs 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 Number Discs task is an integral part of:

  • MWA Space & Logic Years 5 & 6

Green Line
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