Negative Numbers with New Merology

 Notes to support the use of the New Merology IWB resource in teaching negative numbers.

Lessons where students get to utilise their subject knowledge alongside building skills in persistence and logical thinking are always satisfying for both student and teacher.  New Merology is a puzzle that I first encountered in Professor Stewart’s Cabinet of Mathematical Curiosities but is attributed there to Lee Sallows.  Adopted for the classroom students get to practise their skills of adding negative numbers and basic algebra alongside their skills of persistence and logical thinking.   I also use this as an alternative to setting a list of questions to test their understanding of negative numbers.  The children appreciate that there is a purpose and the problem seems less daunting than 20 questions but will actually result in them doing 10s if not hundreds of calculations.

The rules of the puzzle are easy to grasp but the puzzle itself is very difficult to solve:

For the spellings of the numbers zero to twelve assign a unique value to each letter that is consistent across all of the words and that allows each word to add up to the value of its number.

As an example:

Z = 2, E = 1, R = 0 , O = -3

Z + E + R + O = 0

O = -3, N = 3, E = 1 (Notice the consistency with “O” and “E” and that each value is unique.)

O + N + E = 1

This starts off relatively straightforward but quickly you will see conflicts arise.

Start the lesson ensuring each student remembers how to add negative numbers (you can use this scrollable number line to assist with this).  Next show them the New Merology Interactive Whiteboard resource.  As a class discuss the constraints of the puzzle and try a few of the student’s suggestions.  Starting with zero you will usually notice conflicts by the time you reach two or three.  Once the students start to formulate their own ideas of how to solve the problem let them break off into groups or alone to work on the solution.  I would advise against letting them use the IWB resource to work on the problem as that will deprive them of the arithmetic practice and due to its speed of recalculation discourages careful consideration.  It is best used as a demonstration aid only.  The IWB resource has a supporting sheet for laying out the calculations available for download.

Let the class know that periodically you will announce what one of the letters is.  The period of time will depend on how long your session is.  I tend to give one every 10 minutes after the first 15 have elapsed.  The choice of letters you give will depend on how much assistance you wish to give.  Certain letter combinations give away a lot of the puzzle.  For instance, if you make the first two letters “N” and “E” the children can easily work out “I” from nine and “O” from one.  It is good if they make this connection and will show good mathematical thinking but you may not wish to give so much assistance too quickly.

You will find that each letter you give results in groans of dismay as students realising the impact changing one letter has on the current values they have selected both in the creation of duplicates and the changing of the totals. 

With careful choice of letters and timing you should be able to control when the groups get close to solving the problem which will afford them the satisfaction of reaching a solution.  You can test their solutions in a class plenary using the IWB resource.

By the end of the session they will have had lots of practise at the addition of negative numbers in a far more motivating way than answering a list of questions.  While they may never have done any algebra they will also instinctively have done some algebraic reasoning to discover, for instance what I is in N+I+N+E = 9 when they know N and E.  In some ways more importantly they will have had to exhibit persistence, and will have experienced solving a problem that does not have a linear route from start to finish but has false starts and dead ends. 

This lesson, coupled with the IWB resource gives you all you need to teach this lesson and could fill a session of 1 hour or more.

As I first encountered this problem in Professor Stewart’s Cabinet of Mathematical Curiosities I have not included the solution here as I designed this resource to support the book not take anything from it.  It contains many other problems that can be adapted for the classroom.  Note this lesson and resource are in no way connected with the book.

If you use it please add your thoughts on its use for others to learn from in the comments.