New Puzzle, New Numbers

Suppose we were given this as a times problem

Record

Maths 6x = 27

Even knowing the Six times table does not help.

A record of the start of the Six times table goes

Record

Maths 6 x 1 = 6 6 x 2 = 12 6 x 3 = 18 6 x 4 = 24 6 x 5 = 30 6 x 6 = 36

Twenty Seven does not appear in the Six times table.

So there is no solution to the problem. Well yes and no. No when using the numbers we have so far, yes if we invent new numbers. Why not? Numbers are like unicorns, if we can imagine them we can create them.

In The Multiplication Puzzle we saw how division could be used to solve such puzzles. In following blogs we found out more about division and factors and how they behaved.

When we create new numbers to solve these new multiplication puzzles then we need to know what they look like and for them to behave in the same way as numbers we already know.

Let us start with a simpler puzzle.

Record

Maths 2x = 1   x = 1 ÷ 2

and look at this by dividing one apple over two boxes.



We can only do this by splitting the apple into two equal parts .

So the result for each box is .


How about one apple divided between four boxes?



Again we can only do this by cutting the apple into
four equal parts .

and the result for each box is .


How about one apple divided among six boxes ?

We would have to cut the apple into
six equal parts


The result for each box is

Of course the more boxes you divide the apple over the smaller the pieces become and the harder it is to actually cut the apple or draw the picture of the result.

The Game of Maths

In cards One over six boards gives the answer to the puzzle what value was given to

the wildcard when gives the result

where is

We would have to cut the counter into six equal parts, to get on each board .

Now we want to create a new number and a new card that gives the same result as when using just one board.

   Hand                                                                                                     Result

                                                                                          


The new card is and the number is a Sixth or One Sixth. Not very imaginative but it does what it says, One divided into six parts. The name for the number is based on the number of boards, for most numbers just add th to the end

Divide over six boards and get Sixths.
Divide over seventeen boards and get Seventeenths.
Divide over fifty nine boards and get Fifty Ninths.
Divide over sixty boards and get Sixtieths.
Divide over forty four boards and get Forty Fourths.

One, Two, Three and Four are different
Divide over thirty one boards and get Thirty Firsts not Thirty Oneths.
Divide over twenty two boards and get Twenty Seconds not Twenty Twoths
Divide over seventy three boards and get Seventy Thirds not Seventy Threeths.
Divide over two boards and get Halves not Twoths
Divide over three boards and get Thirds not Threeths.
Divide over four boards and get Quarters or Fourths

These new numbers are called fractions and we need to know how to add, subtract, multiply and divide fractions.