Mixed Gift Boxes

A grocer receives two orders for fifteen apples and twenty pears. Using the boxes of apples and pears he has in the shop he puts together one order like this




and one like this



In the first gift box there are five boxes of three apples



And five boxes of four pears



In the second gift box there five boxes containing three apples and four pears

                                    

In both cases there are the same number of apples and pears in the gift box.

Instead of apples and pears we can do the same thing with Threes and Fours.

The Game of Maths

This is the same as arranging a hand of cards in two different ways.

or

and using holders

or which is

In the recording we show that two or more cards held in one holder using brackets.

Record
List           5 x 3 = 15           5 x 4=  20	Maths    5 x 3   +   5 x 4 =15   +   20 = 35

Record
List           5 x (3 + 4) = 5 x 7 = 35                                            35	Maths   5 x (3 + 4) =5 x 7 = 35

We can see that 5 x (3 + 4) = 5 x 3 + 5 x 4

More generally using wildcards

Record
List	Maths

  
         
       

Record
List                             -------------	Maths

We cannot work out a value without being given the declared values for and however we know that   what those values are.

We have found that
is a product of two factors and and we have factorised the number

When dealing with different wildcards think back to The Importance of Counting. Here we saw that all we could do was list ten yachts and twenty crew and could not add the ten and the twenty in any meaningful way.

In the same way because we do not know anything about and and so we must treat them as objects having nothing in common and so we can only put them together as a list