**Addition**

I had 5 apples, and then my brother gave me 2 more apples.

I now have 7 apples.

The process is

**Without the pictures: **

*Starting state (5 apples)*

* Action (add 2 apples)(and bunch them up !)*

* Final state (7 apples)*

**The process is then coded into an equation with the plus sign and the 2 to code the action:
5 + 2 = 7 **

Or with more detail, and the result

Start with 5, add 2, and resulting state is 7

Or the equation with apples

Start with 5 apples, add 2 more apples, and result is 7 apples

**Subtraction**

I have 7 apples now, and then my brother takes 3 of them away.

I end up with 4 apples .

The process is

**Without the pictures:**

*Starting state (7 apples)*

* Action (subtract 3 apples)*

* Final state (4 apples)*

The process is then coded into an equation with a minus sign and the 3 to code the action:

**7 − 3 = 4**

Or with more detail:

Start with 5, add 2, and resulting state is 7

Or the equation with apples:

Start with 5 apples, add 2 more apples, and result is 7 apples

**Both of these equations are read from left to right**

**— – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – — – **

### Counting and adding.

If I start with 1 and then add 1 I get 2. This doesn’t seem to be a problem at all.

If I start with 2 and then add 1 I get 3. This doesn’t seem to be too big a problem.

If I start with 5 and then add 3 I get 8. This starts to get interesting because starting with 2 and adding 5 has exactly the same final state.

It is all to do with counting.

If I count a pile of bricks and there are ten of them I can jumble the bricks around and count them again. I will still have ten of them. The count is “how many” and nothing else.

Consequently the rearrangement has no counting effect, and so any collection of numbers with plus signs and a total of ten can be rearranged, to give the same total of ten.

For example:

**start 2, add 3, add 4, add 1** is the same total as **start 2, add 4, add 1, add 3** (result ten)

We can also swap the start number and the add number with an example:

**start 3, add 5** is **start 4, add 4** and then **start 5, add 3**. Same total.

And finally, with **add** we can combine two adds to get **add 3, add 2 = add 5**

**Counting and subtracting.**

The apple example:

*Starting state (7 apples)*

* Action (subtract 3 apples)*

* Final state (4 apples)*

The starting state*, or start number, is 7*

The action is

*subtract 3*and the final state is

**4**The coded version is ** 7 − 3 = 4**, conventionally,

but more explicitly we can write

*, or even*

**7 −3 = 4**

*7 and −3 = 4***— – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – — – **

### Subtracting several numbers.

Have a look at the following:

**9 and −2 and −4**

The final state is ** 3, **but so is

*9 and***−4 and****−2**Consequently the two subtractions can be swapped.

It’s even better than this: The adds and the subtracts can be put in any order, with the same final state.

Try

* start with 9, add 2, subtract 3, subtract 5, add 4, subtract 2* (final state 5)

or rearranged

*(final state 5)*

**start with 9, add 2, add 4, subtract 3, subtract 5, subtract 2****— – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – — – **

**Properties of operations – a critique**

*Operations and Algebraic Thinking, CCSS grade 1 (italics)*

* Represent and solve problems involving addition and subtraction.*

* 1. ………*

* 2. ………Understand and apply properties of operations and the relationship*

* between addition and subtraction.*

* 3. Apply properties of operations as strategies to add and subtract.*

* Examples:*

* If 8 + 3 = 11 is known, then 3 + 8 = 11 is also known. (Commutative property of*

* addition.) To add 2 + 6 + 4, the second two numbers can be added to make*

* a ten, so 2 + 6 + 4 = 2 + 10 = 12. (Associative property of addition.)*

In mathematics there are numerous **algebraic** structures each consisting of a set of elements equipped with an **operation** that combines any two elements to form a third element. (Wikipedia, adapted, 19th century math)

So we can find an example of this where any two elements in the set of natural numbers can be combined with “+”, with the usual interpretation. (7 + 4 = 11).

More specifically we can use the “commutative property” of addition, a feature of the natural number system, to tell us that 7 + 4 equals 4 + 7 , for example.

**BUT** take a look at the above example, with different numbers:

**7 − 6 + 4 **

Swap **4** and **6**

**7 − 4****+ 6**

So we have to avoid the negative operator.

The conclusion is that the commutative rule applies only when **all** the operators are plus operators.

**Now what are we going to make of this ?????????**

*Table 3. The properties of operations. Here a, b and c stand for arbitrary numbers in a given number system. The **properties of operations apply to the rational number system, the real number system, and the complex number **system. (CCSS)*

*Commutative property of addition is a+ b = b + a*

This is a definition. It states that “a complicated name” is what we know already, and can actually prove by counting. **So it is a complete waste of time for K-7 or further.**

Not only that, but the actions **add 3** and **subtract 4** are the original 17th and 18th century statements, and are **proper algebra**.