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News from Mathnasium of South Beaverton

The importance of the Law of Sameness- No Banapples!

Jun 26, 2018

When you add two apples with three apples you get five apples. When you add two bananas with three bananas you get five bananas.

When you add two apples and three bananas do you get five banapples?

You don’t get five banapples, just as if you add pennies and nickels you don’t get pickles. The reality is that you cannot add two things that have different names.

According to The Law of SAMEness, you can only add or subtract things that have the same name.


There are Four Principles to think about:

Principle 1:

A “thing” can be simple or complexIt could be as simple as a penny in your hand or as complex as the total number of eyes, noses, and fingers of all the people on Earth.

Principle 2:

Every thing has a name.

The name is that thing’s denomination

Principle 3:

Things have a number associated with them. The number can be as general as singular and plural or as specific as a number (i.e. person, several people, twoapples, a half gallon milk, twelve days).  

The number is the quantity of things we have.

Principle 4:

Combining Principles 1, 2, and 3, we learn to think in terms of:

Quantity and Denomination = "How many – of what? 

So, the “name" of today’s game is Denomination, Quantity, and SAMEness. 

Quantity Principle


Every thing in math has a name. This name gives the denomination of the thing, making it unique within a larger group of things.

For example:

Tuna, shark, and salmon are denominations of fish.

Halves, quarters, and thirds are denominations of fractions.

Pennies, nickels, dimes, and quarters are denominations of coins.

After knowing the name of some thing, you should also know the quantity of that thing in the given problem.

For example:

Three sharks                  quantity = 3

Half of a dozen              quantity = ½ and 6

Two hours in minutes   quantity = 120

After distinguishing between the denomination and the quantity associated with each thing, it is important to understand the functionalities of these things. This calls attention to The Law of SAMEness.

Mathnasium illustrates that according to The Law of SAMEness, you can only add or subtract things that have the same name.

Keep the rules of quantity and denomination in mind will enjoy math more because the rules you have learned will make more sense to you! For more practice, work on these problems: 

Larry Example 3