What Is the Law of Sameness?
A Mathnasium teaching principle that states only "like" terms can be combined or compared in an equation or expression
The Law of Sameness is a concept used at Mathnasium to help students understand that in order to combine or compare mathematical quantities, they must belong to the same category. We like to illustrate this like so:
- 1 banana + 1 banana = 2 bananas
- 1 apple + 1 apple = 2 apples
- But what is 1 banana + 1 apple = 2 bananapples?
In math terms, this means that:
- You can only combine like terms. These are the terms that have the same variable (e.g. 3x + 5x = 8x where both terms have the variable x) and the same exponent (e.g. 2y² + 4y² = 6y² where both terms have y²).
- You cannot combine terms that are different. For example, 4x² + 2x cannot be combined because x² and x are not the same, and 5 + 3x cannot be combined because one is a whole number (5) and the other is a variable term (3x).
Whether students are adding terms, comparing values, or solving equations, the Law of Sameness reminds them to ask: “Are these the same kind of thing?”
Let’s look at a few more examples:
- 5a + 8a = 13a: Since both terms have the variable a, they can be combined.
- 2y² + 6y: These terms cannot be combined because y² and y are not the same.
- x² + 3y: These terms cannot be combined because the terms do not have the same variables or exponents.
- 7 + 5=12: Both are whole numbers, so they can be added to make 12.
The Law of Sameness helps keep equations and expressions organized and makes it possible to solve problems correctly.
We use the Law of Sameness when:
- Combining like terms in algebra
- Comparing quantities in word problems
- Matching units in measurement problems
- Organizing values by place value in early arithmetic
When Do Students Learn About the Law of Sameness?
The Law of Sameness is a Mathnasium teaching concept introduced early, typically starting in Grade 1 through our learning materials and methods.
While most students encounter this idea during algebra in traditional classrooms, Mathnasium builds this habit from the beginning to strengthen understanding across all topics.
