# Grade 4/5 Math: Equations, Expressions, and the Equals Sign

October 4, 2017

The words equation and expression have been popping up a great deal lately in 4th and 5th grade.  While they seem interchangeable to students, there are significant differences between them.

An equation says that two things are equal to each other.  An equation always has an equals sign. Here are some examples of equations:

4+3 = 5+2

A = 10

24 = 6 x 4

An expression, on the other hand does not have an equals sign.  It is a mathematical phrase that can contain ordinary numbers, variables (like x or y) and operators (like add,subtract,multiply, and divide), and it expresses a relationship.

A misconception that often comes up with students is that an equation always has an answer.  This is a misunderstanding of what the equals sign means.  We are often conditioned to believe that = means “here comes the answer.”  However, it means that everything on one side has the same value as everything on the other side.  That’s why 4+3 = 5+2 is an equation.  There is no answer, just as there is no question.

This misconception often leads to some incorrect mathematical statements.  For example, a number of students in fifth grade made a common error while showing their thinking for applying two rules in a function.  The rule was multiply by 4 and add 1. If the number in the “in” side of the table was 3,  the students displayed their thinking as follows:

3 x 4 = 12 + 1 = 13

The students were writing as they were thinking, “3 x 4 = 12 and 1 more equals 13.”  However, this equation states that 3 x 4 is the same as 12 +1 which is the same as 13.  It is not true that 3 x 4 is the same as 12 +1.  The way to remedy this would be as follows.  A student could show their thinking by breaking it into two equations:

3 x 4 = 12

12 + 1 = 13

If they want to put their thinking into one equation, they can do this:

(3×4) + 1 = 13

While the parentheses are not required due to the order of operations, it does help make their thinking clear.

This is a conversation we will continue to have throughout the year.  It takes some time to internalize this idea.

Enjoy the rest of your week!