The Language Trap: Decoding "More Than" and "Less Than"

If you’ve ever seen a student read the phrase "5 more than x is 12" and immediately write $5+x=12$, only to see them do the exact same thing for "5 more than x is greater than 12," you aren’t alone.

For many students, word problems are less about logic and more about "keyword hunting." They see "more than" and instinctively reach for the plus sign. They see "less than" and prepare to subtract. The challenge isn't that they don't know the math; it's that they don't recognize the grammar of inequalities.

Here is how to help students distinguish between an operation (addition/subtraction) and a relationship (inequality).

The most powerful tool in a student’s arsenal is the word "is." In the English language, "is" acts as a bridge to a comparison.

• The Operation (Action): "Six more than a number."

  • There is no "is." This is an incomplete thought, a mathematical phrase. It translates to $n+6$.

• The Inequality (Relationship): "Six more than a number is greater than ten."

  • The "is" changes the "more than" from an instruction to add into a statement of comparison.

The Strategy: Have students circle the verb in every word problem. If they find "is," "was," or "will be" attached to the comparative phrase, they are likely dealing with an inequality or an equation, not just an expression.

When students think of "more than" as addition, they are thinking of a destination. When they think of it as an inequality, they need to think of a region.

Ask your students: "If I have more than $5, do I have exactly $6?" The answer, of course, is "Maybe, but I could also have $100."

By using number line sketches in their journals, students can visualize the difference. An operation is a single point moving forward or backward. An inequality is a shaded arrow that covers infinite possibilities.

Teach students to look for limiters. Words like "maximum," "minimum," "at least," and "budget" are red flags for inequalities.

• Addition context: "Sarah has 5 apples and got 3 more." (She is combining items to find a total).

• Inequality context: "Sarah needs more than 5 apples to bake a pie." (5 is the threshold, not a part of a sum).

Give students "Switch-Up" drills. Provide two nearly identical sentences and ask them to write the mathematical equivalent for each:

1. "A number decreased by 10." ($n-10$)

2. "A number is less than 10." ($n<10$)

By placing these side-by-side, students begin to see that the "less than" in the first sentence is an action being performed on the number, while the "is less than" in the second is a boundary the number cannot cross.

Moving students away from keyword hunting requires us to teach them to be "math linguists." When they stop looking for "more" and start looking for the relationship between the values, the confusion between $x+5$ and $x>5$ evaporates. It’s not just about the numbers; it’s about what the numbers are allowed to be.