Table of Contents

Unit 2: Overarching Habits of Mind (MP1 and MP6)

2.3 Attending to Precision (MP6)

Precision in mathematics is often thought of in terms of accuracy of measurements. This is an important part of precision in mathematics, but MP6 “attend to precision” also refers to the way in which we use the language and symbols of mathematics, in particular the clarity of language and use of definitions.

One example of attending to precision is in analyzing the necessity and sufficiency of mathematical statements.

  • Necessity: A necessary condition of a statement must be satisfied for the statement to be true.
  • Sufficiency: A sufficient condition is one that, if satisfied, assures the statement's truth.

Consider the statement "A cat is a mammal." This statement satisfies the necessity condition because the condition of being a mammal is necessary for cats. However, this statement does not satisfy the sufficiency condition because there are mammals that are not cats. In order for the statement "A cat is a mammal" to be precise, the following two conditional statements must be satisfied:

1. If an animal is a cat, then it is a mammal.

2. If an animal is a mammal, then it is a cat.

The first statement is true and satisfies the necessity condition of the original statement. The second statement is not always true since there are mammals that are not cats, so this does not satisfy the sufficiency condition of the original statement. 

This simplified example demonstrates how, in mathematics, students often assume necessary conditions are also sufficient — especially in ways they make their statements. Oftentimes, students focus on conditional statement #1 and ignore #2 when determining a precise definition.