In this lesson, students build on previous work with square roots to learn about irrational numbers. Students recall the definition of rational numbers, numbers that can be expressed as positive or negative fractions, as they search for a number such that .

Students begin by finding positive solutions to equations of the form by finding values of that will make each given equation true. They build on this idea as they try to find a solution to the specific equation in three different ways.

Then students consider a square with area 2 square units and are asked to find its side length. Some students may choose a measuring strategy, while others may choose to use square root notation. By comparing these two strategies, students begin to see that the value of is close to 1.5 (MP2).

Next, students are given a series of values and must test if the square of any of them equals exactly 2. While the values are specifically chosen to get closer and closer to the value of , none of them will equal exactly. Students continue searching, aided by the use of a calculator (without a square root button). They will not be able to find a rational number whose square equals exactly 2, allowing for the introduction of the term irrational number: a number that cannot be expressed as a positive or negative fraction.

Students should not be left with the impression that looking for and failing to find a rational number whose square is 2 is proof that is irrational. Proving this is out of the scope of grade 8, and so ultimately students must just accept it as fact for now.