Lesson objective: Understand that the square roots of some numbers are irrational, but can still be approximated.

Students bring prior knowledge of finding side lengths of squares from 7.G.B.6. This prior knowledge is extended to include irrational square roots as students estimate the side length of squares with given areas. A conceptual challenge students may encounter is how to find the closest possible square root, since it will be impossible to find the exact number which, when squared, equals the given area.

The concept is developed through work with visual models of squares with perfect square roots and irrational square roots, which help students see how an area that falls between two perfect-square numbers will result in a side length that falls between the square roots of those numbers.

This work helps students deepen their understanding of numbers because they discover that irrational numbers are not "nonsense numbers," but represent actual distances.

Students engage in Mathematical Practice 8 (look for and express regularity in repeated reasoning) as they develop an efficient method for approximating an irrational square root. Students will be encouraged to develop a method, such as using tables of values to systematically test out a series of numbers in order, that will work for estimating any irrational square root.

**Key vocabulary:**

- irrational number
- square root