Lesson objective: Extend understanding that irrational numbers represent real distances by reasoning about the locations of numerical and variable expressions on a number line.

Students bring prior knowledge of irrational numbers such as \(\mathrm\pi\) and irrational square roots from 8.NS.A.2. This prior knowledge is extended to include simple expressions with irrational numbers as students reason about how to place \(-\frac12\mathrm\pi\), \(\mathrm\pi^2\), \(\boldsymbol b^2\), and others on a number line. A conceptual challenge students may encounter is deciding how to manipulate the irrational number for \(\mathrm\pi\). Another conceptual challenge is reasoning abstractly about numbers whose values are unknown.

The concept is developed through work with a number line, which reinforces the fact that irrational numbers are measures of length which are located certain distances from 0 on the number line.

This work helps students deepen their understanding of numbers because they see that even though they are estimating the value of these expressions with irrational numbers, the expressions themselves represent actual values.

Students engage in Mathematical Practice 1 (make sense of problems and persevere in solving them) as they struggle with the idea of performing operations (multiplication, squaring) on irrational and unknown numbers.

**Key vocabulary:**

- irrational number
- pi
- squared
- square root