In this Warm-up, students place numbers involving square roots in their approximate location on the number line. In the associated Algebra 1 lesson, students will examine irrational solutions to quadratic equations. This Warm-up helps students get a better sense of the value of these numbers. Students reason abstractly and quantitatively (MP2) when they locate irrational numbers on a number line.

The purpose of this discussion is for students to understand that irrational numbers also have a position on the number line and to get a sense of their approximate location. Here are some questions for discussion:

• “The decimal value for a number like goes on forever and does not repeat. Does this mean the value is infinite? Does it mean the position on the number line moves?” (No, the value has a fixed position on the number line somewhere between 2 and 3. For comparison, the number also has an infinite decimal expansion, but is exactly one-third of the way between 0 and 1. Similarly, has a certain position on the number line, and it does not move.)
• “How can you know that is somewhere between 4 and 5?” (I know that , and , so it must be between those values.)

In this activity, students use the Pythagorean Theorem to find a missing side from a right triangle. In the associated Algebra 1 lesson, students examine irrational solutions to quadratic equations. This concrete example gives students support for thinking about irrational values. Students look for and make use of structure (MP7) when using their knowledge of the Pythagorean Theorem to solve problems.

In this activity, students solve quadratic equations of the form and estimate the value of any irrational solutions. In the associated Algebra 1 lesson, students see irrational solutions for quadratic equations solved using completing the square. Practicing this portion of solving by completing the square can help students focus on other parts of the method for solving.

Students critique a statement or response that is intentionally unclear, incorrect, or incomplete and improve it by clarifying meaning, correcting errors, and adding details (MP3).