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.

The purpose of the discussion is to understand that irrational values have an actual value that can be seen as the length of a side. Select students to share their responses and reasoning. Here are some topics for discussion:

• “Use the first triangle to compare the value of by comparing it to 2, 3, 4, and 5.” (It is greater than 3 since it is the hypotenuse of a triangle with a leg of length 3. It looks a little less than 4 since 2 of the sides of length 2 in that image seem longer than the hypotenuse.)
• “Use the fact that to sketch a line that has length .” (I can draw a right triangle with leg lengths 2 and 10, then the hypotenuse has a length of .)

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).

The purpose of the discussion is to understand the approximate values of irrational numbers by comparing them to integers. Display the first number line, and select students to add the position of their solutions to the image in their approximate position.

After the solutions have been added, display this number line.

Ask students if they would like to move any of their solutions on this updated number line.