Lesson 15

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 use appropriate tools strategically (MP5) 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 get a sense of their approximate location. Ask students,

• “The decimal value for a number like \(\sqrt{5}\) 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 \(\sqrt{5}\) has a fixed position on the number line somewhere between 2 and 3. For comparison, the number \(\frac{1}{3}\) also has an infinite decimal expansion, but is exactly one-third of the way between 0 and 3. Similarly, \(\sqrt{5}\) has a certain position on the number line that does not move.)
• “How can you know that \(\sqrt{20}\) is somewhere between 4 and 5?” (I know that \(\sqrt{16} =4, \sqrt{25} = 5\), and \(\sqrt{16} < \sqrt{20} < \sqrt{25}\), 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.

Ask students what they remember about the connection between the lengths of the sides of the triangle. If it does not come up in discussion, remind students of the Pythagorean Theorem. Here are some examples of right triangles to show as needed.

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. Ask students,

• “Use the first triangle to compare the value of \(\sqrt{13}\) 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 seems longer than the hypotenuse.)
• “Use the fact that \(4 + 100 = 104\) to sketch a line that has length \(\sqrt{104}\).” (I can draw a right triangle with leg lengths 2 and 10, then the hypotenuse has a length of \(\sqrt{104}\).)

In this activity, students solve quadratic equations of the form \((x+a)^2 = b\) 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.

Ask students, “What do you know about the solution to \(x^2 = 18\)?” Ensure that students discuss:

• there are two solutions, \(\pm \sqrt{18}\)
• its value is between \(\pm 4\) and \(\pm 5\)
• that \(\sqrt{18}\) represents an exact value rather than an approximate value such as 4.2426.

Tell students to represent their solutions as exact values such as \(\sqrt{18}\) rather than approximate values.

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.