Lesson 20

In this warm-up, students find where an irrational value should be on a number line. In particular, students try to determine whether \(\sqrt{21}\) is closer to 4 or 5 without the use of a calculator. This increases number sense as students continue to see more irrational values through their work with quadratics. Students model with mathematics (MP4) when they use a number line to locate an irrational number.

Tell students to not use calculators for this activity.

The purpose of the discussion is to get a better idea about irrational values. Select students to share their solution and reasoning. Display these values for all to see:

• \(1 = \sqrt{1}\)
• \(2 = \sqrt{4}\)
• \(3 = \sqrt{9}\)
• \(4 = \sqrt{16}\)
• \(4.5 = \sqrt{20.25}\)
• \(5 = \sqrt{25}\)

Tell students to find the difference between each of the values under the square root and ask them what they notice and wonder about the differences. Focus attention on the idea that the distance between the values under the square root grows at a non-linear rate. For example, even though 20.25 is closer to 16 (\(20.25 - 16 = 4.25\)) than it is to 25 (\(25 - 20.25 = 4.75\)), the square root is exactly between 4 and 5.

In this activity, students examine some properties of rational numbers. In the associated Algebra 1 lesson, students look at solutions of quadratic equations as either rational or irrational. This work supports students by reminding them of the definition of rational numbers and providing practice working with some different ways they may appear.

The purpose of the discussion is for students to recognize some properties of rational numbers. Select students to share their solutions and things they notice about the decimal representation of rational numbers. Ask students,

• “How can a whole number like 12 be written as a fraction to fit the definition of a rational number?” (\(\frac{12}{1}, \frac{24}{2},\) or other equivalent fractions)
• “Do you think \(\sqrt{2}\) is a rational number? Explain your reasoning.” (It is okay if students are unsure about this question at this point. Sample response: I think it is not a rational number since the decimal representation does not appear to repeat or stop.)

In this activity, students approximate the value by comparing it to rational numbers that are near the value. For this activity, students should not use technology to find a decimal approximation for \(\sqrt{2}\) or \(\sqrt{5}\) although they can use technology for other purposes, such as squaring rational numbers to speed the process.

The purpose of the discussion is to notice that irrational numbers have a specific value that can be approximated by rational numbers. Select students to share their responses and reasoning. Ask students, “Does a number like \(\sqrt{2}\) move on the number line?” (No, it has a fixed position.) Explain that our understanding of its exact position on the number line can be updated to a more accurate position as we approximate its value with more decimal places, but there is a single place for the number. Demonstrate this idea using a geometric understanding of \(\sqrt{2}\):

Display the image of a square with side length 1.

Ask students, “How long is the diagonal segment across the square? Explain your reasoning.” (It has length \(\sqrt{2}\) since there are 2 right triangles each with legs of length 1, so by the Pythagorean Theorem, the diagonal line has a length of \(\sqrt{2}\).)

Display the image of the number line that incorporates the square to show the exact position of \(\sqrt{2}\).

Tell students that the work they did in this activity narrowed the window for where \(\sqrt{2}\) could be located on the number line, but there is an exact position for the value which is determined by this geometric interpretation.