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. As students refine their understanding of rational and irrational numbers through this practice, they look for and express regularity in repeated reasoning (MP8).

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. Here are some questions for discussion:

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

In this activity, students approximate the value by comparing it to rational numbers that are near the value. 

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 have an exact value or does it 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 :

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 since there are 2 right triangles each with legs of length 1, so by the Pythagorean Theorem, the diagonal line has a length of .)

Display the image of the number line that incorporates the square to show the exact position of .

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