This is the first of three activities where students investigate the value of . By finding the areas and side lengths of three specific squares, students see that the value of is a bit less than 1.5. In following activities, students will look for a more precise value of .

Monitor for students who:

• Compare lengths directly either by creating a grid ruler or by tracing a segment with tracing paper and bringing it side by side with another segment. Most likely these students will say that the side length of the tilted square is around 1.5 units or a little less.   
• Find the area of the tilted square first and recall the square root notation from the previous lesson. Since the area of the tilted square is 2 square units, its side length can be expressed as units.

The goal of this discussion is to connect the idea that can be approximated by measuring the side length of a square with an area of 2 square units. 

Display strategies from previously selected students for finding the side length of the tilted square. Invite students to briefly describe their approach, then use Compare and Connect to help students compare, contrast, and connect the different approaches. Here are some questions for discussion:

• “How are the two approaches different?” (One is measuring to find an approximate value, and the other one gives us an exact value.)
• “Are there any benefits or drawbacks to one approach compared to the other?” (The measuring strategy is easier and could be faster, but finding the area of the square gives us an exact answer.)
• “How can you use what you know about the largest square and the smallest square to determine the side length of the remaining square?” (The value of is somewhere between 1 and 2 because an area of 2 square units is between 1 and 4 square units.)

This is the second of three activities in which students investigate the value of . The goal is to achieve a more precise value of by checking for possible solutions to , and to revisit the definition of a rational number in preparation for defining an irrational number.

Since is a number that can be multiplied by itself to get 2, it is true that .

The purpose of this discussion is to revisit the definition of a rational number as a number that can be expressed as a positive or negative fraction. Students also add to their understanding that some square roots are rational numbers.

Select students to share their reasoning for why each number is not a solution to . If not brought up in students’ explanations, make sure that they notice that is not a terrible approximation for , since  , and is only a bit larger than 2. An even better approximation for is , since  , which is just a little bit smaller than (which equals 2). Refer to  the display of perfect squares as necessary. 

Remind students that a fraction is a number on the number line that is the result of dividing the unit interval into equal parts and finding the point that is  of them from 0. We can always write a fraction in the form , where and are whole numbers (and is not 0), but there are other ways to write them. For example, 0.4 is a fraction because it is the point on the number line that is the result of dividing the unit interval into 10 equal parts and finding the point that is 4 away from 0. (The same point is a result of dividing the unit interval into 5 equal parts and finding the point that is 2 away from 0.) When we first learned about fractions, we didn’t know about negative numbers. Rational numbers are fractions, but they can be positive or negative. So, is also a rational number.

Here are some examples of rational numbers:

Invite students to share how each of these examples is a number that can be expressed as a positive or negative fraction, particularly the two values represented using square roots.

This is the third of three activities in which students investigate the value of .  Students look for a fraction that can be multiplied by itself where the product is exactly 2. Since there is no such number, this allows for the term "irrational number" to be defined.

Simply looking for rational solutions to and not finding any is not a proof that is irrational. For the purposes of this course, students are asked to take it as a fact that is irrational. They may have opportunities to understand or write a proof that  is irrational in future studies.

The purpose of this discussion is to define an irrational number. Invite previously selected students who found particularly close values to share their number with the class and what strategy they used to find it. Applaud students for their perseverance, but then confess that no such rational number exists because is an irrational number.

Display a number line such as the one shown here for all to see. Tell students that an irrational number is a number that is not rational. That is, it is a number that cannot be expressed as a positive or negative fraction. is one example of an irrational number. It has a location on the number line, and its location can be made precise (it’s a tiny bit to the right of ), but  cannot be found on a number line by subdividing the interval from 0 to 1 into parts and taking of them. We have to define in a different way, such as the side length of a square with area 2 square units.

A number line with 10 evenly spaced tick marks. The first tick mark is labeled 0 and the sixth tick mark is labeled 1. An arrow points to the eighth tick mark and is labeled seven-fifths. A second arrow points to a point slightly to the right of the eighth tick mark and is labeled the square root of 2.