Lesson 3

The purpose of this activity is to elicit strategies and understandings students have for square and cube roots. These understandings help students develop fluency and will be helpful later in this lesson when students will connect square and cube roots to exponents of \(\frac12\) and \(\frac13\). Students might note the negative solutions to the quadratic equations, but it is not necessary to emphasize those solutions at this time if they are not mentioned. Students will explore the symmetry of positive and negative solutions to quadratic equations more deeply in upcoming lessons.

While participating in this activity, students need to be precise in their word choice and use of language (MP6), because they have no algebraic method that they can use to solve and no calculators that could give approximate solutions numerically. They have to be able to recall the definitions of square and cube roots.

Ask students to put away any devices. Tell students that their task is to find the most precise answers they can without a calculator.

Briefly discuss students’ responses. Here are some questions for discussion:

• How did you find the answers that are not integers? (I estimated a value for \(w\) and then cubed it to see how close it was to 19. I used a square root sign to write \(z\) as \(\sqrt7\).)
• How could we tell if our answers to these questions make sense? (We could square or cube them; for example, we know \(y\) is 2 because \(2^3=8\).)
• Could we write \(x\) or \(y\) using radicals? (Yes, because \(5=\sqrt{25}\) and \(2=\sqrt[3]8\).)

Key discussion points are that we can write exact answers using radical signs and that we can check the accuracy of our answers by squaring or cubing, as appropriate.

The purpose of this activity is for students to connect numbers that are a base to the \(\frac12\) power to square roots. First, they use a graph to estimate the value of such numbers. Then, they extend exponent rules to find that the numbers must be square roots of the base. The first number students explore is \(9^{\frac12}\). They will find that this number squares to make 9, which immediately indicates that \(9^{\frac12}\) is either 3 or -3. It is likely that students will not bring up the possibility that \(9^{\frac12}=\text-3\) because the graph suggests \(9^{\frac12}\) should be positive. If students do not bring up the possibility that \(9^{\frac12}=\text-3\), it is not important to mention at this time. In an upcoming lesson, students will study pairs of positive and negative square roots more deeply. They will also be introduced to the convention that the square root of a number is the positive square root.

In the second part of the activity, students study \(3^{\frac12}\) and find that it squares to make 3. This is a less familiar square root, so students will not have an immediate number in mind. Rather, the wording of the task encourages students to think about what \(\left(3^{\frac12}\right)^2=3\) means in terms of roots.

Making graphing technology available gives students an opportunity to choose appropriate tools strategically (MP5).

Tell students that they have worked with integer exponents, but now they will think about what it means to have an exponent of \(\frac12\).

Ask students to share their estimates for \(9^{\frac12}\) and \(3^{\frac12}\), and display their graphs for all to see. Ask students to share their responses to questions 2 and 3. The goal is for students to explain how applying exponent rules means that \(9^{\frac12}=3\). Then, ask students to share their responses to the last question. The important takeaway is that following the exponent rules means that \(b^\frac12\) is another way of writing \(\sqrt{b}\) because \(\left(b^{\frac12}\right)^2=b^{\frac12 \boldcdot 2} = b\).

The purpose of this activity is for students to extend what they learned about square roots and \(\frac12\) exponents to cube roots and \(\frac13\) exponents.

Tell students that they will use the same reasoning as in the previous activity to make sense of numbers to the \(\frac12\) power or \(\frac13\) power. Give a few minutes of quiet work time followed by a brief, whole-class discussion.

The key takeaway is that extending exponent rules to these fractions means that \(b^{\frac12}=\sqrt{b}\) and \(b^{\frac13}=\sqrt[3]{b}\). Select students to share their responses for \(15^{\frac12}\) and \(2^{\frac13}\).

Conclude the discussion by reminding students that these expressions which involve the \(\sqrt{}\) symbol are often called radical expressions because the technical name for the symbol is the radix (which is also where the word root comes from). Expressions like \(\sqrt{15}\) and \(\sqrt[3]{2}\) are examples of radical expressions.

In this activity, students match equivalent expressions involving exponents and radicals. This is the first time they see negative rational exponents, so expect a lot of discussion around those expressions. Students will have more chances to interpret negative exponents, so it is not crucial for them to show mastery at this time.

Monitor for groups who reason about the expressions with negative fractional exponents in different ways. For example, some students may see \(7^{\text- \frac12}\) as \(\left(7^{\text-1}\right)^{\frac12}\), while others will think of the expression as \(\left(7^{\frac12}\right)^{\text-1}\). Ask these groups to be ready to share their reasoning during the discussion.

Arrange students in groups of 2. Give students a few minutes of quiet think time before asking them to share their responses with their partners. Tell students that if there is disagreement, they should work to reach agreement. Follow with a whole-class discussion.

If students aren’t sure how to match the negative rational exponents, ask them if they could make an educated guess based on the other patterns they notice.

Begin the discussion by selecting 2–3 groups to share how they matched the expressions with positive exponents. Next, invite previously identified groups to share how they matched the expressions with negative exponents. While students share their thinking, display any exponent rules referenced for all to see, such as \(\left(b^m\right)^n = b^{m \boldcdot n}\) or \(b^{\text-n} = \frac{1}{b^n}\). It is important for students to understand that fractional exponents follow all the same exponent rules as integer exponents.