This activity introduces the use of notation as a simple way to express the two square roots of a number. Students solve several simple equations by finding square roots and express their solutions using this notation.

Students also see that sometimes the solutions are not rational numbers and can be expressed as exact values using the radical notation rather than using their decimal approximations. For example, the solutions to can be written as rather than , and the solutions to can be written as instead of .

When using the notation for the first time, some students may struggle to remember that it shows a number and its opposite. Encourage those students to continue writing their answers both ways in this lesson to reinforce the meaning of the new notation.

Some students may still struggle to understand the meaning of the square root. Watch for students who evaluate a square root by dividing by 2. Ask these students what the square root of 9 is. If this does not convince students that dividing by 2 is the incorrect operation, demonstrate that we call 3 a square root of 9 because .

Invite students to share their solutions. Record and display the solutions for all to see. Discuss any disagreements, if there are any.

Draw students’ attention to the last three sets of solutions, which are irrational. Ask students to recall the meaning of rational numbers and irrational numbers. Remind students that a rational number is number that can be written as a positive or negative fraction—for example, 12, -7, , , 90.38, or 0.005. Irrational numbers are numbers that are not rational—for example, ,  and . (Students will have more opportunities to review and classify rational and irrational numbers later in the unit, so for now, it suffices that they remember that an irrational number is a number that is not rational.)

Highlight that when the solution is irrational, the most concise way to write an exact solution is, for example, . Writing this in decimal form allows us to write only an approximate solution. For example, in this case, 3.24 and -5.24 are approximate solutions. 

Then, display the variations in writing the solutions to for all to see. Discuss how they are all different ways of writing the same two solutions. Tell students that all of the solutions are correct ways of writing the answer, but some may be clearer than others, as discussed in the Launch.

• and (Be careful when writing the  symbol, or it might look like , which is a different value.)
• and

This activity allows students to integrate several skills and ideas they have learned so far: solving quadratic equations by graphing and by completing the square, using notation, and expressing solutions both approximately and exactly (by using the square root symbol). In solving the equations algebraically and using the notations to express and verify solutions, students practice attending to precision (MP6).

The last equation may be challenging for some students, as it involves fractions and messier-looking solutions. It is not essential that all students get to this equation. They will have a chance to encounter more complicated equations and solutions in upcoming lessons.

Discuss any common struggles or mistakes made when solving the equations. Then, invite students to reflect on the merits and challenges of solving by each method. Ask questions such as:

• “What are some benefits of solving by graphing? What are some drawbacks?” (Benefits: It is quick and straightforward. It shows the solutions or that there are no solutions, even when the equations involve fractions or very large numbers. Drawbacks: It does not give exact solutions. Some tools may require adjusting the graphing window quite a bit to see the solutions.)
• “What are some benefits and drawbacks of solving by completing the square?” (Benefit: It can be used to find exact solutions to any equation. Drawbacks: It can be pretty time consuming. When the equations have fractions or very large or very small numbers, the calculations get complicated and may be prone to error.)