In this activity, students use algebraic reasoning to solve quadratic equations and to identify the solutions as rational or irrational. But first, they inspect the zeros of a corresponding function for hints about whether the solutions might be rational. They notice that it is impossible to know whether a solution is irrational simply by looking at the decimal approximation of a point shown on a graph.

While solutions obtained by algebraic solving can better show the types of number they are, some solutions are difficult to classify because they are combinations of rational and irrational numbers, such as and . Students will investigate these kinds of solutions in the next activity.

When using algebra to solve equations such as , some students may expand rather than use square roots (perhaps to use the quadratic formula to solve). For students who use this approach, check to see if they expanded  correctly. These students may benefit from continued use of a rectangle diagram to expand. Emphasize the need to check their answers against the graphical solution. When their solutions don’t agree, encourage them to find the error in their work.

Invite students to share their responses and explanations. Ask students to compare the zeros they found by graphing and the solutions they found algebraically. Discuss with students:

• “How do the zeros of the functions and , which you found by graphing, compare to the solutions to and ?” (The zeros found by graphing are decimal approximations. They show decimal approximations up to three decimal places, or more, depending on the tool or setting used. The solutions found algebraically can be written as expressions that use the square root symbol.)
• “Is it easier to tell whether the solutions to an equation are irrational by graphing and finding the zeros or by solving algebraically?” (By solving algebraically. Observing the coordinates on a graph doesn’t always help, because it is impossible to tell whether a number is irrational by looking at only a few digits of its decimal approximation.)

Draw students’ attention to the numerical expressions they encountered while solving the equations:

• : This expression is rational because it equals 6.
• : This is irrational because 8 is not a perfect square.
• : is irrational, and 7 is rational. Is the sum rational or irrational?
• : 4 is rational, and is irrational. Is the product rational or irrational?

Tell students that they will now experiment with the sums and products of rational and irrational numbers and investigate what kinds of numbers the sums and products are.

Display the six statements for all to see. Invite students to share their responses. After each student shares, ask others whether they agree or disagree. Ask students who disagree for an explanation or a counterexample. Develop a consensus on what the class thinks is true when we combine numbers by addition and multiplication. Record and display the consensus for all to see.

Explain to students that in an upcoming lesson, they will have a chance to test their conjectures.