Some students might persist in attempting to find examples of two integers that add up to a noninteger. Rather than giving them more time to find examples, encourage them to think about the placement of integers on a number line and what it might imply about the sum or product of any two integers.

In this activity, students use logical reasoning to develop a general argument about why the sum and product of two rational numbers are also rational numbers.

Students first study some numerical examples of adding two rational numbers and articulate why the sums are rational numbers. Next, they reason more abstractly about rational numbers in the form of and (MP2). Students consolidate what they know about the sum and product of integers (that these are also integers) and about rational numbers (that these are fractions with integers in the numerator and in the denominator) to argue that the result of adding or multiplying any two rational numbers must be another rational number (MP3).

Some students may not recognize 0.175, 4.175, and -0.75 as rational numbers. Demonstrate that these numbers can be written as the fractions , , and  (or ).

The second question guides students through the pieces needed to make an argument that the sum of two rational numbers must be rational. Use the discussion to help students consolidate these pieces into a logical and coherent argument:

• If and are fractions, then its sum is . The expressions , and must be integers because they are each a product of integers, and consequently must also be an integer. Because the numerator and the denominator are both integers, the number is a fraction and is therefore rational.

Make sure students see how to construct a similar argument for the product of two rational numbers, as shown in the student response.

As in the previous activity, students are guided through the pieces needed to make a particular argument—that the sum of a rational number and an irrational number must be irrational. Make sure students can consolidate these pieces into a logical and coherent argument:

• Let be the sum of (rational) and (irrational). 
• Suppose is rational. Adding and must produce a rational number because the sum of two rational values is rational. 
• Adding and also gives , or , which is irrational. 
• The sum of and cannot be both rational and irrational, so the assumption that is rational must be false. So (the sum of a rational number and an irrational number) must be irrational.

Make sure students see how to construct a similar argument for the product of a rational number and an irrational number, as shown in the Student Response.

In this optional activity, students investigate how the parameters in a quadratic equation affect the number of solutions and the kinds of solutions. Students are first given the equation and are asked to find a value of that would produce a certain number or kind of solution. To do so, students may take different approaches, each with different degrees of efficiency.

Monitor for these strategies, from less efficient to more efficient:

• Substitute different integers for , and calculate the solutions by rewriting the equation in factored form (if possible), completing the square, or using the quadratic formula.
• Enter the equation in a graphing tool, graph the equation at different integer values of , and identify the -intercepts (and notice when there is only one or when there are none).
• Use a spreadsheet tool to calculate the solutions at various values of (using the quadratic formula), and choose the values of accordingly.

Students then study their findings and make general observations about which values of produce 0, 1, or 2 solutions. Making generalization about the connections between and the number of solutions encourages students to look for and make use of structure (MP7).

The last question is an open-ended task and may be challenging. It prompts students to write original equations that produce specified solutions. To do so effectively and efficiently, students need to make use of structure and any productive strategies seen in the first question and in past work (MP7), and students need to persevere (MP1). If time is limited and if desired, consider returning to this question at another time.

Depending on the strategy used, equations with rational solutions and those with single solutions might be harder to write than those with irrational solutions or no solutions. (The former requires making use of structure, while the latter could be achieved by choosing , and —in —somewhat randomly and checking by graphing that the zeros appear irrational or are nonexistent.) If desired, differentiate the work by assigning certain questions to certain students or groups.

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

Invite previously selected students to share their strategies for finding the right values. Sequence the discussion of the strategies in the order listed in the Activity Narrative. If possible, record and display their work for all to see.

Connect the different responses to the learning goals by asking questions such as:

• “Why do the values of that produce different numbers of solutions come in intervals?” (Looking at the graphs for different values of , it moves the parabola in an arch shape. So, it hits the horizontal axis twice for a while, then at just the right value of , its vertex is on the axis. Then it is not touching the -axis for a while. Finally, at large values of , it touches the axis again at one point and then at two..)
• “Which form: standard, factored, or vertex is easiest to use when writing an equation with a certain type of solution?” (Factored form is easiest because the solutions to an equation written as are and .)

If time permits, consider demonstrating how a graph or spreadsheet technology could be used to help spot patterns and suggest which values of , and in a quadratic equation lead to certain solutions.