This activity prompts students to recognize that the sum of any two integers is always an integer and that the product of any two integers is also always an integer. Students will not be justifying in a formal way why these properties are true. In a future course, when studying polynomials, students will begin considering integers as a closed system under addition, subtraction, and multiplication. 

Some students may simply assert that these are true because they are not able to find two integers that add up to or multiply to make a noninteger. Others may reason that:

• It seems impossible to add two whole amounts and end up with a partial amount in the sum.
• It seems impossible to multiply whole-number groups of whole numbers to make a non-whole number quantity.
• If we use the number line to represent addition of two integers, we will start at an integer and take whole-number steps, so we cannot end up at a point between integer values.

These are valid conclusions at this stage. Later in the lesson, students will use these conclusions to reason about the sums and products of rational numbers.

Select students or groups to share their examples and their challenges in finding examples for the second part of each question. Invite as many possible explanations as time permits for why they could not find examples of two integers that add or multiply to make a number that is not an integer. If no students bring up reasons similar to those listed in the Activity Narrative, ask students to consider them.

Explain that while we have not proven that two integers can never produce a sum or a product that is not an integer, for now we will accept this to be true.

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).

In the previous activity, students justified why the sum and product of two rational numbers are also rational. In this activity, they develop an argument to explain why the sum and product of a rational number and an irrational number are irrational.

Students encounter an argument by contradiction. They learn that we can first assume that the sum is rational. If this is indeed true, when this sum is added to another rational number, the result must also be rational. It turns out that this is not the case (adding this sum to a certain rational number produces the original irrational number), which means that the sum cannot be rational.

In the last part of the activity, students make a similar argument for why the product of a rational number and an irrational number is irrational. They must apply a similar argument to a new situation and use repeated reasoning to generalize the result (MP8).

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).