The purpose of this Math Talk is to elicit strategies and understandings students have for finding a value that will multiply to a certain number. It encourages students to think about multiplication and to rely on the structure of equations to mentally solve problems. The understanding elicited here will be helpful in the associated Algebra 1 lesson when students will need to be able to find missing values to help factor quadratic expressions.

To calculate the missing factor that will result in the needed product, students need to look for and make use of structure (MP7).

The goal of this discussion is to review students’ strategies for solving for a missing factor.

Ask students to share their strategies for each problem. Record and display their responses for all to see. To involve more students in the conversation, consider asking:

• “Who can restate ’s reasoning in a different way?”
• “Did anyone have the same strategy but would explain it differently?”
• “Did anyone solve the problem in a different way?”
• “Does anyone want to add on to ’s strategy?”
• “Do you agree or disagree? Why?”
• “What connections to previous problems do you see?”

In this activity, students find pairs of integers that have a given product and given sum. In the associated Algebra 1 lesson, students factor quadratic expressions in standard form that have a negative constant term. Finding pairs of values with a given negative product and given sum will help students find values to try while factoring. This practice with combining numbers is an opportunity for students to reason abstractly and quantitatively (MP2).

In this activity, pairs of students expand binomial factors into the standard form of a quadratic expression. Each student then compares their original question and solution with their partner to notice what is the same and different. In particular, students should notice how rearranging factors and changing the operation within factors can change the standard form. In the associated Algebra 1 lesson, students factor quadratic expressions involving a negative constant term. The fluency and pattern recognition from this activity should help students recognize similar patterns when factoring.

Students have the opportunity to construct viable arguments and critique the reasoning of others (MP3) when they work with a partner to discuss if their similar problems result in the same or different solutions. Students look for and make use of structure (MP7) when they notice how slight differences in factors result in differences in the equivalent standard form.