In this activity, students revisit the representation of a multiplication chart that has been extended to include negative numbers. Students complete the chart and use repeated reasoning to identify patterns about the sign of the products of positive and negative numbers (MP8).

If necessary, remind students how a multiplication chart works—the factors are listed at the end of the rows and columns, and their products go in the boxes.

In the digital version of the activity, students use an applet to complete the multiplication chart. The applet reveals additional rows and columns when prompted and shows incorrect answers in a different color. This activity works best when students have access to the applet because students can focus on fewer numbers at a time and receive immediate feedback on whether their answers are correct. 

The blackline master has a multiplication chart that also includes the factors 1.5, -1.5, 3.2, and -3.2 so that students can see how the patterns extend to rational numbers that are not integers. Encourage students to complete the rows and columns for the integers first and then come back to 1.5, -1.5, 3.2, and -3.2 later. Directions are included on the blackline master for a way that students can fold their papers to hide the nonintegers while they fill in the integers. If students use this technique, it would be beneficial to demonstrate and walk students through the process of folding their paper.

The goal of this discussion is to emphasize that the product of two negative numbers is a positive number. This is true whether or not the numbers are integers. 

Display a complete chart for all to see, and begin by asking students to share any patterns they see. Record these observations for all to see. Then ask students to explain the ways in which the chart shows that the product of two negative numbers is a positive number. The general argument involves assuming that a pattern observed in a row or column will continue on the other side of 0.

Students sort different expressions during this activity. A sorting task gives students opportunities to analyze representations and structures closely and make connections (MP7). Students recall links between positive fractions and multiplication in preparation to think about division as multiplication by the reciprocal.

Monitor for students who determine a way to tell if two expressions are not equivalent without computing the product. For example, they may decide whether each product would be positive or negative before doing any arithmetic.

The purpose of this discussion is for students to share the strategies they used to find equivalent expressions. Once all groups have completed the card sort, discuss the following: 

• “Which matches were tricky? Explain why.”
• “Did you need to make adjustments in your matches? What might have caused an error? What adjustments were made?”

Ask the previously identified students to share their rationale for identifying those that do not match.

If time allows, consider highlighting the link between multiplying by a fraction and dividing by a whole number. For example, ask students to rewrite a multiplication expression, such as , using division (). It is not necessary for students to know rules for dividing signed numbers at this point, as this will be the focus of future lessons.

This activity gives students an opportunity to practice multiplying signed numbers. The solutions to the problems in each row are the same, so students can check their work with a partner.

The purpose of this discussion is for students to share strategies they used to multiply signed numbers and to share any patterns they noticed. 

Consider asking some of the following questions:

• “Were there any rows for which you and your partner did not get the same answer?”
• “Did you and your partner use the same strategy for each row?”
• “What was the same and different about both of your strategies?”
• “Did you learn a new strategy from your partner?”
• “Did you try a new strategy while working on these questions?”