In this activity, students use addition, subtraction, and multiplication with rational numbers to solve problems related to temperature and money, but they do not initially have enough information to do so. To bridge the gap, they need to exchange questions and ideas.

The Information Gap structure requires students to make sense of problems by determining what information is necessary, and then to ask for information they need to solve it. This may take several rounds of discussion if their first requests do not yield the information they need (MP1). It also allows them to refine the language they use and ask increasingly more precise questions until they get the information they need (MP6).

• “What strategies did you use to help you understand or answer the problem?” (I drew a number line diagram. I rewrote a subtraction problem as an addition problem. I wrote an equation to represent the situation.)
• “Which quantities could be represented by a negative number in each situation?” (temperature getting colder and the cost for the museum tickets)
• "How did you use multiplication in each of these situations?" (The rate the temperature increased or decreased was multiplied by the length of time the temperature was changing. The cost of each ticket was multiplied by the number of students, and the cost of each painting was multiplied by the number of paintings sold.)

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?”