Intent The “hot and cold cubes” sequence of activities offer a model for the operations of integer arithmetic, key tools in high school mathematics. This activity reviews some basics about negative numbers and reaffirms some conventions before students begin to make sense of the “hot and cold cubes” model for integer arithmetic.

Mathematics The IMP program assumes that most students have had prior exposure to negative numbers and have been taught—but may not remember or understand—basic rules for arithmetic with integers. In this set of activities, students are introduced to a model that embodies these rules and serves as a metaphor for thinking about integer arithmetic. Rather than simply reviewing the rules for such arithmetic, this model provides a frame of reference for the rules and will allow students, if necessary, to reconstruct the rules for themselves in the future. The basic operations for natural numbers are defined, at least intuitively, in terms of putting sets of objects together and taking objects away from a set, but this definition doesn’t make sense for negative numbers. In moving from whole numbers to integers, numbers are no longer simply a magnitude, but also a direction. Treating an integer as merely an opposite of a whole number, as do such commonly memorized rules as “subtracting a negative is the same as adding a positive,” does not encourage a more powerful understanding of integer. The hot and cold cubes model emphasizes both the magnitude and the direction of an integer and encourages awareness of the meaning of the operation involved.

Progression The activity begins with discussion of the need to justify solutions when doing integer arithmetic and is followed by a brief review of notation, language, and conventions. Students are then introduced to the model and, in their groups, perform some integer arithmetic to help make sense of the model. They are asked to translate the chefs’ moves (using hot and cold cubes to change the temperature in the cauldron) into integer arithmetic and to translate integer arithmetic into chefs’ moves.

Approximate Time 35 minutes

Classroom Organization Groups

Materials Manipulatives, such as cubes or tiles, in two colors

Doing the Activity Many students have probably been exposed to the basics of computing with negative numbers and may be reluctant to learn another way of thinking about the process. The hot and cold cubes model, however, will help them to understand why the rules work. It offers a frame of reference for the rules and will allow students to reconstruct the rules for themselves in the future. Two quick questions can provide information on students’ prior knowledge. What is the answer to (–3)(–5)? How do you know your answer is right? What is the answer to –3 + –5? How do you know your answer is right? You will likely receive a variety of solutions; record them all on the board. Remind students that they should be able to state why their answers are correct. Some students may be able to apply the rules to find the correct answers, but many will have trouble explaining why the product of two negative numbers is positive while the sum of two negative numbers is negative. The intent of this introductory challenge is to convince students that they still have something to learn about working with negative numbers and that it might be worthwhile to have an approach that doesn’t rely on memorizing rules. Review the notation and terminology of positive and negative numbers. These activities use the “raised sign” notation, such as +5 size 12{ {} rSup { size 8{+{}} } 5} {} and +7 size 12{ {} rSup { size 8{+{}} } 7} {}. These should be read as “positive five” and “negative seven” not “plus five” or “minus seven.” Using clearly defined terminology helps to distinguish between positive and negative numbers and the operations of addition and subtraction. Once these activities are completed, the student book reverts to the standard notation conventions: positive numbers typically include no sign, and negative numbers are denoted with the same symbol as subtraction. Tell students that this convention is typical in most contexts. Also review the following terminology and notation with students. The sign of a number indicates whether it is positive or negative. Zero is considered neither positive nor negative. The numbers in a pair such as +3 size 12{ {} rSup { size 8{+{}} } 3} {} and −3 size 12{ {} rSup { size 8{-{}} } 3} {} are sometimes called opposites. That is, −3 size 12{ {} rSup { size 8{-{}} } 3} {} is the opposite of +3 size 12{ {} rSup { size 8{+{}} } 3} {}, and +3 size 12{ {} rSup { size 8{+{}} } 3} {} is the opposite of −3 size 12{ {} rSup { size 8{-{}} } 3} {}. The word integer refers to a number that is zero, a natural number, or the opposite of a natural number. (The natural numbers are positive whole numbers: 1, 2, 3, 4 and so on.) Thus the set of integers is {. . ., −3 size 12{ {} rSup { size 8{-{}} } 3} {}, −2 size 12{ {} rSup { size 8{-{}} } 2} {}, −1 size 12{ {} rSup { size 8{-{}} } 1} {}, 0 size 12{0} {}, +1 size 12{ {} rSup { size 8{+{}} } 1} {}, +2 size 12{ {} rSup { size 8{+{}} } 2} {}, +3 size 12{ {} rSup { size 8{+{}} } 3} {}, . . .} The number line is a way to picture both positive and negative numbers. By convention, positive numbers are on the right and negative numbers are on the left; numbers are considered to get larger as one moves to the right on the number line. Thus, for example, +5 size 12{ {} rSup { size 8{+{}} } 5} {} > −8 size 12{ {} rSup { size 8{-{}} } 8} {} and −7 size 12{ {} rSup { size 8{-{}} } 7} {} < −3 size 12{ {} rSup { size 8{-{}} } 3} {}. Have students read the introduction to The Chefs’ Hot and Cold Cubes and the first five paragraphs of “The Story.” Introduce students to the manipulatives—such as two colors of cubes or tiles—for representing hot and cold cubes. Ask groups to use their manipulatives to create several cauldrons, each representing a temperature of 0°, to introduce the idea that a hot cube and a cold cube “cancel out” one another. Have groups read the next paragraph (beginning “For each hot cube . . .”) and then create cauldrons for other specific temperatures, such as +5 size 12{ {} rSup { size 8{+{}} } 5} {}° or −3 size 12{ {} rSup { size 8{-{}} } 3} {}°. The idea is for students to get a sense of the cancellation mechanism and to see that a given temperature can be represented in many ways. After this introduction, let students read the rest of “The Story” individually and then work in their groups on the questions. As groups work, you may want to emphasize that the equations and arithmetic expressions focus on the change in temperature and not on the temperature itself. Some students may be confused by the fact that the same notation is used in different ways. For instance, +5 size 12{ {} rSup { size 8{+{}} } 5} {} can mean “add five bunches of a certain number of hot or cold cubes” (as in +5⋅+20=+100 size 12{"" lSup { size 8{+{}} } 5 cdot "" lSup { size 8{+{}} } "20"="" lSup { size 8{+{}} } "100"} {}) or “a bunch containing five hot cubes.” If this comes up, acknowledge that this part of the model is something they may have to pay extra attention to. It is similar to the dual meaning in multiplication of whole numbers, in which 5 . 3 can mean “5 groups with 3 objects in each group” or “3 groups with 5 objects in each group,” with 5 representing either the number of groups or the size of each group.

Discussing and Debriefing the Activity This activity will likely not require a formal debriefing.

Key Questions What is the answer to (–3)(–5)? How do you know your answer is right? What is the answer to –3 + –5? How do you know your answer is right? How might you represent the situation with objects?

Supplemental Activity Positive and Negative Ideas (extension) extends the work with hot and cold cubes and asks students to consider other ways they might model integer arithmetic.