Accentuate the Negative mathematical and problem solving goals:
Accentuate the Negative was created to help students:
- Develop strategies for adding, subtracting, multiplying, and dividing integers
- Determine whether one integer is greater than, less than, or equal to another integer
- Represent integers on a number line
- Model situations with integers
- Use integers to solve problems
- Explore the use of integers in real-world applications
- Compare integers using the symbols =, >, and <
- Understand that an integer and its inverse are called opposites
- Graph in four quadrants
- Set up a coordinate grid on a graphing calculator by naming the scale and maximum and minimum values of x and y
- Graph linear equations using a graphing calculator
- Informally observe the effects of opposite coefficients and adding a constant toy = ax
- Answer questions using equations, tables, and graphs
The overall goal of the Connected Mathematics curriculum is to help students develop sound thematical habits. Through their work in this and other algebra units, students learn important questions to ask themselves about any situation that can be represented and modeled mathematically, such as: What situations in daily life can be represented by positive or negative numbers? How can a meaning be found for operations on negative numbers? Where can such operations be modeled? Is it possible to use “less than” or “greater than” concepts with integers? How are the integers d~fferent from ordinary whole numbers? How are these two sets of numbers alike? Can the coordinate grid be expanded to include negative numbers? Is it possible to make graphs on such grids using a graphing calculator? What patterns will occur in these graphs? How can these patterns be used to find and understand other patterns?
The Mathematics in Accentuate the Negative
Most of your students can add, subtract, multiply, and divide whole numbers and decimals. However, most have not been asked to think about what the operations mean and what kinds of situations call for which operation. Without the development of the disposition to seek ways of making sense of mathematical ideas and skills, students may end up with technical skills but without ways of deciding when and how those skills can be used to solve problems.
One good way to work toward creating the desire to make sense of these ideas is to model such thinking in rich classroom conversation. Asking questions about meaning, about what makes sense, as a regular, expected part of classroom discourse helps focus students on making connections. Exploring new aspects of number in a way that builds on and connects to what they already know is likely to have two good effects. First, students will deepen their understanding of familiar numbers and operations on them. Second, the new numbers—in this case, integers- will be more deeply integrated into students’ own mathematical knowledge and resources.
Students find several things difficult about integers and operations on integers.
- The fact that —27 is less than —12 is contrary to students’ experience with (positive) whole numbers. This understanding requires building mental images and models that allow students to visualize these new comparisons and relationships.
- The operation of subtraction, and especially subtracting a negative, is difficult for students to make sense o[ In this unit, students will have several opportunities to think about what makes sense and why, and they will encounter several representations and models that will help them think more deeply about subtraction.
- The idea that subtracting a negative number gives the same result as adding the opposite of the negative number (adding a positive) is difficult for many students. This understanding must develop over time as students make observations and comparisons between subtraction and addition. Recognizing that these are inverse operations and that addition sentences are related to subtraction sentences helps students to expand their understanding of this concept.
- Multiplying two negatives and getting a positive seems like magic to most students. In fact, the usual ways of giving meaning to multiplication—such as accumulating an amount over and over (repeated addition)—seem of no help in making sense of —12 x 5.
In the unit, we approach these difficult concepts through the use of two basic models and the observation of patterns. The accumulation of evidence from the models is more powerful than that from a single model. In addition, a particular model may be more useful for making sense of parts of the picture, but not the whole picture.
Modeling Addition of Integers
In this unit, students will use the number line to find a strategy for adding integers. Adding positive integers is interpreted as “going to the right”; adding negative numbers is interpreted as “going to the left.” To add ~5 and 7, for example, the strategy is to start at the point marked 0, go right 5 units to the point labeled 5, and then go left 7 units, which puts you at the point labeled —2. Thus, 5 + 7 = —2.
Colored chips can also be used to develop a strategy for adding integers. Using this model requires an understanding of opposites. Two colors of chips are used, one color (such as black) to represent positive numbers and another (such as red) to represent negative numbers. To add ~8 + 5, the strategy is to first put 8 black chips on the chip board. Since the problem involves addition, which means to combine, 5 red chips are added to the board to represent the 5.
Because each chip represents 1 unit, either positive or negative, a black and red chip are thought of as opposites. Two opposite chips make 0 (~1 + 1 0). In this problem, 5 chips of each color can be paired to make zeros. After the paired chips are removed, 3 black chips remain—which represent +3, the sum.
Modeling Subtraction of Integers
Since addition and subtraction are inverse, or opposite, operations, subtraction can be represented as doing the inverse of addition. To calculate +7 — +5, the strategy is to start at 0 and go to the right 7 units to +7 (the same as in the addition strategy). Then, because the operation is subtraction, the direction of the +5 move is reversed (this may also be thought of as going in the opposite direction), going to the left 5 units to +2. Thus, +7 - +5 = +2.
Examining the number line model for subtraction often leads students to observe that the subtraction expression is equivalent to an addition expression (+7 + -5) and to recognize that the strategy of subtracting an integer is equivalent to adding the opposite of the integer (+7 - +5 = +7 + -5). Students will need several experiences with subtraction before they notice these patterns.
On a chip board, subtraction involves removing the number of chips (of the correct color) by the amount indicated after the subtraction sign. To calculate ~7 — ~5, the strategy is to first add 7 black chips to the board (the same as in the addition strategy). Since the problem involves subtraction, 5 black chips are removed, leaving 2 black chips (which represent +2). Removing chips is the opposite of the procedure for addition, in which the amount of the number after the operation sign is added to the board. Thus, +7 - +5 = +2.
A more complex problem is +7 - -5. To do this problem on a chip board, 7 black chips are added to the board. Then, because the operation is subtraction, 5 red chips need to be removed (-5). Since there are no red chips on the board, zeros—in the form of black-red pairs of chips—must be added to the board. In this problem, 5 red chips and 5 black chips must be added. Once there are 5 red chips on the board, they can be removed, leaving 12 black chips. Thus, +7 - —5 = +12.
Modeling Multiplication of Integers
Movement right and left (or up and down) on a number line can be used to model multiplication of integers, but it can be difficult to follow Consider this example: The temperature at midnight is 0 degrees C. If the temperature drops an average of 3 degrees per hour, what is the temperature at 2:00 A.M.?
This problem is asking what the temperature will be 2 hours after midnight if the temperature drops 3 degrees each hour, or 2 x -3. This can be represented on a thermometer, or on a number line as shown below, to demonstrate that -3 + -3 = 2 x -3 = -6.
Making sense of a negative integer times a negative integer is difficult to do with models, but looking at patterns and understanding positive and negative integers can help students to make sense of this concept. For example, consider the following list of number sentences, which asks what the pattern suggests for the product of —5 and —1, and so on.
Division of Integers
Division of integers is developed by relating division to multiplication. Multiplication sentences yield related division sentences and can tell students about the sign of a division problem involv- ing these new kinds of numbers. For example, since we know that
we can write related division sentences:
From these kinds of experiences, students can generalize rules for handling the sign of the quotient in a division problem.