6.EE.2b, 6.EE.5, 6.EE.6, 6.EE.7, 7.EE.4a, 8.EE.7, A-CED.1
There are foundational concepts that students must understand in order to be able to solve equations and simplify expressions. When numbers in the expression or equation are negative, additional consideration must be given. Equations with negatives pose some common problems for students. Solving equations often involves subtracting the same quantity from both sides to simplify the equation. However, this can lead to subtracting a negative from a negative, which confuses students. Likewise, multiplication by a negative can lead to the same kinds of issues.
In addition, there are other difficulties that arise with negatives in equations that are separate from those arithmetic difficulties. Equations are full of other symbols, and it is easy for a student's eye to skip over minus signs while trying to focus on "more important" things like variables. Furthermore, it is not always clear to a student what is an appropriate way to eliminate an unwanted term or factor from an equation when that term or factor includes a minus sign.
|i) 12 - x = 7
ii) 4 – x = 5
iii) -4 – x = 10
iv) -x = 7
|Students can do (i), but the others are beyond reasoning unless they are grounded in integer computation.|
2 – 3x + 6 = 2x + 18
2 – 1x + 6 = 18
|The student has detached the minus sign preceding the 3x. That type of difficulty –identified by Herscovics and Linchevski (1991)—produced the error.|
3x -4 = x+9
3x = x + 5
In order to cancel a negative term, some students use subtraction.
1) Adding a number to its additive inverse:
-3 + 3= 6
2) Adding a positive number to a negative number
-8+ 3= 5 or 11 or -11
3) Subtracting a positive number from a negative
-3 - 8 = -5, 5, or 11
4) Subtracting a positive number from a smaller
3 - 8= 5
5) Subtracting a positive number from 0
0 - 8 = 8 or 0
1) The student has ignored the negative sign.
2) The student thinks addition should give a positive
number or a number looking bigger than the addends.
3) The student thinks that the smaller number should be
subtracted from the larger number. Or, the student has
ignored the sign.
4) The student has ignored the order of terms.
5) The student has ignored the zero assuming it does
(I: Interviewer, S: Student)
|I:||What is a negative number?|
Possible Correct: Smaller than zero, smaller than all positive numbers, to the left of zero, zero minus something, subtracting a larger number from a smaller number
Incorrect: a little, there is no such number, I don’t know, zero
|I:||-8 -3=? What does this expression mean? What does your answer represent? There are different ways of thinking to reach an answer to a question like this. Try to describe your way of thinking.|
Possible Correct: -11, you are taking away a number from a negative number and you will get a “more” negative number. This could represent that you owe money to someone and you borrow more from the same person, so now you owe more money to this same person.
Incorrect: -5; you are taking a positive number away from a negative number so now it is closer to positive number.
It is easier for children to learn to work with positive numbers because children can map their understanding of operations and properties to physical materials. However, developing a similar way of understanding negative numbers is harder because negative amounts do not appear physically in the child’s world.
To help students develop their understanding of negative numbers, teachers can use manipulatives (like integer chips), number lines, and several different metaphors (like money/debt, temperature, time, elevators). These ideas will help students to bridge the gap between the meaning of negative numbers and everyday world. However, manipulatives, number lines and metaphors should be used carefully so that they do not become “unnatural” experiences for students. In other words, activities using manipulatives, number lines and metaphors should be about mathematical ideas to be developed, not the procedures to be followed to use them.
Research indicates that students’ understanding of negative numbers improves when they are aware of the limitations of the models (manipulatives, number lines and metaphors). For example, a money/debt model could be used to develop the meaning of a negative number. A debt of $7 can be represented by -7. But teachers must make sure that students distinguish the ‘-’ in front of the 7 from the subtraction operation. For example, if you owe $2 to your father and $3 to your mother, then your total debt could be found by computing (-2) + (-3) = -5 which means you owe $5. It is possible that teachers represent this situation by -2-3; which is arithmetically correct but creates confusion between the negative sign and the sign for the subtraction operation. This shortcut could confuse students when they try to develop the meaning of the negative sign and the subtraction sign.
The money/debt metaphor can confuse students in problems like (-8) - (-2). Some students tend to translate this expression incorrectly as “You owe $8 and then you owe another $2.” This ignores the fact that the subtraction sign in the middle indicates that you are taking away from your debt, which does not make sense in children’s world.
Having a mental number line is also often considered a component of number sense that helps to build an understanding of negative numbers. Internalization of negative numbers is the stage when a person becomes skillful in performing subtractions. Understanding of the numerical system and relative size of the numbers, including the number zero, is an important part of having a mental number line. Understanding of absolute value is very powerful in this aspect.
Why was –x = 7 the most difficult of the introductory equations? Students consider –x statically just like -4. They treat them as the same. They could not conceive that –x = 7 could give x = -7 because to them, the right-hand side of the original equation would have had to have been -7. Students need to distinguish between the x and the minus sign. Thinking of the negative sign as signaling the opposite of x could help with relational reasoning.
Students could deal with 6x = 7 more easily than -5x = 10. They did not recognize that -5 and x were being multiplied. The negative sign persuades students that there is a subtraction of -5 that needs to be reversed with addition rather than a multiplication by -5. It is essential that students are grounded in variables and notation before solving equations.
Using different models for negative numbers and knowing the pros and cons of each model will help students to understand negative numbers and their operations. Teachers should alternate among models to compensate for the imperfection of each model; however, such alternations should be done by pointing out the issues in the model.
Some commonly used models are:
(i) Debt or owing: We need to be careful as we use this model because not all operations can be modeled naturally in this way. For example, subtracting a negative number from a negative number can become problematic for some children. Instead of the debt or owing model for negative numbers and operations with them, we can use the story of a postman delivering checks or bills to an address and the person at the address keeps track of the bills and checks as they are delivered. In this story line subtracting a negative number from a negative number can be conveyed by delivering a bill to the wrong address and postman comes back to take away the wrong bill.
(ii) Two color integer chips(sometimes called Equilibrium): One color represents positive numbers, the other color represents the negative numbers. With this model most of the operations are modeled naturally; however, for subtracting negatives, pairs of negative and positive chips need to be generated, and this is somewhat artificial. For example, for 4-(-3) four black chips are displayed then 3 pairs of black and white chips are formed to display zeros, then three white chips are taken away to display the value of the operation, 7, with seven black chips.
(iii) Elevators: Numbers are used to represent both position (the third floor) and the action (going up three floors). So it might be hard for students to grasp these two meaning of the number. However, this is a model that aligns well with the number line.
(iv) Time: A scale from B.C. to A.D. could be used to model numbers. However, time is already a hard concept for students to grasp, and adding and subtracting dates does not make much sense.
(v) Temperature: This model may be beneficial to use if children already have a conceptual understanding of temperature. However, if they don’t grasp the temperature idea it might be hard for them to understand what it means to add or subtract temperatures. This is another model that aligns well with the number line for those students who do understand temperature.
(vi) Number line model for operating with negative numbers: In this model addition and subtraction operations indicate which direction you face; for addition you face positive numbers and for subtraction you face negative numbers. The sign of the number indicates how you move on the number line, if you have positive number then you move forward in the direction you are facing and if you have a negative number you move backward in the direction you are facing. For example, for (-7) + (-2) you start at -7 on the number line and since you are adding you face positive numbers. Since you have -2, you move backwards 2 units facing positive numbers and land on -9. This model helps students to operate with negative numbers systematically. However, it is possible for students memorize the steps to produce the answers without really understanding the meaning behind the operation. Thus, providing an alternative model to develop the meaning of negative number is highly suggested.
Other considerations when working with equations that contain negative terms:
● Relational ("the same as") thinking can help students solve the questions posed in i-iv (see the Teaching Strategies section of Algebraic Relations: One and two step equations (x on one side only) for a description of relational versus operational thinking).
● Asking the students to justify their steps forces them to do only steps that they can explain.
● Students should be expected to defend the validity of their solutions for confidence improvement.
● Common errors should be discussed at the appropriate point, both in lessons and in textbooks, especially at the introductory level because it prevents development of bad habits and misconceptions on the part of the student.
● Interviewing students rather than just looking at written work will help uncover misconceptions.
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- What have you learned about your students’ thinking regarding this topic?
- What have you learned that is an effective teaching strategy to help students understand this topic?