Primary mathematics/Negative numbers
- The intended audience for this lesson is educators and parents, whether in traditional or home-schooling environments. For the same lesson, but aimed directly at students, whether children or adults, please see this Wikibooks page.
Introducing Children to Negative Numbers
The concept of negativity is can be difficult to explain to children. This is in part because, from a child’s perspective, there are no obvious examples of negative numbers naturally found in nature. Temperatures below zero are an artificial construct, as is the idea of below sea-level, or floors in a building below the first floor. The concepts of owing money or borrowing candy bars are abstract ideas as well. Children generally do not understand “overnight” what negativity is. However, introducing children to all of these examples will, over time, help them to make the necessary connections that will lead to a strong basic understanding of the concept.
Children should be encouraged to think of their own examples of negativity. Concepts such as backwards and forwards, up and down, give and take, hot and cold, should be explored.
Models and Manipulatives
There are a few mathematical models used to introduce negative numbers to children but the most commonly used one is most certainly the number line..
Because the number line is the one-dimensional component of the Cartesian coordinate system, it should eventually be modeled both horizontally and vertically, and because the x and y axis' of the Cartesian system run left to right and down to up respectively, number lines should be presented this way as well. Connections can then be made between number lines, negative numbers and thermometers, elevators, rulers (for measuring sea-level).
Another way to teach negative numbers is with a set of manipulatives that look like little plus and minus signs (sometimes known as integer tiles). These tiles can be grouped, combined, and swapped around in ways that model the four basic arithmetic operations. They can then be modeled at a higher level of abstraction by students on paper. These manipulatives and their accompanying visual models are normally introduced in the upper primary grades. Below is an example of addition using these tiles:
Note that in this model's answer set, plus signs and minus signs can be grouped to create null sets, whose value is 0. In this particular problem, 2 null sets have been found and circled in the answer. What remains is the sum of the two initial numbers - in this case a single negative tile. Problems should be set up using this method involving two negative numbers, as well as negative and positive numbers where the sum of these numbers is equal to both positive and negative answers.
Other activities and suggestions:
There is also a "red chip/black chip" manipulative, where red chips are used to show negative numbers and black positve numbers. This manipulative can be confusing for students because both the red chips and the black chips exist.
A lesser known manipulative used to teach negative numbers is The ZeroSum Ruler, a description of which can be found on the blog http://zerosumruler.wordpress.com/ The ZeroSum Ruler is a foldable ruler that allows students to add integers of opposite signs wth absolute value. The ZeroSum Ruler also works for subtraction, as "3 - 8", for example, is the sum of 3 and -8.
Give students pictorial representations of using this model and have them identify the associated equation. Have students model three different ways to model the number -5, 3, and 0 using tiles. Give students equations with larger and more complicated numbers until they, in recognizing how cumbersome this model can get, begin to solve the problems by creating their own algorithms or "rules" to speed up the process. In this way, students will understand at a fundamental level why you can just subtract 46.2 from 234.5 and put a negative sign on the answer to solve the problem .
Subtraction Involving Negative Numbers
In the earlier primary grades, students use various manipulatives as they explore subtraction. Candy, money, toys, are all things that children relate to. But because negative numbers are not found in nature, the subtraction of negative numbers should be introduced only when students are comfortable with more abstract models such as number lines and plus and minus tiles. The number line will have the advantage in that students will already be familiar with their use for adding and subtracting positive numbers. In the upper primary grades, students should be exposed to the idea that addition and subtraction are inverse operations.
Because students are usually taught to "walk backwards" to model subtraction on a numberline, in order to model the subtraction of a negative integer, we can tell them to "do the opposite" of "walking backwards", thus modeling the idea that subtracting a negative is the same as adding a positive. But this can be very confusing for students and other models should be introduced to give students a broader understanding.
The operation of subtraction can be best understood by most students in terms of the act of "taking away". Teaching a student to say (and eventually think) the words "take away" whenever they see a subtraction sign is a solid teaching strategy that just happens to translate well to working with negative numbers when using plus and minus tiles.
The following illustration models the problem
The statement "5 negative tiles take away 2 negative tiles" describes the values and the operation. Note that it is important to stress the words "take away".
Students need to become comfortable with this level of problem before they are exposed to problems where the first number in the equation does not permit the taking away of the tiles specified. Consider the problem . It is not possible to "take away" 4 negative tiles when there is only 1 negative tile in the initial set. Students must be taught to model the number -1 in such a way that there are 4 negative tiles available to remove:
Notice that in this illustration, the number -1 has been modeled using three null sets (circled in blue). With the inclusion of these null sets, the value of the number in the rectangle is still -1. Note that the student could put in 10 or 20 null sets and the answer will still be the same. Once students are exposed to this method, they might use more null sets than necessary. This is OK. It is in figuring out for themselves how many null sets are necessary that they truly come to an understanding of the operation.
Students should learn to model problems such as:, , and . After achieving a certain level of proficiency with this model, students soon come to their own personal understanding of what it means when we say that that subtracting a negative is the same as adding a positive.
Multiplying negative numbers
- Multiply a positive number by a negative normally. The result is negative.
- Multiply two negative numbers normally. The result, however, is positive. Think of it as a double negative making a positive.
Dividing negative numbers
- Divide a positive number by a negative normally. The result is negative.
- Divide a negative number by a positive normally. The result is negative.
- Divide a negative number by a negative normally. The result, however, is positive. Think of it as a double negative making a positive.
- This same logic also applies to fractions, where the numerator (top number) and/or denominator (bottom number) is negative. That is, if one of the two numbers is negative, then the fraction is negative. If both numbers are negative, the minus signs cancel, and the fraction is positive.