Represent integers, compare positive and negative values and explore them in context.
Counting numbers describe quantities such as 1, 2, 3 and so on, but many situations also need values below a reference point. Temperatures can be below zero, floors can be below a ground-level floor, and a position can be to the left of a chosen origin. Negative numbers represent the other side of zero, while positive numbers lie on the positive side. Zero separates the two directions and is neither positive nor negative. Together, negative whole numbers, zero and positive whole numbers form the integers.
On a horizontal number line, numbers increase as we move right and decrease as we move left. Zero is the reference point. A number farther to the right is greater, so every positive integer is greater than zero and every negative integer is less than zero. For example, −2 is greater than −5 because −2 lies to the right of −5. This can feel counterintuitive if one compares only the digits: the sign and position matter. Equal intervals on the line represent equal changes in value.
A token model can represent positive and negative quantities with two kinds of tokens. A positive token and a negative token of equal value form a zero pair because their combined value is zero. Adding a zero pair does not change the total, but it can make a subtraction easier to represent. Keep the meaning of each token fixed throughout a calculation. The model supports reasoning about opposite quantities and helps show why adding or removing tokens changes the total in a predictable way.
A negative number is meaningful only in relation to a reference point and a direction. A temperature of −3° is three degrees below zero; a floor marked −1 may mean one level below ground; a balance may use negative values to show an amount owed. The context determines the unit and interpretation. When comparing two values, return to the reference point: −1 is warmer than −4 on a temperature scale, even though 4 is a larger digit than 1. Use a labelled number line to avoid confusion.
Moving right on a number line can model adding a positive integer; moving left can model adding a negative integer. For example, starting at −2 and moving 5 steps right gives 3. Moving 4 steps left from 1 gives −3. Subtraction can be understood as the change from one value to another or as removing tokens in the token model. Draw the starting value, mark the direction and count equal steps. This method makes the sign of the answer visible instead of relying on a memorised rule.
Example 1: Compare −6 and −2. Since −2 is farther right, −6 < −2. Example 2: Starting at 3, add −5 by moving five places left: 3 → 2 → 1 → 0 → −1 → −2, so 3 + (−5) = −2. Example 3: A change from −4 to 2 is six units to the right, because 2 − (−4) = 6. Common mistakes include treating negative signs as decoration, claiming −8 is greater than −3 because 8 > 3, or forgetting the reference point in a real-world situation. Check each answer on a number line.
| Idea | How to use it | Check |
|---|---|---|
| Identify the rule | Describe what changes and what remains fixed. | Test more than one example. |
| Choose a representation | Use a diagram, number line, table or equation. | Labels and units are clear. |
| Explain the answer | Show the operation and each step. | The result fits the question. |
Example A. Read the question and name the relevant mathematical idea. Write the information given before choosing an operation. This prevents a familiar-looking number from being used in the wrong way.
Example B. Show the calculation in small steps. For example, when a calculation involves a product and a sum, compute the product first if the expression requires it, then add. Keep intermediate values visible so an arithmetic slip can be found.
Example C. Check the result using a second representation where possible: estimate it, draw a model, use a number line, or substitute it back into the original relationship. A check is especially useful when a question involves units, scale, direction or a geometric condition.
Negative whole numbers, zero and positive whole numbers.
−2, because it is to the right of −5.
One positive and one equal negative token.
Move left by the required number of equal steps.
The Other Side of Zero builds mathematical understanding through careful observation and clear representations. Identify the key quantities, shapes or relationships before choosing a method. Use diagrams, tables, number lines or equations to make thinking visible. Show calculations in order and keep labels and units consistent. Check whether the result satisfies the definition or rule in the question. Explain the result in a sentence so another learner can follow the reasoning.