The key idea here is that subtracting a negative integer has the same effect as adding a positive integer. For example, ⁺5 – ⁻7 = ⁺12. Rather than memorising a rule such as “two negatives make a positive” (which will mislead them), students can see from this learning experience why this happens.

Using materials

Ask the students to imagine that the bucket is filled with water. The red unifix cubes are units of heat (or “hot cubes”) and the blue unifix cubes are units of cold (or “cold cubes”). Draw a horizontal line on the board or in the modelling book. Place a small triangle at the middle so that the drawing represents a simple balance.

Explain that if there are an equal number of hot cubes and cold cubes (pairs), the situation is represented by 0 (there is a perfect balance between hot and cold). If there are more hot cubes than cold cubes, there is a positive situation or warmer water. If there are more cold cubes than hot cubes, there is a negative situation.

Place 10 red cubes and 10 blue cubes in the bucket. Agree that the starting situation is neutral or 0. Look at the following equations and show us what would happen to the ‘water’ in the bucket if you carry out the operation. Act out the equation with the model to find or prove your answer.

(1) 0 – ⁺2 = ?
(2) 0 + ⁺2 = ?
(3) 0 + ⁻2 = ?
(4) 0 – ⁻2 = ?

For the first equation, listen for explanations and demonstrations that include the following idea/action: starting at neutral and taking away two hot cubes tips the balance to the cold side and the result is described as a ⁻2 situation (that is, there will be two more cold cubes than hot cubes in the bucket).

For the second equation, listen for explanations and demonstrations that include the following idea/ action: starting at neutral and adding two hot cubes tips the balance to the warm side and the result is described as a ⁺2 situation (that is, there will be two more hot cubes than cold cubes in the bucket).

For the third equation, listen for explanations and demonstrations that include the following idea/ action: starting at neutral and adding two cold cubes tips the balance to the cold side and the result is described as a ⁻2 situation (that is, there will be two more cold cubes than hot cubes in the bucket).

For the fourth equation, listen for explanations and demonstrations that include the following idea/ action: starting at neutral and taking away two cold cubes tips the balance to the warm side and the result is described as a ⁺2 situation (that is, there will be two more hot cubes than cold cubes in the bucket).

Discuss with the students what they notice about the answers. Pose new sets of four problems with different starting points (for example, add a positive, add a negative, subtract a positive, subtract a negative).

Examples: The water in the bucket is at a ⁻3 situation, so we have 3 more cold cubes than hot cubes. Now model ⁻3 + ⁺4, ⁻3 + ⁻4, ⁻3 – ⁺4, ⁻3 – ⁻4 … The water in the bucket is at a ⁺5 situation …

Using imaging and number properties

Students will need practice working with the addition and subtraction of integers. The imaging of the bucket model can be especially helpful, as it is easy to imagine a starting point and the way the balance shifts as they add or withdraw elements of heat and cold.

Students who cannot solve these problems by imaging or number properties should fold back to manipulating blocks in the bucket model. They need to realise that it doesn’t matter how many actual cubes are in the bucket, it’s the balance or difference between the two sets of cubes that we are interested in.

Examples: ⁻25 – ⁻34, ⁺42 – ⁻3, ⁻99 – ⁻1 …