Mathematics Developmental Continuum
A Negative Multiplied by a Negative: 4.75
Supporting materials
Indicator of Progress
Success depends on students being able to multiply integers with understanding. Many students find it very surprising and counter-intuitive that a negative number multiplied by a negative number gives a positive number. Unless they have a reason for this, they will feel that they must simply remember arbitrary rules.
Illustration 1
Students who rely on rules without reasons may get them confused. For example, reciting “two negatives makes a positive” can lead to errors like this:
(–2) + (–3) = + 5 or (–2) – (–3) = + 5
Teaching Strategies
This is an instance where students have to work beyond the range of simple physical models. The teaching strategy is to build up basic understanding using concrete models or physical representations, but then to derive meaning from mathematical patterns and logical reasoning instead.
In this case, there are several good models for showing the meaning of a positive number times a negative number, but these models do not stretch to explaining the meaning of negative times a negative. For the latter, students need to derive meaning from mathematical patterns and logical reasoning instead.
Activity 1 A positive multiplied by a negative takes the first step of using concrete or physical models to build preliminary understanding of multiplying a positive by a negative.
Activity 2 A negative multiplied by a negative extends the idea to multiplying a negative by a negative number by generalisation from number patterns.
Activity 1: A positive multiplied by a negative
Several physical models can illustrate why a positive multiplied by a negative is negative. Here are two examples.
Model 1: Represent a positive number as a profit, and a negative number as a loss. Represent multiplication by a positive number as a doubling, trebling etc of the profit or loss.
Students can reason by analogy to work out the meaning of multiplication of a positive and a negative number:
- if three people make a profit of $6, together they make a profit of $18 (because 3 × 6 = 18),
- if three people each make a loss of $6, together they make a loss of $18 (so 3 × (-6) = -18).
Model 2: Represent a positive number by the change when pebbles are added to a bag (which starts with a large unknown number of pebbles inside), and a negative number as the change when pebbles are taken out of the bag. Represent multiplication by a positive number as repetition of the action.
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+ 3 is represented by ADDING 3 pebbles to the bag. 5 × (+3) is ADDING 3 pebbles 5 times. |
-4 is represented by TAKING 4 pebbles from the bag. 5 × (-4) is TAKING 4 pebbles 5 times. |
Activity 2: A negative multiplied by a negative
The models of Activity 1 cannot readily explain multiplication of a negative by a negative. We can’t have -3 people losing money and we cannot take balls out of an urn -3 times. Multiplication by negative is beyond the range of these models. So at this stage, it is best to turn away from the model and use reasoning from patterns or number properties.
List the 2 times table (we can choose any multiplication table) and continuing the pattern with the second factor being below zero. We can see that the pattern of down by 2 continues.
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2 × 5 =
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10
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observe pattern: 'down by 2'
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2 × 4 =
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8
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2 × 3 =
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6
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2 × 2 =
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4
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2 × 1 =
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2
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2 × 0 =
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0
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2 × -1 =
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-2
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'down by 2' pattern continues
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2 × -2 =
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-4
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2 × -3 =
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-6
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2 × -4 =
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-8
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2 × -5 =
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-10
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Now list the -2 times table as far as we know it based on activity 1: a pattern of up by 2 develops.
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-2 × 5 =
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-10
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observe pattern: 'up by 2'
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-2 × 4 =
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-8
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-2 × 3 =
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-6
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-2 × 2 =
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-4
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-2 × 1 =
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-2
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-2 × 0 =
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0
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-2 × -1 =
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?
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?
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-2 × -2 =
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?
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-2 × -3 =
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?
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-2 × -4 =
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?
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-2 × -5 =
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?
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Continue the table, into the negative by negative region, using the pattern of up by 2 .
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-2 × 5 =
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-10
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observe pattern: 'up by 2'
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-2 × 4 =
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-8
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-2 × 3 =
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-6
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-2 × 2 =
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-4
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-2 × 1 =
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-2
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-2 × 0 =
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0
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-2 × -1 =
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2
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use pattern of 'up by 2'
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-2 × -2 =
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4
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-2 × -3 =
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6
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-2 × -4 =
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8
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-2 × -5 =
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10
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Making the pattern of 'up by 2' continue forces us to agree that a negative number by a negative number is a positive.
Observing this pattern does not make a proof, but it can convince students that
a negative number × a negative number = a positive number.
Mathematicians prove this result by logical deduction from basic properties of numbers.
References
This material is adapted from Marston, K. & Stacey, K. (Eds.) (2003). Foundations for Teaching Arithmetic. Version 2. Faculty of Education, University of Melbourne. http://extranet.edfac.unimelb.edu.au/DSME/arithmetic
Other models for negative numbers are also presented there.