This unit provides opportunities to interpret negative quantities in practical situations allowing students experiences to shift their thinking from the idea that “measures” must be positive. Students will have experienced negative numbers in context from primary education and know that the number line extends beyond zero.

Understanding is developed by placing numbers on a number line and using the number line as a representation for addition and subtraction of negative numbers as well as a scaling model for multiplying and dividing. Carefully structured tasks allow students to extend existing understanding of calculations with integer values to include negative numbers. By extending the axioms of positive numbers to negative numbers we will conclude that (−1)×(−1) = 1 and a − (−b) = a + b.

Students should be aware of language, using phrases such as 'subtract negative three' rather than 'minus, minus three'. Laminated sheets with a selection of number lines (labelled and unlabelled) could prove a useful resource especially for demonstrating multiplication and division with negative numbers. 

Interpret negative numbers in a variety of contexts.

Explore movement on an 'extended' number line.                           

To interpret the absolute value of a number as its distance from zero.

To understand how to order negative numbers using inequality notation.                                                                                                    To be able to model addition of a positive number as a translation on the number line.

To use this model to develop strategies for addition.   To be able to model addition of a negative number as a translation on the number line.

To understand that if two numbers are additive inverses they sum to 0.                                                          Subtracting positive numbers is equivalent to the addition of the additive inverse.

Subtracting negative numbers is equivalent to the addition of the additive inverse.

Subtraction is then explored of as the inverse of addition.

Explore multiplication involving at least one positive number.

Visualise multiplication as a stretch by a scale factor.

Negative scale factors are described as a stretch in the opposite direction.

The commutativity of multiplication is explored.           

Multiplication with a negative scale factor is extended to finding the product of two negative numbers.

Division by a positive integer is equivalent to multiplying by the reciprocal.

Further division facts are derived using ‘fact families’.

The axioms introduced in Unit 2 also hold for negative numbers.

Manipulating and simplifying calculations using these axioms.

 Initial exposure to negative numbers from Primary School.

Movement on a number line. To understand how to order positive numbers using inequality notation.                                                                                 

To be able to move comfortably on the extended number line.                                                                    

To be able to model the addition of a positive

number using a number line. 

To be able to find the additive inverse of any number.

Addition with positive and negative numbers.                  

Multiplication of positive integers by 'simple' fractional qualities.                                             Multiplication with a positive scale factor.                    

Multiplication with negative numbers.

Knowledge of fact families with positive numbers.            

Familiarity with the axioms in Unit 2

The four operations involving negative numbers.

Integer Inverse Negative

Do I understand what the inequality signs are ? Am I able to write down the correct inequality into a sum?

 Key questions that students should be able to answer at the end of the 'Topic'