Unit 3: Negative Numbers

Unit 3: Negative numbers

Planning guidance

Unit objectives

Represent and order positive and negative numbers on a number line
Use inequality symbols to compare numbers
Show addition and subtraction on a number line
Apply the four basic operations on positive and negative integers
Calculate with decimals (including negative decimals)
Calculate with fractions (including negative fractions)

There are three task sets provided for this unit, which should take about two weeks to complete. Students should have met negative numbers in primary education and know that the number line extends beyond zero. These tasks will focus on misconceptions students may have when comparing the size of numbers and the four basic operations using negatives. Confusion can arise around the meaning of the minus sign, which is whether it represents a negative number or a subtraction. Teachers should be aware of the language used in relation to this, using phrases such as 'subtract negative three' rather than 'minus minus three'. Number lines are used throughout this unit. Laminated sheets with a selection of number lines (labelled and unlabelled) could prove a useful resource. In particular, they are extremely useful for demonstrating multiplication and division with negative numbers.

Each of the overview pages contains some tasks that need to be completed as a department. These tasks should serve as a starting point for planning since each class is different and will have different needs. After completing the task, we invite you to use the resources and to edit them as needed to plan your own Mathematics Mastery lessons.

Suggested Structure

The actual number of lessons spent on each section will depend on the individual class and the number of lessons available to teachers. We suggest the following based on a total of 6, 8 or 10 lessons over a fortnight:

Section / 6 lessons / 8 lessons / 10 lessons
Ordering and comparing positive and negative numbers / 1 / 1-2 / 2
Adding and subtracting negative numbers / 3 / 4 / 5
Multiplying and dividing negative numbers / 2 / 2-3 / 3

Ordering and comparing positive and negative numbers

Objectives

Identify and represent positive and negative numbers on a number line
Order positive and negative numbers

Students should be encouraged to work with negative numbers in context as much as possible through temperature and depth scales. Even though this section does not cover addition and subtraction, these concepts can be discussed informally such as talking about the number levels a lift needs to go up/down to get from one floor to another. In this way, students will be able to move to addition and subtraction in a much more intuitive way.

When working with number lines, students should be encourage to include decimals and fractions in their answers, this applies to tasks such as those asking students to identify ‘5 numbers between -2 and 3’.

For further guidance, including the Department Tasks for this section, click Ordering and comparing positive and negative numbers under Planning resources on the main page for this unit.

Adding and subtracting negative numbers

Objectives:

Demonstrate addition and subtraction on a number line
Use inequality symbols to compare calculations
Work with negative fractions and decimals (if appropriate)

When adding and subtracting negative numbers it is really vital that students use number lines to develop a sense of the concepts being met rather than learning rules or methods. In particular, when dealing with questions such as 2+ -3, confusing language such as ‘a plus and minus make a minus’ should be avoided. There are lots of ideas on the appropriate ‘Ideas for representation page’ that can support the teaching of this unit, in particular the use of the yin yang symbols to show ‘cancelling’.

This section begins by considering addition and subtraction in the familiar context of temperatures and allows students to appreciate the movement along the number line in terms of getting hotter and colder. The number punch activities allow the development of key concepts and number lines are used throughout to reinforce understanding rather than rote learning. As before, the use of decimals and fractions (if appropriate) should be encouraged.

Towards the end of the section, and depending on the time available, there is opportunity to consolidate earlier work on algebraic notation by practising substitution skills using negative numbers.

For further guidance, including the Department Tasks for this section, click Adding and subtracting negative numbersunder Planning resources on the main page for this unit.

Multiplying and dividing negative numbers

Objectives:

Appreciate the nature of multiplication and division as making numbers x-times bigger/smaller
Multiply and divide negative and positive numbers
Work with fractions and decimals as appropriate

Whilst using number lines is incredibly helpful when working on addition and subtraction, they need to be used in a slightly different way for students to appreciate the nature of multiplication and division when working with negative numbers.

The first few slides in this pack are really to support teachers in getting students to develop the view that multiplication and division make things x-times bigger or smaller. This makes much more sense than trying to look for analogies to positive numbers – for example, it doesn’t make much sense to ask ‘how many -3s are there in 12?’.

Once students grasp the basics, they can be challenged to work with fractions and decimals and to apply their knowledge to algebraic expressions. There is also opportunity here for students to work on developing their overall conceptual understanding of multiplication and division by taking a multiplication equation (such as -5 x 3 = 15) and writing that as division equations.

For further guidance, including the Department Tasks for this section, click Multiplying and dividing negative numbersunder Planning resources on the main page for this unit.