This resource explores the concept of the additive identity property. The additive identity can be expressed in four different ways, with each way being able to be written in two equivalent forms:

a + 0 = a or a = a + 0
0 + a = a or a = 0 + a
a – 0 = a or a = a – 0
a – a = 0 or 0 = a – a

It is effective for students to see each rule written in its alternative form to help them realise that the equals sign does not have to be the penultimate symbol in an equation, but can be near the beginning.

The resource is designed to help in exploring these four different forms, and recording rules or conjectures about adding and subtracting zero. Understanding and applying the additive identity is important for future experiences that involve the solving of algebraic equations.  This idea is explored further in the Algebraic thinking concept map (additive identity).

Possible next steps

Although students seem to have an intuitive understanding of what happens when zero is added to or subtracted from a number, it is important to have them articulate, share and critique these understandings as conjectures or rules about zero. Agreed-upon conjectures or rules can then be recorded on separate pieces of paper for display. The student who comes up with the rule may even have their name attached to it (e.g. "Sally's Rule").

For example, question c) could be used as the basis for discussion about what happens when zero is subtracted from a number and may lead to a conjecture or rule which describes a – 0 = a, e.g.,

" When you subtract zero from a number it doesn't change the number you started with."
[Source: a class of Year 4 students.]

"You can take away zero from a number, and you get that number back."
[Source: a class of 7-8 year old students.)]

When having these discussions it may be necessary to explain that it is the number zero, meaning nil or no quantity that is being explored, rather than zero as a placeholder in numbers such as 500 or 200.
The Algebraic thinking concept map has more information about conjectures.

Students who have grasped the idea of the additive identity can then be encouraged to write their own open number sentences for others to solve using the additive identity rules. Some examples from a class of Year 4 students include:
5 + 500 000 – 500 000 =
254 268 + 59 – 59 =
79 + 11 – 79 + 0 + 79 – 79 =

Students can also write true/false number sentences that involve the additive identity element. Resource Looking at zero explores this further.  Further examples of student number sentences can be found in the Algebraic thinking concept map (number sentences).

As well as using the identity element in their own number sentences, students can also "find the zero" in open number sentences, e.g. 26 + 54 – 54 = or = 33 + 29 – 33 + 62 – 62.
In these situations, students should be identifying the zero and solving the problem without calculation.
Resources Solving problems and Solving simple equations deal with these types of problems.