Number sentences III
Y4 (11/05)  
a) b) c) d)

0 28 0 16 Total score 
very easy very easy very easy very easy
easy 
Common errors  Likely calculation  Likely misconception  
a) c) 
15 33 
15  0 33  0 
Repeats the number appearing twice in the equation. Does not recognise the role of 0 as the additive entity. 
b) d) 
0 0 
28  28 16  16 
Recognises the role of 0 as the additive entity but misinterprets the equation. 
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).
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. Agreedupon conjectures or rules can then be recorded on separate pieces of paper for display. The student who came up with the rule may even have their name attached to it (e.g., "Sally's Rule").
For example, question c) could be used to discuss 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 take away zero from a number and you get that number back." ^{1}
[Source: a class of 78 year old students]
The Algebraic thinking concept map has more information about conjectures. 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. Students who have grasped the idea of the additive identity can then be encouraged to write their own open number sentences using the additive identity rules. Some examples from a class of Year 4 students include:
5 + 500000 – 500000 =
254268 + 59 – 59 =
79 + 11 – 79 + 0 + 79 – 79 =
Students can also write true/false number sentences for others to solve that involve the additive identity. Resource Looking at zero II 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.
Number sentences II is a parallel Level 3 resource, administered to a sample of Year 6 students.
Click on the link for the Algebraic thinking concept map
Figure It Out
Good As Gold (Number Sense and Algebraic Thinking, L23, book 2, p. 12) explores the idea of adding zero.
NEMP
 Report 9: Mathematics 1997. Addition facts, p. 14. The results from this question imply that students had difficulty with addition problems that were presented with the 0 first. A focus on adding numbers to zero is suggested.
 Report 37: Mathematics 2005. Number Helper, p.15, Questions 4 + 5. Just over 50% of Year 4 students were able to identify 0 as the number you can add or take away from 8 and 8 will stay the same.
^{1} Carpenter, T., Franke M. L., & Levi, L. (2003). Thinking Mathematically. Integrating Arithmetic and Algebra in Elementary School. Portsmouth , NH : Heinemann, p. 50.