Looking at zero

Looking at zero

Auto-markingPencil and paperOnline interactive
Overview
Using this Resource
Connecting to the Curriculum
Marking Student Responses
Working with Students
Further Resources
This task is about what happens when you use zero when adding or taking away.

Question 1Change answer

illustration: plus or minus zero
Look at each number sentence. Select if it is True or False.
a)  12 + 0 = 12     truefalse
b)  34 – 34 = 34   truefalse
c)  91 – 0 = 91     truefalse
d)  70 + 0 = 700   truefalse
e)  0 + 87 = 0       truefalse
Task administration: 
This task can be completed with pencil and paper or online (with auto-marking displayed to students).
Copyright: 
Graphic copyright: NZCER
Levels:
2, 3
Curriculum info: 
Maths, Number and Algebra, Equations and expressions
Keywords: 
properties of zero, additive identity, addition, subtraction, inverse, discussion
Description of task: 
Students explore the additive identity by deciding whether addition and subtraction number sentences are true or false.
Curriculum Links: 

This resource can help to identify students' ability to use addition and subtraction as inverse operations.

Learning Progression Frameworks
This resource can provide evidence of learning associated with
Using symbols and expressions to think mathematically, set 3
Using symbols and expressions to think mathematically, set 4
within the Mathematics Learning Progressions Frameworks.
Read more about the Learning Progressions Frameworks.
Answers/responses: 
  Y6 (11/2005)* Y6 (11/2013) online **
a)
b)
c)
d)
e)		 T
F
T
F
F

Total score
All 5 correct
very easy
very easy
very easy
very easy
very easy
 
very easy
very easy
very easy
very easy
very easy
very easy
 
very easy

* Based on a representative sample of 173 Y6 students
** Based on a sample of 65 Y6 students

Diagnostic and formative information: 

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). 

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. "Sufi’s Rule").

For example, question b) could be used to discuss what happens when a number is subtracted from itself and may lead to a conjecture or rule which describes a – a = 0, e.g., 
"If you take the number you started with away from the number you get 0." [Source: a class of Year 4 students]
"A number minus the same number is zero." 1 [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. Using question d) as a discussion point can be useful in helping to clarify this with students. 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 true/false number sentences using the additive identity rules.  Some examples from a class of Year 4 students include:

33 + 8 – 0 = 33 + 8 + 0 (True)
500 + 0 = 500 + 100 – 100 (True)
77 + 502 – 77 = 501 (False)

Students can also write open number sentences for others to solve that involve the additive identity. The resource Number sentences 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 property 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.

1 Carpenter, T., Franke M. L., & Levi, L. (2003). Thinking Mathematically. Integrating Arithmetic and Algebra in Elementary School. Portsmouth, NH: Heinemann, p. 50.

Figure It Out
Good As Gold (Number Sense and Algebraic Thinking, L2-3, 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.

Looking at zero II is a parallel Level 2 resource, administered to a sample of Year 4 students.