Looking at zero II
| Y4 (11/05) | ||
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a) b) c) d) e)
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T
F F T F
Total score
All 5 correct 4 correct |
easy
easy easy easy easy
easy
very easy |
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. Agreed-upon 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., "Joanne'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]
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. Using question d) as a discussion point can be useful in helping to clarify this with students.
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. Resource Number sentences III 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.
Click on the link for the Algebraic thinking concept map
Looking at zero is a parallel Level 3 resource, administered to a sample of Year 6 students.
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.
1 Carpenter, T., Franke M. L., & Levi, L. (2003). Thinking Mathematically. Integrating Arithmetic and Algebra in Elementary School. Portsmouth , NH : Heinemann, p. 50.