Even and Odd Numbers Closure Activity – Lesson Plan
Lesson Overview
This activity is good for introducing the concept for closure. It can also be used to reinforce
the definitions of even and odd numbers. This activity explores whether the even numbers or
the odd numbers are closed under addition and multiplication. We will see that the even
numbers are closed under addition, while the odd numbers are not. This demonstrates that
not all sets of numbers are closed under the same operation. However, both the even
numbers and odd numbers are closed under multiplication. For younger grades, this activity
can be done without the multiplication tasks.
Lesson Materials
• A copy of the Even and Odd Numbers Closure Activity worksheet for each student.
• Optional: manipulatives to model even and odd numbers.
• Optional: a calculator, perhaps for students to add very large even numbers.
Lesson Introduction – Whole Class
1. Review the definition of even and odd numbers. For younger children, you may want to
model these definitions using manipulatives.
2. If appropriate, review addition of whole numbers.
3. Introduce the word “closure”. Closure occurs when something is done to two things of
the same type and the result is the same type of thing. Some non-mathematics examples
include
Apple juice and cranberry juice or both fruit juices. If they are mixed together, the
result – apple-cranberry juice – is still a fruit juice. Thus, mixing fruit juices is
closed.
Hydrogen and oxygen are both gases. But if they are combined in the right way, they
form water (H2O), which is not a gas. Since, combining gases does not always result
in a gas, combining gases is not always closed.
4. Closure in mathematics occurs when you have a set of things, such as a set of numbers,
and when you do an operation on two numbers, the number that results from the
operation is from the same set as the original two numbers. The sets of numbers for this
activity are the even numbers and the odd numbers. The operation is addition.
5. Have students do the Even and Odd Numbers Closure Activity worksheet. You may
want the students to work in groups and bring the class together to discuss the tasks at
various times.
Worksheet Solutions, Discussion, and Extensions
Task 1A
Describe how you can tell that 6 and 8 are even numbers.
The students should work with the definition or definitions given to them in class.
Task 1B
Add 6 and 8. Is their sum an even or odd number? How do you know?
6 + 8 is 14, which is an even number.
Task 1C
Pick two other even numbers and add them. Is their sum even or odd? How do you know?
Answers will vary.
Task 1D
Can you find any two even numbers whose sum is not an even number? Why not?
If students are modeling even numbers with manipulatives, they can show that the sum of two
even numbers can always be modeled with a 2 x n rectangle since the addends are always 2 x n
rectangles. Older students may be able to show that since the addends are divisible by 2,their
sum must be divisible by 2.
Task 1E
Explain why the even numbers are closed under addition.
By the definition of closure, since the sum of two even numbers is always an even number, then
the even numbers are closed under addition. This is a good opportunity to discuss proof. A finite
set of examples is not a proof in mathematics. Some other type of reasoning must be used to
show something is true in all cases. The arguments given in Task 1D are proofs since they apply
to any pair of even numbers.
Task 2A
Predict whether or not you think the odd numbers are closed under addition.
This task is asking students to make a conjecture. You may want to introduce this word and
explain that a conjecture is an educated guess, like a hypothesis in science.
Task 2B
Using the same process you used in Task 1, determine whether or not the odd numbers are closed
under addition. Explain how you determined your answer.
The odd numbers are not closed under addition. For example, 3 + 3 = 6. To prove something is
not true in all cases, only one counterexample is needed. Thus, one counterexample is sufficient
to prove something false, but an example – even many examples – are never sufficient to prove
something true. This is an important concept to understand about reasoning and proof in
mathematics.
Task 3A
Determine whether or not the even numbers are closed under multiplication. Explain how you
determined your answer.
The even numbers are closed under multiplication. The interpretation of multiplication as
repeated addition can be used to prove this assertion. Since an even number plus an even
number is always an even number, then repeatedly adding even numbers will result in an even
number. Again, a few examples is not considered a proof.
Task 3B
Determine whether or not the odd numbers are closed under multiplication. Explain how you
determined your answer.
The odd numbers are closed under multiplication. The area model of multiplication can be used
to prove this. All odd numbers are an even number plus 1. In the model below, one odd factor is
represented by E+1, while the other odd factor is represented by e+1, where E and e are the two
even numbers that are one less than odd factors. When E+1 is multiplied by e+1, the product is
the sum of three even numbers (E x e + E + e) plus one. So, since the product is an even number
plus one, the product must be an odd number.
E
1
e
E x e = even
e
1
E
1
It is interesting to note that two different meanings of multiplication were used to prove the
results in Task 3A and Task 3B.
Task 4
Annie added 214 and 3368 and got 3583. Without adding, how can you use closure to tell that
her answer is incorrect?
Annie added two even numbers and the sum of two even numbers must be an even number since
the even numbers are closed under addition, but 3583 is not an even number. This example
shows that the knowing the closure properties of the numbers can be used in providing quick
reasonableness checks to problem solutions.