Units Multiplication Closure Activity – Lesson Plan
Lesson Overview
This activity is different from the Even and Odd Numbers and Square Numbers closure
activities in that it involves working with finite sets, as opposed to infinite sets such as the
even numbers, and a non-traditional operation called units multiplication. Units
multiplication is simply multiplying two numbers and taking the digit in the ones position of
the product to be the answer. For example, 12 x 7 is 84, so the units multiplication answer to
12 x 7 would be 4. Unlike the previous activities, in this activity students are asked to find
sets of numbers that are closed under the given operation.
Lesson Materials
 A copy of the Units Multiplication Closure Activity worksheet for each student.
 Optional: a calculator, perhaps for students to multiply large square numbers.
Lesson Introduction – Whole Class
1. Define the units multiplication operation as multiplying two numbers and taking the ones
digit of the product to be the answer. For example, 4 x 6 is 24, so the units multiplication
answer of 4 x 6 is 4. Give several other examples until students are comfortable with the
definition.
2. Review the definition of closure. Key points:
 To determine closure, a set of objects and an operation are needed.
 An operation on a set is closed if each possible operation on two objects in the set
results in an object in the set.
 Sometimes it is helpful to organize your work to keep track of all the possible
combinations. Tables are often useful when dealing with a finite set of objects.
3. Explain that in this activity, the set of numbers will be finite sets of numbers and the
operation will be units multiplication.
4. Have the students do the Units Multiplication 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
Determine if the set of numbers 1, 5, and 9 is closed under units multiplication. Show your work
by filling in the table below. Explain how you used the table to determine if this set is closed.
UM
1
5
9
1
1
5
9
5
5
5
5
9
9
5
1
Students should fill in the table as shown. Since the result of the units
multiplication of any two of the numbers in the set {1,5,9} always result in
a number from that set, then the set {1,5,9} is closed under units
multiplication.
Task 1B
It is not necessary to fill in the whole table in Task 1A. Why not? What is the only part or parts
of the table you need to fill in to determine if the set of 1, 5, and 9 is closed under units
multiplication?
UM
1
5
9
UM
1
1
5
9
1
5
5
5
9
9
1
9
9
5
9
5
9
5
5
5
1
It is only necessary to fill in the upper or
lower quadrant of the table since
multiplication is commutative; so units
multiplication is commutative. Therefore,
units multiplication answer to 5 x 9 is the
same as the answer to 9 x 5.
Task 1C
Determine if the set of numbers 1, 3, 7 and 9 is closed under units multiplication. Explain how
you determined your answer.
UM
1
3
7
9
1
1
3
7
9
3
3
9
1
7
7
7
1
9
3
9
9
7
3
1
Students should create a table as shown. Since the units
multiplication answer to any two numbers from the set {1,3, 7,9}
results in a number from that set, then the set {1,3 7,9} is closed
under units multiplication.
Task 2A
The set of numbers 1, 3, and 5 is not closed under units multiplication. What number or numbers
do you need to add to this set to create a set of numbers that is closed under units multiplication?
Show how you determined your answer.
By filling a table as shown, it is clear that 9 needs to be added to the set.
UM
1
3
5
1
1
3
5
3
3
9
5
5
1
5
5
UM
1
3
5
9
1
1
3
5
9
3
3
9
5
7
5
1
5
5
5
9
9
7
5
1
UM
1
3
5
9
7
1
1
3
5
9
7
3
3
9
5
7
1
5
5
5
5
5
5
9
9
7
5
1
3
7
7
1
5
3
9
Once 9 is added to the set and the table, it is clear that 7 needs to be
added to the set.
Once 7 is added to the set and the table, it is clear that the set
{1,3, 5,7,9} is closed under units multiplication.
This task helps students learn how to use tables and a
methodological approach to solving problems involving closure.
Task 2B
Find a set of three even numbers that is closed under units multiplication. Show how you
determined your set is closed.
One solution is the set {1, 4, 6}. Students should create a table to show their set is closed under
units multiplication.
Task 2C
Can you find another set of numbers that is closed under units multiplication? Show how you
found it.
Answers will vary.