# Properties of Multiplication

Commutative   Associative   Distributive   Property of Zero   Identity

Materials: A large quantity of some item to distribute in the classroom.  I used pencils in this activity.  The number of pencils used depends on how big of multiplication problems you are using.  (I.e. 8x15 will require 120 pencils.)

Commutative Property:  8 x 7 = 7 x 8
1. Pick 8 students to stand up.  Give these 8 students 7 pencils.
2. Ask the class “How many pencils have I just given out?” If the students do not know the answer immediately, work through it by using addition of each student’s pencils.

3. The answer is 56.
4. “How do I write this as a mathematical statement?”

5. 8 people x 7 pencils = 56 pencils.  (Write this on the board)
6. Take the pencils back and have 7 different students stand up.
7. Give each of these 7 students 8 pencils.
8. Ask the class “How many pencils have I given out this time? If the students do not know the answer immediately, work through it by using addition of each student’s pencils.

9. The answer is 56.
10. “How do I write this as a mathematical statement?”

11. 7 people x 8 pencils = 56 pencils.  (Write this on the board)
12. Take the pencils back.
13. Repeat this activity with different numbers if desired.
14. With both equations written on the board show the students that no matter how you order the numbers of a multiplication problem, the answer will still be the same.
15. “This is the commutative property.  9 x 12 = 12 x 9.”
16.  “What do you call it when you leave home in the morning and drive to school, then leave school in the afternoon and drive home?” (If your students are familiar with parents making a long drive to work, you might use that example instead).  Entertain any responses.
17. “This is called a COMMUTE. This means to go back and forth between two locations. So for our math problem we can say: ”
 HOME à SCHOOL SCHOOL à HOME 8 x 7 = 7 x 8

Associative Property 4 x (8 x 2) = (4 x 8) x 2
1. Pick 4 students to stand up. Give each of the four students 8 sets of pencils, each set will be 2 pencils.
2. Ask the class “How many pencils have I just given out?” If the students do not know the answer immediately, work through it by using addition of each student’s pencils.

3. The answer is 64.
4. “How do I write this as a mathematical statement?”

5. 4 people x (8 sets x 2 pencils) = 64
6. Take the pencils back from the students.
7. This time, have students stand up in 8 groups of 4.  (You may need to change this problem to suit your class size).
8. Give each person in the groups 2 pencils.
9. Ask the class “How many pencils have I just given out?” If the students do not know the answer immediately, work through it by using addition of each student’s pencils.

10. The answer is 64.
11. “How do I write this as a mathematical statement?”

12. (4 people x 8 groups) x 2 pencils = 64
13. Repeat with different sets of numbers
14. Both equations are on the board, and both equal the same number.  So we can say that:

15. 4 x (8 x 2) = (4 x 8) x 2
16. “This is the associative property of multiplication:  “Factors can be grouped in any order without affecting the product.”
Distributive Property: 4 x 13 = 52
1. Pick 4 students to stand up.  Give these 4 students 15 pencils.
2. Ask the class “How many pencils have I just given out?” If the students do not know the answer immediately, work through it by using addition of each student’s pencils.

3. The answer is 52.
4. “How do I write this as a mathematical statement?”

5. 4 people x 13 pencils = 52 pencils.  (Write this on the board)
6. “We can use a property of multiplication to make the calculations a little easier by breaking up one of our numbers.
7. Go to each of the students that are standing up.  Have them put 10 pencils in the right hand and 3 pencils in the left hand.
8. “We still have the same number of pencils given out, correct?” (just to assure that the students understand you are using the same answer)
9. “How can we represent this situation using a mathematical statement?  We want to be sure our statement includes the 4, 10 and 3?”

10. 4 people have 13 pencils = 4 x 13
4 people have 10 pencils in one hand and 3 in the other = 4 x (10 + 3).  (Write this on the board)

11. “So how would we determine how many pencils were distributed if we didn’t already know it was 52?”
12. Ask the students “How many pencils are in all of the right hands?” If the students do not know the answer immediately, work through it by using addition of each student’s pencils.

13. The answer is 40 pencils in the right hands.
14. “How do we write this as a mathematical statement?”

15. 4 students x 10 pencils in the right hand = 40 pencils (write this on the board)
16. Ask the students “How many pencils are in all of the left hands?” If the students do not know the answer immediately, work through it by using addition of each student’s pencils.

17. The answer is 12 pencils in the left hands
18. “How do we write this as a mathematical statement?”

19. 4 students x 3 pencils in the left hands = 12 pencils (write this on the board)
20. “So how many pencils total are there?”

21. The answer is 52 pencils.
22. “What did we do to get this answer?”  The students will likely say that they did this by adding the two numbers, 40 and 12.  This is fine.
23. “How can we take the two statements we wrote before (pointing to the ones on the board) and put them together to get a mathematical statement representing how we got the total of 52 pencils?”

24. There may be a number of guesses.  The desired answer is:
(4 students x 10 pencils) + (4 students x 3 pencils) = 40 + 12 = 52
25. “Let’s compare the three statements we have for this calculation.”  Point to the following three that should be on the board:

26. 4 x 13 = 52
4 x (10 + 3) = 52
(4 x 10) + (4 x 3) = 52

27. “We have used the Distributive property.  What does the word distribute mean?” Entertain responses.

28. Distribute means to hand out, pass out, give out, circulate, spread around
29. “Which number did we distribute out to the other numbers?”

30. The answer is 4
31. “The distributive property is used to make problems a little easier to work.  One number is split into an addition problem of smaller numbers and the other number is DISTRIBUTED to the numbers of the addition problem.  When using the distributive property, it is easiest if you split the number into numbers that are easy to multiply, such as one digit numbers and multiples of five and ten.”
Property of Zero
1. “If I were to give 0 people in this room all the pencils in my hand, how many pencils would I have given out?”

2. The answer is 0
3. “How would you represent this as a mathematical statement?”

4. 0 people x 26 pencils = 0 pencils
5. “If I were to give 17 people in this room 0 pencils, how many pencils would I have given out?”

6. The answer is 0
7. “How would you represent this as a mathematical statement?”

8. 17 people x 0 pencils = 0 pencils
9. “So if I multiply anything times 0, or 0 times anything, what is my answer going to be?”
10. “The property of zero says that any number times 0 will always equal 0.”
Identity Property
1. “What does the word identity mean?” “What is your identity?” Entertain responses.

2. Your identity is who you are, your self
3. “Do you know what the IDENTITY number is?” Entertain responses.

4. The identity number is 1
5. “Why do you think this is the identity number?” Entertain responses.
6. Have one student stand up.  Give this student 12 pencils.
7. “How many pencils did I give out?”

8. The answer is 12
9. “How do I write this as a mathematical statement?”

10. 1 person x 12 pencils = 12 pencils (write this on the board)
11. Take the pencils back.  Give each student in the class 1 pencil.
12. “How many pencils did I give out? There are 17 people in the class.”

13. The answer is 17
14. “How do I write this as a mathematical statement?”

15. 17 people x 1 pencil = 17 pencils. (Write this on the board)
16. “So when I multiply anything by 1, I get that thing back... the answer is the identity of the number being multiplied by 1.”
17. “The identity property of multiplication is that any number times 1 equals the number.”
18. At the end of the activity, students were allowed to keep the one pencil
created by Sarah Davis, Resident Scientist, TAMU NSF GK-12, September 29, 2004