Commutative Property of Multiplication

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1 Student Probe Solve the following open number sentence. 4 x 9 = x 4 Commutative Property of Multiplication If the student cannot correctly identify the missing number as 9 at all, or if the student has to first compute 4 x 9 then tries to figure out what to multiply by 4 to get that product, he needs to do this lesson. Lesson Description This lesson is intended to help students understand and apply the commutative property of multiplication. Rationale Although the commutative property may seem obvious to adults, it is not necessarily obvious to children. They may see that it works in individual cases, but not understand that it works all the time. Since the commutative property is useful in problem solving, mastering basic facts, and mental math, it is important that students are able to understand the concept and apply it. Children may know that they can change the order of small numbers when they multiply them, but not be sure that they can do the same for very large numbers. This becomes very critical when they start manipulating algebraic expressions and have to apply this understanding to At a Glance What: Applying the commutative property for multiplication Common Core State Standard: CC.3.OA.5. Apply properties of operations as strategies to multiply and divide. Mathematical Practices: Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Look for and express regularity in repeated reasoning. Who: Students who don t understand that it makes no difference in which order two numbers are multiplied. Grade Level: 3 Prerequisite Vocabulary: equal to, product, multiply Prerequisite Skills: equality, multiplication Delivery Format: small group or individual Lesson Length: minutes Materials, Resources, Technology: counters, array cards Student Worksheets: None expressions and equations that have numbers and variables. (It is not so important that children are able to name the property.) Preparation Each student needs at least 50 counters. The teacher needs to prepare array cards for an assortment of products or use the premade ones below.

2 Lesson The teacher says or does Expect students to say or do If students do not, then the teacher says or does 1. Ask student to make a rectangular array using 15 counters. (If student makes a 1 x 15 or 15 x 1 array, ask them to make a different one.) Student should make a 5 x 3 array or a 3 x 5 array. Help student make a correct array. 2. Ask student to describe the dimensions of his array. 3. Compare a 3 x 5 array and a 5 x 3 array. (If only working with one student or if all students make the same array, teacher should make the other array.) Ask student what is alike and what is different about each array. 4. Ask student to write a multiplication number sentence that matches his array and include the total number of counters. 5. Ask student to point out what each numeral in the number sentence stands for. Student should say the array is a 5 x 3 or a 3 x 5 depending on the orientation of his array. Student should say that both arrays have the same number of counters (15). A difference would be that one array has 3 rows with 5 counters in each row and the other array has 5 rows with 3 counters in each row. Student should write 5 x 3 = 15 or 3 x 5 = 15. Five tells how many rows and 3 tells how many columns (or 3 tells how many columns and 5 tells how many rows). Fifteen tells how many there are in all. Ask student to count how many rows his array has. This is one dimension of the array. Ask student to count how many columns his array has. This is the other dimension of the array. Have students count the counters of each array to see that each has 15. Point out the rows and columns on each array. Remind student that the dimensions of the array are the factors in the number sentence and the total number of counters is the product. Model what each number in the equation represents.

3 The teacher says or does Expect students to say or do If students do not, then the teacher says or does 6. Ask student if 3 x 5 = 5 x 3. Ask why or why not. 7. Give student an array card. Ask him how many squares there are in all. (For this step, the array card 6 x 3 is used.) 8. Ask student to write a multiplication number sentence to match the array. 9. Ask student to point out what each numeral in the number sentence stands for. 10. Rotate the array so that the orientation is reversed. For example, if it was a 3 x 6 array, turn it so that it is a 6 x 3 array. Student should say that 3 x 5 = 5 x 3 because each side of the equation is equal to 15. Student should give the correct number of squares. He may count one by one, count by the rows or columns, or use a fact. Example: For the 3 x 6 array, he may (1) count 1, 2, 3, 4, 5 until all are counted, (2) count by 3s or count by 6s, or (3) know that 3 x 6 =18 or 6 x 3 =18. Student should write 6 x 3 = 18 or 3 x 6 = 18. Six tells how many rows and 3 tells how many columns (or 3 tells how many columns and 6 tells how many rows). Eighteen tells how many there are in all. If student says 3 x 5 is not equal to 5 x 3, refer to the two arrays. Identify the dimensions of each array and count the total number of counters in each array. Write the equation 3 x 5 = 15 and the equation 5 x 3 = 15. Since each is equal to 15, we can write 3 x 5 = 5 x 3. Help student count the squares. Remind student that the dimensions of the array are the factors in the number sentence and the total number of counters is the product. Model what each number in the equation represents.

4 The teacher says or does Expect students to say or do If students do not, then the teacher says or does 11. Ask him how many squares there are in all and how he knows. Student should say that there are 18 squares without having to count them again. 12. Ask student to write a multiplication number sentence to match the rotated array. 13. Ask student to point out what each numeral in the number sentence stands for. 14. Ask student if 3 x 6 = 6 x 3. Ask why or why not. 15. Repeat Steps 7 14 above as needed using other array cards. Teacher Notes None Since 3 x 6 is the same as 6 x 3, the number of squares has not changed so there are still 18. Student should write the number sentence with the factors switched. For example, if he wrote 3 x 6 = 18 above, he should write 6 x 3 = 18. Six tells how many rows and 3 tells how many columns (or 3 tells how many columns and 6 tells how many rows). Eighteen tells how many there are in all. Student should say that 3 x 6 = 6 x 3 because each side of the equation is equal to 18. If the student has to count the squares, ask him if any squares have been added or taken away. (No) Since none have been added or taken away, there are still 18. Remind student that the dimensions of the array are the factors in the number sentence and the total number of counters is the product. Model what each number in the equation represents. If student says 3 x 6 is not equal to 6 x 3, refer to the two arrays. Identify the dimensions of each array and count the total number of counters in each array. Write the equation 3 x 6 = 18 and the equation 6 x 3 = 18. Since each is equal to 18, we can write 3 x 6 = 6 x 3.

5 Variations An array is one representation of the commutative property of multiplication. Other strategies to teach the commutative property of multiplication may include repeated addition (e.g =5+5+5), skip counting (e.g. 3, 6, 9, 12, 15 and 5, 10, 15) and equal sets (e.g. 5 sets of 3 and 3 sets of 5). Formative Assessment Give students the following problem: Beth had 7 baskets. She put 6 eggs in each basket. How many eggs were there in all in the baskets? After students have correctly solved this problem, give them the following problem: Samantha made 6 bracelets. She put 7 beads on each bracelet. How many beads did Samantha use in all? Do students have to multiply to solve the second problem or do they know that the answer has to be the same as the first problem? Ask students to solve the following open number sentences: 9 x 8 = 8 x 10 x = 6 x 10 x 13 = 13 x 18 Have students write and solve an open number sentence that shows the commutative property of multiplication. References Russell Gersten, P. (n.d.). RTI and Mathematics IES Practice Guide Response to Intervention in Mathematics. Retrieved 4 26, 2011, from rti4sucess. Sousa, David A., (2008), How The Brain Learns Mathematics, Thousand Oaks, CA, Corwin Press Van de Walle, John A., Karp, Karen S., and Bay Williams, Jennifer M., (2010), Elementary and Middle School Mathematics: Teaching Developmentally, Boston, Allyn & Bacon.

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7 Sample Array Cards

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