Student Probe Commutative Property of Multiplication At a Glance Solve the following open number sentence. 4 x 9 = x 4 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 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: 15 – 20 minutes Materials, Resources, Technology: counters, array cards Student Worksheets: None 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 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. Lesson The teacher says or does… 1.
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Expect students to say or do… If students do not, then the teacher says or does… Ask student to make a Student should make a 5 x 3 Help student make a correct rectangular array using 15 array or a 3 x 5 array. array. counters. (If student makes a 1 x 15 or 15 x 1 array, ask them to make a different one.) Ask student to describe Student should say the array Ask student to count how the dimensions of his is a 5 x 3 or a 3 x 5 depending many rows his array has. This array. on the orientation of his is one dimension of the array. array. Ask student to count how many columns his array has. This is the other dimension of the array. Compare a 3 x 5 array and Student should say that both Have students count the a 5 x 3 array. (If only arrays have the same number counters of each array to see working with one student of counters (15). that each has 15. or if all students make the same array, teacher A difference would be that should make the other one array has 3 rows with 5 Point out the rows and array.) Ask student what counters in each row and the columns on each array. is alike and what is other array has 5 rows with 3 different about each counters in each row. array. Ask student to write a Student should write 5 x 3 = Remind student that the multiplication number 15 or 3 x 5 = 15. dimensions of the array are sentence that matches his the factors in the number array and include the total sentence and the total number of counters is the number of counters. product. Ask student to point out Five tells how many rows and Model what each number in what each numeral in the 3 tells how many columns (or the equation represents. number sentence stands 3 tells how many columns and for. 5 tells how many rows). Fifteen tells how many there are in all. 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. Student should say that 3 x 5 If student says 3 x 5 is not Ask why or why not. = 5 x 3 because each side of equal to 5 x 3, refer to the the equation is equal to 15. 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. 7. Give student an array Student should give the Help student count the card. Ask him how many correct number of squares. squares. squares there are in all. He may count one by one, (For this step, the array count by the rows or columns, card 6 x 3 is used.) 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. 8. Ask student to write a Student should write 6 x 3 = Remind student that the multiplication number 18 or 3 x 6 = 18. dimensions of the array are sentence to match the the factors in the number array. sentence and the total number of counters is the product. 9. Ask student to point out Six tells how many rows and 3 Model what each number in what each numeral in the tells how many columns (or 3 the equation represents. number sentence stands tells how many columns and 6 for. tells how many rows). Eighteen tells how many there are in all. 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. 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 Student should say that there If the student has to count the squares there are in all are 18 squares without having squares, ask him if any and how he knows. to count them again. squares have been added or taken away. (No) Since none Since 3 x 6 is the same as 6 x have been added or taken 3, the number of squares has away, there are still 18. not changed so there are still 18. 12. Ask student to write a Student should write the Remind student that the multiplication number number sentence with the dimensions of the array are the factors in the number sentence to match the factors switched. For sentence and the total rotated array. example, if he wrote 3 x 6 = 18 above, he should write 6 x number of counters is the product. 3 = 18. 13. Ask student to point out Six tells how many rows and 3 Model what each number in what each numeral in the tells how many columns (or 3 the equation represents. number sentence stands tells how many columns and 6 for. tells how many rows). Eighteen tells how many there are in all. 14. Ask student if 3 x 6 = 6 x 3. Student should say that 3 x 6 If student says 3 x 6 is not Ask why or why not. = 6 x 3 because each side of equal to 6 x 3, refer to the the equation is equal to 18. 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. 15. Repeat Steps 7 – 14 above as needed using other array cards. Teacher Notes None 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. 3+3+3+3+3=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. Sample Array Cards