Primary Type: Formative Assessment
Status: Published
This is a resource from CPALMS (www.cpalms.org) where all educators go for bright ideas!
Resource ID#: 36539
Justifying the Commutative Property of Addition
Students work with cubes or color tiles to understand and justify the Commutative Property of addition.
Subject(s): Mathematics
Grade Level(s): 1
Intended Audience: Educators
Freely Available: Yes
Keywords: MFAS, MAFS.1.OA.2.3, Addition, commutative, tiles, cubes, generalize, strategies
Resource Collection: MFAS Formative Assessments
ATTACHMENTS
MFAS_JustifyingCommutativeProperty_Worksheet.docx
FORMATIVE ASSESSMENT TASK
Instructions for Implementing the Task
1. The teacher should ask the student to examine these two equations.
7+2=9
2+7=9
2. The teacher asks, “What do you notice about these two equations?” “How are they alike?” “How are they different?” If the student does not note that the two
addends have been interchanged, the teacher provides prompting questions to lead the student to observe this.
3. Once the student recognizes that the addends were interchanged, the teacher asks, "I noticed that you said the two and the seven switched places. Do you think that
you can always switch those two numbers in an addition equation and still get the same answer?"
4. Next, the teacher should gather a group of approximately nine red cubes and a group of four brown cubes. It is important that the student does not count the cubes in
each group. The teacher says, “I notice we have some red cubes in this group, and some brown cubes in this group, and we don't know how many are in each group.”
5. Then, the teacher asks, “If I want to count the total number of cubes, does it matter if I count the red cubes first or the brown cubes first? Should I get the same
answer no matter which color I count first? Do you think this will always work with any two sets of cubes?”
6. Finally, the teacher asks the student to add 7 + 6. After the student finds the correct answer, ask the student what 6 + 7 is. Observe the student to see if he or she
counts or recalls to determine the answer.
page 1 of 4 TASK RUBRIC
Getting Started
Misconception/Error
The student persists in solving the final problem by counting instead of applying the Commutative Property.
Examples of Student Work at this Level
The student directly models and counts to find the sum of seven and six. In the second problem, the student again directly models and counts to find the sum of six and
seven despite prompting by the teacher to see that it is not necessary to add again if the addends are the same.
Questions Eliciting Thinking
Show the student the equations 5 + 3 = 8 and 3 + 5 = _____. Direct the student to examine both equations, and describe how the two equations are alike and how they
are different. Then, ask the student to complete the second equation.
If I have four pencils and I get six more, how many would I have? Now, if I ask you what is 6 + 4? Could you tell me without counting?
If I know that 9 + 8 = 17, do I need to add to find the answer to 8 + 9?
Instructional Implications
Using cubes, model for students how 2 + 6 equals the same as 6 + 2 and that it is not necessary to add the numbers in the second problem to find its answer.
Have the student hold two pennies in his or her right hand and three pennies in his or her left hand. Ask the student to determine how many pennies there are. After the
student determines that there are five pennies, ask the student to move one penny from his or her left hand to his or her right hand (the student may need some
guidance with left and right). Now, ask the student if the number of pennies has changed. Model for the student how equations can be written from the activity (2 + 3 =
5 and 3 + 2 = 5). Have the student repeat the exercise using a total of seven pennies. Guide the student in writing equations for the numbers of pennies the student
holds in each hand. Continue having the student model with pennies until he or she can see that if two addends are the same in each equation, then their sum is the same
regardless of their order.
Moving Forward
Misconception/Error
The student counts to solve the final problem despite agreeing that the order in which the cubes were counted does not matter.
Examples of Student Work at this Level
The student directly models and/or counts to find 7 + 6 = 13 in the first problem. In the second problem, the student still solves the problem by direct modeling and/or
counting. When the teacher prompts the student to apply the Commutative Property, the student hesitantly agrees that you do not have to add 6 + 7 if you know that 7
+ 6 is 13.
Questions Eliciting Thinking
Show the student the equations 7 + 6 = 13 and 6 + 7 = _____. Direct the student to examine both equations, and describe how the two equations are alike and how
they are different. Then, ask the student to complete the second equation.
If I know that 9 + 8 = 17, do I need to add to find the answer to 8 + 9?
Why do you think we don’t have to add 11 + 3 if we know the answer to 3 + 11?
Instructional Implications
Present two problems to the student in commuted pairs (e.g., 5 + 8 and 8 + 5), and use counting to find the sums. Help the student to see that the addends and their
sums are the same in both sets of problems.
Have the student practice decomposing numbers (both addends unknown problems), and write the decompositions as equations. Have a table for students to record the
decompositions. Ask students to write commuted pairs across from each other in the table.
Write a sum on a card, such as 9 + 6. Ask the student to find a sum that uses the same addends and has the same answer but looks different, such as 6 + 9. Emphasize
that computing the sum is not necessary. Repeat this exercise until the student can routinely respond.
Almost There
Misconception/Error
The student solves the final problem by using the Commutative Property but cannot explain why this property works.
Examples of Student Work at this Level
The student correctly finds the sum of six and seven in the final problem without adding but the student cannot explain why it was not necessary to add to find this sum.
The student is not confident in his or her strategy.
Questions Eliciting Thinking
Another first grade student found that 11 + 3 = 14. When she tried to find the answer to 3 + 11, she counted on from three. Is there an easier way the student could
have found the answer to 3 + 11? How? Why does that work?
If I know that 7 + 5 = 12, do I need to add to find the answer to 5 + 7? Why not?
Why do you think we don’t have to add 6 + 7 if we know the answer to 7 + 6?
Do you think that the order in which we add two numbers matters? Will we always get the same answer regardless of the order in which we add?
page 2 of 4 Instructional Implications
Have the student write pairs of problems in which the Commutative Property can be used to solve the second problem in each pair.
Model for the student how you would explain the Commutative Property (it is not necessary for students to refer to the Commutative Property by name).
Help the student see the similarities between commuted pairs, and help the student explain how the two equations are similar and why the sum is the same for each
equation.
Pose the following problem: Cameron had four pennies and then his mom gave him five more. How many pennies does Cameron have? Ask the student to solve the
problem. Then, show the student a set of equations that both have nine as the sum but the addends are different (3 + 6 = 9 and 5 + 4 = 9). Ask the student to show
you which equation would help solve the problem. Then, ask the student to explain why 3 + 6 = 9 would not help us find out how many pennies Cameron has now.
Provide the student opportunities to apply the Commutative Property to make computations easier. For example, ask the student, “Which expression would you rather
solve, 1 + 12 or 12 + 1?” Have the student explain why.
Got It
Misconception/Error
The student has no misconceptions or errors.
Examples of Student Work at this Level
The student correctly finds the sum in the final problem and can clearly explain why it was not necessary to add to find this sum. The student is able to explain that the
order in which the numbers are added does not matter; the sum will be the same.
Questions Eliciting Thinking
Another first grade student found that 11 + 3 = 14. When she tried to find the answer to 3 + 11, she counted on from three. Is there an easier way the student could
have found the answer to 3 + 11? How? Why does that work?
Do you think that the order in which we add two numbers matters? Will we always get the same answer regardless of the order in which we add?
Does this idea work with subtraction?
Instructional Implications
Show the student a subtraction equation, and ask if we can switch the numbers around and still get the same answer.
Help the student begin to generalize the Commutative Property. Show the student that the Commutative Property can be expressed more generally as a + b = b + a.
Give the student a set of numbers, and ask the student to create equations that involve sums of commuted pairs. Once the student is able to create pairs, extend the
activity, and ask the student to find the sums.
Give the student a problem with three addends, and help the student to apply the Associative Property (as well as the Commutative Property) to compute more easily.
ACCOMMODATIONS & RECOMMENDATIONS
Special Materials Needed:
Justifying Commutative Property worksheet
Cubes or color tiles
SOURCE AND ACCESS INFORMATION
Contributed by: MFAS FCRSTEM
Name of Author/Source: MFAS FCRSTEM
District/Organization of Contributor(s): Okaloosa
Is this Resource freely Available? Yes
Access Privileges: Public
License: CPALMS License - no distribution - non commercial
page 3 of 4 Related Standards
Name
MAFS.1.OA.2.3:
Description
Apply properties of operations as strategies to add and subtract. Examples: If 8 + 3 = 11 is known, then 3 + 8 = 11
is also known. (Commutative property of addition.) To add 2 + 6 + 4, the second two numbers can be added to make
a ten, so 2 + 6 + 4 = 2 + 10 = 12. (Associative property of addition.)
page 4 of 4