Using Mental Math to Add and Subtract

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SWBAT apply a variety of methods (such as breaking apart, compensation, and counting on) to add and subtract whole numbers mentally.

Big Idea

Students can use the break apart, compensation, and counting on methods to add and subtract without paper and pencil.


5 minutes

 For this direct instruction lesson, we will discuss real-world scenarios that the students can relate to their lives.  If they can relate to it, then they understand why they need to know it. By the end of the power point, the students should understand methods that will help them calculate mentally.  This all ties in to CCSS 4.OA.A3 because the students use mental computations to solve problems. This lesson is important because in future lessons, the students will need to know if their answers are reasonable by using mental computations to help them when they estimate.

To capture the students' interest and get them excited about the lesson, I ask them a question.  "What would you do if paper and pencil were no longer allowed in school?"  This should have the students in a buzz. "How would you solve your math problems?"   Call on 2 or 3 students to share their answers.  

"Today, you will learn strategies of how to add and subtract without paper and pencil."

Direct Instruction

10 minutes

To begin my instruction on "Using Mental Math to Add and Subtract", I call the students to the carpet because I want all of my students to be attentive to the lesson.  The power point is displayed on the Smart Board.  I like for my students to interact with me during my lessons.  As we go through the power point, I often stop to ask questions to make sure the students are comprehending what's in the power point.  For example, as we talk about the "break apart method", I may ask a student to explain why we now have 3 addends instead of 2 addends in the problem.  

Going over vocabulary (listed below) at the beginning of the lesson is very important because the students need to know any unfamiliar words before we get into the lesson.  The students must repeat after me as I say the vocabulary words and the definitions.  Throughout the power point, when we come across one of the vocabulary words, I call on a student to tell me what that vocabulary word means.  I believe that the more the students see, hear, and say the vocabulary words, the better they will understand how to use them on their own.


—Breaking apart – a mental math method used to rewrite a number as a sum of numbers to form an easier problem.
—Compensation – adding or subtracting the same number to make the sum or difference easier to find.
—Counting on – An adding strategy
—Commutative Property of Addition – Numbers can be added in any order and the sum remains the same.
—Associative Property of Addition – Addends can be regrouped and the sum remains the same.
—Identity Property of Addition – The sum of any number and zero is that number.


I like to anticipate any misconceptions that the students may have at the very beginning of the lesson.  I address those during direct instruction time to alleviate the problems that they might cause later for the students.  As I go through the power point, I am strategic in asking questions that can help clear up misconceptions.

Possible Misconception(s):

1.  Compensation Method:  Students may think that once they have added or subtracted to find an answer, then they are finished with the problem

Questions to clear up misconceptions:   In this problem did we add or subtract the exact number that was in the problem or did we add or subtract a different number?  If we added or subtracted a different number, what must we do to the answer?  





Group or Partner Activity

20 minutes

To give the students practice with the skiill and to allow them to interact with their classmates, I let the students work as pairs.  I like for my students to work as pairs because they all have the opportunity to be heard.  Sometimes in larger groups, some students hide behind the other students and never share their voices.  

This activity requires the students to solve word problems.  My thought on the students solving word problems for this activity is that I feel that the school work must relate to their everyday lives.  We must put it in a context that they understand.  We must make it relevant to them or they will not see the purpose for doing it.  

 In pairs, the students are given an activity sheet with real world scenarios to solve.   Together, the students solve the problems using the compensation, break apart, and count on methods.  As the students are working in groups, I monitor the groups to listen in on their conversations.  I interject with questions or observations to get the students to think about their answers.  Some questions or observations: 1) Explain the method that you are using, 2) If you add more than the problem tells you to add, then what must you do next to correct this?  What can you add to this number to make a number with a zero in the ones place?   

The directions for the activity are as follows:  

Without using paper and pencil, solve the problem mentally by using the compensation, break apart, or count on method.  Then explain the answer and which method you used to your partner.  Take turns solving the problems.  If your partner solves the problem using the compensation method, then you should solve the same problem using a different method. If your answers are different, then you should both solve the problem with the third method.  In a respectful way, debate any differences that you have with your answers.


If any of the pairs finish early and I feel that they understood the lesson, they can go to the computers to practice using the mental math strategies at the following site:




Independent Activity

10 minutes

After the students complete their collaborative activity, the students work on an independent assignment.  This allows the students to practice the skill on their own.  This lets me know if the students are having trouble with the skill.  

For this activity, the students are given 1 real world problem to solve.  The students solve this problem without working it out on paper and pencil.  Because I set the scenario up earlier of there no longer being paper and pencil used in the classroom, I will have the students use a dry erase board and marker to show me their answer.  (If I don't do this, I know I'm going to have at least one student say, "Mrs. Monroe, I thought there was no paper and pencil allowed at school.)

On the Smart Board the students will have the following problem: 

Teresa's little brother is very popular at school.  He is friends with 38 boys and 26 girls.  How many friends does he have in all?

Directions:  Solve the problem using mental math.  Use the compensation, break apart, or count on method.  Write your answer on the dry erase board. When you finish, turn your white boards over and wait for further directions.

After the students have completed the independent activity, I walk around to each student to assess their answers on the white board.   Any students having difficulty with this skill will receive remediation in small group.



15 minutes

To close the lesson, I bring the class back together as a whole.  I call on 3 students to present their answers.  One student for each method: break apart, compensation, and count on.  I select students based upon the answers that I saw on their dry erase board.  I strategically select students with right and wrong answers to the problem.  This is not to embarass a student, but to allow classmates and myself to question the student to lead them to the right answer.  Most often there is more than one student who may have the same misconceptions about a skill.  This allows for the other students who are still struggling with the skill to hear questions and answers that can help them as well.