Work Backwards

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SWBAT work backwards by using the inverse operation.

Big Idea

Students can find answers to problems by starting with the end result and working backwards using inverse operations.

Whole Class Discussion

15 minutes

In today's lesson, the students learn to solve multi-step word problems involving the four operations by using the strategy of working backwards (4.OA.A3). This is a very important problem-solving strategy that the students must know in order to proficiently solve word problems.

I call the students to the carpet as we prepare for a whole class discussion.  The Working Backwards power point is already up on the Smart board.  I like for my students to be near so that I can have their full attention while I'm at the Smart board.  We begin the lesson with a problem.


Teresa has been collecting money for a surprise birthday party for her mom.  Her brother, Tim, gave her $50. Teresa had to spend part of the money.  She paid $25 for the invitations to the birthday party and $175 for the birthday cake.  Her sister, Rita, gave Teresa $75 for the party.  Teresa now has $450.  How much money did Teresa start with?   

To find the amount of money that Teresa started with, it is sometimes helpful to begin with the last detail given.

Teresa has $450 left.  What did Teresa do last?

Teresa received $75 from Rita.  If she received money, that means she added $75 to the amount of money she has.  If we work backwards, we need to use the inverse operation.

What is the inverse of addition? Student response:  Subtraction.

Next, Teresa spent some money.  Teresa spent $175 for the birthday cake and $25 for the invitations.  If we spend money, we subtract it from what we have.

What is the inverse operation of subtraction? Student response:  Addition.

The first thing that Teresa did was receive $50 from her brother.  When we receive money, we add it to what we have. 

We know that the inverse operation of addition is subtraction.

How much money did Teresa start with? 

Let’s find out.



-   75




+   25


-    50


 Teresa started with $525.00.

Check the answer by starting with $525 and complete each transaction to see if you will end with $450.

Skill Building/Exploration

20 minutes

During this group activity, the students work in pairs.  Each pair has a copy of the Working Backwards.  The students must work together to solve the problems by working backwards. The students are guided to the conceptual understanding through questioning by their classmates, as well as by me.  The students communicate with each other and agree upon the answer to the problem.  This takes discussion, critiquing, and justifying of answers by both students.  As the pairs discuss the problem, they must be precise in their communication within their groups using the appropriate math terminology for this skill (MP6).  As I walk around, I am listening for the students to use "talk" that will lead to the answer. I am holding the students accountable for their own learning.  

As they work, I monitor and assess their progression of understanding through questioning. Some of the questions that I ask:

1.  What detail was given last?

2. What is the inverse operation for this step?

3.  After you found your answer, did you check it by starting at the first step?

Any groups that finish the assignment early, can go to the computer to practice the skill at the following site until we are ready for the whole group sharing.


15 minutes

To close the lesson, I have students share their answers.  This gives those students who still do not understand another opportunity to learn it.  I like to use my document camera to show the students' work during this time. Some students do not understand what is being said, but understand clearly when the work is put up for them to see.

I feel that by closing each of my lessons by having students share their work is very important to the success of the lesson.  Students need to see good work samples (Student Work - Working Backwards) as well as work that may have incorrect information.  More than one student may have had the same misconception.  During the closing of the lesson, all misconceptions that were spotted during  partner sharing will be addressed whole class.  


In the Video for Working Backwards, you will see how this pair of students did not use the inverse operation for the $18 and the $148.  I had to work with these students on identifying the transaction, then finding the inverse operation for that transaction.