How Many Did I Start With? (Working Backwards)

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SWBAT take the information given and work backwards to solve real-world problems.

Big Idea

When you are given the end result and need to find the beginning number, you can work backwards.


5 minutes

In this Introductory Video on Working Backwards, I explain today's objective.

In today's lesson, the students will learn how to work backwards to solve a problem using addition and subtraction.  This aligns with 4.NBT.B4 because the students are adding and subtracting using the standard algorithm.  The students have already learned how to use the inverse operation to "undo"  a calculation.  They will use this skill to help work backwards to find the beginning balance of a bank account.

To get the students started with the lesson, I ask the students a question.  "If I know the total in my bank account at the end of the day, how can I find out how much I had in the bank at the beginning of the day?"  I give the students a few minutes to think about the question.  I encourage my students to always think before they answer. I take a few responses from the students.  I let them know that today we will use what's called working backwards to solve a problem.

Direct Instruction

10 minutes

I call the students to the carpet to begin our whole class discussion on working backwards using addition and subtraction.  I like for my students to be close and to make sure that they are all attentive to the lesson.  The Working Backwards power point is on the Smart board.  We begin the lesson with a problem.


Bank Account Balance:


Beginning:  ?

Ending:  $1,450


What information do you think is needed in order for me to know the beginning balance?

I give the students a few minutes to think about the problem and then call on a few students to respond.  The students had a difficult time answer that question.  I let the students know that in order to know my beginning balance, I need to know all of the transactions that took place that day.

Below are a list of transactions from my bank account:



ATM withdrawal


Gas purchase








To make sure that the students understand bank account information, I explain to them that withdrawals mean that you are taking money out of the bank.  Deposits are when you put money into the bank.

With the transactions, we can work backwards to find the beginning balance.  Do the inverse of the transaction to “undo” the transaction.


+     50   (add)

+     45   (add)

-    100   (subtract)

+      68  (add)

$ 1,513 Beginning Balance   
So the beginning balance is $1,513.


I explain to the students that you must check your answer to make sure it is correct.  To check your answer, take the beginning balance and add deposits and subtract withdrawals.  


     -       50


     -       45


    +      100


     -        68


The answer checks out correctly because it gives us our ending balance.


Group or Partner Activity

20 minutes

I give the students practice on this skill by letting them work together.  I find that collaborative learning is vital to the success of students.  Students learn from each other by justifying their answers and critiquing the reasoning of others (MP3).

For this activity, I put the students in pairs.  I give each group the How Many Did I Start With real-world problem to solve.  The students must decontextualize the problem and represent them symbolically (MP2).  The students must work together to find the beginning amount of a problem by working backwards to find the answer.   They must communicate precisely to others within their groups (MP6).   They must use clear definitions and terminology as they discuss this problem.

As the students work in pairs, I walk around to monitor and listen in on the discussion.  To guide the students' learning, I ask assessing questions to help lead them to the answer.  I find that when you don't give the students the answer, but help guide them to it, they feel more of a sense of accomplishment.

One particular group struggled with whether to add or subtract.  They were really confused because if it was a withdrawal, then they wanted to subtract.  Through questioning I guided them through the task.

1.  What is the inverse of subtraction?

2.  Why should you use the inverse? 

3.  What was the next transaction?  

4.  Did you  check your answer?   

5.  Did you come back to the ending balance?


Independent Activity

10 minutes

To give me a clear understanding of what each student knows, they do an independent assignment.  As a teacher I need to assess students independently to make sure they are all receiving the help they need.  I put a problem on the Smart board for the students to work.  The students use paper and pencil to solve the problem.  I walk around to visually assess the students understanding, keeping track of all students who I will work with in small group for remediation.


Problem (Independent Assignment Working Backwards):

There are 41 people in line now.  Earlier 15 people got waited on and left.  Then 8 new people got in line.  How many people were in line at the very beginning?



The students should get the answer of 48 people started out in line.  All students who do not have this answer will receive intervention in a small group.


10 minutes

To close the lesson, I have two 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, as well as work that may have incorrect information.  More than one student may have had the same misconception.  

Possible Misconceptions:

1.  To add or subtract when you should use the inverse operation

Questions to help with misconceptions:

1.  How would you undo subtraction?  addition?

2.  Why did you add? subtract?

3.  Have you checked you answer?