examples of problem solving using working backwards

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Math Strategies: Problem Solving by Working Backwards

As I’ve shared before, there are many different ways to go about solving a math problem, and equipping kids to be successful problem solvers is just as important as teaching computation and algorithms . In my experience, students’ frustration often comes from not knowing where to start. Providing them with strategies enables them to at least get the ideas flowing and hopefully get some things down on paper. As in all areas of life, the hardest part is getting started! Today I want to explain how to teach  problem solving by working backwards .

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–>Pssst! Do your kids need help making sense of and solving word problems? You might like this set of editable word problem solving templates ! Use these with any grade level, for any type of word problem :

Solve a Math Problem by Working Backwards: 

Before students can learn to recognize when this is a helpful strategy, they must understand what it means. Working backwards is to start with the final solution and work back one step at a time to get to the beginning.

It may also be helpful for students to understand that this is useful in many aspects of life, not just solving math problems.

To help show your students what this looks like, you might start by thinking about directions. Write out some basic directions from home to school:

• Start: Home
• Turn right on Gray St.
• Turn left on Sycamore Ln.
• Turn left on Rose Dr.
• Turn right on Schoolhouse Rd.
• End: School

Ask students to then use this information to give directions from the school back home . Depending on the age of your students, you may even want to draw a map so they can see clearly that they have to do the opposite as they make their way back home from school. In other words, they need to “undo” each turn to get back, i.e. turn left on Schoolhouse Rd. and then right on Rose Dr. etc.

In math, these are called inverse operations . When using the “work backwards” strategy, each operation must be reversed to get back to the beginning. So if working forwards requires addition, when students work backwards they will need to subtract. And if they multiply working forwards, they must divide when working backwards.

Once students understand inverse operations , and know that they must start with the solution and work back to the beginning, they will need to learn to recognize the types of problems that require working backwards.

In general, problems that list a series of events or a sequence of steps can be solved by working backwards.

Here’s an example:

Sam’s mom left a plate of cookies on the counter. Sam ate 2 of them, his dad ate 3 of them and they gave 12 to the neighbor. At the end of the day, only 4 cookies were left on the plate. How many cookies did she make altogether?

In this case, we know that the final cookie amount is 4. So if we work backwards to “put back” all the cookies that were taken or eaten, we can figure out what number they started with.

Because cookies are being taken away, that denotes subtraction. Thus, to get back to the original number we have to do the opposite: add . If you take the 4 that are left and add the 12 given to the neighbors, and add the 3 that Dad ate, and then add the 2 that Sam ate, we find that Sam’s mom made 21 cookies .

You may want to give students a few similar problems to let them see when working backwards is useful, and what problems look like that require working backwards to solve.

Have you taught or discussed problem solving by working backwards  with your students? What are some other examples of when this might be useful or necessary?

Don’t miss the other useful articles in this Problem Solving Series:

• Problem Solve by Drawing a Picture
• Problem Solve by Solving an Easier Problem
• Problem Solve with Guess & Check
• Problem Solve by Finding a Pattern
• Problem Solve by Making a List

So glad to have come across this post! Today, word problems were the cause of a homework meltdown. At least tomorrow I’ll have a different strategy to try! #ThoughtfulSpot

I’m so glad to hear that! I hope you found some useful ideas!! Homework meltdowns are never fun!! Best of luck!

This is really a great help! We have just started using this method for some of my sons math problems and it helps loads. Thanks so much for sharing on the Let Kids Be Kids Linkup!

That’s great Erin! I hope this is a helpful method and makes things easier for your son! 🙂

I’ve not used this method before but sounds like a good resource to teach. Thanks for linking #LetKidsBeKids

I hope this proves to be helpful for you!

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Primary 3 Working Backwards & Its Method

Math heuristics for problem solving, primary 3 working backwards & its method, what is working backwards in math.

The scenario occurs when the quantity data is insufficient to work from the beginning . Working Backwards is a problem-solving strategy in which you start with the end goal and work backward to figure out the steps needed to get there. In other words, instead of starting from the beginning and moving forward, you start from the end and move backward. This strategy is commonly used in math problems that ask you to find a starting value or figure out what happened before a given situation.

How to Solve Math Questions with Working Backwords Method?

Let's take a look at this primary 3 word problem example:.

Watch the tutorial for free!

Sarah had some pens. She bought 34 pens. She then threw away 29 pens as they were spoilt. In the end, she had 64 pens. How many pens did Sarah have at first?

Identify the Concept

We know this is a Working Backwards question as…

Workings Explained

Always remember when we work backwards, everything will be reversed. Example the 2nd sentence – “She bought 34 pens”. We know when we buy things, we will have more. We need to add. However, when we work backwards, instead of adding, we need to subtract.

• We will start drawing the model from the end by drawing a box and label it “End”. Put the end amount “64” in the box.
• Draw arrow to point to the left, draw another box. On top of the arrow write “+29” as “Sarah threw away 29 pens”. Instead of subtract, we need to add. In the box, write “93” (64+29=93).
• Draw another arrow to point to the left, draw another box. On top of the arrow, write “-34” as Sarah bought 34 pens. Instead of adding, we need to subtract. In the box, write “59” (93-34=59). Label the box “At First” or “Before”.

Sarah had 59 pens at first.

We know this is a Working Backwards question as we were not told the number of pens Sarah had but were asked for the number of pens she had at first.

See other problem-solving strategies and methods

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What's Problem Sum?

Problem Sum is a term commonly used in the context of math education, particularly in regions that follow the Singapore Math curriculum . It refers to a type of math problem that typically involves multiple steps and requires students to apply various mathematical concepts and strategies to find the solution. These problems are often word problems that describe a real-world scenario, requiring students to read, comprehend, and analyze the situation before applying the appropriate math operations to solve it.

In essence, a Problem Sum combines elements of arithmetic and logical reasoning , challenging students to think critically and use their problem-solving skills. It is a key component in developing a deeper understanding of math concepts and enhancing analytical abilities in students.

Problem Solving: Working Backwards

Teaching problem solving strategies is as important as teaching computation.  i like to call problem solving, creative thinking, as i believe that give the process a more positive spin.  kids are less intimidated by the word than they are a problem solving as that implies that there is a “problem”.   students need to learn to interpret what they are reading and draw out information.  this can be difficult for some students.  i have found that students’ frustration with working on “story problems” has been where to start and what information do you need.   each month i am tacking a different problem solving strategy this month it is working backwards ., working backwards is sometimes the best strategy and one of the most difficult for students to understand.  i was one of those that didn’t understand this strategy because rarely did i read all the information.  i saw numbers, key words and started putting things together but not always correctly.  also being dyslexic it was hard enough reading the information let alone interpreting it and then working backwards and apply inverse operations.  , when introducing the different types of problem solving strategies; looking for patterns, make a list, draw a picture, guess and check, and working backwards, working backwards can sometimes be a fun one to start first.  giving the students a trick about reading through all the directions is unintimidating and proves a point in an amusing way.  , do you remember the problem where you ask students to read all the directions before starting and then the last question says don’t do the work just sign your name at the top  these worksheets are a great way to get students to read all the directions, take in all the information and then work from there.    .

After doing these worksheets apply working backwards to math.  Our first goal in any problem is to read the whole problem.  Is there an answer provided?  Does the question have key words like equal to, amount left is, leaving ___?  Once you have established that there is an answer to the problem we need to start working backwards which also means using the inverse operation .   In mathematics we are also developing the idea of inverse operation, using the opposite operation than the one given. This is a great math way to show how that works.

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Working Backwards: Heuristic for Problem Solving

Working Backwards is a non-routine heuristic that all pupils learn in primary schools . Many pupils learn this heuristic as early as when they were in primary two. You can easily find this heuristic being included in one of the topics in assessment books .

Though common, this heuristic is in fact one of the toughest in primary Mathematics syllabus . Problem sums that can be solved using Working Backwards are usually wordy with a sequence of events taking place which make them complex. You need endurance and clarity of mind to follow through a series of events that are unfolding in sequence, not to mention in a backward manner.

Without further ado, let’s delve into one typical upper primary problem sum to find out how we can tackle this category of problem sums using the heuristic of Working Backwards .

Xiaoming and Ali were playing a card game using 96 pokemon cards. In the first game, Ali lost 1 5 of his cards to Xiaoming. In the second game, Xiaoming lost 1 3 of his cards to Ali. After the second game, both boys had the same number of pokemon cards. How many pokemon cards did Xiaoming have at first?

Study and Understand the Problem

In this problem sum, there are only two variables – Xiaoming and Ali.

Cards are transferred between Xiaoming and Ali in a series of games, but the total number of cards between them is still the same – internal transfer.

Think of a Plan

There are contextual clues of a typical “Working Backwards” problem sum:

• Final information is given on how a situation ends and you need to find the answer in the beginning.
• There is a focus on sequence of events.

Act on the Plan

Reverse the solution steps by working backwards in a systematic manner using a table. A table will help to organise your working in a more orderly manner and track the steps in sequence.

Final number of cards each person had = 96 ÷ 2 = 48

	XM	A
End	96 ÷ 2 = 48	96 ÷ 2 = 48

Second game

After Xiaoming lost 1 3 of his cards to Ali, he was left with 2 units as 1 unit was won by Ali (Refer to the numerator and denominator). Thus, 2 units = 48

	XM	A
Second game	2 units = 48 1 unit = 24 3 units = 72 XM had 72 cards before second game	96 – 72 = 24 Ali had 24 before second game

After Ali lost 1 5 of his cards to Xiaoming, he was left with 4 units as 1 unit was won by Xiaoming (Refer to the numerator and denominator). Thus, 4 units = 24

	XM	A
First game	96 – 30 = XM had 66 cards at first	4 units = 24 1 unit = 6 5 units = 30 Ali had 30 cards at first

Reflect on my Answer

Work forward with the answer you have.

XM → 66 A → 30

A → of 30 = 6 30 – 6 = 24 XM → 66 + 6 = 72

XM → of 72 = 24 72 - 24 = 48 (correct!) A → 24 + 24 = 48 (correct!)

More Examples of Problems Sums Involving Working Backwards

Try to pick out the contextual clues that tell you that Working Backwards can be used.

P3 Math question

Alan bought some fish. One day, 6 of his fish died. After that, he bought the same number of fish as those which were still alive. He gave away all his fish equally among 8 friends and each friend had 4 fish. How many fish did Alan have at first?

P6 Math question

A MRT train left Bugis station with some passengers. At Lavender station, no passengers alighted and the number of passengers who boarded the train was  1 4 of the original number of passengers in the train. At Kallang station,  2 5 of the passengers alighted and 51 passengers boarded the train. At Aljunied station, 2 3 of the passengers alighted and 24 passengers boarded the train. At Paya Lebar station, all 122 passengers alighted from the train. How many passengers were there when the train left Bugis Station?

Whenever possible, use a table or draw boxes to help you solve Working Backwards questions in an orderly and systematic manner.

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About the Author

Teacher Zen has over a decade of experience in teaching upper primary Math and Science in local schools. He has a post-graduate diploma in education from NIE and has a wealth of experience in marking PSLE Science and Math papers. When not teaching or working on OwlSmart, he enjoys watching soccer and supports Liverpool football team.

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Imagine where you want to be someday. Now, how did you get there? Retrograde analysis is a style of problem solving where you work backwards from the endgame you want. It can help you win at chess -- or solve a problem in real life. At TEDYouth 2012, chess grandmaster Maurice Ashley delves into his favorite strategy.

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Backward Thinking in Problem Solving - Part 1 Introduction

Sometimes, if you do not begin at the end, you end at the beginning (Ginat, 2005).

Backward thinking in problem solving can be observed frequently in my life and design career, either happening to me or some one else. I have learnt many lessons about it and want to write about it here.

Following are the definitions of forward and backward thinking (terms thinking and working can be used interchangably).

Working forwards is the way of approaching a problem by starting from initial state and advancing forwards, towards the goal state (Polya, 1957 in Ginat, 2005).

Working backwards is the way of advancing backwards from the goal state (Polya, 1957 in Ginat, 2005).

The goal state can be defined by you, and presence sometimes is not fully known as you thought.

It is easy to see why people often choose the forward option: from Initial state to Goal state. However, the reverse might be more logical and often optimal. People tend to think they understand what they have at the presence and constrain themselves to that understanding when planning for the future. Without defining the desired goal, it’s easy to limit yourself and miss out ambitious targets.

When designing a new product, people start from it being a sucess and work backward what they need to achieve to reach that goal. However, other people prefer to look at the past performance and current state of product and work forward what success they might reach. We will compare these two ways in the subsequent articles of this series. You have used backward thinking before

Though fancy as it sounds, backward thinking is what you actually have used since you were a kid. Remember the last time you lied to someone? For example, when you didn’t come on time. You started with not coming on time, went backwards to seek excuse and settled with reason that there was a traffic jam. Your friend might have asked why she shared the same route and didn’t see any traffic jam. You started with the fact that she didn’t see the jam and thought backward to seek another explanation.

Math is one of the first domains where we learn how to work backward. Let’s consider two following math problems:

Problem 1: A car park has a maximum capacity of 75 cars. What is 2/3 of its capacity?

Problem 2: A car park currently has 50 cars parking. We know that this is only 2/3 of its maximm capacity. What is its maximum capacity then?

In the first problem, 75 cars is the known initial state, the relation to goal state is operation with 2/3. The solution is times 75 and 2/3, which is working forward.

In the 2nd probem, 50 cars is the known goal state, the relation to maxium capacity (unknown initial state) is 2/3. The solution is to know 1/3 of maximum capacity first and then derive the whole, which is working backward.

Why I care about Backward Thinking?

As more and more people, including me, say we are problem solvers, I reflected and recognized gaps in my knowledge and skills required to solve problems, especially in UX design. I decided to learn problem solving from foundation. I started with which concepts and framework people use to solve problems in case studies. The next step is to undersand those concepts and frameworks. Problems requiring backward thinking like problem 2 mentioned earlier represent many case studies I have read.

After knowing the concepts of backward and forward thinking, I reflected on what happened in my past projects and notice that I used both methods under different circumstances.

In the next article, I’m going to talk about starting project with objectives, instead of problems. Meanwhile, you can start looking at how you solve daily problems from perspective of working forward and backward. Prepare to be surprised!

Ginat, David. “The suitable way is backwards, but they work forward.” Journal of Computers in Mathematics and Science Teaching, vol. 24, no. 1, 2005.

Start with Objectives, not Problems

We are usually given problems to solve, which might make understanding problem further appealing as the immediate step. What we want to start any project with should be objectives.

First meetup in 2016

My first meetup in 2016 is User Experience Singapore (UXSG) no.21