examples of working backwards problem solving

Fun teaching resources & tips to help you teach math with confidence

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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Problem solving. work backwards.

A good mathematics problem will not have an obvious solution. We must consider what information we have been given and what we already know, and how these things may relate to the problem. If we are systematic in our thinking this will often lead us to the solution.

If we don’t seem to be making progress, we can:

• Read the problem again;
• Reread the information given and focus on key information that could be useful;
• Use what we already now to reduce the scope of the problem;
• Record work done carefully, so it is easy to retrace steps and to verify or change the method.

Examples of working backwards to tackle a problem

First we will read the two examples and have a quick think about them and then we will look at how working backwards can help us with each one:

Angle Problem

The following diagram shows an isosceles triangle and a square drawn on a straight line. Find the size of angle A:

Fraction Shaded Problem

The following diagram is of a square with four semicircles drawn inside. What faction of the square is shaded?

Worked Solutions

We can start by adding in the angles that we know about in the square, marking each of them as 90 degrees. Then we can think about the isosceles triangle and the properties of it that we know about. Using this we can calculate that each of the base angles in this triangle must be 70 degrees. Once we have marked each of these angles on, it should be clearer to see what angle A is.

We don’t have a formula for finding this specific area and we don’t know any measurements, so we will have to use letters to represent some of the unknown lengths.

Let’s start by looking at one semicircle, the radius of which we will call r:

Using the formula for the area of a circle, we know that the area of this semicircle is ½πr 2 .

But we don’t want the whole semicircle. We just want the petals. The “white” shape that we need to exclude has a strange shape, but we can tackle the problem instead by excluding a triangle within the semi-circle, which will leave us with “half-petals” from which we can easily find the area of the petals:

Because we have already named the radius of the semicircle as “r”, we can find the area of this “white” triangle, because it has base length 2r and height r, so its area is ½ x 2r x r = r 2 .

So the two red half petals on the diagram above have a combined area or ½πr 2 – r 2 . There are four times these in the whole diagram, so the total area shaded is 4(½πr 2 – r 2 ).

The area of the rectangle is 2r x 2r = 4r 2 .

So the fraction we want is 4(½πr 2 – r 2 ) / 4r 2 . This simplifies to five ½π – 1, which is about 0.57.

The following questions can be solved in different ways. The worked solutions provided afterwards are based on the ways suggested above.

29 Questions of increasing difficulty

Worked Solutions to Questions

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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.

Working Backwards: A Strategic Approach to Productivity

This guide will walk you through the essential elements of using working backwards - the productivity method to keep your team productive and engaged.

In today's fast-paced world, the concept of working backwards has gained significant attention as a strategic approach to productivity. In this comprehensive guide, we will explore the origin, utility, methods, as well as the pros and cons of working backwards, and provide a detailed step-by-step guide for implementing this approach effectively. Additionally, actionable tips, do's and dont's, and frequently asked questions will be addressed to ensure a thorough understanding of this concept. Whether you are an individual striving for personal efficiency, a team leader looking to streamline processes, or an organization aiming for innovation and growth, the principles of working backwards can be a valuable asset. Let's delve into this powerful methodology and uncover how it can revolutionize your approach to problem-solving, planning, and goal attainment.

What is Working Backwards in the Context of Productivity?

In the realm of productivity, the concept of working backwards entails beginning with the end goal in mind and then structuring the steps required to achieve it. This approach involves envisioning the desired outcome and systematically plotting the reverse steps necessary to reach that outcome. It underscores the significance of clear, well-defined objectives as a precursor to formulating the path leading up to them.

By utilizing this methodology, individuals and organizations can align their efforts, resources, and strategies in a manner that is directly linked to achieving the intended results. This approach is particularly pertinent in scenarios where the ultimate goal is clearly defined, but the means to attainment are open to exploration and refinement. Working backwards beckons a shift in perspective, cultivating a mindset that is oriented towards thoroughly analyzing and understanding the end goal before embarking on the journey towards it.

This novel approach to productivity has garnered attention not only in corporate settings but also in personal development, project management, and various aspects of problem-solving.

Origin of Working Backwards

The concept of working backwards finds its roots in diverse domains, particularly in problem-solving methodologies. It has been a fundamental aspect of various innovation and strategic planning processes across different industries. The origin of this approach can be traced back to the techniques employed in fields such as engineering, design thinking, Agile project management, and software development.

In addition, the methodology has been prominently featured in the operational strategies of renowned companies, notably exemplified by its integration into Amazon's product development and strategic decision-making processes. Amazon's adoption of the working backwards approach has propelled its innovative endeavors, playing a pivotal role in the company's success and global impact.

The concept's evolution as a foundational principle in strategic planning and execution underscores its versatile applicability and enduring relevance in diverse contexts.

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Who is Working Backwards For?

Working backwards is a versatile methodology that caters to a broad spectrum of individuals, teams, and organizations seeking to elevate their productivity, streamline their operations, and drive towards well-defined objectives. This approach is particularly beneficial for:

Innovators and Entrepreneurs : Individuals seeking to introduce groundbreaking products or pioneering solutions can benefit from the structured approach offered by working backwards, aiding in the comprehensive delineation of their vision.

Project Managers and Teams : Project managers and teams can leverage this approach to meticulously plan and execute projects, ensuring that every step aligns with the overall project goals.

Strategic Planners and Decision-makers : Professionals involved in strategic planning and decision-making can utilize this methodology to set clear, achievable targets and steer their organizations towards success.

Individuals Pursuing Personal Goals : The process of working backwards can be applied by individuals striving to achieve personal milestones, whether in terms of career progression, skill enhancement, or lifestyle transformations.

The widespread applicability of this approach underscores its relevance across various domains and its potential to drive impactful outcomes.

What are the Pros and Cons of Working Backwards?

Pros of working backwards.

The working backwards approach offers a multitude of benefits, making it an attractive productivity strategy for individuals and organizations alike. Some of the key advantages include:

Clarity and Precision : By commencing with a definitive end goal and working backwards, clarity and precision are fostered throughout the planning and execution phases, ensuring a clear trajectory towards the intended outcome.

Efficient Resource Allocation : This method enables the efficient allocation of resources by aligning them with the specific requirements and milestones identified during the working backwards process.

Risk Mitigation : Working backwards allows for a comprehensive analysis of potential pitfalls and challenges, thereby facilitating proactive risk mitigation strategies as part of the planning process.

Enhanced Innovation : The systematic approach of working backwards nurtures an environment conducive to innovation and unconventional thinking, often leading to breakthrough solutions and novel approaches.

Adaptability and Flexibility : The iterative nature of working backwards allows for adaptability and flexibility, empowering individuals and teams to adjust their strategies and tactics in response to evolving circumstances.

Cons of Working Backwards

While the working backwards methodology offers substantial advantages, it is imperative to consider potential drawbacks, including:

Complexity in Implementation : The detailed nature of this approach may result in a more intricate planning and execution process, potentially demanding additional time and resources.

Potential Over-analysis : There is a risk of over-analyzing the reverse steps, possibly leading to delays in the commencement of actions and initiatives.

Dynamic Environment Management : Adapting the planned steps to accommodate unexpected changes in a dynamic environment may necessitate ongoing evaluation and adjustment.

Striking a balance between the benefits and drawbacks is crucial for effectively implementing the working backwards methodology.

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How to Get Started with Working Backwards

Embarking on the journey of working backwards involves a deliberate and systematic approach, encompassing several key phases. The initial steps include defining the desired outcome, articulating the reverse steps, and committing to iterative refinement. Let's explore these fundamental stages in greater detail.

Step-by-Step Guide for Working Backwards

Step 1: define the end goal.

The primary phase of working backwards involves clearly defining the end goal or desired outcome. At this stage, it's imperative to articulate the specific objectives and results that are to be achieved. This pivotal step sets the foundation for the subsequent phases, guiding the formation of the reverse action plan.

Step 2: Identify Key Milestones

Once the end goal is defined, the next step entails identifying the key milestones and intermediate objectives that collectively lead to the attainment of the ultimate target. This stage involves delineating the critical stages and achievements that mark the progression towards the end result.

Step 3: Outline the Reverse Action Plan

With the end goal and milestones established, the reverse action plan is formulated, detailing the steps required to reach each milestone and, subsequently, the final objective. This involves structuring the sequence of actions in a reverse order, commencing from the last step and progressing towards the initial phase.

Step 4: Iterative Refinement

The iterative refinement stage involves continuous evaluation and refinement of the action plan based on feedback, insights, and evolving circumstances. This iterative process ensures that the plan remains adaptable and responsive to changes.

By systematically traversing through these sequential steps, individuals and teams can effectively embrace the working backwards approach, infusing precision and purpose into their endeavors.

Actionable Tips for Working Backwards

Incorporating working backwards into your productivity arsenal can be enhanced by integrating the following actionable tips:

Embrace Iterative Thinking : Cultivate a mindset that welcomes iterative thinking and planning, facilitating continuous improvement and adaptability.

Leverage Feedback Loops : Establish feedback loops within the planning process to solicit insights and perspectives that can drive refinement and enhancement.

Foster Open Communication : Encourage open communication and collaboration among team members to cultivate a holistic understanding of the working backwards methodology.

Maintain a Clear Vision : Uphold a clear and articulate vision of the end goal to channel efforts and resources effectively throughout the reverse planning process.

Do's and Dont's

Below is a concise representation of the essential do's and dont's to consider when embracing the concept of working backwards:

Do's	Dont's
Clearly define the end goal and outcomes	Avoid becoming excessively rigid in adhering to the predefined reverse plan
Establish clear milestones and objectives	Do not overlook opportunities for flexibility and adaptation throughout the implementation
Foster a collaborative and iterative approach	Refrain from underestimating the value of comprehensive analysis and planning
Encourage open communication and feedback loops	Avoid overcomplicating the reverse planning process by introducing unnecessary intricacies

In conclusion, working backwards offers a powerful and systematic approach to aligning efforts and resources with the attainment of well-defined objectives. By crystallizing the end goal, mapping out the reverse steps, and embracing iterative refinement, individuals and organizations can navigate their journeys with purpose and precision. The methodology's versatility and adaptability render it a valuable asset in diverse domains, empowering innovators, project managers, strategic planners, and individuals pursuing personal goals. Embracing the principles of working backwards can enrich problem-solving, strategic planning, and goal attainment endeavors, charting a course towards impactful and enduring success.

1. What are the primary industries where working backwards is commonly applied?

The working backwards methodology finds extensive application in domains such as product development, strategic planning, project management, and innovation-driven industries.

2. How does working backwards foster innovation and unconventional thinking?

By encouraging a structured yet flexible approach, working backwards provides a conducive environment for nurturing innovative thinking, enabling individuals and teams to explore unconventional solutions and approaches.

3. Can working backwards be applied to personal goal setting?

Absolutely. The methodology can be leveraged by individuals pursuing personal goals, as it allows for a systematic approach to defining objectives and formulating the steps necessary to achieve them.

4. How does iterative refinement contribute to the effectiveness of working backwards?

Iterative refinement ensures that the reverse action plan remains adaptable and responsive to changes, thereby enhancing its relevance and efficacy in dynamic environments.

5. What pivotal role does clear communication play in the success of the working backwards approach?

Clear and open communication conduces to fostering a shared understanding and commitment to the predefined goals and action steps, catalyzing the collective efforts towards success.

By integrating these guidelines and insights into your projects and endeavors, you can harness the power of working backwards to catalyze your journey towards triumph and innovation.

Remember, success is not achieved by chance, but through deliberate strategy and purposeful execution. Embrace the principles of working backwards, and unlock the potential for transformative achievements and enduring excellence.

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Chapter 3: working backwards.

The very name of this strategy sounds confusing to most people. It is a very unnatural way of doing things. When most of us went to school, we were taught to solve mathematical problems in a direct, straightforward manner. And yet, working backwards is the way many real-life problems are often resolved. As a simple example, suppose you had to pick up your child from football practice at exactly 5:00 p.m. At what time should you leave? Well, let us say it take 30 minutes to get to the ballpark. We would better leave a 5-minute safety valve. Okay, then we need to leave 35 minutes earlier, or not later than 4:25 p.m. Without even thinking about it, we were working backwards! Of course, this is a very simple example of this strategy…

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Working backward to solve problems - maurice ashley.

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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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