blake education problem solving working backwards

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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<strong>Problem</strong> <strong>Solving</strong><br />

<strong>Working</strong> <strong>Backwards</strong><br />

Sharon Shapiro Upper Primary<br />

THE PROBLEM<br />

SOLVING PROCESS<br />

It is important that students follow a logical and<br />

systematic approach to their problem solving. Following<br />

these four steps will enable students to tackle<br />

problems in a structured and meaningful way.<br />

STEP 1: UNDERSTANDING<br />

❖ Encourage students to read the problem<br />

carefully a number of times until they fully<br />

understand what is wanted.They may need to<br />

discuss the problem with someone else or<br />

rewrite it in their own words.<br />

❖ Students should ask internal questions such as,<br />

what is the problem asking me to do, what<br />

information is relevant and necessary for solving<br />

the problem.<br />

❖ They should underline any unfamiliar words and<br />

find out their meanings.<br />

❖ They should select the information they know<br />

and decide what is unknown or needs to be<br />

discovered.They should see if there is any<br />

unnecessary information.<br />

❖ A sketch of the problem often helps their<br />

understanding.<br />

STEP 2: STUDENTS SHOULD<br />

DECIDE ON A STRATEGY OR PLAN<br />

Students should decide how they will solve the<br />

problem by thinking about the different strategies<br />

that could be used.They could try to make<br />

predictions, or guesses, about the problem. Often<br />

these guesses result in generalisations which help<br />

to solve problems. Students should be discouraged<br />

from making wild guesses but they should be<br />

encouraged to take risks.They should always think<br />

in terms of how this problem relates to other<br />

problems that they have solved.They should keep<br />

a record of the strategies they have tried so that<br />

they don’t repeat them.<br />

Some possible strategies include:<br />

❖ Drawing a sketch, graph or table.<br />

❖ Acting out situations, or using concrete<br />

materials.<br />

❖ Organising a list.<br />

❖ Identifying a pattern and extending it.<br />

❖ Guessing and checking.<br />

❖ <strong>Working</strong> backwards.<br />

❖ Using simpler numbers to solve the problem,<br />

then applying the same methodology to the<br />

real problem.<br />

❖ Writing a number sentence.<br />

❖ Using logic and clues.<br />

❖ Breaking the problem into smaller parts.<br />

STEP 3: SOLVING THE PROBLEM<br />

❖ Students should write down their ideas as they<br />

work so they don’t forget how they approached<br />

❖ Their approach should be systematic.<br />

❖ If stuck, students should reread the problem<br />

and rethink their strategies.<br />

❖ Students should be given the opportunity to<br />

orally demonstrate or explain how they reached<br />

an answer.<br />

STEP 4: REFLECT<br />

❖ Students should consider if their answer makes<br />

sense and if it has answered what was asked.<br />

❖ Students should draw and write down their<br />

thinking processes, estimations and approach, as<br />

this gives them time to reflect on their<br />

practices.When they have an answer they<br />

should explain the process to someone else.<br />

❖ Students should ask themselves ‘what if’ to link<br />

this problem to another.This will take their<br />

exploration to a deeper level and encourage<br />

their use of logical thought processes.<br />

❖ Students should consider if it is possible to do<br />

the problem in a simpler way.<br />

© <strong>Blake</strong> <strong>Education</strong>—<strong>Problem</strong> <strong>Solving</strong>: <strong>Working</strong> <strong>Backwards</strong><br />

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<strong>Problem</strong> <strong>Solving</strong> <strong>Working</strong> <strong>Backwards</strong> Sharon Shapiro Upper Primary THE PROBLEM SOLVING PROCESS It is important that students follow a logical and systematic approach to their problem solving. Following these four steps will enable students to tackle problems in a structured and meaningful way. STEP 1: UNDERSTANDING THE PROBLEM ❖ Encourage students to read the problem carefully a number of times until they fully understand what is wanted.They may need to discuss the problem with someone else or rewrite it in their own words. ❖ Students should ask internal questions such as, what is the problem asking me to do, what information is relevant and necessary for solving the problem. ❖ They should underline any unfamiliar words and find out their meanings. ❖ They should select the information they know and decide what is unknown or needs to be discovered.They should see if there is any unnecessary information. ❖ A sketch of the problem often helps their understanding. STEP 2: STUDENTS SHOULD DECIDE ON A STRATEGY OR PLAN Students should decide how they will solve the problem by thinking about the different strategies that could be used.They could try to make predictions, or guesses, about the problem. Often these guesses result in generalisations which help to solve problems. Students should be discouraged from making wild guesses but they should be encouraged to take risks.They should always think in terms of how this problem relates to other problems that they have solved.They should keep a record of the strategies they have tried so that they don’t repeat them. Some possible strategies include: ❖ Drawing a sketch, graph or table. ❖ Acting out situations, or using concrete materials. ❖ Organising a list. ❖ Identifying a pattern and extending it. ❖ Guessing and checking. ❖ <strong>Working</strong> backwards. ❖ Using simpler numbers to solve the problem, then applying the same methodology to the real problem. ❖ Writing a number sentence. ❖ Using logic and clues. ❖ Breaking the problem into smaller parts. STEP 3: SOLVING THE PROBLEM ❖ Students should write down their ideas as they work so they don’t forget how they approached the problem. ❖ Their approach should be systematic. ❖ If stuck, students should reread the problem and rethink their strategies. ❖ Students should be given the opportunity to orally demonstrate or explain how they reached an answer. STEP 4: REFLECT ❖ Students should consider if their answer makes sense and if it has answered what was asked. ❖ Students should draw and write down their thinking processes, estimations and approach, as this gives them time to reflect on their practices.When they have an answer they should explain the process to someone else. ❖ Students should ask themselves ‘what if’ to link this problem to another.This will take their exploration to a deeper level and encourage their use of logical thought processes. ❖ Students should consider if it is possible to do the problem in a simpler way. © <strong>Blake</strong> <strong>Education</strong>—<strong>Problem</strong> <strong>Solving</strong>: <strong>Working</strong> <strong>Backwards</strong> 1

Teaching Notes <strong>Working</strong> <strong>Backwards</strong> The strategy of working backwards is used to solve problems that include a number of linked factors or events, where some of the information has not been provided, usually at the beginning of the problem. To solve these problems it is usually necessary to start with the answer and work methodically backwards to fill in the missing information. This strategy is extremely useful in dealing with a situation or a sequence of events.The events occur one after the other and each stage, or piece of information, is affected by what comes next. Students begin at the end, with the final action, and work through the process in reverse order to establish what happened in the original situation. In order to use the strategy of working backwards effectively, students will need to develop the following skills and understanding. USING THE OPPOSITE OPERATION WHEN WORKING BACKWARDS When you are solving a problem by starting at the end and working backwards, any mathematical operations you come across will have to be reversed. This means that if the problem requires you to add something, then when working backwards you must subtract it, or if multiplying when working forwards, you must divide when working backwards. So if the problem the correct way round is – ÷ 8 x 2 = 14 then backwards it will be – Or ? 14 ÷ 2 x 8 = 56 Jack is 35 years younger than Karen. Frank is half of Jack’s age. Jennifer is 17 years older than Frank. If Jennifer is 35 years old, how old is Karen? Jennifer is 35. She is 17 years older than Frank. So using the opposite operation plus becomes minus. So Frank is 35 – 17 = 18 Frank is half Jack’s age so the opposite operation is 18 x 2 = 36 Jack is 35 years younger than Karen so 36 + 35 = 71 Therefore Karen is 71 years old. STARTING WITH THE ANSWER AND WORKING BACKWARDS In a problem where you know the final outcome but don’t know the starting point, beginning at the end of the problem and working backwards is the best way of arriving at a solution. For example, in a dancing competition all the contestants started dancing together.After three minutes half the people were eliminated. During the next ten minutes half of the remaining were eliminated.At the 15 minute mark, half again were eliminated, and at the 20 minute mark, half of those still remaining were eliminated. In the last two minutes one more contestant was eliminated leaving a winner of the competition. How many dancers were there in the beginning? You know that there is one winner and that the number of contestants was halved at certain intervals. Using this information, it is possible to work backwards and find out how many dancers entered the competition. Start with the winner 1 person dancing Last 2 minutes 1+1 = 2 dancers After 20 minutes Double = 4 dancers After 15 minutes Double = 8 dancers After 10 minutes Double = 16 dancers After 3 minutes Double = 32 dancers 32 dancers started. © <strong>Blake</strong> <strong>Education</strong>—<strong>Problem</strong> <strong>Solving</strong>: <strong>Working</strong> <strong>Backwards</strong> 2

• Page 1: Blake’s Topic Bank Problem Solvin
• Page 5 and 6: Teaching Examples Working Backwards
• Page 7 and 8: BLM Working Backwards ★ Understan
• Page 9 and 10: PROBLEM SOLVING TASK CARDS -Working
• Page 11 and 12: PROBLEM SOLVING TASK CARDS -Working
• Page 13 and 14: PROBLEM SOLVING TASK CARDS -Working
• Page 15: Problem 10 Working backwards, Chris

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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:

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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