real world problem solving examples

39 Best Problem-Solving Examples

Chris Drew (PhD)

Dr. Chris Drew is the founder of the Helpful Professor. He holds a PhD in education and has published over 20 articles in scholarly journals. He is the former editor of the Journal of Learning Development in Higher Education. [Image Descriptor: Photo of Chris]

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Problem-solving is a process where you’re tasked with identifying an issue and coming up with the most practical and effective solution.

This indispensable skill is necessary in several aspects of life, from personal relationships to education to business decisions.

Problem-solving aptitude boosts rational thinking, creativity, and the ability to cooperate with others. It’s also considered essential in 21st Century workplaces.

If explaining your problem-solving skills in an interview, remember that the employer is trying to determine your ability to handle difficulties. Focus on explaining exactly how you solve problems, including by introducing your thoughts on some of the following frameworks and how you’ve applied them in the past.

Problem-Solving Examples

1. divergent thinking.

Divergent thinking refers to the process of coming up with multiple different answers to a single problem. It’s the opposite of convergent thinking, which would involve coming up with a singular answer .

The benefit of a divergent thinking approach is that it can help us achieve blue skies thinking – it lets us generate several possible solutions that we can then critique and analyze .

In the realm of problem-solving, divergent thinking acts as the initial spark. You’re working to create an array of potential solutions, even those that seem outwardly unrelated or unconventional, to get your brain turning and unlock out-of-the-box ideas.

This process paves the way for the decision-making stage, where the most promising ideas are selected and refined.

Go Deeper: Divervent Thinking Examples

2. Convergent Thinking

Next comes convergent thinking, the process of narrowing down multiple possibilities to arrive at a single solution.

This involves using your analytical skills to identify the best, most practical, or most economical solution from the pool of ideas that you generated in the divergent thinking stage.

In a way, convergent thinking shapes the “roadmap” to solve a problem after divergent thinking has supplied the “destinations.”

Have a think about which of these problem-solving skills you’re more adept at: divergent or convergent thinking?

Go Deeper: Convergent Thinking Examples

3. Brainstorming

Brainstorming is a group activity designed to generate a multitude of ideas regarding a specific problem. It’s divergent thinking as a group , which helps unlock even more possibilities.

A typical brainstorming session involves uninhibited and spontaneous ideation, encouraging participants to voice any possible solutions, no matter how unconventional they might appear.

It’s important in a brainstorming session to suspend judgment and be as inclusive as possible, allowing all participants to get involved.

By widening the scope of potential solutions, brainstorming allows better problem definition, more creative solutions, and helps to avoid thinking “traps” that might limit your perspective.

Go Deeper: Brainstorming Examples

4. Thinking Outside the Box

The concept of “thinking outside the box” encourages a shift in perspective, urging you to approach problems from an entirely new angle.

Rather than sticking to traditional methods and processes, it involves breaking away from conventional norms to cultivate unique solutions.

In problem-solving, this mindset can bypass established hurdles and bring you to fresh ideas that might otherwise remain undiscovered.

Think of it as going off the beaten track when regular routes present roadblocks to effective resolution.

5. Case Study Analysis

Analyzing case studies involves a detailed examination of real-life situations that bear relevance to the current problem at hand.

For example, if you’re facing a problem, you could go to another environment that has faced a similar problem and examine how they solved it. You’d then bring the insights from that case study back to your own problem.

This approach provides a practical backdrop against which theories and assumptions can be tested, offering valuable insights into how similar problems have been approached and resolved in the past.

See a Broader Range of Analysis Examples Here

6. Action Research

Action research involves a repetitive process of identifying a problem, formulating a plan to address it, implementing the plan, and then analyzing the results. It’s common in educational research contexts.

The objective is to promote continuous learning and improvement through reflection and action. You conduct research into your problem, attempt to apply a solution, then assess how well the solution worked. This becomes an iterative process of continual improvement over time.

For problem-solving, this method offers a way to test solutions in real-time and allows for changes and refinements along the way, based on feedback or observed outcomes. It’s a form of active problem-solving that integrates lessons learned into the next cycle of action.

Go Deeper: Action Research Examples

7. Information Gathering

Fundamental to solving any problem is the process of information gathering.

This involves collecting relevant data , facts, and details about the issue at hand, significantly aiding in the understanding and conceptualization of the problem.

In problem-solving, information gathering underpins every decision you make.

This process ensures your actions are based on concrete information and evidence, allowing for an informed approach to tackle the problem effectively.

8. Seeking Advice

Seeking advice implies turning to knowledgeable and experienced individuals or entities to gain insights on problem-solving.

It could include mentors, industry experts, peers, or even specialized literature.

The value in this process lies in leveraging different perspectives and proven strategies when dealing with a problem. Moreover, it aids you in avoiding pitfalls, saving time, and learning from others’ experiences.

9. Creative Thinking

Creative thinking refers to the ability to perceive a problem in a new way, identify unconventional patterns, or produce original solutions.

It encourages innovation and uniqueness, often leading to the most effective results.

When applied to problem-solving, creative thinking can help you break free from traditional constraints, ideal for potentially complex or unusual problems.

Go Deeper: Creative Thinking Examples

10. Conflict Resolution

Conflict resolution is a strategy developed to resolve disagreements and arguments, often involving communication, negotiation, and compromise.

When employed as a problem-solving technique, it can diffuse tension, clear bottlenecks, and create a collaborative environment.

Effective conflict resolution ensures that differing views or disagreements do not become roadblocks in the process of problem-solving.

Go Deeper: Conflict Resolution Examples

11. Addressing Bottlenecks

Bottlenecks refer to obstacles or hindrances that slow down or even halt a process.

In problem-solving, addressing bottlenecks involves identifying these impediments and finding ways to eliminate them.

This effort not only smooths the path to resolution but also enhances the overall efficiency of the problem-solving process.

For example, if your workflow is not working well, you’d go to the bottleneck – that one point that is most time consuming – and focus on that. Once you ‘break’ this bottleneck, the entire process will run more smoothly.

12. Market Research

Market research involves gathering and analyzing information about target markets, consumers, and competitors.

In sales and marketing, this is one of the most effective problem-solving methods. The research collected from your market (e.g. from consumer surveys) generates data that can help identify market trends, customer preferences, and competitor strategies.

In this sense, it allows a company to make informed decisions, solve existing problems, and even predict and prevent future ones.

13. Root Cause Analysis

Root cause analysis is a method used to identify the origin or the fundamental reason for a problem.

Once the root cause is determined, you can implement corrective actions to prevent the problem from recurring.

As a problem-solving procedure, root cause analysis helps you to tackle the problem at its source, rather than dealing with its surface symptoms.

Go Deeper: Root Cause Analysis Examples

14. Mind Mapping

Mind mapping is a visual tool used to structure information, helping you better analyze, comprehend and generate new ideas.

By laying out your thoughts visually, it can lead you to solutions that might not have been apparent with linear thinking.

In problem-solving, mind mapping helps in organizing ideas and identifying connections between them, providing a holistic view of the situation and potential solutions.

15. Trial and Error

The trial and error method involves attempting various solutions until you find one that resolves the problem.

It’s an empirical technique that relies on practical actions instead of theories or rules.

In the context of problem-solving, trial and error allows you the flexibility to test different strategies in real situations, gaining insights about what works and what doesn’t.

16. SWOT Analysis

SWOT is an acronym standing for Strengths, Weaknesses, Opportunities, and Threats.

It’s an analytic framework used to evaluate these aspects in relation to a particular objective or problem.

In problem-solving, SWOT Analysis helps you to identify favorable and unfavorable internal and external factors. It helps to craft strategies that make best use of your strengths and opportunities, whilst addressing weaknesses and threats.

Go Deeper: SWOT Analysis Examples

17. Scenario Planning

Scenario planning is a strategic planning method used to make flexible long-term plans.

It involves imagining, and then planning for, multiple likely future scenarios.

By forecasting various directions a problem could take, scenario planning helps manage uncertainty and is an effective tool for problem-solving in volatile conditions.

18. Six Thinking Hats

The Six Thinking Hats is a concept devised by Edward de Bono that proposes six different directions or modes of thinking, symbolized by six different hat colors.

Each hat signifies a different perspective, encouraging you to switch ‘thinking modes’ as you switch hats. This method can help remove bias and broaden perspectives when dealing with a problem.

19. Decision Matrix Analysis

Decision Matrix Analysis is a technique that allows you to weigh different factors when faced with several possible solutions.

After listing down the options and determining the factors of importance, each option is scored based on each factor.

Revealing a clear winner that both serves your objectives and reflects your values, Decision Matrix Analysis grounds your problem-solving process in objectivity and comprehensiveness.

20. Pareto Analysis

Also known as the 80/20 rule, Pareto Analysis is a decision-making technique.

It’s based on the principle that 80% of problems are typically caused by 20% of the causes, making it a handy tool for identifying the most significant issues in a situation.

Using this analysis, you’re likely to direct your problem-solving efforts more effectively, tackling the root causes producing most of the problem’s impact.

21. Critical Thinking

Critical thinking refers to the ability to analyze facts to form a judgment objectively.

It involves logical, disciplined thinking that is clear, rational, open-minded, and informed by evidence.

For problem-solving, critical thinking helps evaluate options and decide the most effective solution. It ensures your decisions are grounded in reason and facts, and not biased or irrational assumptions.

Go Deeper: Critical Thinking Examples

22. Hypothesis Testing

Hypothesis testing usually involves formulating a claim, testing it against actual data, and deciding whether to accept or reject the claim based on the results.

In problem-solving, hypotheses often represent potential solutions. Hypothesis testing provides verification, giving a statistical basis for decision-making and problem resolution.

Usually, this will require research methods and a scientific approach to see whether the hypothesis stands up or not.

Go Deeper: Types of Hypothesis Testing

23. Cost-Benefit Analysis

A cost-benefit analysis (CBA) is a systematic process of weighing the pros and cons of different solutions in terms of their potential costs and benefits.

It allows you to measure the positive effects against the negatives and informs your problem-solving strategy.

By using CBA, you can identify which solution offers the greatest benefit for the least cost, significantly improving efficacy and efficiency in your problem-solving process.

Go Deeper: Cost-Benefit Analysis Examples

24. Simulation and Modeling

Simulations and models allow you to create a simplified replica of real-world systems to test outcomes under controlled conditions.

In problem-solving, you can broadly understand potential repercussions of different solutions before implementation.

It offers a cost-effective way to predict the impacts of your decisions, minimizing potential risks associated with various solutions.

25. Delphi Method

The Delphi Method is a structured communication technique used to gather expert opinions.

The method involves a group of experts who respond to questionnaires about a problem. The responses are aggregated and shared with the group, and the process repeats until a consensus is reached.

This method of problem solving can provide a diverse range of insights and solutions, shaped by the wisdom of a collective expert group.

26. Cross-functional Team Collaboration

Cross-functional team collaboration involves individuals from different departments or areas of expertise coming together to solve a common problem or achieve a shared goal.

When you bring diverse skills, knowledge, and perspectives to a problem, it can lead to a more comprehensive and innovative solution.

In problem-solving, this promotes communal thinking and ensures that solutions are inclusive and holistic, with various aspects of the problem being addressed.

27. Benchmarking

Benchmarking involves comparing one’s business processes and performance metrics to the best practices from other companies or industries.

In problem-solving, it allows you to identify gaps in your own processes, determine how others have solved similar problems, and apply those solutions that have proven to be successful.

It also allows you to compare yourself to the best (the benchmark) and assess where you’re not as good.

28. Pros-Cons Lists

A pro-con analysis aids in problem-solving by weighing the advantages (pros) and disadvantages (cons) of various possible solutions.

This simple but powerful tool helps in making a balanced, informed decision.

When confronted with a problem, a pro-con analysis can guide you through the decision-making process, ensuring all possible outcomes and implications are scrutinized before arriving at the optimal solution. Thus, it helps to make the problem-solving process both methodical and comprehensive.

29. 5 Whys Analysis

The 5 Whys Analysis involves repeatedly asking the question ‘why’ (around five times) to peel away the layers of an issue and discover the root cause of a problem.

As a problem-solving technique, it enables you to delve into details that you might otherwise overlook and offers a simple, yet powerful, approach to uncover the origin of a problem.

For example, if your task is to find out why a product isn’t selling your first answer might be: “because customers don’t want it”, then you ask why again – “they don’t want it because it doesn’t solve their problem”, then why again – “because the product is missing a certain feature” … and so on, until you get to the root “why”.

30. Gap Analysis

Gap analysis entails comparing current performance with potential or desired performance.

You’re identifying the ‘gaps’, or the differences, between where you are and where you want to be.

In terms of problem-solving, a Gap Analysis can help identify key areas for improvement and design a roadmap of how to get from the current state to the desired one.

31. Design Thinking

Design thinking is a problem-solving approach that involves empathy, experimentation, and iteration.

The process focuses on understanding user needs, challenging assumptions , and redefining problems from a user-centric perspective.

In problem-solving, design thinking uncovers innovative solutions that may not have been initially apparent and ensures the solution is tailored to the needs of those affected by the issue.

32. Analogical Thinking

Analogical thinking involves the transfer of information from a particular subject (the analogue or source) to another particular subject (the target).

In problem-solving, you’re drawing parallels between similar situations and applying the problem-solving techniques used in one situation to the other.

Thus, it allows you to apply proven strategies to new, but related problems.

33. Lateral Thinking

Lateral thinking requires looking at a situation or problem from a unique, sometimes abstract, often non-sequential viewpoint.

Unlike traditional logical thinking methods, lateral thinking encourages you to employ creative and out-of-the-box techniques.

In solving problems, this type of thinking boosts ingenuity and drives innovation, often leading to novel and effective solutions.

Go Deeper: Lateral Thinking Examples

34. Flowcharting

Flowcharting is the process of visually mapping a process or procedure.

This form of diagram can show every step of a system, process, or workflow, enabling an easy tracking of the progress.

As a problem-solving tool, flowcharts help identify bottlenecks or inefficiencies in a process, guiding improved strategies and providing clarity on task ownership and process outcomes.

35. Multivoting

Multivoting, or N/3 voting, is a method where participants reduce a large list of ideas to a prioritized shortlist by casting multiple votes.

This voting system elevates the most preferred options for further consideration and decision-making.

As a problem-solving technique, multivoting allows a group to narrow options and focus on the most promising solutions, ensuring more effective and democratic decision-making.

36. Force Field Analysis

Force Field Analysis is a decision-making technique that identifies the forces for and against change when contemplating a decision.

The ‘forces’ represent the differing factors that can drive or hinder change.

In problem-solving, Force Field Analysis allows you to understand the entirety of the context, favoring a balanced view over a one-sided perspective. A comprehensive view of all the forces at play can lead to better-informed problem-solving decisions.

TRIZ, which stands for “The Theory of Inventive Problem Solving,” is a problem-solving, analysis, and forecasting methodology.

It focuses on finding contradictions inherent in a scenario. Then, you work toward eliminating the contraditions through finding innovative solutions.

So, when you’re tackling a problem, TRIZ provides a disciplined, systematic approach that aims for ideal solutions and not just acceptable ones. Using TRIZ, you can leverage patterns of problem-solving that have proven effective in different cases, pivoting them to solve the problem at hand.

38. A3 Problem Solving

A3 Problem Solving, derived from Lean Management, is a structured method that uses a single sheet of A3-sized paper to document knowledge from a problem-solving process.

Named after the international paper size standard of A3 (or 11-inch by 17-inch paper), it succinctly records all key details of the problem-solving process from problem description to the root cause and corrective actions.

Used in problem-solving, this provides a straightforward and logical structure for addressing the problem, facilitating communication between team members, ensuring all critical details are included, and providing a record of decisions made.

39. Scenario Analysis

Scenario Analysis is all about predicting different possible future events depending upon your decision.

To do this, you look at each course of action and try to identify the most likely outcomes or scenarios down the track if you take that course of action.

This technique helps forecast the impacts of various strategies, playing each out to their (logical or potential) end. It’s a good strategy for project managers who need to keep a firm eye on the horizon at all times.

When solving problems, Scenario Analysis assists in preparing for uncertainties, making sure your solution remains viable, regardless of changes in circumstances.

How to Answer “Demonstrate Problem-Solving Skills” in an Interview

When asked to demonstrate your problem-solving skills in an interview, the STAR method often proves useful. STAR stands for Situation, Task, Action, and Result.

Situation: Begin by describing a specific circumstance or challenge you encountered. Make sure to provide enough detail to allow the interviewer a clear understanding. You should select an event that adequately showcases your problem-solving abilities.

For instance, “In my previous role as a project manager, we faced a significant issue when our key supplier abruptly went out of business.”

Task: Explain what your responsibilities were in that situation. This serves to provide context, allowing the interviewer to understand your role and the expectations placed upon you.

For instance, “It was my task to ensure the project remained on track despite this setback. Alternative suppliers needed to be found without sacrificing quality or significantly increasing costs.”

Action: Describe the steps you took to manage the problem. Highlight your problem-solving process. Mention any creative approaches or techniques that you used.

For instance, “I conducted thorough research to identify potential new suppliers. After creating a shortlist, I initiated contact, negotiated terms, assessed samples for quality and made a selection. I also worked closely with the team to re-adjust the project timeline.”

Result: Share the outcomes of your actions. How did the situation end? Did your actions lead to success? It’s particularly effective if you can quantify these results.

For instance, “As a result of my active problem solving, we were able to secure a new supplier whose costs were actually 10% cheaper and whose quality was comparable. We adjusted the project plan and managed to complete the project just two weeks later than originally planned, despite the major vendor setback.”

Remember, when you’re explaining your problem-solving skills to an interviewer, what they’re really interested in is your approach to handling difficulties, your creativity and persistence in seeking a resolution, and your ability to carry your solution through to fruition. Tailoring your story to highlight these aspects will help exemplify your problem-solving prowess.

Go Deeper: STAR Interview Method Examples

Benefits of Problem-Solving

Problem-solving is beneficial for the following reasons (among others):

• It can help you to overcome challenges, roadblocks, and bottlenecks in your life.
• It can save a company money.
• It can help you to achieve clarity in your thinking.
• It can make procedures more efficient and save time.
• It can strengthen your decision-making capacities.
• It can lead to better risk management.

Whether for a job interview or school, problem-solving helps you to become a better thinking, solve your problems more effectively, and achieve your goals. Build up your problem-solving frameworks (I presented over 40 in this piece for you!) and work on applying them in real-life situations.

• Chris Drew (PhD) https://helpfulprofessor.com/author/chris-drew-phd-2/ 15 Montessori Values for your Classroom
• Chris Drew (PhD) https://helpfulprofessor.com/author/chris-drew-phd-2/ 24 Montessori Classroom Design Ideas
• Chris Drew (PhD) https://helpfulprofessor.com/author/chris-drew-phd-2/ 21 Cozy Classroom Reading Corners
• Chris Drew (PhD) https://helpfulprofessor.com/author/chris-drew-phd-2/ 25 Math Center Ideas for Teachers

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32 Examples of Real-World Math Problems

• • Published: April 23, 2024
• • Last update: June 14, 2024
• • Grades: All grades

Introduction

8th grade algebra problem:

Farmer Alfred has three times as many chickens as cows. In total, there are 60 legs in the barn. How many cows does Farmer Alfred have? [1]

Does this sound like a real-world math problem to you? We’ve got chickens, cows, and Farmer Alfred – it’s a scenario straight out of everyday life, isn’t it?

But before you answer, let me ask you something: If you wanted to figure out the number of cows, would you:

• Count their legs, or
• Simply count their heads ( or even just ask Farmer Alfred, “Hey Alfred, how many cows do you have?” )

Chances are, most people would go with option B.

So, why do our math books contain so many “real-world math problems” like the one above?

In this article, we’ll dive into what truly makes a math problem a real-world challenge.

Understanding 'Problems' in Daily Life and Mathematics

The word “problem” carries different meanings in everyday life and in the realm of mathematics, which can sometimes lead to confusion. In our daily lives, when we say “ I have a problem ,” we typically mean that something undesirable has occurred – something challenging to resolve or with potential negative consequences.

For instance:

• “I have a problem because I’ve lost my wallet.”
• “I have a problem because I forgot my keys at home, and I won’t be able to get into the house when I return from school.”
• “I have a problem because I was sick and missed a few weeks of school, which means I’ll likely fail my math test.”

These are examples of everyday problems we encounter. However, once the problem is solved, it often ceases to be a problem:

• “I don’t have a problem getting into my house anymore because my mom gave me her keys.”

In mathematics, the term “problem” takes on a different meaning. According to the Cambridge dictionary [2], it’s defined as “ a question in mathematics that needs an answer. “

Here are a few examples:

• If x + 2 = 4, what is the value of x?
• How do you find the common denominator for fractions 1/3 and 1/4?
• What is the length of the hypotenuse in a right triangle if two legs are 3 and 4 feet long?

These are all examples of math problems.

It’s important to note that in mathematics, a problem remains a problem even after it’s solved. Math problems are universal, regardless of who encounters them [3]. For instance, both John and Emma could face the same math problem, such as “ If x + 2 = 4, then what is x? ” After they solve it, it still remains a math problem that a teacher could give to somebody else.

Understanding Real-World Math Problems

So, what exactly is a “real-world math problem”? We’ve established that in our daily lives, we refer to a situation as a “problem” when it could lead to unpleasant consequences. In mathematics, a “problem” refers to a mathematical question that requires a mathematical solution.

With that in mind, we can define a real-world math problem as:

A situation that could have negative consequences in real life and that requires a mathematical solution (i.e. mathematical solution is preferred over other solutions).

Consider this example:

Could miscalculating the flour amount lead to less-than-ideal results? Absolutely. Messing up the flour proportion could result in a less tasty cake – certainly an unpleasant consequence.

Now, let’s explore the methods for solving this. While traditional methods, like visually dividing the flour or measuring multiple times, have their place, they may not be suitable for larger-scale events, such as catering for a wedding with 250 guests. In such cases, a mathematical solution is not only preferred but also more practical.

By setting up a simple proportion – 1 1/2 cups of flour for 8 people equals x cups for 20 people – we can quickly find the precise amount needed: 3 3/4 cups of flour. In larger events, like the wedding, the mathematical approach provides even more value, yielding a requirement of nearly 47 cups of flour.

This illustrates why a mathematical solution is faster, more accurate, and less error-prone, making it the preferred method. Coupled with the potential negative consequences of inaccuracies, this makes the problem a real-world one, showcasing the practical application of math in everyday scenarios.

Identifying Non-Real-World Math Problems

Let’s revisit the example from the beginning of this article:

Farmer Alfred has three times as many chickens as cows. In total, there are 60 legs in the barn. How many cows does Farmer Alfred have?

3rd grade basic arithmetic problem:

Noah has $56, and Olivia has 8 times less. How much money does Olivia have?

Once more, in our daily routines, do we handle money calculations this way? Why would one prefer to use mathematical solution ($56 : 8 =$7). More often than not, we’d simply ask Olivia how much money she has. While the problem can theoretically be solved mathematically, it’s much more practical, efficient, and reliable to resolve it by directly asking Olivia.

Examples like these are frequently found in math textbooks because they aid in developing mathematical thinking. However, the scenarios they describe are uncommon in real life and fail to explicitly demonstrate the usefulness of math. In essence, they don’t showcase what math can actually be used for. Therefore, there are two crucial aspects of true real-world math problems: they must be commonplace in real life, and they must explicitly illustrate the utility of math.

To summarize:

Word problems that are uncommon in real life and fail to convincingly demonstrate the usefulness of math are not real-world math problems.

Real-World Math Problems Across Professions

Many professions entail encountering real-world math problems on a regular basis. Consider the following examples:

Nurses often need to calculate accurate drug dosage amounts using proportions, a task solved through mathematical methods. Incorrect dosage calculations can pose serious risks to patient health, leading to potentially harmful consequences.

Construction engineers frequently need to determine whether a foundation will be sturdy enough to support a building, employing mathematical solving techniques and specialized formulas. Errors in these calculations can result in structural issues such as cracks in walls due to foundation deformation, leading to undesirable outcomes.

Marketers often rely on statistical analysis to assess the performance of online advertisements, including metrics like click-through rates and the geographical distribution of website visitors. Mathematical analysis guides decision-making in this area. Inaccurate analyses may lead to inefficient allocation of advertisement budgets, resulting in less-than-optimal outcomes.

These examples illustrate real-world math problems encountered in various professions. While there are numerous instances of such problems, they are often overlooked in educational settings. At DARTEF, we aim to bridge this gap by compiling a comprehensive list of real-world math problems, which we’ll explore in the following section.

32 Genuine Real-World Math Problems

In this section, we present 32 authentic real-world math problems from diverse fields such as safety and security, microbiology, architecture, engineering, nanotechnology, archaeology, creativity, and more. Each of these problems meets the criteria we’ve outlined previously. Specifically, a problem can be classified as a real-world math problem if:

• It is commonly encountered in real-life scenarios.
• It has the potential for undesirable consequences.
• A mathematical solution is preferable over alternative methods.
• It effectively illustrates the practical utility of mathematics.

All these problems stem from actual on-the-job situations, showcasing the application of middle and high school math in various professions.

Math Problems in Biology

Real-world math problems in biology often involve performing measurements or making predictions. Mathematics helps us understand various biological phenomena, such as the growth of bacterial populations, the spread of diseases, and even the reconstruction of ancient human appearances. Here are a couple of specific examples:

1. Reconstructing Human Faces Using Parallel and Perpendicular Lines:

Archeologists and forensic specialists often reconstruct human faces based on skeletal remains. They utilize parallel and perpendicular lines to create symmetry lines on the face, aiding in the recovery of facial features and proportions. Our article, “ Parallel and Perpendicular Lines: A Real-Life Example (From Forensics and Archaeology) ,” provides a comprehensive explanation and illustrates how the shape of the nose can be reconstructed using parallel and perpendicular lines, line segments, tangent lines, and symmetry lines.

2. Measuring Bacteria Size Using Circumference and Area Formulas:

Microbiology specialists routinely measure bacteria size to monitor and document their growth rates. Since bacterial shapes often resemble geometric shapes studied in school, mathematical methods such as calculating circumference or area of a circle are convenient for measuring bacteria size. Our article, “ Circumference: A Real-Life Example (from Microbiology) ,” delves into this process in detail.

Math Problems in Construction and Architecture

Real-world math problems are abundant in architecture and construction projects, where mathematics plays a crucial role in ensuring safety, efficiency, and sustainability. Here are some specific case studies that illustrate the application of math in these domains:

3. Calculating Central Angles for Safe Roadways:​

Central angle calculations are a fundamental aspect of roadway engineering, particularly in designing curved roads. Civil engineers use these calculations to determine the degree of road curvature, which significantly impacts road safety and compliance with regulations. Mathematical concepts such as radius, degrees, arc length, and proportions are commonly employed by civil engineers in their daily tasks. Our article, “ How to Find a Central Angle: A Real-Life Example (from Civil Engineering) ,” provides further explanation and demonstrates the calculation of central angles using a real road segment as an example.

4. Designing Efficient Roof Overhang Using Trigonometry

Trigonometry is a powerful tool in architecture and construction, providing a simple yet effective way to calculate the sizes of building elements, including roof overhangs. For example, the tangent function is used to design the optimal length of a roof overhang that blocks the high summer sun while allowing the lower winter sun to enter . Our article, “ Tangent (Trigonometry): A Real-Life Example (From Architecture and Construction) ,” provides a detailed use case with two methods of calculation and also includes a worksheet.

5. Designing Efficient Roofs for Solar Panels Using Angle Geometry:

Mathematics plays a crucial role in architecture, aiding architects and construction engineers in designing energy-efficient building structures. For example, when considering the installation of solar panels on a building’s roof, understanding geometric properties such as adjacent and alternate angles is essential for maximizing energy efficiency . By utilizing mathematical calculations, architects can determine the optimal angle for positioning solar panels and reflectors to capture maximum sunlight. Our article, “ A Real-Life Example of How Angles are Used in Architecture ,” provides explanations and illustrations, including animations.

6. Precision Drilling of Oil Wells Using Trigonometry:

Petroleum engineers rely on trigonometric principles such as sines, cosines, tangents, and right-angle triangles to drill oil wells accurately. Trigonometry enables engineers to calculate precise angles and distances necessary for drilling vertical, inclined, and even horizontal wells. For instance, when drilling at an inclination of 30°, engineers can use trigonometry to calculate the vertical depth corresponding to each foot drilled horizontally. Our article, “ CosX: A Real-Life Example (from Petroleum Engineering) ,” provides examples, drawings of right triangle models, and necessary calculations.

7. Calculating Water Flow Rate Using Composite Figures:

Water supply specialists frequently encounter the task of calculating water flow rates in water canals, which involves determining the area of composite figures representing canal cross-sections . Many canal cross-sections consist of composite shapes, such as rectangles and triangles. By calculating the areas of these individual components and summing them, specialists can determine the total cross-sectional area and subsequently calculate the flow rate of water. Our article, “ Area of Composite Figures: A Real-Life Example (from Water Supply) ,” presents necessary figures, cross-sections, and an example calculation.

Math Problems in Business and Marketing

Mathematics plays a crucial role not only in finance and banking but also in making informed decisions across various aspects of business development, marketing analysis, growth strategies, and more. Here are some real-world math problems commonly encountered in business and marketing:

8. Analyzing Webpage Position in Google Using Polynomials and Polynomial Graphs:

Polynomials and polynomial graphs are essential tools for data analysis. They are useful for a wide variety of people, not only data analysts, but literally everyone who ever uses Excel or Google Sheets. Our article, “ Graphing Polynomials: A Real-World Example (from Data Analysis) ,” describes this and provides an authentic case study. The case study demonstrates how polynomials, polynomial functions of varying degrees, and polynomial graphs are used in internet technologies to analyze the visibility of webpages in Google and other search engines.”

9. Making Informed Business Development Decisions Using Percentages:

In business development, math problems often revolve around analyzing growth and making strategic decisions. Understanding percentages is essential, particularly when launching new products or services. For example, determining whether a startup should target desktop computer, smartphone, or tablet users for an app requires analyzing installation percentages among user groups to gain insights into consumer behavior. Math helps optimize marketing efforts, enhance customer engagement, and drive growth in competitive markets. Check out our article “ A Real-Life Example of Percent Problems in Business ” for a detailed description of this example.

10. Analyzing Customer Preferences Using Polynomials:

Marketing involves not only creative advertising but also thorough analysis of customer preferences and marketing campaigns. Marketing specialists often use simple polynomials for such analysis, as they help analyze multiple aspects of customer behavior simultaneously. For instance, marketers may use polynomials to determine whether low price or service quality is more important to hotel visitors. Smart analysis using polynomials enables businesses to make informed decisions. If you’re interested in learning more, our article “ Polynomials: A Real Life Example (from Marketing) ” provides a real-world example from the hotel industry.

11. Avoiding Statistical Mistakes Using Simpson’s Paradox:

Data gathering and trend analysis are essential in marketing, but a good understanding of mathematical statistics helps avoid intuitive mistakes . For example, consider an advertising campaign targeting Android and iOS users. Initial data may suggest that iOS users are more responsive. However, a careful statistical analysis, as described in our article “ Simpson’s Paradox: A Real-Life Example (from Marketing) ,” may reveal that Android users, particularly tablet users, actually click on ads more frequently. This contradiction highlights the importance of accurate statistical interpretation and the careful use of mathematics in decision-making.

Math Problems in Digital Agriculture

In the modern era, agriculture is becoming increasingly digitalized, with sensors and artificial intelligence playing vital roles in farm management. Mathematics is integral to this digital transformation, aiding in data analysis, weather prediction, soil parameter measurement, irrigation scheduling, and more. Here’s an example of a real-world math problem in digital agriculture:

12. Combatting Pests with Negative Numbers:

Negative numbers are utilized in agriculture to determine the direction of movement, similar to how they’re used on a temperature scale where a plus sign indicates an increase and a minus sign indicates a decrease. In agricultural sensors, plus and minus signs may indicate whether pests are moving towards or away from plants. For instance, a plus sign could signify movement towards plants, while a minus sign indicates movement away from plants. Understanding these directional movements helps farmers combat pests effectively. Our article, “ Negative Numbers: A Real-Life Example (from Agriculture) ,” provides detailed explanations and examples of how negative numbers are applied in agriculture.

Math Problems in DIY Projects

In do-it-yourself (DIY) projects, mathematics plays a crucial role in measurements, calculations, and problem-solving. Whether you’re designing furniture, planning home renovations, crafting handmade gifts, or landscaping your garden, math provides essential tools for precise measurements, material estimations, and budget management. Here’s an example of a real-world math problem in DIY:

13. Checking Construction Parts for Right Angles Using the Pythagorean Theorem:

The converse of the Pythagorean theorem allows you to check whether various elements – such as foundations, corners of rooms, garage walls, or sides of vegetable patches – form right angles. This can be done quickly using Pythagorean triples like 3-4-5 or 6-8-10, or by calculating with square roots. This method ensures the creation of right angles or verifies if an angle is indeed right. Our article, “ Pythagorean Theorem Converse: A Real-Life Example (from DIY) ,” explains this concept and provides an example of how to build a perfect 90° foundation using the converse of the Pythagorean theorem.

Math Problems in Entertainment and Creativity

Mathematics plays a surprisingly significant role in the creative industries, including music composition, visual effects creation, and computer graphic design. Understanding and applying mathematical concepts is essential for producing engaging and attractive creative works. Here’s an example of a real-world math problem in entertainment:

14. Controlling Stage Lamps with Linear Functions:

Linear and quadratic functions are essential components of the daily work of lighting specialists in theaters and event productions. These professionals utilize specialized software and controllers that rely on algorithms based on mathematical functions. Linear functions, in particular, are commonly used to control stage lamps, ensuring precise and coordinated lighting effects during performances. Our article, “ Linear Function: A Real-Life Example (from Entertainment) ,” delves into this topic in detail, complete with animations that illustrate how these functions are applied in practice.

Math Problems in Healthcare

Mathematics plays a crucial role in solving numerous real-world problems in healthcare, ranging from patient care to the design of advanced medical devices. Here are several examples of real-world math problems in healthcare:

15. Calculating Dosage by Converting Time Units:

Nurses frequently convert between hours, minutes, and seconds to accurately administer medications and fluids to patients. For example, if a patient requires 300mL of fluid over 2.5 hours, nurses must convert hours to minutes to calculate the appropriate drip rate for an IV bag. Understanding mathematical conversions and solving proportions are essential skills in nursing. Check out our article, “ Converting Hours to Minutes: A Real-Life Example (from Nursing) ,” for further explanation.

16. Predicting Healthcare Needs Using Quadratic Functions:

Mathematical functions, including quadratic functions, are used to make predictions in public health. These predictions are vital for estimating the need for medical services, such as psychological support following traumatic events. Quadratic functions can model trends in stress symptoms over time, enabling healthcare systems to anticipate and prepare for increased demand. Our article, “ Parabola Equation: A Real-Life Example (from Public Health) ,” provides further insights and examples.

17. Predicting Healthcare Needs Using Piecewise Linear Functions:

Piecewise linear functions are useful for describing real-world trends that cannot be accurately represented by other functions. For instance, if stress symptoms fluctuate irregularly over time, piecewise linear functions can define periods of increased and decreased stress levels. Our article, “ Piecewise Linear Function: A Real-Life Example (From Public Health) ,” offers an example of how these functions are applied.

18. Designing Medical Prostheses Using the Pythagorean Theorem:

The Pythagorean theorem is applied in the design of medical devices , particularly prostheses for traumatic recovery. Components of these devices often resemble right triangles, allowing engineers to calculate movement and placement for optimal patient comfort and stability during rehabilitation. For a detailed explanation and animations, see our article, “ The 3-4-5 Triangle: A Real-Life Example (from Mechatronics) .”

19. Illustrating Disease Survival Rates Using Cartesian Coordinate Plane:

In medicine, Cartesian coordinate planes are utilized for analyzing historical data, statistics, and predictions. They help visualize relationships between independent and dependent variables, such as survival rates and diagnostic timing in diseases like lung cancer. Our article, “ Coordinate Plane: A Real-Life Example (from Medicine) ,” offers a detailed exploration of this concept, including analyses for both non-smoking and smoking patients.

Math Problems in Industry

Mathematics plays a vital role in solving numerous real-world problems across various industries. From designing industrial robots to analyzing production quality and planning logistics, math is indispensable for optimizing processes and improving efficiency. Here are several examples of real-world math problems in industry:

20. Precise Movements of Industrial Robots Using Trigonometry:

Trigonometry is extensively utilized to direct and control the movements of industrial robots. Each section of an industrial robot can be likened to a leg or hypotenuse of a right triangle, allowing trigonometry to precisely control the position of the robot head. For instance, if the robot head needs to move 5 inches to the right, trigonometry enables calculations to ensure each section of the robot moves appropriately to achieve the desired position. For detailed examples and calculations, refer to our article, “ Find the Missing Side: A Real-life Example (from Robotics) .”

21. Measuring Nanoparticle Size via Cube Root Calculation:

Mathematics plays a crucial role in nanotechnology, particularly in measuring the size of nanoparticles. Cube roots are commonly employed to calculate the size of cube-shaped nanoparticles. As nanoparticles are extremely small and challenging to measure directly, methods in nanotechnology determine nanoparticle volume, allowing for the calculation of cube root to determine the size of the nanoparticle cube side. Check out our article, “ Cube Root: A Real-Life Example (from Nanotechnology) ,” for explanations, calculations, and insights into the connection between physics and math.

22. Understanding Human Emotions Using Number Codes for Robots:

Mathematics is employed to measure human emotions effectively, which is essential for building smart robots capable of recognizing and responding to human emotions. Each human emotion can be broken down into smaller features, such as facial expressions, which are assigned mathematical codes to create algorithms for robots to recognize and distinguish emotions. Explore our article, “ Psychology and Math: A Real-Life Example (from Smart Robots) ,” to understand this process and its connection to psychology.

23. Systems of Linear Equations in Self-Driving Cars:

In the automotive industry, mathematics is pivotal for ensuring the safety and efficiency of self-driving cars. Systems of linear equations are used to predict critical moments in road safety, such as when two cars are side by side during an overtaking maneuver. By solving systems of linear equations, self-driving car computers can assess the safety of overtaking situations. Our article, “ Systems of Linear Equations: A Real-Life Example (from Self-Driving Cars) ,” provides a comprehensive explanation, including animated illustrations and interactive simulations, demonstrating how linear functions and equations are synchronized with car motion.

Math Problems in Information Technology (IT)

Mathematics serves as a fundamental tool in the field of information technology (IT), underpinning various aspects of software development, cybersecurity, and technological advancements. Here are some real-world math problems encountered in IT:

24. Selecting the Right Word in Machine Translation Using Mathematical Probability:

Theoretical probability plays a crucial role in AI machine translation systems, such as Google Translate, by aiding in the selection of the most appropriate words. Words often have multiple translations, and theoretical probability helps analyze the frequency of word appearances in texts to propose the most probable translation. Dive deeper into this topic in our article, “ Theoretical Probability: A Real-Life Example (from Artificial Intelligence) .”

25. Creating User-Friendly Music Streaming Websites Using Mathematical Probability:

Probability is utilized in user experience (UX) design to enhance the usability of digital products , including music streaming websites. Recommendation algorithms calculate theoretical probability to suggest songs that users are likely to enjoy, improving their overall experience. Learn more about this application in our article, “ Theoretical Probability: A Real-Life Example (from Digital Design) .”

26. Developing Realistic Computer Games with Vectors:

Mathematics is essential in animating objects and simulating real-world factors in computer games. Vectors enable game developers to incorporate realistic elements like gravity and wind into the gaming environment. By applying vector addition, developers can accurately depict how external forces affect object trajectories, enhancing the gaming experience. Explore this concept further in our article, “ Parallelogram Law of Vector Addition: A Real-Life Example (from Game Development) .”

27. Detecting Malicious Bots in Social Media Using Linear Inequalities:

Mathematical inequalities are valuable tools in cybersecurity for identifying malicious activity on social media platforms. By analyzing behavioral differences between real users and bots—such as friend count, posting frequency, and device usage—cybersecurity experts can develop algorithms to detect and block suspicious accounts. Discover more about this application in our article, “ How to Write Inequalities: A Real-life Example (from Social Media) .”

28. Increasing Computational Efficiency Using Algebraic and Rational Expressions

Algebraic and rational expressions are crucial in computer technology for increasing computational efficiency . Every step in a computer program’s algorithm requires valuable time and energy for execution. This is especially important for programs used in various safety and security systems. Simplifying these expressions helps boost computational performance. Our article, “ Algebraic and Rational Expressions: A Real-Life Example (from Computer Technology) ,” explains this in detail.

Math Problems in Legal Issues

In legal proceedings, mathematics plays a crucial role in analyzing data and making informed decisions. Here’s a real-world math problem encountered in legal work:

29. Proving the Reliability of Technology in Court Using Percentages:

In certain court cases, particularly those involving new technologies like autonomous cars, lawyers may need to employ mathematical methods to justify evidence. Percentage analysis can be utilized to assess the reliability of technology in court. For instance, in cases related to self-driving cars, lawyers may compare the percentage of errors made by autonomous vehicles with those made by human drivers to determine the technology’s safety. Delve deeper into this topic in our article, “ Solving Percent Problems: A Real-Life Example (from Legal) .”

Math Problems in Safety and Security

In the realm of safety and security, mathematics plays a vital role in protecting people, nature, and assets. Real-world math problems in this field can involve reconstructing crime scenes, analyzing evidence, creating effective emergency response plans, and predicting and responding to natural disasters. Here are two examples:

30. Reconstructing a Crime Scene with Inverse Trigonometry:

Forensic specialists rely on mathematical methods, such as inverse trigonometry, to uncover details of crimes long after they’ve occurred. Inverse trigonometric functions like arcsin, arccos, and arctan enable forensic specialists to calculate precise shooting angles or the trajectory of blood drops at crime scenes. Dive into this topic in our article, “ Arctan: A Real-Life Example (from Criminology) .”

31. Responding to Wildfires with Mathematical Variables:

Variables in mathematical language are also prevalent in real-world scenarios like firefighting. For instance, when dealing with wildfires, independent variables like forest type, wind speed and direction, rainfall, and terrain slope influence fire spread speed (dependent variable) . By utilizing specialized formulas incorporating these variables, fire protection specialists can accurately predict fire paths and optimize firefighting efforts. Explore this practical application in our article, “ Dependent and Independent Variables: A Real-Life Example (from Fire Protection) .”

Math Problems in Weather and Climate

In the realm of weather and climate, mathematics is crucial for creating mathematical models of Earth’s atmosphere, making weather forecasts, predicting weather patterns, and assessing long-term climate trends. Here’s an example:

32. Forecasting Thunderstorms with Negative Numbers:

Negative numbers play a significant role in forecasting thunderstorms. The probability of thunderstorms is closely tied to the temperature differences between air masses in the atmosphere. By subtracting negative numbers (representing temperature values) and analyzing the results, meteorologists can forecast the likelihood of thunderstorms based on the temperature differentials. Dive deeper into this concept in our article, “ Subtracting Negative Numbers: A Real-Life Example (from Weather Forecast). “

Conclusions

Many students perceive math as disconnected from the real world, leading them to question its relevance. Admittedly, maintaining this relevance is no easy feat—it requires collective effort. Farmer Alfred can’t tackle this challenge alone. However, introducing more real-world math examples into the school curriculum is a crucial step. Numerous studies have demonstrated that such examples significantly enhance students’ motivation to learn math. Our own pilot tests confirm this, showing that a clear connection between real-world problems and math education profoundly influences how young people view their future careers.

Therefore, to increase the overall relevance of math education and motivate students, we must prioritize the integration of real-world examples into our teaching practices. At DARTEF, we are dedicated to this cause, striving to bring authenticity to mathematics education worldwide.

• Vos, Pauline. ““How real people really need mathematics in the real world”—Authenticity in mathematics education.” Education Sciences  8.4 (2018): 195.
• Cambridge dictionary, Meaning of problem in English.
• Reif, Frederick.  Applying cognitive science to education: Thinking and learning in scientific and other complex domains . MIT press, 2008.

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STEM learning is largely about designing creative solutions for real-world problems. When students learn within the context of authentic, problem-based STEM design, they can more clearly see the genuine impact of their learning. That kind of authenticity builds engagement, taking students from groans of “When will I ever use this?” to a genuine connection between skills and application.

Using STEM to promote critical thinking and innovation

“Educational outcomes in traditional settings focus on how many answers a student knows. We want students to learn how to develop a critical stance with their work: inquiring, editing, thinking flexibly, and learning from another person’s perspective,” says Arthur L. Costa in his book Learning and Leading with Habits of Mind . “The critical attribute of intelligent human beings is not only having information but also knowing how to act on it.”

Invention and problem-solving aren’t just for laboratory thinkers hunkered down away from the classroom. Students from elementary to high school can wonder, design, and invent a real product that solves real problems. “ Problem-solving involves finding answers to questions and solutions for undesired effects. STEM lessons revolve around the engineering design process (EDP) — an organized, open-ended approach to investigation that promotes creativity, invention, and prototype design, along with testing and analysis,” says Ann Jolly in her book STEM by Design . “These iterative steps will involve your students in asking critical questions about the problem, and guide them through creating and testing actual prototypes to solve that problem.”

STEM projects that use real-world problems

Here are some engaging projects that get your students thinking about how to solve real-world problems.

Preventing soil erosion

In this project, meant for sixth – 12th grade, students learn to build a seawall to protest a coastline from erosion, calculating wave energy to determine the best materials for the job.  See the project.

Growing food during a flood

A natural disaster that often devastates communities, floods can make it difficult to grow food. In this project, students explore “a problem faced by farmers in Bangladesh and how to grow food even when the land floods.”  See the project .

Solving a city’s design needs

Get your middle or high school students involved in some urban planning. Students can identify a city’s issues, relating to things like transportation, the environment, or overcrowding — and design solutions. See the project here or this Lego version for younger learners.

Creating clean water

Too many areas of the world — including cities in our own country — do not have access to clean water. In this STEM project, teens will learn how to build and test their own water filtration systems.  See the project here .

Improving the lives of those with disabilities

How can someone with crutches or a wheelchair carry what they need? Through some crafty designs! This project encourages middle school students to think creatively  and  to participate in civic engagement.   See the project here .

Cleaning up an oil spill

We’ve all seen images of beaches and wildlife covered in oil after a disastrous spill. This project gets elementary to middle school students designing and testing oil spill clean-up kits. See the project here .

Building earthquake-resistant structures

With the ever-increasing amount of devastating earthquakes around the world, this project solves some major problems. Elementary students can learn to create earthquake resistant structures in their classroom. See the project here .

Constructing solar ovens

In remote places or impoverished areas, it’s possible to make solar ovens to safely cook food. In this project, elementary students construct solar ovens to learn all about how they work and their environmental and societal impact.  See the project here .

Stopping apple oxidization

Stop those apples from turning brown with this oxidation-based project. Perfect for younger learners, students can predict, label, count, and experiment! See the project here .

Advancing as a STEAM educator

The push for STEM has evolved into the STEAM movement, adding the arts for further enrichment and engagement. There are so many ways to embed STEM or STEAM lessons in your curriculum, but doing it well requires foundational knowledge and professional development. Imagine what type of impact you could have on your students and your community if you were supported by a theoretical framework, a variety of strategies, and a wealth of ideas and resources.

You may also like to read

• Teaching STEM: Challenging Students to Think Through Tough Problems
• Professional Development Resources for STEM Teachers
• What is the Washington State STEM Lighthouse Program?
• Characteristics of a Great STEAM Program
• Building a Partnership Between Your School and a STEAM Organization
• The Art of Inquiry in STEAM Education

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Getting smart collective, impact update, 7 real-world issues that can allow students to tackle big challenges.

Ever since I started teaching in 1990, I have been a student voice advocate. Whether it was as a media/English teacher, student leadership advisor or a site leader. I have always believed that students not only have good ideas, but that they may just have new, unique or even better ones. In an effort to find their own voice and place in the world, they may see things that we don’t see or have long been paralyzed to do anything about. In 1999, I saw students address a school’s racial divide and cultural issues by creating a school-wide learning experience (see Harmony at Buchanan High School ). Ever since then, I have believed that projects with real-world outcomes hold some of the greatest potential for helping students become driven, empathetic and engaged citizens. The outpouring of student voice  in the wake of the recent tragedy in Parkland, Florida, is a great example.

When we begin the project design process in PBL, we can start either with a challenging problem or question and then tie it to our standards, or we can start with our standards and connect them to a real-world challenge. This second approach is more foundational to project based learning, for many reasons, including student engagement, student voice, relevance and authenticity. But beyond that, we also do it because this is where jobs are. Jobs are created and grown as we work to address the real problems facing our world and peoples. Our students are ready to tackle the problems facing our world. They have a voice. They have the tools and resources. And they are not afraid to collaborate and form new communities poised for the problem-solving work that needs to be done.

As an educator, parent and advocate for an engaged/empowered citizenry, I could not be prouder of how the students in Parkland, Florida – along with their peers across the nation – have both found their voice, as well as changed the narrative. These students, as well as many others across the nation, are not afraid to collaborate, and use new technologies and form new professional networks in order to address our current and future challenges. Let’s be honest, our best hope of improving the status of our planet’s many issues truly lie with our youth.

With all of this in mind, there are a number of current and ongoing real-world challenges that we currently face (and probably will for a long time). I don’t like the term “problem-solving” in this context, as it implies that we can fix, cure or eradicate a problem or challenge, but by going after our problems with new solutions, we can certainly move progress forward. And in that movement, there is magic. There is innovation. There is change. There is our collective human mission: how can we creatively collaborate, critically think and communicate in ways that make our world a better place to live.

New Pathways Handbook

Over the last few years, we’ve shared hundreds of stories about connecting students to work (and skills) that matters through our blog, podcast, and various publications. To synthesize these key learnings, we compiled the New Pathways Handbook, a great jumping-off point to our numerous resources and launchpad for getting started with pathways.

Our students are ready to exercise their collective voices and create calls to action. The following seven ideas are not ranked, but are rather my go to “top seven” that naturally lend themselves to projects that excite student interest, rely on available resources, and maintain relevance and authenticity. Moreover, they are not subject-specific. Indeed, there are many opportunities for English, science, social science, math and others to connect to these project challenges. They are:

1) Climate Change – Climate Change will have a significant impact on our students’ lives. Indeed, there may not be one issue that will impact them more comprehensively. Students have seen the data and witnessed the changes, and are listening to the science community. They know that this an urgent issue that will affect almost everything, including, but not limited to, weather, sea levels, food security, water quality, air quality, sustainability and much more. Many organizations – such as NASA , The National Park Service , National Center for Science Education , National Oceanic Atmospheric Association  and SOCAN  to name a few – are working to bring climate change curriculum and projects to teachers and students.

2) Health Care  – Since this has become a prominent topic in the national debate, students are becoming aware of the issues in our country related to rising costs, access, quality and equity. They are beginning to understand the importance both individually and societally. Like the aforementioned topic of climate change, students are also (and unfortunately) learning that we are not necessarily leading the world in this area. They know that this problem is connected to profits, insurance, bureaucracy and more, but they also have a fresher sense of how it could be different, and how we could learn from others around the world. The work on this topic, like many others, is being led by our universities. Institutions such as University of Michigan , Johns Hopkins  and Stanford are leading the way.

3) Food Insecurity   – as our students become more aware of their surrounding communities, as well as the peers they interact with daily, they begin to see differences. Differences in socioeconomic status, opportunities for growth, housing, security, support services and more. And since 13 million young people live in food-insecure homes, almost all of our students, as well as educators, know someone who is hungry on a daily basis. This may often start with service-based projects, but can also lead to high quality project based learning complete with research, data analysis, diverse solutions and ultimately a variety of calls to action. If you want to see how one teacher and his students transformed not only their school, but entire community related to food insecurity, check out Power Of A Plant author Stephen Ritz and the Green Bronx Machine .

4) Violence  – This is a natural given current events taking the nation by storm. However, the related topics and issues here are not new. And yes, they are politically charged, but young people care about these issues . They care about their collective safety and futures, but also know something can be done. In addition to the specifics related to school violence and safety, students can study details of how to advocate, organize, campaign and solicit support, learn that this is a complex problem that has many plausible causes, and, perhaps most importantly, hope for progress. They also know that although they are concerned about attending school in safe environments, our society and culture have violence-related problems and issues that they want to see addressed. Following the recent incident in Florida and the subsequent response from students, the New York Times has compiled a list of resources  for educators on this topic.

5) Homelessness  – We often hear the expression “think globally, act locally.” The topic of homelessness has garnered more attention than ever as more and more communities wrestle with a growing homeless population. In addition to opportunities for our students and schools to partner with local non-profit organizations dealing with homelessness, this topic, like others, is also a great way to elicit empathy in our students. We often hear from educators, employers and others that we want to raise adults that are able to solve problems, improve our communities, and have the ability to see beyond themselves. This topic can provide a number of options for helping students develop those skills. Finally, we also have a growing population of homeless students. So, the relevancy and urgency are all there. Many have laid the groundwork for us to address this within our curriculum. Organizations like Bridge Communities , National Coalition For The Homeless , Homeless Hub  and Learning To Give  are some of the many leading the way.

6) Sustainability  – This is an extremely global issue that affects everything from energy, to food, to resources, economics, health, wellness and more. Students are becoming more and more aware that our very future as a species depends on how we address sustainability challenges. They are aware that this challenge requires new ways of thinking, new priorities, new standards and new ways of doing things. Sustainability is all about future innovation. Students have tremendous opportunities to collaborate, think critically, communicate, and be creative when questioning if a current practice, method, resource or even industry is sustainable without dramatic change and shifts. Students who tackle these challenges will be our leaders – business, political and cultural – of the future. Educators and students can find almost infinite resources and partners. A few of these are Green Education Foundation , Green Schools Initiative , Strategic Energy Innovations , Facing the Future  and Teach For America .

7) Education  – It seems that each and every day, more and more of us (though maybe still not enough) are moving closer to realizing that our educational systems are seemingly unprepared to make the big shifts needed to truly address the learning needs of 21st-century students. The related challenges are many – new literacies, skills, economic demands, brain research, technology, outcomes and methodologies. It’s a good thing that more and more people – both inside and outside of education – are both demanding and implementing change. However, one of the continued ironies within education is that we (and I recognize that this is a generalization) rarely ask the primary customer (students) what they think their education should look, feel and sound like. We have traditionally underestimated their ability to articulate what they need and what would benefit them for their individual and collective futures. One of the many foundational advantages of project based learning is that we consult and consider the student in project design and implementation. Student “voice & choice” creates opportunities for students to have input on and make decisions regarding everything from the final product, to focus area within a topic or challenge, and even whom they may partner with from peers to professionals. It’s this choice that not only helps elicit engagement and ownership of learning, but offers opportunities for students to enhance all of the skills that we want in our ideal graduates. As one might guess, there is not a lot of formal curriculum being developed for teachers to lead students through the issue of education reform. This may need to be an organic thing that happens class by class and school by school. It can start as easily as one teacher asking students about what they want out of their education. Some other entry points are The Buck Institute for Education , Edutopia’s Five Ways To Give Your Students More Voice & Choice , Barbara Bray’s Rethinking Learning  and reDesign .

This is not intended to be an exhaustive or comprehensive list. However, these seven broad topics present hundreds of relevant challenges that our students can and should have opportunities to address. If they do, they will not only be more prepared for their futures, but also poised to positively impact all of our futures.

For more, see:

• High Quality PBL Case Study: School21
• In Broward County, Student Voice Impacts the Classroom and Beyond
• Introducing a Framework for High Quality Project Based Learning

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New Designs for School From Teacher to Teammate: Real-World Projects Challenge Students and Teachers Alike

Elizabeth Luna (she/her) Teacher and Department Chair Athens Drive Magnet High School in Raleigh, North Carolina

We’ve all had the experience of truly purposeful, authentic learning and know how valuable it is. Educators are taking the best of what we know about learning, student support, effective instruction, and interpersonal skill-building to completely reimagine schools so that students experience that kind of purposeful learning all day, every day.

Community-based projects are a game-changer in the classroom, when students learn to use design thinking to solve problems, address community needs, and make an impact... and teachers become partners in learning.

After 12 years of teaching, I thought I had mastered connecting curriculum to student interest. However, during the pandemic, the problem of student engagement became glaring as students struggled to find purpose. Faced with learning on their own, they encountered the daunting task of finding meaning in standards that were disconnected from “real life.”

While students struggled to find motivation and purpose, I also faced the need to rethink my view of teaching and learning. There were a lot of questions that ran through my mind. The main one that continued to resurface was, “What actually matters?” With shortened teaching time, on top of the disengagement, I encountered the formidable moment of figuring out the goal of my courses and what I truly wanted students to learn/know. For years, the county and the state have been telling teachers what matters most through the curriculums provided. This pandemic shined a light on the fact that the standards weren’t cutting it.

When I reflected on what my design courses had lost in the online transition, I realized that the community connections and design thinking that I had built in my first few years of teaching had vanished. Classes became flat and task driven, and my students couldn’t “think big” anymore because they didn’t have the space to do so. The new tech standards weren’t enough for my students, and they didn’t work for me either. Teaching a student how to click buttons in an ever-changing and evolving technology-filled world felt pointless and went against every virtue taught to me within my design degree. I needed more and so did they.

“What actually matters?” is a tough question to answer. Upon some reflection, I realized that I have two major goals within my classroom.

First, I want students to become proficient in the design programs so that they can create.

Second, I need students to understand how to use these skills to solve problems, address community needs, and thus, make an impact.

The base curriculum covered the first goal, but the second part was on me to create and develop.

A Design Thinking Classroom

The design process is a procedure with which everyone needs to become familiar in the 21st century and is a necessity to achieve my second goal. My graphic design degree from North Carolina State University taught me the importance of design thinking and its impact on creating change. As a teacher, this training is implemented within my classroom. I explain to students that no matter what subject area they are interested in, the process is applicable. You will always be solving problems and inventing solutions whether they are designed products, business plans, or vaccines; your life is about problem-solving and innovating. There are a lot of design processes out there, from simple to complex, but they all move you through the same base steps.

• Empathizing with the user/ Gathering information
• Defining the problem
• Ideating/ Finding solutions
• Creating and evaluating solutions: Sharing out and receiving critique
• Refining and adjusting those solutions

These steps were the guiding process in my classroom before the pandemic and needed to be reinstated at the forefront of my classroom as students returned to school this year. Rather than teaching the five step process to students as a discrete skill, I recognized the importance of integrating the five steps into every project and assignment, to teach through the five steps. Transforming a classroom to incorporate this type of thinking is complex and time consuming. It forced me to break away from the provided curriculum and enhance it. I had to carve out time within my classes for students to think deeply around problems, to debate, and to develop and express opinions.

For the first time in a long time I saw growth as students pulled away from a task-completion mindset and, instead, focused on seeking feedback and refining ideas. Allowing time for thought and exploration was key to moving students toward being problem-solvers. Students began to turn to each other for opinions and, instead of rejecting the other person’s thoughts, they sought to understand another viewpoint. When they fell short on a technical skill, they didn’t shut down. Instead, they asked for help from me or another student, searched on the internet for support, etc. Because the assignments were structured in a more open-ended manner, students were no longer worried about being seen as “right.” Instead, they focused on creating the best result possible for them at that time.

Building a creative thought process was only half the battle, though. I needed to connect students to the community to engage in purposeful ways. In the past, I have connected to community members that needed graphic design work done. I knew that my students responded better to real-world projects than any construct I could mimic. But as a teacher, I was overwhelmed by the time required to track down community members in need, connect them with students, and figure out the direction and goal of the project. I needed community connection to help my students build their purpose, but figuring out how to make that happen was a roadblock. Then Project Invent came along.

Project Invent would impact my classroom in ways that I did not understand immediately, challenging me as an educator while also providing structure and opportunities for my students. Project Invent was a tangible organization that took the design process I had been working so hard to embed in my classroom and connected it to problem-solving within the community. They were the bridge that I needed to make my second goal come to fruition.

Get the learner perspective on real-world projects from a former student in Ms. Luna’s class! Rethinking Education: Prioritizing Creativity and People over Grades and Rubrics

Real-World Context Is Everything to a Student

My students need connection. I was inundated with statements like, “I am most excited about being able to apply design to group projects and work towards making an impact for someone,” and “getting to do something real.” Project Invent partnered us with two community members that matched our school’s magnet theme. We have a global health theme at our school that teachers are supposed to integrate into curriculum, and some students take part in a STEM Academy which focuses on sustainability. My community partners targeted both of these areas, one from the North Carolina Forestry Division and another with a son that struggled with microcephaly amongst other medical conditions.

In order for students to take full ownership, I handed the reins to them immediately. We reviewed some basic interview etiquette, students did some background research, and then we scheduled our first interview. My students met with the community partners and controlled the interactions from the start. The interviews brought a deeper engagement. I often work to mimic projects and requests they would receive in the workforce so my students are used to getting a ‘real’ project brief that I create for them. However, I have never before put them in front of someone and asked them to listen to everything that person has to say and attempt to deconstruct the client’s biggest needs. In this situation, they were tasked with figuring out what the client’s problems were, empathizing with them, and defining their direction of work. Meeting our clients was a game-changing experience for students. They left their interview sessions discussing every statement said, went back and analyzed the transcripts, and argued about what was most important and what should be the focus. Their passion bloomed before my eyes as these excited students became invested in the people sitting across from them.

Visiting a Site Creates Opportunities to Learn and Understand

Visiting the site of a problem or seeing a situation in-person is essential for kids to "feel" what is going on. Encouraging students to venture out to a space or physically research and experiment with a problem is the key to better understanding. Students come up with more accurate solutions and/or develop new ways to look at things when they can experience the space or problem first hand. My students were sure their solution direction was the right way to go, until they had an opportunity to visit the site of concern. After walking the space, experiencing the different areas, and hearing more from the client, they agreed they’d have to pivot their ideas and throw out their plans. It turns out, they had not solved the most essential problems and they knew their plans weren’t accurate anymore. The feedback from their partner and the chance to immerse themselves into a situation made students aware of possibilities they had not imagined based on their own experiences.

Team Assignments and Collaboration Support Change-Making

I wanted students to self-select the design challenges most interesting to them.Interviews, breakdowns, and problem analysis were done with all students in randomly assigned small groups as well as a whole class. This ensured a basic understanding of both clients for all and would foster a better creative environment as our class progressed through the year. If every student understood the problems at hand, every one could provide valuable feedback and be active supporters in the invention process. As research and problem definition progressed, students began to really gravitate toward one client over the other; natural separations started to form within the classroom.

Even with the class splitting, groups were still too large for each student to have an active voice. They worked through a variety of ideation charts as they narrowed their points of interest with the client, thus further dividing themselves into smaller cohorts. I also began surveying students around different skill sets to ensure that groups were balanced with knowledge/ability as much as possible. However, one student pointed out, “It’s more important that I am passionate about what I am working on and who I am working with than having a technically skilled team. I want to be invested in this.” His statement reminded me that the point of the teams was to push student-passion forward; I needed to be careful about balancing a team around skill and needed to focus more on dedication and client-based interest.

Community Engagement Can Fill the Knowledge/Skill Gap of the Teacher

“I don’t know how” might be the most common phrase uttered by students and myself this year. I vividly remember the point where I looked at my students and said we had hit the point where I no longer had the knowledge base to support them. I am a graphic designer. I like to make things look good and I enjoy the challenge of communication. When we hit questions around feasibility and the prototyping stage, I was way out of my league. I openly admitted this to my students. That moment was transformative as the playing field was leveled. My job was to support them and help them problem-solve. I couldn’t tell them what was right or wrong and, as students realized this, we entered a space where trial and error would lead to a lot of failure and, hopefully, some success. The rhythm of the class changed.

Already engaging with community members as clients, it occurred to me that I could use community members to support my students where my knowledge was scant. I sought help from teachers, friends, college students, and local businesses. Since virtual meetings are so prominent now, the challenge of getting students and professionals together was almost non-existent. It became a natural process: students would hit a question that we genuinely couldn’t answer, we’d debate about what type of person would be able to support us, and I would work alongside my magnet coordinator to find a professional that could help. Product developers could understand their idea and talk to students about feasibility, often giving them parts to look up or products to learn about. Engineers talked about intricate electronic concepts, pointing students to a type of relay or remote. Gardeners and landscapers discussed invasive species and methods of removal, directing us back to specific research articles and studies completed.

Honest and purposeful feedback is absolutely vital for students to know whether an idea is successful, possible, or should be pursued further. Real feedback from invested clients and business professionals encourages students’ refining and adjusting solutions for the benefit of the project. Without client and community insight, the initial design may be "good" but not fulfill all the essential needs. Since feedback came from multiple volunteers spanning different backgrounds, students were exposed to a variety of viewpoints. They learned how to sift through reviews and pull the most essential information. Designers need to take feedback as a gift of new possibilities, not a "put down" of their ideas.

Every conversation a student had, even when it was filled with critique, led to more energy and motivation. Students pivoted concepts, considered issues they hadn’t previously thought about, and had moments of confirmation that I couldn’t give them. Every time, they came back and asked for more. I realized that the engagement with outsiders filled the knowledge gap I faced while also giving the kids meaningful experience with experts in a variety of fields. The cool part was, they came back to me to share their excitement, their success, their questions and concerns. I was now a member of each team; instead of being the one with the answers, I was getting to be part of their support system. I was working alongside them to hypothesize and decide on their next steps as an equal, which so rarely happens in the classroom.

Local professionals want to work with students and, when asked to engage actively and not just as a speaker, they are as fulfilled as the kids. The experience benefits everyone involved. Making time for those interactions is key to keeping students engaged in their work. ‘Rejuvenating’ was the word many community members used. The students brought new hope and excitement to their lives. One professional said, “Your students’ passion is electrifying. I am so impressed with their concepts and work. They will change the world and have the ability to do that because of the opportunities you’re providing them.” His excitement reminded me of what many professionals have said over the years, “Students are the future and I want to work with them and inspire them to create.”

Student Success Is So Much More than Test Scores

As a teacher, my job has been consistently broken down into teaching students standards and technical skills. Because of a need to ‘test’ students to prove their success, my curriculum has lost the applicability, creative thinking, and all of the upper-level verbs in Bloom’s Taxonomy. I’ve come to believe that focusing on new high-tech skills is an archaic choice. The world is continuously evolving and shifting; my job is not to teach a skill that will be outdated in a year to come, but use the skills as a means of production and creation.

With this mindset, students learn by necessity and naturally gain ability as they apply themselves to larger tasks. It is not going to be equal. Every student will not develop success in the same area. Some students end up gaining coding knowledge, others focus on design, some find out their success is in presentation, but all grow in areas that interest them and help direct them in future career choices. Amongst individual accomplishment, there is also a level of success that comes from having to work closely in a team. They’ve learned how to hold each other accountable, how to manage peers and time, and how to celebrate each small victory.

I have enhanced the learning experience to wrap around a student and their natural abilities by building creative spaces, community connection, and providing opportunities in which everyone learns from each other. Beyond the student, I have enhanced my teaching experience, challenging myself to be comfortable with students working on a variety of skills, with non-traditional ways to measure success, and with not always having the control over learning direction. The environment that forms from client-driven projects and community engagement is life-giving to community members, teachers, and students alike.

The Portrait of a Graduate in Practice

New from NGLC! The outcomes that Elizabeth Luna sees from real-world projects are often the competencies expressed in a school or district's graduate portrait. Find out how portraits are inspiring system-wide change where this kind of teaching and learning can thrive.

All photos courtesy of the author.

Elizabeth Luna (she/her)

Teacher and department chair, athens drive magnet high school.

Elizabeth Luna is a graphic design teacher and career tech education department chair at Athens Drive Magnet High School, Center for Medical Sciences and Global Health Initiatives. She also coaches four teams of students through Project Invent . She has led work at the county and school level focused on implementing design thinking in the classroom.

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Grappling With Real-World Problems

Project-based learning can focus on real community issues to combine content and student interests.

Problem-based learning (PBL) is integrated at Two Rivers Public Charter School in Washington, DC, at every grade level—pre-K through eighth grade. Students are presented with a real-world problem, undertake a series of investigations, and create a product that they present to an authentic audience as part of the Expeditionary Learning (EL) Education framework. 

PBL enables the school to reach all learners. “There are multiple entry points,” explains Julia Tomasko, a fourth-grade teacher. “It’s easy to scaffold [PBL] for students who need more support, and the sky’s the limit for extensions.”

In Tomasko’s problem-based unit covering Jamestown, her class looked through primary resources like John Smith’s diary. They discussed representation and how all the primary resources are from the English settlers. Tomasko recalls one of her students asking, “‘Out of all the cultures in the world, which culture do you think needs to have its story told more and have its voice heard?’ I was blown away. That’s not typical fourth-grade thinking, but she was clearly thinking through these ideas in a deep way and wondering how [they] can apply to other things.”

How It's Done

1. Backwards plan. Jeff Heyck-Williams, the director of curriculum and instruction, believes that the perfect problem connects content, student interest, and an authentic context. To guide your planning, he suggests asking:

• What content and skills do my students need to learn?
• What would be proof of their understanding?
• In what contexts will they develop understanding?
• What are my students interested in?
• What are real problems that people in my field—ecology, biology, local history—grapple with that are related to the content I need to teach?
• What is the problem that I want my kids to solve?
• What product will my students create?

“Once you have those big pieces in place, you can start to plan: ‘What are the day-to-day things that I'm going to do to get them to face that problem and then move towards an ultimate solution?’” says Heyck-Williams.

2. Find a problem that’s relevant to your students’ interests and appropriate for their age. “Our youngest kids are working on problems that speak to things in their immediate environment,” explains Heyck-Williams, “but as kids move forward, they work with more philosophical problems outside of their direct community.”

First-grade students roamed school fields to investigate spiders. To discover the truth about spiders and help reduce people’s fear of them, each student created a scientific drawing of a spider and wrote a book exploring their characteristics, like eating mosquitos or bugs that harm people’s gardens. Fourth-grade students were asked how they could improve the quality of their local, polluted river. They created a website to teach kids how to take care of it. Eighth-grade students learned about gene editing, explored the ethics around it, and presented policy briefs to the National Academies of Sciences, Engineering, and Medicine. (See 4 Tips on Teaching Problem Solving [From a Student] .)

3. Be flexible with the product. It’s good to have a product in mind that you can guide your students towards, like creating a book, website, or policy brief. If you want your students to create a website, you can introduce websites as great resources in prior lessons. But the product isn’t the learning goal. Solving the problem and understanding the content is. The product is just the avenue to get there. If your students are excited about another product idea, go with it. When planning, think about the variety of products that your students might come up with to solve the problem, suggests Tomasko. Plan for flexibility.

4. Some lessons will be a flop, and that’s OK. “You think that you’re guiding your kids towards a certain idea,” reflects Tomasko, “and not only do they not come up with that, but sometimes they don’t come up with anything.” When this happens, go back to the planning board and think about how you can reteach that content another way. (See 3 Ways Lesson Plans Flop—and How to Recover .)

5. Start small. “When we first started problem-based learning, it was important for people to see that they could do this in small ways,” says Jessica Wodatch, the executive director of Two Rivers Public Charter School. “It’s really about taking your daily routine and thinking, ‘Where could kids have input? Where could kids be asked to solve a problem?’”

Instead of giving your students directions for an in-class assignment, ask them what they should do. If your students are lining up and it’s noisy, tell them what’s not working and ask them how they can solve it. If you create a birthday chart every year, have your students create it.

“It doesn’t need to be a three-week unit. It can be a little part of your day,” says Wodatch. “Part of the shift is thinking, ‘What can I hand to them? What am I deciding for them that I don’t need to?’ It’s about giving them some of that decision-making power, authority, and choice, and that is where we start to see the problem-based learning live.”

6. Use KWI to help your students problem solve. K: What do your kids already know about the problem? W: What do they need to know in order to solve the problem? I: What ideas do your students have to solve the problem? “Even if your students are solving an open-ended math problem, they can think through: What do they know about the problem, what’s being asked, and what different ideas do they have to solve it? Then you can apply that same structure to a more long-term project like a learning expedition,” says Tomasko.

Allowing students to explore ideas and make mistakes is a key element of problem-based learning at Two Rivers Public Charter School. Wodatch explains, “We want kids getting in the practice of weighing information, grappling with difficult problems that don’t have clear answers, considering different points of view and data, asking for expert opinions, and ultimately coming up with the solution. Those are things that we all do every day of our lives, and we want our kids to do that.”

Two Rivers Public Charter School

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26 Snappy Answers to the Question “When Are We Ever Going to Use This Math in Real Life?”

Next time they ask, you’ll be ready.

As a math teacher, how many times have you heard frustrated students ask, “When are we ever going to use this math in real life!?” We know, it’s maddening! Especially for those of us who love math so much we’ve devoted our lives to sharing it with others.

It may very well be true that students won’t use some of the more abstract mathematical concepts they learn in school unless they choose to work in specific fields. But the underlying skills they develop in math class—like taking risks, thinking logically and solving problems—will last a lifetime and help them solve work-related and real-world problems.

Here are 26 images and accompanying comebacks to share with your students to get them thinking about all the different and unexpected ways they might use math in their futures!

1. If you go bungee jumping, you might want to know a thing or two about trajectories.

https://giphy.com/gifs/funny-fail-5OuUiP0we57b2

Source: GIPHY

2. When you invest your money, you’ll do better if you understand concepts such as interest rates, risk vs. reward, and probability.

3. once you’re a driver, you’ll need to be able to calculate things like reaction time and stopping distance., 4. in case of a zombie apocalypse, you’re going to want to explore geometric progressions, interpret data and make predictions in order to stay human..

Trigger an outbreak of learning and infectious fun in your classroom with this Zombie Apocalypse activity from TI’s STEM Behind Hollywood series.

5. Before you tackle that home wallpaper project, you’ll need to calculate just how much wall paper glue you need per square foot.

6. when you buy your first house and apply for a 30-year mortgage, you may be shocked by the reality of what interest compounded over 30 years looks like., 7. to be a responsible pet owner, you’ll need to calculate how much hamster food to have on hand., 8. even if you’re just an armchair athlete, you can’t believe the math involved in kicking field goals.

Check out this Field Goal for the Win activity that encourages students to model, explore and explain the dynamics of kicking a football through the uprights.

9. When you double a recipe, you’re going to need to understand ratios so your dinner guests don’t look like this.

10. before you take that family road trip , you’re going to want to calculate time and distance., 11. before you go candy shopping, you’re going to have to figure out x trick or treaters times x pieces of candy equals…, 12. if  you grow up to be an ice cream scientist, you’re going to have to understand the effect of temperature and pressure at the molecular level..

https://giphy.com/gifs/ice-lick-cream-3Z1kRYmLRQm5y

Explore states of matter and the processes that change cow milk into a cone of delicious decadence with this Ice Cream, Cool Science activity .

13. Once you have little ones, you’ll need to know how many diapers to buy for the month.

14. because what if it’s your turn to organize the annual ping pong tournament, and there are 7 players at a club with 4 tables, where each player plays against each other player, 15. when dressing for the day, you might want to consider the percent likelihood of rain., 16. if you go into medical research, you’re going to have to know how to solve equations..

Learn more about inspiring careers that improve lives with STEM Behind Health , a series of free activities from TI.

17. Understanding percentages will help you get the best deal at the mall. For example, how much will something cost with 40% off? What about once the 8% tax is added? What if it’s advertised as half-off?

https://giphy.com/gifs/blue-kawaii-pink-5aplc3D2G0IrC

18. Budgeting for vacation will require figuring out how many hours at your pay rate you’ll have to work to afford the trip you want.

19. when you volunteer to host the company holiday party, you’ll need to figure out how much food to get., 20. if you grow up to be a super villain, you’re going to need to use math to determine the most effective way to slow down the superhero and keep him from saving the day..

Put your students in the role of an arch-villain’s minions with Science Friction, a STEM Behind Hollywood activity .

21. You’ll definitely want to understand how to budget your money so you don’t look like this at the grocery checkout.

22. if you don’t work the numbers out in advance, you might at some point regret choosing that expensive out-of-state college., 23. before taking on a building project, remember the old saying—measure twice, cut once., 24. if have aspirations of being a fashion designer, you’ll have to understand geometry in order to make the perfect twirling skirt.

https://giphy.com/gifs/loop-bunny-ballet-yarFJggnH24da

Geometry and fashion design intersect in this STEM Behind Cool Careers activity .

25. Everyone loves a good bargain! Figuring out the best deal is not only fun, it’s smart!

26. if you can’t manage calculations, running the numbers at the car dealership might leave you feeling like this:, you might also like.

21 Essential Strategies in Teaching Math

Even veteran teachers need to read these. Continue Reading

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How to Use Real-World Problems to Teach Elementary School Math: 6 Tips

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When you think back on elementary school math, do you have fond memories of the countless worksheets you completed on adding fractions or solving division problems? Probably not.

Researchers and educators have been pushing for years for schools to move away from teaching math through a set of equations with no context around them, and towards an approach that pushes kids to use numerical reasoning to solve real problems, mirroring the way that they’ll encounter the use of math as adults.

The strategy is largely about setting kids up for success in the professional world, and educators can lay the groundwork decades earlier, even in kindergarten .

Here are some tips for using a real world problem-solving approach to teaching math to elementary school students.

1. There’s more than one right answer and more than one right method

A “real world task” can be as simple as asking students to think of equations that will get them to a particular “target” number, say, 14. Students could say 7 plus 7 is 14 or they could say 25 minus 11 is 14. Neither answer is better than the other, and that lesson teaches kids that there are multiple ways to use math to solve problems.

2. Give kids a chance to explain their thinking

The process you use to solve a real world math problem can be just as important as arriving at the correct answer, said Robbi Berry, who teaches 5th grade in Las Cruces, N.M. Her students have learned not to ask her if a particular answer is correct, she said, because she’ll turn the question back on them, asking them to explain how they know that it is right. She also gives her students a chance to explain to one another how they arrived at a particular solution, “We always share our strategies so that the kids can see the different ways” to arrive at an answer, she said. Students get excited, she said, when one of their classmates comes up with an approach they never would have thought of. “Math is creative,” Berry said. “It’s not just learning and memorizing.”

3. Be willing to deal with some off-the-wall answers

Problem solving does not necessarily mean going to the word problems in your textbook, said Latrenda Knighten, a mathematics instructional coach in Baton Rouge, La. For little kids, it can be as simple as showing a group of geometric shapes and asking what they have in common. Students may go off track a bit by talking about things like color, she said, but teachers can steer them towards thinking about things like how a rectangle differs from a triangle.

4. Let your students push themselves

Tackling these richer, real-world problems can be tougher than solving equations on a worksheet. And that is a good thing, said Jo Boaler, a professor at Stanford University and an expert on math education. “It’s really good for your brain to struggle,” she said. “We don’t want kids getting right answers all the time because that’s not giving their brains a really good workout.” These types of problems require collaboration, a skill that many don’t associate with math, but that is key to how math reasoning works beyond the classroom. The complexity and difficulty of the tasks means that students “have to talk to each other and really figure out what to do, what’s a good method?”

5. Celebrate ‘favorite mistakes’ to encourage intellectual risk taking

Wrong answers should be viewed as learning opportunities, Berry said. When one of her students makes an error, she asks if she can share it with the class as a “favorite mistake.” Most of the time, students are comfortable with that, and the class will work together to figure how the misstep happened.

6. Remember there’s no such thing as a being born with a ‘math brain’

Some teachers believe that certain students are just naturally good at math, and others are not, Boaler said. But that’s not true. “Brains are constantly shaping, changing, developing, connecting, and there is no fixed anything,” said Boaler, who often works alongside neuroscientists. What’s more, many elementary school teachers lack confidence in their own math abilities, she said. “They think they can’t do [math],” Boaler said. “And they often pass those ideas on” to their students.

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10 cool projects created by kids addressing real-world problems

The children of today are the great inventors of the future, and these kids are the ones proving it.

Caitlin Brown

Our children hold the key to determining a brighter future. Over the years, we’ve seen an emergence of notable teens spearheading the way and campaigning for issues they are passionate about.

Take Malala Yousafzai, for example, and her tireless campaigning to provide girls better access to education globally. More recently, Greta Thunberg took a central place on the world stage as the leading voice for climate change activism.

Students from all over the globe continue to create tangible solutions and ideas that address real-world problems, such as the US student who invented a possible way to detect pancreatic cancer , or the student in the UK who founded an organisation that successfully lobbied the government to fund free sanitary products in all English schools and colleges.

Read more about STEM :

• 15 super fun DIY science experiments for kids to try at home
• People with aphantasia are more likely to work in a STEM field
• Why aren't there more women in science?

This year, over 2,000 students across the UK participated in the Youth Industrial Strategy Competition , a national STEM competition organised by the British Science Association in partnership with the Department for Business, Energy and Industrial Strategy.

The competition challenged students to come up with practical projects that address four of the most pressing issues facing society as outlined by the UK Government: Clean Growth, Artificial Intelligence and Data, Ageing Society and the Future of Mobility.

Take a look at some of the most creative ideas that made it through to the finals:

Ava Garside's air quality sensor

First, is the winner of the junior category, 13-year old, Ava Garside from Leeds, who created an air quality sensor in response to the ‘Clean Growth’ category.

The device which you wear collects data and detects the air quality of wherever you are, helping to detect the cleanest and healthiest routes to work or school.

She is now working alongside scientists at the University of Manchester to develop the prototype further in order to fashion it into a small, stylish pin-badge.

Rushil Patel's AI processor

Rushil Patel, a student at St Wilfrid’s Catholic Comprehensive School in Crawley claimed the throne as winner of the senior category.

Rushil developed a project (using a handheld device and glasses) that uses offline artificial intelligence to accurately process a specific object or event in the world around the user, conveying limitless information in the most discrete way possible.

The young inventor hopes it can help people with autism not only identify, but also respond to, perceived emotions.

Llanwern High School's fish-grown lettuce

Taking innovation to the next level, a group of students from Llanwern High School were shortlisted for coming up with a project that uses fish to grow lettuce.

Responding to the challenge of Clean Growth, the students hoped to reduce the need for fertiliser and increase food production by using aquaponics and hydroponics.

This was noted as a possible solution to a side-effect of a potential No Deal Brexit, with a shortage in foodstuffs and fresh vegetables.

The system would be used within a fish tank, with the fish waste providing nutrients for the plants.

The scheme would reduce runoff into rivers and waterways with the same water being used, thereby increasing food production.

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Personal Assistant Dementia app

One student from Lady Eleanor Holles in Hampton, responded to the challenge of ‘Ageing Society’ by designing a Personal Assistant Dementia (PAD) app that will allow patients with early stage dementia to remain independent for as long as possible.

The app uses a combination of smartphone calendars, medication alerts, mealtime prompts and home safety functions.

It would also have contacts, maps and a function to help users find or track missing items around the house, as well as access to emergency services.

Martian habitat to reduce overpopulation

Thinking outside the box (or globe!) year 12 students from Altrincham Grammar School for Girls in Greater Manchester, decided that their solution to the challenge of an ‘Ageing Society’, would be to create a habitat for humans on Mars in order to reduce the effects of overpopulation on Earth!

Mary Webb School and Science College's hydro-electric plumbing solution

Next, is a group of year 11 students from Mary Webb School and Science College in Shrewsbury, who won the intermediate category after designing a project responding to the challenge of Clean Growth.

The team made a hydro-electric plumbing solution designed to reduce energy wastage, carbon dioxide emissions and society's reliance on mass electricity generation.

Their product aims to make use of water flowing through the pipes in our homes, by fitting a device into household water pipes that acts as a mini hydroelectric generator.

Orkbrides Monorail

Addressing the Future of Mobility, year 9 students from St. Peter's Church of England Aided School were praised for designing a project that connects the Scottish islands known as the Outer Hebrides and the Orkney Islands to mainland Scotland through a monorail system known as Orkbrides Monorail.

Tonbridge Grammar School's dye sensitised solar cells

The entry from Tonbridge Grammar School in Kent involved exploring how different purple dyes made from a range of fruits affected the voltage output of a Grätzel cell, in response to the challenge of Clean Growth.

A Grätzel cell is a type of dye sensitised solar cell that is low cost and has little impact on the environment. The students were able to make the cells themselves through Pilkington Glass supplying slides of conducting glass.

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Royal Docks Academy's health shoes

Combatting an Ageing Society, year 10 students from Royal Docks Academy in Newham designed a project that uses vibrogen boots, a type of shoe inspired by power plates that are found in the gym to advance health and wellbeing.

Dundonald High School's rainwater-harvesting bench

And finally, a group of year 9 students from Dundonald High School in Northern Ireland addressed Clean Growth by creating a project that harvests rainwater within the inner-city that can then be filtered and used again for human consumption.

The product is multifunctional in that it also operates as a public bench.

The next generation of inventors

With thousands of participants, multiple finalists and three winners in this competition alone, there is tangible proof that children really do have the enthusiasm, skills and confidence to design and develop real solutions to the challenges we face.

They are our future, and if the future is set to be bright, then we need to continue encouraging them to question the world around them, think creatively and bring ideas to life in the hope of solving societal issues.

Who knows, the next Thomas Edison, Steve Jobs or Marie Curie could be your child, student or relative!

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5 Real Life Algebra Problems That You Solve Everyday

Algebra has a reputation for not being very useful in daily life. In fact, in my experience as a high school math teacher, the complaint that I get the most often is that we don’t spend enough time solving real life algebra problems.

You might be surprised to hear that I understand the frustration that my students experience. Unless we are solving real life algebra problems related to money in some way, algebra can feel very “artificial” or disconnected from real life.

My goal here is to walk you through 5 real life algebra problems that will give you a whole new appreciation for the application of algebra to the real-world. I am excited to help you see how many algebraic equations and algebraic concepts are applicable beyond just algebra word problems in your math class!

What is an Example of Algebra in Real Life?

While it is often seen as an abstract branch of mathematics, there are many real-life applications of algebra in everyday life. Now, it is unlikely that you will be solving quadratic equations while walking your dog, or solving real-world problems with linear equations while you play video games. But you can see examples of real life algebra problems all around you!

A simple example is when you want to quickly determine the total cost of a product including taxes, or the total cost after a discount from the original price. Knowing the total amount of money something will cost is a real-life scenario that everyone can relate to!

Depending on your chosen career path, you may see the use of algebra more often than others (I know I see it a lot in my daily life as a math teacher…!).

For example, if you are a business owner, you may use algebra to determine the number of labor hours to spread amongst your staff, or the lowest price you can sell your product for to break even.

For more uses of algebra, check out my list of 20  examples of algebra in real life !

What is an Example of an Algebra Problem in Real Life?

An algebra problem is a mathematical problem that requires the use of algebraic concepts and strategies to determine unknown values or unknown variables. Much like how the order of operations are required to evaluate numerical expressions, algebra problems require the problem solver to apply a set of rules in order to arrive at a solution.

Real world problems that require the use of algebra usually involve modelling real-life situations with  algebraic formulas . A formula is a specific equation that can be applied to solve a problem. Formulas make it possible to make predictions about a given real-life scenario.

For example, consider the following problem:

You are saving up for a new smartphone and currently have $200 in your savings account. Your plan is to save a certain amount of money each week from your allowance. If the smartphone costs $600, and you want to have enough money to buy it in 8 weeks, how much money should you save each week?

To solve this problem, we first need to use the information provided in the problem to create an equation that models the real-life scenario. Thinking about the problem in terms of variables, we can define T as the total of the savings, and variable x as the amount saved each week.

Since we know that we have a fixed value of 200, we can use the following equation to model this real world problem:

$$T=200+8x$$

This equation says “the total saved is equal to the original $200 plus whatever amount is saved per week, for 8 weeks”.

Substituting the total of the smartphone allows us to begin solving for the unknown variable x. Remember, when solving algebraic equations, you must apply the same operation to both sides of the equation.

$$ \begin{split} T&=200+8x  \\ \\ 600&=200+8x  \\ \\ 600-200 &= 8x \\ \\ 400 &= 8x \\ \\ \frac{400}{8} &= \frac{8x}{8} \\\\ 50 &= x \end{split} $$

Therefore, since x = 50, you should save $50 each week in order to save enough money for the smartphone. For more practice with the algebra used in this solution, check out this free collection of  solving two step equations worksheets !

5 Real Life Algebra Problems with Step-By-Step Solutions

There are so many real-life examples of algebra problems, but I want to focus on 5 here that I believe will convince you of just how applicable algebra is to the real-world! So let’s dig into these 5 real-world algebraic word problems!

Example #1: Comparing Cell Phone Plans

Link is considering two different cell phone plans. Plan A charges a monthly fee of $30 and an additional $0.10 per minute of talk time. Plan B charges a monthly fee of $45 regardless of how much time is used talking. How many minutes of total time talking will make the plans equal in cost?

The best way to start this problem is by writing two equations to represent each scenario. If C represents total cost, and x represents minutes of talk time used, the equations can be written as follows:

• Plan A: \(C=30+0.1x\)
• Plan B: \(C=45\)

Setting the first equation equal to the second equation will allow us to employ algebra to solve for the number of minutes that makes the two plans equal.

$$ \begin{split}  30+0.1x&=45 \\ \\ 30-30+0.1x&=45-30 \\ \\ 0.1x&=15 \\ \\ \frac{0.1x}{0.1}&=\frac{15}{0.1} \\ \\ x&=150 \end{split} $$

Therefore, the two cell phone plans are equal when 150 minutes of total time talking are used.

Example #2: Calculating Gallons of Gas

Zelda is driving from Hyrule to the Mushroom Kingdom, which are 180 miles apart. Her car can travel 30 miles per gallon of gas. Write an equation to represent the number of gallons of gas, G, that Zelda needs for the trip in terms of the distance, d, she needs to travel. Then calculate how many gallons of gas she needs for this trip.

The number of gallons of gas (G) Zelda needs for any trip can be represented by the equation \(G = \frac{d}{30}\). Since the distance between Hyrule and the Mushroom Kingdom is 180 miles, we can substitute 180 into the equation for  d  to determine the number of gallons of gas needed:

$$G=\frac{180}{30}=6$$

Therefore, Zelda needs 6 gallons of gas for her trip.

Example #3: Basketball Players in Action!

A basketball player shoots a basketball from a height of 6 feet above the ground. Unfortunately he completely misses the net and the ball bounces off court. A sports analyst models the path of the basketball using the equation \(h(t) = -16t^2 + 16t + 6\), where h(t) represents the height of the basketball above the ground in feet at time t seconds after the shot. Determine the time it takes for the basketball to hit the ground.

Since we are asked for when the ball hits the ground and  h(t)  is given as the height above the ground, we know that we are looking for the x-intercepts of this quadratic function. We therefore set the equation equal to zero and solve for x. 

Note that we cannot use  trinomial factoring  here since the quadratic is not factorable! Thankfully quadratic equations are solvable using the quadratic formula!

$$ \begin{split}  x&=\frac{-b \pm \sqrt{b^2-4ac}}{2a} \\\\ &=\frac{-16 \pm \sqrt{16^2-4(-16)(6)}}{2(-16)}\\\\ &=\frac{-16 \pm \sqrt{640}}{-32}\\\\ x&=-0.291 \\\\ x&=1.291 \end{split}$$

Therefore, the ball hits the ground after approximately 1.3 seconds. Remember that time cannot be negative, so the first answer is inadmissible and rejected!

Example #4: Saving for a Computer Game

You are saving to buy a new computer game that costs $90. You decide to save up for the computer game by depositing some money into a savings account that earns an annual interest rate of 5% (compounded monthly). You start with an initial deposit of $30 and plan to save for 22 months. Will you have enough to purchase the computer game?

This is an example of a math problem that connects to financial problems people encounter everyday! Since the account you chose earns  interest , we can apply a compound interest formula to help us out here:

$$A=P(1+i)^n$$

In this formula:

• A(t)  is the total amount of money.
• P  is the initial deposit (which is $30 in this case).
• i  is the monthly interest rate (5% annual interest, compounded monthly means that  i  is approximately 0.004167).
• n  is the time that has elapsed (since we are working with months, we multiply by 12)

We can set up our equation and see if our total amount of money is greater than $90:

$$\begin{split}  A(22)&=30(1.004167)^{22 \times 12} \\\\ &=$89.93 \end{split} $$

Remember to always include a dollar sign in your answer and to round to two decimal places when working with money!

Since our answer is approximately equal to $90, we can say that you will have enough money after 22 months! It’s time to get saving!

Example #5: How Many Tickets Did the Movie Theater Sell?

A movie theater charges $10 per ticket for adults and $6 per ticket for children. On a particular day, the theater sold a total of 150 tickets, and the total revenue for the day was $1350. Write a system of equations to represent this real-life scenario and then solve for the number of adult and child tickets sold.

Let’s assume that variable  x  represents the number of adult tickets sold and variable  y  represents the number of child tickets sold. We can set up two linear equations as follows:

• First Equation (the total number of tickets sold): \(x+y=150\) 
• Second Equation (the total revenue from ticket sales is 1350): \(10x+6y=1350\) 

We can use substitution to solve this linear system by rearranging the first equation and substituting it into the second equation. You can catch a quick overview of the substitution process by checking out  this substitution video  on my YouTube channel!

Rearranging the first equation into a different form to solve for  y  results in \(y=-x+150\). Substituting this expression for  y  into the second equation results in: 

$$ \begin{split}  10x+6(-x+150)&=1350 \\ \\ 10x-6x+900&=1350 \\ \\ 4x&=450\\ \\ x&=112.5\\ \\ \end{split}  $$

We then substitute this value for  x  into our expression for  y: 

$$ \begin{split}  y&=-x+150 \\ \\ &=-112.50+150 \\ \\ &=37.5\\ \\ \end{split}  $$

Since we can’t have fractional ticket sales, we can say that approximately 112 adult tickets were sold and 38 child tickets were sold.

Appreciating Real Life Algebra Problems

While algebra is often seen as an abstract topic, I am hopeful that I have shown you just how applicable it can be to real-life situations! Some of these examples you may have even encountered in your own life!

Even if you aren’t drawing up complex equations and solving them while you are playing basketball, combining basic math and problem solving is one of the most important skills people can have in both their work and their lives. 

I hope that I have helped you further your understanding of algebra, while growing an appreciation for the different ways it can be used in your own life!

Did you find this guide to real life algebra problems helpful? Share this post and subscribe to Math By The Pixel on YouTube for more helpful mathematics content!

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Praxis Core Math

Course: praxis core math   >   unit 1.

• Algebraic properties | Lesson
• Algebraic properties | Worked example
• Solution procedures | Lesson
• Solution procedures | Worked example
• Equivalent expressions | Lesson
• Equivalent expressions | Worked example
• Creating expressions and equations | Lesson
• Creating expressions and equations | Worked example

Algebraic word problems | Lesson

• Algebraic word problems | Worked example
• Linear equations | Lesson
• Linear equations | Worked example
• Quadratic equations | Lesson
• Quadratic equations | Worked example

What are algebraic word problems?

What skills are needed.

• Translating sentences to equations
• Solving linear equations with one variable
• Evaluating algebraic expressions
• Solving problems using Venn diagrams

How do we solve algebraic word problems?

• Define a variable.
• Write an equation using the variable.
• Solve the equation.
• If the variable is not the answer to the word problem, use the variable to calculate the answer.

What's a Venn diagram?

• 7 + 10 − 13 = 4 ‍   brought both food and drinks.
• 7 − 4 = 3 ‍   brought only food.
• 10 − 4 = 6 ‍   brought only drinks.
• Your answer should be
• an integer, like 6 ‍  
• a simplified proper fraction, like 3 / 5 ‍  
• a simplified improper fraction, like 7 / 4 ‍  
• a mixed number, like 1   3 / 4 ‍  
• an exact decimal, like 0.75 ‍  
• a multiple of pi, like 12   pi ‍   or 2 / 3   pi ‍  
• (Choice A)   $ 4 ‍   A $ 4 ‍  
• (Choice B)   $ 5 ‍   B $ 5 ‍  
• (Choice C)   $ 9 ‍   C $ 9 ‍  
• (Choice D)   $ 14 ‍   D $ 14 ‍  
• (Choice E)   $ 20 ‍   E $ 20 ‍  
• (Choice A)   10 ‍   A 10 ‍  
• (Choice B)   12 ‍   B 12 ‍  
• (Choice C)   24 ‍   C 24 ‍  
• (Choice D)   30 ‍   D 30 ‍  
• (Choice E)   32 ‍   E 32 ‍  
• (Choice A)   4 ‍   A 4 ‍  
• (Choice B)   10 ‍   B 10 ‍  
• (Choice C)   14 ‍   C 14 ‍  
• (Choice D)   18 ‍   D 18 ‍  
• (Choice E)   22 ‍   E 22 ‍  

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problem-based assignments solve real-world issues.

Thanks to its partnership with publisher Eye on Education, EducationWorld is pleased to present this blog post by Eye On Education's Senior Editor, Lauren Davis .

Last month, I blogged about the differences among problem-based, project-based, and inquiry-based learning . Let’s take a closer look at problem-based learning, where students work on investigating and resolving a real-world problem. I’ve spoken to a lot of teachers who want to try this method in the classroom but aren’t sure how to get started. Nancy Sulla to the rescue! In her book, Students Taking Charge: Inside the Learner-Active, Technology-Infused Classroom , she offers the following clear, step-by-step guidelines for creating problem-based assignments.

Start with the standards . Identify the content, skills and concepts you plan to cover in a specified unit of time, such as a three-week period.

Think application . Ask yourself, “When would someone use this knowledge in the real world? So what? Why?”

Think authenticity and relevancy . Students always want to know why they have to learn something. Authentic assignments show them why! Students are more motivated when they’re given work that is realistic and that reflects something they might actually have to do in the real world, beyond school doors. Here’s what inauthentic vs. authentic assignments look like:

Inauthentic : Create a model of an ecosystem and describe the lifecycle and food-chain relations to it.   Authentic : A local building contractor is planning to bulldoze all of the trees in a nearby copse. These are some local groups appealing to save the trees. Your job is to identify one type of tree indigenous to the area and design a presentation to convince the contractor to spare the trees based on the impact it will have on the local ecosystem (Sulla, 23).

Think open-endedness . In other words, make sure your assignment has no one right answer. For example, if you are teaching topography of a certain area, you could have students create a salt-and-flour map. Students would have to do research and create a replica, and they might have fun doing so, but they’re really just regurgitating knowledge. Instead, "asking a student to determine where the next airport should be built is an open-ended problem that exists in the realms of both the known and unknown. Students research the area, including the location of existing airports. They must then propose a location for the next airport, and substantiate their decision. No one right answer exists, as the 'right answer' is yet unknown" (Sulla, 24).

Think product . Consider how students will present their solution to the problem. “Avoid thinking along the lines of a project, with ‘glue and glitter flying.’ A product could be a poem, a persuasive letter, a webpage, an annotated bibliography, or a series of graphs, as well as the more project-oriented posters, skits and multimedia presentations" (Sulla, 24).

Think content . Make sure you don’t let the bells and whistles of a multimedia presentation overwhelm the actual content. Yes, it’s important for students to present the content in an engaging way, but you don’t want students to spend so much time looking for a fun visual that they’re not really learning the heart of the material.

Here are some additional examples of problem-based learning assignments:

Science example : Some researchers think that we can use certain types of bacteria to clean up radioactive pollution in water. Not everyone agrees. Have students use the scientific method, evaluate data on bacteria, and decide how one bacteria or a combination of them would work effectively as microscopic radioactive pollution eaters. Then have students decide how to present their work (Sulla, 150).

Math example : Indoor ski slopes are gaining popularity. Have students explore existing indoor slopes and develop a plan for their own. They should use their knowledge of coordinate planes, slopes, lines and graphing linear equations. They should design three unique ski slopes for varying abilities, draw them on graph paper and include the corresponding linear equations.

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Sign up for our free weekly newsletter and receive top education news, lesson ideas, teaching tips and more! No thanks, I don't need to stay current on what works in education! COPYRIGHT 1996-2016 BY EDUCATION WORLD, INC. ALL RIGHTS RESERVED. COPYRIGHT 1996 - 2024 BY EDUCATION WORLD, INC. ALL RIGHTS RESERVED. SchoolNotes.com The Educator's Network October 1, 2018 To Solve Real-World Problems, We Need Interdisciplinary Science Solving today’s complex, global problems will take interdisciplinary science By Graham A. J. Worthy & Cherie L. Yestrebsky T he Indian River Lagoon, a shallow estuary that stretches for 156 miles along Florida's eastern coast, is suffering from the activities of human society. Poor water quality and toxic algal blooms have resulted in fish kills, manatee and dolphin die-offs, and takeovers by invasive species. But the humans who live here have needs, too: the eastern side of the lagoon is buffered by a stretch of barrier islands that are critical to Florida's economy, tourism and agriculture, as well as for launching NASA missions into space. As in Florida, many of the world's coastlines are in serious trouble as a result of population growth and the pollution it produces. Moreover, the effects of climate change are accelerating both environmental and economic decline. Given what is at risk, scientists like us—a biologist and a chemist at the University of Central Florida—feel an urgent need to do research that can inform policy that will increase the resiliency and sustainability of coastal communities. How can our research best help balance environmental and social needs within the confines of our political and economic systems? This is the level of complexity that scientists must enter into instead of shying away from. Although new technologies will surely play a role in tackling issues such as climate change, rising seas and coastal flooding, we cannot rely on innovation alone. Technology generally does not take into consideration the complex interactions between people and the environment. That is why coming up with solutions will require scientists to engage in an interdisciplinary team approach—something that is common in the business world but is relatively rare in universities. On supporting science journalism If you're enjoying this article, consider supporting our award-winning journalism by subscribing . By purchasing a subscription you are helping to ensure the future of impactful stories about the discoveries and ideas shaping our world today. Universities are a tremendous source of intellectual power, of course. But students and faculty are typically organized within departments, or academic silos. Scientists are trained in the tools and language of their respective disciplines and learn to communicate their findings to one another using specific jargon. When the goal of research is a fundamental understanding of a physical or biological system within a niche community, this setup makes a lot of sense. But when the problem the research is trying to solve extends beyond a closed system and includes its effects on society, silos create a variety of barriers. They can limit creativity, flexibility and nimbleness and actually discourage scientists from working across disciplines. As professors, we tend to train our students in our own image, inadvertently producing specialists who have difficulty communicating with the scientist in the next building—let alone with the broader public. This makes research silos ineffective at responding to developing issues in policy and planning, such as how coastal communities and ecosystems worldwide will adapt to rising seas. Science for the Bigger Picture As scientists who live and work in Florida, we realized that we needed to play a bigger role in helping our state—and country—make evidence-based choices when it comes to vulnerable coastlines. We wanted to make a more comprehensive assessment of both natural and human-related impacts to the health, restoration and sustainability of our coastal systems and to conduct long-term, integrated research. At first, we focused on expanding research capacity in our biology, chemistry and engineering programs because each already had a strong coastal research presence. Then, our university announced a Faculty Cluster Initiative, with a goal of developing interdisciplinary academic teams focused on solving tomorrow's most challenging societal problems. While putting together our proposal, we discovered that there were already 35 faculty members on the Orlando campus who studied coastal issues. They belonged to 12 departments in seven colleges, and many of them had never even met. It became clear that simply working on the same campus was insufficient for collaboration. So we set out to build a team of people from a wide mix of backgrounds who would work in close proximity to one another on a daily basis. These core members would also serve as a link to the disciplinary strengths of their tenure home departments. Initially, finding experts who truly wanted to embrace the team aspect was more difficult than we thought. Although the notion of interdisciplinary research is not new, it has not always been encouraged in academia. Some faculty who go in that direction still worry about whether it will threaten their recognition when applying for grants, seeking promotions or submitting papers to high-impact journals. We are not suggesting that traditional academic departments should be disbanded. On the contrary, they give the required depth to the research, whereas the interdisciplinary team gives breadth to the overall effort. Our cluster proposal was a success, and in 2019 the National Center for Integrated Coastal Research (UCF Coastal) was born. Our goal is to guide more effective economic development, environmental stewardship, hazard-mitigation planning and public policy for coastal communities. To better integrate science with societal needs, we have brought together biologists, chemists, engineers and biomedical researchers with anthropologists, sociologists, political scientists, planners, emergency managers and economists. It seems that the most creative perspectives on old problems have arisen when people with different training and life experiences are talking through issues over cups of coffee. After all, "interdisciplinary" must mean more than just different flavors of STEM. In academia, tackling the effects of climate change demands more rigorous inclusion of the social sciences—an area that has been frequently overlooked. The National Science Foundation, as well as other groups, requires that all research proposals incorporate a social sciences component, as an attempt to assess the broader implications of projects. Unfortunately, in many cases, a social scientist is simply added to a proposal only to check a box rather than to make a true commitment to allowing that discipline to inform the project. Instead social, economic and policy needs must be considered at the outset of research design, not as an afterthought. Otherwise our work might fail at the implementation stage, which means we will not be as effective as we could be in solving real-world problems. As a result, the public might become skeptical about how much scientists can contribute toward solutions. Connecting with the Public The reality is that communicating research findings to the public is an increasingly critical responsibility of scientists. Doing so has a measurable effect on how politicians prioritize policy, funding and regulations. UCF Coastal was brought into a world where science is not always respected—sometimes it is even portrayed as the enemy. There has been a significant erosion of trust in science over recent years, and we must work more deliberately to regain it. The public, we have found, wants to see quality academic research that is grounded in the societal challenges we are facing. That is why we are melding pure academic research with applied research to focus on issues that are immediate—helping a town or business recovering from a hurricane, for example—as well as long term, such as directly advising a community on how to build resiliency as flooding becomes more frequent. As scientists, we cannot expect to explain the implications of our research to the wider public if we cannot first understand one another. A benefit of regularly working side by side is that we are crafting a common language, reconciling the radically different meanings that the same words can have to a variety of specialists. Finally, we are learning to speak to one another with more clarity and understand more explicitly how our niches fit into the bigger picture. We are also more aware of culture and industry as driving forces in shaping consensus and policy. Rather than handing city planners a stack of research papers and walking away, UCF Coastal sees itself as a collaborator that listens instead of just lecturing. This style of academic mission is not only relevant to issues around climate change. It relates to every aspect of modern society, including genetic engineering, automation, artificial intelligence, and so on. The launch of UCF Coastal garnered positive attention from industry, government agencies, local communities and academics. We think that is because people do want to come together to solve problems, but they need a better mechanism for doing so. We hope to be that conduit while inspiring other academic institutions to do the same. After all, we have been told for years to "think globally, act locally" and that "all politics is local." Florida's Indian River Lagoon will be restored only if there is engagement among residents, local industries, academics, government agencies and nonprofit organizations. As scientists, it is our responsibility to help everyone involved understand that problems that took decades to create will take decades to fix. We need to present the most helpful solutions while explaining the intricacies of the trade-offs for each one. Doing so is possible only if we see ourselves as part of an interdisciplinary, whole-community approach. By listening and responding to fears and concerns, we can make a stronger case for why scientifically driven decisions will be more effective in the long run. Graham A. J. Worthy is founder and director of the National Center for Integrated Coastal Research at the University of Central Florida (UCF Coastal) and chairs the university's department of biology. His research focuses on how marine ecosystems respond to natural and anthropogenic perturbations. Cherie L. Yestrebsky is a professor in the University of Central Florida's department of chemistry. Her research expertise is in environmental chemistry and remediation of pollutants in the environment. 10 Examples Of How We Use Computational Thinking In Real-life The brain has often been compared to that of a computer and that was all because of one mental ability- Computational thinking. In essence, it is a way of solving problems, designing systems, and understanding human behavior that draws on concepts fundamental to computer science. It can also be called a thought process that is applicable to many fields, including science, engineering, medicine, humanities, and business. And it wouldn’t be wrong to say, computational thinking is a set of skills that enables people to think like computers. Hence, many educators are now more than willing to incorporate this form of thinking in regular classrooms.  Even though there are debates regarding its applicability in the education sector, computation thinking exercises a great deal of influence in our everyday lives, especially in today’s tech-driven world. Hence, the article below discusses some real-life areas that have employed the usage of computational thinking.  Computation thinking: A crucial mental skill? Computational thinking is a way of solving problems and an efficient approach to understanding the world around us. It is a valuable skill to have in today’s increasingly technology-driven world. One key aspect of computational thinking is the ability to decompose problems. This means breaking down a large, complex problem into smaller, more manageable pieces that can be tackled individually. By breaking the problem down into smaller parts, we can more easily understand the problem and identify the necessary steps to solve it. An example of using computational thinking to decompose a problem for kids is to have them plan a birthday party. Define the problem: The child wants to plan a birthday party for their friend. Invitations: Who to invite, how to create and send the invitations. Decorations: What decorations to buy or make, how to set up the decorations. Food and drinks: What food and drinks to serve, how to make or order the food and drinks. Entertainment: What games or activities to plan, how to organize and run the games or activities. Inputs: guest list, budget, party theme Outputs: invitations sent, decorations set up, food and drinks prepared, entertainment organized Develop a solution Another important component or process of computational thinking is the ability to recognize and identify patterns. This involves looking for repeating or predictable behaviors or structures within a problem or system.  In the example of planning a birthday party, computational thinking can also be used to recognize and identify patterns. Recognizing patterns in inputs: For example, the child may notice that they always invite the same group of friends to their parties and that they always have a similar budget. This pattern can help them make decisions about who to invite and what decorations to buy. Identifying patterns in outputs: After hosting a few parties, the child may notice that certain games or activities are more popular than others, or that certain foods are always a hit. By identifying these patterns, they can make decisions about what entertainment to plan and what food to serve at future parties. Recognizing patterns in problem-solving: With experience, the child may also notice patterns in their problem-solving process, such as always starting with invitations or always forgetting to plan for drinks. By recognizing these patterns, they can make a plan to address and correct them in the future. Identifying patterns in feedback: After each party, the child may notice certain feedback from guests such as always requesting a certain type of food or activity. By identifying these patterns, they can make a plan to include them in future parties. Logical reasoning is also a key component of computational thinking. This involves using logical arguments and deductive reasoning to come up with solutions to problems. It requires the ability to make inferences, draw conclusions, and evaluate the validity of arguments. In the context of planning a child’s birthday party, this could involve using logical reasoning to determine the best course of action based on a set of constraints and requirements. For example, a child can be assisted in logical reasoning to determine the best location for the party based on factors such as cost, size, and proximity to the child’s home. Additionally, they can use logical reasoning to determine the best date and time for the party based on factors such as the availability of guests and his/her schedule. Once the child has determined the best location, date, and time for the party, he/she can then use logical reasoning to make decisions about the party’s theme, decorations, food, and activities based on the preferences of the child and the guests.  Finally, computational thinking involves the ability to analyze and evaluate the results of one’s work. This includes the ability to test and debug solutions, as well as to critically assess the validity and reliability of one’s findings. In the context of planning a child’s birthday party, analyzing and evaluating results can be used to determine the success of the party and identify areas for improvement. For example, after the party, children can analyze data such as the number of guests who attended, and the total cost, and take feedback from the guests to determine if the party was successful. They can also evaluate the effectiveness of the party by assessing if the party met the goal, for example, the children had fun, the guests were entertained, and the party was within budget. This information can be used to identify areas for improvement, such as reducing costs or increasing the number of guests. Additionally, it can be used to make decisions about future parties, such as whether to have the party in the same location or to try a different location. Overall, computational thinking is a crucial mental skill and a valuable asset in today’s technological landscape that can be applied in a wide range of fields and disciplines, including computer science, engineering, business, and more. Real-life examples of computational thinking From concrete thinking to abstract thinking , each of these has plenty of practical uses that come into use on a daily basis. Similarly, here are 10 real-life examples of how computational thinking, influences various behaviors and daily activities, but may or may not have caught our attention 1. Planning a vacation Computational thinking can be used to help in planning a vacation by breaking down the process into manageable tasks, identifying patterns and commonalities, and analyzing and evaluating results. For example, travelers use abstraction to break down the planning process into smaller tasks such as choosing a destination, determining a budget, and researching accommodations. Further, Generalization is applied by identifying patterns and commonalities between different vacation options, such as cost, climate, and activities. This helps them narrow down their options and make decisions more efficiently.  Logical reasoning to determine the best time to go based on factors such as weather, crowds, and cost. After the vacation, analyzing and evaluating results can be done by assessing the vacation’s success and identifying areas for improvement, such as reducing costs or finding more activities.  2. Designing a building Architects and engineers use computational thinking to design buildings and other structures. They create models and simulations to test the stability and feasibility of different design options. For instance, abstraction can help with breaking down the design process into smaller tasks such as creating a floor plan, determining the structural system, and selecting materials. Generalization can be applied by identifying patterns and commonalities between different design options, such as building codes and regulations, energy efficiency, and aesthetic preferences. Finally, logical reasoning can be used to make decisions such as choosing between different materials based on factors such as cost, durability, and sustainability. Additionally, logical reasoning is used to determine the best building layout based on factors such as functionality, safety, and accessibility. Once the building is designed, analyzing and evaluating results can be done by assessing the building’s performance and identifying areas for improvement, such as reducing energy consumption or increasing the natural light. This information can be used to make decisions about future building designs and to plan them more efficiently. 3. Predicting the weather Predicting the weather using computational thinking is a process that involves using data, models, and algorithms to make predictions about future weather conditions. The process starts with Meteorologists collecting a large amount of data from various sources such as weather stations, satellites, and radars. This data is then analyzed and processed to identify patterns and trends that can be used to make predictions. Next, using the acquired data, mathematical models and algorithms are applied to simulate the weather conditions and make predictions. The outcome is a forecast that predicts the weather for a specific time and location. The predictions are then evaluated for their accuracy using historical data, and any errors or discrepancies are analyzed to identify areas for improvement. 4. Diagnosing diseases Medical professionals use computational thinking to analyze patient data and make diagnoses based on patterns and trends. For example, generalization is applied by identifying patterns and commonalities between different diseases and their symptoms. Next, in the diagnosis,  logical reasoning is used to make decisions such as choosing the best diagnostic test based on factors such as the patient’s symptoms and medical history. Once the diagnosis is made, analyzing and evaluating results is done by assessing the accuracy of the diagnosis and identifying areas for improvement, such as incorporating more data sources or using more advanced models. This information can be used to make decisions about future diagnoses and to improve their accuracy. 5. Detecting fraud Financial institutions use computational thinking to analyze data and identify patterns that may indicate fraudulent activity. In the case of fraud, generalization helps with the identification of different types of fraud, while logical reasoning is used to make decisions such as choosing the best method to detect fraud based on factors such as the type of fraud, the data available, and the resources. Once fraud is detected, analyzing and evaluating results can be done by assessing the effectiveness of the detection method and identifying areas for improvement, such as incorporating more data sources or using more advanced models. This information is now being implemented to make decisions about future fraud detection and to improve their accuracy. 6. Personalizing recommendations Companies like Netflix and Amazon use computational thinking to analyze customer data and make recommendations for products or content that may be of interest. For instance, AI behind companies like Netflix collects data and then uses logical reasoning in their system to suggest content and products. Such companies are always on the look for better recommendation algorithms that use computational thinking.  7. Analyzing social media trends Marketing firms use computational thinking to analyze data from social media platforms and identify trends and patterns that can inform marketing strategies. For instance, the recognition of patterns is the most effective strategy in social media campaigns. Whenever a particular song or video shows engagement, more firms jump on the bandwagon. Finally, they use analytics tools to track their engagement and profits, derived through participation in social media trends.  8. Self-driving cars Self-driving cars are an example of how computational thinking is applied in real-world technology. It uses computational thinking to analyze data from sensors and cameras to navigate roads and make decisions about when to turn, stop, or accelerate.  The problem of safely navigating a self-driving car on the road can be broken down into several smaller problems. Engineers and researchers use a variety of techniques from computer science, such as image processing, machine learning, and control theory to help the car,  perceive and understand its environment, including detecting and identifying other vehicles, pedestrians, and obstacles, determining its position and orientation on the road, and plan a safe and efficient path to its destination, and then control its motion to follow that path. 9. Robotics Computational thinking can be used to help robotics or robots in many ways. Robots are complex systems that require a combination of hardware and software to perform a variety of tasks. For example, in order to make a robot capable of performing a task, such as moving from one point to another, several sub-problems need to be solved, such as its ability to perceive and understand its environment, including detecting and identifying obstacles, and its ability to plan a safe and efficient path to its destination, and then control its motion to follow that path. Additionally, computational thinking is also used in the robot’s decision-making process, which is based on the logical reasoning of the machine.  10. Virtual assistants Virtual assistants, such as Amazon’s Alexa or Google Assistant, use computational thinking in order to understand and respond to user commands and queries. The problem of understanding and responding to user input was broken down into several smaller problems and virtual assistants were equipped with the ability to accurately transcribe spoken words into text,  understand the meaning of the user’s input and extract relevant information, determine an appropriate response based on the user’s input and the current context of the conversation, and convert the text-based response into speech. All of these features were added by carefully decomposing the problem and then formulating solutions as per computational thinking.  Computational thinking in real life is getting its due credit after decades. All thanks to the high computational thinkers that have a variety of advantages. Such thinkers apart from having used computational thinking in the above-mentioned examples, are able to identify the key components of a problem, can easily recognize patterns and relationships in data and use them to make predictions and solve problems, and think abstractly and use abstract models to represent and solve problems. Hence,  computational thinking is one of the most realistic and problem-oriented types of thinking and in real life computational thinking can be a relevant and important skill to possess.  An engineer, Maths expert, Online Tutor and animal rights activist. In more than 5+ years of my online teaching experience, I closely worked with many students struggling with dyscalculia and dyslexia. With the years passing, I learned that not much effort being put into the awareness of this learning disorder. Students with dyscalculia often misunderstood for having  just a simple math fear. This is still an underresearched and understudied subject. I am also the founder of  Smartynote -‘The notepad app for dyslexia’,  Leave a Comment Cancel reply You must be logged in to post a comment. Effective Problem-Solving Techniques in Business Problem solving is an increasingly important soft skill for those in business. The Future of Jobs Survey by the World Economic Forum drives this point home. According to this report, complex problem solving is identified as one of the top 15 skills that will be sought by employers in 2025, along with other soft skills such as analytical thinking, creativity and leadership. Dr. Amy David , clinical associate professor of management for supply chain and operations management, spoke about business problem-solving methods and how the Purdue University Online MBA program prepares students to be business decision-makers. Why Are Problem-Solving Skills Essential in Leadership Roles? Every business will face challenges at some point. Those that are successful will have people in place who can identify and solve problems before the damage is done. “The business world is constantly changing, and companies need to be able to adapt well in order to produce good results and meet the needs of their customers,” David says. “They also need to keep in mind the triple bottom line of ‘people, profit and planet.’ And these priorities are constantly evolving.” To that end, David says people in management or leadership need to be able to handle new situations, something that may be outside the scope of their everyday work. “The name of the game these days is change—and the speed of change—and that means solving new problems on a daily basis,” she says. The pace of information and technology has also empowered the customer in a new way that provides challenges—or opportunities—for businesses to respond. “Our customers have a lot more information and a lot more power,” she says. “If you think about somebody having an unhappy experience and tweeting about it, that’s very different from maybe 15 years ago. Back then, if you had a bad experience with a product, you might grumble about it to one or two people.” David says that this reality changes how quickly organizations need to react and respond to their customers. And taking prompt and decisive action requires solid problem-solving skills. What Are Some of the Most Effective Problem-Solving Methods? David says there are a few things to consider when encountering a challenge in business. “When faced with a problem, are we talking about something that is broad and affects a lot of people? Or is it something that affects a select few? Depending on the issue and situation, you’ll need to use different types of problem-solving strategies,” she says. Using Techniques There are a number of techniques that businesses use to problem solve. These can include: Five Whys : This approach is helpful when the problem at hand is clear but the underlying causes are less so. By asking “Why?” five times, the final answer should get at the potential root of the problem and perhaps yield a solution. Gap Analysis : Companies use gap analyses to compare current performance with expected or desired performance, which will help a company determine how to use its resources differently or adjust expectations. Gemba Walk : The name, which is derived from a Japanese word meaning “the real place,” refers to a commonly used technique that allows managers to see what works (and what doesn’t) from the ground up. This is an opportunity for managers to focus on the fundamental elements of the process, identify where the value stream is and determine areas that could use improvement. Porter’s Five Forces : Developed by Harvard Business School professor Michael E. Porter, applying the Five Forces is a way for companies to identify competitors for their business or services, and determine how the organization can adjust to stay ahead of the game. Six Thinking Hats : In his book of the same name, Dr. Edward de Bono details this method that encourages parallel thinking and attempting to solve a problem by trying on different “thinking hats.” Each color hat signifies a different approach that can be utilized in the problem-solving process, ranging from logic to feelings to creativity and beyond. This method allows organizations to view problems from different angles and perspectives. SWOT Analysis : This common strategic planning and management tool helps businesses identify strengths, weaknesses, opportunities and threats (SWOT). “We have a lot of these different tools,” David says. “Which one to use when is going to be dependent on the problem itself, the level of the stakeholders, the number of different stakeholder groups and so on.” Each of the techniques outlined above uses the same core steps of problem solving: Identify and define the problem Consider possible solutions Evaluate options Choose the best solution Implement the solution Evaluate the outcome Data drives a lot of daily decisions in business and beyond. Analytics have also been deployed to problem solve. “We have specific classes around storytelling with data and how you convince your audience to understand what the data is,” David says. “Your audience has to trust the data, and only then can you use it for real decision-making.” Data can be a powerful tool for identifying larger trends and making informed decisions when it’s clearly understood and communicated. It’s also vital for performance monitoring and optimization. How Is Problem Solving Prioritized in Purdue’s Online MBA? The courses in the Purdue Online MBA program teach problem-solving methods to students, keeping them up to date with the latest techniques and allowing them to apply their knowledge to business-related scenarios. “I can give you a model or a tool, but most of the time, a real-world situation is going to be a lot messier and more valuable than what we’ve seen in a textbook,” David says. “Asking students to take what they know and apply it to a case where there’s not one single correct answer is a big part of the learning experience.” Make Your Own Decision to Further Your Career An online MBA from Purdue University can help advance your career by teaching you problem-solving skills, decision-making strategies and more. Reach out today to learn more about earning an online MBA with Purdue University . If you would like to receive more information about pursuing a business master’s at the Mitchell E. Daniels, Jr. School of Business, please fill out the form and a program specialist will be in touch! Connect With Us More From Forbes Stumped five ways to hone your problem-solving skills. Share to Facebook Share to Twitter Share to Linkedin Respect the worth of other people's insights Problems continuously arise in organizational life, making problem-solving an essential skill for leaders. Leaders who are good at tackling conundrums are likely to be more effective at overcoming obstacles and guiding their teams to achieve their goals. So, what’s the secret to better problem-solving skills? 1. Understand the root cause of the problem “Too often, people fail because they haven’t correctly defined what the problem is,” says David Ross, an international strategist, founder of consultancy Phoenix Strategic Management and author of Confronting the Storm: Regenerating Leadership and Hope in the Age of Uncertainty . Ross explains that as teams grapple with “wicked” problems – those where there can be several root causes for why a problem exists – there can often be disagreement on the initial assumptions made. As a result, their chances of successfully solving the problem are low. “Before commencing the process of solving the problem, it is worthwhile identifying who your key stakeholders are and talking to them about the issue,” Ross recommends. “Who could be affected by the issue? What is the problem – and why? How are people affected?” He argues that if leaders treat people with dignity, respecting the worth of their insights, they are more likely to successfully solve problems. Best High-Yield Savings Accounts Of 2024 Best 5% interest savings accounts of 2024, 2. unfocus the mind. “To solve problems, we need to commit to making time to face a problem in its full complexity, which also requires that we take back control of our thinking,” says Chris Griffiths, an expert on creativity and innovative thinking skills, founder and CEO of software provider OpenGenius, and co-author of The Focus Fix: Finding Clarity, Creativity and Resilience in an Overwhelming World . To do this, it’s necessary to harness the power of the unfocused mind, according to Griffiths. “It might sound oxymoronic, but just like our devices, our brain needs time to recharge,” he says. “ A plethora of research has shown that daydreaming allows us to make creative connections and see abstract solutions that are not obvious when we’re engaged in direct work.” To make use of the unfocused mind in problem solving, you must begin by getting to know the problem from all angles. “At this stage, don’t worry about actually solving the problem,” says Griffiths. “You’re simply giving your subconscious mind the information it needs to get creative with when you zone out. From here, pick a monotonous or rhythmic activity that will help you to activate the daydreaming state – that might be a walk, some doodling, or even some chores.” Do this regularly, argues Griffiths, and you’ll soon find that flashes of inspiration and novel solutions naturally present themselves while you’re ostensibly thinking of other things. He says: “By allowing you to access the fullest creative potential of your own brain, daydreaming acts as a skeleton key for a wide range of problems.” 3. Be comfortable making judgment calls “Admitting to not knowing the future takes courage,” says Professor Stephen Wyatt, founder and lead consultant at consultancy Corporate Rebirth and author of Antidote to the Crisis of Leadership: Opportunity in Complexity . “Leaders are worried our teams won’t respect us and our boards will lose faith in us, but what doesn’t work is drawing up plans and forecasts and holding yourself or others rigidly to them.” Wyatt advises leaders to heighten their situational awareness – to look broadly, integrate more perspectives and be able to connect the dots. “We need to be comfortable in making judgment calls as the future is unknown,” he says. “There is no data on it. But equally, very few initiatives cannot be adjusted, refined or reviewed while in motion.” Leaders need to stay vigilant, according to Wyatt, create the capacity of the enterprise to adapt and maintain the support of stakeholders. “The concept of the infallible leader needs to be updated,” he concludes. 4. Be prepared to fail and learn “Organisations, and arguably society more widely, are obsessed with problems and the notion of problems,” says Steve Hearsum, founder of organizational change consultancy Edge + Stretch and author of No Silver Bullet: Bursting the Bubble of the Organisational Quick Fix . Hearsum argues that this tendency is complicated by the myth of fixability, namely the idea that all problems, however complex, have a solution. “Our need for certainty, to minimize and dampen the anxiety of ‘not knowing,’ leads us to oversimplify and ignore or filter out anything that challenges the idea that there is a solution,” he says. Leaders need to shift their mindset to cultivate their comfort with not knowing and couple that with being OK with being wrong, sometimes, notes Hearsum. He adds: “That means developing reflexivity to understand your own beliefs and judgments, and what influences these, asking questions and experimenting.” 5. Unleash the power of empathy Leaders must be able to communicate problems in order to find solutions to them. But they should avoid bombarding their teams with complex, technical details since these can overwhelm their people’s cognitive load, says Dr Jessica Barker MBE , author of Hacked: The Secrets Behind Cyber Attacks . Instead, she recommends that leaders frame their messages in ways that cut through jargon and ensure that their advice is relevant, accessible and actionable. “An essential leadership skill for this is empathy,” Barker explains. “When you’re trying to build a positive culture, it is crucial to understand why people are not practicing the behaviors you want rather than trying to force that behavioral change with fear, uncertainty and doubt.” Editorial Standards Reprints & Permissions Join The Conversation One Community. Many Voices. Create a free account to share your thoughts.  Forbes Community Guidelines Our community is about connecting people through open and thoughtful conversations. We want our readers to share their views and exchange ideas and facts in a safe space. In order to do so, please follow the posting rules in our site's  Terms of Service.   We've summarized some of those key rules below. Simply put, keep it civil. 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Please read the full list of posting rules found in our site's  Terms of Service. The fascination and complexity of the world’s hardest math problems What math problems could be so challenging and complex that even the most brilliant mathematicians have yet to find a solution . Interesting Engineering traffic_analyzer/iStock    Mathematics has been a fascinating and challenging subject for centuries. From the ancient Greeks to modern-day mathematicians, the pursuit of understanding and mastering math has been a source of intrigue and intellectual curiosity.  But have you ever wondered what the hardest math problem is? What could be so challenging and complex that even the most brilliant mathematicians have yet to find a solution?  This article will explore some of the hardest math problems ever posed and the different approaches mathematicians have used to solve these problems.  So, buckle up and get ready to explore some of the most challenging math problems ever! 5 hardest math problems in the world Mathematics has been around for thousands of years and has contributed to numerous fields, including science, engineering, and finance. However, some math problems have stumped even the most brilliant mathematicians for centuries.  Here are some brutally difficult math problems that once seemed impossible to solve and some that still are. The Poincaré Conjecture The Poincaré Conjecture, proposed by mathematician Henri Poincaré in 1904, is a problem that stumped the mathematics community for nearly 100 years.  It states that every connected, closed three-dimensional space is topologically equivalent to a three-dimensional sphere (S3). To understand what this means, we need to delve into the world of topology. Topology is the study of the properties of objects that remain unchanged when they are stretched, bent, or otherwise deformed. In other words, topologists are interested in the ways that objects can be transformed without tearing or breaking. The Poincaré Conjecture concerns the topology of three-dimensional spaces. A three-dimensional space is a space volume with three dimensions – length, width, and height. A sphere is a three-dimensional object with a round, curved surface.  The Poincaré Conjecture proposes that every simply-connected three-dimensional space (meaning it has no holes or voids) which is closed (meaning it has no edges or boundaries) is topologically equivalent to a three-sphere (S3) — the set of points in four-dimensional space at some fixed distance to a given point. This may seem simple, but it took over 100 years to fully prove the conjecture.  Poincaré later extended his conjecture to any dimension (n-sphere). In 1961, the American mathematician Stephen Smale showed that the conjecture is true for n ≥ 5; in 1983, the American mathematician Michael Freedman showed that it is true for n = 4, and in 2002, the Russian mathematician Grigori Perelman finally completed the solution by proving it true for n = 3. Wikimedia Commons   Perelman finally solved the problem using a combination of topology and geometry. All three mathematicians were awarded a Fields Medal , one of the highest honors in mathematics. Perelman refused his Fields Medal. He was also awarded a million-dollar prize by the Clay Mathematics Institute (CMI) of Cambridge, Mass., for solving one of the world’s most difficult mathematical problems (seven problems dubbed the Millenium Problems), which he also refused. The Poincaré Conjecture has had significant implications in the field of topology and has been described as the “holy grail” of mathematics. It has opened up new research avenues and inspired numerous mathematicians to tackle other challenging problems in the field.  The Riemann Hypothesis The Riemann Hypothesis is a mathematical conjecture proposed by the German mathematician Bernhard Riemann in 1859 that has puzzled mathematicians for over 150 years.  It states that every nontrivial zero of the Riemann zeta function has a real part of ½. The Riemann zeta function is one that can be used to represent the distribution of prime numbers. Prime numbers are only divisible by themselves and 1, such as 2, 3, 5, 7, and 11. The distribution of prime numbers has long been of interest to mathematicians, as understanding their patterns and relationships can lead to new insights into number theory and other areas of mathematics. The Riemann Hypothesis suggests a relationship exists between the distribution of prime numbers and the zeros of the Riemann zeta function. If this relationship is proven to be accurate, it could have significant implications in the field of number theory and potentially lead to discoveries in other areas of mathematics. Despite being considered one of the most important unsolved problems in mathematics, the Riemann Hypothesis is yet to be proven or disproven. Many mathematicians have attempted to solve it, but the conjecture remains elusive.  In 2002, mathematician Michael Atiyah claimed to have proof of the Riemann Hypothesis, but it is yet to be formally accepted by the mathematical community. The hypothesis is another of the seven Millennium Prize Problems set by the Clay Institute. And anyone who can establish the validity or invalidity of the Riemann hypothesis will receive a prize of $1 million. The Collatz Conjecture The Collatz conjecture, also known as the “3n + 1” problem, is a mathematical problem that involves taking any positive integer and repeatedly applying a specific set of rules until you reach the number 1. The rules are as follows: 1. If the number is even, divide it by 2. 2. If the number is odd, triple it and add 1. For example, let’s start with the number 7: 7 is odd, so we triple it and add 1 to get 22 22 is even, so we divide it by 2 to get 11 11 is odd, so we triple it and add 1 to get 34 34 is even, so we divide it by 2 to get 17 17 is odd, so we triple it and add 1 to get 52 52 is even, so we divide it by 2 to get 26 26 is even, so we divide it by 2 to get 13 13 is odd, so we triple it and add 1 to get 40 40 is even, so we divide it by 2 to get 20 20 is even, so we divide it by 2 to get 10 10 is even, so we divide it by 2 to get 5 5 is odd, so we triple it and add 1 to get 16 16 is even, so we divide it by 2 to get 8 8 is even, so we divide it by 2 to get 4 4 is even, so we divide it by 2 to get 2 2 is even, so we divide it by 2 to get 1 We have now reached the number 1, which means we can stop. This sequence of numbers we generated (7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1) is the Collatz sequence for the number 7. The Collatz conjecture states that no matter which positive integer you start with, you will always eventually reach the number 1 if you follow these rules. In other words, the conjecture claims that the Collatz sequence for any positive integer will ultimately reach the number 1. Despite many efforts, the Collatz conjecture has not yet been proven or disproven. It is considered one of the most famous unsolved problems in mathematics and has fascinated mathematicians for many years. One exciting aspect of the Collatz conjecture is that it is very simple to understand and apply. Still, so far, people are yet to be able to solve it, even the most famous mathematicians.  In 2019, mathematician Terence Tao made a breakthrough in the problem, but he subsequently explained that this was only a partial solution. The Collatz conjecture has also been studied in computer science, as it can be used to create efficient algorithms for specific types of calculations. Fermat’s Last Theorem Fermat’s Last Theorem, named after the French mathematician Pierre de Fermat , is a famous statement in mathematics, stating that there are no positive integers a, b, and c that satisfy the equation an + bn = cn for any integer value of n greater than 2.  In other words, it is impossible to find three integers that can be plugged into the equation an + bn = cn such that the equation is true for any value of n greater than 2. Fermat first stated this theorem in the margin of a math book in 1637, but he never provided proof. The theorem remained unproven for over 350 years until Andrew Wiles, a mathematician at the University of Oxford, finally published a proof in 1994. Fermat’s Last Theorem has fascinated mathematicians for centuries because it is such a simple statement that seems to defy logic. It’s hard to believe that there could be no solution to the equation an + bn = cn for any value of n greater than 2, but that’s exactly what the theorem states. So why was it so difficult to prove Fermat’s Last Theorem? Part of the reason is that it involves a type of math called number theory, which deals with the properties of integers. Proving the theorem required a deep understanding of number theory and advanced mathematical techniques like elliptic curves and modular forms. Despite the difficulty of proving Fermat’s Last Theorem, it has significantly impacted mathematics. It has inspired many mathematicians to pursue careers in number theory and led to the development of new mathematical concepts and techniques. andresr/iStock   The Continuum Hypothesis The Continuum Hypothesis is a mathematical problem involving the concept of infinity and the size of infinite sets. It was first proposed by Georg Cantor in 1878 and has remained one of the unsolvable and hardest math problems ever since. The Continuum Hypothesis asks whether there is a set of numbers larger than natural numbers (1, 2, 3, etc.) but smaller than real numbers (e.g., all numbers on the number line). This set of numbers, if it exists, would be known as the “continuum.” One way to understand the Continuum Hypothesis is to consider the concept of “cardinality,” which refers to the number of elements in a set. For example, the set of natural numbers has an infinite cardinality because it contains an infinite number of elements. The set of real numbers also has an infinite cardinality, but it is a larger infinity than the set of natural numbers. The Continuum Hypothesis suggests that no set of numbers has an infinite cardinality between the set of natural numbers and the set of real numbers. In other words, it indicates that no set of numbers is “larger” than the set of natural numbers but “smaller” than the set of real numbers. The Continuum Hypothesis has been the subject of much debate and controversy among mathematicians. Some have argued that it is simply a matter of definition – that the concept of an infinite set is too vague and ambiguous to be proven or disproven.  Others have attempted to prove or disprove the Continuum Hypothesis using various mathematical techniques, but no one has conclusively proven or disproved it. In conclusion, the world’s hardest math problems are indeed the cream of the crop when it comes to challenging the limits of human understanding and problem-solving skills. From the elusive Continuum Hypothesis to the mind-bending Riemann Hypothesis, these problems continue to stump even the most brilliant mathematicians. RECOMMENDED ARTICLES But despite their difficulty, these problems continue to inspire and motivate mathematicians to push the boundaries of what is possible.  Whether or not they are ever solved, they serve as a testament to the boundless potential of the human mind and the ever-evolving nature of our understanding of the world around us.  While some of these problems may never be fully solved, they continue to inspire and drive progress in the field of mathematics and serve as a testament to the vast and mysterious nature of this subject.  As we saw in our recent article , even seemingly complex math problems can have practical and real-world implications.  So the next time you encounter a particularly challenging math problem, don’t be discouraged – you may be on the path to solving one of the hardest math problems in the world! The Blueprint Daily Stay up-to-date on engineering, tech, space, and science news with The Blueprint. By clicking sign up, you confirm that you accept this site's Terms of Use and Privacy Policy ABOUT THE EDITOR Interesting Engineering Interesting Engineering is a diverse group of journalists, videographers, and creators that aims to help the world better understand the art and science of engineering. With a combination of innovative storytelling and bespoke content formats, we cover the latest developments and breakthroughs in engineering, science, and technology. POPULAR ARTICLES Us alert: category 2 hurricane beryl now hits texas with 87 mph winds. transportation Green monster: NASCAR unveils 1st EV racer prototype with 1000kW output Ammo atm ai-powered bullet vending machines introduced in us, mission zero emission: la gets 100,000-pound-carrying electric forklifts, related articles. 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COMMENTS 104 Examples of Real World Problems An overview of real world problems with examples. Real world problems are issues and risks that are causing losses or are likely to cause losses in the near future. This term is commonly used in science, mathematics, engineering, design, coding and other fields whereby students may be asked to propose solutions to problems that are currently relevant to people and planet as opposed to ... 39 Best Problem-Solving Examples (2024) 10. Conflict Resolution. Conflict resolution is a strategy developed to resolve disagreements and arguments, often involving communication, negotiation, and compromise. When employed as a problem-solving technique, it can diffuse tension, clear bottlenecks, and create a collaborative environment. Real-World Math Problems: 32 Genuine Examples 32 Genuine Real-World Math Problems. In this section, we present 32 authentic real-world math problems from diverse fields such as safety and security, microbiology, architecture, engineering, nanotechnology, archaeology, creativity, and more. Each of these problems meets the criteria we've outlined previously. Creative Problem Solving Examples That Solved Real World Problems The Exploring Complex Problems course focuses on how to solve complex, real world problems using creative problem solving methods. It's a deep dive into the Define and Research phases of the human-centred design cycle and demonstrates why remaining in the problem space and iterating between these two phases creates a strong foundation for ideation. STEM Projects That Tackle Real-World Problems Learn how to use STEM design process to solve real-world engineering challenges with your students. Explore examples of projects that address issues such as soil erosion, food growth, urban planning, clean water, and more. 7 Real-World Issues That Can Allow Students To Tackle Big Challenges Jobs are created and grown as we work to address the real problems facing our world and peoples. Our students are ready to tackle the problems facing our world. They have a voice. They have the tools and resources. And they are not afraid to collaborate and form new communities poised for the problem-solving work that needs to be done. Real-World Problem-Solving Projects for Students & Teachers Allowing time for thought and exploration was key to moving students toward being problem-solvers. Students began to turn to each other for opinions and, instead of rejecting the other person's thoughts, they sought to understand another viewpoint. When they fell short on a technical skill, they didn't shut down. 5 Strategies for Aligning PBL to Real-World Problem-Solving In this strategy, students engage with people outside the classroom at the beginning, middle, and end of a project to hear stories that relate to the problem context, receive guidance on the technical aspects of the content they are learning, and ask questions. 5: Groups work together in small bursts of time to solve problems. Grappling With Real-World Problems Overview. Problem-based learning (PBL) is integrated at Two Rivers Public Charter School in Washington, DC, at every grade level—pre-K through eighth grade. Students are presented with a real-world problem, undertake a series of investigations, and create a product that they present to an authentic audience as part of the Expeditionary ... Using Math to Solve Real World Problems But the underlying skills they develop in math class—like taking risks, thinking logically and solving problems—will last a lifetime and help them solve work-related and real-world problems. Here are 26 images and accompanying comebacks to share with your students to get them thinking about all the different and unexpected ways they might ... How to Use Real-World Problems to Teach Elementary School Math: 6 Tips Here are some tips for using a real world problem-solving approach to teaching math to elementary school students. 1. There's more than one right answer and more than one right method. A "real ... 10 cool projects created by kids addressing real-world problems 10 cool projects created by kids addressing real-world problems - BBC Science Focus Magazine. CEMC A PDF document highlighting these real-world applications of mathematics is available for download. Problem 1 with solution. Problem 2 with solution. The CEMC has become Canada's largest and most recognized outreach organization for promoting and creating activities and materials in mathematics and computer science. 5 Real Life Algebra Problems That You Solve Everyday Plan B: C = 45. Setting the first equation equal to the second equation will allow us to employ algebra to solve for the number of minutes that makes the two plans equal. 30 + 0.1x 30 − 30 + 0.1x 0.1x 0.1x 0.1 x = 45 = 45 − 30 = 15 = 15 0.1 = 150. Therefore, the two cell phone plans are equal when 150 minutes of total time talking are used. 10 Real World Problems Effectively Solved by the AI 7. Solving Puzzles is No Longer a Challenge. We all love solving puzzles. However, staying stuck for hours while trying to solve them is something we would never want. For example, I am addicted to Scrabble. However, it used to take a toll me because I could not think of enough words. The most frustrating part was that I used to lose every time ... Algebraic word problems Algebraic word problems are questions that require translating sentences to equations, then solving those equations. The equations we need to write will only involve. basic arithmetic operations. and a single variable. Usually, the variable represents an unknown quantity in a real-life scenario. Problem-Based Assignments Solve Real-World Issues In her book, Students Taking Charge: Inside the Learner-Active, Technology-Infused Classroom, she offers the following clear, step-by-step guidelines for creating problem-based assignments. Start with the standards. Identify the content, skills and concepts you plan to cover in a specified unit of time, such as a three-week period. Creative Problem Solving Examples That Solved Real World Problems Real-world problems are messy and complicated, and they require a creative approach to problem solving. ... Creative Problem Solving Examples Example #1: Adapting Customer Service to Evolving ... To Solve Real-World Problems, We Need Interdisciplinary Science To Solve Real-World Problems, We Need Interdisciplinary Science. Solving today's complex, global problems will take interdisciplinary science. By Graham A. J. Worthy & Cherie L. Yestrebsky. Neil ... 18 Real World Life Problems with Examples: How to Solve 5. Relational. This forms one of the biggest problem areas as real-world problems and examples. In your family, a close friend of yours or your boss or co-worker at work can be a problem with your relationship. Whatever the problem, emotions, and emotions of course play a very important role in solving such problems. App ideas that solve real-world problems Here are some common examples: ... By focusing on solving real-world problems, these app ideas have the potential to make a significant impact and enhance the lives of individuals and communities. 10 Examples Of How We Use Computational Thinking In Real-life 8. Self-driving cars. Self-driving cars are an example of how computational thinking is applied in real-world technology. It uses computational thinking to analyze data from sensors and cameras to navigate roads and make decisions about when to turn, stop, or accelerate. Effective Problem-Solving Techniques in Business Problem solving is an increasingly important soft skill for those in business. The Future of Jobs Survey by the World Economic Forum drives this point home. According to this report, complex problem solving is identified as one of the top 15 skills that will be sought by employers in 2025, along with other soft skills such as analytical thinking, creativity and leadership. Real-World Problems, and How Data Helps Us Solve Them Nov 23, 2023. --. With the constant buzz around new tools and cutting-edge models, it's easy to lose sight of a basic truth: the real value in leveraging data lies in its ability to bring about tangible positive change. Whether it's around complex business decisions or our everyday routines, data-informed solutions are only as good as the ... 14 Business Ideas That Solve Problems to Inspire Your Next Venture 14 businesses that were founded to solve a problem. 1. The ADU Guide. Startup story: "My journey began when I came across the widespread issue of limited housing options. Recognizing the need for ... Stumped? Five Ways To Hone Your Problem-Solving Skills Respect the worth of other people's insights. getty. Problems continuously arise in organizational life, making problem-solving an essential skill for leaders. The fascination and complexity of the world's hardest math problems For example, let's start with the number 7: ... it comes to challenging the limits of human understanding and problem-solving skills. ... complex math problems can have practical and real-world ... assignment essay homework paper phd report review speech thesis