5 steps in problem solving in mathematics

Problem Analysis in Math: Using the 5-Step Problem-Solving Approach

This blog will explore the ways in which problem analysis of student mathematics difficulties can be applied within a problem-solving approach. We will review the core principles of a multi-tier system of support (MTSS) framework, identify the steps within a problem-solving approach, and explore the ways in which problem analysis helps to inform intervention development within the context of mathematics instruction.

Core Principles of an MTSS Framework

A variety of definitions of an MTSS framework exist within the field of education; however, several common principles are apparent and have helped to shape much of the work within this area. The National Association of State Directors of Special Education (Batsche et al., 2006) define eight core principles that capture some of the most important aspects and core beliefs of an MTSS framework:

• We can effectively teach all children.
• Intervene early.
• Use a multi-tier model of service delivery.
• Use a problem-solving model to make decisions within a multi-tier model.
• Use scientific, research-based validated intervention and instruction to the extent available.
• Monitor student progress to inform instruction.
• Use data to make decisions.
• Use assessment for screening, diagnostics, and progress monitoring.

The fourth core principle refers to utilizing a problem-solving model to make decisions. More specifically, educators and administrators should use a clearly defined problem-solving process that guides their team in identifying the problem, analyzing the size and effect of the problem, developing a plan for intervention to address the problem, implementing the plan, and examining the effectiveness of the intervention plan.

5-Step Problem-Solving Approach

The problem-solving approach utilized by the FastBridge Learning ® system includes the following five steps:

Problem identification

• Problem analysis
• Plan development
• Plan implementation
• Plan evaluation

Following this 5-step problem-solving approach helps to guide school teams of educators and administrators in engaging in data-based decision making: a core principle of the MTSS framework.

The first step in this problem-solving approach is problem identification. Christ and Arañas (2014) define a problem as a discrepancy between observed and expected performance. Regularly scheduled universal screening plays an important role in problem identification. The FastBridge Learning ® system offers a variety of screening measures for mathematics, which are summarized in the table below. While the results of regularly scheduled universal screening help to inform whether or not a discrepancy between observed and expected performance exists, problem analysis aims to identify the size and effects of the problem.

Problem Analysis

After a problem has been identified, the problem must be defined through the method of problem analysis. Only after a problem has been sufficiently analyzed and defined, can the significance of a problem be understood (Brown-Chidsey & Bickford, 2016). Problem analysis can occur at both the individual level and the group level.

Problem Analysis at the Individual Leve l

At the individual level, a discrepancy between observed and expected performance may appear within an individual’s universal screening results. FastBridge Learning ® Individual Skills Reports can provide detailed information to support problem analysis. The Individual Skills Report provides a snapshot of a given student’s risk, relative to benchmark goals, and also provides detailed results on an item-by-item basis. This item-by-item analysis provides insight into the skills the student demonstrates and which skills may require additional instructional support.

If the student’s performance is relatively close to the benchmark goal, the problem is likely to be understood as a minor problem. Item-by-item analysis can help a teacher determine if targeted reteaching of specific content may help the student to reach the goal, or if more intensive intervention may be necessary. An example of a FastBridge Learning ® Individual Skills Report for the earlyMath subtest Numeral Identification (Kindergarten) may be seen below.

The above report suggests that the sample student is at “some risk” for difficulty with numeral identification. It also reveals that the student was able to identify the given numerals with 91% accuracy. Additionally, the item-by-item analysis indicates that the student struggled to identify the following numerals: 12, 14, and 19. In this case, problem analysis may suggest that this is a minor problem, which may be remedied through targeted reteaching of the numerals 12, 14, and 19. In contrast, if during problem analysis, an Individual Skills Report for Numeral Identification indicated very low accuracy and a substantial number of misidentified numerals, the results would suggest a more significant problem. Significant problems warrant planning for more intensive intervention.

Problem analysis at the group level

At the group level, problem analysis seeks to determine the size and effects of a problem at the class, grade, or school-wide level. FastBridge Learning ® Group Screening Reports and Group Skills Reports help to provide insight about groups of students at risk for learning difficulties. An example of a Group Screening Report for earlyMath may be seen below.

The Group Screening Report pictured above indicates that approximately 77% of Ms. Horst’s kindergarten class scored at or above the benchmark goal on the earlyMath Composite during the fall benchmark period, while 6% and 18% were identified as being at “some risk” and “high risk” respectively.

All students should demonstrate growth in math achievement throughout the school year. If there are students who met the fall benchmark for mathematics, but not the winter and/or spring benchmarks, problem analysis must occur. In Ms. Horst’s class, we can see by the spring benchmark period, only 12% of her kindergarten class scored at or above the benchmark goal on the earlyMath composite, while 18% and 71% were identified as being at “some risk” and “high risk” respectively.

Problem analysis at the group level may aim to answer questions such as the following:

• What type of mathematics instruction was provided for these students?
• Did other kindergarten classrooms within the school experience a similar pattern of mathematics difficulty?
• With which specific mathematics skills (e.g., decomposing, number sequencing, numeral identification) did the group demonstrate success and difficulty?
• How close to the end-of-year mathematics goals are the students now?

In order to complete a thorough problem analysis, supplemental data about student performance and instructional practices may need to be collected.

For instance, classroom observations and/or a teacher interview would provide insight into the first question. Group Screening Reports for the other kindergarten classrooms in the school could help identify if this is a schoolwide pattern of difficulty or a classroom-specific pattern of difficulty. An analysis of the Group Skills Report for Ms. Horst’s class would provide insight into students’ strengths and weaknesses across the earlyMath Composite’s subtests. Additional, targeted, follow-up assessment using selected FastBridge Learning ® earlyMath subtests could help to gather information about the final question.

The goal of problem analysis is to determine the significance of a problem and to develop a hypothesis about why a problem has occurred. The information gathered during the problem analysis stage is then used to inform the third step of the 5-step problem-solving approach: plan development.

Batsche, G., Elliott, J., Graden, J. L., Grimes, J., Kovaleski, J. F., Prasse, D., Schrag, J., & Tilly, W.D. (2006). Response to intervention: Policy considerations and implementation. Alexandria, VA: National Association of State Directors of Special Education, Inc.

Brown-Chidsey, R. and Bickford, R. (2016). Practical handbook of multi-tiered systems of support: Building academic and behavioral success in schools. New York, NY: The Guilford Press.

Christ, T.J., & Arañas, Y.A. (2014). Best practices in problem analysis. In A. Thomas & J. Grimes (Eds.), Best Practices in School Psychology VI . Bethesda, MD: National Association of School Psychologists.

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What can QuickMath do?

QuickMath will automatically answer the most common problems in algebra, equations and calculus faced by high-school and college students.

• The algebra section allows you to expand, factor or simplify virtually any expression you choose. It also has commands for splitting fractions into partial fractions, combining several fractions into one and cancelling common factors within a fraction.
• The equations section lets you solve an equation or system of equations. You can usually find the exact answer or, if necessary, a numerical answer to almost any accuracy you require.
• The inequalities section lets you solve an inequality or a system of inequalities for a single variable. You can also plot inequalities in two variables.
• The calculus section will carry out differentiation as well as definite and indefinite integration.
• The matrices section contains commands for the arithmetic manipulation of matrices.
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10 Best Strategies for Solving Math Word Problems

1. Understand the Problem by Paraphrasing

2. identify key information and variables, 3. translate words into mathematical symbols, 4. break down the problem into manageable parts, 5. draw diagrams or visual representations, 6. use estimation to predict answers, 7. apply logical reasoning for unknown variables, 8. leverage similar problems as templates, 9. check answers in the context of the problem, 10. reflect and learn from mistakes.

Have you ever observed the look of confusion on a student’s face when they encounter a math word problem ? It’s a common sight in classrooms worldwide, underscoring the need for effective strategies for solving math word problems . The main hurdle in solving math word problems is not just the math itself but understanding how to translate the words into mathematical equations that can be solved.

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Generic advice like “read the problem carefully” or “practice more” often falls short in addressing students’ specific difficulties with word problems. Students need targeted math word problem strategies that address the root of their struggles head-on. 

A Guide on Steps to Solving Word Problems: 10 Strategies 

One of the first steps in tackling a math word problem is to make sure your students understand what the problem is asking. Encourage them to paraphrase the problem in their own words. This means they rewrite the problem using simpler language or break it down into more digestible parts. Paraphrasing helps students grasp the concept and focus on the problem’s core elements without getting lost in the complex wording.

Original Problem: “If a farmer has 15 apples and gives away 8, how many does he have left?”

Paraphrased: “A farmer had some apples. He gave some away. Now, how many apples does he have?”

This paraphrasing helps students identify the main action (giving away apples) and what they need to find out (how many apples are left).

Play these subtraction word problem games in the classroom for free:

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Students often get overwhelmed by the details in word problems. Teach them to identify key information and variables essential for solving the problem. This includes numbers , operations ( addition , subtraction , multiplication , division ), and what the question is asking them to find. Highlighting or underlining can be very effective here. This visual differentiation can help students focus on what’s important, ignoring irrelevant details.

• Encourage students to underline numbers and circle keywords that indicate operations (like ‘total’ for addition and ‘left’ for subtraction).
• Teach them to write down what they’re solving for, such as “Find: Total apples left.”

Problem: “A classroom has 24 students. If 6 more students joined the class, how many students are there in total?”

Key Information:

• Original number of students (24)
• Students joined (6)
• Looking for the total number of students

Here are some fun addition word problems that your students can play for free:

The transition from the language of word problems to the language of mathematics is a critical skill. Teach your students to convert words into mathematical symbols and equations. This step is about recognizing keywords and phrases corresponding to mathematical operations and expressions .

Common Translations:

• “Total,” “sum,” “combined” → Addition (+)
• “Difference,” “less than,” “remain” → Subtraction (−)
• “Times,” “product of” → Multiplication (×)
• “Divided by,” “quotient of” → Division (÷)
• “Equals” → Equals sign (=)

Problem: “If one book costs $5, how much would 4 books cost?”

Translation: The word “costs” indicates a multiplication operation because we find the total cost of multiple items. Therefore, the equation is 4 × 5 = $20

Complex math word problems can often overwhelm students. Incorporating math strategies for problem solving, such as teaching them to break down the problem into smaller, more manageable parts, is a powerful approach to overcome this challenge. This means looking at the problem step by step rather than simultaneously trying to solve it. Breaking it down helps students focus on one aspect of the problem at a time, making finding the solution more straightforward.

Problem: “John has twice as many apples as Sarah. If Sarah has 5 apples, how many apples do they have together?”

Steps to Break Down the Problem:

Find out how many apples John has: Since John has twice as many apples as Sarah, and Sarah has 5, John has 5 × 2 = 10

Calculate the total number of apples: Add Sarah’s apples to John’s to find the total,  5 + 10 = 15

By splitting the problem into two parts, students can solve it without getting confused by all the details at once.

Explore these fun multiplication word problem games:

Diagrams and visual representations can be incredibly helpful for students, especially when dealing with spatial or quantity relationships in word problems. Encourage students to draw simple sketches or diagrams to represent the problem visually. This can include drawing bars for comparison, shapes for geometry problems, or even a simple distribution to better understand division or multiplication problems .

Problem: “A garden is 3 times as long as it is wide. If the width is 4 meters, how long is the garden?”

Visual Representation: Draw a rectangle and label the width as 4 meters. Then, sketch the length to represent it as three times the width visually, helping students see that the length is 4 × 3 = 12

Estimation is a valuable skill in solving math word problems, as it allows students to predict the answer’s ballpark figure before solving it precisely. Teaching students to use estimation can help them check their answers for reasonableness and avoid common mistakes.

Problem: “If a book costs $4.95 and you buy 3 books, approximately how much will you spend?”

Estimation Strategy: Round $4.95 to the nearest dollar ($5) and multiply by the number of books (3), so 5 × 3 = 15. Hence, the estimated total cost is about $15.

Estimation helps students understand whether their final answer is plausible, providing a quick way to check their work against a rough calculation.

Check out these fun estimation and prediction word problem worksheets that can be of great help:

When students encounter problems with unknown variables, it’s crucial to introduce them to logical reasoning. This strategy involves using the information in the problem to deduce the value of unknown variables logically. One of the most effective strategies for solving math word problems is working backward from the desired outcome. This means starting with the result and thinking about the steps leading to that result, which can be particularly useful in algebraic problems.

Problem: “A number added to three times itself equals 32. What is the number?”

Working Backward:

Let the unknown number be x.

The equation based on the problem is  x + 3x = 32

Solve for x by simplifying the equation to 4x=32, then dividing by 4 to find x=8.

By working backward, students can more easily connect the dots between the unknown variable and the information provided.

Practicing problems of similar structure can help students recognize patterns and apply known strategies to new situations. Encourage them to leverage similar problems as templates, analyzing how a solved problem’s strategy can apply to a new one. Creating a personal “problem bank”—a collection of solved problems—can be a valuable reference tool, helping students see the commonalities between different problems and reinforcing the strategies that work.

Suppose students have solved a problem about dividing a set of items among a group of people. In that case, they can use that strategy when encountering a similar problem, even if it’s about dividing money or sharing work equally.

It’s essential for students to learn the habit of checking their answers within the context of the problem to ensure their solutions make sense. This step involves going back to the original problem statement after solving it to verify that the answer fits logically with the given information. Providing a checklist for this process can help students systematically review their answers.

Checklist for Reviewing Answers:

• Re-read the problem: Ensure the question was understood correctly.
• Compare with the original problem: Does the answer make sense given the scenario?
• Use estimation: Does the precise answer align with an earlier estimation?
• Substitute back: If applicable, plug the answer into the problem to see if it works.

Problem: “If you divide 24 apples among 4 children, how many apples does each child get?”

After solving, students should check that they understood the problem (dividing apples equally).

Their answer (6 apples per child) fits logically with the number of apples and children.

Their estimation aligns with the actual calculation.

Substituting back 4×6=24 confirms the answer is correct.

Teaching students to apply logical reasoning, leverage solved problems as templates, and check their answers in context equips them with a robust toolkit for tackling math word problems efficiently and effectively.

One of the most effective ways for students to improve their problem-solving skills is by reflecting on their errors, especially with math word problems. Using word problem worksheets is one of the most effective strategies for solving word problems, and practicing word problems as it fosters a more thoughtful and reflective approach to problem-solving

These worksheets can provide a variety of problems that challenge students in different ways, allowing them to encounter and work through common pitfalls in a controlled setting. After completing a worksheet, students can review their answers, identify any mistakes, and then reflect on them in their mistake journal. This practice reinforces mathematical concepts and improves their math problem solving strategies over time.

3 Additional Tips for Enhancing Word Problem-Solving Skills

Before we dive into the importance of reflecting on mistakes, here are a few impactful tips to enhance students’ word problem-solving skills further:

1. Utilize Online Word Problem Games

Incorporate online games that focus on math word problems into your teaching. These interactive platforms make learning fun and engaging, allowing students to practice in a dynamic environment. Games can offer instant feedback and adaptive challenges, catering to individual learning speeds and styles.

Here are some word problem games that you can use for free:

2. Practice Regularly with Diverse Problems

Consistent practice with a wide range of word problems helps students become familiar with different questions and mathematical concepts. This exposure is crucial for building confidence and proficiency.

Start Practicing Word Problems with these Printable Word Problem Worksheets:

3. Encourage Group Work

Solving word problems in groups allows students to share strategies and learn from each other. A collaborative approach is one of the best strategies for solving math word problems that can unveil multiple methods for tackling the same problem, enriching students’ problem-solving toolkit.

Conclusion 

Mastering math word problems is a journey of small steps. Encourage your students to practice regularly, stay curious, and learn from their mistakes. These strategies for solving math word problems are stepping stones to turning challenges into achievements. Keep it simple, and watch your students grow their confidence and skills, one problem at a time.

Frequently Asked Questions (FAQs)

How can i help my students stay motivated when solving math word problems.

Encourage small victories and use engaging tools like online games to make practice fun and rewarding.

What's the best way to teach beginners word problems?

Begin with simple problems that integrate everyday scenarios to make the connection between math and real-life clear and relatable.

How often should students practice math word problems?

Regular, daily practice with various problems helps build confidence and problem-solving skills over time.

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Problem Solving in Mathematics

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The main reason for learning about math is to become a better problem solver in all aspects of life. Many problems are multistep and require some type of systematic approach. There are a couple of things you need to do when solving problems. Ask yourself exactly what type of information is being asked for: Is it one of addition, subtraction, multiplication , or division? Then determine all the information that is being given to you in the question.

Mathematician George Pólya’s book, “ How to Solve It: A New Aspect of Mathematical Method ,” written in 1957, is a great guide to have on hand. The ideas below, which provide you with general steps or strategies to solve math problems, are similar to those expressed in Pólya’s book and should help you untangle even the most complicated math problem.

Use Established Procedures

Learning how to solve problems in mathematics is knowing what to look for. Math problems often require established procedures and knowing what procedure to apply. To create procedures, you have to be familiar with the problem situation and be able to collect the appropriate information, identify a strategy or strategies, and use the strategy appropriately.

Problem-solving requires practice. When deciding on methods or procedures to use to solve problems, the first thing you will do is look for clues, which is one of the most important skills in solving problems in mathematics. If you begin to solve problems by looking for clue words, you will find that these words often indicate an operation.

Look for Clue Words

Think of yourself as a math detective. The first thing to do when you encounter a math problem is to look for clue words. This is one of the most important skills you can develop. If you begin to solve problems by looking for clue words, you will find that those words often indicate an operation.

Common clue words for addition  problems:

Common clue words for  subtraction  problems:

• How much more

Common clue words for multiplication problems:

Common clue words for division problems:

Although clue words will vary a bit from problem to problem, you'll soon learn to recognize which words mean what in order to perform the correct operation.

Read the Problem Carefully

This, of course, means looking for clue words as outlined in the previous section. Once you’ve identified your clue words, highlight or underline them. This will let you know what kind of problem you’re dealing with. Then do the following:

• Ask yourself if you've seen a problem similar to this one. If so, what is similar about it?
• What did you need to do in that instance?
• What facts are you given about this problem?
• What facts do you still need to find out about this problem?

Develop a Plan and Review Your Work

Based on what you discovered by reading the problem carefully and identifying similar problems you’ve encountered before, you can then:

• Define your problem-solving strategy or strategies. This might mean identifying patterns, using known formulas, using sketches, and even guessing and checking.
• If your strategy doesn't work, it may lead you to an ah-ha moment and to a strategy that does work.

If it seems like you’ve solved the problem, ask yourself the following:

• Does your solution seem probable?
• Does it answer the initial question?
• Did you answer using the language in the question?
• Did you answer using the same units?

If you feel confident that the answer is “yes” to all questions, consider your problem solved.

Tips and Hints

Some key questions to consider as you approach the problem may be:

• What are the keywords in the problem?
• Do I need a data visual, such as a diagram, list, table, chart, or graph?
• Is there a formula or equation that I'll need? If so, which one?
• Will I need to use a calculator? Is there a pattern I can use or follow?

Read the problem carefully, and decide on a method to solve the problem. Once you've finished working the problem, check your work and ensure that your answer makes sense and that you've used the same terms and or units in your answer.

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Intermediate Algebra Tutorial 8

• Use Polya's four step process to solve word problems involving numbers, percents, rectangles, supplementary angles, complementary angles, consecutive integers, and breaking even. 

Whether you like it or not, whether you are going to be a mother, father, teacher, computer programmer, scientist, researcher, business owner, coach, mathematician, manager, doctor, lawyer, banker (the list can go on and on),  problem solving is everywhere.  Some people think that you either can do it or you can't.  Contrary to that belief, it can be a learned trade.  Even the best athletes and musicians had some coaching along the way and lots of practice.  That's what it also takes to be good at problem solving.

George Polya , known as the father of modern problem solving, did extensive studies and wrote numerous mathematical papers and three books about problem solving.  I'm going to show you his method of problem solving to help step you through these problems.

If you follow these steps, it will help you become more successful in the world of problem solving.

Polya created his famous four-step process for problem solving, which is used all over to aid people in problem solving:

Step 1: Understand the problem.  

Step 2:   Devise a plan (translate).  

Step 3:   Carry out the plan (solve).  

Step 4:   Look back (check and interpret).  

Just read and translate it left to right to set up your equation

Since we are looking for a number, we will let 

x = a number

*Get all the x terms on one side

*Inv. of sub. 2 is add 2  

FINAL ANSWER:  The number is 6.

We are looking for two numbers, and since we can write the one number in terms of another number, we will let

x = another number 

ne number is 3 less than another number:

x - 3 = one number

*Inv. of sub 3 is add 3

*Inv. of mult. 2 is div. 2  

FINAL ANSWER:  One number is 90. Another number is 87.

When you are wanting to find the percentage of some number, remember that ‘of ’ represents multiplication - so you would multiply the percent (in decimal form) times the number you are taking the percent of.

We are looking for a number that is 45% of 125,  we will let

x = the value we are looking for

FINAL ANSWER:  The number is 56.25.

We are looking for how many students passed the last math test,  we will let

x = number of students 

FINAL ANSWER: 21 students passed the last math test.

We are looking for the price of the tv before they added the tax,  we will let

x = price of the tv before tax was added. 

*Inv of mult. 1.0825 is div. by 1.0825

FINAL ANSWER: The original price is $500.

Perimeter of a Rectangle = 2(length) + 2(width)

We are looking for the length and width of the rectangle.  Since length can be written in terms of width, we will let

length is 1 inch more than 3 times the width:

1 + 3 w = length

*Inv. of add. 2 is sub. 2

*Inv. of mult. by 8 is div. by 8  

FINAL ANSWER: Width is 3 inches. Length is 10 inches.

Complimentary angles sum up to be 90 degrees.

We are already given in the figure that

x = one angle

5 x = other angle

*Inv. of mult. by 6 is div. by 6

FINAL ANSWER: The two angles are 30 degrees and 150 degrees.

If we let x represent the first integer, how would we represent the second consecutive integer in terms of x ?  Well if we look at 5, 6, and 7 - note that 6 is one more than 5, the first integer. 

In general, we could represent the second consecutive integer by x + 1 .  And what about the third consecutive integer. 

Well, note how 7 is 2 more than 5.  In general, we could represent the third consecutive integer as x + 2.

Consecutive EVEN integers are even integers that follow one another in order.     

If we let x represent the first EVEN integer, how would we represent the second consecutive even integer in terms of x ?   Note that 6 is two more than 4, the first even integer. 

In general, we could represent the second consecutive EVEN integer by x + 2 . 

And what about the third consecutive even integer?  Well, note how 8 is 4 more than 4.  In general, we could represent the third consecutive EVEN integer as x + 4.

Consecutive ODD integers are odd integers that follow one another in order.     

If we let x represent the first ODD integer, how would we represent the second consecutive odd integer in terms of x ?   Note that 7 is two more than 5, the first odd integer. 

In general, we could represent the second consecutive ODD integer by x + 2.

And what about the third consecutive odd integer?  Well, note how 9 is 4 more than 5.  In general, we could represent the third consecutive ODD integer as x + 4.  

Note that a common misconception is that because we want an odd number that we should not be adding a 2 which is an even number.  Keep in mind that x is representing an ODD number and that the next odd number is 2 away, just like 7 is 2 away form 5, so we need to add 2 to the first odd number to get to the second consecutive odd number.

We are looking for 3 consecutive integers, we will let

x = 1st consecutive integer

x + 1 = 2nd consecutive integer

x + 2  = 3rd consecutive integer

*Inv. of mult. by 3 is div. by 3  

FINAL ANSWER: The three consecutive integers are 85, 86, and 87.

We are looking for 3 EVEN consecutive integers, we will let

x = 1st consecutive even integer

x + 2 = 2nd consecutive even integer

x + 4  = 3rd  consecutive even integer

*Inv. of add. 10 is sub. 10  

FINAL ANSWER: The ages of the three sisters are 4, 6, and 8.

In the revenue equation, R is the amount of money the manufacturer makes on a product.

If a manufacturer wants to know how many items must be sold to break even, that can be found by setting the cost equal to the revenue.

We are looking for the number of cd’s needed to be sold to break even, we will let

*Inv. of mult. by 10 is div. by 10

FINAL ANSWER: 5 cd’s.

To get the most out of these, you should work the problem out on your own and then check your answer by clicking on the link for the answer/discussion for that  problem .  At the link you will find the answer as well as any steps that went into finding that answer.

  Practice Problems 1a - 1g: Solve the word problem.

(answer/discussion to 1e)

http://www.purplemath.com/modules/translat.htm This webpage gives you the basics of problem solving and helps you with translating English into math.

http://www.purplemath.com/modules/numbprob.htm This webpage helps you with numeric and consecutive integer problems.

http://www.purplemath.com/modules/percntof.htm This webpage helps you with percent problems.

http://www.math.com/school/subject2/lessons/S2U1L3DP.html This website helps you with the basics of writing equations.

http://www.purplemath.com/modules/ageprobs.htm This webpage goes through examples of age problems,  which are like the  numeric problems found on this page.

Go to Get Help Outside the Classroom found in Tutorial 1: How to Succeed in a Math Class for some more suggestions.

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5 Essential Problem Solving Techniques

• Critical Thinking

In the first post in this series, I talked about the difference between solving problems and problem solving. This week, I will continue my series on problem solving and share five essential problem solving techniques for your problem solving routines.

A strong problem-solving routine is essential for helping students develop their problem-solving strategy toolboxes. Over the years, I have used a variety of routines that have helped my students develop problem-solving strategies and critical thinking skills. (Read more about my favorite routine here !) Through a lot of trial and error, I found several routines that worked well for my students. (I will share more about them next week!) Today, I want to share some trade secrets with you to help you get the most from your problem-solving routines with five essential problem-solving techniques.

Five Essential Problem Solving Techniques

1. share student thinking and strategies..

This is essential! I can’t tell you how many times I have seen teachers give a great problem solving or critical thinking task and then never allow students to share their responses. Sometimes, our students are the best teachers and they can get a message across when we struggle to do so. Also, providing an opportunity for students to talk to other students about their thinking increases math vocabulary and builds communication skills.

After students have had an opportunity to share their thinking with a group member or partner, I encourage you to discuss the task as a class. This gives the teacher an opportunity to reiterate correct thinking, modify incorrect thinking, ask questions, build math vocabulary, and increase students’ communication skills.

Read more about getting started with math talk in the classroom here .

2. Solve non-routine problems.

In an earlier blog post, I emphasized the importance of using non-routine problems with students. Not only are students typically more engaged, but students have the opportunity to use strategies beyond writing an equation/number sentence or drawing a picture. If you’re interested in some fun, non-routine tasks, please check out my Solve It! Friday page.

One of the things many people say they love about math is the fact that there is a right and wrong answer. While there certainly are wrong answers, sometimes, there can be more than one right answer. These types of tasks really stretch some kids’ thinking. They also provide a natural venue for discussion. Students can debate the answers only to discover that more than one works!

3. Discuss efficiency.

During problem-solving experiences, students will often use beautiful and complicated solution strategies to solve problems. While we want to encourage outside-of-the-box thinking, we also want students to attend to efficiency. One way to do this is to have several students share their solutions. They can then discuss what strategies are best for specific types of problems. When discussing difficulty becomes a regular part of your routine, students will begin to utilize their problem-solving strategies in a way that not only gets them to the correct answer but also using an efficient method.

4. Make connections.

Recently, I wrote about making connections as part of my Summer PD series. Read it here ! When students make connects, it deepens their understanding of other content and skills. One way to do this is to connect the problem-solving task to grade-level content and skills. Another way is to have students represent problems in a variety of ways, i.e. pictures, numbers, words, or equations. Each representation is crafted in a specific way, so being able to translate words into an equation or numbers into a picture is a big skill that has many benefits.

5. Use “high ceiling, low floor tasks.”

The term “high ceiling, low floor” refers to a task having multiple entry points to allow all students a way to access the task; however, it also includes ways to extend the tasks for those students who are ready for more of a challenge. These types of tasks increase participation because students can participate at a level that is comfortable for them. Students are also able to showcase what they can do instead of what they are unable to do. Even better, these tasks provide instant opportunities for differentiation because all students can participate in a way that allows them to be most successful.

Using a regular problem-solving routine can help students develop the tools necessary to be powerful thinkers of mathematics; however, in order to get the most from the routines, certain problem-solving techniques must be included. While you may not want to add all of the above techniques to your routine, I encourage you to commit to adding one or two of them this year. I highly recommended starting with “sharing student thinking and strategies.” It’s probably the most important technique of all of the problem-solving routines. It will get you the most bang for your buck!

Sound Off! How do get the most from your problem-solving routine? Which problem-solving techniques do you think are most important?

Shametria Routt Banks

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3 Responses

Hi, can you provide an example of a high ceiling, low floor task? Thank you!

Hi Jen! Great question! The high ceiling, low floor tasks give all students a chance to engage in the task but have places to go to extend the learning for students. One problem that comes to mind is a task where students are asked to find combinations of numbers to achieve a goal, like the following problem: Farmer Brown’s niece Angie is in charge of her uncle’s farm while he is on vacation. He gave her strict instructions to make sure none of the animals ran away. When Angie counted the pigs and chickens, she counted 32 legs. How many pigs and chickens did she count? All students should be able to determine a combination of pigs and chickens; however, what if I added a new condition to say: Angie counted a total of 12 animals. This changes the level of rigor because students are now looking for a specific combination. Some students will struggle with this but others may be ready to tackle it; so, using tasks that have a high-ceiling allow for this flexibility. Check out more high ceiling, low floor tasks here: https://www.youcubed.org/task-grades/low-floor-high-ceiling/ .

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Quadratic inequalities

Here you will learn about quadratic inequalities, including how to solve quadratic inequalities, identify solution sets using interval notation and represent solutions on a number line.

Students first learn about inequalities in 6 th grade with their work in solving linear inequalities. They expand that knowledge as they move through Algebra I and Algebra II.

What are quadratic inequalities?

Quadratic inequalities are mathematical statements that compare a quadratic expression to another value. Similar to linear inequalities, you can solve quadratic inequalities to find the range of values that makes the inequality statement true.

The strategies used to find the solutions of quadratic inequalities are the same strategies used to solve quadratic equations (factoring, using the quadratic formula, completing the square, etc).

Any quadratic equation has 0, 1 or 2 solutions only. Let’s visualize this using the quadratic equation y=x^{2}-4.

Step-by-step guide: Solving quadratic equations

Once you know the exact solutions to the quadratic equation, it is useful to plot these on a number line as shown above.

When solving any inequality, solutions are given in a range or interval, rather than an exact value.

For example, the inequality 4x>12 has the range of solutions x>3 as any value of x greater than 3 that is substituted into 4x will obtain a solution greater than 12. This can be shown on a number line:

You can see how by plotting the values for x on the number line, you divide the number line into two sets of numbers. Those that are greater than 3, and those that are less than or equal to 3. With a quadratic inequality, the number line can be divided into three sets of numbers.

Below are three alternative ways to determine the correct range of solutions for a quadratic inequality. Each refer to the example of y=x^{2}-4 outlined above.

The three alternatives are:

Visualizing solution sets using graphs

• Visualizing solution sets using the coefficient of x^2

Visualizing solution sets using substitution

[FREE] Inequalities Worksheet (Grade 1 to 7)

Use this quiz to check your grade 1 to 7 students’ understanding of inequalities. 10+ questions with answers covering a range of 1st – 7th grade inequalities topics to identify areas of strength and support!

Let’s review the example of the quadratic equation y=x^{2}-4 when y=0. The two solutions of x^{2}-4=0 are x=2 and x=- \, 2 (the points of intersection between the curve y=x^{2}-4 and the straight line y=0.

Now let’s look at the quadratic equation y=x^{2}-4 when y<0. This can be described as the quadratic inequality x^{2}-4<0.

The range of values for x that gives a solution for y<0 is between - \, 2 and 2. This is the interval (- \, 2,2). Representing this on a number line gives:

Notice that both circles are open-circles as the solutions are only less than 2 or greater than - \, 2, and not equal to.

Now let’s look at the quadratic equation y=x^{2}-4 when y\geq{0}. This can be described as the quadratic inequality x^{2}-4\geq{0}.

The range of values for x that gives a solution for y\geq{0} is less than or equal to - \, 2 and greater than or equal to 2. This is the set of solutions (- \, \infty,- \, 2]\cup[2,\infty) where is the union of the two intervals. Representing this on a number line gives:

Now the two circles are closed circles as the solutions are less than or equal to - \, 2 and greater than or equal to 2.

Visualizing solution sets using the coefficient of x²

Recall that when the coefficient of x^2 is positive, the parabola is a “ u -shape” and when the coefficient of x^2 is negative, the parabola is an “ n -shape”.

Let’s look back at the example of the quadratic equation y=x^{2}-4 when y=0. The two solutions of x^{2}-4=0 are x=2 and x=- \, 2 (the points of intersection between the curve y=x^{2}-4 and the straight line y=0.

As the quadratic y=x^{2}-4 has a positive coefficient of x^{2}\text{:}

• If y<0, then the range of solutions will be between two values. For example, x^{2}-4<0 so (- \, 2,2) .
• If y>0, then the range of solutions will be less than one value, and greater than another. For example, x^{2}-4>0 so (- \, \infty,- \, 2)\cup(2,\infty).

If the coefficient of x^{2} is negative, then the solution sets are opposite.

As the quadratic y=- \, x^{2}+4 has a negative coefficient of x^{2}\text{:}

• If y<0, then the range of solutions will be less than one value, and greater than another. For example, - \, x^{2}-4<0 so (- \, \infty,- \, 2)\cup(2,\infty).
• If y>0, then the range of solutions will be between two values. For example, - \, x^{2}-4>0 so (- \, 2,2) .

This is the same if the quadratic inequalities are also “less than or equal to”, and “greater than or equal to”.

The intervals of the two quadratics y=x^{2}-4 and y=- \, x^{2}+4 are given below when y>0 and when y<0.

You can use substitution to determine whether the solution is between two values, or the union of two intervals. Using a number line can help to visualize the correct interval(s).

Let’s look again at the example of the quadratic equation y=x^{2}-4 when y=0. The two solutions of x^{2}-4=0 are x=2 and x=- \, 2 (the points of intersection between the curve y=x^{2}-4 and the straight line y=0.

Now let’s look at the quadratic equation y=x^{2}-4 when y<0. This is the quadratic inequality x^{2}-4<0.

First, solve the quadratic equation x^{2}-4=0.

At this stage, it is not clear whether the solution is the interval (- \, 2,2) or (- \, \infty,- \, 2]\cup[2,\infty) so use substitution to find out which values of x satisfy the inequality x^{2}-4<0.

Substitute x=3 into x^{2}-4<0 to test the validity of the interval (- \, \infty,- \, 2]\cup[2,\infty).

5\nleq{0} and so this value is not within the valid range.

Conversely, substitute x=0 into x^{2}-4<0 to test its validity.

- \, 4\leq{0} and so this value is within the valid range.

The solution set is therefore (- \, 2,2).

Common Core State Standards

How does this relate to high school math?

• High School Algebra – Creating Equations (HSA-CED.A.1) Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions

How to solve quadratic inequalities

In order to solve quadratic inequalities:

Use a strategy to solve quadratic equations to find values of \textbf{x}.

Place interval limits on the number line.

Write the intervals formed by the values of \textbf{x}.

Select a test point from the bounded interval and substitute into the quadratic.

Determine which intervals represent the solution set to the inequality.

Solving quadratic equations examples

Example 1: closed points.

Determine the solution set to the quadratic inequality, x^{2}+4x+3\leq0.

This quadratic factors, so using the factoring method for solving quadratics, factor the quadratic to find the values of x.

The two solutions to the quadratic equation x^{2}+4x+3=0 are x=- \, 3 and x=- \, 1.

2 Place interval limits on the number line.

x=- \, 3 and x=- \, 1 are the limits to the solution set for x^{2}+4x+3\leq{0}. Since the inequality is “less than or equal to” use closed points to plot the values for x.

3 Write the intervals formed by the values of \textbf{x}.

The three intervals are:

4 Select a test point from the bounded interval and substitute into the quadratic.

Let x=- \, 2\text{:}

As - \, 1\leq{0}, this is a true comparison so the interval is part of solution.

5 Determine which intervals represent the solution set to the inequality.

The solution set to the quadratic inequality is [- \, 3,- \, 1].

Note: Graph with number line for reference.

Example 2: open points

Solve the quadratic inequality, x^2+7 x+10>0.

The quadratic can be factored so use the factoring strategy of solving quadratics to find the values of x.

\begin{aligned}&x^{2}+7x+10=0 \\\\ &(x+5)(x+2)=0 \\\\ &x+5=0\text{ so }x=- \, 5 \\\\ &\text{And }x+2=0\text{ so }x=- \,2 \end{aligned}

Since the inequality is “greater than”, use open points to plot the limits for x.

The intervals are:

\begin{array}{c}(- \, \infty,- \, 5) \\\\ (- \, 5,- \, 2) \\\\ (- \, 2,\infty) \end{array}

Let x=- \, 3\text{:}

x^{2}+7x+10=(- \, 3)^{2}+7(- \, 3)+10=9-21+10=- \, 2

As - \, 2\ngeq{0}, this is a false comparison so the interval is not part of solution.

The solution set to the quadratic inequality is (- \, \infty,- \, 5)\cup(- \, 2,\infty).

Example 3: rearrange quadratic

Solve the quadratic inequality, 5x^{2}-27x\leq{- \, 10}.

Before solving the quadratic for the x values, first rearrange the quadratic so that one side has a value of 0. In this case, add 10 to both sides of the inequality.

5x^{2}-27x+10\leq{0}

The quadratic can be factored to find the values of x. So use the strategy of factoring to find the values of x.

\begin{aligned}&5x^{2}-27x+10=0 \\\\ &(5x-2)(x-5)=0 \\\\ &5x-2=0\text{ so }x= \, \cfrac{2}{5} \\\\ &\text{And }x-5=0\text{ so }x=5 \end{aligned}

Since the inequality is “less than or equal to” use solid points to plot the limits of x.

Let x=1\text{:}

5x^{2}-27x+10=5(1)^{2}-27(1)+10=5-27+10=- \, 12

As - \, 12\leq{0}, this is a true comparison so the interval is part of solution.

The solution set to the quadratic inequality is \left[\cfrac{2}{5}, 5\right].

Example 4: two intervals in solution set

Find the solution to the quadratic inequality, 2x^{2}+4x+4>- \, 5x.

Before solving the quadratic for the x values, first rearrange the quadratic so that one side has a value of 0. In this case, add 5x to both sides of the inequality.

\begin{aligned}2x^{2}+4x+4&>- \, 5x \\\\ 2x^{2}+9x+4&>0 \end{aligned}

\begin{aligned}&2x^{2}+9x+4=0 \\\\ &(2x+1)(x+4)=0 \\\\ &2x+1=0\text{ so }x=- \, \cfrac{1}{2} \\\\ &\text{And }x+4=0\text{ so }x=- \, 4 \end{aligned}

Use open points to plot the values of x.

2x^{2}+9x+4=2(- \, 2)^{2}+9(- \, 2)+4=8-18+4=- \, 6

As - \, 6\ngtr{0}, this is a false comparison so the interval is not part of solution.

The solution set to the quadratic inequality is (- \, \infty,- \, 4)\cup\left(- \, \cfrac{1}{2},\infty\right).

Example 5: no c term

Find the solution to the quadratic inequality, 4x^2\geq- \, 28x.

Before solving, make sure one side of the equation is 0. In this case, add 28x to both sides.

\begin{aligned}&4x^2\geq- \, 28x \\\\ &4x^2+28x\geq{0} \end{aligned}

Now you can use one of the strategies for solving quadratics. This quadratic factors, so use that strategy.

\begin{aligned}&4x^{2}+28x=0 \\\\ &4x(x+7)=0 \\\\ &4x=0\text{ so }x=0 \\\\ &\text{And }x+7=0\text{ so }x=- \, 7 \end{aligned}

Use closed points to plot the numbers on the number line since the original inequality is “less than and equal to.”

\begin{array}{c}(- \, \infty,- \, 7] \\\\ {[- \, 7,0]} \\\\ {[0,\infty)} \end{array}

4x^{2}+28x=4(- \, 2)^{2}+28(- \, 2)=16-56=- \, 40

As - \, 40\ngeq{0}, this is a false comparison so the interval is not part of solution.

Since there are two intervals that represent the solution set to this quadratic inequality, use the union symbol in between intervals.

(- \, \infty,- \, 7]\cup[- \, 1,\infty)

Example 6: quadratic formula

Solve the quadratic inequality, 5x^2+6x-12<0.

The inequality is ready to solve, but notice that it does not factor. So, you can use the quadratic formula to find the values of x.

The general form of a quadratic equation is ax^2+bx+c=0.

The quadratic formula is used to solve any quadratic equation.

The quadratic formula is x=\cfrac{- \, b\pm\sqrt{b^2-4ac}}{2a}.

In this question, a=5, \, b=6, and c=- \, 12

Substitute those values into the quadratic formula to find the values for x\text{:}

\begin{aligned}x&=\cfrac{- \, 6\pm\sqrt{6^2-4\times5\times-12}}{2\times5} \\\\ &=\cfrac{- \, 6\pm\sqrt{36-(- \, 240)}}{10} \\\\ &=\cfrac{- \, 6\pm\sqrt{276}}{10} \\\\ x&=\cfrac{- \, 6+2\sqrt{69}}{10}\text{ and }x=\cfrac{- \, 6-2\sqrt{69}}{10} \end{aligned}

At this point, calculate decimals values rounded to the nearest hundredth.

x=\cfrac{- \, 6+2\sqrt{69}}{10}=1.06

x=\cfrac{- \, 6-2\sqrt{69}}{10}=- \, 2.26

Step-by-step guide: Quadratic formula

Use solid points when plotting the values of x on the number line.

\begin{array}{c}(- \, \infty,- \, 2.26) \\\\ (- \, 2.26,1.06) \\\\ (1.06,\infty) \end{array}

Let x=0\text{:}

5x^{2}+6x-12=5(0)^{2}+6(0)-12=0+0-12=- \, 12

The intervals that are the solution set are:

(- \, 2.26,1.06)

Teaching tips for quadratic inequalities

• Start the lesson by assessing if students can recall how to solve linear inequalities, including how to graph them on the number line. This will help guide the lesson.
• Infuse activities such as gallery walks so that students have an opportunity to showcase their work as well as have open dialogue with classmates about how to solve the problems.
• Instead of worksheets, have students practice skills by game-playing.
• Have access to the plot of each quadratic equation so that students can visualize the solution.

Easy mistakes to make

• Forgetting the negative value when taking the square root of a number A common error when calculating the square root of a number is only writing the positive solution. A negative number squared will also give a positive solution. For example: \sqrt{16}=4 and - \, 4.4\times{4}=16 and (- \, 4)\times(- \, 4)=16.
• Not using the original inequality symbol when testing points When substituting test points to check if the intervals work as the solution, substitute those values for x into the original inequality.
• Mixing up way to plot numbers in the number line For example, when the inequality is \leq or \geq, use solid points on the number line because the solution set includes those values of x. When the inequality is < or >, use open points on the number line because the solution set does not include the values for x.

Related quadratic inequalites lessons

• Inequalities
• Greater than sign
• Less than sign
• Linear inequalities
• Solving inequalities
• Graphing inequalities

Practice quadratic inequalities questions

1. Find the solution set to the quadratic inequality, x^{2}+13x+30<0.

The quadratic inequality factors, so use factoring to find the limit values for x.

Plot the values of x on the number line and determine the intervals and the test point you will use.

Let x=- \, 5\text{:}

– \,10<0 so this is a true comparison and this interval is part of the solution.

The solution is the interval: (- \, 10, – \, 3)

Note: Graph and number line for reference.

2. Solve the quadratic inequality, x^{2}+3x-10\leq{0}.

The quadratic inequality factors, so use factoring to find the values for x.

– \, 10\leq{0} so this is a true comparison and this interval is part of the solution.

The solution is: [- \, 5, 2]

3. Find the solution to the quadratic inequality, x^2+6>7 x.

First, make sure one side of the inequality has a value of 0. In this case, subtract 7x from both sides of the inequality.

The quadratic factors, so factor it to find the values of x.

Let x=2\text{:}

– \, 6\ngtr{0} so this is a false comparison and this interval is not part of the solution.

The solution is:

4. Find the solution to the quadratic inequality, 2x^{2}-11x+5<0.

The quadratic factors, so find the values of x by factoring.

– \, 9<0 so this is a true comparison and this interval is part of the solution.

The solution is \left(\cfrac{1}{2}, 5\right)

5. Find the solution to the quadratic inequality, x^2-25>0.

The quadratic is the difference of two perfect squares so it factors.

– \, 25\ngtr{0} so this is a false comparison and this interval is not part of the solution.

6. Find the solution to the quadratic inequality, 1-x^2\leq{0}.

Factor the quadratic to find the two limiting values of x\text{:}

– \, 1\nleq{0} so this is a false comparison and this interval is not part of the solution.

7. Solve the quadratic inequality: 3x^{2}+4x-2\leq0.

The quadratic does not factor, so use the quadratic formula to find the values for x.

Identify the values of a, b, and c and then substitute into the formula.

Substitute these values into the quadratic formula to find the values for x\text{:}

Plot the values of x on the number line with closed points.

– \, 2\leq{0} so this is a true comparison and this interval is part of the solution.

Quadratic inequalities FAQs

The discriminant is helpful when determining the type of solutions you will have for a quadratic equation. So, it can be helpful when solving a quadratic inequality.

Yes, you can sketch the parabola using a table of values and vertex point to determine which points make the quadratic inequality true and not true.

If the quadratic does not factor, you can use the quadratic formula or the strategy of completing the square. Remember, you can use the quadratic formula and completing the square regardless if the quadratic factors or not.

The next lessons are

• Math formulas
• Coordinate plane
• Types of graphs

Still stuck?

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[FREE] Common Core Practice Tests (3rd to 8th Grade)

Prepare for math tests in your state with these 3rd Grade to 8th Grade practice assessments for Common Core and state equivalents.

Get your 6 multiple choice practice tests with detailed answers to support test prep, created by US math teachers for US math teachers!

Using a Plan of Attack for Math Problem-Solving

Spending the majority of their time modeling problems and making sense of relationships in math can help students move beyond a surface-level grasp.

At every grade level, math teachers in the Success Academy Charter Schools Network in New York City ask students to spend the lion’s share of their time during math lessons deeply examining the question they are being asked to solve. Students start by asking themselves questions like, “What are the mathematical relationships in the problem?” “What is this problem asking me to do?” and “How can I model my thinking?” Every classroom even has a formula—a problem-solving plan for math, printed out and displayed on the wall—called the “Plan of Attack,” which includes three parts: using 80 percent of the allotted time to conceptualize the question by reading the problem multiple times, then modeling the relationships and actions in the problems; 10 percent to answer the question by determining a strategy they will use to solve it and then computing; and finally double-checking in the last 10 percent of their time—by rereading the problem, evaluating their own reasoning, and checking computations for accuracy.

First-grade teacher Evelyn Gonzales and eighth-grade teacher Fei Liu both reinforce this strategy during precious class time by working through the problem as a whole with their students first, emphasizing the importance of this step before rushing in to solve. As a result, their students develop a much stronger understanding of the mathematical concepts at hand. “In my classroom, I don’t really care for the answer,” says Liu. “They can double-check once they have the answer. What we really need to focus on is why we set the things up, so that when they see a problem, they have an idea of where to start to think.”

The network led the state for math test scores in the 2023–2024 school year, with with 49 percent of Black and 55 percent of Hispanic students earning fours, the highest possible mark.

See all of Edutopia’s coverage of Success Academy Charter Schools to learn more about the network.

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5 Phases of Problem-Solving using the Six Sigma DMAIC approach

In the world of continuous improvement, the Six Sigma DMAIC methodology stands out as a powerful tool for problem-solving and process optimization.

|| 10 minutes read ||

Overview of the problem and the approach taken

In this blog, we delve into how a team of professionals tackled a daunting $65 million issue using DMAIC – Define, Measure, Analyze, Improve and Control. By leveraging tools such as Pareto analysis, fishbone diagrams, 5 Whys, Gemba walks, and SCAMPER brainstorming sessions, the team was able to identify root causes, generate innovative solutions, and implement sustainable changes.

Join us on this step-by-step guide to see how DMAIC, combined with teamwork and strategic planning, can lead to successful problem resolution and long-term results.

1.    Define: Identifying the problem and setting goals

In the Define phase of DMAIC, the first crucial step is to identify the problem at hand. By defining the problem statement and setting specific, measurable, achievable, relevant, and time-bound (SMART) goals, the team can align their efforts toward a common objective. It is essential to gather relevant  data, engage stakeholders, and establish a baseline for current performance.

Through effective communication and collaboration, the team can ensure everyone is on the same page regarding the issue and the desired outcome.

Stay tuned to learn, how a well-defined problem statement lays the foundation for the success of the DMAIC process.

2.   Measure: Collecting data and analyzing the current state

In the Measure phase of DMAIC, the focus shifts to collecting relevant data and analyzing the current state of affairs. This step involves identifying key metrics, establishing data collection methods, and ensuring data accuracy and reliability. By analyzing the gathered information, the team can gain valuable insights into the root causes of the problem and quantify the extent of the issue. Through rigorous data analysis and interpretation, the team can uncover patterns and trends that will guide future improvement efforts.

Stay tuned to discover how data-driven decision-making plays a crucial role in solving the 65-million-dollar problem using the DMAIC methodology.

3.   Analyze: Identifying root causes and potential solutions

In the Analyze phase of DMAIC, the focus shifts towards identifying the root causes of the problem and exploring potential solutions. This stage involves conducting a thorough analysis of the collected data to pinpoint underlying issues that contribute to the $65 million problem. By utilizing various tools and techniques such as root cause analysis, fishbone diagrams, and statistical analysis, the team can delve deeper into the factors influencing the problem.

Through this meticulous process, the team can gain a comprehensive understanding of the issues at hand, paving the way for informed decision-making and targeted action plans.

Stay tuned to uncover, how the Analyze phase propels us closer to resolving the multi-million-dollar challenge.

4.   Improve: Implementing changes and measuring results

In the Improve phase of DMAIC, the focus is on implementing solutions identified during the Analyze phase to address the root causes of the $65 million problem.

This stage involves developing and executing action plans aimed at improving processes and eliminating inefficiencies. By carefully monitoring and measuring the outcomes of these changes, the team can determine their effectiveness in tackling the problem. Continuous evaluation and adjustment of strategies are key elements of this phase to ensure sustainable improvements.

Stay engaged as we dive into how the Improve phase plays a pivotal role in transforming our approach to resolving this high-stakes challenge.

5.   Control: Monitoring progress and sustaining improvements

In the final phase of DMAIC, Control, the emphasis shifts towards monitoring the progress of implemented solutions to ensure sustained improvements in addressing the $65 million problem. This stage involves establishing control measures and mechanisms to track key performance indicators and verify that the desired outcomes are being achieved consistently.

By setting up regular reviews and audits, the team can identify any deviations or issues early on, enabling prompt corrective action. Maintaining open communication channels and documenting procedures are essential in upholding the gains achieved during the Improve phase.

Join us as we explore how effective control measures solidify the success of DMAIC in resolving complex challenges.

Results and Lessons learned from using  DMAIC

After implementing DMAIC in addressing the $65 million problem, the results were profound. By following the structured approach of Define, Measure, Analyze, Improve, and Control, the team successfully identified root causes, developed effective solutions, and ensured sustained improvements. The financial impact of saving $65 million is significant, showcasing the power of DMAIC in problem-solving.

Key lessons learned from this case study include the importance of data-driven decision-making, collaboration among team members from diverse backgrounds, and the value of maintaining a systematic and structured approach throughout the problem-solving process.

Stay tuned to gain further insights into how DMAIC can be leveraged to overcome complex challenges effectively.

Conclusion and final thoughts

In conclusion, the case study highlighting the successful resolution of a $65 million problem through DMAIC reinforces the effectiveness of this structured problem-solving methodology. The disciplined approach of Define, Measure, Analyze, Improve, and Control proved invaluable in achieving significant cost savings and sustainable improvements. Emphasizing data-driven decision- making, cross-functional collaboration, and systematic problem-solving methods were key takeaways.

As organizations face increasingly complex challenges, leveraging DMAIC can provide a systematic framework for driving positive change and delivering tangible results. By adhering to the principles of DMAIC, businesses can enhance their problem-solving capabilities and achieve long-term success.

If you are interested to achieve similar success stories, write to us!

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OpenAI Announces a New AI Model, Code-Named Strawberry, That Solves Difficult Problems Step by Step

OpenAI made the last big breakthrough in artificial intelligence by increasing the size of its models to dizzying proportions, when it introduced GPT-4 last year. The company today announced a new advance that signals a shift in approach—a model that can “reason” logically through many difficult problems and is significantly smarter than existing AI without a major scale-up.

The new model, dubbed OpenAI o1, can solve problems that stump existing AI models, including OpenAI’s most powerful existing model, GPT-4o . Rather than summon up an answer in one step, as a large language model normally does, it reasons through the problem, effectively thinking out loud as a person might, before arriving at the right result.

“This is what we consider the new paradigm in these models,” Mira Murati , OpenAI’s chief technology officer, tells WIRED. “It is much better at tackling very complex reasoning tasks.”

The new model was code-named Strawberry within OpenAI, and it is not a successor to GPT-4o but rather a complement to it, the company says.

Murati says that OpenAI is currently building its next master model, GPT-5, which will be considerably larger than its predecessor. But while the company still believes that scale will help wring new abilities out of AI, GPT-5 is likely to also include the reasoning technology introduced today. “There are two paradigms,” Murati says. “The scaling paradigm and this new paradigm. We expect that we will bring them together.”

LLMs typically conjure their answers from huge neural networks fed vast quantities of training data. They can exhibit remarkable linguistic and logical abilities, but traditionally struggle with surprisingly simple problems such as rudimentary math questions that involve reasoning.

Murati says OpenAI o1 uses reinforcement learning, which involves giving a model positive feedback when it gets answers right and negative feedback when it does not, in order to improve its reasoning process. “The model sharpens its thinking and fine tunes the strategies that it uses to get to the answer,” she says. Reinforcement learning has enabled computers to play games with superhuman skill and do useful tasks like designing computer chips . The technique is also a key ingredient for turning an LLM into a useful and well-behaved chatbot.

Mark Chen, vice president of research at OpenAI, demonstrated the new model to WIRED, using it to solve several problems that its prior model, GPT-4o, cannot. These included an advanced chemistry question and the following mind-bending mathematical puzzle: “A princess is as old as the prince will be when the princess is twice as old as the prince was when the princess’s age was half the sum of their present age. What is the age of the prince and princess?” (The correct answer is that the prince is 30, and the princess is 40).

“The [new] model is learning to think for itself, rather than kind of trying to imitate the way humans would think,” as a conventional LLM does, Chen says.

OpenAI says its new model performs markedly better on a number of problem sets, including ones focused on coding, math, physics, biology, and chemistry. On the American Invitational Mathematics Examination (AIME), a test for math students, GPT-4o solved on average 12 percent of the problems while o1 got 83 percent right, according to the company.

The new model is slower than GPT-4o, and OpenAI says it does not always perform better—in part because, unlike GPT-4o, it cannot search the web and it is not multimodal, meaning it cannot parse images or audio.

Improving the reasoning capabilities of LLMs has been a hot topic in research circles for some time. Indeed, rivals are pursuing similar research lines. In July, Google announced AlphaProof , a project that combines language models with reinforcement learning for solving difficult math problems.

AlphaProof was able to learn how to reason over math problems by looking at correct answers. A key challenge with broadening this kind of learning is that there are not correct answers for everything a model might encounter. Chen says OpenAI has succeeded in building a reasoning system that is much more general. “I do think we have made some breakthroughs there; I think it is part of our edge,” Chen says. “It’s actually fairly good at reasoning across all domains.”

Noah Goodman , a professor at Stanford who has published work on improving the reasoning abilities of LLMs, says the key to more generalized training may involve using a “carefully prompted language model and handcrafted data” for training. He adds that being able to consistently trade the speed of results for greater accuracy would be a “nice advance.”

Yoon Kim , an assistant professor at MIT, says how LLMs solve problems currently remains somewhat mysterious, and even if they perform step-by-step reasoning there may be key differences from human intelligence. This could be crucial as the technology becomes more widely used. “These are systems that would be potentially making decisions that affect many, many people,” he says. “The larger question is, do we need to be confident about how a computational model is arriving at the decisions?”

The technique introduced by OpenAI today also may help ensure that AI models behave well. Murati says the new model has shown itself to be better at avoiding producing unpleasant or potentially harmful output by reasoning about the outcome of its actions. “If you think about teaching children, they learn much better to align to certain norms, behaviors, and values once they can reason about why they’re doing a certain thing,” she says.

Oren Etzioni , a professor emeritus at the University of Washington and a prominent AI expert, says it’s “essential to enable LLMs to engage in multi-step problem solving, use tools, and solve complex problems.” He adds, “Pure scale up will not deliver this.” Etzioni says, however, that there are further challenges ahead. “Even if reasoning were solved, we would still have the challenge of hallucination and factuality.”

OpenAI’s Chen says that the new reasoning approach developed by the company shows that advancing AI need not cost ungodly amounts of compute power. “One of the exciting things about the paradigm is we believe that it’ll allow us to ship intelligence cheaper,” he says, “and I think that really is the core mission of our company.”

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