the 4 steps for problem solving in math

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The easy 4 step problem-solving process (+ examples)

This is the 4 step problem-solving process that I taught to my students for math problems, but it works for academic and social problems as well.

Every problem may be different, but effective problem solving asks the same four questions and follows the same method.

What’s the problem? If you don’t know exactly what the problem is, you can’t come up with possible solutions. Something is wrong. What are we going to do about this? This is the foundation and the motivation.
What do you need to know? This is the most important part of the problem. If you don’t know exactly what the problem is, you can’t come up with possible solutions.
What do you already know? You already know something related to the problem that will help you solve the problem. It’s not always obvious (especially in the real world), but you know (or can research) something that will help.
What’s the relationship between the two? Here is where the heavy brainstorming happens. This is where your skills and abilities come into play. The previous steps set you up to find many potential solutions to your problem, regardless of its type.

When I used to tutor kids in math and physics , I would drill this problem-solving process into their heads. This methodology works for any problem, regardless of its complexity or difficulty. In fact, if you look at the various advances in society, you’ll see they all follow some variation of this problem-solving technique.

“The gap between understanding and misunderstanding can best be bridged by thought!” ― Ernest Agyemang Yeboah

Generally speaking, if you can’t solve the problem then your issue is step 3 or step 4; you either don’t know enough or you’re missing the connection.

Good problem solvers always believe step 3 is the issue. In this case, it’s a simple matter of learning more. Less skilled problem solvers believe step 4 is the root cause of their difficulties. In this instance, they simply believe they have limited problem-solving skills.

This is a fixed versus growth mindset and it makes a huge difference in the effort you put forth and the belief you have in yourself to make use of this step-by-step process. These two mindsets make a big difference in your learning because, at its core, learning is problem-solving.

Let’s dig deeper into the 4 steps. In this way, you can better see how to apply them to your learning journey.

Step 1: What’s the problem?

The ability to recognize a specific problem is extremely valuable.

Most people only focus on finding solutions. While a “solutions-oriented” mindset is a good thing, sometimes it pays to focus on the problem. When you focus on the problem, you often make it easier to find a viable solution to it.

When you know the exact nature of the problem, you shorten the time frame needed to find a solution. This reminds me of a story I was once told.

When does the problem-solving process start?

The process starts after you’ve identified the exact nature of the problem.

Homeowners love a well-kept lawn but hate mowing the grass.

Many companies and inventors raced to figure out a more time-efficient way to mow the lawn. Some even tried to design robots that would do the mowing. They all were chasing the solution, but only one inventor took the time to understand the root cause of the problem.

Most people figured that the problem was the labor required to maintain a lawn. The actual problem was just the opposite: maintaining a lawn was labor-intensive. The rearrangement seems trivial, but it reveals the true desire: a well-maintained lawn.

The best solution? Remove maintenance from the equation. A lawn made of artificial grass solved the problem . Hence, an application of Astroturf was discovered.

This way, the law always looked its best. Taking a few moments to apply critical thinking identified the true nature of the problem and yielded a powerful solution.

An example of choosing the right problem to work the problem-solving process on

One thing I’ve learned from tutoring high school students in math : they hate word problems.

This is because they make the student figure out the problem. Finding the solution to a math problem is already stressful. Forcing the student to also figure out what problem needs solving is another level of hell.

Word problems are not always clear about what needs to be solved. They also have the annoying habit of adding extraneous information. An ordinary math problem does not do this. For example, compare the following two problems:

What’s the height of h?

A radio station tower was built in two sections. From a point 87 feet from the base of the tower, the angle of elevation of the top of the first section is 25º, and the angle of elevation of the top of the second section is 40º. To the nearest foot, what is the height of the top section of the tower?

The first is a simple problem. The second is a complex problem. The end goal in both is the same.

The questions require the same knowledge (trigonometric functions), but the second is more difficult for students. Why? The second problem does not make it clear what the exact problem is. Before mathematics can even begin, you must know the problem, or else you risk solving the wrong one.

If you understand the problem, finding the solution is much easier. Understanding this, ironically, is the biggest problem for people.

Problem-solving is a universal language

Speaking of people, this method also helps settle disagreements.

When we disagree, we rarely take the time to figure out the exact issue. This happens for many reasons, but it always results in a misunderstanding. When each party is clear with their intentions, they can generate the best response.

Education systems fail when they don’t consider the problem they’re supposed to solve. Foreign language education in America is one of the best examples.

The problem is that students can’t speak the target language. It seems obvious that the solution is to have students spend most of their time speaking. Unfortunately, language classes spend a ridiculous amount of time learning grammar rules and memorizing vocabulary.

The problem is not that the students don’t know the imperfect past tense verb conjugations in Spanish. The problem is that they can’t use the language to accomplish anything. Every year, kids graduate from American high schools without the ability to speak another language, despite studying one for 4 years.

Well begun is half done

Before you begin to learn something, be sure that you understand the exact nature of the problem. This will make clear what you need to know and what you can discard. When you know the exact problem you’re tasked with solving, you save precious time and energy. Doing this increases the likelihood that you’ll succeed.

Step 2: What do you need to know?

All problems are the result of insufficient knowledge. To solve the problem, you must identify what you need to know. You must understand the cause of the problem. If you get this wrong, you won’t arrive at the correct solution.

Either you’ll solve what you thought was the problem, only to find out this wasn’t the real issue and now you’ve still got trouble or you won’t and you still have trouble. Either way, the problem persists.

If you solve a different problem than the correct one, you’ll get a solution that you can’t use. The only thing that wastes more time than an unsolved problem is solving the wrong one.

Imagine that your car won’t start. You replace the alternator, the starter, and the ignition switch. The car still doesn’t start. You’ve explored all the main solutions, so now you consider some different solutions.

Now you replace the engine, but you still can’t get it to start. Your replacements and repairs solved other problems, but not the main one: the car won’t start.

Then it turns out that all you needed was gas.

This example is a little extreme, but I hope it makes the point. For something more relatable, let’s return to the problem with language learning.

You need basic communication to navigate a foreign country you’re visiting; let’s say Mexico. When you enroll in a Spanish course, they teach you a bunch of unimportant words and phrases. You stick with it, believing it will eventually click.

When you land, you can tell everyone your name and ask for the location of the bathroom. This does not help when you need to ask for directions or tell the driver which airport terminal to drop you off at.

Finding the solution to chess problems works the same way

The book “The Amateur Mind” by IM Jeremy Silman improved my chess by teaching me how to analyze the board.

It’s only with a proper analysis of imbalances that you can make the best move. Though you may not always choose the correct line of play, the book teaches you how to recognize what you need to know . It teaches you how to identify the problem—before you create an action plan to solve it.

Chess book to help learn problem solving

The problem-solving method always starts with identifying the problem or asking “What do you need to know?”. It’s only after you brainstorm this that you can move on to the next step.

Learn the method I used to earn a physics degree, learn Spanish, and win a national boxing title

I was a terrible math student in high school who wrote off mathematics. I eventually overcame my difficulties and went on to earn a B.A. Physics with a minor in math
I pieced together the best works on the internet to teach myself Spanish as an adult
*I didn’t start boxing until the very old age of 22, yet I went on to win a national championship, get a high-paying amateur sponsorship, and get signed by Roc Nation Sports as a profession.

I’ve used this method to progress in mentally and physically demanding domains.

While the specifics may differ, I believe that the general methods for learning are the same in all domains.

This free e-book breaks down the most important techniques I’ve used for learning.

Step 3: What do you already know?

The only way to know if you lack knowledge is by gaining some in the first place. All advances and solutions arise from the accumulation and implementation of prior information. You must first consider what it is that you already know in the context of the problem at hand.

Isaac Newton once said, “If I have seen further, it is by standing on the shoulders of giants.” This is Newton’s way of explaining that his advancements in physics and mathematics would be impossible if it were not for previous discoveries.

Mathematics is a great place to see this idea at work. Consider the following problem:

What is the domain and range of y=(x^2)+6?

This simple algebra problem relies on you knowing a few things already. You must know:

The definition of “domain” and “range”
That you can never square any real number and get a negative

Once you know those things, this becomes easy to solve. This is also how we learn languages.

An example of the problem-solving process with a foreign language

Anyone interested in serious foreign language study (as opposed to a “crash course” or “survival course”) should learn the infinitive form of verbs in their target language. You can’t make progress without them because they’re the root of all conjugations. It’s only once you have a grasp of the infinitives that you can completely express yourself. Consider the problem-solving steps applied in the following example.

I know that I want to say “I don’t eat eggs” to my Mexican waiter. That’s the problem.

I don’t know how to say that, but last night I told my date “No bebo alcohol” (“I don’t drink alcohol”). I also know the infinitive for “eat” in Spanish (comer). This is what I already know.

Now I can execute the final step of problem-solving.

Step 4: What’s the relationship between the two?

I see the connection. I can use all of my problem-solving strategies and methods to solve my particular problem.

I know the infinitive for the Spanish word “drink” is “beber” . Last night, I changed it to “bebo” to express a similar idea. I should be able to do the same thing to the word for “eat”.

“No como huevos” is a pretty accurate guess.

In the math example, the same process occurs. You don’t know the answer to “What is the domain and range of y=(x^2)+6?” You only know what “domain” and “range” mean and that negatives aren’t possible when you square a real number.

A domain of all real numbers and a range of all numbers equal to and greater than six is the answer.

This is relating what you don’t know to what you already do know. The solutions appear simple, but walking through them is an excellent demonstration of the process of problem-solving.

In most cases, the solution won’t be this simple, but the process or finding it is the same. This may seem trivial, but this is a model for thinking that has served the greatest minds in history.

A recap of the 4 steps of the simple problem-solving process

What’s the problem? There’s something wrong. There’s something amiss.
What do you need to know? This is how to fix what’s wrong.
What do you already know? You already know something useful that will help you find an effective solution.
What’s the relationship between the previous two? When you use what you know to help figure out what you don’t know, there is no problem that won’t yield.

Learning is simply problem-solving. You’ll learn faster if you view it this way.

What was once complicated will become simple.

What was once convoluted will become clear.

Ed Latimore

I’m a writer, competitive chess player, Army veteran, physicist, and former professional heavyweight boxer. My work focuses on self-development, realizing your potential, and sobriety—speaking from personal experience, having overcome both poverty and addiction.

Follow me on Twitter.

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20 Effective Math Strategies To Approach Problem-Solving

Katie Keeton

Math strategies for problem-solving help students use a range of approaches to solve many different types of problems. It involves identifying the problem and carrying out a plan of action to find the answer to mathematical problems.

Problem-solving skills are essential to math in the general classroom and real-life. They require logical reasoning and critical thinking skills. Students must be equipped with strategies to help them find solutions to problems.

This article explores mathematical problem solving strategies, logical reasoning and critical thinking skills to help learners with solving math word problems independently in real-life situations.

What are problem-solving strategies?

Problem-solving strategies in math are methods students can use to figure out solutions to math problems. Some problem-solving strategies:

Draw a model
Use different approaches
Check the inverse to make sure the answer is correct

Students need to have a toolkit of math problem-solving strategies at their disposal to provide different ways to approach math problems. This makes it easier to find solutions and understand math better.

Strategies can help guide students to the solution when it is difficult ot know when to start.

The ultimate guide to problem solving techniques

Download these ready-to-go problem solving techniques that every student should know. Includes printable tasks for students including challenges, short explanations for teachers with questioning prompts.

20 Math Strategies For Problem-Solving

Different problem-solving math strategies are required for different parts of the problem. It is unlikely that students will use the same strategy to understand and solve the problem.

Here are 20 strategies to help students develop their problem-solving skills.

Strategies to understand the problem

Strategies that help students understand the problem before solving it helps ensure they understand:

The context
What the key information is
How to form a plan to solve it

Following these steps leads students to the correct solution and makes the math word problem easier .

Here are five strategies to help students understand the content of the problem and identify key information.

1. Read the problem aloud

Read a word problem aloud to help understand it. Hearing the words engages auditory processing. This can make it easier to process and comprehend the context of the situation.

2. Highlight keywords

When keywords are highlighted in a word problem, it helps the student focus on the essential information needed to solve it. Some important keywords help determine which operation is needed. For example, if the word problem asks how many are left, the problem likely requires subtraction. Ensure students highlight the keywords carefully and do not highlight every number or keyword. There is likely irrelevant information in the word problem.

3. Summarize the information

Read the problem aloud, highlight the key information and then summarize the information. Students can do this in their heads or write down a quick summary. Summaries should include only the important information and be in simple terms that help contextualize the problem.

4. Determine the unknown

A common problem that students have when solving a word problem is misunderstanding what they are solving. Determine what the unknown information is before finding the answer. Often, a word problem contains a question where you can find the unknown information you need to solve. For example, in the question ‘How many apples are left?’ students need to find the number of apples left over.

5. Make a plan

Once students understand the context of the word problem, have dentified the important information and determined the unknown, they can make a plan to solve it. The plan will depend on the type of problem. Some problems involve more than one step to solve them as some require more than one answer. Encourage students to make a list of each step they need to take to solve the problem before getting started.

Strategies for solving the problem

1. draw a model or diagram.

Students may find it useful to draw a model, picture, diagram, or other visual aid to help with the problem solving process. It can help to visualize the problem to understand the relationships between the numbers in the problem. In turn, this helps students see the solution.

math problem that needs a problem solving strategy

Similarly, you could draw a model to represent the objects in the problem:

2. Act it out

This particular strategy is applicable at any grade level but is especially helpful in math investigation in elementary school . It involves a physical demonstration or students acting out the problem using movements, concrete resources and math manipulatives . When students act out a problem, they can visualize and contectualize the word problem in another way and secure an understanding of the math concepts. The examples below show how 1st-grade students could “act out” an addition and subtraction problem:

The problem	How to act out the problem
Gia has 6 apples. Jordan has 3 apples. How many apples do they have altogether?	Two students use counters to represent the apples. One student has 6 counters and the other student takes 3. Then, they can combine their “apples” and count the total.
Michael has 7 pencils. He gives 2 pencils to Sarah. How many pencils does Michael have now?	One student (“Michael”) holds 7 pencils, the other (“Sarah”) holds 2 pencils. The student playing Michael gives 2 pencils to the student playing Sarah. Then the students count how many pencils Michael is left holding.

3. Work backwards

Working backwards is a popular problem-solving strategy. It involves starting with a possible solution and deciding what steps to take to arrive at that solution. This strategy can be particularly helpful when students solve math word problems involving multiple steps. They can start at the end and think carefully about each step taken as opposed to jumping to the end of the problem and missing steps in between.

For example,

To solve this problem working backwards, start with the final condition, which is Sam’s grandmother’s age (71) and work backwards to find Sam’s age. Subtract 20 from the grandmother’s age, which is 71. Then, divide the result by 3 to get Sam’s age. 71 – 20 = 51 51 ÷ 3 = 17 Sam is 17 years old.

4. Write a number sentence

When faced with a word problem, encourage students to write a number sentence based on the information. This helps translate the information in the word problem into a math equation or expression, which is more easily solved. It is important to fully understand the context of the word problem and what students need to solve before writing an equation to represent it.

5. Use a formula

Specific formulas help solve many math problems. For example, if a problem asks students to find the area of a rug, they would use the area formula (area = length × width) to solve. Make sure students know the important mathematical formulas they will need in tests and real-life. It can help to display these around the classroom or, for those who need more support, on students’ desks.

Strategies for checking the solution

Once the problem is solved using an appropriate strategy, it is equally important to check the solution to ensure it is correct and makes sense.

There are many strategies to check the solution. The strategy for a specific problem is dependent on the problem type and math content involved.

Here are five strategies to help students check their solutions.

1. Use the Inverse Operation

For simpler problems, a quick and easy problem solving strategy is to use the inverse operation. For example, if the operation to solve a word problem is 56 ÷ 8 = 7 students can check the answer is correct by multiplying 8 × 7. As good practice, encourage students to use the inverse operation routinely to check their work.

2. Estimate to check for reasonableness

Once students reach an answer, they can use estimation or rounding to see if the answer is reasonable. Round each number in the equation to a number that’s close and easy to work with, usually a multiple of ten. For example, if the question was 216 ÷ 18 and the quotient was 12, students might round 216 to 200 and round 18 to 20. Then use mental math to solve 200 ÷ 20, which is 10. When the estimate is clear the two numbers are close. This means your answer is reasonable.

3. Plug-In Method

This method is particularly useful for algebraic equations. Specifically when working with variables. To use the plug-in method, students solve the problem as asked and arrive at an answer. They can then plug the answer into the original equation to see if it works. If it does, the answer is correct.

If students use the equation 20m+80=300 to solve this problem and find that m = 11, they can plug that value back into the equation to see if it is correct. 20m + 80 = 300 20 (11) + 80 = 300 220 + 80 = 300 300 = 300 ✓

4. Peer Review

Peer review is a great tool to use at any grade level as it promotes critical thinking and collaboration between students. The reviewers can look at the problem from a different view as they check to see if the problem was solved correctly. Problem solvers receive immediate feedback and the opportunity to discuss their thinking with their peers. This strategy is effective with mixed-ability partners or similar-ability partners. In mixed-ability groups, the partner with stronger skills provides guidance and support to the partner with weaker skills, while reinforcing their own understanding of the content and communication skills. If partners have comparable ability levels and problem-solving skills, they may find that they approach problems differently or have unique insights to offer each other about the problem-solving process.

5. Use a Calculator

A calculator can be introduced at any grade level but may be best for older students who already have a foundational understanding of basic math operations. Provide students with a calculator to allow them to check their solutions independently, accurately, and quickly. Since calculators are so readily available on smartphones and tablets, they allow students to develop practical skills that apply to real-world situations.

Step-by-step problem-solving processes for your classroom

In his book, How to Solve It , published in 1945, mathematician George Polya introduced a 4-step process to solve problems.

Polya’s 4 steps include:

Understand the problem
Devise a plan
Carry out the plan

Today, in the style of George Polya, many problem-solving strategies use various acronyms and steps to help students recall.

Many teachers create posters and anchor charts of their chosen process to display in their classrooms. They can be implemented in any elementary, middle school or high school classroom.

Here are 5 problem-solving strategies to introduce to students and use in the classroom.

How Third Space Learning improves problem-solving

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Explore the range of problem solving resources for 2nd to 8th grade students.

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Former teachers and math experts write all of Third Space Learning’s tutoring lessons. Expertly designed lessons follow a “my turn, follow me, your turn” pedagogy to help students move from guided instruction and problem-solving to independent practice.

Throughout each lesson, tutors ask higher-level thinking questions to promote critical thinking and ensure students are developing a deep understanding of the content and problem-solving skills.

Problem-solving

Educators can use many different strategies to teach problem-solving and help students develop and carry out a plan when solving math problems. Incorporate these math strategies into any math program and use them with a variety of math concepts, from whole numbers and fractions to algebra.

Teaching students how to choose and implement problem-solving strategies helps them develop mathematical reasoning skills and critical thinking they can apply to real-life problem-solving.

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There are many different strategies for problem-solving; Here are 5 problem-solving strategies: • draw a model • act it out • work backwards • write a number sentence • use a formula

Here are 10 strategies for problem-solving: • Read the problem aloud • Highlight keywords • Summarize the information • Determine the unknown • Make a plan • Draw a model • Act it out • Work backwards • Write a number sentence • Use a formula

1. Understand the problem 2. Devise a plan 3. Carry out the plan 4. Look back

Some strategies you can use to solve challenging math problems are: breaking the problem into smaller parts, using diagrams or models, applying logical reasoning, and trying different approaches.

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person with long dark hair sit at a table working at a laptop. 3x+2 and x² equations float in the air signifying that she is working on math problems

$the 4 steps for problem solving in math$

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.

Get step-by-step explanations

Graph your math problems

Practice, practice, practice

Get math help in your language

Timely Resources
Innovative Teaching Ideas
Classroom Voices
Education Trends

Four Principles for Effective Math Intervention

The Institute for Education Sciences, an authority on Response to Intervention (RtI) for math published a groundbreaking report on RtI that outlines a series of intervention recommendations. The report Assisting Students Struggling with Mathematics: Response to Intervention (RtI) for Elementary and Middle schools covers recommendations that were judged to have strong or moderate evidence to support RtI and provide the foundation for effective math intervention. Here are specific strategies that fall under these four key recommendations:

1. Instruction during the intervention should be explicit and systematic.

This recommendation from the report includes providing models of proficient problem solving, verbalization of thought processes, guided practice, corrective feedback, and frequent cumulative review.

Students who have been classified as Tier 2 or 3 in RtI need their instruction to be organized and scaffolded. They lack the numeracy skills and background knowledge to engage in theoretical exercises with math

Nothing is more systematic than the four-step approach to problem solving, first outlined by educator George Polly in 1945:

Understand the problem. Restate the problem, and then identify the information given and the information that needs to be determined.
Make a plan. Relate the problem to similar problems solved in the past. Consider possible strategies, and then select a strategy or a combination of strategies
Carry out the plan. Execute the chosen strategy and perform the necessary calculations. Revise and apply different strategies as necessary.
Look back at the solution. Evaluate the strategy/strategies used for problem solving and then assess if there is a better way to approach the problem.

2. Interventions should include instruction on solving word problems that is based on common underlying structures.

$the 4 steps for problem solving in math$

A stripped-down version of the Gradual Release Model—the “I do, We do, You do” strategy—is effective in all levels of education. As RtI students require as much structure as possible, the strategy gives them an effective way to know what to expect from a lesson. In word problems, not only can “I do, We do, You do” be used to solve problems, it can also be used to have students create their own word problems, reaching synthesis, a higher level of taxonomy. Students should start with sentences that involve a specific math operation and build from there. (McCarney, S. B., Cummins Wunderlich, K., Bauer, A. The Teacher’s Resource Guide.)

3. Intervention materials should include opportunities for students to work with visual representations of mathematical ideas.

This recommendation from the RtI report goes on to say that interventionists should be proficient in the use of visual representations of mathematical ideas. Graphics are important in math instruction, especially as the curriculum becomes more data=based under the Common Core State Standards. RtI students are not excluded from having to be able to read charts, graphs, and other math graphics. Intervention Central , which provides educators with free RtI resources, has a great intervention that uses the Question-Answer Relationship (QAR) to help students break down math graphics. In short, QARs come in four types:

“Right There” questions are found explicitly in the graphic.
“Think and Search” questions are not quite as explicit, but still can be found in the graphic with some close analysis.
“Author and You” questions ask students to compare the data with their own life experiences and opinions.
“On My Own” questions require only the student’s own knowledge and experiences to answer.

4. Interventions at all grade levels should devote about 10 minutes in each session to building fluent retrieval of basic arithmetic facts.

Rehearsal of math facts is a key step in the RtI process because many of the students who fall into Tier 2 or 3 status are missing key pieces of background information, usually in the form of math facts.

Taking ten minutes each day to review facts is typically done with flash cards, which can be tedious and, therefore, does not always prove to be effective. Another strategy, described by Intervention Central, uses flash-card practice that balances “known” facts with “unknown” facts. Unknown facts are modeled by a teacher or tutor and then presented with known facts—those already mastered by the student—in a sequence. Not only does this systematically build recall, it also builds confidence in the student because they are consistently getting cards correct throughout the process.

Using Technology to Boost RtI for Math

Digital curriculum, at its best, incorporates the scaffolds and formative assessment that define successful RtI for math. Personalized learning and ongoing, robust data collection provide a meaningful feedback loop for both the student and teacher that leads to deeper learning and higher achievement. Real-time feedback—while learning is happening—is critical so that students don’t practice new math skills, again and again , incorrectly.

Technological interventions to boost math achievement can be an important part of a successful math intervention , but a commitment to the process is also a necessary part of helping students succeed in math and sharpen their math skills .

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The Discovery Education blog is a free resource for educators to find time-saving teaching strategies and compelling content for their daily lessons.

Full of timely tips, high-quality DE resources, and advice from our DEN community, these posts are meant to entertain and inform our users while supporting educators everywhere with new ways to engage their students in and out of the classroom.

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4 Best Steps To Problem Solving in Math That Lead to Results

Eastern Shore Math Teacher

What does problem solving in math mean, and how to develop these skills in students? Problem solving involves tasks that are challenging and make students think. In teaching through problem solving, learning takes place while trying to solve problems with specific concepts and skills. Therefore, teachers need to provide safe learning spaces that foster a growth mindset in math in order for students to take risks to solve problems. In addition, providing students with problem solving steps in math builds success in solving problems.

$A teacher working on problem solving in math.$

By providing rich mathematical tasks and engaging puzzles, students improve their number sense and mindset about mathematics. Click Here to get this Freebie of 71 Math Number Puzzles delivered to your inbox to use with your students.

Students who feel successful in math class are happier and more engaged in learning. Check out The Bonus Guide for Creating a Growth Mindset Classroom and Students Who Love Math for ideas, lessons, and mindset surveys for students to use in your classroom to cultivate a positive classroom community in mathematics. You can also sign up for other freebies from me Here at Easternshoremathteacher.com .

Have you ever given students a word problem or rich task, and they froze? They have no idea how to tackle the problem, even if it is a concept they are successful with. This is because they need problem solving strategies. I started to incorporate more problem solving tasks into my teaching in addition to making the 4 steps for problem solving a school-wide initiative and saw results.

Bonus Growth Mindset Classroom resources to use to cultivate a growth mindset classroom.

What is Problem Solving in Math?

When educators use the term problem solving , they are referring to mathematical tasks that are challenging and require students to think. Such tasks or problems can promote students’ conceptual understanding, foster their ability to reason and communicate mathematically, and capture their interests and curiosity (Hiebert & Wearne, 1993; Marcus & Fey, 2003; NCTM, 1991; van de Walle, 2003).

$When educators use the term problem solving, they are referring to mathematical tasks that are challenging and require students to think.$

How Should Problem Solving For Math Be Taught?

Problem solving should not be done in isolation. In the past, we would teach the concepts and procedures and then assign one-step “story” problems designed to provide practice on the content. Next, we would teach problem solving as a collection of strategies such as “draw a picture” or “guess and check.” Eventually, students would be given problems to apply the skills and strategies. Instead, we need to make problem solving an integral part of mathematics learning.

In teaching through problem solving, learning takes place while trying to solve problems with specific concepts and skills. As students solve problems, they can use any strategy. Then, they justify their solutions with their classmates and learn new ways to solve problems.

Students do not need every task to involve problem solving. Sometimes the goal is to just learn a skill or strategy.

List of Criteria for Problem Solving in Math

Criteria for Problem Solving Math

Lappan and Phillips (1998) developed a set of criteria for a good problem that they used to develop their middle school mathematics curriculum (Connected Mathematics). The problem:

has important, useful mathematics embedded in it.
requires higher-level thinking and problem solving.
contributes to the conceptual development of students.
creates an opportunity for the teacher to assess what his or her students are learning and where they are experiencing difficulty.
can be approached by students in multiple ways using different solution strategies.
has various solutions or allows different decisions or positions to be taken and defended.
encourages student engagement and discourse.
connects to other important mathematical ideas.
promotes the skillful use of mathematics.
provides an opportunity to practice important skills.

Of course, not every problem will include all of the above. However, the first four are essential. Sometimes, you will choose a problem because your students need an opportunity to practice a certain skill.

The real value of these criteria is that they provide teachers with guidelines for making decisions about how to make problem solving a central aspect of their instruction. Read more at NCTM .

$Resources to Use for Problem Solving Steps in Math.$

Problem Solving Teaching Methods

Teaching students these 4 steps for solving problems allows them to have a process for unpacking difficult problems.

As you teach, model the process of using these 4 steps to solve problems. Then, encourage students to use these steps as they solve problems. Click here for Posters, Bookmarks, and Labels to use in your classroom to promote the use of the problem solving steps in math.

How Problem Solving Skills Develop

Problem solving skills are developed over time and are improved with effective teaching practices. In addition, teachers need to select rich tasks that focus on the math concepts the teacher wants their students to explore.

Problem Solving 4 Steps

Understand the problem.

Read & Think

Circle the needed information and underline the question.
Write an answer STEM sentence. There are_____ pages left to read.

Plan Out How to Solve the Problem

Make a Plan

Use a strategy. (Draw a Picture, Work Backwards, Look for a Pattern, Create a Table, Bar Model)
Use math tools.

Do the Problem

Solve the Problem

Show your work to solve the problem. This could include an equation.

Check Your Work on the Problem

Answer & Check

Write the answer into the answer stem.
Does your answer make sense?
Check your work using a different strategy.

Check out these Printables for Problem Solving Steps in Math .

$Problem Solving steps for Math poster.$

Teaching Problem Solving Strategies

A problem solving strategy is a plan used to find a solution. Understanding how a variety of problem solving strategies work is important because different problems require you to approach them in different ways to find the best solution. By mastering several problem-solving strategies, you can select the right plan for solving a problem. Here are a few strategies to use with students:

Draw a Picture
Work Backwards
Look for a Pattern
Create a Table

Why is Using Problem Solving Steps For Math Important?

Problem solving allows students to develop an understanding of concepts rather than just memorizing a set of procedures to solve a problem. In addition, it fosters collaboration and communication when students explain the processes they used to arrive at a solution. Through problem-solving, students develop a deeper understanding of mathematical concepts, become more engaged, and see the importance of mathematics in their lives.

NCTM Process Standards

In 2011 the Common Core State Standards incorporated the NCTM Process Standards of problem-solving, reasoning and proof, communication, representation, and connections into the Standards for Mathematical Practice. With these process standards, the focus became more on mathematics through problem solving. Students could no longer just develop procedural fluency, they needed to develop conceptual understanding in order to solve new problems and make connections between mathematical ideas.

Engaging Students to Learn in Mathematics Class

Engaging students to learn in math class will help students to love math. Children develop a dislike of math early on and end up resenting it into adult life. Even in the real world, students will likely have to do some form of mathematics in their personal or working life. So how can teachers make math more interesting to engage students in the subject? Read more at 5 Best Strategies for Engaging Students to Learn in Mathematics Class

$Puzzles in Math with Answers on a computer screen.$

Teachers can promote number sense by providing rich mathematical tasks and encouraging students to make connections to their own experiences and previous learning.

Sign up on my webpage to get this Freebie of 71 Math Number Puzzles delivered to your inbox to use with your students. Providing opportunities to do math puzzles daily is one way to help students develop their number sense. CLICK Here to sign up for 71 Math Number Puzzles and check out my website.

Promoting a Growth Mindset

Research shows that there is a link between a growth mindset and success. In addition, kids who have a growth mindset about their abilities perform better and are more engaged in the classroom. Students need to be able to preserve and make mistakes when problem solving.

Read more … 5 Powerful and Easy Lessons Teaching Students How to Get a Growth Mindset

Here are some Resources to Use to Grow a Growth Mindset

Free Mindset Survey
Growth Mindset Classroom Display Free
Growth Mindset Lessons

Growth Mindset in Math Resources on a computer screen.

Using Word Problems

Story Problems and word problems are one way to promote problem solving. In addition, they provide great practice in using the 4 steps of solving problems. Then, students are ready for more challenging problems.

For Kindergarten

Subtraction within 5

For First Grade

Word Problems to 20
Word Problems of Subtraction

Word Problems of Addition and Subtraction on a computer screen.

For Second Grade

Two Step Word Problems with Addition and Subtraction
Grade 2 Addition and Subtraction Word Problems
Word Problems with Subtraction

$Problem Solving in Math with these addition and subtraction word problems with different problem structures. Can be used digitally or as a worksheet.$

For Third Grade

Word Problems Division and Multiplication
Multiplication Word Problems

Use repeated addition to multiply and find the total number of items. See the connection between repeated addition and multiplication when using arrays.

For Fourth Grade

Multiplication Area Model
Multiplicative Comparison Word Problems

Solving Multiplicative comparison word problems on a computer screen.

Resources for Problem Solving

3 Act Tasks
What’s the Best Proven Way to Teach Word Problems with Two Step Equations?
5 Powerful and Easy Lessons Teaching Students How to Get a Growth Mindset
5 Powerful Ideas to Help Students Develop a Growth Mindset in Mathematics

Problem Solving Steps For Math

In mathematics, problem solving is one of the most important topics to teach. Learning to problem solve helps students apply mathematics to real-world situations. In addition, it is used for a deeper understanding of mathematical concepts.

Check out The Free Ultimate Guide for Creating a Growth Mindset Classroom and Students Who Love Math for ideas, lessons, and mindset surveys to use to cultivate a growth mindset classroom.

Start by modeling using the problem solving steps in math and allowing opportunities for students to use the steps to solve problems. As students become more comfortable with using the steps and have some strategies to use, provide more challenging tasks. Then, students will begin to see the importance of problem solving in math and connecting their learning to real-world situations.

$Kids solving word problems.$

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Unlocking the Power of Growth: How to Help Students Develop a Growth Mindset

This guide aims to provide insightful strategies and practical tips for how to help students develop

Try out these more and less kindergarten activities to use with students to compare numbers.

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Students need to learn their numbers to 120 and using missing number activities is a fun way for students to learn their numbers.

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I am always looking for new and fun missing numbers activities. These activities help students practice

Understand teen numbers with these time saving number activities to help students count, identify and represent teen numbers.

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To understand teen numbers, kids need to understand place value. Kids learn to count to 10

Hi, I'm Eastern Shore Math Teacher!

I have been teaching for over 22 years in an elementary school. I help educators plan engaging math lessons and cultivate a positive math culture in their classrooms.

Sign up and I will send you the growth mindset classroom guide and I will help you get your elementary students to love math.

4 Steps in Solving Problems

TLDR This video tutorial outlines a four-step approach to solving math word problems: understanding the problem, making a plan, carrying out the plan, and checking the answer. It emphasizes the importance of comprehending the problem thoroughly before attempting a solution and suggests using strategies such as looking for clues and keywords. The video encourages persistence, advising viewers to reread the problem and reassess their understanding if they encounter difficulties. It also highlights the value of checking one's work to ensure accuracy and reasonableness of the solution.

📚 Understand the problem thoroughly before attempting to solve it.
🔍 Read the word problem carefully, looking for key words and important numbers.
🖊 Highlight or underline crucial information to aid in comprehension.
🧠 Reread the problem multiple times to ensure complete understanding.
📈 Choose an appropriate problem-solving strategy for the type of math word problem.
🔢 Consider whether addition, subtraction, multiplication, division, or a combination is needed to solve the problem.
📝 Use a notebook or scrap paper for calculations; it's okay to make mistakes and start over.
🔄 If stuck, reread the problem to ensure you're on the right track.
✅ Always check your answer to ensure it is reasonable and makes sense within the context of the problem.
🔄 If the answer doesn't make sense, revisit the problem to correct any misunderstandings or calculation errors.

What are the four main steps in solving a math word problem according to the video?

- The four main steps are: 1. Understand the problem, 2. Make a plan, 3. Carry out the plan, and 4. Check your answer.

Why is it important to read a word problem carefully before attempting to solve it?

- Reading a word problem carefully is important because if you start working on a problem without fully understanding it, you may get frustrated and want to give up.

What should you do to understand a word problem better?

- To understand a word problem better, read it carefully once, then read it again looking for important clues like key words and numbers, and underline or highlight important information.

How many times should you read a word problem to ensure you fully understand it?

- You should read a word problem at least three times to ensure you fully understand it.

What is the first step in making a plan to solve a math word problem?

- The first step in making a plan is to choose an appropriate problem-solving strategy.

Where can you learn about problem-solving strategies as mentioned in the video?

- You can learn about problem-solving strategies from the videos labeled 'Problem Solving Strategies' on the site.

What should you do if you get stuck while carrying out your plan to solve the problem?

- If you get stuck, go back and reread the problem to ensure you've understood it correctly and check your calculations.

How do you check if your answer to a word problem is reasonable?

- To check if your answer is reasonable, go back to the problem and see if your answer makes sense and is logical within the context of the problem.

What should you do if your answer does not seem reasonable after checking?

- If your answer does not seem reasonable, go back to step 1 and ensure you've understood the problem correctly, then check your calculations.

What is the advice given in the video for solving math word problems?

- The advice given is to keep solving math word problems with fun online games and quizzes, and always remember to be clever.

Why might highlighting important information in a word problem be helpful?

- Highlighting important information can help you focus on the key details needed to solve the problem and prevent overlooking crucial data.

📚 Introduction to Math Word Problem Solving

This paragraph introduces the viewer to a video tutorial on solving math word problems. It outlines four main steps: understanding the problem, making a plan, carrying out the plan, and checking the answer. The importance of fully understanding the problem before attempting to solve it is emphasized, along with the suggestion to look for key words and numbers. An example word problem involving the ages of Alex, Max, Josie, and Suzie is provided to illustrate the process.

💡 Problem Solving

💡 understand the problem, 💡 make a plan, 💡 carry out the plan, 💡 check your answer, 💡 key words, 💡 important numbers, 💡 problem solving strategies, 💡 reasonable.

Learn the 4 main steps to solving math word problems.

Understand the problem before attempting to solve it.

Read the word problem carefully to identify key words and numbers.

Underline or highlight important information in the problem.

Reread the problem to ensure full understanding.

Choose a problem-solving strategy after understanding the problem.

Watch 'Problem Solving Strategies' videos for different strategies.

Determine if addition, subtraction, multiplication, division, or a combination is needed.

Use 'Look for Clues and Key Words' strategy to know which operations to use.

Work out the word problem using a notebook or scrap paper.

It's okay to make mistakes and start over during problem-solving.

Reread the problem if you get stuck while carrying out the plan.

Check your answer to ensure it makes sense and is reasonable.

If the answer is not reasonable, revisit the problem to ensure correct understanding.

Remember the 4 strategies to solve math word problems.

Pause the video to write down the steps for future reference.

Practice solving math word problems with online games and quizzes.

Always approach problem-solving with cleverness and persistence.

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5 Easy Steps to Solve Any Word Problem in Math

February 27, 2021

Picture this my teacher besties. You are solving word problems in your math class and every student, yes every student knows how to solve word problems without immediately entering a state of confusion! They know how to attack the problem head-on and have a method to solve every single problem that is presented to them.

How Do You Solve Word Problems in Math?

Ask yourself this, what do you think is the #1 phrase a student says as soon as they see a word problem?

You guessed it, my teacher friend, I don’t know how to do this! I think the most common question I get when I’m teaching my math classes, is how do I solve this?

Students see word problems and immediately enter freak-out mode! Let’s take solving word problems in the classroom and make it easier for students to SOLVE the problem!

How to Solve Word Problems Step by Step

There are so many methods that students can choose from when learning how to solve word problems. The 4 step method is the foundation for all of the methods that you will see, but what about a variation of the 4 step method that every student can do just because they get it.

Students are most likely confused about how to solve word problems because they have never used a consistent method over the years. I’m all about consistency in my classroom. Fortunately, in my school district, I get to teach most of the students year after year because of how small our class sizes are. So I’m going to give you a method based on the 4 step method, that allows all students to be successful at solving word problems.

Even the most unmotivated math student will learn how to solve word problems and not skip them!

steps-to-solving-word-problems-in-mathematics

Tips, Tricks, and Teaching Strategies to Solving Word Problems in Math

Going back to the 4 step method just in case you need a refresher. If you know me at all a little reminder of “oh yeah I remember that now” always helps me!

4 steps in solving word problems in math:

Understand the Problem
Plan the solution
Solve the Problem
Check the solution

This 4 step method is the basis of the method I’m going to tell you all about. The problem isn’t with the method itself, it is the fact that most students see word problems and just start panicking!

Why can they do an entire assignment and then see a word problem and then suddenly stop? Is there a reason why books are designed with word problems at the end?

These are questions that I constantly have asked myself over the last several years. I finally got to the point where my students needed a consistent approach to solving word problems that worked every single time.

The first thing I knew I needed to start doing was introducing students to word problems at the beginning of each lesson.

Once students first see the word problems at the beginning of the lesson, they are less likely to be scared of them when it comes time to do it by themselves!

This also will increase their confidence in the classroom. In case you missed it, I shared all about how I increase my students’ confidence in the classroom.

Wonder how increasing their confidence will help keep them motivated in the classroom?

So confident motivated students will see word problems that could be on their homework, any standardized test, and say I GOT THIS!

Steps to Solving Word Problems in Mathematics

We are ready to SOLVE any word problem our students are going to encounter in math class.

Here are my 5 easy steps to SOLVE any word problem in math:

S – State the objective
O – Outline your plan
L – Look for Key Details – Information
V – Verify and Solve
E – Explain and check your solution

Do you want to learn how to implement this 5 steps problem-solving strategy into your classroom? I’m hosting a FREE workshop all about how to implement this strategy in your classroom!

I am so excited to be offering a workshop to increase students’ confidence in solving word problems. The workshop is held in my Facebook Group The Round Robin Math Community. It also will be sent straight to your inbox and you can watch it right now!

If you’re interested, join today and all the details will be sent to you ASAP!

I will see you there!

PS. Need the SOLVE method for your bulletin board for your students’ math journals/notebooks? Check out this bulletin board resource here:

$problem solving bulletin board$

Love, Robin

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Hi, I'm Robin!

I am a secondary math teacher with over 19 years of experience! If you’re a teacher looking for help with all the tips, tricks, and strategies for passing the praxis math core test, you’re in the right place!

I also create engaging secondary math resources for grades 7-12!

Learn more about me and how I can help you here .

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A Visual Guide to Problem-Solving: How to Do A Tape Diagram Using 4 Methods

Tape diagrams, also called bar models or strip diagrams, are awesome visual aids that can make it easier for you to get and solve math problems. They show numbers as rectangles or bars, giving you a clear picture of how things are related and making it simpler to figure things out. In this guide, we will look at different ways how to do a tape diagram , so you can get the hang of this super useful trick for solving problems. Whether you're a student, teacher, or just a parent helping out, getting the hang of tape diagrams can boost your math skills and confidence. Let's jump into the tape diagrams and see how they can make solving math problems a piece of cake.

Part 1. What is a Tape Diagram

Part 2. how to use a tape diagram, part 3. how to do a tape diagram with mindonmap, part 4. how to do a tape diagram with 3 other tools, part 5. faqs on how to do a tape diagram.

A tape diagram is a handy tool in math for illustrating problems with numbers, ratios, and how different amounts are related to each other. It's made up of bars or tapes drawn to look like real-life sizes, with each bar showing a certain value or amount.

Key Points About Tape Diagrams

• It makes it easy to see how numbers and quantities are related, helping you better understand and solve math problems. • They're great for showing the parts representing a different amount. • It is really good for comparing the amount of one thing compared to another, especially when dealing with ratios and proportions. • Breaking down tricky problems into smaller, easier-to-see parts makes solving math problems feel less daunting.

This guide will show you how to draw a tape diagram to tackle math problems. By getting the hang of making and looking at tape diagrams, you can better solve problems and better grasp math ideas.

Figure out what numbers matter in the situation. Decide which number is the total and which is the pieces.

Draw a shape like a rectangle or bar to show the total amount. Then, split the shape into pieces according to the given information.

Write down what each piece is worth or how much it is. Check out the picture to see how the numbers relate.

Use the diagram to make up equations or do the math.

MindOnMap is the best online tool that makes it easy to sort out your thoughts, ideas, and projects visually. Whether you're just throwing out ideas, planning your next move, or trying to figure something out, it's great for making mind maps, flowcharts, and all sorts of diagrams. People from all walks of life, like students, teachers, workers, and teams, love it because it's a simple yet effective way to make complicated stuff easier to understand by breaking it down into clear, easy-to-see diagrams. What sets MindOnMap apart is its easy use, even if you need to improve at making mind maps and diagrams. It has features like working on things together in real-time, picking from different templates, and working on it from anywhere with your device. MindOnMap is perfect for getting things done faster, better-grasping things, and sharing your thoughts with others. Whether you're using it for personal stuff, school projects, or working with a team, MindOnMap makes it a breeze to see and organise all

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Steps on how to use tape diagrams in MindOnMap

Search MindOnMap in the search engine. You can download or use it free online. Arrange your data. Make sure you're clear on what numbers or values you're dealing with, and choose the Flowchart.

Choose a big rectangle or line to show the total amount. Break it down into smaller sections to show the different parts. You can change the colours and themes. After that, you can use brackets to separate it.

Label your data and create a formula to sum it up.

Once you've looked over all the info and possible fixes, click the Save button to save your project.

How do you make a tape diagram? Tape diagrams are super helpful visual aids that make it easier to get and solve math problems. They show numbers as rectangles or bars, clearly showing how things are connected and simplifying the problem-solving process. This guide will look at three go-to tools for making tape diagrams: EdrawMax Online, Creately, and Lucidchart. Each tool has features, so you can pick the one that fits what you need and like the most. By making tape diagrams, you'll get better at solving problems and understanding math concepts more deeply. Let's jump into the tape diagram and see why these tools are awesome.

Option 1. EdrawMax Online

EdrawMax Online is a handy tool for making all sorts of diagrams. It comes with many different templates, shapes, and ways to tweak them. Even though it's not made just for tape diagrams, its ability to do many things makes it a good fit for making these kinds of visual guides.

Search the tool and begin by making a new document.

Choose rectangles or lines from the shape library to show the tape and its sections. Use the tools to make the rectangles the right lengths for your data. Use text boxes to mark the different sections of the tape diagram.

Change the colours, fonts, and other elements to make your tape diagram your own.

Save your work if you are satisfied with your tape diagram.

Option 2. Creately

Creately is a strong drawing tool with many templates, shapes, and customisation options. Although it's not made just for tape diagrams, its flexibility makes it a good fit for creating these visual maps.

Steps on how to do a tape diagram with Creately

Go to the Creately website and create a diagram using a basic rectangular shape and connector to represent the tape and its divisions.

Use the resizing tools to adjust the lengths of the rectangles to match your data. Label the different parts of the tape diagram using text boxes.

Change colours, fonts, and other visual elements to personalise your tape diagram.

You can now save your tape diagram in Creately. Just click Export.

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

A photo illustration of a hand with a glitch texture holding a red question mark.

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.

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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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