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A Guide to Problem Solving

When confronted with a problem, in which the solution is not clear, you need to be a skilled problem-solver to know how to proceed. When you look at STEP problems for the first time, it may seem like this problem-solving skill is out of your reach, but like any skill, you can improve your problem-solving with practice. How do I become a better problem-solver? First and foremost, the best way to become better at problem-solving is to try solving lots of problems! If you are preparing for STEP, it makes sense that some of these problems should be STEP questions, but to start off with it's worth spending time looking at problems from other sources. This collection of NRICH problems  is designed for younger students, but it's very worthwhile having a go at a few to practise the problem-solving technique in a context where the mathematics should be straightforward to you. Then as you become a more confident problem-solver you can try more past STEP questions. One student who worked with NRICH said: "From personal experience, I was disastrous at STEP to start with. Yet as I persisted with it for a long time it eventually started to click - 'it' referring to being able to solve problems much more easily. This happens because your brain starts to recognise that problems fall into various categories and you subconsciously remember successes and pitfalls of previous 'similar' problems." A Problem-solving Heuristic for STEP Below you will find some questions you can ask yourself while you are solving a problem. The questions are divided into four phases, based loosely on those found in George Pólya's 1945 book "How to Solve It". Understanding the problem

• What area of mathematics is this?
• What exactly am I being asked to do?
• What do I know?
• What do I need to find out?
• What am I uncertain about?
• Can I put the problem into my own words?

Devising a plan

• Work out the first few steps before leaping in!
• Have I seen something like it before?
• Is there a diagram I could draw to help?
• Is there another way of representing?
• Would it be useful to try some suitable numbers first?
• Is there some notation that will help?

Carrying out the plan STUCK!

• Try special cases or a simpler problem
• Work backwards
• Guess and check
• Be systematic
• Work towards subgoals
• Imagine your way through the problem
• Has the plan failed? Know when it's time to abandon the plan and move on.

Looking back

• Have I answered the question?
• Sanity check for sense and consistency
• Check the problem has been fully solved
• Read through the solution and check the flow of the logic.

Throughout the problem solving process it's important to keep an eye on how you're feeling and making sure you're in control:

• Am I getting stressed?
• Is my plan working?
• Am I spending too long on this?
• Could I move on to something else and come back to this later?
• Am I focussing on the problem?
• Is my work becoming chaotic, do I need to slow down, go back and tidy up?
• Do I need to STOP, PEN DOWN, THINK?

Finally, don't forget that STEP questions are designed to take at least 30-45 minutes to solve, and to start with they will take you longer than that. As a last resort, read the solution, but not until you have spent a long time just thinking about the problem, making notes, trying things out and looking at resources that can help you. If you do end up reading the solution, then come back to the same problem a few days or weeks later to have another go at it.

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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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Problem Solving Steps

Introduction.

Solving problems is an important part of any math course. Techniques used for solving math problems are also applicable to other real-world situations. When solving problems it is important to know what to look for and to understand possible strategies for solving. As with most things, the more problems you solve, the better you will get at it. As you solve different types of math problems, you will gain a better understanding of the techniques that can be used with each type of problem.

In his book called How to Solve It, George Polya (1887 – 1985), proposed a four-step process for problem solving. In this module, we will take a look at those four common problem-solving principles as well as examples of techniques to be applied when solving problems.

Step 1: Understand the problem

In order to solve a problem, it is important to understand what you are being asked to find.

Strategies for understanding the problem:

• Review the problem. If you are solving a word problem, read through the entire problem.
• Seek to understand all the words used in stating the problem. Look for key words that will help you determine whether you will need to add, subtract, multiply, divide, or use a combination of these functions.
• Determine what you are being asked to find or show.
• Restate the problem in your own words.
• Try drawing a picture or diagram to better understand the problem.
• Make sure you have all of the information you need to solve the problem.

Step 2: Develop a plan

Determine how you will solve the problem. Some problems are solved by using a formula and others require you to develop an equation. Pictures, tables, or charts may also be used.

Keep in mind, you may be solving problems that require multiple steps. When you encounter these, break them down into smaller steps and solve each piece. The more problems you solve, the easier it will get to develop a plan for solving problems. You will begin to learn what techniques work best for each type of problem you solve.

Here are some of the common problem-solving techniques:

• Guess and check
• Make a table or list
• Eliminate possibilities
• Use symmetry
• Consider special cases
• Solve as an equation
• Look for a pattern
• Draw a sketch or a picture
• Solve a simpler problem
• Use a model
• Start from the end - work backward
• Use a formula

Step 3: Carry out the plan

Once you have determined your plan for solving the problem, the next step is putting that plan to work. Use the approach that makes sense for the problem and solve it. In most cases, this step will be easier than actually determining the plan. Having an understanding of basic math (pre-algebra) skills will help as you perform the necessary steps to solve the problem. Memorizing the simple multiplication and division tables at least to 10 can make solving problems much easier as well.

If you find that the problem-solving approach you chose does not work, you will need to go back to step two and choose a new approach. Having patience while carrying out your plan is important. It is not uncommon for mathematicians to have to try multiple approaches when solving problems.

Step 4: Look back and check

Here is where you check your logic. If you solved an equation, fill your answer into the equation and check to make sure it works. If you solved a word problem, consider whether or not your answer makes sense. If the problem asked for the height of a ball in the air and your answer was -10 feet, that does not make sense. Just because you get a number doesn’t mean it is right. It is important to check your answer and see if it logically makes sense. If your answer does not make sense, you should review the approach you chose as well as your math calculations. Many errors are corrected in this final step of the problem-solving process.

Example Problems

Example 1 – Difference in temperature

The hottest temperature ever recorded in Death Valley, CA was 134 degrees on July 10, 1913. The coldest temperature ever recorded there, 15 degrees, occurred earlier that year on January 8, 1913. What was the difference between these record temperatures in 1913? (www.nps.gov)

Step 1: Understand the problem: After reading through the problem, you will find that the problem to solve is clearly stated. You will need to determine the difference in temperature between the hottest and coldest recorded days in Death Valley, CA.

Step 2: Develop a plan: As you were reading the problem, you noted the key word “difference.” To find the difference between two numbers, you will need to use subtraction. You are now ready to set up an equation. You can assign the variable, x, to the unknown. In this case, the unknown is the difference in temperatures.

x = 134 - 15

Step 3: Carry out the plan: Now that you have developed your equation, go ahead and solve it. x = 134 - 15 x = 119 degrees

Step 4: Look back and check: To check your equation, fill the answer back into the equation and make sure it works. Also, consider whether or not your answer makes sense. If you had received an answer that was significantly higher or lower than either of the temperatures in the problem, that would indicate that there may be an error in your calculations. 119 = 134 – 15

Example 2 – Interest Earned

If a savings account balance of $2650 earns 4% interest in one year, how much interest is earned? What will the account balance be after the interest is earned?

Step 1: Understand the problem: After reading through the problem, you recognize that there are actually two problems to solve. You will need to determine the amount of interest on the account balance and you will need to determine what the account balance will be when the interest is earned.

Step 2: Develop a plan: The first problem you need to solve is the amount of interest earned on $2650. To find the amount of interest, you will need to multiply the current account balance by the interest rate. In order to complete the multiplication problem, you will need to change the percent into a decimal. You can assign the variable, x, to the unknown. In this case, the unknown is the amount of interest earned.

Once you determine how much interest will be earned, you can solve the second problem. You will need to add the current balance and the interest earned to determine how much will be in the account once the interest is applied. To establish an equation for the second problem, you can assign a variable, y, to the unknown. You establish this equation to solve the second problem.

y = $2650 + x

Step 3: Carry out the plan: Now that you have developed your equations, go ahead and solve them.

Part 1: x = .04 x $2650 x = $106

Part 2: y = $2650 + x y = $2650 + $106 y = $2756

Step 4: Look back and check: To check your equations, fill the answers back into them. Also, consider whether or not your answers makes sense. If you had received an answer that was lower or significantly higher than the original account balance, that would indicate there may be an error in your calculations.

Part 1: $106 = .04 x $2650

Part 2: $2756 = $2650 + $106

Example 3 - Book Buyers

In a recent sample of book buyers, 70 more shopped at large-chain bookstores than at small-chain/independent bookstores. A total of 442 book buyers shopped at these two types of stores. How many buyers shopped at each type of bookstore?

Step 1: Understand the problem: After reading through the problem, you determine you are asked to find the number of buyers shopping at each type of bookstore.

Step 2: Develop a plan: To solve this problem you will need to assign a variable, x, for one of the unknowns. If x is the number of book buyers shopping at large-chain bookstores, then (x – 70) = the number of book buyers shopping at small-chain/independent bookstores.

To solve the problem, you come up with this equation: x + x - 70 = 442

Step 3: Carry out the plan: Now that you have developed your equation, go ahead and solve it.

x + x - 70 = 442 2x – 70 + 70 = 442 + 70 2x /2= 512 /2 x = 256

After solving the equation, you determine that 256 people shopped at large-chain bookstores. You can plug 256 into the equation representing those shopping at small-chain bookstores (x - 70).

256 – 70 = 186

186 people shopped at small-chain bookstores

Step 4: Look back and check: To check your answers, fill them back into the original problem. Also, consider whether or not your answers makes sense. If the number of small-chain store shoppers was greater than the number of large-chain store shoppers or if the two numbers did not equal 442 that would indicate there was an error in your calculations.

The number of large chain shoppers (256) is 70 more than the number of small-chain store shoppers (186), and the total number of these shoppers (256 + 186) is 442.

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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.
• The graphs section contains commands for plotting equations and inequalities.
• The numbers section has a percentages command for explaining the most common types of percentage problems and a section for dealing with scientific notation.

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What IS Problem-Solving?

Ask teachers about problem-solving strategies, and you’re opening a can of worms! Opinions about the “best” way to teach problem-solving are all over the board. And teachers will usually argue for their process quite passionately.

When I first started teaching math over 25 years ago, it was very common to teach “keywords” to help students determine the operation to use when solving a word problem. For example, if you see the word “total” in the problem, you always add. Rather than help students become better problem solvers, the use of keywords actually resulted in students who don’t even feel the need to read and understand the problem–just look for the keywords, pick out the numbers, and do the operation indicated by the keyword.

This post contains affiliate links, which simply means that when you use my link and purchase a product, I receive a small commission. There is no additional cost to you, and I only link to books and products that I personally use and recommend.

Another common strategy for teaching problem-solving is the use of acrostics that students can easily remember to perform the “steps” in problem-solving. CUBES is an example. Just as with keywords, however, students often follow the steps with little understanding. As an example, a common step is to underline or highlight the question. But if you ask students why they are underlining or highlighting the question, they often can’t tell you. The question is , in fact, super important, but they’ve not been told why. They’ve been told to underline the question, so they do.

The problem with both keywords and the rote-step strategies is that both methods try to turn something that is inherently messy into an algorithm! It’s way past time that we leave both methods behind.

First, we need to broaden the definition of problem-solving. Somewhere along the line, problem-solving became synonymous with “word problems.” In reality, it’s so much more. Every one of us solves dozens or hundreds of problems every single day, and most of us haven’t solved a word problem in years. Problem-solving is often described as  figuring out what to do when you don’t  know what to do.  My power went out unexpectedly this morning, and I have work to do. That’s a problem that I had to solve. I had to think about what the problem was, what my options were, and formulate a plan to solve the problem. No keywords. No acrostics. I’m using my phone as a hotspot and hoping my laptop battery doesn’t run out. Problem solved. For now.

If you want to get back to what problem-solving really is, you should consult the work of George Polya. His book, How to Solve It , which was first published in 1945, outlined four principles for problem-solving. The four principles are: understand the problem, devise a plan, carry out the plan, and look back. This document from UC Berkeley’s Mathematics department is a great 4-page overview of Polya’s process. You can probably see that the keyword and rote-steps strategies were likely based on Polya’s method, but it really got out of hand. We need to help students think , not just follow steps.

I created both primary and intermediate posters based on Polya’s principles. Grab your copies for free here !

I would LOVE to hear your comments about problem-solving!

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Do you tutor teachers?

I do professional development for district and schools, and I have online courses.

You make a great point when you mentioned that teaching students to look for “keywords” is not teaching students to become better problem solvers. I was once guilty of using the CUBES strategy, but have since learned to provide students with opportunity to grapple with solving a problem and not providing them with specified steps to follow.

I think we’ve ALL been there! We learn and we do better. 🙂

Love this article and believe that we can do so much better as math teachers than just teaching key words! Do you have an editable version of this document? We are wanting to use something similar for our school, but would like to tweak it just a bit. Thank you!

I’m sorry, but because of the clip art and fonts I use, I am not able to provide an editable version.

Hi Donna! I am working on my dissertation that focuses on problem-solving. May I use your intermediate poster as a figure, giving credit to you in my citation with your permission, for my section on Polya’s Traditional Problem-Solving Steps? You laid out the process so succinctly with examples that my research could greatly benefit from this image. Thank you in advance!

Absolutely! Good luck with your dissertation!

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Overview of the Problem-Solving Mental Process

Kendra Cherry, MS, is a psychosocial rehabilitation specialist, psychology educator, and author of the "Everything Psychology Book."

Rachel Goldman, PhD FTOS, is a licensed psychologist, clinical assistant professor, speaker, wellness expert specializing in eating behaviors, stress management, and health behavior change.

• Identify the Problem
• Define the Problem
• Form a Strategy
• Organize Information
• Allocate Resources
• Monitor Progress
• Evaluate the Results

Frequently Asked Questions

Problem-solving is a mental process that involves discovering, analyzing, and solving problems. The ultimate goal of problem-solving is to overcome obstacles and find a solution that best resolves the issue.

The best strategy for solving a problem depends largely on the unique situation. In some cases, people are better off learning everything they can about the issue and then using factual knowledge to come up with a solution. In other instances, creativity and insight are the best options.

It is not necessary to follow problem-solving steps sequentially, It is common to skip steps or even go back through steps multiple times until the desired solution is reached.

In order to correctly solve a problem, it is often important to follow a series of steps. Researchers sometimes refer to this as the problem-solving cycle. While this cycle is portrayed sequentially, people rarely follow a rigid series of steps to find a solution.

The following steps include developing strategies and organizing knowledge.

1. Identifying the Problem

While it may seem like an obvious step, identifying the problem is not always as simple as it sounds. In some cases, people might mistakenly identify the wrong source of a problem, which will make attempts to solve it inefficient or even useless.

Some strategies that you might use to figure out the source of a problem include :

• Asking questions about the problem
• Breaking the problem down into smaller pieces
• Looking at the problem from different perspectives
• Conducting research to figure out what relationships exist between different variables

2. Defining the Problem

After the problem has been identified, it is important to fully define the problem so that it can be solved. You can define a problem by operationally defining each aspect of the problem and setting goals for what aspects of the problem you will address

At this point, you should focus on figuring out which aspects of the problems are facts and which are opinions. State the problem clearly and identify the scope of the solution.

3. Forming a Strategy

After the problem has been identified, it is time to start brainstorming potential solutions. This step usually involves generating as many ideas as possible without judging their quality. Once several possibilities have been generated, they can be evaluated and narrowed down.

The next step is to develop a strategy to solve the problem. The approach used will vary depending upon the situation and the individual's unique preferences. Common problem-solving strategies include heuristics and algorithms.

• Heuristics are mental shortcuts that are often based on solutions that have worked in the past. They can work well if the problem is similar to something you have encountered before and are often the best choice if you need a fast solution.
• Algorithms are step-by-step strategies that are guaranteed to produce a correct result. While this approach is great for accuracy, it can also consume time and resources.

Heuristics are often best used when time is of the essence, while algorithms are a better choice when a decision needs to be as accurate as possible.

4. Organizing Information

Before coming up with a solution, you need to first organize the available information. What do you know about the problem? What do you not know? The more information that is available the better prepared you will be to come up with an accurate solution.

When approaching a problem, it is important to make sure that you have all the data you need. Making a decision without adequate information can lead to biased or inaccurate results.

5. Allocating Resources

Of course, we don't always have unlimited money, time, and other resources to solve a problem. Before you begin to solve a problem, you need to determine how high priority it is.

If it is an important problem, it is probably worth allocating more resources to solving it. If, however, it is a fairly unimportant problem, then you do not want to spend too much of your available resources on coming up with a solution.

At this stage, it is important to consider all of the factors that might affect the problem at hand. This includes looking at the available resources, deadlines that need to be met, and any possible risks involved in each solution. After careful evaluation, a decision can be made about which solution to pursue.

6. Monitoring Progress

After selecting a problem-solving strategy, it is time to put the plan into action and see if it works. This step might involve trying out different solutions to see which one is the most effective.

It is also important to monitor the situation after implementing a solution to ensure that the problem has been solved and that no new problems have arisen as a result of the proposed solution.

Effective problem-solvers tend to monitor their progress as they work towards a solution. If they are not making good progress toward reaching their goal, they will reevaluate their approach or look for new strategies .

7. Evaluating the Results

After a solution has been reached, it is important to evaluate the results to determine if it is the best possible solution to the problem. This evaluation might be immediate, such as checking the results of a math problem to ensure the answer is correct, or it can be delayed, such as evaluating the success of a therapy program after several months of treatment.

Once a problem has been solved, it is important to take some time to reflect on the process that was used and evaluate the results. This will help you to improve your problem-solving skills and become more efficient at solving future problems.

A Word From Verywell​

It is important to remember that there are many different problem-solving processes with different steps, and this is just one example. Problem-solving in real-world situations requires a great deal of resourcefulness, flexibility, resilience, and continuous interaction with the environment.

Get Advice From The Verywell Mind Podcast

Hosted by therapist Amy Morin, LCSW, this episode of The Verywell Mind Podcast shares how you can stop dwelling in a negative mindset.

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You can become a better problem solving by:

• Practicing brainstorming and coming up with multiple potential solutions to problems
• Being open-minded and considering all possible options before making a decision
• Breaking down problems into smaller, more manageable pieces
• Asking for help when needed
• Researching different problem-solving techniques and trying out new ones
• Learning from mistakes and using them as opportunities to grow

It's important to communicate openly and honestly with your partner about what's going on. Try to see things from their perspective as well as your own. Work together to find a resolution that works for both of you. Be willing to compromise and accept that there may not be a perfect solution.

Take breaks if things are getting too heated, and come back to the problem when you feel calm and collected. Don't try to fix every problem on your own—consider asking a therapist or counselor for help and insight.

If you've tried everything and there doesn't seem to be a way to fix the problem, you may have to learn to accept it. This can be difficult, but try to focus on the positive aspects of your life and remember that every situation is temporary. Don't dwell on what's going wrong—instead, think about what's going right. Find support by talking to friends or family. Seek professional help if you're having trouble coping.

Davidson JE, Sternberg RJ, editors.  The Psychology of Problem Solving .  Cambridge University Press; 2003. doi:10.1017/CBO9780511615771

Sarathy V. Real world problem-solving .  Front Hum Neurosci . 2018;12:261. Published 2018 Jun 26. doi:10.3389/fnhum.2018.00261

By Kendra Cherry, MSEd Kendra Cherry, MS, is a psychosocial rehabilitation specialist, psychology educator, and author of the "Everything Psychology Book."

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Literal equations

Here you will learn about what a literal equation is, how to solve for a specific variable in a literal equation, and how to use a literal equation to problem solve.

Students first learn about literal equations in algebra 1 and expand that knowledge as they progress through high school mathematics.

What is a literal equation?

Literal equations are equations that have more than one variable in them. Literal equations are represented by formulas and equations. Here are some examples of literal equations.

Notice how the formulas all have more than one variable in them. To solve a literal equation, it essentially means to rearrange the formula or equation to isolate one of the variables.

The steps taken to solve a literal equation are the same as those applied to solving a regular multi-step equation that has one unknown variable and numerical values.

When solving equations, the inverse operation(s) are applied to the equation. When solving literal equations, the inverse operation needs to be applied in order to solve.

Recall the order of operations and their inverses:

Solving a literal equation for a specific variable is also known as changing the subject of the formula or equation.

Let’s look at a side-by-side comparison of solving algebraic equations and literal equations to make important connections.

Common Core State Standards

How does this apply to high school math?

• High School Algebra – Creating Equations (HSA-CED.A.4) Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm’s law V=IR to highlight resistance R.
• High School Algebra – Reasoning with Equations and Inequalities (HSA-REI.B.3) Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.

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How to solve literal equations

In order to solve literal equations:

Use inverse operations to move a number, term, or variable.

Repeat this process until the remaining variable is the subject.

Write the answer.

Solving literal equations examples

Example 1: solve the formula for a variable.

Use the distance formula to solve for t.

To solve for t move the r to the other side of the equation by doing the inverse operation, which is to divide.

2 Repeat this process until the remaining variable is isolated.

The t is isolated so no need to do the process again.

3 Write the answer.

Example 2: solve a linear equation for y

Solve the linear equation for y.

Move the 2x first by doing the inverse operation, which is to subtract 2x from both sides of the equation.

Repeat this process until the remaining variable is isolated.

The y is still not isolated.

Move the 3 to the other side of the equation by doing the inverse operation, which is to divide by 3.

You can also express the answer by dividing both 12 and 2x by 3.

y=4-\cfrac{2}{3}x

Example 3: volume formula

Use the formula for volume of a cylinder to solve for h.

To isolate the h, move the \Pi by doing the inverse operation, which is to divide.

The h is still not isolated, move the r^2 by doing the inverse operation which is division or multiply by the reciprocal.

This can be simplified by multiplying by the reciprocal.

\cfrac{V}{\Pi}\times\cfrac{1}{r^2}=h

h=\cfrac{V}{r^2 \Pi}

Example 4: literal equation with a fraction

Solve the given equation for x.

To solve the equation for x, first move the b by doing the inverse operation, which is to multiply by b.

The x is still not isolated. Move the y by doing the inverse operation, which is to subtract y.

Example 5: literal equation with parentheses

Solve the equation for x.

To start, move the 2 by doing the inverse operation, which in this case is to divide by 2.

The x is still not isolated, so the next step is to move the 3y by doing the inverse operation, which in this case is to subtract 3y.

x=7\cfrac{1}{2}-3y

Note: An alternative strategy to solving the equation for x is to distribute the 2 and then use inverse operations.

Example 6: literal equation with fractions

Given the equation \cfrac{a}{b}+\cfrac{c}{d}=e, solve it for a.

The first step is to move the fraction, \cfrac{c}{d} by doing the inverse operation, which in this case is to subtract \cfrac{c}{d}.

The a is still not isolated. Move the b by doing the inverse operation which in this case is to multiply.

a=be-\cfrac{bc}{d}

Teaching tips for solving literal equations

• Have students make comparisons between the strategies used for solving equations and apply them to literal equations. Also, have students circle or highlight the indicated variable that needs to be isolated to help them make visual connections.
• In order for students to practice problems, have them gameplay using digital platforms or participate in scavenger hunts instead of assigning worksheets.
• Incorporate activities such as having students create and solve their own literal equations.

Easy mistakes to make

• Applying the inverse operations incorrectly The inverse operation used is incorrect. For example, let M=DV. Here, the formula is rearranged to get M-V=D and so the value of V has been subtracted, rather than divided. The correct answer should be \cfrac{M}{V}=D.
• Moving the variables incorrectly Letters are simply moved from one side of the equals sign to the other. For example, A=lw becomes Al=w which is incorrect. Instead, you need to use the inverse operation which is to divide by l, to the right side and left side of the equation to get w=\cfrac{A}{l}.
• Not multiplying or dividing throughout When given the equation y=mx+c, if you divide both sides of the equation by m we should get \cfrac{y}{m}=x+\cfrac{c}{m} whereas students would incorrectly write \cfrac{y}{m}=x+c. When multiplying and dividing by something, every term on each side of the equals sign must be multiplied or divided by it.
• Expanding parentheses When expanding parentheses, all terms within the parentheses must be multiplied by the value on the outside of the parentheses — in other words apply the distributive property. For example, 2(x+3), you get 2x+6 as both of the terms inside the bracket have been multiplied by 2.
• Not using the correct root to undo a power When you have a variable raised to a power (exponent), you must use inverse operations (root) to remove that power. For example, if you have x^4 you need to calculate the 4 th root of the variable to get x. You cannot use the square root to undo the power of 4. x^{4}\rightarrow inverse 4 th root \rightarrow\sqrt[4]{x^4}

Related math formulas lessons

• Math formulas
• Kinematic equations

Practice solving literal equation problems

1. Solve the equation x-2y=10 for y.

To solve the equation for y, means to isolate the y on one side of the equation. First, move the – \; 2y by doing the inverse operation which in this case is to add 2y.

Next, move the 10 by doing the inverse operation, which is to subtract 10.

Finally divide throughout by 2 so that y is isolated.

2. Rearrange the formula, A=s^2 so that s is isolated.

To solve the equation for s, do the inverse operation, which is to take the square root. Undoing a square is to take a square root.

Remember when taking the square root in solving equations, to include both the positive and negative solutions.

3. Solve the equation \cfrac{a}{b}+\cfrac{c}{d}=e, for c.

To solve the equation for c first move \cfrac{a}{b} by doing the inverse operation which in this case is to subtract.

Notice how the c is still not isolated. In order to isolate it, move the d by doing the inverse operation which is to multiply.

4. The formula \cfrac{\theta\Pi{d}}{360}=A represents the arc length A of a sector of a circle with diameter d. Make d the subject of the formula.

To solve the formula, \cfrac{\theta\Pi{d}}{360}=A for d, first move 360 by doing the inverse operation which in this case is to multiply.

Notice how d is still not isolated on one side of the equation. Move the \theta and \Pi at the same time by doing the inverse operation which is division.

5. Solve the equation for y.

To solve the equation, m=\cfrac{y-x}{y} for y, first move the y from the denominator by doing the inverse operation which is to multiply.

The y isn’t isolated and there are two terms that contain y. So, the next step is to move the y from the right to the left by doing the inverse operation which in this case is to subtract y.

In order to only have one y, factor out the y from the left side of the equation.

Now that there is one y, you can solve for it by moving the (m-1).  To move it, do the inverse operation which is to divide.

6. Make b the subject of the formula \cfrac{a}{\sin(A)}=\cfrac{b}{\sin(B)}.

To solve the equation for b move \sin(B) by doing the inverse operation which in this case is to multiply.

Literal equations FAQs

Yes, if the highest exponent in any equation is 2, it’s considered to be a quadratic equation. So, literal equations can be quadratic.

Yes, literal equations can contain an absolute value. The process to solve it would be the same as the steps taken when solving a regular algebraic absolute value equation.

In pre-algebra you are taught how to solve one-step, two-step, and multi-step equations. Your teacher might also expose you to some basic literal equations too.

The next lessons are

• Coordinate plane
• Types of graphs
• Graphing linear equations

Still stuck?

At Third Space Learning, we specialize in helping teachers and school leaders to provide personalized math support for more of their students through high-quality, online one-on-one math tutoring delivered by subject experts.

Each week, our tutors support thousands of students who are at risk of not meeting their grade-level expectations, and help accelerate their progress and boost their confidence.

Find out how we can help your students achieve success with our math tutoring programs .

[FREE] Common Core Practice Tests (Grades 3 to 6)

Prepare for math tests in your state with these Grade 3 to Grade 6 practice assessments for Common Core and state equivalents.

40 multiple choice questions and detailed answers to support test prep, created by US math experts covering a range of topics!

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How to Ace Math Problem Solving

When your kids struggle with their math, it’s time to take a step back and take a deep breath. They need to slow down and take their time. Here’s a step by step guide that will help your kids get through those tough math problems.

We’ll use a grade 3 addition word problem as an example to clarify:

Pinky the Pig bought 36 apples while Danny the Duck bought 73 apples and 14 bananas. How many apples do they have altogether?

Read the problem

Carefully read through the problem to make sure you understand what is being asked.

Pinky the pig and Danny the duck bought apples and bananas. The question is how many apples they have together.

Re-read the problem

Read through the problem again and as you read through it, make notes.

Pinky the pig –36 apples. Danny the duck –73 apples and 14 bananas. How many apples together?

What is the problem asking

In your own words, say or write down exactly what the question is asking you to solve.

The question is asking how many apples the pig and the duck bought together.

Write it down in detail

Go through the problem and write out the information in an organized fashion. A diagram or table might help.

Turn it into math

Figure out what math operation(s) or formula(s) you need to use in order to solve this problem.

The problem wants us to add the number of apples Pinky the Pig and Danny the Duck have together. That means we need to make use of addition to add the apples.

Find an example

Are you still struggling? Sometimes it’s hard to work out the solution, especially if the math problem involves several steps. It’s time to present the problem in an easier way. As teachers and parents we can often help our kids simplify the problem from our own math knowledge. If the problem is a bit harder, there are lots of resources online that you can look up for similar problems that have been worked out on paper or a video tutorial to watch.

In our example, let’s say the double-digit numbers are intimidating our student, so we’re going to simplify the equation for the sake of helping our student understand the operation needed.

Let’s say Pinky the Pig bought 3 apples and Danny the Duck 7 apples and 1 banana. Now, how many apples have they bought together? With 3 apples and 7 apples bought, the total number of apples is 10.

Work out the problem

Now that we have got to the bottom of what is being asked and know what operation to use, it’s time to work out the problem.

Pinky the Pig bought 36 apples. Danny the Duck bought 73 apples. (The 14 bananas do not matter) We need to add up the apples. 36 + 73 = 109

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10.1: George Polya's Four Step Problem Solving Process

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Step 1: Understand the Problem

• Do you understand all the words?
• Can you restate the problem in your own words?
• Do you know what is given?
• Do you know what the goal is?
• Is there enough information?
• Is there extraneous information?
• Is this problem similar to another problem you have solved?

Step 2: Devise a Plan: Below are some strategies one might use to solve a problem. Can one (or more) of the following strategies be used? (A strategy is defined as an artful means to an end.)

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The Algebra Problem: How Middle School Math Became a National Flashpoint

Top students can benefit greatly by being offered the subject early. But many districts offer few Black and Latino eighth graders a chance to study it.

By Troy Closson

From suburbs in the Northeast to major cities on the West Coast, a surprising subject is prompting ballot measures, lawsuits and bitter fights among parents: algebra.

Students have been required for decades to learn to solve for the variable x, and to find the slope of a line. Most complete the course in their first year of high school. But top-achievers are sometimes allowed to enroll earlier, typically in eighth grade.

The dual pathways inspire some of the most fiery debates over equity and academic opportunity in American education.

Do bias and inequality keep Black and Latino children off the fast track? Should middle schools eliminate algebra to level the playing field? What if standout pupils lose the chance to challenge themselves?

The questions are so fraught because algebra functions as a crucial crossroads in the education system. Students who fail it are far less likely to graduate. Those who take it early can take calculus by 12th grade, giving them a potential edge when applying to elite universities and lifting them toward society’s most high-status and lucrative professions.

But racial and economic gaps in math achievement are wide in the United States, and grew wider during the pandemic. In some states, nearly four in five poor children do not meet math standards.

To close those gaps, New York City’s previous mayor, Bill de Blasio, adopted a goal embraced by many districts elsewhere. Every middle school would offer algebra, and principals could opt to enroll all of their eighth graders in the class. San Francisco took an opposite approach: If some children could not reach algebra by middle school, no one would be allowed to take it.

The central mission in both cities was to help disadvantaged students. But solving the algebra dilemma can be more complex than solving the quadratic formula.

New York’s dream of “algebra for all” was never fully realized, and Mayor Eric Adams’s administration changed the goal to improving outcomes for ninth graders taking algebra. In San Francisco, dismantling middle-school algebra did little to end racial inequities among students in advanced math classes. After a huge public outcry, the district decided to reverse course.

“You wouldn’t think that there could be a more boring topic in the world,” said Thurston Domina, a professor at the University of North Carolina. “And yet, it’s this place of incredibly high passions.”

“Things run hot,” he said.

In some cities, disputes over algebra have been so intense that parents have sued school districts, protested outside mayors’ offices and campaigned for the ouster of school board members.

Teaching math in middle school is a challenge for educators in part because that is when the material becomes more complex, with students moving from multiplication tables to equations and abstract concepts. Students who have not mastered the basic skills can quickly become lost, and it can be difficult for them to catch up.

Many school districts have traditionally responded to divergent achievement levels by simply separating children into distinct pathways, placing some in general math classes while offering others algebra as an accelerated option. Such sorting, known as tracking, appeals to parents who want their children to reach advanced math as quickly as possible.

But tracking has cast an uncomfortable spotlight on inequality. Around a quarter of all students in the United States take algebra in middle school. But only about 12 percent of Black and Latino eighth graders do, compared with roughly 24 percent of white pupils, a federal report found .

“That’s why middle school math is this flashpoint,” said Joshua Goodman, an associate professor of education and economics at Boston University. “It’s the first moment where you potentially make it very obvious and explicit that there are knowledge gaps opening up.”

In the decades-long war over math, San Francisco has emerged as a prominent battleground.

California once required that all eighth graders take algebra. But lower-performing middle school students often struggle when forced to enroll in the class, research shows. San Francisco later stopped offering the class in eighth grade. But the ban did little to close achievement gaps in more advanced math classes, recent research has found.

As the pendulum swung, the only constant was anger. Leading Bay Area academics disparaged one another’s research . A group of parents even sued the district last spring. “Denying students the opportunity to skip ahead in math when their intellectual ability clearly allows for it greatly harms their potential for future achievement,” their lawsuit said.

The city is now back to where it began: Middle school algebra — for some, not necessarily for all — will return in August. The experience underscored how every approach carries risks.

“Schools really don’t know what to do,” said Jon R. Star, an educational psychologist at Harvard who has studied algebra education. “And it’s just leading to a lot of tension.”

In Cambridge, Mass., the school district phased out middle school algebra before the pandemic. But some argued that the move had backfired: Families who could afford to simply paid for their children to take accelerated math outside of school.

“It’s the worst of all possible worlds for equity,” Jacob Barandes, a Cambridge parent, said at a school board meeting.

Elsewhere, many students lack options to take the class early: One of Philadelphia’s most prestigious high schools requires students to pass algebra before enrolling, preventing many low-income children from applying because they attend middle schools that do not offer the class.

In New York, Mr. de Blasio sought to tackle the disparities when he announced a plan in 2015 to offer algebra — but not require it — in all of the city’s middle schools. More than 15,000 eighth graders did not have the class at their schools at the time.

Since then, the number of middle schools that offer algebra has risen to about 80 percent from 60 percent. But white and Asian American students still pass state algebra tests at higher rates than their peers.

The city’s current schools chancellor, David Banks, also shifted the system’s algebra focus to high schools, requiring the same ninth-grade curriculum at many schools in a move that has won both support and backlash from educators.

And some New York City families are still worried about middle school. A group of parent leaders in Manhattan recently asked the district to create more accelerated math options before high school, saying that many young students must seek out higher-level instruction outside the public school system.

In a vast district like New York — where some schools are filled with children from well-off families and others mainly educate homeless children — the challenge in math education can be that “incredible diversity,” said Pedro A. Noguera, the dean of the University of Southern California’s Rossier School of Education.

“You have some kids who are ready for algebra in fourth grade, and they should not be denied it,” Mr. Noguera said. “Others are still struggling with arithmetic in high school, and they need support.”

Many schools are unequipped to teach children with disparate math skills in a single classroom. Some educators lack the training they need to help students who have fallen behind, while also challenging those working at grade level or beyond.

Some schools have tried to find ways to tackle the issue on their own. KIPP charter schools in New York have added an additional half-hour of math time to many students’ schedules, to give children more time for practice and support so they can be ready for algebra by eighth grade.

At Middle School 50 in Brooklyn, where all eighth graders take algebra, teachers rewrote lesson plans for sixth- and seventh-grade students to lay the groundwork for the class.

The school’s principal, Ben Honoroff, said he expected that some students would have to retake the class in high school. But after starting a small algebra pilot program a few years ago, he came to believe that exposing children early could benefit everyone — as long as students came into it well prepared.

Looking around at the students who were not enrolling in the class, Mr. Honoroff said, “we asked, ‘Are there other kids that would excel in this?’”

“The answer was 100 percent, yes,” he added. “That was not something that I could live with.”

Troy Closson reports on K-12 schools in New York City for The Times. More about Troy Closson

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How to Solve Maths Problems with Google Circle for Students

Math can be a challenging subject for many students. Between understanding word problems and applying the right formulas, getting stuck is a common experience. Google has introduced a powerful tool called Google Circle, designed to help students solve math problems with confidence.

Read In Short: What Google Circle is and how it can help students solve math problems. How to use Google Circle to break down and solve different types of math problems. Additional tips and tricks to maximize your learning experience with Google Circle

What is Google Circle?

Google Circle is a feature within the Google Search app for Android devices . It allows you to solve math problems directly from your phone or tablet. Instead of simply providing answers, Google Circle focuses on math problem-solving by offering step-by-step guidance. This makes it a valuable tool for understanding the concepts behind the calculations, not just getting the final answer.

How Does Google Circle Work

Using Google Circle is incredibly simple. Here’s how it goes:

Step 1: Open the Google Search app on your Android device.

Step 2: find the math problem you’re working on., step 3: circle the specific problem you want help with using your finger., step 4: google circle will recognize it and use its artificial intelligence to analyze it., how to enable circle to search, step 1: open settings on your phone, step 2: scroll down and select display, step 3: scroll down and select navigation bar, step 4: select the toggle and turn on circle to search, which devices are getting the circle to search update.

Below is the list of android which is getting Google Circle App

• Samsung Galaxy S24, S24 Plus, and S24 Ultra
• Samsung Galaxy S23, S23 Plus, and S23 Ultra
• Samsung Galaxy S23 FE
• Samsung Galaxy Z Fold 5
• Samsung Galaxy Z Flip 5
• Samsung Galaxy Tab S9, S9 Plus, and S9 Ultra
• Google Pixel 8 and Pixel 8 Pro
• Google Pixel 7 and Pixel 7 Pro
• Google Pixel 6 and Pixel 6 Pro
• Google Pixel 7a
• Google Pixel 6a
• Google Pixel Fold
• Google Pixel Tablet

What Types of Math Problems Can Google Circle Solve?

Currently, Google Circle is particularly helpful with math word problems. These can be some of the trickiest problems to tackle, as they require translating real-world situations into mathematical terms.

Google Circle excels at helping students understand the key information in a word problem, identify the relevant mathematical concepts, and apply the appropriate formulas to reach the solution.

Here are some specific areas where Google Circle can be helpful:

• Algebra: Solve for variables, manipulate equations, and understand linear relationships.
• Geometry: Calculate areas, and volumes, and understand geometric shapes and their properties.
• Statistics: Analyze data sets, calculate measures of central tendency, and interpret graphs.

How to Solve Math Probelm Using Google Search

Step 1: activate circle to search.

On your Android smartphone or tablet, activate the Circle to Search feature.

Step 2: Circle the Problem

Find the math problem you’re stuck on and circle it on your screen. This prompts the feature to focus on that particular problem.

Step 3: Get Step-by-Step Instructions

The feature breaks down the problem and provides step-by-step instructions to solve it

Google Circle Vs Photomath Vs Socratic by Google

Tips for getting the most out of google circle.

• Practice makes perfect: Don’t just rely on Google Circle for the answer. Use it as a guide to understand the steps and then try solving similar problems on your own. This will solidify your learning and build your confidence.
• Focus on understanding, not just answers: While getting the right answer is important, the true power of Google Circle lies in its explanations. Pay attention to why each step is taken and how the concepts are applied.
• Don’t be afraid to experiment: Google Circle can handle various math problems. Try circling different parts of a word problem to see how it breaks down the information and guides you toward the solution.
• Take notes: As Google Circle explains the steps, jot down important points or formulas for future reference. This will create a personal study guide tailored to your learning needs.
• Combine it with other resources: Google Circle is a powerful tool, but it shouldn’t be your only resource. Use it alongside your textbook, class notes, or online tutorials for a well-rounded understanding.

Google Circle is a new tool designed to help students to become independent math problem solvers. By using its step-by-step guidance and clear explanations, you can transform your approach to math, gaining a deeper understanding of concepts and building the confidence to tackle even the most challenging problems. So, next time you’re stuck on a math question, don’t hesitate to circle it with Google Circle and watch your problem-solving skills soar!

Google Circle for Students – FAQs

Is google circle available on iphones.

Unfortunately, as of now, Google Circle is only available on Android devices within the Google Search app. However, there might be plans for future expansion to other platforms.

Is Google Circle safe to use?

Google Circle is a feature within the official Google Search app, making it a safe and reliable tool. However, it’s always a good practice to be mindful of the information you share online and only use it for educational purposes.

What if Google Circle can’t solve my math problem?

While Google Circle is constantly learning and expanding its capabilities, it might encounter problems it can’t handle yet. In such cases, don’t hesitate to consult your teacher, a tutor, or online resources for alternative solutions.

Will Google Circle do my math homework for me?

Google Circle is not a magic shortcut. It’s designed to be a learning tool. While it can guide you through the problem-solving process, it’s still important for you to understand the steps and concepts involved.

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