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• 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.
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How to Do Algebra – Practical Tips for Mastering Equations

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Understanding the Basics of Algebra

Solving common algebra problems, avoiding mistakes and building confidence, addition and subtraction, multiplication and division, working with fractions and decimals, graphs and functions in algebra, advanced algebraic concepts, applications of algebra.

As we begin our journey in algebra, it’s essential to foster confidence along the way, which comes from practice and the realization that algebra is a set of tools that empowers us to solve a variety of problems.

We recognize that each algebraic equation is a puzzle waiting to be solved, and just like any puzzle, there’s a method to approach it. Familiarity with the order of operations , the ability to manipulate algebraic expressions, and knowing how to handle fractions and decimals are all part of our toolkit.

Let’s join forces and demystify algebra, turning it from a feared subject into one where we can excel through patience and persistence. Stay tuned as we explore practical tips that will help us maneuver through the world of algebra together.

When we approach algebra, we’re engaging with a branch of mathematics where symbols, often called variables , take the place of numbers. These variables can represent numbers whose values are not yet known, allowing us to solve a wide range of problems.

In algebra, it’s crucial to get comfortable with variables. Think of a variable like a placeholder for a number we’re trying to find. For example, in the equation x – 2 = 4 , x is our variable.

Order of Operations Remembering the order of operations is key to solving algebraic equations. We use the acronym PEMDAS to recall this sequence:

• P arentheses
• M ultiplication and D ivision (from left to right)
• A ddition and S ubtraction (from left to right)

By sticking to this order, we ensure that we solve equations accurately.

Working with Equations Equations are like puzzles where we aim to isolate the variable and find its value. Let’s break this down:

• Identify what’s being added, subtracted, multiplied, or divided to the variable.
• Perform the opposite operation on both sides of the equation.
• Simplify until the variable stands alone with its value revealed.

Here’s a simple table to illustrate these steps with the equation x + 3 = 7 :

By applying these foundational concepts, we set ourselves up for success in solving algebraic problems. Remember, practice is our best friend in becoming proficient with algebra.

When tackling algebra problems , our first step is always to understand the problem at hand. For linear equations , we often have to find the value of a variable that makes the equation true. Let’s look at an example:

Problem: Solve for x: 2x + 3 = 7 Solution: First, subtract 3 from both sides to get 2x = 4. Next, divide both sides by 2 to find x = 2.

The Quadratic equations can be a bit trickier. These involve terms squared (like x²) and often require finding two solutions.

Problem: Solve for x: x² – 5x + 6 = 0 Solution: We can factor this equation to (x – 2)(x – 3) = 0, which gives us the solutions x = 2 and x = 3.

Word problems add another layer of complexity because we must translate a narrative into an equation. Here’s a step-by-step approach:

• Define variables: Let x represent what we’re solving for.
• Set up an equation: Use the given information to form an equation.
• Solve the equation: Use algebraic methods to solve for x.
• Verify your answer: Check if the solution makes sense in the context of the problem.

To ensure we’re solving problems efficiently, we adhere to the following best practices:

• Check for understanding: Ensure we grasp all aspects of the problem.
• Organize information: Write down what we know and what we need to find out.
• Break it down: Tackle the problem in smaller, more manageable steps.
• Check your work: Always review solutions for potential errors.

By following these strategies, we’ll be equipped to handle most algebra problems that come our way!

When we approach algebra, understanding the common pitfalls is as important as learning the concepts themselves. We will identify a few frequent mistakes and share strategies to bolster your confidence in algebra.

Order of Operations: A frequent mistake is not following the PEMDAS/BODMAS rules, where you perform calculations in this order: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). Remembering this acronym helps us avoid computation errors.

Variables and Equations: It’s easy to misinterpret variables or simplify equations incorrectly. We should always double-check our work, ensuring that we combine like terms and distribute coefficients properly.

Quadratic Equations: Solving these can be tricky. Remember to apply formulas and the factoring techniques correctly. Incorrect application could lead to wrong solutions.

To enhance our confidence, we must practice regularly. Working on various problems increases our familiarity with the types of questions that may appear on algebra tests.

Self-Teaching Guide: Make use of guides and textbooks. These resources often contain practice exercises and clear explanations for solving different types of algebra problems.

Use Technology Wisely: Incorporate educational technology where appropriate. Software and online platforms can provide interactive lessons and exercises that adapt to our skill level.

Study Groups: Working with peers can clarify doubts and reinforce learning. If someone else understands a concept, they can help explain it from a student’s perspective.

Regular Review: Finally, revisiting past topics periodically can help cement our understanding and contribute to long-term retention of algebra concepts. This habit ensures we’re always prepared for upcoming exams.

By staying mindful of these tips, we elevate our algebra skills and approach tests with greater assurance.

Exploring Algebraic Operations

In algebra, understanding how to manipulate numbers and variables through basic operations is essential. We’ll guide you through each process step by step.

When we’re dealing with algebraic expressions , addition and subtraction are about combining like terms. Here’s how we do it:

• Identify like terms : These are terms with the same variable raised to the same power.
• Combine them : Add or subtract their coefficients.

Example : (3x + 2x = (3+2)x = 5x) ($7y^2 – 4y^2$ = $(7-4)y^2 = 3y^2)$

For the multiplication of algebraic terms, we use the distributive property and laws of exponents.

• Distributive property : (a(b + c) = ab + ac)
• Multiplying exponents : When bases are the same, we add exponents. $(x^a \cdot x^b = x^{a+b})$

With division :

• Divide coefficients : If $( \frac{8x}{2x} )$, divide 8 by 2 to get 4.
• Subtract exponents : If $( \frac{x^3}{x^2} )$, subtract exponents (3 – 2 = 1), so our answer is $(x^1)$ or simply (x).

In algebra, we often encounter fractions and decimals in equations:

For fractions :

• Find a common denominator for addition and subtraction.
• Multiply straight across the numerator and denominator for multiplication.
• Flip the second fraction and multiply for division (reciprocal).

Example : $( \frac{3}{4} + \frac{1}{4} = \frac{3+1}{4} = \frac{4}{4} = 1 ) ( \frac{1}{3} \cdot \frac{3}{4} = \frac{1 \cdot 3}{3 \cdot 4} = \frac{3}{12} = \frac{1}{4} )$

With decimals , line up the decimal points for addition and subtraction. For multiplication, multiply normally and count decimal places from right to left in the product equal to the sum of the decimal places in the factors.

Practice these operations to build a strong foundation in algebra. Remember, step by step, we can solve even the most complex algebraic expressions.

When we work with functions in algebra, we often represent them visually using graphs. Graphs make understanding and interpreting functions easier because they provide a picture of what’s going on.

Function Notation is how we write functions algebraically. For example, if we have a function f that takes an input x, we write it as f(x) . This notation helps us quickly see how the output is connected to the input.

Function Transformations change the appearance of a graph. Here are the basic types:

To graph a function , we create a set of points where each point represents an input and its corresponding output . By connecting these points, we get the shape of the function, which helps us see patterns like linearity, curvature, and other important behaviors.

When interpreting functions , we look for specific characteristics like:

• The intercepts , where the graph crosses the axes .
• The slope , indicating steepness and direction for linear functions.
• The vertex for quadratic functions, which is the highest or lowest point.

Remember, different types of functions have different features, but once you master interpreting these features on graphs, you’ll gain a deeper understanding of how the function behaves across different values.

In short, familiarizing ourselves with graphing and interpreting functions equips us with the tools to tackle algebra problems more effectively.

In advanced algebra, we dive deeper into more sophisticated topics beyond the foundation we’ve built in basic algebra. To excel in these areas, it’s important to understand the following concepts:

Polynomials : We explore polynomials further by engaging with higher-degree equations and understanding the nuances of factoring polynomials . Factorization is key, allowing us to break down complex polynomials into simpler factors, which simplifies solving the equations.

Ratios and Proportion : We use ratios to compare two quantities and proportions to express that two ratios are equal. These are crucial when solving problems involving scale and in real-world applications such as in mixtures or map reading.

Inequalities : Expanding upon simple inequalities, we encounter systems of inequalities and learn to graph the solutions on a coordinate plane. We also explore absolute value inequalities which require careful attention as they introduce two potential ranges of solutions.

Factorials : Often represented with an exclamation point (n!), factorials are the product of all positive integers less than or equal to n. They are frequently found in permutations and combinations, which play a large role in probability and statistics.

Here’s a quick reference on how to denote various key elements:

• Absolute value : |x|
• Factorial of 5 : 5!
• Polynomial factorization : ( (x – r_1)(x – r_2)…(x – r_n) ), where each ( r ) is a root.

By grasping these concepts, we prepare ourselves for tackling algebraic challenges with confidence. Remember, practice is vital, so work through problems, and don’t hesitate to seek help when needed. Happy solving!

We encounter algebra in many situations, often without even realizing it! Here are a few practical tips on how we can apply algebra in our daily lives.

Budgeting: By creating equations to manage our incomes and expenses, we can keep our finances in check. It’s a practical application of solving for an unknown variable , our savings!

Cooking and Recipes: Altering a recipe? We use ratios and proportions, which are algebraic concepts, to adjust ingredient amounts whilst maintaining taste and consistency.

Shopping Discounts: Calculating discounts and sales prices is a day-to-day use of algebra. A price tag showing 20% off? No sweat, we just multiply the original price by 0.8 to find the new price.

When it comes to learning algebra, resources are plentiful:

Technology Tips: Websites like kastatic.org and kasandbox.org provide exercises and lessons to strengthen our algebra skills without the hassle of a web filter blocking our study. These resources usually load external resources so we can practice unblocked.

Programming: For those of us interested in technology, programming offers a direct connection to algebra. Understanding variables and logic in code requires algebraic thinking.

Projectile Motion: Sports enthusiasts use algebra intuitively when they calculate the best angle and force to throw a ball. It’s a real-world application of systems of equations !

If we need help, there’s a wealth of math lessons and exercises online that can assist us with our algebra studies. Remember, we’re not alone, and there are always more ways to make algebra both fun and relevant to our lives.

In our journey through algebra, we’ve equipped ourselves with an array of strategies to solve equations and understand mathematical expressions. We’ve learned that algebra is not just about finding the value of ‘x’, but about forming connections and developing problem-solving skills that are applicable far beyond the classroom.

Let’s recap the key takeaways:

• Practice regularly : Just like learning a new language, proficiency in algebra comes from consistent practice.
• Build from basics : Ensure your foundational math skills are solid, as they are the bedrock on which algebra stands.
• Understand the principles : Grasping the concepts behind the formulas makes it easier to apply them to different problems.
• Seek help when needed : Don’t hesitate to ask for assistance or use additional resources if concepts are unclear.

Remember, every equation represents a problem that can be dissected and conquered. By now, we should be better at identifying how algebra relates to real-world situations, which improves our analytical thinking.

As we continue to apply these skills in everyday scenarios, the versatility of algebra becomes increasingly clear. Whether budgeting finances, adjusting recipes or analyzing data, the power of algebra enhances our decision-making abilities and helps us approach complex problems with confidence.

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• Probability
• Sets & Set Theory
• Trigonometry

How to Solve Algebra Problems Step-By-Step

• Pre Algebra & Algebra
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• Worksheets By Grade
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Solving Algebra word problems is useful in helping you to solve earthly problems. While the 5 steps of Algebra problem solving are listed below, the following will help you learn how to first identify the problem.

• Identify the problem.
• Identify what you know.
• Make a plan.
• Carry out the plan.
• Verify that the answer makes sense.

Identify the Problem

Back away from the calculator ; use your brain first. Your mind analyzes, plans, and guides in the labyrinthine quest for the solution. Think of the calculator as merely a tool that makes the journey easier. After all, you wouldn’t want a surgeon to crack your ribs and perform a heart transplant without first identifying the source of your chest pains.

The steps of identifying the problem are:

• Express the problem question or statement.
• Identify the unit of the final answer.

Express the Problem Question or Statement

In Algebra word problems, the problem is expressed as either a question or a statement.

• How many trees will John have to plant?
• How many televisions will Sara have to sell to earn $50,000?
• Find the number of trees John will have to plant.
• Solve for the number of televisions Sara will have to sell to earn $50,000.

Identify the Unit of the Final Answer

What will the answer look like? Now that you understand the word problem’s purpose, determine the answer’s unit. For example, will the answer be in miles, feet, ounces, pesos, dollars, the number of trees, or a number of televisions?

Algebra Word Problem

Javier is making brownies to serve at the family picnic. If the recipe calls for 2 ½ cups of cocoa to serve 4 people, how many cups will he need if 60 people attend the picnic?

• Identify the problem:   How many cups will Javier need if 60 people attend the picnic?
• Identify the unit of the final answer: Cups

In the market for computer batteries, the intersection of the supply and demand functions determines the price, p dollars , and the quantity, q , of goods sold. Supply function: 80 q - p = 0 Demand function: 4 q + p = 300 Determine the price and quantity of computer batteries sold when these functions intersect.

• Identify the problem:   How much will the batteries cost and how much will be sold when supply and demand functions meet?
• Identify the unit of the final answer: The quantity, or q , will be given in batteries. The price, or p , will be given in dollars.
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Solving Equations

What is an equation.

An equation says that two things are equal. It will have an equals sign "=" like this:

That equations says:

what is on the left (x − 2)  equals  what is on the right (4)

So an equation is like a statement " this equals that "

What is a Solution?

A Solution is a value we can put in place of a variable (such as x ) that makes the equation true .

Example: x − 2 = 4

When we put 6 in place of x we get:

which is true

So x = 6 is a solution.

How about other values for x ?

• For x=5 we get "5−2=4" which is not true , so x=5 is not a solution .
• For x=9 we get "9−2=4" which is not true , so x=9 is not a solution .

In this case x = 6 is the only solution.

You might like to practice solving some animated equations .

More Than One Solution

There can be more than one solution.

Example: (x−3)(x−2) = 0

When x is 3 we get:

(3−3)(3−2) = 0 × 1 = 0

And when x is 2 we get:

(2−3)(2−2) = (−1) × 0 = 0

which is also true

So the solutions are:

x = 3 , or x = 2

When we gather all solutions together it is called a Solution Set

The above solution set is: {2, 3}

Solutions Everywhere!

Some equations are true for all allowed values and are then called Identities

Example: sin(−θ) = −sin(θ) is one of the Trigonometric Identities

Let's try θ = 30°:

sin(−30°) = −0.5 and

−sin(30°) = −0.5

So it is true for θ = 30°

Let's try θ = 90°:

sin(−90°) = −1 and

−sin(90°) = −1

So it is also true for θ = 90°

Is it true for all values of θ ? Try some values for yourself!

How to Solve an Equation

There is no "one perfect way" to solve all equations.

A Useful Goal

But we often get success when our goal is to end up with:

x = something

In other words, we want to move everything except "x" (or whatever name the variable has) over to the right hand side.

Example: Solve 3x−6 = 9

Now we have x = something ,

and a short calculation reveals that x = 5

Like a Puzzle

In fact, solving an equation is just like solving a puzzle. And like puzzles, there are things we can (and cannot) do.

Here are some things we can do:

• Add or Subtract the same value from both sides
• Clear out any fractions by Multiplying every term by the bottom parts
• Divide every term by the same nonzero value
• Combine Like Terms
• Expanding (the opposite of factoring) may also help
• Recognizing a pattern, such as the difference of squares
• Sometimes we can apply a function to both sides (e.g. square both sides)

Example: Solve √(x/2) = 3

And the more "tricks" and techniques you learn the better you will get.

Special Equations

There are special ways of solving some types of equations. Learn how to ...

• solve Quadratic Equations
• solve Radical Equations
• solve Equations with Sine, Cosine and Tangent

Check Your Solutions

You should always check that your "solution" really is a solution.

How To Check

Take the solution(s) and put them in the original equation to see if they really work.

Example: solve for x:

2x x − 3 + 3 = 6 x − 3     (x≠3)

We have said x≠3 to avoid a division by zero.

Let's multiply through by (x − 3) :

2x + 3(x−3) = 6

Bring the 6 to the left:

2x + 3(x−3) − 6 = 0

Expand and solve:

2x + 3x − 9 − 6 = 0

5x − 15 = 0

5(x − 3) = 0

Which can be solved by having x=3

Let us check x=3 using the original question:

2 × 3 3 − 3 + 3  =   6 3 − 3

Hang On: 3 − 3 = 0 That means dividing by Zero!

And anyway, we said at the top that x≠3 , so ...

x = 3 does not actually work, and so:

There is No Solution!

That was interesting ... we thought we had found a solution, but when we looked back at the question we found it wasn't allowed!

This gives us a moral lesson:

"Solving" only gives us possible solutions, they need to be checked!

• Note down where an expression is not defined (due to a division by zero, the square root of a negative number, or some other reason)
• Show all the steps , so it can be checked later (by you or someone else)

Please ensure that your password is at least 8 characters and contains each of the following:

• a special character: @$#!%*?&

Get step-by-step solutions to your math problems

Try Math Solver

Get step-by-step explanations

Graph your math problems

Practice, practice, practice

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How to Solve an Algebraic Expression

Last Updated: April 6, 2024 Fact Checked

This article was co-authored by David Jia . David Jia is an Academic Tutor and the Founder of LA Math Tutoring, a private tutoring company based in Los Angeles, California. With over 10 years of teaching experience, David works with students of all ages and grades in various subjects, as well as college admissions counseling and test preparation for the SAT, ACT, ISEE, and more. After attaining a perfect 800 math score and a 690 English score on the SAT, David was awarded the Dickinson Scholarship from the University of Miami, where he graduated with a Bachelor’s degree in Business Administration. Additionally, David has worked as an instructor for online videos for textbook companies such as Larson Texts, Big Ideas Learning, and Big Ideas Math. There are 10 references cited in this article, which can be found at the bottom of the page. This article has been fact-checked, ensuring the accuracy of any cited facts and confirming the authority of its sources. This article has been viewed 487,723 times.

An algebraic expression is a mathematical phrase that contains numbers and/or variables. Though it cannot be solved because it does not contain an equals sign (=), it can be simplified. You can, however, solve algebraic equations , which contain algebraic expressions separated by an equals sign. If you want to know how to master this mathematical concept, then see Step 1 to get started.

Understanding the Basics

• Algebraic expression : 4x + 2
• Algebraic equation : 4x + 2 = 100

• 3x 2 + 5 + 4x 3 - x 2 + 2x 3 + 9 =
• 3x 2 - x 2 + 4x 3 + 2x 3 + 5 + 9 =
• 2x 2 + 6x 3 + 14

• You can see that each coefficient can be divisible by 3. Just "factor out" the number 3 by dividing each term by 3 to get your simplified equation.
• 3x/3 + 15/3 = 9x/3 + 30/3 =
• x + 5 = 3x + 10

• (3 + 5) 2 x 10 + 4
• First, follow P, the operation in the parentheses:
• = (8) 2 x 10 + 4
• Then, follow E, the operation of the exponent:
• = 64 x 10 + 4
• Next, do multiplication:
• And last, do addition:

• 5x + 15 = 65 =
• 5x/5 + 15/5 = 65/5 =
• x + 3 = 13 =

Joseph Meyer

To solve an equation for a variable like "x," you need to manipulate the equation to isolate x. Use techniques like the distributive property, combining like terms, factoring, adding or subtracting the same number, and multiplying or dividing by the same non-zero number to isolate "x" and find the answer.

Solve an Algebraic Equation

• 4x + 16 = 25 -3x =
• 4x = 25 -16 - 3x
• 4x + 3x = 25 -16 =
• 7x/7 = 9/7 =

• First, subtract 12 from both sides.
• 2x 2 + 12 -12 = 44 -12 =
• Next, divide both sides by 2.
• 2x 2 /2 = 32/2 =
• Solve by taking the square root of both sides, since that will turn x 2 into x.
• √x 2 = √16 =
• State both answers:x = 4, -4

• First, cross multiply to get rid of the fraction. You have to multiply the numerator of one fraction by the denominator of the other.
• (x + 3) x 3 = 2 x 6 =
• Now, combine like terms. Combine the constant terms, 9 and 12, by subtracting 9 from both sides.
• 3x + 9 - 9 = 12 - 9 =
• Isolate the variable, x, by dividing both sides by 3 and you've got your answer.
• 3x/3 = 3/3 =

• First, move everything that isn't under the radical sign to the other side of the equation:
• √(2x+9) = 5
• Then, square both sides to remove the radical:
• (√(2x+9)) 2 = 5 2 =
• Now, solve the equation as you normally would by combining the constants and isolating the variable:
• 2x = 25 - 9 =

• |4x +2| - 6 = 8 =
• |4x +2| = 8 + 6 =
• |4x +2| = 14 =
• 4x + 2 = 14 =
• Now, solve again by flipping the sign of the term on the other side of the equation after you've isolated the absolute value:
• 4x + 2 = -14
• 4x = -14 -2
• 4x/4 = -16/4 =
• Now, just state both answers: x = -4, 3

Community Q&A

• The degree of a polynomial is the highest power within the terms. Thanks Helpful 9 Not Helpful 1
• Once you're done, replace the variable with the answer, and solve the sum to see if it makes sense. If it does, then, congratulations! You just solved an algebraic equation! Thanks Helpful 7 Not Helpful 3
• To cross-check your answer, visit wolfram-alpha.com. They give the answer and often the two steps. Thanks Helpful 8 Not Helpful 5

You Might Also Like

• ↑ https://www.math4texas.org/Page/527
• ↑ https://www.khanacademy.org/math/cc-sixth-grade-math/cc-6th-expressions-and-variables/cc-6th-combining-like-terms/v/combining-like-terms-2
• ↑ https://www.mathsisfun.com/algebra/factoring.html
• ↑ https://www.mathsisfun.com/operation-order-pemdas.html
• ↑ https://sciencing.com/tips-for-solving-algebraic-equations-13712207.html
• ↑ https://www.mathsisfun.com/algebra/equations-solving.html
• ↑ https://tutorial.math.lamar.edu/Classes/Alg/SolveExpEqns.aspx
• ↑ https://www.mathsisfun.com/algebra/fractions-algebra.html
• ↑ https://math.libretexts.org/Courses/Coastline_College/Math_C045%3A_Beginning_and_Intermediate_Algebra_(Chau_Duc_Tran)/10%3A_Roots_and_Radicals/10.07%3A_Solve_Radical_Equations
• ↑ https://www.mathplanet.com/education/algebra-1/linear-inequalitites/solving-absolute-value-equations-and-inequalities

About This Article

If you want to solve an algebraic expression, first understand that expressions, unlike equations, are mathematical phrase that can contain numbers and/or variables but cannot be solved. For example, 4x + 2 is an expression. To reduce the expression, combine like terms, for example everything with the same variable. After you've done that, factor numbers by finding the lowest common denominator. Then, use the order of operations, which is known by the acronym PEMDAS, to reduce or solve the problem. To learn how to solve algebraic equations, keep scrolling! Did this summary help you? Yes No

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On this page

Basic Algebra

• 1. Addition and Subtraction of Algebraic Expressions
• 2. Multiplication of Algebraic Expressions
• 3. Division of Algebraic Expressions
• 4. Solving Equations
• 5. Formulas and Literal Equations
• 6. Applied Verbal Problems
• al-Khwarizmi Father of Algebra

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Related algebra chapters:

• Factoring & Fractions
• Quadratic Equations
• Exponents & Radicals
• Exponential & Logarithmic Functions
• Polynomial Equations of Higher Degree
• Systems of Equations
• Matrices and Determinants
• Inequalities
• Series and the Binomial Theorem
• Functions and Graphs

This chapter contains elementary algebra tutorials on the following topics:

1. Adding and Subtracting Algebraic Expressions , shows you how to do problems like: Simplify: −2[−3( x − 2 y ) + 4 y ] .

2. Multiplication of algebra expressions , has examples like: Expand (2 x + 3)( x 2 − x − 5) .

3. Division of algebraic expressions , for example: (12a 2 b) ÷ (3ab 2 )

4. Solving Equations , like this one: 5 − ( x + 2) = 5 x .

5. Formulas and Literal Equations , which shows how to solve an equation for a particular variable.

6. Applied Verbal Problems shows why we are doing all this.

What is Algebra?

Algebra is the branch of mathematics that uses letters in place of some unknown numbers.

You've been using algebra since your early schooling, when you learned formulas like the area of a rectangle , with width w , height h :

A = w × h

We used letters to stand for numbers. Once we knew the width and height, we could substitute them into the formula and find our area.

Another one you may have seen is the area of a circle , with radius r :

A = π r 2

As soon as we know the length of the sides, we can find the area.

Literal numbers (the letters used in algebra) can either stand for variables (the value of the letter can change, like the w , h and r in the examples of the area of a rectangle and the area of a circle) or constants (where the value does not change), for example:

π (the ratio circumference/diameter of a circle, value 3.141592.... ) g (the accelaration due to gravity, 9.8 m/s 2 ), e (which has a constant value of 2.781828... ).

And as my students constantly ask...

Why Do We Have to do This?

If we didn't use letters in place of numbers (and used words instead), we would be writing many pages for each problem and it would be much more confusing.

This elementary algebra chapter follows on from the earlier chapter on Numbers.

If you find this chapter difficult...

If you struggle with this chapter, it may be a good idea to go back and remind yourself about basic number properties first, since that's important background.

On with the show

OK, let's move on and learn some basic algebra tips:

1. Addition and Subtraction of Algebraic Expressions »

Problem Solver

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Game Central

How do you solve word problems in math?

Master word problems with eight simple steps from a math tutor!

Author Amber Watkins

Published April 2024

• Key takeaways
• Students who struggle with reading, tend to struggle with understanding and solving word problems. So the best way to solve word problems in math is to become a better reader!
• Mastery of word problems relies on your child’s knowledge of keywords for word problems in math and knowing what to do with them.
• There are 8 simple steps each child can use to solve word problems- let’s go over these together.

Table of contents

• How to solve word problems

Lesson credits

As a tutor who has seen countless math worksheets in almost every grade – I’ll tell you this: every child is going to encounter word problems in math. The key to mastery lies in how you solve them! So then, how do you solve word problems in math?

In this guide, I’ll share eight steps to solving word problems in math.

How to solve word problems in math in 8 steps

Step 1: read the word problem aloud.

For a child to understand a word problem, it needs to be read with accuracy and fluency! That is why, when I tutor children with word problems, I always emphasize the importance of reading properly.

Mastering step 1 looks like this:

• Allow your child to read the word problem aloud to you. 
• Don’t let your child skip over or mispronounce any words. 
• If necessary, model how to read the word problem, then allow your child to read it again. Only after the word problem is read accurately, should you move on to step 2.

Step 2: Highlight the keywords in the word problem

The keywords for word problems in math indicate what math action should be taken. Teach your child to highlight or underline the keywords in every word problem. 

Here are some of the most common keywords in math word problems: 

• Subtraction words – less than, minus, take away
• Addition words – more than, altogether, plus, perimeter
• Multiplication words – Each, per person, per item, times, area 
• Division words – divided by, into
• Total words – in all, total, altogether

Let’s practice. Read the following word problem with your child and help them highlight or underline the main keyword, then decide which math action should be taken.

Michael has ten baseball cards. James has four baseball cards less than Michael. How many total baseball cards does James have? 

The words “less than” are the keywords and they tell us to use subtraction .

Step 3: Make math symbols above keywords to decode the word problem

As I help students with word problems, I write math symbols and numbers above the keywords. This helps them to understand what the word problem is asking.

Let’s practice. Observe what I write over the keywords in the following word problem and think about how you would create a math sentence using them:

Step 4: Create a math sentence to represent the word problem

Using the previous example, let’s write a math sentence. Looking at the math symbols and numbers written above the word problem, our math sentence should be: 10 – 5 = 5 ! 

Each time you practice a word problem with your child, highlight keywords and write the math symbols above them. Then have your child create a math sentence to solve. 

Step 5: Draw a picture to help illustrate the word problem

Pictures can be very helpful for problems that are more difficult to understand. They also are extremely helpful when the word problem involves calculating time , comparing fractions , or measurements . 

Step 6: Always show your work

Help your child get into the habit of always showing their work. As a tutor, I’ve found many reasons why having students show their work is helpful:

• By showing their work, they are writing the math steps repeatedly, which aids in memory
• If they make any mistakes they can track where they happened
• Their teacher can assess how much they understand by reviewing their work
• They can participate in class discussions about their work

Step 7: When solving word problems, make sure there is always a word in your answer!

If the word problem asks: How many peaches did Lisa buy? Your child’s answer should be: Lisa bought 10 peaches .

If the word problem asks: How far did Kyle run? Your child’s answer should be: Kyle ran 20 miles .

So how do you solve a word problem in math?

Together we reviewed the eight simple steps to solve word problems. These steps included identifying keywords for word problems in math, drawing pictures, and learning to explain our answers. 

Is your child ready to put these new skills to the test? Check out the best math app for some fun math word problem practice.

Parents, sign up for a DoodleMath subscription and see your child become a math wizard!

Amber Watkins

Amber is an education specialist with a degree in Early Childhood Education. She has over 12 years of experience teaching and tutoring elementary through college level math. "Knowing that my work in math education makes such an impact leaves me with an indescribable feeling of pride and joy!"

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Do You Understand the Problem You’re Trying to Solve?

To solve tough problems at work, first ask these questions.

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Problem solving skills are invaluable in any job. But all too often, we jump to find solutions to a problem without taking time to really understand the dilemma we face, according to Thomas Wedell-Wedellsborg , an expert in innovation and the author of the book, What’s Your Problem?: To Solve Your Toughest Problems, Change the Problems You Solve .

In this episode, you’ll learn how to reframe tough problems by asking questions that reveal all the factors and assumptions that contribute to the situation. You’ll also learn why searching for just one root cause can be misleading.

Key episode topics include: leadership, decision making and problem solving, power and influence, business management.

HBR On Leadership curates the best case studies and conversations with the world’s top business and management experts, to help you unlock the best in those around you. New episodes every week.

• Listen to the original HBR IdeaCast episode: The Secret to Better Problem Solving (2016)
• Find more episodes of HBR IdeaCast
• Discover 100 years of Harvard Business Review articles, case studies, podcasts, and more at HBR.org .

HANNAH BATES: Welcome to HBR on Leadership , case studies and conversations with the world’s top business and management experts, hand-selected to help you unlock the best in those around you.

Problem solving skills are invaluable in any job. But even the most experienced among us can fall into the trap of solving the wrong problem.

Thomas Wedell-Wedellsborg says that all too often, we jump to find solutions to a problem – without taking time to really understand what we’re facing.

He’s an expert in innovation, and he’s the author of the book, What’s Your Problem?: To Solve Your Toughest Problems, Change the Problems You Solve .

  In this episode, you’ll learn how to reframe tough problems, by asking questions that reveal all the factors and assumptions that contribute to the situation. You’ll also learn why searching for one root cause can be misleading. And you’ll learn how to use experimentation and rapid prototyping as problem-solving tools.

This episode originally aired on HBR IdeaCast in December 2016. Here it is.

SARAH GREEN CARMICHAEL: Welcome to the HBR IdeaCast from Harvard Business Review. I’m Sarah Green Carmichael.

Problem solving is popular. People put it on their resumes. Managers believe they excel at it. Companies count it as a key proficiency. We solve customers’ problems.

The problem is we often solve the wrong problems. Albert Einstein and Peter Drucker alike have discussed the difficulty of effective diagnosis. There are great frameworks for getting teams to attack true problems, but they’re often hard to do daily and on the fly. That’s where our guest comes in.

Thomas Wedell-Wedellsborg is a consultant who helps companies and managers reframe their problems so they can come up with an effective solution faster. He asks the question “Are You Solving The Right Problems?” in the January-February 2017 issue of Harvard Business Review. Thomas, thank you so much for coming on the HBR IdeaCast .

THOMAS WEDELL-WEDELLSBORG: Thanks for inviting me.

SARAH GREEN CARMICHAEL: So, I thought maybe we could start by talking about the problem of talking about problem reframing. What is that exactly?

THOMAS WEDELL-WEDELLSBORG: Basically, when people face a problem, they tend to jump into solution mode to rapidly, and very often that means that they don’t really understand, necessarily, the problem they’re trying to solve. And so, reframing is really a– at heart, it’s a method that helps you avoid that by taking a second to go in and ask two questions, basically saying, first of all, wait. What is the problem we’re trying to solve? And then crucially asking, is there a different way to think about what the problem actually is?

SARAH GREEN CARMICHAEL: So, I feel like so often when this comes up in meetings, you know, someone says that, and maybe they throw out the Einstein quote about you spend an hour of problem solving, you spend 55 minutes to find the problem. And then everyone else in the room kind of gets irritated. So, maybe just give us an example of maybe how this would work in practice in a way that would not, sort of, set people’s teeth on edge, like oh, here Sarah goes again, reframing the whole problem instead of just solving it.

THOMAS WEDELL-WEDELLSBORG: I mean, you’re bringing up something that’s, I think is crucial, which is to create legitimacy for the method. So, one of the reasons why I put out the article is to give people a tool to say actually, this thing is still important, and we need to do it. But I think the really critical thing in order to make this work in a meeting is actually to learn how to do it fast, because if you have the idea that you need to spend 30 minutes in a meeting delving deeply into the problem, I mean, that’s going to be uphill for most problems. So, the critical thing here is really to try to make it a practice you can implement very, very rapidly.

There’s an example that I would suggest memorizing. This is the example that I use to explain very rapidly what it is. And it’s basically, I call it the slow elevator problem. You imagine that you are the owner of an office building, and that your tenants are complaining that the elevator’s slow.

Now, if you take that problem framing for granted, you’re going to start thinking creatively around how do we make the elevator faster. Do we install a new motor? Do we have to buy a new lift somewhere?

The thing is, though, if you ask people who actually work with facilities management, well, they’re going to have a different solution for you, which is put up a mirror next to the elevator. That’s what happens is, of course, that people go oh, I’m busy. I’m busy. I’m– oh, a mirror. Oh, that’s beautiful.

And then they forget time. What’s interesting about that example is that the idea with a mirror is actually a solution to a different problem than the one you first proposed. And so, the whole idea here is once you get good at using reframing, you can quickly identify other aspects of the problem that might be much better to try to solve than the original one you found. It’s not necessarily that the first one is wrong. It’s just that there might be better problems out there to attack that we can, means we can do things much faster, cheaper, or better.

SARAH GREEN CARMICHAEL: So, in that example, I can understand how A, it’s probably expensive to make the elevator faster, so it’s much cheaper just to put up a mirror. And B, maybe the real problem people are actually feeling, even though they’re not articulating it right, is like, I hate waiting for the elevator. But if you let them sort of fix their hair or check their teeth, they’re suddenly distracted and don’t notice.

But if you have, this is sort of a pedestrian example, but say you have a roommate or a spouse who doesn’t clean up the kitchen. Facing that problem and not having your elegant solution already there to highlight the contrast between the perceived problem and the real problem, how would you take a problem like that and attack it using this method so that you can see what some of the other options might be?

THOMAS WEDELL-WEDELLSBORG: Right. So, I mean, let’s say it’s you who have that problem. I would go in and say, first of all, what would you say the problem is? Like, if you were to describe your view of the problem, what would that be?

SARAH GREEN CARMICHAEL: I hate cleaning the kitchen, and I want someone else to clean it up.

THOMAS WEDELL-WEDELLSBORG: OK. So, my first observation, you know, that somebody else might not necessarily be your spouse. So, already there, there’s an inbuilt assumption in your question around oh, it has to be my husband who does the cleaning. So, it might actually be worth, already there to say, is that really the only problem you have? That you hate cleaning the kitchen, and you want to avoid it? Or might there be something around, as well, getting a better relationship in terms of how you solve problems in general or establishing a better way to handle small problems when dealing with your spouse?

SARAH GREEN CARMICHAEL: Or maybe, now that I’m thinking that, maybe the problem is that you just can’t find the stuff in the kitchen when you need to find it.

THOMAS WEDELL-WEDELLSBORG: Right, and so that’s an example of a reframing, that actually why is it a problem that the kitchen is not clean? Is it only because you hate the act of cleaning, or does it actually mean that it just takes you a lot longer and gets a lot messier to actually use the kitchen, which is a different problem. The way you describe this problem now, is there anything that’s missing from that description?

SARAH GREEN CARMICHAEL: That is a really good question.

THOMAS WEDELL-WEDELLSBORG: Other, basically asking other factors that we are not talking about right now, and I say those because people tend to, when given a problem, they tend to delve deeper into the detail. What often is missing is actually an element outside of the initial description of the problem that might be really relevant to what’s going on. Like, why does the kitchen get messy in the first place? Is it something about the way you use it or your cooking habits? Is it because the neighbor’s kids, kind of, use it all the time?

There might, very often, there might be issues that you’re not really thinking about when you first describe the problem that actually has a big effect on it.

SARAH GREEN CARMICHAEL: I think at this point it would be helpful to maybe get another business example, and I’m wondering if you could tell us the story of the dog adoption problem.

THOMAS WEDELL-WEDELLSBORG: Yeah. This is a big problem in the US. If you work in the shelter industry, basically because dogs are so popular, more than 3 million dogs every year enter a shelter, and currently only about half of those actually find a new home and get adopted. And so, this is a problem that has persisted. It’s been, like, a structural problem for decades in this space. In the last three years, where people found new ways to address it.

So a woman called Lori Weise who runs a rescue organization in South LA, and she actually went in and challenged the very idea of what we were trying to do. She said, no, no. The problem we’re trying to solve is not about how to get more people to adopt dogs. It is about keeping the dogs with their first family so they never enter the shelter system in the first place.

In 2013, she started what’s called a Shelter Intervention Program that basically works like this. If a family comes and wants to hand over their dog, these are called owner surrenders. It’s about 30% of all dogs that come into a shelter. All they would do is go up and ask, if you could, would you like to keep your animal? And if they said yes, they would try to fix whatever helped them fix the problem, but that made them turn over this.

And sometimes that might be that they moved into a new building. The landlord required a deposit, and they simply didn’t have the money to put down a deposit. Or the dog might need a $10 rabies shot, but they didn’t know how to get access to a vet.

And so, by instigating that program, just in the first year, she took her, basically the amount of dollars they spent per animal they helped went from something like $85 down to around $60. Just an immediate impact, and her program now is being rolled out, is being supported by the ASPCA, which is one of the big animal welfare stations, and it’s being rolled out to various other places.

And I think what really struck me with that example was this was not dependent on having the internet. This was not, oh, we needed to have everybody mobile before we could come up with this. This, conceivably, we could have done 20 years ago. Only, it only happened when somebody, like in this case Lori, went in and actually rethought what the problem they were trying to solve was in the first place.

SARAH GREEN CARMICHAEL: So, what I also think is so interesting about that example is that when you talk about it, it doesn’t sound like the kind of thing that would have been thought of through other kinds of problem solving methods. There wasn’t necessarily an After Action Review or a 5 Whys exercise or a Six Sigma type intervention. I don’t want to throw those other methods under the bus, but how can you get such powerful results with such a very simple way of thinking about something?

THOMAS WEDELL-WEDELLSBORG: That was something that struck me as well. This, in a way, reframing and the idea of the problem diagnosis is important is something we’ve known for a long, long time. And we’ve actually have built some tools to help out. If you worked with us professionally, you are familiar with, like, Six Sigma, TRIZ, and so on. You mentioned 5 Whys. A root cause analysis is another one that a lot of people are familiar with.

Those are our good tools, and they’re definitely better than nothing. But what I notice when I work with the companies applying those was those tools tend to make you dig deeper into the first understanding of the problem we have. If it’s the elevator example, people start asking, well, is that the cable strength, or is the capacity of the elevator? That they kind of get caught by the details.

That, in a way, is a bad way to work on problems because it really assumes that there’s like a, you can almost hear it, a root cause. That you have to dig down and find the one true problem, and everything else was just symptoms. That’s a bad way to think about problems because problems tend to be multicausal.

There tend to be lots of causes or levers you can potentially press to address a problem. And if you think there’s only one, if that’s the right problem, that’s actually a dangerous way. And so I think that’s why, that this is a method I’ve worked with over the last five years, trying to basically refine how to make people better at this, and the key tends to be this thing about shifting out and saying, is there a totally different way of thinking about the problem versus getting too caught up in the mechanistic details of what happens.

SARAH GREEN CARMICHAEL: What about experimentation? Because that’s another method that’s become really popular with the rise of Lean Startup and lots of other innovation methodologies. Why wouldn’t it have worked to, say, experiment with many different types of fixing the dog adoption problem, and then just pick the one that works the best?

THOMAS WEDELL-WEDELLSBORG: You could say in the dog space, that’s what’s been going on. I mean, there is, in this industry and a lot of, it’s largely volunteer driven. People have experimented, and they found different ways of trying to cope. And that has definitely made the problem better. So, I wouldn’t say that experimentation is bad, quite the contrary. Rapid prototyping, quickly putting something out into the world and learning from it, that’s a fantastic way to learn more and to move forward.

My point is, though, that I feel we’ve come to rely too much on that. There’s like, if you look at the start up space, the wisdom is now just to put something quickly into the market, and then if it doesn’t work, pivot and just do more stuff. What reframing really is, I think of it as the cognitive counterpoint to prototyping. So, this is really a way of seeing very quickly, like not just working on the solution, but also working on our understanding of the problem and trying to see is there a different way to think about that.

If you only stick with experimentation, again, you tend to sometimes stay too much in the same space trying minute variations of something instead of taking a step back and saying, wait a minute. What is this telling us about what the real issue is?

SARAH GREEN CARMICHAEL: So, to go back to something that we touched on earlier, when we were talking about the completely hypothetical example of a spouse who does not clean the kitchen–

THOMAS WEDELL-WEDELLSBORG: Completely, completely hypothetical.

SARAH GREEN CARMICHAEL: Yes. For the record, my husband is a great kitchen cleaner.

You started asking me some questions that I could see immediately were helping me rethink that problem. Is that kind of the key, just having a checklist of questions to ask yourself? How do you really start to put this into practice?

THOMAS WEDELL-WEDELLSBORG: I think there are two steps in that. The first one is just to make yourself better at the method. Yes, you should kind of work with a checklist. In the article, I kind of outlined seven practices that you can use to do this.

But importantly, I would say you have to consider that as, basically, a set of training wheels. I think there’s a big, big danger in getting caught in a checklist. This is something I work with.

My co-author Paddy Miller, it’s one of his insights. That if you start giving people a checklist for things like this, they start following it. And that’s actually a problem, because what you really want them to do is start challenging their thinking.

So the way to handle this is to get some practice using it. Do use the checklist initially, but then try to step away from it and try to see if you can organically make– it’s almost a habit of mind. When you run into a colleague in the hallway and she has a problem and you have five minutes, like, delving in and just starting asking some of those questions and using your intuition to say, wait, how is she talking about this problem? And is there a question or two I can ask her about the problem that can help her rethink it?

SARAH GREEN CARMICHAEL: Well, that is also just a very different approach, because I think in that situation, most of us can’t go 30 seconds without jumping in and offering solutions.

THOMAS WEDELL-WEDELLSBORG: Very true. The drive toward solutions is very strong. And to be clear, I mean, there’s nothing wrong with that if the solutions work. So, many problems are just solved by oh, you know, oh, here’s the way to do that. Great.

But this is really a powerful method for those problems where either it’s something we’ve been banging our heads against tons of times without making progress, or when you need to come up with a really creative solution. When you’re facing a competitor with a much bigger budget, and you know, if you solve the same problem later, you’re not going to win. So, that basic idea of taking that approach to problems can often help you move forward in a different way than just like, oh, I have a solution.

I would say there’s also, there’s some interesting psychological stuff going on, right? Where you may have tried this, but if somebody tries to serve up a solution to a problem I have, I’m often resistant towards them. Kind if like, no, no, no, no, no, no. That solution is not going to work in my world. Whereas if you get them to discuss and analyze what the problem really is, you might actually dig something up.

Let’s go back to the kitchen example. One powerful question is just to say, what’s your own part in creating this problem? It’s very often, like, people, they describe problems as if it’s something that’s inflicted upon them from the external world, and they are innocent bystanders in that.

SARAH GREEN CARMICHAEL: Right, or crazy customers with unreasonable demands.

THOMAS WEDELL-WEDELLSBORG: Exactly, right. I don’t think I’ve ever met an agency or consultancy that didn’t, like, gossip about their customers. Oh, my god, they’re horrible. That, you know, classic thing, why don’t they want to take more risk? Well, risk is bad.

It’s their business that’s on the line, not the consultancy’s, right? So, absolutely, that’s one of the things when you step into a different mindset and kind of, wait. Oh yeah, maybe I actually am part of creating this problem in a sense, as well. That tends to open some new doors for you to move forward, in a way, with stuff that you may have been struggling with for years.

SARAH GREEN CARMICHAEL: So, we’ve surfaced a couple of questions that are useful. I’m curious to know, what are some of the other questions that you find yourself asking in these situations, given that you have made this sort of mental habit that you do? What are the questions that people seem to find really useful?

THOMAS WEDELL-WEDELLSBORG: One easy one is just to ask if there are any positive exceptions to the problem. So, was there day where your kitchen was actually spotlessly clean? And then asking, what was different about that day? Like, what happened there that didn’t happen the other days? That can very often point people towards a factor that they hadn’t considered previously.

SARAH GREEN CARMICHAEL: We got take-out.

THOMAS WEDELL-WEDELLSBORG: S,o that is your solution. Take-out from [INAUDIBLE]. That might have other problems.

Another good question, and this is a little bit more high level. It’s actually more making an observation about labeling how that person thinks about the problem. And what I mean with that is, we have problem categories in our head. So, if I say, let’s say that you describe a problem to me and say, well, we have a really great product and are, it’s much better than our previous product, but people aren’t buying it. I think we need to put more marketing dollars into this.

Now you can go in and say, that’s interesting. This sounds like you’re thinking of this as a communications problem. Is there a different way of thinking about that? Because you can almost tell how, when the second you say communications, there are some ideas about how do you solve a communications problem. Typically with more communication.

And what you might do is go in and suggest, well, have you considered that it might be, say, an incentive problem? Are there incentives on behalf of the purchasing manager at your clients that are obstructing you? Might there be incentive issues with your own sales force that makes them want to sell the old product instead of the new one?

So literally, just identifying what type of problem does this person think about, and is there different potential way of thinking about it? Might it be an emotional problem, a timing problem, an expectations management problem? Thinking about what label of what type of problem that person is kind of thinking as it of.

SARAH GREEN CARMICHAEL: That’s really interesting, too, because I think so many of us get requests for advice that we’re really not qualified to give. So, maybe the next time that happens, instead of muddying my way through, I will just ask some of those questions that we talked about instead.

THOMAS WEDELL-WEDELLSBORG: That sounds like a good idea.

SARAH GREEN CARMICHAEL: So, Thomas, this has really helped me reframe the way I think about a couple of problems in my own life, and I’m just wondering. I know you do this professionally, but is there a problem in your life that thinking this way has helped you solve?

THOMAS WEDELL-WEDELLSBORG: I’ve, of course, I’ve been swallowing my own medicine on this, too, and I think I have, well, maybe two different examples, and in one case somebody else did the reframing for me. But in one case, when I was younger, I often kind of struggled a little bit. I mean, this is my teenage years, kind of hanging out with my parents. I thought they were pretty annoying people. That’s not really fair, because they’re quite wonderful, but that’s what life is when you’re a teenager.

And one of the things that struck me, suddenly, and this was kind of the positive exception was, there was actually an evening where we really had a good time, and there wasn’t a conflict. And the core thing was, I wasn’t just seeing them in their old house where I grew up. It was, actually, we were at a restaurant. And it suddenly struck me that so much of the sometimes, kind of, a little bit, you love them but they’re annoying kind of dynamic, is tied to the place, is tied to the setting you are in.

And of course, if– you know, I live abroad now, if I visit my parents and I stay in my old bedroom, you know, my mother comes in and wants to wake me up in the morning. Stuff like that, right? And it just struck me so, so clearly that it’s– when I change this setting, if I go out and have dinner with them at a different place, that the dynamic, just that dynamic disappears.

SARAH GREEN CARMICHAEL: Well, Thomas, this has been really, really helpful. Thank you for talking with me today.

THOMAS WEDELL-WEDELLSBORG: Thank you, Sarah.  

HANNAH BATES: That was Thomas Wedell-Wedellsborg in conversation with Sarah Green Carmichael on the HBR IdeaCast. He’s an expert in problem solving and innovation, and he’s the author of the book, What’s Your Problem?: To Solve Your Toughest Problems, Change the Problems You Solve .

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When it comes to math, maybe the kids aren’t the problem

The way we are teaching, testing, and requiring mathematics is damaging our society with potentially fatal results. That’s a strong statement, I know, but to get a sense of why, take a moment to think of something that you would find hard or maybe impossible to do — executing 30 pullups, playing a violin concerto, walking a tightrope over an abyss, feeding a snake, solving a complex polynomial equation. 

Now suppose our society decrees that to graduate from high school, you must be able to do that thing or be branded a failure and not earn a diploma. How are you feeling about yourself and your prospects?

Imagine trying your hardest to learn the skills, but the textbooks, online resources and sometimes even teachers confuse you, and on top of that, some exams are so badly designed they are virtually impossible to pass. 

The 2022 results from the National Assessment of Educational Progress showed only 36% of fourth graders and 26% of eighth graders were considered proficient in mathematics. The national average for high school math proficiency is just 38%.

Why are so many young people not achieving math proficiency? Consider how we define “proficiency.” 

For example, the Common Core math standards state that, among other things, high school students should “Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f (0) = f (1) = 1, f (n+1) = f (n) + f (n-1) for n ≥ 1.” And that’s just one example of many.

Suppose your child came to you with that as homework each night, possibly in tears. Now imagine that your child has special needs and deals with dyscalculia , ADD or other challenges.  

One of my sons struggles with ADD and is trying to complete an online financial algebra course offered by his school district to fulfill his mandatory “math pathway” for graduation. To help him succeed, a certified math and science tutor and I have worked with him every day through some very poorly designed online lessons provided by his district. 

We also helped him try to solve problems during a unit exam, but together, we achieved only a score of 32% on the exam. A certified math teacher, and a former statistics instructor with a Ph.D., failed a high school financial algebra exam. Why? Because the lessons were deeply flawed, the exam items were ambiguously worded, there were too many problems for the allotted time and the rules prohibited the use of resources like a financial calculator that one might enlist in real life. 

All that is not an excuse, and this column is not some reflexive “anti-testing” rant. It is, unfortunately, the truth. That leaves us asking, “If we failed the exam, how will most students, or their parents, possibly succeed?”

If you make children and their parents feel like failures, it is predictable that some are eventually going to give up on school and on themselves. Once that happens, they are far more likely to turn to behaviors that are harmful, self-destructive, and, tragically, sometimes fatal. And when significant portions of our population share those feelings, it inevitably harms our society as a whole. It is nothing less than educational and social malpractice to create unrealistic and unnecessary demands for children and then fail to provide them with the best resources to help them succeed.   

A good start would be to insist that policymakers, superintendents, school boards and school principals throughout our state attempt to learn each subject in math through the same texts or online resources students are required to use. I don’t mean just skimming a text or taking the word of someone else, I mean really digging in and trying to learn the material and take the tests firsthand. I doubt most people would accept the challenge and I am certain even fewer would pass the exams. 

That is not because the administrators and teachers aren’t smart, good people — it is because the materials and exams are so poorly designed, and the requirements are so profoundly detached from daily life. But if that is so, which the evidence suggests it is, why are we inflicting such things on our children and with what impacts?

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Video transcript

Easy Finger Math Tricks to Help Kids Solve Problems

While using your fingers isn't the fastest way to recall a multiplication fact while doing a problem, finger math tricks can help kids figure out how to answer the problem at hand — and as they work on their math, they will eventually learn all the facts by repetition.

Note that before your child can understand other finger tricks, they must be able to count by 2s, 5s, and 10s and multiply by 2s, 3s, and 4s.

Quick Finger Math Tricks for Threes and Fours

The tricks for multiplying by threes and fours are really a matter of counting out the answer on your fingers. As your children count out the answer repeatedly, they'll memorize it and then be able to move on to larger numbers.

Multiplying by Three

Did you realize that all of your fingers have three segments? Therefore, you can figure out anything from 3 x 1 to 3 x 10 by counting the segments on each finger. To start:

• Hold up the number of fingers you're going to multiply by 3. For example, if the problem is 3 x 4 — hold up four fingers.
• Count each segment on each finger you're holding up, and you should come up with 12 — which is the correct answer.

Multiplying by Four

Multiplying by four is the same as multiplying by two — twice. To start:

• Hold up the number of fingers to correspond with the number you are multiplying by four. For example, if you are multiplying 4 x 6 — hold up six fingers.
• Count each finger by two, moving from left to right. Then count each finger again, continuing to count by twos, until you've counted every finger twice.

Helpful Hack To keep track of the fingers you've counted twice, sometimes it's easier to put your finger down as you count the first time, and back up as you count the second time.

Finger Math Tricks for Multiplying by 6, 7, 8, and 9

While numbers one through five are easy for most kids to remember, six and up often pose a problem. This handy trick will make it a little easier to work those problems out.

Multiplying 6, 7, 8, and 9 by Hand

To begin, assign each finger a number. For example, your thumbs represent 6, your index fingers each represent 7, etc. This will remain the same throughout the finger math hack.

Your left hand will represent the first number that you are multiplying and your right hand will represent the second number you are multiplying. In this example, we are multiplying 7 x 8. 

To Determine the Part of Your Answer:

• On your left hand, put down the finger that represents the number you are multiplying as well as any fingers whose number value is less than this figure. In this example, you are multiplying 7 x 8, so the left hand will represent 7. You will drop your index finger (number 7) and your thumb (number 6).
• Similarly, the right hand will represent eight, so you will drop down your middle finger (number 8), your index finger (number 7), and your thumb (number 6).
• Now, just multiply the fingers that are still pointed upwards. In this case, you will have three fingers on your left hand and two on your right, so you will multiply 3 x 2 to get 6. This is the first part of your answer!

To Determine the Second Part of Your Answer:

• Keeping your fingers in the same positions, count how many fingers are folded down. In the 7 x 8 example, you should have five fingers folded. 
• You will count each of these in quantities of ten. So, 10, 20, 30, 40, 50.
• 50 is your answer.

To Determine Your Final Answer:

• Add your two numbers together. In this example, you would add 6 + 50, which gives you 56!

Another Finger Math Trick Just for Nine

There is a trick that works separately, just for multiplying by the number nine.

• To start, hold up all ten fingers, with your palms facing you.
• Assign each finger a number, starting with your left-hand thumb and ending with your right-hand thumb. The left-hand thumb will be one, the left-hand index finger will be two, and so on until you reach the number 10 for your right-hand thumb.
• To tackle a problem, put down the corresponding finger of the number you're multiplying by nine. For example, if you are multiplying 9 x 8, you'd put down the eighth finger (which will be on your right hand).
• Count all the fingers to the left of the finger you have folded down. This will give you 7. This is the first digit of your answer.
• Count all the fingers to the right of the finger you have folded down. This will give you 2. This is the second digit of your answer.
• Put the numbers together! Your answer is 72.

Finger Multiplication Tricks Can Make Math Easy and Fun

While the hope is that your kids will eventually memorize their multiplication charts , using some quick hand tricks for multiplication and letting them count things out on their fingers is not a bad way to learn. It keeps frustration at bay since the answer is always a fingertip away, and the repetition of having to figure it out will help cement those facts into their brains.