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Browse old math questions or ask your own. Expert answers. The archives have thousands of answered questions covering all math topics from elementary to college level. 

Active discussion board where you can ask math questions about algebra, calculus, geometry, trigonometry, probability, college math, computer science, physics, and more.

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From pre-algebra on through university-level math. It has experienced staff, is completely free, and as a policy does not give answers away without the asker showing some effort.

A math homework help forum.

Message boards from prealgebra through calculus and SAT/ACT.

Chegg Homework Help offers both free and upgradable services. You can view 2 step-by-step solutions from any book and in any subject for free.

Post homework questions and get answers. Tutors in every subject are standing by. Students can sign up as tutors too!

High school math questions are answered for free.

Key to Algebra offers a unique, proven way to introduce algebra to your students. New concepts are explained in simple language, and examples are easy to follow. Word problems relate algebra to familiar situations, helping students to understand abstract concepts. Students develop understanding by solving equations and inequalities intuitively before formal solutions are introduced. Students begin their study of algebra in Books 1-4 using only integers. Books 5-7 introduce rational numbers and expressions. Books 8-10 extend coverage to the real number system.

TutorHub provides free collaborative student-to-student homework help, combined with optional, private (paid) tutoring. The website has a wide curriculum coverage, with the most popular subject being math.

Over 4,000 free, online video lessons for basic math, algebra, trigonometry, and calculus. Videos also available in Spanish. I've written of MathTV lessons when they used to be offered on CDs.

A free resource for math review material from Algebra to Differential Equations.

An extensive library of algebra lessons.

A comprehensive math tutorial website, covering lots of topics. Also includes a cram sheet generator that generates handy study sheets on various math topics.

A large online library of short video lessons and quizzes. The videos are recorded by several different tutors, who vary in their methods and teaching styles, so you have the chance to learn the same lesson from several perspectives.

An animated, interactive online math dictionary for students which explains over 600 common mathematical terms and math words in simple language.

Online interactive lessons (free) covering high school algebra and calculus courses (including AP calculus).

Common student mistakes in algebra, trig, and calculus explained.

Find out what your students really understand about math. The site contains ssessment tools, and lists of reasoning strategies student use and misuse with whole numbers, fractions, and decimals.

Online tutoring - tutors for any subject. Pricing $29.99 for two hours; trial lessons are also available for $9 per hour (money back guarantee).

Private, In-home SAT Prep to LSAT Prep in New York, New Jersey, Cleveland, Miami and Los Angeles.

A free home tutoring marketplace. Tutors post profiles for free and students contact them directly for free. Limited revenue is generated from the sale of featured listings for certain tutors. The website has a very clean, user-friendly layout.

Origins Tutoring emphasizes a personalised, holistic tutoring approach that combines not only subject knowledge, critical reasoning, and problem solving skills, but also study skills development and mentorship. Based in the NYC.

An online marketplace that allows students to browse private tutor profiles for over 90 subject areas.

A nationwide tutor-student matching service. Search for local in-home math tutors in your area.

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The Hechinger Report

Covering Innovation & Inequality in Education

Why schools are teaching math word problems all wrong

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CENTRAL FALLS, R.I. — When Natalia Molina began teaching her second grade students word problems earlier this school year, every lesson felt difficult. Most students were stymied by problems such as: “Sally went shopping. She spent $86 on groceries and $39 on clothing. How much more did Sally spend on groceries than on clothing?”

Both Molina, a first-year teacher, and her students had been trained to tackle word problems by zeroing in on key words like “and,” “more” and “total”  — a simplistic approach that Molina said too often led her students astray. After recognizing the word “and,” for instance, they might mistakenly assume that they needed to add two nearby numbers together to arrive at an answer.

Some weaker readers, lost in a sea of text, couldn’t recognize any words at all.

“I saw how overwhelmed they would get,” said Molina, who teaches at Segue Institute for Learning, a predominantly Hispanic charter school in this small city just north of Providence.

So, with the help of a trainer doing work in Rhode Island through a state grant, Molina and some of her colleagues revamped their approach to teaching word problems this winter — an effort that they said is already paying off in terms of increased student confidence and ability. “It has been a game changer for them,” Molina said.

Perhaps no single educational task encompasses as many different skills as the word problem. Between reading, executive functioning, problem solving, computation and vocabulary, there are a lot of ways for students to go wrong. And for that reason, students perform significantly worse overall on word problems compared to questions more narrowly focused on computation or shapes (for example: “Solve 7 + _ = 22” or “What is 64 x 3?”).

If a student excels at word problems, it’s a good sign that they’re generally excelling at school. “Word-problem solving in lower grades is one of the better indicators of overall school success in K-12,” said Lynn Fuchs, a research professor at Vanderbilt University. In a large national survey , for instance, algebra teachers rated word-problem solving as the most important among 15 skills required to excel in the subject.

Teacher takeaways

• Don’t instruct students to focus mainly on “key words” in word problems such as “and” or “more” 
• Mix question types in any lesson so that students don’t assume they just apply the same operation (addition, subtraction) again and again
• Teach students the underlying structure — or schema — of the word problem

Yet most experts and many educators agree that too many schools are doing it wrong, particularly in the elementary grades. And in a small but growing number of classrooms, teachers like Molina are working to change that. “With word problems, there are more struggling learners than non-struggling learners” because they are taught so poorly, said Nicole Bucka, who works with teachers throughout Rhode Island to provide strategies for struggling learners.

Too many teachers, particularly in the early grades, rely on key words to introduce math problems. Posters displaying the terms — sum, minus, fewer, etc. — tied to operations including addition and subtraction are a staple in elementary school classrooms across the country.

Key words can be a convenient crutch for both students and teachers, but they become virtually meaningless as the problems become harder, according to researchers. Key words can help first graders figure out whether to add or subtract more than half of the time, but the strategy rarely works for the multi-step problems students encounter starting in second and third grade. “With multi-step problems, key words don’t work 90 percent of the time,” said Sarah Powell, a professor at the University of Texas in Austin who studies word problems and whose research has highlighted the inefficacy of key words . “But the average kindergarten teacher is not thinking about that; they are teaching 5-year-olds, not 9-year-olds.”

Many teachers in the youngest grades hand out worksheets featuring the same type of word problem repeated over and over again. That’s what Molina’s colleague, Cassandra Santiago, did sometimes last year when leading a classroom on her own for the first time. “It was a mistake,” the first grade teacher said. “It’s really important to mix them up. It makes them think more critically about the parts they have to solve.”

Another flaw with word problem instruction is that the overwhelming majority of questions are divorced from the actual problem-solving a child might have to do outside the classroom in their daily life — or ever, really. “I’ve seen questions about two trains going on the same track,” said William Schmidt, a University Distinguished Professor at Michigan State University. “First, why would they be going on the same track and, second, who cares?”

Schmidt worked on an analysis of about 8,000 word problems used in 23 textbooks in 19 countries. He found that less than one percent had “real world applications” and involved “higher order math applications .”

“That is one of the reasons why children have problems with mathematics,” he said. “They don’t see the connection to the real world … We’re at this point in math right now where we are just teaching students how to manipulate numbers.”

He said a question, aimed at middle schoolers, that does have real world connections and involves more than manipulating numbers, might be: “Shopping at the new store in town includes a 43% discount on all items which are priced the same at $2. The state you live in has a 7% sales tax. You want to buy many things but only have a total of $52 to spend. Describe in words how many things you could buy.”

Schmidt added that relevancy of word problems is one area where few, if any, countries excel. “No one was a shining star leading the way,” he said. 

In her brightly decorated classroom one Tuesday afternoon, Santiago, the first grade teacher, gave each student a set of animal-shaped objects and a sheet of blue paper (the water) and green (the grass). “We’re going to work on a number story,” she told them. “I want you to use your animals to tell me the story.”

“ Once upon a time,” the story began. In this tale, three animals played in the water, and two animals played in the grass. Santiago allowed some time for the ducks, pigs and bears to frolic in the wilds of each student’s desk before she asked the children to write a number sentence that would tell them how many animals they have altogether.

Some of the students relied more on pictorial representations (three dots on one side of a line and two dots on the other) and others on the number sentence (3+2 = 5) but all of them eventually got to five. And Santiago made sure that her next question mixed up the order of operations (so students didn’t incorrectly assume that all they ever have to do is add): “Some more animals came and now there are seven. So how many more came?”

One approach to early elementary word problems that is taking off in some schools, including Segue Institute, has its origins in a special education intervention for struggling math students. Teachers avoid emphasizing key words and ask students instead to identify first the conceptual type of word problem (or schema, as many practitioners and researchers refer to it) they are dealing with: “Total problems,” for instance, involve combining two parts to find a new amount; “change problems” involve increasing or decreasing the amount of something. Total problems do not necessarily involve adding, however.

“The schemas that students learn in kindergarten will continue with them throughout their whole career,” said Powell, the word-problem researcher, who regularly works with districts across the country to help implement the approach. 

In Olathe, Kansas — a district inspired by Powell’s work — teachers had struggled for years with word problems, said Kelly Ulmer, a math support specialist whose goal is to assist in closing academic gaps that resulted from lost instruction time during the pandemic. “We’ve all tried these traditional approaches that weren’t working,” she said. “Sometimes you get pushback on new initiatives from veteran teachers and one of the things that showed us how badly this was needed is that the veteran teachers were the most excited and engaged — they have tried so many things” that haven’t worked.

In Rhode Island, many elementary schools initially used the strategy with students who required extra help, including those in special education, but expanded this use to make it part of the core instruction for all, said Bucka. In some respects, it’s similar to the recent, well publicized evolution of reading instruction in which some special education interventions for struggling readers  — most notably, a greater reliance on phonics in the early grades — have gone mainstream.

There is an extensive research bas e showing that focusing on the different conceptual types of word problems is an effective way of teaching math, although much of the research focuses specifically on students experiencing difficulties in the subject. 

Molina has found asking students to identify word problems by type to be a useful tool with nearly all of her second graders; next school year she hopes to introduce the strategy much earlier.

One recent afternoon, a lesson on word problems started with everyone standing up and chanting in unison: “Part plus part equals total” (they brought two hands together). “Total minus part equals part ” (they took one hand away) .

It’s a way to help students remember different conceptual frameworks for word problems. And it’s especially effective for the students who learn well through listening and repeating. For visual learners, the different types of word problems were mapped out on individual dry erase mats.

The real work began when Molina passed out questions, and the students— organized into the Penguin, Flower Bloom, Red Panda and Marshmallow teams — had to figure out which framework they were dealing with on their own and then work toward an answer. A few months ago, many of them would have automatically shut down when they saw the text on the page, Molina said.

For the Red Pandas, the question under scrutiny was: “The clothing store had 47 shirts. They sold 21, how many do they have now?”

“It’s a total problem,” one student said.

“No, it’s not total,” responded another.

“I think it’s about change,” said a third.

None of the students seemed worried about their lack of consensus, however. And neither was Molina. A correct answer is always nice but those come more often now that most of the students have made a crucial leap. “I notice them thinking more and more,” she said, “about what the question is actually asking.”

This story about word problems was produced by The Hechinger Report , a nonprofit, independent news organization focused on inequality and innovation in education. Sign up for the Hechinger newsletter .

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Confused about math how one expert teacher used questions to support learning.

June 20, 2024 | By Tarik Buli

• Math teachers often must support students in working through their confusion as they learn conceptual mathematics.
• Math teachers can support their students to grapple with confusion by applying their knowledge of their students and their own knowledge base, to select and sequence questions that can push their students’ thinking.
• Math teachers can leverage their culture and empathy, and build a repertoire of questioning strategies to plan responses in advance to student confusion in the moment.

At Digital Promise, we aim to ensure that historically and systematically excluded (HSE) learners engage in meaningful learning experiences that prepare them for the future. Mathematics classrooms in particular can help HSE learners develop the necessary skills, mindsets, and behaviors that lead to well-being, agency, and economic security. To ensure students have postsecondary opportunities and access to lucrative STEM careers, it is essential that they develop their mathematics literacy (Moses & Cobb, 2002).

One way educators can help students develop their mathematics literacy is by building their conceptual understanding of the subject. Discussion-based classrooms, where students frequently explain what they think and understand about math, enable students to bridge from what they already know to what they are in the process of learning. However, there are many opportunities for them to become confused. In our research work to support mathematics educators, we’ve explored how an expert teacher supports HSE students to learn challenging mathematics by helping them work through their confusion as they grapple with new concepts.

Meet Ms. Collier

Ms. Collier is a Black teacher who grew up on the south side of Chicago and attended Chicago Public Schools. As an educator for more than 16 years, Ms. Collier had an explicit goal to educate and empower Black students. She explains, “I want to teach Black students. Very specifically, I want to teach people who look like me.” Her purpose is to be “a positive representation of a Black person teaching math,” so that her students can “see people that look like them educated and being able to educate them.”

Ms. Collier realizes that her students often get confused in math class, and she works very hard to support them in understanding ambitious and complex mathematics concepts. Below are some key ways Ms. Collier responds to student confusion to support their conceptual understanding.

How Does Ms. Collier Handle Student Confusion?

Ask students specific questions..

Below is an excerpt of a discussion between Ms. Collier and her students during a virtual class session. As the class is reviewing a task, a student, Jordan, becomes confused. Ms. Collier responds to his confusion by asking him questions rather than explaining how to find answers.

Ms. C: Okay. So, somebody tell me what you thought the rule was. Share with me your input/output diagram. What’d you come up with? What’d you come up with? Jordan.

Jordan: Oh, the input is 16…And then you uh, uh, uhhhh…

Ms. C: So let’s think about it this way. If you were squaring a number, what would you do? Jordan turns audio off. Turn your mic on. Don’t be tapping out. Turn that microphone on.

Jordan: Multiply by itself.

Ms. C: You would multiply by itself. Right? So, if I’m looking for the opposite, what would I do?

Jordan: Divide by itself?

Ms. C: Divide by itself, right? But I don’t know what itself is technically, right? Like, that’s kinda odd. That’s weird. So, the opposite of a square, you can’t say divide by itself when you don’t know what itself is. Okay? Does anybody know the terminology for the inverse of a square? The inverse of squaring a number is what, Sandra?

Sandra: The square root.

Know the students and the content deeply.

Ms. Collier sequenced her questions to Jordan in such a way that 1) she knew that he could answer each of them, and 2) she knew would lead him toward understanding what she wanted him to learn next. Because she knows both Jordan and his knowledge deeply—as well as the scope of the mathematics she was teaching—Ms. Collier knew which question from her repertoire to ask next. Ms. Collier’s depth of knowledge allowed her to bridge between Jordan’s current and intuitive understanding to new mathematical knowledge.

Tap into cultural resources and empathy to show high expectations.

In her teaching, Ms. Collier regularly taps into her identity, personality, and cultural resources during instruction. We see a glimpse of this through her discussion with Jordan in how she used culturally situated language (“don’t be tapping out”) to discourage Jordan from opting out of the discussion. After Jordan persisted in the discussion, Ms. Collier prompted him to use his understanding of how to square a number to propose a rule for identifying the number whose square is 16. Here, she supports Jordan in recognizing the inverse relationship between the square and square root, without telling him directly. After Jordan responds, “Divide by itself?”, Ms. Collier employs empathy by acknowledging that she too recognizes it sounds “kinda odd,” while affirming Jordan’s reasonable intuition, which supports the class’s collective understanding in making sense of the rule in the problem.

Ms. Collier explained why she has high expectations for her students—such as refusing to allow Jordan to opt out of the problem and pushing students to express their ideas with mathematically precise language:

“It goes back to knowing these kids and knowing what they’re capable of and their maximum potential. I ask my children all the time, why be average when you could be extraordinary? … I want you to go above and beyond because you’re capable of doing so. And if you’re going to go above and beyond in mathematics … you want to use the language that people use because that is the expectation.”

Ms. Collier shows high expectations for her students because she personally believes her students have the capacity to meet them.

Responding To Student Confusion? Try These Strategies

• Use your knowledge of what your students know to select and sequence which questions you should ask when they experience confusion. It might be hard to know the right question to ask in the moment, so anticipating how your students might respond and building a repertoire of questioning strategies before you teach the lesson can provide you with flexibility when asking students questions in the moment.
• Tap into your own culturally authentic resources, as well as those of your students, to build meaningful connections with them and the content. Listen to this podcast interview from Kids Math Talk with Dr. Margaret (Peg) Smith, who co-wrote “ Five Practices for Orchestrating Productive Mathematics Discussions ,” for five practices to support teachers in facilitating authentic classroom discussions.

Sign up for our newsletter to stay updated on our work about supporting students’ mathematics learning.

References:

Moses, R., & Cobb, C. E. (2002). Radical equations: Civil rights from Mississippi to the Algebra Project. Beacon Press.

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Schools are teaching math word problems all wrong. But some educators have found a better way.

In central falls, r.i., teachers are trying new strategies that move away from focusing on “key words,” the traditional, simplistic approach that often leads younger students astray.

CENTRAL FALLS, R.I. — When Natalia Molina began teaching her second grade students word problems earlier this school year, every lesson felt difficult. Most students were stymied by problems such as: “Sally went shopping. She spent $86 on groceries and $39 on clothing. How much more did Sally spend on groceries than on clothing?”

Both Molina, a first-year teacher, and her students had been trained to tackle word problems by zeroing in on key words like “and,” “more” and “total” — a simplistic approach that Molina said too often led her students astray. After recognizing the word ‘and’, for instance, they might mistakenly assume that they needed to add two nearby numbers together to arrive at an answer.

Some weaker readers, lost in a sea of text, couldn’t recognize any words at all.

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“I saw how overwhelmed they would get,” said Molina, who teaches at Segue Institute for Learning, a predominantly Hispanic charter school in this small city just north of Providence.

So, with the help of a trainer doing work in Rhode Island through a state grant, Molina and some of her colleagues revamped their approach to teaching word problems this winter — an effort that they said is already paying off in terms of increased student confidence and ability. “It has been a gamechanger for them,” Molina said.

Perhaps no single educational task encompasses as many different skills as the word problem. Between reading, executive functioning, problem solving, computation and vocabulary, there are a lot of ways for students to go wrong. And for that reason, students perform significantly worse overall on word problems compared to questions more narrowly focused on computation or shapes (for example: “Solve 7 + _ = 22″ or “What is 64 x 3?”).

If a student excels at word problems, it’s a good sign that they’re generally excelling at school. “Word-problem solving in lower grades is one of the better indicators of overall school success in K-12,” said Lynn Fuchs, a research professor at Vanderbilt University. In a large national survey , for instance, algebra teachers rated word-problem solving as the most important among 15 skills required to excel in the subject.

Yet most experts and many educators agree that too many schools are doing it wrong, particularly in the elementary grades. And in a small but growing number of classrooms, teachers like Molina are working to change that. “With word problems, there are more struggling learners than non-struggling learners” because they are taught so poorly, said Nicole Bucka, who works with teachers throughout Rhode Island to provide strategies for struggling learners.

Too many teachers, particularly in the early grades, rely on key words to introduce math problems. Posters displaying the terms — sum, minus, fewer, etc. — tied to operations including addition and subtraction are a staple in elementary school classrooms across the country.

Key words can be a convenient crutch for both students and teachers, but they become virtually meaningless as the problems become harder, according to researchers. Key words can help first graders figure out whether to add or subtract more than half of the time, but the strategy rarely works for the multi-step problems students encounter starting in second and third grade. “With multi-step problems, key words don’t work 90 percent of the time,” said Sarah Powell, a professor at the University of Texas in Austin who studies word problems and whose research has highlighted the inefficacy of key words . “But the average kindergarten teacher is not thinking about that; they are teaching 5-year-olds, not 9-year-olds.”

Many teachers in the youngest grades hand out worksheets featuring the same type of word problem repeated over and over again. That’s what Molina’s colleague, Cassandra Santiago, did sometimes last year when leading a classroom on her own for the first time. “It was a mistake,” the first grade teacher said. “It’s really important to mix them up. It makes them think more critically about the parts they have to solve.”

Another flaw with word problem instruction is that the overwhelming majority of questions are divorced from the actual problem-solving a child might have to do outside the classroom in their daily life — or ever, really. “I’ve seen questions about two trains going on the same track,” said William Schmidt, a University Distinguished Professor at Michigan State University. “First, why would they be going on the same track and, second, who cares?”

Schmidt worked on an analysis of about 8,000 word problems used in 23 textbooks in 19 countries. He found that less than one percent had “real world applications” and involved “higher order math applications .”

“That is one of the reasons why children have problems with mathematics,” he said. “They don’t see the connection to the real world … We’re at this point in math right now where we are just teaching students how to manipulate numbers.”

He said a question, aimed at middle schoolers, that does have real world connections and involves more than manipulating numbers, might be: “Shopping at the new store in town includes a 43% discount on all items which are priced the same at $2. The state you live in has a 7% sales tax. You want to buy many things but only have a total of $52 to spend. Describe in words how many things you could buy.”

Schmidt added that relevancy of word problems is one area where few, if any, countries excel. “No one was a shining star leading the way,” he said.

In her brightly decorated classroom one Tuesday afternoon, Santiago, the first grade teacher, gave each student a set of animal-shaped objects and a sheet of blue paper (the water) and green (the grass). “We’re going to work on a number story,” she told them. “I want you to use your animals to tell me the story.”

“Once upon a time,” the story began. In this tale, three animals played in the water, and two animals played in the grass. Santiago allowed some time for the ducks, pigs and bears to frolic in the wilds of each student’s desk before she asked the children to write a number sentence that would tell them how many animals they have altogether.

Some of the students relied more on pictorial representations (three dots on one side of a line and two dots on the other) and others on the number sentence (3+2 = 5) but all of them eventually got to five. And Santiago made sure that her next question mixed up the order of operations (so students didn’t incorrectly assume that all they ever have to do is add): “Some more animals came and now there are seven. So how many more came?”

One approach to early elementary word problems that is taking off in some schools, including Segue Institute, has its origins in a special education intervention for struggling math students. Teachers avoid emphasizing key words and ask students instead to identify first the conceptual type of word problem (or schema, as many practitioners and researchers refer to it) they are dealing with: “Total problems,” for instance, involve combining two parts to find a new amount; “change problems” involve increasing or decreasing the amount of something. Total problems do not necessarily involve adding, however.

“The schemas that students learn in kindergarten will continue with them throughout their whole career,” said Powell, the word-problem researcher, who regularly works with districts across the country to help implement the approach.

One recent afternoon, a lesson on word problems started with everyone standing up and chanting in unison: “Part plus part equals total” (they brought two hands together). “Total minus part equals part” (they took one hand away).

It’s a way to help students remember different conceptual frameworks for word problems. And it’s especially effective for the students who learn well through listening and repeating. For visual learners, the different types of word problems were mapped out on individual dry erase mats.

The real work began when Molina passed out questions, and the students— organized into the Penguin, Flower Bloom, Red Panda and Marshmallow teams — had to figure out which framework they were dealing with on their own and then work toward an answer. A few months ago, many of them would have automatically shut down when they saw the text on the page, Molina said.

For the Red Pandas, the question under scrutiny was: “The clothing store had 47 shirts. They sold 21, how many do they have now?”

“It’s a total problem,” one student said.

“No, it’s not total,” responded another.

“I think it’s about change,” said a third.

None of the students seemed worried about their lack of consensus, however. And neither was Molina. A correct answer is always nice but those come more often now that most of the students have made a crucial leap. “I notice them thinking more and more,” she said, “about what the question is actually asking.”

This story about word problems was produced by The Hechinger Report, a nonprofit, independent news organization focused on inequality and innovation in education. Sign up for the Hechinger newsletter .