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## Free Math Worksheets — Over 100k free practice problems on Khan Academy

Looking for free math worksheets.

You’ve found something even better!

That’s because Khan Academy has over 100,000 free practice questions. And they’re even better than traditional math worksheets – more instantaneous, more interactive, and more fun!

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High school statistics, ap®︎/college statistics, precalculus, differential calculus, integral calculus, ap®︎/college calculus ab, ap®︎/college calculus bc, multivariable calculus, differential equations, linear algebra.

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## Frequently Asked Questions about Khan Academy and Math Worksheets

Why is khan academy even better than traditional math worksheets.

Khan Academy’s 100,000+ free practice questions give instant feedback, don’t need to be graded, and don’t require a printer.

## What do Khan Academy’s interactive math worksheets look like?

Here’s an example:

## What are teachers saying about Khan Academy’s interactive math worksheets?

“My students love Khan Academy because they can immediately learn from their mistakes, unlike traditional worksheets.”

## Is Khan Academy free?

Khan Academy’s practice questions are 100% free—with no ads or subscriptions.

## What do Khan Academy’s interactive math worksheets cover?

Our 100,000+ practice questions cover every math topic from arithmetic to calculus, as well as ELA, Science, Social Studies, and more.

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Khan Academy is a nonprofit with a mission to provide a free, world-class education to anyone, anywhere.

## Want to get even more out of Khan Academy?

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## Browse Course Material

Course info, instructors.

- Prof. Haynes Miller
- Dr. Nat Stapleton
- Saul Glasman

## Departments

- Mathematics

## As Taught In

Learning resource types, project laboratory in mathematics.

Next: Revision and Feedback »

In this section, Prof. Haynes Miller and Susan Ruff describe the criteria for good mathematical writing and the components of the writing workshop .

A central goal of the course is to teach students how to write effective, journal-style mathematics papers. Papers are a key way in which mathematicians share research findings and learn about others’ work. For each research project, each student group writes and revises a paper in the style of a professional mathematics journal paper. These research projects are perfect for helping students to learn to write as mathematicians because the students write about the new mathematics that they discover. They own it, they are committed to it, and they put a lot of effort into writing well.

## Criteria for Good Writing

In the course, we help students learn to write papers that communicate clearly, follow the conventions of mathematics papers, and are mathematically engaging.

Communicating clearly is challenging for students because doing so requires writing precisely and correctly as well as anticipating readers’ needs. Although students have read textbooks and watched lectures that are worded precisely, they are often unaware of the care with which each word or piece of notation was chosen. So when students must choose the words and notation themselves, the task can be surprisingly challenging. Writing precisely is even more challenging when students write about insights they’re still developing. Even students who do a good job of writing precisely may have a different difficulty: providing sufficient groundwork for readers. When students are deeply focused on the details of their research, it can be hard for them to imagine what the reading experience may be like for someone new to that research. We can help students to communicate clearly by pointing out places within the draft at which readers may be confused by imprecise wording or by missing context.

For most students, the conventions of mathematics papers are unfamiliar because they have not read—much less written—mathematics journal papers before. The students’ first drafts often build upon their knowledge of more familiar genres: humanities papers and mathematics textbooks and lecture notes. So the text is often more verbose or explanatory than a typical paper in a mathematics journal. To help students learn the conventions of journal papers, including appropriate concision, we provide samples and individualized feedback.

Finally, a common student preconception is that mathematical writing is dry and formal, so we encourage students to write in a way that is mathematically engaging. In Spring 2013, for example, one student had to be persuaded that he did not have to use the passive voice. In reality, effective mathematics writing should be efficient and correct, but it should also provide motivation, communicate intuition, and stimulate interest.

To summarize, instruction and feedback in the course address many different aspects of successful writing:

- Precision and correctness: e.g., mathematical terminology and notation should be used correctly.
- Audience awareness: e.g., ideas should be introduced with appropriate preparation and motivation.
- Genre conventions: e.g., in most mathematics papers, the paper’s conclusion is stated in the introduction rather than in a final section titled “Conclusion.”
- Style: e.g., writing should stimulate interest.
- Other aspects of effective writing, as needed.

To help students learn to write effective mathematics papers, we provide various resources, a writing workshop, and individualized feedback on drafts.

## Writing Resources

Various resources are provided to help students learn effective mathematical writing.

The following prize-winning journal article was annotated to point out various conventions and strategies of mathematical writing. (Courtesy of Mathematical Association of America. Courtesy of a Creative Commons BY-NC-SA license.)

An Annotated Journal Article (PDF)

This document introduces the structure of a paper and provides a miscellany of common mistakes to avoid.

Notes on Writing Mathematics (PDF)

## LaTeX Resources

The following PDF, TeX, and Beamer samples guide students to present their work using LaTeX, a high-quality typesetting system designed for the production of technical and scientific documentation. The content in the PDF and TeX documents highlights the structure of a generic student paper.

Sample PDF Document created by pdfLaTeX (PDF)

Sample TeX Document (TEX)

Beamer template (TEX)

The following resources are provided to help students learn and use LaTeX.

LaTeX-Project. “ Obtaining LaTeX .” August 28, 2009.

Downes, Michael. “Short Math Guide for LaTeX.” (PDF) American Mathematical Society . Version 1.09. March 22, 2002.

Oetiker, Tobias, Hubert Partl, et al. “The Not So Short Introduction to LaTeX 2ε.” (PDF) Version 5.01. April 06, 2011.

Reckdahl, Keith. “Using Imported Graphics in LaTeX and pdfLaTeX.” (PDF) Version 3.0.1. January 12, 2006.

## Writing Workshop

Each semester there is a writing workshop, led by the lead instructor, which features examples to stimulate discussion about how to write well. In Spring 2013, Haynes ran this workshop during the third class session and used the following slide deck, which was developed by Prof. Paul Seidel and modified with the help of Prof. Tom Mrowka and Prof. Richard Stanley.

The 18.821 Project Report (PDF)

This workshop was held before students had begun to think about the writing component of the course, and it seemed as if the students had to be reminded of the lessons of the workshop when they actually wrote their papers. In future semesters, we plan to offer the writing workshop closer to the time that students are drafting their first paper. We may also focus the examples used in the workshop on the few most important points rather than a broad coverage.

- Download video

This video features the writing workshop from Spring 2013 and includes instruction from Haynes as well as excerpts of the class discussion.

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- How it works
- Homework answers

## How to Do Math Homework Correctly

If you are facing a textbook filled with fractions, square roots or calculus then you know how demanding and overwhelming the homework assignments can become. If you need to get a good grade in your math class, but aren’t able to add up the right formulas, then you will want to find a different way to get assistance with your Math homework .

Understanding how to follow specific guidelines with your math assignments and making sure you are able to prepare your homework in the correct manner can then lead you into higher scores and better results in your class. Students that are faced with specific struggles can follow specific guidelines to achieve higher results.

To make sure you are able to do your homework correctly when preparing for math assignments, you will want to keep some basic rules in mind. When you begin to solve problems in any math topic, you want to start by following the guidelines that you have been given while moving through the steps of the topic. This will ensure that you are able to get the right results with the math you are doing. You should combine this with re-checking your answers to make sure the process that you have followed fits with the correct results. If you are still having trouble understanding your math homework, then you can receive assistance and help from professionals. There are several that understand how to work through formulas and that have a high level of expertise with homework assignments. If you can’t work through the process of a problem or if you don’t understand the guidelines, then professional assistance will help you to move forward with your needs.

No matter what level of math you are at, you can always begin to add up to the right answers. Following basic guidelines to solve problems and receiving help and assistance from a professional can then lead you into achieving an A in any class. The different steps to take with math formulas will allow you to not only get the highest grades but also to have a better understanding of how to solve math problems.

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## Choose Your Test

Sat / act prep online guides and tips, the 5 best homework help websites (free and paid).

Other High School , General Education

Listen: we know homework isn’t fun, but it is a good way to reinforce the ideas and concepts you’ve learned in class. But what if you’re really struggling with your homework assignments?

If you’ve looked online for a little extra help with your take-home assignments, you’ve probably stumbled across websites claiming to provide the homework help and answers students need to succeed . But can homework help sites really make a difference? And if so, which are the best homework help websites you can use?

Below, we answer these questions and more about homework help websites–free and paid. We’ll go over:

- The basics of homework help websites
- The cost of homework help websites
- The five best homework websites out there
- The pros and cons of using these websites for homework help
- The line between “learning” and “cheating” when using online homework help
- Tips for getting the most out of a homework help website

So let’s get started!

## The Basics About Homework Help Websites–Free and Paid

Homework help websites are designed to help you complete your homework assignments, plain and simple.

## What Makes a Homework Help Site Worth Using

Most of the best sites allow users to ask questions and then provide an answer (or multiple possible answers) and explanation in seconds. In some instances, you can even send a photo of a particular assignment or problem instead of typing the whole thing out!

Homework help sites also offer more than just help answering homework questions. Common services provided are Q&A with experts, educational videos, lectures, practice tests and quizzes, learning modules, math solving tools, and proofreading help. Homework help sites can also provide textbook solutions (i.e. answers to problems in tons of different textbooks your school might be using), one-on-one tutoring, and peer-to-peer platforms that allow you to discuss subjects you’re learning about with your fellow students.

And best of all, nearly all of them offer their services 24/7, including tutoring!

## What You Should Should Look Out For

When it comes to homework help, there are lots–and we mean lots –of scam sites out there willing to prey on desperate students. Before you sign up for any service, make sure you read reviews to ensure you’re working with a legitimate company.

A word to the wise: the more a company advertises help that veers into the territory of cheating, the more likely it is to be a scam. The best homework help websites are going to help you learn the concepts you’ll need to successfully complete your homework on your own. (We’ll go over the difference between “homework help” and “cheating” a little later!)

You don't need a golden piggy bank to use homework help websites. Some provide low or no cost help for students like you!

## How Expensive Are the Best Homework Help Websites?

First of all, just because a homework help site costs money doesn’t mean it’s a good service. Likewise, just because a homework help website is free doesn’t mean the help isn’t high quality. To find the best websites, you have to take a close look at the quality and types of information they provide!

When it comes to paid homework help services, the prices vary pretty widely depending on the amount of services you want to subscribe to. Subscriptions can cost anywhere from $2 to $150 dollars per month, with the most expensive services offering several hours of one-on-one tutoring with a subject expert per month.

## The 5 Best Homework Help Websites

So, what is the best homework help website you can use? The answer is that it depends on what you need help with.

The best homework help websites are the ones that are reliable and help you learn the material. They don’t just provide answers to homework questions–they actually help you learn the material.

That’s why we’ve broken down our favorite websites into categories based on who they’re best for . For instance, the best website for people struggling with math might not work for someone who needs a little extra help with science, and vice versa.

Keep reading to find the best homework help website for you!

## Best Free Homework Help Site: Khan Academy

- Price: Free!
- Best for: Practicing tough material

Not only is Khan Academy free, but it’s full of information and can be personalized to suit your needs. When you set up your account , you choose which courses you need to study, and Khan Academy sets up a personal dashboard of instructional videos, practice exercises, and quizzes –with both correct and incorrect answer explanations–so you can learn at your own pace.

As an added bonus, it covers more course topics than many other homework help sites, including several AP classes.

Runner Up: Brainly.com offers a free service that allows you to type in questions and get answers and explanations from experts. The downside is that you’re limited to two answers per question and have to watch ads.

## Best Paid Homework Help Site: Chegg

- Price: $14.95 to $19.95 per month
- Best for: 24/7 homework assistance

This service has three main parts . The first is Chegg Study, which includes textbook solutions, Q&A with subject experts, flashcards, video explanations, a math solver, and writing help. The resources are thorough, and reviewers state that Chegg answers homework questions quickly and accurately no matter when you submit them.

Chegg also offers textbook rentals for students who need access to textbooks outside of their classroom. Finally, Chegg offers Internship and Career Advice for students who are preparing to graduate and may need a little extra help with the transition out of high school.

Another great feature Chegg provides is a selection of free articles geared towards helping with general life skills, like coping with stress and saving money. Chegg’s learning modules are comprehensive, and they feature solutions to the problems in tons of different textbooks in a wide variety of subjects.

Runner Up: Bartleby offers basically the same services as Chegg for $14.99 per month. The reason it didn’t rank as the best is based on customer reviews that say user questions aren’t answered quite as quickly on this site as on Chegg. Otherwise, this is also a solid choice!

## Best Site for Math Homework Help: Photomath

- Price: Free (or $59.99 per year for premium services)
- Best for: Explaining solutions to math problems

This site allows you to t ake a picture of a math problem, and instantly pulls up a step-by-step solution, as well as a detailed explanation of the concept. Photomath also includes animated videos that break down mathematical concepts to help you better understand and remember them.

The basic service is free, but for an additional fee you can get extra study tools and learn additional strategies for solving common math problems.

Runner Up: KhanAcademy offers in-depth tutorials that cover complex math topics for free, but you won’t get the same tailored help (and answers!) that Photomath offers.

## Best Site for English Homework Help: Princeton Review Academic Tutoring

- Price: $40 to $153 per month, depending on how many hours of tutoring you want
- Best for: Comprehensive and personalized reading and writing help

While sites like Grammarly and Sparknotes help you by either proofreading what you write via an algorithm or providing book summaries, Princeton Review’s tutors provide in-depth help with vocabulary, literature, essay writing and development, proofreading, and reading comprehension. And unlike other services, you’ll have the chance to work with a real person to get help.

The best part is that you can get on-demand English (and ESL) tutoring from experts 24/7. That means you can get help whenever you need it, even if you’re pulling an all-nighter!

This is by far the most expensive homework site on this list, so you’ll need to really think about what you need out of a homework help website before you commit. One added benefit is that the subscription covers over 80 other subjects, including AP classes, which can make it a good value if you need lots of help!

## Best Site for STEM Homework Help: Studypool

- Best for: Science homework help
- Price: Varies; you’ll pay for each question you submit

When it comes to science homework help, there aren’t a ton of great resources out there. The best of the bunch is Studypool, and while it has great reviews, there are some downsides as well.

Let’s start with the good stuff. Studypool offers an interesting twist on the homework help formula. After you create a free account, you can submit your homework help questions, and tutors will submit bids to answer your questions. You’ll be able to select the tutor–and price point–that works for you, then you’ll pay to have your homework question answered. You can also pay a small fee to access notes, lectures, and other documents that top tutors have uploaded.

The downside to Studypool is that the pricing is not transparent . There’s no way to plan for how much your homework help will cost, especially if you have lots of questions! Additionally, it’s not clear how tutors are selected, so you’ll need to be cautious when you choose who you’d like to answer your homework questions.

## What Are the Pros and Cons of Using Homework Help Sites?

Homework help websites can be a great resource if you’re struggling in a subject, or even if you just want to make sure that you’re really learning and understanding topics and ideas that you’re interested in. But, there are some possible drawbacks if you don’t use these sites responsibly.

We’ll go over the good–and the not-so-good–aspects of getting online homework help below.

## 3 Pros of Using Homework Help Websites

First, let’s take a look at the benefits.

## #1: Better Grades Beyond Homework

This is a big one! Getting outside help with your studies can improve your understanding of concepts that you’re learning, which translates into better grades when you take tests or write essays.

Remember: homework is designed to help reinforce the concepts you learned in class. If you just get easy answers without learning the material behind the problems, you may not have the tools you need to be successful on your class exams…or even standardized tests you’ll need to take for college.

## #2: Convenience

One of the main reasons that online homework help is appealing is because it’s flexible and convenient. You don’t have to go to a specific tutoring center while they’re open or stay after school to speak with your teacher. Instead, you can access helpful resources wherever you can access the internet, whenever you need them.

This is especially true if you tend to study at off hours because of your extracurriculars, work schedule, or family obligations. Sites that offer 24/7 tutoring can give you the extra help you need if you can’t access the free resources that are available at your school.

## #3: Variety

Not everyone learns the same way. Maybe you’re more of a visual learner, but your teacher mostly does lectures. Or maybe you learn best by listening and taking notes, but you’re expected to learn something just from reading the textbook .

One of the best things about online homework help is that it comes in a variety of forms. The best homework help sites offer resources for all types of learners, including videos, practice activities, and even one-on-one discussions with real-life experts.

This variety can also be a good thing if you just don’t really resonate with the way a concept is being explained (looking at you, math textbooks!).

Not so fast. There are cons to homework help websites, too. Get to know them below!

## 3 Cons of Using Homework Help Websites

Now, let’s take a look at the drawbacks of online homework help.

## #1: Unreliable Info

This can be a real problem. In addition to all the really good homework help sites, there are a whole lot of disreputable or unreliable sites out there. The fact of the matter is that some homework help sites don’t necessarily hire people who are experts in the subjects they’re talking about. In those cases, you may not be getting the accurate, up-to-date, and thorough information you need.

Additionally, even the great sites may not be able to answer all of your homework questions. This is especially true if the site uses an algorithm or chatbot to help students…or if you’re enrolled in an advanced or college-level course. In these cases, working with your teacher or school-provided tutors are probably your best option.

## #2: No Clarification

This depends on the service you use, of course. But the majority of them provide free or low-cost help through pre-recorded videos. Watching videos or reading info online can definitely help you with your homework… but you can’t ask questions or get immediate feedback if you need it .

## #3: Potential For Scamming

Like we mentioned earlier, there are a lot of homework help websites out there, and lots of them are scams. The review comments we read covered everything from outdated or wrong information, to misleading claims about the help provided, to not allowing people to cancel their service after signing up.

No matter which site you choose to use, make sure you research and read reviews before you sign up–especially if it’s a paid service!

## When Does “Help” Become “Cheating”?

Admittedly, whether using homework help websites constitutes cheating is a bit of a grey area. For instance, is it “help” when a friend reads your essay for history class and corrects your grammar, or is it “cheating”? The truth is, not everyone agrees on when “help” crosses the line into “cheating .” When in doubt, it can be a good idea to check with your teacher to see what they think about a particular type of help you want to get.

That said, a general rule of thumb to keep in mind is to make sure that the assignment you turn in for credit is authentically yours . It needs to demonstrate your own thoughts and your own current abilities. Remember: the point of every homework assignment is to 1) help you learn something, and 2) show what you’ve learned.

So if a service answers questions or writes essays for you, there’s a good chance using it constitutes cheating.

Here’s an example that might help clarify the difference for you. Brainstorming essay ideas with others or looking online for inspiration is “help” as long as you write the essay yourself. Having someone read it and give you feedback about what you need to change is also help, provided you’re the one that makes the changes later.

But copying all or part of an essay you find online or having someone write (or rewrite) the whole thing for you would be “cheating.” The same is true for other subjects. Ultimately, if you’re not generating your own work or your own answers, it’s probably cheating.

## 5 Tips for Finding the Best Homework Help Websites for You

Now that you know some of our favorite homework help websites, free and paid, you can start doing some additional research on your own to decide which services might work best for you! Here are some top tips for choosing a homework help website.

## Tip 1: Decide How You Learn Best

Before you decide which site or sites you’re going to use for homework help, y ou should figure out what kind of learning style works for you the most. Are you a visual learner? Then choose a site that uses lots of videos to help explain concepts. If you know you learn best by actually doing tasks, choose a site that provides lots of practice exercises.

## Tip 2: Determine Which Subjects You Need Help With

Just because a homework help site is good overall doesn’t mean that it’s equally good for every subject. If you only need help in math, choose a site that specializes in that area. But if history is where you’re struggling, a site that specializes in math won’t be much help. So make sure to choose a site that you know provides high-quality help in the areas you need it most.

## Tip 3: Decide How Much One-On-One Help You Need

This is really about cost-effectiveness. If you learn well on your own by reading and watching videos, a free site like Khan Academy is a good choice. But if you need actual tutoring, or to be able to ask questions and get personalized answers from experts, a paid site that provides that kind of service may be a better option.

## Tip 4: Set a Budget

If you decide you want to go with a paid homework help website, set a budget first . The prices for sites vary wildly, and the cost to use them can add up quick.

## Tip 5: Read the Reviews

Finally, it’s always a good idea to read actual reviews written by the people using these homework sites. You’ll learn the good, the bad, and the ugly of what the users’ experiences have been. This is especially true if you intend to subscribe to a paid service. You’ll want to make sure that users think it’s worth the price overall!

## What’s Next?

If you want to get good grades on your homework, it’s a good idea to learn how to tackle it strategically. Our expert tips will help you get the most out of each assignment…and boost your grades in the process.

Doing well on homework assignments is just one part of getting good grades. We’ll teach you everything you need to know about getting great grades in high school in this article.

Of course, test grades can make or break your GPA, too. Here are 17 expert tips that’ll help you get the most out of your study prep before you take an exam.

Ashley Sufflé Robinson has a Ph.D. in 19th Century English Literature. As a content writer for PrepScholar, Ashley is passionate about giving college-bound students the in-depth information they need to get into the school of their dreams.

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## How to Start an Assignment

Last Updated: January 29, 2024 Fact Checked

This article was co-authored by Michelle Golden, PhD . Michelle Golden is an English teacher in Athens, Georgia. She received her MA in Language Arts Teacher Education in 2008 and received her PhD in English from Georgia State University in 2015. There are 8 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 107,463 times.

Getting started on an assignment or homework can often times be the hardest step. Putting off the assignment can make the problem worse, reducing the time you have to complete the task and increasing stress. By learning how to get started and overcome the urge to procrastinate, you can get your assignments done on schedule and with less stress, opening up more free time.

## Restructuring Your Assignment

- For example, you might research areas of a report that you find most interesting before moving on to other areas.
- If your math assignment has different types of questions, try doing those that you enjoy the most before moving on to the others.
- You might also try tackling smaller or easier tasks first so you can cross a few items off your list. Seeing that you've already made progress may help you feel motivated to continue.

- Promise yourself that you will meet your goal of working for five minutes on the assignment.
- Once you get started, you may find that you don't want to stop working. Otherwise, you can take a break and come back to the assignment, knowing you're at least five minutes closer to finishing than you were before.

- Try to set reasonable periods of time that you know you can meet. For example, you might set aside two hours on a Friday to dedicate to your assignment. If you don't have that much time all at once, try to carve out a few 20- or 30-minute blocks.
- You may or may not wish to continue working after your time limit has gone by.
- Have a realistic understanding of how fast you can write and plan your schedule accordingly.

- It can help to read the assignment as soon as you get it and then ask any questions you might have.
- If you're not sure if you understand the assignment, try rewriting it in your own words or explaining it to someone else. If you find you can't or have a lot of questions, you may need more information.
- You should have an overview of the assignment, understand the main task, and understand the technical and stylistic requirements.
- Look for important words in the instructions to understand the assignment. These words might include define, explain, compare, relate, or prove.
- Keep your audience in mind and write a paper that would best deliver information to them.

- Goals that are too big or not well defined can be difficult to start working towards.
- Smaller and well defined goals can seem easier to achieve than larger ones.
- For example, you could break a research paper down into several smaller tasks: 1) do preliminary research, 2) write an outline, 3) draft an introduction, 4) draft body paragraphs, 5) write conclusion, 6) revise. Each of these is much more do-able on its own.

## Changing Your Focus

- You might want to go for a quick walk after working for a set amount of time.
- Try reading a website or book that you enjoy for a few minutes after working.
- Alternatively, try a quick burst of exercise before setting to work. Exercise releases feel-good chemicals called endorphins and can also help boost your memory. [8] X Research source

- Instead of dreading your work, focus on how good it will feel to make progress. You won't have it hanging over your head. You can actually enjoy the weekend instead of feeling guilty.
- Keeping your eye on long-term rewards can help you stay motivated to finish your assignment.

- Avoid moving your workspace constantly.
- Don't get lost on tangential research.
- Don't take constant breaks to get a snack.

- For every hour you waste procrastinating, you can limit how much television you watch that night.
- If you waste too much time procrastinating, you might deny yourself a favorite snack later on.

## Community Q&A

## You Might Also Like

- ↑ https://www.psychologytoday.com/blog/solving-unsolvable-problems/201408/4-steps-stop-procrastinating
- ↑ https://www.psychologytoday.com/blog/friendship-20/201405/the-surefire-first-step-stop-procrastinating
- ↑ http://writingcenter.unc.edu/handouts/procrastination/
- ↑ https://kidshealth.org/en/teens/homework.html
- ↑ http://writingcenter.unc.edu/handouts/understanding-assignments/
- ↑ https://open.alberta.ca/dataset/ab22ff64-3358-4387-9761-8c58878a6b84/resource/3ee38320-17e4-46f9-b24f-c95f9f345eb9/download/ipp7.pdf
- ↑ http://well.blogs.nytimes.com/2013/08/07/how-exercise-can-help-us-learn/
- ↑ https://kidshealth.org/en/teens/happy-life.html

## About This Article

To start an assignment, try working on the most enjoyable or easiest parts of the assignment first to get the ball rolling. Even if no part of the assignment seems enjoyable or easy, set a timer and try to make yourself work for at least 5 minutes, which is usually enough time to build momentum and overcome procrastination. You can also try breaking your assignment up into smaller, more manageable tasks and scheduling yourself regular breaks so it doesn't seem as overwhelming. To learn how to stay positive and avoid procrastination while working on your homework, scroll down! Did this summary help you? Yes No

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## Unit 1: Numbers and operations

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## Math Assignment

What is an annotated bibliography?

Assignments in math courses are usually a list of problems to be solved. Each question may be of various difficulties and types, which are marked and returned to the student to be used as a source of feedback. Math assignments are designed to provide opportunities for 'doing math' and to consolidate students’ understanding of the content. The questions often come from the most recent week of learned material, but some questions may require students to synthesize concepts from further back. This is because learning math is cumulative by nature; you continuously build upon what you've learned before. Because math assignments are typically used as a checkpoint for understanding, they tend to weigh less than other assessments in terms of grade, therefore, it's important to treat them as learning opportunities instead of a tool for maximizing course grades.

Click on the Timeline for a visual representation of the timeline. Click on the Checklist for a document containing the checklist items for a math assignment.

According to your start and end dates ( 2024-05-31 to 2024-06-06 ), you have 6 days to finish your assignment.

## Step 1: Understand the questions Complete by Fri May 31, 2024

Read through the assignment once it’s released and understand the questions. Read the questions early to prime your brain; too many students start too late.

- For some courses, assignments come out before or as content is covered, so having the questions in your mind as you see the content can help prime you to actively learn the content and will also help you know where to look for relevant material in the lecture notes and/or videos. Active learning of content means you are asking yourself questions as you are learning, actively monitoring your understanding of the topic, and anticipating next steps.
- Read all the questions to estimate the range of tested material, make note of anything you don’t understand. Understanding the question correctly reduces the chance of getting stuck.

The hardest thing being with a mathematician is that they always have problems.

– Tendai Chitewere

## Step 2: Gather materials and review Complete by Sat Jun 01, 2024

- Gather required materials (lecture notes and/or videos, textbook sections).
- Study the material (definitions, concepts, solved problems, etc.) until you are familiar with it. Identify connections across concepts/topics and practice additional problems. Regularly reviewing your material will also help you to find relevant information quickly if you get stuck on a question.

## Step3: Solve questions and get help Complete by Wed Jun 05, 2024

Part a: attempt to solve each question.

- Rule of thumb for time allocation: one hard question may take significantly longer than all easy questions combined. Consider how long you’ve taken to solve all the easy questions. It may take at least twice as long to solve one hard question. Try all easy questions as soon as possible to gauge how long the rest could take.
- Try giving yourself a break for a day before attempting them again to allow time for your mind to continue working subconsciously.
- Still stuck? Check out Math problems: What to do when you're stuck for more strategies and tips.
- Keep track of unsolved questions and what stages you were stuck at. Don’t throw away any work in progress.Your instructor or TA can provide better help if you can narrow down on the issue.

The only way to learn mathematics is to do mathematics.

– Paul Halmos

Mathematics is not a deductive science -- that's a cliche. When you try to prove a theorem, you don't just list the hypotheses, and then start to reason. What you do is trial and error, experimentation, guesswork.

– Paul Halmos

## Part B: Get help on unsolved questions

- The sooner you ask for support, the more time you and the instructor/TA will have for follow-up conversations.
- Focus on asking questions to help you close the gap between your understanding, and the understanding you need to complete the question. Questions such as “Can I get a hint on #4?” or “What is the answer?” are not typically helpful in increasing your problem-solving abilities.
- For questions you were challenged by, but were able to solve after receiving help from the instructor or TA, make sure that you fully understand how to come up with the solution in case a similar question is asked on a test.
- Many students find it helpful to study together. Make sure you understand what your instructor’s expectations are around group work (if you’re not sure, ask!). Be mindful about how much you post about an assignment solution on the course’s discussion board and if you’re in doubt, consider posting privately. Familiarize yourself with the academic integrity standards expected in your courses.

The goal of seeking help is to try to find out how to approach questions that you are stuck on. From this extra help you still may not see the complete picture of how to solve the problem. To see this may require that you go back to Part A , now with the instructor/TA’s suggestions in mind.

I have not failed. I've just found 10,000 ways that won't work.

– Thomas Alva Edison

## Step 4: Check your solutions Complete by Thu Jun 06, 2024

- It’s important that you spend some time away from your solutions before you check them. For problems you did solve, try to figure out if you can check your own answer like you would have to on a test (i.e. don’t look up answers in a book or online).
- You could ask the instructor/TA for guidance, but avoid questions such as “Did I do this right?” Instead, ask yourself whether you are asking the right questions about your solution.
- Remember not to focus only on the final answer, but also on how you communicated your answer. Ask yourself if a typical student in the class could follow along with what you’ve written.
- Remember to cite any external sources that you used in your work, including work completed with others (if this was permitted).

## Step 5: Review Marked Assignments Complete by Thu Jun 06, 2024

After receiving your marked assignment check over incorrect questions as well as those you got right. Review posted solutions and read them critically. Often, there are different approaches to a problem and you may learn about some of these new approaches through this review.

## Help Articles

Math assignments, learner help center dec 5, 2022 • knowledge, article details.

Before you submit your assignment, you'll be able to preview your answer to see what the grader will see.

You can use basic symbols for addition (+), subtraction (-), multiplication (*), division (/), and power/exponential (^).

You need to use (*) for multiplication every time. For example, 3x should be 3*x instead.

## Function Syntax

Math assignments accept the following basic and advanced functions.

You can enter these math functions in the form of function(value). For example, the square root of 17 should be entered as sqrt(17). The cosine function of 5x should be entered as cos(5*x).

## Basic numeric functions

- abs (absolute value)
- gcd (greatest common denominator; 2 arguments only, enter three or more arguments as gcd((a, b, c, d)) or gcd([a, b, c, d]) e.g. a list/tuple)
- lcm (least common multiple; 2 arguments only, enter three or more arguments as gcd((a, b, c, d)) or gcd([a, b, c, d]) e.g. a list/tuple)
- sign (returns -1 if negative, 0 if zero, 1 if positive)
- exp (exponent)
- ln (natural logarithm, base e)
- log (common logarithm, base 10)
- root (2-argument, where root(a, 2) is equivalent to sqrt(a))
- sqrt (square root)

## Trigonometry

- Basic trigonometric functions: cos, cosh, cot, coth, sin, sinh, tan, tanh
- Trigonometric inverses: acos, acosh, acot, acoth, asin, asinh, atan, atan2 (2-argument arctangent), atanh
- deg (converts radians to degrees)
- rad (converts degrees to radians)

## Complex numbers

- arg (returns phase in radians of a complex number)
- im (get imaginary part of a number)
- re (get real part of a number)

## Other functions

- erf (error function)

## Allowed constants

Math assignments recognize the following common constants:

- Catalan (Catalan's constant)
- E (note uppercase)
- EulerGamma (Euler-Mascheroni constant)
- GoldenRatio
- I (sqrt(-1) - note uppercase)
- J (sqrt(-1), same as I)
- oo (infinity)
- zoo (complex infinity)

Learners can also use the Equatio extension on Google Chrome to write equations more easily. Equatio allows you to hand-write or speak equations.

To use Equatio:

- Install the Equatio Chrome extension and turn it on in your browser. You may also want to pin the extension to your Chrome bar for easy access in the next step.
- Open the math assignment you

- Using the Equatio toolbar, create your math expression by typing it, hand-writing it, or using speech-to-text. The math expression will be rendered in the course item.

For tips on getting started with Equatio or additional syntax help, we recommend visiting Texthelp's official website and support articles .

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## Young Students Gravitate to Math. How Teachers Can Build on That Curiosity

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Zachary Champagne’s 3rd and 4th graders figure out early on that this math class will be different when their teacher tells them: “I don’t care about the answer.”

The goal is to shift his elementary students’ thinking from some numerical endgame toward the problem-solving process itself. In his more than two decades as a classroom teacher and math researcher, Champagne has found this strategy can be a balm for math anxiety, spur students’ creativity, and pique their curiosity about a subject many find boring and irrelevant.

Telling students the answer doesn’t matter—or throwing it out early on, then working backwards, another of Champagne’s go-to strategies—"can reframe the way we think about mathematics,” said Champagne, who teaches at The Discovery School, a private school in Jacksonville, Fla.

“If we’re thinking about math where the solving is actually the interesting, important part, it frees kids from the stigma of ‘I’m not good at this because I always get things wrong,’” said Champagne, who spent more than a decade working in Florida’s Duval County public schools and served as a math researcher at Florida State University.

This problem-solving or open-ended approach, which emphasizes flexible thinking and real-world situations, is a powerful strategy for engaging the youngest learners in math . Kindergarten through 5th grade is an important time for building students’ skills, confidence, and interest in math—the critical building blocks for middle- and high-school-level math and science, experts say.

Though the approach has been around for decades, districts are striving to incorporate more real-world problem-solving into math class in recent years. California, for instance, recently adopted a controversial framework that puts a heavy emphasis on the approach . And there’s new urgency to get students motivated in math as federal data show students’ math achievement plummeting.

The vast majority of educators—92 percent—say students are more motivated to learn math and science if teachers employ a problem-solving approach, according to a survey of 1,183 district and school leaders and teachers, conducted by the EdWeek Research Center in April. Despite the fact that this approach is highly popular among educators, many have not been trained in how to use it, the same survey found.

## Does using real-world problems motivate students?

The Canadian province of Quebec has been using a problem-based approach for decades—and it helps students connect with math and understand how to use it in the real world, said Krista Muis, a professor at McGill University in Montreal, who has studied student perceptions of the teaching strategy.

“When you look at the motivational profiles of students who are just given traditional word problems, or more standard types of math problems, or math content, their motivation is really low when it comes to the value of what they’re learning,” Muis said. “The main question they ask is, ‘why should I care? How is this relevant to me? How am I ever going to use this?’”

But when students tackle common real problems—a favorite of Muis’ asks elementary schoolers to map out the trick-or-treating route that nets the most Halloween candy—they get excited.

“They see the value in it,” Muis said. “And they’re fun problems. They can do them in groups together collaboratively, they can do them individually.”

Quebec students’ higher motivation in math may explain why the province outperforms the rest of Canada—and the United States— on the Trends in International Mathematics and Science Study or TIMMS , Muis said.

In 2019, the most recent year the test was given, Quebec’s 4th graders didn’t perform statistically differently from their U.S. counterparts in math. But 8th graders from the province scored significantly better than their U.S. peers . One reason may be the increased motivation to learn math that Muis believes stems from exposing students to a problem-solving approach early on.

To be sure, a problem-based or open-ended approach to teaching math is often pitted against more traditional, procedural methods—think of the math worksheets full of equations without context.

But many experts and educators see value in exposing students to both strategies.

“I think, really, these things can mutually support one another. And both are necessary,” said Julia Aguirre, a professor and the faculty director of teacher certification programs at the University of Washington Tacoma. “I think we can all agree that a math class that’s only about worksheets would not be a very fulfilling or interesting math class.”

## Promote young students’ natural curiosity and creativity

The approach is most effective when teachers apply it to students’ existing interests.

That’s especially important for elementary school students, who start school with a natural curiosity that often dissipates by the time they get to high school, said Molly Daley, a regional math coordinator for Education Service District 112, which serves about 30 districts near Vancouver, Wash.

Thinking about “math is a universal human behavior, and people of all ages engage in math for their own purposes,” Daley said.

Students are using math when they play games and make crafts, she said, or even just look at the landscape.

For instance, a preschool teacher might take a picture of the classroom shoe rack and ask students questions like: How many shoes are there? What patterns do you notice? What shapes do you see?

“If we can honor the math that kids are doing beyond the classroom, then we’re more likely to create a mathematical connection and really allow every person to see how math is not just useful but enjoyable,” Daley said.

In Champagne’s mixed age classroom of 3rd and 4th graders in Florida—which he co-teaches with another educator—students turn to math early in the day, the time when younger students tend to be most able to focus on the subject, in Champagne’s view.

Champagne typically kicks off with a 10- or 15-minute math routine as a warm-up. That might be a “number talk” in which Champagne will put an equation on the board, say 29 plus 15, and then students will solve the problem in their heads.

They’ll spend the next few minutes comparing strategies for finding a solution. One student might have added 30 plus 15 and subtracted one, while another might have added 9 and 5, then 30.

The exercise is aimed at promoting flexibility and the idea that there are multiple ways to solve a problem, Champagne said. It lets students know: “I don’t have to revert to just one strategy. I can think about it in different ways,” he said. The idea is to gives students a chance to use their creative thinking skills in math class.

Students still learn the basics, but lessons are structured so that students can see how seemingly simple problems play out in different, real-world contexts. For instance, if students are learning about dividing with remainders, they may consider how four people can share 31 balloons. In that case, each person gets seven balloons, with three left over.

But what if it were 31 dollars instead of balloons? How does that change the answer? Or what if 31 people needed to get somewhere in four cars? How could they divide up?

Problems can also get more complex—and interdisciplinary—as students advance in elementary school.

## Teachers need more training in the problem-solving approach

Tackling big problems with no clear answer is another way to engage elementary school students in math.

Last school year, Aguirre worked with Janaki Nagarajan’s 3rd graders outside Seattle on a project involving a real-life problem with salmon the students had raised and planned to release.

Inexplicably, the fish began dying. So Nagarajan’s students used mathematical modeling to estimate how quickly they were losing salmon, answering questions like: Will we have enough salmon for each student on release day? What can we do if we don’t? Students worked on the problem in groups, and then presented their answers. The class voted on the solution they thought would work best.

The project was “really engaging,” said Nagarajan. She believes that students will be motivated to learn math if they “feel the skills have some purpose outside the classroom.”

But she thinks that many teachers don’t know how common procedures learned in math class could be applied in the real world, so they struggle to make those connections for their students.

Nagarajan began teaching in Renton, a different, Seattle-area school district this school year, largely because it provides more support for teachers to use the real-world problem-solving approach in elementary school math.

Though the approach was encouraged in her previous school, Nagarajan said her new district uses a curriculum that embraces problem-solving and provides coaches who can help her implement the strategy.

Professional development in the problem-solving approach remains uneven. About one in five educators said they “completely agree” that their districts have offered deep and sustained professional development into how to teach math and science from a problem-solving perspective, while just over 40 percent said they disagree—at least somewhat—that they’ve been offered that level of support.

That professional development can be particularly important for elementary school teachers who typically “aren’t math specialists, right? They are generalists,” said Muis of McGill University. “Often, teachers who are not comfortable with mathematics don’t necessarily understand it fully themselves. And so when you bring in complexity that scares them. And then you see teachers kind of stepping back going, ‘I can’t really teach this, I don’t really know what I’m doing.’”

And the approach requires teachers to respond to what students see or notice, which can be stressful for some, Daley said.

“We can get too hyper focused on ‘this is my goal’” in a particular lesson, she said. That can look like: “We’re learning about fractions, but the student made a comment about multiplication. I gotta ignore that.’”

Teachers need to learn not to be afraid if students go off script, Daley said. A problem-solving approach is about “creating more space for students’ ideas and students’ thinking versus just letting your own dominate.”

Making that shift isn’t easy. But if teachers are successful, they positively shape their students’ relationship with math, potentially for years, Daley said.

“Especially with younger learners, when we’re following their lead, that’s how we’re going to tap into their connection and their motivation to engage with math and build up their sense of themselves as a mathematician,” she said.

Coverage of problem solving and student motivation is supported in part by a grant from The Lemelson Foundation, at www.lemelson.org . Education Week retains sole editorial control over the content of this coverage.

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For everyone whose relationship with mathematics is distant or broken, Jo Boaler , a professor at Stanford Graduate School of Education (GSE), has ideas for repairing it. She particularly wants young people to feel comfortable with numbers from the start – to approach the subject with playfulness and curiosity, not anxiety or dread.

“Most people have only ever experienced what I call narrow mathematics – a set of procedures they need to follow, at speed,” Boaler says. “Mathematics should be flexible, conceptual, a place where we play with ideas and make connections. If we open it up and invite more creativity, more diverse thinking, we can completely transform the experience.”

Boaler, the Nomellini and Olivier Professor of Education at the GSE, is the co-founder and faculty director of Youcubed , a Stanford research center that provides resources for math learning that has reached more than 230 million students in over 140 countries. In 2013 Boaler, a former high school math teacher, produced How to Learn Math , the first massive open online course (MOOC) on mathematics education. She leads workshops and leadership summits for teachers and administrators, and her online courses have been taken by over a million users.

In her new book, Math-ish: Finding Creativity, Diversity, and Meaning in Mathematics , Boaler argues for a broad, inclusive approach to math education, offering strategies and activities for learners at any age. We spoke with her about why creativity is an important part of mathematics, the impact of representing numbers visually and physically, and how what she calls “ishing” a math problem can help students make better sense of the answer.

What do you mean by “math-ish” thinking?

It’s a way of thinking about numbers in the real world, which are usually imprecise estimates. If someone asks how old you are, how warm it is outside, how long it takes to drive to the airport – these are generally answered with what I call “ish” numbers, and that’s very different from the way we use and learn numbers in school.

In the book I share an example of a multiple-choice question from a nationwide exam where students are asked to estimate the sum of two fractions: 12/13 + 7/8. They’re given four choices for the closest answer: 1, 2, 19, or 21. Each of the fractions in the question is very close to 1, so the answer would be 2 – but the most common answer 13-year-olds gave was 19. The second most common was 21.

I’m not surprised, because when students learn fractions, they often don’t learn to think conceptually or to consider the relationship between the numerator or denominator. They learn rules about creating common denominators and adding or subtracting the numerators, without making sense of the fraction as a whole. But stepping back and judging whether a calculation is reasonable might be the most valuable mathematical skill a person can develop.

But don’t you also risk sending the message that mathematical precision isn’t important?

I’m not saying precision isn’t important. What I’m suggesting is that we ask students to estimate before they calculate, so when they come up with a precise answer, they’ll have a real sense for whether it makes sense. This also helps students learn how to move between big-picture and focused thinking, which are two different but equally important modes of reasoning.

Some people ask me, “Isn’t ‘ishing’ just estimating?” It is, but when we ask students to estimate, they often groan, thinking it’s yet another mathematical method. But when we ask them to “ish” a number, they're more willing to offer their thinking.

Ishing helps students develop a sense for numbers and shapes. It can help soften the sharp edges in mathematics, making it easier for kids to jump in and engage. It can buffer students against the dangers of perfectionism, which we know can be a damaging mindset. I think we all need a little more ish in our lives.

You also argue that mathematics should be taught in more visual ways. What do you mean by that?

For most people, mathematics is an almost entirely symbolic, numerical experience. Any visuals are usually sterile images in a textbook, showing bisecting angles, or circles divided into slices. But the way we function in life is by developing models of things in our minds. Take a stapler: Knowing what it looks like, what it feels and sounds like, how to interact with it, how it changes things – all of that contributes to our understanding of how it works.

There’s an activity we do with middle-school students where we show them an image of a 4 x 4 x 4 cm cube made up of smaller 1 cm cubes, like a Rubik’s Cube. The larger cube is dipped into a can of blue paint, and we ask the students, if they could take apart the little cubes, how many sides would be painted blue? Sometimes we give the students sugar cubes and have them physically build a larger 4 x 4 x 4 cube. This is an activity that leads into algebraic thinking.

Some years back we were interviewing students a year after they’d done that activity in our summer camp and asked what had stayed with them. One student said, “I’m in geometry class now, and I still remember that sugar cube, what it looked like and felt like.” His class had been asked to estimate the volume of their shoes, and he said he’d imagined his shoes filled with 1 cm sugar cubes in order to solve that question. He had built a mental model of a cube.

When we learn about cubes, most of us don’t get to see and manipulate them. When we learn about square roots, we don’t take squares and look at their diagonals. We just manipulate numbers.

I wonder if people consider the physical representations more appropriate for younger kids.

That’s the thing – elementary school teachers are amazing at giving kids those experiences, but it dies out in middle school, and by high school it’s all symbolic. There’s a myth that there’s a hierarchy of sophistication where you start out with visual and physical representations and then build up to the symbolic. But so much of high-level mathematical work now is visual. Here in Silicon Valley, if you look at Tesla engineers, they're drawing, they're sketching, they're building models, and nobody says that's elementary mathematics.

There’s an example in the book where you’ve asked students how they would calculate 38 x 5 in their heads, and they come up with several different ways of arriving at the same answer. The creativity is fascinating, but wouldn’t it be easier to teach students one standard method?

A depiction of various ways to calculate 38 x 5, numerically and visually. | Courtesy Jo Boaler

That narrow, rigid version of mathematics where there’s only one right approach is what most students experience, and it’s a big part of why people have such math trauma. It keeps them from realizing the full range and power of mathematics. When you only have students blindly memorizing math facts, they’re not developing number sense. They don’t learn how to use numbers flexibly in different situations. It also makes students who think differently believe there’s something wrong with them.

When we open mathematics to acknowledge the different ways a concept or problem can be viewed, we also open the subject to many more students. Mathematical diversity, to me, is a concept that includes both the value of diversity in people and the diverse ways we can see and learn mathematics. When we bring those forms of diversity together, it’s powerful. If we want to value different ways of thinking and problem-solving in the world, we need to embrace mathematical diversity.

## Python Operators Cheat Sheet

- python basics
- learn python

Discover the essential Python operators and how to effectively use them with our comprehensive cheat sheet. We cover everything from arithmetic to bitwise operations!

If you’ve ever written a few lines of Python code, you are likely familiar with Python operators. Whether you're doing basic arithmetic calculations, creating variables, or performing complex logical operations, chances are that you had to use a Python operator to perform the task. But just how many of them exist, and what do you use them for?

In this cheat sheet, we will cover every one of Python’s operators:

- Arithmetic operators.
- Assignment operators.
- Comparison operators.
- Logical operators.
- Identity operators.
- Membership operators.
- Bitwise operators.

Additionally, we will discuss operator precedence and its significance in Python.

If you're just starting out with Python programming, you may want to look into our Python Basics Track . Its nearly 40 hours of content covers Python operators and much more; you’ll get an excellent foundation to build your coding future on.

Without further ado, let's dive in and learn all about Python operators.

## What Are Python Operators?

Python operators are special symbols or keywords used to perform specific operations. Depending on the operator, we can perform arithmetic calculations, assign values to variables, compare two or more values, use logical decision-making in our programs, and more.

## How Operators Work

Operators are fundamental to Python programming (and programming as a whole); they allow us to manipulate data and control the flow of our code. Understanding how to use operators effectively enables programmers to write code that accomplishes a desired task.

In more specific terms, an operator takes two elements – called operands – and combines them in a given manner. The specific way that this combination happens is what defines the operator. For example, the operation A + B takes the operands A and B , performs the “sum” operation (denoted by the + operator), and returns the total of those two operands.

## The Complete List of Python Operators

Now that we know the basic theory behind Python operators, it’s time to go over every single one of them.

In each section below, we will explain a family of operators, provide a few code samples on how they are used, and present a comprehensive table of all operators in that family. Let’s get started!

## Python Arithmetic Operators

Arithmetic operators are used to perform mathematical calculations like addition, subtraction, multiplication, division, exponentiation, and modulus. Most arithmetic operators look the same as those used in everyday mathematics (or in spreadsheet formulas).

Here is the complete list of arithmetic operators in Python:

Most of these operators are self-explanatory, but a few are somewhat tricky. The floor division operator ( // ), for example, returns the integer portion of the division between two numbers.

The modulo operator ( % ) is also uncommon: it returns the remainder of an integer division, i.e. what remains when you divide a number by another. When dividing 11 by 4, the number 4 divides “perfectly” up to the value 8. This means that there’s a remainder of 3 left, which is the value returned by the modulo operator.

Also note that the addition ( + ) and subtraction ( - ) operators are special in that they can operate on a single operand; the expression +5 or -5 is considered an operation in itself. When used in this fashion, these operators are referred to as unary operators . The negative unary operator (as in -5 ) is used to invert the value of a number, while the positive unary operator (as in +5 ) was mostly created for symmetrical reasons, since writing +5 is effectively the same as just writing 5 .

## Python Assignment Operators

Assignment operators are used to assign values to variables . They can also perform arithmetic operations in combination with assignments.

The canonical assignment operator is the equal sign ( = ). Its purpose is to bind a value to a variable: if we write x = 10 , we store the value 10 inside the variable x. We can then later refer to the variable x in order to retrieve its value.

The remaining assignment operators are collectively known as augmented assignment operators . They combine a regular assignment with an arithmetic operator in a single line. This is denoted by the arithmetic operator placed directly before the “vanilla” assignment operator.

Augmented assignment operators are simply used as a shortcut. Instead of writing x = x + 1 , they allow us to write x += 1 , effectively “updating” a variable in a concise manner. Here’s a code sample of how this works:

In the table below, you can find the complete list of assignment operators in Python. Note how there is an augmented assignment operator for every arithmetic operator we went over in the previous section:

## Python Comparison Operators

Comparison operators are used to compare two values . They return a Boolean value ( True or False ) based on the comparison result.

These operators are often used in conjunction with if/else statements in order to control the flow of a program. For example, the code block below allows the user to select an option from a menu:

The table below shows the full list of Python comparison operators:

Note: Pay attention to the “equal to” operator ( == ) – it’s easy to mistake it for the assignment operator ( = ) when writing Python scripts!

If you’re coming from other programming languages, you may have heard about “ternary conditional operators”. Python has a very similar structure called conditional expressions, which you can learn more about in our article on ternary operators in Python . And if you want more details on this topic, be sure to check out our article on Python comparison operators .

## Python Logical Operators

Logical operators are used to combine and manipulate Boolean values . They return True or False based on the Boolean values given to them.

Logical operators are often used to combine different conditions into one. You can leverage the fact that they are written as normal English words to create code that is very readable. Even someone who isn’t familiar with Python could roughly understand what the code below attempts to do:

Here is the table with every logical operator in Python:

Note: When determining if a value falls inside a range of numbers, you can use the “interval notation” commonly used in mathematics; the expression x > 0 and x < 100 can be rewritten as 0 < x < 100 .

## Python Identity Operators

Identity operators are used to query whether two Python objects are the exact same object . This is different from using the “equal to” operator ( == ) because two variables may be equal to each other , but at the same time not be the same object .

For example, the lists list_a and list_b below contain the same values, so for all practical purposes they are considered equal. But they are not the same object – modifying one list does not change the other:

The table below presents the two existing Python identity operators:

## Python Membership Operators

Membership operators are used to test if a value is contained inside another object .

Objects that can contain other values in them are known as collections . Common collections in Python include lists, tuples, and sets.

## Python Bitwise Operators

Bitwise operators in Python are the most esoteric of them all. They are used to perform bit-level operations on integers. Although you may not use these operators as often in your day to day coding, they are a staple in low-level programming languages like C.

As an example, let’s consider the numbers 5 (whose binary representation is 101 ), and the number 3 (represented as 011 in binary).

In order to apply the bitwise AND operator ( & ), we take the first digit from each number’s binary representation (in this case: 1 for the number 5, and 0 for the number 3). Then, we perform the AND operation, which works much like Python’s logical and operator except True is now 1 and False is 0 .

This gives us the operation 1 AND 0 , which results in 0 .

We then simply perform the same operation for each remaining pair of digits in the binary representation of 5 and 3, namely:

- 0 AND 1 = 0
- 1 AND 1 = 1

We end up with 0 , 0 , and 1 , which is then put back together as the binary number 001 – which is, in turn, how the number 1 is represented in binary. This all means that the bitwise operation 5 & 3 results in 1 . What a ride!

The name “bitwise” comes from the idea that these operations are performed on “bits” (the numbers 0 or 1), one pair at a time. Afterwards, they are all brought up together in a resulting binary value.

The table below presents all existing bitwise operations in Python:

## Operator Precedence in Python

Operator precedence determines the order in which operators are evaluated in an expression. Operators with higher precedence are evaluated first.

For example, the fact that the exponentiation operator ( ** ) has a higher precedence than the addition operator ( + ) means that the expression 2 ** 3 + 4 is seen by Python as (2 ** 3) + 4 . The order of operation is exponentiation and then addition. To override operator precedence, you need to explicitly use parentheses to encapsulate a part of the expression, i.e. 2 ** (3 + 4) .

The table below illustrates the operator precedence in Python. Operators in the earlier rows have a higher precedence:

## Want to Learn More About Python Operators?

In this article, we've covered every single Python operator. This includes arithmetic, assignment, comparison, logical, identity, membership, and bitwise operators. Understanding these operators is crucial for writing Python code effectively!

For those looking to dive deeper into Python, consider exploring our Learn Programming with Python track . It consists of five in-depth courses and over 400 exercises for you to master the language. You can also challenge yourself with our article on 10 Python practice exercises for beginners !

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- Our Mission

## Setting Up High-Impact Tasks in Elementary School Math Centers

Allowing students to select math centers based on interest instead of skill level provides opportunities for them all to grow.

In our work as math coaches and consultants, we are often asked to help teachers structure small group practice time. Teachers who are required to implement “what I need” (WIN) groups or small group math time have questions about how to put it into practice so that the wide range of students’ needs are being met. We invite you to consider how prioritizing equity-based principles and providing high-impact tasks can offer a path to differentiating instruction, deepening skills and concepts, and strengthening problem-solving.

## differentiation in math centers

Elementary teachers are often encouraged to use math workshop (also called math center time or math rotations) to differentiate. In this model, teachers create structures for small groups of students to move from one task to another in timed rotations to complete activities that the teacher has assigned and prepared for them. In this model, there is almost always a “teacher table,” where the teacher works closely with small groups of students on “what they need.”

This model can send unintended messages to students about what it means to be a successful, competent learner of mathematics. It can result in a math classroom that is hierarchical and leveled rather than one that supports students with multi-abilities. Students may begin at early ages to feel the impact of being identified and tracked . So what is the alternative?

## designing better Tasks for math centers

Math centers offer ideal opportunities to go deep with the mathematics . The choice of tasks and the ways that teachers interact with students during the workshop impact essential equity standards. We prioritize activities or games with a low floor and high ceiling and that have a high cognitive demand with multiple solution strategies. These games and activities encourage students to make conjectures, to reason through multiple solutions, and to practice important mathematical and problem-solving skills .

One example we often begin with is counting collections . The task focuses on significant mathematics, and yet the directions are uncomplicated. Students choose a collection of items and then figure out how many items are in the collection . The complexity and rigor come from students having to figure out how to count and then how to represent their count. Students develop essential skills such as counting, sorting, grouping, and problem-solving.

Role of the teacher: Instead of having a teacher be stationary at a table where they supervise and lead students through a task, they move around the classroom listening to the students as they engage in the activities. They press on ideas, nudge and notice how students respond and interact. They ask probing questions to help surface mathematical ideas, and they take notes on what they observe and how students respond. Then they use their observations to assess student understanding and inform planning.

Teachers can still gather a small group of students together to bring forward some aspect of their work. In the example of counting collections, we sometimes bring together students to practice ideas related to one-to-one correspondence or extend skills related to number sense.

## Grouping Students for centers

We plan tasks for math centers that allow us to leverage multiple competencies among learners and challenge spaces of marginality. In small groups or partnerships, students with different strengths learn with and from each other as they collaborate on activities. We see the variation in student abilities within a group as benefiting all group members, as it allows for greater richness of ideas and knowledge mobility . Students with varying skills and solution strategies work side-by-side using each other as equal thought partners who are able to engage in the mathematics as sense-makers.

Many math games, like the classic “compare” games , offer the kind of richness that makes them well-suited as tasks that leverage the multiple competencies of our students during math workshop. For instance, when they play “multiplication compare,” the game directions are routine: Draw the number of cards needed, figure out the product, and compare it to the product of your partner’s hand. We use sentence starters to support partner talk for all learners and develop protocols for partner decision-making .

The success of multi-ability groups depends on how the teacher establishes an equitable math learning community and how that community is nurtured and maintained throughout the year. We pair students randomly in order to disrupt any narrative that only certain kinds of learners are capable of engaging in deep mathematics. Random groupings position all learners as competent and capable.

Role of the teacher: Following a class period of math rotations, we debrief with our students , not only about the mathematical content they have been engaged in, but also about aspects of their group work and interactions. We help students acknowledge and describe how a partner’s solution offered something new and productive to consider.

## Affirming Learners’ Math Identities

Because we see math centers as opportunities to position all students as competent, we prioritize student agency and expand access for all . We select games and activities that support students’ independence, interdependence, and decision-making. Choice is an essential part of the math workshop we are advocating for. When we reposition our students as competent independent learners and give them opportunities to make mathematical decisions, students will rise to the challenge in ways that surprise and excite their teachers as they develop more positive identities as math learners.

Role of the teacher: As we interact with students during math workshop, we press on important mathematical ideas and help shape how students view mathematics and how they view their relationship to mathematics. For example, during “counting collections,” we might say, “You have shown one way to count this quantity. Is there another way you and your partner could count this collection and represent it so that other people would know easily how many items there are? Mathematicians often make several attempts at representing their ideas in order to communicate them clearly to others. See what you might come up with for a second attempt.”

Or, after observing students at a table playing “multiplication compare,” we might say, “I noticed that your group had a few ways of figuring out who had the greater product. When we meet at the end of workshop time, it would be helpful for your classmates to hear your ideas. Why don’t you talk together now about how you might present your ideas to the whole group?”

## What’s Next?

We believe it’s time to reconsider some of the assumptions and expectations that educators typically bring to the design and implementation of math workshop time. We suggest prioritizing the development of positive math identities for all students by providing opportunities for access, agency, collaboration, and independence by giving them choice and voice.

We have found that when we rethink the role of the teacher, the kinds of mathematically rich tasks we offer, and the way in which we group students, math workshop can become integral to the creation of equitable math classrooms and be a place for students to develop strong habits of mind alongside math competencies.

## IMAGES

## VIDEO

## COMMENTS

When you write in a math class, you are expected to use correct grammar and spelling. Your writing should be clear and professional. Do not use any irregular abbreviations or shorthand forms which do not conform to standard writing conventions. Mathematics is written with sentences in paragraphs.

Organizing the Report. The formal report is written to summarize an analysis or investigation of problems of interest. The elements that should be included in your report are: This set of instructions is closely based on a similar set distributed to his students by Prof. Keith Howard. Some of the wording is blatantly stolen from his handout.

When trying to solve a problem or do an exercise it is acceptable to break all these rules. What is important is that they are followed when writing up the solution for someone else to read. An example In a geometry course I stated the cosine rule. Cosine rule: Suppose that a triangle has edges of length a, b and cwith the angle opposite aequal ...

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This will ensure that you are able to get the right results with the math you are doing. You should combine this with re-checking your answers to make sure the process that you have followed fits with the correct results. If you are still having trouble understanding your math homework, then you can receive assistance and help from professionals.

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For example, you could break a research paper down into several smaller tasks: 1) do preliminary research, 2) write an outline, 3) draft an introduction, 4) draft body paragraphs, 5) write conclusion, 6) revise. Each of these is much more do-able on its own. Method 2.

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Up next for you: Course challenge Test your knowledge of the skills in this course. Start Course challenge. Square roots review. Community questions. Learn eighth grade math—functions, linear equations, geometric transformations, and more. (aligned with Common Core standards)

The purpose of this video is to give you a quick basic lesson on how to access and complete an assignment. We will explore the Course Assignment panel of the...

Math assignments are designed to provide opportunities for 'doing math' and to consolidate students' understanding of the content. The questions often come from the most recent week of learned material, but some questions may require students to synthesize concepts from further back. This is because learning math is cumulative by nature; you ...

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To create an assignment from a specific content page please follow the below steps: Step 1: browse our library until you find material that you want your students work on. This could be a video, an article, or an exercise. Step 2: Once you are viewing a piece of Khan Academy content, click the Assign button near the top right-hand corner of the ...

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Augmented assignment operators are simply used as a shortcut. Instead of writing x = x + 1, they allow us to write x += 1, effectively "updating" a variable in a concise manner. Here's a code sample of how this works: # Initial assignment. x = 10. print(x) . # output: 10. # Augmented assignment operator: subtraction. x -= 2.

Math centers offer ideal opportunities to go deep with the mathematics. The choice of tasks and the ways that teachers interact with students during the workshop impact essential equity standards. We prioritize activities or games with a low floor and high ceiling and that have a high cognitive demand with multiple solution strategies.