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Everyone struggles with homework sometimes, but if getting your homework done has become a chronic issue for you, then you may need a little extra help. That’s why we’ve written this article all about how to do homework. Once you’re finished reading it, you’ll know how to do homework (and have tons of new ways to motivate yourself to do homework)!

We’ve broken this article down into a few major sections. You’ll find:

• A diagnostic test to help you figure out why you’re struggling with homework
• A discussion of the four major homework problems students face, along with expert tips for addressing them
• A bonus section with tips for how to do homework fast

By the end of this article, you’ll be prepared to tackle whatever homework assignments your teachers throw at you .

So let’s get started!

How to Do Homework: Figure Out Your Struggles 

Sometimes it feels like everything is standing between you and getting your homework done. But the truth is, most people only have one or two major roadblocks that are keeping them from getting their homework done well and on time. 

The best way to figure out how to get motivated to do homework starts with pinpointing the issues that are affecting your ability to get your assignments done. That’s why we’ve developed a short quiz to help you identify the areas where you’re struggling. 

Take the quiz below and record your answers on your phone or on a scrap piece of paper. Keep in mind there are no wrong answers! 

1. You’ve just been assigned an essay in your English class that’s due at the end of the week. What’s the first thing you do?

A. Keep it in mind, even though you won’t start it until the day before it’s due  B. Open up your planner. You’ve got to figure out when you’ll write your paper since you have band practice, a speech tournament, and your little sister’s dance recital this week, too.  C. Groan out loud. Another essay? You could barely get yourself to write the last one!  D. Start thinking about your essay topic, which makes you think about your art project that’s due the same day, which reminds you that your favorite artist might have just posted to Instagram...so you better check your feed right now. 

2. Your mom asked you to pick up your room before she gets home from work. You’ve just gotten home from school. You decide you’ll tackle your chores: 

A. Five minutes before your mom walks through the front door. As long as it gets done, who cares when you start?  B. As soon as you get home from your shift at the local grocery store.  C. After you give yourself a 15-minute pep talk about how you need to get to work.  D. You won’t get it done. Between texts from your friends, trying to watch your favorite Netflix show, and playing with your dog, you just lost track of time! 

3. You’ve signed up to wash dogs at the Humane Society to help earn money for your senior class trip. You: 

A. Show up ten minutes late. You put off leaving your house until the last minute, then got stuck in unexpected traffic on the way to the shelter.  B. Have to call and cancel at the last minute. You forgot you’d already agreed to babysit your cousin and bake cupcakes for tomorrow’s bake sale.  C. Actually arrive fifteen minutes early with extra brushes and bandanas you picked up at the store. You’re passionate about animals, so you’re excited to help out! D. Show up on time, but only get three dogs washed. You couldn’t help it: you just kept getting distracted by how cute they were!

4. You have an hour of downtime, so you decide you’re going to watch an episode of The Great British Baking Show. You: 

A. Scroll through your social media feeds for twenty minutes before hitting play, which means you’re not able to finish the whole episode. Ugh! You really wanted to see who was sent home!  B. Watch fifteen minutes until you remember you’re supposed to pick up your sister from band practice before heading to your part-time job. No GBBO for you!  C. You finish one episode, then decide to watch another even though you’ve got SAT studying to do. It’s just more fun to watch people make scones.  D. Start the episode, but only catch bits and pieces of it because you’re reading Twitter, cleaning out your backpack, and eating a snack at the same time.

5. Your teacher asks you to stay after class because you’ve missed turning in two homework assignments in a row. When she asks you what’s wrong, you say: 

A. You planned to do your assignments during lunch, but you ran out of time. You decided it would be better to turn in nothing at all than submit unfinished work.  B. You really wanted to get the assignments done, but between your extracurriculars, family commitments, and your part-time job, your homework fell through the cracks.  C. You have a hard time psyching yourself to tackle the assignments. You just can’t seem to find the motivation to work on them once you get home.  D. You tried to do them, but you had a hard time focusing. By the time you realized you hadn’t gotten anything done, it was already time to turn them in. 

Like we said earlier, there are no right or wrong answers to this quiz (though your results will be better if you answered as honestly as possible). Here’s how your answers break down: 

• If your answers were mostly As, then your biggest struggle with doing homework is procrastination. 
• If your answers were mostly Bs, then your biggest struggle with doing homework is time management. 
• If your answers were mostly Cs, then your biggest struggle with doing homework is motivation. 
• If your answers were mostly Ds, then your biggest struggle with doing homework is getting distracted. 

Now that you’ve identified why you’re having a hard time getting your homework done, we can help you figure out how to fix it! Scroll down to find your core problem area to learn more about how you can start to address it. 

And one more thing: you’re really struggling with homework, it’s a good idea to read through every section below. You may find some additional tips that will help make homework less intimidating. 

How to Do Homework When You’re a Procrastinator  

Merriam Webster defines “procrastinate” as “to put off intentionally and habitually.” In other words, procrastination is when you choose to do something at the last minute on a regular basis. If you’ve ever found yourself pulling an all-nighter, trying to finish an assignment between periods, or sprinting to turn in a paper minutes before a deadline, you’ve experienced the effects of procrastination. 

If you’re a chronic procrastinator, you’re in good company. In fact, one study found that 70% to 95% of undergraduate students procrastinate when it comes to doing their homework. Unfortunately, procrastination can negatively impact your grades. Researchers have found that procrastination can lower your grade on an assignment by as much as five points ...which might not sound serious until you realize that can mean the difference between a B- and a C+. 

Procrastination can also negatively affect your health by increasing your stress levels , which can lead to other health conditions like insomnia, a weakened immune system, and even heart conditions. Getting a handle on procrastination can not only improve your grades, it can make you feel better, too! 

The big thing to understand about procrastination is that it’s not the result of laziness. Laziness is defined as being “disinclined to activity or exertion.” In other words, being lazy is all about doing nothing. But a s this Psychology Today article explains , procrastinators don’t put things off because they don’t want to work. Instead, procrastinators tend to postpone tasks they don’t want to do in favor of tasks that they perceive as either more important or more fun. Put another way, procrastinators want to do things...as long as it’s not their homework! 

3 Tips f or Conquering Procrastination 

Because putting off doing homework is a common problem, there are lots of good tactics for addressing procrastination. Keep reading for our three expert tips that will get your homework habits back on track in no time. 

#1: Create a Reward System

Like we mentioned earlier, procrastination happens when you prioritize other activities over getting your homework done. Many times, this happens because homework...well, just isn’t enjoyable. But you can add some fun back into the process by rewarding yourself for getting your work done. 

Here’s what we mean: let’s say you decide that every time you get your homework done before the day it’s due, you’ll give yourself a point. For every five points you earn, you’ll treat yourself to your favorite dessert: a chocolate cupcake! Now you have an extra (delicious!) incentive to motivate you to leave procrastination in the dust. 

If you’re not into cupcakes, don’t worry. Your reward can be anything that motivates you . Maybe it’s hanging out with your best friend or an extra ten minutes of video game time. As long as you’re choosing something that makes homework worth doing, you’ll be successful. 

#2: Have a Homework Accountability Partner 

If you’re having trouble getting yourself to start your homework ahead of time, it may be a good idea to call in reinforcements . Find a friend or classmate you can trust and explain to them that you’re trying to change your homework habits. Ask them if they’d be willing to text you to make sure you’re doing your homework and check in with you once a week to see if you’re meeting your anti-procrastination goals. 

Sharing your goals can make them feel more real, and an accountability partner can help hold you responsible for your decisions. For example, let’s say you’re tempted to put off your science lab write-up until the morning before it’s due. But you know that your accountability partner is going to text you about it tomorrow...and you don’t want to fess up that you haven’t started your assignment. A homework accountability partner can give you the extra support and incentive you need to keep your homework habits on track. 

#3: Create Your Own Due Dates 

If you’re a life-long procrastinator, you might find that changing the habit is harder than you expected. In that case, you might try using procrastination to your advantage! If you just can’t seem to stop doing your work at the last minute, try setting your own due dates for assignments that range from a day to a week before the assignment is actually due. 

Here’s what we mean. Let’s say you have a math worksheet that’s been assigned on Tuesday and is due on Friday. In your planner, you can write down the due date as Thursday instead. You may still put off your homework assignment until the last minute...but in this case, the “last minute” is a day before the assignment’s real due date . This little hack can trick your procrastination-addicted brain into planning ahead! 

If you feel like Kevin Hart in this meme, then our tips for doing homework when you're busy are for you. 

How to Do Homework When You’re too Busy

If you’re aiming to go to a top-tier college , you’re going to have a full plate. Because college admissions is getting more competitive, it’s important that you’re maintaining your grades , studying hard for your standardized tests , and participating in extracurriculars so your application stands out. A packed schedule can get even more hectic once you add family obligations or a part-time job to the mix. 

If you feel like you’re being pulled in a million directions at once, you’re not alone. Recent research has found that stress—and more severe stress-related conditions like anxiety and depression— are a major problem for high school students . In fact, one study from the American Psychological Association found that during the school year, students’ stress levels are higher than those of the adults around them. 

For students, homework is a major contributor to their overall stress levels . Many high schoolers have multiple hours of homework every night , and figuring out how to fit it into an already-packed schedule can seem impossible. 

3 Tips for Fitting Homework Into Your Busy Schedule

While it might feel like you have literally no time left in your schedule, there are still ways to make sure you’re able to get your homework done and meet your other commitments. Here are our expert homework tips for even the busiest of students. 

#1: Make a Prioritized To-Do List 

You probably already have a to-do list to keep yourself on track. The next step is to prioritize the items on your to-do list so you can see what items need your attention right away. 

Here’s how it works: at the beginning of each day, sit down and make a list of all the items you need to get done before you go to bed. This includes your homework, but it should also take into account any practices, chores, events, or job shifts you may have. Once you get everything listed out, it’s time to prioritize them using the labels A, B, and C. Here’s what those labels mean:

• A Tasks : tasks that have to get done—like showing up at work or turning in an assignment—get an A. 
• B Tasks : these are tasks that you would like to get done by the end of the day but aren’t as time sensitive. For example, studying for a test you have next week could be a B-level task. It’s still important, but it doesn’t have to be done right away.
• C Tasks: these are tasks that aren’t very important and/or have no real consequences if you don’t get them done immediately. For instance, if you’re hoping to clean out your closet but it’s not an assigned chore from your parents, you could label that to-do item with a C.

Prioritizing your to-do list helps you visualize which items need your immediate attention, and which items you can leave for later. A prioritized to-do list ensures that you’re spending your time efficiently and effectively, which helps you make room in your schedule for homework. So even though you might really want to start making decorations for Homecoming (a B task), you’ll know that finishing your reading log (an A task) is more important. 

#2: Use a Planner With Time Labels

Your planner is probably packed with notes, events, and assignments already. (And if you’re not using a planner, it’s time to start!) But planners can do more for you than just remind you when an assignment is due. If you’re using a planner with time labels, it can help you visualize how you need to spend your day.

A planner with time labels breaks your day down into chunks, and you assign tasks to each chunk of time. For example, you can make a note of your class schedule with assignments, block out time to study, and make sure you know when you need to be at practice. Once you know which tasks take priority, you can add them to any empty spaces in your day. 

Planning out how you spend your time not only helps you use it wisely, it can help you feel less overwhelmed, too . We’re big fans of planners that include a task list ( like this one ) or have room for notes ( like this one ). 

#3: Set Reminders on Your Phone 

If you need a little extra nudge to make sure you’re getting your homework done on time, it’s a good idea to set some reminders on your phone. You don’t need a fancy app, either. You can use your alarm app to have it go off at specific times throughout the day to remind you to do your homework. This works especially well if you have a set homework time scheduled. So if you’ve decided you’re doing homework at 6:00 pm, you can set an alarm to remind you to bust out your books and get to work. 

If you use your phone as your planner, you may have the option to add alerts, emails, or notifications to scheduled events . Many calendar apps, including the one that comes with your phone, have built-in reminders that you can customize to meet your needs. So if you block off time to do your homework from 4:30 to 6:00 pm, you can set a reminder that will pop up on your phone when it’s time to get started. 

This dog isn't judging your lack of motivation...but your teacher might. Keep reading for tips to help you motivate yourself to do your homework.

How to Do Homework When You’re Unmotivated 

At first glance, it may seem like procrastination and being unmotivated are the same thing. After all, both of these issues usually result in you putting off your homework until the very last minute. 

But there’s one key difference: many procrastinators are working, they’re just prioritizing work differently. They know they’re going to start their homework...they’re just going to do it later. 

Conversely, people who are unmotivated to do homework just can’t find the willpower to tackle their assignments. Procrastinators know they’ll at least attempt the homework at the last minute, whereas people who are unmotivated struggle with convincing themselves to do it at a ll. For procrastinators, the stress comes from the inevitable time crunch. For unmotivated people, the stress comes from trying to convince themselves to do something they don’t want to do in the first place. 

Here are some common reasons students are unmotivated in doing homework : 

• Assignments are too easy, too hard, or seemingly pointless 
• Students aren’t interested in (or passionate about) the subject matter
• Students are intimidated by the work and/or feels like they don’t understand the assignment 
• Homework isn’t fun, and students would rather spend their time on things that they enjoy 

To sum it up: people who lack motivation to do their homework are more likely to not do it at all, or to spend more time worrying about doing their homework than...well, actually doing it.

3 Tips for How to Get Motivated to Do Homework

The key to getting homework done when you’re unmotivated is to figure out what does motivate you, then apply those things to homework. It sounds tricky...but it’s pretty simple once you get the hang of it! Here are our three expert tips for motivating yourself to do your homework. 

#1: Use Incremental Incentives

When you’re not motivated, it’s important to give yourself small rewards to stay focused on finishing the task at hand. The trick is to keep the incentives small and to reward yourself often. For example, maybe you’re reading a good book in your free time. For every ten minutes you spend on your homework, you get to read five pages of your book. Like we mentioned earlier, make sure you’re choosing a reward that works for you! 

So why does this technique work? Using small rewards more often allows you to experience small wins for getting your work done. Every time you make it to one of your tiny reward points, you get to celebrate your success, which gives your brain a boost of dopamine . Dopamine helps you stay motivated and also creates a feeling of satisfaction when you complete your homework !  

#2: Form a Homework Group 

If you’re having trouble motivating yourself, it’s okay to turn to others for support. Creating a homework group can help with this. Bring together a group of your friends or classmates, and pick one time a week where you meet and work on homework together. You don’t have to be in the same class, or even taking the same subjects— the goal is to encourage one another to start (and finish!) your assignments. 

Another added benefit of a homework group is that you can help one another if you’re struggling to understand the material covered in your classes. This is especially helpful if your lack of motivation comes from being intimidated by your assignments. Asking your friends for help may feel less scary than talking to your teacher...and once you get a handle on the material, your homework may become less frightening, too. 

#3: Change Up Your Environment 

If you find that you’re totally unmotivated, it may help if you find a new place to do your homework. For example, if you’ve been struggling to get your homework done at home, try spending an extra hour in the library after school instead. The change of scenery can limit your distractions and give you the energy you need to get your work done. 

If you’re stuck doing homework at home, you can still use this tip. For instance, maybe you’ve always done your homework sitting on your bed. Try relocating somewhere else, like your kitchen table, for a few weeks. You may find that setting up a new “homework spot” in your house gives you a motivational lift and helps you get your work done. 

Social media can be a huge problem when it comes to doing homework. We have advice for helping you unplug and regain focus.

How to Do Homework When You’re Easily Distracted

We live in an always-on world, and there are tons of things clamoring for our attention. From friends and family to pop culture and social media, it seems like there’s always something (or someone!) distracting us from the things we need to do.

The 24/7 world we live in has affected our ability to focus on tasks for prolonged periods of time. Research has shown that over the past decade, an average person’s attention span has gone from 12 seconds to eight seconds . And when we do lose focus, i t takes people a long time to get back on task . One study found that it can take as long as 23 minutes to get back to work once we’ve been distracte d. No wonder it can take hours to get your homework done! 

3 Tips to Improve Your Focus

If you have a hard time focusing when you’re doing your homework, it’s a good idea to try and eliminate as many distractions as possible. Here are three expert tips for blocking out the noise so you can focus on getting your homework done. 

#1: Create a Distraction-Free Environment

Pick a place where you’ll do your homework every day, and make it as distraction-free as possible. Try to find a location where there won’t be tons of noise, and limit your access to screens while you’re doing your homework. Put together a focus-oriented playlist (or choose one on your favorite streaming service), and put your headphones on while you work. 

You may find that other people, like your friends and family, are your biggest distraction. If that’s the case, try setting up some homework boundaries. Let them know when you’ll be working on homework every day, and ask them if they’ll help you keep a quiet environment. They’ll be happy to lend a hand! 

#2: Limit Your Access to Technology 

We know, we know...this tip isn’t fun, but it does work. For homework that doesn’t require a computer, like handouts or worksheets, it’s best to put all your technology away . Turn off your television, put your phone and laptop in your backpack, and silence notifications on any wearable tech you may be sporting. If you listen to music while you work, that’s fine...but make sure you have a playlist set up so you’re not shuffling through songs once you get started on your homework. 

If your homework requires your laptop or tablet, it can be harder to limit your access to distractions. But it’s not impossible! T here are apps you can download that will block certain websites while you’re working so that you’re not tempted to scroll through Twitter or check your Facebook feed. Silence notifications and text messages on your computer, and don’t open your email account unless you absolutely have to. And if you don’t need access to the internet to complete your assignments, turn off your WiFi. Cutting out the online chatter is a great way to make sure you’re getting your homework done. 

#3: Set a Timer (the Pomodoro Technique)

Have you ever heard of the Pomodoro technique ? It’s a productivity hack that uses a timer to help you focus!

Here’s how it works: first, set a timer for 25 minutes. This is going to be your work time. During this 25 minutes, all you can do is work on whatever homework assignment you have in front of you. No email, no text messaging, no phone calls—just homework. When that timer goes off, you get to take a 5 minute break. Every time you go through one of these cycles, it’s called a “pomodoro.” For every four pomodoros you complete, you can take a longer break of 15 to 30 minutes.

The pomodoro technique works through a combination of boundary setting and rewards. First, it gives you a finite amount of time to focus, so you know that you only have to work really hard for 25 minutes. Once you’ve done that, you’re rewarded with a short break where you can do whatever you want. Additionally, tracking how many pomodoros you complete can help you see how long you’re really working on your homework. (Once you start using our focus tips, you may find it doesn’t take as long as you thought!)

Two Bonus Tips for How to Do Homework Fast

Even if you’re doing everything right, there will be times when you just need to get your homework done as fast as possible. (Why do teachers always have projects due in the same week? The world may never know.)

The problem with speeding through homework is that it’s easy to make mistakes. While turning in an assignment is always better than not submitting anything at all, you want to make sure that you’re not compromising quality for speed. Simply put, the goal is to get your homework done quickly and still make a good grade on the assignment! 

Here are our two bonus tips for getting a decent grade on your homework assignments , even when you’re in a time crunch. 

#1: Do the Easy Parts First 

This is especially true if you’re working on a handout with multiple questions. Before you start working on the assignment, read through all the questions and problems. As you do, make a mark beside the questions you think are “easy” to answer . 

Once you’ve finished going through the whole assignment, you can answer these questions first. Getting the easy questions out of the way as quickly as possible lets you spend more time on the trickier portions of your homework, which will maximize your assignment grade. 

(Quick note: this is also a good strategy to use on timed assignments and tests, like the SAT and the ACT !) 

#2: Pay Attention in Class 

Homework gets a lot easier when you’re actively learning the material. Teachers aren’t giving you homework because they’re mean or trying to ruin your weekend... it’s because they want you to really understand the course material. Homework is designed to reinforce what you’re already learning in class so you’ll be ready to tackle harder concepts later.

When you pay attention in class, ask questions, and take good notes, you’re absorbing the information you’ll need to succeed on your homework assignments. (You’re stuck in class anyway, so you might as well make the most of it!) Not only will paying attention in class make your homework less confusing, it will also help it go much faster, too.

What’s Next?

If you’re looking to improve your productivity beyond homework, a good place to begin is with time management. After all, we only have so much time in a day...so it’s important to get the most out of it! To get you started, check out this list of the 12 best time management techniques that you can start using today.

You may have read this article because homework struggles have been affecting your GPA. Now that you’re on the path to homework success, it’s time to start being proactive about raising your grades. This article teaches you everything you need to know about raising your GPA so you can

Now you know how to get motivated to do homework...but what about your study habits? Studying is just as critical to getting good grades, and ultimately getting into a good college . We can teach you how to study bette r in high school. (We’ve also got tons of resources to help you study for your ACT and SAT exams , too!)

These recommendations are based solely on our knowledge and experience. If you purchase an item through one of our links, PrepScholar may receive a commission.

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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Articles & Advice > Student Life > Blog

Top Tips for Juggling Schoolwork, Exercise, and Activities

Balancing homework, extracurriculars, and a social calendar can leave little time for fitness. Check out these tips from real students on how to make it work!

by CollegeXpress

Last Updated: Mar 16, 2023

Originally Posted: Dec 3, 2014

As we sat here at the CollegeXpress headquarters, perfectly sedentary, munching on leftover Halloween candy (yes, still!), and wondering why we can barely make it up a flight of stairs, we got to thinking about health and fitness. Specifically, we wanted to know what our users think about diet and exercise and how they maintain a healthy lifestyle along with all the other stuff they have on their plates. That's why we teamed up with O2MAX Fitness to survey our readers and get some answers.

Seven thousand students turned out for the survey—thanks again, all 6,977 of you!—and we learned a ton from their responses. We got so much great information, so we’re sharing some of our favorite responses to the question “ How do you balance your fitness and other activities with your schoolwork?”  If you’re struggling with your schedule, first, know you’re not alone, and second, keep reading for some great tips and inspiration from high school and college students just like you. 

On working out and sports

• “Being involved in sports actually helps me a lot because after practice, I know I have less time to procrastinate and have to get started on my school work.”
• “Sports have always been a part of my life. It's as routine as brushing my teeth. Including fitness in my routine is not an option; it's a necessity.”
• “I find that physical activity actually helps me focus. After coming home from practice or a workout, I know I need to focus on school instead of filling my free time with things that will suck me in and distract me.”
• “I use my sports activity to clear my head and to focus on something physical while still stimulating my mind. Right when I get home from my practice I hit the books.”
• “I do my workouts just before I shower each night so I am more tired before I go to bed.”
• “I make at least a 30 minute block of time to workout because I know it's great for my body. Also, after I work out I focus on school a lot better.”
• “I try to eat healthy, and I walk around my campus between classes. I take walks outside whenever I have time.”
• “I bike to school and home incorporating fitness into my everyday life.”
• “I try to find ways to multitask, such as running from building to building on campus rather than slowly walking.”
• “I either do it right when I wake up because I like to get it out of the way, but sometimes when I had a late night studying or if I have early morning classes, I'll do it right when I get back to my room. However, if some days I feel way too tired I know my body needs a break and I'll take a nice stretching or yoga day or a rest day.”
• “If I have minutes at the end of the day then I run for 20 minutes; it just depends on how much time I have. If I have say an hour, I might lift weights.”
• “I watch TV and exercise at the same time.”
• “I try to stretch and do yoga every night before bed and I make sure I have time to play tennis at least once a week by getting my schoolwork done ahead of time to accommodate for fitness time.”
• “Its super hard, but I try to do some easy workouts in my dorm, whether it be push ups or crunches when I first get home from class, and then when I finish my homework days later I try to make it to the gym”

On completing schoolwork

• “It is very hard. I get so much homework that there is almost no time to work out.”
• “School comes first. I'm a student before an athlete and my coach understands that. Mostly I go to school. Go to practice. Then come home and do my homework until late. Then get up and finish it before school. This process is then repeated for every school day after that.”
• “Schoolwork comes first. However, I will do some exercising between assignments so that I stay awake and mentally active.”
• “After I finish practice I go straight home and do my homework. No TV, phone, computer, or any distractions.”
• “I make time for what's important and recognize when something must take precedence. My schoolwork comes first, as it should, but I know that I need to take breaks and exercise is a good stress relief.”
•  “Schoolwork takes priority. I'm here primarily to learn and, on top of that, I have a scholarship that I need to maintain. But fitness is important, nonetheless. If I can't make it to the gym or don't want to, I'll take a walk or work out in my dorm room.”
• “School always comes first, but I always go to practice, never miss it. I put equal effort, time, and dedication into both.”
• “I have a study hall at school three times a week so that helps. Also, I do homework on the bus rides on the way to my games. I try to concentrate on schoolwork, though.”

On using your time wisely

• “First, I do anything I have to do for school; after I'm done with that, I do my fitness routine until the sun sets.”
• “I make sure to use my weekends wisely and plan my days out so I'm able to fit practices and a certain amount of homework in.”
• “I have really good time management skills. I work on homework any chance I get.”
• “I try to limit myself in certain things, so I have an everyday schedule that helps me have a balanced routine everyday.”
• “I do as much work during class as I can and when I'm not practicing something during a rehearsal I do homework. Then I usually stay up later to finish it.”
• “Make a schedule and stick to it. It's hard at first, but after awhile it becomes routine. I use my weekend for relaxation or to study for a challenging test.”
• “Sometimes I slack on the sleep, but I make both my sport and my schoolwork a priority, so I try very hard not to miss practice, and I never miss an assignment. I'll do almost whatever it takes to get both done, but schoolwork will always be the higher priority.”
• “I balance my activities by practicing before school and completing my school work in the afternoon after school. I do not get more than 6 hours of sleep per night, however.”
• “Plan, plan, and plan.”
• “Try and get as much done in school, so that I don't have much for homework out side of school.”
• “I exercise while I study. If I am reading for school, I will read a passage, then think about it while exercising and then return to reading or studying.”
• “I have a study hall and do as much as possible during class, as well as carefully manage my time.”
• “I schedule what I'll be doing at what time the night before. I find that once I go to the gym to workout my mind is a lot more clear and there's a sense of accomplishment that motivates me to go on with my other activities for the day.”
• “I do pretty well with time management, but the key is getting schoolwork done early whenever possible because you don't know what the next days entail.”
• “I stay up late to finish homework or do almost all projects on the weekends because I don't have time for it all during the week. I do manage my time very well though.”
• “Don't waste any time, hardly at all, ever. Stay focused on my goals and be a little selfish every once in a while. Remind myself that I wanted to be busy and I'm living happily and just how lucky I am to be living.”

Looking for more advice on staying healthy and organized at school? Browse tons of great articles in our Student Life section !

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Time Management Exercise

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Do you find yourself rushing to complete your homework assignment at the last moment? Are you always starting your homework when you're supposed to be going to bed? The root of this common problem may be time management.

This easy exercise will help you identify the tasks or habits that take time away from your studies and help you develop more healthy homework habits.

Keeping Track of Your Time

The first goal of this exercise is to get you to think about how you spend your time . For instance, how much time do you think you spend on the phone per week? The truth may surprise you.

First, make a list of common time-consuming activities:

• Talking on the phone
• Listening to music
• Watching TV
• Playing games/surfing web
• Spending time with family

Next, jot down an estimated time for each one. Record the amount of time that you think you devote to each of these activities per day or week.

Make a Chart

Using your list of activities, create a chart with five columns.

Keep this chart on hand at all times for five days and keep track of all the time you spend on each activity. This will be tough sometimes since you probably spend a lot of time going rapidly from one activity to the other or doing two at once.

For example, you may watch TV and eat at the same time. Just record the activity as one or the other. This is an exercise, not a punishment or a science project. Don't pressure yourself!

Once you have tracked your time for a week or so, take a look at your chart. How do your actual times compare with your estimates?

If you are like most people, you may be shocked to see how much time you spend doing things that are unproductive.

Does homework time come in last place? If so, you're normal. In fact, there are many things that ​ should take more time than homework, like family time. But surely there are some problem areas that you can identify as well. Are you spending four hours a night watching TV or playing video games?

You certainly deserve your leisure time. But to have a healthy, productive life, you should have a good balance among family time, homework time, and leisure time.

Set New Goals

When tracking your time, you may find that you spend some time on things you just can't classify. Whether we're sitting on the bus staring out the window, waiting in line for a ticket, or sitting at the kitchen table gazing off in the distance, we all spend time doing, well—nothing.

Look over your activity chart and determine areas you could target for improvement. Then, start the process over again with a new list.

Make new time estimates for each task or activity. Set goals for yourself, allowing more time for homework and less time on one of your weaknesses, like TV or games.

You will soon see that the mere act of thinking about how you spend your time will bring about a change in your habits.

Suggestions for Success

• Don't work alone. Some of us need support to stick to something. A little competition with a friend always makes things more interesting. Work with a friend, compare notes, lists, and charts. Make a game of it!
• Include your parent. Get your mom or dad involved and have them keep track of the time they waste. Now that might be interesting!
• Negotiate a reward system . Whether you work with a friend or a parent, work out a system for rewarding yourself for progress. If working with a friend, you could agree to provide lunch or dinner for the time-saving winner each week.If working with a parent, you could negotiate an extended curfew for every increased minute devoted toward homework. Perhaps you could even substitute dollars for minutes. The possibilities are endless!
• Have a party for reaching a goal. Even if you're working on your own, you could promise yourself a party as a reward for reaching a specific goal.
• Make it a class project. This would be a great project for an entire class. The teacher or group leader could keep track of progress with a flow chart. When the class reaches a goal as a group—it's party time!
• How Much Time Should I Spend Studying in College?
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• Top 10 Healthy Homework Habits
• 20 Tips for Success in High School
• 8 Benefits of Time Management
• How to Find Time to Exercise in College
• 7 Time Management Tips for Finals Week
• 4 Tips for Completing Your Homework On Time
• Spring Break Guide for College Students
• How to Reduce Your Financial Stress in College
• 11 Genius Productivity Tips You Haven't Tried
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Exercises: Calculus (OpenStax)

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These are homework exercises to accompany OpenStax's "Calculus" Textmap.

• 1.E: Functions and Graphs (Exercises) These are homework exercises to accompany Chapter 1 of OpenStax's "Calculus" Textmap.
• 2.E: Limits (Exercises) These are homework exercises to accompany Chapter 2 of OpenStax's "Calculus" Textmap.
• 3.E: Derivatives (Exercises) These are homework exercises to accompany Chapter 3 of OpenStax's "Calculus" Textmap.
• 4.E: Applications of Derivatives (Exercises) These are homework exercises to accompany Chapter 4 of OpenStax's "Calculus" Textmap.
• 5.E: Integration (Exercises) These are homework exercises to accompany Chapter 5 of OpenStax's "Calculus" Textmap.
• 6.E: Applications of Integration (Exercises) These are homework exercises to accompany Chapter 6 of OpenStax's "Calculus" Textmap.
• 7.E: Techniques of Integration (Exercises) These are homework exercises to accompany Chapter 7 of OpenStax's "Calculus" Textmap.
• 8.E: Differential Equations (Exercises) These are homework exercises to accompany Chapter 8 of OpenStax's "Calculus" Textmap.
• 9.E: Sequences and Series (Exercises) These are homework exercises to accompany Chapter 9 of OpenStax's "Calculus" Textmap.
• 10.E: Power Series (Exercises) These are homework exercises to accompany Chapter 10 of OpenStax's "Calculus" Textmap.
• 11.E: Parametric Equations and Polar Coordinates (Exercises) These are homework exercises to accompany Chapter 11 of OpenStax's "Calculus" Textmap.
• 12.E: Vectors in Space (Exercises) These are homework exercises to accompany Chapter 12 of OpenStax's "Calculus" Textmap.
• 13.E: Vector-Valued Functions (Exercises) These are homework exercises to accompany Chapter 13 of OpenStax's "Calculus" Textmap.
• 14.E: Differentiation of Functions of Several Variables (Exercise) These are homework exercises to accompany Chapter 14 of OpenStax's "Calculus" Textmap.
• 15.E: Multiple Integration (Exercises) These are homework exercises to accompany Chapter 15 of OpenStax's "Calculus" Textmap.
• 16.E: Vector Calculus (Exercises) These are homework exercises to accompany Chapter 16 of OpenStax's "Calculus" Textmap.
• 17.E: Second-Order Differential Equations (Exercises) These are homework exercises to accompany Chapter 17 of OpenStax's "Calculus" Textmap.

Thumbnail: The logarithmic spiral of the Nautilus shell is a classical image used to depict the growth and change related to calculus. (GNU Free Documentation License, Version 1.3 and CC- SA-BY 3.0; Wikipedia).

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20+ creative alternative homework ideas for teachers

When giving homework, it must always be based on learning goals your students have to reach, just like in your lessons. But it’s sad to see that lots of teachers are using homework as extra lesson time. Of course, as a teacher, you’re on a clock. But that doesn’t mean your students have to suffer from it and keep working on those boring textbooks and worksheets at home.

Consider goals like attitudes, real-life experiences, and practice, physical exercise, social encounters, creative solutions, and philanthropy as crucial as your lesson goals. These are things students don’t just pick up in your classroom. These are things they pick up in life.

In this blog post, I’ll give you some innovative homework ideas that will engage your students more. These alternatives to traditional homework will thereby also teach your students new things that can’t be taught in the classroom. You will find a variety of homework ideas: online and offline.

I will mention homework alternatives for primary school and high school. Some of these ideas can be changed a little bit, so they are the perfect fit for the right audience.

20 Creative homework ideas

You can divide homework tasks into the following themes or categories:

• Crafts & arts
• Outdoor activities & outings
• Games and activities
• Physical activities
• Digital or computer activities
• Philanthropy & social work

💡 Good to know : all the ready-to-use homework activities are created with BookWidgets . You can easily create activities like these yourself or duplicate an activity below for free, edit it if needed, and share it with your students. You can do so in the examples separately, or you can find all the homework examples in the BookWidgets Blog group folder .

Crafts and arts homework

1. prepare a dish from a recipe book.

2. Make a board game

3. Create a birdhouse

4. Transform a fictional book character into a hand puppet

Outdoor homework activities and outings

5. coupon game.

Students can also go grocery shopping with their parents. Here, they have to read the ingredients of the products and help their parents choose the healthiest products for the best prices, figure out the best deal between the sizes of items, …

6. Visit the zoo

7. Visit the local dumping ground or container park

8. Build a tree house

Games and activities as homework

9. bookwidgets games.

10. Minecraft

11. Play Cards

12. Play Zoo Tycoon or Rollercoaster Tycoon

Physical homework activities

13. rope skipping.

Many rope-skipping songs let your students do different tricks while rope-skipping. This is an excellent opportunity for homework as well. Ask your students to transform a rope skipping song into a song with lesson content. Let them count or spell or even sum up the different states or capitals. To engage their lifestyles even harder, you can additionally give them the assignment to create a TikTok in which they are jumping and singing.

Click here to see how you can get Tiktok more involved in the classroom.

14. Walking quest

If there aren’t any walking quests in the neighborhood, you could ask your students to create a walking quest like this for their fellow students. What a fun day it will be!

15. Obstacle Quiz

In order for students to answer the questions, they have to run and pass a challenging parkour. This is a fun homework exercise, and in the end, it’s a great lesson starter or lesson end.

16. Swimming games

After the activity, they can fill out an Exit Slip:

Digital or computer homework activities

17. create a picture album.

This teaches them to handle the online software, add pictures and write without spelling mistakes. And of course, creating memories is so much fun!

18. Video job application

19. Your life in 10 minutes - video

20. Email pen-pals

Is it still too complicated? Read the messages from your students, before they send them, and provide them with some feedback.

Philanthropy and social homework

21. grow a community garden.

22. Help in a retirement home

23. Help at a homeless shelter

24. Collect litter

Here’s another homework tip: Don’t call homework “homework”. Call it a challenge. Homework has become a negative word for students, and I bet they start rolling their eyes as you even mention the word.

Still looking for more inspiration? Check out the blog on short films and lesson activities that spice up your Google Classroom . Tip: even if you don’t use Google Classroom, there is a lot of inspiration back here.

Above you have read single assignments. But, you also have the option to involve your homework in a project. Find out more here .

So, as I mentioned earlier, there are many fun alternatives to traditional homework. Now it’s up to you to apply this in the classroom as well. In this folder , you will find all the examples you have come across.

Which idea do you or perhaps your students like the most? Let us know on Twitter . Of course, there are many more alternatives. If you have other ideas, you are always welcome to share it with other teachers in our Facebook group .

One more thing: don’t forget to say hi👋 on LikedIn .

Join hundreds of thousands of subscribers, and get the best content on technology in education.

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Lesson 1: Lines, Ellipses and Boxes

Lines: homework.

Hate to read? No problem. We can read the text to you .

This feature is available completely free . While we will be keeping an eye on how feasible it is to eat the costs associated, we hope to keep it that way.

Just click "Listen with Drawabox"!

Don't forget!

Do not grind . Complete the recommended amount of each exercise to the best of your ability and move on. You'll have ample time to keep practicing and improving your skills when doing these exercises as warmups later.

Do not rush . Give yourself the time you need to plan your lines out, to prepare appropriately, and to demonstrate your current best. Every mark is independently important. Once the ghosting method is introduced in the Ghosted Lines exercise, use it consistently for all your structural marks .

Read all of the instructions carefully , multiple times if necessary. While I've tried to organize this content in as digestible a manner as I can, there is a lot of information here and there is no way you'd absorb all of it at once.

Take breaks! One day you're going to be a beast who can draw for hours without breaking a sweat, but right now, you need to be patient with yourself and take it all one step at a time.

Homework and exercises

The homework assignment for this section is as follows:

2 filled pages of the Superimposed Lines exercise

1 filled page of the Ghosted Lines exercise

2 filled pages of the Ghosted Planes exercise

Each exercise in the list above links off to more detailed instructions on how they're done.

NOTE : These three exercises constitute just one section of this lesson. You should hold onto your homework until you're done all three sections (lines, ellipses, boxes) , and only submit for feedback once they're all done . You may feel hesitant or uncertain about whether or not you're doing things correctly - that's normal, but it's best you push past the urge to second-guess yourself. Focus on reading the instructions carefully, complete the work, and submit once all of it is complete. This will give others a solid body of work on which to base their feedback, giving you a more useful analysis of what you're doing well and what you may not understand.

All the assigned work for this section should be done in ink, using fineliners/felt tip pens as described here . In a pinch, I will accept work done in ballpoint, but only if the situation is dire. This is an exception only for this lesson as students get started.

Of course, if you're not planning on making use of the official critique program , then this is a recommendation rather than a hard requirement.

The Science of Deciding What You Should Draw

Right from when students hit the 50% rule early on in Lesson 0, they ask the same question - "What am I supposed to draw?"

It's not magic. We're made to think that when someone just whips off interesting things to draw, that they're gifted in a way that we are not. The problem isn't that we don't have ideas - it's that the ideas we have are so vague, they feel like nothing at all. In this course, we're going to look at how we can explore, pursue, and develop those fuzzy notions into something more concrete.

Getting Your Work Critiqued

Having your work reviewed by others is critical, as those who are just starting out aren't in a position to properly judge their own work, and won't be for quite some time. Don't be afraid to show your struggles - it's by analyzing your mistakes that we can help you grow. Perfect homework is not what we're looking for; we just need it to be complete .

There are currently two places you can get your work critiqued by the community - Reddit and our Discord Chat Server .

Both of these are completely free .

Private Patreon Critiques

If you are interested in receiving extra help, I critique the work of those who support Drawabox on Patreon .

All of these private critiques are done through reddit, in specific threads where students post their work as a comment, including a link to their work (often hosted on Imgur , though most image hosts are okay).

My requirements are more strict than the free community critiques:

• You must complete the lessons in order, in their entirety, starting from lesson 1
• All work for the lesson must be completed - that means all exercises in the lesson, not just those in a given section
• You may only move onto the next lesson once the previous has been marked complete
• The work must be done in the tools recommended in the lesson

The minimum pledge for this lesson is $5.00/month . The orange button above will take you to the reddit thread for this lesson, you can post a link to your work there and I'll be notified. Once I catch the submission, I'll add it to this backlog spreadsheet .

Pledges are collected at the beginning of the following month, but you may start submitting your work immediately. If you're a new patron, I'll be reaching out to your shortly to collect your reddit username.

Part One: The Basics

An overview of basic skills - both technical and conceptual - with exercises that you will incorporate into your regular warmups for a long time to come. no matter how skilled or experienced you are, start at lesson 0..

Challenges and Drills

A series of drills that fit into the lessons at various times. these should not all be completed after lesson 2, but rather will be listed as recommended next steps or prerequisites as you follow the numbered lessons in order..

Part Two: Constructional Drawing

An exploration of how complex objects can be broken down into their fundamental components, then rebuilt from simple forms. we look at this concept of constructional drawing by applying it to many different topics - the focus is not on learning how to draw that specific subject matter, but rather to tackle construction from different perspectives..

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11.15: Homework Exercises

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• Page ID 10216

• Roland Stull
• University of British Columbia

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11.15.1. Broaden Knowledge & Comprehension

B1. From hemispheric weather maps of winds near the tropopause (which you can access via the internet), identify locations of major global-circulation features including the jet stream, monsoon circulations, tropical cyclones and the ITCZ.

B2. Same as the previous exercise, except using water-vapor or infrared image loops from geostationary satellites to locate the features.

B3. From the web, find a rawinsonde sounding at a location in the trade-wind region, and confirm the wind reversal between low and high altitudes.

B4. Use a visible, whole-disk image from a geostationary satellite to view and quantify the cloud-cover fraction as a function of latitude. Speculate on how insolation at the Earth’s surface is affected.

B5. Download a series of rawinsonde soundings for different latitudes between the equator and a pole. Find the tropopause from each sounding, and then plot the variation of tropopause height vs. latitude.

B6. Download a map of sea-surface temperature (SST), and discuss how SST varies with latitude.

B7. Most satellite images in the infrared show greys or colors that are related to brightness temperature (see the legend in whole-disk IR images that you acquired from the internet). Use these temperatures as a function of latitude to estimate the corresponding meridional variation of IR-radiation out. Hint, consider the Stefan-Boltzmann law.

B8. Download satellite-derived images that show the climatological average incoming and outgoing radiation at the top of the atmosphere. How does it relate to the idealized descriptions in this chapter?

B9. Download satellite-derived or buoy & ship-derived ocean currents for the global oceans, and discuss how they transport heat meridionally, and why the oceanic transport of heat is relatively small at mid to high latitudes in the N. Hemisphere.

B10. Use a satellite image to locate a strong portion of the ITCZ over a rawinsonde site, and then download the rawinsonde data. Plot (compute if needed) the variation of pressure with altitude, and discuss how it does or doesn’t deviate from hydrostatic.

B11. Search the web for sites where you can plot “reanalysis data”, such as the NCEP/NCAR reanalysis or any of the ECMWF reanalyses. Pick a month during late summer from some past year in this database, and plot the surface pressure map. Explain how this “real” result relates to a combination of the “idealized” planetary and monsoonal circulations.

B12. Same as B11, but for monthly average vertical cross sections that can be looped as movies. Display fields such as zonal wind, meridional wind, and vertical velocity, and see how they vary over a year.

B13. Capture a current map showing 85 kPa temperatures, and assume that those temperatures are surrogates for the actual average virtual temperature between 100 and 70 kPa. Compute the thermal wind magnitude and direction for a location assigned by your teacher, and see if this theoretical relationship successfully explains the wind shear between 100 and 70 kPa. Justify your reasoning.

B14 Capture a current map showing the thickness between 100 and 50 kPa, and estimate the thermal wind direction and magnitude across that layer.

B15. Use rawinsonde soundings from stations that cross the jet stream. Create your own contour plots of the jet-stream cross section for (a) heights of key isobaric surfaces; (b) potential temperature; and (c) wind magnitude. Compare your plots with idealized sketches presented in this chapter.

B16. What are the vertical and horizontal dimensions of the jet stream, based on weather maps you acquire from the internet.

B17. Acquire a 50 kPa vorticity chart, and determine if the plotted vorticity is isentropic, absolute, relative, or potential. Where are positive-vorticity maxima relative to fronts and foul weather?

B18. Calculate the values for the four types of vorticity at a location identified by your instructor, based on data for winds and temperatures. Namely, acquire the raw data used for vorticity calculations; do not use vorticity maps captured from the web.

B19. For the 20 kPa geopotential heights, use the wavy pattern of height contours and their relative packing to identify ridges and troughs in the jet stream. Between two troughs, or between two ridges, estimate the wavelength of the Rossby wave. Use that measured length as if it were the dominant wavelength to estimate the phase speed for baroclinic and barotropic waves.

B20. Confirm that the theoretical relationship between horizontal winds, temperatures, vertical velocities, and heights for baroclinic waves is consistent with the corresponding weather maps you acquire from the internet. Explain any discrepancies.

B21. Confirm the three-band nature of the global circulation using IR satellite image movie loops. In the tropics, compare the motion of low (warm) and high (cold) clouds, and relate this motion to the trade winds and Hadley circulation. In mid-latitudes, find the regions of meandering jet stream with its corresponding high and low-pressure centers. In polar regions, relate cloud motions to the polar cell.

B22. Are the ocean-surface current directions consistent with near-surface wind directions as observed in maps or animations acquired from the internet, given the dynamics describe for the Ekman spiral?

11.15.2. Apply

A1(§). For the “toy” model, make a graph of zonally-averaged temperature (°C) vs. latitude for the altitude (km) above ground level (AGL) given here:

A2(§). For the “toy” model, make a graph of zonallyaveraged ∆T/∆y (°C km –1 ) vs. latitude for the altitude (km AGL) given here:

A3. Estimate the annual average insolation (W m –2 ) at the following latitude:

A4. Estimate the annual average amount of incoming solar radiation (W m –2 ) that is absorbed in the Earth-ocean-atmosphere system at latitude:

A5. Using the idealized temperature near the middle of the troposphere (at z = 5.5 km), estimate the outgoing infrared radiation (W m –2 ) from the atmosphere at the following latitude:

A6. Using the results from the previous two exercises, find the net radiation magnitude (W m –2 ) that is input to the atmosphere at latitude:

A7. Using the results from the previous exercise, find the latitude-compensated net radiation magnitude (W m –2 ; i.e., the differential heating) at latitude:

A8. Assuming a standard atmosphere, find the internal Rossby deformation radius (km) at latitude:

A9. Given the following virtual temperatures at your location (20°C) and at another location, find the change of geostrophic wind with height [(m s –1 )/ km]. Relative to your location, the other locations are:

A10. Find the thermal wind (m s –1 ) components, given a 100 to 50 kPa thickness change of 0.1 km across the following distances:

A11. Find the magnitude of the thermal wind (m s –1 ) for the following thickness gradients:

A12. For the toy model temperature distribution, find the wind speed (m s –1 ) of the jet stream at the following heights (km) for latitude 30°:

A13. If an air parcel from the starting latitude 5° has zero initial velocity relative to the Earth, then find its U component of velocity (m s –1 ) relative to the Earth when it reaches the following latitude, assuming conservation of angular momentum.

A14. Find the relative vorticity (s –1 ) for the change of (U , V) wind speed (m s –1 ), across distances of ∆x = 300 km and ∆y = 600 km respectively given below.

A15. Given below a radial shear (∆M/∆R) in [(m s –1 )/ km] and tangential wind speed M (m s –1 ) around radius R (km), find relative vorticity (s –1 ):

A16. If the air rotates as a solid body of radius 500 km, find the relative vorticity (s –1 ) for tangential speeds (m s –1 ) of:

A17. If the relative vorticity is 5x10 –5 s –1 , find the absolute vorticity at the following latitude:

A18. If absolute vorticity is 5x10 –5 s –1 , find the potential vorticity (m –1 ·s –1 ) for a layer of thickness (km) of:

A19. The potential vorticity is 1x10 –8 m –1 ·s –1 for a 10 km thick layer of air at latitude 48°N. What is the change of relative vorticity (s –1 ) if the thickness (km) of the rotating air changes to:

A20. If the absolute vorticity is 3x10 –5 s –1 at 12 km altitude, find the isentropic potential vorticity (PVU) for a potential temperature change of ___ °C across a height increase of 1 km.

A21. Find the horizontal circulation associated with average relative vorticity 5x10 –5 s –1 over area (km 2 ):

A22. For the latitude given below, what is the value of the beta parameter (m –1 s –1 ):

A23. Suppose the average wind speed is 60 m s –1 from the west at the tropopause. For a barotropic Rossby wave at 50° latitude, find both the intrinsic phase speed (m s –1 ) and the phase speed (m s –1 ) relative to the ground for wavelength (km) of:

A24. Plot the barotropic wave (y’ vs x’) from the previous exercise, assuming amplitude 2000 km.

A25. Same as exercise A23, but for a baroclinic Rossby wave in an atmosphere where air temperature decreases with height at 4°C km –1 .

A26(§). Plot the baroclinic wave (y’ vs x’) from the previous exercise, assuming amplitude 2000 km and a height (km):

A27. What is the fastest growing wavelength (km) for a baroclinic wave in a standard atmosphere at latitude:

A28. For the baroclinic Rossby wave of exercise A25 with amplitude 2000 km, find the wave amplitudes of the:

• vertical-displacement perturbation
• potential-temperature perturbation
• pressure perturbation
• U-wind perturbation
• V-wind perturbation
• W-wind perturbation

A29(§). For a vertical slice through the atmosphere, plot baroclinic Rossby-wave perturbation amount for conditions assigned in exercise A28.

A30. Find the latitude-weighted a·u’ momentum value (m s –1 ) for air that reaches destination latitude 50° from source latitude:

A31. Suppose the ____ cell upward and downward speeds are ___ and ___ mm s –1 , respectively, and the north-south wind speeds are 3 m s –1 at the top and bottom of the cell. The cell is about __ km high by ___ km wide, and is centered at about ___ latitude. Temperature in the atmosphere decreases from about 15°C near the surface to –57°C at 11 km altitude. Find the vertical circulation.

A32. Find the friction velocity at the water surface if the friction velocity (m s –1 ) in the air (at sea level for a standard atmosphere) is:

A33. Find the Ekman-spiral depth scale at latitude 50°N for eddy viscosity (m 2 s –1 ) of:

A34(§). Create a graph of Ekman-spiral wind components (U, V) components for depths from the surface down to where the velocities are near zero, for near-surface wind speed of 8 m s –1 at 40°N latitude.

11.15.3. Evaluate & Analyze

E1. During months when the major Hadley cell exists, trade winds cross the equator. If there are no forces at the equator, explain why this is possible.

E2. In regions of surface high pressure, descending air in the troposphere is associated with dry (nonrainy) weather. These high-pressure belts are where deserts form. In addition to the belts at ±30° latitude, semi-permanent surface highs also exist at the poles. Are polar regions deserts? Explain.

E3. The subtropical jet stream for Earth is located at about 30° latitude. Due to Coriolis force, this is the poleward limit of outflow air from the top of the ITCZ. If the Earth were to spin faster, numerical experiments suggest that the poleward limit (and thus the jet location) would be closer to the equator. Based on the spins of the other planets (get this info from the web or a textbook) compared to Earth, at what latitudes would you expect the subtropical jets to be on Jupiter? Do your predictions agree with photos of Jupiter?

E4. Horizontal divergence of air near the surface tends to reduce or eliminate horizontal temperature gradients. Horizontal convergence does the opposite. Fronts (as you will learn in the next chapter) are regions of strong local temperature gradients. Based on the general circulation of Earth, at what latitudes would you expect fronts to frequently exist, and at what other latitudes would you expect them to rarely exist? Explain.

E5. In the global circulation, what main features can cause mixing of air between the Northern and Southern Hemispheres? Based on typical velocities and cross sectional areas of these flows, over what length of time would be needed for the portion 1/e of all the air in the N. Hemisphere to be replaced by air that arrived from the S. Hemisphere?

E6. In Fig. 11.4, the average declination of the sun was listed as 14.9° to 15° for the 4-month periods listed in those figures. Confirm that those are the correct averages, based on the equations from the Solar & Infrared Radiation chapter for solar declination angle vs. day of the year.

E7. Thunderstorms are small-diameter (15 km) columns of cloudy air from near the ground to the tropopause. They are steered by the environmental winds at an altitude of roughly 1/4 to 1/3 the troposphere depth. With that information, in what direction would you expect thunderstorms to move as a function of latitude (do this for every 10° latitude)?

E8. The average meridional wind at each pole is zero. Why? Also, does your answer apply to instantaneous winds such as on a weather map? Why?

E9. Can you detect monsoonal (monthly or seasonal average) pressure centers on a normal (instantaneous) weather map analysis or forecast? Explain.

E10. Figs. 11.3a & 11.5a showed idealized surface wind & pressure patterns. Combine these and draw a sketch of the resulting idealized global circulation including both planetary and monsoon effects.

E11. Eqs. (11.1-11.3) represent an idealized (“toy model”) meridional variation of zonally averaged temperature. Critically analyze this model and discuss. Is it reasonable at the ends (boundaries) of the curve; are the units correct; is it physically justifiable; does it satisfy any budget constraints (e.g., conservation of heat, if appropriate), etc. What aspects of it are too simplified, and what aspects are OK?

E12. (a) Eq. (11.4) has the 3 rd power of the sine times the 2 nd power of the cosine. If you could arbitrarily change these powers, what values would lead to reasonable temperature gradients (∆T/∆y) at the surface and which would not (Hint: use a spreadsheet and experiment with different powers)?

(b) Of the various powers that could be reasonable, which powers would you recommend as fitting the available data the best? (Hint: consider not only the temperature gradient, but the associated meridional temperature profile and the associated jet stream.) Also, speculate on why I chose the powers that I did for this toy model.

E13. Concerning differential heating, Fig. 11.9 shows the annual average insolation vs. latitude. Instead, compute the average insolation over the two-month period of June and July, and plot vs. latitude. Use the resulting graph to explain why the jet stream and weather patterns are very weak in the summer hemisphere, and strong in the winter hemisphere.

E14. At mid- and high-latitudes, Fig. 11.9 shows that each hemisphere has one full cycle of insolation annually (i.e., there is one maximum and one minimum each year).

But look at Fig. 11.9 near the equator.

• Based on the data in this graph (or even better, based on the eqs. from the Solar & Infrared Radiation chapter), plot insolation vs. relative Julian day for the equator.
• How many insolation cycles are there each year at the equator?
• At the equator, speculate on when would be the hottest and coldest “seasons”.
• Within what range of latitudes near the equator is this behavior observed?

E15. Just before idealized eq. (11.6), I mentioned my surprise that E 2 was approximately constant with latitude. I had estimated E 2 by subtracting my toy-model values for E insol from the actual observed values of E in . Speculate about what physical processes could cause E 2 to be constant with latitude all the way from the equator to the poles.

E16. How sensitive is the toy model for E out (i.e., eq. 11.7) to the choice of average emission altitude z m ? Recall that z m , when used as the altitude z in eqs. (11.1-11.3), affects T m . Hint: for your sensitivity analysis, use a spreadsheet to experiment with different z m and see how the resulting plots of E out vs. latitude change. (See the “A SCIENTIFIC PERSPECTIVE” box about model sensitivity.)

E17(§). Solve the equations to reproduce the curves in figure:

E18. We recognize the global circulation as a response of the atmosphere to the instability caused by differential heating, as suggested by LeChatelier’s Principle . But the circulation does not totally undo the instability; namely, the tropics remain slightly warmer than the poles. Comment on why this remaining, unremoved instability is required to exist, for the global circulation to work.

E19. In Fig. 11.12, what would happen if the surplus area exceeded the deficit area? How would the global circulation change, and what would be the end result for Fig. 11.12?

E20. Check to see if the data in Fig. 11.12 does give zero net radiation when averaged from pole to pole.

E21. The observation data that was used in Fig. 11.14 was based on satellite-measured radiation and differential heating to get the total needed heat transport, and on estimates of heat transport by the oceans . The published “observations” for net atmospheric heat transport were, in fact, estimated as the difference (i.e., residual) between the total and the ocean curves. What could be some errors in this atmosphere curve? (Hint: see the A SCIENTIFIC PERSPECTIVE box about Residuals.)

E22. Use the total heat-transport curve from Fig. 11.60. At what latitude is the max transport? For that latitude, convert the total meridional heat-flux value to horsepower.

E23. For Fig. 11.15, explain why it is p’ vs. z that drive vertical winds, and not P column vs. z.

• Redraw Figs. 11.16 for downdraft situations.
• Figs. 11.16 both show updraft situations, but they have opposite pressure couplets. As you already found from part (a) both pressure couplets can be associated with downdrafts. What external information (in addition to the pressure-couplet sign) do you always need to decide whether a pressure couplet causes an updraft or a downdraft? Why?
• For the thermal circulation of Fig. 11.17(iv), what needs to happen for this circulation to be maintained? Namely, what prevents it from dying out?
• For what real-atmosphere situations can thermal circulations be maintained for several days?
• Study Fig. 11.18 closely, and explain why the wind vectors to/from the low- and high-pressure centers at the equator differ from the winds near pressure centers at mid-latitudes.
• Redraw Fig. 11.5a, but with continents and oceans at the equator. Discuss what monsoonal pressures and winds might occur during winter and summer, and why.
• Redraw Fig. 11.19, but for the case of geostrophic wind decreasing from its initial equilibrium value. Discuss the resulting evolution of wind and pressure fields during this geostrophic adjustment.
• Redraw Fig. 11.19, but for flow around a lowpressure center (i.e., look at gradient winds instead of geostrophic winds). Discuss how the wind and pressure fields adjust when the geostrophic wind is increased above its initial equilibrium value.

E28. How would the vertical potential temperature gradient need to vary with latitude for the “internal Rossby radius of deformation” to be invariant? Assume constant troposphere depth.

E29. In the Regional Winds chapter, gap winds and coastally-trapped jets are explained. Discuss how these flows relate to geostrophic adjustment.

E30. At the top of hurricanes (see the Tropical Cyclones chapter), so much air is being continuously pumped to the top of the troposphere that a highpressure center is formed over the hurricane core there. This high is so intense and localized that it violates the conditions for gradient winds; namely, the pressure gradient around this high is too steep (see the Forces & Winds chapter).

Discuss the winds and pressure at the top of a hurricane, using what you know about geostrophic adjustment. Namely, what happens to the winds and air mass if the wind field is not in geostrophic or gradient balance with the pressure field?

E31. In the thermal-wind relationship (eqs. 11.13), which factors on the right side are constant or vary by only a small amount compared to their magnitude, and which factors vary more (and are thus more important in the equations)?

E32. In Fig. 11.20, how would it change if the bottom isobaric surface were tilted; namely, if there were already a horizontal pressure gradient at the bottom?

E33. Draw a sketch similar to Fig. 11.20 for the thermalwind relationship for the Southern Hemisphere.

E34. In maps such as Fig. 11.21, explain why thickness is related to average temperature.

E35. Redraw Fig. 11.22 for the case cold air in the west and warm air in the east. Assume no change to the bottom isobaric surface.

E36. Copy Fig. 11.24.

• On your copy, draw the G 1 and G 2 vectors, and the M TH vector at point B. Confirm that the thermal wind relationship is qualitatively satisfied via vector addition. Discuss why point B is an example of veering or backing.
• Same as (a) but calculate the actual magnitude of each vector at point B based on the spacing between isobars, thickness contours, or height contours. Again, confirm that the thermal wind relationship is satisfied. (1° latitude = 111 km)

E37. Using a spreadsheet, start with an air parcel at rest at the tropopause over the equator. Assume a realistic pressure gradient between the equator and 30° latitude. Use dynamics to solve for acceleration of the parcel over a short time step, and then iterate over many time steps to find parcel speed and position. How does the path of this parcel compare to the idealized paths drawn in Fig. 11.26d? Discuss.

E38. In the thunderstorms at the ITCZ, copious amounts of water vapor condense and release latent heat. Discuss how this condensation affects the average lapse rate in the tropics, the distribution of heat, and the strength of the equatorial high-pressure belt at the tropopause.

E39. Summarize in a list or an outline all the general-circulation factors that make the mid-latitude weather different from tropical weather.

E40. Explain the surface pressure patterns in Figs. 11.31 in terms of a combination of idealized monsoon and planetary circulations.

E41. Figs. 11.31 show mid-summer and mid-winter conditions in each hemisphere. Speculate on what the circulation would look like in April or October.

E42. Compare Figs. 11.32 with the idealized planetary and monsoon circulations, and discuss similarities and differences.

E43. Based on Figs. 11.32, which hemisphere would you expect to have strong subtropical jets in both summer and winter, and which would not. What factors might be responsible for this difference?

E44. For the Indian monsoon sketched in Fig. 11.33, where are the updraft and downdraft portions of the major Hadley cell for that month? Also, what is the relationship between the trade winds at that time, and the Indian monsoon winds?

E45. What are the dominant characteristics you see in Fig. 11.34, regarding jet streams in the Earth’s atmosphere? Where don’t jet streams go?

E46. In Figs. 11.35, indicate if the jet-stream winds would be coming out of the page or into the page, for the:

• N. Hemisphere,
• S. Hemisphere.

E47. Although Figs. 11.36 are for different months than Figs. 11.32, they are close enough in months to still both describe summer and winter flows.

• Do the near-tropopause winds in Figs. 11.36 agree with the pressure gradients (or height gradients) in Figs. 11.32?
• Why are there easterly winds at the tropopause over/near the equator, even though there is negligible pressure gradient there?

E48. Describe the mechanism that drives the polar jet, and explain how it differs from the mechanism that drives the subtropical jet.

E49. In Fig. 11.37b, we see a very strong pressure gradient in the vertical (indicated by the different isobars), but only small pressure gradients in the horizontal (indicated by the slope of any one isobar). Yet the strongest average winds are horizontal, not vertical. Why?

E50. Why does the jet stream wind speed decrease with increasing height above the tropopause?

E51. a. Knowing the temperature field given by the toy model earlier in this chapter, show the steps needed to create eq. (11.17) by utilizing eqs. (11.2, 11.4 and 11.13). b. For what situations might this jetwind-speed equation not be valid? c. Explain what each term in eq. (11.17) represents physically.

E52. Why does an air parcel at rest (i.e., calm winds) near the equator possess large angular momentum?

What about for air parcels that move from the east at typical trade wind speeds?

E53. At the equator, air at the bottom of the troposphere has a smaller radius of curvature about the Earth’s axis than at the top of the troposphere. How significant is this difference? Can we neglect it?

E54. Suppose that air at 30° latitude has no eastwest velocity relative to the Earth’s surface. If that air moves equatorward while preserving its angular momentum, which direction would it move relative to the Earth’s surface? Why? Does it agree with real winds in the general circulation? Elaborate.

E55. Picture a circular hot tub of 2 m diameter with a drain in the middle. Water is initially 1.2 m deep, and you made rotate one revolution each 10 s. Next, you pull the plug, allowing the water depth to stretch to 2.4 m as it flows down the drain. Calculate the new angular velocity of the water, neglecting frictional drag. Show your steps.

E56. In eq. (11.20), why is there a negative sign on the last term? Hint: How does the rotation direction implied by the last term without a negative sign compare to the rotation direction of the first term?

E57. In the Thunderstorm chapters, you will learn that the winds in a portion of the tornado can be irrotational. This is surprising, because the winds are traveling so quickly around a very tight vortex. Explain what wind field is needed to gave irrotational winds (i.e., no relative vorticity) in air that is rotating around the tornado. Hint: Into the wall of a tornado, imagine dropping a neutrally-buoyant small paddle wheel the size of a flower. As this flower is translated around the perimeter of the tornado funnel, what must the local wind shear be at the flower to cause it to not spin relative to the ground? Redraw Fig. 11.43 to show what you propose.

E58. Eq. (11.25) gives names for the different terms that can contribute toward vorticity. For simplicity, assume ∆z is constant (i.e., assume no stretching). On a copy of Fig. 11.44, write these names at appropriate locations to identify the dominant factors affecting the vorticity max and min centers.

E59. If you were standing at the equator, you would be rotating with the Earth about its axis. However, you would have zero vorticity about your vertical axis. Explain how that is possible.

E60. Eq. (11.26) looks like it has the absolute vorticity in the numerator, yet that is an equation for a form of potential vorticity. What other aspects of that equation make it like a potential vorticity?

E61. Compare the expression of horizontal circulation C with that for vertical circulation CC.

E62. Relate Kelvin’s circulation theorem to the conservation of potential vorticity. Hint: Consider a constant Volume = A·∆z .

E63. The jet stream sketched in Fig. 11.49 separates cold polar air near the pole from warmer air near the equator. What prevents the cold air from extending further away from the poles toward the equator?

E64. If the Coriolis force didn’t vary with latitude, could there be Rossby waves? Discuss.

E65. Are baroclinic or barotropic Rossby waves faster relative to Earth’s surface at midlatitudes? Why?

E66. Compare how many Rossby waves would exist around the Earth under barotropic vs. baroclinic conditions. Assume an isothermal troposphere at 50°N.

E67. Once a Rossby wave is triggered, what mechanisms do you think could cause it to diminish (i.e., to reduce the waviness, and leave straight zonal flow).

E68. In Fig. 11.50 at point (4) in the jet stream, why doesn’t the air just continue turning clockwise around toward points (2) and (3), instead of starting to turn the other way?

E69. Pretend you are a newspaper reporter writing for a general audience. Write a short article describing how baroclinic Rossby waves work, and why they differ from barotropic waves.

E70. What conditions are needed so that Rossby waves have zero phase speed relative to the ground? Can such conditions occur in the real atmosphere?

E71. Will Rossby waves move faster or slower with respect to the Earth’s surface if the tropospheric static stability increases? Why?

E72. For a baroclinic wave that is meandering north and south, consider the northern-most point as the wave crest. Plot the variation of this crest longitude vs. altitude (i.e., x vs. z). Hint: consider eq. (11.40).

E73. Use tropopause-level Rossby-wave troughaxes and ridge-axes as landmarks. Relative to those landmarks, where east or west is: (a) vertical velocity the greatest; (b) potential-temperature deviation the greatest; and (c) vertical displacement the greatest?

E74. In Figs. 11.51 and 11.53 in the jet stream, there is just as much air going northward as there is air going southward across any latitude line, as required by mass conservation. If there is no net mass transport, how can there be heat or momentum transport?

E75. For the Southern Hemisphere: (a) would a direct circulation cell have positive or negative CC? (b) for each term of eq. (11.51), what are their signs?

E76. Compare definitions of circulation from this chapter with the previous chapter, and speculate on the relevance of the static stability and Earth’s rotation in one or both of those definitions.

E77. Consider a cyclonic air circulation over an ocean in your hemisphere. Knowing the relationship between ocean currents and surface winds, would you anticipate that the near-surface wind-driven ocean currents are diverging away from the center of the cyclone, or converging toward the center? Explain, and use drawings. Note: Due to mass conservations, horizontally diverging ocean surface waters cause upwelling of nutrient-rich water toward the surface, which can support ocean plants and animals, while downwelling does the opposite.

11.15.4. Synthesize

S1. Describe the equilibrium general circulation for a non-rotating Earth.

S2. Circulations are said to spin-down as they lose energy. Describe general-circulation spin-down if Earth suddenly stopped spinning on its axis.

S3. Describe the equilibrium general circulation for an Earth that spins three times faster than now.

S4. Describe the spin-up (increasing energy) as the general circulation evolves on an initially non-rotating Earth that suddenly started spinning.

S5. Describe the equilibrium general circulation on an Earth with no differential radiative heating.

S6. Describe the equilibrium general circulation on an Earth with cold equator and hot poles.

S7. Suppose that the sun caused radiative cooling of Earth, while IR radiation from space caused warming of Earth. How would the weather and climate be different, if at all?

S8. Describe the equilibrium general circulation for an Earth with polar ice caps that extend to 30° latitude.

S9. About 250 million years ago, all of the continents had moved together to become one big continent called Pangaea , before further plate tectonic movement caused the continents to drift apart. Pangaea spanned roughly 120° of longitude (1/3 of Earth’s circumference) and extended roughly from pole to pole. Also, at that time, the Earth was spinning faster, with the solar day being only about 23 of our present-day hours long. Assuming no other changes to insolation, etc, how would the global circulation have differed compared to the current circulation?

S10. If the Earth was dry and no clouds could form, how would the global circulation differ, if at all? Would the tropopause height be different? Why?

S11. Describe the equilibrium general circulation for an Earth with tropopause that is 5 km high.

S12. Describe the equilibrium general circulation for an Earth where potential vorticity isn’t conserved.

S13. Describe the equilibrium general circulation for an Earth having a zonal wind speed halfway between the phase speeds of short and long barotropic Rossby waves.

S14. Describe the equilibrium general circulation for an Earth having long barotropic Rossby waves that had slower intrinsic phase speed than short waves.

S15. Describe the nature of baroclinic Rossby waves for an Earth with statically unstable troposphere.

S16. Describe the equilibrium general circulation for an Earth where Rossby waves had no north-south net transport of heat, momentum, or moisture.

S17. Describe the equilibrium general circulation for an Earth where no heat was transported meridionally by ocean currents.

S18. Describe the equilibrium ocean currents for an Earth with no drag between atmosphere and ocean.

S19. Suppose there was an isolated small continent that was hot relative to the surrounding cooler ocean. Sketch a vertical cross section in the atmosphere across that continent, and use thickness concepts to draw the isobaric surfaces. Next, draw a planview map of heights of one of the mid-troposphere isobaric surfaces, and use thermal-wind effects to sketch wind vectors on this same map. Discuss how this approach does or doesn’t explain some aspects of monsoon circulations.

S20. If the Rossby wave of Fig. 11.50 was displaced so that it is centered on the equator (i.e., point (1) starts at the equator), would it still oscillate as shown in that figure, or would the trough of the wave (which is now in the S. Hem.) behave differently? Discuss.

S21. If the Earth were shaped like a cylinder with its axis of rotation aligned with the axis of the real Earth, could Rossby waves exist? How would the global circulation be different, if at all?

S22. In the subtropics, low altitude winds are from the east, but high altitude winds are from the west. In mid-latitudes, winds at all altitudes are from the west. Why are the winds in these latitude bands different?

S23. What if the Earth did not rotate? How would the Ekman spiral in the ocean be different, if at all?

• EXPLORE Random Article

How to Exercise While Doing your Housework

Last Updated: December 12, 2022 Approved

This article was co-authored by Errol Ismail . Errol Ismail is a Personal Trainer and the CEO and Co-Founder of Maestro Training. With nearly 10 years of experience, Errol specializes in helping individuals incorporate exercise into their daily lives by making it convenient and creating a community of support and encouragement. Before starting his own company, he honed his craft at at Equinox, one of the most prestigious gyms in the USA. wikiHow marks an article as reader-approved once it receives enough positive feedback. In this case, 100% of readers who voted found the article helpful, earning it our reader-approved status. This article has been viewed 73,038 times.

Busy people may not find time in their schedules to do the recommended 150 to 175 minutes of moderate cardiovascular exercise per week. A UK study found that the average British woman spends 143 minutes per week cleaning their house. If you are the main caretaker of your own house or another residence, then you may be able to count regular cleaning and chores as part of your exercise time. Many household chores are considered medium intensity cardiovascular or strength-training exercises. Plan your chores in a way that allows you to do them in succession to increase heart rate and work your major muscle groups. Read more to find out how to exercise while doing your housework.

Indoor Housework Workout

• Create a playlist that lasts at least 30 minutes. This can serve as your timer. Playing your favorite songs in succession will also increase the endorphin-boosting power of music. Try dance, Latin, hip hop and fast rock songs. Go to bpmdatabase.com to check if your favorite song has 120 BPM or more .

• You burn approximately 190 calories per hour vacuuming. Increase the calories you burn by performing single leg lunges every 2 to 3 minutes. Take a large step forward with your right leg and lunge until your right knee is at a 90 degree angle. Hold the position for 3 to 10 seconds and then step back. Repeat with your left leg.

• Mopping burns approximately 195 calories per hour. Make sure you switch arms as you sweep, mop or rake. You should reach and pull back with your right and left sides evenly. Favoring 1 side will create a muscle imbalance and may lead to pain. Get on your hands and knees and scrub intensely. This cleaning exercise works your shoulders and upper arms intensely.

• You burn approximately 500 calories per hour walking up and down stairs. Increase your strength-training by squatting whenever you pick up objects. Lean slightly back and pick up an object when your arms are close to the floor. Bending over causes intense strain on your lower back. Squatting builds the quadriceps, hamstring and ab muscles, helping you help your back when done repeatedly.

• Repeat this exercise 10 times and do 2 to 3 sets with 1 minute resting between sets.

• Return to the original position, rest for a few seconds and repeat until you are done folding your laundry.

• You burn approximately 180 calories per hour cleaning windows. You work approximately the same calories per hour dusting the house, which also works your upper and lower arm muscles.

Outdoor Chores Workout

• This exercise burns approximately 230 calories per hour.

• Painting burns between 250 and 300 calories per hour if done on large surfaces, like walls.

Expert Q&A

• Stretch all your major muscle groups after you end your housework workout. Take a shower after you have stretched and rehydrated with plenty of water. Thanks Helpful 0 Not Helpful 0
• Use these approximate calorie counts to calculate the calories you burn. Divide the actual minutes spent doing the task by 60 (1 hour). Multiply this figure by the calorie count to figure out how many calories you are actually burning. You can establish daily workouts according to the time, calories or intensity you prefer. Thanks Helpful 0 Not Helpful 0
• Avoid all bending if you have lower back or joint problems. Weeding, vacuuming and kneeling on hard surfaces may not be recommended for these chronic conditions. Thanks Helpful 7 Not Helpful 1

Things You'll Need

• Flexible clothing
• Athletic shoes (that can get dirty)
• Mop, broom or rake
• Sponges or cloths

You Might Also Like

• http://www.weightlossresources.co.uk/exercise/tips/housework_workouts.htm
• http://www.agingcare.com/Articles/caregiver-exercises-147273.htm

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Couples Therapy: Homework Exercises For Communication And Bonding

A common part of couples therapy for many is partaking in exercises outside of sessions, which are often known as “homework” exercises. Below, we’ve compiled some easy and helpful homework exercises that might aid you in your journey to strengthen your relationship, including gratitude lists, relationship check-ins, journaling, and more. These homework exercises can be tailored to specific romantic relationship goals, so feel free to get creative with how you use them. It can also be a good idea to complete these exercises under the guidance of a licensed mental health professional, which you can do in person or online.

Improve communication and strengthen your bond

Couples therapy is not just for couples experiencing challenges; anyone can benefit from improving relationship skills, bettering communication, and strengthening their bond with their partner. Researchers have looked into the impact of communication on relationships and found evidence that the better a couple communicates with each other, the more long-lasting and fulfilling their relationship is likely to be. 

What to expect in couples therapy

Couples therapy can go beyond the therapist’s office. Professional relationship therapists often recommend homework to their clients, such as the exercises included in this list. Relationship counseling is often the most effective when both members of the couple are willing to put in the work, both inside and outside of the therapist’s office. 

When you first attend couples therapy, the therapist will likely get to know you as a couple and then may want to talk to you individually. After determining your goals as a couple and as individuals, the therapist can support you in reaching those goals with evidence-based therapeutic interventions and relationship-building exercises. 

Depending on the therapist’s approach to counseling, they may use a variety of techniques, such as behavioral experiments, cognitive behavioral therapy, dialectical behavior therapy, psychotherapy, and homework. 

Best couples therapy homework exercises

Below are a few of the best couples therapy homework exercises recommended by couples therapists. You can use these couples therapy homework exercises at home with your partner, but they may be more impactful when combined with talk therapy from a licensed professional. A therapist can guide you through couples therapy exercises with tips for how to get the most out of the experience. 

Letter writing

If you go to couples counseling, you might practice letter writing as a homework assignment. Writing a letter to your partner can have many benefits, such as making it easier to express things that are hard to say out loud. By writing the letter, you may also discover underlying feelings, beliefs, or thoughts that can shift your perspective on a particular issue or subject. 

A great writing prompt for couples therapy that you can try at home is writing a letter to your partner describing all their best qualities and what you love about them. Then, you can have them write the same letter from their perspective, describing what they love about you and your relationship. This couples therapy exercise can help you better understand your partner’s views and recognize their best qualities. 

Gratitude lists

Making gratitude lists can be an ongoing homework exercise for couples to reflect on the positive things in their relationship and better appreciate each other. A gratitude list normally involves writing everything you are grateful for, or, in this case, what you are grateful for in your relationship. This exercise can be all about recognizing what the other person does that makes you happy and showing appreciation for them. 

For two weeks or more, write down a few things that made you happy or went well that day. When you look back on your list of positive moments, it may reveal patterns in the relationship. Talking with your partner about the list and noticing the patterns together can help you determine what is working in your relationship and how you can create joyful memories together. 

Studies show that displaying gratitude can have powerful benefits for mental health and well-being, such as reducing symptoms of anxiety and depression. Giving thanks has been found to make people happier overall, including in their relationships. 

One study on the benefits of gratitude for couples found that mutually expressing gratitude not only tended to make couples feel happier, but also made it easier to express what they wanted to change in their relationships. Positive encouragement from being shown gratitude could also help couples engage in more relationship “maintenance,” such as spending time together, checking in with each other, and being more responsive to each other’s needs. 

Scrapbooking memories

A fun and romantic couples therapy activity to try is making a scrapbook out of your memories together. Instead of a scrapbook, you could create a picture book with handmade drawings, collages, or online cut-outs. This can be an opportunity to assemble a cohesive memory book for you to reflect on your relationship together. 

If you have mementos, like romantic letters from the beginning of the relationship, or sentimental items, such as your first concert tickets together, you can put them in your memory book. You can also print out photos from your favorite memories together. Create the narrative of your love story in your scrapbook, then label the images with context about the memories and how you each feel about them. 

Icebreakers 

When you are first getting started with couples therapy, doing some fun and easy exercises like asking each other “icebreaker” questions may be beneficial. You may be surprised to find that there are things you do not know about your partner, even if you have been in a long-term relationship. These icebreaker questions are generally meant to be light-hearted and get couples comfortable before they delve into more challenging communication exercises. 

• What is your favorite childhood memory?
• Is there a funny story from your past you can share with me? 
• Would you ever want to be famous?
• What is your favorite love song?
• Do you know any random or useless facts?
• What was the first album or CD you purchased?
• If you had to eat one food for the rest of your life, what would it be?
• If you were a candy, what kind would you be?
• What is your favorite thing to do when you are home alone?

Relationship journaling

Studies have shown that journaling can be a beneficial therapeutic device for tapping into your inner feelings, thoughts, and beliefs. Journaling together as a couple can be a homework exercise that bonds you closer through shared communication. You may choose to journal weekly or daily. 

For journaling to be most effective in a therapy setting, the couple should be honest with themselves and their partners about their feelings. Recording your feelings, thoughts, experiences, and goals in a journal can help you better understand what is going on in your partner’s inner world. Reflecting on journal entries can make patterns in the relationship more transparent, showing you what is working versus what you may need to adjust. 

The journal can be used to reflect on positive memories together and as a safe way to express frustrations about the relationship or things that you wish were different. The journal entries can also be reviewed with your therapist during couples therapy sessions later on, helping you create a plan for how to best support each other in reaching your relationship goals. 

Relationship check-in

Around once a week, try to set aside 30 minutes to an hour to discuss your relationship and check in with each other about your feelings. You may also want to record your check-ins in your relationship journal or with couples therapy worksheets so you can reflect on them later. There are couples therapy toolkits that you can download for free online, or your therapist may provide resources. 

Checking in with each other can be an important step in maintaining healthy relationships, ensuring that you are both on the same page communication-wise and not letting anything slip through the cracks.

Benefits of online therapy

These homework exercises for building bonds and improving communication may be the most impactful when combined with guidance from a licensed therapist. You can find highly qualified couples therapists online with therapy platforms like BetterHelp. Couples with busy schedules or travel limitations may have an easier time accessing online couples therapy than traditional therapy at an office. 

Effectiveness of online therapy

One 2022 study examined the effectiveness of online and in-person couples therapy and found that internet-based therapeutic interventions could be just as effective as traditional sessions. However, online sessions typically offered the benefit of being more accessible and appealing to couples who may not otherwise attend. The data collected by the researchers in this study showed overwhelmingly positive results for the majority of participants. 

• Navigating Sympathy Vs. Empathy In Interactions Medically reviewed by Andrea Brant , LMHC
• De-Escalation Techniques To Defuse Conflicts in Relationships Medically reviewed by Julie Dodson , MA
• Relationships and Relations

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45+ Powerful Therapy Exercises For Clients, Couples & Groups

By identifying and sharing appropriate therapy exercises with clients, therapists help clients learn to manage existing problems and gain self-help skills for use going forward (Nelson-Jones, 2014).

In this article, we share many of our favorite free therapy exercises and suggest situations and groups where they may be best placed. Why not review them and reflect on their potential to boost engagement while supporting growth and development in individual, couple, and group settings?

Before you continue, we thought you might like to download our three Positive CBT Exercises for free . These science-based exercises will provide you with detailed insight into positive Cognitive-Behavioral Therapy (CBT) and give you the tools to apply it in your therapy or coaching.

This Article Contains

How to use exercises in therapy, 19 popular therapy exercises to try, 14 exercises and activities for couples, 7 group therapy exercises for your sessions, 5 simple therapeutic writing exercises, resources from positivepsychology.com, a take-home message, frequently asked questions.

A vital aspect of therapy is for the counselor or therapist to “collaborate with clients to achieve change and then for clients to maintain that change” long after treatment ends (Nelson-Jones, 2014, p. 52).

Mental health practitioners must understand the skills their clients need to develop, demonstrate how they can be implemented, and engage them in performing structured activities and homework tasks.

While it is essential that the counselor form a solid therapeutic bond with their client, it is similarly crucial that they identify and share powerful therapy exercises that support them in replacing their old self-defeating ways with more helpful, better skills (Nelson-Jones, 2014).

Unsurprisingly, research recognizes a significant connection between completing therapeutic homework and treatment outcomes (Mausbach et al., 2010).

The following exercises can be empowering when working with clients experiencing anxiety.

6 Exercises for managing anxiety

While the effects of anxiety can be catastrophic and far reaching, therapeutic interventions can be highly successful in helping clients redirect their minds away from “worry and negative self-appraisals and toward greater acceptance of internal states” (Crowley et al., 2017, p. 130).

The following therapy exercises will help:

• Event Visualization Worksheet Detailed imagination of a future event or challenge offers a safe and controlled environment for reducing concern and anxiety and gaining confidence without risk of failure.
• Tackling Anxious Thoughts Clients can learn how to notice anxious and irrational thoughts and find more helpful and rational alternatives.
• Managing Panic This is a valuable worksheet for identifying triggers and sources of panic and anxiety and recognizing associated feelings and behavior.
• Anxiety Record Maintaining a record of the causes of anxiety can be enlightening and empowering.
• Best and Worst When working with children, creating a Venn diagram can be a helpful visual representation of their anxieties versus potentially positive outcomes.
• Labeling Your Emotions Giving names to feelings can help children identify and understand their anxiety without forming guilt or engaging in judgment.

6 Best exercises for depression

Depression can be helped by understanding its causes and triggers while building a resilient mindset that increases positivity, improves stress recovery, and maintains flexibility in challenging environments (Waugh & Koster, 2015).

The following worksheets are valuable tools for use when working with clients experiencing depression or at risk of future episodes:

• Recognizing Rumination Persistent negative thinking is a key risk factor for depression. This template helps identify unhelpful thoughts and how they interfere with daily living.
• Guilt and Shame Emotions That Drive Depression Feeling guilt and shame can push an individual deeper into their depression. The clients answer questions in this exercise to help identify and better understand such emotions.
• Depressive Thought Worksheet for Teens Young people can benefit from reflecting on situations causing negative thinking and changing them into more realistic thoughts.
• What Is Depression? A Fact Sheet for Teenagers This valuable and insightful fact sheet explains what depression is, its signs and symptoms, potential behavioral changes, and how to seek help.
• My Depression Story Use this worksheet with clients to create a timeline of their lives to understand the key moments that shaped their perspective.
• Unhelpful Thinking Styles Our underlying thought patterns can worsen our depression. Share this worksheet with clients to identify unhelpful thinking styles and how to reconstruct them more positively.

7 Helpful exercises for building self-esteem

While poor self-esteem may emerge early in life, it can also develop in adulthood, caused by a combination of negative self-beliefs, harsh feedback, and challenging environments (Orth & Robins, 2019).

The following helpful exercises can boost clients’ self-esteem and challenge harsh self-evaluations:

• Designing Affirmations Positively focused self-affirmations can reinforce our self-identity and outcomes related to meaningful personal values.
• The Self-Esteem Checkup This valuable tool offers clients insight into their degree of self-love, self-respect, and confidence in their capabilities.
• Understanding Self-Confidence This worksheet helps teens, adolescents, and adults familiarize themselves with the mental and bodily experiences associated with self-confidence.
• My “Love Letter” to Myself Use this worksheet with clients to help them identify their best traits, abilities, and talents and consider how they have benefited them and others in their lives.
• Things I Like About Me This worksheet helps children and teenagers see the beauty resulting from their uniqueness. Use this worksheet to encourage them to understand all they can do, how they treat others, and what they like about themselves.
• Self-Esteem Journal for Adults Journaling can promote positive self-reflection and enhance self-esteem. Ask the client to complete the questions and then reflect on their thinking patterns, feelings, and emotions.
• Track and Measure Success We are all much better at remembering what we did wrong rather than our successes. Ask clients to keep a copy of what went well and review it before future challenges.

Download 3 Free Positive CBT Exercises (PDF)

These detailed, science-based exercises will equip you or your clients with tools to find new pathways to reduce suffering and more effectively cope with life stressors.

Download 3 Free Positive CBT Tools Pack (PDF)

By filling out your name and email address below.

Inevitably, couples disagree. However, when differences become irreconcilable, couples therapy can help regain trust, rebuild communication, and strengthen relationship bonds (Greiger, 2015).

The following exercises and activities are powerful tools for use with clients to support them on their journey.

30 Questions to ask couples

• Conflict Resolutions Checklist This valuable set of 10 questions ensures both partners have taken the steps to reduce or remove conflict.
• Valuing My Partner These five questions support clients in seeing their partners in a more positive (and realistic) light.
• Relationship History and Philosophy Questionnaire These 11 questions encourage couples to rediscover their admiration and love for one another through revisiting their shared history.
• Marital Conflict Questionnaire Use this four-question sheet to recognize and understand multiple conflicts the couple is experiencing.

4 Couples exercises to build trust

• Anger Exit and Re-Entry Routines This exercise builds relationship trust by identifying when conversations become heated and when it is time to exit, cool down, or re-enter.
• Imago Workup Identifying each other’s needs, desires, and past experiences can enhance understanding, encourage vulnerability, and boost trust in any relationship.
• Things I Love These are 10 prompts to be answered by each partner to encourage more robust, fruitful connections and strengthen the couple’s bond.
• Good Qualities Ask couples to work through the four prompts and capture and share good qualities, cherished memories, what they appreciate, and how the other person shows they care.

3 Exercises couples can perform at home

• From My Way to Our Way Couples may find living together challenging. This helpful exercise encourages partners to find a middle ground between two possibly very different views of everyday activities.
• Turning You Into I Worksheet “You” statements can sound critical and judgmental. In this exercise, each partner focuses on using the “I” pronoun to express how they feel and thereby support empathy.
• 10 Tips for Coping With Your Partner’s Upset While listening to a partner’s pain is sometimes difficult, this exercise can help avoid becoming overwhelmed.

3 Exercises for long-distance relationships

• Active Listening Reflection Worksheet Being apart can significantly strain a relationship. Each partner will benefit from improving their active listening skills to boost understanding and reduce the likelihood of misinterpretation of what is being communicated on a call, video chat, or in person.
• Traps to Avoid and Tips for Success Conflicts can often be avoided — or at least managed better — by learning the mistakes we make in our communication and following these six tips for conflict resolution.
• Effective Communication Reflection Worksheet Provide clients with this helpful worksheet to encourage them to reflect on their communication and how it might be improved.

The following exercises support group-based therapy in children and adults:

• Telling an Empathy Story Telling someone else’s story can be a powerful way to understand their perspective while developing empathy. This five-step worksheet helps group members focus on feelings and what it’s like to be in someone else’s situation.
• What I See in You We rarely see ourselves as others do. In this exercise, the group takes turns offering compliments to an individual member, which they then repeat back using the pronoun “I.”
• Nudge Interventions in Groups A group environment creates a powerful opportunity to identify, explore, and discuss small changes that can have significant behavioral outcomes.
• Group Boundary-Setting Exercise This exercise provides an opportunity to practice using body language and speech to set boundaries with others in a group setting.
• Creating an Empathy Picture Helpful for multiple age groups, this exercise encourages members to reflect on and understand another person’s feelings.
• Support Group Evaluation Form It is vital to assess the appropriateness of interventions performed continuously within a group setting to ensure their suitability.
• Group Counseling Permission Form This is a helpful form for parents to give their consent for their children to attend group counseling.

“Writing about traumatic, stressful or emotional events has been found to result in improvements in both physical and psychological health.”

Baikie & Wilhelm, 2005, p. 338

Simply capturing our thoughts, emotions, and concerns regularly — perhaps daily in a journal — has been shown to boost our moods and improve our overall sense of psychological wellbeing (Baikie & Wilhelm, 2005).

The following five exercises encourage clients to self-reflect and then capture how they feel and think digitally or on paper:

• Gratitude Journal Writing down daily everything that we are grateful for and learning from the challenges we face provides a powerful exercise for boosting our focus on the good things in life.
• Who Am I? Stopping to reflect and answer questions about ourselves increases self-awareness and self-knowledge. This two-part writing activity can be used in individual and group settings.
• Self-Love Journal These 10 self-love writing prompts encourage self-inquiry while identifying ways to introduce more self-directed compassion and kindness.
• Self-Love Sentence Stems Completing these 20 self-love partial sentences can boost self-awareness and self-kindness in clients who tend toward self-criticism.
• Reverse the Rabbit Hole Capturing worries and potentially positive and negative outcomes on paper can make clients’ concerns more manageable.

17 Science-Based Ways To Apply Positive CBT

These 17 Positive CBT & Cognitive Therapy Exercises [PDF] include our top-rated, ready-made templates for helping others develop more helpful thoughts and behaviors in response to challenges, while broadening the scope of traditional CBT.

Created by Experts. 100% Science-based.

As we have already seen, we have many resources for therapists working with individuals, couples, and groups to support multiple issues and concerns while promoting overall wellbeing.

More extensive versions of the following tools are available with a subscription to the Positive Psychology Toolkit© , but they are described briefly below.

Building the rituals of connection

Regular, relationship-focused habits can help foster more productive communicative behavior in a relationship and can offer emotional significance.

The four steps include:

• Step one – understanding various ritual types and timings, such as when parting, showing affection, and arranging date nights
• Step two – identifying specific actions for inclusion in each ritual
• Step three – planning how and when they should take place
• Step four – reflecting on the positive emotions that arose from each ritual and recognizing their importance

A strengths versus weakness focus

We often devote more time to our weaknesses than our strengths. The following two steps can be performed in a group setting to improve awareness regarding the importance of strength awareness and focus.

• Step one – Divide the group into three subgroups, as follows.

– Group 1 (weakness focus) spends time reflecting on challenging aspects of their jobs that drain their energy. – Group 2 (strength focus) discusses the highlights of their job. – Group 3 (observers) keeps an eye on the other two groups, noting their distinctions and dynamics.

• Step two – After 15 minutes, regroup. The “weakness” and “strength” groups share what they discussed first. Then, the observers point out the contrasts in energy, mood, and behavior between the two.

This exercise supports participants as they introspectively analyze their strengths and weaknesses, all while fostering group communication and collaboration.

If you’re looking for more science-based ways to help others through CBT, check out this collection of 17 validated positive CBT tools for practitioners. Use them to help others overcome unhelpful thoughts and feelings and develop more positive behaviors.

Therapy exercises are powerful tools for therapists and counselors working with individuals, couples, and groups. Such interventions, performed as homework between sessions, are linked to successful treatment outcomes (Mausbach et al., 2010).

The article shares many free therapy exercises and interventions grounded in research that support working with various psychological challenges, including complicated relationships, anxiety, depression, and self-esteem issues.

Such therapeutic exercises have the potential to foster meaningful change in your clients, equipping them with the tools to manage immediate challenges and the skills to solve issues in the future and after therapy. In doing so, they support and encourage individuals to participate actively in their healing and growth.

Besides the free therapy exercises highlighted, we offer various resource packs available on our website that underpin successful client outcomes. As therapists and counselors, you can use these activities and exercises as they are or tailor them to your clients’ specific needs and situations, ensuring you provide the best support for a positive therapeutic outcome.

We hope you enjoyed reading this article. For more information, don’t forget to download our three Positive CBT Exercises for free .

Ed: Updated November 2023

When talk therapy doesn’t meet a client’s needs, a more active approach, such as drama therapy , can be helpful. Role-play and storytelling can be powerful tools for treating young people experiencing behavioral challenges, older clients facing age-related issues, and anyone with social and emotional difficulties (Boila et al., 2020).

Typically, stabilizing mental health involves a multifaceted approach. Individuals seeking help benefit from actively engaging in therapy and creating personal treatment plans, including recognizing strengths and setting personal goals.

Counseling offers therapeutic support and learning skills to help clients form solid connections with others and adopt a positive mindset by reframing negative thoughts, practicing gratitude, and focusing on successful outcomes (Dixon et al., 2016; Jacob, 2015).

• Baikie, K., & Wilhelm, K. (2005). Emotional and physical health benefits of expressive writing. Advances in Psychiatric Treatment , 11 (5), 338–346.
• Bandelow, B., & Wedekind, D. (2022). Internet psychotherapeutic interventions for anxiety disorders: A critical evaluation. BMC Psychiatry , 22 (1).
• Boila, V., Klettke, L., Quong, S., & Gerlitz, C. (2020). Raising the curtain on drama therapy: Healing benefits for youth and older adults. Behavioural Sciences Undergraduate Journal , 3 (1), 45–50.
• Crowley, M. J., Nicholls, S. S., McCarthy, D., Greatorex, K., Wu, J., & Mayes, L. C. (2017). Innovations in practice: group mindfulness for adolescent anxiety: Results of an open trial. Child and Adolescent Mental Health , 23 (2), 130–133.
• Dixon, L. B., Holoshitz, Y., & Nossel, I. (2016). Treatment engagement of individuals experiencing mental illness: review and update. World Psychiatry , 15 (1), 13–20.
• Dwyer, L. A., Hornsey, M. J., Smith, L. G. E., Oei, T. P. S., & Dingle, G. A. (2011). Participant autonomy in cognitive behavioral group therapy: An integration of self-determination and cognitive behavioral theories. Journal of Social and Clinical Psychology , 30 (1), 24–46.
• Greiger, R. (2015). The couples therapy companion: A cognitive behavior workbook . Routledge.
• Jacob, K. S. (2015). Recovery model of mental illness: A complementary approach to psychiatric care. Indian Journal of Psychological Medicine , 37 (2), 117–119.
• Lenz, A. S., Hall, J., & Bailey Smith, L. (2015). Meta-analysis of group mindfulness-based cognitive therapy for decreasing symptoms of acute depression. The Journal for Specialists in Group Work , 41 (1), 44–70.
• Mausbach, B. T., Moore, R., Roesch, S., Cardenas, V., & Patterson, T. L. (2010). The relationship between homework compliance and therapy outcomes: An updated meta-analysis. Cognitive Therapy and Research , 34 (5), 429–438.
• Nelson-Jones, R. (2014). Practical counselling and helping skills . Sage.
• Orth, U., & Robins, R. W. (2019). Development of self-esteem across the lifespan. In D. P. McAdams, R. L. Shiner, & J. L. Tackett (Eds.), Handbook of personality development (pp. 328–344). Guilford Press.
• Waugh, C. E., & Koster, E. H. (2015). A resilience framework for promoting stable remission from depression. Clinical Psychology Review , 41 , 49–60.

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Homework 2 - Spring 2024

This homework covers the material from lectures 7 to 11

Due date: June 1st, 10pm Waterloo time.

PDF version of Homework 2 .

LaTeX template , in case you want to write it in LaTeX.

Required Exercises

Problem 1 - fingerprinting (20 points).

Consider the problem of deciding whether two integer multisets $S_1$ and $S_2$ are identical (that is, each integer occurs the same number of times in both sets). This problem can be solved by sorting the two sets in $O(n \log n)$ time, where $n = |S_1| = |S_2|$. In this question, you will devise 2 faster randomized algorithms for this problem.

You can assume that the multisets $S_i$ only have integers of bit complexity $w$, that $n \leq 2^w$, that integer operations of $O(w)$-bit integers can be executed in $O(1)$ time (RAM model), and that a prime with $O(w)$-bits can be found in $O(n)$ time.

Use polynomial identity testing to give a $O(n)$ time algorithm for the problem above.

Use hashing to give an algorithm which runs in expected $O(n)$ time for the problem above.

Your algorithms for both parts should succeed with probability $\geq 2/3$.

Problem 2 - Randomized Matching (10 points)

Given a randomized algorithm $\mathcal{A}$ for testing the existence of a perfect matching in a graph $G$, give a randomized algorithm for finding a perfect matching in $G$. Analyze the running time and success probability of your algorithm on graphs with $n$ vertices and $m$ edges, given that the algorithm $\mathcal{A}$ runs in time $T(n,m)$, and has failure probability $\leq 1/n$.

Your algorithm should have success probability $\geq 2/3$, and it should definitely be guaranteed to stop after $\textsf{poly}(n,m) T(n,m)$ time.

Problem 3 - Sublinear-time Algorithms (15 points)

Sublinear-time algorithms for connectedness in graphs with bounded degree.

Given a graph $G$ of maximum degree $d$ (as adjacency list ), and a parameter $\epsilon > 0$, give an algorithm which has the following behavior: if $G$ is connected, then the algorithm should output PASS with probability $1$, and if $G$ is $\epsilon$-far from connected (at least $\epsilon \cdot n \cdot d$ edges must be added to connect $G$), then the algorithm should output FAIL with probability at least $3/4$. Your algorithm should look at a number of edges that is independent of $n$, and polynomial in $d, \epsilon$.

For this problem, when proving the correctness of your algorithm, it is ok to show that if the input graph $G$ is likely to be passed, then it is $\epsilon$-close to a graph $G_0$ which is connected, without requiring that $G_0$ has degree at most $d$.

Problem 4 - Markov Chains (15 points)

• Show that if a Markov chain with transition matrix $P$ is irreducible and has a state $i$ such that $P_{i,i} > 0$, then it is also aperiodic.
• Let $a, b$ be positive integers and consider the Markov chain with state space $$ \{ (i,j) \mid 0 \leq i \leq a-1, \ \ 0 \leq j \leq b-1 \} $$ where $i,j$ are integers, and the following transition mechanism: if the chain is in state $(i,j)$ at time $t$, then at time $t+1$ it moves to $( (i+1) \bmod a, j)$ with probability $1/2$ or to $( i, (j+1) \bmod b)$ with probability $1/2$.

Show that this Markov chain is irreducible, and show that it is aperiodic if, and only if, $\gcd(a,b) = 1$.

• Consider a chessboard with a lone white king making random (king) moves, meaning that at each move, he picks one of the possible squares to move to, uniformly at random. Is the corresponding Markov chain irreducible and/or aperiodic? If so, what is the stationary distribution?

Problem 5 - Random Walks (10 points)

Consider a Markov chain on the vertices of a triangle: the chain moves from a vertex to one of the other two vertices with probability $1/2$ each. Find the stationary distribution and $\varepsilon$-mixing time of this Markov chain. Also, find the probability that the chain is at the starting vertex after $n$ steps, for $n \geq 1$.

Suppose that we alter the transition probabilities of the Markov chain in the previous question as follows: $$ p_{12} = p_{23} = p_{31} = 2/3, \ \ \ \ p_{21} = p_{32} = p_{13} = 1/3.$$ Find the stationary distribution and the $\varepsilon$-mixing time of this Markov chain. Also, find the probability that the chain is at the starting vertex after $n$ steps, for $n \geq 1$.

Problem 6 - Linear Programming (15 points)

Let $S = \{ x \in \mathbb{R}^n \mid A x \leq b \}$ and $T = \{ x \in \mathbb{R}^n \mid B x \leq c \}$, where $A \in \mathbb{R}^{m \times n}$, $b \in \mathbb{R}^m$, $B \in \mathbb{R}^{t \times n}$ and $c \in \mathbb{R}^t$. Given $A, B, b, c$ as an input, give a polynomial time algorithm for the problem of checking whether $S \subset T$.

Problem 7 - LP Duality (15 points)

For a directed graph $G(V, E)$, with $2$ special vertices $s, t \in V$, let $c_e$ be the capacity of edge $e \in E$ and consider the max-flox LP: $$ \max \sum_{P \ s\text{-}t \text{ path of } G} f_P $$ subject to $$ \sum_{e \in P} f_P \leq c_e \quad \forall e \in E, $$ $$ 0 \leq f_P \quad \text{ for all paths } P. $$ The first constraint says that the total flow on all paths from $s$ to $t$ is at most the capacity of each edge, and the second constraint says that the flow on each path is non-negative.

Give the dual of this LP and explain how to interpret the dual LP as a minimum cut problem.

Practice Problems

You are not required to submit the solution to these problems, but highly encouraged to attempt them for your own knowledge sake. :)

Problem 1 - Matching

Let $G(U, V, E)$ be a bipartite graph, and let $X$ be the matrix of indeterminates given from the adjacency matrix of $G$, as we saw in lecture. Show that the size of the maximum matching in $G$ is equal to the rank of $X$ over the field $\mathbb{F}(X)$.

Hint: you can use the fact that the rank of $X$ as above is the same as the max rank of any evaluation of $X$ over $\mathbb{F}$.

Problem 2 - Probability Amplification

In the distinct element algorithm we saw in lecture, we showed that with probability at least $2/3$ we have $(1-\varepsilon) D \leq Y \leq (1+\varepsilon) D$, where $Y$ is the answer that our algorithm returns. Suppose we run $k$ independent copies of the algorithm in parallel and obtain the estimates $Y_1, \ldots, Y_k$ and return the median $Y’$ of $Y_1, \ldots, Y_k$ as our answer. Prove that $(1-\varepsilon) D \leq Y’ \leq (1+\varepsilon) D$ with probability $1-\delta$ when $k = O(\log 1/\delta)$

This technique of amplifying the success probability is very general and useful when one would like to have an algorithm with arbitrarily high success rate, given only an algorithm with success probability being a constant strictly greater than 1/2.

Problem 3 - Karger’s Algorithm

One advantage of Karger’s random contraction algorithm for the minimum cut problem is that it can be used to output all minimum cuts. In this question, we assume Karger’s algorithm as a black box, which when given a graph $G$ as an input, for any minimum cut $(S, \overline{S})$, can be used to output the minimum cut $(S, \overline{S})$ with probability at least $2/n(n - 1)$ in time $O(n^2),$ where $n$ is the number of vertices in the input graph. Explain how Karger’s algorithm can be used to output all minimum cuts and analyze its running time to output all minimum cuts with success probability at least $0.9999.$

Problem 4 - Testing Sortedness in Sublinear Time (courtesy of Nathan Harms)

In this question we mean that an array is sorted if it is sorted in increasing order .

Let $A$ be an array of $n$ integers, and let $\varepsilon \in (0,1)$. We say that $A$ is $\varepsilon$-far from sorted if every sorted array $B$ has at least $\varepsilon n$ entries that are different from $A$. In other words, at least $\varepsilon n$ entries of $A$ must be modified to turn it into a sorted array.

An $\varepsilon$-tester for sortedness with query complexity $q$ is a randomized algorithm that queries a sequence $i_1, \ldots, i_q$ of indices from $A$ and outputs YES or NO as follows:

• if $A$ is sorted, the algorithm output YES with probability 1
• if $A$ is $\varepsilon$-far from sorted, the algorithm outputs NO with probability at least $2/3$.

Question 1 : the simplest idea for a tester is the random adjacent pair tester , which chooses a random index $i \in \{ 1, \ldots, n-1 \}$, queries $A[i]$ and $A[i+1]$ and rejects if $A[i] > A[i+1]$. Repeat this procedure $q$ times and reject if at any query we output reject.

Give an example array $A$ that is $1/2$-far from sorted, where this algorithm with $q=1$ repetitions outputs no with probability at most $1/(n-1)$. Conclude that the tester must repeat at least $q = \Omega(n)$ times, so the query complexity in this case is $\Omega(n)$

Question 2: Another idea is to choose $q$ independently, uniformly random indices $i_1, \ldots, i_q$ and output NO if the subsequence $A[i_1], A[i_2], \ldots, A[i_q]$ is not sorted.

Give an example of an array that is $1/2$-far from sorted where this algorithm requires $q = \Omega(\sqrt{n})$ queries.

Hint: remember the birthday paradox

Question 3: Assume each entry of $A$ is distinct. Consider the set $G$ of good indices of $A$, where index $i$ is good if using binary search on $A$ to find $A[i]$ will successfully find the index $i$.

• For any set $G = \{ i_1, \ldots, i_g \}$ of good indices, show that the subsequence $A[i_1], \ldots, A[i_g]$ is sorted.
• Show that, if $A$ is $\varepsilon$-far from sorted, then $|G| \leq (1-\varepsilon)n$.
• Show that there is an $\varepsilon$-tester for sortedness that uses $O(\frac{1}{\varepsilon} \log n)$ queries.

Hint: consider choosing a random index $i ~ \{1, \ldots, n\}$, querying $A[i]$, then trying to use binary search to find element $A[i]$.

Problem 5 - Mixing Time of Random Walks

Compute the mixing time (both upper bounds and lower bounds) of the dumbbell graph: the graph on $2n$ nodes that consists of two complete graphs on $n$ nodes joined by a single edge, where $n \geq 3$.

Hint: try breaking up the graph into 4 pieces which converge to the stationary distribution in the appropriate time, and then show that any other probability distribution converges quickly to one of these pieces.

Alternative Hint: you can solve this by using elementary probability calculations

Problem 6 - Random Walks (more advanced material)

A cat and a mouse each independently take a random walk on a connected, undirected, non-bipartite graph $G$. They start at the same time on different nodes, and each makes one transition at each time step. The cat catches the mouse if they are ever at the same node at some time step. Show an upper bound of $O(m^2 n)$ on the expected time before the cat catches the mouse, where $n$ is the number of vertices and $m$ is the number of edges of $G$.

Problem 7 - Linear Programming

There are $n$ terminals on a circuit board. Each terminal is either positive or negative. A wire will be attached to each terminal. It must be the case that for every pair of terminals of opposite polarity the two wires must be long enough so that they can be made to touch.

E.g., if there’s a positive terminal at point $a = (0, 0)$, and a negative terminal at $b = (0, 10)$, and another negative terminal at $c = (10, 0)$ then one solution is to put wires of length $5$ at all three terminals. Or to put a wire of length $10$ at $a$ and length zero at the others.

Or, for any $x \in [0, 10]$, wires of length $\geq 10 - x$ at $a$, and length $\geq x$ at both $b$ and $c$.

So given the locations of the terminals (and their polarities), the problem is to compute the shortest total length of wires needed to satisfy the stated requirement.

• Consider the following specific instance of the problem. All the terminals are along the $x$-axis. There are positive terminals at $0, 110, 111, 112$. And there are $2$ negative terminals at $99$ and $100$.

What’s the optimal solution to the problem?

(You don’t need to prove your solution is optimal.)

• Formulate this problem as an LP. (Illustrate your LP by writing this for an example with two positive and two negative terminals.)
• Write the dual of the LP in part (2.). (Again, write down this dual for the two terminal example in part (2.).)
• Show how to compute an optimal solution to the dual problem of part (3.), without using a general LP solver.

Hint: see Seidel’s 2D algorithm for LPs

Problem 8 - Linear Programming

Suppose we are given a sequence of $n$ linear inequalities of the form $a_i x + b_i y \leq c_i$, where $x, y$ are variables and $a_i, b_i, c_i \in \mathbb{R}$. Collectively, these $n$ inequalities describe a convex polygon $P$ in the plane.

• Describe a linear program whose solution describes the largest axis-aligned square that lies entirely inside $P$
• Describe a linear program whose solution describes the maximum-perimeter axis-aligned rectangle that lies entirely inside $P$
• Describe a linear program whose solution describes the largest circle that lies entirely inside $P$

Problem 9 - Linear Programming

• Describe a polynomial time algorithm to compute two interior-disjoint axis-aligned squares with maximum total perimeter that lie entirely inside $P$
• Describe a polynomial time algorithm to compute two interior-disjoint axis-aligned rectangles with maximum total perimeter that lie entirely inside $P$

Problem 10 - Linear Programming Duality (basic Helly’s theorem)

Given a finite set of halfspaces $H_i := \{ x \in \mathbb{R}^d \mid a_i^T x \leq b_i, \ a_i \in \mathbb{R}^d, \ b_i \in \mathbb{R} \}$, where $1 \leq i \leq n$ such that any set of $d+1$ halfspaces have a non-empty intersection, prove that there is a point lying in the intersection of all these halfspaces.

Last updated on May 20, 2024

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