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Unit 1: Algebra foundations

Unit 2: solving equations & inequalities, unit 3: working with units, unit 4: linear equations & graphs, unit 5: forms of linear equations, unit 6: systems of equations, unit 7: inequalities (systems & graphs), unit 8: functions, unit 9: sequences, unit 10: absolute value & piecewise functions, unit 11: exponents & radicals, unit 12: exponential growth & decay, unit 13: quadratics: multiplying & factoring, unit 14: quadratic functions & equations, unit 15: irrational numbers, unit 16: creativity in algebra.

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Frequently Asked Questions about Khan Academy and Math Worksheets

Why is khan academy even better than traditional math worksheets.

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

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

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

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Answered Problems By Math Tutors

A soccer coach wants to know how many hours a week his players spend training at home. He has 20 players and he decides to ask the first 4 players to arrive at Monday's soccer practice how many hours they spend training per week. He then calculated that they spend an average of 10 hour per week. Therefore, he assumed that all the players train 10 hours per week. Is this an example of a simple random sample? 

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B Yes, because each student had an equal chance of being selected. 

C No, because he did not sample every soccer player. 

D Yes, the minimum number of students sampled need to be four for it to be a simple random sample. 

The figure shows the graph of f. 

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The diagram shows two rectangles, A and B. 

All measurements are in centimetres. 

The area of rectangle A is equal to the area of rectangle B. 

Find an expression for y in terms of w. 

George's page contains twice as many typed words as Bill's page and Bill's page contains 50 fewer words than Charlie's page. If each person can type 60 words per minute, after one minute, the difference between twice the number of words on Bill's page and the number of words on Charlie's page is 210. How many words did Bill's page contain initially?

 Bill's page initially contained        words. 

Simplify 2sin(5x)cos(3x) - sin(2x) to one an expression containing one trigonometric function. Then graph the original function and your simplified version to verify they are identical. Enclose arguments of functions in parentheses. For example, sin(2x). 

2sin(5x)cos(3x)-sin(2x)=

A health psychologist was interested in the effects of vitamin supplements on the immune system. Three groups of adults were exposed (in a highly ethical way) to the cold virus; one group took no supplements for a week before exposure, another had vitamin C supplements, and a third had multivitamins (excluding C). The severity of the cold was measured as a percentage (0% = not contracted, 100% very severe symptoms). The psychologist also measured the number of cigarettes that each person smoked per day, as smoking suppresses the immune system. The psychologist was interested in the differences in the severity of the illnesses across different vitamin groups accounting for cigarette usage. What technique should be used to analyse these data? 

A. Two-way repeated-measures ANOVA 

B. Two-way independent ANOVA 

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Google’s Circle to Search will help you with your math homework

Circle to search on android can now help students learn to solve mathematical word problems thanks to google’s new learnlm model..

By Umar Shakir , a news writer fond of the electric vehicle lifestyle and things that plug in via USB-C. He spent over 15 years in IT support before joining The Verge.

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Google is enhancing Android’s Circle to Search — the feature that lets you literally circle something on your Android phone’s screen to search it on Google — with a new ability to generate instructions on how to solve school math and physics problems .

Using an Android phone or tablet, students can now use Circle to Search to get AI assistance on mathematical word problems from their homework. The feature will help unpack the problem and list what the student needs to do to get the correct answer. According to Google, it won’t actually do the homework for you — only help you approach the problem.

Over the past year, the use of AI tools like ChatGPT has become a hot topic in the field of education, with plenty of concern over how students can and will use it to get work done quickly. Google, however, is explicitly positioning this as a feature to support education, potentially walking around some of the concerns about AI doing all of the work for students.

Later this year, Circle to Search will also gain the ability to solve complex math equations that involve formulas, diagrams, graphs, and more. Google is using LearnLM, its new AI model that’s fine-tuned for learning, to make the new Circle to Search abilities work.

Circle to Search first launched on  Samsung’s Galaxy S24 series in January and then on the Pixel 8 and 8 Pro later the same month. It’s one of the star new features of Android, and although iOS users can’t yet circle their math homework for help, anything is possible .

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Softening the sharp edges in mathematics.

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

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

“Mathematics should be flexible, conceptual, a place where we play with ideas and make connections," says Professor Jo Boaler. (Photo: Robert Houser Photography)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Click to enlarge: A depiction of various ways to calculate 38 x 5, numerically and visually. (Image: Courtesy of Jo Boaler)

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

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

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

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Android's Circle to Search can now help students solve math and physics homework

It can show step-by-step instructions on how to solve a range of math and physics problems..

Google has introduced another capability for its Circle to Search feature at the company's annual I/O developer conference, and it's something that could help students better understand potentially difficult class topics . The feature will now be able to show them step-by-step instructions for a "range of physics and math word problems." They just have to activate the feature by long-pressing the home button or navigation bar and then circling the problem that's got them stumped, though some math problems will require users to be signed up for Google's experimental Search Labs feature.

The company says Circle to Search's new capability was made possible by its new family of AI models called LearnLM that was specifically created and fine-tuned for learning. It's also planning to make adjustments to this particular capability and to roll out an upgraded version later this year that could solve even more complex problems "involving symbolic formulas, diagrams, graphs and more." Google launched Circle to Search earlier this year at a Samsung Unpacked event, because the feature was initially available on Galaxy 24, as well as on Pixel 8 devices. It's now also out for the Galaxy S23, Galaxy S22, Z Fold, Z Flip, Pixel 6 and Pixel 7 devices, and it'll likely make its way to more hardware in the future.

In addition to the new Circle to Search capability, Google has also revealed that devices that can support the Gemini for Android chatbot assistant will now be able to bring it up as an overlay on top of the application that's currently open. Users can then drag and drop images straight from the overlay into apps like Gmail, for instance, or use the overlay to look up information without having to swipe away from whatever they're doing. They can tap "Ask this video" to find specific information within a YouTube video that's open, and if they have access to Gemini Advanced, they can use the "Ask this PDF" option to find information from within lengthy documents.

Google is also rolling out multimodal capabilities to Nano, the smallest model in the Gemini family that can process information on-device. The updated Gemini Nano, which will be able to process sights, sounds and spoken language, is coming to Google's TalkBack screen reader later this year. Gemini Nano will enable TalkBack to describe images onscreen more quickly and even without an internet connection. Finally, Google is currently testing a Gemini Nano feature that can alert users while a call is ongoing if it detects common conversation patterns associated with scams. Users will be alerted, for instance, if they're talking to someone asking them for their PINs or passwords or to someone asking them to buy gift cards.

Catch up on all the news from Google I/O 2024 right here !

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Circle to Search is the new cheat sheet for your Math and Physics homework

Published on May 14, 2024

• Circle to Search can now help students with homework, making it easy to search for instructions and answers without leaving the digital homework screen.
• Later this year, Circle to Search will be able to solve even more complex problems involving symbolic formulas, diagrams, graphs, and more.

Circle to Search was one of the flagship features of the Galaxy S24 series . It lets users easily access Google Search on practically any screen. The feature then expanded to the Pixel 8 series and then to other Samsung Galaxy phones and Pixel devices. If you have Circle to Search on your phone, you can now use it to help with your homework.

As part of its Google I/O 2024 announcements, Google has announced that Circle to Search can now help students with homework. When students need help, they can open their camera or their digital homework on their phones or tablets and circle the question they are stuck on. Circle to Search will then present a step-by-step instruction to solve their homework question.

Subjects covered include physics and match word problems. This will make searching far easier, as you no longer have to leave your digital homework screen or input the question yourself. The fine print says that right now, this is available for some math word problems when opted into Search Labs.

Google says that later this year, Circle to Search will be able to solve even more complex problems involving symbolic formulas, diagrams, graphs, and more. This will be made possible through LearnLM, Google’s new family of AI models fine-tuned for learning.

Circle to Search is currently available on over 100 million devices, and Google aims to double its reach to 200 million devices by the end of 2024. With new features like homework help, Circle to Search becomes even more useful, and it is available only on (select) Android devices.

Do you use Google or other search engines to help with your homework? Will you now switch to Circle to Search for your homework? Let us know in the comments below!

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Circle to Search is now a better homework helper

It’s a teacher’s worst nightmare: The AI is doing the kids’ homework. At the Google I/O 2024 developer conference on Tuesday , the company announced that its AI-powered Circle to Search feature, which allows Android users to get instant answers using gestures like circling, will now be able to solve more complex problems across psychics and math word problems. 

The new capabilities are made possible thanks to Google’s new family of AI models for learning, LearnLM . 

The addition expands the capabilities of the new search feature, first introduced at Samsung’s Unpacked event in January . Circle to Search is designed to make it more natural to engage with Google Search from anywhere on the phone by taking some action — like circling, highlighting, scribbling, or tapping. But it’s also a way to ensure users engage with Google Search over other information-retrieval tools in the age of AI.

Today, Circle to Search will be able to better help kids with their homework directly from supported Android phones and tablets. Now if they get stuck on a particularly hard problem, they’ll be able to use Circle to Search to pull up step-by-step instructions that guide them through solving it. Google says the feature will be able to handle problems such as those with symbolic formulas, diagrams, graphs, and more.

Since launching earlier this year, Circle to Search has expanded to more Samsung and Google Pixel devices, making the feature available to more than 100 million devices in total. Google says it expects to double that number by year-end. 

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Android's Circle to Search can now help students solve math and physics homework

G oogle has introduced another capability for its Circle to Search feature at the company's annual I/O developer conference, and it's something that could help students better understand potentially difficult class topics . The feature will now be able to show them step-by-step instructions for a "range of physics and math word problems." They just have to activate the feature by long-pressing the home button or navigation bar and then circling the problem that's got them stumped, though some math problems will require users to be signed up for Google's experimental Search Labs feature.

The company says Circle to Search's new capability was made possible by its new family of AI models called LearnLM that was specifically created and fine-tuned for learning. It's also planning to make adjustments to this particular capability and to roll out an upgraded version later this year that could solve even more complex problems "involving symbolic formulas, diagrams, graphs and more." Google launched Circle to Search earlier this year at a Samsung Unpacked event, because the feature was initially available on Galaxy 24, as well as on Pixel 8 devices. It's now also out for the Galaxy S23, Galaxy S22, Z Fold, Z Flip, Pixel 6 and Pixel 7 devices, and it'll likely make its way to more hardware in the future. 

In addition to the new Circle to Search capability, Google has also revealed that devices that can support the Gemini for Android chatbot assistant will now be able to bring it up as an overlay on top of the application that's currently open. Users can then drag and drop images straight from the overlay into apps like Gmail, for instance, or use the overlay to look up information without having to swipe away from whatever they're doing. They can tap "Ask this video" to find specific information within a YouTube video that's open, and if they have access to Gemini Advanced, they can use the "Ask this PDF" option to find information from within lengthy documents. 

Google is also rolling out multimodal capabilities to Nano, the smallest model in the Gemini family that can process information on-device. The updated Gemini Nano, which will be able to process sights, sounds and spoken language, is coming to Google's TalkBack screen reader later this year. Gemini Nano will enable TalkBack to describe images onscreen more quickly and even without an internet connection. Finally, Google is currently testing a Gemini Nano feature that can alert users while a call is ongoing if it detects common conversation patterns associated with scams. Users will be alerted, for instance, if they're talking to someone asking them for their PINs or passwords or to someone asking them to buy gift cards. 

Catch up on all the news from Google I/O 2024 right here !