A global education is one that incorporates learning about the cultures, geographies, histories, and current issues of all the world’s regions. It emphasizes the interconnectedness and diversity of peoples and histories.
The chapter revisits the concept of numbers, starting from natural numbers, whole numbers, and integers, and introduces students to new types of numbers such as rational and irrational numbers.
Importance: Understanding the classification of numbers is foundational for higher mathematics. It forms the basis of algebra, geometry, and advanced calculus.
2. Real Numbers
Definition: Real numbers include all rational and irrational numbers.
Decimal Representation:
Rational numbers have either terminating or recurring decimals.
Irrational numbers have non-terminating, non-recurring decimals.
Importance: These laws simplify calculations and are essential for algebraic expressions and equations.
5. Representation of Real Numbers on the Number Line
Concept: Real numbers can be represented on the number line using successive magnification.
Visualization: Techniques like zooming in on the number line help locate irrational numbers like 2\sqrt{2}.
6. Rationalization
Definition: The process of converting an expression with an irrational denominator into a rational form by multiplying numerator and denominator with a suitable conjugate.
CBSE Class 10, 12 Sample Papers 2025 has been released. The download links is given here.
Central Board of Secondary Education has released CBSE Class 10, 12 Sample Papers 2025. The sample question papers for classes 10 and 12 for the current academic session 2024-25 is available to candidates on the official website of CBSE Academic at cbseacademic.nic.in.
CBSE Class 10, 12 Sample Papers 2025 released, download links here
As per the official notice, the Board issues Sample Question Papers (SQPs) and Marking Schemes (MS) for classes X and XII to provide a broad template to serve as a guide for ensuring uniformity and proper coverage of the curricula.
CBSE Class 10, 12 sample question papers give a broad understanding about the question paper design which should be used for classroom teaching and learning activities with an overall focus on promoting the application of concepts in real-life.
CBSE Class 10, 12 Sample Papers 2025: How to download
To download the sample papers for Class 10, 12 candidates can follow these simple steps given below.
Visit the official website of CBSE Academic at cbseacademic.nic.in.
Click on CBSE Class 10, 12 Sample Papers 2025 link available on the home page.
A new PDF file will open where the Class 10 and Class 12 sample papers links are attached.
Click on those links and the sample paper will be displayed on the screen.
Now click on the subject you want to check the sample paper for.
Once done, download the page and keep a hard copy of the same for further need.
Meanwhile, CBSE has started the submission of list of candidates for Class 10, 12 examination 2025. The submission of data started on September 5, 2024 on Pariksha Sangam website. Only those students will be allowed to appear for Class 10, 12 board examination in 2025, whose names will be submitted through online process of submission of LOC. For more related details candidates can check the official website of CBSE.
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News/Education/Board Exams/ CBSE Class 10, 12 Sample Papers 2025 released at cbseacademic.nic.in, download links here
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UP NEET UG 2024 Counselling Round 2 schedule released. The registration will begin on September 9, 2024.
Medical Education and Training, Uttar Pradesh has released UP NEET UG 2024 Counselling Round 2 schedule. The complete schedule is available on the official website of UP NEET at upneet.gov.in.
UP NEET UG 2024 Counselling: Round 2 schedule out, registration begins Sept 9(Representational image)
The registration process begins on September 9 and will end on September 13, 2024. The last date for payment of the application fee is September 13, at 2 pm. The merit list will be displayed on September 14, 2024.
The choice filling will begin on September 14 and will conclude on September 18, 2024. The Round 2 seat allotment result will be announced on September 19, and candidates can download the allotment letter from September 20 to September 25, 2024.
UP NEET UG 2024 Counselling: How to apply for Round 2
To apply for the Round 2 counselling process, candidates can follow the steps given below.
Visit the official website of UP NEET at upneet.gov.in.
Click on UP NEET UG 2024 Counselling Round 2 registration link available on the home page.
Enter the registration details and click on submit.
Once done, fill the application form and make the payment of application fee.
Click on submit and download the confirmation page.
Keep a hard copy of the same for further need.
New candidates who want to participate in Round 2 counselling will have to pay ₹30000 as security money for government seats and ₹2 lakh for private medical college seats. For private sector dental colleges, it is mandatory to deposit Rs. 1,00,000/-.
Un-allotted/admitted/not-reported candidates from the first round will have to participate in the second round of counselling. For those candidates participating in the second round of counselling who had not registered in the first round of counselling, it will be mandatory for such candidates to register online on the website by depositing the registration fee of Rs. 2000/-. For more related details candidates can check the official website of UP NEET.
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News/Education/Admission News/ UP NEET UG 2024 Counselling: Round 2 schedule out at upneet.gov.in, registration begins September 9
Prime Minister Narendra Modi has said efforts were being made to recognise bright teachers practising innovative methods as he took an apparent swipe at past governments, saying he would not talk about what used to happen earlier.
Prime Minister Narendra Modi said on Friday that the efforts were being by the government to recognise bright teachers practising innovative methods. (Photo Credit: ANI)
In an interaction with teachers who were conferred National Teachers’ Awards, he said they could prepare young students for ‘Viksit Bharat’.
The interaction took place on Friday while its video was shared on Saturday.
Modi suggested that the teachers could involve the students in selecting the top 100 tourist destinations in the country by encouraging them to learn about different places they might visit as part of education tours.
The teachers awarded were selected after a long process as their efforts can be useful in the new National Education Policy (NEP), Modi said while noting that there would be many others doing excellent work.
He told the teachers to take their students to nearby universities and watch sporting events as the experience could fire up their dreams.
As many as 82 teachers from around the country were selected for the award. They included 50 teachers by the Department of School Education & Literacy, 16 by the Department of Higher Education and 16 by the Union Ministry of Skill Development and Entrepreneurship, an official statement said.
Modi suggested that the awardee teachers connect with each other on social media and share their best practices so that everyone could learn, adapt and benefit.
He highlighted the impact of the NEP and spoke about the significance of attaining education in the mother tongue.
The prime minister suggested that the teachers teach local folklore to their students in different languages so that they learn multiple languages and also get exposure to India’s vibrant culture.
Indian Institute of Technology (IIT) Kanpur’s Startup Incubation and Innovation Centre (SIIC) took part in the three-day Armed Forces Festival ‘Sashakt aur Surakshit Bharat: Transforming the Armed Forces’ inaugurated by Chief Minister Yogi Adityanath in Lucknow on Friday, September 6.
Uttar Pradesh CM Yogi Adityanath checking defence technology displayed by a SIIC-incubated startup during the Armed Forces Festival 2024.
The event witnessed an impressive display of military might and innovation, showcasing modern technology-based military weapon systems, including tanks, infantry weapons, and artillery guns, a press release informed.
Demonstrations during the program included slithering, para jumps, and insertion of special forces via helicopters.
Additionally, there were other highlights such as paramotors, microlight flying, fighter aircraft flypasts, and horse, dog, and motorcycle displays.
As part of the program, Animesh Mishra, the Manager of SIIC, IIT Kanpur, led a group of startups incubated under SIIC, all making significant contributions to the defence sector.
Some of the technological innovations highlighted were the Alakh, Sabal 10/20, and Vibhram, a suite of surveillance and logistics drones developed by EndureAir Systems Pvt. Ltd which have been designed to enhance operational capabilities.
The event also featured the Maraal Solar UAV by Maraal Aerospace Pvt. Ltd., an indigenously designed and developed solar-powered UAV with extended endurance and cyber-attack resilient features.
Likewise, Dream Aerospace Pvt. Ltd. showcased its Green Monopropellant Thruster named the Atom thruster, which utilizes a HAN-based non-toxic green fuel available in various thrust levels starting from 1N.
Other startups that featured their technological innovations included VU-Dynamics Pvt. Ltd., Space Philic Pvt Ltd., and Cyethack Solutions Pvt Ltd.
It is worth mentioning here that the products of these startups have been developed under the SIIC and aim to significantly contribute to the defence sector.
Prof. Deepu Philip, Professor-in-Charge of SIIC at IIT Kanpur, expressed his pleasure at the technological innovations brought by the startups to the defence sector.
He said that these young companies have showcased technologies that can significantly enhance the capabilities of armed forces.
Early on in Math Mind: The Simple Path to Loving Math, Shalinee Sharma experiences a moment of doubt. After 13 years at Bain & Company, she cofounds Zearn, an educational nonprofit. Zearn wants to offer great math teaching free to any child. But Sharma and her cofounders immediately encounter an obstacle––one that has frustrated many an educational newcomer: there is no agreement about what great math teaching looks like. “There was no manual for what I wanted to do,” she writes. “I had to go a different way.”
This was in 2012. Since then, Zearn has evolved into a free digital mathematics curriculum and learning platform. Some teachers might pair students with a Chromebook and headphones for supplemental “Zearn time.” It can also be used as a standalone resource with self-paced videos, practice, and games. About 10 percent of elementary teachers across the country report using Zearn once or more a week, according to a 2023 RAND survey. Quite a few math problems have been solved on its platform—over 14 billion.
Math Mind: The Simple Path to Loving Math by Shalinee Sharma Avery, 2024, $28.00; 288 pages.
In the past 12 years, Sharma has visited “thousands of classes” and reached some conclusions. Chief among these is that school is badly broken. “Our math education system makes learning math a hellish experience,” she writes. “Most kids hate math.” Her experience as CEO of Zearn suggests this suffering is needless. All kids can not only learn math but also love math, if supported properly by parents and educators. “It’s time that the adults get together and make that happen.” But how?
Sharma’s first big idea comes courtesy of none other than Carol Dweck, the Stanford psychology professor whose research on the benefits of cultivating a growth mindset in students now pervades U.S. classrooms. Sharma devotes a chapter to the importance of growth mindset and the related concept of stereotype threat. When teachers get these wrong, it can be disastrous for kids. Take Mr. Rockhill, Sharma’s mathematics teacher, who on the first day of high school melodramatically placed 18 desks out for 20 students. “At least two of us, he said—but probably more—wouldn’t make it,” Sharma recalls, creating a scene I thought existed only in fiction.
Next comes her favorite recommendation, the one she mentions to a “fabulously successful investor” who asks what Zearn has divined from its database of millions of students. Sharma tells the investor that it has learned to offer pictures (“especially brightly colored ones”) to students when they’re stuck. Educators typically refer to this (minus the colors) as the “Concrete, Pictorial, Abstract” approach, with roots in Jerome Bruner’s 1960s theories of instruction, which then inspired Singapore’s influential math curriculum work in the 1980s.
What else? She’d like to see students encouraged to solve problems using their own ideas. While affirming the importance of algorithms, she takes a stand for creative calculation, what she calls “easier problems.” Why not allow kids to solve 30 × 18 by first finding 30 × 20, then subtracting 30 × 2? This would combat what Sharma considers dominant myths—that speed is all that matters in math and that there’s only a single way to solve each problem. She’d also like an end to “long worksheets with unrelated problems” and a reorientation around “meaningful practice.”
Sharma’s final idea is “trying a different way” and is illustrated by a debate with Steve Levitt, the Freakonomics economist. Zearn’s analysts had found that, after a student commits an error, it is better to offer an easier approach to the grade-level question rather than send them back to shore up foundational skills. A skeptical Levitt challenged the analytics team to prove it, leading to a quasi-experimental study supporting the Zearn approach. “We can only guess at the reason,” Sharma writes.
Shalinee Sharma
I don’t have to guess. What child wants to return to 4th-grade content while working on a 6th-grade question? It may not have been obvious to Levitt, Sharma, or the Zearn data analysts, but I doubt anyone who has worked with children would find this even the least bit mysterious.
Sharma herself has not worked in schools. Raised in Buffalo, New York, by refugees fleeing the Partition of India, at age eight she dreamed of joining the American Red Cross. From then it was on to Bain and Zearn, we learn, where she sometimes visits classrooms and speaks with children. Her admiration for great teachers comes through frequently in Math Mind, and I don’t doubt its sincerity.
As a teacher, I know our profession is far too quick to dismiss outsiders as unrealistic. But the issue here isn’t pie-in-the-sky pedagogy as much as conventionality. Sharma’s long journey with Zearn has brought her to some of the most common ideas circulating among math educators: growth mindset, visuals, student strategies along with meaningful practice, and supporting students with grade-level standards. Is that all it takes?
Math Mind is aimed at parents and novice educators, so conventionality per se isn’t necessarily a problem. But Sharma means these to be transformational ideas. (We are now in the second decade of Dweck in schools with not much to show from it.) Zearn itself has come out seeming decidedly non-revolutionary in research studies. A Johns Hopkins evaluation led by Jennifer Morrison found that while teachers and students enjoyed working with Zearn, its impact on achievement was not statistically significant. Other studies, such as Shirin Hashim’s, found the curriculum to offer positive but quite modest results.
The major question lurking here is whether something like a “simple path to loving math” really exists. (At one point in the book, an “Ivy League professor” voices a similar concern, to Sharma’s shock.) Educators know there’s only so much that great teaching can do. For one, math is tasked with educational gatekeeping, caught in what historian David Labaree describes as our desire to provide universal access to social advantage—an impossibility. Sharma likewise describes loving math as an “exclusive world” that every child could freely join. But given the role math plays socially, access to this sort of exclusivity is anything but simple.
An irony is that school outsiders—economists, policy wonks, CEOs, and others—are more likely to promote the magic of great teaching than teachers themselves do. School staff are confronted daily with forces that thwart even the wisest, kindest instruction. A partial list includes immense early disadvantages that only grow, an inability to learn in conventional settings, the endless demand for increased rigor at ever younger ages, and brutal competition for distinction at the top. Great teaching navigates these obstacles but can never eliminate them.
“From Sorting to Teaching,” the final chapter of Math Mind, is where Sharma comes closest to grappling with these tensions. Tracking along with testing, she writes, creates an academic hierarchy. Most students learn their place in the pecking order and lose a love for the subject. It’s a fair point. But what can be done? Many educators, facing this same dilemma, end up offering a radical solution. Following educator Jo Boaler, they sometimes call to end tracking. Others propose decoupling mathematics and high-stakes testing. Some demand significant reform of the math curriculum. These ideas are not all to my liking, but at least they’d meet the moment.
In any event, this is not where Sharma lands because Math Mind, ultimately, is a case for educational technology. Great teaching can foster a love of math, but if you get a lousy teacher, you’re stuck. That’s where Zearn comes in, able to deliver videos and practice questions that “expand what was once the domain of a sorted few.” As if we haven’t heard this one before! The promise that technology will equalize schooling is, at this point, definitively unfulfilled. “It’s time to design and build a new system to teach everyone math,” Sharma writes. You might be disappointed to learn that what she’s describing, in the end, is just an app.
A previous version of this review incorrectly identified Sharma’s former employer.
Such subcontractual partnerships – while presenting several benefits for students and higher education institutions (HEIs) alike – also present risks, according to an Office for Students (OfS) insights document published on September 3.
These include staff incentivising recruiting and retaining students over the quality of courses, students with insufficient English skills being told these are enough to join a course and not being given the extra support they might need and even reports of students paying for falsified English language tests.
A subcontractual partnership is when a lead provider – for example, a university – uses a third-party provider to deliver all or part of a course on its behalf. Students pay their fees to the lead provider, which keeps a percentage and passes the rest on to the delivery partner.
The report warned that “without appropriate oversight”, the “arms-length delivery” of such partnerships “presents significant risks to students, taxpayers and the higher education sector”.
It said these risks are worsened when:
The lead provider has subcontractual arrangements with multiple delivery partners.
Such arrangements involve a large number (for example, more than 1,000) of the lead provider’s students.
The students involved are a significant proportion of those the lead provider registers.
Such agreements are a rapidly growing area in the sector, with students taught in the programs doubling since 2019/20 to over 138,000 in 2022-23 – representing over 5% of students in the sector, according to the report. And in some cases, lead providers now teach more students through subcontractual partnerships that directly on their own campuses.
But “there is evidence that [it] is not always true for students in such arrangement” that they are “treated fairly, receive a high quality education and get the outcomes they deserve”, it added.
In line with the growing number of students taught in subcontractual partnership courses, the UK‘s education watchdog the OfS is casting a beady eye over these deals. “This is to ensure that universities and colleges have robust governance and oversight of these arrangements, in the interests of ensuring positive outcomes for students and taxpayers, and the reputation of the higher education sector,” its report said.
“We invite leaders in universities and colleges that already have subcontractual arrangements, or are considering entering them, to consider the suggestions in this brief on how to ensure effective governance and oversight. While universities and colleges must contact us if any material concerns arise with a subcontractual partnership, we can also advise at an earlier stage, including before an arrangement is made.”
There are also serious risks to public money where these arrangements are not managed properly
David Smy, OfS
It continued: “Some of the more striking examples we have seen of such arrangements leading to strong positive outcomes include collaborations with performing arts and media colleges, and further education and sixth form colleges.”
“From our analysis…we can see that many benefits can be achieved through subcontractual partnerships, for both students and higher education providers…But partnerships need robust management and oversight if they are to achieve these benefits, and deliver for students and taxpayers,” said David Smy, the OfS’s deputy director for enabling regulation at the OfS.
“In these financially challenging times for higher education providers, it’s more important than ever that they recognise that business models that rely heavily on subcontractual partnerships carry additional risks, and these risks must be effectively managed.”
He continued: “There are also serious risks to public money where these arrangements are not managed properly. This can include universities and colleges receiving public funding for students who may not be genuinely studying on the course their tuition fees are funding, and students who may be receiving other payments they’re not entitled to. We continue to work closely with the Department for Education and the Student Loans Company to ensure public funding is protected.
‘We hope this Insight brief will be a useful resource for universities and colleges that are, or are contemplating, working through subcontractual partnerships, and a reminder of the importance of good management and effective governance. Lead universities should seriously consider whether they should offer courses in this way if they cannot manage partners and public money effectively while ensuring that courses are delivering positive outcomes for students.”
