“Find my age” puzzles (but with all true facts).

Posted on February 3, 2025 by b.pate

And (spoiler alert) the solutions. (Click the image to enlarge it.)

January 2025 (with bad drawings)

Posted on January 31, 2025 by b.pate

Happy 2025! This is a special year, because — mathematically speaking — it is a year when you and I are alive, a year when lifeless matter has coalesced into living bodies on the wet skin of a rocky sphere in a vibrant corner of an inconceivable cosmos.

Also, it’s a special year because 2025 = 45² = (1+2+3+4+5+6+7+8+9)² = 1³ + 2³ + 3³ + 4³ + 5³ + 6³ + 7³ + 8³ + 9³.

So, good times no matter how you slice it.

Finnished

I got to go to Helsinki this month, thanks to my translator/publisher/pal/Finnisher Juha Pietiläinen.

His company Terra Cognita publishes big-time science books by big-name authors on socially urgent topics, so that Finnish readers can join the global conversation in their native tongue. So, yes, my stick-figure jokes are a perfect fit. No further questions please.

Additional thanks to Rosebud Books, who hosted Juha and me in conversation on Saturday. I got to read aloud the Dr. Seuss parody from my new book, swapping verses with Juha’s Finnish version, so the audience could hear how he handled the impossible task.

(Juha got applause. I did not. This is fair.)

Signed Copies

This month, while making my rounds of the local bookstores, I took a few moments to sign the available copies of my books.

Sorry, did I use the first-person singular there? Bad habit. My daughters did most of the signing.

Autographed versions of Math for English Majors are now in abundant supply at Next Chapter.

Going Dutch

To my honor, the first pages from Math for English Majors (originally published at Lit Hub) were translated into Dutch and printed in Belgium’s paper of record, De Standaard.

At the editor’s invitation, I added a few paragraphs, which I offer in English here:

We all know the stereotype: there are math people, and “not math” people. The separation begins early in childhood. By adulthood, the two belong to wholly separate intellectual traditions—which might as well be different species, evolved on different planets, with no common language except for disapproving noises and warlike glares.

Why, then, has the union of math and literature been so fruitful?

One possibility is that the two are complements. Like sweet and salty, math and literature are opposite flavors that pair well together. “The union of the mathematician with the poet,” said William James, “fervor with measure, passion with correctness, this surely is the ideal.” Perhaps math’s clarity and coolness is the perfect balance for poetry’s intensity and heat.

Another possibility—the one I tend to favor—is just the opposite: that math and literature are not as different as they appear. “Mathematics,” said the mathematician Oswald Veblen, “is one of the essential emanations of the human spirit, a thing to be valued in and for itself, like art or poetry.” Mathematics and literature are both expressions of human imagination.

Sofia Kovalevskaya, the first woman to receive a PhD in mathematics, put it even more sharply. “It is impossible to be a mathematician,” she said, “without being a poet in the soul.”

Math Books I Enjoyed in 2024

For a while (2018, 2019, 2020, 2021) I compiled my favorite books of the year into a post. I cannot imagine where I got the energy for such an undertaking.

Instead, I offer a quick rundown of the mathier highlights from my 2024 reading.

First, if you’re looking for literature:

The MANIAC, by Benjamin Labatut. A 20th-century monster story. The monster is John von Neumann, and the world of artificial intelligence that he helped to create. The story is narrated in the voices of those who knew him.

Luminous, by Greg Egan. Egan’s sci-fi short stories are more mathematically sophisticated than a lot of math textbooks. (And don’t worry, the literary sophistication is higher, too.) A note for Ted Chiang fans: the title story makes a fascinating pairing with Chiang’s “Division By Zero.”

Second, if you’re looking not for literature per se, but good books about it:

Once Upon a Prime, Sarah Hart. A tour of connections between literature and mathematics. One of my new favorite works of math popularization. It’s a book I long intended to write, and I’m glad Hart got there first, because she did a better job than I would have.

Much Ado About Numbers, by Rob Eastaway. Best enjoyed as a kind of trivia book, with mathematician Eastaway guiding us through Shakespeare’s language, time, and world.

Third, if you’re looking for mathematical puzzles, I’ve already sung the praises of a few books, but for even more in the same vein, consider:

Seven Games, by Oliver Roeder. Those seven games: chess, checkers, go, bridge, poker, scrabble, and backgammon. Published a few years ago, but still a good backdoor exploration of artificial intelligence, in all its powers and limitations.

The Puzzler, by A.J. Jacobs. Cheeky, breezy nonfiction about the many worlds of puzzles. Ranges from crosswords to jigsaws to Smullyan-esque logical traps. Lots of good puzzles throughout.

And finally, if you’re looking for good ol’ nonfiction:

Trefethen’s Index Cards, by Lloyd N. Trefethen. Since youth, mathematician Trefethen has kept a sort of diary in the form of index cards. This book draws a few hundred from across the decades, with frank, wistful observations on math, life, and culture.

The Emergence of Probability, by Ian Hacking. A historical tour de force that I wrote about in a recent post.

Numbers Don’t Lie, by Vaclav Smil. A clear-eyed, iron-hearted, quantitatively-driven survey of the 21st century landscape.

Parting Puzzle(s).

These six come from a New Year’s email sent by my father Jim Orlin. A renowned researcher in network flows, he is widely considered one of the most adorable of all Orlins.

2025 is a special year in that it is the square of an integer.

1. What was the last year (prior to 2025) that was a square of an integer?

2. What is the next year (after 2025) that is a square of an integer?

Also, 2025 has 15 divisors.

3. What was the last year (prior to 2025) that has 15 divisors?

4. What is the next year (after 2025) that has 15 divisors?

The following is a bit surprising, I think.

5. What was the last year (prior to 2025) that has 16 divisors?

6. What is the next year (after 2025) that has 16 divisors?

The illusory consensus of math reform.

Posted on January 20, 2025 by b.pate

When my first book Math with Bad Drawings came out, I kept having a strange experience.

“Yes!” a stranger would say, pointing at the stick figure on the cover. “We need more of this.”

My enthusiastic supporter had, until 30 seconds prior, never heard of the book. They still hadn’t cracked the spine. I was honored by the support, but it was obvious that I hadn’t yet earned it. For all they knew, I had written 376 pages of vulgar limericks. What convinced them that I was on the right path?

I’ve come to see this as a basic dynamic in math education reform: an illusory spirit of consensus. Clearly math education needs more something. But more what?

One popular answer: “more data science.” Let’s renounce all of those fusty, old-fashioned trigonometric formulas. Instead, fill math class with 21^st-century virtues: data visualization, probabilistic thinking, and statistical literacy. This isn’t necessarily about new pedagogy; it’s about new content. It’s about re-centering math class on techniques that will cash out, on skills that students might actually apply.

Another popular answer: “more student-centered.” Let’s banish all tasks mechanical and rote. Instead, fill math class with open-ended puzzles, inquiry-based learning, and creative projects. This isn’t necessarily about new content; it’s about new pedagogy. It’s about re-centering math class on the voices, the ideas, and the ingenuity of the students themselves.

I see value in both of these approaches (and many others besides). My first book, Math with Bad Drawings, with its long sections on statistics and probability, was largely about data literacy. My second book, Change is the Only Constant, was a human-centered take on calculus.

But when it comes to systemic change, the two are orthogonal at best, and opposites at worst.

Should we dismantle our regime of standardized tests? Or are they a vital tool for measuring the success of a new curriculum?

Should math education impart deep experiences of beauty? Or practical, wage-raising skills?

Should students pursue open-ended, creative thinking? Or develop concrete knowledge with immediate applications?

Is math a liberal art, akin to art or music? Or is it a practical craft, like computer programming or home economics?

Should economic productivity be the guiding principle for secondary education? And if not, then what should be?

I wouldn’t endorse either vision in its purest form. Math education pursues a hodgepodge of goals: quantitative literacy, humanistic growth, preparation for STEM careers. The educators I admire most don’t subscribe to easy dichotomies. They value student voices and factual knowledge, algebraic fluency and open-ended exploration. Myself, I am comfortable with only two universal claims about math education: there are always tensions, and there are always tradeoffs.

That’s why I worry about the illusory consensus around reform. How can we make wise tradeoffs if we don’t acknowledge the tensions?

The solution, I think, is simple: show your work.

Show us what a good lesson (or unit, or year) looks like, and explain why.

I’m trying to do this in my own teaching now: to resolve all these contradictory possibilities in a way that works for my students. It’s not easy! There are days when I do an adequate job, and days when I do quite the opposite.

Anyway, take this as a pledge: in 2025, I’d like to show more of my work. I’m teaching two classes this term: Intro to Statistics, and Liberal Arts Mathematics (a kind of escape hatch for students who don’t want to take college algebra as their final required math class). I approach them both with a muddle of overlapping values, and a desire to do whatever works best for my students’ learning. More forthcoming.

Puzzles of strange coinage (increasing in difficulty).

Posted on January 13, 2025 by b.pate

A survey of recent works in Oulipo.

Posted on January 6, 2025 by b.pate

“Oulipans are rats who build the labyrinth
from which they will try to escape.”
–Raymond Queneau

It brings me no joy to report the rebirth (or the renewed undeadness) of the zombie literary movement known as OuLiPo.

Oulipo’s first birth came in 1960, from the vibrant and idle minds of Raymond Queneau and François Le Lionnais. They dubbed it ouvroir de littérature potentielle (“workshop of potential literature”). A self-conscious experiment in applying strict mathematical constraints to art, its results (such as George Perec’s La Disparition, a novel that avoids the letter e) were spectacles of virtuosity, triumphs of ingenuity, and, at their very best, passable works of art. No coincidence, I say, that the name “potential literature” stands opposed to actual literature.

I believed this volcano had gone dormant. I was wrong. The last year witnessed four eruptions. I offer brief comments on each species of ash.

In Sarah Hart’s CENTIPEDES: 100-LEGGED FICTIONS (Flatiron Books, $24.95) the self-imposed constraints are punishingly tight. In each of the book’s 100 works of micro-fiction, you will find: (1) a single sentence, (2) precisely 100 words long, (3) with no words repeated.

The rules are so constrictive they threaten to cut off all blood flow. Fortunately, they do not. An example to illustrate the form:

After Amber’s dad left, her anger found tiny outlets everywhere: shoving in lunch lines, rude names flung at unsuspecting classmates, graffitied desks, smoldering glares when Miss Gillow pressed with “Are you feeling okay?” or “Is there anything wrong, Amber?” – but whenever his name came up, those hazel eyes suddenly pacified, the fighter’s stance softened, that mouth effected neutral smiles, and a demure young lady emerged, only vaguely curious about Father’s whereabouts; these reversals vexed counselors, threw them off-balance, like firefighters reaching an inferno’s site to find no smoke, nor heat, not even sparks, just houses collapsing amidst silent winter air.

Quibbles, of course, abound: I question whether “Amber” and “Amber’s” should count as distinct; I note the tonal awkwardness of “Father’s”; and I lament that the semicolon after “whereabouts” is writhing in existential pain, denied its rightful identity as a period.

But in such games, rule-bending is inevitable. And to Hart’s credit, these stories actually function as stories. (Never a guarantee in the labyrinths of Oulipo.)

Another of Hart’s centipedes:

First, Melissa left him, pledging love and reunion someday soon; second, she threw herself into the crucible of New York; third came two grueling years given over to worshipping deadlines, answering emails through lunch, hallucinating PowerPoint, day-dreaming Excel, squeezing selfish “errands” (like dental surgery) among more pressing corporate duties; fourth, there were occasional visits home, giddy weekends enjoyed mostly in bed, sleeping late together, pondering no future further than breakfast; fifth, finally, a permanent return, plagued by halting conversations, dodging topics that frightened them, such as whether their flame still smoldered, or if it had quietly spent its last fuel.

The constraints manage to serve the art, imbuing the story-sentences with nervous energy, as if they are navigating around some unutterable thought. The clauses search and twist, forbidden to return home, scrambling onward until an inevitable collapse.

One more illustration:

Aged eighteen, bored and restless, he took to sea: working odd jobs on ships, shunning routine, quitting when promotion beckoned; slipping between continents, self-medicating insomnia, pouring paychecks into liquor bottles; leaving no traces or friends among port cities, only half-concerned lovers, their faces scarcely more permanent in memory than Etch-a-Sketch portraits; until one day the trade-winds died, depositing his ship at a leeward harbor, whose piers were crowded with strange waterlogged fragments, forgotten decades earlier but improbably carried here, along fate’s currents, confronting him now as pale, bloated symbols of what neither sailor nor soul can ever really escape: oneself.

Great literature? Certainly not. But Hart’s work is never dull and only occasionally forced. Those merits alone place it in the higher ranks of Oulipo.

RANDOM INCIDENTS (MIT Press, $20), a collection of poems coauthored by Ben Orlin and Hendree Milward, takes up a different Oulipo tradition: randomness as the seed of creativity. Each cluster of poems is inspired by a random sample drawn from a different aspect of our digital world — for example, a randomly chosen pair of YouTube channels. The poets then seek to weave meaning from the fragments.

Note: I say that they seek to weave meaning. I do not say that they succeed.

Most indicative, and most abysmal, are the Wikipedia-inspired poems. The authors used Wikipedia’s “random article” button three times per poem, and then stitched the three disparate topics together. At best, the stitches are horrifyingly obvious. At worst, there are no stitches at all, just a loose Frankenstein of juxtaposed limbs, held together by inertia alone.

(One senses that Milward is the more capable poet, and that Orlin is responsible for the more severe embarrassments, but that is cold comfort. If you mix chicken soup with house paint, the result belongs neither on spoons nor on walls.)

I suppose I must inflict an example upon you. Here is one of their more valiant efforts, titled “Name-Giving.” (If only the whole book met this modest standard of quality.)

Wikipedia #6:
Name-Giving

1.

In the place we call Romania,
by the river they call Buhai,
there is a persistent little trickle
of water along the ground,
and since we cannot help
but give names to things
the trickle is called Pârâul Întors:
the inner creek.

Without the name, it’d still be inner.
Without the name, it’d still be a creek.
And the name will not keep it
from drying up.

2.

There are certain sunflowers
called “chamomile,”
and certain of these
called “mayweed.”
But this was not good enough
for scientists, who decided
the specimens were better suited
by the name Tripleurospermum.

Flowers must look different to them.

3.

In Washington DC,
on the 8^th of December, 2008,
film critics gathered to assign names.
Slumdog Millionaire they called “Best Film.”
Danny Boyle they called “Best Director.”
Mickey Rourke they called “Best Actor.”

Maybe such names really work,
like magic.
Maybe, for the next year,
Mickey Rourke really was
the Best Actor on Earth,
until they gave the name
to someone else.

Beyond Wikipedia, Orlin and Milward sample other digital universes: posts on Reddit, photos on Instagram, videos on TikTok. While these poems are no better, I admire the ingenuity of the sampling methods: in the absence of a “random post” button, the authors cleverly simulated random walks across the platform, tapping into the “dark matter” of unseen posts, those disfavored and concealed by the algorithms.

As sociology, the results are illuminating; as mathematics, effective; and as poetry, dire beyond redemption.

In short: a characteristic work of Oulipo.

Gizem Karaali’s AUTOBIOGRAPHY IN THE FORM OF INTEGER SEQUENCES (Milkweed, $17.99) is a poetic riff on the Online Encyclopedia of Integer Sequences. It probes the individual personalities (and the shared impersonality) of numbers, the extent to which they can (and cannot) be imbued with life.

Titles of some entries:

“Apartment numbers at which I have lived, in chronological order”

“Hours I spent on my phone each day for the last year”

“Ages at which I felt a phase of my life had ended”

“Ages at which, looking back now, I feel a phase of my life ended”

“Page numbers of the poems that have changed me” (Karaali provides no information on the books or the poets, only the pages)

“Number of my parents still alive on my birthday each year” (The sequence begins with a long string of 2’s, but reader and writer both know that the sequence cannot rise, only descend, sooner or later, in two terrible steps)

Some numbers are freighted with heavy-handed significance (“funerals attended each year of my adulthood”). Others are obscure and trivial (“number of times my friend Ruby said the word ‘anarchic’ in successive conversations last month”). Others seem to be saturated with hidden meanings, accessible only to the poet, while others must be inscrutable even to her, the inhuman data forming a kind of anti-autobiography (or, more properly, auto-anti-biography).

The collection works because Karaali herself does not seem settled on the basic questions. What can numbers do? What can they not do? Karaali offers tentative answers but no single thesis. This is, I suspect, the formula for successful Oulipo: careful calculation plus unyielding skepticism of the enterprise itself.

Fourth and finally, Rob Sleezer’s ON CONCISION (Princeton University Press, $14.95) is not necessarily a work of Oulipo proper, but rather, the kind of literary prank that Queneau et al might appreciate.

The book is structured as a sort of tedious page-a-day calendar, with each leaf imparting a famous person’s thoughts on concision. Invariably, our heroes are in favor (ardently in favor!) of expressing oneself concisely.

What are the proper proportions of a maxim? A minimum of sound to a maximum of sense.

Mark Twain

I have only made this letter longer because I have not had time to make it shorter.

Blaise Pascal

Just as a good editor of prose ruthlessly prunes out unnecessary words, so a designer of statistical graphics should prune out ink.

Edward Tufte

The quotations themselves are unfailingly concise. Sleezer is interested in form, and how it mirrors (or undercuts) meaning. The book’s epigraph (yet another quote, the only one not concision-themed) is from Jim Propp: “Form without meaning is incomplete, but even before meaning attaches itself to form, form can point the way towards meaning.”

Alas, I cannot stop yet. You must experience the full soporific effect of the repetition:

You know that I write slowly. This is chiefly because I am never satisfied until I have said as much as possible in a few words, and writing briefly takes far more time than writing at length.

Carl Friedrich Gauss

Not that the story need be long, but it will take a long while to make it short.

Henry David Thoreau

Concision in style, precision in thought, decision in life.

Victor Hugo

Vigorous writing is concise. A sentence should contain no unnecessary words, a paragraph no unnecessary sentences, for the same reason that a drawing should have no unnecessary lines and a machine no unnecessary parts.

Strunk & White

I began as a naturalistic painter. Very quickly I felt the urgent need for a more concise form of expression.

Piet Mondrian

French parents teach their kids to be concise and amusing, to keep everyone listening.

Pamela Druckerman

Examine every word you put on paper. You’ll find a surprising number that don’t serve any purpose.

William Zinsser

The catalogue goes on. And on. And on. All this concision adds up to something quite bloated. I’m offering here only a sample of Sleezer’s onerous compendium—but it must be sizable sample, or else the punchline will fail to land.

In all pointed sentences, some degree of accuracy must be sacrificed to conciseness.

Samuel Johnson

Eliminate the unnecessary so that the necessary may speak.

Hans Hofmann

The most valuable of all talents is never using two words when one will do.

Thomas Jefferson

The concise man makes one think; the verbose bores.

Edouard Manet

There is no truth which cannot be given in fifty words; the truth is always concise.

Barry Malzberg

Most directors subscribe the principle that less is more, and the best direction is the most concise direction.

Joshua Marston

A story should, to please, at least seem true,
Be apropos, well told, concise, and new:
And whenso’er it deviates from these rules,
The wise will sleep, and leave applause to fools.

Benjamin Stillingfleet

It will be my earnest aim that The New York Times give the news, all the news, in concise and attractive form.

Adolph Ochs

The more alien and strange a world or situation is, the more concise you have to be if you want the reader to follow you.

Karin Tidbeck

By the book’s halfway point, the effect becomes almost unbearable. All these paeans to concision—none original, none deeper than the skin of an apple, none saying anything the others do not already encompass. A stack of paper squandered on the monotonous recitation of the same platitude.

Then, finally, you reach the last and longest quotation.

Concision means you have to be able to say things between two commercials. Now that’s a structural property of our media—a very important structural property which imposes conformism in a very deep way, because if you have to meet the condition of concision, you can only either repeat conventional platitudes or else sound like you are from Neptune. That is, if you say anything that’s not conventional, it’s going to sound very strange. For example, if I get up on television and say, “The Soviet invasion of Afghanistan is a horror,” that meets the condition of concision. I don’t have to back it up with any evidence; everyone believes it already so therefore it’s straightforward and now comes the commercial. Suppose I get up in the same two minutes and say, “The U.S. invasion of South Vietnam is a horror.” Well, people are very surprised. They never knew there was a U.S. invasion of South Vietnam, so how could it be a horror? They heard of something called the U.S. “defense” of South Vietnam, and maybe that it was wrong, but they never heard anybody talk about the U.S. “invasion” of South Vietnam. So, therefore, they have a right to ask what I’m talking about. Copy editors will ask me when I try to sneak something like this into an article what I mean. They’ll say, “I don t remember any such event.” They have a right to ask what I mean. This structural requirement of concision that’s imposed by our media disallows the possibility of explanation; in fact, that’s its propaganda function. It means that you can repeat conventional platitudes, but you can’t say anything out of the ordinary without sounding as if you’re from Neptune, a wacko, because to explain what you meant—and people have a right to ask if it’s an unconventional thought—would take a little bit of time…. Here, our media are constructed so you don’t have time; you have to meet the condition of concision. And whether anybody in the public relations industry thought this up or not, the fact is that it’s highly functional to impose thought control.

Noam Chomsky

Sleezer’s book, amusing as it is, suffers a limitation common to satiric works. It punctures the myth of concision, pops the overinflated balloon—but in the silence that ensues, it has nothing else to say. The straitjacket of concision is thrown off; but one wishes Sleezer (or Chomsky, or Twain, or anyone!) would chime in with the wisdom that concision had withheld from us.

Ah well. I suppose it’s fun to see balloons (and constraints) pop.

On Amazon, Sleezer’s book is listed (among other categories) as a work of “Inspirational Quotations.” Perhaps just a marketing blunder, or perhaps another layer to the satire, but I choose to read it in earnest. What better form of inspiration than to declare that all your prior inspiration was truisms and lies?

Delta math login

Posted on August 27, 2024 by b.pate

Delta Math: An Online Program

You can make your own projects with Delta Math, an internet-based tool that’s free. You can distribute an assignment to the group once it has made. The pupils receive immediate input along with detailed guidance on how to resolve any issue. The kids adore this fantastic tool, which is great to use in a math lesson! After making an account, users can begin creating tasks that match their classroom. Additionally, they will learn how to distribute homework to other instructors. Whether you’re a teacher trying to enhance your students’ math skills or vice versa. Give your pupils extra help in delta math. It might be a useful tool.

How Delta Math Access by teachers?

Teachers can construct courses and assign pupils particular issue types using Delta Math. Teachers can design their own unique problems that are self-graded in the Plus edition. They choose particular pupils or student groups. This is well suited for instructional and educational differentiation. The site records information and can be by educators. Access learning statistics, evaluate their development, and generate ad hoc assignments for extra practice and corrective work. By adding questions from the website library, with thousands of pre-made problems. Teachers can construct new ones. Students receive comprehensive, step-by-step instructions on how to answer each query.

Steps for Delta Math Login

Go to the official site for Delta Math
Browse to the authorized Delta Math website using a public web browser.
Select Your Position: There are two ways to log in when you get to the Delta Math portal. I am a teacher and a student. Choose the one that defines your position.

Learner Login:

To indicate that you are a student, select “I’m a pupil.”
Enter the special class login that your teacher sent you.
Request your teacher to give you a student identification number if you don’t already have one.
To gain access to your Delta Math regard, select the “Go” option.

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Enter your login credentials to access your Delta Math educator profile.
To register as a new teacher, select “Sign Up.”
Getting Around Your Considerations: Your dashboard can be once you’ve logged in. Examine the materials that are accessible, set goals, and get training.

Features of Delta Math

No limits on assignments
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1800+ pre-made issue categories
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Grade-by-auto
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Proof of students’ development and learning
Techniques to prevent cheating
Provide and duplicate the tasks.

Creating Assignments in Delta Math Login

In Delta Math, creating an assignment is a straightforward and uncomplicated process. After logging in, choose Create/edit. Then Generate basic assignment from your teacher login. Give the task a name and choose the class you wish to provide it to. Then, choose the talents you wish to assign by clicking on the talents section. After that, select a date by clicking on the Due date tab. After finishing, select “Create Project.” You can create as many projects as you like using your instructor account. You can attach movies, choose a particular problem subtype. Create your own challenges using the Plus edition. They will use their credentials to access shared activities.

Significance of Delta Math Login

Performance Monitoring

Users of Delta Math can check their advancement. You can keep track of your results once you log in. Establish goals for your own arithmetic skill and note your areas of weakness.

Tailored Education:

Delta Math customizes its content for each user. Students can get tasks and test challenges. Their requirements and ability levels by logging in. Teachers can also keep an eye on their pupils’ development and offer focused assistance.

Reactions and Evaluations:

When a project or assessment is finished, Delta Math offers thorough feedback. Students and teachers can recognize strengths and problems. With the aid of this evaluation. Encourage a more effective method of studying.

Libraries of Resources:

The website provides a vast collection of educational resources, practice questions, and tests. Users can access these resources at their convenience with the help of Login Delta Math.

Project Accessibility:

With Delta Math, teachers can generate assignments for their pupils. They can assign particular assignments, keep track of students’ progress, and check their work by signing in.

Fixing Problems with Delta Math Login:

In the event that you have any trouble accessing Delta Math. Take into account these technical suggestions:

Verify the Connection to the Internet:

To prevent network problems, make sure that your internet link is steady.

Check Your Certifications:

Verify again that the class code, login credentials, and password you supplied are accurate.

Speak with Support:

On their official site, Delta Math has a support site in case you have any more login issues. For help, get in touch with their support staff.

Browser Interoperability:

Make sure the web type you are using is suitable. For the optimal user interface, Google Chrome is by Delta Math.

Empty the Browser Store:

Final Words

With a vast range of mathematical resources and tools, Delta Math is an invaluable tool for both instructors and learners. Exposure to a plethora of materials, tracking of advancement, and a customized learning experience are all possible with a Delta Math login. Through the straightforward procedures described in this article. Learners and teachers alike can use Delta Math and improve their ability to compute. Delta arithmetic can help students who want to achieve academic success in arithmetic or teachers who want to deliver exceptional math instruction.

The post Delta math login appeared first on The Mathematics Master.

Mathematics and Social Sciences

Posted on August 14, 2024 by b.pate

Mathematics in the Social Sciences

The analysis of numerals, evidence, and models is as mathematics. Math is in the social sciences to create quantitative representations of real-world occurrences. A procedure, an output, and the input settings are all part of mathematical structures. These elements enable the description of occurrences by mathematical models using input data. Recognize connections between various event components and plane predictions. Social sciences are a group of disciplines. They include politics, the past, sociology, finance, and so forth. They research human societies and interpersonal relationships. Specifying these criteria can be difficult. Social science data and criteria are not measurable. If an economist wants to know how relationships between various genders are by cultural standards.

The Social Sciences’ Linguistic

Any scientific subject, including social research, is on supreme mathematics and alphabet. In the fields of politics, sociology, finance, or mental health, for example. Every field uses a different vocabulary and terminology. To explain and examine social events. With its accuracy and generality, mathematics offers a universal language. It cuts over linguistic and cultural divides. the models, ideas, and terminology used in mathematics. Social researchers have the ability to convey intricate principles and hypotheses.

In the Social Sciences, Math Applications

Social scientists can better understand social problems and queries. Concern communal living by using mathematics.
Analysts examine market dynamics through the use of mathematical frameworks.
Mathematical structures are by sociologists to address inquiries. On the interactions between people and human society.
Calculated models are by political researchers. To analyze how voters respond to certain situations.
To explain historical events and comprehend historical techniques, historians use mathematical diagrams.

Finance

Macroeconomics, the study of financial circumstances, is in economics, which examines market relations. The study of personal economic preferences is microeconomics. Accounting represents corporate and individual actions. As market situations using computational models. Economists can also establish equilibria and tie many financial variables. It is to one another using supreme mathematics numbers. Based on the trends seen in those models, make forecasts. The model is one example of this type. It states that the equilibrium price is when supply and demand are exactly equal. One may predict the economy using these models. These forecasts relate to trends in economic indicators. Like jobs as well as financial periods like boom-bust phases.

Study of Social Networks

Comprehending social networks is essential to sociology and psychological studies. Recognize the impact, behavior, and dissemination of information of people. The idea of graphs in the supreme mathematics chart in mathematics offers a foundation. It is for social network analysis and visualization. Researchers can map links, and identify important players. Find trends through social network study. It exists in a range of social formations, including organizations and partnerships. It can highlight the nodes of impact. Show how information propagates throughout a network. even forecast the chance of specific group actions.

The Psychology of Math

Therapists use mathematical methods to comprehend human thought processes and actions. Mathematical models and data are in mathematical psychology. It develops theories regard thinking, decision-making, and mental functions. Mathematical concepts are the foundation of psychometric evaluations. It is like personality assessments and IQ testing. They quantify and analyze individual variations. Furthermore, psychologists can investigate subjects. They like thinking, recall, and learning with the aid of modeling math.

Modeling

In Economics

The interconnectedness of the economy was a concern of classical economics. The cost of manufacturing, or the price of inputs, determines the price of goods. Output is what consumers are willing to pay at the time. Calculus was by Cournot to determine the prices and elements in his model. He developed the quantity demanded versus price formula.
F(p) = D

Predict the evolution of the Marshallian flexibility theory. They propose that using the demand difference to yield
F ′(p) + F(p) = 0.

Cournot went on to examine the most issues.

In Sociology

It is possible to represent implications from the standpoint of the game. Faden studied situations in which offenders and victims each followed their own logic. When there are no police, they collaborate. When there are cops and criminals, they collaborate because of the victims division. Such modeling is the tip of the iceberg in a complicated societal issue. There may be differences in three population sizes and distributions. The relationships between them, and their driving forces. For example, if we concentrate on the conflict. It is for survival between two groups, let’s say victims and criminals. The likelihood of the latter will increase as the former increases. It produces a basic system of simultaneous equations like

( dfrac{1}{v} = f(c) and (dfrac{dv}{dt}) )

( f(v)= (dfrac{dc}{dt})dfrac{1}{c} )

In Psychology

Fechner’s Law is an ancient example of an inner observer model in psychology. Ernst Heinrich Weber’s research is creating this legislation. In a few of his tests, he gave participants weights to hold and asked them to report any increases in weight. Fechner suggested noting discernible variations in experience because feelings are immeasurable. The stimulus, R, can be as pounds of weight. Note the constant point (c), at which a discernible change has about by the trigger.

( c = dfrac{dR}{R} )

This partnership takes the shape of

( dfrac{dR}{R} = c(dS) )

This suggests that all stimulus alterations are uniform and aggregateable. Thus, the equation can be to quantify feeling.

Difficulties and Ethical Issues

There is no denying of the relationship between mathematics and the social sciences. But there are also difficulties. Careful thought should be to data privacy, and ethical issues. The possibility that mathematical models could reinforce assumptions. To overcome these issues, investigators need to be thorough. They have to work toward mathematical techniques in the social sciences. It is in an appropriate and conscientious manner.

Final Words

Evidence to this is the union of mathematics and social sciences. It is an interdisciplinary structure and capacity to the interwoven reality in which we live. The useful tools that mathematics offers are its accuracy and inclusiveness. They are by the supreme mathematics and social scientists. To investigate, dissect, and comprehend a variety of social processes.

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Geometric Flower

Posted on March 6, 2024 by b.pate

Why do so many biological phenomena follow this pattern that has long baffled scientists? The association of geometric flower can be random. In other circumstances, the ratio exists as a result of the evolution of that specific development pattern. For a population to survive, this is the best strategy.

These universal symbols are to be in flower extracts. It connects with the deepest aspects of the human mind. Thus bringing our altered selves to light. The fact is that arithmetic has a huge impact on many aspects of our lives. While you are not required to perform complex computations on a daily basis. The advantages of having a firm grasp of the basics are considerable. One such field in which math is essential is floral design. Do you think that’s absurd? Let’s investigate.

What Does Sacred Geometry Means?

The fundamental geometry of nature is to as sacred shape. It is proportions, fractals, recurring patterns, and forms in nature. It is as the symmetry of awareness according to Spirit Studies. Its central thesis is that all awareness, including human one, exists. It only uses sacred geometry as a foundation. We can start to perceive and comprehend where we have come from, where we are right now, and where we go because it is.

Flower Symmetry: Flower of Life

Another holy geometric shape is the blossom of life. It serves as the creation’s emblem. It is by first forming a round, following it to its edge, and creating another. Each circle is the same size and starts one radius apart from the others. The flower of life contains both male and female power, despite the fact that it appears to be a female force. Male energy makes up the ends, whereas female energy makes up the curves. 61 circles form the entire flower of life. However, most of the significance is in the 13 rings of the plant of existence.

Why does geometry exist in flowers?

Geometry permeates our environment. If you pay serious attention, you’ll see patterns: some flowers have three petals, some have five, a spiral expanding, etc. The two items are flowers and foliage, both of which have distinctive shapes. Even rarely discover the difference between face flowers, linear, and auxiliary elements. The shape or design of symmetrical flowers is on geometric concepts. For instance, many flower crowns in the plant kingdom grow with a specific arrangement of petals, sepals, and swirls. They line up with the Fibonacci sequence of figures.

General Natural Patterns

The patterns of some flower symmetry are the following:

Harmony

The contemplation or spin of a pattern part to create a close repetition.

Spirals

A constant curve that widens (or narrows) around a center point.

Fractals

Repeating motifs on smaller areas

Tessellations

Tiles repeated in patterns over something flat

Purpose of Flowers Symmetry

A flower’s purpose is to draw visitors. As a result, these flowers have modified their look and selection in response to their clients. Flower symmetry has evolved over several eras, according to evolutionists. It might range from being bilateral or radial to even being unbalanced. Additionally, researchers discovered that bilateral flowers made it easier for pollinators to get down. When the Fireweed plant’s bottom petals were to be more symmetrical. More nectar was by it.

Flowers with Geometry

Daisy (Ligulate)

It’s simple to clarify. The strap-like leaflet of Compositae ray flowers is as a ligule. So a flower that resembles a daisy is one that has ligulate flowers. The trend today is for petals that resemble outer rays. Actinomorphic flowers are ligulate blooms.

Tomato Flowers (Rotate)

Whirl flowers are quite typical. The petals of these flowers aren’t merged; if they are, it’s only into a very short vessel. It gives them their peculiar appearance. No fusion or Corolla lobes. Rotate florals are actinomorphic.

Lilies (Stellate)

Fascinating flowers have stellate petals. They are align and have a star form. The fact that these actinomorphic blossoms are up of tepals rather than petals is what makes them particularly intriguing. Some flowers have sepals and petals that are identical in comparison. The sepals of many flowers are green and striking, unlike the flowers.

Bedstraw (Cruciform)

Floral known as cruciform are cross-shaped and have four petals. Their symmetry is actinomorphic. This shape is present in many flowers. Even though they are little white numerals that we miss.

Buttercup (Cyanthiform)

A cyanthiform flower has a bowl- or cup-like form. It possesses actinomorphic or radial shapes. The bowl-like shape is by the upward-curving lobes.

Harebell (Campanulate)

Campanulate denotes a bell-like or rounded structure. All bell plants are, in reality, Campanulate. Their family is as the Campanulaceae for this reason. The corolla funnel or collar is longer, although they are still symmetrical. Actinomorphic florals include campanulates.

Blueberry (Urceolate)

When they look down, urceolate flowers resemble tiny urns or jars. They feature a circular corolla bulb and tidy petals at the tip. They are achromatic. Particularly the bell heather, urceolate blooms are well represented by the warmth.

Mint Flowers (Labiate and Bilabiate)

Flowers called labiate have mouths or are winged. The flowers resemble an opening with an upper and lower lip in the shape of a neck. They have a lower lip that is larger and lengthy, with joined corolla lobes. Flowers with labia display bilateral or zygomorphic uniformity. Two lips are on bilabiate flowers. Both of these are merged.

Final Words

From leaves to vegetation, creatures to flowers, bugs to fruits, harmony is all around us. The geometric flower of the natural world promotes peace and equilibrium while also adding to its appeal. It is pollination that gives the structures their variety and elegance. It makes pollinators flock to flowers. Although not all plants are symmetrical to be lovely. Those that do have a way of drawing our attention. It appears that the explanation for this is that our brains want to look for patterns. Particularly in a world that sometimes seems somewhat confusing.

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Trigonometric Ratios Worksheet Answers

Posted on March 4, 2024 by b.pate

The Greek words trigonon, “triangle,” and metron, “measure,” combine to form the word trigonometry. In the third century BC, the Hellenistic civilization gave birth to science. through the use of geometry in astronomical studies. While Indian scholars created the earliest tables of figures. They are for trigonometric ratios worksheet answers. It is also known as trigonometric operations as a sine. The Greeks focused on chord computations.

Throughout the past, trigonometry has been in geography, mapping, space mechanics, and sailing. There are several alternatives for trigonometry. Trigonometric solutions are rewritten using these trigonometric equations to make them simpler. Find a more practical way to express them, or work out a formula. Sine, cos, tan, cosec, sec, and cot are six crucial trigonometric operations. The trigonometry-related questions will be in this post.

What Do You Understand by the Word Trigonometry?

Thought to be one of the most crucial areas of maths is trigonometry. Trigonometry is a term that fuses the words “trigonon” and “metron.” It is the examination of how a right triangle’s sides and angles interact. Utilizing methods and opinions, aids in identifying and assessing the right-angled triangle’s unknown sizes. In the basics of trigonometry, angles measured, and problems involving angles. Sine, cosine, and tangent are the three fundamental operations of trigonometry. These three fundamental ratios or functions cotangent, secant, and cosecant—can be to construct further significant trigonometric equations. The fundamental ideas of trigonometry are upon these parameters.

Father of Trigonometry

The study of the correlations between triangular side dimensions and angles is trigonometry. Hipparchus is the inventor of trigonometry. The Greek mathematical expert Hipparchus discovered trigonometry in the second decade BC. He created the first trigonometric table to resolve several circular trigonometric problems.

What are Trigonometric Ratios?

The values of each trigonometric operator are as ratios. They are focusing on the right-handed triangle’s edge trigonometric ratio worksheets. A right-angled triangle’s side lengths in relation to any of its acute vertices. They are to as a specific angle’s trigonometric ratios. The right triangle’s three sides are as follows:

The broadest side is the hypotenuse.
Side that is perpendicular to the right angle
The bottom side that runs parallel to the angle

How to Find Trigonometric Ratios?

The corresponding trigonometric equations, where a, b, and c are the lengths of the sides. They are in the preceding illustration. They are to the known angle A to produce these ratios:

The sine value (sin) is the ratio of the side across from the position to the hypotenuse.

(Sin A= dfrac{Opposite Side}{Hypotenuse Side}) = (dfrac{a}{c})

The cosine value is the ratio between the hypotenuse and the neighboring side. It is the part of the triangle related to the right angle.

(Cos A= dfrac{Adjacent Side}{Hypotenuse Side}) = (dfrac{b}{c})

The tangent value is the proportion of the opposing leg to the adjoining line.

(Tan A= dfrac{Opposite Side}{Adjacent Side}) = (dfrac{a}{b})

The hypotenuse of a right triangle is the side that faces away from the (90)-degree curve. It is the largest half of the triangle. One of the two sides that are to angle A and the neighboring leg is to the side of angle A. The opposing side is to as being perpendicular to angle A. The terms perpendicular and base are to describe the reverse and neighboring sides.

Table Formula for Trigonometry

	( 0º )	( 30º )	( 45º )	( 60º )	( 90º )
( sin )	( 0 )	( dfrac{1}{2} )	( dfrac{1}{sqrt{2}} )	( dfrac{sqrt{3}}{2} )	( 1 )
( cos )	( 1 )	( dfrac{sqrt{3}}{2} )	( dfrac{1}{sqrt{2}} )	( dfrac{1}{2} )	( 0 )
( tan )	( 0 )	( dfrac{1}{sqrt{3}} )	( 1 )	( sqrt{3} )	undefined
( cosec )	undefined	( 2 )	( sqrt{2} )	( dfrac{2}{sqrt{3}} )	( 1 )
( sec )	( 1 )	( dfrac{2}{sqrt{3}} )	( sqrt{2} )	( 2 )	undefined
( cot )	undefined	( sqrt{3} )	( 1 )	( dfrac{1}{sqrt{3}} )	( 0 )

Uses of Trigonometric Ratios

The following are a few uses for trigonometry:

Determining the peaks of large mountains or skyscrapers
Calculating the length between the beach and the sea
Measurement of the separation between two heavenly bodies
Calculating the output of power from solar cells at various orientations
Expressing several physical parameters, including electromagnetic and mechanical motions, etc.

Trigonometric Worksheet Answers

Example 1: A man is looking at a tree that is (57) feet tall. He determined that the tree’s shadow was (26) feet long. Can you assist him in estimating the angle of altitude of the sun from the top of the shadow?

Solution:

Let x be the angle of height so,

(tan x= dfrac{57}{26}= 2.1923)

( x= tan^{-1}(2.1923))

Or (x= 65.4) degree

Example 2: If the hypotenuse of a right-angled triangle ABC is 5 cm, the base is 3 cm, and the perpendicular is 4 cm, and if ACB =, then determine the tan, sin, and cos angles.

Solution:

Given values

Hyp. AC= (5) cm

Base BC= (3) cm

Perp. AB= (4) cm

tan = (dfrac{4}{3})

sin = (dfrac{4}{5})

cos = (dfrac{3}{5})

Final Words

The connection is using trigonometric ratios. Such as sine, cosine, tangent, cotangent, secant, and cosecant. Trigonometric ratios worksheet answers have been throughout the piece. It is how to calculate them and how to solve a few related issues. Learners may access the site to learn more about trigonometry and associated topics. We have provided detailed explanations.

FAQS

SOH CAH TOA: What is it?

The acronym SOH CAH TOA can be used to recall the trigonometric ratios equation so that:

SOH: The proportion of the opposite side to the hypotenuse is the sine function.

CAH: Cosine is a function of the side nearest to the hypotenuse, or CAH.

TOA: The ratio of the opposing and neighboring sides is known as the tangent.

What connection does there exist between sin, cos, and tan?

The relationship of the sine and cosine functions is what determines tangent functions.

Tan equals ( dfrac{sin}{cos}).

How are cotangent, secant, and cosecant calculated?

The cotangent is the base/perpendicular ratio of the next and divergent sides.

Hypotenuse to neighboring side ratio is known as secant (Hypotenuse/Base).

The cosecant (Hypotenuse/Perpendicular) is the percentage of the hypotenuse to the opposing side.

How can you determine which is near or opposite?

The portion that is other than the angle you are attempting to solve for is known as the opposing side. The end next to the angle you are calculating for is the neighboring side.

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AP Calculus AB vs BC

Posted on February 27, 2024 by b.pate

Regarding math, two prominent options often emerge: AP Calculus AB and AP Calculus BC. These courses are your gateway to college-level mathematics, but what sets them apart? Let’s dig into the differences between AP Calculus AB vs BC to help you make an informed decision that sets you on the path to mathematical success.

Understanding AP Calculus AB and AP Calculus BC

AP Calculus AB is an introductory college-level course that delves into differential and integral calculus concepts and applications. The course culminates in an AP Exam designed to evaluate your understanding of theoretical concepts and problem-solving skills through applying calculus principles.

On the other hand, AP Calculus BC mirrors AP Calculus AB’s exploration of these concepts and applications but adds several additional topics. AP Calculus BC covers more ground than its counterpart, even though both courses demand the same core skills.

AP Calculus AB vs. AP Calculus BC: Key Differences

While both AP Calculus AB and BC share a foundational curriculum, several significant distinctions set them apart:

Depth and Breadth: AP Calculus BC explores calculus more comprehensively, encompassing a broader range of topics than AP Calculus AB. BC includes infinite sequences and series, parametric equations, polar coordinates, and vector-valued functions. This broader coverage makes BC a more challenging and rigorous course.

Course Duration: Calculus AB is designed to cover one semester of college-level calculus over one year. This encompasses definite integrals, derivatives, and the fundamental theorems of calculus. Conversely, Calculus BC covers a full year of college-level coursework in one year, containing all of the AB material and additional topics.

Prerequisite Considerations: Your choice between AB and BC may depend on your performance in precalculus, as both courses require it as a prerequisite. If you find precalculus relatively easy, BC may be a viable choice. However, if precalculus demands significant effort, AB may be a more suitable option to avoid overloading yourself with coursework.

Future Goals: Your career aspirations play a role in your decision. It could be the right fit if you’re eyeing fields like medicine or engineering and feel confident managing the BC workload. A high AP Calculus BC exam score may even grant you more college credit than the AB exam, depending on your university’s policy.

Preparation for the AP Calculus AB Exam

Preparing for the AP Calculus AB exam involves various strategies:

Flashcards: Create or find flashcards to review class topics. Carry them with you for quick study sessions during free moments.

Practice Tests: Take AP calculus ab practice exam to learn from mistakes and access detailed explanations for common questions.

Question Format Familiarization: Familiarize yourself with the exam’s question format to answer quickly and efficiently.

Study Guides: Utilize study guides provided by your AP calculus teacher or study with peers to enhance your preparation.

How to Utilize an AP Calculus Practice Exam

To make the most of an AP calculus AB practice exam:

Diverse Questions: Look for practice exams with diverse questions, correct answers, and detailed explanations. This helps you learn from your mistakes and become a more efficient studier.

Exposure to Questions: The more practice tests you take, the more questions you expose yourself to, providing insight into the test’s format.

Preparing for the AP Calculus BC Exam

Preparing for the AP Calculus BC exam follows a structured approach:

First Semester: Focus on answering free-response and multiple-choice questions on topics covered in your class.

Second Semester: Take and score a complete AP calculus BC practice exam, analyze your results, address weak areas, take another practice exam, and adjust your study plan accordingly.

FAQs

Why Use an AP Calculus AB Score Calculator?

An AP Calculus AB score calculator is wise because it leverages official scoring data from past College Board exams to offer precise and up-to-date insights. Understanding the components of a (3, 4,) or (5) AP Calculus AB score can significantly boost your confidence and readiness, making exam preparation smoother and less stressful.

Why Use an AP Calculus BC Score Calculator?

Utilizing an AP Calculus BC score calculator is a strategic move to help you plan effectively for your upcoming exam. These calculators use official scoring materials from prior College Board exams, ensuring the accuracy of the information provided. Equipped with the knowledge of what constitutes a (3, 4,) or (5) AP Calculus BC score, you’ll be better prepared to excel on the exam, alleviating unnecessary anxiety.

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