News & Updates
October 15, 2023
Daily Life Skills That Can Better Arithmetic Skills In Young Minds
Children who are being taught math have an opportunity to strengthen their arithmetic skills through varying activities and daily tasks that require similar skills that their lessons do.
The spectrum of comprehension in young minds is truly boundless, but every student needs a little extra boost from time to time.
The content of mathematics only gets more difficult as the education continues. Therefore, it’s important to recognize that children will need resources and help to supplement their in-school work. This can be represented through homework, in-class reviews, tutoring, and even fun online games.
Beyond the actual extra math help and skills to better their learning, there are things others can do at home. Guardians, tutors, babysitters, and other adults in a student’s life can aim to incorporate real-life scenarios that’ll have children subconsciously strengthening their arithmetic.
You can accentuate the arithmetic in various life skills, and it’s easier for the student to comprehend because it isn’t on a worksheet staring back up at them necessarily; it’s an activity they may want to participate in and not realize it uses some of the same skills they learn at school.
Children best explore and absorb their learning through play and hands-on activities. So, utilize their preferred playtime activities to exemplify easy math problems and solutions so they’re more inclined to take on the challenge!
It’s essential to remember that every age has a different level of basic knowledge and skills to have, so be sure to calibrate your arithmetic strengthening activities with the actual coursework the child is working on in their academics.
Here are some ways to better your student’s arithmetic skills:
- Include them in household activities that require numbers or quantitative measuring! This can take on the form of measuring baking ingredients, counting toys out of a toy box, grouping small amounts of objects, and more! If your child is older, start adding basic subtraction or addition into playtime to encourage organization, but also give them the sense of control they like to have over their choice of play!
- Yes, technology can sometimes have its benefits when exposed to young minds. There is a wide array of math and learning-oriented apps and games to download, especially with characters and storylines your child may adore! If you find one your child pays utmost attention to, allot a little bit of time every day (no more than an hour) for them to play the math game. Including a fun concept, the child grasps will further motivate them to participate in the arithmetic work.
- This may be a bit of a given, but never stop remaining in contact with the child’s teacher! If they have separate teachers for separate subjects, be sure to get the direct contact information to ensure direct communication. You can even update the teacher on what work is done at home to reiterate that you want to be on the same page as the child’s school happenings!
- Timekeeping is essential in maximizing time and work management, but you can even include it in playtime! Depending on how young the student is, distinguish the numbers the big and small hands should be on when their playtime begins and finishes. That way, the child is learning basic time-telling and utilizing their surroundings to construct a happy productive environment. Setting time boundaries also encourages young students to complete their work more efficiently.
Now, there are some older students beyond basic arithmetic in their schoolwork that they may not need to count blocks or toys to strengthen their skills. However, arithmetic is all around us and should be worked on a little bit every day regardless!
If your child is in an upper elementary or middle school grade, money is a great real-life example of basic arithmetic. It also gets them to grasp a very prevalent life skill they’ll eventually handle on their own in adolescence and adulthood!
This would be a great opportunity if the child earns allowance for small home chores and tasks; at the end of a month or week, lay out the allowance and group together the currency by the amount and that’ll incorporate basic multiplication!
Regardless of the age or level of skill, there is one simple task that’ll overall strengthen their confidence in completing math. No child learns the same way, or the same place, so it’s very important that you are NOT comparing the student to others’ successes and failures.
There is such a thing as healthy competition, however, a child may not grasp the concept as effectively if they’re focused on simply “beating” another kid out when they do not have to. A child’s mind is elastic but can grasp even the smallest of negative behaviors, so constitute a constructive yet positive environment so they not only better their math but their human skills!
October 8, 2023
The History of Geometry as a Branch of Mathematics in Classical Antiquity
Math has been around longer than you think. Here’s a little inside scoop to the early stages of one of our most common math practices.
As we all know, many disciplines within Mathematics cater to particular scenarios, theories, and areas of quantitative knowledge. One of the more common and foundational disciplines aside from algebra would be geometry!
Like algebra, geometry is one of the oldest branches of mathematics still being utilized today! Merriam Webster’s formal definition of Geometry is articulated as “a branch of mathematics that deals with the measurement, properties, and relationships of points, lines, angles, surfaces, and solids”. Many of us initially learn the concept of geometry through shapes and graphs, but it seeps into so many more configurations! The purpose of geometry is to determine spatial relationships in our real-life scenarios and environments.
Where did geometry come from, you ask? Well, the term geometry originates from the two Greek phrases “gēo” (“Earth”) and “metron” (measurement). The Greek historian Herodotus (484-425 BC) was the one to credit Egypt with the subject’s origination. However, the Babylonian, Chinese and Hindu civilizations were the first to put it into practice; it was passed to the Greeks and Egyptians who popularized it through their obvious technological and cultural advancements.
If you’re familiar with the Babylonians, they were the ones to utilize clay tablets as an early form of documentation. Some of the tables found on said tablets indicate the need for square roots, the area of various polygons, even reciprocals. Not only that but it is believed the Babylonians were also the first to calculate measurements of a circle, specifically the circumference. This instigated the long process of discovering the infamous, infinite number of pi (π).
Unlike the Babylonians, the Egyptians kept records on papyrus scrolls. The Egyptians utilized geometry primarily for land surveying and construction; that’s how the Pyramids came into fruition! These groups of people were cognitively able to create and maintain streamlined systems of production! Geometry is such an elemental factor in the construction of our lives, it’s no wonder many ancient civilizations made great use of it as well.
The early Greeks were the ones to adopt geometry as a more rigid process. Greek philosopher Thales of Miletus (620-546 BC) was initially credited with bringing math from Egypt to Greece. After inhabiting the subject from their predecessors and neighboring communities, the early Greeks began prioritizing reasoning over results. Their emphasis on logic is incredibly representative in many influential works by early Greeks mathematicians.
One of the most infamous Greek documentations of math is known as Euclid’s Elements. It is a collection of 13 books the mathematician Euclid (appx. 300 B.C.) contrived. This series was intended to exemplify the functionality of many geometrical strategies. Plane geometry, geometric algebra, the geometry of a circle, elementary number theory, and proportions are just a few of the many subjects covered.
Euclidian math clenched the reins on math education for many centuries, yet many other Greek mathematicians were eager to contribute newfound knowledge. Infamous figures such as Pythagoras and Archimedes were able to provide foundational parts of geometry we still use today!
Because geometry was a perfect strategy to determine the location and environmental factors, the Greeks incorporated it into other practices such as astronomy. They philosophized so much about the universe they orbited, therefore the Greeks felt compelled to calculate what we now recognize as our solar system.
The most famous mind in early astronomy and classical antiquity is Ptolemy (circa 2nd century B.C.). Ptolemy was an Egyptian astronomer well known for his advancements in the model of our universe. He argued that our Earth was the center of the universe, thus formulated the geocentric solar system; this is what’s currently known as the Ptolemaic system.
The application of geometry in astronomy consumed much of Greek thought. Their passion and prioritization of logic over result fueled much of their fire for many of their geocentric theories. Much later on, Greek cosmologists were the ones to apply the practicality of geometry to the Earth’s measurements and its orbital cycle in terms of time, location, even season.
Many centuries passed before other civilizations throughout Asia and Europe were able to get their grip on geometry. Many geocentric Greek theories were debunked, and other communities took it upon themselves to study astronomy for results more than reason. Geometry has even gone on to evolve into different kinds, such as analytical geometry and progressive geometry.
Universally applying these dimensional techniques was to not only grow humankind’s knowledge but the many empires. Without the study of these dimensional elements in our ever-growing reality, much of our structural and technological integrity would be lacking today.
October 1, 2023
Through the early mathematical and philosophical learnings made by classical and ancient civilizations, we’ve come to better acquaint ourselves with the machinations of our Universe.
It’s no secret that much of our knowledge of the Universe was closely related to the fundamental discoveries both in Math and Science. Astronomy has proven especially resourceful in how we’ve been able to comprehend our place in the never-ending cosmos.
Despite the few early misconceptions, there have been many astronomers, mathematicians, and philosophers who’ve produced profound learnings we still apply today. Some of the most influential minds in math and astronomy derived from classical and ancient civilizations such as Greece, Rome, Babylonia, Egypt, even China, and India.
There’s distinctive material evidence to prove Babylonia was among the first to adopt mathematical practices when studying celestial occurrences. Back when they utilized cuneiform tablets (a pictographic system) for writing and recording, the Babylonians left behind various charts to indicate they were geometrically calculating astronomical placements!
The Babylonians, around 300 to 400 BC, started to use math as a way to calculate what is now known as the Zodiac chart! They divided the path of the Sun, Moon, and planets equally into 12 “phases”, then named them based upon nearby constellations. This was one of their many ways to compute where the planets, Sun, and Moon were which helped them eventually decipher what time of year it was!
It was one of the first and more advanced exemplifications of math being applied to astronomical affairs, but certainly not the last. Their early strategies centered around mathematical astronomy helped them calculate predictions of the position and track of Jupiter’s orbit!
The Babylonians, as well as the Egyptians, went on to adopt early astronomical calculations to formulate advanced calendars. They were the civilizations to primarily adopt astrology to better understand the machinations of our Universe. Unlike the earlier Mesopotamian communities, the Romans and Greeks used math and astronomy for other purposes.
Early Romans were very familiar with celestial beings! Ptolemy, a prominent Egyptian philosopher (when Egypt was under Rome’s rule) was most known for formulating an Earth-centric depiction of our solar system. Ptolemy’s findings were so heavily inspirational to the Romans, they even created mythological figures to represent their understanding of the planets!
The ancient Romans interpreted the role of the seven planets (they knew at the time): the Moon, the Sun, Mercury, Venus, Jupiter, Mars, and Saturn. It was the Greeks who ended up naming the planets, however, the Romans applied their own gods’ names to fit their narrative. We typically refer to these planets as their Roman names:
- Mercury (aka Mercurius): named after the god of commerce, eloquence, travelers,
- Venus: named after the goddess of love for its bright light and softer appearance
- Mars: for the god of war, rightfully named due to the planet’s bold red color
- Jupiter: the biggest planet in the system named for the head Roman deity, the god of sky and thunder, and King of the Roman gods in their mythology
- Saturn: the father of Jupiter, god of agriculture. Fun fact: based on Ptolemy’s model, Saturn is also named after Saturday (Saturn’s day)
Once the other planets in our solar system were discovered, they also received names in conjunction with Roman mythology. Uranus was named for the Roman personification of the sky, while Neptune was named after the god of the sea for its beautiful blue-green hue.
The Romans weren’t just accredited with naming the solar system; they created our current and most used yearly calendar! Before they adopted a prominent amount of Greek astronomy, the Romans paid meticulous attention to the placements of stars and planets in the sky to determine the lunar cycle. This aided them in growing a ten-month cycle to the twelve-month cycle.
Much of our current understanding of astronomy would be nowhere without the Greeks. They were able to provide many other civilizations with specific findings of more minute planetary aspects at the time- for instance, our own Earth. Because the helio-centric depiction of our solar system didn’t arise until the discoveries of Copernicus (1473-1543), the Greeks took observational astronomy to try and detail more about Earth.
One of Greece’s innovative philosophers and mathematicians, Pythagoras, pushed many discoveries forward with his work. However, he based many of his revelations on mathematical perfectionism rather than genuine quantitative reasoning. For instance, Pythagoras was the first to propose that the Earth was spherical not because it made “sense”, but because the sphere is considered a “perfect” 3D shape.
Pythagoras was not the only Greek to hypothesize major astronomical functions. Along came Aristarchus of Samos (310-230 BC) with his much more realistic contributions revolving around the Earth’s role in our Universe. He was the inspiration for Copernicus’s work.
Aristarchus focused mostly on the movement of the Earth and its size concerning both the Sun and Moon through eclipses! He procured three core premises that helped him articulate his findings.
While observing a lunar eclipse, Aristarchus confirmed through geometrical analysis that the size of Earth’s shadow on the moon further proves that the Sun is of greater size than the Earth. Although Aristarchus made true statements about the measurements of our planet & its surroundings, he still followed the inaccurate geocentric model of our solar system (rather than the heliocentric one).
Many many years passed before math and astronomy were applied to the RIGHT depiction of our solar system. Thanks to Copernicus and his proposal of an accurate heliocentric model, we were able to better detail why our Earth experiences the natural occurrences it does and how it affects other planets in our system.
It took quite a long time, and many other astronomers, to apply relevant math to confirm our place in the Universe. However, without the ancient and classical civilizations passionate about investigating our vast world, we would not have a strong basis to formulate our findings.
August 27, 2023
How the first recorded female mathematician Hypatia paved way for women in math and philosophy
The Great Alexandria
Many of us have been educated on humankind’s earliest civilizations. One of the more infamous yet mysterious communities we have collected knowledge on was Alexandria. It was a port city located in present-day northern Egypt, and one of the greatest Mediterranean cities to exist in our recorded history.
It was founded in approximately 331 BC by Alexander the Great, a fierce Macedonian ruler and military genius. Alexander left to pursue a takeover of nearby Persia, so Alexandria was left to be ruled by the Ptolemaic Dynasty for nearly three centuries; a famous ruler within this dynasty includes queen Cleopatra VII. Alexandria was known as the greatest city to ever exist and would later become a major hotspot for early Christianity, a center for religious turmoil from clashes between faiths.
A highly regarded figure that emerged from this time was true icon Hypatia.
Who was Hypatia?
Hypatia (355-415 AD) was one of the first well known feminine figures in philosophy, astronomy, and math.
“First” is a loosely applicable term in this discussion since the TRUE first female mathematician was Pandrosion, but Hypatia was the first to be well recorded and depict historical accuracy about one of the world’s most prolific civilizations.
She lived during the time of ancient Alexandria, but existed amidst an incredibly disorderly and violent era in the empire’s existence. Due to the turbulent chain of events that occurred during and after her time, it greatly accentuates Hypatia’s success to defy traditional odds inflicted upon Alexandrian women.
She was absolutely set up for greatness; her father was a well known mathematician, Theon of Alexandria (335-405 AD). It was most likely that she was taught and instructed by her father, which shows through the similarities in their separate works. Theon was most known for aiding in the preservation of Euclid’s Elements, a thirteen book collection encapsulating mathematical theories and proofs. Much of his work molded her own, as she took great effort in preserving the historical intellect of Greek heritage, especially in the fields of math and philosophy.
A well-loved Pagan, Hypatia was quite tolerant towards those of Christian faith.This is incredibly important being that she existed in a time where the Christians, Jews, and Pagans were all experiencing conflict with one another. Her well-composed demeanor helped establish a reputable relationship with the elite upper class; this dynamic even bled into responsibilities she held later in her life.
Hypatia’s realms of expertise were heavily male dominant fields of intellect, work, and discourse. Therefore, a woman entering that space with confident and productive contributions challenged the conventionality of Alexandria’s gender roles and dynamics.
What did she DO exactly?
Hypatia was credited with writing commentaries for various texts. She constructed a commentary focusing on Diophantus’ Arithmetica. This is a partially survived thirteen volume text harping on the development of number theory through equations. Her other prominent commentary revolved around Apollonius of Perga’s Conics. Conics is a dissertation about conic sections and their influence on modern and ancient analytical geometry.
Her commentaries greatly exemplified her high level of intelligence and understanding of math as well as philosophy. In fact, she was so well-versed that she became a very well-known
teacher at the Neoplatonic school of Alexandria; she taught philosophy and astronomy. This school grounded their particular education in the teachings of infamous figures Plato and Aristotle. It catered largely to both Christian and Pagan students, and many who were of the Pagan faith became loyal pupils and friends of Hypatia.
Aside from her educational endeavors, Hypatia also spent time constructing various tools for use in her fields, such as the hydrometer and astrolabes. Even though she didn’t invent them, she showed great familiarity with their functionality.
Hypatia’s great relationships with her pupils influenced the effects of historical events on her life. The religious divide in Alexandria was very tumultuous and chaotic, and possibly led to the burning of the Great Library of Alexandria. There were books from the Library believed to be kept in the Serapeum, a temple of the Greco-Roman god Serapis, led by Saint Theophilus of Alexandria. That temple was eventually destroyed as well due to its ties to the Great Library.
Coincidentally, Theophilus was associated with one of Hypatia’s devoted pupils, Synesius. These mutual ties temporarily permitted Hypatia to continue her work until her death. Once the theologian St. Cyril ascended into his power, he continued his uncle Theophilus’s work by enabling the violence against non-Christian civilians. Hypatia was gruesomely murdered by a group of fanatical Christians; Hypatia was a well-known Pagan and fell victim to their rioting.
Despite her demise, Hypatia remains a powerful feminist icon and influential mathematician. She represented great resilience, female empowerment and a perseverance to break the mold that’s been repeatedly inflicted upon feminine figures in the past and present.
August 20, 2023
How ancient Greece birthed an entire foundation of thought for modern day technology & intellect
Origins of Mathematical Knowledge
A multitude of classical and archaic communities paved the way for modern life and thought. Civilizations such as Rome, China, India, Egypt, Mesopotamia, Persia, and many more were able to start an array of practices- from SCRATCH. They were not privileged with many of the resources we have today, so they relied on futuristic thought and hard labor to create a life they were happy with (and proud of).
One of the fundamental empires is Greece, a civilization rich with philosophical thought, groundbreaking strategy, and a jubilant social nature. They provided a historical miscellany of concepts and contraptions, which lays ground for many of the practical modalities we modernized in order to build our own civilization. Humankind embodies many Greek contributions on an intellectual basis, and we don’t even know it!
There’s so much to dive into when exploring ancient Greek culture. Despite the ecosystem of knowledge they’ve nourished, there is a core contribution that initially planted the seed of their impact + influence on modern day thought.
Major Contributions from Ancient Greece
One of ancient Greece’s most influential contributions primarily involves the school of thought. Logic, philosophy, and academia was one of their strong suits, for many schools of thought were born and flourished. They all lay a foundation for other areas of thought to incubate. However, one of their greatest subjects of impact is mathematics.
Math is the backbone for many other areas of knowledge we’ve used such as Science, Astronomy, Architecture & Engineering, Warfare, even Agriculture. The Greeks needed to concoct a logical methodology to formulate tools, tricks, and processes that would build their empire as efficiently as possible.
LONG before modern technology, math was a slightly more laborious process. For instance, everything was written or through word of mouth; for instance, ancient civilizations kept charts and tables on clay tablets or papyrus scripts. And because the value of thought was so potent, the exchange was that much more impactful!
Between 685-525 BCE, (before the common era), Egypt’s ports along the Nile river opened up to Greek trade, breaking the barrier of interaction between them. With the migration of people & goods, both verbal and written communication acted as the vehicle to carry Egyptian ideas about math. That explains how and why much of Greek mathematics was adopted from the nearby civilization.
Egypt and their neighbors, like Mesopotamia and Ionia, had some of the finest math in the world. They utilized calculations for engineering purposes, to build structures for living and business such as the Great Pyramids or boats for trade & transportation. Unlike the others, the Greeks took these mathematical calculations to create practical applications for effective life skills.
Rigor was a major characteristic of Greek math. It was meticulous, exact, and at times super specific. They spent much of their effort contemplating deeper connotations behind the math they were working on. Even the word theorem evolved from the Greek word theoreo, which translates to “I contemplate”. Therefore, Greek math was intricately rooted in the association between mathematical review and analytical scrutiny.
Here are some of the most common contributions ancient Greek math gave to modern and even Western thought:
Ratios of a Triangle: Many of us have dabbled with the Pythagorean theorem, a tool proposed by one of ancient Greece’s most impactful mathematicians, Pythagoras. The 3:4:5 triangle was easily understood as a right triangle, but the Greeks were incredibly interested in the specificities of this abstract thought unlike their Egyptian benefactors.
Eventually, they expanded on it further by trying to calculate the longest side of the triangle (the hypotenuse) by calculating the similarity between the two smaller sides. This cracked open an intellectual revolution!
Numerical System: One of the most popular and widely-used tools the Greek created was their base system. By picking one core number, they formulated a number system for real-life usage that was easily divisible; this helped especially with fractions and proportions.
The ancient Greeks used the base number 60, which wasn’t as difficult to apply as we think. It’s a moderately divisible number with lots of other divisible factors, which made it a pretty flexible system to work with.
Square Root: This was an idea concocted a tad after the Pythagorean theorem swept civilization. With the new theorem begged a new question: if two sides of a right triangle are 1 unit, and the diagonal side equates to the square root of the two sides, what is it’s exact calculation?
After trying to find the square root of 2 and realizing it was irrational, this opened a world of questions regarding the square root of all numbers and what made them rational versus irrational.
Geometry: This is an entire discipline with math that works with the properties and relationships between lines, points, shapes, surfaces, and higher dimensional figures. With Greek architecture and engineering came the need for deep understanding of shapes and their dimensional properties.
Because it was a time of practicality, the Greeks were really using geometry as a logistical science to calculate land measurements. This was also a practice that originated with the Egyptian mathematical perspective; how do you think they built the Pyramids so beautifully?
Proofs: Known as one of the most tedious and difficult techniques to master, proofs are arguments based in inference and math logic to assure the answer to a problem is correct! Other theorems and math techniques can be applied to verify the validity of the proof, encouraging the practice of deductive reasoning with logic.
The first mathematical proof was credited to another Greek math icon, Thales of Miletus. He also proposed proofs that concerned ALL mathematical shapes and figures, not just the abstract ones! His contributions kickstarted the discussion of what the Universe was made of.
July 23, 2023
How the Contributions of One Alexandrian Mathematician Influenced the Course of Human Thought
What Started As A Thought…
Soon blossomed into an entire movement of thought and education.
We’ve all been taught about ancient Greece as well as their intellectually rich culture. They birthed and molded so many foundational leaders of thought in realms such as math, philosophy, religion, even art and architecture.
One of the most contributors to the evolution of math is Euclid. Euclid was known and referred to as the “father” or “founder” of Geometry. Geometry is a very distinct field within mathematics responsible for explaining the relationships of planes and objects. We all start to integrate geometrical knowledge when we learn the properties of shapes, points, lines, and the connection between them all!
Euclid is the namesake of Euclidean geometry, which is the basis of plane geometry like his published works allude to. But we’ll get into that later!
What we know of Euclid derives from a summary of famous mathematicians contrived by Greek philosopher Proclus (410-485 CE). Very little detail is actually known about Euclid’s life, however, has been hypothesized and framed by key events: his birth, death, and prominent mathematical contributions.
Euclid was born approximately 325 BCE and hails from the great Alexandria, a prominent civilization in Egypt. It is believed he passed there as well about 265 BC.
He taught in Alexandria at the time of Ptolemy I Soter (367/366-283/282 BC), the Macedonian ruler of Egypt.
In the summary procured by Proclus, it describes Euclid stating:
Not much younger than these [pupils of Plato] is Euclid, who put together the “Elements”… for Archimedes, who followed closely upon the first Ptolemy makes mention of Euclid, and further they say that Ptolemy once asked him if there were a shorted way to study geometry than the Elements, to which he replied that there was no royal road to geometry…
Euclid of Alexandria is typically mistook for Euclid of Megara, who lived 100 years before he was even born! And even though much of his existence remains in question, there is absolutely no doubt that he provided one of the most revolutionary pieces of work for math to evolve in the following 2000 years.
Euclid’s Contributions + Euclidean Geometry
What did Euclid ultimately contribute to math?
He wrote a collection of work known as the Elements. Euclid’s Elements is a compilation of postulates, proposals, and rules of geometry. There are five postulates introduced at the beginning of this collection, assuming the existence of points and lines and how they relate to one another.
The five postulates state:
– A straight line segment can be drawn to connect any two points.
– Any straight line segment can extend indefinitely in a straight line.
– Given any straight line segment, a circle can be drawn having the segment as a radius and one endpoint (of the segment) as the circle’s center.
– All right angles are congruent (equal).
– If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.
The fifth postulate defines what’s known as a parallel postulate, and it has not been proven yet despite many attempts.
You’ll probably find many of these postulates as basic rules introduced to you when you learned what geometry was! Elements essentially highlights the fundamentals of Euclidean geometry, which is taught in secondary education.
There are thirteen books in total, highlighting definitions and propositions revolving around the theory of geometry, proportions, circles, number theory, geometric algebra, and solid figures. Euclid dives into great detail for each subject he accentuates with his writing to ensure the comprehension of how they all correlate to one another.
It’s one of the oldest surviving mathematical publications known to humankind, which is why it’s been carried into our prevalent education today.
Why Is It Important?
Subcategories of math such as geometry and number theory help us develop a wide array of applications. For instance, geometry helps us develop and understand spatial awareness as well as relationships. This helps us create modern structures that hold up, and it also helps us find shapes for functional inventions. Without the fundamental knowledge of circles, polygons, and solid shapes, we wouldn’t have half the stuff we use on a daily basis!