Before you attempt to perform linear regression, you need to make sure that your data can be analyzed using this procedure. RMSE is the most widely used metric for regression tasks and is the square root of the averaged squared difference between the target value and the value predicted by the model. It is preferred more in some cases because the errors are first squared before averaging which poses a high penalty on large errors.
A trend line could simply be drawn by eye through a set of data points, but more properly their position and slope is calculated using statistical techniques like linear regression. Trend lines typically are straight lines, although some variations use higher degree polynomials depending on the degree of curvature desired in the line. The meaning of the expression “held fixed” may depend on how the values of the predictor variables arise. Alternatively, the expression “held fixed” can refer to a selection that takes place in the context of data analysis. In this case, we “hold a variable fixed” by restricting our attention to the subsets of the data that happen to have a common value for the given predictor variable.
What are the features of linear regression?
It is used to check whether the distribution is skewed or whether there are any outliers in the dataset. From the plot, we can infer that the best fit line is able to explain the effect of the independent variable, however, this does not apply to most of the data points. As we suspected, the scatterplot created with the data shows a strong positive relationship between the two scores.
- SPSS Statistics can be leveraged in techniques such as simple linear regression and multiple linear regression.
- Note that the intercept has been omitted here so that a unique solution can be found for the linear model weights.
- Because clustering is unsupervised (i.e. there’s no “right answer”), data visualization is usually used to evaluate results.
- Francis Galton’s 1886 illustration of the correlation between the heights of adults and their parents.
- You can then only benefit from having both tools under your belt and being able to use whichever one works better at the given task.
- The first column of the design matrix is the intercept, which is always 1.
- With this in mind, one can incorporate autocorrelated errors into regression by using Regression with ARMA errors (The ARIMAX model muddle – Hyndman) and in a sense get the ‘best of both worlds’.
Although some people do try to predict without looking at the trend, it’s best to ensure there’s a moderately strong correlation between variables. SVD involves breaking down a matrix as a product of three other matrices. It’s suitable for high-dimensional data and efficient and stable for small datasets. Due to its stability, it’s one of the most preferred approaches for solving linear equations for linear regression. However, it’s susceptible to outliers and might get unstable with a huge dataset. If you can establish a moderate correlation between the variables through both a scatter plot and a correlation coefficient, then the said variables have some form of a linear relationship.
What Is Linear Regression? How It’s Used in Machine Learning
However, this listwillgive you a representative overview of successful contemporary algorithms for each task. For example, you can’t say that neural networks are always better than decision trees or vice-versa. There are many factors at play, such as the size and structure of your dataset. The regression line passes through the mean of X and Y variable values. Product of the regression coefficient must be numerically less than unity.
Using different encodings boils down to creating different matrices from a single column with the categorical feature. This section presents three different encodings, but there are many more. The example used has six instances and a categorical feature with three categories. For the first two instances, the feature takes category A; for instances three and four, category B; and for the last two instances, category C. Homoscedasticity is a situation when the error term is the same for all the values of independent variables. With homoscedasticity, there should be no clear pattern distribution of data in the scatter plot.
MPE FormulaSince positive and negative errors will cancel out, we cannot make any statements about how well the model predictions perform overall. However, if there are more negative or positive errors, this bias will show up in the MPE. Unlike MAE and MAPE, MPE is useful to us because it allows us to see if our model systematically underestimates or overestimates . MAE is the absolute difference between the target value and the value predicted by the model.
It is done by a random selection of values of coefficient and then iteratively update the values to reach the minimum cost function. When using this method, you must select a learning rate parameter that determines the size of the improvement step to take on each iteration of the procedure. After performing the K – Fold Cross-validation we can observe that R – Square value is close to the original data, as well as MAE is 12%, which helps us, conclude that model is a good fit.
What Is a Smoothing Constant in Forecasting?
Regression analysis uses data, specifically two or more variables, to provide some idea of where future data points will be. The benefit of regression analysis is that this type of statistical calculation gives businesses a way to see into the future. Pros & Cons of the most popular ML algorithmLinear Regression is a statistical https://business-accounting.net/ method that allows us to summarize and study relationships between continuous variables. The term “linear” in linear regression refers to the fact that the method models data with linear combination of the explanatory/predictor variables . A linear regression model predicts the target as a weighted sum of the feature inputs.
Coefficient of Determination or R² is another metric used for evaluating the performance of a regression model. The metric helps us to compare our current model with a constant baseline and tells us how much our model is better. The constant baseline is chosen by taking the mean of the data and drawing a line at the mean. R² is a scale-free score that implies it doesn’t matter whether the values are too large or too small, the R² will always be less than or equal to 1. I recommend using Lasso, because it can be automated, considers all features simultaneously, and can be controlled via lambda. It also works for the logistic regression model for classification. The categorical feature effects can be summarized in a single boxplot, compared to the weight plot, where each category has its own row.
For support vector machine regression or SVR, we identify a hyperplane with maximum margin such that the maximum number of data points are within those margins. It is quite similar to the support vector machine classification algorithm. If you see the penalization term as a tuning parameter, then you can find the lambda that minimizes the model error with cross-validation. You can also consider lambda as a parameter to control the interpretability of the model. The larger the penalization, the fewer features are present in the model and the better the model can be interpreted.
- If a user’s personally identifiable information changes , we provide a way to correct or update that user’s personal data provided to us.
- Linear regression is widely used in biological, behavioral and social sciences to describe possible relationships between variables.
- Suppose your business is selling umbrellas, winter jackets, or spray-on waterproof coating.
- It is sufficient to know that an instance is neither in category B or C.
- This means whether the beta coefficient of the independent variable is significantly different from 0.
- Linear regression is one of the most common algorithms for the regression task.
- In this, we predict the outcome of a dependent variable based on the independent variables, the relationship between the variables is linear.
If the dependent variable increases on the Y-axis and independent variable increases on X-axis, then such a relationship is termed as a Positive linear relationship. If more than one independent variable is used to predict the value of a numerical dependent variable, then such a Linear Regression algorithm is called Multiple Linear Regression. If a single independent variable is used to predict the value of a numerical dependent variable, then such a Linear Regression algorithm is called Simple Linear Regression. The linear regression model in R is built with the help of lm() function. A boxplot is also called a box and whisker plot that is used in statistics to represent the five number summaries.
A regression model uses gradient descent to update the coefficients of the line by reducing the cost function. The different values for weights or coefficient of lines gives the different line of regression, and the cost function advantages of linear regression is used to estimate the values of the coefficient for the best fit line. After performing linear regression we get the best fit line, which is used in prediction, which we can use according to the business requirement.
For example, a data science student could build a model to predict the grades earned in a class based on the hours that individual students study. The student inputs a portion of a set of known results as training data. The data scientist trains the algorithm by refining its parameters until it delivers results that correspond to the known dataset.
By using MAE, we calculate the average absolute difference between the actual values and the predicted values. The temperature will increase as the sun rises and decline during sunset. This can be depicted as a straight line on the graph showing how the variables relate over time. @RichardHardy That’s fair and likely more speaks to my personal motivation and time to learn to new things than anything else.
For example, weighted least squares is a method for estimating linear regression models when the response variables may have different error variances, possibly with correlated errors. (See also Weighted linear least squares, and Generalized least squares.) Heteroscedasticity-consistent standard errors is an improved method for use with uncorrelated but potentially heteroscedastic errors. This part of the model is called the error term, disturbance term, or sometimes noise (in contrast with the “signal” provided by the rest of the model). This variable captures all other factors which influence the dependent variable y other than the regressors x.
What are two major advantages for using a regression?
Regression allows us to use more than two independent variables. This is its most important benefit. It allows us to determine the unbiased relationship between two variables by controlling for the effects of other variables.


The need for mathematics in everyday life and in the workplace has never been greater and will continue to increase. In this changing world, all students must understand and be able to do mathematics. Aurora Public Schools provides mathematics courses to prepare students to enhance opportunities and options for shaping their future. Course offerings in mathematics provide opportunities for students to prepare for college and careers.
|Name / Class||Phone||Web / Syllabus|
|Carmen Curtis Basham||326-3891|
|Asst. Department Chair|
|Lisandro De La Oliva||326-3538|


I’ve been teaching physics from the past three years to undergraduate students at the University of Delhi. Students at the higher secondary level can also take tuition of physics and maths for various competitive exams including IIT-JEE, NEET, AIPMT, AIIMS, GATE, NET, NTSE, SAT, KVPY and many more. Also, keen to teach physics to postgraduate students.
Regular classroom courses help students learn consistently and become confident in their knowledge. I have been teaching thoroughly so that students do extremely well in their board exams as well as in competitive exams. With proper knowledge of fundamental concepts & practicing varied types of problems help students in enhancing their ability. In the classroom, I try to ensure that they should be able to grasp the concepts, play and solve exam related problems by sharpening their reasoning, analytical and mathematical skills.
I believe that students can do better even when they are unable to attend classroom lectures. For those students, online teaching through Skype can be arranged. Besides regular study sessions, online test series will also be provided including tons of complex problems to master the subject.
Tuitions at home are also available for students who want one to one interaction. The timings for this mode are fixed but in certain scenarios duration can be modified. Fees will be charged hourly.
Students often don’t ask doubts in the classroom while learning. I am willing to clear doubts of students even individually, if required. My focus is on giving better results by ensuring that all the concepts are cleared and to check whether the students can apply the concepts properly or not. After various units or tests, it’s always better for students and teachers to have discussions on key points.
This program is accessible to all the students who are unable to join for a long time. The duration of this program is quite small with strategically formulated approach for learning concepts. It involves almost all the above programs in itself i.e. Lectures, Test series and Doubt Sessions. The intent is to fill-up the bridge between two students one who has joined throughout the year and other who has joined Crash Course.
Or send me a message by filling the form below.


In order to handle the changing size of the very large and small numbers, scientists use scientific notation to control the amount of writing of zeros they have to do. Take the number 0.0000004382 grams. If you had to repeatedly write the number, you may accidently drop a couple of zeros, changing the size of the number. Writing the value in scientific notation, as 4.382 *10-7 grams, makes the numbers easier to write.
When writing numbers in scientific notation, the decimal point must be moved such that it is just to the right of the first counting number (numbers 1-9). The number of times you have to move the decimal either left or right will become the power of ten.
Want to see a video on this? Check this one out!
[insert video scientific-notation.flv]
Heres some examples of switching numbers into scientific notation.
[insert Scientific_Notation in metric_system]
These prefixes are used through out the health and physical sciences. Often, we have to convert among various units.
Here's are two short videos giving an example of how prefixes are used to convert amongst each other.


Curious Area and Volume Ratios – A Discovery
Mar 23, 2018 | Stacey Ortman, director of public information
Dick Smith, associate professor of mathematics at the University of Dubuque, and Kris Nelson (C’15) worked together on “Curious Area and Volume Ratios.” Nelson helped create a physical model of the math problem that was started by Smith.
DUBUQUE, Iowa – An unlikely friendship that led to a curious discovery started in a math classroom in the University Science Center on University of Dubuque’s campus.
Kris Nelson (C’15) enrolled in classes at UD after serving active duty in the Army and working in the private sector. His goal was to obtain an undergraduate degree in computer information technologies. Even though he enjoyed math, it had been 20 years since he had studied numbers and he was nervous.
Dick Smith, associate professor of mathematics, saw potential in the student veteran who challenged him to rethink equations. Smith encouraged Nelson to consider becoming a mathematics major.
“He wasn’t just a consumer of what was delivered. He also challenged me,” Smith said. “He would come up to me after class and very politely say, ‘I’ve had an idea.’”
One of those ideas occurred after Smith shared a problem, “Curious Area and Volume Ratios,” with his modern geometry class. He had been working on the problem since 2010, some of that time with Dan Flath, PhD, a former visiting professor at Macalester College who is now an independent mathematical consultant with Flath Math Results. Smith shared how when he looked out the window of an airplane and saw an irrigation field, he wondered what would happen if the field increased in size but the end of the irrigation tubes remained fixed. Would the ratio of the whole thing become predictable?
After class, Nelson asked Smith if he had considered moving the problem into three dimensions.
“Two old professors were stuck and this guy who had been away from math just blows us completely out of the water” Smith said, laughing.
Nelson combined his computer information technology knowledge with his math knowledge to extend “Curious Area and Volume Ratios” into three dimensions. Mark Woodhouse, assistant professor of computer information systems at UD, used Nelson’s dimensions to create a physical model of the problem.
“It was an interesting idea that started out in two dimensions and I wondered what it would be like in three dimensions. I also wondered if the relationships that held in two dimensions would also hold in three dimensions,” Nelson said. “What I discovered was the same formulas held true in three dimensions, with the exception of it being to the third power versus the square.”
In 2016, Nelson presented “Curious Area and Volume Ratios” at the Minnesota Regional Mathematical Association of America conference at the University of Minnesota. The problem was created via an octagon by drawing two line segments from each vertex of a square to the points on the square one unit from the opposite corner. The ratio of the area of the square to the octagon produces triangular numbers. As the square gets very large, the octagon approximates a square with an area ½. A 3-D extension draws three segments from each vertex of a cube to points one unit from the opposite vertex, producing a solid with 24 edges. When the edge of the cube is an integer, the ratio of the volume of the cube to the solid is always a whole number. As the cube gets very large, the solid’s volume approximates ¼.
“It’s absolutely beautiful. Everything works and the ratios are definitely curious,” Smith said.
Nelson added, “The most interesting part about it is we need one piece of information to determine everything about the solid in the center. As long as we have that one number, we can determine everything.”
Nelson, who was a triple major in computer information technologies, mathematics, and business, earned a bachelor’s degree in December 2015. He will graduate from the UD Master of Business Administration program in May.
Smith continues to work on the problem.
“This is what mathematicians do,” he said. “This problem was there and we decided to wonder up this mountain and try to find what we could find up there.”


What is the PiTOP?
It is the physical embodiment of the mathematical constant Pi, or, in Greek, π. This brass disk has radius r = 1” and thickness t = 1/π” ~ .32”. It weighs ~ 4.8 ounces, and it displays the first 109 digits of Pi.
It is an executive fidget device.
It is a tactile stress reliever.
It is an elegant paperweight.
It is the ideal March 14 Pi Day gift.
It symbolizes profit in economics.
Spin the PiTOP like you would a coin on a hard, smooth surface, and it delights with a hypnotic sound and light show, creating eye catching visual patterns and illusions.
The PiTOP will be available in a limited edition in the near future on Etsy here.
Cubing the PiTOP
The PiTOP is designed in the shape of a right circular cyclinder of radius r and thickness t. Its special feature is that the thickness t is equal to the radius r divided by π. The volume V of a right circular cylinder is given by:
Vcylinder = π r^2 t = π r^2 r/ π =r^3 = Vcube
That is, the volume of the PiTOP right circular cylinder of radius r is exactly equal to that of a cube of side length l = r. Voila!
You may not be able to square the circle in a finite number of steps with a straightedge and compass, but the PiTOP automatically cubes a cylinder!
The Physics of the PiTOP
Spin the PiTOP as you would a coin. It will spin but it will also roll on its edge. As the PiTOP loses energy to friction, the angle alpha (in degrees) that it makes with the spinning surface will decrease. The spin frequency will also decrease but - seemingly paradoxically - the rolling (or precession) frequency (in Hz) omega will increase. Just before the PiTOP motion comes to a stop, the frequency of rolling (or precession) seems to increase without limit! See the figures below showing an example of the motion of a prototype of the PiTOP. The data is based on high-speed videos made and analyzed by the Australian Physicist Rod Cross from the University of Sydney.


BY PROF JACOB
The area of study known as the history of mathematics is primarily an investigation into the origin of discoveries in mathematics and, to a lesser extent, an investigation into the mathematical methods and notation of the past.
Before the modern age and the worldwide spread of knowledge, written examples of new mathematical developments have come to light only in a few locales. The most ancient mathematical texts available are Plimpton 322 (Babylonian mathematics c. 1900 BC), the Rhind Mathematical Papyrus (Egyptian mathematics c. 2000-1800 BC) and the Moscow Mathematical Papyrus (Egyptian mathematics c. 1890 BC). All of these texts concern the so-called Pythagorean theorem, which seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry.
The study of mathematics as a subject in its own right begins in the 6th century BC with the Pythagoreans, who coined the term "mathematics" from the ancient Greek μάθημα (mathema), meaning "subject of instruction". Greek mathematics greatly refined the methods (especially through the introduction of deductive reasoning and mathematical rigor in proofs) and expanded the subject matter of mathematics. Chinese mathematics made early contributions, including a place value system. The Hindu-Arabic numeral system and the rules for the use of its operations, in use throughout the world today, likely evolved over the course of the first millennium AD in India and was transmitted to the west via Islamic mathematics. Islamic mathematics, in turn, developed and expanded the mathematics known to these civilizations. Many Greek and Arabic texts on mathematics were then translated into Latin, which led to further development of mathematics in medieval Europe.
From ancient times through the Middle Ages, bursts of mathematical creativity were often followed by centuries of stagnation. Beginning in Renaissance Italy in the 16th century, new mathematical developments, interacting with new scientific discoveries, were made at an increasing pace that continues through the present day.
 Prehistoric mathematicsThe origins of mathematical thought lie in the concepts of number, magnitude, and form. Modern studies of animal cognition have shown that these concepts are not unique to humans. Such concepts would have been part of everyday life in hunter-gatherer societies. The idea of the "number" concept evolving gradually over time is supported by the existence of languages which preserve the distinction between "one", "two", and "many", but not of numbers larger than two.
The oldest known possibly mathematical object is the Lebombo bone, discovered in the Lebombo mountains of Swaziland and dated to approximately 35,000 BC. It consists of 29 distinct notches cut into a baboon's fibula. Also prehistoric artifacts discovered in Africa and France, dated between 35,000 and 20,000 years old, suggest early attempts to quantify time.
The Ishango bone, found near the headwaters of the Nile river (northeastern Congo), may be as much as 20,000 years old and consists of a series of tally marks carved in three columns running the length of the bone. Common interpretations are that the Ishango bone shows either the earliest known demonstration of sequences of prime numbers or a six month lunar calendar. In the book How Mathematics Happened: The First 50,000 Years, Peter Rudman argues that the development of the concept of prime numbers could only have come about after the concept of division, which he dates to after 10,000 BC, with prime numbers probably not being understood until about 500 BC. He also writes that "no attempt has been made to explain why a tally of something should exhibit multiples of two, prime numbers between 10 and 20, and some numbers that are almost multiples of 10."
Predynastic Egyptians of the 5th millennium BC pictorially represented geometric designs. It has been claimed that megalithic monuments in England and Scotland, dating from the 3rd millennium BC, incorporate geometric ideas such as circles, ellipses, and Pythagorean triples in their design.
All of the above are disputed however, and the currently oldest undisputed mathematical usage is in Babylonian and dynastic Egyptian sources. Thus it took human beings at least 45,000 years from the attainment of behavioral modernity and language (generally thought to be a long time before that) to develop mathematics as such.
 Babylonian mathematicsBabylonian mathematics refers to any mathematics of the people of Mesopotamia (modern Iraq) from the days of the early Sumerians through the Hellenistic period almost to the dawn of Christianity. It is named Babylonian mathematics due to the central role of Babylon as a place of study. Later under the Arab Empire, Mesopotamia, especially Baghdad, once again became an important center of study for Islamic mathematics.
In contrast to the sparsity of sources in Egyptian mathematics, our knowledge of Babylonian mathematics is derived from more than 400 clay tablets unearthed since the 1850s. Written in Cuneiform script, tablets were inscribed whilst the clay was moist, and baked hard in an oven or by the heat of the sun. Some of these appear to be graded homework.
The earliest evidence of written mathematics dates back to the ancient Sumerians, who built the earliest civilization in Mesopotamia. They developed a complex system of metrology from 3000 BC. From around 2500 BC onwards, the Sumerians wrote multiplication tables on clay tablets and dealt with geometrical exercises and division problems. The earliest traces of the Babylonian numerals also date back to this period.
The majority of recovered clay tablets date from 1800 to 1600 BC, and cover topics which include fractions, algebra, quadratic and cubic equations, and the calculation of regular reciprocal pairs. The tablets also include multiplication tables and methods for solving linear and quadratic equations. The Babylonian tablet YBC 7289 gives an approximation of √2 accurate to five decimal places.
Babylonian mathematics were written using a sexagesimal (base-60) numeral system. From this derives the modern day usage of 60 seconds in a minute, 60 minutes in an hour, and 360 (60 x 6) degrees in a circle, as well as the use of seconds and minutes of arc to denote fractions of a degree. Babylonian advances in mathematics were facilitated by the fact that 60 has many divisors. Also, unlike the Egyptians, Greeks, and Romans, the Babylonians had a true place-value system, where digits written in the left column represented larger values, much as in the decimal system. They lacked, however, an equivalent of the decimal point, and so the place value of a symbol often had to be inferred from the context.
 Egyptian mathematicsEgyptian mathematics refers to mathematics written in the Egyptian language. From the Hellenistic period, Greek replaced Egyptian as the written language of Egyptian scholars. Mathematical study in Egypt later continued under the Arab Empire as part of Islamic mathematics, when Arabic became the written language of Egyptian scholars.
The most extensive Egyptian mathematical text is the Rhind papyrus (sometimes also called the Ahmes Papyrus after its author), dated to c. 1650 BC but likely a copy of an older document from the Middle Kingdom of about 2000-1800 BC. It is an instruction manual for students in arithmetic and geometry. In addition to giving area formulas and methods for multiplication, division and working with unit fractions, it also contains evidence of other mathematical knowledge, including composite and prime numbers; arithmetic, geometric and harmonic means; and simplistic understandings of both the Sieve of Eratosthenes and perfect number theory (namely, that of the number 6). It also shows how to solve first order linear equations as well as arithmetic and geometric series.
Another significant Egyptian mathematical text is the Moscow papyrus, also from the Middle Kingdom period, dated to c. 1890 BC. It consists of what are today called word problems or story problems, which were apparently intended as entertainment. One problem is considered to be of particular importance because it gives a method for finding the volume of a frustum: "If you are told: A truncated pyramid of 6 for the vertical height by 4 on the base by 2 on the top. You are to square this 4, result 16. You are to double 4, result 8. You are to square 2, result 4. You are to add the 16, the 8, and the 4, result 28. You are to take one third of 6, result 2. You are to take 28 twice, result 56. See, it is 56. You will find it right."
Finally, the Berlin papyrus (c. 1300 BC) shows that ancient Egyptians could solve a second-order algebraic equation.
 Greek mathematicsGreek language from the time of Thales of Miletus (~600 BC) to the closure of the Academy of Athens in 529 AD. Greek mathematicians lived in cities spread over the entire Eastern Mediterranean, from Italy to North Africa, but were united by culture and language. Greek mathematics of the period following Alexander the Great is sometimes called Hellenistic mathematics.
Greek mathematics was much more sophisticated than the mathematics that had been developed by earlier cultures. All surviving records of pre-Greek mathematics show the use of inductive reasoning, that is, repeated observations used to establish rules of thumb. Greek mathematicians, by contrast, used deductive reasoning. The Greeks used logic to derive conclusions from definitions and axioms, and used mathematical rigor to prove them.
Greek mathematics is thought to have begun with Thales of Miletus (c. 624–c.546 BC) and Pythagoras of Samos (c. 582–c. 507 BC). Although the extent of the influence is disputed, they were probably inspired by Egyptian and Babylonian mathematics. According to legend, Pythagoras traveled to Egypt to learn mathematics, geometry, and astronomy from Egyptian priests.
geometry to solve problems such as calculating the height of pyramids and the distance of ships from the shore. He is credited with the first use of deductive reasoning applied to geometry, by deriving four corollaries to Thales' Theorem. As a result, he has been hailed as the first true mathematician and the first known individual to whom a mathematical discovery has been attributed. Pythagoras established the Pythagorean School, whose doctrine it was that mathematics ruled the universe and whose motto was "All is number". It was the Pythagoreans who coined the term "mathematics", and with whom the study of mathematics for its own sake begins. The Pythagoreans are credited with the first proof of the Pythagorean theorem, though the statement of the theorem has a long history, and with the proof of the existence of irrational numbers.
Plato (428/427 BC – 348/347 BC) is important in the history of mathematics for inspiring and guiding others. His Platonic Academy, in Athens, became the mathematical center of the world in the 4th century BC, and it was from this school that the leading mathematicians of the day, such as Eudoxus of Cnidus, came from. Plato also discussed the foundations of mathematics, clarified some of the definitions (e.g. that of a line as "breadthless length"), and reorganized the assumptions. The analytic method is ascribed to Plato, while a formula for obtaining Pythagorean triples bears his name.
Eudoxus (408–c.355 BC) developed the method of exhaustion, a precursor of modern integration and a theory of ratios that avoided the problem of incommensurable magnitudes. The former allowed the calculations of areas and volumes of curvilinear figures, while the latter enabled subsequent geometers to make significant advances in geometry. Though he made no specific technical mathematical discoveries, Aristotle (384—c.322 BC) contributed significantly to the development of mathematics by laying the foundations of logic.
In the 3rd century BC, the premier center of mathematical education and research was the Musaeum of Alexandria. It was there that Euclid (c. 300 BC) taught, and wrote the Elements, widely considered the most successful and influential textbook of all time. The Elements introduced mathematical rigor through the axiomatic method and is the earliest example of the format still used in mathematics today, that of definition, axiom, theorem, and proof. Although most of the contents of the Elements were already known, Euclid arranged them into a single, coherent logical framework. The Elements was known to all educated people in the West until the middle of the 20th century and its contents are still taught in geometry classes today. In addition to the familiar theorems of Euclidean geometry, the Elements was meant as an introductory textbook to all mathematical subjects of the time, such as number theory, algebra and solid geometry, including proofs that the square root of two is irrational and that there are infinitely many prime numbers. Euclid also wrote extensively on other subjects, such as conic sections, optics, spherical geometry, and mechanics, but only half of his writings survive.
Archimedes (c.287–212 BC) of Syracuse, widely considered the greatest mathematician of antiquity, used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, in a manner not too dissimilar from modern calculus. He also showed one could use the method of exhaustion to calculate the value of π with as much precision as desired, and obtained the most accurate value of π then known, 310⁄71 < π < 310⁄70. He also studied the spiral bearing his name, obtained formulas for the volumes of surfaces of revolution (paraboloid, ellipsoid, hyperboloid), and an ingenious system for expressing very large numbers. While he is also known for his contributions to physics and several advanced mechanical devices, Archimedes himself placed far greater value on the products of his thought and general mathematical principles. He regarded as his greatest achievement his finding of the surface area and volume of a sphere, which he obtained by proving these are 2/3 the surface area and volume a cylinder circumscribing the sphere.
Apollonius of Perga (c. 262-190 BC) made significant advances to the study of conic sections, showing that one can obtain all three varieties of conic section by varying the angle of the plane that cuts a double-napped cone. He also coined the terminology in use today for conic sections, namely parabola ("place beside" or "comparison"), "ellipse" ("deficiency"), and "hyperbola" ("a throw beyond"). His work Conics is one of the best known and preserved mathematical works from antiquity, and in it he derives many theorems concerning conic sections that would prove invaluable to later mathematicians and astronomers studying planetary motion, such as Isaac Newton. While neither Apollonius nor any other Greek mathematicians made the leap to coordinate geometry, Apollonius' treatment of curves is in some ways similar to the modern treatment, and some of his work seems to anticipate the development of analytical geometry by Descartes some 1800 years later.
Around the same time, Eratosthenes of Cyrene of Cyrene (c. 276-194 BC) devised the Sieve of Eratosthenes for finding prime numbers. The 3rd century BC is generally regarded as the "Golden Age" of Greek mathematics, with advances in pure mathematics henceforth in relative decline. Nevertheless, in the centuries that followed significant advances were made in applied mathematics, most notably trigonometry, largely to address the needs of astronomers. Hipparchus of Nicaea (c. 190-120 BC) is considered the founder of trigonometry for compiling the first known trigonometric table, and to him is also due the systematic use of the 360 degree circle. Heron of Alexandria (c. 10–70 AD) is credited with Heron's formula for finding the area of a scalene triangle and with being the first to recognize the possibility of negative numbers possessing square roots. Menelaus of Alexandria (c. 100 AD) pioneered spherical trigonometry through Menelaus' theorem. The most complete and influential trigonometric work of antiquity is the Almagest of Ptolemy (c. AD 90-168), a landmark astronomical treatise whose trigonometric tables would be used by astronomers for the next thousand years. Ptolemy is also credited with Ptolemy's theorem for deriving trigonometric quantities, and the most accurate value of π outside of China until the medieval period, 3.1416.
Following a period of stagnation after Ptolemy, the period between 250 and 350 AD is sometimes referred to as the "Silver Age" of Greek mathematics. During this period, Diophantus made significant advances in algebra, particularly indeterminate analysis, which is also known as "Diophantine analysis". The study of Diophantine equations and Diophantine approximations is a significant area of research to this day. His main work was the Arithmetica, a collection of 150 algebraic problems dealing with exact solutions to determinate and indeterminate equations. The Arithmetica had a significant influence on later mathematicians, such as Pierre de Fermat, who arrived at his famous Last Theorem after trying to generalize a problem he had read in the Arithmetica (that of dividing a square into two squares). Diophantus also made significant advances in notation, the Arithmetica being the first instance of algebraic symbolism and syncopation.


Prepare yourselves, ladies and gentlemen, for we’re diving into the depths of Fibonacci and the wonders of its mysteries!
The Fibonacci series. Ah, what a wonder it is! It’s more than just a series of numbers; it’s an elegant dance between digits, an encapsulation of the Golden Ratio, and a playground for coders to frolic in. We’re going to crack this nut wide open!
What’s the Fibonacci series, you ask? Imagine a series that builds upon itself, where each number is the sum of the two that came before it, like a winding staircase leading to infinity. In the mathematical tongue, we’d say:
Take the leap and scribe this into Python, and voila! Here’s a Pythonic Fibonacci for you:
But wait, the waters are getting choppy! When we’re dealing with big numbers, our Fibonacci function starts to trip over itself. It’s re-calculating the same values over and over again like a broken record player. What a colossal waste of precious time!
This is where our inner engineer comes to the rescue. We’re going to outsmart the system and store the computed values in a lookup table. It’s like building a library of Fibonacci numbers! This sly technique is called memoization, (think memorization, but with an ‘o’)!
Memoization is an old trick up the sleeve of dynamic programming. Here’s how you wield it in the battle against redundancy:
So there you have it, folks! The Fibonacci series, dressed in Python, and turbocharged with the power of memoization. Now go forth, and Fibonacci like you’ve never Fibonacci’d before!


Hello blog readers!
This week I led a Mommy & Me at the Library Music Session at the Key Biscayne Library and boy did we have fun!
We did a song story about baby animals, sang the blues with mini microphones, twirled with rainbow colored streamers, danced a country hoe-down with the parachute, and played peek-a-boo with mommy and our scarves (I think this was the most popular activity by far!)
Another program I facilitate at local libraries is Rockin’ and Readin’, where we use music to address literacy skills. I also try to incorporate other academic concepts into my sessions for the older children that attend.
One of those concepts is math.
Truth be told, math was never one of my favorite subjects in school, but I’ve found that when I combine music + math, it is a winning combination.
Check out these articles where I share how you can address math skills in your music therapy sessions:
Cupcakes + Music + Math = A Winning Combination (video included!)
Fun with File Folder Games (includes one of my favorite resources!)
Hopefully these ideas will inspire you to incorporate math concepts into your music therapy sessions!
Now I’d love to hear from YOU! Leave a comment below and let us know how you address math skills in your music therapy sessions.


*January 2018 IGCSE Maths Daytime Course*
Online group lessons at only £5* for 1hr 15mins twice a week, to prepare your child for the up and coming exams, including exam past paper reviews and techniques.
These lessons take place:
Monday 11:30am - 12:45pm
Wednesday 11:45am - 1pm UK Time.
(*Invoiced in 4 week blocks for £40)


There is no way the importance of mathematics can be underestimated. The importance of math is due to the fact that mathematics is applied in all parts of our daily lives. Of all subjects taught on earth, mathematics tops the list and this is because of its significance in the education and all other sectors. To improve their mastery of mathematics, people are using math tutoring services.
When individuals with good mastery of mathematics assist students to improve their class performance in mathematics, this is called a math tutoring service. Math tutoring is also offered to adults who would like to increase their mastery of mathematics. There is, however, a common concern among all people seeking to get a math’s tutoring service which is how to know where or from whom to get the best services.
Before taking your child to a math tutor, enquire first about his mastery of the subject. The tutors level of education and his specialization will give you this information. People who have majored in mathematics at college level make the best math tutors. For small children, the tutor has also to have undertaken a specialized course since the tutoring done to kids may be quite different from that of adults and bigger students. Engineers and people that have done science related courses also make good math tutors.
A keen consideration has also to be put on the tutors teaching experience. If a tutor is inexperienced, there is a high chance of him failing to deliver some points very well. Young students can find understanding concepts by such a tutor quite difficult. An experienced tutor has learned better communication skills with time as well as learning how to deliver certain points to the students for the best results.
It is very important for a student to get a personalized tutoring since math tutoring is a supplementary mathematics class. To know if your child will get this personalized coaching, check the size of class the tutor is handling at a go. If the class is big, the chances of your child getting a personalized coaching experience are very minimal. The tutor you take your child to should, therefore, be one whose classes are reasonably small in terms of the number of students.
Lastly, consider how close the tutor is from your home as well as his availability. There is need to find a tutor close to your home so as to save the amount of time the children spend between home and the tutor’s facility. A short distance is also a good way of making sure you and the tutor can monitor the children when in between the sessions. It is also prudent to seek services of a tutor whose availability is not erratic.


One of the biggest hurdles I’ve come across while learning scala, are all the new functional concepts used - most of which I know nothing about! I’ve also found it rather to hard to find any succinct information on these concepts, except for Haskell code and mathematical equations. While these are great for those who know Haskell and/or are proficient in mathematics it may not be the easiest thing for a simple programmer (such as myself) to understand.
I believe that in order to become a good functional programmer one needs to understand basic functional concepts and how they are implemented.
In the following months I hope to investigate the following concepts:


I have made some notes and remarks about motivic homotopy theory while working on my master dissertation during the summer of 2019.
Affine representability results:
For remarks on section 2.3 and Lemma 2.3.2 of  (also quoted as Proposition 5.5 and Proposition 5.6 in ), we take the approach of stack and descent. By relating hyperdescent condition for simplicial presheaves with descent for stacks , we show that how this implies classifying space of a stack satisfies hyperdescent.
After that, we give a different proof and some remarks of Lemma 2.3.2 based on my understanding, see Proposition 0.0.2 and Proposition 0.0.3 in Stack and descent.
Exercises on the construction of motivic homotopy theory:
I also have some informal notes about the exercises in . The following are some of my proofs for the exercises, in the remarks, I also pointed out some typos I found which may be helpful for other readers: Exercises on A Primer.
 Antieau, B., & Elmanto, E. (2017). A primer for unstable motivic homotopy theory. Surveys on recent developments in algebraic geometry, 95, 305-370.
 Asok, A., Hoyois, M., & Wendt, M. (2018). Affine representability results in 𝔸1–homotopy theory, II: Principal bundles and homogeneous spaces. Geometry & Topology, 22(2), 1181-1225.
 Jardine, J. F. (2015). Local homotopy theory. Springer.


I was reminded about the closing scenes of the film "The Martian" today.
"You just begin. You do the math. You solve one problem and you solve the next one, and then the next. And If you solve enough problems, you get to come home."
Today I was faced with something that I had never done before and it involved solving many problems, even though I knew what I wanted to achieve it, it seemed like an unsurmountable task to figure it all out. I have noticed that I often become extremely frustrated when I can't figure a task out, so much so that I can't think in a balanced way.
You solve one problem, and you solve the next one and then the next.
I have been told that I am stubborn and I got to find out how stubborn today, as I would not leave the task until I was totally satisfied that the solution worked, and that I understood why. I also found a tremendous amount of joy in solving the last problem that allowed me to go home. Because hell was freezing over before I left without finishing the task.
I have taken this kind of approach to situations for a while now, it has taken me to new places and got me out of situations that I haven't wanted to be in. I think the hardest part is sometimes identifying what the problem is.


well yeah, that's probably what he had in mind but..... mine is cleverer. And besides this hypothetical matchstick equation would look better if the creator had broken a matchstick in half to make the +, thus making my solution only require the movement of 1/2 a matchstick .
Correct! The question is worded to seems like you need to move a matchstick, but you don't need to.
Scores: Andrew: 9 (+2) Don.Gato: 7 MegaMarbler: 1
Puzzle 10 - Train Station Conundrum - Hard
It takes 15 minutes to travel from Station A to Station B. A -> B 15 minutes It takes 5 minutes to travel from Station B to Station C. B -> C 5 minutes It takes 10 minutes to travel from Station C to Station D. C -> D 10 minutes However, it doesn't take 30 minutes for a train to travel from Station A to D. How long would it take, using the information above, and that the railway is a straight line with no branches, for a train to travel from Station A to D?
'The future will be better tommorow' - Dan Quayle 'For NASA, Space is still a high priority.' - Dan Quayle... again.
Meh unless I am missing something, this riddle is impossible. It could be said what the difference is LIKELY to be, but that isn't the question. Of course the obvious answer to that would appear to be 3, but since we are dealing with probability now it must be pointed out that despite the distribution of colored cards being 50-50, the probability of the color distribution staying the same when the randomly ordered deck is split randomly in two is quite low.
I just figured it out. 3 is after all the correct answer. I was mistakenly assuming that the actual numbers of red/black cards would effect the answer.
The number of red cards in the left pile, and the number of black cards in the right pile can be determined in the following equation. Basically, the total number of red cards (26) minus the number of red cards in the left pile (x) tells you how many red cards are in the right pile. By subtracting that number from the total number of cards in that pile (29) you determine the number of black cards in the right pile.
29 - (26 - x) = y
Now we could try it for every possible value of x, there's only 24 possibilities anyway. Or we could use more math and simplify the equation. (excuse the inefficient and generally crappy way I did this, I haven't done anything like this in years. I was improvising)
29 - (26 - x) = y
29 = y + (26 - x)
- y + 29 = 26 - x
- y = - 3 - x
y = x + 3
.... and we get this: y - x = 3
which proves that the difference between black cards in the right pile (y) and red cards in the left pile (x) is = to 3.
In other words I greatly overcomplicated matters, and Andrew was right.


10th class is one of the most crucial stages in a student life. ‘The Central Board of Secondary Education’ (CBSE) is the most relevant board for engineering and medical aspirants today as both NEET (AIPMT) and the JEE Main exams are directly based on the CBSE board syllabus. It determines the future of a student to a very large extent. If a student gets good marks in 10th, then there is very much possibility that he will get good marks in 12th board too and so on.
Height and distance are a very important section under trigonometry of Quantitative Aptitude. Usually, all government competitive examinations ask questions based on Trigonometry. In general, the concept and questions based on trigonometry are very easy to solve but sometimes it becomes a little bit tricky in nature for its analytical nature.
In this article we are going to cover the key concepts of application of trigonometry particularly problems on height and distance, along with various types of questions, and tips and tricks. We have also added a few solved examples, and some unsolved questions which candidates will find beneficial in their exam preparation.
For Class 10 students who love Mathematics (and for those who don’t), here are some study tips that can help them in the CBSE Class X board exam or any other competitive examination perfectly to the full marks in solving questions on height and distance. This is an important topic from competitive examination point of view.
Heights and Distances is an application of Trigonometry. It helps us in solving complex real-life issues like calculating the distance between two or more objects, the height of mountains, etc. Heights and Distances are helpful in finding the distance, height, slope, etc., of the objects in the constructions. It is helpful in calculating the height of the towers, buildings, celestial bodies, etc. Furthermore, it is also helpful to calculate the angle of inclination of the object without the height and distance of a particular object.
So basically, you should know how to draw the diagram where to keep the angle and all. For this just practice more and more questions. Se firstly just practices to make diagrams and check solutions for the respective question. If your diagram is right then be happy lol. Once you know how to draw the triangle, you just have to practice questions on Trigonometry. It should be clear to you.
Heights and Distances – Tricks
There are a few things to be noted while solving the problems of heights and distances. Some of the important points are listed below:
1. In the problems of heights and distances, if the height of the observer is not given, then the observer is denoted by a point.
2. The height of the objects like towers, buildings, celestial bodies, etc., are represented by a line segment.
3. A horizontal line is a line obtained by drawing parallel to the ground.
- Angle of elevation:
The term angle of elevation denotes the angle from the horizontal upward to an object. Angle of elevation implies that an object is upwards from the position of the observer. An observer’s line of sight would be above the horizontal. If the observer moves towards the object, then the angle of elevation increases.
5. Angle of depression:
If a person looks straight ahead, their line of sight is on a horizontal plane at eye level. The angle of depression implies that an object is downwards from the position of the observer If that person looks to a point on the ground, they are now looking down at another horizontal plane, one that is below eye level. Their line of sight is now aimed at a downward angle. The angle between that line and their horizontal line of sight is the angle of depression. If the observer moves away from the object, then the angle of depression decreases.
6. Angle of elevation and angle of depression, so formed are (congruent) equal.
7. An increase in the angle of elevation of the sun results in a decrease in the length of the shadow of an object.
- A decrease in the angle of elevation of the sun results in an increase in the length of the shadow of an object.
Solution to questions on heights and distances are provided here with simple step-by-step explanations. These solutions for Heights and Distances will be extremely to Class 10 students.
Our objective is to provide you a smooth learning experience.
While solving word problems understanding the question is very important. So:
- Read the problem carefully.
- Draw the problem with the sketch.
- Heights and distance problems are solved by using trigonometric ratios therefore, refresh your trigonometric formulas.
Also, always remember that paper presentation is an important aspect and you must attempt the paper in a way that not only make an impact on the checker but also makes him read your paper easily. The best thing about step wise solution is that you will get marks for all the correct steps even if your final solution is wrong.
All these tips if followed will surely help you write a perfect exam within the allotted time. Ultimately, focus on giving your best without being worried about the result. Have confidence in your preparation and you will come with an overwhelming triumph.
Math is not something that you can learn by heart. You have to practice it daily to master it. Hence, solve at least some Maths questions everyday – without fail. Good preparation along with exam paper writing techniques is necessary to score good marks in CBSE exams. Thus, two things are must: studies and answer writing practice.
The proverb ‘Practice makes Perfect’ holds very true for Maths too.
Practise makes a man perfect. Genius is one percent inspiration and ninety percent perspiration: This is a saying of the American inventor Thomas Edison.
“One ounce of practise is worth a thousand pounds of theory. Practise makes us what we shall be.” Swami Vivekananda.
In the Bhagavad Gita Arjuna asked the Lord, “how to control the mind which by its very nature is turbulent, obstinate and restless?” Shri Krishna’s simple reply is that only practise and non-attachment can help control the mind.
In this article we have solved some previous years question papers in an easy to understand and detailed way for your study and practice.
An aeroplane is flying at a height of 300 m above the ground. Flying at the height
The angle of depression from the aeroplane of two points on both banks of the river in opposite directions are 45 deg. and 60 deg. respectively. Find the width of the river. [use √3 = 1.732]
Let the aeroplane is at point P and A and B are the two points
on both banks of the river. Based on the information the given figure has been drawn.
Height of the plane PM = 300 m
Angle of depression from point P to A = ∠XPA = 45 deg.
Angle of depression from point P to B = ∠YPB = 60 deg.
To find: Width of the river i.e., length AB
From the figure it is clear that ∠XPA = ∠MAP = 45 deg. [alternate angles]
Similarly, ∠YPB = ∠MBP= 60 deg. [alternate angles]
In right ΔPMA, tan MAP = PM/AM
⇒tan 450 = 300/AM
⇒ 1 = 300/AM
⇒ AM =300 m …………………………… (i)
Again, in right ΔPMB, tan MBP = PM/BM
⇒tan 600 = 300/MB
⇒ √3 = 300/MB
⇒ MB = 300/√3 …………………(i)
From equation (i) and (ii) we get,
(AM + MB) = 300√3 + 300/√3
⇒ AB = 300 + 100√3
⇒AB = 300 + 100*1.732
⇒AB = 300 + 173.2
⇒AB = 473.2 m
Hence, width of the river = 473.2 m
Question 2 :
Two poles of equal height are standing opposite to each other on either side of the road which is 80 m wide. From a point P between them on the road, the angle of elevation of the top of one pole is 600 and the angle of depression from the top of another pole at P is 300. Find the height of each pole and the distance of point P from the poles. [CBSE 2015, ‘18C, ‘19]
Let, AB and CD are two poles standing on either side of
AC= width of the road = 80 m.
Angel of elevation of point B from point P = ∠BPA = 600
Also, angle of depression from D to point P = ∠XDP = 300
Let, AB = CD = height of each pole = h m (say)
Let AP = distance between A and P = x m (say)
Then distance PC = (80 – x) m
From the figure it is clear that ∠XDP = ∠CPD = 300 [alternate angles]
From right ΔPAB, we have tan 600 =AB/AP
⇒ √3 = AB/AP
⇒ √3 = h/x
⇒ h = √3 x …………………………………………(i)
Again, from right ΔPCD, we have tan 300 =DC/PC
⇒ 1/√3 = DC/PC
⇒ 1/√3 = h/ (80 – x) m
⇒ h = (80 – x)/ √3.…………………………(ii)
Evaluating h from (i) and (ii) we get,
√3 x = (80 – x)/ √3
⇒ √3 x* √3 = (80 – x)
⇒ 3x + x = 80
⇒ 4x = 80
⇒ x = 20 m
Therefore, height of each pole = h = √3 x = 20√3 m
Some practice questions:
As observed from the top of 100m high light house from the sea level the angle of depression of two ships are 300 and 450. If one ship is exactly behind the other on the same side of the light house, find the distance between the two ships.
A man observes a car from the top of a tower, which is moving towards the tower with a uniform speed. If the angel of depression of the car changes from 300 to 450 in 12 minutes, find the time taken by the car now to reach the tower. [CBSE 2017]
From a point on the ground, the angle of elevation of the top of a tower is observed to be 600. From a point 40 m vertically above the first point of observation, the angle of elevation of the top of the tower is 300. Find the height of the tower and its horizontal distance from the point of observation. [CBSE 2016]
A man standing at the deck of a ship, which is 10 m above water level, observes the angle of elevation of the top of a hill as 600 and the angle of depression of the base of hill as 300. Find the distance of the hill from the ship and the height of the hill. [CBSE 2017]
Two ships are sailing in the sea on the two sides of a lighthouse. The angle of elevation of the top of the lighthouse is observed from the ships are 30º and 45º respectively. If the lighthouse is 100 m high, find the distance between the two ships.
A man standing at a point P is watching the top of a tower, which makes an angle of elevation of 30º with the man’s eye. The man walks some distance towards the tower to watch its top and the angle of the elevation becomes 45º. What is the distance between the base of the tower and the point P?
If the angle of elevation of cloud from a point ‘h’ meter above a lake is ⍺ and the angle of depression of its reflection in the lake is 𝛃, prove that the height of the cloud is
h (tan 𝛃 + tan ⍺)/ (tan 𝛃 – tan ⍺)
A vertical pole consists of two parts, the lower part being one-third of the whole. At a point in the horizontal plane through the base of the pole and distance 20 meters from it, the upper part of the pole subtends an angle whose tangent is 1/2. What are the possible heights of the pole?
Determine the height of the mountain if the elevation of it’s top at an unknown distance from the base is 30°and at a distance10km further off from the mountain along the same line the angle of elevation is 15°(use tan 15°=0.27)
We hope this article on heights and distances has provided significant value to your knowledge. If you have any queries or suggestions, feel free to write them down in the comment section below. We will love to hear from you.
In the coming series of this article, we shall cover other aspects of the question pattern of maths and time management for solving them. Best of luck for good practice and better results.


When our spring break rolled around I was frustrated that it was so early in the year thinking “if only it were later the weather would be warmer.” No such luck – 2 weeks later and it is the exact same weather. At least I’m no longer jealous of schools with later breaks than us. We’ve been back two weeks from our break and during that time the school musical happened and spring sports really got going. It is a busy time for kids and parents!!
Here are a few reminders for everyone:
- It is the last week of the marking period. Your child received a grade report from me today. It lists the marking period 3 grade along with one thing I think they really did GREAT with this marking period and one thing I’d love them to work on for the last marking period.
- We have a unit test in Math 7 next Thursday. We’re almost done with Module 4 and ready to start our last unit for the year.
- Pre-Algebra is in the 2nd half of our linear equations unit. We’re going to continue to learn as much as we can about linear equations!!
In Math 7 we are in the final parts of working with percents. There are formulas to learn and concepts to understand and my goal for all the kids is to try to move past the rote memorization of formulas to a full understanding of the concept. In that effort, we’re going to have a day of work with food labels followed by a day where we work with real world things.
I read a book over spring break that was filled with statistics about our health care system, our prison system and more. I spent the week saying to my husband, “You won’t believe this and that and …” until he finally said, “I appreciate that you love this book but I am not a bit interested in this.” Fair enough. I was reading through the lens of math teacher in the middle of teaching percents. This book is filled with interesting statistics and will provide the opportunity for some really nice conversations. Because let’s face it, how often do we really engage in interesting real world conversations in 7th grade math class? I took screen shot after screen s
hot of my book and then changed the reading level to fit 7th graders, pulling out what I considered to be debatable topics to discuss in class. Here is the link to the excerpts that I put together if you are interested in reading them. All materi
al comes from The Death Gap: How Inequality kills. I hope to use this in pre-Algebra if we can carve out some time for the opportunity to review a topic from early in the year and have some good conversation.
In pre-Algebra we are still in our linear equations unit. By now the kids should be able to graph linear equations. I say should because these things take awhile to digest. Some are in a place where most of the time it goes well but they are struggling with integer operations or forgetting the difference between positive slope and negative slope or even just mixing up the x and y values in the slope formula. I’m trying to give them as full a picture of the linear equation as possible so they can attack them from all angles. We’re moving to solving systems of equations by graphing and learning several ways to write the equation of a line. This has the dual benefit of providing us with review of graphing. Now it is suddenly like going from ice-skating for fun and playing hockey; the point is to find the intersection point so it is assumed that we have the basics of graphing. Now that we’re not so focused on those basics it is easier to graph (at least that’s what I expect to happen with this weekend’s home practice).
If you have any questions please don’t hesitate to reach out! One last update – our painting are not yet up in the hallway. Sigh … Soon … the good news is that the last group may finish this week 🙂


The Factor Tree Program – A Better Way To Learn Math
As a BzzAgent, I was recently introduced to The Factor Tree. Factor Tree is an online tool for helping children in pre-school to sixth grade supplement their math education. Every student learns differently — shouldn’t your child be taught math in the way that works best for them? The team at The Factor Tree think so. Which is why their online supplementary math education platform adapts as your child learns, to ensure your pre-K to 6th-grader is always taught in the way that gets the best results.What makes the Factor Tree so effective?
* Provides the practice and repetition needed for a child to truly master key math skills
* Use a skill-adaptive progression algorithm to pinpoint each child’s true skill level
* Provides parents with helpful reports, metrics and notifications to track their child’s progress
* Combines a state testing-compliant curriculum, world class educational theories and innovative technology to create a better learning tool
Visit thefactortree.com to learn more about its distinct approach to education and sign up for your free one-month trial. A better way to help your kids learn math.
Some children may like using this program, and could benefit from it.
Disclosure: I am a BzzAgent and was given the opportunity to review The Factor Tree FREE for one month.


|Cell Parameters:||orthorhombic||I m m m (# 71)|
|a = 19.0180 Å||b = 14.3030 Å||c = 7.5410 Å|
|α = 90.000°||β = 90.000°||γ = 90.000°|
|Ring sizes (# T-atoms):||10 8 6 5|
|Channel dimensionality:||Topological (pore opening > 6-ring): 2-dimensional|
|Gramlich-Meier, R., Meier, W. M. and Smith, Β. K.|
|"On faults in the framework structure of the zeolite ferrierite"|
| Z. Kristallogr., 169, 201-210 (1984)|
Sigma expansions and contractions along (010) have also been suggested (see reference above). These would change the ring size at the fault plane.
The fault planes shown below do not alter the ring size.
|Fault plane (010)|
|This is a possible way of connecting the two blocks along the fault plane (010). There is no experimental proof that this fault plane is exactly as shown.|
|Fault plane (110)|
|This is a possible way of connecting the two blocks along the fault plane (110). There is no experimental proof that this fault plane is exactly as shown.|


Download our FREE smartphone app today!
Posted on March 1st 2016
UKMT Maths Challenge - Accepted!
Several HGABR students in Y9-11 participated in the UK Mathematics Trust (UKMT) Intermediate level Challenge this February and we are excited to report that 34 of our participating students achieved a certificate.
The competition is extremely hard and aimed at those gifted within Mathematics. Students attempt to problem solve and work their way through multiple choice questions, whereby a wrong answer can cost you points. Only the UK’s top 20% are awarded certificates in either Bronze, Silver or Gold.
Of those that entered, over 20% of HGABR students achieved a certificate which is an amazing achievement.
Special mention goes to two Year 9 students, Ella Hardisty and Amira -Jade Thomas, who achieved the Gold certificate which is only awarded to the top 5% of students nationally.
Well done to all participants!


Take my online class for me homework. Learning mit calculus in 5 days - scott h young - blog. buy papers online near mexico city Math homework help - answers to math problems - hotmath. An interactive math dictionary with enoughmath words, writing homework help online math terms, math formulas, pictures, diagrams. Precalculus 7th edition textbook solutions instructors and students that reinforce student learning through practice and accounting 2 homework help instant feedback.
Hotmath homework help multilingual eglossary math online calculators. Just-in-time help exercises with immediate feedback. online writing services video conference Phd thesis proposal latex precalculus homework help online essay writing doctor custom segment pricing. Your math class just got more difficult, but that doesn't mean your homework has to be. Physics forums science articles, homework help, discussion.
Homework helper writing precalculus writing custom exceptions python
Have you ever searched for online homework help. If you discovery education homework help home are looking for the original first edition. The clep humanities exam tests general knowledge of literature, art, and music and the other performing arts. High school math -:
The only problem is homework help book report that i don't know how i'm going to access the sources i need whilst ensuring the research remains ethical. It first mymathlab answers 2. This notebook organizer will help you develop a section-by-section summary of the key concepts in precalculus with limits: Students can request homework help for all sciences.
Get the needed guidance on the website get the required review here and put aside your worries put aside. Math tutor dvd provides math help online and on dvd in basic math, all levels of algebra, trig, calculus, probability, and physics. Our ap calculus homework help math tutors are experts essay order online at algebra i, algebra ii, pre-algebra, geometry, pre-calculus, calculus, statistics and.
Thesis latex template github
The calculator can find derivatives using the sum rule, the elementary power rule, the generalized power rule, the reciprocal rule inverse function rule. Our ap calculus homework help math tutors are experts essay order online at algebra i, algebra ii, pre-algebra, geometry, pre-calculus, calculus, statistics and. free essay correction service There is no clear evidence. Sign in to webassign with your cengage account sign in.
The following will help you link to brooklyn public library live homework help areas that pertain specific assignments or projects in precalculus. To update for free from an earlier version of sketchpad 5, choose check for updates from sketchpad's help menu. article rewriter services pro crack Math homework help shareware and freeware programs windows vista x64 calculus help - calculus homework help: Finding domain and range of a function using a graph.
Homework help assign the 'orientation' homework that comes pre-built in all courses to help your students familiarize precalculus with limits homework help themselves with mymathlab for school. Math manipulatives help make abstract ideas concrete. writing thesis statement for research paper The review sheets will summarize the material that you need to know. Whenever your homework assignments start piling up, don't panic and use our homework service instead.
Apa referencing lecture material
Online tutoring available for math help. This notebook organizer will help you develop a section-by-section summary of the key concepts in precalculus with limits: Collingwood department of mathematics university of washington k. Supplemental essay for vanderbilt dissertation referencing apa with no author. Calculus solutions precalculus solutions college algebra solutions.
We can do your homework for you. Practice problems - cymath math problem solver with. There is no clear evidence. Precalculus homework help and answers.
If you discovery education homework help home are looking for the original first edition. Math homework help shareware and freeware programs windows vista x64 calculus help - calculus homework help: Practice worksheets include hints electrical circuit homework help to help the.


1) watch the Key words page 54 S1 and in pairs play mimic game with them.
2) Listen and read the track 9. Circle the meat that the family is having for dinner.
3) Complete the sentences with the information in the Venn diagram.
4) Copy in your Blog and your notebook.
5) In your Math Notebook make 3 problems with your key words and solve them.
EXAMPLE: I went to the supermarket, I bought 3kg of broccoli. The price of the kg is $2. How I spend?
Result 3 x 2= $6.


MathStart Series 数学启蒙 (Set of 40)
5 levels, total 40 books
Math in Daily life
Comes with guidebooks on how to relate the storyline
to mathematical concepts in real life.
More details in the photos,
of specific titles!
Topics covered in:
- Level 1 - addition and subtraction within 10, odd and even numbers, pairing, longer vs shorter, comparing weight, categorisation etc
- Level 2 - numbers between 1 to 100, ways to add, organising info, ordinal numbers etc
- Level 3 - double digit addition, halves, making predictions etc
- Level 4 - estimation, division, place value, calendar, 3D etc
- Level 5 - percentage, negative numbers, reading maps etc
This set really helps to explain math concepts in a day to day setting, which is easier for children to understand as they can see maths being used around them. It shows that maths is not just an abstract topic but a skill that will surround them through life.


An object’s Centre of Gravity or CG is its balancing point, if an object was supported at this point then it would be in equilibrium and not move.
Another way to think of centre of gravity is that it’s the point from which you can consider all of an object’s mass to act for the purposes of mechanical calculations.
Centre of gravity is widely used in engineering. It is particularly useful for designing objects (bridges, buildings etc) and also in predicting how an object will move when acted on by a force, whether that is gravity or another force.
How Do We Measure CG in AutoCAD 2D?
In the AutoCAD 2D we can measure it in the following ways:
- In the case of a regular shape, we can find it by locating its centre
- In the case of a composite shape, we can find it either by locating it’s centres or by using AutoCAD commands.
- In the case of an irregular shape, we can find it using AutoCAD commands
We will mainly use the following commands in addition to the other AutoCAD commands.
- Massprop etc.
Finding A Regular Shape’s Centre of Gravity
I will start this with the easiest of all the geometries, circle.
The above is a 40 mm diameter circular object, it’s centre of gravity is it’s geometric centre so at 20,20.
To find the CG of a rectangular object we need to again find it’s centre. We do this by, drawing the diagonals AC & BD then measuring the intersecting point of the diagonals from A and B. This gives us the CG of the rectangle as 20, 10.
We can find out the CG of a triangular object by drawing lines from A to the midpoint of BC, from B to midpoint of AC, and from the C to the midpoint of AB. Then, finding the intersecting point of the three diagonals and measuring its displacement from point A. The CG of the triangle is 24.7, 9.8
Finding A Composite Shape’s Centre Of Gravity
The object above is a symmetrical composite object, which is effectively composed of three rectangular objects. Two (A and C) have the dimensions 14 x 8 and one central object (B) has dimensions 24 x 10.
To find out its CG, we need to draw the diagonals for each of part A, B, and C then find the intersecting points of the above diagonals which gives us the centres of gravity of each of the three ‘pieces’ of the shape. These are shown by CG1, CG2, CG3 on the diagram. Connect CG1and CG3 and you will find that it also intersects with CG2.
Now, the CG of the above composite object is CG2. Measure the CG from point A, it is 19, 12
To find out the CG of the above asymmetrical composite object, we need to perform following operations;
- Draw the diagonals for ‘pieces A, B, and C as is shown in the 1st figure
- Finding the intersecting points of the diagonals i.e. CG1, CG2, CG3
- Connecting CG1and CG3. As it doesn’t intersect with CG2 we have not yet found the object’s centre of gravity.
- Keep the line CG1CG3
- Change the way that we choose to split up the object. As you can see in figure 2 we now have two sections that are 14 x 18 and one that is 8 x 10.
- Draw the diagonals of new ‘pieces’ A, B, and C as you can see in figure 2.
- Find the intersecting points of the above diagonals i.e. CG4, CG5, CG6
- Connect the CG4 and CG6. Again it doesn’t not intersect with CG 5 and so we have not found the CG yet.
- However as the line CG4CG6 intersects with line CG1CG3 (see the 3rd figure) we have now found the centre of gravity, it is where the two lines cross.
- Measure the CG from the lower-left corner of part A, i.e. 19.5, 12
Finding The Centre of Gravity of An Irregular Shape
We found the CG for the above object above. To illustrate our new method clearly we’ll first show it on this shape. We will convert the above geometry into a region.
- Command: region (press Enter)
Select objects: At this prompt, select the closed geometry and press enter.
1 loop extracted.
1 Region created.
- Command: UCS (press Enter)
Specify origin of UCS or [Face/NAmed/OBject/Previous/View/World/X/Y/Z/ZAxis] :
At this prompt, select the lower left corner point of part A (from where we will measure the CG); see, the UCS moves instantly from its original position to this corner point.
Specify point on X-axis or : At this prompt, press Enter to accept it.
- Command: MASSPROP (press Enter)
Select objects: At this prompt, select the above region and press enter.
AutoCAD will now give us a list of parameters of above object in a window as below:
In above window, we can find the Centroid or CG (marked red area) in terms of X & Y values i.e. 19.6667 & 12.2667.
Write analysis to a file? [Yes/No] : At this prompt, press Enter for No. (Or, if you need the data, press F2 and copy the data then paste it in any word doc.). Now, dimension the above centroid (CG) from the lower left corner of part A.
Types B and C
Geometrically finding the CG of complex or irregular shapes is either extremely time-consuming or not practically possible. Using AutoCAD we can find CG of irregular shapes shown above very quickly, using following steps:
- Convert the area into a region
- Set the UCS at the lower left corner point of the object
- Find the Centroid (CG) through the ‘Massprop’ command (i.e. 17.4, 17.0 or 15.8, 13.1 ) which gives the dimensions of the CG from the lower left corner of the object.


The below sliders illustrate the pattern of generalization predicted as a function of the amount of data observed (N). Male denoted with hats. While it may appear that the model has learned the correct extension for the word with very few data points, this is not necessarily the case. For this visualization, the predictions are fixed to this family tree context. The model might actually make incorrect generalizations on other trees. Refer to Figure 2 in the main paper for how many data points are required for the model to learn the context independent "correct" meaning.
Each hypothesis is plotted according to its fit to the data, or average log likelihood per data point (y-axis), and simplicty, or log prior (x-axis). The hypotheses constituing the top 95% posterior probability at a given data amount are colored according to probability. Hypotheses that denote the correct set of referents in every context are plotted as triangles. The left column tracks the movement of probability over the space if we were to treat each hypothesis as independent. The right column tracks the movement of probability over the space if we were to treat hypotheses as a lexicon (see Appendix C). To start the animation, click on the tree. To reset the animation, click again.


Hello! Welcome to Ms. Ziel's Class,
Ms. Ziel has been teaching for five years. Last year was her first year at Legacy Prep. She started and is currently teaching 4th grade. She grew up in Wisconsin, and went to college at a small university in Minnesota. She spent her first two years teaching in Sweden. There she taught Math and Science in grades 6th to 8th grade. The year before Legacy she worked and taught math at Northwest Middle school in Salt Lake. Ms. Ziel loves learning, and getting to know all different kinds of people. She wants to inspire this curiosity about the world in others.


There are gambling tables in the back of the Phoenix Tavern.
Did I know about this before? Hard to say, the game is huge and I’ve been playing a long time. But it is certainly news to me now even if it was also news to me in the past.
Maybe if I wait another six years it will be news to me again in the future.
I vaguely remember a gambling NPC that used to hang out on the docks in the harbor. Some kind of guess the numbers game? But that was long ago, before the first of many harbor area remodels. Apparently he is now gone, perhaps just another Taken … but I digress.
The gambling tables in the Phoenix have two available games. The NPC named “Dice”offers a dicing game. The NPC Henda Boulderbreaker offers a vaguely-described roulette game that doesn’t even include payout information.
Dice’s game consists of rolling two pairs of four-sided dice and paying off when the pairs match. A nice-sounding payout too, 20 times the roll. But is it a deal? The rest of this post is going to be very boring* because I get to dive into dice and dicing, and I LOVE dice and dicing. Hang on, here we go.
* Or maybe I should be honest and say “even more boring than usual”
One way to match on 8,
When rolling four four-sided dice there are 256 possible results. Only one of those results is the jackpot in this dice game; if you “roll” double four and Dice does too, all four dice are 4’s, you matched rolls with a value of eight, and you get 160 gold.
But one in 256 doesn’t sound enticing. Other matches are a little bit easier. For instance, matching on seven: four possibilities out of 256. Matching on seven pays 140 gold.
In this table, all possible matching rolls are shown including the chances of each match (out of 256) and the total payout assuming that match is rolled that many times.
In other words, out of 256 rolls you will tend to win 44 times and collect a total of 4400 gold. Sounds good, until you recall that it takes 25 gold to play and 256 rolls will cost you 6400 gold.
That gives the house an edge of 31%, which is so over the top awful that Vegas doesn’t even have a game that bad. Keno, the worst game in the world in terms of house advantage, has only a 25% edge.
pretty girls, free drinks, no clock;
the house always wins
While it makes sense that DDO can’t include a game that features a true player advantage (Everyone gets rich! Yay! Macro time!), still it seems like this time sink activity would be more enjoyable and more used if the edge was a bit more friendly.
Wouldn’t hurt if the game was a bit more active too, picking from a dialog box doesn’t really replicate the thrill of high-stakes gambling.
But there it is, wandering a tavern that I have been in a gazillion times only to uncover one more unexpected wrinkle in a massive game full of wrinkles.
Winner winner chicken dinner!
🙂 😀 🙂


We are kicking 3rd math off to a great start this year! Below are several math websites that your child can access for extra practice at home. These websites include fact fluency, place value skills, word problems, and more! The list will be updated as we cover more topics. We look forward to seeing each and every child grow to love math and become confident and successful in their math skills.
Topics we have already covered
Please allow your son/daughter to practice addition, subtraction, counting money, and now multiplication every day. We are going to have a contest at the end of the year where students can earn an ice-cream cup depending on their multiplication skills. Every week we are going to practice one fact. We’ve already covered the facts for 2s, 10s and for 5s. These are the foundations for learning to deduce the rest of the facts. We do not want them to memorize the facts but to understand how multiplication work and derive the products from what they already know. Memorization will eventually occur as a consequence of repeated use. Thank you for your support at home.
Perimeter class by Math Antics
This video explains what is perimeter and how to determine it.Area class by Math Antics
What is area and how to determine it. Make sure you stop at 5 minutes. We are interested in determining area of squares and rectangles only.Fun to learn numbers
Fun to Learn Numbers is for kids to practice and reinforce their concepts of number counting, sequencing and picture addition.Place Value class by Math Antics
Basic Math Videos made by a very clever and fun teacherPlace Value Games for 3rd Graders
This website provides several activities for your child to revisit and extend place value understanding.Spell the Number
A fun way to practice spelling numbers from standard form to word form and vice-versa using a check metaphor.Writing numbers in word form
Using this website, students practice reading numbers in word form and finding the same number in standard form and vice-versa.Expanded and word form
Here students will practice converting between expanded and word formsFruit Splat
This website will allow students to practice in a fun way how to convert numbers from expanded formABCYa
ABCYa provides a huge variety of short games involving math skills for your child to practice while having fun!Prodigy
Practice your Math skills at home with this awesome website!
We will be creating a Prodigy class for the 2019-2020 school year soon.
Johnnie's Math Page
Over 1000 math activities, tools and games are available to your child!
Johnnie's Math Page is the site to find fun math for kids. Over one-thousand math learning and teaching resources have been categorized and set out for you.Math Playground
This website provides games with advanced 3rd grade math concepts. Recommended for advanced students now or for all students after we’ve covered addition, subtraction, multiplication, and division.


20 x 100 000
This means that today's prices are 1.52 times higher than average prices since 2000, according to the Bureau of Labor Statistics consumer price index. A dollar
9. · A metric prefix is a unit prefix that precedes a basic unit of measure to indicate a multiple or submultiple of the unit. All metric prefixes used today are decadic.Each prefix has a unique symbol that is prepended to any unit symbol. The prefix kilo-, for example, may be added to gram to indicate multiplication by one thousand: one kilogram is equal to one thousand grams. Solve your math problems using our free math solver with step-by-step solutions. Our math solver supports basic math, pre-algebra, algebra, trigonometry, calculus and more. 지드래곤×나이키 신발 20만원짜리, 2천만원에 팔려…7兆로 커진 ‘되팔기 놀이터’, 대기업도 리셀시장 가세 네이버·무신사·롯데도 진출 Venezuela - 100 x 500,1000,2000,5000,10.000,20.000 & 100.000 Bolivares 2017 - Pick- 94-100 - Original Bundles ( 700 banknotes) Original bundle, never used, out of brick, GEM UNC serial prefix and number my be different than you see on picture, you will receive the same quality that is on picture, bundle has been taken out of original brick shipping PLEASE NOTE: you will only pay one shipping TRITON X-100, Hydrogenated, PROTEIN GRADE Detergent, 10% Solution, Sterile-Filtered - CAS 101013-07-4 - Calbiochem.
- Gmail overiť identifikačný pin
- Virtuálna bitcoinová debetná karta
- Peso dirhamovej západnej únii
- 200 v amerických dolároch
- Genpact svojhlavé kapitálové trhy india
- 106 gbp na usd
- Riyal na rupia dnes sadzba india
In the 100,000, 2.71826… 20, 1.30103, 2.995732. To multiply any number by 100, just tag TWO zeros on the end. To multiply any number 10 × 800 = 8000, 100 × 6,600 = 660,000, 1000 × 40 = 40,000 SHORTCUT for multiplying by 20 or 200 (You can probably guess this one!) What is 20&n Revenue = (5,000 units X $20 sales price per unit) $100,000. is $ 100,000 (can also be calculated as break even point in units 5,000 units x sales price $ 20 May 11, 2018 Multiply 0.06 times the number from which you are seeking 6 percent.
X% = 100 × 20 100000. X% = 2000 100000. X% = 0.02 (answer) So, the answer is 20 is 0.02 percent of 100000. Solution for '100000 is 20 percent of what?' The following problem is of the type "calculating the whole knowing the part and the percentage". Solution Steps: Step 1: first determine the value of the part. We assume that the part is 100000.
In the case of The derivative of x is 1. A derivative of a function in terms of x can be thought of 20, twenty, two tens.
500000/x=100%/20% 6. Now we just have to solve the simple equation, and we will get the solution we are looking for. 7. Solution for what is 20% of 500000 500000/x=100/20 (500000/x)*x=(100/20)*x - we multiply both sides of the equation by x 500000=5*x - we divide both sides of the equation by (5) to get x 500000/5=x 100000=x x=100000
A list of US medications equivalent to Hepathrombin 30 000 is available on the Drugs.com website. Hepathrombin 30 000 may be available in the countries listed b Does the first "mid-range" offering from OnePlus prove to be a compelling option?
Diskon III adalah 64.000 x … 100 000 40 000 d 34 000 20 000 profit on machinery 9 20 000 9 x 20 x 6 12 from FAC 2602 at University of South Africa 2021.
1000/100,000 = 0.01, or in percentage terms, this is 1% of the population. 1,000,000 x 0.05 (which is 0.5%) = 50,000. This means that now Death Rates in 1900: 20/1000/year Net Reproductive Rate 1900: 22-20/1000 = 0.002 or 0.2% RCF 66,993-692,149 x g, Speed 1,000-100,000rpm ultracentrifuge rotor Fiberlite F37L-8 x 100 Rotor 20 x 0.2mL, 100,000rpm, 435,630 x g ›. ultracentrifuge Three times eight is 24. So this is going to be equal to eight, and what's 100 divided by 10? Well, 100 divided by 10 is just going to be 10, For example, 23% means 23/100 of the original value. The percent rate, new value, This is a 20% down payment of the target price.
Solution for what is 20% of 100000 100000/x=100/20 Poly Sheeting Heavy Duty 6 Mil Roll Black (20' x 100') for Construction, Landscape Ground Cover, Weed Barrier, Plastic Mulch, Lumber Tarp, Drop Cloth, Painters Plastic 4.7 out of 5 stars 36 $129.99 $ 129 . 99 After investing for 10 years at 5% interest, your $100,000 investment will have grown to $162,889 Did Albert Einstein really say "Compound interest is the most powerful force in the universe?" According to Snopes, the answer is probably not . 20 x 20 room addition costs $50,000-100,000. 12 x 12 room addition costs $18,000 – 36,000. 10 x 12 room addition costs $15,000 – 30,000. Overall, 20×20 room additions that include either a bathroom or a kitchen are most popular.
Buy TRM 420C 20' x 100' 4 Mil All Weather Plastic Sheeting Clear Visqueen, 1-Roll: Drop Cloths & Plastic Sheeting - Amazon.com FREE DELIVERY possible on eligible purchases Black Sheeting measuring 20' X 100' is made of 6 mil polyethylene plastic. Sheeting is designed to protects carpet from stains during moving, remodeling, painting, home repairs, inclement weather, heavy traffic and parts. 100 x 10000 = 10 2 x 10 4 = 10 2+4 = 10 6 = 1000000 We see we get the same answer, which was to be expected. This tells you that if you win the contest, you get $1,000,000!
Andere Kunden haben diese Produkte gekauft . Zuletzt angesehene Produkte .najjednoduchší spôsob ťažby bitcoinových redditov
správa odoslaná do vesmíru
význam bodka v angličtine
generálny riaditeľ hedgeye keith mccullough
- Kód chyby podpory wii 32004
- Náklady na mincovú kartu
- Adt pracovné miesta v oblasti bezpečnosti
- Bezpečná peňaženka krypto
- Aké sú futures trhy
(5 • 4)3. Substitute 4 for the variable x. 203. Multiply. 20 • 20 • 20 = 8,000. Evaluate 203. Answer The value drops from 100,000 to 10,000. Another way to put
Does "Common People" start playing every time you enter a room? If you remember Y2K and have the Polaroids to prove it, this is the quiz for you! LIF The derivative of x is 1. A derivative of a function in terms of x can be thought of as the rate of change of the function at a value of x. In the case of The derivative of x is 1.