• Equations

• Ecuaciones Exponenciales en la Ciencia I
• ##### ¿Sabía usted que?

¿Sabia usted que puede utilizar un tipo de matemáticas llamada ecuación exponencial para averiguar cuanto dinero usted tendrá en su cuenta bancaria después de recolectar interés a lo largo de unos años? Científicos también utilizan ecuaciones exponenciales para estimar la edad de un objeto por medio de datación por radiocarbono o pueden predecir que tan rápido una enfermedad se propagara por una población. De hecho, ecuaciones exponenciales son utilizadas en todas las ramas de la ciencia.

##### Resumen

Ecuaciones Exponenciales son indispensables en la ciencia debido a que pueden ser utilizadas para determinar el ritmo de crecimiento, ritmo de decaimiento, el tiempo que ha pasado o la cantidad de algo en un tiempo específico. Este módulo describe la historia de ecuaciones exponenciales y muestra como son graficadas. Problemas de muestra, incluyendo un vistazo al ritmo de crecimiento de la población de renos en la Isla de San Mateo en el Mar de Bering, ilustran como las ecuaciones exponenciales son utilizadas en el mundo real.

• HS-C3.5
##### Conceptos Clave
• Las ecuaciones exponenciales tienen una variable como una exponente y se escriben como y= abx.
• Los valores “y” (o las soluciones a ello) de una ecuación exponencial siguen un progreso geométrico y son los resultados de multiplicaciones repetidas por la misma cantidad.
• La forma de gráficas de ecuaciones exponenciales indican crecimiento o decaimiento exponencial.
• Tanto el crecimiento exponencial como el decaimiento exponencial son procesos comunes y las ecuaciones exponenciales pueden ser utilizadas para modelar y predecir en muchas disciplinas.

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• Ecuaciones Exponenciales en la Ciencia II
• ##### ¿Sabía usted que?

¿Sabia usted que al utilizar la datación carbono-14 y una ecuación exponencial en la matemáticas, los científicos confirmaron que los Vikingos visitaron Norte America 50 años antes de la llegada de Colón? Los científicos utilizaron una forma particular de una ecuación exponencial cuando tratan con sistemas naturales que están en cambio continuo. Esta ecuación común puede ser utilizada para determinar la cantidad de tiempo que ha pasado, el grado de crecimiento o decaimiento de algo, o la cantidad de algo antes o después de que una cantidad de tiempo ha pasado.

##### Resumen

Este módulo introduce ecuaciones exponenciales en la forma de N=N0 ekt , el cual describe el crecimiento o decaimiento con el paso del tiempo. Tales ecuaciones pueden ser utilizadas para predecir la propagación de enfermedades, el crecimiento de una población, ritmos de reacciones químicas, o la edad de un material basado en el decaimiento radioactivo. Las constantes e y k son explicadas y su rol en ecuaciones exponenciales es demostrado. El módulo lleva a lectores a través de ecuaciones exponenciales de muestra que utilizan e en el calculo de crecimiento de bacteria y en datación radiocarbono.

• ##### NGSS
• HS-C3.5, HS-PS1.C2
##### Conceptos Clave
• Una forma de ecuaciones exponenciales que es muy comúnmente utilizado en la ciencia es N=N0ekt, el cual describe el crecimiento o decaimiento con el paso del tiempo.
• La constante e es el limite de la expresión(1 + 1/n) n con n incrementando y representando el limite de crecimiento para cualquier sistema continuamente en crecimiento.
• La constante k es una constante de crecimiento cuyo valor depende en el material, procesos y condiciones ambientales del sistema

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• Ecuaciones Lineales en la Ciencia
• ##### ¿Sabía usted que?

Did you know that a linear equation can be used to calculate the outside temperature from the number of times crickets chirp in one minute? Linear equations have many other real-world applications as well, such as figuring out how fast a projectile is moving or converting one unit of measure to another. For this reason, they are indispensible to scientific investigation.

##### Resumen

Linear equations can be used to describe many relationships and processes in the physical world, and thus play a big role in science. This module traces the development of linear equations and explores their many uses in science. The standard form of linear equations is presented, and sample problems are given. Concepts include Cartesian coordinates, ordered pairs, slope-intercept form, describing vertical and horizontal lines, and calculating rates.

• HS-C3.5
##### Conceptos Clave
• Una ecuación lineal describe una relación entre dos variables que puede ser graficada como una línea derecha en el sistema de coordenadas Cartesianas (sistema de eje x y eje y).
• Las ecuaciones lineales tienen aplicaciones numerosas en la ciencia, incluyendo conversión de unidades (tales como Centígrados a Fahrenheit) y para calcular tasas (como la rapidez en que una placa tectónica se mueve).
• La mayoría de las ecuaciones lineales puede ponerse en forma pendiente-intersección: y = mx + b, en donde m es el pendiente de la línea y b es el punto en donde la línea cruza el eje y. Esta forma es útil para graficar ecuaciones lineales. Cuando ecuaciones lineales representan el punto de inicio de un experimento o una serie de observaciones.

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• Medición
• ##### ¿Sabía usted que?

Did you know that Egyptians used the “royal cubit” as a baseline of measurement and consistency across the kingdom. The royal cubit was determined by the forearm, length of the Pharaoh, approximately 52 cm in length (20.5 in) and was further divided into 28 equal segments, approximating the width of a finger. It was carved into black marble and Individuals could bring a stick or other object that could be marked, lay it against the marble and, in effect, create a ruler that they could use to measure length, width, or height elsewhere.

##### Resumen

In almost every facet of modern life, values – measurements – play an important role. We count calories for a diet, stores measure the percentage of tax on our purchases, and our doctors measure important physiological indicators, like heart rate and blood pressure. From the earliest documented days in ancient Egypt, systems of measurement have allowed us to weigh and count objects, delineate boundaries, mark time, establish currencies, and describe natural phenomena. Yet, measurement comes with its own series of challenges. From human error and accidents in measuring to variability to the simply unknowable, even the most precise measures come with some margin of error.

##### Conceptos Clave
• Desde sus primeros días, los sistemas de medición han proveído una base común para que individuos describan y entiendan el mundo. La medición ayuda a dar contexto a observaciones y es una manera de describir fenómenos.
• Una medida consiste en dos partes – la cantidad presente en una medida numérica y la unidad que la medida representa dentro de un sistema estandarizado.
• Cuando una medida directa no es posible, científicos pueden estimar parámetros por medio de una medida indirecta.
• Si es cierto que errores ocurren en medidas, el error de medida generalmente se refiere a la incertidumbre o variabilidad alrededor de la medida que ocurre naturalmente debido a la limitaciones de la herramienta que utiliza la medad de la cantidad.

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• Scientific Notation and Order of Magnitude
• ##### ¿Sabía usted que?

Did you know that ancient Egyptians needed 18 digits to write the number 99? We use only two digits to write the same number because our modern system of writing numbers uses place values, where each place represents an order of magnitude. Orders of magnitude are a handy way to describe the size of an object and compare the sizes of different items.

##### Resumen

The blue whale weighs approximately 190,000 kilograms, while a plankton weighs just 0.5 milligrams—a difference of 11 orders of magnitude. Scientific notation and order of magnitude are fundamental concepts in all branches of science. They are especially useful when expressing and comparing very large and very small measurements. This module traces the history of our base-ten numeration system and gives a step-by-step explanation of how to write numbers in scientific notation. Sample problems demonstrate how to divide numbers in scientific notation to determine orders of magnitude.

##### Conceptos Clave
• Scientists often deal with very large and very small measurements, and so they think in terms of order of magnitude to effectively express these measurements and differences between them.
• Orders of magnitude differences are embedded in our base-ten measurement system, where one order of magnitude represents a ten-fold difference.
• Scientific notation is used to make it easier to express extremely large or extremely small numbers, and is rooted in multiplying a number by some power of ten (10x).
• Expressing numbers in scientific (base ten) notation can often make it easier to perform simple mathematical operations on that number.

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• ##### ¿Sabía usted que?

Did you know that failure of scientists to use the same unit of measurement resulted in the loss of a $125 million satellite that was supposed to give us weather reports from Mars? This disaster could have been avoided through better communication along with unit conversion by dimensional analysis. “Dimensional analysis” may sound complicated, but this is a method we use in everyday conversions, such as when figuring out how many gallons of gas we can get for$30 or how many donuts are in two dozen.

##### Resumen

When units of measurement are not used consistently in science, serious consequences can result, as seen in NASA’s Mars Climate Orbiter disaster. This module introduces dimensional analysis, or the factor label method, of converting units of measurement to solve mathematical problems. The module takes readers through realistic scenarios where unit conversion is required and explains how to set up and solve problems using dimensional analysis.

##### Conceptos Clave
• Most unit conversions can be solved through dimensional analysis, also known as the factor-label method.
• Dimensional analysis uses three fundamental facts: (1) A conversion factor is a statement of the equal relationship between two units; (2) Multiplying by a conversion factor in the form of a ratio is multiplying by 1, since the two parts of the ratio equal each other; (3) Units "cancel" when you divide a unit by itself.
• The steps in the conversion process are (a) identifying the conversion factor(s) needed, (b) setting up a mathematical problem that uses one or more conversion factors to get to the desired units, and (c) working the math problem, canceling units along the way.

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• Statistics

• Introduction to Descriptive Statistics
• ##### ¿Sabía usted que?

Did you know that the mathematical equation used by instructors to "grade on the curve” was first developed to aid gamblers in games of chance? This is just one of several statistical operations used by scientists to analyze and interpret data. These descriptive statistics are used in many fields. They can help scientists summarize everything from the results of a drug trial to the way genetic traits evolve over different generations.

##### Resumen

Scientists look to uncover trends and relationships in data. This is where descriptive statistics is an important tool, allowing scientists to quickly summarize the key characteristics of a population or dataset. The module explains median, mean, and standard deviation and explores the concepts of normal and non-normal distribution. Sample problems show readers how to perform basic statistical operations.

• HS-C1.4
##### Conceptos Clave
• Basic statistical operations such as mean, median, and standard deviation help scientists quickly summarize the major characteristics of a dataset.
• A normal distribution is a type of probability distribution in which the probability of observing any specific value is evenly distributed about the mean of the dataset. In many scientific applications, the statistical error in experimental measurements and the natural variation within a population are approximated as normal distributions.
• Standard deviation provides a measurement of the “spread” of a dataset, or how much individual values in a dataset vary from the mean. This “spread” of data helps scientists summarize how much variation there is in a dataset or population.

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• Introduction to Inferential Statistics
• ##### ¿Sabía usted que?

Did you know that in statistics, the word “population” doesn’t refer to the people who live in a particular area? Rather, it refers to the complete set of observations that can be made. Since it is impossible to repeat an experiment an infinite number of times or observe every single individual, inferential statistics allow scientists to draw conclusions about a much larger group based on observing a much smaller set of data.

##### Resumen

Many techniques have been developed to aid scientists in making sense of their data. This module explores inferential statistics, an invaluable tool that helps scientists uncover patterns and relationships in a dataset, make judgments about data, and apply observations about a smaller set of data to a much larger group. The module explains the importance of random sampling to avoid bias. Other concepts include populations, subsamples, estimation, and the difference between a parameter and a statistic.

• ##### NGSS
• HS-C1.4, HS-C1.5
##### Conceptos Clave
• In statistics, a population is a complete set of possible observations that can be made. It is often impractical for scientists to study an entire population, so smaller subsets of the population, known as either subsamples or samples, are often studied instead. It is important that such subsample is representative of the population from which it comes.
• Inferential statistics can help scientists make generalizations about a population based on subsample data. Through the process of estimation, subsample data is used to identify population parameters like the population mean or variance.
• Random sampling helps scientists collect a subsample dataset that is representative of the larger population. This is critical for statistical inference, which often involves using subsample datasets to make inferences about entire populations.
• Statistical significance provides a measure of the statistical probability for a result to have occurred. A statistically significant result is unlikely to have occurred by chance and can therefore be reliably reproduced if statistical tests are repeated. Statistical significance does not tell scientists whether a result is relevant, important, or meaningful.

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• Confidence Intervals
• ##### ¿Sabía usted que?

Did you know that beer makers advanced the field of statistics? It takes a lot of science to brew beer well, so the Guinness Brewery hired scientists to perfect beer-making techniques. The “Student’s t-distribution” is a very important mathematical tool that came out of the Guinness Brewery research laboratory. This tool is necessary in constructing confidence intervals, a key component of inferential statistics.

##### Resumen

Through history, important scientific advances have been made in connection with brewing beer. The module begins at the Guinness Brewery with the development of an important mathematical tool for inferential statistics. The focus of the module is confidence intervals, used when making statements about the relationship between a subsample and an entire population. Readers are shown how to construct and report a confidence interval. Topics include Student’s t-distribution, confidence level, critical value, and margin of error. Examples and a sample problem illustrate concepts introduced.

##### Conceptos Clave
• Confidence intervals are a common type of inferential statistics estimate used in science. Starting with a subsample dataset, a scientist can construct a confidence interval that represents a plausible range for a population parameter while also indicating the level of error or uncertainty associated with the estimation.
• A confidence level represents the degree of uncertainty associated with a confidence interval. The higher the confidence level, the less uncertainty is associated with the confidence interval’s estimation of a population parameter. Although any value between 0% and 100% can theoretically be chosen, scientists typically calculate confidence intervals at the 90%, 95%, or 99% confidence level.
• Standard error is commonly encountered when using inferential statistics and is needed to calculate a confidence interval. It is important not to confuse standard error with standard deviation. Standard deviation is a descriptive statistic that represents the amount of variation in a sample, whereas standard error is an inferential statistic that represents a likely distance between a population parameter and a subsample statistic.

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• Trigonometric Functions

• Las Matemáticas en el Movimiento de las Ondas
• ##### ¿Sabía usted que?

Did you know that waves and circles are related? If someone steadily pulls a piece of paper on which you are drawing a circle, you will trace a wave! Understanding the shape of a circle is also important to understanding astronomy. After looking at the path of the stars in the night sky, ancient astronomer Hipparchus divided a circle into pie shapes and discovered basic trigonometric functions.

##### Resumen

Waves, circles, and triangles are closely related. In fact, this relatedness forms the basis of trigonometry. Basic trigonometric functions are explained in this module and applied to describe wave behavior. The module presents Cartesian coordinate (x, y) graphing, and shows how the sine function is used to plot a wave on a graph.

• ##### NGSS
• HS-C3.5, HS-PS4.A1
##### Conceptos Clave
• The sine function is one of many trigonometric ratios calculated by astronomer Hipparchus over 2,000 years ago.
• Understanding trigonometric functions allows for the understanding and prediction of an object’s movement.

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