• Equations

• Exponential Equations in Science I
• ##### Did you know?

Did you know that you can use a type of math problem called an exponential equation to figure out how much money you will have in your bank account after collecting interest over a number of years? Scientists also use exponential equations to estimate the age of an object through carbon dating or to predict how quickly a disease is likely to spread through a population. In fact, exponential equations are used in every branch of science.

##### Summary

Exponential equations are indispensable in science since they can be used to determine growth rate, decay rate, time passed, or the amount of something at a given time. This module describes the history of exponential equations and shows how they are graphed. Sample problems, including a look at the growth rate of the reindeer population on St. Matthew Island in the Bering Sea, illustrate how exponential equations are used in the real world.

• HS-C3.5
##### Key Concepts
• Exponential equations have a variable as an exponent and take the form y= abx.
• The y-values of (or solutions to) an exponential equation follow a geometric progression and are the result of repeated multiplication by the same amount.
• The shape of graphs of exponential equations indicate exponential growth or decay.
• Exponential growth and decay are both common processes and exponential equations can be used to model and predict in many disciplines.
• Exponential Equations in Science II
• ##### Did you know?

Did you know that by using carbon-14 dating and an exponential equation in math, scientists confirmed that Vikings visited North America 50 years before Columbus arrived? Scientists use a particular form of exponential equation when dealing with natural systems that are continuously changing. This common equation can be used to determine the amount of time that had passed, the degree of growth or decay of something, or the amount of something before or after an amount of time has passed.

##### Summary

This module introduces exponential equations of the form N=N0 ekt, which describe growth or decay over time. Such equations can be used to predict the spread of a virus, the growth of a population, chemical reaction rates, or the age of a material based on radioactive decay. The constants e and k are explained, and their role in exponential equations is demonstrated. The module takes readers through sample exponential equations that use e in calculating bacteria growth and in radiocarbon dating.

• ##### NGSS
• HS-C3.5, HS-PS1.C2
##### Key Concepts
• A form of exponential equation that is very commonly used in science is N=N0ekt, which describes growth or decay over time.
• The constant e is the limit of the expression (1 + 1/n)n with increasing n, and represents the limit of growth for any continuously growing system.
• The constant k is a growth constant, whose value depends on the material, process, and environmental conditions of the system.
• Linear Equations in Science
• ##### Did you know?

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.

##### Summary

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
##### Key Concepts
• A linear equation describes a relationship between two variables that can be graphed as a straight line on the Cartesian coordinate system (x- and y-axis system).
• Linear equations have numerous applications in science, including converting units (such as degrees Celsius to Fahrenheit) and calculating rates (such as how quickly a tectonic plate is moving).
• Most linear equations can be put into slope-intercept form: y = mx + b, where m is the slope of the line and b is the point where the line crosses the y-axis. This form is useful for graphing linear equations. When linear equations in this form are used in science, b often represents the starting point of an experiment or series of observations.
• Measurement
• ##### Did you know?

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.

##### Summary

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.

##### Key Concepts
• Since their earliest days, systems of measurement have provided a common ground for individuals to describe and understand their world. Measurement helps to give context to observations and a means to describe phenomena.
• A measurement consists of two parts – the amount present or numeric measure, and the unit that the measurement represents within a standardized system.
• When direct measurement is not possible, scientists can estimate parameters through indirect measurement.
• While errors do occur in measurement, measurement error generally refers to the uncertainty or variability around a measure that occurs naturally due to the limitations of the tool we are using to measure the quantity.
• Scientific Notation and Order of Magnitude
• ##### Did you know?

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.

##### Summary

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.

##### Key Concepts
• 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.
• Unit Conversion
• ##### Did you know?

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.

##### Summary

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.

##### Key Concepts
• 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
• ##### Did you know?

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.

##### Summary

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
##### Key Concepts
• 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.
• Introduction to Inferential Statistics
• ##### Did you know?

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.

##### Summary

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
##### Key Concepts
• 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.
• Confidence Intervals
• ##### Did you know?

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.

##### Summary

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.

##### Key Concepts
• 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

• Wave Mathematics
• ##### Did you know?

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.

##### Summary

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
##### Key Concepts
• 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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