Science 10: Mr. McPhee Name: ______________

Unit 1: Scientific Thought and SkillsContents:

Part 1: What is Science1

• Science 1• The Scientific Method

1• Controlled (Fair) Experiments

3• Natural Laws and Scientific Theories

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Part 2: The Scientist’s Tool Kit6

• Scientific Skills 6

Part 3: Data and Data Analysis10

• Data 10• How to Report, Display and Interpret Data

10• Correlation Types

11• Correlation vs. Causation

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Part 4: Math Supplement14

• Significant Figures14

• Calculations with Significant Figures18

• Scientific Notation21

• The Metric System26

• Graphing Guidelines 28

Part 1: What is Science?Science:Considering this is a science class, I think that is very important for us to ask an important question:

What is Science?

This can be a very complicated question, but we are going to give a very simple answer at first, and then dig deeper. Simply stated science is the process of finding out how the things work by testing them. In other words, using observable evidence, logic and experimentation to draw conclusions about the world (and universe) around us. An idea can only be considered scientific if it is supported by evidence and can be confirmed, repeatedly, by experimentation.

Science is continually evolving and changing. As new evidence arises scientific “facts” and ideas need to be changed and updated, or in extreme cases completely thrown out. This is not a weakness of science. In fact this is the true strength and beauty of scientific thinking: Science is self-correcting (through peer-review). Incorrect or incomplete ideas eventually fall apart under continued testing and experimentation.

There are many examples of scientific ideas that have been proven incorrect (or incomplete) over time, a few examples are:

• The four element theory• Early atomic models• Miasma Theory• Preformationism• Caloric Theory

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• Parts of Dalton’s atomic theory• Geocentric Universe/Solar system• Newtonian Mechanics• Inheritance of acquired physical traits

The Scientific MethodWhen scientists develop or try to develop a new theory, they must follow a process known as the scientific method. Only knowledge that has been found using the scientific method can become part of science. The scientific method ensures that the ideas of science are based on the most up-to-date evidence and are tested and retested. In this way incorrect ideas are discarded and as we learn science continues to evolve.

The scientific method is basically a fancy name for “figure out what happens by trying it.”

In the middle ages (400-1300) western “scientists” were called “philosophers.” These were scholars or thinkers who decided what was “correct” by arguing and debating with each other. Little or no experimentation was done.

There is evidence of actual experimental science, including systematic observations and data collection in the Middle East in the early 500s, and in China circa 1000-1300. Unfortunately, most of the collected histories of science focus on the work done in Europe in this period.

In Europe it was not until the Renaissance which began in the 1300s and ended in the 1500s that scientists like Galileo Galilei and Leonardo da Vinci started using experiments instead of argument to decide what really happens in the world.

In 1620 Sir Francis Bacon published Novum Organum which outlined the earliest rigorous statement of what we now call the scientific method.

It is important to note that these “steps” are not prescriptive. Science is creative and fluid in nature. Think of these more as a rough guideline of how scientific reasoning occurs.

Notice something interesting. Make careful observation of that something. Wonder how that interesting something works or relates to other things. Perhaps even come up with a possible explanation, this is called a hypothesis. Think of a way to test how that relationship can be tested. Design and perform an experiment that will have different outcomes depending on the

parameter(s) being tested. Repeat the experiment, varying your conditions in as many controlled ways as you can. If possible, come up with a model that explains and predicts the behavior you observed. Share your model, your experimental procedures, and your data with other scientists.

Some of these scientists may:

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a. Look at your experiments to see whether the experiments really can distinguish between the different outcomes.

b. Look at your data to see whether the data really do support your model.c. Try your experiments or other related experiments themselves and see if the results

are consistent with your model.d. Add to, modify, limit, refute (disprove), or suggest an alternative to your model.

This process is called “peer review”. If a significant number of scientists have reviewed your claims and agree with them, and no one has refuted your model, your model may gain acceptance within the scientific community. At this point your model can be called a scientific theory.

Make predictions based on the theory and test those predictions.

Notice that the hypothesis is only an optional step. It is possible (and sometimes useful) to have a hypothesis before performing an experiment, but an experiment is just as valid and just as useful whether a hypothesis was involved or not.

In order to highlight the fact that the scientific method is NOT a set of steps, rather a guideline, consider the following example of an “experiment” that follows the scientific method.

Observing. An ancient mariner has noticed that many ships that leave to explore never return. The mariner also notices that the surface of the ocean appears to be flat.

Asking a question/making a hypothesis about the world around us. “Is the Earth flat?” or “I think that the Earth is flat”

Making logical predictions about the future based on the hypothesis. “If the Earth is flat it must have an edge”

Design and perform controlled experiments to test the predictions. “I will sail my boat, in a straight-line path, to the edge of the world. Once there I will look over the edge and throw a tomato into the abyss”

Make observations and collect data. “I have sailed my boat for weeks and months, in a straight line and have still not reached an edge” and “After a very long time of sailing and walking and sailing, all in a straight line, I have returned to my original starting point.”

Analyze the data from your experiment. “It seems that perhaps the Earth has no edges”

Draw Conclusions. “If there are no edges the Earth cannot be flat” Do your observations support the hypothesis? Repeat steps 2-7 until you have a hypothesis that works.

Controlled (Fair) Experiments:The entire goal of a controlled scientific experiment is “FAIRNESS”. Imagine if you wanted to test which car had the best fuel economy. It would not be fair to test the cars with some of them towing a heavy trailer and the others not, or to have one diving up a steep hill and the others on flat land. In both of those examples we are introducing some other factor (the trailer or the hill) or variable to the test.

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For a “fair” test you should only have two variables. In this case 1. The different cars and 2. The fuel economy. The test should strive to keep all other factors equal or controlled for. For example: The speed at which the cars are driven, the outside temperature, the driving style, air-conditioning off or on…

The two variables we are testing for are called the independent variable and the dependent variable. All the other variables are called controlled variables. The biggest challenge in designing a good, fair experiment is identifying and correctly controlling for as many variables as possible.

Independent Variable: One of the two variables the experiment is designed to test for. The independent variable is the parameter that the experimenter DIRECTLY MANIPULATES or changes.

Dependent Variable: One of the two variables the experiment is designed to test for. The dependent variable is the parameter that the experimenter needs to observe and measure to see how it has changed. Also known as the responding variable.

A controlled experiment is designed to test how changes to the independent variable affect the dependent variable.

Controlled Variables: These are all of the factors, other than the independent variable, that could cause changes in the dependent variable. For the experiment to be fair the experimenter must keep these variables constant.

Experimental Control: The accepted “standard” or “normal” to which the results are compared. This is sometimes called a “baseline”. The standard should be included in the experiment as the experimental control. For example, if a scientist is testing how different fertilizers added to water affect plant growth, plain water would be the experimental control.If a scientist is testing a medication, there should be a placebo group that does not receive the medicine.

Natural Laws and Scientific TheoriesIn science we will often talk about theories and laws. It is very important to understand that in science these terms have different meanings in science than they do in everyday language.

A natural law (often just abbreviated to law) is a statement of what happens. Natural laws describe things that we know DO happen, but we do not necessarily know WHY they happen. Natural laws are confirmed by careful observation and repeated scientific experimentation. Natural laws can often be summarized by a mathematical statement.

In everyday language a theory is usually an idea that someone has that is often based on pure speculation. In fact, you will often hear things like:

“Oh well, it’s just a theory…”

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or

“Here’s a crazy theory…”

But in science theory has a much different and much stricter definition:

A scientific theory is a model that attempts to explain why or how something happens. A theory provides some mechanism to explain a law. In order to be accepted as a scientific theory the idea must:

Be backed by evidence. Lots and lots and lots of rigorously confirmed evidence. Explain a collection of related observations Successfully predict the outcomes of related experiments Have never been disproven by scientific methods

Laws and theories can change over time. They are based upon the best available evidence at the time. As new technologies and techniques allow for more and better observations, scientists may need to adapt, revise or even completely abandon a law or theory.

If a theory fails to pass the peer review process, the theory is deemed to be flawed. If the theory is flawed, it must be either modified to explain the new results or discarded completely.

If a theory fails to make accurate predictions, it is flawed. That doesn’t mean it is wrong and needs to be thrown out, just that it may need to be tweaked or added to. It might be wrong, but further testing is required.

Often laws are discovered and described first, and theories later provide a mechanism to explain the law.

Examples of Theories and Laws:

The Theory of Evolution by Natural Selection not only states that species change over long periods of time, but also provides an explanation of how this occurs based on competition, variation and heredity. There has never been an experiment that has contradicted the ideas of natural selection, but thousands of experiments have confirmed it.

Newton’s Law of Gravity states that all massive objects attract all other massive objects. Further the strength of the attraction is based on their masses and the distance between them. It is a law and not a theory because the Law of Gravity does not explain why masses attract each other.

Fg=GMmd2

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The Law of Conservation of Energy states that in any and every physical process the total amount of energy before and after the process must be the same. This is commonly stated as “Energy can neither be created nor destroyed, but can only change form”. The law states that energy is conserved but makes no attempt to explain why.

∑U=∑U o

Atomic Theory states that matter is made of atoms, and that those atoms are themselves made up of smaller particles. The interactions between the particles that make up the atoms (particularly the electrons) are used to explain certain properties of the substances. This is a theory because it gives an explanation for why the substances have the properties that they do. For example atomic theory explains the properties of solids, liquids and gases. Atomic theory explains diffusion. Atomic theory explains heat, conduction and convection. Atomic theory explains electricity and many more phenomena.

It is crucial to understand that both theories and laws must stand up to intense scientific scrutiny. Both laws and theories are backed by overwhelming evidence. Both laws and theories are subjected to rigorous testing and retesting. Both laws and theories are subject to change.

Science evolves. New observations, new technologies, new insights all require that we adapt and update out knowledge. This does not make the old science wrong. This is not a weakness in science, it is, in fact, the main strength of scientific thinking. This continual testing, revising and updating is what makes science such a powerful tool. Incorrect concepts, assumptions and ideas that cannot be supported with evidence cannot survive under scientific scrutiny. Ultimately, the scientific method always tends to lead toward truth.

Part 2: The Scientist’s Tool KitScientific SkillsAll scientists must make use of certain skills in order to do science. Some of those skills are:

observing (qualitative and quantitative) predicting classifying measuring estimating making and models designing and performing experiments organizing and displaying data

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analyzing data graphing

Many of these skills are not specific to science. However, to do good science these skills need to be practiced and honed. The single most important thing that a scientist needs is curiosity.

We make use of many of these skills nearly every day.

Observing:Observing is simply the act of using our senses, and sometimes additional tools, to gather information about the word around us. We look, listen, touch, smell and taste the world around in order to gain knowledge. In science we need to use these same skills to get information about some event.

Observations can be either qualitative or quantitative.

Qualitative: Information that is non-numerical. Some examples of qualitative observations are the softness of your skin, the level of pain you experience and the color of your eyes. Qualitative data can sometimes include data that could be determined numerically. For

example, if you say “There are a lot of people here” or “She is tall” or “the water is cold”

or even “the density of alcohol is lower than the density of water” those are all qualitative observations.

Quantitative: Information that is numerical. Simply stated if can summarized with a number, it is quantitative. Quantitative data is less open to interpretation and opinion and is often considered “hard data” as opposed to “soft data”. Using the examples above qualitative

data would be “There are 114 people here” or “She is 194cm tall” or “the water is 6.0oC” or “the density of alcohol is 790g/L, while water is 1000g/L”.

Quantitative observations include measurements, counts and estimates.

PredictingPredicting is making use of prior knowledge to determine what will happen in the future. It is important to understand that we are not talking about some sort of superstitious fortunetelling or soothsaying. In the scientific sense we are using established scientific principles to determine a future occurrence. For example, during the summer if there have been dry conditions, we can predict that a lightning storm will trigger forest fires. During winter weather we can predict that there will be more car accidents.

Predictions can be used as very strong evidence for a scientific idea. A classic example is the prediction of the existence, mass and orbit of Neptune in 1845 by Urbain de Verrier using Isaac Newton’s Law of Gravitation. Radio waves were predicted by James Maxwell in 1862, he also predicted the existence of X-rays and other forms of electromagnetic radiation.

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In science a prediction often takes the form of: “I believe this is true. If this is true, then this should happen…” It is very important that the prediction is testable. If the prediction is wrong, the underlying scientific theory needs to be reconsidered.

ClassifyingClassification is process of grouping things according to similar properties. Classification helps scientists to make sense of the world around us. Some common examples of scientific classifications are:

Chemical Classification: Chemical can be grouped in many ways according to the ways they behave and react. For example, acids and bases, metals and non-metals, organic and inorganic.

Taxonomy: Grouping of organisms into categories like: Plants and animals, mammals and birds, canine and feline, et cetera.

Subjects: Even science itself is categorized into sub-topics: Biology, chemistry, physics, engineering, geology, astronomy, astrophysics, biochemistry and so on.

Of course, classifying is very common outside of science, too. Just look around Burnaby North; there are all sorts of classifications: teachers/students, grade 8/9/10/11/12, North building/South building and so on.

MeasuringTo make a measurement means to use a tool in order to determine a precise bit of information about an object. The tool we use is a measurement instrument like a scale or a ruler or a stopwatch. The tool could be more advanced like a photogate or a Geiger counter or a sphygmomanometer. When we take a measurement, we are comparing the object to some standard, like the meter or the second or the kilogram. There are different systems of measurement, but science uses the Systѐme International. This system uses the gram, meter and second as the standard mass, length and time units.

When we make measurements, we need to worry about accuracy and precision.

Accuracy: Accuracy describes how close a measured value is to the “real” or “actual” value. The difficulty is that often scientists are measuring quantities for which they do not know the actual value. For example, how do we determine the mass of the Sun? How can we know if our result is “correct”?

Precision: Precision describes how consistent a collection of measurements is. Precision can be described how close together a group of measurements are to one another. Whenever a quantity is measured there is some uncertainty in the measurement. The amount of detail that you can CONFIDENTLY and CONSISTANTLY state in a measurement is the precision. This is determined by your measurement device, your technique and the quantity being measured.

The precision of a number can be seen by the number of significant figures that are in the reported value or often as an uncertainty (more on this later).

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In simplest terms the PRECISION tells you how many decimal places you know the measurement to.

EstimatingAn estimate is an approximation of a measurable value. In science we generally try to avoid using estimates whenever possible, preferring to take actual measurements. However, at times estimates are necessary. For example, a biologist may need to estimate the population of salmon in a stream. For many reasons it may be impossible to actually count each individual salmon and so careful estimates must be made. An estimate is not a guess. An estimate is based upon careful observation and prior knowledge.

Whenever we need to talk about future events, the further into the future the more we will need to estimate. Imagine you need to tell someone how long it will take you to walk from this classroom to the library. Now imagine you need to estimate how long it will take you to get to UBC on public transit. Because the time to UBC is much greater there is a greater chance for error in the estimate. Weather forecasts and predictions about climate change are examples of this type of estimation in science.

ModelsModels: Models are used to study objects or systems that are difficult or impossible to study directly. Models can be used to study very large objects, like the solar system. Models may be used to study very small objects like molecules. Models are used to design and test technologies like the design of an airplane wing. Think about a model showing the Earth – a globe. Until 2005, globes were always an artist’s representation of what we thought the planet looked like. The first known globe to be made (in 150BC) was not very accurate. The globe was constructed in Greece and only showed a small amount of land in Europe. It did not include Australia, China or North and South America. It was however, a sphere. As the amount of knowledge built up over hundreds of years, the model improved. In 2005, the first globe using satellite pictures from NASA was produced. By the time a globe made from satellite images was produced, there was no noticeable difference between the representation and the real thing.

Models are also very important in allowing scientists to make simplifications of complex systems to start to understand how thing work. Once the basics are understood more complexity can be added to the model.

ExperimentingExperimentation is one of the defining features of science. At the most basic level experimenting is simply figuring out what happens by trying it. Every time a child takes apart a toy or

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intentionally drops something on the floor just to see what happens they are conducting a rudimentary experiment.

Experiments are how scientists can generate reliable observations and data. Experiments are the way that scientists create the evidence that is needed to make informed conclusions and to develop sound (good, well-reasoned) theories. Of course, to produce accurate and reliable data, the experiment must be well designed as described on page 3.

Experimental design can be extremely difficult. At this level we will focus on the very basics. To design an experiment you must, of course, first have a question. Then you must determine the dependent and the independent variables. Next you must identify as many controlled variables as possible. It is important that the variables you have identified are measurable and well-defined. If you are looking at plant growth, you must define how growth will be measured: Height, mass, amount of fruit…You must be clear what data is to be collected, and how it will be measured and how it will be analyzed.

It is important that scientists trust their data. If an experiment is well designed, and the data produced is not what is expected the experimenter should not alter or ignore that data. Only considering data that agrees with your ideas and ignoring data that contradicts what you believe is known as confirmation bias and is one of the biggest problems that faces science, and how people understand it.

Part 3: Data and Data Analysis10

Data:All scientific observation and experimentation must produce consistent, testable and repeatable data. Once a scientist has collected sufficient data, they must organize, analyze, interpret and ultimately draw conclusions and make predictions based upon the information they have.

It is very important that a scientist remains unbiased in the collection and analysis of their data. What this means is that the scientist/researcher should not decide what “should happen” before conducting the experiment. Of course, the scientist may have some hunch or may be expecting a certain result, but they must not allow that to affect the process of data collection and interpretation.

Data: is any information that has been collected through observation. This data falls into two main categories as mentioned earlier:

Qualitative Data: information that can't actually be measured. Some examples of qualitative data are the softness of your skin, the level of pain you experience and the color of your eyes. However, try telling Photoshop you can't measure color with numbers.

Quantitative Data: information about an object or event that is numeric. Quantitative data includes any measurements, counts or numerical estimates about an object or event. Examples include height, mass, temperature, time, speed, frequency and volume.

How to Report, Display and Interpret Data:Qualitative data is often given in descriptive paragraph form. Depending on the nature of the experiment, the qualitative data may be summarized in a data table. In general, qualitative data is not displayed in graphs and cannot be rigorously analyzed using mathematical techniques.

Quantitative data is generally displayed in Data Tables, Diagrams and in Graphs. Qualitative data is extremely valuable as it can be analyzed using mathematical techniques. Depending on the type of data collected there can be different types of graphs. The two main types of graph used in science are the bar graph and the scatter plot/best fit curve.

Bar graphs are usually used to compare variables when one is numerical (QUANTITATIVE) and the other is not (QUALITATIVE). For example, the amount of rainfall (QUANTITATIVE) compared to the month of the year (QUALITATIVE). Another example might be the number of chocolate bars sold (QUANTITATIVE) by each member of a soccer team (QUALITATIVE) in a fund raiser. Sometimes a bar graph can be used to compare two numerical variables if one or both is discreet, for example the number of hotdogs sold at each different price point.

Scatter plots with best fit curves are usually used to compare two variables that are both numeric and continuous. For example the mass of a liquid (QUANTITATIVE) compared to the volume of the liquid (QUANTITATIVE). Another example is time (QUANTITATIVE) for a sugar cube to dissolve vs temperature of water (QUANTITATIVE).

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Correlation types: If two numerical values are somehow connected, like speed of a water wave and

depth of the water, they are said to be correlated. Look at the word co (together) related (connected)

If both variables generally increase or decrease together (i.e. either both increase or both decrease at the same time) the correlation is called a POSITIVE CORRELATION.

If one variable generally decreases while the other increases the correlation is called a NEGATIVE CORRELATION.

No correlation Strong positive Strong negativecorrelation (non-linear) correlation (non-linear)

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Correlation vs. Causation: A correlation between two variables simply tells us that the variables change

together. This can sometimes be a little bit misleading.

A correlation between variables, however, does not automatically mean that the change in one variable is the cause of the change in the values of the other variable.

Causation indicates that one event is the result of the occurrence of the other event; i.e. there is a causal relationship between the two events. Causal relationships can be difficult, or even impossible to prove.

Correlation is very often mistaken to imply causation, but this is false and can be very dangerous. This is not a new problem as the Latin phrase “cum hoc ergo propter hoc” which translates more literally to “with this therefore because of this” states this exact misconception.

Consider the following bar graph:

There is a LOT wrong with this graph, but hopefullywe can all agree that although appears to be a strong correlation, we can certainly not attribute any causation!

Another classic example that persists despite overwhelming scientific evidence is the idea that being cold causes an individual to catch a “cold”. Obviously, the terminology isn’t helping anyone here.

We know that colds are caused by viruses. Being cold does not expose you to any extra viruses. Also, unless you are hypothermic, being cold does not suppress the action of your immune system. However, we also know, at least anecdotally, that there are more people sick when the weather is colder than when it is warmer. There seems to be a correlation between temperature and sickness. So what is going on?

Another famous example is miasma theory which stated that disease was caused by bad smells, like those near rotting flesh or sewage. It was believed that you could protect yourself with the heavy use of perfumes. Now, while it is true that exposure to the bad smells of rotting flesh and sewage will certainly increase infection rates (correlation) the bad smells are not making people ill (causation). In fact we know that the same bacteria that create the bad smells cause the infection. So perfume won’t help you!

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Here’s another graph:

This is an example of how the misconception can be dangerous. If we were to believe that organic foods cause autism it could lead people to stop looking for the true causes. This is similar to the anti-vax movement.

When you have two (or more) data points that ‘line up just right’ (or, that correlate with one another) there are six ways of looking at the data:

1. A directly causes B2. B directly causes A3. A causes C, which in turn causes B or B causes C, which in turn causes A.4. Both A and B are consequences of a common cause5. Some combination of 1, 2, and 3 may be in place (which normally describes a self-

reinforcing system, such as the predator-prey system, where one intrinsically affects the other and vice versa.

6. A and B aren’t related, and the correlation is a coincidence.

The major difficulty is trying to determine what the other factors, or variables, are that could be changing along with the variables you are studying. Scientists attempt to establish causation by designing and performing controlled experiments. A controlled experiment is one in which the two variables we are testing for are isolated. The scientist attempts to control for (not allow to change) as many other variables as possible. In this way we can hopefully establish an actual, or at least probable causal link.

For the following examples try to categorize the correlation as one of the 5 types above.

1. Year over year throughout the world, but particularly in the United States and Canada there is a very strong correlation between ice-cream sales and deaths by drowning.

2. Cancer researchers have noticed a strong correlation between the number of years a patient has smoked (5 or more cigarettes per day) and the rates of lung cancer.

3. Between 1986 and 2016 The number of electric vehicles sold, and the per capita consumption of mozzarella cheese had a strong positive correlation.

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4. Since 1976 there has been a strong negative between rates of diabetes and average family income.

Part 4: Math Supplement (Mathy Bits)Significant Figures (Sig Figs)It is important in science to know to what precision a value is being reported. For instance, if some says that the shoes that they bought cost “one hundred dollars” does that mean exactly $100.00? Could the real cost be $98.74 or $122.14? How can we know? In science there is a set of rules to determine the number of digits in a reported value that are known to be reliable. It is important to know these rules when reading a number and when reporting a number.

When we make and record a measurement it is important that we report the value with the correct precision, and so the correct number of sig figs. If we are measuring a static object, the precision is determined by the measuring device. The general rule of thumb is:

“Report all known digits and then YOU MUST estimate the final digit.”

This means the reported value will be given to one decimal point beyond the smallest division on the measuring device.

In a measurement all the measured digits INCLUDING THE ONE ESTIMATED DIGIT are considered to be significant.

EXAMPLE 1: What is the measurement shown below?

We can see that the measurement is between 6.3cm and 6.4cm.

We need to ESTIMATE the final digit.

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If we look closely, we can see that the measurement is closer to 6.3cm than to 6.4cm. We will report this measurement as 6.32cm. (It would be possible to estimate this as 6.31cm or 6.33cm, the final digit is AN ESTIMATE).

The measurement is precise to 0.01cm. This tells you the PLACE VALUE of the ESTIMATED DIGIT. The 2 in 6.32 occupies the hundredths ( 1

100 or .01) place.

We CANNOT estimate beyond one digit. For example you cannot report the measurement to be 6.325cm as this has two digits (6.325cm) beyond the final certain digit.

THE FINAL DIGIT REPORTED IN THE MEASUREMENT THE ESTIMATED DIGIT.

The length of the book is 6.32cm. All of these digits are considered to be trustworthy, and they are known as SIGNIFICANT FIGURES. The number 6.32cm has 3 significant figures.

When we see the measurement 6.32cm we need to understand that what we know is that the measurement is between 6.3cm and 6.4cm, and that it is estimated to be 6.32cm.

EXAMPLE 2:You are given a measured value for the mass of a motorcycle. The mass is given as 226kg.

Look at the number: 226kg

The final digit 6 is the estimated digit. This means we know FOR CERTAIN that the measurement is between 220kg and 230kg. All reported digits including the estimate are considered to be precise. This number is precise to the ONES or to 1kg.

From this we can determine that the mass is between 220 kg and 230 kg The final digit 6 is AN ESTIMATE. This measurement is precise to 1 kg This number has 3 significant figures

EXAMPLE 3:You are told a cannon-ball flies 194.2m from where it is fired to where it lands.

From this we know the cannon ball travels between 194 m and 195 m The final digit, 2 , is AN ESTIMATE. This measurement is precise to 0.1 m This number has 4 significant figures

EXAMPLE 4:The length of a bus is 11.48m.

From this we can determine that the length is between 11.4 m and 11.5 m16

The final digit 8 is AN ESTIMATE. This measurement is precise to .01 m This number has 4 significant figures

EXAMPLE 5:A water balloon filled with cooked spaghetti noodles has a mass of 1.2kg

From this we can determine that the mass is between kg and kg The final digit is AN ESTIMATE. This measurement is precise to kg This number has significant figures

EXAMPLE 6:The speed of a car is determined to be 28m/s.

From this we can determine that the speed is between m/s and m/s The final digit is AN ESTIMATE. This measurement is precise to kg This number has significant figures

EXAMPLE 7:What mass is indicated by the arrow below?

The mass is between 2400g and 2500g. We need to ESTIMATE ONE final digit. Our final answer should be precise to 10g. The arrow is much closer to 2400g than 2500g. My estimated digit needs to be the TENS digit. I estimate 2420g.

From this we know the mass is between ___________g and _____________g The digit, _______, is AN ESTIMATE. This measurement is precise to _____________g This number has significant figures

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2400g 2500g 2600g

The rules for deciding the number of significant figures in a given measurement are:

1. Non-zero digits are significant unless indicated otherwise. 2. Zeros between two significant digits are significant (for instance 205mL has three sig. figs.) 3. Placeholder zeros are not significant.

a. Zeros that lead a number, or come at the end of a number in order to show place value are not significant. For instance 0.003m has only one significant figure. The leading zeros are holding place value to show that the 3 represents 3 “one thousandths”, as opposed to 0.03m where the 3 represents 3 “one hundredths”. b. Zeros at the end of a number that serve to hold place value are not significant. For instance 1200m contains two significant figures. The zeros at the end are to indicate that the 2 represents 2 “hundreds” and the 1 represents 1 “thousand”, as opposed to 120 where the 2 represents 2 “tens” and the 1 represents 1 “hundred”.

4. Zeros after the decimal point and after the first non-zero digit (i.e. “final zeros”) are significant.

For instance 3.200cm has four significant figures.

Examples:

A. 123km Non-zero digits are significant (Rule 1), so all digits are significant. 3 significant figures.

B. 2001cm Non-zero digits are significant (Rule 1). The zeros in this measurement are between two non-zero, and thus significant figures, and so are also significant (Rule 2). 4 significant figures.

C. 0.0230s The two leading zeros are placeholders, and so are not significant (Rule 3a). The zero at the end is significant (Rule 4). 3 sig figs.

D. 1102200g Non-zero digits are significant (Rule 1). The zero between the 1 and 2 is significant (Rule 2). The final 2 zeros are placeholders, and so are not significant (Rule 3b). 5 sig figs

E. 20.0N Non zero digits are significant (Rule 1). The final zero is significant (Rule 4). The other zero is between two sig figs, and so is also significant (Rule 2). 3 sig figs

F. 1000000 Non zero digits are significant (Rule 1). All of the zeros are placeholders and so are not significant (Rule 3b). 1 sig fig.

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Calculations with Significant Figures:When we perform calculations, we need to be very careful with our significant figures. Significant figures indicate the precision of a measurement. When we use these measurements to find other values, we need to know how precise the calculated results are. Sig figs can be difference between whether the bridge stands or falls.The following rules apply to calculations with significant figures:

1. Addition and SubtractionSums and differences of values are known only as well as the least PRECISE input value.

Example 1.

9400m + 822.2m = 10222.2m.

However, this answer has an INCORRECT PRECISION.

The least precise input is 9400m, this is precise to 100m, thus the answer should be reported precise to 100m, or 10200m

So, with proper significant figures:

9400m + 822.2m = 10200m

To see why, look at the operation:

The number 9400m means that the measurement is between 9000m and 10000m. To the best estimate we have written 9400m. It is precise to 100m. The digit 4 is an ESTIMATE, the trailing zeros have no measurement value, they are place holders. The real value could be 9395m or 9428m or 9387.9m, we don’t know exactly!

The number 822.2m means that the measurement is between 822m and 823m and has been estimated to be 822.2m. It is precise to 0.1m. All of the digits in this number are significant.

When we add (or subtract) we line up the decimal points like so…These zeros are INSIGNIFICANT. Their actual value could be anything!

9 4 0 0 . 0 m

+ 8 2 2 . 2 m

0 2 2 2 . 2 m These values were determined by adding INSIGNIFICANT zeros. They are

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not to be trusted!

This is the last column containing ALL significant figures.

So the answer should be reported as 10200m.Example 2.

78.6s + 59s

7 8 . 6 s

+ 5 9 . 0 s

13 7 . 6 s

The final 6 is not significant, HOWEVER it is still used to help us round to find the final answer.

The final answer should be given as 138s

Example 3.

418g + 950g – 99.576g

The least precise number here is 950g. It is precise to 10g. Thus our final answer must be given precise to 10g.

418g + 950g – 99.576g = 1268.43g

= 1270g

2. Multiplication and Division (Including exponents and roots)

Products and quotients of values have the same number of significant figures as the input value with the least number of significant figures.

Example 1:

29cm x 953cm = 27637 cm2

However this answer has an incorrect precision. The input with the least significant figures is 29cm, which has two sig figs. Thus the final answer should have two sig figs and be reported as 28000cm2.

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The explanation for this rule is too complicated to go through here. So just trust me.

Example 2.

200m3.00 s = 66.6666666666666666666666666666666666666667m/s. Clearly this is TOO

MANY digits.

200m has one sig fig, 3.00s has three sig figs, so the answer should be given with one sig fig.

200m3.00 s = 70m/s

Example 3.

(6.6cm)2=43.56cm2 but 6.6 has only 2 sig figs so the answer must be rounded to 2 sig figs:

(6.6cm)2=44cm2

3. ConstantsExact physical and mathematical constants and defined conversion factors (π, 1kg = 1000g) integer coefficients (like the 2 in “C=2πr”), and the 10 in scientific notation (2.3x105m) have an infinite number of significant figures.

4. RoundingTo avoid round-off errors, carry at least 2 extra or insignificant figures in intermediate calculations until the end of the final calculation (Usually I keep all of the digits in my calculator). Insignificant figures (excess digits) are rounded off in the final step of the calculation(s).

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Scientific Notation:Exponents:

Exponents are a sort of mathematical shorthand. If we have a number, any number, or a variable (which is like a number in disguise), multiplied by ITSELF repeatedly we can use exponents to save time (and graphite).

Ex. 2(2)(2)(2), which is 2 multiplied by itself four times, can be written 24.

a(a)(a)(a)(a)(a)(a)(a) can be written a8.

5(5)(5)(5)(5)(5)(5)(5)(5)(5)(5)(5)(5)(5) can be written 514.

You can imagine what writing out 688 would be like without exponents.

Powers of 10

All of this applies to any number, but we are interested in powers of ten. Consider the following powers of 10. You can check all of these on your calculator if you wish.

104 = 10x10x10x10 = 10 000103 = 10x10x10 = 1000 102 = 10x10 = 100101 = 10 = 10100 = 1 = 110-1 = 1

10 = 0.1

10-2 = 110×10 = 0.01

10-3 = 110×10×10 = 0.001

10-4 = 110×10×10×10 = 0.0001

Notice that if you count the number of zeroes in the decimal form it tells you the exponent. This is because our whole number system is based on tens. We use a BASE 10, or DECIMAL number system This can also be thought of as how many places we need to move the decimal point.

Base 10 (Decimal) Numbers:

Consider the number 62150. This number is sixty-two thousand one hundred fifty. You may recall doing something like the following in elementary school.

ten thousands thousands hundreds tens ones

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6 2 1 5 0

Let’s do the same thing in a slightly more mathy way:

ten thousands thousands hundreds tens ones

6 2 1 5 0

104 103 102 101 100

This number is sixty thousand + two thousand + one hundred + fifty. This number is six groups of ten thousand + two groups of one thousand + one

group of one hundred + five groups of ten. This number is 6x104 + 2x103 + 1x102 + 5x101 + 0x100 .

Each number in any decimal number represents some power of ten. The zero at any position in the number is very important. If we left off the “ones” (because, after all, there are none of them) the number would be 6215, which is very different. We need the zero to serve as a ‘place holder’ to show that the 5 represents groups of ten and not groups of one.

What power of ten each digit represents is determined by its position relative to the decimal point. The first digit to the left of the decimal point tells you how many groups of one there are, the next digit to the left tells you how many groups of ten, the next to left tells you how many hundreds, one more left; thousands, then ten thousands, hundred thousands, millions etc. Digits to the right of a decimal tell you many tenths, hundredths, thousandths etc.

We can use this to write some numbers in a simple short-hand.

Scientific Notation

Primarily, scientific notation is simply a shorter way to write numbers that have a lot of zeroes at the beginning or the end. The number is written as a lead number 1 and 10 (1≤ N ¿ 10) multiplied by the appropriate power of ten.

Decimal (Standard) Form Expanded Form Scientific Notation

500 5 x 100 5 x 102

200000 2 x 100000 2 x 105

0.004 4 x 0.001 4 x 10-3

0.0000000007 7 x 0.0000000001 7 x 10-10

This is particularly useful for very large and very small numbers.

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Now look at this number: 4008.2045Here is what it means in our decimal system.

4 0 0 8decimal

point

2 0 4 5Thousan

dsHundreds Tens Ones Tenths Hundredt

hsThousand

thsTen-

Thosandths103 102 101 100 10-1 10-2 10-3 10-4

What we can do in the decimal system is re-write the number by figuring the HIGHEST power of ten in the number, then writing the number as a decimal between 1 and 10 (1≤ N ¿ 10), multiplied by that power of ten.

The highest power or 10 in 4008.2045 is 103. We can re-write this number as: 4.0082045 x 103.

How about 250 000 000? The highest power of ten is 109. We can re-write this as 2.5x109.

How about 0.0000457? The highest power of ten is 10-5 . We can re-write this as 4.57X10-5

For each of these examples if you look at the largest power of ten in the number you can figure out what power of ten you need to use in scientific notation.

In the number 560 the “5” is in the hundreds (102) position, the 6 is in the tens (101) position, the 0 is in the ones (100) position. So the largest power of 10 in this number is 102. We can thus write 520 as 5.2 x 102.

In the number 0.037 the 3 is in the hundredths (10-2) position and the 7 is in the thousandths (10-3) position. So the largest power of ten is 10-2 (-2 > -3). We can thus write 0.037 as 3.7 x 10-2

Use your calculator to confirm the following:

12 = 1.2 x 10560 = 5.6 x 100 = 5.6 x 102

0.037 = 3.7 x 0.01 = 3.7 x 10-2

238400 = 2.384 x 100000 = 2.348 x 105

0.00099 = 9.9 x 0.0001 = 9.9x10-4

As a shortcut all we need to do is count how far we need to move the decimal point. Look at the last two examples:

238400: How far do we need to move the decimal to get it between the 2 and 3?

23840024

Five places, to the left. That means we can write this as 2.384x105

0.00099: How far do we need to move the decimal to get it between the first and second 9?

0.00099

Four places, to the right. This number can be written as 9.9x10-4

So…

Decimal (Standard) Form Expanded Form Scientific Notation

560 5.6 x 100 5.6x102

204000 2.04 x 100000 2.04x104

0.00000000074 7.4 x 0.0000000001 7.4x10-10

More Examples:

604500 Here the 6 represents the largest power of ten. It represents 6 x 105. So we can rewrite this as 6.045 x 105

7770000000000000000 This is annoying. You have to count out to figure out how big this number is. When we do we see that the largest power of 10 is 1017. So 7.77x1017

0.009032 Same rule applies. Which digit represents the LARGEST power of ten? That’s right the 9. What power is it? It represents 9 one thousandths, or 9x10-3, so we can rewrite this number as 9.032x10-3

124x104 This number is NOT in scientific notation. The lead is not between 1 and 10, so what shall we do? Deal with the lead first. 124=1.24x102, thus, 124x104 = 1.24x102x104=1.24x106

Practice:

Write the following in scientific notation:

A. 20 000 B. 0.005 C. 0.00008 D. 900000000000 E. 0.224 F. 135 G. 300250000H. 0.00087 I. 1500 J. 363000 K. 0.000000000044 L. 85 M. 0.0077

N. 13 000 000

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5 4 3 2 1

1 2 3 4

Write the following in scientific notation:

A. 12x103 B. 130x104 C. 0.06x106 D. 0.15x10-5 E. 255x10-10

Write the following in standard form:

A. 2.5x103 B. 2.5x10-3 C. 6x106 D. 4.0079 x 103 E. 8.87x10-7

Scientific Notation and your Calculator

Any scientific (or graphing) calculator will have a scientific notation mode. That means the calculator will put numbers into scientific notation for you. While that is very nice, you will be expected to do it yourself without a calculator. Take some time right now to find where the scientific notation mode is on your calculator. You may need to search through some settings menu somewhere. Look for a key that says FSE or perhaps SCI. Often it is above a key, so you will need to use the second function key, 2ndF. Go ahead, I’ll wait.

Even in scientific notation mode, you still must know how to properly input the numbers into your calculator. Again, your calculator is your friend and it provides you with a short cut. In order to enter the number 6.23x1018 into your calculator do the following:

1. Punch in 6.232. Find the key on your calculator that says one of the following: “EXP” “EE” “x10x”“x10y”3. Punch that key4. Punch in 18

There done. Now try the following using your calculator, the correct answer is given so you can check your work.

1. 1.226x104(8.55x109) Answer: 1.04823x1014

2. 5.5x10-9(6.11x1018) Answer: 3.3605x1010

3. 3.7x105 + 9.9x10-15 Answer: 3.7x105

4. 6.98x10-5 ÷ 1.0x10-9 Answer: 6.98x104

5. 4πx10-7 ÷ 3.2x1022 Answer: 3.926990817x10-29

6. 55.2×104

2πAnswer: 8.78535286x104

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The Metric System:The metric system is a system of measurement which uses a set of BASE UNITS and a system of prefixes based on POWERS of TEN to create a very convenient measurement system. The reason this is so convenient is that our number system is also based on powers of ten. In this way you can convert between metric measurements by simply moving the decimal point in one direction or the other.

The metric system is the official system of measurement in nearly every country in the world, the most notable exception being the United States.

The metric prefixes are as follows. You should be quite familiar with many of them, but some are likely new to you.

Name Abbreviation Meaning MeaningGiga G 1 000 000 000 109 BillionMega M 1 000 000 106 MillionKilo k 1 000 103 Thousand

Hector h 100 102 HundredDeka da 10 10 Ten

B A S E U N I TDeci d 1

10 = 0.1 10-1 Tenth

Centi c 1100

= 0.01 10-2 Hundredth

Milli m 11000

= 0.001 10-3 Thousandth

Micro µ1

1000 000 =

0.00000110-6 Millionth

Nano n1

1000 000 000 =

0.00000000110-9 Billionth

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So from the tables above we can see that the unit of length that we know as the kilometer(km) is the same as 1000 meters, a kilogram is 1000 grams a kiloliter is 1000 liters. The unit of time known as a centisecond is equal to 1

100 of a second, or 10-2 of a second.

What makes the metric system so popular, and the only system used by scientists (even American ones) is the ease it provides in converting units from small to large or vise versa.

For now, we will focus on the prefixes from milli to kilo (and everything in between) and how to do conversions between units with those prefixes.

To remember the prefixes in order, you can use the following sentence:

King Henry Doesn't [Usually] Drink Chocolate Milk

The first letters of the words stand for the prefixes, with "Usually" in the middle standing for the "unit", being meters, grams, or liters. Many memory phrases omit the "Usually", and consequently students forget where the basic unit goes, messing up their conversions. Leave the "Usually" in there so you can keep things straight:

kilo-  hecto-  deka-  [unit]  deci-  centi-  milli-Since each step is ten times or one-tenth as much as the step on either side, we have:

1 kilometer = 10 hectometers = 100 dekameters = 1000 meters                 = 10 000 decimeters = 100 000 centimeters = 1 000 000 millimeters

Alternatively, we have:

1 milliliter = 0.1 centiliters = 0.01 deciliters = 0.001 liters = 0.000 1 dekaliters               = 0.000 01 hectoliters = 0.000 001 kiloliters

The point here is that you move from one prefix to another by moving the decimal point one place, filling in, as necessary, with zeroes. To move to a smaller unit (a unit with a prefix some number of places further to the right in the listing), you move the decimal place to the right that same number of places, and vice versa. Together with the prefix sentence ("King Henry..."), this makes conversion between the different metric sizes very simple.

Convert 12.54 kilometers to centimeters.

How many jumps is it from "kilo-" to "centi-"? Five, to the right.

So I move the decimal point five places to the right, filling in the extra space with zeroes:

You don't have to make a loopy arrow like I did, but the loops help you keep track of the steps that you're counting and make it really easy to see where to add the zeroes, if you need to. In this case, after moving the decimal point and adding the zeroes, I get:

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12.54 km = 1 254 000 cm = 1.254x106 cm

Convert 457 mL to hL.   Copyright © Elizabeth Stapel 2006-2008 All Rights Reserved How many jumps is it from "milli-" to "hecto-"? Five, to the left.

Graphing GuidelinesLine GraphUse a line graph when comparing two sets of numerical data. Line graphs are extremely useful in spotting patters or trends in a data set. They also allow us to “interpolate” or determine unmeasured values between our data and to “extrapolate” or predict values beyond our data range.

1. Title. The title of the graph should be simple and clear. Normally the title is simply Dependent vs. Independent. Just to be clear, you don’t actually write “Dependent vs. Independent” at the top but the actual names of your variables, i.e. Mass vs. Volume, or Height vs. Hours of Daylight.2. Label the axes. The graph has a vertical y-axis and a horizontal x-axis. Write the name of the DEPENDENT variable on the y-axis and the name of the INDEPENDENT variable on the x-axis. Be sure to include the units! If you are graphing data that you have not collected, the independent variable is usually the first variable in the data table. You can also normally spot that the independent variable is usually “nicer numbers” as these values are chosen by the experimenter.3. Choose a scale to number each axis being sure that you scale:

a. Fits ALL of your data. If your maximum mass is 550g, your scale must go to at least 550g. You must also start from zero.

b. Is easy to use. The divisions should be numbers that are easy to count off. If each division is 2.75 it will be difficult to plot a point like 10.

c. Fills more than half of your graph.d. Is evenly spaced. Each square must be the same as each other

square.4. Plot the data. Mark each point with a clear +.5. Draw a line/curve of best fit. The line/curve DOES NOT have to start at the origin! It does not need to be a straight line! Be sure to ignore any bad data, called outliers.

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7. Pat yourself on the back for a job well done.Example 2:

A chemistry student does an experiment to determine how the reaction rate is affected by the concentration of one of the reactants. The reaction they are conducting is written below:

HCl + Ca(OH)2 → H2O + CaCl2

The data is summarized below:

Concentration of HCl

(M)Reaction Time (s)

0.50 251.0 15

30

0 20 40 60 80 100 120

0

5

10

15

20

25

30

35

40

45

Braking Distance vs. Speed

Speed (km/h)

Bra

king

dis

tanc

e (m

)

0 2 4 6 8 10 12 140

5

10

15

20

25

30

Axis Title

Axis

Title

Speed (km/h)

Braking Distance

(m)10 12.120 15.330 18.140 21.450 16.560 27.270 30.080 32.990 36.2

100 39.5

2.0 103.0 8.35.0 76.0 6.78.0 6.3

10.0 6.012.0 6.114.0 6.0

Complete the graph above, following the guidelines on the previous page.

310 2 4 6 8 10 12 140

5

10

15

20

25

30

Time for Reaction vs. Concentration of HCL

Concentration of HCl (M)

Tim

e fo

r Rea

ction

(s)

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