Calculating the Crust:
What You Really Need to Know About Math in Science
Baseball. American’s pastime. Here is a brief description of a baseball game.
At the start of a game both teams are out warming up. Then both teams come in. The nine people from one team go out and one person from the other team goes up and everyone else is in and down. When the one guy is out he goes down and someone else comes up. When three guys from the in team are out then the teams switch and all the out guys go in and the in guys go out. After each team is up nine times and a total of 54 guys are out the game is over and the winning team rushes out and the losing team stays in.
How easy is that description to understand? Well numbers can be just as confusing. It’s often said that statistics lie and liars use statistics. The truth is numbers are very powerful communication tools, and the purpose of this activity is to make sure you understand the best way to use and present numbers.
Introduction to Estimation
Estimation is an important skill that we all use every day. Can I stay in bed 5 minutes longer? How much milk does my cereal need? Is the bus late or am I? But estimation is also an important math and science skill. It might not seem very scientific because we think of science as being accurate and precise and isn’t estimating just a form of guessing?
Estimating may seem unpredictable and unreliable but many scientists use estimation in their work. How many birds are in that tree? How much of the moon is showing? Which mountain shows more erosion? In many cases these things could be measured accurately, but sometimes it is simply more practical to estimate because measuring accurately would be too time consuming, expensive, or difficult. This is the value of estimation. Estimating is also a good way of using math sense to analyze whether an answer “feels” right.
Estimate how long it will take to read a page in a book ______
Is the distance between two classrooms best measured in mm, m, or km ______
What unit of measurement is appropriate for the mass of a pencil ______
Will a soda bottle hold 100 ml of water ______
Tips for Estimating
1. Concentrate on the first digit of each number ... it will have the biggest impact on the answer. You can then look at the other digits to make small adjustments to your answer.
2. Round the numbers up or down before the calculation.
3. Check the number of zeros in your calculation!
4. When adding a lot of similar numbers, look at the numbers and choose an average, and then multiply that average by how many numbers
5. When doing division, change the numbers to fit in with the multiplication tables
6. Group numbers together that will be easy to work on
7. With decimals, percents and fractions try to think what the number means. Think: is it close to 1? Close to half? Close to zero?
8. A percentage is like a decimal: 10% is 0.1, 50% is 0.5, etc.
9. Also, a fraction might be close to zero, close to half or close to one.
Introduction to Accuracy, Precision, and Uncertainty
Imagine that you are at archery class one day and are shooting arrows at a target. On your first try you shoot the pattern shown in Figure A. On your second try you shoot the pattern shown in Figure B. Trial 3 is Figure C and Trial 4 is Figure D. For each pattern describe how you might explain how well you shot to your parents when you get home from school.
A:
B:
C:
D:
There is no such thing as a perfect measurement. Each measurement contains a degree of uncertainty due to the limits of instruments and the people using them. In laboratory exercises, students are expected to follow the same procedure that scientists follow when they make measurements. Each measurement should be reported with some digits that are certain plus one digit with a value that has been estimated. This is called a significant figure.
For example, if a student is reading the level of water in a graduated cylinder that has lines to mark each milliliter of water, then he or she should report the volume of the water to the tenth place (i.e. 18.5 ml.) This would show that the 18 mls are certain and the student estimated the final digit because the water level was about half way between the 18 and 19 mark.
Two concepts that have to do with measurements are accuracy and precision.
The accuracy of the measurement refers to how close the measured value is to the true or accepted value. For example, if you used a balance to find the mass of a known standard 100.00 g mass, and you got a reading of 78.55 g, your measurement would not be very accurate. One important distinction between accuracy and precision is that accuracy can be determined by only one measurement, while precision can only be determined with multiple measurements
Precision refers to how close together a group of measurements actually are to each other. Precision has nothing to do with the true or accepted value of a measurement, so it is quite possible to be very precise and totally inaccurate. In many cases, when precision is high and accuracy is low, the fault can lie with the instrument. If a balance or a thermometer is not working correctly, they might consistently give inaccurate answers, resulting in high precision and low accuracy.
You must strive for both accuracy and precision in all of your laboratory activities this year. Make sure that you understand the workings of each instrument, take each measurement carefully, and recheck to make sure that you have precision. Without accurate and precise measurement your calculations, even if done correctly, are quite useless
One way of mathematically comparing accuracy and precision is by using percent error. Percent error is a measurement of the accuracy of the measurement. It is calculated using the following formula:
Percent error is a positive number when the experimental value is too high and is a negative number when the experimental value is too low.
Introduction to Graphing and Statistics
Graphs – they are perhaps the most important math skill you will use in science class because they make it so easy to communicate and interpret data. And that data is analyzed using statistics. Taken together, graphs and statistics are powerful tools that can help explain difficult scientific concepts, or mislead the unwary to a total misunderstanding of science.
For each of the following, what kind of graph might you use to display the information:
% of Different Chemicals Found in the Ocean ______
Change in Water Temperature Over 24 Hours ______
Temperature of 100 different water samples ______
Average Ocean Depth of the Five Oceans ______
Identify the statistic associated with each formula or example:
X + Y + Z/3 ______
1 1 2 3 4 5 5 5 5 6 7 ______
2 4 1 4 5 2 5 3 4 4 5 ® (4) ______
Statistic Tips
Statistic / Description / Advantages / DisadvantagesMean: / · Aka Average.
· The sum of a set of data divided by thenumber of data.
· Use the mean to describe the middle of a set of data that does not have an outlier. / · Most popular measure in fields such as business, engineering and computer science.
· It is unique - there is only one answer.
· Useful when comparing sets of data. / · Affected by extreme values (outliers)
Median: / · The middle value, or the mean of the middle two values, when the data is arranged in numerical order.
· Think of a "median" being in the middle of a highway
· Use the median to describe the middle of a set of data that does have an outlier. / · Extreme values (outliers) do not affect the median as strongly as they do the mean.
· Useful when comparing sets of data.
· It is unique - there is only one answer.
/ · Not as popular as mean
Mode: / · The value (number) that appears the most.
· It is possible to have more than one mode,and it is possible to have no mode.
· If there is no mode-write "no mode", do not write zero (0) .
· Use the mode when the data is non-numeric or when asked to choose the most popular item. / · Extreme values (outliers) do not affect the mode. / · Not as popular as mean and median.
· Not necessarily unique - may be more than one answer
· When no values repeat in the data set, the mode is every value and is useless.
· When there is more than one mode, it is difficult to interpret and/or compare
Graphing Tips
Graph / Description / Advantages / DisadvantagesPie Chart / A pie chart displays data as a percentage of the whole. Each pie section should have a label and percentage. A total data number should be included. / · Visually appealing
· Shows percent of total for each category / · No exact numerical data
· Hard to compare 2 data sets
· "Other" category can be a problem
· Total unknown unless specified
· Best for 3 to 7 categories
· Use only with discrete data
Bar Graph / A bar graph displays discrete data in separate columns. A double bar graph can be used to compare two data sets. Categories are considered unordered and can be rearranged alphabetically, by size, etc. / · Visually strong
· Can easily compare two or three data sets / · Graph categories can be reordered to emphasize certain effects
· Use only with discrete data
Line Graph / A line graph plots continuous data as points and then joins them with a line. Multiple data sets can be graphed together, but a key must be used. / · Can compare multiple continuous data sets easily
· Interim data can be inferred from graph line / · Use only with continuous data
Scatterplot / A scatterplot displays the relationship between two factors of the experiment. A trend line is used to determine positive, negative, or no correlation. / · Shows a trend in the data relationship
· Retains exact data values and sample size
· Shows minimum/maximum and outliers / · Hard to visualize results in large data sets
· Flat trend line gives inconclusive results
· Data on both axes should be continuous
· Did you select the correct type of graph to represent your data?
· Is your graph correctly labeled and include UNITS ?
§ X-axis: manipulated variable (independent variable) or what you chose
§ Y-axis: responding variable (dependent variable) or what you measured
· Does your graph have an APPROPRIATE and GOOD title? / · Is the scale appropriate?
§ Is each of the lines numbered correctly and consistently?
§ Is the width of each column (bar graph only) the same?
§ Does the graph maximize the use of the page?
· Is the graph visually appealing?
· Does your graph require a key?
· Does your graph accurately represent your data
Putting It All Together
Introduction
Basalt is an extrusive igneous rock that is typically dark in color. It is the most common of the lava rocks and is widespread in many parts of the world. Basalt is composed of pyroxene and plagioclase feldspars.
Granite is the best known of the igneous rocks because of its popularity in the building industry – it has become very fashionable to have kitchen countertops and other home accessories constructed from granite. Granite is an intrusive igneous rock and is lighter in color than basalt. Granite is composed of feldspars, quartz, mica, and hornblende.
Assignment
Determine which rocks are continental in origin and which rocks come from the ocean floor.
Hints
· It’s all about density. One rock is more dense than the other.
· Begin by estimating which rock is more dense.
· Follow up with measurements and calculations.
· Accepted average density of basalt is 3.0 g/ml.
· Accepted average density of granite is 2.7 g/ml.
Product
Prepare a poster report (one per group) and present your poster to the class.
You must use each of the following for some part of your investigation and include the results in your poster:
· Estimating
· Accuracy and Precision
· Percent Error
· Graphing
· Statistics