We often see advertisements about how good or how effective some products and services are in our society. Many people are easily swayed by a 99.9% guarantee. Here's some food for thought:
If 99.9% is good enough then....
-12 newborns will be given to the wrong parents daily
-114,500 mismatched pairs of shoes will be shipped/year
-18,322 pieces of mail will be mishandled/hour
-2,000,000 documents will be lost by the IRS this year
-2.5 million books will be shipped with the wrong covers
-Two planes landed at Chicago's O'Hare airport will be unsafe every day
-315 entries in Webster's Dictionary will be misspelled
-20,000 incorrect drug prescriptions will be written this year
-880,000 credit cards in circulation will turn out to have incorrect
cardholder information on their magnetic strips
-103,260 income tax returns will be processed incorrectly during the
year
-5.5 million cases of soft drinks produced will be flat
-291 pacemaker operations will be performed incorrectly
-3056 copies of tomorrow's Wall Street Journal will be missing one
of the three sections
-A typical day would be 24 hours long (give or take 86.4 seconds)
To understand statistics, one must have a good understanding of the statistical averages: Mean, median, and mode.
The Mean
The mean is the sum of all data points, divided by the number of data points. For example, the mean of 10, 12, 13, 13, 13, 14, 25 is approximately 14.28 (the sum of those 7 numbers is 100, and when divided by 7 the result is 14.28). This is a good statistic to use when the data set is relatively normal, which means that it is fairly evenly distributed from the low end to the high end, with some clumping near the middle. However, when one or more data points are much higher or lower than the rest, the mean is pulled towards those extremes. This leads to reports of the "average" being much higher or much lower than what the majority of data points actually are. The mean is pulled towards the tail when the data is skewed.
The Median
The median is the middle data point among a set of numbers. Using the previous example of 10, 12, 13, 13, 13, 14, 25, the median would be13 because when the data set is in order, 13 is in the exact middle of the list. This is a good statistic when the data is skewed, meaning there are data points that are extremely higher or lower than the majority of the data points.
The Mode
The mode is the most common data point in a set. In the previous example, since 13 occurs 3 times, 13 would also be the mode. This is a good statistic to use when the data is heavily skewed.
Normal Curves and Skewed Curves
Below is a picture of 3 curves. These represent data sets, with the most common data points under the highest part of the bump, and the less common points near the tail ends. The middle picture is a regular normal curve, where the mean, median, and mode are the same point. A negatively skewed data set, the left picture, represents data with extremes on the low end. A positively skewed data set, the right picture, represents data with extremes on the high end.
Notice that in a skewed distribution the mean, median, and mode are pulled further apart from each other. This is how people lie with statistics. If they want you to believe that the majority of people have more positive results in a negatively skewed data set they would use the mode or median instead of the mean. If they want you to believe people have more positive results in a positively skewed data set, they would use the mean instead of the median or mode.
Lying with Statistics
Consider my example from earlier: 10, 12, 13, 13, 13, 14, 25. If these were the hourly wages for 7 people in a company, this would be a positively skewed data set, because 25 is so much higher than the other data points. If a company wants you to believe their average starting salary is higher than the majority of people in the company actually make, the company would report the mean (14.28) instead of the median or mode (13). Now consider these data points to be the number of accidents a company had each month. They would probably report the "average" to be 13, the median or mode number, instead of the mean of 14.28, to make their average accidents per month sound less serious, which completely fails to take the 25 into account and might cause some to overlook it.
Here's a funny little video I found. It's a bit lame, but the tune is catchy. Might be helpful for junior high students.
I found a great video clip about visualizing data. David McCandless makes some great points about the media, environmentalism, and politics. He does a great job of combining math with art to create something meaningful and informative which puts current events into perspective.
The other thing I like about the video clip is that he demonstrates how easy it is to deceive people using data and statistics.
Three very important concepts are taught in elementary school arithmetic classes: Greatest Common Factor (GCF), Least Common Multiple (LCM), and fractions. However, many students I have worked with have a very poor conceptualization of these ideas by the time they reach Algebra. Fractions are used continuously throughout Algebra and beyond, and most teachers don't have the time in the curriculum to go back and teach a full lesson on fractions. Students often shut down when they see fractions, but they aren't as scary as they seem.
Fractions are used in many aspects of life, such as chemistry, baking, economics, business, statistics, etc. Fractions are used all the time, but people often don't think in terms of fractions. This is where many students get lost. I wanted to post a quick review of fractions as a reminder for students who have forgotten. Before we get into fractions, I'm going to review GCF and LCM, because these are necessary for converting fractions to workable forms.
GCF
The Greatest Common Factor (GCF) is the largest factor that two numbers share. The easiest method I've used in finding the GCF between two numbers is to draw a factor tree. For example:
Example 1: Find the GCF of 20 and 42
The factors of 20 and of 42 can be represented as factor trees, like so:
20 42
/ \ / \
4 5 6 7
4 and 5 can be multiplied together to get 20, so those are factors of 20. 6 and 7 can be multiplied to get 42, so those are factors of 42. However, to find the GCF, we have to factor the factors down to prime numbers.
20 42
/ \ / \
4 5 6 7
/ \ / \
2 2 2 3
The bottom number on each branch is a prime factor of the number we stared with. Therefore, the prime factorization of 20 is 2*2*5. The prime factorization of 42 is 2*3*7. The only factor that both prime factorizations share is 2, therefore the GCF(20,42) = 2.
Practice with 1) GCF(5, 7) 2) GCF(10, 20) 3) GCF(30, 385) Answers are at the end of the rant.
Check out this GCF video
LCM
The Least Common Multiple (LCM) is the combination of the prime factorizations of two numbers, without repeating shared factors. I also use factor trees for LCM. For example:
These trees show us the prime factorization of 24 is 2 * 2 * 2 * 3, or (2^3) * 3. The prime factorization of 120 is 2 * 2 * 2 * 3 * 5 or (2^3) * 3 * 5. Both factor trees contain (2^3) and 3. 120 also contains 5. To find the LCM, we multiply all shared factors together and all non-shared factors together.
The shared factors are (2^3) and 3. The non-shared factor is 5. (2^3) * 3 * 5 = 120. Therefore, the LCM(24, 120) = 120.
Practice problems. 1) LCM(30, 63) 2) LCM(8, 24) 3) LCM (6, 15) Answers are at the end of the rant.
Check out this LCM video
Fractions
When working with fractions, there are four simple rules to keep in mind:
1) To add or subtract fractions, they must have the same denominator, found by using the LCM. Then the numerators are added together, and the denominator stays the same.
2) To multiply fractions, multiply the numerators together to form the new numerator, and multiply the denominators together to form the new denominator.
3) To divide fractions, use the reciprocal of the fraction you are dividing by, then multiply as normal.
4) To simplify fractions, find the GCF of the numerator and denominator, then divide both the numerator and denominator by the GCF.
Quick review: The number on top of a fraction is the numerator and the number on bottom is the denominator.
Fraction Examples:
Adding Fractions
To add the fractions (1/3) and (1/6) we must find a common denominator. The numerator for both fractions is 1. The denominator of the first fraction is 3 and the denominator of the second fraction is 6. We find the LCM(3, 6) to find a common denominator. LCM(3, 6) = 6. Therefore, (1/6) is in the form it needs to be in, but (1/3) is not.
3 had to be multiplied by 2 to get the LCM of 6. Therefore, we multiply the numerator and denominator of (1/3) by 2 to get a common denominator. 1 * 2 = 2 and 2 * 3 = 6. Therefore, (1/3) = (2/6)
Now we add (2/6) + (1/6) = (3/6). We add the numerators of 2 and 1 to get 3 as the new numerator. The denominator of 6 is unaffected when adding or subtracting.
(3/6) can be simplified by finding the GCF of 3 and 6. GCF(3, 6) = 3. Therefore, dividing the numerator of 3 by 3, we get 1. Dividing the denominator of 6 by 3, we get 2. The simplified fraction is then (1/2).
Adding Fractions practice problems: 1) (1/3) + (1/3) 2) (1/4) + (1/5) 3) (2/7) + (7/8) Answers are at the end of the rant.
Here's a video for adding fractions:
Multiplying Fractions
Example: Multiply (2/5) by (3/7). Multiplying is one of the easiest operations to perform on fractions. Simply multiply the numerators together, then the denominators together.
The numerators are 2 and 3, multiplying them together gets a new numerator of 6. The denominators are 5 and 7. Multiplying them together gets 35. The new fraction is then (6/35).
Can (6/35) be simplified? Find the GCF(6, 35) = 1. This means that (6/35) is in its simplest form.
Multiplying Fractions practice problems: 1) (3/5) * (4/5) 2) (2/3) * (3/2) 3) (1/8) * (7/8) Answers are at the end of the rant.
Here's another video, for multiplying videos. He demonstrates the cross-canceling method.
Dividing Fractions
Example: Divide (3/5) by (4/5). To perform this division, we first use the reciprocal of (4/5). This means simply switch the numerator and denominator. Therefore, the reciprocal of (4/5) is (5/4).
Now we multiply (3/5) by the reciprocal of (4/5), which is (3/5) * (5/4) = (15/20).
Can (15/20) be simplified? What is the GCF(15, 20)? The prime factorization of 15 is 3 * 5. The prime factorization of 20 is 2 * 2 * 5. The only shared factor is 5, therefore GCF(15, 20) = 5.
Now we divide both the numerator and denominator by the GCF. 15 divided by 5 is 3. 20 divided by 5 is 4. Therefore, (15/20) in simplest form is (3/4).
I hope this has been a helpful review. If anyone would like further clarification, send me a comment and I'll respond when I can. Here is a helpful website for teaching fractions visually: http://www.teachingfractions.co.uk/
GCF practice problem answers:
Answers:
1) GCF(5,7) = 1 because both 5 and 7 are prime numbers, the only factor that they both share is 1. Any number can be factored as 1 * itself. For example 1 * 5 = 5 and 1 * 7 = 7. 1 is a factor of all numbers except 0.
2) GCF(10, 20) = 10. The prime factorization of 10 is 2 * 5. The prime factorization of 20 is 2 * 2 * 5. Since both prime factorizations contain a 2 and a 5, the GCF is 2*5 = 10.
3) GCF(30, 385) = 5. The prime factorization of 30 is 2 * 3 * 5. The prime factorization of 385 is 5 * 7 * 11. The only number both prime factorizations share is 5.
LCM practice problem answers:
1) LCM(30, 63) = 1890. This is because the prime factorization of 30 is 2 * 3 * 5. The prime factorization of 63 is 7 * 9. These two prime factorizations do not share any factors, so the LCM must include all factors from both prime factorizations. Therefore LCM(30, 63) = 2 * 3 * 5 * 7 * 9 = 1890.
2) LCM(8, 24) = 24. The prime factorization of 8 is 2 * 2 * 2 or (2^3). The prime factorization of 24 is 2 * 2 * 2 * 3 or (2^3) * 3. Both prime factorizations contain (2^3). 24 contains a factor of 3, but 8 does not. Therefore, the LCM(8, 24) = (2^3) * 3 = 24.
3) LCM(6, 15) = 30. The prime factorization of 6 = 2 * 3. The prime factorization of 15 = 3 * 5. Both prime factorizations contain 3. The first contains 2, the second contains 5. Multiplying the factors that are shared by the factors that are not shared, we get LCM(6, 15) = 3 * 2 * 5 = 30.
Adding Fractions practice problem answers:
1) To add (1/3) + (1/3) check that the denominators are the same. The denominator of both fractions is 3. Therefore, simply add the numerators 1 + 1 = 2. Therefore (1/3) + (1/3) = (2/3)
2) To add (1/4) + (1/5) we need a common denominator. The LCM(4,5) = 20. To make (1/4) have a denominator of 20, multiply the numerator and denominator by 5. (1/4) = (5/20). To make (1/5) have a denominator of 20, multiply the numerator and denominator by 4. (1/5) = (4/20). Now add (5/20) + (4/20) = (9/20). Check that (9/20) is in simplest form, which it is because GCF(9, 20) = 1.
3) To add (2/7) and (7/8), find LCM (7, 8) = 56. Therefore (2/7) = (16/56) (by multiplying 2 by 8 and 7 by 8). (7/8) = (49/56) (by multiplying the 7 by 7 and 8 by 7). Now add (16/56) + (49/56) to get (65/56). Is (65/56) in simplest form? GCF(65, 56) = 1. (65/56) is an improper fraction, but is acceptable most of the time in Algebra.
Multiplying Fractions Practice Problem answers:
1) To multiply (3/5) by (4/5), multiply the numerators, 3 and 4, together to get 12. Multiply the denominators 5 and 5 together to get 25. The answer is then (12/25). Is (12/25) in simplest form? GCF(12,25) = 1, so yes it is.
2) To multiply (2/3) by (3/2), multiply the numerators, 2 and 3, to get 6. Multiply the denominators 3 and 2, to get 6 again. Therefore, the answer is (6/6). A fraction is another way of signifying division, and anything divided by itself is 1. Therefore, (6/6) in simplest form is just 1.
*note, (2/3) and (3/2) are reciprocals of each other, because their numerators and denominators are the same numbers, only reversed. Whenever you multiply reciprocals, the answer is always 1.
3) To multiply (1/8) * (7/8), multiply the numerators 1 and 7 to get 7, and the denominators 8 and 8 to get 64. Therefore, the answer is (7/64). Since 7 is prime and is not a factor of 64, this is in simplest form.
Dividing Fractions practice problem answers:
1) Dividing (1/2) by (4/5) is the same as multiplying (1/2) by the reciprocal of (4/5), which is (5/4). Multiplying (1/2) by (5/4), multiply the numerators 1 and 5, and multiply the denominators 2 and 4. The answer is then (5/8), which is in simplest form.
2) Before trying to calculate large numbers, look for similarities in the two fractions. To divide (27/56) by (27/56), notice that they are both the same number. Any number divided by itself is 1. Therefore, (27/56) / (27/56) = 1.
3) (3/4) / (5/6) is the same as (3/4) * (6/5), since (6/5) is the reciprocal of (5/6). Multiply the numerators 3 and 6 to get 18. Multiply the denominators 4 and 5 to get 20. Therefore, (3/4) / (5/6) = (3/4) * (6/5) = (18/20). The GCF(18, 20) = 2, so dividing 18 by 2 and 20 by 2 gives the simplest form of (18/20) to be (9/10), which is the final answer.
This is the end of my rant.
Friday, September 17, 2010
"While the true meaning of mathematics might lie hidden from students and remain unappreciated by the general population, mathematics is, nevertheless, the bedrock of our modern world," (p. 16-18) (Roman, 2004). Science, technology, business, medicine, and many other fields rely heavily on math. Math describes and quantifies the world, it helps give us the proper perspective of our place in the universe, and it helps us relate to the world around us.
"Simple mathematics helps us understand such basic things as a mortgage used to buy a house--analyzing how we must compromise between the size of the mortgage, the interest on the loan, and the time span of the loan. Make the wrong choices and buyers soon find themselves unable to afford the monthly payments," (p. 16-18) (Roman, 2004). Whether or not students plan to use higher-level math in their career, the ability to understand money, economics, and loans requires knowledge of algebra, specifically compound interest and exponents. Working out a monthly budget requires addition, subtraction, multiplication, division, and even basic algebra at times.
Statistics are all around us. We are constantly inundated with statistics. What was Babe Ruth's batting average? What percentage of votes did Obama recieve? What chances do I have of winning the lottery? How many miles to the gallon does my car get? I could go on, but the point is that the world is full of statistics, and statistics are an aspect of math. People can be easily fooled by statistics as well, if they don't have a decent understanding of them. I could tell you 96% of people lose weight on a particular diet to get you to buy my products, but I could omit the fact that the people studied were also exercising and changing their lifestyles. People are led into poor financial decisions every day due to a lack of mathematical understanding.
A future blog will go more into the practical nature of math in everyday economics. Until then, that's the end of this rant.
A friend of mine suggested that I call my blog Ryan's Math Rant, because I have the tendency to rant about math when I get going. My first real post, which I plan to start tomorrow, will be about the importance of math in modern education.
Math is an extremely important aspect of modern technology, engineering, science, etc., but is it really necessary for everyone? I often hear the argument that most people don't use math beyond basic arithmetic after high school, so what is the purpose of algebra and beyond for those who don't plan to go into a math oriented career? I plan to start my blog by exploring these questions in further detail. Until then, this is the end of my first rant.
