## Thursday, June 28, 2012

### Classifications of triangles

There are two ways to classify triangles. One is by looking at the largest of the three angles.

Obtuse: The largest angle is greater than 90°.
Right: The largest angle is exactly 90°.
Acute: The largest angle is less than 90°.

Make sure that when you are looking at the angles, you only consider the largest angle.  Every triangle must have at least two acute angles, but we are not interested in all the angles in this classification, only the largest one.

The other classification system deals with the relationships between the angles.

Equilateral: In an equilateral triangle, all three angles are equal.  Since they have to add up to 180°, the angle are 60°, 60° and 60°.
Isosceles: At least two of the angles are equal.  Technically, an equilateral triangle is a special case of isosceles, but isosceles also includes 70°-70°-40°, 40°-40°-100°, 32.2°-32.2°-113.6° and any of an infinite set of three angles where two of the angles are the same.
Scalene: All three angles are different.

If instead of giving the three angles of a triangle, a triangle can be defined by giving the three side lengths.  Not any three positive numbers can be the lengths of the sides of a triangle, because they must conform to the triangle inequality. The simplest way to say it in English is that the two short sides must add up to at least the equal of the long side.  If we call the side lengths r, s and t, where t is the longest, if r + s = t, the drawing would not be of a triangle, but instead a line segment with a point on the line segment that should be the apex of the triangle.  In class, I called this the degenerate case. As a student noted, we can think of it as a "squished triangle".

The easiest formula for the triangle inequality does not force us to find the long side. The formula is

r + s >= t >= | r - s |

The straight line brackets indicate absolute value. The editor for the blog does not have the "greater than or equal to" sign, so I have to type >= to signify this.

Example: 15, 6, 7 are NOT the sides of a triangle. The simplest way to state why is that 6+7 < 15, but I could also try to plug the numbers into the triangle inequality in any order

15 + 6 > 7.  This part is fine.
7 < |15 - 6| = 9 This is where the inequality breaks down, because 7 is less than 9, the difference between 15 and 6.

If the three lengths r, s, t can be the sides of a triangle, both parts of the inequality will be true.  If the lengths don't work, one of the inequalities will fail. If we have three lengths that do make a legitimate triangle, we can figure out the classifications as follows.

Equilateral:  All three sides are the same length.
Isosceles: At least two side lengths are the same.
Scalene: All three side lengths are different.

For the other classification system, we rely on the Pythagorean Theorem.  Here, we need to identify the longest side of the three.  We are used to using a, b and c for the lengths in the Pythagorean Theorem, where c is the longest side.

If a² + b² = c², we have a right triangle.
If a² + b² > c², we have an acute triangle.
If a² + b² < c², we have an obtuse triangle.

Here is a link to all the classification of triangles posts, which will also include practice problems for other ideas including Heron's formula.

### Order of operations

Many students learn the order of operations in math as an acronym (PEMDAS) or as the phrase Please Excuse My Dear Aunt Sally. This stands for Parentheses, Exponents, Multiplication, Division, Addition, Subtraction.  It would actually be better written as

P: Parentheses
E: Exponents
MD: Multiplication and Division
AS: Addition and Subtraction

In fact, multiplication does NOT come before division. Likewise, addition does NOT come before subtraction.  If you have done all the parentheses and exponents and there are both multiplications and divisions to be performed, they should be done from left to right. Likewise, if everything else has been completed and it's time to do additions and subtractions, perform the operations from left to right.

Here are some numbers in a string of operations. I will do the operations step by step with the next operation to perform in bold, and an explanation of why.

sqrt((6-2)² + (10-4)²)
sqrt((6-2)² + (10-4)²) Inside parentheses first, so 6-2 becomes 4.
sqrt(4² + (10-4)²) The second set of inside parentheses first, so 10-4 becomes 6.

sqrt( + 6²) We need to simplify what is inside the parentheses. Exponents before addition, so 4² = 16
sqrt(16 + ) Now the other exponent, 6² = 26
sqrt(16 + 36) Now addition is the last thing inside the parentheses, 16 + 36 = 52
sqrt(52) ~= 7.211.  52 = 2*2*13, so the simplified square root is 2*sqrt(13)

## Tuesday, June 26, 2012

### Square roots

Squares and square roots are what we call inverse operations.  If we take a number x and square it, the value is x². If we take the square root of x², we get the value x. A square, of course, is a geometric shape with four equal sides and four 90° angles. If we know the length of the side and call it s, the area is s², measured in square units. If instead we know the area is a number we call A square units, the side length for all the sides is the square root of A, which I will write on the blog as sqrt(A).

I will repeat this many times in class. When we are doing these kinds of problems, squares give us answers that are measured in square units which represent surface area, but square roots produce answers that are measure in regular units, which is to say lengths, one dimensional measure.

The Pythagorean Theorem is one of the central ideas in math, and it deals with the sides of a right triangle, which means a triangle that has one right angle. (A right angle measures 90°.) It is standard to name the three sides a, b and c, where c is the long side, always opposite the right angle. We call the long side the hypotenuse and the short sides are the legs.  The formula connected to the theorem is

a² + b² = c²

We can re-arrange the letters in the two following ways

a²  = c² - b²

or

b² = c² - a²

What this means is that though there are three numbers in the formula, if you have the values of any two of them, you can find the value of the third. In math, we say this set of numbers has two degrees of freedom.  ("Degrees of freedom" is not the same as "degrees", the units we use to measure angles.) You are "free" to use any two numbers for the lengths a and b, but once those are chosen, we can find c by using the formula and the square root operation, which means c is determined by a and b and can't be just any old number.  Likewise, you are "free" to pick and two numbers and call them b and c (or a and c), provided that c is bigger than the other number.  But once you have chosen two numbers out of the three, the third number is forced to be a specific value by using the formula.

A number is a perfect square or a square number if it is the square of a whole number.  Here are the perfect squares less than or equal to 100.

0² = 0  The square root of 0 is 0.
1² = 1  The square root of 1 is 1.
2² = 4  The square root of 4 is 2.
3² = 9  The square root of 9 is 3.
4² = 16  The square root of 16 is 4.
5² = 25  The square root of 25 is 5.
6² = 36  The square root of 36 is 6.
7² = 49  The square root of 49 is 7.
8² = 64  The square root of 64 is 8.
9² = 81  The square root of 81 is 9.
10² = 100  The square root of 100 is 10.

If a number is not a perfect square, its square root will not be a whole number. In fact, it will be an irrational number, a number that cannot be written as a fraction p/q. This means that as a decimal representation, we never get a pattern of digits that repeats forever. For example, the square root of 2 is represented on your calculator as 1.414213562... Any representation of a square root that has a finite number of digits is only an approximation.  It is common to round square roots to three places after the decimal, which is to say the nearest thousandth. When I write an approximation on the blog, I will use the symbols "~=".

Example: sqrt(2) ~= 1.414

The square root of 2 is the number whose square is exactly 2. but 1.414 is not the exact value.

1.414² = 1.999396

Besides approximating square roots, which we will be doing with calculators, there is also the idea of simplifying square roots, which will involve factoring a number down to its prime factorization.

Here are some practice problems for simplifying square roots, a method we learned today in class.

Here are practice problems for using the Pythagorean Theorem to find the length of the third side of a right triangle if you already know the lengths of the other two sides.

Here are some more practice problems using the Pythagorean Theorem.

Here are two sides of a right triangle.  In these problems, c is always the hypotenuse and a and b are the short sides, also called the legs.  Find the missing side using the Pythagorean Theorem, write it as a square root in simplified form and give the approximation to the nearest thousandth.

1)  a = 7, b = 6, c = _________

2) b = 6, c = 7, a = __________

3)  a = 8, b = 6, c = _________

4) a = 6, c = 8, b = __________

The answers are in the comments.

## Sunday, June 24, 2012

### Logarithmic scales

Monday in class, there will be a handout concerning logarithms and their practical uses, notably in the Richter scale which measures earthquakes and the bel system which measures sound. Besides the handout, here are a few more practice problems.

## Thursday, June 21, 2012

### The basics of scientific notation

Scientific notation can be used to represent any positive number. Usually, we only use it for really small numbers (close to zero) or really big numbers (more than a billion), but the method will work for anything.

Scientific notation is the form a x 10^b, where a is a number between 1 and 10 (1 is included but 10 isn't) and b is an integer, which means a whole number or the negative of a whole number.

The letter a is called the significand and b is the exponent or sometimes the order of magnitude.

Example with a "regular sized" number.

The number of feet in a mile is 5,280.  I want to write it as a number between 1 and 10 times a power of 10. 5,280 is a four digit number, so the nearest power of 10 is 1,000 or 10^3. If I divide 5280/1000, I get 5.28.  The mechanical way to do this is to move the decimal place on 5280. over three places to the left.

5280. =
528.0 x 10^1 =
52.80 x 10^2 =
5.280 x 10^3 or 5.28 x 10^3 (Your calculator will not write trailing zeros after a decimal.)

The last line is scientific notation because 1 < 5.28 < 10. 5280, 528 and 52.8 are too big to be significands.

One foot is 1/5280 of a mile. Type this into your calculator and you get 0.000189394..., a decimal which goes on beyond our calculators limits.  Let's round this to 0.000189.  This is called rounding to three significant digits, which means how many digits we have after we get past the leading zeros.

Because this is less than one, we are going to have to multiply by 10 raised to a negative exponent to get the scientific notation.

0.000189 =
0.00189 x 10^-1 =
0.0189 x 10^-2 =
0.189 x 10^-3 =
1.89 x 10^-4

We would say this number in scientific notation as "one point eight nine times ten to the negative fourth".

Your calculator only has ten places for digits, so a number like 25,000,000,000 (twenty five billion) is too big to be written in regular notation. If you type it in, the TI-30XIIs will display 2.5 x 10^10, or "two point five times ten to the tenth power".

Multiplying numbers in scientific notation

If I have two numbers in scientific notation, here is how I multiply them together.

1) Multiply the significands.
2) Add the exponents.
3) If the product from step 1 is more than 10, divide it by ten (move the decimal place one to the left) and add 1 to the exponent.

Example.

(6.1 x 10^7) x (4.3 x 10^9)  This is 61,000,000 x 4,300,000,000.

1) 6.1 x 4.3 = 26.23
2) 7 + 9 = 16
3) This gives us 26.23 x 10^16. 26.23 is too big to be a significand, so we divide it by 10 and multiply 10^16x10, which gives us 10^17.  The answer in scientific notation is 2.623 x 10^17.

### Changing a repeating decimal to a fraction in lowest terms.

Using Blogger editing, it's a little tricky to put a bar over a group of numbers. Instead I am going to put brackets [] around the repeating part. To write .16161616...., I'll type .[16] and to type .1666666..., I'll type .1[6].

Sorry if it's a little hard to read.

Method for changing from repeating decimal to fraction in lowest terms.

1. Call the repeating decimal x.
2. Multiply x by a power of ten that has as many zeros as there are digits in the repeating part.
3. Subtract x from the bigger number, which will cancel out the repeating part.
4*. IF the subtraction gives you a decimal number, multiply by some power of ten so you get (whole number times) x = (some other whole number)
5. Divide both sides of the equation by the number multiplying x.
6. Reduce the fraction to lowest terms.

Examples.
Example #1: .[16] = 0.1616161616....
1. x = 0.16161616...
2. Because the repeating part has two digits, multiply x by 100 to get 100x = 16.16161616....
3. 100x - x = 16.1616161616... - 0.1616161616...
which reduces to 99x = 16.
step 4 isn't needed.
5. x = 16/99, which is reduced to lowest terms.

Example #2: .1[6] = 0.166666....
1. x = 0.16666...
2. Because the repeating part has one digit, multiply x by 10 to get 10x = 1.6666....
3. 10x - x = 1.66666... - 0.166666...
which reduces to 9x = 1.5
4. 1.5 isn't a whole number, so multiply by 10 on both sides to get 90x = 15.
5. x = 15/90, which is not in lowest terms.
6. 15/90 = 5/30 = 1/6.

Practice problems.

a) Find the fraction for .[35]
b) Find the fraction for .3[5]
c) Find the fraction for .3[54]

Answers in the comments.

Click on this link for more practice problems with solutions.

## Wednesday, June 20, 2012

### Percent and other scales based on powers of 10

Scales based on powers of 10: The most famous scale base on powers of ten in percentage, which really means "per 100". It is much more common to see "53% of the people agree with the president's plan" than ".53 of the people..." or "53 out of every 100 people...". Technically, all those phrases are saying the same thing, but percentage is the most popular.

To get a number based on a power of 10 scale, you take the small number, divide it by the big number and multiply by the power of ten, so it is  small/big*scale. Sometimes we need greater precision because the proportions are so small, the small number is tiny in comparison to the big.

When I ask a class what is the legal limit for blood alcohol while driving, invariably someone will say "point oh eight" and most people will agree. But .08 is wrong; .08 = 8%, and the correct answer is .08% = .0008. I don't blame the students. The number is badly represented and it is an easy mistake to make. Let's take a look at the number on other scales of 10.

.08 out of 100 is the same as
.8 out of 1,000 or
8 out of 10,000 or
80 out of 100,000

80 parts out of 100,000 is a tiny proportion. To give an idea, ounce of pure alcohol mixed into ten gallons of blood would give you 78 parts out of 100,000, and most people have between a half gallon and a gallon and a half of blood in their body, between 4 and 12 pints. The amount of alcohol in a person's blood stream that is over the legal limit is about the same amount of alcohol as found in a capful of mouthwash used after brushing your teeth.

We will look at the per 100,000 scale for another type of statistic, measurements of mortality rates.

Here are the number of homicides in some local cities in 2007.

Oakland: 124 homicides
Richmond: 28 homicides
San Francisco: 98 homicides

Clearly, comparing these numbers is misleading, because we know these cities have very different numbers of citizens, so the standard way to measure these statistics is the per 100,000 population scale, which we find by the formula

small/big x scale

which in this case is

(# of homicides)/(city population) x 100,000

Oakland's population in 2007 is estimated at 415,000, Richmond at 106,000 and San Francisco at 825,000, so the murder rates on this standard scale are as follows

Oakland: 124/415000 * 100000 = 29.9
Richmond: 28/106000 * 100000 = 26.4
San Francisco: 98/825000 * 100000 = 11.9

So even though more people were murdered in San Francisco than in Richmond in 2007, the murder rate in Richmond was over twice as high, because Richmond has barely 1/8 of the population of San Francisco. (note: The trends for the three cities this decade are going in different directions. Oakland's murder rate is on the rise, while Richmond's is falling and San Francisco's has stayed about the same.)

Practice problems: (answers given in comments)
1) Here are the homicide numbers for Oakland, Richmond and San Francisco from 2004.

Oakland: 96 homicides, 399,000 population
Richmond: 40 homicides, 99,000 population
San Francisco: 96 homicides, 775,000 population

Find the murder rates from these years, rounded to the nearest tenth per 100,000 population and rank them from lowest (1st) to highest (3rd).

### Fractions, decimals and percents.Prime factorization and all factors.

Decimals, fractions and percents

There are three standard ways to represent rational numbers, fractions, decimals and percents. Many people are most comfortable with decimals because that is the way a calculator displays a number, but all three of the methods have their advantages and disadvantages.

Example: The number .735 is a decimal.  It is common to call this "point seven three five", but its proper name would be "seven hundred thirty five thousandths". That is because it has three places after the decimal, which is the thousandth place.

Easy rule: how many ever places there are after the decimal, write a fraction, "1" over "1" followed by that many "0".

One place after the decimal corresponds to 1/10 or one tenth.
Two places after the decimal corresponds to 1/100 or one hundredth.
Three places after the decimal corresponds to 1/1000 or one thousandth.

Following this rule, .735 = 735/1000.  You should be able to see that both of these numbers are divisible by 5, so this is not in lowest terms. 735/1000 = 147/200, and this is in lowest terms.  (If you have the TI-30XIIs, you can type 735 [A b/c] 1000 [enter], and the answer will be 147/200.)

The word "per" is from Latin and means "out of". It always represents division. "Percent" is "out of 100". The decimal .735 is equal to 73.5%.  Technically we multiply .735 by 100 to get 73.5, then by putting the percent sign at the back we have divided by 100, so the number is the same, only the representation has changed.  The mechanical rule we are usually taught is.

Decimal to percent:  Move the decimal point two places to the right and put a percent sign behind it.
Percent to decimal:   Move the decimal point two places to the left and remove the percent sign.

That means 33% = .33 and 3% = .03.  As fractions .33 =33/100 and .03 = 3/100.

Prime factorization and all factors of a number

A factor of a number n is a number that divides into n evenly.

Example: 4 is a factor of 12 because 12/4 = 3, which can also be stated as 4x3 = 12.
Counterexample: 5 is not a factor of 12 because 12/5 = 2.4, a decimal and not a whole number.

Every number has a prime factorization, which means we break it down to a product of prime numbers. A prime can only be divided evenly by 1 and itself. (By definition, 1 is not a prime. It is called a unit.)

Here is a list of the primes less than 50: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, ...
There are infinitely many primes, so this list goes on forever.

Find the prime factorization of 45.
Step 1: Find two numbers other than 1 and 45 that multiply to 45.  An easy pair is 5x9.
Step 2: 5 is prime, but 9 is not. That means we can find a pair of numbers (not 1 and 9) that multiply to 9, in this case 3x3. 3 is prime, so 45 = 5x3x3 is the prime factorization of 45.

Find all factors of 45.
When making a list of all factors, it makes sense to write them in pairs. I recommend starting with 1 and the number itself.

1 and 45

Now find the next biggest number after 1 that divides evenly into 45.  2 won't work because 45/2 = 22.5.  3 does work, because 45/3 = 15.

3 and 15

4 doesn't work but 5 does.

5 and 9.

You can try 6, 7 and 8 if you like, but none of them work.  9 works, but it's already on our list, so we can stop.

All the factors of 45 in order are 1, 3, 5, 9, 15 and 45.

Practice

a) Change .09 into a percent.
b) Change .09 into a fraction in lowest terms.
c) Change 24% into a decimal.
d) Change 24% into a fraction in lowest terms.
e) Change 3/8 into a decimal.
f) Change 3/8 into a percent.

Answers in the comments.

For practice on factorization, click on this link.

## Saturday, June 16, 2012

### Practice problems for first homework of Summer 2012

Change these numbers in Hindu-Arabic to Roman numerals.

648 = ______

15,781 = _____

Change these numbers in Roman numerals to Hindu-Arabic.
__
IXCDXLVII = ____

DCXCIV = ____

Find the approximation of 7/37 to the following number of decimal places and find the exact repeating decimal representation.

7/37 to the nearest tenth = _____
7/37 to the nearest hundredth = _____
7/37 to the nearest thousandth = ____

Repeating decimal representation = _____

Find the prime factorization of 48.

Find all the factors of 48.

Add or subtract these times, given in minutes and seconds.  If the minutes are 60 or more, change the answer to hours:minutes:seconds.

27:25 + 18:58 + 21:44 = __________

17:21 - 9:46 = __________

Answers in the comments.

### Syllabus for Summer 2012

Math 15: Math for Liberal Arts Summer 2012
Instructor: Matthew Hubbard
Email: mhubbard@peralta.edu
Text: no required text. If you want a text, personal recommendations can be made
Class website: http://mathlibarts.blogspot.com/
Class hours MTWTh: 10:00 am - 12:05 pm, G-211
Office hours: Math lab G-201
TTh 9:20-9:50 am (also available by appointment)
Scientific calculator required (TI-30IIXs, TI-83 or TI-84 recommended)

Important academic schedule dates:
Last date to add, if class is not full: Sat., June 23
Last date to drop class without a "W": Thurs., June 28
Last date to withdraw from class: Thurs., July 18

Holidays:
Wednesday, July 4: Independence Day

Midterm and Finals schedule:

Half Midterm 1________Thursday, June 28
Full Midterm__________Thursday, July 5
Half Midterm 2________Thursday, July 19
Comprehensive Final ___Thursday, July 26

Quiz schedule (most Tuesdays and Thursdays) no make-up quizzes given
First week: 6/19 and  6/21
Second week: 6/26
Third week: 7/3
Fourth week: 7/10 and 7/12
Fifth week: 7/17
Sixth week: 7/24

Grading Policy
Homework to be turned in: Assigned every Tuesday and Thursday, due the next class
(late homework accepted at the beginning of next class period, 2 points off grade)
If arranged at least a week in advance, make-up midterm can be given.

The lowest two scores from homework and the lowest two scores from quizzes will be removed from consideration before grading.

Grading system
Quizzes 25%* best 2 out of three of these grades
Midterm 1 25%* best 2 out of three of these grades
Midterm 2 (two half midterms combined) 25%* best 2 out of three of these grades
Homework 20%
Lab 5%
Final 25%

Anyone who misses less than two homework assignments and gets a higher percentage score on the final than the weighted average of all grades combined will get the final percentage instead deciding the final grade.

Academic honesty: Your homework, exams and quizzes must be your own work. Anyone caught cheating on these assignments will be punished, where the punishment can be as severe as failing the class or being put on college wide academic probation. Working together on homework assignments is allowed, but the work you turn in must be your own, and you are responsible for checking its accuracy. If I see multiple homework assignments turned in with the exact same wrong answers, I will give a warning. If it happens a second time, the student will get a 0 on the assignment and it will be counted towards the grade.

Class rules: Cell phones and beepers turned off, no headphones or text messaging during class
You will need your own calculator and handout sheets for tests and quizzes. Do not expect to be able to borrow these from someone else.

Student Learning Outcomes

• Analyze an argument for validity using simple rules of logic, and if invalid identify the type of mistake made.
• Compute, with sophisticated formulas, such quantities as interest payments for amortized loans.
• Interpret patterns and draw inferences from them.

Students with disabilities
The Disabled Students Program Services (DSPS) should have your academic accommodation with the instructor. After the first day, I will accept these accommodations electronically or by hard copy on paper. If you need academic accommodation and have not yet applied, please call 510-464-3428 for an appointment.

Exam policies
Quizzes will be closed book and closed notes. Some information you will be expected to remember, other formulas and information will be provided. No sharing of calculators is allowed. You are responsible for knowing how to use your calculator to find answers.

The reciprocal relationship

The teacher will be on time and prepared to teach the class.
The students will be on time and prepared to learn.

The teacher will present the material to the best of his ability.
The students will absorb the material to the best of their ability. They will ask questions when topics are not clear.

The teacher will do his best to answer the questions the students ask about the material, either by repeating an answer with more details included or by taking a different approach to the material that might be clearer to some students.
The students will understand if the teacher feels a topic has been covered enough for the majority of the class and will accept questions being answered outside the class, either in extra time or through written communication.

The teacher will do his best to keep the class about the material. Personal details and distractions that are not germane to the class should not be part of the class.
The students will do their best to keep the class about the material. Questions that are not about the topic should be avoided. Distractions like cell phones and texting are not welcome when the class is in session.

The teacher will give assignments that will help the students master the skills required to pass the course.
The students will put in their best efforts to complete the assignments.
When the assignments are completed, the teacher will make every effort to get the assignments graded and back to the students in a timely manner, by the next class session whenever possible.

The teacher will present real life situations where the skills being learned will be used when they exist. In math, sometimes a particular skill is needed in general to solve later problems that will have real life applications. Other skills have the application of “learning how to learn”, of committing an idea to memory so that committing other ideas to memory becomes easier in the long run.

The student has the right to ask “When will I use this?” when dealing with mathematical topics. Sometimes, the answer is “We need this skill for the next skill we will learn.” Other times, the answer is “We are learning how to learn.” Both of these answers are as valid in their way as “We will need this to understand perspective” or “We use this to balance our checkbooks” or “Ratios can be used to figure out costs” or other real life applications.