Notes for the week of Dec. 7 to Dec. 10

The finals are next week. Here is the schedule.

For the Monday-Wednesday class that meets in the Fieldhouse:
The final is on Wednesday, December 16th from 10:00 to noon.

For the Tuesday-Thursday evening class that meets in G-207:
The final is on Tuesday, December 15 at the usual class time, 5:30-6:45.

You are allowed two pages of notes, front and back (Four pages total)
You need your own calculator that is NOT on a cell phone.
I will not lend anyone a calculator for the final.
 

Two methods for finding outliers

There are two methods for finding outliers, numbers that are far away from "the middle". If we count the middle as the median, we use the five number summary to find the threshold for high and low outliers. If we have the average, we will need to calculate the standard deviation for a sample sx, and find the z-score. With a calculator, this is very simple, but it can also be done by hand with small sets.

The five number summary

 Here is a list of the number of win for the Pac-12 teams as of November 29, 2015. Obviously, the length of the list is 12.

8, 7, 6, 4, 4, 0, 6, 6, 5, 4, 3, 1

We need to put the list in order, either top-to-bottom or bottom-to-top. Since the 8 is the first number on the list, let's go top-to-bottom.

8, 7, 6, 6, 6, 5, 4, 4, 4, 3, 1, 0

The five number summary are the high and low values - very easy, and the Quartiles, Q3, Q2 and Q1. We already know how to get Q2, because it is the median. Q3 is the median of the top half of the data and Q1 is the median of the bottom half of the data. Because there are 12 items on the list, it splits into the top six and the bottom six, and the median is the average of the two middle values.

8, 7, 6, 6, 6, 5 || 4, 4, 4, 3, 1, 0

The median Q2 is (5+4)/2 = 4.5

Q3: For the top half, the median is between the first 6 and the second 6, so it is 6.

Q1: For the bottom half, the median is between the 4 and the 3, so the median is (4+3)/2 = 3.5

High = 8
Q3 = 6
Q2 = 4.5
Q1 = 3.5
Low = 0

Next we get the IQR = Q3 - Q1, which in our instance is 6 - 3.5 = 2.5

The high threshold for outliers is Q3 + 1.5*IQR, or 6 + 1.5*2.5 = 6 + 3.75 = 9.75.  This threshold is above 8, so 8 is not an outlier.

The low threshold for outliers is Q1 - 1.5*IQR, or 3.5 - 1.5*2.5 = 3.5 - 3.75 = -0.25.  This threshold is just barely below 0, so 0 is not an outlier.

The z-score method

We know how to take z-scores if we have the average and standard deviation, but here we are going to have to compute the average and standard deviation instead of them being given. Average isn't hard by hand with smallish data sets, and if you have a calculator that is set up for statistics, both the standard deviation and average are given to you as quickly as you can input the set. If you don't have a calculator. Here is what we need to do.

1. Find the sum of the list, which we will call sum(x).

In our case, it's 8+7+6+6+6+5+4+4+4+3+1+0 = 54

2. Find the sum of the squares of the list, which we will call sum(x²) 

In our case, it's 64+49+36+36+36+25+16+16+16+9+1+0 = 304

3. Then we get  sum(x²) - [sum(x)]²/n

This is 304 - 54²/12 = 304 - 243 = 61

4. The standard deviation is the square root of the value from step 3 divided by n-1.

sqrt(61/11) ~= 2.35487881..., which we can round to 2.35.

The average is 54/12 = 4.5 

To be a high outlier, we need a z-score over 2. To be a low outlier, we need a z-score under -2.


z(8) = (8-4.5)/2.35 ~= 1.48936..., which isn't above 2, so it's not an outlier.

z(0) = (0-4.5)/2.35 ~= =1.91489..., which isn't below -2, so it's not a low outlier, but it was close.

With this particular set, our two methods agreed there were no outliers. The methods sometimes disagree. We can have sets with just high outliers, just low outliers, outliers in both directions or no outliers at all.

Mean, median and modeZ-scores and proportions

A link to a post on using the z-score tables.

A link to the statistics blog about mean (average), median and mode.

Notes on the metric system

The United States uses a measurement system originated in Great Britain, which is sometimes called the Imperial system or the customary system. The relationships between measurements aren't very user friendly, and even Americans aren't very good at remembering all of them, according to tests I gave students at the beginning of classes years ago.

Here are some examples of the stuff you have to remember.

Length
12 inches = 1 foot
3 feet = 1 yard
5,280 feet = 1 mile

There are also odd measurements that very few people use any more, like rods, fathoms, furlong and nautical miles.

Volume

Of all the parts of the system This one makes some sense because the next biggest named measurement is almost always a factor of 2 away, either 2, 4 or 8 times bigger. Let's start with the fluid ounce (fl. oz.) as the basic unit.

8 fl. oz. = 1 cup (8 fl. oz.)
2 cups = 1 pint (16 fl. oz.)
2 pints = 1 quart (32. fl. oz.)
4 quarts = 1 gallon (128 fl. oz.)

Yay, powers of two! Unfortunately it breaks down as we get smaller than a fluid ounce.

2 tablespoons = 1 fl. oz.
3 teaspoons = 1 tablespoon

Another way in the system to measure volume is the cubic inch. This is not related nicely to the fluid ounce, as 1 cubic inch = 0.554413... fluid ounces.

Weight
16 ounces = 1 pound or lb.
2,000 lbs. = 1 ton

There are also little used units like a hundredweight (100 lbs.), a stone (14 lbs.) and a long hundredweight (8 stone or 112 lbs.). When we get smaller than an ounce, the standard is a grain. 7,000 grains is a pound, which means 437.5 grains is equal to an ounce.

The metric system

Unlike the customary system, which was thrown together over time, the metric system was made all at the same time, so length is related to volume and volume is related to weight using water as the standard thing we will weigh. All you need to learn for most measuring are three words and three prefixes for most stuff.

Length 
The standard length unit is the meter or m. 10,000,000 meters is the distance from the equator to the North Pole.

The important prefixes here. For long distances that Americans would measure in miles, the metric system uses kilometers or km. Kilo means "1,000", so this is 1,000 meters.

For short distances Americans would measure in inches, the metric system uses centimeters or cm, which is 1/100 of a meter.

For even smaller lengths where Americans would use fractions of inches, the metric standard is a millimeter, which is 1/1000 meters.

Volume
The standard measurement for volume is the liter or L. It is based on a cube 1/10 or a meter (or 10 cm) on each side. The liter is just slightly larger than a quart.

When dealing with very large volumes, the unit is still the liter.

For smaller volumes, the standard unit is the milliliter or mL. This is a cube one centimeter on each side, so it is also also called a cubic centimeter or cc.

Weight
The basic unit of weight is a gram, which is the weight of a cc of water at about 4 degrees Celsius, or about 39.2 degrees Fahrenheit. The temperature was chosen as the lowest point where at normal pressure, water shows no sign of freezing.

The gram is very small, so when measuring things the customary system would measure in pounds, the metric system uses kilograms or kg. A kilogram is slightly more than 2 pounds, and for very heavy weights that Americans would measure in tons, the metric system users resort to metric tons, which are 1,000 kg. (technically, this would be a megagram, but this word is almost never used.)

As small as the gram is, some small measurements like medicine doses are doled out in milligrams or mg. When medicine is measured, not even Americans use ounces or grains, though when I was a kid, grains were used on aspirin packages. One gram is about 15 grains and sometimes there was confusion, leading to massive overdoses or underdoses.

Conversion values

We only need a single conversion number, and depending which direction the conversion is in, we either multiply by the conversion number or divide by it. Here is an example.

1 inch = 2.54 centimeters (cm)

If we have 40 inches, the 40 is related to the 1, so 40/1 = x/2.54. With fractions, we cross-multiply to get 40*2.54 = x*1, so x = 101.6 cm. If we are asked to round to the nearest whole number, it would be 102 cm.

If instead we have 40 cm, the fraction would be x/1 = 40/2.54 = 15.7480315... inches, which could round to 16 inches, 15.7 inches or 15.75 inches, depending on the level of rounding.

Here are some of the standard numbers used for conversion.

Length, middle distances: 1 meter = 3.2808 feet or 39.37 inches
Length, long distances: 1 kilometer = 0.6214 miles

Weight, small weights: 1 gram = 0.03527 ounces
Weight, medium weights: 1 kilogram = 35.27 ounces = 2.2046 pounds
Weight, large weights: 1 metric ton = 2,204.6 pounds = 1.1023 metric tons

Volume, small measures: 1 milliliter = 0.033814 fluid ounces
Volume, medium to large measures: 1 liter = 33.814 fluid ounces = 1.0567 quarts
Volume, liters to cubic inches or cubic feet: 1 liter = 61.0237 cubic inches = 0.0353 cubic feet

Compound interestEffective interest ratesDoubling periods and half lifesThe Consumer Price IndexMonthly payments on loans

A lot of the math of "the real world" is linear, but some of it involves exponents. The best known interest rate situations are those of a savings account. Here is the equation; instead of using superscripts for exponents, I use the symbol ^, which is the symbol for exponentiation on most of the TI calculators, though some have an "x rasied to the y" button.

Compound interest rate equation

A(t) = P(1+r)^t

A(t) is the amount in the bank at time t
P is the original principle invested 
r is the rate of interest, usually given as a percent


Effective rate compound interest rate equation

Some saving institutions will give the rate as "4% compounded quarterly". This is slightly different from 4% a year. If we use n as the number of compounding periods in a year, the equation changes to 

A(t) = P(1+r/n)^nt

4% compounded quarterly

In the stated example (1+.04/4)^4 = (1.01)^4 = 1.04060401, so the rate r = .04060401, usually rounded to the nearest hundredth of a percent at 4.06%. The difference is small but noticeable over long stretches of time.


$1000 invested at 4% over 20 years is $2,191.12
$1000 invested at 4% compounded quarterly over 20 years is $2,216.72

Doubling periods and half lifes

How long will it take at 4% to turn $1000 into $2000. This amount of time is called the doubling period and is found by the formula log(2)/log(1 + r)

In our case, that would be log(2)/(1 + .04) = 17.6729.....

If the bank is actually compounding annually, the amount only changes on the anniversary of your putting the money in.

After 17 years you have (1.04)^17 = $1,947.90
After 18 years you have (1.04)^18 = $2,025.82

The Consumer Price Index

This week, students got a green handout sheet with the consumer price index number for the years 1950 to 2015.  The simplest way to use it goes as follows.

How did prices change from 1960 to 1980? We use the two CPI values.

CPI(1960) = 29.6
CPI(1980) = 82.4

82.4/29.6 = 2.78378.....

29.6/82.4 = .35922...

What these numbers mean is that on average, a $10 item in 1960 sold for 10*2.78378... = $27.84 in 1980, while a $10 item in 1980 would have sold for 10*.35922... = $3.59.

We can use this to figure out the cost of living increase in any given year by dividing the CPI for that year by the CPI for the previous year.  For example, the rate in 1975 would be

CPI(1975)/CPI(1974) = 53.8/49.3 = 1.09127789... This is 1 + rate, so rounded to the nearest tenth of a percent we would have 9.1% and to the nearest hundredth of a percent it would be 9.13%.

Let's look at the number CPI(1980)/CPI(1960) = 2.783783... This gives us how much prices increased in the 20 year period from 1961 to 1980. To get the average increase over those 20 years, we take (2.783783)^(1/20) = 1.05252... This is 1 + rate, so the average rate = .05252... or 5.3% if rounded to the nearest tenth of a percent and 5.25% rounded to the nearest hundredth.



Monthly payments on loans

If you know the interest rate r, the number of years y you will take to pay off a loan and the amount you wish to borrow A. The formula for the monthly payment is

Monthly payment = A(1 + ry)/(12y)

We can use this to figure out monthly payments or, if we know how much we are willing to pay a month, the rate and the numbers of years we can get the loan for, conversely we can get the amount A with the formula

A = 12y(Monthly payment)/(1 + ry)

Practice problems

a) If you put $1,000 in the bank invested at 4.5%, how much will you have in 10 years?


b) If you put $1,000 in the bank invested at 4.5%, how much will you have in 15 years?


c) what is the effective rate of 5.5% compounded monthly?

d) what is the effective rate of 5.5% compounded daily?

e) A radioactive isotope loses 0.1% of its radioactivity every year. What is the half-life, rounded to the nearest tenth of a year?

f) The worst of inflation in last century in the United States happened in the 1970s and early 1980s. Find the average inflation rates for the following presidential terms. (Remember to start with the year prior to the start of the term.

1. Nixon (1969-1974)
2. Ford (1974-1976)
3. Carter(1977-1980)
4. Reagan(1981-1988)

g) If you borrow $20,000 at 6% and have a ten year loan, what is the monthly payment?


Answers in the comments.

Solving simultaneous equations

Here is a link to practice problems for simultaneous equations. The coin problems all use elimination instead of substitution. Here are the same coin problems using substitution, with the answers in the comment below.

a) 100 coins, all quarters and pennies, total = $13.72

b) 100 tickets, all children ($6) and adult ($12), total =$912

c) 100 coins, all quarters and nickels, total = $8.20

d) 100 coins, all dimes and pennies, total = $3.79

e) 100 coins, all dimes and nickels, total = $7.85

f) 100 coins, all quarters and dimes, total = $14.65

Two points on the plane define a slope (or a slope is undefined)

Here are previous posts dealing with the definition of slope and practice problems for finding the definition of a line that passes through two points (x1, y1) and (x2, y2).

Tiliing the plane

Here is a link to a previous blog I wrote and pictures of some of the regular tilings, either single shape or mix and match.

And here is some fun I had with the Penrose tiles back when I bought them in 2011.

Notes for the week of October 12 through 15

Notes for area of a triangle defines by three points and the distance formula.

Notes for interior angles sums of polygons and the measure of regular angles.

More practice on the interior angles sums and regular angle measurements.

Classifying triangles defined by three vertices in the plane.

In class, we have worked with three points in the plane where one is the origin (0, 0) and the other two are labeled (a, b) and (c, d).  In the notes here on the blog, the points are given coordinates with subscripts such as (x1, y1) and (x2, y2). The formulas are the same, only the letters used in the formulas are different.

The area for the triangle define by (0, 0), (a, b) and (c, d) is given by the formula


Area = ½| ad - bc |

This formula is relatively simple compared to Heron's formula, with no square roots necessary. On the other hand, we will need to do three square roots to get the lengths of the three line segments for both classification methods.  Let's look at an example.

Example # 1: (0, 0), (6, 3), (7, 1)

Area: ½| (6)(1) - (3)(7)| = ½| 6 - 21| = ½| - 15 | = ½(15) = 7.5

Distances:

Between (0, 0) and (6, 3): When (0, 0) is involved the subtractions are easy and we get

Distance = sqrt((0-6)² + (0-3)²) = sqrt(6² + 3²) = sqrt(36 + 9) = sqrt(45)

Usually, I'd ask to have this simplified, but this problem just leave the answer as the square root of 45, it will make the next steps easier.

Between (0, 0) and (7, 1) 

Distance = sqrt((0-7)² + (0-1)²) = sqrt(7² + 1²) = sqrt(49 + 1) = sqrt(50)

Between (6, 3) and (7, 1)   

Distance = sqrt((6-7)² + (3-1)²) = sqrt((-1)² + 2²) = sqrt(1 + 4) = sqrt(5)

Now we have the distances, the classifications can be done.

Equilateral, isosceles or scalene?  The numbers are all different sqrt(45), sqrt(50) and sqrt(5), so the triangle is scalene.

Obtuse, right or acute? Here we need to square the distances, which is easy because they are all written as square roots, so the squares are 45, 50 and 5.  We need to compare the sum of the two short sides squared versus the long side squared and in this case

45 + 5 = 50.

This is a right triangle.

Final answer: Area of 7.5 square units, a scalene right triangle.

Example #2: (3, 2), (4, 1), (1, 7)

Here, we don't have a point at the origin, so we will subtract a value of one given vertex from all of the vertices.  That will give us three new points, but the triangle defined by our new points will have the same area and classifications.  I will choose to subtract (3, 2), but using (4, 1) or (1, 7) would also work.

New vertex #1: (3, 2) - (3, 2) = (0, 0)
New vertex #2: (4, 1) - (3, 2) = (1, -1)
New vertex #4: (1, 7) - (3, 2) = (-2, 5)

Area = ½| (1)(5) - (-1)(-2)| = ½| 5 - 2| = ½| 3 | = ½(3) = 1.5


Between (0, 0) and (1, -1):

Distance = sqrt((0-1)² + (0-(-1))²) = sqrt(1² + 1²) = sqrt(1 + 1) = sqrt(2)

Usually, I'd ask to have this simplified, but this problem just leave the answer as the square root of 45, it will make the next steps easier.

Between (0, 0) and (-2, 5) 

Distance = sqrt((0-(-2))² + (0-5)²) = sqrt(2² + 5²) = sqrt(4 + 25) = sqrt(29)

Between (1, -1) and (-2, 5)   

Distance = sqrt((1-(-2))² + ((-1)-5)²) = sqrt(3² + (-6)²) = sqrt(9 + 36) = sqrt(45)

Now we have the distances, the classifications can be done.

Equilateral, isosceles or scalene?  The numbers are all different sqrt(2), sqrt(29) and sqrt(45), so the triangle is scalene.

Obtuse, right or acute? Here we need to square the distances, which is easy because they are all written as square roots, so the squares are 2, 29 and 45.  We need to compare the sum of the two short sides squared versus the long side squared and in this case

2 + 29 = 31 < 45.

This is an obtuse triangle.


Final answer: Area of 1.5 square units, a scalene obtuse triangle.

Notes for the week of October 5 through 8

Classification of triangles, including the triangle inequality and Heron's formula, which will give us the area of a triangle defined by side lengths, here using the letters u, v and w.

Finding a line segment of the length of the square root of any number.

The easiest thing is to find the square root of any given odd number. To find the square root of any given even number, find the square root of the odd number one less than it. For example, if I want to find sqrt(44), it will be a two step process, where first we find the sqrt(43).



Step 1: 43/2 = 21.5, so the whole numbers closest to splitting 43 in half are 22 and 21. It's easy to prove that 22² - 21² = 43, so we make a triangle with the hypotenuse = 22 and the long leg = 21.  The short leg will be sqrt(43).

Step 2: Now we make a new right triangle by extending the leg of length 21 by 1. Now we have a new right triangle with legs of length sqrt(43) and 1, and the hypotenuse will be sqrt(44).

Notes for the October 1 class

The new topic covered on Thursday night was the Pythagorean Theorem. Here is a link to the posts on the topic.

For the Monday-Wednesday class: Notes for September 21 and 23Also the notes for September 29 for the Tuesday Thursday class

Scales based on other powers of 10 besides 100 (percent) and murder rates per 100,000.

Fractions, decimals and percents.

Changing repeating decimals to fractions in lowest terms.

Approximating and simplifying square roots.

Rounding decimals.

Let's consider 3/7. If you type 3/7 into the calculator, you'll get 0.428571429. If we wanted to write the fraction with a bar over the repeating part, the answer would be
 ______
.428571 

Usually, fractions like this are written as decimals rounded to the nearest tenth, hundredth or thousandth.  Let's do all three versions here.

3/7 rounded to the nearest tenth.
Step 1: Erase everything after the first digit, which gives us .4
Step 2: Because the left most digit we erased is a 2, we leave the .4 alone.
3/7 ~= .4 rounded to the nearest tenth.

3/7 rounded to the nearest hundredth.
Step 1: Erase everything after the second digit, which gives us .42
Step 2: Because the left most digit we erased is an 8, we change .42 to .43, rounding up.
3/7 ~= .43 rounded to the nearest hundredth.

3/7 rounded to the nearest thousandth.
Step 1: Erase everything after the third digit, which gives us .428
Step 2: Because the left most digit we erased is a 5, we increase .428 to .429.
3/7 ~= .429 rounded to the nearest thousandth.

Rounding error.

If we multiply 7 × 3/7, we get 3 exactly.  If we multiply 7 by a rounded version of 3/7, we won't get exactly 3. If we rounded the approximation off, when we multiply we will be below the target. If we round the approximation up, we will be above the target.

7 × .4 = 2.8, so we are 0.2 below 3.

7 × .43 = 3.01, so we are 0.01 above 3.

7 × .429 = 3.003, so we are 0.003 above 3.

For the Tuesday-Thursday class; Notes for Sept. 22

Here is a link to the stories about five computer scientists of the 20th Century.

Posts on Roman numerals

Here are links to previous practice problems dealing with Roman numerals.

Here are a few more practice problems, with answers in the comments.

702 = ______

2,598,960 = _____

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

DCVII = ____

Notes for September 10 (Tues.-Thurs. class) and September 14 (Mon.-Wed. class)

Notes on logarithms. While there are other logarithmic scales including pH, we will be focusing on the bel system (usually written in decibels, or tenth of bels, so that 7.1 bels is written as 71 decibels or 71 dB) and the Richter scale, always written to the nearest tenth, like 4.5 or 3.0.

Notes for September 8 (Tuesday-Thursday) and September 9 (Monday-Wednesday)

The Monday-Wednesday class meets only one day this week, so the notes are split up.

Notes on contingency tables and probabilities based on contingency tables.

Notes on scientific notation.

Links to posts on set theory and Venn Diagrams

Here are links to the posts on set theory, which is a mathematical logic system like our work with AND, OR and NOT, which are directly analogous to intersection, union and complement, respectively.

A pictorial representation of set theory is called Venn Diagrams and you can click on the link to read them.

Notes on decimal, binary and hexadecimal representation and on mathematical logic

Here are links to previous posts on changing between decimal (base 10), binary (base 2) and hexadecimal (base 16).

Here are links to previous posts on mathematical logic.

Link to the life stories of four famous mathematicians

Click on this link to read the life stories of Archimedes, Newton, Euler and Gauss.

Topic list for comprehensive final on 7/23

You get two 8.5" x 11" pages of notes, front and back, which means four pages total.

These are the possible topics for the 100 point test.
Binary, decimal hexadecimal
Logical operators, and ^, or v, not ~
Decimals and percents, changing fractions into repeating decimals and vice versa.
Scales other than percents
Scientific notation and rounding to significant digits.
Roman numerals
Fractions in lowest terms
Prime factorizations and all factors
Time: adding subtracting, converting
100 coin problems
simultaneous equations
bels/decibels and Richter scale
Classifications of triangles given angles
Classifications of triangles given sides
Heron’s formula
Area of a triangle with points (0, 0), (a, b) and (c, d)
Distance between points
Pythagorean Theorem
Sets, Venn diagrams, contingency tables and probabilities
Slope of a line and lines with undefined slope
Point-slope form, slope-intercept form
Sum and product problems
Stats: mean, median, mode, five number summary, IQR, outliers
standard deviation, z-scores, z-scores to proportions less than a z-score, reverse lookup table, proportions between two z-scores for a normally distributed set

If something says "formula" or "form", You can expect that formula to be on the test. For things that take methods, like changing from Richter to amount of energy or looking up z-scores, those will not be explained, so they should be in your notes.
 

Finding percentiles from the z-score list. Finding the proportion at an exact number on a list of a normally distributed set

Finding percentiles from the z-score list. 
 One of the things we can do with the z-score to proportion list is find a z-score that corresponds to the cut-off point for a particular percentile. Here are two examples at the 8th percentile and the 80th percentile.

The 8th percentile. because 8 < 50, we will look on the negative z-score side of the sheet. What we want to find is the two positions on the table where the values go from .08xx to .07xx. you will find then in the -1.4 row at the first two positions.

-1.40 -> 0.0808 (8 above 0.0800)
-1.41 -> 0.0793 (7 below 0.0800)

Because 7 < 8, -1.41 counts as the closest to the two, but we should check to see if the average -1.405 would be better. We look at far/close which is 8/7 ~= 1.142857... and if it's less than 3 (it is), we will use the average,so the z-score for the 8th percentile is -1.405.



The 80th percentile. because 80 > 50, we will look on the positive z-score side of the sheet. What we want to find is the two positions on the table where the values go from .79xx to .80xx. you will find then in the 0.8 row in the middle.

0.84 -> 0.7995 (5 below 0.8000)
0.85 -> 0.8023 (23 above 0.8000)

Because 5 < 23, 0.84 counts as the closest to the two, but we should check to see if the average 0.845 would be better. We look at far/close which is 23/5 = 4.6 and if it's more than 3 (it is), we will use the closest instead of the average,so the z-score for the 80th percentile is 0.84.

Finding the proportion at an exact number on a list of a normally distributed set. Consider the SAT values for average mu_x = 500 and sigma_x = 100. SAT scores are always round to the nearest 10, so if we want to find out what percentage of scores are at 600, we have to look at the proportion at 605 and 595.

Raw score 605 become the z-score (605-500)/100 = 105/100 = 1.05, which corresponds to 0.8531

Raw score 595 become the z-score (595-500)/100 = 95/100 = 0.95, which corresponds to 0.8289.

We subtract the small proportion from the big, .8531 - .8289 = .0242. This means about 2.42% of SAT takers will get exactly 600 on one section of the SAT.
 

Links to the five number summary, IQR and outlier material

This link leads to my statistics blog and several examples of the five number summary and how to use it to define outliers.
 

Finding x and y when we are given their sum and product

If we think about a rectangle, the sum of the two adjacent sides is half the perimeter and their product is the area. If we are given the sum and product, can we find the two side lengths? This is the original problem that was being considered when the quadratic formula was derived nearly 900 years ago, though many civilizations before this had something like the idea that is about to be presented.

x + y = sum
xy = product

Let's call half the sum a, which stands for average. Since we are given the sum, a is a known quantity as well. It might be that x = y = a, but generally what is true is we can find a number d, which stands for difference such that

x = a + d
y = a - d

This means the product is the difference of squares, (a + d)(a - d) = a² - d².

Using this, here is how we solve a given problem.

Problem: The sum is 14 and the product is 24.

Solution method: The average a = 7, so we need to solve for d.

(7 + d)(7 - d) = 7² - d² = 49 - d²= 24


Subtract 24 from each side to get

25 - d² = 0

Add d² to both sides to get

 d² = 25.

Take the square root to get d = 5. We use the numbers 7 and 5 in the following way.

7 + 5 = 12
7 - 5 = 2

12 + 2= 14
12 x 2 = 24

With this problem, you could have used guessing and checking to try to find two numbers that added up to 14 (13 and 1, 12 and 2, 11 and 3, no wait, go back to 12 and 2) where the product was 24. let's do one where guessing and checking isn't an option.

Problem: The sum is 15 and the product is 45.

Solution: The average is 7.5 and a² = 56.25. So 56.25 - d²= 45.

Subtracting, we get 11.25 - d²= 0 or d² = 11.25 = 45/4.

Taking square roots, we get d = sqrt(45/4), which simplifies to 3sqrt(5)/2, which rounds to 3.354. Our two numbers are

7.5 + 2sqrt(5)/2 and 7.5 - 2sqrt(5)/2, which round to 10.854 and and 4.146.

More problems of this type.

Problem: The sum is 20 and the product is 40.


Problem: The sum is 40 and the product is 20.


Problem: The sum is 10 and the product is 40.


Problem: The sum is 40 and the product is 10.

Links to material on homework 9, July 13 and 14 2015

Practice to find the equation of a line given two points.

Working with complex numbers.

Links to Venn Diagrams and contigency tables

This link is to a post from my statistics blog about Venn Diagrams and contingency tables.

Prep for midterm 2

Midterm 2 will be based on homeworks 4 through 7 and the biographies of the five British logicians, Boole, Babbage, De Morgan, Dodgson and Lovelace. The most recent posts on this blog from the June 28th post about Richter scale up through the logician biographies will be of the most use in studying.

Links for July 6 and 7 lectures, homework 7 and the five logicians

Here is a link to the life stories of the five British logicians.

Here are practice problems for the interior angle sum and the regular polygon angle measure.

Here are practice problems for changing between different time units.

Here are practice problems for set theory.

If you need more practice, e-mail me and I'll make some more.


 
 

Links to practice problems for distance between two points and area of a triangle defined by three points, one of which is (0, 0)

Here is the link to the practice problems.

More practice
Here are four points. Distances should be given in simplified square root form and approximated to the nearest thousandth,

(0, 0), (6, 3), (10, 1), (-4, -5)

a) Distance from (0, 0) to (6, 3)
b) Distance from (0, 0) to (10, 1)
c) Distance from (0, 0) to (-4, 5)
d) Distance from (6, 3) to (10, 1)
e) Distance from (6, 3) to (-4, -5)
f) Distance from (10, 1) to (-4, -5)

g) the area of the triangle defined by the points (0, 0), (6, 3) and (10, 1)
h) the area of the triangle defined by the points (6, 3), (10, 1) and
(-4, -5)

answers in the comments.
 

Links to triangle classification with angles

Other problems are included, but you can find problems like the homework due on 1 July 2015 on these pages.

Simultaneous equation practice with Kramer's rule

6x + 5y = 18
2x - y = -4

3x + 7y = 12
2x + 5y = -18

Answers in the comments.

100 coin (or ticket) problems

a) 100 coins, all quarters and pennies, total = $13.72

b) 100 tickets, all children ($6) and adult ($12), total =$912

c) 100 coins, all quarters and nickels, total = $8.20

d) 100 coins, all dimes and pennies, total = $3.79

e) 100 coins, all dimes and nickels, total = $7.85

f) 100 coins, all quarters and dimes, total = $14.65

Answers in the comments

The Richter scale: From two readings, the relative strength and vice versa

In class last week, we learned how to find out how much stronger one quake is compared to another given the two Richter scale readings. For example, on June 28, the strongest quake in the U.S. was a 3.4 in Oklahoma, while the strongest in North America was a 5.4 in Niltepec, Mexico. What is the difference in levels of energy? Here are the steps.

Step 1: Subtract little from big. In our case, 5.4 - 3.4 = 2.0.
Step 2: multiply difference by 1.5. 2.0 * 1.5 = 3.0.
Step 3: Raise 10 to the power of the answer from Step 2: 10^3.0 = 1,000. The Mexican quake was 1,000 times stronger than the Oklahoma quake.

Let's ask the question in the opposite direction. Let's say we have a reading for a quake and we know another quake was x times stronger. Again, it will be a three step process, but now we will take the inverse of our three steps above in reverse order. Let's say we have a quake 350 times stronger than the one in Niltepec.  Here are our steps.

Step 1: Take the log of the strength multiplier. Log is the inverse of raising 10 to a power, just like addition is the inverse of subtraction and division is the inverse of multiplication. log(350) = 2.544...,
Step 2: divide the answer from Step 1 by 1.5 and round this answer to the nearest tenth. 2.544/1.5 = 1.696..., which rounds to 1.7. We round to the nearest tenth because the Richter scale rounds to the nearest tenth.
Step 3: Add the answer from Step 2 to the Richter reading we know. In this case, it would be 5.4+1.7 = 7.1, the reading of the stronger quake. If instead we were told a quake was 350 times weaker than Niltepec, it would be 5.4-1.7 = 3.7

Here are some practice questions. The answers are in the comments.

1. 16 times stronger than a 6.1
2. 250 times stronger than a 6.7
3. 8 times weaker than a 5.8

Link to the computer science biographies

Click on this link for the biographies of the five computer scientists. There will be questions on them on the midterm tomorrow.

Links to notes for 22 and 23 June

Notes on changing from decimal to binary and back, binary to hexadecimal and back and decimal to binary and back.

Notes for bitwise operators using binary notation.

Links to notes for 17 and 18 June 2015

Notes for percent and other scales based on powers of 10.

Practice for rates and scientific notation.

Links to all scientific notation posts.

Topics not covered in class that will be covered on Thursday and on homework 2.


Changing repeating decimals to fractions in lowest terms from 2012.

Changing repeating decimals to fractions in lowest terms from 2011. (Different practice problems.)
 


 
 

Links to homework 1, due Wednesday 17 June 2015

Link for prime factorization and all factors, with practice examples.

Roman and Hindu-Arabic numerals, fractions and decimals.

Practice for Roman to Hindu-Arabic and vice versa, fractions and decimals.

Practice for time conversion.

The following problems are in the form minutes:seconds. If the answer gives more than 60 minutes, write the answer as hours:minutes:seconds. Answers to the question below are in the comments.

 18:34
- 6:58


 42:23
 41:07
 43:01
 42:18
+42:17
 
 

Link to the four great mathematician biographies

Here is the link to the biographies of Archimedes, Newton, Euler and Gauss.

Syllabus for Summer 2015

Math 15: Math for Liberal Arts Summer 2015 (L1 30451)
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 9:25-9:55 am 3:05-3:35 pm (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 21
Last date to drop class without a "W": Sat. June 21
Last date to withdraw from class: Tues., July 23

Holidays: No holidays this session



Midterm and Finals schedule:

Midterm 1__________Thursday, June 25
Midterm 2__________Thursday, July 9
Comprehensive Final ___Thursday, July 23

Quiz schedule (most Tuesdays and Thursdays) no make-up quizzes given

First week: 6/16 and 6/18                       

Second week: 6/23

Third week: 6/30 and 7/2

Fourth week: 7/7

Fifth week: 7/14 and 7/16

Sixth week: 7/21

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 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.


Anyone with a class grade of 97% out of all work before the final (this grade will be given on the next to last day) does not have to take the final. That grade is worth an A.

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.