Notes for Homework 12, due May 8

Notes for Roman numeral to Hindu-Arabic numeral conversion

Notes for repeating decimals to fractions

Notes for percent, decimals and fractions over 100
 

Notes for Homework 11a, due May 1

Notes on the metric system vs. customary

Changing Celsius to Fahrenheit

F = 9/5 C + 32

Changing Fahrenheit to Celsius

C = 5/9(F - 32)

Link to biographies of five British logicians

Link to the biographies of Babbage, Boole, de Morgan, Dodgson and Lovelace

Notes for Homework 10, due April 17

Notes for interest rates on investment, the Consumer Price Index and paying back loans
 

Notes for Homework 9, due April 10

Notes on mean, median and mode

Notes on the five number summary and outliers

Notes on z-scores, normal distribution and proportions

Notes for Homework 8, due March 28

Notes on determinants

A 2x2 matrix is an array of four numbers put in two rows and two columns, such as

| a c |
| b d |

The determinant of this matrix is ad - bc, the product of the main diagonal minus the product of the opposite diagonal. If you use Kramer's rule to solve a pair of simultaneous equations, you will need to know how to calculate determinants.

For example, let's consider the following set of equations.

3x - 2y = 7
2x + 4y = 12

As an augmented matrix , these equations become

|  3  -2  : 7 |
|  2   4  : 12|

Matrix_1 is

| 3 -2 |
| 2  4 |

The determinant is 3(4) - 2(-2) = 12 - -4 = 16
 

Matrix_x is

| 7 -2 |
| 12  4 |

The determinant is 7(4) - 12(-2) = 28 - -24 = 52
  Matrix_y is

| 3  7  |
| 2 12 |

The determinant is 3(12) - 2(7) = 36 - 14 = 22

From here, Kramer's rule then has x = 52/16 = 13/4 or 3 1/4. The value for y = 22/16 = 11/8 = 1 3/8
 
  Notes on solving simultaneous equations
 

Notes for Homework 7, due March 20

Notes for contingency tables

Notes for coin problems

Notes for slope, point-slope and slope-intercept
 

Notes for Homework 6, due March 13

Notes on rates other than percent

Notes on set theory and Venn diagrams
 

Link to biographies of Archimedes, Newton, Euler and Gauss.

Link to the biographies

Notes for Homework 5, due Feb. 27

Notes on area for a triangle defined by three points

Notes on the distance formula

Notes on interior angles of polygons and regular angles
 

Notes for Homework 4, due Feb. 20

Notes for classifying triangles by angles and by side lengths

Notes on creating isosceles triangles


Notes on Heron's Formula (lots of practice with answers in the comments)
 

Notes for Homework 3, Due Feb. 13

Notes on Richter scale and bels/decibels


Notes on simplifying and approximating square roots

Notes for Homework 2, due Feb. 5

Notes on adding and subtracting time, and notes on converting time

Notes on scientific notation
 

Notes for Homework 1, due Tues., Jan 30

Notes on rounding

Notes on binary, decimal and hexidecimal

Notes on the logical operators
 

Notes for Homework 13, due, Mon., Dec 4

Notes on Roman numerals

Notes on repeating decimals to fractions and vice versa

Notes on decimals, fractions and percents
 

Notes for Homework 12, due Mon., Nov 27

Notes on the metric system

Changing Celsius to Fahrenheit

F = 9/5 C + 32

Changing Fahrenheit to Celsius

C = 5/9(F - 32)




 

Notes for Homework 11, due Nov. 13

Notes for CPI table, interest on savings, effective interest rate and interest on loans
 

Notes for Homework 10, due Mon. Nov. 6

Notes on mean, median and mode

Notes on the five number summary, IQR and the calculation of low and high outlier thresholds

Notes on z-scores and table lookup
 

Notes for Homework 9, due Monday, Oct. 30

Notes on simultaneous equations and Kramer's Rule

Notes for Homework 8, due Oct. 23

Notes for contingency tables

Notes for coin problems

Notes for slope, point-slope and slope-intercept 

Notes for Homework 7, due Mon., Oct 16

Notes on set theory and Venn diagrams

Notes on rates based on powers of ten, most often on the per 100,000 scale

History lesson for Wednesday quiz

Here is a link to the biographies of Von Neumann, Turing, Hopper, Dijkstra and Knuth.

Here is a link to NASA's page on Katherine Johnson.

Notes for Homework 6, due Monday Oct. 2 (not accepted late)

Notes for triangles defined by three points: area and classification

Notes for interior angle sum and regular angle formula fro a polygon with n sides

Notes for Homework 5, due Sept. 25

Classifying Triangles by the three angles -or- by the three side lengths


Finding the area of a triangle using Heron's Formula

Notes for Homework 4, due Sept. 18

Practice for bels and decibels, also the Richter scale

More on the Richter scale

Simplifying and approximating square roots

Notes for Homework 3, due MONDAY, SEPT. 11

Notes for decimal part of a day to hours, minutes and seconds.

Notes for scientific notation

Fractional part of a day

A year is defined by how long it takes a planet to make a single orbit around the sun. An Earth year is 365.2422 days. To make up for the decimal part we have a leap year every four years and skip the leap year if the number of the year is divisible by 100. This means we had leap years in 2004, 2008, 2012 and 2016, but we did not have one in 2000. What this lesson is about is to take the decimal part of this time and turn it onto hours, minutes and seconds.

Step 1. Multiply the decimal part by 24 to get the number of hours.
In this case, .2422 x 24 = 5.8128, which means 5.8128 hours.

Step 2. If there is still a decimal part, multiply it by 60 to get the number of minutes. We know know a year is 365 days, 5 hours and some number of minutes. That number is 60 x .8128 = 48.768

Step 3. If there is still a decimal part, multiply it by 60 to get the number of seconds.
We know know a year is 365 days, 5 hours, 48 minutes and some number of seconds. That number is 60 x .768 = 46.08. In these problems, it is okay to round to the nearest tenth of a second, so the final answer is 365 days, 5 hours, 48 minutes and 46.1 seconds.

Let's do another example. A year on Venus is 224.65 Earth days.

Step 1. .65 x 60 = 15.6, so the year on Venus is 224 days, 15.6 hours.

Step 2. 60 x .6 = 36, so that makes the year on Venus 224 days, 15 hours and 36 minutes. There is no more decimal part, so we don't have to add any seconds to our answer.

Here are two more practice problems. The answers are in the comments.

1. It takes Mercury 87.969 Earth days to travel around the sun. Write this number in days, hours, minutes and seconds.

2. It takes Mars 686.98 Earth days to travel around the sun. Write this number in days, hours, minutes and seconds.

Notes for Homework 2, due WEDNESDAY, SEPT. 6

Notes on hexadecimal, binary and decimal

Notes on logic with 1s and 0s

Notes on time problems

Notes for Homework 1, due August 28

Notes on rounding

Notes on significant digits

Notes on binary to decimal and vice versa

Link to the biographies of Archimedes, Newton, Euler and Gauss

Click on this link to read a post on the four great mathematicians listed above

Notes for Homework 9, due Monday, July 24

Notes on the five number summary method for determining outliers

Notes on z-scores and normal distribution
 

Notes for Homework 8, duy Wed. July 19

A quick reminder: this homework will not be accepted late because the answer sheet will be given out at the end of class to help study for Thursday's midterm.

Notes on scientific notation

Notes on Set Theory

Notes on Venn Diagrams

Notes on contingency tables and probabilities

Math history: Five Briitsh logicians

 Here is a link to stories about Boole, Babbage, Dodgson, de Morgan and Countess Lovelace

Notes for Homework 7 due Monday, July 17

Notes on simultaneous equations

Notes on using elimination in coin problems

Notes for Homework 6, due Wednesday July 12

Practice problems for distance.

Practice problems for slope, point-slope and slope intercept.

Notes for Homework 5, due Tuesday July 11

Notes for classifying triangles by angles or by side lengths

Note for finding area using Heron's formula
 
 

Link to the history of five computer scientists

A link to the biographies of Von Neumann, Turing, Hopper, Dijkstra and Knuth.

Notes for Homework 4, due Monday July 3

Notes on decibels and bels (measuring sound) and the Richter scale (measuring earthquakes)

Notes on square roots, approximate and simplified, also Pythagorean Theorem problems

Notes on rates at different scales, notably death rates per 100,000 people

 

Notes for Homework 3, due June 28

Notes on interest rates on investments, interest rates on loans and the Consumer Price Index

Notes on rounding to significant digits
 

Notes for Homework 2, due June 26

Notes for converting from binary (base 2) to decimal (base 10) and binary to hexadecimal (base 16)

Notes for the logical operators AND (^), OR (v) and NOT (~)

Converting repeating decimals to fractions in lowest terms

Time problems

Roman numerals

Notes for Homework 1, due June 21

Notes on fractions, decimals and percents, also on primes and prime factorizations.

Notes on repeating decimals and rounding decimals.

Notes on adding and subtracting time.
 

Link to the stories of four famous mathematicians

The biographical data on Archimedes, Newton, Euler and Gauss.

Notes for Homework 7, due March 27

Contingency tables and probabilities

Two points: Find slope, then the point-slope, then slope-intercept

Coin problems
 

Notes for Homework 6, due March 20

Notes on Set Theory and Venn Diagrams

Notes on interior angle sums and regular angles

Notes on tilings of the plane

Notes on the area of a triangle given three points on the plane
 

Link to the biographies of five computer scientists

Link to the biographies of John Von Neumann, Alan Turing, Grace Hopper, Edsger Dijkstra and Donald Knuth.

Notes for Homework 5, due Monday March 6

Note for classification of triangles by angle measurement and side length

Area by Heron's formula
 

Notes for Homework 4, due Wed. Feb. 22

Because of the Monday holiday, homework is due on Wednesday. There will be no quiz this week.

Notes on the Richter scale and bels and decibels, abbreviated B and dB.

Notes on square roots and the Pythagorean Theorem.

Notes for Homework 3, due Monday, Feb. 13

Practice with decimals and percents

Practice with scales of powers of ten other than percent

Practice with scientific notation
 

Notes for Homework 2, due Monday, Feb. 6

Notes on Hex to Binary and vice versa.

Notes on logical operators AND (^), OR (v) and NOT (~).

Notes on time conversion and adding and subtracting time.

Notes for Homework 1, due Monday, January 31

Links to posts on rounding.

Links to posts on significant digits.
 
Links to posts on decimal to binary and binary to decimal.
 

Link to the biographies of Archimedes, Newton, Euler and Gauss

Click on this link to a post that has the biographies of the four guys I discussed Monday night, the information that will be on the quiz on Wednesday.

Notes for Tuesday-Thursday homework due on Nov. 29

Notes on interest, compound interest, effective rates, half life and loans

Notes for homework of Tuesday Thursday class due Nov. 22

Notes on frequency tables and the five number summary

Notes on frequency tables, n (length of list) and sum(x)

Notes on the shared birthday problem

Notes on binomial distributions (on our homework, making free throws)

Notes on using the normal distribution system (z-scores and look-up tables) to find the percentage between two z-score values (on our homework, males listed at a certain height, on the website, females listed at a certain height. Only the mu and sigma are different.)

Notes for Homework 10

I made a mistake labeling this homework assignment, calling it Homework 11. It is actually Homework 10 for the Tuesday-Thursday class.

These notes are from my statistics blog. The links contain a lot of information that we won't get to in Math for Liberal Arts. You should be able to search in your web browser for words like "mean", "median" and "mode", etc.

Notes on mean, median and mode

Notes on the five summary and outliers

Notes on raw scores to z-scores to proportions, also percentiles to z-scores to raw scores

Link to the biographies of five computer scientists, questions on in the in-class section of midterm 2

Five computer scientists

Notes for Homework 9

Notes on simultaneous equations (repeated from last week)

Notes for Homework 8

Notes on slope, point-slope and slope-intercept formulas

Notes on simultaneous equations
 

Notes for Homework 7

Notes on the distance formula and classifying triangle defined by three points on the plane

Notes on interior angle sums for polygons and the measure of the regular angle

Notes for tilings of the plane
 
 

Notes for Homework 6

Notes on classifying triangles by angles and by side lengths in both systems

Area of a triangle using Heron's Formula, area of a triangle defined by three points

Notes for the five British logicians of the 19th Century

Link to biographies of Lovelace, Dodgson, DeMorgan, Boole and Babbage.

Notes for Homework 5

 Notes on the Pythagorean Theorem and simplified square roots

Notes on bels, decibels and the Richter scale

Notes for Homework 4

 Notes on prime factorization and all factors of a positive number, also notes on fractions, percents and decimals

Notes on scientific notation

Notes on scales based on powers of 10

Notes for Homework 3

Notes for Venn Diagrams

Notes for contingency tables

Notes for probabilities based on contingency tables

Notes on converting time and adding and subtracting times

Notes on percents, decimals and fractions
 

Notes for Homework 2

Notes on Hexadecimal to Binary

Notes on logical operators - AND (^)   OR (v)   NOT (~)  IMPLIES (=>)

Notes of Set Theory

Notes for Homework 1

Notes on rounding and rounding error with fractions.


Notes on binary and decimal representation of numbers.


Notes on rounding to significant digits. 


The link to rounding to significant digits deals with rounding a number great than 1. For example 2^16 = 65,536.

65,536 rounded to one significant digit = 70,000

65,536 rounded to two significant digits = 66,000
65,536 rounded to three significant digits = 65,500

Let's consider rounding a number less than 1.

2^(-16) = 0.000015258789...

If we were asked to round this to the nearest thousandth, we would get 0.000. It's never a good idea to round a number that isn't zero to zero. Doing this means we aren't thinking at the right scale. Rounding to significant digits ensures we will never round a non-zero number to zero.

0.000015258789... rounded to one significant digit = 0.00002
0.000015258789... rounded to two significant digits = 0.000015
0.000015258789... rounded to three significant digits = 0.0000153

In most cases except scientific papers, rounding to three significant digits is considered sufficient.

Link to the stories of the four famous mathematicians

Short biographies of Archimedes, Newton, Euler and Gauss.

Important dates for Fall 2016

Add and drop dates
Last day to add: Sunday, September 4
Last day to drop class without a "W": Sunday, September 4
Last date to drop class with a "W": Friday, November 8

Holidays
Labor Day: Monday, September 5
Thanksgiving: Thursday, November 24

Test dates for Monday-Wednesday class
Midterm 1: Wed., Oct. 5
Midterm 2: Wed., Nov. 9
Comprehensive Final: Wed, Dec. 14 8:00-10:00 am (note time change)

Test dates for Tuesday-Thursday class
Midterm 1: Thurs., Sept. 29
Midterm 2: Thurs, Nov. 3
Comprehensive Final: Tues., Dec. 13 (normal class period)

Notes for Homework 11

Paying Loans

Converting from Customary to Metric and Back

Notes for Homework 10

Finding outliers: the five number summary and the z-score method

Using the look-up sheet for normal distribution.
 

Notes for Homework 9

Slope, point-slope, slope-intercept

Solving simultaneous equations

Solving simultaneous equations: Elimination

Solving simultaneous equations: Kramer's Rule 

Notes for Homework 8

Set Theory

Venn Diagrams

Contingency Tables

Link to the post on the British mathematicians who started mathematical logic

Link to post about Boole, de Morgan, Babbage, Countess Lovelace and Dodgson.

Notes for Homework 7

Pythagorean Theorem problems

Interior angle sum and regular angle measure

Time problems

Set theory
 

Notes for Homework 6

Area and classification of a three vertex triangle.

Second post on the same topic.
 

Notes for Homework 5

Simplifying and approximating square roots.

Classification of triangles.

Heron's formula.

Notes for Homework 4

Links to posts about the common logarithmic scales, the Richter scale and the bel/decibel system.

Link to the post on five computer scientists

Here is a link to the biographies of five computer scientists: John Von Neumann, Alan Truing, Grace Hopper, Edsger Dijkstra and Donald Knuth.

Notes for Homework 3

Posts on binary, decimal and hexadecimal

Posts on logical operators

Notes for Homework 2, handed out June 23, due June 27

Decimals and percents.

Repeating decimals to fractions in lowest terms.

Scales other than percent.

Scientific notation.
 

 

Notes for homework 1, June 21

Practice for Roman numerals.

Practice for all the problems.

Notes for 20 June 2016: Biographies of four famous mathematicians

Click her for a post about Archimedes, Sir Isacc Newton, Leonhard Euler and Karl F. Gauss.

Notes for May 9 and 11

The metric system and the customary system.

Notes for Homework due May 9

Frequency tables: mean and mode

Frequency tables: five number summary

Probability of making r shots out of n attempts.

The Birthday Problem.

Notes for homework due May 1

Notes on Five Number Summary and the method of finding outliers.

Normally distributed sets: z-scores associated with proportions and vice versa.

The last part of the homework wasn't covered in class, but it will be discussed on Monday.

Notes for homework due April 25

Practice problems for simultaneous equations.

Practice problems for conversion of time.
 

Topics for second midterm

The second midterm will have a take-home section and an in-class section. Some topics may appear in both parts. You are allowed a page of notes, front and back of a regular 8.5" x 11" piece of paper.

Homework 6 
 Interest rates and compounding
Half-life of isotopes
Paying back loans: Amount over life of loan
Playing back loans: Amount per month
Maximum amount you can get as a loan given monthly payment, interest rate and length of the loan

Homework 7
Triangles defined by angles
Classification system #1: Largest angle – obtuse, right or acute
Classification system #2: Relations between angles – scalene, isosceles, or equilateral
Triangles defined by side lengths
The triangle inequality
Both classifications by side lengths: Variations on the Pythagorean Theorem
Triangles defined by three points on the plane, one of the points (0, 0)
Distance between points
Both classifications by three points (distance is key for both)

Homework 8
Three points on the plane where none is (0, 0)
Finding the slope (m) between two points (x1, y1) and (x2, y2)
What it means when the slope is undefined: the formula x = k, some constant value
Point slope formula: y – y1 = m(x – x1)
x and y will remain variables m, x1 and y1 will become constants
Slope-intercept: y = mx + b

Homework 9
Tilings of the plane
Interior angle sum for a polygon with n sides (n-gon): sum = 180(n – 2)°
Regular angle sum for a polygon with n sides (n-gon): = 180(n – 2)°/n = (180–360/n)°
Coin problems

Notes for April 4th and 6th

Notes for regular angles and interior sum of angles and tiling the plane.
 

Practice problems for 100 coins or 100 tickets. (Our homework doesn't always have 100 coins, but there is nothing magical about the number, the methods shown will always work.)

Notes for slope, point-slope and slope-intercept

Links to explanation of slope and practice problems for point-slope and slope-intercept.

Notes on TrianglesDefined by anglesDefined by side lengthsDefined by three points on the plane

Classification of triangles based on the three angles. Because the sum must be 180°, only two of the three are needed to find the third. 

Practice problems for area and classification based on three side lengths and the Triangle Inequality, which determines of three lengths can be the sides of a triangle.

Practice problems for area and classification of triangles based on three points in the plane.

Notes for March 7 and 9

Notes for interest on savings, half life computation and paying back loans.

The syllabus posts at the bottom can be ignored.

Note for February 22 and 24

Links to percents and scales based on other powers of 10.

Links to note on rounding and rounding error.

Links to changing repeating decimals into fractions.

Links to simplifying square roots, including the Pythagorean Theorem practice.

Notes on Roman numerals and practical logarithms

A link to Roman numeral posts.

Links to posts on practical logarithms.

Practice Set #1.

Practice Set #2.

Notes on probabilities from contingency tables and scientific notation

A link to probabilities from contingency tables.

Scientific notation: A link to the basics
 

Links to set theory practice problems

Click on this for set theory practice problems.

Links to binary, decimal and hexadecimal conversionLinks to the logical operators AND (^) OR (v) and NOT(~)

Here is the link to previous posts about conversion from decimal to binary and hexadecimal numbers systems, including practice problems.

Here are posts with practice problems for using bitwise logical operators.
 

Link to the story of four great mathematicians

Here is a link to the life stories of Archimedes, Newton, Euler and Gauss.

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