Assignments for Mathematics for Liberal Arts

Day Date Activity Assignment (reading assignments are due the day they appear on the calendar)

Mon 1/14 Welcome Read the preface and chapter #1 for Wednesday

Wed 1/16 Primitive Counting

Explain how the children's game of "musical chairs" works, using the notion of one-to-one correspondence. How else is the idea of one-to-one correspondence used in life? We can sometimes use it to avoid counting!
Do a little web research on the Ishango bone. Where did it come from? How long ago was it made? What was it made from?
Use the method of "primitive counting" described in class to write the representation of

89
96

What numbers are represented as

1,0,0,1,0,1,1
1,1,1,0,0,1

What special sorts of numbers are represented as

1
1,0
1,0,0
1,0,0,0
1,0,0,0,0
etc.?

Fri 1/18 Learn to perform the "counting card trick" that was described in class. Try to figure out why it works....

Mon 1/21 MLK Day

Wed 1/23 Prime and Composite Numbers Chapters 2 (and also 25, but 2 is more important). Homework:

Using pennies, rocks, turnips, or whatever, discover the first composite number that can be expressed as three distinctly different rectangles (not counting the "line of pennies" rectangle).
The number 12 can be expressed as two distinctly different rectangles (rotations of a rectangle don't count as "distinctly different"). What is the first triangular number that can be expressed as exactly two distinctly different rectangles?
What are the next three numbers, after 12, that can be expressed as exactly two distinctly different rectangles?
We talked about squares, triangles, rectangles: how should we think about those numbers called "cubes" in terms of pennies? What are some of the first cubes, and what would they look like represented using pennies?
What are the hexagonal numbers? (A regular hexagon has six equal sides, and six equal angles.) Can you see a hexagon in the image below? Pick out a single penny near the middle, and look at its neighbors:

Fri 1/25

Use trees to write the prime factorizations of the following:

15015
1763

You could play the "3 card trick" with only 6 or 7 cards, instead of 52 cards -- but it's not so impressive, is it? Work the card trick backwards, starting from 3 cards. Do you see that you could have started with either 6 or 7 cards? In either case, the three cards would have been numbered 2,4,6. Continuing to work backwards, what would come in the next step, depending on whether you began 6 or 7 cards?

Mon 1/28 Babylonian/Mayan Math Chapter 6

Wed 1/30 Read this on-line intro to Mayan Math

Write the following numbers (written in our system) in the Babylonian and Mayan systems:

1763
15015
3661

How would you tap out your name in Morse code?

Fri 2/1

Mon 2/4 Bases

Write the following numbers in each of the bases 2, 8, and 16:

42
63
1024
1984
2013

Try to figure out how Lewis Carroll was using unusual bases to do the strange math in chapter two of Alice in Wonderland? On about the third page of chapter two (after the graphic of "Giant Alice watching Rabbit run away"), Alice starts speaking out some bizarre equations:
"Let me see: four times five is twelve, and four times six is thirteen, and four times seven is -- oh dear! I shall never get to twenty at that rate!"
Find some reasonable bases (you'll need several!) to make the "calculations" work out....

Wed 2/6 binary (base 2)

Use primitive counting to write the string for the following numbers of sheep:

37
97
163

Write each number in binary:

37
97
163

How do you write each of these Mersenne Primes in binary?

3=2^2-1
7=2^3-1
31=2^5-1
127=2^7-1
2^{57,885,161}-1 (don't try this one on a piece of paper!:)

How many sheep? Here are the primitive counting strings, you priest you:

11011
101011
100100001

Check out these other base videos:

Schoolhouse Rock strikes again: Little Twelve Toes (guess the base!)
Tom Lehrer strikes, too: New Math -- a lesson in base-8 arithmetic

Fri 2/8 Egyptian Multiplication

Please visit the webpage at this link and read it over for next time.
Demonstrate Egyptian multiplication by multiplying:

23*79
81*123

Mon 2/11 Egyptian Math

Try another Egyptian multiplication, and make sure that you're ready for the upcoming test:

183*47

Demonstrate Egyptian division by dividing:

9/4
13/7

Read Chapter 5, p. 29.

Wed 2/13 Demonstrate Egyptian division by dividing the following using both methods, if you can! That is, using the multiplication table backwards, and using the unit fraction table. One method may be easier than the other!

9/4
13/7
13/17

Fri 2/15 Review

Mon 2/18 Exam I

Wed 2/20 Greek Math Chapters 4 and 12

Fri 2/22 Pythagorean Theorem Finish off the worksheet that was distributed in class:

If the longest side is longer than the Pythagorean theorem would suggest, then the triangle is obtuse;
If the longest side is shorter than the Pythagorean theorem would suggest, then the triangle is acute.
Also see if you can follow our proof of the Pythagorean theorem, as presented on the last page.

Mon 2/25 Irrational numbers

If a right triangle has legs of length 1 and x, what is the length of the hypotenuse?
On a sunny and windy day, a student flies a kite on the level college green just to relax. Her kite takes off and soars. She lets all 150 feet of the string out and attracts a crowd of onlookers. A spectator 90 feet away from the student notices that the kite is directly above him. Unlike a real kite this math-question kite has the string going in a perfectly straight line from the student to the kite. How high is the kite above the ground?
A sailboat has a tall mast. The backstay (the heavy steel cable that attaches the top of the mast to the back, or stern, of the sailboat) is made of 130 feet of cable. The base of the mast is 50 feet from the stern of the boat. How tall is the mast?
(Problems from The Heart of Mathematics)
Read the pages (also from The Heart of Mathematics) at this link: http://www.nku.edu/~longa/classes/mat115_resources/docs/Heart/IrrationalHeart/. Then show that the ${\sqrt{5}}$ is irrational.

Wed 2/27 Indians, Chinese, and Pascal

Write the next line in the Chinese version of Pascal's Triangle.
Write the next line in the Jewish/Arabic/Hindu version of Pascal's Triangle.
Pascal's triangle contains numbers like the triangular and the tetrahedral numbers. What would be call the numbers after the tetrahedral numbers? Explore these numbers: 1, 5, 15, 35, 70, 126, ....

Fri 3/1 Fibonacci and his Nim Play Nim with someone, and see if you can always win:

Try it with 18 counters. Is it better to go first or second?
Try it with 21 counters. Is it better to go first or second?
Try it with 28 counters. Is it better to go first or second?
Try it with 34 counters. Is it better to go first or second?

Read the biography of Leonardo Pisano Fibonacci. You'll see references to some of the mathematics we've already done in this course!

Mon 3/4 Fibonacci Nim Same assignment as before, only now you know the strategy. See if you can always win!

Wed 3/6 Graphs Chapters 3, 24, and 26
Homework (This one is due: Monday, 3/18 -- a good job on this will replace a quiz): I want you to visit this website and turn your own image into a Fibonacci spiral. Choose an appropriate image, then create and then email me a copy of your own spiral image. Make one that everyone will enjoy, as I will create a gallery of images. There may also be prizes for the winners.
Make sure that you send me the final image, and not just a link (that won't do anything!). Right click on the image, and save it to your computer -- then paste the image into your email. That's the easiest way, I think.
One more thing: the image won't last on the server for very long, so download it right away.

Fri 3/8

Mon 3/11 Spring Break

Wed 3/13 Spring Break

Fri 3/15 Spring Break

Mon 3/18 Please read this short piece on graphs; the other readings for this unit have examples of graphs, which I will point out after our general discussion of the history of graphs.

Wed 3/20 More Graphs: Example from our readings Homework:

Draw the complete graph with 8 vertices. How many edges are there?
Draw all the distinctly different simple graphs with five vertices. There are a lot! How many? Use symmetry as much as you can to avoid double counting them. Can you see any patterns in how they're created?

Fri 3/22 Exam II

Mon 3/25 More Graphs

Check out this page and relate it to Euler's Konigsberg problem. Try some of the puzzles yourself.
Write down the update equations (see p. 195 of our text) for this "toy web": You might rename the vertices 1=x, 2=y, 3=z, 4=w, for example, to make writing the equations easier (so that they look more like the equations in the text).

Wed 3/27 Algebra Chapters 7 and 10; here's a bit of homework:

Tchapo was five times older than Thad at one time, then, four years later, Tchapo was only three times older. How many years older is Tchapo than Thad?
If you add the age of a man to the age of his wife, the result is 91. He is now twice as old as she was when he was as old as she is now. How old is the man and his wife?
Is the answer from the previous problem an example of an "icky relationship", according to chapter 7?

Fri 3/29 More Algebra

Mon 4/1

What's your non-icky relationship range?
At what age do the two ages x and y equal each other?
What range do you get for a 12-year old? In what sense does that make sense?

Wed 4/3 Geometry and Topology Chapter 27

Twist a band in two different ways:

four times, and
five times,
and cut the band in "two". What objects result? Describe them exactly, specifying their number of half twists, how they're connected, etc.

Describe exactly what you get if you cut a thrice-twisted band in thirds (as we did in class to the Mobius band, which is only once-twisted). You need a pretty long band to do this!

Make sure that you can make the two distinctly different Mobius bands (they're mirror images of each other). Symmetry is a very important mathematical idea.
Relate the following logo to twisted bands (e.g. Mobius bands):

Is the following recycling symbol correct (i.e. Mobius) or not? How can you tell? The true recycling symbol is not symmetric under rotation: rotate it, and it looks different.

Find two examples of the recycling symbol on nationally known products, one Mobius and the other not Mobius. Which products know what they're doing?
Think about what kind of math would look good (and play well) on the Mobius Music Box. I'll take suggestions, and then either compose some myself, or let you try....

Fri 4/5 Probability Chapter 23; Please read this chapter from Tony Crilly's "50 Mathematical Ideas You Really Need to Know". It gives the basics of probability.

Mon 4/8 I've added another probability reading on the assignment page: this chapter about the birthday problem from Tony Crilly's "50 Mathematical Ideas You Really Need to Know".) Hopefully this will help you understand this remarkable result!

What is the probability that 5 randomly chosen people will have a common birthday?
What is the probability of getting exactly one even die on a roll in a fair game of craps?
What is the probability of tossing three heads in a row on three tosses of a fair coin?
What is the probability that two throws of an ordinary fair die will give two different results? that three tosses will give three different results? that six tosses will give six different results? that seven tosses will give seven different results?

Wed 4/10 Conditional Probability Do the "plant problem" from chapter 23 again, but change the numbers: assume that

${P(D|W')=.75}$
${P(D|W)=.14}$
${P(W')=.40}$
That is, the plant's not ailing as badly, it seems (probabilities of death have gone down); but your friend is more unreliable than ever! Find the probability that your plant dies.

Fri 4/12 More Conditional Probability Now retry the breast cancer problem: suppose that

${P(B)=.012}$
${P(T|B)=.95}$
${P(T|B')=.03}$

Now solve for these probabilities:

${P(B|T)}$
${P(T)}$
${P(T\cap B)}$

Mon 4/15 Statistics Chapter 22

Wed 4/17

Fri 4/19 Exam III

Mon 4/22 Fractals and Infinity Chapters 8, 16, and 30

Wed 4/24 Fractals
Please read this chapter from Tony Crilly's "50 Mathematical Ideas You Really Need to Know". It gives the basics of fractals.
Homework: give the following a try, to prepare for the statistics quiz on Friday: Distribution practice
Also try my funhouse mirror generator: make your own funhouse mirror image, using my web interface and your own image, and I'll post them to our website.

Fri 4/26 Fractals Try these homework problems.
Create your own examples of

a stick fractal, and
an area fractal.
You'll need to

Define the simple rule (e.g. how does a stick turn into other sticks?)
Apply the rule at least twice, so that we can begin to see "the world within the world"

Mon 4/29 Infinity

Wed 5/1 More infinity (can that be?) Homework:

What is the power set of the four Beatles, ${\{John, Paul, George, Ringo\}}$ ?
What does the 6th line of Pascal's triangle tell us about the kinds of sets in the power set of a set with five elements?
Which is bigger: all integers (including the negative ones) or the even natural numbers?
Which is bigger: all prime numbers, or all Fibonacci numbers?

Fri 5/3 Logo Day

Website maintained by Andy Long. Comments appreciated.

Day	Date	Activity	Assignment (reading assignments are due the day they appear on the calendar)
Mon	1/14	Welcome	Read the preface and chapter #1 for Wednesday
Wed	1/16	Primitive Counting	Explain how the children's game of "musical chairs" works, using the notion of one-to-one correspondence. How else is the idea of one-to-one correspondence used in life? We can sometimes use it to avoid counting! Do a little web research on the Ishango bone. Where did it come from? How long ago was it made? What was it made from? Use the method of "primitive counting" described in class to write the representation of 89 96 What numbers are represented as 1,0,0,1,0,1,1 1,1,1,0,0,1 What special sorts of numbers are represented as 1 1,0 1,0,0 1,0,0,0 1,0,0,0,0 etc.?
Fri	1/18		Learn to perform the "counting card trick" that was described in class. Try to figure out why it works....
Mon	1/21	MLK Day
Wed	1/23	Prime and Composite Numbers	Chapters 2 (and also 25, but 2 is more important). Homework: Using pennies, rocks, turnips, or whatever, discover the first composite number that can be expressed as three distinctly different rectangles (not counting the "line of pennies" rectangle). The number 12 can be expressed as two distinctly different rectangles (rotations of a rectangle don't count as "distinctly different"). What is the first triangular number that can be expressed as exactly two distinctly different rectangles? What are the next three numbers, after 12, that can be expressed as exactly two distinctly different rectangles? We talked about squares, triangles, rectangles: how should we think about those numbers called "cubes" in terms of pennies? What are some of the first cubes, and what would they look like represented using pennies? What are the hexagonal numbers? (A regular hexagon has six equal sides, and six equal angles.) Can you see a hexagon in the image below? Pick out a single penny near the middle, and look at its neighbors:
Fri	1/25		Use trees to write the prime factorizations of the following: 15015 1763 You could play the "3 card trick" with only 6 or 7 cards, instead of 52 cards -- but it's not so impressive, is it? Work the card trick backwards, starting from 3 cards. Do you see that you could have started with either 6 or 7 cards? In either case, the three cards would have been numbered 2,4,6. Continuing to work backwards, what would come in the next step, depending on whether you began 6 or 7 cards?
Mon	1/28	Babylonian/Mayan Math	Chapter 6
Wed	1/30		Read this on-line intro to Mayan Math Write the following numbers (written in our system) in the Babylonian and Mayan systems: 1763 15015 3661 How would you tap out your name in Morse code?
Fri	2/1
Mon	2/4	Bases	Write the following numbers in each of the bases 2, 8, and 16: 42 63 1024 1984 2013 Try to figure out how Lewis Carroll was using unusual bases to do the strange math in chapter two of Alice in Wonderland? On about the third page of chapter two (after the graphic of "Giant Alice watching Rabbit run away"), Alice starts speaking out some bizarre equations: "Let me see: four times five is twelve, and four times six is thirteen, and four times seven is -- oh dear! I shall never get to twenty at that rate!" Find some reasonable bases (you'll need several!) to make the "calculations" work out....
Wed	2/6	binary (base 2)	Use primitive counting to write the string for the following numbers of sheep: 37 97 163 Write each number in binary: 37 97 163 How do you write each of these Mersenne Primes in binary? 3=2^2-1 7=2^3-1 31=2^5-1 127=2^7-1 2^{57,885,161}-1 (don't try this one on a piece of paper!:) How many sheep? Here are the primitive counting strings, you priest you: 11011 101011 100100001 Check out these other base videos: Schoolhouse Rock strikes again: Little Twelve Toes (guess the base!) Tom Lehrer strikes, too: New Math -- a lesson in base-8 arithmetic
Fri	2/8	Egyptian Multiplication	Please visit the webpage at this link and read it over for next time. Demonstrate Egyptian multiplication by multiplying: 2379 81123
Mon	2/11	Egyptian Math	Try another Egyptian multiplication, and make sure that you're ready for the upcoming test: 183*47 Demonstrate Egyptian division by dividing: 9/4 13/7 Read Chapter 5, p. 29.
Wed	2/13		Demonstrate Egyptian division by dividing the following using both methods, if you can! That is, using the multiplication table backwards, and using the unit fraction table. One method may be easier than the other! 9/4 13/7 13/17
Fri	2/15	Review
Mon	2/18	Exam I
Wed	2/20	Greek Math	Chapters 4 and 12
Fri	2/22	Pythagorean Theorem	Finish off the worksheet that was distributed in class: If the longest side is longer than the Pythagorean theorem would suggest, then the triangle is obtuse; If the longest side is shorter than the Pythagorean theorem would suggest, then the triangle is acute. Also see if you can follow our proof of the Pythagorean theorem, as presented on the last page.
Mon	2/25	Irrational numbers	If a right triangle has legs of length 1 and x, what is the length of the hypotenuse? On a sunny and windy day, a student flies a kite on the level college green just to relax. Her kite takes off and soars. She lets all 150 feet of the string out and attracts a crowd of onlookers. A spectator 90 feet away from the student notices that the kite is directly above him. Unlike a real kite this math-question kite has the string going in a perfectly straight line from the student to the kite. How high is the kite above the ground? A sailboat has a tall mast. The backstay (the heavy steel cable that attaches the top of the mast to the back, or stern, of the sailboat) is made of 130 feet of cable. The base of the mast is 50 feet from the stern of the boat. How tall is the mast? (Problems from The Heart of Mathematics) Read the pages (also from The Heart of Mathematics) at this link: http://www.nku.edu/~longa/classes/mat115_resources/docs/Heart/IrrationalHeart/. Then show that the ${\sqrt{5}}$ is irrational.
Wed	2/27	Indians, Chinese, and Pascal	Write the next line in the Chinese version of Pascal's Triangle. Write the next line in the Jewish/Arabic/Hindu version of Pascal's Triangle. Pascal's triangle contains numbers like the triangular and the tetrahedral numbers. What would be call the numbers after the tetrahedral numbers? Explore these numbers: 1, 5, 15, 35, 70, 126, ....
Fri	3/1	Fibonacci and his Nim	Play Nim with someone, and see if you can always win: Try it with 18 counters. Is it better to go first or second? Try it with 21 counters. Is it better to go first or second? Try it with 28 counters. Is it better to go first or second? Try it with 34 counters. Is it better to go first or second? Read the biography of Leonardo Pisano Fibonacci. You'll see references to some of the mathematics we've already done in this course!
Mon	3/4	Fibonacci Nim	Same assignment as before, only now you know the strategy. See if you can always win!
Wed	3/6	Graphs	Chapters 3, 24, and 26 Homework (This one is due: Monday, 3/18 -- a good job on this will replace a quiz): I want you to visit this website and turn your own image into a Fibonacci spiral. Choose an appropriate image, then create and then email me a copy of your own spiral image. Make one that everyone will enjoy, as I will create a gallery of images. There may also be prizes for the winners. Make sure that you send me the final image, and not just a link (that won't do anything!). Right click on the image, and save it to your computer -- then paste the image into your email. That's the easiest way, I think. One more thing: the image won't last on the server for very long, so download it right away.
Fri	3/8
Mon	3/11	Spring Break
Wed	3/13	Spring Break
Fri	3/15	Spring Break
Mon	3/18		Please read this short piece on graphs; the other readings for this unit have examples of graphs, which I will point out after our general discussion of the history of graphs.
Wed	3/20	More Graphs: Example from our readings	Homework: Draw the complete graph with 8 vertices. How many edges are there? Draw all the distinctly different simple graphs with five vertices. There are a lot! How many? Use symmetry as much as you can to avoid double counting them. Can you see any patterns in how they're created?
Fri	3/22	Exam II
Mon	3/25	More Graphs	Check out this page and relate it to Euler's Konigsberg problem. Try some of the puzzles yourself. Write down the update equations (see p. 195 of our text) for this "toy web": You might rename the vertices 1=x, 2=y, 3=z, 4=w, for example, to make writing the equations easier (so that they look more like the equations in the text).
Wed	3/27	Algebra	Chapters 7 and 10; here's a bit of homework: Tchapo was five times older than Thad at one time, then, four years later, Tchapo was only three times older. How many years older is Tchapo than Thad? If you add the age of a man to the age of his wife, the result is 91. He is now twice as old as she was when he was as old as she is now. How old is the man and his wife? Is the answer from the previous problem an example of an "icky relationship", according to chapter 7?
Fri	3/29	More Algebra
Mon	4/1		What's your non-icky relationship range? At what age do the two ages x and y equal each other? What range do you get for a 12-year old? In what sense does that make sense?
Wed	4/3	Geometry and Topology	Chapter 27 Twist a band in two different ways: four times, and five times, and cut the band in "two". What objects result? Describe them exactly, specifying their number of half twists, how they're connected, etc. Describe exactly what you get if you cut a thrice-twisted band in thirds (as we did in class to the Mobius band, which is only once-twisted). You need a pretty long band to do this! Make sure that you can make the two distinctly different Mobius bands (they're mirror images of each other). Symmetry is a very important mathematical idea. Relate the following logo to twisted bands (e.g. Mobius bands): Is the following recycling symbol correct (i.e. Mobius) or not? How can you tell? The true recycling symbol is not symmetric under rotation: rotate it, and it looks different. Find two examples of the recycling symbol on nationally known products, one Mobius and the other not Mobius. Which products know what they're doing? Think about what kind of math would look good (and play well) on the Mobius Music Box. I'll take suggestions, and then either compose some myself, or let you try....
Fri	4/5	Probability	Chapter 23; Please read this chapter from Tony Crilly's "50 Mathematical Ideas You Really Need to Know". It gives the basics of probability.
Mon	4/8	I've added another probability reading on the assignment page: this chapter about the birthday problem from Tony Crilly's "50 Mathematical Ideas You Really Need to Know".) Hopefully this will help you understand this remarkable result!	What is the probability that 5 randomly chosen people will have a common birthday? What is the probability of getting exactly one even die on a roll in a fair game of craps? What is the probability of tossing three heads in a row on three tosses of a fair coin? What is the probability that two throws of an ordinary fair die will give two different results? that three tosses will give three different results? that six tosses will give six different results? that seven tosses will give seven different results?
Wed	4/10	Conditional Probability	Do the "plant problem" from chapter 23 again, but change the numbers: assume that ${P(D\|W')=.75}$ ${P(D\|W)=.14}$ ${P(W')=.40}$ That is, the plant's not ailing as badly, it seems (probabilities of death have gone down); but your friend is more unreliable than ever! Find the probability that your plant dies.
Fri	4/12	More Conditional Probability	Now retry the breast cancer problem: suppose that ${P(B)=.012}$ ${P(T\|B)=.95}$ ${P(T\|B')=.03}$ Now solve for these probabilities: ${P(B\|T)}$ ${P(T)}$ ${P(T\cap B)}$
Mon	4/15	Statistics	Chapter 22
Wed	4/17
Fri	4/19	Exam III
Mon	4/22	Fractals and Infinity	Chapters 8, 16, and 30
Wed	4/24	Fractals Please read this chapter from Tony Crilly's "50 Mathematical Ideas You Really Need to Know". It gives the basics of fractals.	Homework: give the following a try, to prepare for the statistics quiz on Friday: Distribution practice Also try my funhouse mirror generator: make your own funhouse mirror image, using my web interface and your own image, and I'll post them to our website.
Fri	4/26	Fractals	Try these homework problems. Create your own examples of a stick fractal, and an area fractal. You'll need to Define the simple rule (e.g. how does a stick turn into other sticks?) Apply the rule at least twice, so that we can begin to see "the world within the world"
Mon	4/29	Infinity
Wed	5/1	More infinity (can that be?)	Homework: What is the power set of the four Beatles, ${\{John, Paul, George, Ringo\}}$ ? What does the 6th line of Pascal's triangle tell us about the kinds of sets in the power set of a set with five elements? Which is bigger: all integers (including the negative ones) or the even natural numbers? Which is bigger: all prime numbers, or all Fibonacci numbers?
Fri	5/3	Logo Day