All posts by profstange

For Monday, February 6th

For Monday:

Proof for Feedback

Combinatorial Proof


  1. Notice that in our last class, we used the so-called “Addition Principle”, which is that if we break a count into exclusive (non-overlapping) cases, then we sum the results to obtain the full count.  For example, if we want to count subsets of size 3 or 4, then we can count subsets of size 3 and subsets of size 4 (two separate sub-problems), and add the results.
  2. Review the “combinatorial proof” we did at the end of last class (class notes in the “Materials Archive” tab above).   It’s an important example proof.  If you are unfamiliar with summation (sigma) notation, or just need a refresher, check out this explanation.
  3. Consider the binomial coefficients {{n}\choose{k}} and {{n}\choose{n-k}}.  Check for n=4 and k=1 that these are equal.  Using the formula for the binomial coefficient (last theorem in last class) to show that these are equal (for all 0 <= k <= n).  Use a counting argument (combinatorial proof) to show that these are equal (for all 0 <= k <= n).
  4. Do your proof for feedback.
  5. With remaining time, check out some practice problems from Hammack.  Section 3.3 #5, 7, 10; Section 3.4 #3, 9, 10, 14;  3.5, #3, 5, 8, 10, 17, and/or any others that interest you.  (A comment on Hammack:  I don’t teach counting the way he does.  I prefer not to subdivide counting into many “types” (e.g. lists, subsets) of problems because I believe that then the student ends up less adaptable to novel problems.  But you may still find it helpful to read Hammack.)

For Friday, Feb 3

For Friday:

Proof for Feedback:

Counting orderings


  1. Here are solutions to the last proof for feedback.
  2. How many ways can you order 5 students?
  3. How many ways can you pick 5 students from 27 students, ordered?  (Meaning the ordering of the students matters.)
  4. How many ways can you pick 5 students from 27 students, unordered?  (Meaning the ordering of the students doesn’t matter.)  To approach this, try to use the solution to the last problem, but adjust for overcounting.  Hint:  how many times did you count each solution?
  5. Fix n and m in general (assume m is greater than or equal to n, both positive integers).  How many ways can you pick n students from m students, unordered?
  6. Think about the rest of the problems on your counting worksheet.  We will discuss these in class, but get a head-start pondering them.  Also, an apology that the worksheet does not involve any non-binary people (there are lots of girls/boys and women/men).  I’ll have to add some more problems!
  7. Finally, I would appreciate if you would include a copy of your self-evaluation sheet with this submission.  Along with our first in-class test, I’d like to review participation in the class and this will help me see how people are doing.  If you have any comments on your participation/comfort/success/feelings about the class, please feel free to include these.  It will help me get a view of how things are going.

Due Wednesday, Feb 1

For Wed:

Proof for feedback:



  1. You have a proof test on Wednesday.  (See “Grading” tab for all the grading details.)  To study, I suggest reviewing example proofs (we’ve done several in class now) by trying them yourself and them comparing to the author’s solutions.  It’s ok for solutions to differ!  Here are some exercises from Hammack that may be good (and have solutions in the back):
        1. Use the definition of an “odd number” to be an integer of the form 2k+1 where k is another integer.  Show that if x is odd, then x^2 is odd.  The solution to this is written in great detail on pages 119-120 of Hammack.
        2. Chapter 8, Exercise 1.
        3. Chapter 7, Exercise 7. (This is an “if and only if” which means it goes both ways:  if A then B and if B then A.  Therefore we can state this as two theorems:  Suppose x, y \in \mathbb{R}.  If (x+y)^2 = x^2 + y^2 then x = 0 or y=0.  Furthermore, if x = 0 or y=0 then (x+y)^2 = x^2 + y^2.  So it’s actually two review problems in one!
        4. Examples 8.5, 8.6 on page 160.  This uses the symbol \mid for “divides” (Definition 4.4 on page 116).
  2. Announcements:
        1. I have posted office hours (“About” tab above)
        2. I have posted some solutions to Jan 27 daily post (“Materials Archive” tab above)
  3. Try Hammack Exercises from Section 3.2, particularly 1, 3, 7, 9 (answers are in the back), more if you want more practice with the multiplication principle.

Due Monday Jan 30th

Due Monday:

Proof for Feedback:

Cups proof


  1. Be aware that weekly wednesday in-class assessments begin next week.  You will write a proof in class.  Schedule is on the Grading tab above.
  2. Note:  I encourage collaboration on your daily posts!  Talking about math is great practice.  But make sure you are using a strict strategy to make sure you are writing your own solutions and not relying too much on other people (which I would consider copying).  For example, if you discuss a proof and work out a solution together, then finish that conversation before you start writing up your solution, and write up your own solution in silence, without using written notes, just from your own brain.  This is an important strategy to make sure you are internalizing what you learn.  If you get stuck, that’s good!  You’ve got an opportunity to struggle and learn.  It’s ok to go back to discussion, but give yourself some space between the collaborative phase and private writing phase.  Even better, take a 15 minute break between.  Be honest with yourself and only give yourself full credit on your self-evaluation sheet if you are using an effective strategy of this sort.
  3. Solutions to last proof problem are here (we discussed in class).
  4. Do your proof for feedback, which is about the cups problem we discussed in class on Friday.  If you forget what we talked about in class Friday, here’s a reminder/hint:  it turns out you can’t get from an even number of cups turned up to an odd number of cups turned up.  Why is that?  Try to write the most beautiful proof you can, whether using what I said in class or another method.
  5. Counting!  Here’s the counting sheet we started in class Friday.  First off, please write out all 12 possible burritos in the first problem (you can use abbreviations).  Make sure you really truly understand why the answer is 12 = 2 \cdot 2 \cdot 3.  Not sure?  Let’s chat!
  6. For general counting problems, my method is to imagine the process of creating each burrito (or triathlon team or bagged cat situation, etc.), and break it down into phases.  The formula for the answer is often a reflection of this process (just as the burrito problem can be thought of as progressing along the burrito bar making choices one-by-one).  Keep this in mind and do problems 1-6 on the worksheet (or as many as time permits).  Write out your reasoning, not just the answer.  You may wish to avoid doing problems 7 and above since I will leave time in class on Monday to work on these problems further (although I should never tell you not to think about mathematics!).
  7. Don’t forget to keep filling in your self-evaluation sheet.

Due Friday, January 27

For Friday:

Proof for Feedback:

Subset proof


  1. Please feel encouraged to work on daily posts together if you like.  You can find friends on discord or during group work in class.  The only rule is that when you are writing up solutions, you are writing your own understanding, not copying.  (The reason for this rule is that you will learn more.)
  2. Log into canvas and check out the feedback on your daily posts so far.  We have a grader who is giving feedback on the proof for feedback.  I’m also dropping in and making some comments here and there.
  3. While you are in canvas…. If you think you would like to make use of office hours regularly this semester, please fill out the when2meet link on canvas I can choose some times that will work for those who will attend (you can use your name, or you can use a fake name and let me know).  If you think you would like to make use of office hours only occasionally, I am happy to set a special time when needed, so you can skip filling out the form.
  4. Take a look at these example proof solutions to the proof from last daily post.  Think about how to improve your proofs.  If you have questions about your proofs or my solutions, please ask them on discord.  (I’m curious what you think of ChatGPT’s proof?  Discuss on discord with me if you have some thoughts on this.)
  5. Don’t forget your proof for feedback above.  You might choose to put it off until you read Section 1.3 (next item) about set containment.
  6. Read Section 1.3 of Hammack.  This is about subsets.  He gets a little wild, and most of his exercises are about sets of sets.  You should try as many of exercises A and B from that chapter as you like (if you want to try exercises C, you may need to dip into Section 1.2, which we haven’t done yet).
  7. Give an example of three sets A, B, and C, such that A \subseteq B, B \subseteq C, C \not\subseteq A, and C \subseteq B.  Optional: ask a friend to double-check your answer on discord.
  8. Give an example of two sets A, and B, such that A \in B, and A \subseteq B.  (Yes, you read that right.)
  9. Use set-builder notation to describe the set of all integers that are both perfect squares and perfect cubes.
  10. Use set-builder notation to describe the set of all sets with cardinality four.
  11. Use set-builder notation to describe the set of all integers which are divisible by seven.  Do this two essentially different ways.
  12. Return to the socks problem and the cups problem from the last daily post and see if you can solve them and give proofs.  We will discuss these in class, so give them a go!

Due Wed Jan 25

Proof for Feedback:

Task list:

  1. Here is the solution sheet from our last daily post (sudoku question).  Please take a look at the grader’s comments on your previous daily post submission and ask for any clarification if needed.
  2. Return to your textbook reading from last daily post and make a study sheet for this topic.  I have some advice on a study sheet in the “Study Sheets” tab at the top of this website.  Leave a little time to do the “proof for feedback” above, so it’s ok if you don’t finish this completely.
  3. Record three specific examples of “active reading” you can do in Section 1.1.  (We talked about active reading in class today.)
  4. Study Example 1.2 in Hammack, and make sure you understand it well.  Ask me questions if needed!
  5. Please do the “Proof for Feedback” above, which the grader will give feedback on.  Always place this at the beginning of what you hand in on canvas, please (it makes grading easier).
  6. Do as many more problems from Section 1.1 as you feel you need to learn the material.  If you have any “aha” moments, add them to your study sheet.
  7. Here are some fun puzzles/proofs for whatever time remains:
      1. Suppose you have 12 socks in a drawer.  They are all identical except for colour.  Four of them are green, four of them are red, and four of them are black.  (There’s no difference between a ‘right’ or ‘left’ sock.)  You pull socks out from the drawer at random, one by one.  The lights are off and you can’t see.  How many socks must you pull out before you are certain you have a pair?  Write up a full, carefully written proof of your answer.
      2. There are seven cups on a table, all standing upside down.  You are allowed to turn over any four of them in one move (you must turn over exactly four).  Is it possible to eventually have all the cups facing right-side up?  What if there are n cups?  What if you are allowed to turn over m of them?  Write up a proof of the best result you can prove.
      3. How many sudoku solutions are there to an nxn simplified sudoku (as defined in the last daily post).  Can you prove it?
  8. Please fill out your self-evaluation form for the first few days of the course (linked in the “Materials Archive” page above) and put a copy of it in today’s submission.  That will allow me to make sure everyone understand how the system works.

Due Mon Jan 23

Hi all!

Welcome to your first daily post.  Between every pair of lectures, there’s a daily post, to which you should devote 1 hour of concentrated study time.  This is a homework assignment, but you get full credit if and only if you put in 1 hour of worthwhile studying and useful effort toward the assignment.  Your grade does not depend on how much of the post you finish or how correct your answers are.  You will track your progress with this self-evaluation sheet.  You will get feedback and/or check your own solutions.

For each daily post, I will give you tasks and problems to do.  In isolated cases where it is clearly to the benefit of your studying, you may choose to skip problems or spend your 1 hour on what is most important for advancing your understanding of the course (please explain if so).  But in most cases, your task is to work through the tasks posted here, which will solidify the material of the last lecture and prepare you for the next lecture.  You do not need to finish everything: if you have put in 1 hour of honest, concentrated, struggling effort to understand (definitely no TV in the background, no copying other solutions, no mindless googling; you must be working your brain), then you get full credit.

When you are finished, you will hand in a record of your work to the daily dropbox on canvas.  Our grader will return feedback on one practice proof.  The grader appreciates if this is at the top/beginning of the work you hand in, so it is easy to find.  But you don’t need to work on it first (in fact, in many cases it makes sense to do the other tasks first, so read them all before beginning).

Your proof for feedback:

Sudoku Proof

Your tasks for today:

  1. Read all the syllabus content on our website (all the tabs at the top of the website).  This describes the course.  Some small changes have occurred in the lead-up to today’s lecture.
  2. Make sure you have figured out how to get on discord (instructions inside canvas).  Discord is our online communication tool; it pops up a notification on my phone when you ask a question and I try to be as available as possible.  You can also work with your peers on here.
  3. Write down your definitions and theorems from today’s activity (see instructions and polyhedron sheet), in the best language you feel you can.  This is a writing exercise.  How can you be as clear as possible?
  4. Do your practice proof for feedback (sudoku — there’s a link above in the post).  (Please also indicate if you’d be willing to have your solution shared for discussion in class, anonymously.)
  5. Read Chapter 1 of Hammack (link to PDF in upper left of website), up to the end of Section 1.1.  In class on Monday, we will discuss effective strategies for reading a mathematics textbook, but for now just try to find your own strategies to read effectively and actively, so that you are exercising your brain and engaging as you go.
  6. Record for yourself 3 strategies you discovered or used during the reading to help yourself learn and engage during the reading.
  7. With what time remains (if any), do exercises for Section 1.1.  You may choose which exercises to do.  Solutions to odd numbered problems are available in the back, and I won’t be writing up solutions to even-numbered ones (although I will check anything you ask me to check or discuss anything during office hour).  In class on Monday, we will discuss effective strategies for making use of these exercises in independent work.
  8. Put all your work together into a pdf file, with your proof for feedback at the top/front.  Hand this in on canvas (there’s a box marked with the due date 1/23).

Welcome to Math 2001, Section 4, Fall 2023!

Welcome all!

I’m really looking forward to meeting you!

To prepare for the course, please do the following:

  • Please log in to canvas and find the discord invitation.  Discord will be our real-time course communication tool.
  • In the menu items above, you will find the course syllabus content, so please visit each of those pages for further information about the course.
  • In particular, there are two official textbooks: