All posts by profstange


Hi all!

First, thank you all for a wonderful semester.  I really enjoyed having you all as students.  Thank you for your interesting presentations and great questions and ideas throughout the semester.  I learned a lot.

A few final reminders:

  • Your grades are now complete for the semester in canvas.  The number-to-letter conversion is now set to match my expected conversion.  I may adjust this slightly, but I have promised not to make it harsher, only kinder.
  • The final proof quiz or other left over work can be picked up in my office hours.
  • Office hours are expanded:
    • Thursday 12:30-2 pm
    • Friday noon-2 pm
    • no office hours on the weekend
    • you can earn one badge this week in office hours (come prepared)
  • The final exam will resemble 1/2 badges quizzes (short answer questions) and 1/2 proof quizzes.  Studying from these materials throughout semester is your best preparation.
  • Here’s what we covered in depth in Hammack:
    • Sets: 1.1-1.6
    • Logic: 2.1-2.10
    • Counting: 3.1-3.4
    • Direct Proof: 4.1-4.5
    • Contrapositive Proof: 5.1-5.3
    • Proof by Contradiction: 6.1-6.4
    • Non-Conditional Proof: 7.1, 7.3-7.4
    • Proofs involving Sets: 8.1-8.3
    • Disproof: 9.1-9.3
    • Induction: 10.1-10.3
    • Relations: 11.1-11.5
    • Functions: 12.1-12.6
    • Cardinality of Sets: 13.1-13.3
    • The sections not listed above are also included in the class in a weak sense, but we did not study them in detail.  In other words, a few points on the final exam may be given for very complete reading of the textbook, including these sections, or I may draw novel examples from these sections (sometimes synthesis badges were inspired by these sections).
    • We also covered some topics that don’t appear in Hammack, most notably graph theory, and lots of interesting examples for induction, etc.  So please also review your class notes and handouts.
    • Finally, since I was asked, nothing about elliptic curves will be on the exam.
  • The final exam is in our regular room.  This information should show in your and also via the registrar’s list of times.  It is:
    • Section 001:  Saturday May 5, 4:30 pm.
    • Section 002:  Sunday May 6, 1:30 pm.
  • I wish you a wonderful summer.  Check out the links in Fun, and be in touch down the road to let me know where you’ve ended up.

Wednesday, May 2nd

Wednesday is our last class!

Monday, April 30th

For Monday:

  • Revisit the Combinatorial Proof Worksheet and try to do the next two problems (so, 1-4 at least).  Solutions to 1 and 2 are available here.
  • Study counting by doing Hammack’s exercises in Sections 3.1-3.3, completing any you haven’t yet done.
  • Campus FCQs had some error and questions changed on Tuesday.  Please log in and make sure your FCQs are the way you want them:  I greatly appreciate your feedback.
  • Monday there will be a badges quiz (same badges as last time).
  • Wednesday there will be a final proof quiz on Combinatorial Proof.
  • Next week you can earn a badge in office hour.  There will be office hours at least 1-2 pm on Wed and Thur.  I can expand these as needed; please email.
  • I was missing a few presentations in canvas and these are being added in.  There is still an opportunity to present if you wish.  See Grading for info on how grading is computed.  Currently canvas is not correctly computing grades.
  • The Association for Women in Mathematics, CU Boulder Chapter, is hosting a study session Thursday, May 3, 5-7pm in ECCR 135. This study session is primarily aimed at undergraduates, graduate students are welcome to come, enjoy some free pizza and snacks, and talk to/get help from other graduate students in preparing for your final exams.

Friday, April 27th

Reminders and announcements:

  • Office hours this week are Thursday 12:30-2:30 and Friday 1-2.  Remember you can earn a badge in office hour, by preparing as described in this old post.
  • The last badges quiz is Monday.  Plan which ones you will study for.  All information is updated in canvas.
  • Groupwork is due Friday.  I’m posting some example combinatorial proofs here.
  • You can present on Friday if you wish.  I have posted my records of presentations/scribing/assignments for your participation grade on canvas.  Out of concern for sufficient remaining time for presentations, I’m dropping one of the 6 requirements for the grade.  See Grading.
  • See the last post for grading changes and quiz schedule changes, as well as exam times.
  • Please complete your online FCQs.  I read these very carefully and reflect on all your feedback.  The department and university also use these to evaluate faculty.  They really do have an impact, so I very much appreciate your thoughtful responses.

For Friday:

Wednesday, April 25th

Late note:

My apologies, but I have to move today’s (Wednesday’s) 1-2 pm office hour to Friday 1-2 pm.  I am also expanding Thursday’s office hour.  Please email me if you need to set up a separate time.  Just to confirm, this week I have office hour:

Thursday 12:30-2:30 pm
Friday 1-2 pm

My apologies about this.

For Wednesday:

  • I will administer online FCQs on Wednesday.  I’ll set aside 10 minutes at the end of class for you to log in and fill out FCQs on your phone, tablet or computer (I will leave the room).  FCQs are very important and I strongly encourage you to fill them out.  I use them to improve my teaching, and my department and university use them to evaluate me as a faculty member.  If you miss class, you can also fill them out anytime at
  • Note the grading changes below.  I have been careful to make sure these do not penalize you for my miscalculation; please contact me if needed.
  • Revisit the homework from the last daily post if you have not finished it.
  • Read Hammack, Section 3.3.  Try Exercises 1-5.
  • Recall the definition 3.2 on page 74 of Hammack, of “n choose k”.  Try to write a combinatorial proof that (n choose k) is the same as (n choose (n-k)).  Don’t write this as an algebraic proof (it is possible to prove it by writing out both formulas as in Fact 3.3).  Instead, explain why both quantities count the same thing, and must therefore be equal.
  • There are a lot of useful things in the last daily post:  review it (exam schedule, info on badges in office hour, consider presenting this week, etc.)

Important Grading Changes:  I miscalculated and didn’t realize we are missing one last friday.  Here are the adjustments:

  • There will be only one more proof quiz, on Wednesday May 2nd.  Your grade will be best 7 of your 13 efforts, overall.
  • This week’s group assignment is the last assignment.  For participation, your 5% presentation/participation/groupwork grade in the class will be based on 6 items instead of 7:  if you present at least 2 times, and scribe at least 2 times for a group, and hand in at least 2 decent efforts at the individual works on canvas.  (Your grade will otherwise be proportional to how many of the 6 tasks above you have done decently.)
  • These adjustments are now changed in the Grading page.

Monday, April 23rd

Have a great weekend, all!

For Monday:

  • Return to the Counting Worksheet and work on 1-12 if you aren’t sure about your answers yet.  We’ll finish taking this up (and badges quiz questions) in class Monday.
  • To help with Counting, check out Hammack, Examples 3.2 and 3.3.  You can read Chapter 3 as needed to work up to these examples.
  • Try the following problems from Hammack:  Section 3.1, 2,4,5,7; and Section 3.2, 3,7,8.
  • Be aware this coming week is the last groupwork week.  It will be posted soon.
  • The Monday badges quiz will include everything except Sets, Logic and Proofs.  In other words, it will be Counting, Relations, Functions, Modular Arithmetic and Synthesis.
  • Next week’s Friday Proof Quiz will be a Combinatorial Proof (count something two different ways to prove a formula).
  • We will focus on Counting and Combinatorial Proof all next week.
  • You may earn one badge in office hour next week (any badges except Synthesis).  This is an excellent way to help your grade!  Please prepare as described in the original post.  Tip:  choose a badge you are having trouble getting full credit on, or is no longer on the quiz.  You can check which badges you have on canvas.  You can schedule a time (email me) if the regular hours aren’t good for you.
  • Here’s your final exam schedule.

Friday, April 20th

For Friday:

  • Individual Work is due on canvas.
  • We will do some examples of proofs with sets and/or take up the individual work.
  • The proof quiz will be something to do with sets.  You can practice with the individual work due Friday, and with Chapter 8, Examples 8.8, 8.9, 8.11, 8.12, and Exercises 6, 7, 21, 22.

Wednesday, April 18th

For Wednesday:

  • Please attempt the counting worksheet, 1-12.  (Problem 13 is a challenge problem, you can sleep on it if you want.)  We’ll take this up in class.
  • Reminders:
    • You can earn any badge in office hour (preparation required: details here).
    • Individual work due Friday (under Resources/Groupwork).  Hand in on canvas.
    • I believe I have a list of those interested in presenting one more time.  If you’re willing to do this this coming Friday (this week), please let me know.  I may take 1-2 presentations to make sure we aren’t too overloaded on the last day.  Remember: you can present on anything mathematically relevant, including groupwork and individualwork you haven’t presented before.

Monday, April 16th

For Monday:

  • Please read Hammack, Section 11, up to the end of 11.3.  This is review of all we have covered about relations.
  • Please read Hammack, Section 11.4, The Integers Modulo n.  Do Exercises 1–8.
  • Badges quiz:  Last chance for Logic and Proofs badges.  Any badges can be earned in office hour.  I will also add Modular Arithmetic badge, and Counting Badges (these are the only ones remaining).  The Counting Badges are certainly premature:  we may or may not start that topic on Monday.  But if you want to try them out, Chapter 3 of Hammack is the place to start.  There are only three more badges quizzes.
  • Meanwhile, I suggest you use the weekend to review anything that needs reviewing to bring your badges score up.  Hammack has exercises with solutions in the back which are very good preparation — just look up the relevant topic.
  • Be aware there will be an individual work due Friday; it has been posted.
  • The grading change to the 5% participation score (as described in this post) has been updated in the “Grading” tab above.

Friday April 13th

For Friday:

  • Presentation day.  You can present any proofs that you wish, including proofs from group work or individual work.  The requirement is only that it be mathematically interesting and pertinent to the class.  Even if you have presented twice, you can volunteer, but those needing presentation credit will be given priority.  If you want to present something, it helps if you email me so I can get the computer set up ahead of time with the correct things (but it’s not strictly required).
  • I will now open office hour badges to any badges you want to earn, whether they are active on the in-class quizzes or not.  One per week.
  • I will ask you to prove something with functions in the abstract.  To study, I suggest you read the handouts about function having an inverse if and only if it is bijective (with slots, and filled).  The type of problems I may ask are similar to that, so study those carefully.