Feb 12

It’s a perennial problem in math classes: Its review day for the test and the professor asks “are there any questions?” only to be met with silence.

Of course there are many reasons students might not ask questions.  Probably the main ones are:

They haven’t started reviewing yet, so they don’t know what questions they have.

They have mostly mastered the material and don’t need to ask questions (this is probably a very small minority).

They aren’t comfortable to ask a question.


Of course there are many reasons why a student might not be comfortable enough to ask a question: thinking that asking is a sign of weakness, worry their their question is dumb, fear that they are dumb or others will think they are, because the professor is an authority figure, stereotype threat,  a past traumatic experience asking a question in a class, or difficulty formulating a coherent question to name a few.

Whatever the reason for the lack of questions, students need to ask questions in order to succeed mathematically.  Therefore, in an attempt to avoid the awkward staring at each other while the students don’t ask any questions, I decided to try a completely different review activity in my Business Calculus class this week.  The activity was called “Find someone who.”  I got the idea from the book Choosing to See: A Framework for Equity in the Math Classroom by Pamela Seda and Kyndall Brown.  They learned about it from the book Kagan Cooperative Learning.

In the “Find someone who” activity, the professor prepares a worksheet formatted like a bingo card/ tic-tac-toe board.  The students are asked to move around the room asking their classmates to explain a problem to them.  Once the student has understood how to solve the problem, the explainer signs that square on their worksheet.

This activity can be adapted for any topic.  Mine was the review for the test on the basics of derivatives (including limit intuition and secant lines).  You can access the PDF  here: FindSomeoneWhoTest1Review .  To tantalize you, I’ve also included a picture of the worksheet.

A picture of the handout for the activity described in this post

Handout for Find Someone Who


The energy in the room during this activity was fantastic!  At first most students only wanted to talk to the people next to them, but with some gentle prodding and some affirmations, the students got into it.  I got to hear fun snippets of conversation like one student explaining the chain rule to another, “and then you stuff it back in…”.

As a review for the test, this activity allowed students to solve problems (as most reviews do), but also allowed them to explain concepts and procedures which helps the explainer to learn and remember the material.

Besides serving as a review for the test, the activity served a few other functions.  Students got/had to ask questions in a non-threatening environment–everyone had to ask questions, so there was no stigma attached to asking.  Hopefully, this will also have a longer lasting effect of normalizing asking for help and increasing student willingness to ask questions of me and each other.  On a related note, the activity helps to “include others as experts” which is one of the pillar’s of Seda and Brown’s titular framework for equity. As the name suggests, this pillar is all about establishing that the teacher is not the one and only source of knowledge–other students and the student’s own self also bring valuable knowledge.

At the end of the class period, I asked students to write me a couple of sentences about how they thought the activity went.  I tried to encourage honest responses by being completely honest with them. I said, “I’ve never done an activity like this before; I read it in a book, so I want to know how it worked from your perspective.”  Most students were enthusiastic in their positive review of the activity.  The students also identified many benefits of the activity, such as:

Getting to know classmates

Teaching classmates helped me remember the steps

Greater confidence for the upcoming test

Hearing the steps out loud and in different ways

Experiencing the value of learning from peers

Awareness of weak spots in knowledge of the material, which will help with studying


I was delighted that the students were extremely engaged and that they were able to articulate some of the many benefits of the activity for themselves. This activity powerfully demonstrated that our classroom is a safe space for asking questions.



Pamela Seda and Kyndall Brown, Choosing to See: A Framework for Equity in the Math Classroom (San Diego, CA: Dave Burgess Consulting Inc., 2021) https://www.daveburgessconsulting.com/books/choosing-to-see/

Spencer Kagan and Miguel Kagan, Kagan Cooperative Learning (San Clemente, CA: Kagan Publishing, 2009)  https://www.kaganonline.com/catalog/cooperative_learning.php#BKCLW


May 18

This is a problem I normally give at the end of semester in calculus 1 as part of a worksheet composed entirely of application problems from various sections of the book (Some of the other problems from this worksheet appear else where in the blog.  Links to them are at the bottom of this post).  This kind of mixed review can be very powerful because the students have to figure out which of the many techniques they have learned apply to the situation.  This is a skill that employers recruiting at our university’s career fairs have told me the are seeking.  I include the references on the worksheet that I hand out, not just the questions.

A variation on this problem (using function composition and similar triangles instead of related rates) is suitable for use in a precalculus course.  The precalculus variation will be appearing as [4].


Here’s the problem:

(Hint: In this problem you need to be very careful with units). The Maldives is an example of a poor low-lying country which will be hit soon and hard by the effects of global climate change including sea level rise [3], [2].  The Maldives is located south-southwest of India.  It consists of approximately 1,190 islands and is the lwest country in the world with a maximum elevation of 2.4 m.  The Maldives has a total area of 298 km squared and a coastline of 644km [2].  Since the actual shape of the Maldives is complicated, we will use a simpler shape to compute with and assume this is a good stand-in for the real islands.  We will pretend the nation is one triangular prism as pictured below.  The width of the base is 1 km and the length of the base is 320 km.  The height of the prism is 2.4m.

a) Convert all units to kilometers and label the picture with the measurements in kilometers.  What is the map area of the prism in kilometers squared?   By map area, in this problem, we mean the area of the rectangle you would see by looking down on the prism from above.


b) At what rate is the area changing when the elevation is 2meters if the sea level is rising at a rate of 4.3325 mm per year?

c) The Intergovernmental Panel on Climate Change (IPCC) predicts between 0.28 and 0.98 m of sea level rise by 2100 [1].  Some reputable sources predict up to 2 m of sea level rise  by 2100 [3].  What is the annual rate of sea level rise in mm per year for each of these estimates?  What is the rate of change of the area for each of these estimates?



[1] J. A. Church, et. al. “Sea Level Change“. In: Climate Change 2013: The Physical Science Basis. Contribution of Working Group I to the Fifth Assessment Report of the Intergovernmental Panel on Climate Change. Cambridge University Press, Cambridge, United Kingdom and New York, NY, USA. 2013.
Full text available at http://www.climatechange2013.org/images/report/WG1AR5\_Chapter13\_FINAL.pdf

[2] The CIA World Factbook, https://www.cia.gov/library/publications/the-world-factbook/geos/mv.html, accessed on May 6, 2015.

[3] K. Dow and T. E. Downing, The Atlas of Climate Change: Mapping the world’s greatest challenge, 3rd ed.
University of California Press, Berkeley,  2011.

[4] Archey, Dawn. “Sea Level Change and Function Composition.” In Mathematics and Social Justice Modules for the Classroom.  Eds. Karaali, Gizem and Khadjavi, Lily. to appear.


Other problems in the Calculus 1 mixed review worksheet included:




Sep 22

Here are two problems I used in my calculus 1 class, just after teaching them the definition of the derivative and discussing average and instantaneous rates of change.


  1. My car has a digital read out with the average fuel economy.  I used to have a car which also had the instantaneous fuel economy.
    What is the difference between these two fuel economies?  Under what circumstances is each more useful?  How can I use the average fuel economy feature to approximate my instantaneous fuel economy?  (Your answer should be  2-5 complete sentences).
  2. Read the article at
    http://www.slate.com/blogs/bad_astronomy/2015/09/03/ice_loss_greenland_and_antarctica_lost_5_trillion_tons_since_1992.html .
    You can use this QR code to access it

    1. What is the average rate of change of the mass of Antarctica’s ice?
    2. What is the average rate of change of the mass of Greenland’s ice?
    3. Looking at the graphs, do you think it is reasonable to talk about the instantaneous rate of change for these quantities?  Why or why not?
      Then answer the following questions. Give all answers in complete sentences with units.


Apr 13

The FoxTrot Strip from November 6, 2005 contained a “numerical word search”

You can find the strip here.

There are seven clues which are arithmetic problems, but there is also the derivative of 43981 x , the definite integral of x^3 from 0 to 16, and the sum from k equals 0 to 47 of k squared.    I asked my calculus students to perform all these calculations.

Apr 13

I asked my calculus students to read this xkcd strip from http://xkcd.com/872/

Fairy Tales

Then I asked them:

a) What is $\lim_{x\rightarrow \infty} (x )$?
b) Explain what might have been weird about the $\lim_{x\rightarrow \infty} (x )$ little pigs.

Mar 23

Randall Munroe drew a lovely comic strip graphing the effects of proximity to a cat on the human intellect:


Cat Proximity

(You can view it here: http://xkcd.com/231/).

In differential calculus I asked students to describe the graphs using their knowledge of calculus.  Statements about both the first and second derivative are expected.

In Intermediate Algebra here at UDM we study qualitative graphs, so I also asked my Algebra students to describe the graph.  These students have to say things like the intelligence of the human is decreasing, but at an ever faster rate.

One caution with this comic: some students do not know what “inanity” means, so I use this problem on the review sheet rather than the exam.

Mar 18

In this FoxTrot strip Paige has been cramming for her math final and thus is answering all questions in mathematical notation.


(Its the May 22, 2005 strip, if the link doesn’t work.)

In precalculus I asked the students:

1) What temperature will it be tomorrow?  Is it going to be hot tomorrow?

(The answer is sin^{-1}(1) =90 degrees, so it will be hot).

2) What does the girl (Paige) want for snack?

(The answer is 3.14159265359…, so she wants pie).

In calculus of sequences and series, you could also ask what the discount was in the fourth panel, which requires computing the sum of an infinite geometric series.

In an algebra class one could ask what time Paige went to bed (\sqrt{121}= 11:00) or What is on TV? (4!=24).

Mar 16

In this FoxTrot strip Jason is “helping” Paige do her homework without even looking in the book.  He claims the answer to question 36 is lim_{x \rightarrow \infty} \frac{x+2}{x-2}.

I asked my students to evaluate this limit, but think this wasn’t a very good use of the comic strip.  If you have a better idea, please comment or email me.


(If the link doesn’t work try going to  http://www.amureprints.com/reprints/sphinx_search , entering the keyword “question” and checking the FoxTrot box.  It was the first hit when I used this search.  The fact that it is from September 13, 2001    may also help you find it.)

Feb 26

Here’s a question I asked my students about limits using http://www.xkcd.com/103/:

Moral Relativity

1) Using the graph in the comic strip, find

\lim_{speed  \rightarrow c} Rationalization.

2) Write one or two complete sentences explaining what the limit I just had you compute has to do with the text of the comic strip.

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