Solutions to Exercises in Math Textbooks

I read a very thought-provoking blog post on the pros and cons of having answers in the back of math textbooks. The article and comments on the article are worth reading.

Rings and polynomials


5 Ways to go Beyond Recitation

Most students will encounter recitation in a math class during their academic career. How can math professors make the experience more meaningful? MAA Teaching Tidbits blog has 5 ways educators can enhance the student experience during recitation.

  1. Focus on getting students to do the work instead of doing it for them.
  2. Incorporate group work into your sessions.
  3. Get students to communicate what they understand to each other and to the class.
  4. Have students relate mathematics to their own experiences.
  5. Cultivate an environment where failure is ok and experimentation is encouraged.

Full article:

My Favorite One-Liners: Part 114

In this series, I’m compiling some of the quips and one-liners that I’ll use with my students to hopefully make my lessons more memorable for them.

I’ll use today’s one-liner whena step that’s usually necessary in a calculation isn’t needed for a particular example. For example, consider the following problem from probability:

Let X be uniformly distributed on \{-1,0,1\}. Find \hbox{Cov}(X,X^2).

The first step is to write \hbox{Cov}(X,X^2) = E(X \cdot X^2) - E(X) E(X^2) = E(X^3) - E(X) E(X^2). Then we start computing the expectations. To begin,

E(X) = (-1) \cdot \displaystyle \frac{1}{3} + 0 \cdot \displaystyle \frac{1}{3} + 1 \cdot \displaystyle \frac{1}{3} = 0.

Ordinarily, the next step would be computing E(X^2). However, this computation is unnecessary since E(X^2) will be multiplied by E(X), which we just showed was equal to 0. While I might calculate E(X^2) if I thought my class needed the extra practice with computing expectations, the answer will not ultimately affect the final answer. Hence my one-liner:

To paraphrase the great philosopher The Rock, it doesn’t matter what E(X^2) is.

P.S. This example illustrates that the covariance of two dependent random variables (X and X^2) can be zero. If two random variables are independent, then the covariance must be zero. But the reverse implication is false.

My Favorite One-Liners: Part 113

I tried a new wisecrack when teaching my students about Euler’s formula. It worked gloriously.


My Favorite One-Liners: Part 112

This was also the story of my childhood.


My Favorite One-Liners: Part 111

I tried a new wise-crack in class recently, and it was a rousing success. My math majors had trouble recalling basic facts about tests for convergent and divergent series, and so I projected onto the front screen the Official Repository of all Knowledge ( and searched for “divergent series” to “help” them recall their prior knowledge.

Worked like a charm.

My Favorite One-Liners: Part 110

I overheard the following terrific one-liner recently. A teacher was about to begin a lecture on exponential growth. His opening question to engage his students: “What does your bank account have to do with bacteria… other than they both might be really tiny?”

My Favorite One-Liners: Part 109

I tried a new joke in class recently; it worked gloriously.

I wrote on the board a mathematical conjecture that has yet to be proven or disproven. To emphasize that nobody knows the answer yet despite centuries of effort, I told the class, “If you figure this out, call me and call me collect,” writing my office phone number on the board.

To complete the joke, I said, “Yeah, this is crazy. So here’s my number…”

I thoroughly enjoyed my students’ coruscating groans before I could complete the punch line.

Engaging students: The quadratic formula

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission again comes from my former student Megan Termini. Her topic, from Algebra: the quadratic formula.

green line

D1. What interesting things can you say about the people who contributed to the discovery and/or the development of this topic?

The Quadratic Formula came about when the Egyptians, Chinese, and Babylonian engineers came across a problem. The engineers knew how to calculate the area of squares, and eventually knew how to calculate the area of other shapes like rectangles and T-shapes. The problem was that customers would provide them an area for them to design a floor plan. They were unable to calculate the length of the sides of certain shapes, and therefore were not able to design these floor plans. So, the Egyptians, instead of learning operations and formulas, they created a table with area for all possible sides and shapes of squares and rectangles. Then the Babylonians came in and found a better way to solve the area problem, known as “completing the square”. The Babylonians had the base 60 system while the Chinese used an abacus for them to double check their results. The Pythagoras’, Euclid, Brahmagupta, and Al-Khwarizmi came later and all contributed to what we know as the Quadratic Formula now. (Reference A)

green line

A2. How could you as a teacher create an activity or project that involves your topic?

A great activity that involves the Quadratic Formula is having the students work in groups and come up with a way to remember the formula. It could be a song, a rhyme, a story, anything! I have found a few examples of students and teachers who have created some cool and fun ways of remembering the Quadratic Formula. One that is commonly known is the Quadratic Formula sung to the tune of “Pop Goes the Weasel” (Reference B). It is a very catchy song and it would be able to help students in remembering the formula, not just for this class but also in other classes as they further their education. Now, having the students create their own way of remembering it will benefit them even more because it is coming from them. An example is from a high school class in Georgia. They created a parody of Adele’s “Rolling in the Deep” to help remember the Quadratic Formula (Reference C). It’s fun, it gets everyone involved, it engaging, and it helps student remember the Quadratic Formula.

green line

E1. How can technology (YouTube, Khan Academy [], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic?

Technology is a great way of engaging students in today’s world. Many students now have cell phones or the school provides laptops to be used during class. is a great website for students to use to learn about the quadratic formula and great way to practice using it. They show you why the formula works and why it is important to know it because not all quadratic equations are easy to factor. There are a few examples on there and then they give the students a chance to practice some random problems and check to see if they got the right answer. This website would be good for student in and out of the classroom (Reference D). Khan Academy is another great way for students to learn how to use the quadratic formula. They have many videos on how to use the formula, proof of the formula, and different examples and practices of applying the quadratic formula (Reference E). Students today love when they get to use their phones in class or computers, so technology is a great way to engage students in learning and applying the quadratic formula.



A. Ltd, N. P. (n.d.). H2g2 The Hitchhiker’s Guide to the Galaxy: Earth Edition. Retrieved September 14, 2017, from
B. H. (2011, April 04). Retrieved September 14, 2017, from
C. E. (2013, January 13). Retrieved September 14, 2017, from
D. The Quadratic Formula. (n.d.). Retrieved September 14, 2017, from
E. Worked example: quadratic formula (negative coefficients). (n.d.). Retrieved September 14, 2017, from