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?”

## All posts in category **Precalculus**

# My Favorite One-Liners: Part 110

*Posted by John Quintanilla on April 23, 2018*

https://meangreenmath.com/2018/04/23/my-favorite-one-liners-part-110/

# 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.

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)

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.

E1. How can technology (YouTube, Khan Academy [khanacademy.org], 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. Coolmath.com 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.

References:

A. Ltd, N. P. (n.d.). H2g2 The Hitchhiker’s Guide to the Galaxy: Earth Edition. Retrieved September 14, 2017, from https://h2g2.com/approved_entry/A2982567

B. H. (2011, April 04). Retrieved September 14, 2017, from https://www.youtube.com/watch?feature=youtu.be&v=mcIX_4w-nR0&app=desktop

C. E. (2013, January 13). Retrieved September 14, 2017, from https://www.youtube.com/watch/?v=1oSc-TpQqQI

D. The Quadratic Formula. (n.d.). Retrieved September 14, 2017, from http://www.coolmath.com/algebra/09-solving-quadratics/05-solving-quadratic-equations-formula-01

E. Worked example: quadratic formula (negative coefficients). (n.d.). Retrieved September 14, 2017, from https://www.khanacademy.org/math/algebra/quadratics/solving-quadratics-using-the-quadratic-formula/v/applying-the-quadratic-formula

*Posted by John Quintanilla on April 13, 2018*

https://meangreenmath.com/2018/04/13/engaging-students-the-quadratic-formula-4/

# Pascal’s Triangle and a British game show

So this happened on the popular British game show “University Challenge” on Monday, April 2. This game show pits teams of four from various British universities and is a severe test of the breadth and depth of their knowledge of many fields, including mathematics. A contestant’s response to one math question, asking for the seventh row of Pascal’s triangle, took the UK by storm this week (start at the 26:42 mark of the video below).

Twitter immediately went ablaze. Amazingly, a write-up of this encounter made it into the Times of London, one of the world’s most venerated newspapers (as opposed to the tawdry English tabloids). The above link requires a subscription; here’s a photo of page 13 from the April 4 edition:

I must admit that I’m a little amused by the amount of press that this little encounter received. When I was a kid, I memorized the first few rows of Pascal’s triangle simply from working with it so often, so when a family member told me about this story earlier this week, I knew the answer to the question instantly. I suspect that’s exactly what the contestant did here. (Whether I could have gotten the answer right under the pressure of a quiz show and a national TV audience, on the other hand, is another matter entirely.)

I have a theory as to why this appeared to be a mighty feat of mental arithmetic. The audience may have thought that he was adding the numbers quickly, but I’m guessing that the real purpose of the introductory clause “If 1,1 is the second row of Pascal’s triangle…” is to label that row as the second row instead of the first row (following the usual convention of starting the row and column counts with 0.)

*Posted by John Quintanilla on April 6, 2018*

https://meangreenmath.com/2018/04/06/pascals-triangle-and-a-british-game-show/

# Engaging students: Simplifying rational expressions

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 comes from my former student Peter Buhler. His topic, from Algebra II/Precalculus: simplifying rational expressions.

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

One activity that could be performed when introducing rational expressions is to demonstrate the reason for simplifying. Before teaching students to simplify, instead ask them to evaluate the expressions given various x values. As they struggle through the painstaking process of taking squares, distributing, multiplying, adding and subtracting as they attempt to evaluate the rational expression, take note of how long it may take the students. Then have several students share their method. Following the student sharing, show your efficient method that allows you to simplify the expression before beginning to evaluate.

This not only shows the students that it is quicker, but it often provides more accurate answers to the process that must be taken to “cancel” the terms and then evaluate. Students should be more willing to participate in the following lesson on simplification due to the desire to do less work. This could also be an opportunity to discuss why it is often helpful to look for “shortcuts” or tools that can be used to simplify long or tricky problems into something manageable, even by high school students.

B2. How does this topic extend what your students should have learned in previous courses?

This topic actually extends several previous topics seen in middle school mathematics. One of these topics is reducing fractions. This actually builds on the topic of finding the greatest common factor (GCF), which students learn in elementary school. To reduce a fraction, students find a GCF from both the top and bottom of the fraction, and then simply eliminate that factor leaving the expression in a simplified form. This could be utilized to introduce the idea of simplifying rational expressions, as students will likely be familiar with reducing fractions to their most simplified form.

This can also be applied to multiplying by fractions, as the GCF can be pulled out of the top and bottom of the fractions and simplified, making the multiplication of the fraction simpler. One last possible application could be in solving proportions, as students are typically taught to simplify the proportions before attempting to solve. The common theme in all of these is simplifying in order to make a problem easier and is a more efficient process for most students.

D2. How was this topic adopted by the mathematical community?

There are many advanced applications of simplifying rational expressions. One such function is the Pade approximant, which is an approximation of a rational function of a given order. It was created by Henri Pade in 1890 and has been used to model certain rational functions. While this is certainly an advanced rational expression, it still holds true as there is a polynomial on the top and the bottom, which can be factored and simplified.

Rational functions have also been commonly used to model certain equations in STEM field such as functions of wave patterns for molecular particles, various forces in physics, and other fields that take mathematical ideas and apply them to a science. As a teacher introducing the topic of simplifying these expressions, one could display various applications of these functions and how they are used in a day-to-day setting. Students should be able to see beyond the cut-and-dry steps of simplifying the expressions and understand the implications beyond what they are doing.

References:

http://blog.mrmeyer.com/2015/if-simplifying-rational-expressions-is-aspirin-then-how-do-you-create-the-headache/

https://en.wikipedia.org/wiki/Rational_function

*Posted by John Quintanilla on March 30, 2018*

https://meangreenmath.com/2018/03/30/engaging-students-simplifying-rational-expressions/

# I Have a Tan

*Posted by John Quintanilla on February 9, 2018*

https://meangreenmath.com/2018/02/09/i-have-a-tan/

# Babylonian trigonometry

An interesting article that I read on Babylonian mathematics.

*Posted by John Quintanilla on February 2, 2018*

https://meangreenmath.com/2018/02/02/babylonian-trigonometry/

# Engaging students: Dividing fractions

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 comes from my former student Kerryana Medlin. Her topic, from Pre-Algebra: dividing fractions.

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

One of the more practical uses of dividing fractions is cooking. Anybody who has baked in the past will know that sometimes one does not possess the proper measuring cup for the job and that they have to crunch some numbers. (This happens a lot when in college.)

The basic idea behind the activity is to ask the students to follow a recipe using a 1/3 cup measuring cup and a teaspoon. This will also allow them to practice dividing whole numbers by fractions, which strengthens to concept as well. They will be reminded that a whole number can be expressed as the number over one.

The ingredient list would be as follows:

Treats:

5-6 cups of rice cereal

1 cup of marshmallow fluff

1/3 cup of sprinkles

Buttercream:

½ cup unsalted butter

1 ½ cups powdered sugar

1 ½ teaspoons of vanilla extract

1-3 teaspoons of milk

They would be asked to figure out how many 1/3 cups each component would take. This would also help the students to use the skill of adding fractions (1 and ½ being 3/2) before dividing. The recipe would ultimately make rice cereal treats with icing on top (enough for the entire class). This is envisioned as an activity in which the students work either individually or in small groups to do the calculations and then come together as a class to provide answers and give me the proper amount of ingredients to put into the recipe.

**How does this topic extend what your students should have learned in previous courses?**

Dividing fractions involves prior knowledge from fractions, generally. If dividing by flipping the dividend and then multiplying the resulting two fractions, the student must use their knowledge of multiplication of fractions and inverses, assuming that they have learned anything about inverses at this point. If the student is taught to find the greatest common denominator first, then they will use their knowledge of greatest common denominators and basic division to find the quotient. They will also be reminded of the concept of whole numbers being expressed as fractions in this topic.

**How did people’s conception of this topic change over time? **

Originally, division of fractions would have been thought of in terms of practical use only and was likely conceptual since the symbolism of fractions was not the clearest. An example of fraction systems that were more difficult to comprehend, would be the Egyptian system, since they would add together unit fractions to represent non-unit fractions, unless it was fraction that had a repeating unit fraction, such as 2/7 = 1/7 + 1/7 (Weisstein). When symbols became clear, the division was done by taking the fractions, finding their common denominator, then dividing the numerators and denominators, leaving the quotient. The Babylonians mostly used the method of taking the inverse of the divisor and then multiplying by the dividend (O’Connor and Robertson, 2000). This is still a popular method. Today we can do either, but some believe that doing this operation algebraically might be better for students because thinking about division of fractions in only a practical sense will stifle their imagination (Ahia and Fredua-Kwarteng, 2006).

**References: **

Jamie. (2016). Birthday Marshmallow Cereal Treats*. My Baking Addiction*. Retrieved from

https://www.mybakingaddiction.com/birthday-marshmallow-cereal-treats/

Ahia, Francis and Fredua-Kwarteng, E.. (2006) Understanding Division of Fractions: An Alternative View.

Retrieved from http://files.eric.ed.gov/fulltext/ED493746.pdf

O’Connor, J. and Robertson E.. (2000). An overview of Babylonian mathematics. Retrieved from

http://www-history.mcs.st-and.ac.uk/HistTopics/Babylonian_mathematics.html

Weisstein, Eric. (n.d.). Egyptian Fraction. *MathWorld. *Retrieved from

http://mathworld.wolfram.com/EgyptianFraction.html

*Posted by John Quintanilla on November 27, 2017*

https://meangreenmath.com/2017/11/27/engaging-students-dividing-fractions-3/

# Day After Thanksgiving

*Posted by John Quintanilla on November 24, 2017*

https://meangreenmath.com/2017/11/24/day-after-thanksgiving/

# Engaging students: Ratios and rates of change

*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.

This student submission again comes from my former student Kelly Bui. Her topic, from Algebra: ratios and rates of change.

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

The activity I created would involve having the entire class make Rice Krispies treats as either groups or partners. The recipe I linked below calls for 6 cups of Rice Krispies, but for the sake of the activity, each table will receive 1 ½ cups of the cereal. Every table will receive the original recipe and determine how many large marshmallows they will need and how many tablespoons of butter they will need for the recipe to be modified to using only 1 ½ cups of cereal. This activity will allow students to use the ratios to convert measurements, such as 40 marshmallows / 6 cups of Rice Krispies. After a group finishes their calculations and finds the ratio of each ingredient in respect to the amount of cereal, they can begin making their Rice Krispies treats. To extend this to a project, the Rice Krispies Treats activity can be done in class and students will be assigned to find a recipe which involves either using the ratio to create a smaller or larger serving of the recipe.

How can this topic be used in your students’ future courses in mathematics or sciences?

Ratios are introduced in middle school when we compare a part to another other part or part to whole. Students also begin to grasp that 12 inches / 1 foot is a relationship between two quantities because there are 12 inches per foot. We also see the use of ratios in high school chemistry when converting units. A simple ratio we first learn is that density is the ratio of mass to volume. This can then be extended, for example, when students begin to solve for the number of moles of an element given its mass in grams. Before teaching a chemistry class that 1 mole = 6.022×10^23, instructors could begin with simple conversions of the length of a state in miles and converting that length into inches. Once students understand the process and the concept that we are taking one unit and converting it to another unit, it will be easier to apply it to more complex situations in chemistry.

As a class, to get into the process of using ratios to convert units, the students can make their own conversion ratios with different objects to model this relationship. For instance, 4 fire extinguishers are the length 1 lab table and 8 lab tables are the length 1 school bus, and based on these ratios, students must find the length of a school bus in terms of fire extinguishers. This activity will allow the students to use objects they see every day and create a relationship among them.

How has this topic appeared in pop culture (movies, TV, current music, video games, etc.)?

Shark Tank is a show which involves 5 or 6 sharks (investors) and entrepreneurs that go into the “shark tank” to pitch their ideas seeking one or more partners who will invest in their business. Most entrepreneurs seek a money amount for an amount of stake in their company or business. If the entrepreneurs are lucky, they will get a deal with one or more of the sharks. In the video below, Aaron Krause pitches his product, the “Scrub Daddy” in which he asks for a $100,000 investment for a 10% equity in his company. We see the topic of ratios appear in this business-related show because 10% equity of $100,000 means he values his company at $1 million, in other words our ratio is 10% / $100,000. This ratio can be used to find the value of the company at 100%. In addition, the sharks also like to know the breakdown of the cost per unit. In this video, Mr. Krause states that it takes $1.00 to create a scrub daddy and he sells it for $2.80 wholesale. This gives the sharks the knowledge of how much they would earn for 1 Scrub Daddy. Given the sharks are willing to negotiate, like in the video, Lori gets 20% equity of the company. For each $2.80 / 1 Scrub Daddy, she will earn $0.56.

*Posted by John Quintanilla on November 13, 2017*

https://meangreenmath.com/2017/11/13/engaging-students-ratios-and-rates-of-change-2/

# Stay Focused

From Kirk Cousins, quarterback of the Washington Redskins:

Sometimes our guests ask why I have this hanging above my desk. It’s an old high school math quiz when I didn’t study at all and got a C+… just a subtle reminder to me of the importance of preparation. If I don’t prepare I get C’s!

*Posted by John Quintanilla on August 23, 2017*

https://meangreenmath.com/2017/08/23/stay-focused/