It turns out that Matlab can be used to optimally place ornaments on a Christmas tree: https://github.com/tsangiotis/christmasSpirit

# Matlab and Optimal Placement of Christmas Ornaments

*Posted by John Quintanilla on December 3, 2016*

https://meangreenmath.com/2016/12/03/matlab-and-optimal-placement-of-christmas-ornaments/

# Mathematical Present Wrapping (Part 2)

Yesterday’s post of mathematically wrapping presents was tongue in cheek. This one is elegant and a nice application of principles from geometry.

Source: http://lifehacker.com/the-mathematical-way-to-wrap-presents-of-various-shapes-1748323319

*Posted by John Quintanilla on December 2, 2016*

https://meangreenmath.com/2016/12/02/mathematical-present-wrapping-part-2/

# Mathematical Present Wrapping (Part 1)

*Posted by John Quintanilla on December 1, 2016*

https://meangreenmath.com/2016/12/01/mathematical-present-wrapping-part-1/

# Engaging students: Finding points on the coordinate plane

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 Lucy Grimmett. Her topic, from Pre-Algebra: finding points on the coordinate plane.

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

As a hands-on learner, I love activities that require me to be up and moving. I have always heard about the human coordinate plane. The teacher creates a life size coordinate plane on the floor of their classroom. The teacher would label the x and y axis and place contact paper on the floor marking coordinate points. Students would walk into class and write their name on a point. Later they would return to their point and the activity begins. The teacher or another student would stand at the origin and ask how she would get to “insert name” point. Students would discover the coordinates of their spot or point. They would then return to their seats, after the activity, and complete a journal entry in their notebook. The teacher would then have notes and discussion with the class. During note taking time I would tell student about the elevator idea. The idea is that you must walk to the elevator before you can go up or down. This is a great reminder for students to remember what order the plot the numbers in, and what direction.

See this link for more detailed instructions: http://www.cpalms.org/Public/PreviewResourceLesson/Preview/49870

How can this topic be used in your students’ future courses in mathematics and science?

Plotting points and finding points on a coordinate plane is a very necessary topic for many courses. If you cannot find points adequately on a coordinate plane then students will not be able to graph equations of lines through plugging in. They will not be able to find slope of a line. They will not be able to graph vectors in both mathematics and physics. There are endless needs for plotting points on a graph. When you move into polar coordinates in mathematics, it is necessary to know how to plot points as well. Polar coordinates then lead into trigonometry, which leads into graphing equations of trig functions, which leads into calculus, which also requires graphing equations and using the graph to visualize where the tangent line would be on the coordinate plane. Being familiar with coordinate points and how to find them and plot them is going to be a lesson student’s take with them forever. Even if they are an art major, they will still used coordinates.

How has this topic appeared in the news?

The idea of plotting points is used in graphs, charts, diagrams and many other visual aids in the news. The news is constantly using these visual aids to make data look more dramatic. Another idea is during the weather segment, meteorologist have to mark the temperatures on the city. Now it is very computerized, but back in the day, they had to use latitude and longitude to find the city, which is very similar to plotting points on a coordinate plane. Another idea is stock news. This news is typically on websites rather then broadcasted segments, however, stocks are a great idea of graphing points. You merely plot the day or time and the rate at which the stock was increasing or decreasing. The points are then connected with lines to show how it the stock goes up and down. This is a good idea when students start to learn about slope of lines as well.

*Posted by John Quintanilla on November 30, 2016*

https://meangreenmath.com/2016/11/30/engaging-students-finding-points-on-the-coordinate-plane-3/

# Engaging students: Order of operations

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 Lisa Sun. Her topic, from Pre-Algebra: order of operations.

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

Given that my students have knowledge on the topic of Order of Operations, I will provide them a project where they must apply their knowledge and present it in front of their peers. Students will each receive a number from me and they must create a mathematical problem, an equation, using all of PEMDAS (Parentheses, Exponents, Multiplication, Division, Addition, and Subtraction).

Students will then present individually in front of their peers at the board. The presenter’s role is to be the teacher. To have the ability to clearly vocalize his/her thought process to achieve their given number with the use of PEMDAS. As each student presents, the audience will be following along on a sheet of paper where they must also solve the equation that the presenter created. This paper must be turned in along with their own project to document that they were paying attention. The audience’s role is to be the grader. To make sure that the presenter’s use of PEMDAS was correct to achieve the number that was given to them. If the presenter’s use of PEMDAS is incorrect, I will select an audience member to explain. The presenter will then have to come present their project again to me before or after school so that I can make sure there is no misconception regarding the Order of Operations.

To help motivate the students’ to be precise with their project, I would state that if all students were able to display their use of PEMDAS correctly, everyone would receive 5 extra points on the upcoming test. I believe that this project would be great for students to strengthen their knowledge on Order of Operations. As they are taking up on their roles as the grader, they are physically and mentally reinforcing their knowledge by solving problems after problems. As the teacher, they are verbally reinforcing their knowledge.

__How has this topic appeared in pop culture?__

Figure 1: Pokémon Center Lego

Pokémon Go is the craze among society today and I believe it would be fitting to engage the class with both Pokémon and Legos. I would present this to the class, preferably one that is physically available to the class and ask the following questions:

- When building this Lego figure, do you think procedures need to be followed sequentially?
- What happens if they are not? (Display to the class what the Lego figure would look like).

I would discuss why doing things in order is important tying it with Orders of Operations. Display a problem with Orders of Operations but solve it by not following PEMDAS (Parentheses, Exponents, Multiplication, Division, Addition, and Subtraction) and state that the solution comes out to be incorrect. Similar to how the Lego that was built in the wrong order didn’t match up with the picture on the box.

I believe YouTube can be a great learning tool in the classroom when it comes to engaging students. People of all demographics post helpful tools on this site that are so easily relatable to students today. Below is a video of a PEMDAS rap song.

I will be playing this PEMDAS rap song as students are walking into class to quickly engage the students. Once class has officially started, I would play this video again as students are reading the lyrics and following along the two examples the video provided. This video is to aid the students to remember the Orders of Operations by the use of PEMDAS (Parentheses, Exponents, Multiplication, Division, Addition, and Subtraction). To engage the students even more, I would have the students sing along the chorus. “Parentheses first, exponents next, multiplication and division in the same step. Addition and subtraction, if you got the nerve, from left to right, first come first serve”. Hopefully, this song will be catchy enough for the students to have it be stuck in their head for a while.

References:

http://www.purplemath.com/modules/orderops.htm

http://www.pbslearningmedia.org/resource/mgbh.math.oa.ooo/order-of-operations/

*Posted by John Quintanilla on November 29, 2016*

https://meangreenmath.com/2016/11/29/engaging-students-order-of-operations-5/

# Engaging students: Fractions, percents, and decimals

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 Kim Hong. Her topic, from Pre-Algebra: fractions, percents, and decimals.

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

I think making the students create a foldable, a short and quick project, would be a good and concrete activity for teaching fractions, decimals, and percents. Each flap is a topic. There is a definition and example. On the back of the foldable the students could create a table going between fractions, decimals and percents with many “harder” values.

The foldable is portable and quick, and can be a helpful and quick resource.

The students can also draw pictures inside the flaps. E.g A pizza and its slices to show fractions.

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

This topic can be used in a students’ future course when they come across proportions and rates. They could see proportions when it appears in physics such a changes in time and speed. They could see rates of change when it appears in calculus involving derivatives. These values are factions that can be changed to decimals and percents because everything is a part of a whole.

Also, fractions, which are numbers over a whole, are the same as the term rational quantities. Rational quantities are numbers that can be written as a ratio that is a fraction. There is a subset of the Reals that are called the Rationals. In advanced logic and math courses, students will be able to work with this subset of the Reals.

How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic? *Note*: It’s not enough to say “such-and-such is a great website”; you need to explain in some detail why it’s a great website.

I found this really awesome website the students could play around with for the first minutes of class to get their juices flowing. Basically the objective of the game is to group the equal values in circles. There is a check answer option as well.

It starts off very simple with very easy mental math and then with each level, the difficulty increases.

http://www.mathplayground.com/Decention/Decention.html

*Posted by John Quintanilla on November 28, 2016*

https://meangreenmath.com/2016/11/28/engaging-students-fractions-percents-and-decimals-2/

# Engaging students: Reducing fractions to lowest terms

*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 comes from my former student Madison duPont. Her topic, from Pre-Algebra: reducing fractions to lowest terms.

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

Reducing fractions to lowest terms can be applied to future mathematics topics such as ratios and proportions, and scientific topics such as chemistry or physics. Ratios can be represented as fractions and are not typically reduced to lowest terms because they represent relationships of two subjects using numbers. Being able to reduce these ratios can help students better identify the underlying relationship and apply this relationship to other aspects of the math problem, such as problems using unit price or map scales. Proportions relate to the concept of reducing fractions to lowest terms when using cross-multiplication. Having both sides of the proportion reduced to lowest terms makes the cross-multiplication much easier to compute and derive a final reduced answer. Chemistry uses fractions reduced to lowest terms with topics, like stoichiometry, that use potentially small and large numbers in several ratios that are multiplied together to obtain a final converted and reduced answer. Physics often uses ratio-like formulas and problems that are applied to real-world scenarios, which typically require fractions reduced to lowest terms because answers like miles per one hour are the goal. All of these topics use concepts of reducing fractions to lowest terms to more easily accomplish problems using a series of fractional computations, or to get an answer that is in terms of a single unit or most reduced so that it makes sense to real-world application.

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

This topic extends previously learned topics such as concepts of unique prime factorizations, greatest common divisor, manipulating fractions, and multiplication facts. The concept of unique prime factorizations greatly aids students in finding the greatest common divisor, which is used to find the greatest factor of the value of both the numerator and denominator. Next, manipulation of fractions is used to properly divide the numerator and denominator by the greatest common divisor. This process of dividing both parts of the fraction utilizes multiplication facts as well to determine what the answer to the division problem on both the top and bottom of the fraction would be. These previously learned concepts are all subtle and important applications when reducing fractions to lowest terms.

How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic? *Note*: It’s not enough to say “such-and-such is a great website”; you need to explain in some detail why it’s a great website.

This video reminded me of many students that I have tutored or encountered in classrooms that were determined that a calculator was all they needed when doing math. Applied to reducing fractions to lowest terms, this video is extremely relevant in displaying that technology cannot be the only source of intelligence when thinking mathematically. Reducing fractions with extremely large numbers or numbers that do not have well-known factors can seem exhausting or impossible. Punching several factors of the numerator and denominator into a calculator attempting to reduce numbers with each common factor, and then not being sure of whether the fraction appearing on their screen is truly in the most reduced form surely indicates the technology is not the only way of solving the problem. Many students hop on a procedural escalator when beginning varying types of problems (in addition to reducing fractions to lowest terms) using memorized steps, punching calculator buttons, feeling comfortable, until suddenly—there is a horribly unattractive fraction halting their progress. This is when using mathematical problem solving skills such as reducing the numerator and denominator by the greatest common divisor or checking to see that the numerator and denominator are relatively prime becomes pertinent. Using these conceptual skills can save someone that is stuck waiting for a calculator to do the work for them, or that has given up on finishing a problem because it seems impossible or difficult, from thinking they are incapable of working out a problem efficiently and successfully. This video highlights the importance of being capable of knowing when it is time to take the effort to climb the stairs to reach your destination.

References:

“Stuck on an Escalator” Video link:

https://www.youtube.com/watch?v=VrSUe_m19FY

found via Google video search

*Posted by John Quintanilla on November 27, 2016*

https://meangreenmath.com/2016/11/27/engaging-students-reducing-fractions-to-lowest-terms/

# Engaging students: Standard Deviation

*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 comes from my former student Jillian Greene. Her topic, from statistics: standard deviations.

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

An activity that I’ve seen presented to introduce the idea of standard deviation requires students to explore the information given to them before actually being taught the math behind standard deviation. As the students settle into their seats, prompt them to work with their shoulder partner and help to measure the width of their left thumbnail (or length of their pointer finger, width of their hand, etc.) and write it on a sticky note. Once the data is collected, the students will calculate the mean of all of the measurements. The mean is then written on the board in the center, and the students are asked to go and stack their post-it notes in either the center if they are perfectly the mean, or on the right or left if it’s bigger or smaller, respectively. Have them find the mean of the distances of each measurement from the mean. When they discover this should be zero, have them discuss with each other and then in the big group what that means. If time provides, it might even be fun to ask deeper understanding questions like what would happen if everyone last half of their thumbnail, or what if just Student A’s thumbnail tripled in size. This will provide a meaningful sequitur into the sometimes confusing world of standard deviation and distances from the mean.

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

http://www.dailymotion.com/video/x3lc0rx

This is a full episode of Everybody Loves Raymond but the clip in reference starts at about 8:45 and lasts a minute or so.

This clip shows a scenario where the couple, Ray and Deborah, is comparing their scores on an IQ test (a very common use for standard deviation). Deborah comments on how her score is very close to Ray’s, being only 15 points higher. The brother that proctored the exam corrected her by saying that 15 points is a standard deviation higher and puts her in a “whole new class” of genius. Have students discuss and explain what it means for Deborah to be one standard deviation higher. Use the information given in the episode (100 is average, 115 is one standard deviation higher) to construct the bell curve for IQ scores. Then use the bell curve to introduce percentiles. Since Ray is the average, center-of-the-bell score, then he is in the 50th percentile. The students can then attempt to discover on their own (or with a group) what percentile Deborah’s score puts her in.

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

Standard deviation is a topic that pervades almost all sciences. In biology classes, students are asked to student the weather and climate of various habitats. In differentiating between the two, one must look at the overall picture. If the student is presented with the information that place A and place B both have average temperatures of 60 degrees, this information might not be good to take as face value. Place A might have a range from 40 to 80 degrees throughout the year while place B might range from 0 to 100 and then have one or two extremely hot outliers that even the average out to 60. Looking at not only the skew of the bell curve, but also what the standard deviation is for each place, might save a student from forgetting to bring a fan to hypothetical place B, or writing that that the climate of that place is cool year round. In addition to biology, standard deviation is a very necessary operation in psychology, which is a very statistics-based science. This can easily be seen in representing IQ scores how we found earlier!

*Posted by John Quintanilla on November 26, 2016*

https://meangreenmath.com/2016/11/26/engaging-students-standard-deviation/

# Engaging students: Square roots

*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 Jessica Martinez. Her topic, from Algebra: square roots.

How has this topic appeared in the news?

There is a (sort of) holiday for square root days; sort of because square root days only come 9 time every century and this year we celebrated 4/4/16. Since it’s not as frequent as Pi Day, it’s a lesser known “holiday”, but even then, it still pops up in the news. I found this online article for a UK news site that described other square root-related fun facts in history. It also included a post from Good Morning America with the hashtag #squarerootday, which gave me this idea: I would like to encourage my students to participate in all of the fun square root-related activities that celebrate this day (if there was one that school year). The founder of square root day has suggestions that include but are not limited to: square dancing, drinking root beer out of square glasses, or even taking a drive on route 66. In the days leading up to this fantastic math-related day, I would consider giving my kids an extra credit point for posting a picture of themselves doing something square root related on the class twitter with the tag #squarerootday (or a post on some other class social media). If there wasn’t a square root day during that academic year, I still think it would be fun to tell my students about this holiday.

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

My students should have already learned about perfect squares and their multiplication tables up to 12 or 13, at least. For a simple refresher, I could have my students color/highlight perfect squares on multiplication tables. Then taking the square root of something is the inverse of creating perfect squares, unless what’s under the square root sign isn’t a perfect square. Then what’s under the radical is something that they need to divide into its prime factors so that they can simplify. My students should have also at least learned about prime numbers, if not prime factoring. A way to solve square roots would be pairing up the prime factors under the square root so that you can “take it out” from under the radical; for my students, I could have them think of the square root sign as a jail cell, and the only way that the numbers could “get out” of the cell is if they had a “prime partner” to escape with (i.e. a pair of 2s, a pair of 3s etc.).

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

So one of the oldest records of square roots in history would be The Old Babylonian tablet YBC 7289, which dates back anywhere from 2000-1600 BC. It depicts a square with two diagonals drawn and on the diagonals are numbers; when they are calculated, you get a very close approximation of the square root of 2 for the diagonal. Their value for the square root of two was about 1.41421297; I could have my students quickly calculate the square root of two (about 1.41421356) and mention to my students that this is pretty impressive for a civilization without modern day technology. The fact that they used clay tablets for math calculations shows how little they had to work with. Yet Babylon was also one of the most famous ancient cities in Mesopotamia; it’s mentioned multiple times in the bible and they were pretty advanced in mathematics for their area, despite the lack of resources we have today. They used a sexagesimal number system, which is base 60; they could solve algebra problems and work with what we now call Pythagorean triples; they could also solve equations with cubes.

References

A Visual Approach to Simplifying Radicals (A Get Out of Jail Free Card). (2012, January 15). Retrieved September 09, 2016, from https://reflectionsinthewhy.wordpress.com/2012/01/15/a-visual-approach-to-simplifying-radicals-a-get-out-of-jail-free-card/

Babylon and the Square Root of 2. (2016). Retrieved September 09, 2016, from https://johncarlosbaez.wordpress.com/2011/12/02/babylon-and-the-square-root-of-2/

Buncombe, A. (16, April 4). Square Root Day: There are only nine days this century like this. Retrieved September 09, 2016, from http://www.independent.co.uk/news/world/americas/square-root-day-there-are-only-nine-days-this-century-like-this-a6967991.html

Fowler, D., & Robson, E. (n.d.). Square Root Approximations in Old Babylonian Mathematics: YBC 7289 in Context. Historical Mathematica, 366-378. Retrieved September 9, 2016, from https://math.berkeley.edu/~lpachter/128a/Babylonian_sqrt2.pdf.

Mark, J. J. (2011, April 28). Babylon. Retrieved September 09, 2016, from http://www.ancient.eu/babylon/

*Posted by John Quintanilla on November 25, 2016*

https://meangreenmath.com/2016/11/25/engaging-students-square-roots-3/

# Thanksgiving recipe

*Posted by John Quintanilla on November 24, 2016*

https://meangreenmath.com/2016/11/24/thanksgiving-recipe/