Source: https://xkcd.com/2303/

A brief explanation can be found at https://www.explainxkcd.com/wiki/index.php/2303:_Error_Types.

Source: https://xkcd.com/2303/

A brief explanation can be found at https://www.explainxkcd.com/wiki/index.php/2303:_Error_Types.

*Posted by John Quintanilla on October 30, 2020*

https://meangreenmath.com/2020/10/30/error-types/

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 Eduardo Torres Manzanarez. His topic, from Algebra: fitting data to a quadratic function.

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

One interesting project that could be done to invoke quadratic modeling is for students to develop a model that fits a business’ data of labor and output. The basic model of labor and output for a given company can be modeled by a quadratic function and it can be used to determine important figures such as the maximum output, minimum output, maximum labor, and minimum labor. The following image is an example of such a relationship.

In general, people would think that the more labor and resources used at the exact same time results in more product. If you have more product produced, then you accumulate more profit. These ideas are not wrong to be thought of but a key aspect that is missed in the thought process is that of land or otherwise known as workspace. The more employees you hire, the more space required so that these individuals can produce but space is limited just like any other resource. Lack of space inhibits production flow and therefore decreases product, decreases profits, and increases cost through increased wages. All of this does not occur until you pass the maximum of the model. So, both of these behaviors are shown and exhibited by a quadratic function. Students can realize these notions of labor and production by analyzing data of various companies. An activity that could show such a relationship in action is having one student create a small particular product such as a card with a particular design and produce as many as they can in a certain amount of time, with certain resources, and a workspace. Record the number of cards produced. Next, have two students create cards with the exact same time, resources, and workspace and record the amount produced. As more students are involved, the behavior of labor and production will be shown to be direct and then inverse to each other. The final piece for this activity would be for students to find realize what function seems to have the same shape as the data on a graph and for them to manipulate the function so that it fits on the data. Turns out the function will have to be a quadratic function.

B1) How can this topic be used in your students’ future courses in mathematics or science?

Fitting data onto a quadratic function is useful in analyzing behavior between variables. In various mathematical courses, data is provided but in science usually one must come up with data through an experiment. Particularly there are many situations in physics where this is the case and relationships have to be modeled by fitting data onto various functions. Doing quadratic modeling and even linear modeling early on is a good introduction into other models that are used in the many fields of science. Not every experiment is recorded perfectly and hence there can never be a perfect model. Through analytical skills presented in this topic, it scaffolds students to find a model for bacteria growth, a model for velocity, a model for the position of an object, and a model for nuclear decay in the future and what to expect the behavior of these models to be. This topic in combination with limits from calculus builds onto piece-wise models for probability and statistics.

E1) How can technology be used to effectively engage students with this topic?

Technology such as graphing calculators, Excel, Desmos, and TI-Nspires can be used to create the best model possible based on least-squares regression. This technology is engaging in developing models, not because of the lack of convoluted math that deals with squaring differences but rather the focus on analyzing particular models such as a quadratic model. They could be engaging for students when students can input particular sets of data they find interesting and need a way to model it. Furthermore, students can use technology to develop beautiful graphs that can be easily interpreted than rough sketches of these models. TI-Nspire software can be used by a teacher to send a particular data set to students and their own TI-Nspires. Students can then insert a quadratic function on the graphing application and manipulate the function by changing its overall shape by the mouse cursor. This allows students to dictate their own particular models and allows for comparison between models as to which is more accurate for particular data.

References

https://study.com/academy/lesson/production-function-in-economics-definition-formula-example.html

*Posted by John Quintanilla on August 24, 2020*

https://meangreenmath.com/2020/08/24/engaging-students-fitting-data-to-a-quadratic-function-2/

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 Johnny Aviles. His topic, from Pre-Algebra: making and interpreting bar charts, frequency charts, pie charts, and histograms.

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

I would create a project where my students would make and interpreting bar charts, frequency charts, pie charts, and histograms. First, I would begin by using the class as data by asking them questions and use a specific chart for each question. For example, I would ask “who here is Team iPhone? Team Android? or who doesn’t care?” Essentially, I will be separating the class in select groups based on their preference of phone. I will then create a pie chart of the class based on their choice. I then would do more examples of the other charts and explain the purpose of each one and when to use it. After some more examples and practice for them to familiarize themselves with the charts, I will assign the project. I would then divide the class into 4 groups and evenly assign a chart to each student to find a real-world example to apply and create their own specified chart that they’ll present. (I divide the class to ensure that every chart gets represented.) The purpose of the project is for all the students to not only be exposed to all the charts but to also apply them and understand the use for each one.

B1. How can this topic be used in your students’ future courses in mathematics or science?

In terms of mathematics, bar charts, frequency charts, pie charts, and histograms are very essential forms of data. These charts are widely used in nearly every future math or science course of students. As appose of a large spreadsheet of data that is hard to interpret, this topic provides a more organized and visual way to provide that collected data and to find useful information. A great example of using this topic is statistics. a spread sheet in given and then transformed in the form of a histogram that would give information of its distribution. With this chart, one can find things such as mean and standard deviation. Statistics also test hypothesis that require data to decide whether or not a certain drug would be effective based on data from frequency charts or histograms. These charts are also widely used in science. They can record the population of a given species, growth of bacteria in a given time, surveys, etc. There are endless possibilities in which these graphs can be applied in students’ future subjects.

C3. How has this topic appeared in the news?

With the vast categories the news covers, there are many examples where bar charts, frequency charts, pie charts, and histograms have been used. The news is for the common people and the common person has socially acquired a short attention span. The news can’t just give a sheet of numbers and expect people to know what it means and let alone look at it. These charts are provided for everyone to be given vast amounts of data gathered in aesthetically pleasing chart that can be quickly interpreted. The weather uses data from previous years to predict what we could be facing in terms of temperature and rain on any given month or season. Sports are all stats that have been recorded and can predict the outcomes of future games and players stats. When a top new story unravels, news channels are quick to look up stats that relate to story and compare data for the viewer. These charts appear in the news frequently and are vital to be comprehended to future students.

*Posted by John Quintanilla on July 27, 2020*

https://meangreenmath.com/2020/07/27/engaging-students-making-and-interpreting-bar-charts-frequency-charts-pie-charts-and-histograms/

Despite its hopelessly flawed methodology, U.S. News & World Report continues to sell magazines with its lists of Top 25 or Top 100 universities in various categories. Some universities who don’t play along, like Reed College, have long suspected that their rankings are penalized. So I enjoyed this press release from Reed College about statistics students who reverse-engineered the rankings to measure the magnitude of this penalty. The results are startling: while Reed was officially ranked #90, the formula should have them at about #38. In one glaring example, the magazine underestimated the college’s financial resources by over 100 spots even though this information the magazine could have obtained this information from free government databases instead of their survey.

*Posted by John Quintanilla on July 3, 2020*

https://meangreenmath.com/2020/07/03/students-find-glaring-discrepancy-in-us-news-rankings/

Source: https://www.xkcd.com/2118/

*Posted by John Quintanilla on March 6, 2020*

https://meangreenmath.com/2020/03/06/how-to-annoy-a-statistician/

This awful pun is just in time for Valentine’s Day.

Source: https://www.facebook.com/NeuroNewsResearch/photos/a.479172065434890/2989385557746849/?type=3&theater

*Posted by John Quintanilla on February 14, 2020*

https://meangreenmath.com/2020/02/14/my-favorite-one-liners-part-116/

I’ve linked to a number of articles about the misuse of *p*-values. Recently, I read a nice article in the October/November 2019 issue of MAA Focus summarizing a conversation between the Executive Directors of the Mathematical Association of America and the American Statistical Association about the ASA’s call to eliminate the use of *p*-values. Per copyright, I can’t copy the entire article here, but let me quote the lead paragraph:

In March 2016, the American Statistical Association took the extraordinary step of issuing a Statement on

p-Values and Statistical Significance. This spring, the association went even further, publishing a massive special issue of its journal The American Statistician entitled Statistical Inference in the 21st Century: A World Beyondp<0.05. The lead editorial in that special issue called for the end of the use of the concept of statistical significance.

It’s going to be a while before entrenched statistics textbooks catch up with this new standard of professional practice.

Here’s an NPR article on the issue: https://www.npr.org/sections/health-shots/2019/03/20/705191851/statisticians-call-to-arms-reject-significance-and-embrace-uncertainty

Other articles cited in the MAA Focus article:

- http://sciencealert.com/statisticians-argue-it-s-time-to-ditch-significance-in-science-and-replace-it-with-uncertainty
- http://psychologytoday.com/us/blog/the-athletes-way/201903/rethinking-p-values-is-statistical-significance-useless
- http://theness.com/neurologicablog/index.php/get-rid-of-statistical-significance
- http://sciencenews.org/article/statisticians-standard-measure-significance-p-values
- http://vox.com/latest-news/2019/3/22/18275913/statistical-significance-p-values-explained

*Posted by John Quintanilla on February 3, 2020*

https://meangreenmath.com/2020/02/03/the-end-of-statistical-significance/

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 Phuong Trinh. Her topic, from Algebra: defining a function of one variable.

How have different cultures throughout time used this topic in their society?

The understanding of functions is crucial in the study of both math and science. Not only that, some functions, especially function with one variable, are often used by everyone in their daily life. For example, a person wants to buy some cookies and a cake. The person will need to figure how much it will cost them to buy a cake and however many cookies they want. If the cost of the cake is $12, and the price for each cookie is $1.50, the person can set up a function of one variable to find the total cost for any number of cookies, expressed as c. The function can be written as f(c) = 1.50c + 12. With this function, the person can substitute any number of cookies and find out how much they would spend for the cookies and cake. Aside from the situation given by this example, function with one variable can also be used in various different scenarios.

What interesting (i.e., uncontrived) word problems using this topic can your students do now? (You may find resources such as http://www.spacemath.nasa.gov to be very helpful in this regard; feel free to suggest others.)

Function with one variable can be used in many real life situations. Word problems can be derived from every day scenarios that the students can relate to.

Problem 1: John is transferring his homework files into his flash drive. This is the formula for the size of the files on John’s drive S (measured in megabytes) as a function of time t (measured in seconds): S (t) = 3t + 25

How many megabytes are there in the drive after 10 seconds?

This problem allows the students to get familiar with the function notation as well as letting the students work with a different variable other than x.

Problem 2: (Found at https://www.vitutor.com/calculus/functions/linear_problems.html )

“A car rental charge is $100 per day plus $0.30 per mile travelled. Determine the equation of the line that represents the daily cost by the number of miles travelled and graph it. If a total of 300 miles was travelled in one day, how much is the rental company going to receive as a payment?”

Besides giving the students practice with finding a solution from a function, this problem let the students practice setting up the equation. This also shows the students’ understanding of the subject.

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

There are multiple resources that can be used to help the students understand what a function is as well as how they should approach a problem with function. One of the resources can be found at coolmath.com. The layout of the website makes it easy to locate the topic of “Functions” under the “Algebra” tab. By comparing a function with a box, Coolmath defines a function in a way that can be easily understood by students, while also showing how a function can be thought of as visually. The site also provides the explanation for function notation with visuals and examples that are easy to understand. On Coolmath, the students will also have the chance to practice with randomly generated questions. They can also check their answers afterward. On other hands, the site also provides definitions and explanations to other ideas such as domain and range, vertical line tests, etc. Overall, coolmath.com is great to learn for students in and out of the classroom, as well as before and after the lesson.

http://www.coolmath.com/algebra/15-functions

References:

“Linear Function Word Problems.” Inicio, www.vitutor.com/calculus/functions/linear_problems.html.

“Welcome to Coolmath.” Cool Math – Free Online Cool Math Lessons, Cool Math Games & Apps, Fun Math Activities, Pre-Algebra, Algebra, Precalculus, www.coolmath.com/algebra/15-functions.

*Posted by John Quintanilla on June 7, 2019*

https://meangreenmath.com/2019/06/07/engaging-students-defining-a-function-of-one-variable-2/

*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 submission comes from my former student Chris Brown. His topic: how to engage students when teaching box and whisker plots.

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

My all-time favorite TV show as a child was Pokémon. This show is still a staple amongst the young and even adult generation of today. The activity that I have created, was designed to take place after a formal lesson over how to create Box and Whisker plots. For this activity, students will be given a labeled bar graph of the Pokémon Type Distribution for generations 1 through 6 of Pokémon, which I have listed an online data source below. The students will be tasked with identifying the top 7 Pokémon types and creating a Box and Whiskers plots for each of those types. They will then go through and analyze the consistency of the creation of Pokémon for that specific type and then compare contrast this same box plot to any other box plot of their choice. The students will then make predications for the number of Pokémon for each of the top 7 Pokémon types, for generation 7 and base their reasoning in the box plots they created. Then the student will finally research the type distributions for the 7^{th} generation of Pokémon, and discuss how the actual number compares to their prediction.

This is the online source for the type distributions for generations 1 – 6:

https://plot.ly/~powersurge360/6.embed

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

From my experience, Box Plots are first taught in the early middle school years, in 6^{th} or 7^{th} grade. When constructing box plots by hand, in its essence, box plots require knowledge of how to order sets of numbers from least to greatest; an understanding and ability to find the maximum, minimum, median, and mean of a data set; and lastly, critical thinking and analytic skills developed from general course content. Box plots allow students to combine each of these skills to effectively analyze data sets with ease and compare different data sets with precision and accuracy. If any or all of these skills are not quite up to par, students will have an opportunity to develop them through box plots as they spend time creating them. For all students no matter their level, they will still gain better insight on how to properly analyze data and grow as analytical thinkers as they take the represented data and turn it into meaningful interpretations.

How can technology be used to effectively engage students with this topic?

In a classroom, I personally believe that Desmos is a wonderful online tool that can aid students in the understanding of how box and whisker plots function, and also a great place to check their work. Desmos, which is linked below, gives students the ability to list as many data points as they need to, and concurrently creates a box plot as they do so. In this way, students are able to see how singular data points can skew the data in significant and insignificant amounts. What I also love about Desmos is that, the list of data points does not have to be in any kind of order, so students do not have to worry about that tedious step! Desmos also lists the 5-point summary in two different places, on the box plot itself, and also on a drop-down menu, which is super convenient. Lastly, I love how Desmos also displays the mean of the data set as well, students can calculate the skew of the data, and definitively determine how it is skewed. This is a super visual, and interactive tool that will allow the student to manipulate box plots so seamlessly they will not be focused on the tediousness of the setup and solely on the concept.

The link to the Desmos setup is here: https://www.desmos.com/calculator/h9icuu58wn

*Posted by John Quintanilla on April 19, 2019*

https://meangreenmath.com/2019/04/19/engaging-students-box-and-whisker-plots-2/

The following article was recommended to me by a former student: https://towardsdatascience.com/statistics-for-people-in-a-hurry-a9613c0ed0b. It’s synopsis is in the opening paragraph:

Ever wished someone would just tell you what the point of statistics is and what the jargon means in plain English? Let me try to grant that wish for you! I’ll zoom through all the biggest ideas in statistics in 8 minutes! Or just 1 minute, if you stick to the large font bits.

*Posted by John Quintanilla on January 4, 2019*

https://meangreenmath.com/2019/01/04/statistics-for-people-in-a-hurry/