## All posts tagged **function**

# Vertical line test

*Posted by John Quintanilla on March 29, 2019*

https://meangreenmath.com/2019/03/29/vertical-line-test/

# Engaging students: Computing the composition of two functions

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 Alexandria Johnson. Her topic, from Algebra II/Precalculus: computing the composition of two functions.

The following link is to a worksheet over composition of functions. The worksheet allows students to explore composition of functions without outright telling them what composition of functions is. Instead, the students are working on real world problems about shopping in a store that is having a 20% sale with mystery coupons. In the worksheet, students explore whether or not it matters which discount is applied first and the equations that go along with each scenario. This worksheet is interesting because it approaches composition of functions in an explorative way and it is using a real-world situation students in high school may find relatable, which can help hook students that are math-phobic.

https://betterlesson.com/community/document/1326462/going-shopping-student-materials-docx

Computing the composition of two functions requires prior knowledge of basic operations and combining like terms. This topic will expand upon their knowledge of basic operations by applying them to functions. Students will be able to add, subtract, multiply, and divide functions. Students should be able to use the distribution property; this is important when students are writing (fog)(x) and (gof)(x). During this topic, students should be able to expand upon their knowledge of creating functions from real world problems, which can be seen in the worksheet from the link above.

Musical composition is a way this topic can appear in high culture. Musical composition is the process of combining notes, chords, and melodies in a particular way. Arranging the notes, chords, or melodies in different ways can change the composition. Function composition is the combining of different functions f(x) and g(x) in different ways like addition, subtraction, multiplication, and division. Order usually matters in function composition just like in musical composition. If you have several band students, or musically inclined students, this would be a good hook to grab students interest.

Work cited

https://betterlesson.com/community/document/1326462/going-shopping-student-materials-docx

*Posted by John Quintanilla on December 31, 2018*

https://meangreenmath.com/2018/12/31/engaging-students-computing-the-composition-of-two-functions/

# Heat Index

*Posted by John Quintanilla on August 27, 2017*

https://meangreenmath.com/2017/08/27/heat-index/

# Monotonic Monotony

*Posted by John Quintanilla on August 26, 2017*

https://meangreenmath.com/2017/08/26/monotonic-monotony/

# Hilbert’s Infinite Hotel Paradox

TED-Ed made a very good video describing the Infinite Hotel Paradox, a thought experiment to describe how injective (one-to-one) functions can be used to examine countably infinite sets.

*Posted by John Quintanilla on August 25, 2017*

https://meangreenmath.com/2017/08/25/hilberts-infinite-hotel-paradox/

# Engaging students: Finding the domain and range of a function

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 Brittany Tripp. Her topic, from Precalculus: finding the domain and range of a function.

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

One of my favorite games growing up was Memory. For those who haven’t played, the objective of the game is to find matching cards, but the cards are face down so you take turns flipping over two cards and have to remember where the cards are so when you find the match you can flip both of the matching cards. To win the game you have to have the most matches. I think creating an activity like this, that involves finding domain and range, would be a really fun way to get students’ engaged and excited about the topic. You could place the students in pairs or small groups and give each student a worksheet that has a mixture of functions and graphs of functions. Then the cards that are laying face down would contain various different domains and ranges. In order to get a match you have to find the card that has the correct domain and the card that has the correct range for whatever function or graph you are looking at. You could increase the level of difficulty by having functions, graphs, domains, and ranges on both the worksheet and the cards. This would require the students to not only be able to look at a graph of a function or a function and find the domain and range, but also look at a domain and range and be able to identify the function or graph that fits for that domain and range.

These pictures provide an example of something similar that you could do. I would probably adjust this a little bit so that the domain and ranges aren’t always together and provide actual equations of functions that the students’ must work with as well.

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

Finding the domain and range of a function is used and expanded on in a variety of ways after precalculus. For instance, one way the domain and range is used in calculus is when evaluating limits. An example is the limit of x-1 as x goes to 1 is equal to zero, because when looking at the graph when the domain, x, is equal to 1 the range, y, is equal to zero. Finding domain and range is something that is applied to a variety of different type of functions in later courses, like when looking at trigonometric functions and the graphs of trigonometric functions. You look at what happens to the domain of a function when you take the derivative in calculus and later courses. You work with the domain and range of different equations and graphs in Multivariable calculus when you are switching to different types of coordinates such as polar, rectangular, and spherical. There are also multiple different science courses that use this topic in some way, one of those being physics. Physics involves a lot of math topics discussed above.

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

I found a website called Larson Precalculus that technically is targeted toward specific Precalculus books, but exploring this website a little bit I found that is would be a super beneficial tool to use in a classroom. This website has a variety of different tools and resources that students could use. It has book solutions which if you weren’t actually using that specific textbook could be a really helpful tool for students. This would provide them with problems and solutions that are not exactly the same to what they are doing, but similar enough that they could use them as examples to learn from. This website also includes instructional videos that explain in depth how to tackle different Precalculus topics including finding domain and range. There are interactive exercises which would give the students ample opportunities to practice finding the domain and range of graphs and functions. There are data downloads that give the students to ability to download real data in a spreadsheet that they can use to solve problems. These are only a few of the different resources this website provides to students. There are also chapter projects, pre and post tests, math graphs, and additional lessons. All of these things could be used to engage students and help advance and deepen their understanding of finding domain and range. The only downfall is that it is not a free resource. It is something that would have to be purchased if you chose to use it for your classes.

References:

http://esbailey.cuipblogs.net/files/2015/09/Domain-Range-Matching.pdf

*Posted by John Quintanilla on July 5, 2017*

https://meangreenmath.com/2017/07/05/engaging-students-finding-the-domain-and-range-of-a-function-2/

# My Favorite One-Liners: Part 72

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.

In calculus, the Intermediate Value Theorem states that if is a continuous function on the closed interval and is any number between and , then there is at least one point so that $f(c) =y_0$.

When I first teach this, I’ll draw some kind of crude diagram on the board:

In this picture, is less than while is greater than . Hence the one-liner:

I call the Intermediate Value Theorem the Goldilocks principle. After all, is too low, and is too high, but there is some point in between that is just right.

*Posted by John Quintanilla on April 13, 2017*

https://meangreenmath.com/2017/04/13/my-favorite-one-liners-part-72/

# Engaging students: Defining a function of one variable

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 Matthew Garza. His topic, from Algebra: defining a function of one variable.

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

Being able to define a function of one variable is necessary for creating a model that describes the most basic phenomenon in math and science. In math, understanding these parent functions is crucial to understanding more complicated functions and, by considering some variables as temporarily fixed, multivariable equations and systems of equations can be easier to understand. In science, we often observe functions of a single variable. In fact, even if there are multiple variables coming into play, a good lab will likely control all but one variable, so that we can understand the relationship with respect to that single variable – a function.

Consider in science, for example, the ideal gas law: PV = nRT, where P is pressure, V is volume, n is the quantity in moles of a gas, R is the gas constant, and T is temperature. This law, taught in high school chemistry, is not taught from scratch. The proportional, single-variable functions that make up the equation are observed individually before the ideal gas law is introduced. Students will probably be taught Boyle’s, Charles’, Gay-Lussac’s, and Avogadro’s laws first. Boyle’s law states pressure and volume are inversely proportional (for a fixed temperature and quantity of gas). This law can be demonstrated in one lab by clamping a pipette with some water and air inside, thus fixing all but two variables. Pressure is applied to the pipette and the volume of air is measured using the length of the air column in the pipette. Students must then evaluate volume V as a function of the single variable pressure P. It should be noted that the length of the air column is measured, while the diameter of the pipette is fixed, thus volume must be calculated as a function of the single variable length. Understanding the single variable, proportional and inversely proportional relationships is crucial to understanding the ideal gas law itself.

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.

Generally speaking, Khan Academy has great videos to help understand math concepts. Although it’s a little dry, this “Introduction to Functions” video is clear, concise, and touches on several ideas that I was having trouble tying in to every example. This introductory video begins with the basic concept of a function as a mapping from one value to another single value. The first examples it uses are a piece-wise function and a less computational function that returns the next highest number beginning with the same letter. At first I didn’t like that these functions were discontinuous, but this actually gives something else to discuss. The video links back prior knowledge, explaining that the dependent variable y that students are familiar with is actually a function of x, and represents the two in a table. The last couple minutes of the video address the fundamental property that a function must produce unique outputs for each x, or it is a relationship.

Source: https://www.khanacademy.org/math/algebra/algebra-functions/intro-to-functions/v/what-is-a-function

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

One idea might be to examine any function in which time is the independent variable. Basic concepts of motion in physics can supplement an activity – Have groups evaluate position and speed with respect to time of, say, a marble or hot wheels car rolling down a ramp. Using a stop watch and marking distance on an inclined plane, students could time how long it took to reach certain points and create a graph over time of displacement. This method might result in some students graphing time as a function of displacement, which could lead to an interesting discussion on independence and dependence, and why it might be useful to view change as a function of time.

Technology could supplement such a lesson as to avoid confusion over whether distance is a function of time or vice versa. Using motion sensor devices to collect data, such as the CBR2, students can use less time collecting and plotting data and more time examining it. Different trials resulting in different graphs can lead to discussion on how to model such motion as a function of time – letting an object sit still would result in a constant graph, something rolling down an incline will give a parabolic graph (until the object gets too close to a terminal velocity).

To add variety, students can examine what a graph looks like if they move toward and away from the CBR2 or try to reproduce given position graphs. This may result in the same position at different times, but since an object can be in only one position at a given time, the utility of using position as a function of time can be represented. Sporadic motion, including changes in speed and direction (like moving back and forth and standing still) also allow discussion of piecewise functions, and that functions don’t necessarily have to have a “rule” as long as only one output is assigned per value in the domain.

*Posted by John Quintanilla on December 27, 2016*

https://meangreenmath.com/2016/12/27/engaging-students-defining-a-function-of-one-variable/

# Engaging students: Finding the domain and range of a function

*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 Esmeralda Sheran. Her topic, from Precalculus: finding the domain and range of a function.

I found that Free Math Help and Khan Academy are both interactive websites that help students learn how to find domain and range of functions. If I were to have a lesson on how to find domain and range of functions I would have my students use the Free Math Help website to explore the concept of domain and range. Using the Free Math Help website a student can input any type of function that they come up with to see what the graph looks like, the steps of how to find the domain/range, and how the domain and range correspond with the graph. I could choose to have students come up with their own functions and they could experiment with expression that are not functions just so they can share some findings they came up on their own. Conversely I could make handouts with a variety of functions both continuous and discrete, expression that are not functions so that I could manage their learning in a way that they can see different graphs and their corresponding domain and ranges. Also I could give them a series of functions with different translations based off of one main parent function.

Then using Khan Academy website I could perform an active elaborate in which the students see a graph and then must give the corresponding domain and range intervals. I can walk around to each student to see what they have recorded and ask them to provide a justification for their answer or explain what properties the graph has that gives the domain and range they come up with. However I chose to structure the activities the students will be able to observe and discuss the changes in the domain and range interactively by using either Khan Academy or Free Math Help.

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

Knowing how to find domain is fundamental to most any mathematical course proceeding and not excluding pre-calculus. Once students are able to understand how to find the domain and range of a function they are able to learn deeper concepts used in calculus, discrete mathematics, and real analysis. Once in calculus students are expected to use domain and range in order to complete derivative problems specifically pertaining to finding critical points like the maximum/minimum and to describe the function as it changes from interval to interval. Understanding domain and range is also important when students must contrive and solve a definite integral from analyzing a graph or data. Then in discrete mathematics students must apply what they have learned from domain and range in the past to understand what preimage and codomain means and how they relate to the domain and how they differ from range. Apart from the regular mathematic courses, physics, differential equations and similar course also have applications of derivatives and integrals that require previous knowledge on how to find the domain and range of a function.

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

- I would have the students create maps such as the ones from The Emperor’s New Groove using colored pencils and paper provided in class.
- The instructions for the activity would be:
- Leave an inch of blank space on the bottom of the page and the left edge.
- Then create your own chase scene
- Using two different characters
- Make sure your chase can pass the vertical line test.

- Then with rulers use the centimeter side to mark your x and y axis
- Now you must find the length and height of each of your chase scenes
- instead of writing 7 cm long; 5 cm high use interval notation [2,9];[1,5]

This activity will help students connect domain and range to being the span of the function’s graph and the possible input and output values. It will be engaging because a kid’s movie is tied into the activity. Also the students can work independently and creatively, which is something different than what they are used to doing in the average classroom. After this activity we could move on to a more in depth discussion of the domain of discrete and discontinuous functions.

References:

The Emperor’s New Groove – Disney Movie

Free Math Help interactive website

http://www.freemathhelp.com/domain-range.html

Khan Academy interactive website

https://www.khanacademy.org/math/algebra/algebra-functions/domain-and-range/e/domain_and_range_0.5

*Posted by John Quintanilla on June 20, 2016*

https://meangreenmath.com/2016/06/20/engaging-students-finding-the-domain-and-range-of-a-function/

# Error involving countable numbers in Glencoe Algebra 2 (2014)

Errors in textbooks happened when Pebbles Flintstone and Bamm-Bamm Rubble attended Flintstone Elementary, and they still happen on occasion today. But even with that historical perspective, this howler is a doozy.

This was sent to me by a former student of mine. It appears in the chapter study guide for Section 2.1 of Glencoe’s Algebra 2 textbook (published in 2014), presumably as an enrichment activity for students learning about the definitions of “one to one functions” and “onto functions.”

- For starters, is most definitely a countable set, and so there is a one-to-one and onto correspondence between integers and rational numbers. See https://proofwiki.org/wiki/Rational_Numbers_are_Countably_Infinite or http://www.homeschoolmath.net/teaching/rational-numbers-countable.php for details.

- The above “proof” is only a blatant assertion, without any justification, either formal or informal, for why the authors think that the statement is false.
- The ordering of the rational numbers in the way listed above is actually reasonably close to the listing that actually does produce the one-to-one correspondence between and .

- Just above Example 2 was Example 1, which gives the correct proof that there’s a one-to-one correspondence between and . If the authors had double-checked this proof in any reputable book, they should have also been able to double-check that their Example 2 was completely false.

The full chapter study guide can be found here (it’s on the last page): http://nseuntj.weebly.com/uploads/1/8/2/0/18201983/2.1relations_and_functions.pdf

Reactions can be found here: https://www.reddit.com/r/math/comments/3k1qe6/this_is_in_a_high_school_math_textbook_in_texas/

Reference to this can be seen on page 10 of the teacher’s manual here: http://msastete.com/yahoo_site_admin1/assets/docs/Chpte2-1.25882808.pdf

*Posted by John Quintanilla on April 24, 2016*

https://meangreenmath.com/2016/04/24/error-involving-countable-numbers-in-glencoe-algebra-2-2014/