# 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.

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.

# 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 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

# 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.”

Just how bad is this mistake?

• 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 $\mathbb{Q}$ and $\mathbb{Z}$.

• Just above Example 2 was Example 1, which gives the correct proof that there’s a one-to-one correspondence between $\mathbb{Z}$ and $\mathbb{N}$. 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

# Another poorly written word problem (Part 8)

Textbooks have included the occasional awful problem ever since Pebbles Flintstone and Bamm-Bamm Rubble chiseled their homework on slate tablets while attending Bedrock Elementary. But even with the understanding that there have been children have been doing awful homework problems since the dawn of time (and long before the advent of the Common Core), this one is a doozy.

There’s no sense having a debate about standards for elementary mathematics if textbook publishers can’t construct sentences that can be understood by students (or their parents).

On its face, problems 11 and 12 don’t look so bad. For #11, the appropriate inequality is

$1400 + 243 + w \le 2000$

$1643 + w \le 2000$

$w \le 357$

For #12, the inequality is

$7 + g \le 15$

$g \le 8$.

These indeed are the answers that the textbook is expecting. However, both answers are wrong because both $w$ and $g$ have to be positive. So the answers should be $0 \le w \le 357$ and $0 \le g \le 8$. Which would be no big deal — except that these problems appeared before compound inequalities were introduced. (Notice that problems 7 through 10 only contain a single inequality.)

So, in a nutshell, the correct answers for these problems require skills that students have not yet learned at the time that they would attempt these problems.

# Another poorly written word problem (Part 6)

Textbooks have included the occasional awful problem ever since Pebbles Flintstone and Bamm-Bamm Rubble chiseled their homework on slate tablets while attending Bedrock Elementary. But even with the understanding that there have been children have been doing awful homework problems since the dawn of time (and long before the advent of the Common Core), this one is a doozy.

There’s no sense having a debate about standards for elementary mathematics if textbook publishers can’t construct sentences that can be understood by students (or their parents).

This one makes my blood boil. According to its advocates, the whole point of the Common Core standards was to increase the rigor in secondary mathematics. However, this one is SIMPLY WRONG.

The textbook does correctly note that the proper definition of a function is a set of ordered pairs. The “correct” answer, according to the textbook, is answer G — the plotted points do not match the ordered pairs.

However, answer H is also wrong. The textbook would have students believe that order is important when listing the elements of a set. However, order is not important — the domain of $\{-3, 1, -1, 3\}$ is the same as $\{-3, -1, 1, 3\} or$latex \{3, -3, -1, 1\}$. This is standard mathematical notation — in an ordered pair (or ordered $n-$tuple), the order is important. For a set, the order is not important. Specifying that the domain is $\{-3,-1,1,3\}$ and the range is $\{2,5,8,11\}$ does not uniquely determine the function. In fact, there are 24 different functions that have this domain and range (where we distinguish between the range of a function and its codomain). In other words, in trying to be clever about properly defining a function and showing different representations of a function, the textbook promotes a misconception about sets… which makes me wonder if the textbook’s attempt at trying to be ultra-careful about the definition of a function is really worth it. # Meat-A-Morphosis: An Introduction to Functions The video below was posted by my colleague Jason Ermer, founder of Collaborative Mathematics. From the YouTube description: A cartoon about the proverbial mathematical “function machine”. I was part of the creative team of mathematics teachers (including Patty Hill and Michael Word) who created this cartoon. It was originally (and still is) a component of the Honors Algebra 1 curriculum at Kealing Middle School in Austin, Texas. I take full responsibility for the spelling error. (Can you find it? 🙂 Enjoy. # Did chaos cause mayhem in Jurassic Park? I’ll happily link to this very readable introduction to chaos theory and the butterfly effect: http://plus.maths.org/content/did-chaos-cause-mayhem-jurassic-park A sampling: Suppose that we want to predict the future state of a system — the weather, for example — that is sensitive to initial conditions. We could measure its current state, and then iterate the system’s governing function on that seed value. This would yield an answer, but if our measurement of the system’s current state had been slightly imprecise, then the true result after a few iterations might be wildly different. Since empirical measurement with one hundred percent precision is not possible, this makes the predictive power of the model more than a few time-steps into the future essentially worthless. The popular buzz-word for this phenonemon is the butterfly effect, a phrase inspired by a 1972 paper by the chaos theory pioneer Edward Lorenz. The astounding thing is that the unpredictability arises from a deterministic system: the function that describes the system tells you exactly what its next value will be. Nothing is left to randomness or chance, and yet accurate prediction is still impossible. To describe this strange state of affairs, Lorenz reportedly used the slogan Chaos: When the present determines the future, but the approximate present does not determine the approximate future. Chaotic dynamics have been observed in a wide range of phenomena, from the motion of fluids to insect populations and even the paths of planets in our solar system. # Engaging students: Finding domain and range 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 Michelle Nguyen. Her topic, from Precalculus: finding domain and range. What interesting word problems using this topic can your students do now? Problem: Joe has an afterschool job at the local sporting goods store. He makes$6.50 an hour. He never works more than 20 hours in a week. The equation s(h)=6.5h can be used to model this situation, where h represents the number of hours Joe works in a week . What is the appropritate domain and range for this problem?

Students will be able to state the domain has to be from 0 to 20 because Joe never works over 20 hours and he can not work negative hours. With the range, the students would have to plug in 20 into the equation and get 130. The range will not exceed 130 because the maximum hours Joe will work is 20 hours. The students will know that Joe cannot be able to earn negative money either. Because of this, students will be able to identify that the range of this problem is from 0 to 130.

https://secure.lcisd.org/schools/HighSchools/FosterHighSchool/Faculty/Math/KarenKlobedans/Algebra2/images/Notes%209-2%20Domains%20and%20Ranges%20from%20Word%20Problems.pdf

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

After learning about the definition of domain and range, I would use a matching activity to assess the students’ knowledge about the topic. For example, I would have different graphs on different cards and their domain and range on another card. The students would shuffle the cards and then find their matching pairs. By doing this, the students would have to discuss with their group or partner about why their domain and range card matches with their graph card. Students will be able to identify the range and domain that would make sense to them and be able to back up their conclusion with what they know about domain and range.

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

Finding the domain and range can be an extension of learning functions. Students have been exposed to functions and their graphs already before this topic is introduced. With the knowledge of functions, students are able to find the domain and range with a graph given. Since they are able to do that, students have prior knowledge to the meaning of x-axis and y-axis. Domain and range is just another word for x and y axis. The students have already been exposed to graphs of different functions and the students have learned how to make their own graph if only an equation is given. Students will most likely make a table with coordinates to graph their graph. With this knowledge, they are able to use it to find the domain and range of a function.

# Engaging students: Computing inverse 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 comes from my former student Derek Skipworth. His topic, from Algebra II: computing inverse functions.

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

In essence, an inverse function is supposed to “undo” what the original function did to the original input.  Knowing how to properly create inverse functions gives you the ultimate tool for checking your work, something valuable for any math course.  Another example is Integrals in Calculus.  This is an example of an inverse operation on an existing derivative.  A stronger example of using actual inverse functions is directly applied to Abstract Algebra when inverse matrices are needed to be found.

C. How has this topic appeared in high culture?

The idea of inverse functions can be found in many electronics.  My hobby is 2-channel stereo.  Everyone has stereos, but it is viewed as a “higher culture” hobby when you get into the depths that I have reached at this point.  One thing commonly found is Chinese electronics.  How does this correlate to my topic?  Well, the strength of the Chinese is that they are able to offer very similar products comparable to high-end, high-dollar products at a fraction of the costs.  While it is true that they do skimp on some parts, the biggest reason they are able to do this is because of their reverse engineering.  Through reverse engineering, they do not suffer the massive overhead of R&D that the “respectable” companies have.  Lower overhead means lower cost to the consumer.  Because of the idea of working in reverse, “better” products are available to the masses at cheaper prices, thus improving the opportunity for upgrades in 2-channel.

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

A few years ago, there was a game released on Xbox 360 arcade called Braid.  It was a commercial and critical success.  The gameplay was designed around a character who could reverse time.  The trick was that there were certain obstacles in each level that prevented the character from reversing certain actions.  To tie technology into a lesson plan, I would choose a slightly challenging level and have the class direct me through the level.  This would tie into a group activity where the students are required to calculate inverse functions to reverse their steps (like Braid) and eventually solve a “master” problem that would complete the activity.  This activity could be loosely based off a second level that could wrap up the class based off the results that each group produced from the activity.

http://braid-game.com/

# Formulas for arithmetic and geometric sequences (Part 1)

I’m not particularly a fan of memorizing formulas. Apparently, most college students aren’t fans either, because they often don’t have immediate recall of certain formulas from high school when they’re needed in the collegiate curriculum.

While I’m not a fan of making students memorize formulas, I am a fan of teaching students how to derive formulas. Speaking for myself, if I ever need to use a formula that I know exists but have long since forgotten, the ability to derive the formula allows me to get it again.

Which leads me to today’s post: the derivation of the formulas for the $n$th term of an arithmetic sequence and of a geometric sequence. This topic is commonly taught in Precalculus but, in my experience, is often forgotten by students years later when needed in later classes.

An arithmetic sequence is specified by two numbers: the first term and the common difference between terms. For example, if the first term is $16$ and the common difference is $3$, then the sequence begins as

$16, 19, 22, 25, 28, 31, 34, \dots$

If the first term is $29$ and the common difference is $-4$, then the sequence begins as

$29, 25, 21, 17, 13, 9, 5, 1, -3, \dots$

For those of us old enough to remember, our favorite arithmetic sequences came from Schoolhouse Rock:

Let’s discuss the first arithmetic sequence, whose first seven terms are:

$16, 19, 22, 25, 28, 31, 34, \dots$

How do we get the $8$th term? That’s easy: we just add $3$ to $34$ to get $37$.

How to we get the $100$th term. That’s easy: we just add $3$ to the $99$th term.

Oops. We don’t know the $99$th term. To get the $99th$ term, we need the $98$th term, which in turn requires the $97$th term. Et cetera, et cetera, et cetera.

The trouble (so far) is that an arithmetic sequence is recursively defined: to get one term, I add something to the previous term. Mathematically, the arithmetic sequence is defined by

$a_n = a_{n-1} + d$,

where $d$ is the common difference. This can be very intimidating to students when seeing it for the first time. So, to make this formula less intimidating, I usually read this equation as “Each next term in the sequence is equal to the previous term in the sequence plus the common difference.”

It would be far better to have a closed-form formula, where I could just plug in $100$ to get the $100$th term, without first figuring out the previous $99$ terms.

To this end, we notice the following pattern:

• Second term: $19 = 16 + 3$
• Third term: $22 = 19 + 3 = 16 + 3 + 3 = 16 + 2 \times 3$
• Fourth term: $25 = 22+ 3 = 16 + (2 \times 3) + 3 = 16 + 3 \times 3$
• Fifth term: $28 = 25+ 3 = 16 + (3 \times 3) + 3 = 16 + 4 \times 3$
• Sixth term: $31 = 28+ 3 = 16+ (4 \times 3) + 3 = 16 + 5 \times 3$
• Seventh term: $34 = 31 + 3 = 16 + (5 \times 3) + 3 = 16 + 6 \times 3$

It looks like we have a pattern, so we can guess that:

• One hundredth term = $16 + (100-1) \times 3 = 313$

In general, we have justified the closed-form formula

$a_n = a_1 + (n-1)d$,

where $a_1$ is the first term, and $d$ is the common difference.  In words: to get the $n$th term of an arithmetic sequence, we add $d$ to the first term $n-1$ times. (This may be formally proven using mathematical induction, though I won’t do so here.)

A closed-form formula for a geometric sequence is similarly obtained. In a geometric sequence, each term is equal to the previous term multiplied by a common ratio. Mathematically, the geometric sequence is recursively defined by

$a_n = a_{n-1}r$,

where $r$ is the common ratio. For example, if the first term is $3$ and the common ratio is $2$, then the first few terms of the sequence are

$3, 6, 12, 24, 48, dots$

By the same logic used above, to get the $n$th term of an geometric sequence, we multiply $r$ to the first term $n-1$ times. Thus justifies the formula

$a_n = a_1 r^{n-1}$,

which may be formally proven using mathematical induction.