Rational Functions

We are studying Rational Functions, and I was looking for technology activities which would help students visualize the graphs of the functions and deepen their understanding of the concepts involved.  Previously, I had taught algebraic and numerical methods to find the key features of the graphs (asymptotes, holes, zeros, intercepts), then students would sketch by hand and check on the graphing calculator.  I wanted to capitalize on technology’s power of visualization* to give students timely feedback on whether their work/graph is correct, and avoid using the grapher as a “magic” answer machine.  I also wanted to familiarize students with the patterns of rational function graphs—in the same way that they know that quadratic functions are graphed as “U-shape” parabolas.

Here are three ideas:

Interactive Sliders

Students can manipulate the parameters in a rational function using interactive sliders on a variety of platforms (Geogebra, TI-Nspire, Transformation Graphing App for TI-84+ family, Desmos).  Consider the transformations of these two parent functions:

eq1 to become  eq3and

eq2 to become    eq4

Each of these can be explored with various values for the parameters, including negative values of a.

Here are screenshots from Transformation Graphing on the TI-84+ family:

Another option is to explore multiple x-intercepts such aseq5.

This TI-Nspire activity Graphs of Rational Functions does just that:


In a lesson using sliders, on any platform, I use the following stages so students will:

  1. Explore the graphs of related functions on an appropriate window.  Especially for the TI-84+ family, consider using a “friendly window” such as ZoomDecimal, and show the Grid in the Zoom>Format menu if desired.  Trace to view holes, and notice that the y-value is indeed “undefined.” capture-6
  2. Record conjectures about the roles of a, h, and k and how the exponent of x changes the shape of the graph.  This Geogebra activity has a “quick change” slider that adjusts the parent function from  eq1  to eq2.capture-geogebra-rationals
  3. Make predictions about what a given function will look like and verify with the graphing technology (or provide a function for a given graph).

A key component of the lesson is to have students work on a lab sheet or in a notebook or in an electronic form to record the results and summarize the findings.  Even if your technology access is limited to demonstrating the process on a teacher computer projected to the class, require students to actively record and discuss.  The activity must engage students in doing the math, not simply viewing the math.


A Desmos activity reminiscent of the classic GreenGlobs, MarbleSlides-Rationals has students graph curves so their marbles will slide through all of the stars on the screen.  If students already have a working understanding of the parent function graphs, this is a wonderful and fun exploration.

The activity focuses on the same basic curves, and it also introduces the ability to restrict the domain in order to “corral” the marbles.  Users can input multiple equations on one screen.

I really liked how it steps the students through several “Fix It” tasks to learn the fundamentals of changing the value and sign of a, h, k and the domains. These are followed by “Predict” and “Verify” screens, one where you are asked to “Help a Friend” and several culminating “Challenges”.  Particularly fun are the tasks that require more than one equation.

On one challenge, students noticed that the stars were in a linear orientation.


Although it could be solved with several equations, I asked if we could reduce it to one or two.  One student wondered how we could make a line out of a rational function.  Discussion turned to slant asymptotes, so we challenged ourselves to find a rational function which would divide to equal the linear function throw the points.  Here was a possible solution:


Asymptotes & Zeros

Finally, I wanted students to master rational functions whose numerator and denominator were polynomials, and connect the factors of these polynomials to the zeros, asymptotes, and holes in the graph.  I used the Asymptotes and Zeros activity (with teacher file) for the TI-84+ family.  It can also be used on other graphing platforms.

Students are asked to graph a polynomial (in blue below) and find its zeros and y-intercept.  They then factor this polynomial and make the conceptual connection between the factor and the zeros.  Another polynomial is examined in the same way (in black below).  Finally, the two original polynomials become the numerator and denominator of a rational function (in green below).  Students relate the zeros and asymptotes of the rational function back to the zeros of the component functions.


I particularly liked the illumination of the y-intercept, that it is the quotient of the y-intercepts of the numerator and denominator polynomials.  We had always analyzed the numerator and denominator separately to find the features of the rational function graph, but it hadn’t occurred to me to graph them separately.

A few concluding thoughts to keep in mind: any of these activities can work on another technology platform, so don’t feel limited if you don’t have a particular calculator or students don’t have computer/internet access.  Try to find a like-minded colleague who will work with you as you experiment with technology implementation, so you can share what worked and what didn’t with your students (and if you don’t have someone in your building, connect with the #MTBoS community on Twitter).  Finally, ask good questions of your students, to probe and prod their thinking and be sure they are gaining the conceptual understanding you are seeking.


*The “Power of Visualization” is a transformative feature of computer and calculator graphers that was promoted by Bert Waits and Frank Demana who founded the Teachers Teaching with Technology professional community.  More information in this article and in Waits, B. K. & Demana, F. (2000).  Calculators in Mathematics Teaching and Learning: Past, Present, and Future. In M. J. Burke & F. R. Curcio (Eds.), Learning Math for a New Century: 2000 Yearbook (51–66).  Reston, VA: NCTM.

All of the activities referenced in this post are found here.  More available on the Texas Instruments website at TI-84 Activity Central and Math Nspired, or at Geogebra or Desmos.

For more about the Transformation Graphing App for the TI-84+ family of calculators, see this information.

GreenGlobs is still available! Check out the website here.

Where Are You? I’ll Meet You There


I was so proud.  I had created a great technology activity to use in my Algebra 2 class, complete with a well-thought out lab sheet for students and their partners to work through and document their learning.  It was an exploration of slopes of parallel and perpendicular lines, with students being guided to “discover” the concepts involved.*

Questions for assessment involved different levels of cognitive demand, including creating their own sets of equations, paying attention to mathematical structure, and writing an explanation of their process.  The graphing calculators were ready and the students worked diligently through the class period.  The lesson was a success—everyone demonstrated their understanding of the mathematical objectives.

So what was the problem?  I asked a few students on their way out of class if they enjoyed the calculator lab activity since it was different from our “regular” routine.  They told me: “It was fine.  But Mrs. Campe, we all already knew about slopes of parallel and perpendicular lines.”

I had failed to properly pre-assess my students’ understanding of the concepts. I had wasted a full class period to cover something they had already mastered, when, instead, I could have been moving forward or exploring some other problem more deeply.  I didn’t check where my students were in their understanding before launching into my “great” activity.

Similar things can happen in my one-on-one work with students.  Since I am not in their classroom with them, when students arrive for a work session, I have to rely on them to tell me what their lesson and unit topics are.  Sometimes I go down a path that veers away from what they have done in class.  Some students resist conceptual explanations, wanting only the quickest route to the answer.  I have to push them to realize that learning the “why” behind a procedure helps them understand when and how to use it, and the conceptual background makes their learning more durable and leads to more success in math class.**

So what have I learned from these situations?

1.  It is vitally important to pre-assess and utilize formative assessment to know where my students are.  Class time is at a premium and I want to use it wisely.

2.  Don’t rely on students’ self-report of their understanding; require them to demonstrate their capabilities by doing problems, explaining a process, and answering “why” questions.

3.  Don’t use technology just because I have it.  It must further the lesson objectives and enhance student understanding.  The same warning goes for “fun” or “cool” lesson activities.

4.  Reflect on your lessons: ask yourself what went well and what needs improving so mistakes don’t get repeated.  And discuss with your colleagues, local and virtual. You will find lots of support in the MTBoS; one teacher commented to another on Twitter just last night: “Thx! I am always looking to improve my teaching!”

Mistakes, obviously, show us what needs improving. Without mistakes, how would we know what we had to work on?     

Peter McWilliams


Notes & Resources:

The technology lab activity on Parallel and Perpendicular Lines is here.  It was written for the TI-84+ family of calculators, but any graphing technology may be used.

*This lab activity is a “Type 1” investigation structure in that it guides students toward the desired mathematical knowledge, in contrast to a “Type 2” inquiry which encourages more open exploration.  Both types of lesson structures are effective, so match the level of exploration with your objectives.  More about this in McGraw, R. & Grant, M. (2005).  Investigating Mathematics with Technology: Lesson Structures That Encourage a Range of Methods and Solutions. In W. J. Masalski & P. C. Elliott (Eds.), Technology-Supported Mathematics Learning Environments: 67th Yearbook (303-318). Reston, VA: NCTM.

Another dimension useful in analyzing a lesson is type of teacher questioning.  “Funneling” questions guide students through a math activity to a predetermined solution strategy, while in “Focusing” interactions, the teacher listens to students’ reasoning and guides them based on where they are and what strategies they are employing, rather than how the teacher might solve the problem.  More in Herbel-Eisenmann, B. A. & Breyfogle, M. L. (2005). Questioning Our Patterns of Questioning.  Mathematics Teaching in the Middle School, 10(9): 484-489.

**Connecting new knowledge to what you already know (elaboration), building conceptual structures (mental models) and practicing what to do when (discrimination skills) are among the strategies for successful learning discussed in Make It Stick (Brown, Roediger & Mc Daniel, 2014).  See this website for more.

As a final thought, my title is misleading, because I don’t just want to meet the students where they are and stay there, I want to plan for appropriate challenges to take them beyond their current understanding.  There is great value in productive struggle, and choosing lesson components within the students’ “Zone of Proximal Development”.

Exams Ahead!

The students with whom I work are heading into midterm exams.  For some of them, the mathematics concepts and procedures come easily, and others have to work harder to feel confident in their understanding.  All of them can benefit from diligent preparation, and although a few still resist, here is the advice I am giving:


  1. Do the whole review packet and check your work against the solution key.  It isn’t optional. Show your mathematical thinking so you can analyze your process.
  1. Review past tests and quizzes, looking at both the questions you got wrong and the correct ones.  Even if you got it right in earlier in the semester, make sure you remember how to do it now.  Re-do the questions, don’t just “read it over”.
  1. The more practice and review you can do BEFORE you get to the review session, the more productive the time will be.  Mark the ones you get wrong and/or don’t know how to do so you have a list of questions ready.  Don’t wait til the last minute to start studying.
  1. Keep track of important formulas, graphs, examples & concepts on a self-created study guide.  Do it as you go through the review packet.  If you need to memorize something, write it out each time you use it until you know it. If you need to be able to solve something without a calculator, practice it that way.
  1. Make use of other resources: if your teacher has a website, go back to unit review sheets and solution guides from the semester. Check another teacher’s website from your school if your teacher doesn’t have one.  Work with a classmate, but don’t merely divide up the work: make sure you both can complete the problems.  Utilize Khan Academy, YouTube & Google.
  1. Cumulative exams are challenging; scores are often somewhat lower than your typical quiz/test scores have been.  However, remember to be confident in the things you know—yes it is a big job to prepare, but you can do it!
  1.  Take care of yourself physically over these weeks:
  • eat right and choose healthy snacks (think protein/fiber not sugar)
  • stay hydrated (more water less soda)
  • stay active because it helps relieve stress and is good for your brain: go to sports practice, work out or run or shoot hoops, or even just take the dog for a brisk walk during a study break
  • wash your hands frequently, etc. so you don’t get the bugs that will inevitably be going around
  • get enough sleep: it is far more important for your brainpower to sleep an extra hour than cram an extra hour.

Good luck!



Function Operations

Using Multiple Representations on the TI-84+

Algebra 2 students are studying function operations and transformations of a parent function.  My student had learned about the graph of eq1 and how it gets shifted, flipped, and stretched by including parameters a, h, and k in the equation.

Now he was faced with this question: how to graph  the equation in #58:


It didn’t fit the model of  eq3new-copy  so it wasn’t a transformation of the absolute value parent function.  He knew how to graph each part individually, but didn’t know how to graph the combined equation.  The TI-84+ showed him the graph with an unusual shape—not the V-shape he expected.

TIP: use the alpha2  button to access the shortcut menus above the  yequals-key, window-key, zoom-key and trace-key  keys. The absolute value template is used here.

“Why does the graph look like this?” he wanted to know. We decided to break up the equation into two parts, using ALPHA-TRACE to access the YVAR variable names.* The complete function is found by adding up the two partial functions.


Then we looked at a table of values, to get a numerical view of the situation.  I remind my students that if they are unsure how to graph a particular function, they can ALWAYS make a table of X-Y values as a backup plan—it isn’t the quickest method to graph, but is sure to work.  To get the Y-values of the combined function, add up the Y-values for the partial functions, since sum-function.

Initially, we “turned off” Y3 by pressing ENTER on the equals sign, so we could view the partial functions in the TABLE. I asked the student what he thought the values in the next column should be.


He mentally added them up, and then we verified his thinking by activating Y3 and viewing the table again.capture-5

To further illuminate the flat portion of the graph, we changed the table increment to 0.1 in order to “zoom in” on those values.

TIP: While in the table,  press plus-key to change the increment table-increment , or press 2nd WINDOW to access the TBL SET screen.capture-6Success! The TI-84+ provided graphical and numerical representations that deepened our understanding of the algebraic equation. This task had challenged the student, because it didn’t fit the parent function model he had learned, but he built on his knowledge of function operations to solve is own problem and help some classmates as well.  One of our approaches to learning is to “use what you know.”**


*You can use function notation on the home screen to perform calculations with any function from the Y= screen. Access the YVARs from ALPHA-TRACE.

**Much has been written about Classroom Norms. See Jo Boaler’s suggestions here and my messages to students here.

For more about transformations on parent functions, see this information about the Transformation Graphing App on the TI-84+ family of calculators.

Setting the Stage

How can we prepare our students for a successful school year?  I work with students mostly one-on-one, and with the school year under way, I’ve been thinking about this issue a great deal.  Whether your perspective is classroom teacher, coach, tutor, or parent, there are valuable preliminaries that can help enhance the learning environment.  I try to set the stage with a few important messages I want my students to hear.


First and foremost, I tell students to “Bring Your A Game” to their math learning.  This means that they can’t just let the math class happen TO them, but instead they have to actively ENGAGE with the material in class and for homework.  Take down the teacher’s notes and examples, and if something seems confusing, mark it to ask about later.  Put in full effort to make sure you’ve done the work (even if the teacher doesn’t check it*).  Work slowly and mindfully through the problems, and watch out for “avoidable errors” (my preferred term for “silly mistakes”).  Know what things tend to trip you up: for example, double- check negative signs and units on answers.  Don’t wait for the test; when you get a quiz back, figure out what went wrong so you can get it right next time.

Students tend to think that each new course brings all new math, but I remind them that they can rely on the fact that all of the math they have ever learned is still true! “Use what you know” about the + and – signs of the four quadrants to help with the unit circle in trig, and for calculus, build the difference quotient definition of the derivative on the formula for slope.  This also helps when you don’t know where to start on a word problem or a geometry proof.  And making connections between mathematical concepts deepens understanding.

Just as an athlete might train by doing different types of workouts (rather than more repetitions of the same exercise), math students can learn something better by trying to “find more than one way” to a solution.  I try to highlight alternative methods of solving by posing questions:

  • What would happen if you didn’t distribute first  to solve this equation? –15 = 3(x – 8)
  • Is this student’s reasoning correct?
  • Find the error in this work.
  • Can you solve this (geometry problem) another way?**

I constantly tell my students to “show your mathematical thinking.” One obvious reason for this is so you can get partial credit for wrong answers, if the teacher grades that way.  But the main purpose is that showing your steps makes it more likely that your work will BE correct.  You can reflect on your thinking process through a problem, which helps improve retention and understanding, and enables you to explain it to others when needed (and study for a test later).  If you have poor handwriting, make it readable (enough).  If you are using a calculator or doing some arithmetic in your head, sketch a graph or jot down what you are computing so your process is clear.  A friend’s daughter summed it up this way: “I feel it is important to show my work. My teacher and I can see what I’m thinking. It’s hard to see it if it’s only in my head.”

Another key to success is to “know your resources.”  Does your teacher have a website?  Is there an online textbook?  Who is in your class you can study with?  Don’t forget to search Google, YouTube, and Khan Academy if you are stuck on something and can’t proceed.  I remind students that getting help from a tutor isn’t the only way to get assistance.  Go to your teacher for extra help because they can see your effort and they can focus you in on the key concepts.

A final message is to “persevere even when it gets hard.”  This is more than just SMP #1 talking, this is about being confident that you can do it and being willing to grow your mathematical skills.  One teacher at my son’s Open House night spent time coaching us parents that the class was going to be challenging (and explained why), and gave strategies students could use to facilitate their understanding.  She said “if you aren’t getting at least a few things wrong, then you aren’t learning something new.”  I try to reinforce a growth mindset in my students and a belief that they can achieve the goals they have set for themselves.

I think it’s going to be a very good year…


* Students often don’t put in effort if there isn’t a grade attached. As a teacher, if you want to promote habits of doing homework, showing mathematical thinking, checking results, extending to more than one solution etc., try to consider this in your grading scheme.  See Wilson, Linda. “What Gets Graded Is What Gets Valued.” Mathematics Teacher 87, no. 6 (September1994): 412–14.

** This summer I worked for awhile on the Art Of Problem Solving Geometry course with an online group.  I was impressed by the focus on finding more than one solution: “Doing a problem two different ways is an excellent way to check your answer” and “When you can’t find an answer right away, try finding whatever you can—you might find something that leads to the answer! Better yet, you might find something even more interesting than the answer.”

These messages about approaches to learning are one version of Classroom Norms. Read more here about one teacher’s use of Jo Boaler’s suggested list.  And here is a consideration of how to change the norms that students may have become accustomed to, in order to increase student engagement.

Summer Assignment!

With the academic year winding down (and already finished for some), take a moment to think about what you will be doing this summer.  I’m sure you are planning to relax and refresh, but don’t neglect recharging your professional batteries…

READ A BOOK.  Is there one on your teaching bookshelf that you’ve been meaning to read?  Here are a few that I’ve read recently or have on my “Pedagogy and Learning” list for this summer:

  • Make It Stick: The Science of Successful Learning by Brown, Roediger & McDaniel
  • 5 Practices for Orchestrating Productive Mathematics Discussions by Smith & Stein
  • Embedding Formative Assessment by William & Leahy
  • Mathematical Mindsets by Boaler

If you have the opportunity, find a “buddy” or group and read together.  Try one chapter a week and discuss in person or via email.  Our Teachers Teaching with Technology cadre of instructors did book “discussion chats” this past year.  Here are some ideas and prompts to organize your comments on Embedding Formative Assessment and 5 Practices (thanks Jennifer!)

And since it is summer, I’m also planning to do some fun math reads.  Consider these, or maybe there are others you have your eye on (tell me in the comments!). The last two have the advantage that each chapter is a stand-alone essay, which is especially good if your attention span is shortened by summer distractions.

  • How to Bake Pi by Cheng
  • The Man Who Knew Infinity by Kanigel (now a major motion picture!)
  • Here’s Looking at Euclid by Bellos (also a math columnist for The Guardian)
  • The Joy of X by Strogatz (originally an essay series for The New York Times)

One more suggestion for your reading list is to catch up on an NCTM journal article you meant to read this year but didn’t have time to.  There are free previews of some articles on the website if you aren’t yet a member.

LEARN SOMETHING NEW (anything! Doesn’t have to be math-related.)  If you come back to school in the fall and share your experience with your students, they will see you as a learner and it may encourage their efforts.  Here are some ideas:

Watch a webinar.  Did you miss one this year you meant to join?  On-demand recordings are easily paused so you can take notes or try a problem yourself.  For example, Texas Instruments has an archive of their free webinars here.

Go to a workshop/class/conference. There are plenty of these available in person and online.  Your department or district may have local offerings.  The TI Teachers Teaching with Technology PD workshops are listed here, and include “virtual” workshops as well.  Jo Boaler’s YouCubed organization offers an online course for teachers “How to Learn Math” [info here].

For my Connecticut and New England colleagues, two great opportunities are nearby.  The T3 Northeast PD Summit is June 22 & 23 in New Britain, covering both the TI-84+ and TI-Nspire [info here and sign up here].  And the Geogebra Institute of Southern CT is holding their 4th annual conference in New Haven on August 16 [info here], including a pre-conference workshop August 15 for beginners.

MAKE A PLAN.  One of the best things for me about being a teacher is the chance to revise and improve my teaching practice on a regular basis.  Some years I taught the same course to more than one class, so each lesson got two or three tries in the same day (or week).  I reflected on how it went with the first group and made adjustments and improvements for the next class.  Remembering the details for the next school year is harder to do, so I would make quick notes right in my lesson plan to capture the changes I’d like to try in the future.  Here is the reminder I used as the last item on my “Lesson Plan Template”:border self eval

If you have notes from this year about lessons you’d like to modify, find them before leaving the building for the summer.

In 1994, Steve Leinwand wrote an article in Mathematics Teacher  “Four Teacher-Friendly Postulates for Thriving in a Sea of Change”.  One of them has resonated with me ever since then:  “It is unreasonable to ask a professional to change much more than 10% a year, but it is unprofessional to change by much less than 10% a year.”  While many of us try to change and improve our teaching each year (or are asked/mandated to implement new practices), change is challenging and daunting.  Leinwand suggests that teachers consider changing about 10% of what they do each school year, a very reasonable amount (just one new lesson every two weeks, or one unit out of the ten you teach).  What will be your 10%?  What topic/class/unit needs work?  Get a jump start this summer on something you’d like to teach differently than you’ve taught it before.

So, take this as your summer assignment: PICK SOMETHING you intend to do to learn and grow professionally this summer, along with your plans to decompress and have fun.  Let me know how it goes.  Have a wonderful summer!


Leinwand, Steven. (1994).  Four Teacher-Friendly Postulates for Thriving in a Sea of Change. Mathematics Teacher 87(6):392–393. [Reprinted in 2007 during 100th anniversary of MT, with commentary by Cathy Seeley: Mathematics Teacher 100(9):580-583.]

More on  Recreational math books here: Math-Frolic.

Problems With Parentheses

I have been noticing lately that my students are making mistakes involving the use of parentheses.  Sometimes parentheses are overused and other times they are missing, and errors are also made while using calculator technology.  Using symbols and notation correctly is part of SMP #6, “Attend to Precision”, and is also a component of mathematical communication, since so much of math is written in symbols.  I want my students to be efficient and accurate in their work, and I hope their notation supports their conceptual understanding. So I’ve been contemplating the purposes of parentheses…

Purpose #1: To Provide Clarity with Negative Integers.  Negative integers can be set off with a pair of parentheses for addition and subtraction, as in these examples, but the expression’s value is unchanged if the parentheses are not used:

  1.    (–4) + 6 = 2
  2.    6 – (–4) = 10

With an exponent on a negative integer, however, the parentheses are essential.  We are working on sequences and series in Algebra II.  When a geometric sequence has a negative common ratio, the explicit formula has a negative number raised to an exponent:

  1.    The sequence  2, –6, 18, –54, … has explicit formula  An = 2·(–3)n-1

To convince my Algebra II students that the parentheses are required, consider  –32  vs.  (–3)2  on the TI-84+ calculator:

Capture 1

The calculator executes the order of operations: exponents are evaluated before multiplication. Since the negative sign actually represents –1 times 32, the 32  is evaluated first.  Although I prefer students to focus on conceptual understanding and not merely procedural rules, I say to “always use parentheses for a negative base”.

Purpose #2: To Specify the Base for Exponentiation.  Another class is studying exponents and logs, and students notice that using parentheses has mathematical meaning for the result.

  1.    (2x)3    vs.   2x3
  1. Each component of the fraction within the parentheses gets raised to the power; these are all different (and the TI-Nspire CAS handles them nicely):

04-28-2016 Image004

Attending to precision is essential for students, and by doing three similar but different problems as a set, they get practice analyzing how the notation changes the results.

Purpose #3: To Properly Represent Fractions.  Fractions generally don’t need parentheses when written by hand, and I’m direct with students about my strong preference for a horizontal fraction bar rather than a diagonal bar when writing fractions on paper or on the board.

Complications can occur when students try to enter the fraction into a calculator without using a fraction template.  Pressing the DIVIDE button to create the “slash”, as in 3/4, has the advantage of connecting a fraction with the operation of division but the drawback of the diagonal bar.  For anything more complex than a simple fraction, parentheses are needed to “collect” the numerator and denominator so that the fraction is computed correctly.   For example:

  1. Find the mean of these three test scores: 85, 96, 77.

Capture 4

  1. Graph a rational function

Capture 3

Thankfully, fraction templates are readily available, so errors using parentheses are avoided.   On any TI-84+, set the mode to “MathPrint” and press ALPHA and Y= to access the template.  On a TI-Nspire, press CTRL and DIVIDE or select the fraction from the template palate.

This was especially useful for finding the sum of the following geometric series; notice the error on the first try due to missing parentheses, and then the corrected version:

And the calculator comes to the rescue! I encourage students to enter complicated expressions all at once.  Making separate entries for each part is taking a risk:

Capture 7

Purpose #4: To Indicate Multiplication. Probably the area in which I am observing the most “overuse” of parentheses is for multiplication.  At some time before students reach me in high school, they have been taught that in addition to using the × symbol to multiply, they can also use • , a raised dot. A third alternative is to use parentheses to indicate multiplication, especially for negative integers or to distribute multiplication over addition:

  1.    (–4)( –6) = 24
  2.    2x(x + 5) = 2x2 + 10x

I’ve seen some students “over-distribute” if they rely on parentheses instead of the raised dot for multiplication:

11.       (–4)(x)(3x2)  should be –12x3 ; however what if a student “distributes” the –4?

over distribute

[One more pet peeve of mine: when students utilize the × symbol for multiplying even when using the variable x.  I strongly suggest that once they are in Algebra I, students should “graduate” to the raised dot  •  to symbolize multiplication.]

When using the Chain Rule in Calculus, students sometimes make the error of “invisible parentheses” and then lose them entirely in their subsequent algebraic simplification.

  1. Find the derivative of (3x2 – 4x + 5)–2

invisible parenth

Notice the missing parentheses for (6x – 4) and how the error carries through.

Purpose #5:  Operator Notations.  My final category of parentheses usage is as part of the notation of certain function operators.  Students are familiar with using parentheses in function notation f(x), where the independent variable x is the input for the function expression.  Other functions such as logs and trig functions can use parentheses to set off their “arguments”, and the calculator supports this use by providing the left parenthesis.  Entering the right parenthesis is optional on the TI-84+, but a good practice for students:

Capture 5

If students get in the habit of using the parentheses, it enables them to correctly apply the “expand to separate logs” and “condense to a single log” rule.  Here the parentheses are not “needed” to indicate the argument of the log, but helpful for this student.

  1. Solve each equation:

And in these last two examples, the parentheses helps the student get the correct result:

  1. Expand to separate logs:


  1. Condense to a single log:


One final note: I want my students to harness the power of parentheses to support their conceptual understanding and mathematical accuracy.  Being precise about notation is not about “doing it my way” but instead about doing it in a way that helps them grasp the purpose of the symbols they use to clearly communicate their mathematical thinking.


For more about the “loss of invisible parentheses”, ambiguous fractions and other common math errors, see this site.

For one teacher’s approach to using parentheses to evaluate function values, read this blog post: An Algebraic Oath.

And here is one teacher’s elegant and simple definition of parentheses: Parenthetically Speaking.

Leap Years & License Plates

Thoughts on Divisibility and Counting

I am always in search of good numbers. When I park in a commercial parking lot, I look for a space with a number that is divisible by 3. I like addresses and phone numbers which involve multiples (such as xxx-1696 since 16 x 6 = 96). I enjoy mathematical dates, like 11/11/11, 10/11/12 or 3/5/15.

calendarSo the year 2016 has been a good one so far, numerically speaking. In February, I noted 2/4/16 and 2/8/16 and 2/14/16 (you can do the math). And on 2/29/16, otherwise known as “leap year day,” I started thinking about how to determine if a given year is a leap year.

Leap years occur every four years in order to synchronize the calendar with the astronomical seasons. To be precise, they occur in every year that is exactly divisible by 4, however years that are multiples of 100 are NOT leap years, unless they are multiples of 400. And how can one look at a number and determine if it is a multiple of four? Simply examine the tens and the units digits; if that two-digit number is a multiple of four, then the larger number is also a multiple of four. So 2016 is divisible by 4, but 2014 is not.

The rules for divisibility are taught at a variety of math levels. I remember learning the rules for divisibility during a seventh grade math unit on bases other than 10. My children learned them in fifth grade while doing factors, multiples and prime numbers. The Common Core State Standards doesn’t mention them by name, but begins discussion of division in grade 3, and factors, multiples, and primes are contemplated in grades 4 and 6.

In my work with middle school and high school students, I find great variation; some students are skilled at divisibility rules and others are surprised to hear about them. Everyone knows how to tell if a number is divisible by 2 or 5 or 10. Many of my students think multiples of 3 should work the same way (they should end in 3 or 6 or 9) and although they may know the “finger trick” for the first ten multiples of nine, they haven’t considered how to test larger numbers. The rules for divisibility by 3 and 9 rely on the sum of the digits (in contrast to the rules for 2, 4, 5, and 10).

My favorite divisibility rule is the one I learned most recently; while doing KenKen puzzles, I wanted a method other than long division to determine divisibility by 7. The technique is this:

  1. Chop off the units’ digit and double it.
  2. Subtract this from the remaining number.
  3. Continue this process until the result is 1 or 2 digits.
  4. If the final number is divisible by 7, then the original number was divisible by 7 [Note that 0 is divisible by 7].

For example: 3052

  1. Take the 2 and double it to get 4.
  2. 305 – 4 = 301
  3. Take the 1 and double it to get 2.
  4. 30 – 2 = 28 which is divisible by 7.

In the same manner, 2016 can be shown to be divisible by 7. So 2016 is a very good year: it is not only a leap year divisible by 4, but is also divisible by 3, 7 and 9.

Can we use technology to help students test for divisibility? On any calculator, we can simply divide each proposed factor. For more efficiency, use the TI-84+ table with the function Y1 = (number)/x and scroll through the table looking for whole number results.


This can be impractical for very large numbers, but on the TI-Nspire there is a factor command which comes to the rescue.


The question of why divisibility rules work is a fruitful exploration for students. For more on proving divisibility rules, see the resources below.

Connecticut_license_plateIn my search for multiples of 7, I have a new place to look: license plates. In Connecticut, they recently converted the license plate format to two letters followed by five numbers, such as AB-12345. I can check these 5-digit numbers for divisibility by 7 while stopped at a traffic light.

Which leads me to the question of license plate formats and the Fundamental Counting Principle. Connecticut used to have license plates with three numbers followed by three letters: 123-ABC. When those ran out, they briefly used a format with one number, four letters, and finally one number, such as 1ABCD2, at first without a hyphen and later with one inserted after the third character, like 1AG-HJ2. I disliked those plates because they were hard to remember (a 3-character chunk is more memorable to me if it is all letters or all numbers). Thankfully, the new seven-character plates appeared last year even though the sequence hadn’t been exhausted.

How many possible license plates are generated by each of the designs? Multiply the number of ways to select each character to determine the total:LLcorrect counting

Old style:         123-ABC:        10*10*10*26*26*26

Interim style:   1ABCD2:         10*26*26*26*26*10

New style:       AB-12345        26*26*10*10*10*10*10

The interim style, with 45.7 million plates, has 2.6 times as many plates as the old style.  The new style has 67.6 million possible plates which is about 3.8 times as many plates as the old style.  And with a one-in-seven probability that the number is divisible by 7, there will be about 9,657,143 multiples of seven out there.

Now that’s a lot of license plates! In the meantime, I’m looking forward to the next great date coming soon: 4/9/16.  Not only is it a progression of perfect squares (when will that happen again?) but 492016 is a multiple of 7.


For more on why divisibility rules work, see the following:

  1. Math Forum: Explaining the Divisibility Rules [link]
  2. James Tanton: Divisibility Rules Galore! [link]

For more about using fingers to multiply and why the tricks work, see Kolpas, Sidney J., “Let Your Fingers Do the Multiplying”, Mathematics Teacher, 95(4), April 2002.



Body Benchmarks!

Body Benchmarks: Analyze the Data and See How You Measure Up!

I first used this activity in an Algebra I class when I wanted students to have a hands-on experience gathering data and modeling it with a linear function. Later, I used this as an “all-ages” station at our elementary school’s Family Math Night. Let’s explore this problem situation and look at low-tech and high-tech methods to analyze this real-life data.

Here are the Instructions: [link to file]

  1. Write your name on the Record Sheet.
  2. Use the measuring tape to measure your forearm in inches from elbow to tip of middle finger.
  3. Record forearm measurement on the Record Sheet next to your name.
  4. Use the measuring tape to measure your height (if you don’t know it, in inches.)
  5. Record height measurement on the Record Sheet next to your name.
  6. Place a mark on the Group Graph that corresponds to your Forearm Length (Horizontal Axis) and Height (Vertical Axis).

There are some interesting discussions that can occur during data collection and recording. Why did we use inches? How precise should our measurements be (to nearest inch, half-inch, quarter-inch)? Are there some measurement activities that are better served by the metric system? Why did we choose to put Forearm Length on the X-axis and Height on the Y-Axis (is one variable clearly the independent variable and the other depends on it, or does it not matter)? What scale did we use on the large graph paper and did I graph my data point accurately? Is there any point that is clearly an outlier? [Not all of these topics came up in every class setting; are there other conversations you experienced or think are important for the teacher to orchestrate?]


Once the data is collected, the group graph shows a positive correlation that could be modeled by a linear function. A low-tech way to find a line of best fit is to graph the data on graph paper, and use a ruler or dry spaghetti/linguine pasta to approximate the line with this criteria: “Follow the trend of the data points, and have about half the points above and half below the line”.

Then select two points (must they be data points? Or just graph grid intersections?) and find the equation of the line (point-slope form or slope-intercept form?). Don’t stop at the equation… what does this formula do for us? Can we predict someone’s height if we know their forearm length? Can you describe the shape of the graph, and how does this relate to the equation? What other questions do you want to ask about this situation?

When technology is available in the classroom, we can use a high-tech approach to analyze the Body Benchmarks data. On the TI-84+ family of graphing calculators (including 83+, 84+, and the color devices 84+C and 84+CE), enter the data into L1 and L2. Then turn on the StatPlot and choose an appropriate window. [What else do you want to discuss with your students… do you have them choose a window or use ZoomStat? Are the students to be responsible for knowing the key presses?]

Then we can analyze the data using the options in the StatCalc menu. I used go directly to the LinReg choice to perform a linear regression. Then I recently discovered that at the bottom of this menu, there is option D:Manual-Fit. This is a high-tech version of the dry pasta best fit line! Note that this feature is available on ALL the TI-84+ graphing calculators, even though my images below are of the color devices.

When Manual-Fit is chosen, you are prompted to designate a location to store the equation. Press ALPHA-TRACE and select Y1. Then arrow down to “Calculate” and press ENTER.

On the graph, move the cursor to place the first point to model the line and press ENTER. If desired, the STYLE of the line can be changed by pressing GRAPH and choosing a new color or line style. Then move the cursor to the second point and press ENTER.

Now, you may have noticed that the line I have chosen is a bit below my set of data, although my slope is a reasonably good fit. BEFORE you are done, you can edit the two parameters M and B in the y = Mx + B equation. Simply enter the new value into the highlighted parameter. When you are happy with your line of fit, press GRAPH to select DONE.


Body Benchmarks Instructions and Record Sheet [click here for file]

Body Benchmarks Data that I used in these screenshots [click here for file]

Body Benchmarks Questions [click here for file]

Step-by-step overview of basic STAT EDIT, STAT PLOT, and STAT CALC functions [click here for file]

Extensions: What other measurements can you take and analyze? Consider armspan, foot length, hand size, and head or wrist circumference.

For more information on hands-on lab activities that generate linear data, see the book: Algebra Experiments 1 Exploring Linear Functions (1992) by Mary Jean Winter & Ronald J. Carlson. [link]