# How Else Can We Show This?

What I love about using calculator technology in my teaching is the “Power of Visualization” and the opportunity to examine math through different lenses.  The multiple representations available on TI graphing calculators—numeric, algebraic, graphical, geometric, statistical—allows me to push my students to approach problems in more than one “right way.”  By connecting these environments and making student thinking visible when we dig into a mathematical situation, we support students in productive struggle and deepen their understanding.*

Read my post on the TI BulleTIn Board Blog for two scenarios in which my students and I pursue multiple pathways to show and make sense of the mathematics at hand (with demonstration videos!)

### How Else Can We Show This?

Read the entire post at the above link, and here is a quick summary:

1. Riding the Curves and Turning the Tables: studying quadratic and polynomial functions.

VIDEO 1  Using different forms of quadratic functions to reveal graph features.

VIDEO 2  Using the graph-table split screen to see numerically what is happening at key points.

2. Absolute Certainty: solving absolute value equations and inequalities.

VIDEO 3 Using the graphical environment to support an algebraic solving procedure.

*Connecting mathematical representations and supporting productive struggle are two of the high-leverage mathematical teaching practices discussed in NCTM’s Principles to Actions: Ensuring Mathematical Success for All (2014).

# Action-Consequence-Reflection Activities for GeoGebra

When I choose to use technology in my math teaching, I want to be sure that the technology tool supports the learning, and helps students to develop conceptual understanding.  The Action-Consequence-Reflection cycle is one structure that I use towards this goal.  I’ve written about Action-Consequence-Reflection activities before, in this post and this post, and I recently had an article published in the North American GeoGebra Journal, “Using Action-Consequence-Reflection GeoGebra Activities To Make Math Stick.”

In the Action-Consequence-Reflection cycle, students

• Perform a mathematical action
• Observe a mathematical consequence
• Reflect on the result and reason about the underlying mathematical concepts

The reflection component is, in my view, the critical component for making learning deeper and more durable.  The article includes the following six activities that use the cycle to help “make the math stick” for students.  Each of the GeoGebra applets is accompanied by a lab worksheet for students to record their observations and answer reflective questions.

EXPLORING GRAPHS & SLIDERS:

The first two activities use dynamic sliders so that students can make changes to a function’s equation and observe corresponding changes on the graph.

In Power Functions, students control the exponent n in the function $f\left(x\right)=x^n$, and can toggle between positive and negative leading coefficients.

In Function Transformations, students investigate the effects of the parameters a, h, and k on the desired parent function.

INTERACTIVE VISUALIZERS:

Using the power of visualization to deepen understanding, the Domain and Range applet highlights sections of the appropriate axis as students manipulate linear and quadratic functions.

UNDERSTANDING STRUCTURE:

In the Rational Functions activity, students explore how the algebraic structure of functions relates to important graph features. The handout includes extensions allowing investigation of other rational function scenarios not already covered.

INVESTIGATING INVARIANTS:

The last two activities have students looking for invariants—something about the mathematical situation that stays the same while other things change.

In Interior & Exterior Angles, students investigate relationships among the angles of a triangle and form conjectures about the sums that do and don’t change as the shape of the triangle changes.

In Right Triangle Invariants, the applet links the geometry figure to a numerical table of values, and students discover several invariant properties occurring in right triangles.

PLANNING FOR REFLECTION:

Simply using these robust technology activities will not guarantee student learning and conceptual understanding; it is imperative that we as teachers plan for reflection by including focusing questions, discussion of students’ mathematical thinking, and clear lesson summaries with the activity.  Use the provided lab worksheets or adapt them for your needs.  Capitalize on the power of the Action-Consequence-Reflection cycle to make the math stick for your students’ success!

Notes and Resources:

This post contains excerpts from the full article (pdf available here) from Vol 7 No 1 (2018): North American GeoGebra Journal.

The North American GeoGebra Journal (NAGJ) is a peer-reviewed journal highlighting the use of GeoGebra in teaching and learning school mathematics (grades K-16). The website for the NAGJ is here.

My GeoGebra Action-Consequence-Reflection applets are in this GeoGebra book, or they can found by entering “kdcampe” into the GeoGebra search box.  Thanks to Tim Brzezinski, Marie Nabbout, and Steve Phelps for their assistance with some of the GeoGebra applets.

# Table Techniques

How might we use the graphing calculator Table feature to build conceptual understanding and support procedural knowledge?  Here are some ideas…

A. Dynamic Tables is one way I use the Action–Consequence–Reflection cycle in my teaching.  We generate a table so that students perform a mathematical action, observe a consequence, and reflect upon the mathematical meaning in order to build conceptual understanding.

In Algebra, when students learn to distinguish linear vs. exponential growth*, we enter the simple equations Y1 = 2x and Y2 = 2x on the TI-84+ family of calculators (TI-84+CE shown here).

To observe the growth of the functions numerically, adjust the Table Settings:  press 2nd WINDOW for [TBLSET] and set the Independent variable to AUTO and the Dependent variable to ASK (see above right).  This will allow students to generate the Y-values one at a time, rather than have them appear all at once.  Press 2nd GRAPH to view the table and press ENTER to generate each value, moving down each column.  View the video here.

How does the Y-value change as you move down each column?  Students should use mathematical language to describe what they observe.  Can you tell where the graphs would intersect?  Which equation grows faster?

Alternatively, set the Independent variable to ASK and the Dependent variable to AUTO.  In this setup, enter the X-value and both Y-values will fill in.  I use this when students are comparing two scenarios to see which grows faster.

Another example of Dynamic Tables is to build understanding of negative and zero exponents.  Enter the following into Y1: (the fraction is used to force fractional results in the table; make sure Answers are set to “Auto” under MODE).  Set Independent to ASK and Dependent to AUTO, then enter X-values of 5, 4, 3, 2, and 1 in the table.

• What mathematical process is happening to the X-value in each new row?
• Subtracting 1 each time, which means subtracting 1 from the exponent.
• What mathematical process is happening to the Y-value?
• Dividing by 2 each time.

Then ask, what do you predict will happen when X = 0?  When X = –1? When X = –2?  Change the base to 3 or 5 or 10 and observe.  Students can now explain what to do when a base is raised to a negative or zero exponent.

Here is what we wrote on the board as we explored the table on the calculator.

B. Noticing Invariants

Tables can also be used to notice Invariants—quantities, shapes or locations that do not change even though other things are changing.  Set up the table starting at 0 with an increment of 15, AUTO-AUTO and use Degree mode.  Examine the table for Y1 = sin(x) and Y2 = cos(x).  What do you notice?  Are any values equal?  Why does this occur?

My students and I don’t just examine the table, we also look at a geometric figure of a right triangle with side lengths that are easy to compute with, such as 3, 4, and 5.  Determine the sine and cosine of each angle and discuss how this relates to the table values.

Next, add another expression into Y3 as shown below left: use (Y1)2 = (Y2)2 to represent sin2x + cos2x, pressing ALPHA TRACE to access the Y-variables.  What does the table display?  Why is this so?  Again, refer to the right triangle figure—can you explain why this property is known as the “Pythagorean Identity”?

C. Aids for Factoring and Simplifying

So far we’ve used the table as a tool for inquiry; now we turn to using it as an aid for computation, number sense, and procedural fluency.  For factoring trinomials and simplifying radicals, students need to determine the numerical factors of a number.  When the number is large, or the student needs some scaffolding support, enter into Y1 the number divided by X and view the table (TblStart=1, ΔTbl=1).

For example, Y1 = 72/X  and the table clearly shows which numbers are factors and which are not, depending on whether a decimal remainder results (note that the “slash” version of the fraction bar forces decimal results).

For students needing assistance remembering perfect squares, cubes, or other powers, enter those functions into Y= and view the table.

When simplifying radicals into exact form, combine the two techniques to find perfect squares, cubes, etc. that are factors of the radicand value.‡

Of course, in both of these examples, students could simply enter calculations on the home screen until they hit upon the “right” divisor.  The table has the advantage of systematically presenting the information in one place.

D. Generating Sequences

If an explicit formula for a sequence is known, simply enter it into Y= and set the table to start at 1 with an increment of 1.  For example, the sequence 2, 5, 8, 11, … has explicit formula an = 2 + 3(n – 1).  In function mode, x is used in place of n.

This can also be accomplished in Sequence Mode.  nMin is the starting term number, u(n) is the explicit formula for term un, and u(nMin) is the value of the first term u1.  Note that the symbol n is found on the    key in sequence mode.  Here is the same sequence as above.

Although Sequence mode isn’t necessary for explicit formulas, it is very useful to generate a sequence recursively**.  This time, express u(n) in terms of the previous term u(n–1).  The u(n–1) variable is found by pressing ALPHA TRACE (or type it directly with the u above the 7 key).

Back in Function mode, I’ve also discovered that helpful sequences can be created with the Table.  When Precalculus students studied the Binomial Theorem, they often wrote out several rows of Pascal’s Triangle rather than use the nCr values for the appropriate power.  The table comes to the rescue: enter nCx into Y1, with the numerical value of the power for n.  Begin the table at 0 and increment by 1, and the appropriate row of Pascal’s Triangle is displayed.

Whether you use the table to enable investigation and inquiry, or use it to support numerical and procedural fluency, take these Table Techniques to your classroom!

Notes and Resources:

♦◊♦ This blog post was revised and expanded and published in the May 2019 issue of NCTM’s Mathematics Teacher Journal. If you can’t access, contact me directly. ♦◊♦

*The complete activity using Dynamic Tables to explore Linear and Exponential Growth is here.

**Recursive sequences can also be generated directly on the Home screen of the TI-84+ family, as an alternative to Sequence Mode.  Simply enter the value of the first term, then perform the recursive operation on the ANS, and press enter for the second term.  Finally, press ENTER as many times as desired to generate the sequence.  Below left is the same sequence discussed above; below right is the sequence based on paying off a $500 credit card bill with 24% annual interest and monthly payment of$75.

‡Thanks to Fred Decovsky for this suggestion.

# That Voice In Your Head

When I work with students one-on-one, I get a unique window into their thinking.  Everyone has a test this week, including several students who are taking the AP Calculus exam.  As we are preparing, I’ve noticed the constant push-pull of the conceptual vs. procedural  debate, because students need to finish by the end of the hour with me, and go away knowing “how to do it.”  I know that giving them conceptual background will help make their learning more durable, but some of them resist going beyond the procedure and don’t welcome the “why does it work that way” explanation.

I’ve found myself with students referring to “that voice in your head” in order to get them to communicate their mathematical thinking, to connect the new knowledge to past related topics, to think about the underlying concepts for each process and help them build structures to support their understanding.

Here’s what I want that voice to be saying:

1. What Does It Look Like?

Knowing what the graphs of various function families look like allows for easy transformations using parameters.  This week we transformed graphs of log functions and rational functions:

In addition, students found limits of functions without technology, based on what they knew of the nature of the graphs.

To find this limit

it is helpful to know these graphs:

To find this limit, think about the end behavior and how to determine horizontal asymptotes for rational functions:

2. What Am I Looking For?

A colleague noted recently that math is all about the verbs: solve, simplify, evaluate, and so on.  When students pay attention to what they are being asked to do, the process follows easily.

For example, while solving equations, students are looking for the variable, which is located in different places in linear, quadratic, exponential, logarithmic and rational equations.  Finding where the variable is now can guide students to a process for solving: inverse operations, factoring and zero product property, converting between exponential and log forms, condensing to a single log, finding a common denominator to clear fractions, etc.

This is an example of an exponential function in a quadratic format; relating prior knowledge of quadratics and “looking for x” enabled the student to solve successfully.

3. What Are The Tools In My Toolbox?

When faced with a problem, think about what tools are available.  With rational expressions and equations, students begin by factoring, and then often (but not always) find a common denominator.

One of my students could easily add these fractions with an LCD:

But had trouble solving this equation:

She had learned one strategy for the expression, and then there was a “new” strategy for the equation that involved multiplying through by the LCD to “clear fractions”.  She couldn’t keep track of which factors remained and was suceptible to errors.

Instead, we built on the strategy of creating common denominator fractions; once all the fractions have the same denominators, she can work with only the numerators and solve successfully:

We also used this strategy to simplify a complex fraction; we created a single fraction in both the numerator and denominator, then remembered that a fraction means DIVIDE:

The AP Calculus students also think about their toolbox when faced with an integral problem: what are the integration strategies they can use as tools?

• Do I know an antiderivative? (Can I simplify algebraically to make one)?
• Is there a known geometric area I can find?
• Is part of the integrand the derivative of another part? (U-substitution)
• Can a trig relationship help me rewrite the integrand?
• Does the integrand contain a product? (Integration by parts)
• Is this a definite integral on the calculator section? Use the calculator!

4. Where Are The Trouble Spots?

When finding the domain of a function, focus on the numerical values that “cause trouble.”  Where should we look for trouble in these?

Finding limits and derivatives of piecewise functions also puts students on the hunt for trouble, and certain integrals need special treatment due to discontinuities:

5. What Should I Write Down?

Write enough to show your mathematical thinking to a teacher/reader who doesn’t know you.  Write enough to be clear and get it right.  No bonus for doing it in your head on multiple-choice, and there is definitely a penalty for doing too much in your head and getting it wrong.  And on free-response questions on the AP (and most questions on teacher-created tests), you need to show work that supports your conclusion.

6. Does My Answer Make Sense?

Even if calculators aren’t available, students can estimate square roots, logs, and other results.  For example,

In word problem situations, does the answer make sense?  If Amy can do the job in 4 hours and Josh can do it in 6 hours, together they should take less time than either of them working alone.  And don’t forget appropriate units if a problem is situated in a real context.

7. How Are These The Same/Different?

Analyzing the small differences between examples helps students home in on important features.

What intercepts and asymptotes will these functions have in common?

(Calculus) What are the different requirements and results of the Intermediate Value Theorem and the Mean Value Theorem?  What is the difference between average rate of change and average value of a function?

Finally, I ask my students how they are feeling about the material:  Are you finding this unit easy or hard?  What parts are more difficult for you?  If you find this to be challenging, you need to put on your thinking cap.  Saying “I can’t do it” gets in the way of your understanding; instead, say “I can’t do it YET, I’m learning” and focus on the MANY things that you do know.  Don’t overthink the easy things or overlook the tough details.  Being confident is an important ingredient for your success.

You’ve got this.

NOTES & RESOURCES:

Two Geogebra applets for transformation of functions are found here (multiple parent functions) and here (rational functions).

More on “easy” and “hard” labels and their impact on students in these blog posts: “Things Not To Say” and “The Little Phrase That Causes Big Problems”.

# 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:

to become  and

to become

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 as.

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.”
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    to .
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.

MarbleSlides–Rationals

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.

NOTES & RESOURCES:

*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.

# 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 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    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   button to access the shortcut menus above the  , , and   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 .

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.

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  to change the increment , or press 2nd WINDOW to access the TBL SET screen.Success! 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 his own problem and help some classmates as well.  One of our approaches to learning is to “use what you know.”**

NOTES & RESOURCES:

*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.

# 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:

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):

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.

1. Graph a rational function

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:

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?

[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

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:

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.

NOTES & RESOURCES:

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.