Category Archives: Algebra

Action-Consequence Advantage!

Using Technology to Make Math Stick

How might we enable students to grasp mathematical concepts and make their learning durable?  One approach is to use the sequence of Action-Consequence-Reflection in lesson activities:

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

The ACTION can be on a graph, geometric figure, symbolic algebra expression, list of numbers or physical model.  Technology can be used in order to have a quick and accurate result or CONSEQUENCE for students to observe.

The REFLECTION component is the most important part of this sequence; without this, students might not pay attention to the important math learnings we intended for the lesson.  They might remember using calculators, computers, ipads, or smartboards, but not recall what the tech activity was about.  And if they did learn the concept in the first place, the process of reflection helps make the learning stick—it is one of the cognitive techniques shown to make learning more successful.*

Students can reflect in many ways: record results, answer questions, discuss implications with classmates, make predictions, communicate their thinking orally or in writing, develop proofs and construct arguments.  The intended (or unexpected) learnings should be summarized either individually or as a class in order to solidify the concepts, preferably in a written form. **

In my one-on-one work with students, we often fall into the “procedural trap” in which my students just want to know “what to do” and don’t feel that the “why it works” is all that important (I’ve written about this before here). Also, our time is limited with many topics to cover.  But this past week, I was able to sneak in some Action-Consequence-Reflection with two students because they had mastered the prior material and were getting ahead on a new unit. It was a great opportunity to have them discover a concept or pattern for themselves, far better than simply being told it is true.

For each of these, I used a simple REFLECTION prompt:  What do you observe?  What changes?  What stays the same?

Student #1: Polynomial Function End Behavior, Algebra 2 (or PreCalculus)End Behav QWe used a TI-84+CE to investigate the polynomials.  We began with the even powers on a Zoom Decimal window, and my student noticed that higher powers had “steeper sides”.  I asked “what are the y-values doing to make this happen?” and we noticed the y-values were “getting bigger faster”.  We then used the Zoom In command to investigate what was going on between x = –1 and x = 1, and noticed that higher powers were “flatter” close to the origin because their y-values were lower.

 

Why did this happen? Take x = 2 and raise it to successive powers, and it gets bigger. Take x = ½ and raise it to successive powers and it gets smaller. We confirmed this with the table:

Setting the Table features to “Ask” for the Independent variable but “Auto” for the Dependent made the table populate only with the values we wanted to view. Fractions can be used in the table, and we used the decimal value 0.5 for clarity. Another observation was that the graphs coincided at three points: (1, 1), (0, 0) and (–1, 1).

In fact, none of this was what I “intended” to teach with the lesson, but it was mathematically interesting nonetheless, and my student already had a deeper appreciation for the graph’s properties. We moved on to the odd powers, and the same “steep” vs. “flat” properties were observed:

AC end beh 6

 

Then I asked my student to consider what made the graph of the even powers different from the graph of the odd powers, and what about them was the same, finally getting around to my “lesson”.  He noticed that even powers had a pattern of starting “high” on the left and ending “high” on the right, while odd powers started “low” on the left but ended “high” on the right.  We made predictions about the graphs of x11 and x12, to apply our understanding to new cases.† Then we moved on to the sign of the leading coefficient: when it is negative, our pattern changed to even powers “low” on the left and “low” on the right, with odd powers “high” on the left and “low” on the right.

All of this took just a few minutes, including the detour at the beginning that I wasn’t “intending” to teach.

Student #2: Interior and Exterior Angles of a Triangle, Geometry

Int Ext 1

We began with a dynamic geometry figure of a triangle which displayed the measurements of the three interior angles and one exterior angle.  I asked my student: “What do you notice?” and “what do you want to know about this figure?”  We dragged point B around to make different types of triangles.

This motivated my student to wonder about the angle measures; he was familiar with Linear Pair Angles, so he noticed that ∠ACB and ∠BCD made a linear pair and stated they would add up to 180°.  I then revealed some calculations of the angle measures, which dynamically update as we changed the triangle’s shape (the 180° in the upper right is the sum of the 3 interior angles).  What changes? What stays the same?

Int Ext 6

We noticed that both sums of 180° were constant‡ no matter how the triangle was transformed, but the sum of the 2 remote interior angles kept changing, and in fact, matched the measure of the exterior angle.  My student recorded his findings in his notebook, and then I asked, “can we prove it?”.  It was easy for us to prove the Exterior Angle Theorem based on his previous knowledge of the sum of the interior angles and the concept of supplementary linear pairs.  This student loved that he had “discovered” a new idea for himself, without me just telling him.

Even though these concepts are relatively simple, I feel that using a technology Action-Consequence activity made the learning more impactful and durable for my students, and I believe it was more effective than just telling them the property I wanted to teach. It took us a few more minutes to explore the context with technology than it would have to simply copy the theorems out of the book, but it was worth it!

 


Notes and Resources:

*For a full elaboration of cognitive science strategies for becoming a productive learner (or designing your teaching to enhance learning), see Make It Stick: The Science of Successful Learning by Peter C. Brown, Henry L. Roediger & Mark A. McDaniel, 2014.  Website: http://makeitstick.net .

**Written summaries allow students to Elaborate and Reflect on the learning, two more of the cognitive strategies.  In addition, by insisting that students record the results of the work, the teacher sends the message that the technology investigation comprises important knowledge for the class.

†Making predictions is a way to formatively assess my students’ understanding.  It also is a form of Generation, another cognitive strategy that makes the learning more durable.

‡This is an example of an invariant, a value or sum that doesn’t change, which are often important mathematical/geometric results.

Here are the two activities discussed in the post. Note that the Geogebra file for Power functions has the advantage of having a dynamic slider, but students won’t view the graphs at the same time, so won’t notice the common points or the graph properties between –1 and 1.

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Searching for Structure

Recently, I read on Twitter some teachers’ frustration with students who just want to know the quick procedures to do the math at hand and don’t have much interest in the meaning of the underlying concepts.  I often come across this dilemma in my one-on-one work with students; in this tutoring role I especially feel the pressure to teach the “how-to” for an upcoming test and don’t always have the time to explore the “why” with the student.  I wrote a bit about this tension before in this post.

Another conversation on Twitter was specific to Algebra 2, about how to build on past knowledge even when some/all of the students seem not to remember that past knowledge.  How might we deepen students’ understanding and not simply retread the procedures?

I was faced with these dual dilemmas when I worked with a student this week reviewing Complex Numbers and Quadratic Equations for an upcoming test.  My approach: pay attention to mathematical structure.

A. Fractions involving imaginary numbers:

imag numbers

These three examples, examined together, allowed us to explore how to handle a negative value in the radicand (“inside the house”) and also how to handle a two-part numerator* with a one-part denominator.  Once the imaginary  i  was extracted and the radical simplified as much as possible, we took a look at when we could and couldn’t “simplify” the denominator.

I wanted to help my student avoid the common mistake of trying to “cancel”** when you can’t.  We used structure to explain.  When there is a two-part numerator and a one-part denominator, you can do one of 3 things:

“Distribute the denominator” to make two separate fractions.  I find this is the most reliable routine to avoid mistakes.

distribute-denominator-e1507861388968.jpg

Divide “all parts” by a common factor.

Divide all

Factor a common factor (if any) in the number, then simplify with the denominator.

factor-common.jpg

{*I wasn’t sure if the numerators qualify as “binomials” since they are numeric values, but my student and I discussed how they have two terms on top and one term on bottom, which can be challenging to simplify.  This structure will be encountered later when solving equations using the quadratic formula.}

{**I have avoided the use of “cancel” since I became more familiar with the “Nix the Tricks” philosophy of using precise mathematical language and avoiding tricks and “rules that expire”.  See resources below for more on this.}

B. Operations with complex numbers:

Again, we looked at a set of three problems to examine structure, which leads us to the appropriate procedures:

operations2

What is the same and different about #10 and #11?  What operation is needed in each?  Which is easier for you?

What is the same and different about #11 and #12?  What do you call these expressions:  4 – 5i and 4 + 5i ?  If you notice this structure, how does the problem become easier?

This led to a fruitful discussion of combining “like terms”, what is a “conjugate”, and whether it mattered if the multiplication was done in any particular order.  My student had been taught to always list answers for polynomials in order of decreasing degree, as in x2 – 2x + 1, so he was writing any  i2  terms first.  This isn’t wrong, but the rearranging of the order of the multiplication could have caused a mistake, so we talked about whether  is a variable or not, and when might it be helpful to treat it like a variable.

By noticing the structure of conjugates and why they are used, we got away from merely memorizing math terminology and instead added to conceptual understanding.

C. Using the discriminant

The discriminant is one of my favorite parts of the quadratic equations unit!  Students must pay attention to the structure of a quadratic equation (is it in the standard form ax2 + bx + c = 0 ?) before using the discriminant to give clues about the number and type of solutions.

discriminant

Rather than memorize what the discriminant means, look at where it “lives” in the quadratic formula.  It is in the radicand, which is why a positive value yields real solutions and a negative value does not.  The radical follows the ± , which is why nonzero discriminants give two solutions (either a real pair or a complex pair).  And the two solutions are conjugates of each other, something that I hadn’t really thought about when I got real solutions using the quadratic formula.  (And there is a nice surprise when you examine the two parts of the “numerical conjugates” and relate them to the graph of the quadratic. See note below for more on this.)

conjugates

I recently read this post about one teacher’s success having students evaluate the discriminant first, then tackle the rest of the quadratic formula.  Her strategy integrates the use of the discriminant with quadratic formula solving, instead of making it a stand-alone procedure.


Calculator Note: when evaluating the Quadratic formula on the TI-84+ family of calculators, use the fraction template  Untitled  to make the calculator input match the written arithmetic.  Press ALPHA then Y= for the fraction template, or get it from the MATH menu.  Then edit the previous entry for the second solution (use the UP arrow to highlight the previous entry and press ENTER to edit).  Here is #26 from above:

Capture 1

Another Calculator Note: the TI-84+ family in a+bi mode can handle the addition and multiplication questions #10-12, so if you are assessing student proficiency on these skills, have them do it without using the calculator.  The color TI-84 Plus CE can operate with an imaginary number within the fraction template, such as questions #4-6 and anything with the quadratic formula.  You can use either the  i  symbol (found above the decimal point) or a square root of a negative number.

Capture 2 arrow

However the B&W TI-84+ can’t use an imaginary number within the fraction template.  Use a set of parentheses and the division “slash” for this to work:

Capture 1 BW


Notes and Resources:

Nix the Tricks website and book: nixthetricks.com.  And lest you think that “Distribute the denominator” is yet another trick, consider this:  The fraction bar is a type of grouping symbol (like parentheses) and it indicates division.  Dividing is equivalent to multiplying by the reciprocal.  So the “distribute the denominator” work for #4 above is also this: Capture

Three articles about “Rules That Expire” have been published in the NCTM journals.  Currently all three are available as “FREE PREVIEWS” on the website.

“Look for and make use of structure” is one of the Standards for Mathematical Practice (SMP #7) in the Common Core State Standards found here: http://www.corestandards.org/Math/Practice/

The “nice surprise” about the solutions to a quadratic equation written as “numerical conjugates” and their relationship to the quadratic graph was pointed out to me by Marc Garneau.  His post here gives more detail and a student activity to go with it.

Thanks as always to the #MTBoS and #iTeachMath community on Twitter for great conversations!

That Voice In Your Head

question 2When 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 capture 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:

Screen Shot 2017-05-09 at 9.50.12 PM


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.

IMG_5386


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: rat expr

But had trouble solving this equation: rat eq

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:

IMG_5396

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:

IMG_5397

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?

domain functions

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?

File May 09, 11 50 40 PMWrite 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,

estimate

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?

IMG_5399

(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?


8. How Do I Feel About This?

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:

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:

screen-shot-2017-02-22-at-11-48-07-am

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.

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.

marbleslides_-rationals22-question

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:

marbleslides_-rationals22

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.

capture-4

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 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:

screen-shot-2016-10-05-at-1-38-41-pm

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.

capture-2

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.

capture-7-new

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 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:

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:

Expand

  1. Condense to a single log:

Condense

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.