# 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.

# Looking Below the Surface

This week, I came across the treasure trove of problems available at ssddproblems.com, created and curated by Craig Barton.  SSDD stands for “Same Surface, Different Deep” and each set of problems contains four questions that have a similar presentation (a common image, shape, or context) but where the deep mathematical structures of the problems are very different¹.

SSDD problems are the essence of “Interleaving Different Practice”, one of the techniques that enhances learning from the book Make It Stick: The Science of Successful Learning². Page 85 explains how interleaving fosters conceptual understanding:

The trouble with interleaving is that it can actually impede initial learning, which is one reason teachers might not gravitate to the technique; however, the research “shows unequivocally that mastery and long-term retention are much better if you interleave practice than if you mass it.” (p. 50) [massed practice is focused, repetitive practice of one thing at a time.]

Craig’s SSDD problems are based on extensive cognitive science research showing that this strategy of interleaved practice is beneficial to learning by helping students discriminate between problem types and choose appropriate strategies to solve based on the underlying deep structure.  He says this is in contrast to the “auto-pilot” approach that can be common with students: they see a right triangle and jump right to the Pythagorean Theorem, whether or not it is useful.  Here is one of his question sets involving right triangles:

Students working on “auto-pilot” is one consequence of how we often teach math topics: practicing similar types of problems in a given lesson, then moving on to another type, then another.  When students are faced with cumulative tests or spiral review problems, they might not remember what they had “mastered” earlier.

Of course, I wanted to get in on the fun!  Craig tweeted yesterday that in the prior 5 days, his website grew from 80 question sets to 200 sets.  He has a helpful template available for creating an SSDD powerpoint slide. I scanned the list of topics on the website, looking for something fruitful that could also capitalize on technology and connect various representations, and didn’t yet have several sets posted.  I developed this question set on Quadratic Graphs:

My process was this: I first brainstormed the types of questions that could get asked about a quadratic graph. By the time Algebra 1 or Algebra 2 students are done with a unit on quadratics, they have covered a lot of ground, and can be confused by different forms of quadratic functions or when to use various solving techniques.

I decided that my focus would be on examining features of the graph to write different forms of the quadratic function, or use the graph’s features to solve equations and real-context problems.

It was challenging to choose one graph that could work for all these purposes:

1. Determine Vertex Form from a graph (and with a ≠ 0 to increase cognitive demand)
2. Determine Factored Form from a graph (required relating zeros of the function to its factors, again with a ≠ 0)
3. Use a graph to Solve an Equation (either find points of intersection with technology, or solve using quadratic formula; including simplifying radicals to exact form or finding decimal approximations)
4. Interpret a Graph in a Real Context (with different scales on horizontal and vertical axes)

I took screenshots from TI-84+CE SmartView Emulator software³.  The color and graph grid makes it possible for students to gather information by inspection and not have to rely on the calculator to analyze the graph’s features.  I’m pleased with how the problems turned out; any feedback is welcome.

You can find my set on ssddproblems.com under the topic “Equation of a Quadratic Curve” or this direct link and the original powerpoint is here.  One caveat to keep in mind about the problem sets on the SSDD site is that Craig works in England; there are some differences in vocabulary between American math and British maths, with both represented on the site.

I am excited about diving in to the rich resource at ssddproblems.com and looking below the surface for deeper learning!  Join me…

Notes and Resources:

1. There is much more background explanation about the SSDD problems and their research base on Craig’s website. They were introduced in his book “How I Wish I Taught Maths: Lessons Learned from Research, Conversations with Experts and 12 Years of Mistakes” (2018).

2. The book “Make It Stick: The Science of Successful Learning” by Brown, Roediger & McDaniel (2014) has a companion website MakeItStick.net.  The authors advocate for the strategies of spaced and interleaved retrieval practice, elaboration, generation, reflection, and calibration in order to optimize learning.  Another helpful blog post by Debbie Morrison about using these techniques in your teaching is here.

3. More information about the TI-SmartView CE Emulator Software for the entire TI-84+ family can be found on the TI Website here.  Currently there is a 90-day free trial available.  Take advantage of color demonstrations for your class, save work in progress, and insert screenshots and keystrokes in your handouts and assessments.

4. Michael Pershan wrote this blog post with his thoughts on the SSDD problem sets. He focused on geometry diagrams, for which I think the SSDD sets are extremely well suited.

# Great Thinking!

One of the bonuses of working with students one-on-one is that I can get a window into their mathematical thinking by asking questions and having them “narrate” their work as they proceed.  Several of my recent conversations demonstrate unique and flexible thinking that helped my students work through computations effectively.

How might we enable this type of visible thinking in our classrooms?  I tried these prompts:

• Tell me what you were thinking.
• What would you do to compute this if you didn’t have a calculator?
• What math operation did you do first?
• Can you explain the steps you did in your mind?

What follows are some observations of “great thinking” by my students, who used flexible number sense for success.

A. Using number sense for easier multiplying or dividing

B. Fantastic Fraction Flexibility

Here, noticing the multiplier is easier than solving the proportion directly:

In this question about slope, the student noticed that the first denominator must be 2 if the fractions are to be equal.

It is hard to find perfect square factors of large numbers; this strategy employs factors and the student never found the original product.

Here, the student used an alternative to the traditional process of rationalizing the denominator.

D. Factors and Multiples

Rather than set up proportions, this student noticed multiples

I’ve avoided the term “cancel” for some time, so here we talked about “dividing away common factors”.

E. Other interesting choices for computation and order of operations

The formula for area of a trapezoid can be thought of as “height * average of the bases”.  This student also applied binomial multiplication to compute without a calculator.

Here the typical order of operations would do the distributive property first.  Instead, the student saved a step.

Also on my mind as I write this are three things I have read this week.  First, James Tanton addresses a multiplication mistake made by an education official in this blog post, forgetting what 7 x 8 is.  Tanton suggests that the “best answer” in this case would have been “This is a tricky one. But I do have in my head that 7×7 is 49, so 7×8 is seven more than this: 56” because it highlights the thought process and downplays the memorization aspect of number facts.  His discussion is thought provoking (and he applies the same idea to trig identities too!).

The second item I was contemplating was in Sarah Carter’s Monday Must Reads post; she and I both liked David Sladkey’s “No eraser allowed” technique of insisting students leave their mathematical thinking available for the teacher to see.

The third concept informing my thinking is the book study I’ve been doing with colleagues from Teachers Teaching with Technology (#T3Learns) on “Visible Learning for Mathematics” (2017) by Hattie et al.  Hattie describes a spiral relationship between surface learning, deep learning, and transfer learning that enables students to achieve.  He notes that surface learning is NOT shallow learning (p 29) but is instead “made up of both conceptual exploration and learning vocabulary and procedural skills that give structure to ideas” (p 104) that “sets the necessary foundation for the deepening knowledge” (P 131) on the path to understanding. Techniques such as number talks, guided questioning, worked examples (including accurate work and work with mistakes), and highlighting metacognitive strategies all enhance the process of surface learning for students, according to Hattie.

I constantly remind students to “Show Your Mathematical Thinking”—which is the updated version of “Show Your Work.” By asking them to “narrate” their thinking, I am focusing on my students’ surface learning to build their foundation of skills and tools on their learning journey.

#### 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)We 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 number sense 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:

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

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?

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.

# 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:

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.

Divide “all parts” by a common factor.

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

{*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:

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.

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.)

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    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:

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.

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:

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:

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

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.