# 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).

# Testing Tips: Using Calculators on Class Assessments

If you’ve been using TI graphing calculators in your teaching, you may have contemplated how to implement the calculators for in-class testing.  Whether you are giving a short quiz, a chapter test, or end-of-term exam, read my post on the TI BulleTIn Board Blog for some tips for how to use TI calculators successfully on class assessments.

### Testing Tips: Using Calculators on Class Assessments

There is much more in the full post, but here is a summary:

• Determine the Objectives: decide which math skills and problems you will assess with and without the calculator.
• Separate the Sections: separate the calculator and non-calculator problems into two sections.
• Set up the Handhelds: to be sure the calculators are useful tools for students and don’t interfere with assessing their math knowledge, set up the handhelds for security and equity.
• Electronic Quizzes with TI-Nspire CX Navigator: take advantage of electronic quizzes if your classroom has the TI-Nspire Navigator System.

# Testing Tips: Using the TI-84+ on the SAT

The fall dates for the SAT and PSAT tests are around the corner.  The TI-84 Plus CE and the entire TI-84 Plus family of graphing calculators are approved for use on the Math–Calculator section of these College Board tests. Read my post on the TI BulleTIn Board Blog for tips on how to leverage your TI-84 Plus for success on test day…

Good luck!

# 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.

# 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!

# Fun Features of the TI-84+ Family

Recently, I was a panelist for a TI Calculators Webinar that happened to follow a new Operating System release for the TI-84+CE color calculator and its accompanying Apps and computer software.  The new calculator features were not our primary focus, but since OS 5.3 was just released, we tried to highlight them when appropriate. This post summarizes a few features of the new OS version 5.3; I want to share in more detail these fun features that will help my teaching and my students.

1.New shortcut key for the fraction template. I use the fraction template    to make the calculator inputs match the math notation.  It has been available for the entire TI-84+family for several operating systems.  Press ALPHA-Y1 to access the shortcut menu for the fraction and mixed number templates. On the color versions of the TI-84+, it is also found in the MATH>FRAC menu.

And now, there is an even faster way on the TI-84+CE: press

2. Piecewise Function Graphing now allows entry that looks like the textbook! Press MATH and select B. piecewise. Choose the number of pieces and then enter them using the inequality operators in the TEST menu (2nd MATH) or the CONDITIONS submenu.  Discuss with students whether it matters which piece comes first in the piecewise definition.

You might notice that in Y= the viewable area is limited when entering a piecewise function.  If desired, you can enter the piecewise function on the Home Screen and store it to one of the graphing function variables.  You will need to start and end the piecewise function with the quotation symbol (press ALPHA and the PLUS button).

Don’t have a CE but want to graph piecewise functions?  Instructions are found HERE and this method can prompt a rich discussion of how to use a logical operator in a math expression (the inequality statement will represent 1 if it is TRUE and 0 if it is FALSE).  Try TRACING the piecewise graph.  Where does the Y-value exist and not exist?

I like using piecewise functions on the calculator: it enables a graphical confirmation of a complicated algebraic expression, and that helps my students build their understanding.  It also gives us access to many real contexts for problem-solving.

3. Updated Transformation Graphing App. The newest Transformation Graphing App allows TWO transformable functions in Y1 and Y2 plus you can graph other static functions in Y3 to Y0 and up to three StatPlots.  You still use the four parameters A, B, C, D and you can now access several stored parent functions for lines, quadratics, cubics and trigonometry.

I like to challenge students to “Match the Graph” with a static function graph whose equation I haven’t shared (see below left, can you transform the blue graph to match the pink one?).  Or put data in a StatList and transform a function for a best fit curve.

It is now easier to move back and forth between the graph and the SETUP screen, and the “TrailOn” feature has moved to this more logical location [it was available in prior versions of Transformation Graphing, accessed from the FORMAT menu by pressing 2nd ZOOM.] See above right for the trail of quadratics.

And the entire Transformation Graphing process is faster than it was before (be patient if the calculator seems slow to respond and check for the “busy” indicator in the upper right corner of the color calculator screens).  HERE is a “How To” file for the Transformation Graphing App.

4. Easy storage of a Tangent Equation. When drawing a Tangent line to a graph, a MENU can be accessed using the GRAPH button (before entering a point of tangency) to store the equation directly to Y=.  This makes it easier to examine the equation of the tangent line in relation to the equation of the original function.

Entering the X-value for the point of tangency can be done by entering a value with the numeric keypad, or tracing to the desired point.  I prefer using the ZOOM DECIMAL window for “friendly” trace increments.

5. Finally: updating to a new OS is easy. You need a USB-calculator cable or a calculator-to-calculator cable.  Instructions are HERE and the OS is a free download from the TI WEBSITE.  Use the (free) TI-Connect™ CE Software or link to an already updated calculator.  For the new OS 5.3 for the CE calculator, there is a “Bundle” file that updates the OS and the Apps at the same time.

*Keep in mind that all the feature enhancements have alternatives for older OS versions. And even if your classroom has a mix of TI-84+ family devices, the TI-SmartView™ emulator computer software displays 3 distinct environments, so you can demonstrate in the CE emulator to take advantage of the features for the whole class.

6. Save Emulator State is back!! (I almost forgot to mention this) The TI-SmartView™ CE software NOW has the ability to “Save Emulator State” so the teacher can save work where you are when the bell rings at the end of the period and you can open it back up when that class returns the next day.  Or you can prepare your SmartView calculator in advance with your functions, lists, programs, etc. so that everything is loaded efficiently when you need it.  Instructions are HERE for using Emulator States.

Enjoy the new and old fun features! Reach out to me if you have any questions.

Resources:

The  “Making Math Stick” Webinar On-Demand is HERE and the activities discussed are HERE.  The new calculator features were not our primary focus, but since OS 5.3 was just released, we tried to highlight them when appropriate.

A great summary of some not-so-new features of the TI-84+ Family was written by my webinar co-panelist John LaMaster and is found HERE.

# 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.

# Body Benchmarks!

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

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

Here are the Instructions: [link to file]

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

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

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

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

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

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

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

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

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

Materials: