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.