Powerful Pairs

Last month, Jenna Laib (@JennaLaib) tweeted about whether there was a name for pairs of fractions that added up to 1, such as ¼ and ¾.  And Susan Russo (@Dsrussosusan) wondered what to call numbers that add up to 100%.

That led me to think about all the fabulous number pairs we teach to our students; read on for these and other Powerful Pairs in math class!

1. Addend Pairs

Jenna’s original idea came from a common use in elementary classrooms: what pairs of numbers add up to 10? This is a key concept for students, and knowing the relationships 1+9, 2+8, 3+7, 4+6, 5+5 and all the reverse orderings is one of the content standards beginning in Kindergarten*.  It is a foundational skill that so much more mathematics is built on.

Later, students explore the addend pairs that make 100, first with groups of 10 (e.g. 70+30) and later for any integer between 0 and 100.  Other “landmark numbers” might be explored as well (some teachers call these “number bonds”).

Jenna didn’t immediately have a name for the pair of fractions whose sum = 1, so she used “complementary” with some students but asked the math twitter community (#MTBoS and #iTeachMath) what we thought.  Suggestions included: unit pairs, unit partners, unit pair fractions (which are not necessarily unit fractions), and unit complement.  Susan’s query for pairs like 85% and 15% that we might use in percent decrease problems, yielded “complementary percentages” and the quirky “cent-lement”.

I think “complement” is a good term to use here, even though it has some other mathematical meanings.  In probability, two events are complementary if their probabilities add up to 1 (because one event only happens when the other does not); this is actually exactly what Susan was asking about.  

2.  Zero Pairs

Numbers that are opposites have the same absolute value but different signs, so their sum is zero, thus the name zero pairs.  This may seem mundane, but this is one of the “inverse operations” that allows us to solve algebra problems such as X + 5 = 12 or X – 3 = 10.  We “undo” the operation that is shown by creating a “zero pair” (and I prefer these terms instead of “move the number to the other side”).

Howie Hua (@Howie_Hua) uses the property that a number and its opposite add up to zero in arithmetic problems; check out his video here. You can use this property to make computations like 117 + 58 easier; just put in a zero pair of + 2 – 2 like this:

3. Factor Pairs

Factor pairs are pairs of numbers that multiply to equal a certain product number. We can model products with area tiles** and notice that there are 3 ways to factor the number 12, for example. 

Prime numbers will have only 1 pair of factors, and square numbers will have one factor pair in which the numbers are the same (the square root). 

Factor pairs are powerful tools for students to use when doing arithmetic and algebra.  One of my go-to computation strategies is to decompose into known factor pairs and regroup conveniently (see below).  In algebra, knowing factor pairs helps with factoring trinomials, especially if one of the coefficients is prime.

4.  Conjugate Pairs

Conjugate pairs are numbers or binomial expressions that “play nicely together”†.  They are of the form A + B and A – B, meaning that the first terms are the same and the second terms are opposites.  They crop up in several places in high school mathematics.

Algebra 2 and PreCalculus students learn about complex conjugates a + bi and a – bi , with equal real parts and opposite imaginary parts.  When multiplied, the product of complex conjugates is the real number a2 + b2 with the imaginary part eliminated.

Other conjugates work in a similar way.  Conjugates involving a radical when multiplied will remove the radicals, which is helpful if you happen to be rationalizing a denominator. 

I’ve started discussing conjugates with my Algebra 1 students so they have an easier time recognizing them‡. The conjugates (a + b)(a – b) are the basis for the “difference of two squares” binomial pattern a2 – b2.  We called the conjugates the “add-subtract pattern” at first and noticed that multiplying conjugates yields two terms that cancel away (“add to zero”) so we can save ourselves some of the distributing steps we usually do (the O and the I in “FOIL” if you use that terminology).

The two solutions to a quadratic equation will be conjugates, which is easiest to see when using the quadratic formula to solve.  The common use of ± instead of writing separate solutions can mask the idea (for some students) that the two solutions will be equal distances from the line of symmetry x = –b/2a.

5.  Geometry Pairs

Our final set of powerful pairs are some wonderful angle pairs in Geometry. Here we use the term “complements” again, since Complementary Angles are a pair that add up to 90°. Complementary angles can be made by partitioning a right angle, but they also appear separately, such as the two acute angles in a right triangle (angles A and B in the triangle below).

Supplementary Angles are a pair that add up to 180°.  If supplementary angles are adjacent (having a common side) then they are called Linear Pair Angles like these angles 1 and 2, making a Straight Angle.  Linear Pair angles and Vertical Angles are two useful angle pairs that can be assumed from a diagram without other markings§, astute Geometry students know.  Non-adjacent supplements pop up inside trapezoids and parallelograms (or anywhere parallel lines are cut by a transversal).

I wondered if there was a name for angles that add up to 360° since that is another useful pair that shows up when we study circles. Indeed, there is! They are called either Conjugate Angles or Explementary Angles¤: each one is the Explement of the other and create a Complete Angle.  When two arcs of a circle have the same endpoints and don’t overlap (one is the major arc, the other is the minor arc), they are called Conjugate Arcs and add up to 360° because they create a complete circle. Minor arc IJ and major arc IKJ are conjugate arcs in the circle below.

What a list we’ve covered of pairs of numbers that add to 1, 10, 100%, 90°, 180°, 360° and more! Even if you don’t remember all the names for these powerful pairs, the important thing is to be able to use them productively for doing math!

Notes & Resources

The original tweet from Jenna is here and from Susan is here.

*Addend pairs that equal 10 are a Kindergarten standard: CCSS-M-K.OA.4 For any number from 1 to 9, find the number that makes 10 when added to the given number.

**These factor pairs are modeled with number tiles available on the Mathigon Polypad at https://mathigon.org/polypad. Their game Factris is a great way to practice factor pairs visually, but be warned, it’s addictive!

† This phrasing came from Beth Hentges (@bethhentges) who noticed that the Spanish verb for playing is part of the word conjugate in this tweet.

‡ The difference of two squares pattern has always been part of the Algebra 1 curriculum, but the binomials that they factor into are rarely called conjugates at this level. I have been using the term both as a preview for later math classes, and also for its power to highlight the useful pattern.

§ Vertical angles and linear pair angles are two of the most useful angle pairs in geometry because these relationships hold true anytime lines intersect. When students are studying angles formed by parallel lines and a transversal, they need to learn several new angle relationships. I find it helpful to distinguish between the properties that are true even when the lines aren’t parallel, from the properties that are only true when the lines are parallel.

¤ These and other terms were defined in The Penguin Dictionary of Mathematics 2nd Edition, edited by David Nelson (Penguin: 1998). Another helpful source is The Words of Mathematics: An Etymological Dictionary of Mathematical Terms used in English by Steven Schwartzman (MAA: 1994).

I ❤️ Candy Math!

It’s that time of year again, when Valentine’s Day candy is all over store shelves. Grab some “conversation hearts” – SweeTart™ Hearts are my favorite – and explore exponential models with your students!

Candy Conversation Hearts

First, let’s model exponential growth.  Begin with 5 conversation hearts in a small cup for each student or group.  The independent variable is “Toss Number” and the dependent variable is “Total Candy” as shown in the data collection table below.  Before any tosses have happened, we have 5 candies, so fill in 5 in the top row.

Toss the candy (gently) onto a flat surface.  Count the number of candies with writing FACE UP, and add that many more to the original 5.  So if 2 hearts show writing, add 2 more candies to the cup, and record 7 total for toss number 1.  Toss the new total candies onto the table, count the number with writing face up, add in that number of candies, and record the new total.

Continue tossing and adding until you have used up your candy supply.

Once the data is collected, create a scatterplot on your technology of choice (or graph paper). Students can use the general exponential model with sliders to model the data.

Let students experiment with different values of A and B to get a good fit. If desired, use an exponential regression to generate the equation that models the simulation data.

Data, exponential model & sliders in Desmos
Data, exponential model & sliders in GeoGebra
Scatterplot & exponential model on TI-84+CE (left) & TI-Nspire CX (right)

What is the “right” model equation? Theoretically, half of the candies will land face up, so there is 50% rate of increase expected, so the base should be 1 + 0.5 or 1.5.  The initial amount is 5, giving the theoretical equation

Discuss with students how close their simulation model is to the theoretical equation. How could we improve our experimental results?


The second part of the activity is “Disappearing Desserts!” to explore exponential decay. Begin with 40 candies, combining groups of students if necessary.  This time, remove the candies that have writing face up, continuing for several tosses until the candy is depleted or the data chart is filled up.  The theoretical model would be

since there is a 50% expected rate of decrease, so the base should be 1 – 0.5 or simply 0.5.

If you’d like to try a different probability experiment, use dice. Add in or take away dice with a certain number, 6 perhaps. The expected rate of increase or decrease would be ±1/6, so the base of the exponent would be 7/6 in the growth scenario and 5/6 in the decay experiment.

Here is the student activity sheet for both the CANDY TOSS and DISAPPEARING DESSERTS explorations, along with a brief teaching guide. Be sure to discuss the results and activity questions with the class afterwards to solidify understanding and clarify misconceptions.


If you don’t have candy1, or don’t want to do the activity that way, you can also use a random number generator on your technology platform of choice.  If you are working with a whole class, be sure to set the “random seed” first so that all the students don’t generate the same set of numbers!2

The random number command for each technology platform is:

  • All TI Calculators: RandInt(lower bound, upper bound, number of trials).
  • Desmos: enter the list of potential values in square brackets: [lower bound,…,upper bound].random.
  • GeoGebra: RandomBetween(lower bound, upperbound).

And there are many random number generators available on the internet. Many simulations are possible using a random number generator that aren’t easy to do in a physical experiment: roll 1000 dice, perhaps?

For further details on each technology platform, check out the notes & resources below. Enjoy a sweet Valentine’s Day ❤️, and hope that your math students’ knowledge grows exponentially!

SweeTarts™ Conversation Hearts

Notes and Resources

1 Variations include using coins, dice, polyhedral dice, spinners, other candies with writing on one side (like M&Ms), etc.

2 Most calculator and computer random number generators are “Pseudo Random Number Generators” because they have an algorithmically-determined list of numbers that is nearly indistinguishable from random. However, they default to the same beginning number, so it is important to “seed” the random command so it generates a number from somewhere else in the list. Imagine my surprise when I didn’t do this, and my entire Algebra 2 class generated the same “random number” after I had explained why randomness was important!

Here are some more details on lists, scatterplots, sliders, regression equations, and random numbers on each of these technology platforms:

Dividing with Zero?

Understanding Zeros in Fractions & Slopes

How might we help students make sense of fractions containing zeros? The difference between and can be confusing.

Everyone knows from elementary school that you “can’t divide by zero” but that rule might not be enough to help learners interpret fractions and slopes that include a zero in the numerator or denominator.

Let’s take a quick look at the “sharing” interpretation of division, then turn to technology for numerical and graphical representations of the amazing ZERO.

Sharing Pizzas

We can interpret a fraction as division of two numbers with a sharing metaphor1. I like to use the idea of money or pizzas with my students.

If you have $10, you can share that among 5 people: $10 \div 5 = \frac{10}{5} = $2 each.

But if you have no money, zero dollars divided by 5 people means everyone gets $0.

\frac{0}{5} = $0.

Disappointing, but mathematically sensible, and we can carry out the division.

With pizzas, we can divide 1 pizza into 4 pieces, 10 pieces, or 1 piece, so those fractions can all be calculated. But if the pizza delivery doesn’t show up, we have 0 pizzas divided among the 4 people in my house, so we each get 0:

\frac{\text{0 pizzas}}{\text{4 people}} = 0

And what about ?

We can’t divide 1 pizza into 0 pieces. It doesn’t make sense: it is “undefined.” If we were dividing money, $10 divided into a certain number of piles (10 piles, 5 piles, 2 piles, or 1 pile, for example) can all be done, but we cannot divide $10 into ZERO piles.

\frac{10}{0} = undefined

Looking at Fractions Numerically

Now, let’s look at fractions numerically by finding decimal equivalents. Begin with these fractions on your calculator or computer technology. As you decrease the numerator down to 0 what do you notice?

Next, try these fractions. What happens as the denominator gets smaller?2

Dividing by zero results in an “undefined” message in Desmos, the TI-84+ family, and TI-Nspire, while GeoGebra displays the \infty symbol for infinity.

This numerical investigation demonstrates that when 0 is in the numerator, the fraction’s value = 0. When 0 is in the denominator, the fraction’s value is undefined. Students can also observe how the fraction’s value decreases when the numerator decreases, but the denominator has the opposite impact: the fraction’s value increases when the denominator decreases.

Zeros in Slopes

Slopes of lines give another window into how zero operates in a fraction. A common point of confusion for Algebra 1 students is the slopes of horizontal and vertical lines, and how these relate to the fraction \frac{\Delta Y}{\Delta X} used for slope.

By graphing the line and changing the numerator and denominator values, we can visualize how slope is impacted when either number is zero. A graph grid and optional “slope triangle” enable students to make mathematical sense of slope, and to understand slope = 0, undefined slope, and negative values for either \Delta Y or \Delta X.

I try to use phrasing and terminology that supports mathematical sense-making, so I say “slope equals zero” or “slope is undefined” rather than the ambiguous “zero slope” or “no slope.” I also avoid the phrase “rise over run” because it isn’t clear which way the line “runs,” and “rise” can seem contradictory for a negative slope. Instead, I emphasize the slope triangle showing a positive or negative move in the X direction, and a positive or negative move in the Y direction.

Here is the slope exploration in Geogebra: Zero & Slope Fractions

In Desmos Activity Builder: What is Zero All About?

On TI-Nspire: Slope Fractions

This student lab handout What is Zero All About? can be used to guide the exploration on any of the platforms, and concludes by prompting students to make some “notes to my future self”3 about how to handle fractions involving zeros, and the slopes, equations, and graphs of the various types of lines.

This version of the student lab handout has specific instructions for TI-84+ and TI-Nspire calculators, including accessing the fraction templates. On the TI-84+ family, use Transformation Graphing to change the values in the slope fraction.

Dividing with zero… nothing to be afraid of!

Notes and Resources:

Two great posts about why you can’t divide by zero:

Zero is amazing in some other ways! Check out Howie Hua’s TikTok explanations of how “adding zero” (the property a + 0 = a) is a great strategy for computations and why 0 × a = 0 shows why a negative times a negative equals a positive. Howie’s Adding Zero TikTok and Howie’s Multiplying by Zero TikTok. (@Howie_Hua)

Plenty of people much smarter than me have shared videos on why we can’t divide by zero, and other uncertainties/indeterminate forms involving zero. Here are a few good ones:

Dividing by zero really is can be dangerous. In Zero: The Biography of a Dangerous Idea, Charles Seife tells the story of the warship USS Yorktown that was disabled in the water in 1997 when its computer code tried to divide by zero. It took engineers two days getting the failure fixed and the ship back under way.

1The “Sharing Interpretation” of Division is also known as Partitive Division. This page http://www.sfusdmath.org/partitive-and-quotitive-division.html has some further explanation.

2In Infinite Powers, Steven Strogatz uses this technique to explain “The Sin of Dividing By Zero” pages 14-15.

3The phrasing “note to my future self” is adapted from Peter Liljedahl’s Building Thinking Classrooms in Mathematics. Peter prompts students to write notes “to my future forgetful self” as a motivation to turn note-taking into a thinking activity rather than mindless copying.

Technology comments: All of the tech platforms can give decimal equivalents for fractions.

  • Desmos has a fraction symbol to toggle between answer formats.
  • GeoGebra displays either or as the toggle button; the \approx is displayed next to a decimal result, even for exact decimal equivalents. This might be confusing to students who interpret the symbol to mean “approximately equal to”.
  • vs.
  • On TI-84+ and TI-Nspire calculators, set mode for answers to “decimal”, or sneak a decimal point into any part of a fraction to force a decimal result while in AUTO mode.

A Back-to-School Tour of the TI-84

As another school year begins, here is a compilation of some of my recent posts and other resources about the features of the TI-84+ family of calculators, so you are ready for success with your students this fall. Whether you have the color TI-84+CE devices or gray-scale models, you will find helpful hints for using TI-84 family technology in your math class.

What’s My Window? TI-84 Tips for Algebra 1

  • Get acquainted with some of the options for a graph viewing window on the TI-84+ calculator.

How to Use the TI-84+CE for Geometry

  • Use the built-in Cabri Jr. App on the TI-84+ to bring powerful dynamic technology to your geometry class.

Function Fundamentals: TI-84 Tips for Algebra 2

  • See how you can perform function operations and explore compositions and inverses.

Logs & Complex Numbers: TI-84 Tips for PreCalculus

  • How might we use the TI-84+ family to build understanding of logs and complex numbers?

Cool Calculus: TI-84 Tips for AP Calculuscoming soon!

Programming Power!

Check out the new Python programming language preloaded on TI-84+CE calculators, or use TI-Basic on any device. The “10 Minutes of Code” lessons engage students with short activities that help spark interest in coding, computer science and robotics.

Families of Functions Video Library

This resource has hundreds of short video lessons that teach students to graph any of 16 parent functions and their transformations. Explore and choose what works for your students for practice, remediation, or flipped classroom lessons. No devices needed!

Webinar: A Back-to-School Tour of Your TI-84

Check out the recording or the resource documents, which include tips on using the TI-SmartView Emulator Software for both face-to-face and online classes.

Have a great school year!

Logs & Complex Numbers

TI-84 Tips for PreCalculus

We all know that calculators can do computations, but did you know that they can be used as a tool for inquiry, helping students to build understanding?  Let’s take a closer look at logarithms and complex numbers on the TI-84 Plus CE* graphing calculator.

What is a Logarithm?

When I introduce logs, I want students to grasp the concept that a log question is asking for the power on the given base.  I use the logBASE command, which is accessed by pressing the Alpha key then Window on all TI-84 Plus models (also available in the Math menu).  We go through a series of calculations and I ask students what they think might be happening with this command. 

Students can create a log evaluation example of their own and share with a partner, answering them with pencil and paper before confirming on their TI-84 Plus.  We can then examine more advanced cases and make predictions: what will be the next result on the screen below?  Can you explain why?

Next, we can explore numbers that are not exact powers of the base: how does log27 compare to log28 and log29?  What is your prediction for log217?

Finally, we explain the use of the button that simply says log and what base it uses.  What do you think the whole number part of the answer represents?  What do you expect for the next result?  Are you surprised?

Notice and Wonder for Log Laws

Students often struggle with the laws of logarithms, since they don’t “work” the way one might expect.  Rather than me presenting the laws for students to memorize, I have students take a look at these two calculator screens. What do you notice? What do you wonder?

Examples like these help students build understanding of log laws.  I ask students to jot down observations in their notebooks before discussing with a partner or the class; this gives everyone a chance to think before ideas are shared.  We discuss common mistakes with the log laws and try them out on the calculator to show that they are false; for example, log(3•5) does not equal log(3) • log(5).

We summarize the actual log laws for the class and have students write them down to avoid any misconceptions that might have arisen during the exploration.  I have found that having students develop the log laws before being told what they say makes it easier to remember them.

Complex Number Mode

Press the Mode button and select a+bi number mode to enable computations involving the imaginary number i.  First, what happens when you try to square root a negative number (and how is this different from what happens in real number mode)?

Then we move on to computations with complex numbers, accessing the number i by pressing 2nd then the decimal point

Inquiry with Imaginary Numbers

There are many interesting patterns using the imaginary number i, and the calculator enables students to explore powers of i.  Once they have experienced the pattern with examples, they are ready to make predictions based on the pattern and formulate a strategy for finding higher powers. 

Conjugates are often used for complex number computations.  What is a conjugate?  What is the sum of a complex number and its conjugate?  What is their product?  Can you explain why this happens?  Why do you think we use complex conjugates when dividing complex numbers?

A final note: many of these computations are those that I want students to be able to do without a calculator.  Once students have solidified their understanding of the process through the calculator exploration, they can practice the skills on their own.  I make sure to have a non-calculator page for tests and quizzes, along with a section where calculators are permitted.

*All the features discussed in this post are available on the entire TI-84 Plus product line, including color and grayscale models.

This post is part of a series on the TI-84+ calculator family. Previous posts are here and here.

Function Fundamentals

TI-84 Tips for Algebra 2

Yvars menu access with Alpha-Trace

Function notation is available on the TI-84 Plus family of graphing calculators! Help students understand functions using the graphing calculator to connect symbolic, numerical, and graphical representations. Read on to see how you can perform function operations and explore domain, range, compositions, and inverses with your Algebra 2 students.

Using Function Notation

The Y-variables Y1, Y2, Y3, etc. are accessed by pressing Alpha then Trace.  Enter one or more functions on the Y= screen, then return to the home screen to evaluate function values numerically and perform operations with the stored functions.  Try a pair of functions that are inverses to see what happens with their composition.

Another numerical use of function notation is for modeling and problem solving.  Once students have found a regression model or other equation, enter it into Y1.  Then use this to answer questions by evaluating particular function values on the home screen.

Graphing Functions

Enter a pair of linear functions on the Y= screen, then explore what happens when you add or subtract these functions.  Why do the graphs of the sum and difference functions look like this?  A table can help illuminate what is going on here.  Ask students to explain this result numerically (with evidence from the table) and algebraically from the function expressions.

Try multiplying or dividing functions and examine the graphs.  Are the graphs what you would expect from the underlying functions?  When studying rational functions, pay attention to the zeros and Y-intercepts of the numerator and denominator functions, and compare them to the zeros and Y-intercepts of the rational function.  Why does the hole appear in the graph at that location?

Graphing Composition of Functions

You can easily graph the composition of two functions.  Enter them in Y1 and Y2, then deactivate them by highlighting the equals symbol and pressing Enter.  In Y3, enter Y1(Y2(x)) as shown.  What is the equation of the composition?  What happens if you graph Y2(Y1(x))?  Is composition commutative?

Challenge your students to create two functions with a particular composition: can you create  y = 2(x + 1)2  or  y = 2x2 – 1, for example?  Then explore what happens when you take the absolute value of another function or the square root of another function; what is the domain and range of the original function and the domain and range of the composition?  For example, Y1 = x2 + x – 2 in blue; Y2 = absolute value in red, and Y3 = square root in black.

Inverses of Functions

Use a square window to graph inverses: select either ZSquare or ZDecimal  from the Zoom menu.  If you haven’t done so already, turn on the GridLine option by pressing 2nd Zoom for Format.  Enter Y1 = x2 and Y3 = x with a dashed graphing style.  To draw the inverse, press 2nd Pgrm for the Draw menu and select DrawInv.  Enter the Y1 variable (Alpha Trace) and the color (press Vars and right arrow twice to select from the COLOR options). 

Is the inverse a function?  What should be the restriction on the domain of Y1 in order to create an inverse function?  You can restrict the domain of a function by dividing by the restriction so those values will be undefined.  Access the inequality symbols from the Test menu (2nd Math).  Then find the inverse algebraically and enter it into Y2.  If desired, check the Table of values to confirm that the domain and range have been exchanged.

Functions can be fun!  Take advantage of function notation on the TI-84 Plus CE to help your Algebra 2 students build their skills and understanding.

*All the features discussed in this post are available on the entire TI-84 Plus product line, however the grayscale models do not have color options in the DRAW menu.

This post is part of a series on the TI-84+ calculator family. The previous post is here.

What’s My Window?

TI-84 Tips for Algebra 1

Zoom menu

When you graph a function on the TI-84 Plus CE graphing calculator*, you have many built-in choices for the viewing window.  Don’t leave it to chance… get acquainted with the options so you choose the best window for the graphing job!

Begin by checking out the Window button to see how it describes a graphing window.  Xmin, Xmax, Ymin, and Ymax set the boundary values for the axes, and Xscl and Yscl give the distance between the tick marks on each axis.  Each of these can be set so that your graphing window is suitable for your needs, showing the important features of a function or the complete data set of a scatterplot.

When my students are going to graph a function or scatterplot, I start by asking them “What is an appropriate window?” so they plan ahead BEFORE pressing Graph.  The line y = 2x – 3 will display fully on a standard window, but what about y = 12x – 37 ?  My students know to change Ymin and Ymax to see more of this graph, but if they forget to adjust the scaling on the Y-axis, their graph might look like the one below with an unreadable Y-axis.  Instead, set Yscl = 5 or 10 to be able to estimate the Y-intercepts.

Some Standard Windows

The Standard window (item 6 in the Zoom menu) is a great place to start for many Algebra 1 graphs, since it covers -10 to 10 for both axes.  And Quadrant1 works well when a graphing situation requires only positive values; it is item A in the Zoom menu, found by scrolling down to the second screen.

You might notice there is another menu available under the Zoom key; use the right arrow to go to the MEMORY submenu.  I use Zprevious to return to the previous window if my changes didn’t work out the way I expected.  And if there is a custom window I want to use again, I can store it and recall it later.

Square Windows

The downside of the Standard and Quadrant1 windows is that the spacing of the scales on the X and Y axes are not equal.  The easiest way to see this is to turn on the GridLine option by pressing 2nd Window for the Format menu.  What do you notice about the graph?  The gridlines create rectangles, not squares, and this distortion can be confusing for students studying perpendicular lines, for example, since they don’t appear to be perpendicular in this viewing window. 

The solution is to select a “square window” in which the axis tick mark spacing is the same for the X and Y axes.  Two choices are ZSquare which adjusts your current window so the spacing is equalized; and ZDecimal , a particular square window that I use all the time (below).  These are items 5 and 4 in the Zoom menu.

The Decimal window is my favorite!  It has a reasonable domain and range to use for many Algebra 1 graphs and there is no skew distortion.  The Trace operation uses the very friendly increment of 0.1 for the x-values so the displayed values don’t have lots of decimal digits (notice the TraceStep value in the Window screen).  With the gridlines showing, my Algebra 1 students can clearly visualize the slope of a line and count “rise” boxes and “run” boxes.

Another wonderful situation for using the Decimal window is to display a function that has a hole.  This rational function clearly has a visible hole at x = 2, and notice that the Trace command shows no value for Y because it is undefined.  If your function’s hole is out of the screen’s range, adjust Ymin and Ymax while keeping the domain as is.

Zoom Fit and Zoom Stat

These two items in the Zoom menu help users quickly arrange the window to display the important points of a graph or scatterplot.  ZStat (item 9 in the Zoom menu) creates an appropriate window for data in the current stat lists, a time-saving feature.  I still require students to decide on a good domain and range for a given data set before using ZStat, because it is an important skill for statistical literacy. 

ZFit (item 0 in the Zoom menu) fits the range to the selected domain: it adjusts Ymin and Ymax to display all Y–values generated by the selected Xmin and Xmax.  For example, the graph of y = -3x2 – 10x +15 on the Decimal and Standard windows doesn’t display either the vertex or y-intercept.  Using ZFit allows students to see the whole graph at once; they can check the Window for accurate interpretation of domain and range, and access the Calc menu (2nd Trace) to analyze key points.

One last tip as you embark on your graphing adventure: you can press the On button to cancel a graph if you don’t want it to finish, or press Enter to pause and restart a graph.  Happy graphing!

*If you use a grayscale model of the TI-84 Plus family, you will notice some domain and range differences due to screen pixel density.  All of the viewing windows discussed in this post are options for the entire TI-84 Plus product line.

Super Sevens!

Today is July 14, 2021, which can be written 7-14-21…. so I’m calling it “Sevens Day”!  Here are some wonderful things about the number SEVEN and related numbers:

  • 7 is the number of days in a week, 14 is the number of days in a fortnight, and 28 is the number of days in a lunar month.
  • 7 is the start of an arithmetic progression of six prime numbers: 7, 37, 67, 97, 127, 157. Sadly 187 breaks the sequence since it is a multiple of 11 and 17.
  • There is only one way that 7 can be decomposed into 3 unique addends, and they are consecutive multiples of 2: 1 + 2 + 4 = 7. This is a useful fact for KenKen and Kakuro puzzles! Other decompositions of 3 addends all include repeated numbers: 3 + 1 + 3 or 2 + 3 + 2 or 1 + 5 + 1.

Divisibility* by 7 is something I learned when doing more advanced KenKen puzzles. The technique I use is this:

  1. Chop off the units’ digit and double it.
  2. Subtract this from the remaining number.
  3. Continue this process until the result is 1 or 2 digits.
  4. If the final number is divisible by 7, then the original number was divisible by 7 [Note that 0 is divisible by 7].

For example: 3052

  1. Take the 2 and double it to get 4.
  2. 305 – 4 = 301
  3. Take the 1 and double it to get 2.
  4. 30 – 2 = 28 which is divisible by 7.

Another version of a divisibility test for 7s was discovered** by Chika Ofili, a 12-year-old Nigerian boy, in 2019. His version is:

  1. Chop off units’ digit and multiply by 5
  2. Add this to the remaining number
  3. Continue… if the result is divisible by 7, then the original was divisible by 7.

This has come to be called “Chika’s Test”.  It might be faster than my other method, because it is faster to get to a recognizable multiple of 7 ( a number between 0 and 70) by adding multiples of 5 than subtracting multiples of 2. And adding numbers is sometimes simpler than subtracting numbers. Read more about Chika’s test here.

Why do these methods both work? For example, take the number with digits ABCD.  Chop off units’ digit D, this leaves ABC which has value 100A + 10B + C.

Adding 5D vs. subtracting 2D is only changing the result by a multiple of 7, so if the number was divisible by 7, it still is. If it wasn’t, it still isn’t.

[100A + 10B + C + 5D] – [100A + 10B + C – 2D] = 7D.

Seven is also a special number for mathematical Symmetry: there are exactly 7 Frieze Patterns. This page gives a nice visual with some simple explanations: Frieze Patterns.

To dive more deeply, check out Paula Beardell Krieg’s (@PaulaKrieg) series of 10 posts, including lots of artwork and models. Start here for an introduction, and this link gives access to all of the posts.

Paula includes links to some great digital GeoGebra applets from Steve Phelps (@MathTechCoach). Here is one to play with.

Fraction Patterns with Sevenths: Any math teacher will tell you that fractions with 7 in the denominator do not have simple decimal conversions. The repeating decimals aren’t as “friendly” for students as those that happen with thirds, sixths, or ninths.

But the repeating pattern is quite beautiful, with 6 different digits in the pattern:

1/7 = 0.142857… = 0.142857142857142857142857

What’s even cooler to see is that all the fractional sevenths have the same 6 digits in the same order, each one starting with the next higher value (of 1, 2, 4, 5, 7, 8):

2/7 = 0.285714…

3/7 = 0.428571…

4/7 = 0.571428…

5/7 = 0.714285…

6/7 = 0.857142…

Read more about these “Magic Sevenths” here.

Heading over to Geometry, the regular polygon with 7 sides (known as a heptagon or septagon) is the smallest polygon that can’t be constructed with a ruler & compass alone.

This week, Becky Warren (@becky_k_warren) shared the method of “tricking” GeoGebra into rotating a segment exactly 360/7° to create a heptagon.

One more fun fact I uncovered this morning has to do with right triangles and Pythagorean triples.  If a and b are the two shorter sides of a Pythagorean Triangle (a right triangle with all integer sides, that is, a set of lengths that comprise a Pythagorean Triple), then one of these will be divisible by 7:  a or b or a+b or a-b.  Check it out with your favorite Pythagorean Triples!

Have a wonderful Sevens Day!!


* More on Divisibility and math-y dates in my post Leap Years & License Plates; there are some nice references on divisibility rules at the bottom.

**Chika’s divisibility test wasn’t a new discovery (it is listed in the Penguin Dictionary referenced below, for example), but it was new to him and quite a feat for a 12-year-old!

Some of these facts are from The Penguin Dictionary of Curious and Interesting Numbers by David Wells (1986).

While searching for images for this post, I learned about electronic 7-segment displays, used in many applications to display numbers (and some letters). Did you have a digital clock that looked like this?

Geometry Grab Bag!

Using Geometry with TI Technology

When students interact with dynamic geometry software, they can construct, measure, and move figures to explore properties, confirm theorems, and visualize geometric situations.  Lessons using this technology allow students to create their own geometry knowledge and are also more fun than reading properties and theorems out of a textbook.  Most importantly, engaging students in doing the math (not simply viewing the math) makes their learning deeper and more durable.

In the GEOMETRY GRAB BAG webinar, we took a look at Exploring A Situation, Confirming Properties, Constructing Un-mess-up-able* Figures, and Capturing Data using the geometry capabilities of TI Calculator technology (both TI-84+ family and TI-Nspire).  Here is a quick overview; be sure to check the links at the bottom of the post for all the resources and the webinar replay.

A. Getting Started

Here are quick HOW-TO guides for geometry on TI-Nspire and the Cabri Geometry App on TI-84+.  For many pre-made activity files, students just need to know how to grab and drag a geometric figure, then observe and reflect on the results.  In a dynamic geometry environment, creating a figure such as a triangle is like creating infinitely many examples of triangles to explore, since measurements update as the figure is moved.

B. Explore A Situation

A pre-made file allows students to focus on the implications of the geometric scenario, rather than on construction details.  Instructions for linking devices and transferring files are included in the webinar materials.

Lines & Transversals asks students to compare and contrast situations when something is true or not true. Targeted questions on the student lab sheet help clarify angle pairs that are sometimes OR always congruent and supplementary.

Triangle Midsegments is a nice exploration to “get your feet wet” with constructing and measuring. I like to ask students “what do you want to measure?” to give them control of the investigation. The most important step of the activity is to DRAG AND OBSERVE; how do the measurements change? Provide a template for students to record sketches, measurements and conjectures, to hold them accountable for mathematical thinking.

Area Formulas in TI-Nspire is a great visualization to help students gain understanding of area relationships. I ask my students: “Why does it work?” and “Can you explain it another way?”

Areas of triangle & parallelogram

C. Confirming Properties

The dynamic geometry activity can help students discover properties and confirm them with measurements and calculations. This can lead to justification and formal proof if desired in your curriculum objectives.

The Pythagorean Theorem can be visualized by building squares on the sides of a right triangle. The Converse helps students determine if a triangle is acute, right, or obtuse.

Coordinate Reflections takes advantage of the coordinate plane that “lives within” each geometry page in Cabri Jr. and TI-Nspire. Students can explore many properties of transformations besides the changes to coordinates.

Coordinates of reflected figures

D. Constructing Figures

There is an important distinction between drawing and constructing figures in dynamic geometry environments. When a figure is accurately constructed — sometimes called “un-mess-up-able”– it is guaranteed to retain its specific properties no matter how it is dragged on the screen.  If a figure is merely drawn to look like a particular diagram (for example, drawing a “right” triangle without using the perpendicular tool), its properties won’t stay valid as the figure is manipulated.

Parallelogram Properties has students first constructing a parallelogram, then measuring its parts to find out important properties.  You can extend the learning to other types of quadrilaterals or explore less common theorems with the power of dynamic geometry, allowing even struggling students to go beyond the basic topics.

E. Capture Carnival

Data capture is a huge asset within TI-Nspire.  Once variables are defined, values can be captured manually or automatically.  Then students can create a scatterplot, develop an algebraic model, or perform regressions as desired.  If you are using Cabri Jr. on the TI-84+ family, you can still collect data by hand, then store in the Stat Lists for further examination.

Chords in Circles is an example of a geometric figure that generates an interesting function relationship.  Other data collection activities are included in the webinar materials.

F. Tips for Successful Teaching: Online & Face-To-Face

  • Keep your TI-84 SmartView Emulator software or TI-Nspire Premium software on top of your working document (Word, Google Docs, Smart Notebook, etc.)
  • Use screenshots frequently so students can keep up and catch up. Annotate these for asynchronous students.
  • Use color and motion to highlight ideas and address misconceptions directly.
  • Think about what preliminaries are important for your students to be successful: needed definitions, labels on points (or not), dragging points before taking down data, reminders to “make different kinds of triangles,” etc.
  • Provide templates for students to RECORD observations and REFLECT on the geometry concepts.
  • SUMMARIZE the results for the class, either yourself or by students, since this ensures that important math concepts don’t get lost in the technology activity, and it helps make the learning more durable.

Using dynamic geometry technology has greatly enhanced my geometry teaching! Dive into these resources to see what the power of geometry technology can do for your students.

Notes and Resources:

View the recording of the GEOMETRY GRAB BAG webinar HERE.  The supporting materials are HERE.

For more activities, go to https://education.ti.com/en/activities, choose your device and search by topic.

This webinar was based on two prior blog posts on the T3Learns Blog:

How to Use the TI-84 Plus CE for Geometry  and  How to Use the TI-Nspire CX for Geometry.

*The term “Un-mess-up-able Figure” is from the CME Project’s Geometry textbook published by Pearson (2009: Cuoco, et. al.), and is defined as: A figure that remains unchanged when you move one point or other part of the figure.

Quarantine Queries

Being in quarantine had me wondering about math-y words starting with Q.  Here is my quest to find out their meaning and history.


Quarantine refers to a forty-day period in which ships were required to stay isolated before passengers and crew could go ashore during the Black Death epidemic in the 1300s.  The word is related to the Venetian/Italian words quarantena or quarantino, meaning “forty days”, derived from the Italian word quaranta, and similar to the words for 40 in French and other languages.

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Let’s next look at the number 4, and the Latin quadri- gives us quadrilateral (4-sided figure), quadrangle (4-angled figure), quadrillion (4th power of 1 million, or 1024, see below), and quadruple (to multiply by four).  Notice that quadratic is missing from this list; more on that below.

1024 = 1,000,000,000,000,000,000,000,000

A quadrangle is a plane figure in which segments connect 4 non-collinear points, and it has some interesting math properties to explore. If the points are connected in cyclical order, a convex or concave quadrilateral is the result, otherwise the figure is called a crossed, non-simple, butterfly, or bow-tie quadrilateral (and many high school geometry texts do not consider this to be a quadrilateral at all).

quadrangle 4

If we construct the 6 lines connecting the 4 points in all possible pairs, we create a “complete quadrangle”.  The 3 extra points of intersection (that are not vertices) are called diagonal points.  The midpoints of the sides, along with the 3 diagonal points, all lie on a conic called the Nine-point conic

Nine Point Conic

Check out this GeoGebra visualization; the nine-point conic seems to be an ellipse when the quadrangle ABCD is concave and a hyperbola when the quadrangle ABCD is convex or a non-quadrilateral “bow-tie” shape.  What else do you notice?

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box plot TI

A related Latin root is quartus or fourth.  Taking this to mean one-fourth (¼) gives us quarter, quartile, and quart, whereas a fourth degree polynomial is a quartic function.  Similarly, the Latin quintus means fifth, and yields the words quintic, quintile, quintillion, and quintuple.

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So, what about quadratic, which feels like it should have to do with four, but instead is a polynomial of degree two?  It comes from the underlying Latin word quadratum which means “square”.  The Greeks and Romans understood the abstract quantity x2 as a square with side x.  That’s why something raised to the second power is said to be squared or quadratic.  The related word quadrant, from quadrare (“to make square”) is one of the four “square” regions of the Cartesian plane.  And graph paper is sometimes called quadrille ruled, based on the French word for “small square”.Capture

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Two last Q words relating to math are quantity (from the Latin quantus meaning “how much” or “how great”) and quotient (Latin quotiens = how often, how many times).  So the quotient is the quantity that tells how many times one number fits into another number.


Notes & Resources:

I’ve had the idea for this post rattling around in my brain for more than forty days, and thankfully, Ed Southall’s (@edsouthall) presentation for MathsConf23 From Abacus to Zero: The Etymology of the Words of Mathematics has spurred me on to write it.  The full virtual conference recordings are at this link.

  • spec-17-covEd helpfully suggested a few books that detail the meaning of mathematical words.  Much of my information is from The Words of Mathematics: An Etymological Dictionary of Mathematical Terms used in English by Steven Schwartzman (1994), The Mathematical Association of America.
  • The ebook is available here [on 50% sale through the summer!]


¹ This image of the Nine-Point Conic is from Weisstein, Eric W., “Nine-Point Conic.” From MathWorld–A Wolfram Web Resource. https://mathworld.wolfram.com/Nine-PointConic.html.  More about the Nine-Point Conic and Complete Quadrangles can be found on Wikipedia here and here.

Note that a Complete Quadrilateral (right below) is a different figure (and is the dual of the complete quadrangle, left below); read more about this at Cut The Knot here and Wolfram MathWorld here.complete quadrangle and quadrilateral

² Latin quod erat demonstrandum “which was to be shown”.  Typically used at the end of a proof to show that the proposition in question had been proved.