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.

◊  ◊  ◊  ◊  ◊

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?

◊  ◊  ◊  ◊  ◊

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.

◊  ◊  ◊  ◊  ◊

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

◊  ◊  ◊  ◊  ◊

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.


Easy Angles

Check out my entry in “The Big Lock-Down Math-Off” from The Aperiodical Read both posts and vote for your favorite. I share some angle-measuring tools you can easily create when you find yourself without a protractor.  Plus I finish with a business card surprise!

Big Lock-Down Math-Off Match 23

Voting has ended, and I am happy to report I won the match.  THANKS!!  Full post below.


Many of us are stuck at home these days; what if you need to measure angles and just don’t have a protractor? You can quickly distinguish acute and obtuse angles with one of the right angles you carry around all the time – the angle formed by your thumb and index finger – but that is hardly a precise instrument for measuring! In this pitch, I will share with you some easy angle tools that you can create with only a piece of paper.

Right angles and 45°

The corner of a sheet of paper is a more exact right angle than that created by our fingers, and I have suggested this to my students when they are first learning to visualize acute angles that are smaller than 90° versus obtuse angles that are bigger.

To create an easy 45° angle, fold a sheet of paper diagonally so one edge aligns with the adjacent edge, bisecting the right angle:


Now you can measure 90° and 45° angles, as well as determine whether your angle is larger than 90°, smaller than 45°, or between those values. For more precision, fold your folded edge to meet the same side edge as before, and this bisection of the 45° creates 22.5° and 67.5° demarcations as well.


This bisecting strategy works, but the angle sizes created aren’t especially useful (except for defining the intervals). Let’s create some commonly used angles next.

More Useful Angles

After 45° and 90°, arguably the most mathematically useful angle sizes are 30° and 60°.

A. Begin with a square piece of paper – create this by doing steps 1 and 2 above, and then cutting off the rectangle section at the top. No scissors? Simply fold the top edge down over the triangle, press your finger along the fold, reverse the fold and press firmly again, then tear carefully along the crease line.


B. Unfold the square, then fold the paper in half vertically: fold the right edge over to the left edge to make a crisp vertical crease down the middle of the square. Unfold again.


C. Fold the upper left corner into a triangle, by folding it down towards the lower part of the vertical crease.
• Aim that corner approximately 2/3 of the way down the vertical crease
• Visualize the angles being created in the upper right of your paper – the folded-over piece should look equal to the visible portion of the underneath paper.
• You have created a triangle that has angles measuring 30°, 60°, and 90°.
Can you explain why the 30° angle has that measure?


D. Fold the bottom left corner up to form a second triangle. Fold until the left edge of the paper lines up with the creased edge of the first triangle.
• This is also a 30°-60°-90° triangle. You can check the 60° angle is the same size as the one in the layer below that was formed in step C, and if you unfold all the way, it is clear that 3 equal angles equal 180° because they make the straight edge of the paper.


E. Fold the bottom right corner up, so the right edge of the paper meets the edge of the first 30°-60°-90°. Then tuck the corner under the second 30°-60°-90° triangle.


F. The Final Product is a 45°-60°-75° triangle! The 15° angle was created by bisecting a 30° angle, and the 75° (30° + 45°) is its complement. Use this “protractor” to measure these special angles, and estimate the size of angles that fall between measurements. Ask students to unfold and label other angles.


A Business Card Special Triangle?

Take a standard US business card, which measures 3.5 inches by 2 inches, and fold the upper right corner down to meet the lower left corner. Next fold the upper left corner over to meet the folded edge, and finally, the corner that is pointing down folds underneath the rest. What is the result? Can you see the equilateral triangle and the 30°-60°-90° triangle?

It’s a great cocktail party trick, but is it actually an equilateral triangle? Here is the construction; the angle marked θ is half of one of the triangle’s angles. Does it equal 30° as expected for half an angle in an equilateral triangle? Sadly, no, the inverse tangent of 2 ÷ 3.5 ≈ 29.74°. Close enough to visualize, but not exact! Check out the GeoGebra visualization.

Screen Shot 2020-06-22 at 10.40.11 PM

I hope you’ve enjoyed this pitch about folding your own protractor from paper. And if you really need a protractor to borrow, I’ve got plenty!



Notes & Resources:

1.  The ProRadian protractor in the top photo (bottom right) is one of 3 models invented by Jennifer Silverman (@jensilvermath). More information at ProRadian.net.

2.  The small clear protractor in the bottom photo is available at a great price from Didax. I love it for use with my students because of its small size and durable, transparent plastic. You can decide which of these deals is the better buy! [I do not have any affiliate relationship with Didax.]

Screen Shot 2020-06-22 at 11.41.46 PM

Puzzle Pastimes

In the current circumstances of staying home during the Covid-19 pandemic, I’ve spent a lot of time doing puzzles.  Some are jigsaw, some are pencil & paper, and all of them have got me thinking about the math ideas that the puzzles generate.

We have a 1000-piece rectangular jigsaw puzzle going on our kitchen counter. We begin by searching for edge and corner pieces (a method recommended by Jennifer Fairbanks @JenFairbanks8 in her puzzle post here.)

I started to wonder how many pieces would be along the long and short edges of the rectangle, and then… how many different ways can 1000 pieces be arranged in a rectangle? It turns out my puzzle had 40 pieces along the long edge and 25 pieces along the short edge.

How many edge pieces are there (single-edge) and how many corner pieces? What math did you use to count them?  I figured it this way:

puzzle pieces

  • There are 4 corner pieces
  •  Long side of 40 pieces – 2 corner pieces = 38 edge pieces
  • Short side of 25 pieces – 2 corner pieces = 23 edge pieces
  • Edge pieces = 2*38 + 2*23 = 122

Then there must be 874 interior pieces: I can subtract corners and edges from 1000, or multiply 38 long rows by 23 short rows to figure this out.

Screen Shot 2020-05-20 at 4.07.29 PM

Another math-y feature of the puzzle is that its dimensions are 29 inches by 20 inches. I wondered about the approximate area of each piece.

Along the long edge, 29 inches ÷ 40 pieces = 0.725 inches/piece; along the short edge, 20 inches ÷ 25 pieces = 0.8 inches/piece.

How can we calculate the approximate area of each piece in square inches? [I like to use inch2 as a more mathematically meaningful unit when working with students.]

Even though 0.725 * 0.8 is hard to compute in my head or by hand, I’ve learned how to expand my thinking with Pam Harris’s weekly #MathStratChat on Twitter (Pam tweets at @pwharris) .

Here are some ways I think about this calculation. First, I believe that “Fractions Are Your Friends” so I use fractions instead of decimals. This method uses the strategy of decomposing 8 into its factors 2*4 and then multiplying 725 by these friendly numbers.


Another method is to go back to the fractions that originally created the decimals. I divided by the common factor of 20 (I try to avoid the wording “cancel out”). Next, I multiply by 2/2 which is a fraction with value = 1, so it changes the form of my fraction without changing the value.


A third method is to use a ratio table to multiply 725 * 8. I find annotations helpful to understand the math thinking behind the table. I needed to subtract to finish my table, and so I had two alternative strategies for computing 72 – 14: first, I “removed friendly numbers” and second, I “made the first number nicer and then compensated with an inverse operation.”


A final method I considered was to try to multiply the decimals directly (yikes!) by using a “double-and-half” strategy.  Since doubling (multiplying by 2) and halving (dividing by 2 or multiplying by ½ ) are inverse operations, they undo each other. I used an additional inverse operation in the last step to help me think about multiplying by 0.1; I find this to be a more mathematically meaningful approach than “move the decimal”.

And YES! If each puzzle piece has area = 0.58 inch2, then the 1000 piece puzzle has area of 580 inch2 which makes sense for dimensions of 20 inch x 29 inch.

For those teachers who ask students to convert between units, I use a “unit analysis” structure. If we know that there are 2.54 cm in 1 inch, do we multiply or divide? This technique uses the concept that a fraction whose numerator and denominator are equal has a value of 1. So both of these fractions have the same value, and we choose appropriately to create the units we need (and cancel away the units we don’t need).

IMG_4090 IMG_4090-1

This result is close to the area if we had multiplied the centimeter dimensions given, but not exact due to rounding off (another good discussion to have with students!)

Some of our finished puzzles and work-in-progress:

Jigsaw puzzles aren’t my only puzzle-y way to pass the time at home! There are many, many opportunities to engage with math-y puzzles online, and here are some I’ve found fun:

1.   Matt Parker’s Math Puzzles:  the “stand-up mathematician” (@standupmaths on Twitter) is posting video puzzles every week or two at http://think-maths.co.uk/maths-puzzles and then solution videos within a few days. The puzzles are fun and accessible, and each one has taken me some time to play around with it. I love that there are many ways to solve and there isn’t an immediately obvious solution path. The team at Think Maths is keeping score, with points for correct solutions, bonus speed points, and partial credit for wrong answers.  Here are some of my solution props:

2.   I do quick daily puzzles from The Times Mindgames and Sunday Times Brain Power and @YohakuPuzzle on Twitter. These don’t require much time commitment and are great fun! Catriona Shearer (@CShearer41) writes geometry problems on Twitter, and many of her puzzles rely on simple and common geometry principles. Here are two recent ones I’ve solved:

3.   It is very satisfying to play around with physical objects, and I’ve been exploring puzzles with Jenga blocks and Panda Squares, both posed by David K. Butler (@DavidKButlerUoA). David’s full set of Jenga Views are available here.

4.   And finally, there are plenty of people and websites that are posting daily or weekly puzzles, or have archives of puzzles available. Here are some that I have encountered:

  • Alex Bellos sets a puzzle on alternate Mondays in The Guardian: Monday Puzzle.
  • Chris Smith (@aap03102 on Twitter) is posing “Corona Conundrums” on YouTube.
  • Dan Finkel (@MathForLove) has all his puzzles here.
  • Art of Problem Solving has puzzles on their website.
  • Project Euler has a huge problem archive available, registration is free.
  • Zach Wissner-Gross (@xaqwg) posts “The Riddler” at FiveThirtyEight.com.

So if you are looking for something to pass the time, get puzzling!


The puzzles shown are Pomegranate ArtPiece Puzzles. They are high quality and we are enjoying the variety of artwork they depict. The puzzles pictured above are:

  • Frank Lloyd Wright: Saguaro Forms and Cactus Flowers, 1928
  • Diego Rivera: Detroit Industry, North Wall (detail), 1933
  • Birds & Flowers: Japanese Hanging Scroll

More can be found at Pomegranate’s website and at many online retailers.

Screen Shot 2020-05-23 at 2.10.37 PM

Jigsaw puzzles have become a popular pastime during the pandemic, and have been hard to find at stores and online merchants. The New York Times published an article April 8, 2020 about the buying boom and the manufacturing process. [Photo credit: Roderick Aichinger, NYTimes]

Have you ever opened a jigsaw puzzle and found two or more pieces still connected? What do you do? Dave Richeson (@divbyzero) found the definitive answer with this Twitter poll:

Screen Shot 2020-05-20 at 6.15.07 PM

Summer Math Refreshments


In my previous post, I set out an ambitious Reading List of math and education related reads.  So far, I’ve made only fair progress; because of daily life but also because of other opportunities for fun and enjoyment with math online and in person.  Read on for some diversions and refreshments to include with your summer pursuits.

Online “Events”:

Two online opportunities that I enjoyed immensely last summer are back again for 2019.  First is the #MathPhoto19 weekly photo sharing on Twitter.  Erick Lee (@TheErickLee) is hosting weekly prompts asking for photos on all sorts of math-y topics, such as Circles, Estimation, and even Beauty.  “The Coffee Porch” is my favorite summer location for my math diversions (and this week’s entry for #Lines).


Stay up to date on twitter with #MathPhoto19 and check out the archive of previous years at mathphoto19.wordpress.com.

The second event is the return of the Big Internet Math-Off organized by the folks at Aperiodical.com.  Last summer, sixteen mathematicians shared a fun math(s) pitch in a short blog post and/or video format.  Every topic was captivating, from origami and hexaflexagons, to airplane seating, phantom parabolas, mathematical modeling, and more.  The only downside was that every face-off resulted in an interesting mathematician going home, so there is a new format this year.

This time around, there is a “group stage” so every participating math person can give three presentations. Then on to the semi-finals and finals.  The full list of sixteen “players” and schedule is here, and the fun begins on July 1.  Follow on twitter using #bigmathoff  and @aperiodical.wallchart-2019-1-1024x724-border


Listening to math conversations via podcasts is another way to enjoy math on-the-go, whether you are walking the dog in the early morning like me, traveling to a vacation spot, or even while cooking or working around the house.  I’m catching up on some that I’ve missed from Mr. Barton Maths Podcasts with Craig Barton (@mrbartonmaths) and Estimation 180 with Andrew Stadel (@mr_stadel) and his math minions.

I’m not alone looking for good listens; this thread from @JennSWhite on Twitter gives some more suggestions (too many to list here, so click through!).  Consider loading up a Global Math Department webinar podcast, or Make Math Moments That Matter makemathmoments.com/podcast and enjoy.  And Craig Barton has recommended the Odds And Evenings podcast for “cracking puzzles and 100% math goodness”.

Blogs (Reading &/or Writing):

Summer is a great time to reflect on the teaching year that has gone by, and one way to do this is to dust off the neglected blog and write about some great teaching and learning experiences you meant to share along the way.  What will you do again?  What needs changing?  What difficulties did you face and/or overcome?  Many teachers in the #MTBoS* community have commented on Twitter that they plan to catch up on their writing (and the challenge of remembering what happened during the academic year!).  And even if you aren’t writing, catch up on reading the blogs you bookmarked during the year when there wasn’t time to process.

Chats & Discussions:

Take part in the many chats and discussions happening on Twitter… about books you are reading or how you solved a math problem.  Here are some to check out:

#MathStratChat with Pam Harris @pwharris every Wednesday evening.

#NecessaryConditions and #MathRecessChat for slow chat discussions on those books with a weekly schedule this summer.

inf powersThere is also a summer reading group chat on the book Infinite Powers by Steven Strogatz (@stevenstrogatz).  Learn more here.

The hashtag #EduRead is being used as well for discussing educational books by teachers on Twitter.  Or just search #MTBoS or #iTeachMath to find others thinking about the same things as you; use #ElemMathChat #MSmathchat #GeomChat #Alg1chat #PreCalcChat etc. to specify your audience.

Puzzle Play:

I love puzzles and I spend many hours engaged with them, especially when they have a math or logic angle needed to solve.  I recently discovered the daily MindGames puzzles from The Times UK [website here] along with their book collections.  Cell Blocks is a great visual brainteaser, whose object is to divide the grid into square or rectangular blocks, each containing one number and made up of that many cells (image shows a solution; puzzle starts without any of the dark boundaries).CellblockMy current favorite is Suko**, which is a 3-by-3 array for the digits 1 through 9.  The number in each circle must equal the sum of the four surrounding cells, and the total of the colored cells must match the color totals given outside the array.

For example, in this puzzle from www.transum.org , the three blue squares add up to 17, and the four lower left squares add up to 23.  Thus the value of the green square in that section must be 6.

The two red squares sum to 12, and the four lower right squares sum to 24, so the two blue squares in the middle column must also equal 12.  Since the three blue squares add up to 17, that leaves 5 for the lower left blue square.  And so on.

And if this isn’t enough to keep you going, one more puzzle idea is to try the Daily Math & Logic Challenges from @brilliantorg.

Whatever math you teach, summer is a wonderful time to reflect, refresh, and recharge.  Hope you enjoy these math entertainments!

Notes and Resources:

#MTBoS stands for “Math Twitter Blog-o-Sphere”, the online community of math teachers on Twitter.  Ask questions, find resources, & discuss issues about teaching math, and follow @ExploreMTBoS for more.

**  Suko, was created by Jai Kobayaashi Gomer (@kobayaashi2018), of Kobayaashi Studios, in late 2010. Website with information on Suko & other puzzles is here.

I’ve found these websites with interactive Suko puzzles, listed above and here again: TIMES Mindgames, the Sunday TIMES Brain Power, and www.transum.org.  Printing is also an option.  Warning, these puzzles are habit-forming!