# 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:.

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…

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!!

Notes:

* 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?”

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.

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.

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

*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).

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

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?

◊  ◊  ◊  ◊  ◊

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”.

◊  ◊  ◊  ◊  ◊

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.

Q.E.D.²

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.

• Ed 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.

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.

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.]

# 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:

• 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.

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).

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!

Notes:

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.

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:

# 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.

Podcasts:

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”.

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.

There 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).My 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!

Summer is around the corner (and already here for some educators) and it is a great time to dive into some reading that you have been meaning to get to throughout the school year.  Here is my list:

### Teaching Practices:

The first two books on my list will be discussed this summer in Twitter Slow Chats.  This is is a great motivator for me to get reading and join a thought-provoking discussion with other educators.  This chat format is very accessible since you can be part of the discussion anytime during the week; there is not a particular time of day or week that you must be available.

Necessary Conditions by Geoff Krall (@geoffkrall) will be discussed starting May 27, with one chapter each week through August.  Use the hashtags #NecessaryConditions and #StenhouseMath to join in.

Math Recess by Sunil Singh (@Mathgarden) and Christopher Brownell (@cbrownLmath) will be on a faster pace, starting May 29 and finishing up in early July.  Use the hashtag #MathRecessChat.

Routines for Reasoning by Grace Kelemanik (@GraceKelemanik), Amy Lucenta (@AmyLucenta), & Susan Creighton is my third book focused on teaching practices. This past winter Connecticut’s ATOMIC math organization did a book study on the book which I facilitated.  The prompts and discussions were open to all and still available here. You can download a sample chapter from this link. I am overdue writing a blog post containing my thoughts on the book and our discussion, but I will link it here as soon as I get to write it.

In case you missed them, some other wonderful books on teaching practices include Becoming the Math Teacher You Wish You’d Had by Tracy Zager (@TracyZager), and Motivated by Ilana Horn (@ilana_horn).

### Cognitive Science:

I am extremely interested in the power of applying cognitive science research to education practices, and I’ve been trying to use these ideas in my work with students. I’m looking forward to the upcoming release of Teaching: Unleash the Science of Learning by Pooja Agarwal (@PoojaAgarwal) and Patrice Bain (@PatriceBain1). Here is a link to their planned summer book discussions.

Others in this genre that I’d recommend are Make It Stick: The Science of Successful Learning by Brown, Roediger, & McDaniel; Why Don’t Students Like School by Daniel Willingham and How I Wish I’d Taught Maths by Craig Barton (@mrbartonmaths). Also check out the great cognitive science resources and podcasts on his website.

### Equity & Race

It is becoming increasingly clear that issues of social justice and equity cannot be sidelined away from discussions of pedagogy and math teaching practice.  We teach our students within the context of their lived experiences, background, and perceived privilege, and it is urgently important that we consider this factor in our teaching. Therefore, I have several books on my list this summer that consider equity, privilege and race.

• This Is Not A Test by Jose Luis Vilson (@TheJLV)
• Multiplication is For White People by Lisa Delpit
• White Fragility: Why It’s So Hard for White People to Talk About Racism by Robin DiAngelo
• Building Equity: Policies and Practices to Empower All Learners by Smith, Frey, Pumpian, Fisher (ASCD)
• Blind Spot: Hidden Biases of Good People by Mahzarin Banaji & Anthony Greenwald

If you are diving into some of this work, check out the Twitter account @ClearTheAirEdu, and use the hashtag #ClearTheAir.  Val Brown (@ValeriaBrownEdu) has posted a complete discussion guide for White Fragility here, and there are other discussions and resources on the Clear The Air website.

### The Joy and Influence of Math:

This category includes a range of books, all touching on the power of math and its influence on us as individuals and on society.

• Infinite Powers by Steven Strogatz (@stevenstrogatz)
• The Art of Logic in an Illogical World by Eugenia Cheng (@DrEugeniaCheng)
• Hello World: Being Human in the Age of Algorithms by Hannah Fry (@FryRsquared)
• Mind and Matter:A Life in Math and Football by John Urschel (@JohnCUrschel)
• Math with Bad Drawings by Ben Orlin (@benorlin)

### Puzzles and Fun:

I am a huge fan of number and geometry puzzles for entertainment, and I buy way too many books in this category. Some of these book authors post puzzles on Twitter (for free!), and people post their various solutions and methods.

So, as school winds down and summer days stretch out ahead of you, please join me in picking a few books for reading, learning, and enjoyment.  There’s something for everyone on the list!

Notes:

My post “Summer Assignment” from 2016 also has book recommendations.

Another list of math-related books is from Math Frolic here.

I could not fit this into another category, but wanted to mention Adding Parents to the Equation: Understanding Your Child’s Elementary School Math by Hilary Kreisberg (@Dr_Kreisberg) and Matthew Beyranevand (@MathWithMatthew).

On Twitter last week, Kristen asked, “What are your grading categories and what percent of the students’ final grade comes from each?”  Kristen is a 7th grade math teacher and was looking for suggestions because she wants to change her approach.

When I was in the classroom full time, I used a grading system that I had revised and honed over several years, that worked for me and my students.  Before I describe its details and justifications, let me first say that each school and classroom has its own considerations, so every teacher should do what works for your situation.  Also, I mostly taught high school math, so some of this would be different for middle school and 9th grade.¹

POINTS NOT PERCENTAGES:

In my math classes, every test, quiz, lab activity, or hand-in assignment was worth a certain amount of points.  Quizzes were usually 10–40 points, Tests 50–100, Labs/Hand-ins were 20–50.  To find a student’s average, add up all the points and divide.

Students need to show the mathematical thinking (“show your work”) in order to get full credit.  If they show good work, but have the wrong answer, they might earn 4 out of 5 points.  If they have the right answer but without supporting work, they only earned 1–2 points.  Any mistake that is carried through to later parts of a problem without making new errors does not get new deductions (akin to the AP exam free response question grading).

Each question on a test or quiz is worth whatever the mathematics warrants, from 1 to 10 points.  Thus every similar question throughout the marking period has similar weighting, it doesn’t matter if it is on a “test” or a “quiz”.  This is one reason I changed from “Tests 50%, Quizzes 30%” type system; in that setup, the same question on a test can be worth much more than on a quiz.  Also, I sometimes only had two tests in a marking period, which didn’t seem worthy of half of the student’s grade.

Lab activities and hand-in assignments meant that a student’s grade did not just depend on timed “higher-stakes” assessments.  Students with test anxiety could demonstrate their knowledge in another way, with less stress and time pressure.  I usually had one of these every week or two throughout the marking period.

HOMEWORK:

I checked homework daily, for completeness (not accuracy). During the few minutes I took to get around the room, students were discussing the homework with partners or small groups, checking answers from a key, and resolving misunderstandings.  I marked down if homework was complete, incomplete, or not done.

The student’s homework results moved their grade up or down from their class average in a range from +2 to –5 percentage points.  My reasoning was that doing homework consistently helped their class average be as good as it could be, and homework was essential to success.  If a student did their homework all the time, their grade was increased by 2 percentage points; if they missed many assignments, they were penalized by losing up to 5 points. Many students counted on those two added points and worked hard to earn them.

I had felt that other systems for grading homework weren’t equitable.  For example, if the homework was worth 100 points (perhaps 3 points per night), students who had been running a 90% might have their grade go up 3 points, but students with a 70% might get a 10 point boost.  With the +2 to –5 range, every student got the same impact for doing (or not doing) their homework.

OTHER CATEGORIES:

As for Class Participation or Notebooks, these seemed to be hard to capture with a grade, and often created extra record-keeping work for me.  Students might have viewed them as easy ways to bring their grades up, but I generally did not attach any grades to them.  If I valued notebooks or taking notes for a particular class, I might grade it as a lab, especially in middle school or 9th grade when I was trying to build habits for success for the rest of high school.  In some classes, we did a quarterly portfolio as a way to summarize, consolidate, and reflect upon the learning.²

Other commentators in the twitter discussion pointed out that a teacher might value engagement in discussion, or seeking help, or collaboration with other students. Consider using a rubric (shared in advance with your students) to promote the student habits you desire.  Here is one on “Class Participation” along with a record sheet for students to analyze their contributions (thanks to Carmel Schettino @SchettinoPBL) and here is one on “Student Work Habits”.

Other types of Formative Assessment don’t fall into my grading scheme, because they are formative… the information being gathered helps steer my teaching and gives the student feedback on their learning progress.  Nearly everything that happens in the classroom is part of formative assessment, helping all of us calibrate where we are on the learning journey.³

The decision whether to do Test Corrections or Retakes is a much larger discussion, but basically I did not give retakes or give points back for corrections.  My experience while teaching high school was that if students expected a guaranteed option for a retake, they didn’t always take responsibility for being prepared in the first place.

There were some times when everyone bombed an assessment, and usually that means I didn’t do the job as the teacher.  We would reteach, review, reflect, and then take a second version of the test that was averaged with the first.  I wanted to send the message that the first (poorer) grade doesn’t go away.

EXTRA CREDIT/BONUS QUESTIONS:

When a good opportunity arose, I would put a bonus question on an assessment or give extra credit for an optional part of a lab activity.  The points earned for these things accumulated as a separate “bonus quiz” for each student, rewarding them for doing more math extensions on our current work.

If a bonus was worth 5, you got 5/5 in your bonus quiz.  By the end of the quarter, students had bonus quizzes worth anywhere from 1/1 to 35/35, and some had none.  The bonus quiz didn’t fill holes of points lost elsewhere, but helped boost your average on the margins.

I’ve seen teachers who have successfully added ON bonus points (or included a grade such as 5/0).  This method allows bonus knowledge to make up for mistakes, which I have tried when coursework is very difficult and/or class averages are very low.  But if class averages are doing well, the 5/0 method results in averages greater than 100%.  It also might imply that you can make up for not knowing/doing some math last month by doing an extra project this month, and I wanted to make the point that all our work is important and students can’t avoid some work while still earning top grades.

MISCELLANEOUS:

I did NOT give pop quizzes, because I felt that to be a punitive practice (kind of like, “Gotcha! You’re not prepared”) that could be used by teachers to combat other issues, such as poor behavior or not doing homework.  Students always knew in advance what a test or quiz would cover, and most classes had designated review time in the day(s) prior.

Whatever your school’s grade scheme (letter grades, numerical grades up to 100%, 4.0 GPA, etc.) decide in advance what your cut-offs and rounding routines will be.  I had a firm “.50 and higher rounds up, .49 and lower rounds down” policy, which meant that a student with an 89.48 did not get the A– for the quarter.  If this feels unfair to you, decide in advance what you would do; options are to “borrow” the .02 from the coming quarter or to be lenient if you feel that particular student has earned the higher grade level.

MOST IMPORTANT REMINDERS:

Whatever your grading system is, perhaps the most critical thing is to be prompt returning graded items with feedback.  The learning process is a partnership between me and my students, and if I delay or deny feedback, I’m not doing my half of the job.  When students wait days before getting a quiz back, they cannot learn from mistakes on concepts that are the foundation for new material.  Often, the same topics will be on the upcoming test, and I want my students to benefit from having the quiz to study from.

If you decide to change your grading system part way through the year, be honest with students about the changes and your rationale for making them.  Discuss with them the incentives you want your system to provide.  Linda Wilson wrote a 1994 Mathematics Teacher article, “What Gets Graded is What Gets Valued” and that is true to a large extent, for better or worse.  I found that if I didn’t check or grade homework, my students wouldn’t do it; so if I valued that practice, I needed to include it in my grading structure.

Ralph Pantozzi (@mathillustrated) notes that whenever people are given a metric that they will be judged upon, they behave so that they perform well against *that standard*.  His advice is to “make your system revolve around students doing the math you value” so that they will work to achieve those goals.  Well said.

Notes:

¹For younger students, I used +3 for homework (which I also did for HS when material was very difficult).  I required test corrections in some middle school classes as a homework assignment, to place clear value on understanding what went wrong and what can be done differently to avoid errors in the future.

²Portfolios took some time, but were worthwhile in both Algebra 2 and PreCalculus.  A good resource for portfolios is Mathematics Assessment: Myths, Models, Good Questions, and Practical Suggestions edited by Jean Kerr Stenmark (NCTM, 1991). These same classes also did writing in Math Journals with a few prompts each marking period. A nice summary of how to use writing is Marilyn Burns’ article “Writing In Math” in Educational Leadership (ASCD, October 2004).

³Thanks to Steve Phelps @giohio and Martin Joyce @martinsean for these points about Formative Assessment.  I have also found the book Mathematics Formative Assessment by Page Keeley & Cheryl Rose Tobey (Corwin, 2011) to be helpful.

# How Else Can We Show This?

What I love about using calculator technology in my teaching is the “Power of Visualization” and the opportunity to examine math through different lenses.  The multiple representations available on TI graphing calculators—numeric, algebraic, graphical, geometric, statistical—allows me to push my students to approach problems in more than one “right way.”  By connecting these environments and making student thinking visible when we dig into a mathematical situation, we support students in productive struggle and deepen their understanding.*

Read my post on the TI BulleTIn Board Blog for two scenarios in which my students and I pursue multiple pathways to show and make sense of the mathematics at hand (with demonstration videos!)

### How Else Can We Show This?

Read the entire post at the above link, and here is a quick summary:

1. Riding the Curves and Turning the Tables: studying quadratic and polynomial functions.

VIDEO 1  Using different forms of quadratic functions to reveal graph features.

VIDEO 2  Using the graph-table split screen to see numerically what is happening at key points.

2. Absolute Certainty: solving absolute value equations and inequalities.

VIDEO 3 Using the graphical environment to support an algebraic solving procedure.

*Connecting mathematical representations and supporting productive struggle are two of the high-leverage mathematical teaching practices discussed in NCTM’s Principles to Actions: Ensuring Mathematical Success for All (2014).

# Action-Consequence-Reflection Activities for GeoGebra

When I choose to use technology in my math teaching, I want to be sure that the technology tool supports the learning, and helps students to develop conceptual understanding.  The Action-Consequence-Reflection cycle is one structure that I use towards this goal.  I’ve written about Action-Consequence-Reflection activities before, in this post and this post, and I recently had an article published in the North American GeoGebra Journal, “Using Action-Consequence-Reflection GeoGebra Activities To Make Math Stick.”

In the Action-Consequence-Reflection cycle, students

• Perform a mathematical action
• Observe a mathematical consequence
• Reflect on the result and reason about the underlying mathematical concepts

The reflection component is, in my view, the critical component for making learning deeper and more durable.  The article includes the following six activities that use the cycle to help “make the math stick” for students.  Each of the GeoGebra applets is accompanied by a lab worksheet for students to record their observations and answer reflective questions.

EXPLORING GRAPHS & SLIDERS:

The first two activities use dynamic sliders so that students can make changes to a function’s equation and observe corresponding changes on the graph.

In Power Functions, students control the exponent n in the function $f\left(x\right)=x^n$, and can toggle between positive and negative leading coefficients.

In Function Transformations, students investigate the effects of the parameters a, h, and k on the desired parent function.

INTERACTIVE VISUALIZERS:

Using the power of visualization to deepen understanding, the Domain and Range applet highlights sections of the appropriate axis as students manipulate linear and quadratic functions.

UNDERSTANDING STRUCTURE:

In the Rational Functions activity, students explore how the algebraic structure of functions relates to important graph features. The handout includes extensions allowing investigation of other rational function scenarios not already covered.

INVESTIGATING INVARIANTS:

The last two activities have students looking for invariants—something about the mathematical situation that stays the same while other things change.

In Interior & Exterior Angles, students investigate relationships among the angles of a triangle and form conjectures about the sums that do and don’t change as the shape of the triangle changes.

In Right Triangle Invariants, the applet links the geometry figure to a numerical table of values, and students discover several invariant properties occurring in right triangles.

PLANNING FOR REFLECTION:

Simply using these robust technology activities will not guarantee student learning and conceptual understanding; it is imperative that we as teachers plan for reflection by including focusing questions, discussion of students’ mathematical thinking, and clear lesson summaries with the activity.  Use the provided lab worksheets or adapt them for your needs.  Capitalize on the power of the Action-Consequence-Reflection cycle to make the math stick for your students’ success!

Notes and Resources:

This post contains excerpts from the full article (pdf available here) from Vol 7 No 1 (2018): North American GeoGebra Journal.

The North American GeoGebra Journal (NAGJ) is a peer-reviewed journal highlighting the use of GeoGebra in teaching and learning school mathematics (grades K-16). The website for the NAGJ is here.

My GeoGebra Action-Consequence-Reflection applets are in this GeoGebra book, or they can found by entering “kdcampe” into the GeoGebra search box.  Thanks to Tim Brzezinski, Marie Nabbout, and Steve Phelps for their assistance with some of the GeoGebra applets.