Woman Superhero T-Shirt Challenge

Autumn asked, “Why don’t you wear a girl superhero T-shirt for our videos, dad?”  And I said, “Why not?  Absolutely!”  We started searching for a woman superhero T-shirt for men but ran into trouble:  There are a lot of women superhero T-shirts for women, but very few for men that are perfect for our video series.  So we need your help!

If you know of an excellent woman superhero T-shirt for men, please share it with us in the comment section below.  If you include your name (you don’t have to), we will thank you in the video when I wear your T-shirt!

CHANNEL: Growing up with Eureka
© 2015 Autumn Baldridge and Scott Baldridge

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Mathematician Clayton Shonkwiler talks about Polymer Science

Intended Audience: Everyone, and especially teachers who want to show to their students a mathematician explaining the motivation behind their own research.

In this episode we meet Clayton Shonkwiler, a mathematician from Colorado State University, who talks to us about applications of geometry and topology to the study of random polygons and polymer science.

Clayton discusses some ideas related to his paper with Jason Cantarella and Tetsuo Deguchi, Probability Theory of Random Polygons from the Quaternionic Viewpoint.  Here is the first paragraph of the abstract to their paper.

We build a new probability measure on closed space and plane polygons. The key construction is a map, given by Knutson and Hausmann using the Hopf map on quaternions, from the complex Stiefel manifold of 2-frames in n-space to the space of closed n-gons in 3-space of total length 2. Our probability measure on polygon space is defined by pushing forward Haar measure on the Stiefel manifold by this map. A similar construction yields a probability measure on plane polygons which comes from a real Stiefel manifold.

While the video above is for a general audience, Clayton’s paper is not (it’s written for other mathematicians).  However, ambitious high-school students may still enjoy looking at it to see what advanced theorems and proofs look like.

Students, parents, teachers and mathematicians alike will also enjoy visiting Clayton’s website http://shonkwiler.org/, which features stunning mathematical art inspired by the beautiful mathematics that arises in his research and teaching.  Some highlights from Clayton’s portfolio are shown below.

If you enjoy what you see, please be sure to Like our Facebook page.



Parts of a Whole:






Follow me @ScottBaldridge or like my Facebook page at www.fb.com/scottjbaldridge.


CHANNEL: Geometry and Topology Today
© 2015 Scott Baldridge and David Shea Vela-Vick
Supported by NSF CAREER grant DMS-0748636 and NSF grant DMS-1249708

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Variables made easy

Intended Audience:  Teachers, prospective teachers, and parents (public, private, homeschool).


In this video, 6-year-old Autumn explains that a variable is a slot that you can put a number into.  The slot is usually represented on paper as a letter (such as x) or a mark (such as ___).  Here’s the definition of a variable symbol:

VARIABLE. A variable symbol is a symbol that is a placeholder for a number from a specified set of numbers. The set of numbers is called the domain of the variable.

A variable is a placeholder for “a” number; this number does not “vary.” An (unfortunately) common Textbook School Mathematics description of variable in the U.S. textbooks is, “A variable is a quantity that varies.” How does “a quantity” vary? (No, really, explain how a quantity like the length of a football field varies!)  It is no surprise to me that students in the U.S. don’t understand descriptions that don’t make any sense.   However, the description, “a placeholder for a number,” is about a single, non-varying number: “A thing” is much more concrete to students than, “A thing that could be this thing or that thing or maybe that thing over here; it varies.”

The beauty of the correct description of variable (and a point that needs to be made over and over to our students) is that it is the person who is using the variable who has ultimate control over what number they wish to insert into the placeholder.   The power to choose the number they insert into the placeholder rests in the will of the student, not in the variable itself!  The power to choose (and possibly start over and choose again) is what “vary” in “variable” means.

Okay, here are some notables about the video above that I like:

  • Around the 1 minute mark, I say “are going to be numbers,” and Autumn follows up with a slightly strange sounding addendum remark, “or a number.”  Autumn is actually clarifying my statement here: We don’t insert more than one number into a slot at a time.  Instead, we insert only one number into a slot and that same number is inserted into every instance of the slot.
  • The unit language that we have been using throughout this video series (and which runs all throughout Eureka Math) is present in the way we speak of arithmetic with variables as well.  So, for example, “3 tens plus 2 tens is 5 tens” that students say in early grades is the same as “3 slots plus 2 slots is 5 slots” that Autumn says in this video.  This makes the link between arithmetic and algebra much more obvious—in fact the two statements are exactly the same if we insert the number 10 into the slot!
  • The algebraic expressions discussed in this video are:  3x+2x, 11x+6x, 15x-8x, 3(7x), x^3, x^5, x^{10} (and Autumn mentions x^{100}).
  • Autumn explains that 3(7x) = 21x, and shows that if you insert 2 for the slot,  then the expression becomes a numerical expression 21 \cdot 2, or 42.  Of course, we have the power to vary:  we could start over and insert 5 into the slot instead, and then 3(7x)=21x becomes 3(7\cdot 5) = 21\cdot 5 or 105.  (Once we insert a number in for x, the number goes into all instances of x at the same time.)

Finally, we hope you enjoy laughing (with us) at the blooper that runs after the credits.  We cut this segment from the main video because it was a rare “mental typo” that didn’t contribute to understanding the main point of the video.   In general, in the “Growing up with Eureka” videos, we actually make a point of showing the real errors because Autumn and my discussions of those errors can lead to a better understanding of the concept for the viewer (cf. the discussion of “42 slots” in this video).

Variables as Slots

As always, comments are welcomed.

CHANNEL: Growing up with Eureka
© 2015 Autumn Baldridge and Scott Baldridge
Partially supported by NSF CAREER grant DMS-0748636

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Building the ⟨Sci|State⟩ Studio—a weekend in time lapse.

The SciState Studio is finished and we’re all ready for production! In this video, we get a behind the scenes look at some of the finer details of our studio construction. As we embarked on this adventure, one of the first things we learned was that high-quality audio is an essential component of any successful video series. With this in mind, we have outfitted our studio with acoustic insulation panels and installed various other echo reducing components throughout. We also constructed an awesome plexiglass “light board,” which is similar to a traditional whiteboard but doesn’t require one’s back to face the camera. We’re really proud of everything we’ve produced and hope you agree!


We should point out that it takes a real expert (thank you Justin Reusch!) to set up the lighting for the video stage.  You can see how that came out by watching our “Testing out the Green Screen” video.  As a point of comparison, note the terrible sound in the “Testing out the Green Screen” video—that is what the studio sounded like before we built the acoustic insulation panels.

As always, comments are welcome below.

CHANNEL: Geometry and Topology Today
© 2015 Scott Baldridge and David Shea Vela-Vick
Supported by NSF CAREER grant DMS-0748636 and NSF grant DMS-1249708

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Something fun: What it is like to be the son of Captain Derivative


Above is a picture of my stepdad, Captain Derivative, doing integration exercises for yet another day in the vector field against his arc enemy, the pathological and degenerate Prime Matrix.

Some people have asked me, “What is it like being the son of a differential operator who can wield power series with such ease?” Well, when I was young and still very near my initial value, I used to oscillate rapidly between believing whether his exponential powers were real or imaginary–often shifting my phase over time with some frequency. But then he removed a discontinuity from a complex surface right before my very eyes, and I saw the proof of his limitless analysis: I converged upon the realization that my own scientific skepticism had been irrational. From that moment on (t=5), I would often help him from his secret power base by relaying coordinates of tangents he could approach in the Cartesian plane so he could do battle against those divergent improper integrals.

Go Captain Derivative, we are with you (or at least, within an epsilon ball of you). May the functions you differentiate always be smooth!

(My stepfather, Fred Reusch, is the calculus teacher at my old high school in Rockford, Michigan.  Today was “Super-hero Day” at the school, and he came dressed as “Captain Derivative.”  I think he would wear this outfit to school everyday if he could! :-) )

CHANNEL: That’s News To Me
© 2015 Scott Baldridge

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Adam Saltz speaks on an annular refinement of the transverse element in Khovanov homology

Intended Audience: Research mathematicians, professors of mathematics, graduate students in mathematics, and advanced undergraduate students in mathematics.

In this 1 hour episode, we see a presentation by Adam Saltz, a mathematician and graduate student at Boston College, on a new invariant of transverse knots in links coming from Khovanov homology.

In the talk, Adam discusses some of the details contained in his paper with Diana Hubbard, An annular refinement of the transverse element in Khovanov homology. Here is the abstract to their paper:

We construct a braid conjugacy class invariant κ by refining Plamenevskaya’s transverse element ψ in Khovanov homology via the annular grading. While κ is not an invariant of transverse links, it distinguishes some braids whose closures share the same classical invariants but are not transversely isotopic. Using κ we construct an obstruction to negative destabilization (stronger than ψ) and a solution to the word problem in braid groups. Also, κ is a lower bound on the length of the spectral sequence from annular Khovanov homology to Khovanov homology, and we obtain concrete examples in which this spectral sequence does not collapse immediately. In addition, we study these constructions in reduced Khovanov homology and illustrate that the two reduced versions are fundamentally different with respect to the annular filtration.

This video and paper are aimed at mathematicians, graduate students and undergraduates with lots of experience in topology.  However, high school students who think they are potential math geniuses may still enjoy looking at it to see what advanced theorems and proofs look like.

As always, comments are welcome!

Adam Saltz

CHANNEL: Geometry and Topology Today
© 2015 Scott Baldridge and David Shea Vela-Vick
Supported by NSF CAREER grant DMS-0748636 and NSF grant DMS-1249708

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Testing out the new Green Screen (Plus a bonus 14×16 calculation)

This is a test video that Shea and I did to test the new green screen. Shea is spoofing a bit with me—acting all surprise at my ability to calculate 14×16 in my head. He looses it slightly near the end.  (Research mathematicians do a lot more than mental math although we are almost constantly doing mental math: it is just math that is a lot harder than simple arithmetic.) Check out Shea laughing at the end.

The test video is to try the different types of backgrounds we can now use with the new studio setup (and green screen). You can vote which background you like the most here:

This is just an informal poll.  You can comment below if you have a better idea for a background.  For example, Shea’s son wanted whales swimming.  Maybe not quite professional enough for math videos…

Thanks especially goes to Justin Reusch, who came all the way from Austin, Texas to set up—and explain how to set up—the lighting for this type of shoot. You can see his work in the video on our faces (the back-light halo effect on our heads, the side lights and shadows on our faces, etc. ). Thank you, Justin!  Thanks also goes to Hang, who kept the kids busy while we put together the shot.  You can hear the kids in the background playing during the video: like I said, this was just a test video.

Scott and Shea

CHANNEL: Geometry and Topology Today
© 2015 Scott Baldridge and David Shea Vela-Vick
Supported by NSF CAREER grant DMS-0748636 and NSF grant DMS-1249708

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Powers of 2

Intended Audience:  Teachers, prospective teachers, and parents (public, private, homeschool).


In this video, 6-year-old Autumn explains that the meaning of “2 to the power of 4” is a product of four factors of 2:

2^4=2\cdot 2 \cdot 2\cdot 2.

This is the beginning of exponentiation and it is easy to understand—the power (in this case, 4) tells us how many factors of the base (2) there should be.  I usually tell my college students tongue-in-cheek that, “Mathematicians are laaaaazzzzzy. We came up with the notation 2^{30} because we got tired of writing all 30 factors of two out!”

There are a few issues to watch out for when introducing a child to exponents (some of which I mentioned in the video):

  • At this level, after students understand multiplication, powers of 2 are no harder or easier than learning addition or multiplication facts.
  • However, it is a new operation and one that is easily confused with multiplication.  The statement, “2 times 4,” sounds a lot like, “2 to the 4th.”  Furthermore, the processes for evaluating the expressions are are similar but for powers we use multiplication instead of addition: “2 times 4” is describing a number of addends while “2 to the 4th” is describing a number of factors.

    For these reasons, you (as the parent or teacher) should be very deliberate about asking, “What is 2 to the power of 4?” to help your child/students understand that you are asking for something very different than “2 fours.”

  • Definitely stick with “powers of 2” until students are comfortable with the meaning of exponents and can confidently tell you what 2^1, 2^2, \ldots, 2^{10} are.  Remember, you are trying to help students learn the meaning of the operation, not memorize a bunch of numbers. (If you feel you can move on, go to powers of 10 next, and then to powers of 3.)

    Once they are comfortable moving back and forth between 2^7 and 128 the real fun can begin—learning the properties of exponentiation like a^m\cdot a^n=a^{m+n}.  Don’t worry about this for now; Autumn and I will show you how easy and pleasurable it is to learn some of these properties in another video.  For now, just concentrate on learning the powers of 2 themselves (up to 10 or 12).

  • Finally, it is much easier to remember the powers of 2 if you use “tags” or “pins,” i.e., memorizing a couple of easy-to-remember powers of 2, and using them to quickly figure out the rest.  I recommend: 2^5=32, 2^8=256, and 2^{10}=1024.  Then one can quickly find 2^{11}=2^{10}\cdot 2= 1024\cdot 2=2048, which is just an easy double.  You will see the “story” Autumn and I used to remember 2^8=256 in the video.

Check back soon for a new video on the properties of exponentiation.  In the meantime, enjoy learning the powers of 2 with your child/students!

Autumn and Powers of 2

As always, comments are welcomed.

CHANNEL: Growing up with Eureka
© 2015 Autumn Baldridge and Scott Baldridge
Partially supported by NSF CAREER grant DMS-0748636

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An article on how Eureka Math began.

Leigh Guildry wrote a nice piece for The Town Talk newspaper, “How did Eureka Math start? 2 writers answer.”  She traces the steps that I, Robin Ramos, Nell McAnnelly, and Lynne Munson (director of Great Minds) took to develop the Eureka Math/Engage NY curriculum.

She starts with recent test results of students from Rapides Parish School System who were using Eureka Math:

“Rapides [Parish School System] students progressed at almost double the national average on benchmark tests by Discovery Education. Students gained an average of 141 scale score points during the 2014–15 school year, compared to 73 points for students in districts nationally, according to eureka-math.org. The district more than doubled the average U.S. gains in middle school grades (6–8) and in kindergarten.”

Pretty impressive! Congratulations go to students and their teachers.

The article goes on to interview Pam Goodner, who was the lead writer who worked with me in creating the 12th grade “Precalculus Course.”  (Thank you, Pam, for your hard work.)

Louisiana Bayou, JR Meeker (1884) wiki commons

You can follow Leigh Guidry on twitter at @Leigh_TownTalk.

CHANNEL: That’s News to Me
© 2015 Scott Baldridge

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Electric Girls: Shaping Role Models in Technology

Check out the video below, then visit their website to learn more about the program.  I need to see if there is a similar program in Baton Rouge for my daughter Autumn.


From the website:

“Electric Girls is a 12-week education program for girls ages 9-14 in New Orleans. Using a mentorship structure, we teach girls to become leaders and role models in STEM (Science, Technology, Engineering, and Math). Girls come away with a new set of hard skills (soldering, drilling, building circuits, etc.) and soft skills (perseverance, curiosity, leadership, self-motivation).”

Watch for Maya Ramos in the video.  (She is Robin Ramos’ daughter.  Robin is a good friend and colleague, and the lead writer/teacher of A Story of Units.)  Maya studies music in New Orleans when she isn’t building electrical circuits.  You can listen to Maya’s piano playing in her band “Spare Change” by visiting their Facebook page.


CHANNEL: That’s News to Me
© 2015 Scott Baldridge

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The Secrets of my Teaching Success: James Tanton

Note from Scott Baldridge:  I recently asked James Tanton, “What are some of the reasons for your success as a teacher?”  I consider James to be a fabulous high school teacher and a good friend, and was interested in his response.  To see why so many people think highly of James, I recommend that you visit James’ websites www.JamesTanton.com and www.gdaymath.com and follow him on twitter: @JamesTanton (and click his suggested links below).  James responded with the following 12 points. I felt they were so well-articulated and compelling that I couldn’t let these gems remain just between the two of us.  Thank you, James!

The Secrets of my Teaching Success

by James Tanton

I have been asked:

What are some of the reasons for your success as a teacher?

And I sat down recently to try to give a serious answer to this question.

But I first need to point out that I disagree with the underlying premise of this question: I am not sure that my success means that I am a “good” math teacher by the usual standards of teaching. People may be shocked to learn that I always arrange the tables in my classes and workshops so that everyone is facing the board. I lecture, I only lecture, and all is focused on the teacher in the room. I use the board extensively, and there is absolutely no technology anywhere in sight as I teach. And I don’t do anything innovative in the classroom – seriously, zero, zip, zilch on the innovation front.

So this ego-full, self-focused piece is my attempt to answer the question as to why some people seem to think I am a successful teacher despite the above. It comes as twelve points.

Success 1I have a cute accent.

I am serious in that I think this a big part of my success in the classroom.

I was raised in Australia with a British father and as a result my accent is something confusing: Australians think I am British, Brits know I am Australian, and everyone else is confused as to what I am. But my accent seems to be extremely pleasing to the American ear and I am fully aware that it works to my full advantage in my American life.

Success 2I treat everyone like adults – even kids.

I always assume everyone just does the right thing.

Well, I need to qualify that. We are all human and I know we make silly mistakes when under stress and pressure and so might slip on doing the right thing every now and then. But that is the learning process for all this.

So I assume people just do the right thing, and if they don’t, will learn from the goof and just not do it again.

I had one blatant, silly act of cheating as a college professor: I received two identical, word for word, silly error for silly error, homework papers. My response was to give one paper an A+ and the other a C- and never say a word. It turned out that the two authors never said a word either and it never happened again.

I once gave a lecture on cheating 101, general pieces of basic advice on how to get away with things. (If you’ve copied someone’s paper, don’t hand it at the same time as your partner-in-crime: make sure your papers sit in different parts of the pile. Photocopying someone’s answers is just a ludicrous idea. Don’t copy the same spelling and obvious math mistakes – perhaps insert a few more of your own. Don’t wear a baseball cap during an exam – the rim points the same way you are looking. And so on.)

When I moved to high-school teaching I was flabbergasted at the idea of “needing to remove the temptation of cheating” for our students. This incorporated ideas such as erecting screens between seats during class quizzes and having students being supervised while they do make-up tests. Where are students meant to learn about the wrongs of cheating and making those first-time silly mistakes? Plus the insult to students assuming they can’t be adult about all this! I did none of the things I deemed insulting to students.

There was one time when young Jenna was looking over at someone’s paper during a quiz. I just walked up behind her and quietly whispered: “Just be careful where your eyes go during a quiz.” End of issue. I just ask students “to do the right thing” when it comes to finishing up a test at home. Even if they slip, there is an emotion that accompanies the wrong doing that sits and lingers, and contending with that emotion is the learning experience. (Plus students, by and large, do do the right thing!)

Success 3I am quirky and I like to play with ideas.

Here’s a tiny piece of quirkiness that illustrates the power of playfulness:

In learning about permutations we start by counting the number of ways to rearrange letters in words or, better yet, in our names: the letters of JIM can be arranged 6 ways (3!), the letters of JAMES 120 ways (6!). But a name like BOB or DANA represents a problem. (Brute force gives 3 and 12 ways, respectively.) The problem is worse for the word CHEESE. So we need to figure out a reasonable way to handle repeated letters.

As ideas develop we go from CHEESE to CHEESES to CHEESIEST, and when we have the hang of it, we go straight to CHEESIESTESSNESS, the quality of being the cheesiest of all the cheeses. People just seem to love the “word” cheesiestessness and the whole lesson sticks.

I also love squine and cosquine (www.jamestanton.com/?p=605). I love to ask how many degrees there are in a Martian circle (www.jamestanton.com/?p=633).  And I love quirky words from the history of math: vinculumobelusradix, and so on (www.jamestanton.com/?p=1258).

Success 4: I think hard about “what’s really going on” and “why anyone cares.”

I think I am good at thinking deeply about stuff and can cut through all the usual surrounding clutter. That’s why my lecture style works, I think: what I ramble on about is de-cluttered content and so sustains interest.

Plus I do the quirky, straight to the heart-of-the-matter, lectures. Exploding Dots is a prime example. (www.gdaymath.com/courses/exploding-dots/)

Success 5I break every 37 ½ minutes.

I once read a paper early in my career that said that the average attention span of an adult or near-adult audience member sitting through a lecture-style presentation is 37½ minutes. I’ve taken that as a literal fact, and have made it a universal law in my teaching. I tell this little story at the start of my courses and workshops and we religiously have a break at the 37½ minute mark, even if it is only a 45 minute class!  

Success 6I know some history of math.

I want math to be the human story that it is. I share the tales of the backs and forths and the struggles of developing ideas leading to how we see and use them today.

Success 7I am not at all afraid to make mistakes. Even whopper of ones.

It is a vital and genuine part of math to be human in your relationship with it. I don’t need to be seen as the expert. But I do need to model what it means to engage with mathematics as a human being.

Success 8I seem to be good at helping people feel it is okay not to know.

After all, I know very little myself. The message I give is that it is completely okay not to know something, but it is not okay not to want to find out.

This notion is tied into the use of the word should, as in “you should know this” or “students should know.” Should statements often have a feeling of judgement attached to them and they induce unpleasant sinking feelings in the gut for the recipients. I avoid making should comments.

And usually these statements are moot: even if students should know how to distribute a negative sign by grade 9 and your students don’t, it is irrelevant – it just means that you need to talk about distributing the negative sign with that class. (Try something like 1.4 of www.gdaymath.com/courses/astounding-power-of-area/.)

But there is another aspect of these “should” comments that worries me. As one’s mathematical sophistication grows one starts to see former concepts in a new light. Subtleties and hidden assumptions become clear and previously comfortable topics become uncomfortable and shaky. The idea that, for example, by the end of middle school students should be comfortable with fractions is ludicrous to me. Fractions are actually very hard and a thinking high-school student really should revisit them and be uncomfortable with them! (www.jamestanton.com/?p=1461.) (Did I just use the word “should”?)

Success 9I think I am good at recognizing “hazy” thinking.

You know when you are lecturing or teaching on content that you really haven’t quite properly sorted out for yourself. You can do the work, you can explain the piece, but you know you don’t really “get it,” the heart of it, that is. I have lectured while in this state too, it happens, but I share my emotional state with the audience. I like to think it helps students recognize hazy thinking when it happens to them too. Hazy thinking is a call to go for a walk, to mull on the idea, and to ask “What’s really going on with this topic?”

Success 10: I incorporate in my courses moments of “looking back” as part of pushing forward.

Here’s a piece that illustrates what I mean by this. www.jamestanton.com/wp-content/uploads/2012/03/Curriculum-Essay_August-2015_Undergraduate-Courses-for-Teaching1.pdf

Success 11I have a PhD from Princeton of all bleedin’ places!

People seem to think that means I know my stuff. Hmm. That perception certainly contributes to my teaching success.

Success 12I am not obsessed about assessment.

I just want students to prove to me that they get it in the end. If it takes a while before they do and grades are lousy during that period, no worries, get it in the end we can ignore all that. This notion seems to be an anathema in high-school world – of all places!

(If I am forced to think about assessment in high-school teaching, I think this way: http://www.jamestanton.com/?p=968.)

Tanton_Photo Nov 2014

As always, please feel free to comment below!

CHANNEL: Engineering School Mathematics
© 2015 James Tanton

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Something fun: Have you ever used a ruler to measure a Snafoose?

Have you ever used a ruler to measure a snafoose?
I’ve never, never-ever, tried to measure a snafoose.
Certainly not on a goose.
Or near a boar on the loose.
No, I never tried to measure a snafoose.

But I have used a ruler, why, I have used it a lot!
I have used it to find the distance between this point and thot!
Between two and three,
or three and eight, as I was toght,
to find the distance between four-thirty-three-point-four and two-point-two to the naught.

I’ve measured here and there, in good days and bad,
measuring always had a way of making me feel glad.
I’ve measured while eating,
I’ve measured while preening,
why I’ve even measured while dancing the twirl-e-bop-de-careening.

And here’s what I’ve learned, if you permit me to spin:
the distance between naught and three is just three again.
Now it seems almost like—like you get that for free,
so maybe it should be no surprise:
it is also the same to the negation of three!

Absolute value does not need to be an absolute bore,
It just comes from measuring, measuring and measuring some more.
And when it’s finally brought up, in algebra, with letters,
kids who’ve spent time with rulers,
will best even those…who should know better.

— Scott Baldridge, 2007

I wrote this silly little poem in 2007.  (I can safely claim to be a non-poet.)  I stumbled across it in an email recently looking for another email and thought I would share it.  Note that even then, years before there was anything called “Common Core,” I was advocating that elementary students use rulers, protractors, beakers, weighing scales, etc. to build an intuitive understanding of units.  At the time I wrote this little poem, I did not realize that a few short years later I would be writing an entire curriculum, A Story of Units, based upon manipulating units.

(Check back later for a picture of a Snafoose that Autumn is designing!)


CHANNEL: Growing up with Eureka
© 2015 Autumn Baldridge and Scott Baldridge
Partially supported by NSF CAREER grant DMS-0748636

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What is the sum 1+2+3+4+…+24+25?

Intended Audience:  Math teachers, prospective teachers, and parents (public, private, homeschool).

In this video, 6-year-old Autumn shows how to find the sum


This problem actually has a rich history involving one of the greatest mathematicians who ever lived: Carl Friedrich Gauss. Please share Gauss’ fascinating story with your children or students. The story goes like this:  When Gauss was seven, his elementary instructor gave what the teacher thought was an extra-pointless exercise of adding up the first (say) 25 numbers just to keep his students busy for an hour.  Almost immediately, young Gauss threw down his slate and declared, “There it lies.”  When the hour was up, the teacher inspected Gauss’ answer and found it to be correct.  An interesting article investigating the full story can be found here.

Autumn’s solution to this question was probably very similar to how Gauss solved it.  In the video, I mention that Autumn ran up stairs and came down 10 minutes later with the solution.  Here is her actual work (click Autumns_Work for a .pdf version):


In this picture you can see her thinking: Pairing up 1+25, 2+24, 3+23, etc. and keeping track of which terms she summed on the line below (you can also see the left over 13 at the end).  I think what happened next is that she recognized (in her mind) that

26\times 12 + 13 = (25\times 12 + 12) +13 = 25\times 12 +25 = 13\times 25,

because she writes “same as 13 x 25.”  Regardless, you can see her calculating the

25 \times 10 = 250

just as she did in the video, and you can see me checking her work underneath (and actually showing her another mental math technique–look!).

For Teachers who teach Eureka Math:  This sum is called a finite series because it is a sum of a finite sequence of numbers.  We explore this particular series in many different places in the high school curriculum, but a particularly interesting discussion around this series occurs in Lesson 8 of Module 3 of Grade 9 (Algebra I).  In that discussion we show how the sum can be visualized as “triangles:”

Triangular Numbers–Grade 9, Module 3, Lesson 8 of Eureka Math/EngageNY

This visualization quickly leads (through pictures!) to the general formula of the sum of the first n positive integers:


Plugging 25 in for n shows that S(25)=25\times 13, which is what Autumn derived.  If you are teaching Lesson 8 (or any of the other lessons where this series shows up), think about sharing this video of Autumn with your class as a way to stimulate a discussion with your students.


As always, comments are welcomed!  In particular, we are trying out different places to do math.  The math in this video was done on the hood of a Porsche 914.  We have friends with all kinds of cool sports cars who may be willing to let us borrow them for an afternoon, so look for more “Math on the hood of sports cars” soon (especially if we get positive feedback to do more videos like this).

Adding the first 25 numbers

CHANNEL: Growing up with Eureka
© 2015 Autumn Baldridge and Scott Baldridge
Partially supported by NSF CAREER grant DMS-0748636

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Eureka Math Curriculum: A Breakout Hit

In his opinion article at US.News & World Report, Common Core’s First Breakout HitRobert Pondiscio discusses how the EngageNY curriculum (both the English Language Arts and Mathematics curricula) is being warmly received by school districts and states all across the country:

“I recently obtained data from the New York State Education Department showing that while EngageNY units, lessons and curriculum modules have been downloaded nearly 20 million times as of early May, more than half of those users have been outside of New York. EngageNY may be quietly emerging as Common Core’s first breakout hit.”

Why might Eureka Math/EngageNY be a breakout hit?  There are, of course, a number of reasons.  I hope and think that one of the reasons is because teachers have been searching for and finally found a mathematics curriculum that actually works for them–that they are seeing marked improvement in their students’ understanding of mathematics like no other curriculum before, and that news is spreading across the country.

I think teachers are the real breakout hits.

Plato's_Academy_mosaic_from_PompeiiPlato’s Academy

I encourage my readers to follow Robert Pondiscio on Twitter @rpondiscio.  Of course, feel free to follow me on Twitter @ScottBaldridge or like my Facebook Page to get updates.

CHANNEL:  That’s News to Me

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In Memory of Dr. Dolores Margaret Richard Spikes

Mathematics is…mathematics.  It may seem cold and impersonal at times—appearing not to be driven by human wants or desires but by the precise statements of assumptions and propositions that lead to the proofs of theorems.  This is just not so! Math is, after all, as human as art.  And yet, the seemingly impersonal nature of the discipline is actually one of its greatest assets.  Mathematicians’ shared intensity to irrefutable argument is one of the reasons why we as a group love to celebrate the beautiful mind of anyone who discovers and proves a new theorem.

Today I wish to celebrate the mathematical genius of Dr. Dolores Margaret Richard Spikes, a Ph.D. alumna of Louisiana State University who died last week.  Help celebrate one of her mathematical accomplishments with me: take a moment to read and absorb the abstract to her 1971 Ph.D. thesis:

Title: Semi-Valuations and Groups of Divisibility

From the abstract:  This paper gives procedures for constructing a class of groups of divisibility of rings (not necessarily domains) which properly includes the class constructed by Ohm.  Toward that end, we first extend the concept of a semi-valuation of a field to rings which may contain zero-divisiors.  The notion of a composite of two valuations of fields is then extended to the notion of a composite of two semi-valuations of total quotient rings (which may not be fields), and the construction of this composite is then related to an exact sequence of semi-value groups.  Necessary and sufficient conditions for this sequence to be lexicographically exact are given.

Dr. Spikes also made history in 1971 by becoming the first black graduate to receive a doctorate in mathematics from Louisiana State University.  She was only the 19th African American woman to earn a Ph.D. in mathematics at any university.  Her thesis work stands on its own and is a true testament to her intellect, but the circumstances of the times surrounding her achievement makes it all the more remarkable.  Thank you for your contributions to mathematics, Dr. Spikes.

I highly encourage everyone to read more about Dr. Spike’s rich and impactful life by visiting her Wikipedia page and reading the excellent reference articles there.  Also, check this out.

spikes-dolores(photo: SUNO)

CHANNEL: That’s News to Me
© 2015 Scott Baldridge

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A Quick Comparison of a State Assessment and Eureka Math

by Scott Baldridge and Ben McCarty

Intended Audience: Parents, teachers, and other educators involved in the Eureka Math/EngageNY Curriculum.

In this article, we simply discuss types of math problems from a state assessment and then show similar problems from the Eureka Math curriculum.  We focus on the new TCAP Achievement Test (called TNReady Math), mainly because one of the authors of this article (Ben McCarty) is an assistant professor of mathematics at the University of Memphis.  The format of this post is simple: we will discuss features of the TCAP and then show examples that match those features in the Grade 3 Eureka Math curriculum.

Example 1: Language

From the old Grade 3 TCAP sample test:

True Number Sentence Problem

and a similar problem from Eureka Math Grade 3 (Module 1, Lesson 13):


Note the similarity in the language of “number sentence” between the two problems. Language like this will very likely continue to be used in the new TNReady assessments. Students using Eureka Math will be prepared for that language.  This aspect is particularly important for fractions in 3rd and 4th grade where Eureka Math uses the exact same language (like “unit fraction”) as specified in the Tennessee Academic Standards.


Example 2: Explain-Your-Reasoning Problems

As explained in the “Seven Things You Need to Know About TNReady Math” document, the new TNReady assessments will no longer be all multiple-choice and instead will have a variety of formats:

3.     TNReady will replace the state’s multiple choice only test in math and will include a variety of questions.

Eureka Math teaches a variety of answer formats that should mesh well with the types of questions asked on TNReady.   Eureka Math also gives insight and guidance to the teacher in how to model ways to answer these problems (cf. the teacher-student vignette on the left hand side):


This problem was taken from Grade 3, Module 3, Lesson 11.  This one explanation describes three different models/strategies that students can use in explaining their answer: (1) The tape diagram at the top, (2) the use of letters to represent unknowns, and (3) several mental math strategies for finding 72-28.  Taken altogether, and enacted in daily use over the entire year, these explanation strategies become part of the bread-and-butter methods students can use to answer TNReady assessment problems.

Example 3: Multi-step Problems

Also explained in the “Seven Things You Need to Know About TNReady Math” document, the new TNReady assessments will have many more multistep problems than the old assessments:

4.     TNReady will ask students to solve multi-step problems, many without using a calculator, to show what they know.

Multistep problems are one of the prominent features of the Eureka Math curriculum; it truly exceeds other curricula in preparing students for multistep problems.  The problem in Example 2 above is one such example, but for good measure, here is another example:

Grade 3 Word Problem with Tape Diagram

The tape diagram (the picture between the two paragraphs) helps students convert this multistep problem into pictures they can use to solve the problem.  The TNReady assessments will expect students to be fluent with answering word problems using tape diagrams and other models because they are part of the standards at every grade.  Compared to the mostly-one-step problems of the old TCAP, students using the Eureka Math curriculum will be ready to excel on the new TNReady assessments.

Seven Things for TNReady Math Full


Scott Baldridge
Distinguished Professor of Mathematics,
Louisiana State University
Lead Writer and Lead Mathematician, Eureka Math/EngageNY Curriculum
ScottBaldridge.net (This article and other Engineering School Mathematics articles can be found at this website)

Ben McCarty
Assistant Professor Mathematics
University of Memphis
Mathematician, PK-5, EngageNY Mathematics Curriculum

CHANNEL: Engineering School Mathematics
© 2015 Scott Baldridge and Ben McCarty

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Meet Mathematician Jeremy Van Horn-Morris

Intended Audience: College students, and high school students who think they may be potential math geniuses.

In this episode we meet Jeremy Van Horn-Morris, a mathematician from the University of Arkansas, who talks to us about some geometric and visual tools mathematicians use to understand questions in classical physics concerning the motion of particles.

Jeremy discusses some of the motivation behind his paper with Kenneth Baker and John Etnyre, Cabling, contact structures and mapping class monoids.  Here is the abstract to their paper:

In this paper we discuss the change in contact structures as their supporting open book decompositions have their binding components cabled. To facilitate this and applications we define the notion of a rational open book decomposition that generalizes the standard notion of open book decomposition and allows one to more easily study surgeries on transverse knots. As a corollary to our investigation we are able to show there are Stein fillable contact structures supported by open books whose monodromies cannot be written as a product of positive Dehn twists. We also exhibit several monoids in the mapping class group of a surface that have contact geometric significance.

While the video above is for a general audience, Jeremy Van Horn-Morris’s paper is not (it’s written for other mathematicians).  However, high school students who think they are potential math geniuses may still enjoy looking at it to see what advanced theorems and proofs look like.

Students and mathematicians alike will also enjoy visiting Kenneth Baker’s blog, Sketches of Topology.  The post, Its full of surfaces, provides a stunning visualization and description of the open book decomposition coming from the trefoil knot, which was mentioned by Jeremy the end of our interview.  Some additional posts containing phenomenal depictions of open book decompositions can be found here and here.

As always, comments are welcome!

Jeremy Van Horn-Morris

CHANNEL: Geometry and Topology Today
© 2015 Scott Baldridge and David Shea Vela-Vick
Supported by NSF CAREER grant DMS-0748636 and NSF grant DMS-1249708

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Skip Counting with Fractions

Intended Audience:  Grades 3-6 math teachers, prospective teachers, and parents (public, private, homeschool).

In this video, 6-year-old Autumn skip counts by fractions 1/2, 1/3, and 1/5.  There are a number of ways to skip count by a fraction.  Here are some of them used in Eureka Math/EngageNY:

  1. 1/4   2/4   3/4   4/4   5/4   6/4   7/4   8/4   9/4 …
  2. 1/4   2/4   3/4   1   1 1/4   1 2/4   1 3/4   2   2 1/4 …
  3. 1/4   1/2   3/4   1   1 1/4   1 1/2   1 3/4   2   2 1/4 …
  4. 1/4   1/2   3/4   1   5/4   3/2   7/4   2   9/4 …

Autumn is doing the second skip counting technique above (the fourth is the hardest, which is why it shows up in later grades—try it with 1/6).  The beauty of the second skip counting technique is that

  • it emphasizes the whole unit “…, 3 fourths, ONE, ONE and 1 fourth, ONE and 2 fourths, …”
  • it emphasizes the repeating pattern of important fractional units (1/4, 2/4, 3/4)  between each whole unit.

That doesn’t mean the other skip counting techniques are not important! They all have a role to play in a curriculum.  For example, the first skip counting technique emphasizes that counting fractions is just like counting whole numbers but in a different unit.  That is, “1 fourth, 2 fourths, 3 fourths, 4 fourths, 5 fourths,…” is just like “1 apple, 2 apples, 3 apples, 4 apples, 5 apples…”

Obviously, Autumn already knows a lot about fractions.  I apologize for not showing how to develop the concept of a fraction (maybe another video?).  This process takes a long time and is carefully developed in the Eureka Math/EngageNY curriculum.  You can find out more about how we do this in the curriculum by reading Chapter 6 of “Elementary Mathematics for Teachers” that I co-authored with Thomas Parker.

Regardless, there are many things that parents can (and often already do!) with their children to help them get ready for fractions.  These things include very sensible activities like using a tape measure or cooking cups where the notion of fraction just naturally manifests itself, “Honey, measure out 1/3 cup of sugar please.”  Early on, “1/3” is basically only an adjective modifying the noun “cup;” it references a particular measuring cup, but even so it does bring up a nice way to have a discussion about meaning of those fractions with your children.  Tape measures are also great, “What are those marks between 1 inch and 2 inches on the ruler? Between 5 inches and 6 inches? What could they mean?”

Surprisingly to me, one of the main paths that Autumn came to understand fractions was from reading to her.  Here’s the story:  I started reading full-length novels to her starting when she was 2 years old (stories like Narnia, Lord of the Rings, Harry Potter, Watership Down, etc.).  These are thick books that are close-to or over 1000 pages each.  As we read each book, I started (rather by accident) to discuss with her the fraction of the book that we had read, “Look, honey, we are 2 thirds of the way through!”  An unanticipated-but-nice feature of thick books is that it is very easy to split the book’s pages into thirds, fourths, fifths, sixths by separating the pages with your fingers.  Since the books were all of different thicknesses, over time Autumn came to see the main issues in defining fractions:  to establish the whole unit and the relationship of the fractional unit to that whole unit (cf. how fractions are developed in grade 3 of the CCSS).

As always, comments are welcomed!

Skip Counting with Fractions

CHANNEL: Growing up with Eureka
© 2015 Autumn Baldridge and Scott Baldridge
Partially supported by NSF CAREER grant DMS-0748636

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Ben McCarty plays “Swing that Hammer”

Mathematicians have many talents! In this episode, mathematician Ben McCarty plays and sings the song, “Swing that Hammer.” Ben is a professor of mathematics at the University of Memphis, and the lead mathematician for grades PK-5 of the Eureka Math/EngageNY curriculum.  He coauthored the article “Fluency without Equivocation.”

Ben is here at LSU this week working on a new theorem with me on “special Lagrangian cones,” a type of object that is helpful in studying mirror symmetry from theoretical physics.  I was able to cajole him into playing a song while I recorded.  Ben’s favorite instrument is the banjo, which is probably why his email address is “banjoben.”  In this video, Ben is playing a Breedlove Pro Series C25/CRH guitar.  Enjoy!


CHANNEL: Geometry and Topology Today
© 2015 Scott Baldridge and David Shea Vela-Vick
Supported by NSF CAREER grant DMS-0748636

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Remarks on the History of Ratios

by James Madden

The idea that a ratio is a pair of magnitudes is in Euclid (fl. 300 BC), Elements, Book V.  It is interesting to note that Euclid says that a ratio is the relationship between two magnitudes, not the pair itself.

Greek mathematics did not have the explicit concept of equivalence relation, but the “conceptual grammar” of Greek math is most easily understood by us if we describe it using the idea that they recognized certain equivalence relations “implicitly.”  Consider Euclid’s definition of angle:

“the inclination to one another of two lines in a plane which meet one another and do not lie in a straight line”

Clearly, Euclid means something that can be the same in two different pairs of lines.  This pair of lines and that pair of lines may have the same inclination, and if so, they are equivalent. The same thinking applies to ratio: two different pairs of magnitudes may stand in the same ratio. The genius of ancient Greek mathematics is to produce an operational definition for sameness. Book V, Definition 5 states:

“Magnitudes are said to be in the same ratio, the first to the second and the third to the fourth, when, if any equimultiples whatever are taken of the first and third, and any equimultiples whatever of the second and fourth, the former equimultiples alike exceed, are alike equal to, or alike fall short of, the latter equimultiples respectively taken in corresponding order.”

One thing that I find interesting about the discussion on the notations of ratios—and it’s an issue that occurs often in the analysis of school math—is the contrast between the thing itself, and the means by which we make reference to it. The most basic example is the distinction between the written symbol and the thing the symbol refers to: the number 6 (which, according to Plato, the soul observes before birth—but you can take its ontological status to be what you prefer) and the numeral 6, which is a mark on paper.

As for ratio, is it a number? A pair of numbers—or pair of quantities—by means of which we refer to a number? Or an equivalence class of pairs of numbers? One can find advocates for each of these conceptual images.  The Progressions Document on Ratios and Proportional Relationships has chosen to view a ratio as an ordered pair of real numbers, not an equivalence class of such pairs, nor any of the other options.

Because of history, the word “ratio” has many meanings. I haven’t even mentioned the “Rule of Three” and the way that has shaped the way we talk about proportion. We can acknowledge that there have been many different traditions.  But if we don’t agree on a simple, direct language, we will wind up like the poor servant in the comical fairy tale of the “Master of All Masters.”  What we need is a firm grasp of an idea and the ability to provide an account of how we are using our words.  There is no need to preserve odd notions from Textbook School Mathematics.

For the purposes of presenting math clearly, we must attach meanings to words in only one way.  We can control our own classrooms and the language used there, but we cannot do anything about the fact that in the world there is a mixture of habits, traditions and perspectives, and everyone will surely encounter that at some point in some way or another.  After students have mastered one way of talking about things, they may find it convenient or even necessary to consider or use other ways.

[Please feel free to leave comments about anything said here in the comment section below.]


CHANNEL: Engineering School Mathematics
© 2015 James Madden

Posted in Engineering School Mathematics | Tagged , , , , , | 2 Comments

Subtraction Problems with Kittens

Intended Audience:  Grades K-5 teachers, prospective teachers, and public/private and homeschool parents.

In this video, 6-year-old Autumn shows different methods for subtracting in the context of the word problem, “If there are XXXX kittens in a barn and YYYY are adopted, how many are left?” Watch as the question degrades quickly!

The first question, answered by finding 17-8, is done using the number bond “8 is 1 and 7.” First, take away 7 from 17 to get 10, then take 1 more to get 9.


This is one of the “bread-and-butter” methods of Eureka Math because it also helps teach place value (subtract to 10, then subtract the rest).  To prepare students to use this method (including Autumn!), a lot of work done in PK-1 centers around 10 frames:


This one picture shows many number relationships all at once.  It corresponds to the “hand number line” in the “Learning to Multiply, Part I” video (e.g., the top row corresponds to the left hand).  It shows the number bond “9 is 5 and 4” (1 left hand and 4 right-hand digits).  Most important for the subtract method that Autumn used, it shows the number bond “10 is 9 and 1” (note the empty space can be counted too!).  Autumn has done enough work with 10 frames that this picture is one of the pictorial representations she can visualize when doing subtraction calculations.

The answer to the second problem, 53-18, is solved using a different method. In this problem, Autumn sees that 18 is close to 20, and that 20 is easy to take away from 53: 53-20=33. She took 2 too many, though, so adds those 2 back in to get 35. As an exercise, try to draw this on a number line yourself.

The third and final method shows up in answering 114-96.  Autumn imagines 96 and 114 on a number line. She then knows that the difference is just the distance between the two numbers, which is easily found by backing up 14 to go from 114 to 100, and then another 4 from 100 down to 96:


The total distance is 18, which means:


The final question is just the third method used again, and in this case, it is even easier to see: 1017-999 = 17+1 = 18.

Finally, let’s talk about the question, “If there are 1017 kittens in a barn, and 999 are adopted, how many are left?”  In the Eureka Math curriculum, this is what I started calling (and which the writers have come to affectionately use as part of their vernacular):

Completely Ridiculous Artificial Problems

If used in isolation, the 1017-999 word problem in the video is absolute C-R-A-P.  It’s so ridiculous that every student would see it as artificial.  The writers of Eureka worked very hard to not inadvertently write C-R-A-P because it sends the very negative message that “math is only useful for ridiculous, artificial problems.”  Sadly, one of the reasons many students get turned off to math is due to all the C-R-A-P in the standard Textbook School Mathematics (TSM) curricula in the U.S.  If a large enough percentage of math problems are C-R-A-P, students are likely to judge that the entire enterprise of mathematics is ridiculous and artificial as well.

But, as this video shows, one can delve into the world of C-R-A-P if the teacher is honest with their students that the problem is ridiculous and made-up.  In the video, we build up to the C-R-A-P problem by starting with a reasonable question (barns often have 10-100 cats due to so many mice eating grain), and slowly making the problem worse.  The C-R-A-P problem then helps students understand what is a reasonable math question and what isn’t (while having a bit of fun at the same time).  Enjoy Autumn’s expressions as the problems get more ridiculous.

Follow me on twitter @ScottBaldridge or like my Facebook page www.facebook.com/scottjbaldridge to get updates to this website.

As always, comments are welcomed!


CHANNEL: Growing up with Eureka
© 2015 Autumn Baldridge and Scott Baldridge
Supported by NSF CAREER grant DMS-0748636

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Ratios, ordered pairs versus points, proportional relationships, and proportions

My NCTM talk last Friday (April 17, 2015) generated quite a bit of social media discussion. I had a Twitter discussion with Bowen Kerins and Bill McCallum that was very interesting, but I thought there were a few (twitter-induced?) misunderstandings that I’d like to clear up.

Ratio Definition and Notation

What is the context in which the notation (2,5) is used for describing a ratio?


As I said in my talk, we must go back to definitions.  The definition of ratio I used for the Eureka Math/EngageNY was based directly upon the progressions document “6-7, Ratios and Proportional Relationships.”  The progressions documents are a set of companion documents to the Common Core Math Standards.  While the progressions documents are not the actual standards (and are in complete-but-still-draft form), they provide guidance in creating curricula that meet the Common Core Math Standards.  Here is a picture of the definition of ratio from that document (page 13):


This definition needs a bit of translation to write it without the notation A:B.  “Pair” in this definition means “ordered pair,” which is given by the order of A and B in the notation A:B.  A literal restatement of the definition of ratio above without notation is:

ratio is an ordered pair of non-negative numbers, which are not both zero.  

Note: Neither the progressions document’s definition nor the restated definition mentions  equivalence classes of ordered pairs of numbers.  That is, 2:5 is a different ratio from 4:10. This distinction between 2:5 and 4:10 is important and useful pedagogically (for example, it makes it easy and natural to refer to “a set of equivalent ratios” as a grouping of many different-but-equivalent ordered pairs, as is done over-and-over in the progressions document).

Now let’s talk about ways to notate ordered pairs.  When talking about ratios, it is common to notate an ordered pair of numbers 2 and 5 by 2:5.  But here is another perfectly valid way to notate the same ordered pair: (2,5).  In fact, the notation (2,5) is the most commonly accepted mathematical way to notate an ordered pair of numbers (cf. here for equivalent definitions and notation of ordered pair).  In a middle school curriculum we actually want both notations and other notations as well (for example, a column/row in a ratio table) to describe a ratio, depending upon context of course.  I promise to explain why below but let’s look at the confusion first.

Point versus Ordered Pair Confusion

I think the possible confusion generated on twitter and my talk may have occurred because people were substituting “point” in their mind for “ordered pair.”  The ordered pair (2,5) corresponds to a point in a coordinate plane—but, it is only a correspondence: Ordered pairs are generically different than geometric points.  Mathematically, an ordered pair is a general term for a set of two objects in a given order (again, see definitions here).  For example, the notation (M,N) where M and N are two 3×3 matrices is also an example of an ordered pair in mathematics.  Thus, an ordered pair does not automatically mean it is a point in a plane!  In the presence of a coordinate plane, however, it is safe to blur (and we often do) the distinction and refer to the ordered pair of two numbers as a point.

Here is where I must apologize to Bowen and other attendees of my talk:  I was very, very careful about this distinction throughout the talk but I did not make that distinction explicit.  I referred to the ordered pair (2,5) as an ordered pair.  I did not say that ratios (as ordered pairs) were geometric points until we got to the slides that showed the graph of a proportional relationship.  The graph puts us in the context of a coordinate plane where it becomes safe to blur the distinction between an ordered pair and a point.

Thus, Bill McCallum is absolutely correct when he said in a tweet:

“A ratio is an ordered pair in a certain context; I wouldn’t say [the point] (2,5) is a ratio without context.”  (The phrase “the point” was part of another tweet that Bill was commenting on.)

I too wouldn’t say the “the point (2,5)” is a ratio without context.  Of course, Bill McCallum would probably also say, and I would agree, that one only really uses the notation (2,5) for ratios in the context of proportional relationships, which we will talk about next.

Proportional Relationship Definition

We are getting closer to the moment where we can explain why having multiple notations for ratio is so very useful.  But first we need to clear up another possible confusion about what a proportional relationship is according to the draft progressions document.  One of the questions asked on Twitter was,

“Is a proportional relationship a set of equivalent ratios? … I’m confused.”

Here’s a picture of the definition in the progressions document (page 14):


The two definitions are synonymous: Set is another word for collection, ratios are (ordered) pairs of numbers, and two ratios are in this set if they are equivalent.  Mathematically, we are just using synonyms to say the same thing.  You can read more about proportional relationships here.

Why it is useful to have multiple notations for ratio

With the definition of proportional relationship understood, we are finally ready to see the huge benefit of having different but equally valid ways to notate ratios.  Sometimes it is useful to write a ratio as 2:5, like when we write  a single ratio in a word problem.  But when writing down a proportional relationship, it is useful to write a set of equivalent ratios as

{(2,5), (4,10), (6,15), (8,20), …},

and because of that notation, it is even easier to see what to do with this set of ratios when graphing it in a coordinate plane.  In grade 6 and 7 of the Eureka Math curriculum, proportional relationships like {(2,5), (4,10), (6,15), (8,20), …} are initially written as ratio tables.  But there is an important teaching sequence that goes from ratio tables to ordered pairs to plots of points of a graph of a proportional relationship, and the use of the (2,5) notation helps facilitate this transition without getting bogged down in ugly pedantic semantics about notation.


While we are at it, let’s clear up one more thing that came up as a question during and after the talk: the term “proportion” and the difference between “equal” and “equivalence.”  What is a proportion?  For two ratios with well-defined values, a proportion is a statement of equality between the values of the ratios (i.e., an equation).  If you do a search of the progressions document you will see that this is exactly how the term proportion is used in each and every case.  Why use the values?  Because of the difference between when two ratios are equal and when they are equivalent:

  • For numbers a,b,c,d, the statement a:b=c:d is true if and only a=c and b=d are true.  Example:  2:5=2:(4+1), but 2:5≠4:10.
  • For numbers a,b,c,d, the ratios a:b and c:d are equivalent if there is a number r such that a=rc and b=rd.  Example:  2:5 is equivalent to 4:10, and 2/5 = 4/10.

By using values we get around the need for having two different meanings for the equal sign with regards to ratios (see my post here about how important it is to use the equal sign consistently).  For the brave-of-heart:  Mathematicians have special notation to get around this problem with special notation for the “class of equivalent ratios,” see the use of [2:5] in the introduction to Projective Space.


Overall, it’s my opinion that the progressions document writers got the conceptual image of ratio essentially correct (for many pedagogical reasons not listed in this post, actually), but they could have been a little bit more clear about how they were using the word “pair” in the progressions Document.  Hopefully this will be cleaned up in the final version of the progressions document (which is still in draft form)–maybe by removing the notation from the definition of ratio (to make the definition notation independent) and using the term “ordered pair” instead of just pair.

As I said in the talk, I certainly empathize with teachers who have thrown up their hands at some point and said, “6.RP.A.1 doesn’t make any sense.”  But the main point of my talk was that if you understand the conceptual images and definitions that the CCSS and progressions writers were using, then it does make sense!

[Please feel free to leave comments about anything said here in the comment section below.]

Follow me on twitter @ScottBaldridge or like my Facebook page www.facebook.com/scottjbaldridge to get updates to this website.

CHANNEL: Engineering School Mathematics
© 2015 Scott Baldridge

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A big NCTM thanks and next year’s talk

Thanks to everyone who came to my talk at NCTM, especially on a Friday afternoon during happy hour: You are some hardcore rule-breakers! (See question#4 here)  It was a joy to make so many new friends.  I hope you liked the talk and got something out of it. In fact, look for my next post soon that will do a deeper analysis of the terms ratio, ordered pair, and proportional relationship.


Next Year’s Talk:

My 6-year-old daughter’s pleasurable learning antics has inspired me to consider a talk where we “do math” together on stage at next year’s NCTM meeting, and show off some of the techniques used in Eureka Math.  You can see some of her pleasurable learning in the videos below (she is my co-teacher in this series).  Let me know if you would like to see Autumn in the comment section at the end of this post, or feel free to suggest a topic for me to speak about.  The deadline is coming up quick, so let me know soon!

Follow me at @ScottBaldridge or like my Facebook page www.facebook.com/scottjbaldridge to get updates to this website.

CHANNEL: That’s News to Me

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I’m at the NCTM meeting this week and would like to meet you!

Want to talk with the lead writer and mathematician of the Eureka Math/EngageNY curriculum?  Here’s your chance to do so at the NCTM national meeting.


I’m scheduled to be at the Eureka Math booth #1308 & #1309 at the following times:

  • Thursday: 11:00–2:00 pm
  • Friday: 9:30–11:00 am, and 2:00–3:00 pm.  UPDATE: I can no longer meet at 2:00pm.  I may be there later, but I should be in 104C a little after 3pm.

You can’t miss the Eureka Math booth—it’s the one with the classroom-like feel and the cool video graphics on the wall.  Definitely come by and share with me your stories about students learning.

IMPORTANT:  Don’t miss my talk on Friday from 3:30-4:30pm in Room 104C (BEC) on the

Mathematical Secrets behind the Common Core State Standards 

Abstract:  Have you ever read a CCSS standard and wondered, “What was the thought behind that standard?” Hear the mathematical meanings behind some of the ratio, rate, and function standards, why they are important, and how those meanings can lead to effective teaching innovations that will help your students to see math as a coherent whole that makes sense.

Presentation Format: General Interest/All Audiences Session
Grade Band Audience: General Interest/All Audiences

FAQ about my talk on Friday:

(1)  I’m an elementary teacher.  Should I attend your talk Scott?

Answer: Absolutely!  In this talk I will describe how vitally important your work is in A Story of Units (grades PK-5) for helping middle school students understand ratios and rates.

(2) I’m a high school teacher.  What’s in it for me?

Answer: Well, converting quantities into measurements, and measurements into numbers is a major step towards studying real-valued functions with real number domains, which is the main theme of A Story of Functions (grades 9-12).  Read my article here for more info.  Plus, rates are the first step towards differential calculus—yes, it’s that important (we won’t be talking about calculus though).

(3) I’m a middle school teacher.  Help!  What exactly is a proportional relationship?  A unit rate?

Answer: These questions are at the heart of the math content of my talk.  The talk will help you look at middle school and A Story of Ratios (grades 6-8) in a whole new way.

(4) Is this talk going to be boring?

Answer: I have a simple test that you can take to determine whether or not you will find my talk boring. To take the test, just follow this one, simple instruction: Stop reading this paragraph right now–not another word.  Couldn’t stop could you?  You are still reading this paragraph, aren’t you?  I fully have your attention now and you couldn’t stop even if I asked you to again, which I won’t.  And here’s the great news–we just got rid of all those mindless, instruction-following, boring people who did stop reading.  The rest of us rule-breakers are now guaranteed to have a good time at my talk!

Follow me at @ScottBaldridge or like my Facebook page www.facebook.com/scottjbaldridge to get updates all this week.

CHANNEL: That’s News to Me

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Meet Mathematician Aaron Lauda

Intended Audience: Everyone, and especially teachers who want to show to their students a mathematician explaining the motivation behind their own research.

In this episode we meet Aaron Lauda, a mathematician from the University of Southern California, who shows us how to represent complicated expressions and equations using pictures. Enjoy! In fact, Aaron has provided more artwork at his website.  Go check it out.

Aaron explains the motivation behind his paper with Mikhail Khovanov, “A diagrammatic approach to categorification of quantum groups I.”  Here is the abstract to their paper:

To each graph without loops and multiple edges we assign a family of rings. Categories of projective modules over these rings categorify U−q(𝔤), where 𝔤 is the Kac-Moody Lie algebra associated with the graph.

While the video above is for a general audience, Aaron Lauda’s paper is not (it’s written for other mathematicians).  However, high school students who think they are potential math geniuses may still enjoy looking at it to see what advanced theorems and proofs look like.

Aaron Lauda

CHANNEL: Geometry and Topology Today
© 2015 Scott Baldridge and David Shea Vela-Vick
Supported by NSF CAREER grant DMS-0748636

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‘Eureka Math’ Embeds Real-World Problems in PreK-12 Mathematics Lessons

Jessica Hughes quotes me in her article:

‘Eureka Math’ Embeds Real-World Problems in PreK-12 Mathematics Lessons.

The article discusses how Eureka Math is a new curriculum for delivering Science, Technology, Engineering, and Mathematics (STEM) education in the United States.

Jessica Hughes asked insightful questions that went straight to the heart of what the curriculum is all about.  You can keep up with Jessica at the Center for Digital Education by following the twitter account: @centerdigitaled.

CHANNEL: That’s News to Me



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Pleasurable Adding

Intended Audience:  Grades K-2 math teachers and parents.

In this video, 6-year-old Autumn shows different techniques for adding two numbers. Originally the plan was to concentrate on just one technique, but working with a child who already knows multiple techniques means that you just have to “go with the flow.”

The technique we were going to show is based upon the associative property. In the first problem, 7+8, Autumn breaks 8 into 3 and 5 and groups the 7 and 3 together to get 10. The answer is then simple: 10+5 or 15. This method shows off the Associative Property in algebra because we are changing the 3’s association with 5 to an association with 7. This “re-associating” is done symbolically with the parentheses below:

7+8 = 7+(3+5) = (7+3)+5 =10+5 = 15.

For teachers, parents, and students using the Eureka Math curriculum, your students practice this technique through the use of number bonds (the bond has 8 in the “whole” circle, and 3 and 5 in the two “part” circles).

Yes, it’s first grade yet Eureka Math is already preparing your children to be successful in algebra in middle school! Of course, we are not burdening students with words like “associative property” at this stage in their learning.

This is the same technique used in the last problem: 999+64. Autumn takes 1 from 64 and associates the 1 with 999 to get 1000. The answer is then easy:

999+64 = 999+(1+63) = (999+1)+63 = 1000+63 = 1063.

The second technique shows up in the second problem: 6+7. In doing this problem, Autumn says that, since 6+8=14, then 6+7=13 . She knows that 6+7 must be one less than 6+8. Enjoy the look of surprise on my face—I was definitely not ready for that response.

The third technique is easy and was used for 27+12 and 232+232. Autumn realized that, since there was no regrouping, she could add “2 tens + 1 ten = 3 tens” and “7 ones + 2 ones = 9 ones” to get 3 tens 9 ones, or 39. (One does not have to start in the farthest right ones place as in column addition.) I had to help her with the meaning of the digits, like when she said, “2+1=3” and I encouraged her to say “2 tens + 1 ten = 3 tens.”

Mathematically, this technique uses the “Any-order property,” which just means we can arrange addends in a sum in any order with any grouping we want. (It is just repeated applications of the Commutative Property and Associative Property). Symbolically,

27+12 = (20+7)+(10+2)=(20+10)+(7+2)=30+9=39,

The final technique is a combination of the previous techniques when Autumn finds 57+58:


Autumn, however, doesn’t do this problem exactly like that. It’s hard to tell from the video, but the method she used to find 7+8 was to break each number into 5 plus a number and use the fact that 5+5=10:

7+8=(5+2)+(5+3)=(5+5)+(2+3) = 10+5 = 15.

You get a hint that this is her method when she says “2+3=5.”

As always, comments are welcomed!

GUWE Adding

CHANNEL: Growing up with Eureka
© 2015 Autumn Baldridge and Scott Baldridge
Supported by NSF CAREER grant DMS-0748636

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Common Core, Eureka Math shake up Louisiana classrooms

Jessica Williams quotes me in her article, “Common Core, Eureka Math shake up Louisiana classrooms” from March 13, 2015.   The article talks about the difference between the Common Core State Standards and curricula that satisfy the Common Core State Standards.

Here’s a quote from her article:

Still, Common Core and Eureka aren’t identical, Munson said. Standards are guidelines for what children should learn and know; curriculums are a way to get there. “Do they meet the standards? Yes,” Munson said. “But they are far from the same thing, and they’ve never been the same thing.”

Jessica Williams has a “nothing but the facts” reporting style that I like and appreciate very much.  She can be followed on twitter at @jwilliamsNOLA.

CHANNEL: That’s News to Me



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Interview with Thomas Lam on electrical networks

Intended Audience: Everyone, and especially teachers who want to show to their students a mathematician explaining the motivation behind their own research.

In this episode we meet Thomas Lam, professor of mathematics at the University of Michigan, who studies electrical networks (among other topics).  Thomas gives a great introduction to one of the main problem in electrical networks as well as an application of electrical networks to medicine.  At the end, I ask Thomas when he knew he wanted to become a mathematician.   Did you know that there is a “math olympics” for high school students?

The main problem presented in this video is the motivation behind several papers about electrical networks, including the paper, “Electrical networks and Lie theory,” by Thomas Lam and Pavlo Pylyavskyy.  Here is the abstract to their paper:

We introduce a new class of “electrical” Lie groups. These Lie groups, or more precisely their nonnegative parts, act on the space of planar electrical networks via combinatorial operations previously studied by Curtis-Ingerman-Morrow. The corresponding electrical Lie algebras are obtained by deforming the Serre relations of a semisimple Lie algebra in a way suggested by the star-triangle transformation of electrical networks. Rather surprisingly, we show that the type A electrical Lie group is isomorphic to the symplectic group. The nonnegative part (EL_{2n})_{\geq 0} of the electrical Lie group is a rather precise analogue of the totally nonnegative subsemigroup (U_{n})_{\geq 0} of the unipotent subgroup of SL_{n}. We establish decomposition and parametrization results for (EL_{2n})_{\geq 0}, paralleling Lusztig’s work in total nonnegativity, and work of Curtis-Ingerman-Morrow and de Verdi\`{e}re-Gitler-Vertigan for networks. Finally, we suggest a generalization of electrical Lie algebras to all Dynkin types.

While the video above is for a general audience, Thomas Lam’s paper is not (it’s written for other mathematicians).  However, high school students who think they are potential math geniuses may still enjoy looking at it to see what advanced theorems and proofs look like.

CHANNEL: Geometry and Topology Today
© 2015 Scott Baldridge and David Shea Vela-Vick
Supported by NSF CAREER grant DMS-0748636

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EdReports.org’s Review of Eureka Math

Here’s the quote from Education Week that summarizes the report in a nutshell:

In all, just one curriculum series stood out from the pack. Eureka Math, published by Great Minds, a small Washington-based nonprofit organization, was found to be aligned to the Common Core State Standards at all grade levels reviewed.  edweek.org

The full report can be found at edreports.org.  A graphic showing the overall alignment for all curricula can be found here.

As the lead writer and lead mathematician of the Eureka Math curriculum, I just want to take this opportunity to thank all of the wonderful writers who worked so hard on this project.  Let’s keep listening to the teachers and parents who are using our curriculum, and use their advice to make it even better.

I also agree with Lynne Munson, President and Executive Director of Great Minds, when she says:

The teachers who wrote Eureka Math have so much to be proud of today.  Indeed, Eureka is exemplary because the people who wrote it are extraordinary.  Eureka is the result of a historic collaboration between teachers and mathematicians, who know the standards, the math, and the best practices for teaching students.  Every student deserves a great math education, because every child is a great mind in the making. —Lynne Munson

CHANNEL:  That’s News to Me

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