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 mathematically-rigorous 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.

Proportions

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.

Conclusion

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!

CHANNEL: Engineering School Mathematics

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!

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!

CHANNEL: That’s News to Me

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.

CHANNEL: Geometry and Topology Today

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

Jessica Hughes quotes me in her article:

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

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:

57+58=(50+7)+(50+8)
=(50+50)+(7+8)
=100+(7+(3+5))
=100+((7+3)+5)
=100+(10+5)
=100+15
=115

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.”

CHANNEL: Growing up with Eureka

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

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

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

The Mardi Gras Problem

Intended Audience: Grades 4 – 8 math teachers and/or home school parents, or anybody who is interested in Mardi Gras parades!

We enjoy going to family-friendly Mardi Gras parades in Louisiana. These parades are exciting events filled with floats where riders throw beads and stuffed animals to children.  At the 2015 “Krewe of Artemis” parade in Baton Rouge, Autumn and I wondered:

That question lead to “The Mardi Gras Problem” video.  The video can be used as part of an enrichment activity that investigates the following topics in mathematics: measurement, scale drawings, ratios and proportional relationships, estimation, and length conversions.

The steps for doing this activity are:

STEP 1.  Show the video.  It gives a 3 minute introduction of a family-friendly Mardi Gras parade “Krewe of Artemis” and sets up the problem.

STEP 2.  Hand out the picture of the 2015 parade route and one of the two Google maps below.  About half of the students should get the first map Google map and half of the students get the second map.

STEP 3.  Let the kids think about a plan of attack on their own.  Talk to individuals or small groups and offer hints for starting the problem.  If necessary, after 3-4 minutes, suggest to the whole group that they might start by redrawing the route on the Google map.  Ask: What information does the Google map provide that the parade route map doesn’t?  How can you use that information?

STEP 4.  Offer students rulers or better yet, suggest they use the edge of a cardboard stock to create a ruler such that the distance between adjacent marks is the length of the “200 feet” segment at the bottom of the Google map (see “answer” picture below).  Let students work for the next 10-20 minutes estimating parade route length by measuring the number of “200 feet” lengths using their ruler (Answer: about 76.5 units of “200 feet” lengths.)

STEP 5.  Guide students to calculate the length of the parade in feet:  If 1 unit is 200 feet, then 76.5 units is 15,300 feet.  Use either tape/arrow diagrams or rate calculations (depending on your grade) to explain how to do this.

STEP 6. Remember to ask why students who used Map 1 got the same number of feet as students who used Map 2.  Have a delightful discussion of proportional relationships and scale drawings!

STEP 7. Using the conversion, 5280 feet = 1 mile, convert the number of feet to miles to get approximately 2.9 miles.  Conclude that 3 miles is a good estimate for the length of the parade.