# Rational Function Fan Fair

Sometimes when planning a unit, I browse through the Desmos Activity Builder.  When searching for Rational Functions, I came across Dylan Kane’s Building Rational Functions Activity.  Excellent.  I now had a muse.  Here is what I came up with for college algebra:

I like to gush over my students when they do awesome stuff, and this was no exception.  I love it when my classroom is abuzz with sense-making conversations.  I feel like this activity helped students become more comfortable with the structure of rational functions and how that equation structure is reflected in the graphs. Thanks, Dylan, for inspiring some awesome thinking in my class today.

# Watching Solitaire in Silence

Remember Windows Solitaire? I have fond memories playing this fantastic digital distracter with my high school beau on his brand new Gateway computer.  We would take turns striving for success in this card-clicking frenzy, the other watching and waiting patiently for the deck to empty.

But have you ever watched someone play solitaire on the computer?  It is so…what word comes to mind?  Frustrating?  Infuriating?  Aggravating, perhaps?  And why is that?

Check out this screenshot:

What if the player was about to click on that blue, flowery deck of cards…would you be fighting the urge to save them from their potentially game-ending error of failing to move the sequence beginning with the six of spades to its rightful place atop the seven of hearts?  Or would you idly sit by and let them to figure out that solitaire is won by carefully searching for card moves before drawing from the deck?  Would you make any suggestions for improving their game once failure was inevitable?

I think this solitaire analogy is a lot like teaching.   I realized fully today why the “productive struggle” is so hard to sustain and perhaps why teachers so often fall back on traditional methods of delivering information to students:  Watching people struggle without intervening is difficult. Just as it’s natural to want to smooth out the path for our children, it’s also tempting to do the same for our students.  It’s just easier (and so much faster) to zip Maria’s (my daughter) coat or buckle her seat belt or pick up her toys.

As a simple, mathematical example, imagine one of your students is attempting to solve a quadratic equation. They start off like this:

Being the savvy algebra teacher you are, you can anticipate the error that the student is most likely going to make.  You’ve seen it hundreds, if not thousands of times.  Your inner teacher voice might be thinking, “For the love of humanity, Herbie (not your real name), set the dang thing equal to zero!  Quadratic formula!  IT’S GOT A SONG, FOR GOODNESS SAKE!”

Instead, you do not impede their solving and let them continue on their merry, algebraic way.

Re-enter teacher voice in your head, “Now look what you did, Herbie.  You’ve gone and…wait…one of those answers is right.  Great.  Now we’ve really got issues.”

So what do we do about this?  Clearly the student needs some redirection and the teacher’s role is to guide the learning.  But had we intervened during earlier steps, we rob this student of a golden opportunity for brain growth.  Plus, we deprive the rest of the class the chance to learn from the misconception.  Even more, what a fantastic extension we have here:  why did the student get part of the problem correct and part incorrect?

In summary, we deny students the opportunity to learn from mistakes if we  prevent them from making mistakes in the first place.

Related Side Note:  I’m currently reading The Gift of Failure by Jessica Lahey.  Her introduction about her son’s shoelace-tying trials seems strikingly familiar.  And I can use this antedote as a reminder when encountering the zippers, and the seat belts, in addition to quadratic equations.

# This is Our Theorem – College Algebra

“We came up with a theorem once at my old school.  The teacher has it in a frame behind his desk.”

This statement from one of my college algebra students made me both elated and sad at the same time.  Thrilled because this is the type of mathematics I believe all students should have the chance to engage in on a regular basis.  Disappointed because this type of discovery happens so infrequently in American mathematics classrooms that the incident warranted a sacred place on the wall of this teacher’s room.

In College Algebra, part of today’s learning objective was to define a polynomial function and determine some key features.  I have the awesome types of students that if I were to write down the surly definition and features of a polynomial function onto the whiteboard, each would follow in lock-step and write it in their notebooks solidifying it’s place among mathematical obscurity.

Today, we were going to break that cycle with something different.

But I needed to know where they were at, so I had them write down what they knew about a polynomial function.

After some discussion and leading questions, we were sure that linear, quadratic, cubic, quartic, x^5, x^6, and so on were all polynomial functions.  Awesome. We weren’t, however, as sure about functions including negative exponents, roots, sin/cos, or algebraic fractions.

What makes this group we are sure about special?  Last week, we spent a considerable amount of time on features of functions including domains, end behavior, intercepts, intervals, symmetry, and turning points.  In their groups, I had them examine the graphs of these alleged “polynomials” through the lens of the features of functions.

Two similarities emerged as significant:

Questions:  Was this true of all polynomial functions?  And if both conditions were not met, could we exclude it from our known polynomial functions?  Hiding my initial excitement, I then had them look at our list of “questionable” functions. For example, did “y = 9 + 1/x” meet each of these two criteria?

Christopher Danielson suggested that my class give this new theorem a name, so we could refer back to it with ease:

“Class. We have found that all polynomials blah blah blah…” [while writing the statement of the theorem on the board.]  In mathematics, when we have an important finding like this, and when all mathematicians have agreed the finding is true, it gets a name.  Sometimes it is named for a person, such as ‘Fermat’s Last Theorem’; sometimes it is named for what it says, as in ‘The Triangle Inequality’.  But that name makes it possible to refer to it going forward. It helps us to remember and to use the thing we figured out. So we need to name our theorem. Who has a name they’d like to suggest?”

Alas, the excitement of naming the theorem will have to wait until tomorrow.

# Conceptual Function Foundation Follow-up

One of my favorite parts about teaching is having the opportunity to see growth in myself and my students.  I love when a lesson I have used with success previously gets even better the next time around, especially when it is a lesson that exemplifies my teaching philosophy.

Last trimester, I used the New Visions for Public Schools’ Algebra 2 functions unit to help build a conceptual foundation.  I wrote about that experience here.

Day 1:  To start off this unit on functions, they were in pairs:  one partner facing the projector screen and one facing the back of the room.  I then drew an arbitrary function on an unlabeled set of axes.  The person facing the screen needed to use words only (no pointing, no gestures) to help their partner draw the graph.  The person drawing was not allowed to ask questions, just draw what they hear.  For example:

Follow-up question (before they turn and look): What could you tell your partner to help them improve their graph?

Afterward, we talked about what descriptions were helpful.  My goal was to turn those descriptors such as “hills” and “curvy lines” into more specific function features.  Classwork (with their group of 4) involved looking at specific graphical examples to define end behavior, turning points, positive and negative intervals, etc.

Day 2:  Gave students a set of 36 graphs.  They needed to sort the graphs into exactly four groups based on their function feature.  I then followed up by having them choose one graph that best represented each group.  Some examples of student work:

Next week, we will look at their groupings and decide which ones highlight important features of the graphs.  Then, we will see if we can add some specificity and some real-world.  Seriously, I love the way this progression helps my students make sense of the function features.  It sure beats standing at the front and going through examples.

# Use All the Methods!

This is from Illustrative Mathematics (the people over there do wonderful work.  Plus they are lovely.) The problem I posed to my college algebra class was this:

I had them try it on their own and as I circulated the room, I noticed about 3 methods:  taking the square root, putting the problem in standard form and then factoring, or putting in standard form then applying the quadratic formula.  After clearing up errors and misconceptions, I was confident we understood that the answers were x = 3 and x = 9.

What now?  We could gather up all of the methods they came up with and make a lovely list.  Or we could take a look at the method that students almost 100% of the time ignore/forget/dismiss:  graphing.

Step 1:  Understand what a “solution” looks like graphically.

We separated the equation into two quadratic functions which both were equal to y.  Now we had a system of equations and this group knows that systems of equations have solutions at points of intersection.

Step 2:  Without a graphing calculator, sketch each of these two functions to approximate how they cross. “Expect to be wrong and give it a go anyway so that we can all learn from each other.”

Step 3: Examine some of our solutions.  As I expected, only about 2 students had a solid understanding of where y = (2x-9)^2 sat on the xy-plane.

I categorized the errors into three groups:

A.  The negative 9 means the graph is shifted down.

B.  Our answers when we solved were x = 3 and x = 9, so this graph must cross there.

C.  I don’t want to be wrong so I’m only graphing y = x^2.

Step 4: Look at the graphs of some similar quadratics like y=(x – 3)^2 and see if that thinking applies here.

*I feel like here is where understanding happened.

Step 5:  Take the gridlines and axes off of the Desmos graphs and find the points of intersection.

Interesting to see them “know” that x=3 and x = 9 from solving this equation algebraically somehow applied here, but they weren’t sure how.

Eventually we arrived at (3,9) and (9,81).

My Take-Away(s)

We explored a method that most (if not all) students don’t think of when asked to solve an equation:  Graphically.  But I think it’s an important one when trying to figure out how the pieces of functions and algebra fit together.  Yes, we could practice factoring, the quadratic formula, and completing the square all day long.  But in the end, know those individual methods doesn’t give my students an idea of how those solutions connect with the actual functions they represent.

This problem made me think about what we tell students when we explain methods of solving equations.  Any time we show a student a method, we are inexplicitly stating that this method has higher status than any other.  Giving students an opportunity to solve a problem using their prior knowledge is important to the learning process.   Their way of solving isn’t always going to be algebraic and building from where they are at is vital to creating a foundation of understanding.  If they start with “guess and check,” help them build structure from that rather than insist that the algebraic method of solving is superior.  In the case of the Problem above, any algebraic method was probably the most efficient but it isn’t always.

For example, what about this problem from Nrich.  Would your students approach it algebraically first?  Or is there foundation elsewhere?

# App Review: Osmo Numbers #tmwyk

Since it was Christmas, and we finally had some free time, I decided to bust out something she had gotten earlier this month: Osmo Numbers.  I have a feeling that given the success of other Osmo products, the popularity of this one might surge.  We had tried Newton, Masterpiece, and Tangrams and kinda liked them.  I was hopeful.  Optimistic even.  (I think you see where this post is going).

Their contains the following video, which addresses math anxiety and illustrates the power of turning math into a game.

Here are some quotes from reviewers:

I couldn’t agree more:  we (educators and parents) create math anxiety in our students/children by insisting that math has one right answer. Often we convey that there is one right way to arrive at that answer as well. Consequently, children grow up believing math is about rules and procedures rather than creativity and exploration.

Reviewing the App

Description (from website): Kids arrange physical tiles, including dots and digits, to make numbers and complete levels. Add by putting more tiles, subtract by removing tiles and multiply by connecting tiles together. Experimenting becomes fast and intuitive.

The first section (Count) is decent.  Children have multiple dot tiles (like dice) that contain either 1, 2, or 5 dots.  They need to arrange tiles to total the numbers in the bubbles to make them pop.  It was interesting to me to see my daughter, Maria, make sums in different ways.

I was also intrigued that she always made numbers between 10 and 20 using two fives and then the remainder.

Things went downhill quickly.

In the “Add” section this is 1 + 4:

However, in the “Connect” section, this is 14:

Furthermore, in the “Multiply” section, this is 1 x 4:

Why is this a (HUGE) problem?  The design of each of these stages reinforces the belief that mathematics is about isolated rules rather than connected ideas.

That’s worth reiterating:  Math should be about connecting ideas and not about isolated rules.

The app boasts “when kids get the idea that there are multiple good ways to solve a problem, math becomes creative and fun.” I’m fighting my knee-jerk sarcastic-response mode big time in order to keep this professional.  All kids are really doing here is finding different ways to recall facts.  And as they progress through the app, the meanings of their representations change.  This doesn’t help build math confidence and break down math anxiety.  It sturdies the foundational thinking that math concepts don’t relate.

Here’s a great example of that.  Maria is frustrated that this doesn’t equal 10 like it did yesterday. But there isn’t anything to differentiate addition from multiplication besides the fact the tiny title (that she can’t read anyway) says so.

There is no visual representation of 5×5 being 25.  No arrays, no area models, no dots.

Why am I making this such a big deal?  This app, and many like it are being touted as game-changing, revolutionary, and brilliant. But when I search through the Osmo Makers page, I see no teachers.  In fact, I see five years of actual classroom experience, none of it in a public school and none math-specific.

I admire the goal to “revamp the unimaginative worksheet.” But I’d rather it replaced with a better worksheet than with something that tears down conceptual foundations of how numbers build and replaces it with fact-fluency.  Sorry, Osmo.  This product is a giant fail.  I’m returning to Talking Math with Kids where math doesn’t just claim to be creative. It actually is.

# Somewhere between Concrete Sequential and Abstract Random

It occurred to be relatively early in during the trimester this past fall that my college algebra students (generally) have no idea what I mean when I say “quadratic function.”  This isn’t because they have never heard it, learned it, or used it.  But that technical of a term simply has not stuck around in their long term memory.

So, similar to linear functions, we start with a pattern:

from youcubed.org and visualpatterns.org

I then had them make posters including how it is growing, what the 10th, 100th, 0th, and -1st cases would look like, table, graph, expression, and relationship to the pattern.  Instead of one large poster, they use 4 smaller pieces of paper and tape them together.  That way each group member can contribute simultaneously.

I noticed:

• It is difficult for students to describe how an irregular shape is growing.
• It is even more difficult for them to describe something abstract like the -1 case.
• Many of them expressed the overall growth as “exponential.”
• Most could easily see the two rectangles formed and determine the dimensions with respect to “n.”

I wondered:

• If they could connect the work with patterns to other quadratics.
• How to have a meaningful discussion around the “exponential growth” issue.

Their homework was to answer similar questions for this pattern from You Cubed’s Week of Inspirational Math:

Spoiler alert: the rule for the pattern is f(n) = (x + 1)^2 or f(n) = x^2 + 2x + 1

So where do we go from here, two days before Winter Break?  My goals are to review some specifics on quadratic functions and simultaneously help the students make connections between different representations.  I know what I must do.  I must channel my inner Triangleman.

[Backstory:  Christopher Danielson and I go way back. At least to 2014. Maybe even 2013.  Seriously though, I strive to organize my college algebra class the way Professor Danielson describes in his blog.  I have picked his brain on more than a few occasions and he is gracious enough to give me advice in certain curricular areas. In short, his philosophy titled “They’ll Need it for Calculus” is the foundation of my College Algebra course. ]

Ok, back to room C118.

Me: Write down everything you know about the function y = (x+1)^2

(Most write down the expanded form, some start to graph, but not many)

Me: What other ways can we represent this function?

Students: Tables! Graphs! Pictures! Words! Patterns! Licorice!

Me: Sweet!  Let’s do all of that, minus the Licorice.

(I give them a few minutes to create a table and a graph.)

Me: NOW, write down everything you know about this function.

I circulate and hand each group a half sheet of paper.

Me: Write down the most important thing on your groups list.

At first I wasn’t really concerned what exactly they wrote down, but how they defended their choice. Then I came across this in all of my classes:

We came up with a pleasing list of attributes of a positive parabola that included vertex placement, end behavior, leading coefficients, and rate of change.

Next up for discussion: Parabolas grow exponentially.

Me: Turn to your partner and tell them whether you agree or disagree with this statement and defend your choice.

Students: Yes, words, words, words.

Me: Ok, now the other partner, say how you know something grows exponentially.

Students: Multiplied every time, more words, blah blah blah.

So we agreed that 2, 4, 8, 16, 32, 64… is an example of something that grows exponentially.

Me: Numerically, how can we tell how something is growing?

Students: (eventually) Rate of Change!

We came up with this table and agreed that these two functions were definitely NOT growing in a similar way.

Now on to helping them understand what it means for something to grow quadratically…