Categories

## What’s Going On in This Graph

Today the New York Times Learning Network dropped the first “What’s Going On in This Graph?” (WGOITG) of the new school year. This feature started last year as a monthly piece, but now expands to a weekly release. In WGOITG, an infographic from a previous NYT article is shown with the title, and perhaps some other salient details, stripped away – like this week’s graph…

Challenge your students to list some things they notice and wonder about the graph, and visit the NYT August post to discover how teachers use WGOITG in their classrooms. Here are some ideas I have used before with my 9th graders:

• Have groups work in pairs to write a title and lede (brief introduction) to accompany the graph.
• Ask tables to develop a short list of bullet points facts which are supported by the graph, and share out on note cards.
• Have students consider how color, sizing, scaling are used in effective ways to support the story (note how the size of the arrows play a role in the graph shown here). This is a wonderful opportunity to think of statistics beyond traditional graphs and measures.

Invite your students to join in the moderated conversation, which drops on Thursday. Have your own favorite way to use WGOITG? Share it in the comments!

Categories

## How I Stumbled Into Math Modeling Without Even Realizing It.

We started a unit on counting principles this week in my 9th grade honors class – permutations, combinations – eventually leading to the binomial theorem.  Since my  classes had used Desmos Activity Builder a few times and were familiar with the need to enter a 5-character code to start an activity, I planned to ask the following question as a class opener:

How many different 5-character DesmosActivity Builder codes exist?

This problem would have likely met my intended goal of having kids think about the fundamental counting principle in a real-world context.  It also would have taken about 10 minutes of class time, and have been forgotten about by the next day.  It felt like I was missing an opportunity to develop a deeper discussion.  A slight tweak to the question added just the right layer:

Activity codes for Desmos Activity Builder currently have 5 characters, as shown here.  When will Activity Codes need to expand to 6 characters?

And now we have a problem which requires a bit more than a quick calculation.  To start, I asked students to work in their teams to make a list of information they would need to help solve this problem.  This was not easy or comfortable for them – but a preliminary list of questions emerged from group discussions:

• How many 5-character codes are there?
• Are codes used less on weeekends and summers?
• Can letters repeat in codes?
• How many codes a day are used?

This was a good start to set kids in motion to think about how to solve the problem.  I’m hoping they will think about new questions or revise their questions as we go along…the class did not disappoint!

HOW MANY CODES ARE THERE?

As kids worked, clarifying questions came up – some of which I just didn’t know the answer to, and hadn’t really thought about:

Mr. L, are there any zeroes in codes? Kids might confuse them with the letter O.

Mr. L, I don’t see any L’s in the codes?

Excellent observations, and restrictions we need to think about in our calculation. A tweet to the Desmos crew lent some clarity, and added more restrictions!

Thank for the intel, Eli!

HOW MANY CODES PER DAY ARE USED?

This was tricky for my class. To help, I reminded students that when we started the semester, codes were 4 characters.  When did the Desmos 5-character era begin?  A quick scroll through my history (shown here) provides some info. After further interrogation from my class, I shared that Activity Builder started around July of last year with 4-character codes.  Add this to our bucket of helpful info.

SHARING IS CARING

Writing a draft solution was the next task for students.  But instead of turning it in to me immediately, I formed class teams of 3 where students shared their drafts and ideas.  I used this opportunity to build teams of students who I observe don’t often interact or chat.  From here, I gave students another day to think about their explanation – keeping in mind that there are no right answers to this question, only answers we can defend. But it still feels like we are missing a key piece in this problem……

DID WE MISS ANYTHING?

The next morning as students were mingling before the bell, I looked across the room at the laptop of Jacob – one of my more insightful, but also introverted, students:

It’s the mother lode!

The google trends graph for student.desmos.  Yes! Yes! Yes!  Stop everything kids, we need to talk!  Jacob – tell us all about this graph. How does this new info factor into our estimates?  What should we do with it?  Is this going to continue?  And with this, I gave the class an extra day to think about their responses, share, and dig deeper.  And while many students simply estimated a growth rate by doubling or tripling their computed rate (this is fine with me), I am getting some responses which far exceed my expectations – like Jacob, who developed a growth function and evaluated integrals (did I mention this is a 9th grade class????)

Yep, this was definitely better than my originally intended problem!

Categories

## Class Opener – Day 74 – PolyGraphs

My 9th graders have only about 2 weeks left with me before their final exam. Most of them will move on to Algebra 2 next semester, so my strategy with them has been 2-pronged: ensure we are produtive with new material and put them in a “happy place” to make a seamless tranisiton to Algebra 2. With a unit review today, and a pre-holiday-break quiz Monday, this was a perfect time to test-drive the new FREE PolyGraph activity from my friends at Desmos, along with the awesome work of Dan Meyer and Christopher Danielson. The Parabola activity sounded perfect for my class, though there are also activities featuring linear functions, rational functions and hexagons.

My freshmen have limited understanding of quadratic functions. While we have encountered some useful vocabulary regarding parabolas in my class (intercepts, vertex, domain and range), these students have not had a formal unit in graphing them yet. I was curious if students could transfer what they already knew to a new scenario. I was tempted to do a quick review of vocabulary before sending kids to the lab, but thought better of it. I want gut reactions.

In the activity, one student acts as the “picker” and chooses 1 parabola from a set of 16. The “guesser” then asks yes or no questions to help narrow down which parabola was chosen. “I don’t know” is also available as an option, if the question is not clear.

Between games, students are given challenges to help guide their understanding of vocabualry and “good” leading questions.  I found these “intermission” questions to be extremely helpful, and noticed that the quality of the questions students asked improved after participating in them.

Some obeservations about my students in this activity, which we did for about 30 minutes.

• Students didn’t have vocabulary to describe parabolas which “open up” (a>0) versus those which “open down”. The question “is it a smiley face or a frowny face?” was used by more than one student and led to some side discussions of what this meant.
• Students also recognized that parabolas could have different widths, and describing the differences between these was more challenging. Questions like “Is it wide?” or “Is it narrow?” are helpful for identifying some extreme curves, but without a baseline for what a “regular” curve looks like, this leads to some confusion over which parabolas should be eliminated.
• In the first round, few student mentioned the x or y-axis in their questions. Later, I noticed these became valuable tools for elimination.
• Questions which attempted to use the vertex showed mixed success. “Is the vertex positive?” is unclear, but these attempts improved with more game plays. Similarly, attempts to describe domain or range often needed more work.
• Students can be sneaky, and mine are no exception. Some students attempted to bypass mathematical conversation by asking “Is it in the top row?”.  Nice try – until they realize the parabolas are mixed up. Also, sometimes students were assigned to play against students sitting right next to them. Not ideal, but workable.

Here’s where I would go with this, if my next unit was on a formal discussion of quadratics: copy the student-developed phrases like “smiley face” nto a document. As we encounter those ideas formally in graphs, develop more math-specific language, match them up with the student descriptors, and improve the document. I want students to take ownership of their descriptions, and allow for their self-generated language. Hopefully, this builds richer connections to the vocabulary.

At the end of each class, I had students complete a Google Form evaluation. I appreciate the feedback from students who took this task seriously!