Teacher Showcase: Digital AP Math with Greg Christy

In AP Statistics and AP Calculus, Northview’s Greg Christy uses digital tools to help students understand the concepts behind the equations.

Northview math teacher Greg Christy opens his AP Statistics class with a simple question: “Have you all had your tetanus shots?” After one lost soul asks, “What’s a tetanus shot?” Greg begins distributing thumb tacks. He jokes a little about this lesson being “the most dangerous thing you’ve done today, maybe all year,” but he quickly moves past these quips to instructions for the lesson. Students will use these thumb tacks to explore the move from randomness to probability, a chapter 14 lesson. And while doing that, they will use simple, yet powerful spreadsheet tools in Google Sheets to calculate.

Here’s the sequence: Greg has created a spreadsheet and shared it with all of his students, giving them edit access. He displays it on the SmartBoard and they open it on their Chromebooks. Then, the students toss the tacks at their desks for a trial of twenty tosses, laughing as they do so, but recording the results carefully. On the shared sheet, they record the percentage of “tips up” landings in the column next to their name. A second trial of twenty tosses follows with a second entry. As the students enter data, it appears on the class screen live, as does another number, the long-run relative frequency of “tips up” landings. Greg has designed the sheet to calculate the average percentage of these landings, and as the numbers keep flowing in, the percentage evens out to about 63%.

The practical test and collaborative data recording have given Greg the introduction he needs to discuss several concepts in probability and statistics, such as trial, outcome, sample space, Law of Large Numbers, empirical probability, and theoretical probability. All the while, he goes back to the Google sheet, explaining the meaning behind the numbers. He adds to that discussion by pulling up graphic representations through histograms and Google’s new “Explore” feature in sheets. Using the latter, students can see how trials over a space of time even out to a nearly flat line. The numbers in the original sheet become empirical probability, and the tossing of tacks becomes a conceptual reality for these students.

By using Google sheets and the collaborative spreadsheet power they provide, Greg has brought students into the development of large number data to yield conclusions regarding probability. Rather than reading these results passively, or watching a demonstrated trial, the students participate in the work in real time with immediate impact. Greg extends that engagement when he shows students how to generate long lists of random numbers on their own sheets with the “randombetween” function. Even this little trick helps students visualize the concept. As Greg says, “It helps them to make a connection conceptually with what they’re doing analytically.”

Greg guides AP Calculus students through digital graphing.

That connection does not just happen in AP Statistics, though. It also happens in AP Calculus. The day before the tack lesson, Greg was helping his AP Calculus students visualize the concept underlying polynomial function rate of change. Students are familiar with these concepts (e.g. derivative, concave up/down, decreasing/increasing) and are ready to graph the rate of change of a function based on its graph. Today, Greg does not rely on Google, which is too general for calculus. Instead, he relies on NCTM’s Illuminations resources.

Interactive Calculus Tool
The NCTM Illuminations Interactive Calculus Tool

At the SmartBoard, Greg models the Interactive Calculus Tool available in Illuminations. Students follow on their Chromebooks as he graphs the function and then tries to anticipate the derivative. Once he has shown them, he asks that they create their own function graphs to trade with partners for derivative graphing. The students love this step, and one hands his Chromebook to the student behind him with a smile and a warning: “This is brutal.” Everyone tries to graph the derivatives for the polynomials given to them as Greg circulates and answers questions. As the class progresses, students work a problem described on the handout Greg gave them.

Greg provides guidance to two students as they work through graphs on NCTM’s Illuminations utility.

So what’s the big deal here, and is it big? These two lessons might beg a math teacher to ask how going digital with either Google or Illuminations has significantly improved the learning experience for these AP students. According to Greg, “We do a lot with analytical equations. That’s by hand. Our technology allows us to take those equations and create graphs of those. That’s all based on analytical methods, but with those analytical methods, the students don’t get an intuitive feel for what’s happening.” As he explained before, the students are better able to grasp the concepts when manipulating them through dynamic digital tools. The concepts, which may be elusive when navigating complex equations, appear clearly on the screen. On the AP Calculus test, Greg explains that the test asks “questions on a conceptual basis . . . There’s not a lot of it that’s just rote mechanical operations. Even though the testing environment is traditional, the conceptual foundation still needs to be there. It does transfer. If it didn’t, I wouldn’t do it.”

These digital tools are not simply relevant to Greg’s AP classes. NCTM offers Illuminations content to a wide range of math classes and skill levels. Google Sheets can help student visualize mathematical relations with similar variety. Greg has simply shown how digital interaction provides easy and powerful tools for working with numbers in a hands-on, engaging fashion.

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Follow these links for more information!

NCTM Illuminations Home Page

NCTM Illuminations YouTube Channel

NCTM Illuminations Function Matching Tool (Excellent tool for a wide range of secondary math classes)


Teacher Showcase: Offering an Entire Course in One Google Sheet with Lauren Clark

Read about Lauren Clark’s “Argumentation Writing Guide,” a 21st century approach to Chromebook learning!

Lauren Clark’s Timberstone seventh graders seem to like their own jokes. They set up for class with smiles on their faces and laughs in their voices. Mostly, those laughs come from their own punchlines, usually incomprehensible to anyone else. It’s a junior high personality that is as delightful as it is quirky, and Lauren matches it with her own jokes and smiles. After the trading of fun energy, though, it’s time for work. Lauren directs all attention to the Google Slides presentation on the board, which announces the agenda for the day: “Hooks in Argument Writing.”

Lauren’s students are developing argument compositions based on Scholastic Scope material she has collected over years. The source articles present pro and con positions on some interesting topics: Should we live forever? Should we ban competitive eating? Is being bored good for us? Should we deport Justin Bieber? Before today, students chose their topics, read the articles, and began topic development worksheets for the composition. Lauren’s process to this point has demonstrated traditional approaches. Her students read photocopied materials and hand wrote responses to the worksheet. That traditional approach, however, is about to feel a boost that will push it into a perfect blend of traditional and digital.

Students follow Lauren’s introduction from their seat.

Class begins with a Google Slides presentation that students can watch on the SMARTBoard and access on their Chromebooks simultaneously. The presentation organizes their materials and presents notes on concepts. It even offers an enticing video on the difference between topic sentences and thesis statements, produced by Shmoop. While students handwrite on their development worksheets, they glance at the Chromebook screens. This blending clearly marks ownership. The screen is Lauren; the paper is them.

Check out the Google Slides presentation!
One of the many engaging Scholastic Scope argument topics
One of the many engaging Scholastic Scope argument topics

But, the digital activity of these seventh graders jumps as the class continues. To carry the students through the concepts required to compose an argument, Lauren has designed her own “Argumentation Writing Guide,” where the real transformation of the class is located. The guide is a deceptively simple Google sheet. In reality, though, it is an entire course in itself. On the grid, each row lists a different writing concept connected to writing arguments, everything from argument language and strong evidence to proper MLA citation. On the same row, next to each concept, Lauren has linked Internet resources for independent student learning. So, if a student is struggling with how to write strong topic sentences, she can go to row 13 and click on any of the three supporting resources: two text resources and one interactive resource. If another student was struggling with plagiarism, he could find help in any of four resources, including text, video, and interactive lessons. Lauren’s AWG collects online resources from a variety of sources to appeal to a variety of different learning styles and ability levels. With it, students can customize their learning. Today’s lesson introduces the guide to ground the students in this independent approach to writing.

Check out the Argumentation Writing Guide!
ARGUMENTATION Student Writing Guide Google Sheets
Lauren’s packed and rich Argumentation Writing Guide

Traditionally, students learned concepts by listening to teacher lecture or reading a textbook. Application helped reinforce, but the one-dimensional approach of lecture/textbook meant that students would be limited to repeated teaching approaches and a small range of resources. They would also be forced to endure long lectures on concepts that they may grasp quickly, or miss extended discussions on concepts everyone else got when they did not. These are the typical problems with a “One size fits all” approach. Lauren’s AWG offers an opportunity for students to learn and reinforce the concepts they need to study in the format that works for them. Really, it’s an elegant and powerful resource that can transform a limited, teacher-centered environment into a multi-faceted, student-centered one. And even better, any teacher could adapt it to any skills-based or standards-based environment, regardless of content area.

Lauren’s class continues as she moves past the introduction of the AWG and into “Hooks,” techniques for generating interest in the opening of an argument. It’s listed on line 24 of the AWG. (When she refers to a hook as a device, one of her seventh grade jokesters complains that “My device has lost power.” He laughs. Lauren stares back with a furrowed brow.) Since the hook is a new concept, Lauren will require all students to learn it, saving differentiated instruction for later. Right now, students watch videos and complete Google Forms-based quizzes to check understanding. The flow of the class begins to shift as students work at their own pace, accessing resources as they wish. While they do so, Lauren never sits still. She floats around the room, answering individual student questions and providing direction as students work. The class time flies.

Lauren helps students as they work at their own pace.
Lauren helps students as they work at their own pace.

The success of blended learning here is evident, but that success spreads beyond this one moment. Truly, to grasp the impact of Lauren’s approach one would need to watch the development over time. These seventh graders are moving into a learning environment where they will work at their own pace, reflect on areas of strength and weakness, and collaborate for learning. They will own their work. It’s all because Lauren developed her Argumentation Writing Guide, a foundation of resources and technique that created an environment for independent student learning. Moving from the textbook to the AWG has meant moving from limitations to possibilities.

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PD Session: Learn How to Navigate in Drive!

Do you feel lost when someone starts talking about Google Docs, Drive, Forms, Slides, and . . . uh . . . all the rest? Relax and come to this session!

Presented by Marilyn Waite and Darla Omey

Last presented on October 19, 2015 as part of the Sylvania Fall Inservice Day; Next presentation to be announced!

From the ad: “Do you feel lost when someone starts talking about Google Docs, Drive, Forms, Slides, and . . . uh . . . all the rest? Relax and come to this session!”

We will work on basic principles in Google products, such as sharing documents, organizing drive, converting Microsoft Office files, and more. This is the session for beginning users and those that want to make sure they know the basics.  Session participants may want to bring a flash drive or have access to Word files so those files can be uploaded to Drive.

Couldn’t make the session?  Check out this video from Google Education on You Tube!

Session Materials

Google Help Page

Google Apps Learning Center

Google Apps Learning Center for Google Docs