Describing Graph Transformations with Input / Output diagrams

I regularly stress to my grade 10 and 11 students that when transforming a function it is imperative to dilate before translating in any direction. In other words, if a function is both stretched and translated horizontally the stretch must happen first. Ron Watkins’ demonstration of Input / Output diagrams at OAME2014 has helped to see how rearranging the equation that describes a function can translate to rearranging the kind and order of transformations

An Input / Output diagram shows the sequence of operations applied on a function’s argument, x, to arrive at its value g(x). Naturally this sequence follows order of operations. What is interesting is that transformations are made clear through the diagram. Look first at the parent function indicated by the red arrow, which in this case is f(x) = x². Moving towards the right are the vertical transformations: stretch by factor 3 then translate 5 up. Following the arrows towards the left are the horizontal transformations: compress by factor one half, translate 4 right.

 Often students don’t see why horizontal transformations on the parent function are related to the inverses of operations seen in the function’s equation.  The diagram shows that to go from the input of the parent function (red arrow) to the horizontal coordinate x requires inverse operations, while going from the output of the parent function to g(x) does not.

Why I’m excited about Google Apps for Education

After several years of consideration and deliberation, Google Apps for Education has been launched throughout the YRDSB. Google Apps is a collection of services including the Google Docs collaborative suite of office applications, 30 GB of Google Drive cloud storage, email, calendaring, website creation, and more. Every student and staff member now has a Google Apps for Education account with the username and password synchronized to their school computer login.
With Google Apps comes many new possibilities:

• Students can collaborate on documents, spreadsheets, drawings, and presentations from any location at any time.
• Teachers can see students’ work as it develops, and give feedback by adding comments.
• Teachers can see which students contribute to a collaborative document by viewing the document’s revision history.
• Documents cannot be forgotten at home or lost on a corrupted USB drive.
• Departments can share a folder of common resources to access them anywhere, and even offline using Google Drive Sync
• All students now have a Google Account, so can use that to easily access services like Khan Academy.
• All students now have an email address under the gapps.yrdsb.ca domain, so they can now enrol services like Prezi’s Enjoy Edu which are free for educational users only.
• Shared calendars which can be added to mobile devices or embedded in Moodle courses can help with communication with departments, classes, clubs, and teams.

What you should do right now

This year, resolve to change your Passwords

My family's online presence started in 1987 when we signed up for the pre-Internet communication provider CompuServe. After dialing in with our modem, we logged in to the service with a nine digit username and a password supplied by CompuServe. Twenty five years later that password is still in use....
I must have hundreds of accounts with passwords on different web sites, and I expect you are the same. Our passwords protect information that ranges in value from inconsequential to critical banking and email details. Wired magazine senior writer Mat Honan discovered first hand this summer how vulnerable our password protected services are when hackers compromised his email accounts and remotely deleted everything on his iPhone, iPad, and MacBook. Lacking proper backups, he lost significant personal data including every picture he had ever taken of his 18 month old daughter. Most photos were later retrieved (at considerable expense) by a forensic disk recovery lab, but Honan was left shaken. Reflecting on the incident in December's edition of Wired magazine, he concludes that passwords alone are a flawed way to authenticate identity in a networked world filled with inexpensive and powerful computers. Though it might be a little frightening, I highly recommend reading the full article, or listening to last Friday's interview on Q.
Even if Honan is right that "the age of the password has come to an end," the reality is that we keep using passwords dozens of times a day. We need incorporate practices that give as much protection as possible until more robust authentication methods gain wide-spread use. Here are five things to consider as you decide how to manage your passwords:

iPad Case Studies

During the month of November, six staff at my school experimented with a set of ten iPad's borrowed from our board's technology lending library. Here are stories of two ways they were used.

Zooming Around

Grade 9 students spend a lot of time working with straight lines in math class. They draw lines that model real life scenarios, write equations to represent lines, and look at how pairs of lines intersect. Looking at the equations of two lines, students should be able to say if the lines intersect, of the lines are parallel so do not intersect, or if they are coincident. Graphing sets of equations of lines by hand can help students to connect algebraic and visual representations, but graphing this way has the physical limitation of drawing accuracy. One teacher used iPads with her grade 9 classes to overcome this limitation and investigate the intersection of lines that would be tricky to graph by hand.

The App	Safari Browser to use the Desmos online graphing calculator A home screen icon linking directly to http://www.desmos.com made the website easy to find and made it appear to opens like an app.
The Approach	Arrange students in groups of three. Give students pairs of lines to graph with Desmos. Students zoom in and zoom out to investigate how the lines overlap. Students record their observations on a template.
The Verdict	The teacher liked that zooming manipulation was tactile and directly with the graph, rather than clicking a button. This was good to give students a sense of large and small numbers through zooming. Students appeared engaged - groups were all huddled over the workspace with the iPad.

Explain it to me

One of our geography teachers continues look for are ways to encourage her students to learn in collaborative ways. With a built in microphone and a touch sensitive screen, an iPad is a good tool for students to explain ideas through words and gestures. By recording these explanations, they can be shared with classmates to help them understand concepts or with the teacher to demonstrate learning.

The App	EduCreations is a whiteboard which records sound, writing, typing, and movement. Recordings can be played back on the device or through a website at a later date.
The Approach	Arrange students in groups of three. Give students a discussion question from the textbook. Students discuss their answer to the question in their group and find supporting images online. Students record their explanation and upload the resulting video.
The Verdict	Some groups of three worked well, but in others one worked, one assisted, and one was distracted. More devices would be better. Overall quality of work did not improve, but student work was more accessible because students could look at the work from other groups online after class Lack of ability to edit recordings in EduCreations was disappointing. Because of this there was no way for the teacher or students to correct errors without redoing completely. Students were excited to use iPads, and asked to use them in later classes.

Being Less Helpful* with SMART Notebook

Sometimes being less helpful would be a good thing. A "helpful" friend who shares the highlights and score to a game you've PVR'd is an unwelcome friend. Similarly, giving our students all the information we think they need can diminish curiosity and bypass opportunities to build inference and thinking skills.

Cover Up

Using white ink to cover information in SMART Notebook can be used to encourage curiosity, while building inference and thinking skills.
 
Help students to develop their ability to infer meaning from text by covering key words and short phrases, and then ask students to suggest what might fill the blanks. Ask students to write their suggestions on the board to complete the document. When finished, read the document together to see if the text reads smoothly and coherently. Finally, use the eraser tool to reveal the underlying text so that students can compare their inference with the original writing.
Many textbooks and resources turn problem solving opportunities into pattern matching activities. Sometimes these scenarios and the questions that follow have the possibility to spark curiosity and prompt thinking, but too often instead of scaffolding the question in a light and differentiated way, the resource builds an elevator to "the answer" where all that is required of us and our class has the equivalent complexity to walking and pressing a button. By covering up as much information as possible we can open the range of approaches and help students to develop their thinking and problem solving skills.

The Technique

To cover text with white ink:
1. Pick up a pen from the tray
2. Press the Colour button to display the palette
3. Press the white square in the lower right
4. Press the Line Style button to the right of the colour button to choose a wider pen thickness.
5. Draw with white ink over the items you would like to conceal.

Sources:

• Michelle Grainger: Markville SS Digital Literacy strategies for OSSLT (bit.ly/Pf3fla)
• Dan Meyer: Math class needs a makeover (bit.ly/QwwOM5)
• Foundations of Mathematics 10, McGraw-Hill Ryerson

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[*]"Be Less Helpful" is Dan Meyer's slogan. For more of his work see: http://blog.mrmeyer.com

Getting into the "Real-World" with SMART Ink

Internet connected classroom computers link us and our students with the world. A web browser is a powerful teaching and learning tool providing access to current information, real data sets, rich media, interactive applets, and collaborative authoring tools. When used with a web browser, you might think of your SMART Board then as a centrally located computer interface, with the added benefits of easy annotation and capture.

Introducing SMART Ink

With the release of SMART Notebook 11, the "ink layer" has been replaced with a new program called SMART Ink.

Write Anywhere

SMART Ink is activated when a computer which has SMART Board Tools running is connected to a SMART Board or another SMART interface like an Airliner. A small toolbar is added to the title bar of each program. Picking up a pen adds a pen colour indicator to the SMART Ink tool bar to show that it is ready for writing. You can write or draw anywhere on the program window. Writing on the screen does not block interaction with the program like it did with the old "ink layer," and ink is associated with the underlying program so that it follows the location of the program window even when minimized. Unfortunately, ink does not scroll with the underlying program, so annotations will become misaligned on a web page if you scroll to read more.

Insert Writing as Text

Through SMART Ink, handwriting can be inserted as text in any application. First write the phrase to insert on the application. Next select the writing by returning the pen to the tray and touching the writing.[1] Make sure that the cursor is at the location you would like to insert the text by touching that place. Finally touch the icon beside the writing to recognize and insert the text. Unfortunately SMART Ink handwriting recognition is not as accurate as within SMART Notebook.

Choose a variety of Tools

Different pens and additional tool like a highlighter can be selected from the SMART Ink toolbar at the top of each window, or from the floating toolbar at the edge of the screen. The floating toolbar can be shown by choosing "Show Floating Tools" from the icon: in the tray in the lower right of the screen.

SMART Ink Setbacks

There two key features that were present in the old "ink layer" system that have now been removed: Ink can only be used outside of SMART Notebook when a SMART device is attached, and a few tools like the Magic Pen no longer work outside of SMART Notebook.

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[1] On boards with Touch Recognition enabled it is not necessary to put the pen back before selecting the writing because the board can sense the difference between the size of a pen tip and finger.

Blended Learning

Learning that incorporates both face to face classroom settings and online environments is known as "Blended Learning." Because there are many ways to blend these environments, blended learning may look quite different to match with the needs of different classes. For example, a teacher using a blended learning environment might:

• post reference, remedial and enrichment materials for students on a class Moodle
• use classroom computers or a lab for students to learn at their own pace through self-directed online activities
• use online quizzes for formative assessment
• ask students to share their learning with classmates online through a Moodle forum
• encourage the use of Google Docs for group tasks so that students can learn to collaborate remotely and be accountable for their own work.
• use webquests to help students work with primary Internet sources in an inquiry format

Why use Blended Learning

As we prepare students for success both at high school and after they move beyond there are many reasons to consider enhancing how we employ Blended Learning in our teaching practice. Here are three:

1. Differentiated Learning: Learning in an online environment can often be more individualized because students may be able to choose from alternative formats like interactive activities, and progress through material at their own pace. Additionally, some students who are hesitant to share their views or ask questions in a physical group environment will participate online.
2. Authentic Learning: Linking to real world stories and data is a natural fit with online environments. We can also help students learn to find and evaluate Internet sources when seeking information. Working with real scenarios can contribute to student engagement and is a valuable life skill.
3. Lifelong Learning: In many workplaces, as well as college and university, our students will need to learn and collaborate in online environments. We can make a strong contribution to our students' success in the coming decades is to build their Digital Literacy, and I believe this is a responsibility we share regardless of subject area in the way we all contribute to foundational literacy and numeracy.

Moodle as a tool for Blended Learning

Moodle 2 offers several features that make it a great tool to support blended learning. As always, resources can be linked and curated by the teacher making it an excellent anchor for a class' online elements. The new "Activity Completion" feature makes a Moodle course more engaging by revealing different and perhaps more appropriate activities based on the success or completion of other activities. The workshop activity builds self and peer assessment into the writing process. Enhancements in the quiz and feedback activities, especially in the ways they are created, make them good tools for adding interactivity and individual feedback online.