Visualizing Mathematical Functions

One data visualization tool that would be helpful for students to use in a math classroom is Desmos graphing calculator.  I could use Desmos in a lesson about graphing sinusoidal functions.  I might begin the lesson by instructing my students to log on to Desmos and simply modify the equation y=sinx in any way that they wish.  I would instruct them write down what they observe as they do so.  After they have had some time to explore, I might provide a worksheet for some more guided practice.  The worksheet might ask them to do things such as add a constant to the equation y=sinx and observe what happens.  Students should notice that the graph is moved up by whatever constant they added, like in the image below.

The worksheet might also ask them to multiply the sine curve by a constant.  When doing so, students should notice that the sine curve gets taller.

The use of Desmos graphing calculator allows students to quickly and easily see how modifying the equation y=sinx changes the graph.  Once students have any idea of different modifications change the graph, I would introduce formal vocabulary for the modifications.  For example, adding a constant creates a vertical shift and multiplying by a constant changes the amplitude of the sine curve.

Image credit: Dylan Ng

 

Are you a Loyalist or a Patriot?

Revolutionary War Reenactment #1

In fifth grade, the social studies standards generally cover the founding of North America, the lead up to the Revolutionary War, and the war itself.  In some exploring of these data visualizations websites, I think I would want to use the US News Map website to help students visualize what attitudes different populations of people had regarding the Revolutionary War and the separation of the American colonies from Great Britain.  This website would be a great way for students to see where exactly different groups of people lived based on their political opinions expressed in newspapers of the day.  They can search for key words such as “Loyalist” and “Patriot” and this website will show them where the terms were used most heavily in newspapers in different regions.  I think this would be a great resource to just let my kids explore and discover different patterns for themselves instead of me telling them something like: “Most Loyalists lived in New England…” With this, they can discover that for themselves, as well as get the opportunity to read some primary source documents from the time period.

Can a flipped lesson actually work in a traditional classroom?

 

Luckily in my student teaching placement, we have a lot of access to technology, but students are not able to take that technology home with them that would be necessary for a truly flipped approach to be successful.  The lesson I would have students do is this Google Slides presentation that is also interactive!  The best part of this Slides presentation is that it is interactive, and has places for students to type and move arrows around on the screen built into the presentation, making it much more engaging than a traditional PowerPoint presentation.  You would somehow have to share this presentation with students, with Google Classroom or a class blog of some kind being the best option for sharing a Google Slides presentation.

This lesson has five stages: Engagement, Exploration, Explanation, Elaboration, and Evaluation.  The first part of the lesson, Engagement, has students watch 2 videos that that explains how energy moves in a food chain.  This is a good way to introduce new material to students in this flipped format because they can go back and re-watch any parts of the video that they missed, or did not initially understand.  Following the video, there is a place for students to record any words that students did not know when they watched the videos.  These are both good videos for introducing a new topic to students, as they put the academic vocabulary that goes along with Ecosystems and food chains into very kid-friendly language and the videos are interesting and engaging for students to watch on their own.

The next three parts of the activity, Exploration, Explanation, and Elaboration all have students doing a variety of different activities that have to do with Food Chains.  The one that I like the best that I would want to use as an evaluative tool instead of the test at the end of the presentation is the choice board that comes with the Elaboration section of the activity.

This has students show me in a way that works best for them what they learned about Food Chains during this lesson that they completed on their own.  This activity would also be a good way to tell me where specific students’ misunderstandings were, and would be a good way to plan how to differentiate instruction when students come into class the next day.

For the in-class part of this flipped lesson, I would plan some sort of extension activity that has students work together to build their own food chain or food web, and I could help individual students with misunderstandings surrounding the topic for those who need that individual scaffolding.

Predicting the Next Big Blockbuster

To add some spice to the class and really relate to the kids while teaching them about trends and graphing data, I’d love to use the bookworm movies site. Just browsing the site and playing with it was really cool. So cool, in fact, that I would definitely want my kids to play around with it for a while and see what they would think about it. By the way, this would definitely work well with a class/unit/lesson dealing with collecting data, analyzing trends, etc.

 

Here’s how the lesson would look like (coming to you live from my brain):

 

5 Minutes: Take a survey of the class concerning their favorite movie/genre of movies. We would use this data to come up with a graph. From this data, we would also list the different stereotypes (archetypes?) that are common throughout that genre of movie, ie hero always saves the day, the nerdy girl is always asked out by the popular guy, red coats die first, stormtroopers always miss, etc. etc.

15 Minutes: The next part of the lesson involves introducing the students to the website bookworm: movies and having them search those phrases. Using this website, the students are able to filter the results they get.

In the image above, you can see just how specific students will be able to be when searching for phrases and/or individual words. During this introduction, the teacher/facilitator will give the students 2-3 example searches to test out on the site, just so that the students become familiar with the site and its features. The students will then be turned loose and left to their own devices; searching for words/phrases that either pertain to their favorite movie genre, or just phrases/words in general that they’d like to see data on.

5 Minutes: Yoga break! Students will be asked to turn away from their screens for some body moving time or optional class yoga. Studies show that students start to lose interest (basically shut their brains off) 30 minutes into a lecture/class/whatever it is that they’re doing and they’re not particularly interested in. So be proactive and get your students active!

Remainder of class: The remaining class time will be set aside for discussion. After the yoga/movement break, students will go back to the data they’ve looked up and share it with the class or in groups (whatever tickles the teacher’s fancy). The class will then be asked the following questions:

  1. Were there words/phrases that you tried to look up and couldn’t find? Why do you think that was?
  2. What did you notice about the trends in the data and the time period during which they occurred?
  3. Pick a graph comparing at least 2 phrases. (Pretend they picked the one below)

What do you predict will happen to this graph in the next 5 years? 10 years? How do you think the usage of these words/phrases will change over time?

 

 

And That’s a Wrap! : Students will then record their findings in either a journal or on a separate piece of paper and turn that in.

Aaaaaaaaaaaaaaand cut! You’ll be the coolest teacher around with this lesson. Guaranteed. (But not really. Don’t quote me on that. I don’t know the dynamics of your class and how they might react to this lesson. Just try it man. Nike.)

 

 

 

Featured image: Cinema Entertainment Film Reel Movie Projector on MaxPixel

 

Class 12: Google Hackathon

In today’s class we will explore the ever-changing world of Google apps for education. I’ve created a Google site with a section devoted to Google tools – docs, forms, MyMaps, slides, sites and more. After spending some time exploring the resources, students will have the chance to create an activity using a Google app.

Assignments
  1. Use a Google tool to design an activity. Create a blog post that explains the intent of the activity and includes an embedded version of the Google tool.
  2. Prepare for using iBooks Author to showcase your critical thinking lesson. Check this page for getting your content ready for using iBooks Author

Image credit: Google photos icon link

Newsflash: Oldschool Authors Preferred Writing About the Devil Than God. Wonder Where They Ended Up, Eh?

I would have my students use Ngram to write about trends in literature pertaining to the use of God and the Devil. They would have to differentiate between actual religious texts and literature and make assumptions on the time period based on what they find. I really like this website because it shows somewhat perfectly what was written during a certain specific time period. I am free to incorporate history in the lesson due to time being such an important thing while using Ngram. This will also help the student’s research skills immensely–for they will have to interpret information based off of the graph that Ngram shows.

One thing I do not like about Ngram is how cluttered it is when looking at the books used in the graph. I wish there was a better way to sort through each book presented. I think that a better algorithm would help student’s research immensely. Other than that, I think it is a fantastic tool for seeing what words were used during certain specific times, as well as seeing how popular they were.

I find it absolutely fascinating that the word “devil” appears more in the 19th century than the word “god”. I am not surprised by it, but still fascinated. I think if a student had little knowledge of the literature being put out during the 19th century, they would be shocked by this fact. This is why Ngram is a great tool. It combines academic research and purely satisfying facts–the kind of satisfaction one gets when mindlessly reading Wikipedia.

I hope that one day the program will be updated so that more-than-three-word-phrases can be looked at. That should be a pretty easy thing to do, right? If that were the case, then everyone would be able to be a linguist! We’d be able to look at phrases and see where they were from and how much they were used. I think it is necessary for Ngram to develop an algorithm that would allow us to do this.

**The website is not allowing me to add a picture to featured image**

Are you destined to be unhealthy?

I remember using Gapminder World in High School. I specifically remember using it sophomore year when (as a grade) we were participating in a model United Nations. It gave teams the opportunity to learn large amounts of information quickly about both their country and other countries. It was a great tool and I was excited to plan a lesson around it after last class.


Picture a science health class in a 1-1 school (which is where I dream to be eventually). The students have just finished a unit on different global diseases and this is their final project and assessment for the unit.  In groups of two, students will pick a disease that Gapminder World has data on. Once they pick the disease they want to learn more about for their project the objective is to determine what are major influencing factors for that disease. For example, if I picked Malaria then I would chose different factors such as GDP, Education, Poverty and observe their influence on Malaria. Once I have manipulated and analyzed the different graphs the students will chose 1-2 factors that they want to research more.

The students will research the disease itself if it was not covered during the unit within the class. Finding information such as: how you get it, symptoms, and is there a cure?  They will also research the affect and impact that the one-two different factors had on their disease. The students will get multiple days to accomplish this. At the end they will have created a poster, presentation, video or some method of representing their group research. In class there will be a gallery walk where students will set up their project and get to walk around the classroom and see everyone’s research.

Lesson Objectives:

Students will be able to manipulate and interpret data of a health disease to determine important factors.

Students will be able to research their disease and important factors after analyzing and interpreting the data.

Final Product:

A representation of the research on both the disease and its important factors. Could be a video, poster board, presentation, etc.


For an assignment such as this one I wanted to do a student led project. I wanted students to be invested in what they were researching so I let them chose their own disease. I also wanted to let students use their creativity and imagination in presenting their findings. Often assignments and projects in classes tend to be very scripted so I wanted a project with more freedom.

I also wanted the students to be able to play around with the program and learn how it worked along with the different methods it has for representing data!

Math and Fun: Not an Oxymoron in a Flipped Classroom

In middle school and high school, I heartily looked forward to science classes each day.  They were a chance to ‘experiment’ – to play, test, follow curiosity – and yet, somehow, through this, learn.  What a radical idea: the combination of natural interest and discovery and education!  But all sarcasm aside, teachers are testing more ways to make learning fun, memorable, and meaningful.  Labs need not be limited to a science classroom.  Indeed, flipped learning as a strategy for any class attempts to take advantage of technology to give students the best of both worlds: interesting lessons and non-drudgery homework in the form of class ‘explorations’ or other active learning strategies.  There is huge potential in using technology for this purpose, and as teachers, we can continue to learn by trial and error, using our imaginations (and the scope of the internet) to craft new lesson sequences and see what is effective.

Here is my attempt at creating such a lesson for a high school trigonometry or Pre-Calculus class learning about the Unit Circle.

Learning Objective

Students will be able to label the Unit Circle on their own, using intuition.  They will be able to identify the patterns that the trig functions follow.  Furthermore, they will be able to explain how we derive trig values from special triangles on a circle of unit radius.  Finally, when given a major angle measure, they will be able to return the corresponding sine, cosine, and tangent values, or vice versa, given trig values to match with angle measures.

Digital Resources for Flipped Elements

This lesson will work off of Bloom’s Taxonomy.  Students will begin with “homework” to do the lowest level of thinking to memorize the basic values of the unit circle.  They will have at their disposal multiple tech tools to do this, as students learn in a variety of ways.

Visual learners might want to utilize these programs on Desmos and Geogebra, which help students to connect the dots between shapes/spaces and trig values.

 

Desmos Unit Circle Exploration: I especially like the use of colors on Desmos – never hurts to make math prettier and more appealing!

 

Geogebra Unit Circle Exploration: Note how there is a slider which helps students visualize the effect of radian measure on trig value.

Students more inclined to numerical or algebraical thinking might want to use a chart (especially an interactive one like this), which shows patterns among trig values to ease memorization.

Clicking “Show” or “Hide” reveals the corresponding trig value in the chart, making this like a matching game.

The objective of this beginning stage is to gain comfort with the idea of radians, triangles inscribed in a circle, and the connection between angles and trig values.  At this point, students should develop a base knowledge of trig values in relation to certain radian measures, but I would not expect them to understand the significance of these values.

Active Learning Strategies – Interactive, Collaborative, Digging Deep

The next part of the blended lesson is the in-class work, which here will consist of active student participation in a “lab” to get to the root of the importance of the unit circle.  Students would work in groups, following a lab procedure that would lead them through activities looking at recognizing and questioning the qualities of the circle.  At sporadic points throughout the lab, I might draw the class together to make important points or to give them hints on how to proceed, but their background knowledge should give them the foundation to proceed more or less independently.  Finally, at the end, we would come together as a class to review our findings and discuss how this new material will benefit us in the coming unit.

How Will the Lesson Flow?

Using a flipped lesson sequence frees up time to focus on deeping students’ learning in class.  By intentionally guiding students’ focus from basic to more advance studies of the circle, my hope is that they will not feel overwhelmed by the unit circle and instead give each aspect of it its due diligence and attention; ultimately, this should pay off in gaining both a general knowledge and a more application-based ability concerning the unit circle.

Why Use the Blended Model for this Lesson?

The goal of this lesson is for students to develop an intuitive understanding of the unit circle in order to be fully prepared to deal with all things trig-related in the following units.  The unit circle is hugely important and often not covered in due depth.  It may seem that math teachers’ insistence on knowing by heart the values on the unit circle is overboard, but really – they are not deceiving you in telling you that knowing this material like the back of your hand will pay you back several fold as you continue in math.  More critical even than memorizing the values of the circle and a much more thorough way of learning the circle, in fact – is being able to derive these values.  From what do they stem?  How are radians related to measures of a circle?  How are trig functions tied to coordinate points?  How do we convert between trig functions, or even undo trig functions with inverses?  These are the deeper questions that students should be able to answer.

Featured image: Bob B. Brown

What’s Your Sine?

1. Learning Objective

Students will be able to find the derivatives of sinusoidal functions.

2.  Digital Resource(s)

Before coming to class, students will watch an online video that I will create of me proving \frac{d}{du}sin(u) = cos(u) and \frac{d}{du}cos(u) = -sin(u).

3. Active Learning Strategies

Once students have an idea where the equations come from, they can use the freed up class time to solve problems where they are asked to find the derivatives of sinusoidal functions.  One active learning strategy that I might employ is Numbered Heads Together.  I will split students into groups of three or four and have them collaboratively work on problems that I will provide.  One randomly selected student from each group will then provide a brief explanation of how their group went about solving the problem.  Students who would have difficulties solving the problems on their own will have the benefit of group members to help them out.

4. Lesson Flow

The video explains to students where the equations \frac{d}{du}sin(u) = cos(u) and \frac{d}{du}cos(u) = -sin(u) come from.  I will begin class by demonstrating to students how we can use these equations to find the derivatives of various sinusoidal functions.  The example problems will require the use of techniques taught in previous lessons such as the power rule, the product rule, and the chain rule.  After the example problems, students will be split into groups for the Numbered Heads Together activity where they will solve some problems in groups.  Finally, students will be given some homework problems to work on individually for the last part of class.

5. Benefit for Students

As a math teacher, I dislike presenting a formula or equation to students and asking that they believe it on blind faith.  I really believe that students benefit from seeing where formulas come from and why they work the way they do.  However, in many classes (especially calculus), there are simply too many equations to offer a formal proof for each one during class time.  By flipping this lesson, students are able to see where the equations \frac{d}{du}sin(u) = cos(u) and \frac{d}{du}cos(u) = -sin(u) come from, and I won’t have to use class time to prove the equations.  Instead, students can use class time to collaboratively work on problems where they will apply the equations.  This is preferred to having students work on problems at home, where they would not have a teacher or classmates to help them out if they get stuck.

Image credit: Creativity103

Class 10: Teaching with Data Visualizations

Quite often edtech tools are used by the teacher rather than the students and don’t do much more than make things prettier.

Think: Teacher at Smartboard as replacement for the overhead.

New digital technologies allows us to “see” information in new ways.

Think: Students analyzing text using Wordle

Many apps and websites can be a great tool to introduce the research method – form a hypothesis, gather and analyze data, revise hypothesis (as needed), draw conclusions, assess research methods. Working in teams students can easily pose research questions, run the data, revise and assess their research strategy. Students can quickly make and test predictions. They can then present and defend their conclusions to other classroom groups. All skills called for by the new Common Core standards.

In today’s class we will explore a sampling of free online data visualization tools that can be used in the classroom. Students will be asked to incorporate one of these tools into a lesson design.

GapMinder World: manipulate moving bubble graphs, select x and y axis from a variety of data sets

 NGram Viewer: online research tool that allows you to quickly analyze the frequency of names, words and phrases -and when they appeared in the Google digitized books. Ideas for classroom use Books Ngram Viewer.  For more advanced searches using NGram Viewer click here.

Chronicling America has digitized newspapers from across America from 1836-1922. You can search word frequency here.   Search Chronicling America and visualize the results across space and time at USNewsMap project

Bookworm: a collection of simple and powerful way to visualize trends in repositories of digitized texts.

  • Movies: dialogues of movie and TV shows
  • ArXiV: science publications
  • US Congress: bills, amendments, and resolutions (by political party)

Timelapse: is a global, zoomable video that lets you see how the Earth has changed over the past 32 years.

Metrocosm: All the World’s Immigration Visualized in 1 Map

Combine multiple online tools for research: For example, Black History in America:
Map of White Supremacy mob violence here
Mapping the “Negro Travelers’ Green Book” here

Assignment

Choose one or more of these digital tools (or use a favorite our yours) and blog about how you would use it in an activity, lesson or unit. A few things to keep in mind:

  1. Be sure you design a lesson that allows your students to be using the tool
  2. Be sure to include an embed of the tool (if possible) or at least a screen shot.
  3. Blog post due: 3/30

Image credits:
Header: AdobeSpark public domain
Insert: Teaching with a SMART Board / Flickr

There can never be too many labs….

1) Learning Objective

Students will be able to demonstrate knowledge of cellular respiration through a tennis ball lab experiment through data analysis and graphing.

2) Digital Resources Used

The first part of the lesson would be the students on their devices at home watching a video that I created through EDpuzzle. The video would be an introduction and full overview of the process of cellular respiration, I would add checkmarks of questions that the students had to answer and they couldn’t skip ahead they have to watch all of the video. I most likely would not grade these questions but see it as a form of informal assessment for me to know where my students are at (it would all depend on where in the unit we are at). This would free up what would be classroom lecture time to insert a lab which I wish I could always do more of!

3) Active Learning Strategies

For me the best active learning strategies are getting students involved in a lab experiments in class. Currently I do not have the abilities to operate a flipped classroom so I can only do a small amount of labs, but in my opinion labs are one of the best hands-on and active way for students to learn content. The lab that I would do is a tennis ball lab that connects to cellular respiration. The students are in groups and they have to squeeze a tennis as hard as they can for 30 seconds take a rest and do it again. The students have to do this five times in a row and keep track of how many times they can fully squeeze the tennis ball. The connection is that over time your muscles fatigue with has to do with cellular respiration, glycogen storage, and anaerobic fermentation.

4) Lesson Flow

The lesson will flow in the direction of starting with the video and questions at home so that when students come into class they have received and reviewed the content, therefore they are ready to begin the lab. I would most likely do a review at the beginning of the class of content that pertained to the lab and then the rest of the class period would be devoted to the students lab time. A block class period would be perfect to also start a discussion and analyze the data however it could be done over a class period and a half or two full periods.

5) Benefit to Students

This format allows students to be exposed to the material outside of class so that by the time they come into class they know what they know and what they do not know. This allows time for more questions and one-on-one time with students in class. The content offloading also allows for more hands-on, minds-on activities in the class, which for me means LABS LABS LABS. Labs are wonderful opportunites for students to demonstrate their knowledge and learning in a more active format. It also helps students that struggle with learning content from a screen or a person another way to learn material.

Image Attribution: Allison Meier

Sample write up: Where I’m from

This  is a model blog post that demonstrates a write up for the Blended / Flipped Lesson assignment. Bold face is your assignment. Followed by how I would have written it up.

This write up is based on a lesson I used at the beginning of my Alaskan History and Culture course in the MAT program at the University of Alaska SE in summer 2016. See my assignments here:  Where I’m From and Google MyMaps lesson: Place

Here’s the work done by Jimmy Andrew – one of my students from a Yup’ik village of 300 people – Kwigillingok Alaska.

Learning objective – content and or skills students will know or be able to do by end of the lesson.

This was the second class of course and it served multiple objectives:

  1. My primary objective was to introduce the idea of place-based education with a poem and personal reflection on a place they were already familiar with. (later they would learn about another place – Alaska)
  2. I wanted to give the students a relatively simple tech-based assignment (Using HaikuDeck and creating a WordPress post) to build confidence for more elaborate tech assignments later in the course
  3. I need to free up class time so that students could get individually logged into their new WordPress accounts

Digital resource(s) you’ll use for flipped / blended elements.
Note: it’s not necessary to develop the digital elements – you can just describe them.

  1. I designed and posted the lesson to our WordPress blog so that everyone had access to assignment and resources.
  2. I made a few how-to videos describing how to create a Haiku Deck account, create a presentation, and embed it in Word Press.
  3. I made few how-to videos on using Google MyMaps
  4. I reused a collection of how-to videos that I had created for how to use WordPress

Active learning strategies employed with freed up class time.

Students used the time to read the poem for inspiration and get to work designing their own personal “Where I’m From.” Others elected to try the second design option using Google MyMaps using a lesson called Place. While the Haiku Deck approach was more visual poetry, MyMap used geographic tools and a spacial approach to design a Google map tour. Once I was done meeting with individual students to create their WordPress accounts, I was free to move around getting to know students better and assisting them on specific questions.

How the digital resource integrates into other instructional elements of lesson – what’s the flow of the lesson?

The digital content was created and posted to the web in advance. At the end of the previous class I asked students to read both lesson options (HaikuDeck  and MyMaps) in advance and think which lesson they want to try and how they will respond. The day of the class, most of our 3-hour session was spent with students working with HaikuDeck or MyMap. Then they began turning their work into their first blog post. Many students were excited to “show off”  their creations with one another.

Benefit for student content mastery, collaboration or learning workflow – Why is it worth it flip / blend some of the content.

Using these digital resources enabled students to begin the course with an easy “tech skills win.” Every student was successful in completing the assignment and posting it on WordPress. I was able to re-use my WordPress how-to videos originally made for another course. Along with the how-tos for HaikuDeck and MyMaps, I was freed up to handle the management of new WordPress accounts and spend some one-on-one time with all my new students.