Two Criteria

Come up with two ways of ranking characters, such as “morality” and “physical strength”. Pick some characters that have different levels of these characteristics.

Finally, ask students to place each character on a graph, with x- and y-axes representing the two criteria. Here’s an example:

I used the following characters:

• The Wicked Witch of the West, The Wizard Of Oz
• Gaston, Beauty and the Beast
• Mr. Incredible, The Incredibles
• Katniss Everdeen, Hunger Games
• Ron Weasley, Harry Potter
• Superman, Superman
• The Penguin, Batman

My Justification

The Wicked Witch isn’t physically powerful, she relies on an army of monkeys and is easily defeatd by a bucket of water.

She is, however, more powerful than The Penguin, due to her magic. Gaston is certainly the most physically powerful of the three characters with low morality.

Superman and Mr. Incredible are equally powerful, but Mr. Incredible has several moral lapses that move him down the y-axis.

Neither Katniss nor Ron are physically strong, but both have high levels of morality. I put Katniss a bit lower, due to her controversial love triangle

Quadrants

Let’s increase the math connection by labeling the four quadrants.

• Quadrant I: Heroes – Powerful, morally strong characters.
• Quadrant II: Underdogs – Physically weak, but morally strong characters.
• Quadrant III: Masterminds – Physically weak, evil characters.
• Quadrant IV: Brutes – Powerful, evil characters.

The Best Part

Did you disagree with my placement? Great because your students will disagree also, and they’ll get into a fascinating discussion about character analysis.

Other Criteria

Looking for some more ways to graph characters?

• fear
• pride
• courage
• knowledge
• happiness

Naturally your students are going to come up with even better ideas for graph characters.

The Product

Let students pick five characters from a huge list. Have students create graphs using cutout characters, printed graphs with color printers, or hand-illustrated creations. Include brief explanations for each character.

If you’re up for the challenge, let students chose a third criteria and build 3D graphs to hang around the room.

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The paradox content imperative is a blast to expose students to. Here are three famous paradoxes to delight and confound your deep thinkers:

1. This sentence is false.
2. In a town, there is one male barber. Every male in town keeps himself clean-shaven. Some men shave themselves. The rest go to the barber. The barber shaves only those men in town who do not shave themselves. Does the barber shave himself?
3. There are two sign-makers, Cellini and Bellini. Cellini always writes false signs, and Bellini always writes the truth. A sign says “Cellini wrote this.” Is this possible?

My favorite student comment after working through the these: “My brain just exploded!”

Here’s a less mind-blowing bonus from Yogi Berra: “Nobody goes there anymore. It’s too crowded.”

Do you have some fun paradoxes to share with students? Let me know at ian@byrdseed.com!

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I have a class set of HG Wells’ The Time Machine. It was easily affordable, a classic of the genre, and a well-known title because of its recent movie adaptation.

The problem? It was written in the late 19th century in England and is simply above most of my students’ (admittedly high) independent reading levels.

However, this book was definitely within their instructional reading level, so I turned this lit study into a read–aloud. I read the book out loud (with several selected edits) to my gifted 6th graders over the course of two months.

Here were some of the amazing benefits…

Exposure To An Important Author

Students learned about a science-fiction pioneer who created War of the Worlds and The Invisible Man. Wells even coined the term “time machine.”

Major Thematic Discussion

Time travel!

Talk about a concept to blow students’ minds. The novel opens with a discussion of time as the fourth dimension. The protagonist explains how he has learned to move in the fourth dimension just as we can move in the other three dimensions. I loved to see students struggle with this concept and try to explain it to their peers.

The concept of traveling through time opened up heady discussions about paradoxes. Can you change the past? What does that do to the present? What if you killed your own grandfather!?

We discussed other famous works with parallel themes:

Unintentional Consequences

In Back To The Future, Marty must travel back in time and help his parents fall in love… only his mom develops a crush on him!

Learn a lesson about the present

In A Christmas Carol, Scrooge learns about his own behavior by traveling to his past as well as the future. Similarly, in It’s A Wonderful Life, George Bailey learns about his own importance by traveling to an alternate future in which he never existed.

Fix the past

There are numerous accounts of time travelers going back to stop a terrible event. In The Twilight Zone episode Back There, a man attempts to stop Lincoln’s assassination, with surprising consequences.

These fascinating themes open up some great creative writing options.

Time’s Influence On Creators

After finishing the novel, we the watched the 1960 film version of The Time Machine.

Students were aghast at the strange changes. In the book, the Time Traveler jumps directly to the year 802,701. However, in the film, he stops off at both World Wars, then experiences a 1966 atomic attack on London, then finally jumps to 802,701.

Why would the filmmakers change the story in such a strange way?

We discussed the concept that creators’ own time period influences their work. War was on the mind of people in the late 50s. Naturally, their movie was shaped by this influence.

HG Wells, however, was influenced by the events of the late 19th century in England. His story commented on the effects of communism, technology, and the emerging “easy life.”

This set up an avenue for analysis: how do authors’ own experiences affect their writing? We could spend months on a big, juicy idea like this, and there’s no way the anemic selections from our anthology could support such a discussion.

DIY Special Effects

The film’s effects are humorous to our eyes, but won an oscar in their day. Buildings grow and fall, plants wither and are reborn, the sun swiftly flies across the sky. The evolution of fashion is shown via a department store window’s mannequin.

I asked students to group up and develop a scene to show the world changing through stop motion.

Student ideas included:

• hand drawn animation
• LEGO stop motion
• time lapse photographs of the neighborhood

I bought $20 worth of green fabric to serve as a green screen.

They used technology such as:

• their mobile phones and iPod Touches
• the iPhone and iPad app iMotion HD
• Flipnote Studio, an app for the Nintendo DSi

Once they made their clip, iMovie provides simple drag and drop green screen removal.

Naturally, students can create their own soundtrack. Our lab has Garageband on the computers, which integrates beautifully with iMovie. And publishing the movie is a snap by exporting to a movie hosting site like YouTube or burning DVDs.

We’re still working on it, so a sample is on its way

Taking On The Challenge

Rarely do I see my class confounded by a story. It was awesome to really work through this book with them and take them to a level beyond their independent ability.

Try a tough book with your group and see what happens!

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The Little Prince

It all started with The Little Prince, an incredibly deep novel that’s easily readable and clocks in at 100 pages long. The class read it as a novel study, but I felt like we barely scratched the surface of its layered meanings.

Sure, we could have studied this novel until the end of the year, but there’s so many other great novels to expose students to. So how do we continue to take advantage of a novel’s depth?

Bring A Character Along

I decided to invite The Little Prince over to our next novel study by reusing his unique point of view. So, as we read From The Mixed Up Files of Mrs. Basil E. Frankweiler, students evaluated how The Little Prince would judge Claudia, our new protagonist.

At a recent training, my friend and colleague Linda Gillen discussed her class’ recent interest in philosophy, and it occurred to me that the Little Prince’s “point of view” is really his “philosophy.”

Practical Philosophies

The word “philosophy” conjures up images of wizened men in togas arguing about impossibly abstract topics. But teachers have philosophies about education, our students have philosophies about homework, and parents have philosophies about raising their children. A philosophy can be practical and concrete. Let’s use philosophies to push our students’ to higher levels of thinking.

Identifying Characters’ Philosophies

I explained that a philosophy is a character’s big idea towards an ethical issue. The Little Prince believes that spending time with the things you love is important. This philosophy drives his actions and interactions.

Begin with a couple examples: I used my imaginary character, the extremely mean teacher named Mr. Boid, as an example, contrasted with Ms. Frizzle from The Magic School Bus.

• Mr. Boid’s philosophy is to punish children until they do what he asks.
• Ms. Frizzle’s philosophy is to give students exciting, real-life science opportunities, (even if it puts students in mortal danger).

Then I asked students to identify other characters’ philosophies:

• The Brick Pig: Hard work pays off.
• The Giving Tree: Giving is the most important action, even if it means sacrifice.
• The Time Traveler: Satisfying curiosity is even more important than personal safety.

Now students can evaluate characters using several philosophies, which leads to interesting conflicts, ambiguities, and questions:

The Brick Pig would approve of Claudia’s plan to run away, because she has put so much hard work into planning the escape, but The Giving Tree would not approve, since her plan is selfishly motivated.

But we can go even further!

Bringing In The Big Guns

What about the great thinkers? Philosophers like Socrates, Aristotle, Confucius, and Descartes could also serve as points of view. I defined their beliefs as succinctly as possible:

• Socrates: Evil is caused by ignorance.
• Aristotle: Friendships with selfish purposes lead to problems.
• Confucius: Elders should be kind to youth, and youth should be respectful to elders.
• Descartes: Begin by doubting all things.

Now students can apply these philosophies in their evaluation of characters:

Socrates would not approve of Claudia’s plan because she is acting out of ignorance of how much her parents love her, but Aristotle would approve since she’s strengthening her friendship with her brother.

The best part is when students’ opinions differ. Some might apply Aristotle’s philosophy differently, arriving at a conflicting conclusion.

Thinking At A Higher Level

With this simple introduction, you will expose students to hugely influential thinkers, give them insight into the “meaning of life,” and broaden their perspectives when analyzing characters. After a few lessons, students will be discussing the morality of character’s decisions using the philosophies of The Little Prince, Socrates, and Confucius.

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Here’s how I differentiated the Houghton Mifflin comprehension skill of “Compare & Contrast” for my gifted sixth grade students, who have been successfully comparing and contrasting since kindergarten.

Bring In The Classics

The Houghton Mifflin (HM) selection is from artist Chuck Close’s biography. Although HM suggests comparing and contrasting Chuck in the beginning and the end of the story, I opted to bring in brief biographies of other artists.

We read through Van Gogh’s biography from Simple Wikipedia on one day and then followed up with Claude Monet’s bio the day after that. Both were edited slightly for content and to make them a bit shorter.

Naturally, when discussing these other artists, I made a quick powerpoint of their most famous art so students could visualize The Potato Eaters and Water Lilies.

After reading the selection and these two bios, students had information about three significant artists to compare and contrast.

Up-Level The Wording

“Compare and Contrast” inevitably leads to simple, easy responses like “they’re both men” or “one is from France and one is from America.” Creating more specific guidelines combats this problem, and drives students to a deeper level of thinking. I asked students to:

Compare and Contrast the Effects of Each Artists’ Hardships

This simple addition led to a fascinating analysis of tragedy’s tendency to inspire, as well as artists’ varying abilities to overcome these difficulties. It can also serve as a great chance to speak with students about overcoming their own difficulties.

An Interesting Product

A Venn Diagram or Double-Bubble Map is a not much of an authentic product. Instead, students created a haiku summing up their discoveries about artists’ hardships. They had to condense their understanding of this vast idea into a mere seventeen syllables. The tight structure led to some very diverse statements about artists.

To publish their poems, students wrote the finished haikus in the center of a blank piece of paper, and then used cross-hatching to shade in a spotlight effect. The corners of their page were darkest, leading to a bright center where the words were located. Everything was done in simple black and white. Displayed together, the poems created a beautiful image.

In The End

Students went far beyond Houghton Mifflin’s expectation of filling out a worksheet. The class:

• was exposed to two classic artists
• delved into the concept of art’s relationship with hardships
• condensed many details into a single generalization
• worked with a specific type of poetry
• learned a practical art technique

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If you’re attempted to differentiate your math program through preassessment, I’m sure you’ve stumbled across students who have already demonstrated mastery of an upcoming unit.

Typically, we try to come up with something deep and meaningful for these students to work on while we instruct the class. This, however, is a tricky problem with no simple solution. As I’ve mulled it over, I’ve thought of four qualities to look for when developing a project:

• Authentic Data
• Interesting Conflict
• Expert Perspective
• Enticing Product

Finding Numbers

A project has to have some juicy, real-world data to get things going. I consider this the “seed” of the project. I listed 33 places to find data here, but here are some favorites:

• IKEA furniture
• Census Data
• Box Office Mojo

Here’s an image of some US census data that I’ve been looking at:

It’s a seed of an idea. There are tons of numbers, but where’s the interesting hook? Where’s the drama to draw our students in?

Conflict!

I’m a Californian, and frequently travel to Texas. So when I noticed that California is the top state by population, and Texas is second, I knew I had a hook.

I graphed the census data to look for some interesting patterns:

Bingo! Texas is catching up!

Now we’ve got some conflict to build upon that might entice a student in California (or Texas!). How long will Texas need to catch up? How fast must California grow to keep Texas at bay?

Let’s look for an expert’s perspective to further develop this project.

Think Like A…

I like to give students a specific lens to view the conflict through. Here’s Wikipedia’s list of academic disciplines, if you’re looking for inspiration.

I settled on “Thinking Like A Tourism Geographer” after stumbling across Texas’ official tourism site as well as California’s.

Here’s how I’d set it up:

> California’s board of tourism is concerned with Texas’ growing population. You’ve been hired to analyze the population data and increase the demand to move to California.

First students develop a multiline graph to show the changing populations of Texas and California. They should look for patterns and (hopefully) arrive at the same conclusion I did.

Next, they would follow the pattern, and determine about how long California had until Texas took the crown.

Note, I would not give them my multiline graph. That was purely for my own inspiration.

We could also ask students to show Texas’ population each year as a percent of California’s. These specific math requirements are up to you, and should be based on your students’ grade levels and abilities.

Then, students would determine how to highlight California’s best features to convince people to move, ensuring The Golden State’s dominance over Texas.

Digging into this idea, students could cross disciplines and use:

• geographical points of interest
• weather and climate information
• locations of historical importance

Product

The icing on the cake is to give students an interesting, authentic product to communicate their findings and ideas.

Depending on their skills and interests, students might create:

• a website
• a commercial (either a skit or a video)
• a brochure

In the end, be sure to give your students a chance to show their classmates what they’ve created, and display it proudly appropriately around the class.

A Flexible Pattern

Maybe you’re a Texan. This project could easily be framed around helping Texas quickly catch up to California’s population. In a smaller state? You could make it work there also. Maybe your conflict is breaking into the top ten, or catching up to the state above you.

These four steps are a nice, flexible framework for developing math projects for the needs of your advanced learners. While they’re off working independently, you are free to spend time helping the rest of your class with grade-level materials.

Finally, expect to change these projects around every year. At the least, your product options should be modified to reflect changing technology and the unique interests of your students.

Good luck, and let me know if you come up with something juicy!

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It’s becoming increasingly obvious that students need instruction in online behavior and consequences. Unfortunately, we’re stuck with textbooks that feature “internet” lessons like the one seen above. We have to prepare our students for an online world that our curriculum isn’t even aware of.

Day One: Reputation

Spend the initial lesson establishing the concept of reputation.

Ask students for definitions, then offer a dictionary definition: “The beliefs or opinions that are generally held about someone or something.”

Explain that teachers have reputations like: funny, strict, exciting, and creative. Ask how teachers get their reputations. Let students share thoughts. Discuss that teachers acquire their reputations through actions. Further, these reputations hold for years and years. Younger students know the “mean” and “funny” teachers through older siblings.

Form a generalization: “Reputation is built on actions and survives over time.”

Day Two: Historical Figures

Today we’ll integrate reputation with historical figures.

For example, Martin Luther King, Jr. has a reputation for bravery. He gained this reputation through the Montgomery Bus Boycott and the March On Washington.

Ask students to identify reputations of other historical figures. Require evidence that supports these reputations. Give them a list of character traits to promote specific word choice. You’ll need some resources for students to read. Simple Wikipedia is often great for this.

Form up trios and give students jobs like reader, writer, and presenter to keep groups running smoothly.

Some examples (with contemporary celebrities to spice things up!):

• Mother Theresa
• Ghandi
• George Washington
• Selena Gomez
• LeBron James

Conclude with group presentations.

Day 3: Online Reputations

Now for the meat! Let’s connect reputation to the online world.

Begin with this Microsoft-sponsored survey:

75 percent of U.S. recruiters and human-resource professionals report that their companies require them to do online research about candidates, and many use a range of sites when scrutinizing applicants — including search engines, social-networking sites, photo- and video-sharing sites, personal Web sites and blogs, Twitter and online-gaming sites.

Break this down with your students. Most companies use the internet to learn about potential employee’s online reputations. The kicker?

Seventy percent of U.S. recruiters report that they have rejected candidates because of information found online

Discuss, and then continue with this New York Times article on cyberbullying:

Girls from Benjamin Franklin reflected about their cyberdramas: “We had so many fights in seventh grade,” one girl said. “None of them were face-to-face. We were too afraid.” … Another concurred. “It’s easier to fight online, because you feel more brave and in control,” she said. “On Facebook, you can be as mean as you want.”

This snippet (slightly edited) raises the stakes:

“I had a 45-year-old father crying in my office,” Mr. Orsini said. “He kept asking, ‘Why would someone do this to my son?’”

A Facebook page had sprung up about the man’s son, who was new in town… nearly 50 children piled on, many of them readily identifiable.

Discuss how easy it is to develop a poor online reputation. Bring it back to the Microsoft study: people will see our actions, and the consequences are severe!

Hopefully someone will point out that one could just delete incriminating evidence. Tell them to stay tuned for tomorrow!

Day Three: The Internet Never Forgets

Ask your class: can we delete online actions?

“Yes! Of course! It’s easy!”

Oh. How do we know things online are really deleted?

The fact is, we don’t know. We trust Facebook and the rest that our data is really gone. But guess what?

CNN reported that photos “deleted” in 2009 were still accessible on Facebook in 2012.

When we first investigated this phenomenon in 2009, we discovered that photos “deleted” from Facebook seemingly never go away if you have a direct link to the image file on Facebook’s servers.

Three years later:

I reached out to Facebook once again, looking for an answer as to why this is still going on nearly three years after the company first promised it was “working” on the issue.

This information should shock your students. A clean reputation is not just a click away.

Now show The Wayback Machine to your class. This site saves webpages over time as an act of historical preservation… even if we don’t want them preserved!

Here are some funny examples to share that reinforce that the internet never forgets:

• Yahoo circa 1996
• Google from 1999
• Apple in 1997

This evidence should work to open students’ eyes that internet activities are more permanent than we realize. Our online actions should be carefully considered – even more so than offline actions, which are rarely recorded in such detail.

Day Four: Back To Historical Figures

Finally, consider how great historical figures would use the internet if they had access in their time. Bring Martin Luther King, Jr. back as an example. With a reputation for bravery and a passion for equality, how would he use modern internet services?

• YouTube: broadcast the brutality in Birmingham
• Blog: update his followers on progress towards equality
• Twitter: organize rallies and demonstrations.
• Facebook: create a Page where others could join and support each other

To connect with Presidents’ Day, I used several US presidents as our subjects. I gathered information on a dozen presidents (using this book). Trios chose one and went through the packet using the same jobs from earlier.

They noted the president’s reputation and actions that led to his reputation. Then they brainstormed potential online activities for their president. Finally, groups presented their rather insightful ideas.

Wrap Up

I used this procedure as an entry point to our own class website. It also tied in nicely with a holiday and, most importantly, brought to light the importance of maintaining a good reputation even when online.

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Symbolism, a mainstay of literature discussion, seems too abstract and ephemeral to teach to younger students. However, with a well-constructed lesson and a plethora of examples, students will quickly get the hang of symbolic representation. We’ll finish this unit up with some great pixel-art and computer work.

Rings

I begin by taking off my wedding band and discussing it with my students. I ask them what it stands for.

Quickly we get some outstanding examples:

• love
• marriage
• my wife
• forever (due to its circular shape)
• the ring finger connects to the vein of love

What if I lost my ring? Does that mean my marriage is gone? Is my love for Mrs. Byrd gone? Of course not! The ring is not actually love.

But many students know that a lost wedding ring could put me in some hot water. So a symbol has strong emotional connections to the idea it symbolizes.

Does a ring always symbolize love?

My favorite counter-example is the Ring of Power from Lord of the Rings. The Ring symbolizes quite the opposite: uncontrollable power. Anyone who obtains the ring becomes consumed by its unquenchable thirst for more power.

Liberty

Next, I display an image of the Statue of Liberty. What does it stand for?

• liberty (duh)
• freedom
• America
• friendship (especially with France)
• hope

Then I ask if anyone has ever seen a movie or image with the statue destroyed or damaged. My 6th graders had tons of examples. We talked about how filmmakers use a damaged Statue of Liberty to show the destruction of liberty and freedom.

I use the classic image from Planet of the Apes to show how the Statue of Liberty symbolizes Earth and its destruction.

Students might also enjoy discussing the symbolism in flags or other national symbols.

The Desert

Symbols do not have to be specific items. A desert is a common symbol, often representing difficulty. When we see a character in the desert, they are possibly going through a hardship or trial.

Bringing It Back To Characters

Throughout the year, we read stories that include symbolism. Hatchet‘s Brian owns a hatchet which symbolizes his hope to survive. Some students said it symbolized man versus nature. Nice!

Some quick symbolism from Pixar movies your students might be familiar with:

• Nemo’s “lucky fin” symbolizes the baraccudda attack that destroyed his family.
• On the bottom of Woody’s boot, Andy’s scrawled name symbolizes their bond and love.
• Mr. Incredible’s room of newspaper clippings represents his longing for another time.

Computer Icons

Our last topic of discussion is the symbols that represent computer programs. We discuss Firefox’s image of the globe surrounding by a fox. Safari is a compass. Keynote is a podium. Garageband is a guitar.

Creation

I explain that we are going to generate similar icons to represent a character. We identify a character’s main traits and their motivation. We think about the dialog, thoughts, and actions that show these traits and motivations.

Then students consider items that share these traits.

At first, students might try to simply draw the character or an object they hold. This is not enough. We want a very abstract symbol.

Consider a simple story such as The Three Little Pigs. The straw pig demonstrates laziness. What other objects show laziness? A straw house is too literal. We want something abstract. A recliner? A bed? A can of soda and some fast food? Now we’re talking!

After students have identified suitable items, I let them start creating their pixel art. We use a small grid. This keeps non-artists in a nice rigid framework, and challenges artists to be creative within the framework. Plus, all kids (and many adults) love pixely artwork because of its connections to video games.

Here is the worksheet (as a PDF) I used to structure this process.

Pixel Art

Naturally, our computers are too locked down to install any cool pixel art programs, so I repurposed Microsoft Excel. Students made square-sized cells and learned to change their background colors, in effect painting with a spreadsheet.

This could be done in the spreadsheet of your choice, including Excel, Numbers, and online with Google Docs.

In fact, here’s an incredible example of collaborative spreadsheet art using Google Docs:

Have fun and let me know how it goes!

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Photo by Jonny Keelty

Area, perimeter, and circumference cause more confusion among students than necessary.

The problem is that we dive in with formulae before students have their bearings. Let your students get their hands dirty with geometry. They’ve got to play with the shapes and see why these magical formulae work.

Beginning adders and subtractors work with manipulatives before they delve into abstract arithmetic. Older students are still novices at geometry. Give them a chance to touch the math and have some fun.

Area’s Relationship to Perimeter

Let’s set up this lesson by creating some conflict around a common misconception.

Students, my square has a perimeter of four meters. I’m going to double it. Will my area double also?

No! Doubling perimeter does not double area. But I bet your students think it does.

Introduce this as an inductive exploration. Set up a pattern by starting with a perimeter of four square feet.

Stop here, because several interesting patterns have arisen. Ask for observations. Here are a couple:

• At 4 m per side, perimeter and area intersect.
• The area increases much faster than the length of the perimeter (you could graph this and discuss linear versus exponential growth).
• Sides increase by one, perimeters increase by four, and area increases by three, five, seven, then nine! I wonder if it continues…

Show how dramatic the increase is with larger numbers:

Squares Versus Rectangles

Continue to expand their perceptions of the relationship between area and perimeter. Ask, “Will a perimeter of 20 m always lead to an area of 25 sq m?”

Uncertain nods

Let’s build some rectangles instead of squares and surround them with 20 m of fence. Ask for all the combinations that give 20 m:

Look at the remarkable difference in areas! A square is nearly three times more efficient than a skinny rectangle.

What if you included decimals? How low could the area go? A 0.1 m by 9.9 m rectangle has a shockingly small area.

But is this always true?

Try it with some other perimeters. Lead students to the generalization: a square uses perimeter more efficiently than any other rectangle.

And The Circle Rolls In

So, the square may be the most efficient rectangle, but is it the most efficient of all shapes?

Roll a circle in and have a competition.

Will a circle with circumference 20 m beat the square’s area of 25 sq m?

If your students have gotten their feet wet in algebra, they can find the radius of a circle by starting with the circumference.

Find the radius of the circle:

1. c = πd (formula of circumference of a circle)
2. 20 = πd (circumference given)
3. 20 / π = d (divide both sides by π
4. ~ 6.37 = d (approximate radius)

Halve that diameter to get a radius of ~ 3.185 m.

Using the radius, find the area of the circle:

1. A = πr² (formula of area of a circle)
2. A = π * 10.144 (square the radius we found)
3. A = ~ 31.85 (approximate area of the circle)

So the square was king of the quadrilaterals with 25 sq m, however the circle handily beats that with an area of nearly 32 sq m.

But… Why?

Now, if you’ve got a bunch of gifted minds in your classroom, someone is going to ask the age-old question: why?

Why is the circle so efficient?

The short answer: given the same perimeter, a regular figure with more sides will cover more area.

But don’t just tell them! Show it using this calculator. Slowly (and dramatically!) build a table like so:

I bet they’ll be disappointed when you stop at the decagon. Feel free to go further, or give it as some optional homework.

So a hexagon will beat a square, an octagon will beat a hexagon, and so on.

Guess which shape has the most sides? That’s right, the circle with infinity sides will always win.

Finally

While you’re discussing polygons, please play this video for your students: Nonagon.

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Photo by lydia shining brightly

We’ve just experienced the solar new year, and now many cultures will celebrate the lunar new year.

Looking at calendars throughout history is an interesting way to explore multiple perspectives. Nothing is as ingrained in our lives as seven day weeks, twelve months, and 365 days per year. Challenge your students’ perceptions by investigating various solar, lunar, and lunisolar calendars.

Solar vs Lunar

Begin by exploring the advantages of a solar calendar and a lunar calendar.

A solar year occurs once every 365 days, 5 hours 49 minutes 19 second (or about 365.242 days). The Gregorian calendar, the internationally accepted calendar, follows a solar year and your students should be familiar with the Earth’s annual trip around the sun.

Meanwhile, a lunar calendar follows lunar cycles, which average 29.530589 days (thank you Wikipedia). Twelve of these cycles result in a lunar year that’s about 354.37 days long.

The Islamic calendar is a pure lunar calendar of twelve months.

Lunisolar

However, very few cultures have used a true lunar calendar, instead opting for a combo lunisolar calendar.

For example, the Egyptians followed a calendar of twelve lunar months, each with thirty days. Since this only equals 360 days, they added five more days at the end to line up with the solar year.

Adding extra days to connect lunar and solar calendars is common across many cultures. These days are known as intercalary days.

The Aztecs had a calendar which lasted for eighteen, twenty day periods, plus five extra days known as “nameless days” or nemontemi.

Non-Standard Years

Not everyone used a 365 day year. Various Mesoamerican cultures used a 260-day “ritual” calendar that had no relation to the sun, moon, or seasons.

Weeks and Days

We take our seven day week for granted, but there have been a variety of week lengths throughout history.

The Egyptians used a ten day week. There were exactly three weeks in their thirty day months. The French calendar also used ten day weeks during the French Revolution.

China also had periods where they used ten day weeks.

The Romans used an eight day week, with days named A though H.

The Soviet Union went with five day weeks, labeled with colors or Roman numerals. This only lasted a couple years before they moved onto six day weeks, and then, eventually, the seven day week.

Create

So what on earth can we do with all of this information?

This might fit into a calendar-creation activity. I have my students design their own civilizations, to connect with our study of ancient civilizations.

I would ask them these questions about their civilization’s calendar system:

1. How many months will you have?
2. How many days will be in each month?
3. Will this equal 365 exactly, or will you use intercalery days?
4. What will your months be named?
5. What will your days be called?
6. Explain the the significance of the names of your months and days.

Going Further

The various calendar systems are fascinating and could turn into an independent study for interested students.

Social Studies

• Study the origins of the Gregorian calendar.
• Explore the transition from the Julian to Gregorian calendar.
• Examine the popularity of certain calendars in geographical regions.
• Investigate the influence of various religions on calendars.
• Digging into the month’s names opens up interesting avenues, and connects with Ancient Roman history (Simple Wikipedia is a student-friendly resource for this).
• Learn other language’s names for the days and months.

Math

• Investigate the combinations of months and days that equal 365. For example, ten 36 day months gets us to 360.
• Add in weeks to the previous idea and try to make everything a nice, round number. How many options are there?
• Determine how many 354 day lunar years it takes to line back up with 365 day solar years. How many 260 day years?

Science

• Develop calendars for different planets in the solar system (Jupiter’s solar year consists of 10,475.8 Jovian days).
• Explore the seasons’ impact on calendars (Rome considered winter a monthless period!)

Please let me know if you come up with a great idea using calendars.

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