Carnival of Probability

This past week, my awesome partner, Hannah, and I hosted a Carnival of Probability for our 7th graders in our school's multi-purpose room. Let's get to it:

Station 1: Spinner 1

Pick a side. Will the spinner (arrow) land on the 5 or the 15? If you choose wisely, you win that many tickets.

Station 2: Spinner 2

No need to pick a section. You get three spins. Land on the 1, you get one ticket. Land on the 100, you win 100 tickets. BAM!

Station 3: Rolling a die

Pick a game. Grab the six-sided die and roll. 

• Roll an even number, win 8 tickets.
• Roll a multiple of 3, win 15 tickets.
• Roll a one, win 30 tickets. 

Station 4: Rolling two dice

Pick a game. Grab two dice and roll. 

• Roll two sixes, win 45 tickets.
• Roll an even number and a five, win 45 tickets.
• Roll two multiples of three, win 32 tickets.
• *Roll the same number on our twelve-sided dice (pictured below), win 70 tickets.

See the blue 12-sided die? There's a 12-sided die inside. Cool, right?!

Station 5: Bag of letters

Pick a game. Reach inside the bag of 26 chips chips (pictured above) and pick one chip. 

• Pick a vowel, win 25 tickets.
• Pick a consonant, win 21 tickets.
• Pick the first initial of your first name, win 40 tickets.

Station 6: Ball toss

Take a ball. Toss it in the direction of the cups. If it goes in any white cup, win 3 tickets. If it lands in the red cup, win 35 tickets. Just like that!

Station 7: Deck of cards

Decide on a game. Pick a card or two. Win tickets!

• Pick a red or black card, win 1 ticket.
• Pick a red card and a queen, win 40 tickets.
• Pick a Jack, Queen, or King and win 22 tickets.

Station 8: Coin toss

• Flip it once and land on heads, you win 8 tickets.
• Flip it twice and land on heads both time, you win 20 tickets.

You might ask, "Did you really pass out all those tickets?"

The simple answer is, "No!" You don't think we're that crazy, do you? We assigned one to two students as captains for each game. The game captains gave each contestant one ticket with the number won written on it. 

The math: we spent a few days leading up to the carnival talking about probability and identifying "and" statements along with "or" statements. Station 4 was all "and" probabilities which made your chances of winning more challenging. The 12-sided dice game had a less than one percent chance of winning. Station 7 and 8 had a couple of "or" probabilities. Our goal here was for students to have a concrete introduction and application of probability. To enter the carnival, students had to represent the probability of each game as a fraction, decimal, and percent. 

The students had a blast. It was fascinating to hear from them about the carnival the following day. They explained which games were the "easiest" and "hardest." The games (as you can see from the pictures) were nothing fancy, but they did the job. Students will get to cash in their tickets for prizes such as candies, pencils, pens, and Expo Dry Erase Markers. What would your carnival look like? Please share.  Head over here for the paperwork/handouts we used. 

Carni,

633

180 Ways to Use Estimation 180

New challenge: Find 180 ways to use Estimation 180. Wait a minute. That's seems a little extreme, yet I still love the idea. Here's one use I came up with this week: inequalities.  Click on any picture to enlarge.

Me: Hey guys, here's a piece of paper. Fold it into fourths for me like this.

Me: Great. Now, who remembers how tall I am?

I know, silly question, right? If these guys don't know my height by now, I'll send them back a few grades. Then I show them this picture and ask: What's the height of Mrs. Stadel compared to Mr. Stadel?

I take about 4-6 guesses and write them on the whiteboard. It’s been awhile since we’ve done this specific estimation challenge in class. Many have forgotten my wife’s height. Great! However, it’s quite obvious her height is less than mine.

Me: Let me step back. Let’s all look at these guesses. Is it safe to say that all of your guesses put her height less than mine?

Class: Yes.

Me: Okay, so before I reveal the answer let’s put this in our notes for today.

Me: Before I reveal the answer, who can remind me why we used an open circle?

Student: Because we know her height is not the same as yours.

Reveal Mrs. Stadel's height. Wait for it... Wait for it... Here it comes... Student responses: 

“Yes, I was right!” 

“Ohh, I was close.”

“Ohh, I was off by an inch.”

Next up, our beloved Mr. Meyer.

“Woah! He’s tall!”

“Someone is actually taller than you, Mr. Stadel?”

Again, toss 4-6 guesses up on the whiteboard. Step back. What do we notice? Yes, all the guesses should be greater than my height.

Walk through the notes on this new section with more student confidence and participation.

Ask students:

• What type of circle should we use this time?
• If we’re talking about guesses that are greater than my height, how will that affect the inequality symbol?
• What are ways to remember this inequality as greater than?
• Which direction will we shade now?
• Someone give me a variable we can use for Mr. Meyer's height.

Reveal answer. Same responses as my wife, but usually a different kid was right this time.

Okay, great. Now what? What about those other two inequalities, right?

Me: Before I show you this next picture, last year the 6^th grade English teacher (@mrkubasek) at my old school read this novel with his students and came across these two books he found interesting. I found it interesting too and took a picture so we could talk about it in math.

Show picture.

Me: How many pages in the book on the LEFT?

No need to write anything down. Just get 6-8 guesses out loud. Don’t spend much time here. Reveal the answer. Give some math love [one clap on three: 1, 2, 3, CLAP!] to the closest student. Seriously, it’s pure gut instinct here people.  

NOW, show this picture and ask how many pages in the book on the RIGHT.

They don't know it yet, but you just broke their brains for a bit. Yup, you’ll get a lot of guesses below 307. But wait for it. I guarantee in a class of 35-40 students like mine, one student will say 307. If you do, treat it like every other guess you've gotten.

...and if no one guesses 307, step back after about 6-8 guesses and...

Me: You know all of your guesses make sense to me. I'm curious though guys. This is the same novel here, right? What if? I mean, WHAT IF? What if these books had the same amount of pages? Do you think that's possible? Do I have your permission to add it to your guesses?

Step back again. Look at all those beautiful guesses. 

Me: So if I'm looking at this right, you guys think the number of pages could be equal to 307, or could be less than 307? I wonder how we could represent this mathematically?

BAM! Focus here on the circle. Why are we shading it in this time? What's up with that line under the less than inequality?

Me: Okay, before I reveal the answer, someone remind me why we shaded in the circle. 

Reveal!

Brains broken! Now repair. 

Me: What's up with that? Anyone have any ideas/theories why they have the same amount of pages?

Alright. That fourth and final inequality. Here are the goods. Repeat all the other moves from above. 

Initial guess of one bar.

Take some guesses out loud. Reveal the answer.

Toss up new picture.

Write down some guesses on the board. Play up the "what if" again. Complete the notes. Ask a few questions before revealing the answer.

BAM!

That was fun. Boy, I wish it didn't have to end. But it did. Okay, let's find other ways to use Estimation 180 in the curriculum. The full lesson will soon be is available on the lessons page at Estimation 180.

180 ways,

957

Presentations & Workshops

Last month I added the Lessons page to Estimation 180 so you can quickly access lessons I've made. I will continue to add lessons as I make them and host them in that space.

This month, I'm adding a Presentations & Workshops page to the site. Since last November, I've been fortunate to work with some amazing math teachers at conferences and workshops. I've learned a lot and have truly enjoyed doing math with teachers as we share instructional strategies and lessons. My goal is to help support math teachers in strengthening their instructional tool belt for the Common Core classroom. 

I'm excited about this new chapter. Drop me a line if you're interested.

PD,
945

Explain that, please.

Recently, I've given a few teacher workshops/conferences and have had the luxury of reflecting on teacher moves as I facilitate a lesson with the attendees. One of the many things we talk about are teacher responses to students.

Me: Did anyone hear me say, "No. That's wrong. You're wrong. I don't like your answer."

Attendees: No.

Me: Right. Instead, you'll hear me say things like, "Can you explain what you did here? Explain that, please. I noticed you did [this] here, please share how you got [that]. I'm curious how you came up with that. Walk me through what you did."  

I tell teachers that I'm taking the emphasis away from right versus wrong answers and placing an interest on the student's thought process and problem-solving. I continue with teachers:

Me: By telling a student they're wrong, a student can have the tendency to shutdown [I make the sound effect of a machine shutting down, "BOOOOvvvvvvvv"]. By asking a student to explain things, it shows that I'm more interested in how they arrived at their answer. 

As teachers, we know a student can be told they're wrong and it's easy for them to give up. On the flip side, when we validate a kid by telling them they're right, the student can also shut down and never reach the higher levels of Depth of Knowledge.

Recently, a workshop attendee asked me how I respond to students who have nailed the answer to a 3 Act task. First, I have them explain their problem-solving plan to me. Second, I question any details that were unclear, encourage them to be more precise, or have them explain their units of measurement. Third, I ask them if they feel confident in their answer after explaining it to me. Fourth, I validate them by simply saying, "That makes sense to me."

I don't tell them they're correct. I treat them just like as if they got the answer wrong. If that doesn't satisfy them, I respond with, "We'll find out soon if you're correct, but that (their explanation and work) makes sense to me." At this point, I offer them an extension to the task. I'd like to talk more about this later, but usually the extension revolves around the students creating something with the new knowledge or skills they have just recently gained.

After all that, please add your favorite lines when questioning students to this Google doc. I think it's also helpful we create a list of lines we avoid using with students as they explore math.

Here are a few people with other stellar teacher moves/lines to support students.
Max Ray: 26 Questions You Can Ask Instead
Dan Meyer: You Don't Have To Be The Answer Key
David Cox: Creating A Culture Of Questions
Steve Leinwand: Accessible Mathematics

BOOOOvvvvvvvv,
645

Estimation 180 has Lessons!

Head over to Estimation 180 and you'll see this lovely new option in the menu bar.

LESSONS!

That's right! 

LESSONS!

I've added a "Lessons" page with many lessons I've created, sorting them by their CCSS. I'd like to thank Dan Meyer and Robert Kaplinsky for their friendly suggestions (nudging) to tag my lessons in an attempt to make it easier for other teachers to find and use. Plus, I'm tired of my lessons collecting digital dust and hope that teachers can find and use them.

I was honored to give a workshop for teachers in my district today. The workshop became the motivating factor for making this Lessons page. Right now, most of the lessons are 3 Act lessons that can be found at Dan's 101qs.com A few other lessons are ones I've blogged about. However, I have added two test pages at Estimation 180 where the entire lesson is available for teachers to use. Right now. At Estimation 180.

Pay close attention to my File Cabinet and Stacking Cups lesson PAGES!.

These two full-on lessons are ready for you and your students. You'll see all three acts, teacher notes, student work, student handout (if you like/need), and downloadable videos. Let me know if you have any thoughts, advice, or questions.

I hope this "Lessons" page is useful and/or better than that silly unorganized spreadsheet I've got lingering. You'll notice a few links are under construction, but many links deliver the goods. Check in often for updates.

LESSONS!
1023

P.S. Thanks to Fawn, Nathan, Robert, and Eric for your feedback.

Your Eyes Are Amazing

This Centrum television commercial caught my ear for a few reasons. I tracked it down on the Internet tonight and edited it for Act 1. You can find the entire lesson here at 101qs.com. It's a quick little lesson for Math 6 (6.RP.3d).

Act 1:

Question: How many football fields is 10 miles?

Act 2:
I'm not giving much information for Act 2 as I'm leaving this part of the mathematical modeling process up to the teachers and the students (mainly students). I think there's an essential part to the classroom discussion and I hint at it with the following questions (if necessary) left in the teacher notes:

• Ask students, "What information would be helpful here?"
• Ask students, "How are football fields measured and with what unit of measurement?"
• Allow your students to decide the length of a football field.

I'd like you to do the math right now. Go ahead. I'll wait. It won't take you long.

10 miles. How many football fields is that?

Act 3:

Wait!
Timeout!
Is this commercial's math wrong???

Should it be 176 football fields or 146 football fields?

What did you use as your football field length? Did you use 100 yards? Did you account for the end-zones being 10 yards each, making the total length of the football field 120 yards?

On a related note, I'm a little surprised the Centrum didn't use 100 yards so they could claim 176 football fields for a more dramatical pitch in their commercial. I also think it's fun to talk about what it would take for human eyes to actually see that candle 10 miles away. Darkety-dark-dark probably. No light pollution. No obstructions. Maybe a desert? No bright moon (which the commercial includes for some weird reason).

I'll be using this with my sixth graders this year when we get to conversions. It's a fun little task. Let me know if you have anything to add.

Candle Eyes,
1059

Fun With A Dot and A Line

Who would have thought a dot and a line would be so much fun?

[Update] Fun With A... series
Fun With A Sticky
Fun With A Name Tent

I gave my 6th grade students a pre-assessment a week ago Monday. They bombed on questions like this:

Here's today's launch (Round 1):
Me: We're going to have a little competition. Who can draw the best reflection of this point across this line in the middle of your paper?
I handed each student a paper with this at the top.

My kids we're doing some cool things as they attempted to reflect the given point across the middle vertical line. I'll recreate some of them for you.

Julio used a long pencil to line up the point and measured the distance from the point to the vertical line so he could put a point equidistant on the other side.

Jason measured in from the edges of the paper.

Silvana folded her paper down the vertical line and did something on the back.

I asked each group (of four) to pick what they considered the best "reflection" from their group and bring it up to the document camera. We first eliminated some contestants by eye-balling their point and narrowed it down to 5 reflections. I said,
"These all look pretty good, but I feel there's gotta be a more accurate way to determine who has the best reflection here. I need your help guys. How do you think we can determine the winner?"

They thought...

Student: "We can fold the paper over and see whose dot lines up with the first dot."

I try that, but they quickly see I have trouble making a good (accurate) fold.

Student: "Can we measure how far the point is?"

I ask: "What do you mean? Can you please explain?"

Student: "Like how far is each point from the line?"

Me: "Which line?"

Student: "The one in the middle that goes like this. [holds up arm in a vertical manner]

Me: "Let's do it!"

I grab my trusty blue stencil and line up the original. Students watch me measure the original point. 7 centimeters.

Me: Okay, looks like our winner has to be the closest to 7 centimeters. Let's find out.

We get down to two contestants. Anthony has 6.4 centimeters and Stephanie has 6.5 centimeters. Stephanie edged him out by 0.1 centimeters. However, I noticed his point was better aligned with the original... so I threw that out to the class. They settled for a tie.

Round 2
Okay, who can now draw the best reflection of the original point across the horizontal line?
Same rules: pick the best reflection from your group, but it can't be the same person as in Round 1. Our target: 3.2 centimeters.

For a long time, we had a tie between Miguel and Luis. Miguel's point was 3.0 centimeters from the line and Luis' point was 3.4 centimeters from the line. Then, here it came, the last contestant. Jason hands me his paper and I measure it to be 3.1 centimeters. OUR WINNER! Jason is our winner!

Queue the direct instruction and mathematical vocabulary. It became really annoying to keep saying this line and this line. We have already talked about the x-axis and y-axis, so it was easy to convince my students to use those terms. We went into some practice questions that looked like the first picture in this post:
I do, we do, you do!

And now we end class with our final competition: a double reflection. I'd like you to reflect it first across the y-axis and then across the x-axis. Who can draw the best double-reflection?

Fun with a dot and a line. That's an understatement. I think we all had A LOT of fun. Who would've thought?

I'm finding more and more success with these types of lessons. I've been trying to design lessons that have a simple visual, ask a simple question, are geared toward some type of competition and/or game, require students to keep each other accountable, students are checking the answers with me as opposed to me telling them the answers, and fun. I'm trying to keep a simple checklist going. How's this for a start? Anything else to add?

Dot,
515 +1 dot

Little Mr. Sunshine

I'm honored to receive some rays from the sunshine of Shauna Hedgepeth and John Stevens. Like Christopher Danielson, I don't forward those silly emails of inspirational PowerPoints, cats in meadows, Bigfoot sightings, and other pyramid emails. However, I am compelled to do my part (or some of it) and respond to the two math tweeps who threw sunshine my way. If you're not familiar with sunshine, click on any of the three links above and read their posts so I don't have to retype it.

Act 1: Acknowledge the nominating blogger(s)
Hedge is totally awesome! She loves her students. Don't even argue that. Email her some thoughts, ideas, and/or jokes and her replies have a sincere, lovable vibe to them. She makes me laugh with all her A.D.D. talk, but I'm not sure how true that is. Wait, yes, I do. Just read one of her blog posts. They're awesomely funny. Where she might be intimidated by my height, I think I would be equally intimidated (if not more) by her love and knowledge of stats. Stats is definitely one of my shortcomings as a math person. I look forward to meeting her at NCSM in April 2014.

John Stevens reminds me of a professional juggler. Not necessarily a clown juggler. However, recently he did dress up in a chef's outfit during a presentation. This guy juggles and contributes to many cool parts of math education.  Check out his blog for iPad resources, his CUE Rockstar status, presentations, #CAedchat and his up and coming Would You Rather site. When you meet him in person, he's a very personable and interested guy. Sit down and have a beer with him. Keep rocking John!

Act 2: Eleven Facts
1) I've never broken a bone (knock on wood). Never.
2) I don't drink coffee. However, I love iced tea.
3) I'm a philosophy major.
4) I've only used my passport once and that was to Canada. I hope to travel to other countries in the future.
5) I enjoy doing yard work.
6) I can't remember the last time I've slept passed 6:30 a.m.
7) My iTunes library would take 36 days, 22 hours, 55 minutes, and 5 seconds to play from start to finish.
8) I'm actually terrible at estimation (just kidding).
9) My In-N-Out order is:
1 Protein-style Cheeseburger with grilled onions
1 Animal-style Double-Double
1 light-well french fries
1water cup (with water in it)
10) I love/have the board game Othello, but I have no one to play against.
11) Newcastle and Rogue

Act 3: Answer questions from the people in Act 1
Hedge's Q's:
1) If you had to pick one area/concept of math that is your "jam", what would it be?
Spatial reasoning.

2) True, but there's at least one student that sticks out in my mind that I feel I failed. Do you have one?
Yes. Actually, many students from my 2012-2013 school year. See this recent post for a visual. I went overboard with too many questions, not enough validation and too little direct instruction. Avoid this cocktail.

3) Twenty years from now, what's something kids will probably remember about you (phrase, moment, habit, characteristic, etc.)?
That guy sure took a lot of pictures standing in front of stuff so we could estimate its height. He worked hard and expected us to work hard.

4) What's something that you'd like to "fix" about yourself in your current job?
I'd like to learn more Spanish.

5) Name a movie title that describes you and why.
Tommy Boy. I don't need to explain.

6) Which tweep would you love to have a conversation with over a beverage?
My top 3 tweeps I haven't done this with are Hedge, Christopher Danielson, and Steve Leinwand. By the way, just one beverage? What kind of nonsense is that? I'm excited as it looks like I'll meet Hedge and Triangleman at NCSM this year! WOOOOOHOOOOO!

7) If you couldn't teach your specific subject, what else would you teach?
Home economics and culinary. I'd learn how to be a better cook and I could eat.

8) Everybody has a song they can dance/jam out to. What's yours?
Venice Queen by the Red Hot Chili Peppers.

9) Who would you love to see karaoke at TMC14 and why?
Steve Leinwand. First he, wouldn't need a microphone. Second, he can probably rap faster than any rap artist, alive or dead. Third, what jam would he pick? Definitely NOT a ballad.

10) What's one thing (item, app, software, etc.) that you love so much that you can't imagine doing your job without it?
Apple's Keynote or Pages. Don't even ask me to use PowerPoint.

11) If you could job shadow one tweep for a week, who would it be and why?
Michael Pershan. Having met Michael and been intimidated by his fountain of intellectual curiosity, I'd love to see how he facilitates a classroom. Runner-up: Fawn Nguyen. Maybe I could steal her Brad Fulton books when she's not looking.

John's Q's:
1) Why do you teach?
Am I supposed to be teaching? I'm still learning and I expect my students to learn with me. This might help explain.

2) If you didn't teach, what would you do for a living instead?
Playing music in crappy bars while making guitars at a guitar factory.

3) Money being no obstacle, where would you like to visit? Why?
Europe. Most of it. See my passport comment above.

4) Kids always ask who your favorite student is.  Describe the characteristics of yours.
My favorite student is the one who perseveres, is open to making mistakes, takes responsibility along with the risks, and can see that math isn't about printed number on a paper.

5) What is your favorite board game and why?
See fact #10 above.

6) What is the most frustrating component of education right now?
Grades.

7) Would you rather buy a Mac or a PC?
See the answer to Hedge's question #10. Mac!
John, you had to sneak in a "Would you rather?"

8) What is your favorite book?
Early John Grisham stuff.

9) If you had to choose blogging with no way to share it (ex. via twitter) or tweeting with no way to elaborate (ex. via a blog), which would you choose?
Blogging.

10) Who is your hero?  Why?
Wow! Too many to name. My grandfather (hero for being a role model of a man). My sister (a family and best friend hero). Dan Meyer (a math hero who saved me from ruining my teaching career).

11) What is the most exciting part about your job?
I get to make mistakes. I get to create my own lessons. I get to explore number sense and logic with non-adults. My job doesn't define who I am.

The Sunshine stops here. Thanks for reading. I'm not torturing 11 other tweeps. It was fun.

Sunshine,
243

Being the Answer Key (or not)

After reading through the Pimm Quotes that Dan selected, some type of bittersweet emotion about teachers being the answer key was rekindled within me. I left a few comments/questions and I appreciate Dan's timely and thoughtful responses.

For the following questions, I'll define "yourself" to include you, your students, and your classroom culture.

• Where would you place yourself today?
• Where would you place yourself at the beginning of this year?
• Where would you place yourself last year? 
• Where would you place yourself during your first year?
• Where would you like to be placed at the end of this year? 
• Where would you like to be next year?

*I'll share mine at the end.

There's way more to talk about here. I did not fully capture the essence of Pimm's quote, Dan's response, nor my own thoughts. I just wanted to toss this around as is.

I challenge you to blog about this too.

• Reflect. 
• Label your axes how you want. 
• Should it just be two axes? 
• Either way, create/add some type of visual.

Stadel this school year:

Stadel last school year (I went overboard and it was not enjoyable for anyone.):

AVOID quadrant IV

Stadel in the past (pre #MTBoS):

Stadel as a rookie:

Answer key,

609

Daily Something [WCYDWT]

#WCYDWT

What can you do with this?

Why would a teacher use this?

How would you use this in your class?

What would you add? subtract? replace? other?

How would you use the information from student performance on this?

How could this be used as a pre-assessment? an assessment? an intervention?

Take a few minutes to complete each day. What do you notice?

Hint:
*  **  ***  ****  *****  ******  *******  ********  *********

Answer as many or as few of these questions... or feel free to add your own.

This:

Thanks in advance!
843

Piles of Tiles

I was at my parents' house for Christmas and came across this game (older than me) in a closet full of board games. Made by The Cootie Company back in the 70's, I give you Op Tile.

There's a lot going on here; game boards, tiles, dice, cards, bell-bottoms, shag carpeting. Instead of typing up the directions, amuse yourself with these:

The tiles look like plastic jello. 

The cards offer some opportunities for strategy throughout the game.

There are many things I like about this game, even though I have never played it. I like the spatial reasoning component, the challenge of placing tiles depending on what you roll with the dice (and the order in which you have to place them), and the demand for strategy that the cards present. I didn't like reading through all the directions to discover all the nuances. Some parts of the game are not intuitive. However, I really like what is intuitive: placing the tiles on your game board in the best way possible to cover the most area (square units) of the game board earning the most amount of points.

I brought the box home to create some adaptations for my students. Here's phase 1: Piles of Tiles. Having recently blogged about weekly POPS, Piles of Tiles will become an additional option for the first P (Patterns/Puzzles) in POPS. In phase 1, I'd like my students to play around with the Piles of Tiles puzzle like this:

I'll pass out this sheet (maybe two) to students and have them cut out all of the figures. They can keep their cut-outs in a plastic sandwich bag.

The student game board will look like this. 

Students can use their cut-outs to fill the 12x12 board with the specific tiles, found in the table at the bottom of the page. Once they have their solution, they can outline each figure within the game board and specify which section refers to figure A, B, C, and so on. They can use colored pencil or crayons to keep each similar figure the same color. For you detailed people, I made the grid so it prints each square unit as 1 cm by 1 cm. Therefore, you get a total of 144 square centimeters. AWESOME!

Students can stay organized and also submit the following to me:

All the Piles of Tiles goods (blank templates) can be found in my weekly POPS folder.

My goals (right now) are to get students to:

• manipulate shapes through rotations and translations
• build their spatial reasoning
• recognize there are multiple solutions
• organize their data
• have a better understanding of area

I'm open to suggestions or feedback, so please let me know.

Since we're on the topic of puzzles, ThinkFun has some great puzzles (as I've mentioned before). Go over to their site and check out these Big Games group activities to use with students. Many of the activities are puzzles you might have seen on paper somewhere, but they rewrote them as group activities to foster collaboration, spatial reasoning, and problem-solving amongst students.

Piles of Tiles,
1032

J.T.A. my F.I.L.

Today I experienced a bittersweet-simultaneous-challenging-weakening-strengthening perspective on the following:

• There are some really amazing, big-hearted, significantly good-for-humanity people that come into your life (and the lives of others): make the most of the time you have with them.
• There can be a blurry (sometimes blinding) difference between how remarkable somebody was, and if given more time, what could have been. Don't let the latter blur the significance of the former.
• Life is more enjoyable celebrating it.

R.I.P.  J.T.A