Posted by: rdkpickle | 04.08.2017

log properties

Just wrapped up a unit on logarithms in Algebra 2 Honors. (p.s. It’s Friday and I’m officially on spring break!!!) [note: I didn’t finish this post when I started it, and now it’s Saturday and my break is officially almost over!!!!] In teaching logarithms for what is maybe the 8th year in a row, I managed to stumble on a different way to talk about the log properties that was like, okay, duh.

Let me preface this by saying I do think the way we talk about mathematics out loud can create or reinforce misconceptions for students about how math works. The best example I can point to is using “cancel” as an all-purpose term that students interpret as “these things disappear!” rather than taking the time and care to point out when inverse operations undo each other (“divides to one” or “adds to zero”). [See Nix the Tricks for more]

I feel the same way about sloppily using the word “distribute” when I’m not talking about the distributive property – especially given how quickly students make the mistake of trying to “distribute” operations. (i.e. log(X+Y) = log(X) + log(Y))

So, this year when I started talking about log properties, I was more careful about the way I spoke. After first noticing and then proving the properties by making the connection to exponent properties, I fell into a habit of describing them the same way I describe the exponent ones. That is, if we expanded log(XY) to log(X) + log(Y), I’d say “because when you multiply two powers, you add their exponents.” Ditto log(X/Y) = log(X) – log(Y) (“when you divide two powers, you subtract the exponents”) and log(A^C) = C*log(A) (“when you raise a power to a power, you multiply the exponents”).

Might seem like a small shift, but I caught myself a time or two reverting back to really imprecise language like “so the multiplication here becomes addition of the two logs” and realized how irritating and confusing this sounds. What’s happening with these log properties isn’t a magical swap of one operation for another, and these aren’t algebraic rules that students should feel like they are memorizing. Instead, these properties are a natural extension of ideas the students are really familiar and comfortable with – exponents.

Posted by: rdkpickle | 03.23.2017

today, the future

Got to do some fun things in class today. Credit to Jamie for curating these (and every other element of the awesome, so fun and challenging A2H curriculum):

Screen Shot 2017-03-22 at 10.42.26 PM.png

Screen Shot 2017-03-22 at 10.25.45 PM.png

In addition to the mathematical challenges the day posed, I felt challenged by the time of year and all of the time-sensitive, human needs I ought to be attending to as we send home feedback in the form of grades and comments to our students as well as recommendations for courses for next year.

The following excerpt from this article, which I read to prep for a Strategic Plan working group meeting last week, felt especially timely:

The same principle applies: Do whatever you can to take instinct out of consideration and rely on hard data. That means, for instance, basing promotions on someone’s objectively measured performance rather than the boss’s feeling about them. That seems obvious, but it’s still surprisingly rare. Be careful about the data you use, however. Using the wrong data can be as bad as using no data. Let me give you an example. Many managers ask their reports to do self-evaluations, which they then use as part of their performance appraisal. But if employees differ in how self-confident they are—in how comfortable they are with bragging—this will bias the manager’s evaluations. The more self-promoting ones will give themselves better ratings. There’s a lot of research on the anchoring effect, which shows that we can’t help but be influenced by numbers thrown at us, whether in negotiations or performance appraisals. So if managers see inflated ratings on a self-evaluation, they tend to unconsciously adjust their appraisal up a bit. Likewise, poorer self-appraisals, even if they’re inaccurate, skew managers’ ratings downward. This is a real problem, because there are clear gender (and also cross-cultural) differences in self-confidence. To put it bluntly, men tend to be more overconfident than women—more likely to sing their own praises. One meta-analysis involving nearly 100 independent samples found that men perceived themselves as significantly more effective leaders than women did when, actually, they were rated by others as significantly less effective. Women, on the other hand, are more likely to underestimate their capabilities. For example, in studies, they underestimate how good they are at math and think they need to be better than they are to succeed in higher-level math courses. And female students are more likely than male students to drop courses in which their grades don’t meet their own expectations. The point is, do not share self-evaluations with managers before they have made up their minds. They’re likely to be skewed, and I don’t know of any evidence that having people share self-ratings yields any benefits for employees or their organizations.

[emphasis mine]

Designing a Bias Free Organization, Harvard Business Review

I want my students to know I see them and value the ways they give themselves to our mathematical endeavors each day. I worry about recommendations (Honors, not Honors) and the messages they send to students, intended or not. I worry about the different ways in which different student groups receive these messages, based on internalized oppression and a lifetime of being told they probably won’t measure up.

My day ended with two other colleagues – sharing, venting, laughing, and making ourselves wholly vulnerable over drinks and dinner. Our upcoming spring break will be energizing and so needed. The work will continue, soon enough.

Posted by: rdkpickle | 03.13.2017

favorite problems 3 – dartboard

[previously: favorite problems 1 – pirates, favorite problems 2 – triangles]

This one came from the “#MTBoS,” before it was named such a thing.

These were early days, when I was still kinda geeking out over the fact that I was tweeting with k8, and Dan, and Sam. There was this energy about the whole thing for me – I’m a competitive person, and realizing there was this group of whip-smart, motivated, brilliant teachers to learn from and look up to – I wanted to impress, to be impressive.

I’m pretty sure I could use my internet skills to figure out who first posed the square dartboard problem, but it’s a Monday night with a long week ahead and you’ll forgive my delinquency, here. What’s more important is what I haven’t forgotten – that I spent a long Greyhound bus ride to New York City sketching notes in a small Moleskine notebook (the same one I used to sketch tiny maps of Metro stops + the cross streets I’d find upon riding the escalator up up up and emerging onto the D.C. sidewalk). The problem posed was a challenge. I wanted to be in with the in crowd, wanted to solve the problem and share that I’d done so. Brain buzzing, headphones on, I jotted down my ideas, ignoring the seatmate eyeing my mathematical scribbles with furtive, skeptical glances.

The question: “There is a square dartboard, and a dart has equal probability to land anywhere on the board. What is the probability that the dart will land closer to the center than to the edges?”

It wasn’t until the ride back that I solved the thing. I don’t think I tweeted about it. I did, however, save the work. It was posted proudly in the “cubes” at my first school, traveled with me to Nashville to get laminated and hung near my desk. Still have it, somewhere. Years and miles separate me from the girl who experienced this problem as a test of her authenticity as a math nerd, who saw a solution as signal that I was worth being invited to the table.

Still something kinda magical about it, though.

So. Scrolling down invites spoilers.



[Worth mentioning – I posed this problem to students, years ago. Kids I didn’t even teach, who hung out in my room to talk math and after an hour or so filled my whiteboard with the work below. None of us back down from a challenge. All of us want to feel like we earned our way to the table.]


[[edit: @dandersod points me to the original question, as posed by infinigons. his work: here. “you were the nerds i wanted to accept me!”]]

Posted by: rdkpickle | 01.02.2017


Some MA students made an awesome documentary about teachers (and i’m in it). I should probably post this here!

Posted by: rdkpickle | 01.02.2017

slow down, cont’d

Some more words from other people about social media, multitasking, slowing down;

“At saturation level, social media reduces the amount of time people spend in uninterrupted solitude, the time when people can excavate and process their internal states. It encourages social multitasking: You’re with the people you’re with, but you’re also monitoring the six billion other people who might be communicating something more interesting from far away. It flattens the range of emotional experiences.


When we’re addicted to online life, every moment is fun and diverting, but the whole thing is profoundly unsatisfying. I guess a modern version of heroism is regaining control of social impulses, saying no to a thousand shallow contacts for the sake of a few daring plunges.”

Intimacy for the Avoidant“- David Brooks

And more:

“A restlessness has seized hold of many of us, a sense that we should be doing something else, no matter what we are doing, or doing at least two things at once, or going to check some other medium. It’s an anxiety about keeping up, about not being left out or getting behind.

I think it is for a quality of time we no longer have, and that is hard to name and harder to imagine reclaiming. My time does not come in large, focused blocks, but in fragments and shards. The fault is my own, arguably, but it’s yours too — it’s the fault of everyone I know who rarely finds herself or himself with uninterrupted hours. We’re shattered. We’re breaking up.

It’s hard, now, to be with someone else wholly, uninterruptedly, and it’s hard to be truly alone. The fine art of doing nothing in particular, also known as thinking, or musing, or introspection, or simply moments of being, was part of what happened when you walked from here to there, alone, or stared out the train window, or contemplated the road, but the new technologies have flooded those open spaces. Space for free thought is routinely regarded as a void and filled up with sounds and distractions.”

The Encyclopedia of Trouble and Spaciousness – Rebecca Solnit

I felt a bit of discomfort with the uninterrupted time afforded me over the recent break, never seeming to make use of it the way I’d like. On the other hand, when school is in session, I am constantly fighting the feeling of fragmented multitasking in response to a busy schedule. Do you feel this tension around being fully present in your work and relationships? Do your students? What seems to work for you to reclaim space for daring plunges?

Posted by: rdkpickle | 10.06.2016

slow down.

Was going to make some lists of where I’ve been and what’s on deck. Typed and deleted a few things.

It’s hard to be fully present. I was going to add a lot of qualifiers to that sentence like “it’s hard to be fully present when an inbox full of emails demands attention” and “it’s hard to be fully present when your phone promises an easy distraction to even a moment of boredom” and “it’s hard to be fully present when your mental to-do lists are shuffling out your ears and littering the lawn” but I needn’t bother. It’s hard to be fully present. Period.

Last week at school a speaker came to talk to the students about smartphones and the effect they are having on teenagers’ development of social and emotional intelligence. Her talk really stuck with me, as I thought about all of the ways I distract myself from what’s around me, all of the ways I disengage with the challenge of being truly present, and all of the ways the things that I think help me stay connected (twitter, facebook, instagram, email, texts, etc.) — also harm me. To that end, I’ve tried to be more intentional about when and where I carry my cell phone with me, how many times in a work day I check my email (I never leave it “open.” I respond quickly once I’ve read a message, but I designate times to read and respond), and how much screen time I allow myself in the evening before bed.

I checked twitter two nights ago after a busy, packed, fantastic weekend with a dear friend, saw all of the good stuff you all have been up to, and immediately experienced so much guilt and self-doubt about how I choose to spend my time. It’s this strange, competitive (embarrassing) FOMO around public professional successes. “I’m not blogging enough!” “I’m not tweeting enough!” “I didn’t make the time to read that book/article/blog post yet.” “I’m not curating mathematical art shows.” “I didn’t apply to be a Desmos fellow.” “Dan Meyer didn’t give me a shoutout in his recent blog post.” Dumb dumb dumb dumb dumb. Dumb brain. Comparison is the thief of joy.

Here are recent successes that exist wholly outside the digital domain: I’ve slowed down. I’ve still had my share of disjointed, less-than-ideally paced classes this year, but on the whole the entire vibe in my classroom has been calm, focused, productive, positive. My interactions with students feel more deliberate, my conversations with colleagues less rushed. My classes are a story with a beginning, middle, and end (and – I am not the main character in this story.) In the lunch line, rather than checking my notifications on my smartphone, I try to find someone to talk to about how their day is going. As I walk around campus, I am comfortable. I check in with students I taught last year, and really sink into those conversations. I work hard as a team member on my shared courses, refining curriculum and executing lessons that allow students the chance to authentically engage in meaningful mathematics. I am busy, but I feel like I have a good sense of the work that is worth my time, and I put as much of my effort as I can into that work. Then, I put boundaries and parameters around the rest of it in order to preserve my mental and emotional health.

I know there will never be enough time. But the time that we have is a gift. Trying oh-so-hard to slow down, and enjoy it.





Posted by: rdkpickle | 08.27.2016

first days of a2h

A quick post!

The school year is officially underway, with one week in the books. Because we are on a block schedule, this means I’ve seen each of my classes twice.

This year, I’m teaching 1 section of Geometry, 1 section of Precalculus, and picked up 2 sections of Algebra 2 Honors. I’m super excited about my schedule, the mix of old and new, and the opportunity to teach an honors class at MA.

I kicked off my Algebra 2 Honors class with an awesome “Think of 2 Numbers” puzzle that came from the teacher who previously taught the course. He generously shared his entire (amazing) curriculum with me, and I’m so looking forward to digging into the cool problems and explorations he has refined over his 10 years with the course. Here’s the puzzle:

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Fun, right?

After playing with this puzzle long enough for the class to make a conjecture about how to figure out the 2 numbers (and putting a student on the spot to figure out MY two numbers), we did brief introductions, then moved into our next activity.

I decided to use this NRICH “Algebra Match” activity that has students silently, in groups of four, trade away cards with mathematical expressions until each group member has a matching set of 4 cards. The rules?

  • No one can talk or give non-verbal signals to other members of the team.
  • Each member of the team starts with four cards in front of them.
  • The cards in front of each person should be visible to everyone.
  • Team members can only give cards; they cannot take cards from someone else.
  • Each team member must have at least two cards in front of them at all times.

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Here is one of my classes busy at work on the activity.


After each group successfully matched the cards, I asked them to take a moment to reflect on the experience by responding to a few prompts. We discussed the first few together, and then I went home and read the answers to the last two (more personal) questions, collecting some of my favorite responses to share back to the class on the second day. Here are the questions I posed, and some of the responses I got:

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What are some of the benefits of group work in math class? What are some of the challenges?

D Block responses


(p.s. not on the board, but on a student’s paper. “Group work is challenging when people are stubborn. It’s even more challenging when stubborn people are incorrect.” lololol)

E Block responses


(p.s. not on the board, but nearly half of the students in this class wrote that a challenge with group work is “being on the same page” or “getting everyone on the same page.”)


What do YOU need to be successful during group work? From your peers? From me?

D Block responses

  • “I need space to work things out before writing down my ideas to be successful in a group. My peers would need to be willing to be wrong, but also willing to correct me if I’M wrong.”
  • “I need to stay focused and finish my stuff.”
  • “Nothing. I’m very easy-going and working in groups has always been easy for me.”
  • “I need for everyone to be engaged and trying to do the work.”
  • “Time to fully discuss with group mates. Hear out everyone’s ideas – not shut someone down.”
  • “Time to fully work through and discuss the problem.”
  • “A friendly atmosphere and teammates not significantly above or below my level. Clear directions and help if all group members agree we’re stuck.”
  • “Communication.”
  • “I need to communicate with my group mates and make sure I’m on the same page as them. I need my peers to be open to going over things and explaining problems if I need help.”
  • “Be prepared and focused, fully committed.”
  • “I need to make sure that I both listen and share ideas.”
  • “Patience and focus from all the partners. Also, not rushing if even only one person is behind.”

E Block responses

  • “People who fully listen to other ideas but also participate”
  • “I need a diverse group of learners. I need some support because I usually take on a larger workload and get stressed or overwhelmed.
  • “Good communication and time management skills (for myself). Willingness to share and collaborate (from peers).”
  • “I really like being in groups that are quiet until everyone is finished and then they discuss.”
  • “Time to understand the concept and not just follow along.”
  • “Enough time.”
  • “The time to think through problems first.”
  • “I need to be able to distribute ideas and have them heard.”
  • “To be successful, I need peers that work at about the same speed as me.”
  • “I need supportive peers who are understanding, patient, and positive. From you I need mostly the same things, support being paramount.”
  • “I need my peers to have a positive attitude, and be willing to share their ideas as well as listen to mine.”
  • “The most important thing to me in group work is that everyone is kind and respectful to everyone and their opinions and ideas.”
  • “Just no need for me to feel embarrassed if I am one of the few in the group that doesn’t get it.”
  • “Lots of communication. Arguments are okay if they’re managed. Time to work on our own (in groups.)”
  • “Group members who are open to opinions and not stubborn. Everyone should have a turn to talk.”


What individual goals do you have for yourself as a mathematician during group work this year?

D Block responses

  • “I want to let others participate (in the past, I wasn’t very willing to let others do much for fear they would mess it up) during group work this year.”
  • “Stay on track and always keep focused on finding the answer”
  • “Collaborating effectively and sharing ideas to help each other along.”
  • “I want to try to help other people.”
  • “Having more confidence in group work and share my ideas. Don’t be afraid of taking risks and making mistakes.”
  • “I want to collaborate better as well as develop my skills with algebra.”
  • “Know when to step down. Fully understand everyone’s thinking, not only my own.”
  • “I hope to actually understand the material instead of only memorizing answers.”
  • “Be helpful and humble and learn to see things in different ways.”
  • “Try to become more of a leader and take charge while solving problems instead of following what other people figure out.”
  • “I hope that I can keep up to the pace of my peers and help them out too.”

E Block responses

  • “To say more. Last year I was frequently hesitant to speak up and sometimes missed my time to shine!”
  • “Share ideas I’m unsure about/take risks.”
  • “Sometimes in group work I can get too bossy or take on too much leadership, or (depending on the class) I can also be scared to say anything, so I really want to find a happy medium for sharing my ideas.”
  • “I would like to become better at solving problems faster.”
  • “Be an active participant/leader.”
  • “I hope to grow confident as a mathematician and perform the best I can. I hope to be able to help my peers and not just be someone who waits to be told something or have something explained to me.”
  • “I want to feel more confident in my answers, and also feel more at ease with asking for help from my peers.”
  • “More confident in sharing my ideas (if I’m not sure they’re right.)”
  • “Step up when I need to and step back to get other people involved.”
  • “I want to put as much effort and work into team work as I can, and have a positive attitude.”


To be honest, I wasn’t sure how this activity or the debrief afterwards was going to go, so I was pleasantly surprised to see that the task was sufficiently challenging (mostly because of the “no talking” requirement, although it did give me a sense of which students might be a little rustier with their algebra or need to rely on scratch paper vs. working in their heads) AND that they really dug into the discussion afterwards in a way that I think helped establish some class norms for group work that will pay off throughout the year.

At the end of class, we returned to the “Think of 2 Numbers” trick, where (again, stealing Jamie’s idea) the students had to prove algebraically why the trick worked for homework.

That was all just Day 1! Day 2 was even cooler! I’ve already written quite a bit and want to get back to enjoying my Saturday (it’s a beautiful day in the Bay Area!), but here’s the problem we tackled to start class on Day 2 and OH BOY did the students take it even farther than I had expected.

This is going to a fantastic group, and a fantastic year. I’m in love.

Posted by: rdkpickle | 07.16.2016

mary had an off-task-teacher (#descon16)

With 2 trips immediately following both Descon and TMC, I know that if I don’t blog quickly with these few nuggets I’m in danger of never blogging at all. So, quickly:

Desmos has been working to make its graphing calculator fully accessible for visually impaired and blind students. As of now, if you enable voiceover on your device (command + F5 on mac), Desmos will read the expressions in a way that “sounds like math” (ex: reading sin as “sine” instead of s-i-n) and gives students clues as to where they might be in the expression (ex: subscript, superscript, reading the opening and closing of grouping symbols). It is also capable of “playing” a graph to give a pitched audio representation of whatever has been graphed.

[Predictably] I immediately set out to use the feature to write a song, ignoring all other directions about what we were “supposed to be doing.” James and I worked on it during our morning work time, with serious insight and help from Chris and Sam to get us over a few hurdles. I loved the experience of playing with something that I had no idea how it worked by experimenting (pitches compared to y-values, time compared to x-values, etc.), trying and tossing out ideas that ended up not working, and I ended up learning some new things about how to use Desmos (lists of functions, anyone?) along the way.

Here’s our final version of “Mary had a little lamb.”

james and rachel

Desmos also premiered a few new things which others have written about – a new layout on the landing page, bundles of selected+sequenced activities with notes for teachers about how each lesson builds on the preceding ones, and the “labs” option which enables teachers to use the brand new marbleslides and card sort features for activity builder lessons. The MTBoS has already set to work creating quite a few activities to test out the new features, and I can’t wait to think about how I might incorporate these in my own lessons for the year ahead.

Posted by: rdkpickle | 05.05.2016

“i did it a different way.”

Are there more exciting words?

I’m at the board after some time spent in groups working on the warmup, going over a few answers and trying to clarify big ideas. They’ve stopped their work to tune in – quiet moments with eyes on me are so much rarer these days. Some kind of magic has happened in this happy, focused A block Geometry class this year. They work hard. They persist. This has so little to do with me – no one has told them they can’t be good at math, no one has told them it’s boring. So they believe they are capable, they expect to enjoy math class, and they are and they do. (I hope!) I give them good problems and explorations designed by the Geometry team and they set to work. I am so secondary to the work they do in my room.

My inexperience teaching Geometry actually feels like a gift – this combined with 3 preps at a new school (which means I can’t obsessively overplan each lesson) – means that the questions they ask sometimes surprise me; the solutions they offer often delight me.

I’m at the board, I’ve solicited an approach to a problem and written it on the board; it’s what I predicted, I’m moving on. M stops me. I can’t even remember if she raised her hand but honestly it’s better if she didn’t, let’s all just imagine she shouted at me. “Rachel, my group did it a different way.”

[“Rachel – let me share this with the class. Rachel – I know we’re all smarter than you, you tell us that all the time, let me prove it for the 30th time this week.”]

I pause (we all pause.) It’s that moment in the music where the wind players breathe before the downbeat.

3 examples of “I did it a different way” from my students from this week:




[Note about that last one that could probably be its own post. In reflecting upon “their way” and why it felt so startling, I’ve come to the conclusion that I am deeply a product of 1) learning math through rote, procedural practice in my high school classes, and 2) of the technology that was available to me when I learned this particular concept. I had a TI-83 calculator in high school – both common and natural log were at my disposal, but I didn’t have the ability to evaluate logs of any base (without using change of base formula) – so I couldn’t type something like the example above in directly to get a decimal approximation. My students with TI-84s can. Of course the two answers are algebraically equivalent. I would never in a million years have used this approach.]


Posted by: rdkpickle | 05.01.2016

favorite problems 2 – triangles

[previously: favorite problems 1 – pirates]

Two summers ago at Twitter Math Camp 14, I saw Dylan Kane (@math8_teacher) present a “My Favorite” about a problem he had enjoyed pondering during his summer adventures. You can read his post about it on his blog here. I believe the original problem came from the Five Triangles blog (which is now password protected. 😦 )trianglesGiven three “lined up” congruent equilateral triangles, each with area 20 cm^2, what is the area of the shaded region?

Dylan was using the problem to tell a story, but I couldn’t help myself and immediately started sketching out ideas for solutions. If you’ve been to TMC, you know the sheer volume of good ideas, interactions with rockstar teachers, and mathy things to ponder is overwhelming, so after my initial attempt at a solution I got distracted and set the problem aside for the rest of the conference.

Once I finished driving home from Jenks, the first thing I did that night was sit down and start thinking about the problem again. I told my sister Lindsey about it and she got in on the fun too. Here we are, lounging in my room past midnight doodling triangles and getting stumped.


The week got busy as preparations for the 2014-2015 school year got underway, and I think I forgot about the problem for a bit. Lucky for me, I mentioned the problem to Dawson over brunch one Sunday, and by mid-afternoon he was texting me about his solution.

[Note: Don’t read ahead if you don’t want to get spoiled on the problem!]

Dawson: Solved the triangle problem (I think/hope)! It’s great!! Although I thought it was leading me toward a different answer than I ultimately got

Rachel: i think i did too
did you do it in terms of 60 units sq?
can we compare answers?
and i agree…. i was almost annoyed at what it turned out to be
(if i did it right)

Dawson: Heading out to run errands
Happy to discuss later but…

Rachel: okay no worries!

Dawson: With each eq triangle 20 (total of 60)
I got 50/3

Rachel: ME TOO!

Dawson: Thought I was headed to 20 and that would have been awesome

Rachel: haha, yes.
okay let’s chat later about methods. i’m relieved we got the same thing though!

Dawson: Me too! Talk soon
Extensions to the triangle problem are amazing
I knew part of what was happening after 4 triangles…but 5 triangles really clarified it for me

Rachel: oh goodness, dawson. i guess i have to do that now…

Dawson: No no would love to talk about whatever you have later on

[Note: Do you see what he did there?!?! Fast-forward to “later on.”]

Rachel: what did you get for 4, 5? did you do it in terms of 20 sq units per triangle? or just proportion of one whole triangle?
both are illuminating, but proportion of one whole triangle has a nicer pattern… i think… if i’m doing it right
(2n-1) / 6 * area of 1 triangle??

Dawson: Proportion of one triangle

Rachel: n = number of triangles

Dawson: Yes

Rachel: sweet

Dawson: Did you look at each individual small triangle?

Rachel: just one, then scaled down area
like my work for the first one was 2/3 + (1/2)^2 * (2/3)
and the second 3/4 + (2/3)^2 * (3/4) + (1/3)^2* (3/4)

Dawson: Ok…

Rachel: once i had thoroughly convinced myself the first shaded region was 2/3, i was good and set

Dawson: If you start with 5 eq triangles of area 20…
What’s the area of each individual part?
That’s what got me hooked

Rachel: when you look at the individual pieces is there something pretty?

Dawson: YES

Rachel: i just continued on… 4/5 + (3/4)^2 * (4/5) + (2/4)^2 * (4/5) + (1/4)^2 * (4/5)

Dawson: I think we did this WAY differently
At least in approach
Take your (3/4)^2 * 4/5 and simplify

Rachel: 9/20…. okay okay give me a sec

Dawson: You bet…I needed a long time!!

give me ANOTHER second

Dawson: Yes!!!!!!!
It’s awesome

Rachel: holy cow what is going on??? so 4 triangles is 9/12+4/12+1/12???

Dawson: Yes
It’s beautiful

Rachel: holy cow
and those are triangular numbers on the denom??
does that continue?

Our text conversation turned into an email thread that included Dylan, Justin Lanier (@j_lanier), Nathan Kraft (@nathankraft1), Glenn Waddell (@gwaddellnvhs), Mimi Yang, Mark Greenaway, Jed Butler (@MathButler), and Tina Cardone (@crstn85), all of whom sent out their own version of the solution. It’s completely fascinating to me how many different approaches people took. People employed so many different strategies from using similar triangles and scale factor of sides/area, using ratios of sides in special right triangles and the formula for area of a triangle, writing equations of lines and finding points of intersection, to using trig and law of sines.

Several folks tackled the extension Dawson prompted above, and some worked on generalizing the problem for congruent isosceles triangles, or even congruent scalene triangles.

I have a folder where I’ve saved all of the submitted solutions so far, and I’m still collecting more, if you want to share! Mine and Dawson’s are both posted below. Not quite as high-tech as Justin’s typed-up solution, or Jed’s impressive iMovie solution, but they get the job done. I especially enjoy Dawson’s for his use of colored pens, and the running self-deprecating commentary throughout. Enjoy. 🙂



triangle problem solution dawson 1

triangle problem solution dawson 2

One last thing: I used this problem as a “warm up” problem in a math department meeting a few months ago. One of my colleagues emailed me with his solution after the meeting, along with what might be my favorite quote in recent memory.

“It took me about 15 minutes beyond the mtg, and yes, after one sees it, one feels slightly ashamed, as with all good puzzles.

Indeed. 😀

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