Category: geometry

Indirect Measurement Lab

Probably my second-favorite Algren line. The man knew of what he spoke. (And yes, I know Hammond is not Chicago. But we share a border, and weather, so I’m granting myself poetic license.)

Been planning an indirect measurement lab for my geometry classes that included an outdoor component. Keeping an eye on the long-term forecast, I flip-flopped the lab and our unit quiz to days I felt had a better chance of sunshine, and thus, shadows.

Whiff.

Friday’s effort with two of my classes was a strictly indoor affair, lack of sufficient sunshine calling for Plan B.

So instead we measured student heights in the classroom then went to the stairwell on my side of the building and set our sights on what appeared to be a 20-25 foot high wall.

We used the Measure app on their phones rather than the DIY clinometers that I built a million years ago the first time I taught geometry. If those tools still exist they are buried in a box in my basement. And that definitely made our data a little shaky. Getting angle measurements with the level feature on the Measure app is definitely an inexact science. No one was really close to the actual (18.5 foot) height of the wall. And we didn’t have the similar triangles/proportions results to compare.

It was good enough for a Friday and gave them a chance to dig back in their notes for trig application problem set-up. I also asked them to reflect on possible sources of error.

Walking in this morning I saw abundant sunshine and hoped for outside project time but we got off to a sluggish start taking each others’ height mesurements in my 2nd hour. Not enough time to try to get all the angle and distance measurements done whether outside or inside. We stepped outside again for third hour but there just was not enough sunshine so ducked back inside.

One last class, getting underway at about 12:40. My twitter bio doesn’t say “Stubborn Jackass” for nothing. So back out we went. Just in time to get a sliver of sun, enough to measure a lamppost shadow and student shadows before the sun ducked back behind the clouds. We got what we needed though. Took our angles of elevation and horizontal distance to the lamppost under the cloud cover and headed back in to crunch some numbers.

The experience made an awesome dry run for the quiz next block. I wrote a couple questions that mimic this activity so I’m hopeful that second rep in quick order will spark their memories.

And honestly, just formative-assessing-by-walking-around it was clear we could use the practice on using similar triangles and trigonometry to perform indirect measurement. But that’s literally my job to tease that out.

We’ll take the first half of class to review next time, quiz the second half, and we’ll see if hands-on opportunities make the learning last.

Because it might feel like October out there but next Monday it’s May already and we’ve got 16 blocks left and there’s work to do.

Out Of The Shadows

As a longtime Region guy, I appreciate having four seasons. During the years we lived in Vegas, where there’s 320 days of sunshine a year, I used to pine for just one sorta crappy cloudy day every now and then, just for old times’ sake and for my mental equilibrium.

But perpetual sunshine had its benefits when I was planning outdoor activities in math that required shadows. My mentor teacher turned me on to an indirect measurement lab where his kids determined the height of the streetlamps in the drive in front of our school using similar triangles.

I had my kids outside pairing up and measuring heights and shadows, then we came back in and the math to figure the height of the lamppost was the first question on their quiz. (Surprise!) It worked out super-well.

Fast forward 20 years and here I am hoping to take my geometry kids outside for an indirect measurement lab next week. But sunshine in the Region in April is a dicey proposition at best. Case in point: Last week: 80F and glorious. Bike rides and T-bones on the grill and hanging outside with the puppies.

Dogwwods are in bloom by me.

By Sunday tho? Overnight snow in the forecast.

But I’m willing to risk it. Put together the paperwork for the lab before I left the building Friday (doc here – feel free to copy/remix/improve). Also: I’ll need to be flexible.

We are very large Pixar fans in this house

I originally wanted to quiz Friday-Monday and do the lab Tuesday-Wednesday but it looks like we have a better chance of sun (and thus, shadows) on Friday and Monday. So here we go. Flip-flop the lab and the quiz, right? Honestly I put a couple questions on the quiz that mimic the lab so that’s actually probably better from a measuring learning POV.

They’ll partner up, brainstom objects on campus they could indirectly measure, measure each others’ height (top of head and to eyes), then head outside and measure each others’ shadow length.

They’ll select one of the objects they brainstormed to measure, then determine the length of that object’s shadow, as well as their distance from it.

Having all the data we’ll head back inside and do some calculations, determining the height of their object both by similar triangles and by trigonometry. Then they’ll contrast and compare the results and the process.

I’m hopeful that the chance to apply some of the math they’ve encountered will make the learning more permanent. And any day we can get outside and get hands-on with math is a good day. So whatever day we get sunshine, that’s the plan.

I’ll let you know how it goes.

Oh BTW the way the tortilla soup was fabulous.

Can I Treat This Like A Triangle?

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My Algebra 1A freshmen were very freshman-y today. Retaking a class they hate, last day before a four-day weekend, quiz day. Recipe for disaster. OK maybe “disaster” is a strong word but nobody’s packaging up today’s sixth hour and sending it to the Teaching Hall of Fame. The bell rang to end class and I took a minute to walk the room and straighten desks and pick up left-behind review pages and pencils and snack wrappers, feeling like a not-good teacher.

We’ll try again Tuesday.

But take your minute to sulk (a good playlist helps) and then get your head straight because geometry is coming in like four minutes. I had a singleton Grey Day ahead of the long weekend. We just started our Right Triangles & Trigonometry unit, two sections in, so we had done the Pythagorean Theorem and Special Right Triangles. We had our ups and downs, in about equal measure. What I need for this day is a Three-Act. Taco Cart has been a go-to for years, with the added benefit of being an application of the Pythagorean Theorem. Perfect for my Never-Ending Quest to fill in my students’ pandemic-related foundational holes.

We started off with a Would You Rather rates bellringer since we’re going to be thinking about distance and time.

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Then rolled right into the Desmosified version of the Taco Cart activity.

We did our noticing and wondering, made predictions, asked for more information, then I turned them loose in pairs to math this out. And in my last class of the day, it happened: “Mr. Dull, can I treat this like a triangle? And do that a-2 plus b-2 is c-2 thing?”

Hell yes you can. A right triangle even. Let’s go.

We’ve been kind of hammering away at using the principles of a right triangle to solve problems when we have a vertical and horizontal distance. And she connected the dots. It was epic. It’s super-important to me that we can take the training wheels off and let my kids ride on their own. (Pro-tip that was passed on to me by a wise older dad: take the pedals off your kids’ bike when they are learning to ride without training wheels. They’ll learn balance first, and the next thing you know, we ride at dawn!)

With some nudging many of the students produced solid work. Some quality questions in there too. “Were the tacos worth it?” “Do they really walk slower in sand?” “Don’t they have a car?”

Meet them where they’re at , walk with them to get started, and magic will happen.

So that’s how you recover from a disaster of a sixth-hour class. And how you roll into a long weekend. With singing, obvi.

Oh, and I’m not giving up on the freshmen. Gonna tweak my Algebra II survey/data project a little bit and turn them loose on that this week. Bribe them with a quiz grade. Do some math and give some hope. But that sounds like a whole ‘nother post.

Road Trip

We are three-quarters of the way through the school year and almost halfway though my building’s conversion to a project-based learning model. From the jump it’s been a team effort, planning together and taking turns suggesting activities. This unit one of my geometry colleagues was struck with a PrBL inspiration at a faculty meeting and quickly worked up a rough draft of an activity to share. I went and tweaked it a little (because we collaborate like that) and away we go.

Because Spring Break seems like an ideal time for an Imaginary Road Trip.

Our kids began by locating Hammond on the map, then identifying two other cities they have lived in, have visited, have family in, want to visit, know from music or movies or books, whatever.

They placed points on the map for each city, connected the dots, and then measured out the straight-line distance between each city. Using the scale on the map, they wrote and solved a proportion to find the mileage.

Cool. But roads between cities don’t typically go in straight lines. So off to Google maps we go to find the actual driving distance.

We grooved to wide-open spaces driving tunes while we worked.

Next up, let’s attach some real-ish world math to our project. I grabbed up some data from my last cross-country trip to see my Army MP son and had them write and solve proportions to determine how many gallons of gas they’d need for the trip and how much they would spend on fuel. We wrapped it up with my kids calculating the average cost of a gallon of gas on this trip, and their miles per gallon.

My geometry students are part of that cohort that spent their formative algebra years either learning remotely or wearing masks in rows and columns in class, and many tell me they feel shaky still when it comes to the Xs and Os of algebra 1. That’s fine. It’s always been my job to find those gaps and help fill them in. So if we beat proprtions to death these last few weeks, trust me when I say we needed to.

Spring break is here. I know because I stretched out for just like five minutes after dinner yesterday and woke up an hour later when there was just enough light coming through the window that I wasn’t 100% sure if it was 7:00 pm or 7:00 am the next morning.

I needed to find the juvenation machine. For that purpose, ain’t but one place to go. Or maybe two.

Mrs. Dull told me I seemed very relaxed at River St. Joe. It’s that obvious…

Never happier than I’m I’m by water. It was good to be on my feet and moving, good to be by myself, good to hear frogs and birds and leaves and nature sounds (and I am the king of loving trucks and busses and horns and city sounds).

Much needed.

But even in the midst of My Happy Place™, math is never faw away.

“What do you notice? What do you wonder?

Sometime in the next couple of days I’ll post my quarter grades, make sure work clothes are clean and gas tank is full and lunches are prepped. But until then I can squeeze in a road trip or two. Imaginary or otherwise.

Teaching To The Student

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Today is the first day of Daylight Saving Time. It is unquestionably Mrs. Dull’s least favorite day of the year. For her that lost hour of sleep is a thing she can never get back and it feels like its been unfairly taken, the same reaction you have as a kid to dropping an ice cream cone on the ground.

For me, it’s one of the unmistakable markers of the coming of spring. Truth be told, I’ll be dragging tomorrow morning too but there’s a 7:00 sunset scheduled for Monday of spring break and that re-energizes me for the last 9 weeks of school. I’m very Vitamin D-responsive. I’m more motivated to get my dogs and myself out for evening walks when it’s light out later. I’ll put the patio chairs back out and spend some quiet time sitting outside after dinner. It’s mentally and physically healthy for all of us.

Everybody out there trying to get what they need, and some folks need something different. For Cath, we’ll balance out the sleep deprivation some night soon with an impromptu pizza dinner on the beach at sunset.

My district revealed its testing numbers a few weeks ago. There was good news and bad news: we are improving but still lagging the state-wide average. And breaking down our students’ I-Ready and PSAT results, we have got significant work to do.

Our principal shared out the results at last month’s faculty meeting, and added a note of guidance: if we have a large portion of our students reading at well below grade level, and we are giving text-based assessments, we can’t be too surprised that our students are struggling grade-wise in our classes. His suggestion: can we consider and implement other ways we can give our students to show us what they know?

Amen. Preaching to the choir right there.

I’m a long-time project-as-assessment guy. Especially when Algebra II gets weird.

Last week we finished triangle similarity and took a Desmos quiz on the topic, with so-so results. I followed it up with the Capture-Recapture goldfish lab and made that a quiz grade. It checks plenty of my favorite boxes: collaborative, crunching numbers, real-world application, a quick snack while mathing. They are after all The Snack That Smiles Back™.

If you want the definitive write-up and docs, Julie Reulbach is my go-to. And you’ll probably dig the BBC video that serves as the hook.

There was some quality math on display and plenty of productive table-talk, and a little competition (which group came closest to the actual number of goldfish in the bag) never hurts.

They showed me they can set up and solve proportions, which is a major objective for the unit. That was my motivation for making it a quiz grade. As a former colleague of mine likes to say, “you learn it, you earn it”. For many of my students, it helped balance out their score on the more traditional quiz. Which seems eminently fair to me, and is aligned with the philosophy our school leadership is espousing.

Win-win. I can teach to the test, or I can teach to the student.

Kind of an easy decision, as I understand it.

Now let’s make it throught this last week of the quarter, enjoy spring break, light the Weber and bathe in the soft light of a late-March sunset.

What You Need

I’ve got six sections of geometry this year. So I get plenty of reps with each lesson, plenty of opportunities to learn and reflect and adjust. I’ve joked for years that the class that gets the 1.0 version of the lesson should get a 2-3% score bump for being test subjects.

So this tweet caught my eye this week:

The second half of unit 3 has been filled with stops and starts. Testing, the Natural Helpers retreat, an e-learning day for family-teacher conferences, more testing. It’s taken like two weeks to cover two sections. So it felt like the right time to build in a review day/work day before a project-based activity to wrap up the unit. I didn’t want to just throw a worksheet at them so I went with one of my go-tos. Grudgeball.

A classic. Excellent for review days. And as an added bonus it shines a bright light on areas of need and allows for plenty of remediation. And because 20 brains are better than one™, there’s a lot of ways for my kids to get math help.

This week was a study in contrasts. Hoo, did my second hour ever crush it. Petty and cutthroat and ultra-competitive. That had the effect of keeping the kids super-dialed in on working the problems and amped up the collaboration. Kids were helping each other get started or get unstuck, there was fabulous table-talk math going on. I’d have paid money to have my admin stop by for a walkthrough at that moment.

But the bookends of my fifth hour Tuesday and my third hour Wednesday, well those classes we needed a different approach. I had intro’d the activity, built it up, revealed the first exercise slide, and… crickets.

I couldn’t just turn them loose and wait to see which table group worked out their problem correctly, first. They needed help just getting started. I can dig that. After 18 months of pandemic-era remote teaching, many of my students come to me with holes where algebra skills belong. Constant spiral review is pretty much a fact of life. So regarding Grudgeball, it was time for an audible, obviously. The game turned into basically an 80-minute small group tutoring session, plenty of re-teaching interspersed with whole-class review of a handful of selected problems.

Not at all what I planned, or what I hyped the day up to be. But was what my students needed. Especially because I’m going to ask them to use many of these same skills to complete their PrBL next block.

It’s probably the greatest benefit of being “seasoned”. I recognize when my classes have different needs from whatever I have planned, and I have enough confidence to pivot seamlessly to Plan B while keeping a long-term goal in mind.

Pushing me to differentiate between the needs of my six sections of geometry? I love Grudgeball even more now.

Flip The Switch. Ho Ho Ho…

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Our A/B block schedule gifted me with a singleton Red Day this week. It will pair up with an e-learning day on Election Day next week. But for this day, I needed a project.

To keep my classes in synch I’ve been using those days as one-off activity days. This year it’s not just a change of pace, it keeps everyone sharp as we continue our transition to project-based learning. As an added bonus I got a partner in crime. I shared my slide deck for the project with my geometry teammates, my instructional coach, and my department chair. My next door neighbor teacher was willing to go along for the ride.

Like many major cities, one radio station in Chicago changes its format to 24/7 holiday music every year during November and December. It’s become a bit of a parlor game to guess when the FM Lite will flip the switch. The earliest date? November 2, back in 2006 and 2007. Here’s the history, and a graphic look at the ratings bump.

Seems like a good opportunity to make predictions (based on admittedly limited information – what kid listens to radio, much less pays any kind of attention to when a radio station changes formats), explore/research the topic, then analyze data using appropriate tools. So math, yes, but also pushing our way up Bloom’s taxonomy, right?)

I’ve been using this activity with my in-person classes for a few years. The slide deck has evolved a bit since 2019 and I’m pretty happy with its current form but that doesn’t mean you couldn’t remix it and make it better.

I should point out that we don’t play Christmas music during the activity, and I’m upfront with my students that the “guess the date” is really just a timely way to generate data. I could have asked them the day of the month their birthday falls on or the last two digits of their street address or something else.

And the soundtrack? The 2023 version of my student-generated Friday Playlist.

Students start by guessing the date they think the switch will occur, and give a justification for why they think the station will change that day. Then I ask, what if we had more info, would that help you make a more accurate guess?

I share a Robert Feder column with some of the recent history of the changeover date. I years past I used this time to go into the spreadsheet that holds their responses and add a column that contains the number of days after Halloween to the date they guessed, computed manually by me.

Then this year I learned that there is a DAYS function in Sheets that will automate that process. So sweet.

What can we do with this data set?

Contrast and compare, duh. I pulled the data from a group of classes last year and used that as the comparison.

Students go to Desmos, create a dotplot of the data, insert link to their dotplot as well as a png into a slide.

“What’s that big gap around Day 40 about, you guys? And why the big spike between Days 25-30?”

I assume it’s been a minute since they interpreted dot plots so I point them to a quick refresher. Then a brief reflection (“Think about a time you made a decision and did some research first. What was the decision and how did it work out?) and we’re good to go.

The Which One Doen’t Belong makes for an outstanding setup to the activity. I tell my kids that WODB serves multiple purposes in our class:

  • Instant SMP 3 (taking a position and defending that position)
  • An opportunity to use appropriate math vocabulary in context
  • An opportunity to practice noticing subtle similarities and differences

I obviously can’t use this activity in exactly this format for an e-day activity. That group will get a chance to make a prediction (the flip will probably have already happened by then but who knows?) then they’ll get a slide with the two dotplots and do the contrast and compare. Reflection too. Those are the money parts anyway.

My Red Day kids? We’ll hang out together on Halloween, prove some lines are parallel, maybe sneak in a Nevada Day reference, and groove to Bow Wow Wow.

See what I did there?

Because on Tuesday, Mariah awaits…

Math Marketing 101

Had a chance last week to do some introspection, pondering strengths and weaknesses. Always kind of a loaded question, but one of my strengths is that I know my weaknesses. Or one of them anyway. I’m not the salesman in the family. We need someone convinced to do something, well I’m gonna hand that call off. Packaging tho? I’m not bad. The marketing part of sales I kind of get. I know just enough to be dangerous.

And there’s definitely times I feel like high school math needs better marketing.

Our first quarter ends 0n Thursday. Coincidentally it is also PSAT Week and Homecoming Week. The kind of week that is made to order for a makeup day on the only two regular class days before fall break. (Background on amnesty days here.)

The options for students on Amnesty Day includes quiz corrections, daily work makeups on MathXL, and a playlist of alternate assignments. The idea is for students to display their learning, and to make up the points. But are points enough incentive? Maybe I need a pot sweetener? Something… extra?

I want to give my students different (and ideally, interesting) ways to show their learning of the concepts of the quarter. So I work hard to source or craft a variety of opportunities with various entry points and various levels of complexity and creativity. But on the last regular block before testing, career day, and then taking a week off for fall break, work is not at the top of my students’ list of good times. But every ear perks up at the words “extra credit”.

OK then Mr. Marketer Guy, you want your kids to do your Amnesty Day work? Give it a catchy name. Like “Amnesty Day Extra Credit Playlist”.

The options today:

As a side note, we’ve super-into drag and drop activities as of late. Used this video to learn the basics:

And yeah, I know a drag and drop is basically a DOK 1 e-worksheet. Good as a check for understanding tho.

And in retrospect I’d have added some type of reflection question to the logic form because it didn’t go as deep mathematically maybe as it could have. More of an extension to the Law of Syllogism activity we did a few weeks ago. Also, the actual Barbie Zipline activity is totally worth the two or three days of classtime. It might make an awesome PrBL activity one of these years. (I still have all the materials, and I spent way too much on them to have it all sit in a bag in my basement.)

Another day. But for this day I needed a group of activities to set the hook (as Stevens would say) and get my students engaged on a day when both learning and points were on the table.

I think we accomplished that. I still don’t get 100% participation on Amnesty Days. Extra credit is “extra” by definition. Optional. But I think I get more takers than I would if I just called this “Alternate Assignment Day”. Part of my job is piquing my students’ interest in whatever I’m trying to get them to do, right?

We watch a little too much HGTV in my house, and those real estate people are kind of skilled at marketing notes. So if my mom & dad’s 50’s ranch is now a “mid-century modern” gem, I can call today’s playlist whatever I want.

Prove It

We’re about to begin the Reasoning and Proof unit in geometry, and judging by the looks on the faces of some of my students when I introduced it Wednesday and Thursday, well, its reputation certainly precedes it.

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Our curriculum map skips over logic statements and inductive reasoning, but back in the day I started doing a little writing activity as an application of the Law of Syllogism and it still holds up pretty well, and allows us to ease into a challenging subject.

I’ve been fortunate that my schedule has varied from year to year, toggling back and forth between Algebra II and Geometry the last 7 years or so. It keeps things fresh. I’m a believer in “teach 20 years but don’t teach one year 20 times”. But I also subscribe to the theory that in this business you keep what works and throw out the rest.

Thus, the If You Give a Mouse a Cookie project made its grand reappearance in Room 247.

(After an online convo this week I realized I hadn’t written about this one yet, so here we go. Take it and make it awesome-er if you want).

Student handout here. When I resurrected this activity after a long layoff teaching Algebra 1 and engineering, I couldn’t find the doc I had made back in 20-oh-whatever. A quick search online turned up this one, which I edited for my purposes. I don’t know who originally made it tho, so if it’s you, thanks.

We start the day with a video bellringer:

In a Google form I ask students to write down two “if-then” statements they heard in the story.

Then we talk about “cause and effect” – what happens if you stay up late binge watching a show? You are tired in the morning. What happens when you are tired in the morning? You hit snooze on your alarm. What happens if you hit snooze on your alarm? You wake up late. What happens if you wake up late? You miss the bus. And so forth.

All my kids can relate. In one class a student interjected, “so, you mean a chain of events?” Exactly. Let’s go.

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The assignment is to write a story in the style of If You Give A Mouse A Cookie. Eight linked if-then statements that tell a cohesive, school-appropriate story, with a title/author/cover illustration.

Stick figures are definitely allowed.

My handout provides a lot of scaffolding – separate boxes for each of the eight sentences and then one more for the end where they link the hypothesis of the first sentence to the conclusion of the last, sometimes to great comedic effect. I make it a small quiz grade (15 points) because this unit does kind of grind up kids on the actual proofs quiz, and IYGAMAC serves as balance grade-wise. Plus, just about everyone can pinpoint that cascade of interrelated events in their own lives and the next thing you know they’ve got 8 good sentences. Some of them tap into their deep well of creativity and hilarity ensues.

It’s a good day. We get creative, we write in math, my check for understanding at the end of class usually reveals that everyone gets the idea of a syllogism, so we connect the activity back to the vocabulary. Checks a lot of boxes.

In an A/B block schedule I have to be careful about taking a day to step out of the curriculum map. But this activity has a high ROI in terms of understanding building an argument of factual statements. Worth the cost.

If I had four hours to chop down a tree, I’d spend the first two hours sharpening the axe

Not said by who you think said it, by the way

Monday we can go back to furrowing our brows over proving angles congruent or whatever.