Transversals Lesson: Street Views

The following Problem Based math lesson covers the concept of transversals crossing parallel lines and their angle relationships. The scenario of the task predicated on needing to determine “safe” and “troublesome” intersections in town. Intersections that are closer to right angles are deemed “safe,” while intersections with extreme angles result in limited-vision turns. But that leaves a lot of student-centered wiggle room to define what a safe turn is.

So we start with a concept attainment exercise. I’ve also heard it referred to as “definition and non-definition.” We say that these are “safe” intersections.

These are “troublesome” intersections.

After we make some concrete definitions, students then set off to find examples of such intersections in their town.

Having grown up in Austin, TX, I’m well aware of some of these “troublesome” intersections with extreme angles.

Students are required to produce the following:

  • Develop a mathematical definition for “safe” turns and “troublesome” turns using these examples. Please use specific and accurate definitions and vocabulary.
  • Find examples of “safe” and “troublesome” intersections in our town using a map tool (such as google maps or a physical map). Be sure to include examples that include consecutive intersections, such as the ones with two blue lines above. 
  • On the examples you find, clearly label the turn angles going every which way. And be sure to identify patterns in the angles.

Ideally, through investigation, measurement, ingenuity, and/or intuitiveness, students will recognize the angle relationships involved. Specifically, we want to highlight the relationship between alternate exterior angles, alternate interior angles, and same side interior angles.

To round things out, here is a rubric on which I’ll assess the student work.

Here’s a google doc version of the rubric.

The full lesson, including the setup and the requirments are below. (google doc)

My question for you: how would you modify this task? And, for bonus points, can you find examples of troublesome intersections in your town? Post ’em if ya got ’em!

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Greetings students of New Tech High,

As a city planner, the safety of road intersections are of utmost interest to me. I write to you to help the city identify “safe” turns and “troublesome” intersections in our town so we may identify measures to make intersection safer. The following are “safe” turns.

These are intersections that have seen very few traffic accidents.

The following are “troublesome” turns.

The sharp turns reduce peripheral visibility and result in more accidents with pedestrians and other cards than you’d expect.

I require the following tasks:

  • Develop a mathematical definition for “safe” turns and “troublesome” turns using these examples. Please use specific and accurate definitions and vocabulary.
  • Find examples of “safe” and “troublesome” intersections in our town using a map tool (such as google maps or a physical map). Be sure to include examples that include consecutive intersections, such as the ones with two blue lines above. 
  • On the examples you find, clearly label the turn angles going every which way. And be sure to identify patterns in the angles.

Thank you in advance,

Geoff Krall

City Manager

Geometric Constructions Task: Pizza Delivery Regions

The following task is probably best suited for the a beginning unit on Geometric Constructions or use of a compass and straight edge. However, you may also wish to use it as a fun review as students reenter your classrooms in January, groggy from two weeks of sleeping late.

The task is adapted from the NCTM Illuminations task “Dividing a Town into Pizza Delivery Regions” (membership required). Students are asked to split a town into delivery regions based on proximity. We’re hoping to get students to use a compass and straight edge to construct overlapping circles.

I adapted it by making it local to my area (using a simple search of “Domino’s Pizza” on Google Maps. I then split it into a warm up in which students identify which Domino’s Pizza four specific locations would go to. From there, students are given a larger map and asked to generalize the regions (that’s where the constructions and tools come in). The other adaptation is simply a solicitation for the map and a written explanation for that map.

Here’s the task: [Pizza Delivery Regions]

And in the spirit of good assessment practices, here’s the rubric I’ll use.

[Pizza Delivery Regions Rubric]

Feel free to share in the comments how you’d adapt the task even further. And if you do so and implement the task, feel free to email me (gmkrall@gmail.com) and share some student work. It’s always appreciated.

As an aside, I’m starting to feel a bit odd about how often I blog about pizza….

Routines, Lessons, Problems, and Projects: the DNA of your math classroom

This blog post introduces a new mini-series from Emergent Math: Routines, Lessons, Problems, and Projects.

all four icon

In my time in math classrooms – my own and others’ – I’ve developed a rough taxonomy of activities. Think of these as the Four Elements of a math class: the “Earth, Air, Fire, Water” of math as it were. Or perhaps think of these as the Nucleic Acid sequence (GATC) that creates the “DNA” of your math classroom. Or the Salt, Fat, Acid, Heat of a class. 51dtoe1qufl-_sx260_Speaking of which, the author of Salt, Fat, Acid, Heat, Samin Nosrat, suggests “if you can master just four basic elements … you can use that to guide you and you can make anything delicious.” While I’m certainly not the first to think about teaching-as-cooking, I’m compelled by the way Nosrat distills cooking into four essential elements. I’d similarly posit if you can master these four elements of math instruction – Routines, Lessons, Problems, and Projects – and apply them in appropriate doses at appropriate moments, you can craft lessons and an entire course year for maximum effectiveness and engagement.

Let’s define our terms – after which we’ll criticize them.

Routines – Routines are well-understood structures that encourage discourse, sensemaking, and equity in the classroom. A teacher may have many different types of routines in her toolbelt and utilizes them daily.

Lessons – Lessons include any activity that involves transmitting or practicing content knowledge. Lessons can vary from whole class lectures to hands-on manipulative activities.

Problems – Problems are complex tasks, not immediately solvable without further knowhow, research or decoding of the prompt. Problems can take anywhere from one class period to three or four class periods.

Projects – Projects apply mathematical knowhow to an in-depth, authentic experience. A project occurs over the course of two to four weeks. Ideally, projects are outward facing, community based, and/or personally relevant.

These definitions may not be perfect. I’d encourage you to come up with better (or at least more personalized) definitions and toss ’em in the comments. I reserve the right to change these definitions throughout this mini-series.

To be sure, these four elements often blur and lean on each other: you might teach a lesson within a project. You may employ a routine while debriefing a problem. Many times I’ve been facilitating one of Andrew’s Estimation180’s as a routine and it wound up leading to a full on investigation (which we’ll call a “lesson,” I suppose). Is Which One Doesn’t Belong? a routine or a lesson? Or maybe it’s a problem. It honestly probably depends on how you facilitate it.

Most of the time, however, you’ll be able to walk into a classroom and identify which one of these four things are occurring. If students are engaged in some sort of protocol, they’re in a routine. If the teacher is standing at the front of the class demonstrating something, we’re looking at a lesson. If students are engaged in a complex task, we’re probably in a problem. And if students are creating something over the course of days or weeks, we’re probably in a project.

But why bother with such distinctions?

Perhaps I’m overly interested in taxonomy, but I find it helpful to sort things into categories (perhaps it’s a character flaw).

The real answer to the question of “why bother with such distinctions” is that I was trying to describe the difference between a “traditional” math classroom and a more “dynamic” one. Both of these terms are meaningless, even if they do connote what I’m trying to convey: traditional = bad; dynamic = good. Traditional classes are ones where teachers are lecturing most of the time. Dynamic classrooms are ones where kids are working in groups most of the time. But even that’s not a sufficient clarification: good classrooms employ all kinds of activities, including lectures, including packets.

So it began as an attempt to describe the ideal classroom juxtaposed against a teacher-centric one. A teacher-centric classroom might employ lessons 85% of the time, while a dynamic classroom might employ lessons 55% of the time (I’m making these numbers up entirely).

Then I began to find it challenging to talk about Projects vs. Problems. In my work I’m often asked to describe an ideal classroom: wall-to-wall Project Based Learning (PBL) or Problem-Based Learning (PrBL) or a mixture of both? And how often ought we actually teach in a PBL or PrBL learning environment? How does an Algebra 1 class differ from an AP Stats course?

I’m not going to answer these questions for you, but I hope that this framework will equip you with the vocabulary to design your best math class.

And just like halfway through my adolescence, they discovered a fifth taste (“umami”) we can’t discuss these four elements without the thing that binds classes together: active caring. Perhaps it’s backwards, but we’ll conclude this mini-series with a discussion about active caring and how it’s essential. The best routines, lessons, problem, and projects in the world are moot to a classroom without caring. I suppose it’s a bit too on-the-nose to make a Captain Planet reference with the fifth planeteer’s power being “heart” but that works well as a metaphor if we’re looking for a fifth, I suppose.

One last metaphor: you know those sound boards they have to mix songs? Those ones with a million knobs? And in every movie about a band there’s always a really cool scene where the band is killing this one song and the sound engineer slowly pushes those levers up while bobbing his head and looking at the producer all knowingly? That’s kind of what we’re doing here: playing with the knobs and seeing what it sounds like. We want to get better at each of these instruments individually, and put them together to make beautiful music. Or food. Or genes.

Coming up in this mini-series:

  • Routines: the driving beat of your class
  • Lessons: the stuff we envision, only better
  • Problems: then a miracle occurs
  • Projects: what they’ll remember in 20 years
  • Active Caring: the essential ingredient

Stop Thief!, The Fugitive and introducing equations of circles

When I was a kid, we had this super high-tech board game called Stop Thief!. The gist was this: someone committed a crime somewhere on the game board, which was rife with jewelry displays, unattended cash registers and safes. Your job as the detective was to identify where the thief was. The location of the thief was tracked by a phone looking device that calls to mind those old Radio Shack commercials with car phones. After each turn, the invisible thief would move some number of spaces away from the crime scene. The phone made these noises indicating where he could be – opening a door, climbing through a window, breaking glass. Based on these clues and the number of turns that elapse, you’d try to identify where he was.

stop-thief-2
From http://www.retroist.com

Fast-forward a few years. We all remember this scene from The Fugitive:

These are the artifacts that were going through my head as I designed this lesson, linking the pythagorean theorem and equations of circles. In it, students must overlay a circle to establish a “perimeter” (side note: shouldn’t Tommy Lee Jones have used the term “circumference?”).

While this task only starts from the origin, you could quickly modify it to have other starting points, which would allow students to explore what the equation of a circle looks like when you center it wound non-origin points. I’d expect that to occur the next day or later in the lesson as part of the debrief.

Feel free to tweak it to make it better. The desmos graph is linked below, along with a couple word handouts.

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(Note: a version of this task will appear in my forthcoming book from Stenhouse Publishers, Necessary Conditions.)

The set-up: a crime has been committed and it’s up to the students to establish a perimeter based on how much time has elapsed. After using the pythagorean theorem a few times to identify buildings the thief could be hiding in.

PrBL - RunningFromTheLaw-01

Given the time that’s passed and typical footspeed, the criminal could be anywhere up to 5 kilometers from the crime scene.

Which of the buildings above could he be in?

[Desmos Graph]

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Additional resources:

Running From the Law

Running from the Law Student Handout

Counting Idling Cars: An Elementary Math Project Based Learning Unit

pickup

I’m sitting in my car, waiting to pick up my son from school. It’s too cold to wait outside  this time of year so I keep the heat on, the engine running, and continue listening to the Dunc’d On Basketball Podcast, the nerdiest podcast about basketball out there. I’m also quite anti-social, so I prefer to sit in the car, rather than, like, talk to people.

The driver of the car in front of me is doing the same (presumably, minus the podcast listening), ditto for the car behind of me. Maybe they’re reading “The Pickup Line,” an e-mag specifically for parents who sit in the car, waiting to pick up their kid from school. It occurs to me: boy there are a lot of cars idling in front of the school right now. I’d guess about 40. But y’know, someone should really count these up.

I get typically get to the school about 10 minutes before the release bell rings and I’m sort of in the middle of the pack of idling cars. I’d guess it’s about the average of when most cars arrive, again, most of which are idling. While I don’t conduct this environment-destroying practice all year long – when the weather is nice I’ll get out and check my phone, rather than talk to other parents – I practice it for maybe half the school year. That’s about 80 days or so.

80 days x 10 minutes of idling per day. Boy, 800 minutes of idling seems like a lot doesn’t it? And if there are indeed 40 cars at my son’s school, averaging a similar amount of idling time, we’re looking at 32,000 minutes of car idling. That’s over eight days of just idling.

We have a train that goes through town and we have signs encouraging us to turn off our car, rather than sit there idling, while we wait for the train to pass through. And I sometimes follow that instruction! I should probably follow it more often and more aggressively. But what about idling in the school pickup line? Or along the side of the school for us anti-socialites?

  • How much gas are we wasting?
  • How much Carbon Monoxide are we putting in the air?
  • How much gas waste / CO is the entire town/state/country contributing?
  • Would it be better to just switch off the car and start it later?

Boy, oh boy, someone oughta do the research on this…

What about at your school? How much gas is wasted in a day, week, or school year? Could students do the research? Could they create an awareness campaign for reducing gas waste (and presumably promoting cleaner air at their school)? Seems like something a bunch of go-getter students could handle.

If this scenario interests you and these questions intrigue you, consider adapting it to fir your school.

Update 8/25/2019: Stay tuned on this post or subscribe to follow ups. I like this project idea so much I’m going to be developing it and adding resources and sample project calendars.

Post-It Problem: Grades 2-3

If imitation is the purest form of flattery, then Graham should be pretty darned flattered. I imitated (read: stole) his The Big Pad problem for slightly younger grades. Graham’s task necessitates fractions, which was a bit further down the line for my intended audience, roughly grades two or three. In this task, the giant Post-It is 15 inches x 15 inches and the small Post-Its are 3 in x 3 in. Enjoy!

Screen Shot 2017-04-21 at 2.16.34 PM.png

(Coming soon: a 8 in x 6 in Post-It Problem for grades 4-5, with additional commentary)

Act 1

Facilitation:

  • Watch the video. What do you notice? What do you wonder?
  • How many small post-its will it take to fill up the big post-it?
  • What do you know? What do you need to know in order to solve the problem?

Act 2

Post-It Problems 15x15 Act 2.png

Post it Problems Act 2 Post its.png

Act 3

Systems of Linear Inequalities: Paleontological Dig

(Editor’s note: the original post and activity mistook Paleontology for Archaeology. Archaeology is the study of human made fossils; paleontology is the study of dinosaur remains. The terminology has since been corrected and updated. Thanks to the commenters for the newfound knowledge.) 

Here’s an activity on systems of inequalities that teaches or reinforces the following concepts:

  • Systems of Linear Equations
  • Linear Inequalities
  • Systems of Linear Inequalities
  • Properties of Parallel and Perpendicular Slopes (depending on the equations chosen)

In this task students are asked to design four equations that would “box in” skeletons, as in a paleontological dig.

DOC version: (paleo-dig)

This slideshow requires JavaScript.

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Facilitation

  • Give students the entry event and instructions. Have one student read through it aloud while others follow along.
  • Consider getting started on the first one (Unicorn) as a class. Should our goal be to make a really large enclosed area or a smaller one?
  • Students may wish to start by sketching the equations first, others may chose to identify crucial points. Answers will vary.
  • If you have access to technology, you may wish to have students work on this is Desmos. Personally, I prefer pencil and paper. Here’s the blank graph in Desmos:  https://www.desmos.com/calculator/y1qkrfnsw2
  • For students struggling with various aspects of the problem , consider hosting a workshop on the following:
    • Creating an equation given a line on a graph
    • Finding a solution to a system of equations
  • Sensemaking:
    • Did students use parallel and perpendicular lines? If so, consider bubbling that up to discuss slopes.
    • Who thinks they have the smallest area enclosed? What makes them think that? Is there any way we can find out?
    • Let’s say we wanted to represent the enclosed area. We would use a system of linear inequalities. Function notation might be helpful here:
      • f(x) < y < g(x) and h(x) < y < j(x) (special thanks to Dan for helping me figure this notation out in Desmos!)

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=== Paleontological Dig ===

Congratulations! You’ve been assigned to an paleontological dig to dig up three ancient skeletons. Thanks to our fancy paleontology dig equipment, we’ve been able to map out where the skeletons are.

Your Task: For each skeleton, sketch and write four linear functions that would surround the skeleton, so we may then excavate it.

Check with your peers: Once you have it, compare your functions to your neighbors. Their answers will probably be different. What do you like about their answers?

Optional: For the technologically inclined, you may wish to use Desmos. (https://www.desmos.com/calculator/y1qkrfnsw2)

Challenge: What’s the smallest area you can make with the four functions that still surround each skeleton.

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https://www.desmos.com/calculator/6v0rb28bsl?embed