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….

Why don't students feel like they're learning? (when they actually are)

It’s something we’ve all experienced: we’ll lecture and feel like students got it. Maybe they’ll even do well on the practice problems we assign them. Then the next day or the next week we try something a bit more open ended – a problem based lesson, a 3-Act Task, an Open Middle task – and it feels … unresolved. The practice problems we assign after the task don’t look like the ones we just did, so naturally they are completed at a lower success than via our rock solid lecture.

According to research, however, it’s possible our rock solid lecure wasn’t as effective as we (or our students) thought. And it’s just as possible the more “active instruction” was more effective for student learning. In fact that’s what this recent study suggests. From Measuring actual learning versus feeling of learning in response to being actively engaged in the classroom by Deslauriers et al. (2019)

“Students in active classrooms learned more (as would be expected based on prior research), but their perception of learning, while positive, was lower than that of their peers in passive environments.”

“[W]hen students experience the increased cognitive effort associated with active learning, they initially take that effort to signify poorer learning. That disconnect may have a detrimental effect on students’ motivation, engagement, and ability to self-regulate their own learning.”


I wrote tangentially about this phenomenon in Necessary Conditions, dubbing it the “cycle of low level tasks”:

To be blunt, rote tasks are a drug for both teacher and student. With every passing lesson where students are rewarded for demonstrating low-level thinking and under- standing, it becomes more challenging to wean them off rote tasks. Making matters worse is an obsession with standardized test scores and coverage of standards, which contribute to the cycle of low-level tasks. I’ve seen the cycle countless times. It goes like this:

  1. Students exhibit low performance on standardized assessment.
  2. Teacher decides to use complex tasks to enhance rigor.
  3. Students struggle with complex tasks: the task feels aimless
    or students are highly reticent when it comes to tackling the task.
  4. The teacher follows up the seemingly failed experiment with more rote instruction.
  5. The rote instruction is seemingly well received.
  6. Students exhibit low performance on standardized assessment.

A lecture, like low level tasks, can lull a class into offering false positives about student knowhow.

In many ways, this article is reassuring for those of us who promote more active learning over teacher-driven lectures. But before we pat ourselves on the back too much and say “seee?!!!!” we should eat a slice of humble pie and recognize that if students don’t feel like they’re learning, that’s almost as bad as not learning. As the authors write, “As the success of active learning crucially depends on student motivation and engagement, it is of paramount importance that students appreciate, early in the semester, the benefits of struggling with the material during active learning.” So I read the study as a call to action to make “active instruction” more public in its knowledge transfer. Here are a few ways to make sure students are learning and know that they’re learning.

  1. Celebrate what students did and recap what you intended students to do. When you have a complex task that students took in another direction, first off: that’s great! That probably resulted in more permanent learning if they had done it the way you intended. Afterward, share the approach you took to solve the problem and reiterate that it’s pretty cool that they chose a different route.
  2. Make your follow up problems more aligned to the task. Call an “audible,” if you will. Right after the complex task or the student centered activity, let students demonstrate their learning via a handful of practice problems that are related to the concept you just tackled. In a past blog post I called it an “audible.”
  3. Ask students to rate themselves on a rubric before and after the activity. You’ll be amazed at how the simple act of self-assessing against a rubric will communicate growth for many students. Here’s an example of a Problem Solving Rubric (again, from Necessary Conditions, but modified from my employer, New Tech Network).

Over the course of a day, a week, a unit, or a grading period, ask students to rate themselves on their skill. Once at the beginning, once toward the end. They’ll be able to see their own individual growth and you’ll be able to have excellent data on how students are experiencing your class.

The research paper itself contains a few additional recommendations for making students more aware of the benefits of active learning activities and I recommend you check those out as well.

How do you communicate to students that the struggle is indeed worth it?

Five steps to plan a problem based lesson

Far from a simple undertaking, incorporating more inquiry in your class is a challenging process. You and your students may have to unlearn some of the tendencies you’ve built up over the years. So I hope you don’t take this “five steps” post as a flippant, “it’s so easy” post. The opposite is true: problem based learning can require more effort than your typical lesson planning. I’ve broken down my approach here.

Step 1: Find the problem. Modify if necessary.

Easy start, huh? Just go through every problem set in every resource on the World Wide Web and find a problem. Should take no more than two to three hundred hours. Or you could follow my approach. 

I start with the standard, then I go searching for problems. That search begins with my own Common Core Curriculum Maps, where I find my trustiest repositories of problems from bloggers and organizations that have made their resources freely available. I’m so grateful for these caches of math tasks. 

Another place I go searching is the problem set in the associated section in the textbook (a gasped hush falls over the audience). The application section in typical textbook problems sets are actually decent places to find the kernel of quality problems. Textbook problems typically take a bit of adaptation, but you can rest assured they’re at least aligned with the intended standard.

From Necessary Conditions: Teaching Secondary Math with Academic Safety, Quality Tasks, and Effective Facilitation

Step 2: Plan the outcomes and assessment

How are you going to assess students? And, if the problems are group tasks, how are you going to ensure that individual students know the material and had a chance to contribute. I have to go-tos: 

For the groupwork, I use a rubric; one that assesses both the collaboration of the students and the content put on paper (or poster paper).

For students’ individual content knowledge, I give a few practice problems. Depending on how much time I expect after we share out our problem solutions, I may find three or four. Or, better yet, use David’s “Choose Your Own Problems” method. 

Step 3: Plan the launch

How are you going to launch the problem? I always start with some sort of problem decoding routine. I want to make sure that every student understands the problem and have a few immediate next steps the moment we turn to groupwork.

I also want to consider students who might need a few extra moments to process the problem. Sure, I could be talking about students with IEPs, but even students without designated modifications may need extra time to process. I need extra time to process, often in writing, before I’m ready to engage. Consider how you’ll structure your problem launch so all students can engage. 

The strategy I used most often was to give students the problem the day before. Their “homework” was to read through the problem and identify any of the following: unknown vocabulary, important information, ideas for solving, drawing the scenario, etc. This way the day of the problem based lesson was not the first time students see the problem. They can come prepared with ideas and questions. 

Step 4: Prepare the scaffolding

Many problem based lessons require some sort of scaffold during or after the problem. Sometimes the scaffold may be a whole class lecture, other times it might be a small workshop. It’s also possible your scaffolding might be in the form of “hint cards.” These would be little hints to get kids unstuck as they progress through a problem. These hints could be in the form of questions: 

From Necessary Conditions: Teaching Secondary Math with Academic Safety, Quality Tasks, and Effective Facilitation

Step 5: Identify students and skills so as to promote academic status

One of the (if not the) benefits of problem based learning is that complex tasks afford the opportunity to demonstrate ingenuity, creativity, and camaraderie in a way that a rote, teacher-centric lesson cannot. And while you’ll certainly find opportunities in the moment, it helps to plan ahead as well: what are the types of mathematical thinking may come up during the problem? Who are the specific students in your class that you’d like to offer a confidence boost to?

Most lessons plans are very task and agenda driven. I’d encourage you to bucket some space on your lesson plan template for assigning academic status. Think about students you haven’t connected with in a while. Examine your biases. I’ve attempted to do so on the lesson plan template from Necessary Conditions. You can use this template or not, but I do suggest having a place to capture this planning.

And “voila!

There’s certainly so much more than goes into planning a problem based lesson, but hopefully this will give you an insight into my planning process. For some of those additional things, check out my selected blog posts and problem based learning pathway

Math Mindset and Attitudes Survey

The start of the school year offers a unique time in the academic calendar to obtain some baseline data on how your students view themselves as mathematicians and the discipline of math itself. Most beginning-of-year info sheets solicit information about students’ passions and/or guardians’ phone numbers. This year, I encourage you to ask some questions about math itself. Encourage kids to be honest: What do they like about the subject? What do they dislike? How do they (or do they) view themselves as mathematicians?

The following is a sample card from the Academic Safety Card Set based on Necessary Conditions.

I encourage a mix of free-write prompts and quantitate prompts. Here is a link to a Math Mindsets and Attitudes Survey I created and had a few schools implement.

This mix of free response and Agree / Disagree / Neutral/Not Sure gave us good, actionable data in the form of word jumbles and data around students’ mathematical identity.

You can access the PDF of the survey and facilitation guide from the companion website to my book, Necessary Conditions.

If you’d like the google form of the survey, send me a message and I can make a copy and share it. I’m also happy to help you break down and make sense of the data.

Specifics before Strategies

In this blog post, we’ll explore how to get specific with math or non-math classroom issues before we develop strategies. We’ll also see an example of how to build a rubric from the ground up.

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“My kids just won’t work together.”

This (or something like it) is a common complaint I hear during professional development as I encourage teachers to facilitate rich tasks via small student groups. It’s an understandable pushback and also an unhelpful one. If you’re struggling with an issue in the classroom – whether it be groupwork, late work, or anything – the first step is to Get Specific.

Let’s review the opening statement with some questions and commentary:

My kids…” Which kids? All of them? Some of them? Nelson Muntz? How many and which kids are we talking about? Is there a commonality between these students? 

“…won’t…” What do you mean by “won’t”? They don’t want to? They don’t know how? What are they doing instead of working together? 

“…work together.” What does it mean to “work together?” Does it mean to check each other’s work? Discuss a problem before they move on to the next? What structures have you provided to help them work together? What prompts, aside from “work together” have you provided to help students know what it means to work together?

I provide this example to demonstrate how many of us are vague with our comments, when what we need is specificity. We can’t get better when the issue is vague and nebulous.

Sticking with the “work together” situation… this is a common issue among students (and adults). I have two children who hate groupwork more than anything in the world. It’s natural to want to work individually. Some of us are even wired that way. I have a difficult time collaborating with peers far beyond “you do this, I’ll do that.” So let’s drill down.

What are the specific things you want students to do while working together?

  • Do you want students to offer words of encouragement?
  • Do you want students to check the answer with their peers before the move to the next problem?
  • Do you want everyone in the group to share an idea?
  • Do you want students to divvy up the work?

If any of these are the case, say so and don’t just say “work together.”

(***Rubric sense starts tingling***) 

In fact, let’s create a small rubric for this. 

Rubric Example

Let’s take the question around checking answers with their peers before moving to the next problem. 

What is the behavior your want? Well, we just answered that in the previous sentence. Let’s make that the PROFICIENT column. 

What is the current state of students? Let’s say students currently are working entirely individually such that they aren’t checking each other at all. That’s no good. Let’s put that in the far left column.

Now, what would be a stretch goal for students? What would it look like if students were really, really checking in on each other? Maybe: Checks with peers and makes sure everyone understands before proceeding.

And now to fill in the gap: what’s the midpoint between “doesn’t check” and “checks”? I’ll toss in the modifier “occasionally” but I’m guessing the more seasoned rubric developers may have better ideas. “Occasionally” isn’t terribly descriptive. Maybe we should be specific with language like “once or twice.” But that’s what I got and we’re in the middle of a PD session right now. 

Now we have a small rubric on Corroborates Solutions With Peers which is an aspect of collaboration (not the whole thing). We can even use it throughout the year – every time we work on a problem set as a class. We can get better at it. We can improve the rubric as well (let’s go ahead and change it to “once or twice” and “all”).

Because we were able to get specific about the behavior we’re trying to assess, we can now communicate and scaffold towards it. You’ll be able to document with some reliability how many students are at what specific level of this specific aspect of collaboration. I’ll admit: I haven’t offered any strategies in this blog post to treat the initial issue around students working together. But once we have specifics – and rubrics are a great way to get specifics – we can start addressing the problem areas and celebrate the bright spots.

I’d encourage you to check out some of New Tech Networks rubrics around Collaboration, Agency, and Communication for other (better) exemplars. 

To wrap up this with a meta-comment, I’m realizing more and more that I don’t often know what the second step is, but the first is to understand.

Your Student Portfolio System Begins Now

As we transition back into School Mode, I’d like to offer a brief encouragement to use this school year to establish a system of student portfolios. If you’d like a “why” around this, I’ll point you to my Shadowcon Talk from a couple years ago.

If you’d prefer not to watch a video, here are the highlights:

  • Student portfolios allow students to demonstrate and realize their own growth over time (ok, just watch the first minute and a half of the video, up until “Damn, I’ve grown!”)
  • Rich tasks provide better data about what students know and can do than standardized test scores
  • Rich tasks better reflect our instruction and, as any follower of this blog or my twitter feed knows……

Provided you think of it and plan a bit before the school year starts, facilitating a portfolio system is not too difficult. Here’s what you need:

  • Six to ten rich problem-solving tasks
  • A calendar
  • A place to store student work
  • A tool to assess and/or have students self-reflect
  • A couple hours to collect the above items

Let’s take each one by one.

Six to ten rich problem solving tasks. In other words, Portfolio Problems. There are many places to find such tasks. I’ve started by asterisking problems in my Problem-Based Curriculum Maps that I think are worthy of a student portfolio. But I’m sure there are also excellent assessment items in your textbook. Yes, that’s right, your textbook. 

If it helps, consider this scoring guide for Quality Tasks card for a quick check on whether or not a task is worthy. (From Necessary Conditions.)

A calendar. Put the tasks on the calendar now. Every 4-6 weeks block off a couple of days for a Portfolio Problem. You can change them later, but if they’re on the calendar, they’ll get deployed. If they aren’t, they won’t, as other seemingly more urgent business pops up. You can also build in twenty minutes of reflection and share-out time the following class period. 

A place to store student work. Your options here are a file cabinet from an Army surplus store or Google Drive. (There are dozens of other options for physical work and digital work, but these are my go-to’s).

A tool to assess and/or have students self-reflect. After each problem you and/or your pupils will need to assess their work in the moment. Ideally, a you’d use a rubric with common indicators throughout the year. New Tech Network has Math rubrics (and a plethora of others, including Collaboration, Communication, and Agency) that work nicely. But feel free to use your own. 

A couple hours to collect the above items. This is why we’re doing this now. Hopefully you have a couple hours of individual or departmental planning time built in to your in-service before the year starts. The most effective thing you can do with these precious hours is identify nowmonths in advance – the problems you’d like to serve as Portfolio Problems. Once you have those problems identified and on the calendar, there’s no stopping you. 

Here is a related table from Necessary Conditions.