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 requirements 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,

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.

• 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.

Geoff Krall

City Manager

## 4 thoughts on “Transversals Lesson: Street Views”

1. chrissmith180954163 says:

Thanks for this, Geoff. I’ll give this one a share in the newsletter at some point! Great stuff. All the best for 2020… Chris

2. Like this a lot.

For the first time last semester I got students looking at the angle relationships in nonparallel lines. They found cool relationships with the angle between the lines that made the parallel theorems feel like special cases.

3. Rachel says:

I like the idea of this, but have a couple of questions. At what point in the unit would you do this assignment? Would the students know vocabulary such as consecutive interior, alternate exterior, etc.?

Also what do you mean by your third bullet point in the required tasks? Would you encourage students to label [apparently] congruent angles with the same color or the like? I’m thinking through how this would transition into the next lesson.

Thanks!

1. Hi Rachel, these are some great questions. Personally, I’d consider rolling this out towards the beginning of a unit. We would build our vocabulary around interior, opposite exterior, etc. from this investigation.

I think you’re right to point out that third bullet is a bit oddly worded. I do mean exactly as you say: label the angles using proper notation; perhaps even using like colors and symbols to represent congruent angles. I think I’ll go back and make an edit to clear up that language. Thanks tremendously for voicing the question! It’ll make the task better going forward!