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Locating a Warehouse

In this Illustrative Mathematics task, students are asked to place a warehouse equidistant from three roads.

Students started on paper. Some used their ruler, some their compass, and some folded their paper. I was surprised how few thought to solve a simpler problem (equidistant from two roads), since we did that the lesson before with the fire hydrant. But I think I am beginning to recognize that problem solving itself is a practice – you have to practice it to get better at it, and you have to think about what you are doing and whether something you have done before might be helpful. I had a college professor who talked about a “bag of tools” – each time we would learn some new method, he would remind us to store that method in our bag of tools to consider using the next time we had to solve a problem.

Many students tried the centroid (point of concurrency for the medians).

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Just for the record, I use the Reflector App through my computer & iPad to display student work on my Promethean Board so that everyone can see it and so that we can write on it as needed.

Some folded the roads on top of each other through each point of intersection to come up with the angle bisectors.

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Some students tried the circumcenter again. So we showed that the circumcenter isn’t necessarily equidistant from each of the sides of the triangle.

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We moved to TI-Nspire where students constructed the incenter of the triangle and then the inscribed angle. Students changed the triangle to observe the location of the incenter. Can the location of the incenter give us insight into the type of triangle like the location of the circumcenter?

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We ended with the straightedge and compass construction for an angle bisector and paid attention to what is congruent in the result.

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And so the journey continues…

 
 

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Locating a Warehouse

Two more days of geometry have come and gone. On Monday, we proposed another task from Illustrative Mathematics, Locating Warehouse. In this task, students are asked to determine a location for a warehouse that is equidistant from three “roads”.

1. Hopefully, students are beginning to note that “solving a simpler problem” is a good problem-solving strategy. Where could you locate a warehouse that is equidistant from two roads? (These roads do intersect, but it also might be interesting to consider where the warehouse should be when they don’t intersect.)

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2. We are finding that some of our students have a weak geometry vocabulary. A few of our high school geometry students really didn’t know what we meant by the midpoint of a segment. We don’t know yet the results, but we are trying to present vocabulary in context. We are trying to anticipate the vocabulary words needed for a lesson so that as teachers, we can call out those words during the lesson. We are having students keep a list of terms as they happen in the lesson. And then we are trying to summarize important terms at the end of the lesson. We are trying to “attend to precision”, but we are doing it by example, instead of explicitly teaching vocabulary and notation from the traditional first chapter of a geometry textbook. So when a student said yesterday that any point on the perpendicular bisector of a line is equidistant to its endpoints, together we refined the statement to any point on the perpendicular bisector of a segment is equidistant to its endpoints. We all knew what he meant the first time, but the act of correcting the statement provided us an opportunity to enter into the mathematical practice of attending to precision.

After exploring two tasks that gave us a reason to know about the perpendicular bisector and the angle bisector, we got out the compasses and straightedges on Wednesday to construct them on paper. I have had students do the construction in the past, but this year, we paid more attention to the results. After the students constructed the perpendicular bisector, we asked the question “what things are equal”? Things might not have been the best word to use…but we wanted them to think past segments with equal measure and on to angles with equal measure, and possibly even congruent triangles. We then let them follow the directions themselves to construct the angle bisector (although some of them weren’t ready to follow the directions themselves) and again asked the question “what things are equal”. Performing the constructions in class took longer than expected, but we are trying to “ease the hurry syndrome” and let the needed exploration happen, even if it means we are already behind…

 
 

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