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Category Archives: Angles & Triangles

Assessing the Centroid of a Triangle

The centroid of a triangle is often called the balancing point of the triangle. It is the point at which the medians of the triangle intersect.

Students used technology to explore the relationship between the vertices of a triangle in the coordinate plane and the vertices of the centroid.

If your students knew the relationship between the vertices of a triangle and the vertices of the centroid, how would you expect them to answer the following question? (I included this question on an end of unit assessment.)

The vertices of a triangle are (a,b–c), (b,c–a), and (c,a–b). Prove that its centroid lies on the x-axis.

A few of my student responses are below.

What learning opportunities could I have provided in class to better prepare my students for this question without just giving them a similar problem?

And so the journey to provide meaningful learning episodes that prepare students to answer questions they haven’t seen before continues …

 
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Posted by on August 22, 2016 in Angles & Triangles, Geometry

 

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MP8: The Centroid of a Triangle

We had been working on a unit on Coordinate Geometry.

How do you give students the opportunity to practice “I can look for and express regularity in repeated reasoning”?

SMP8 #LL2LU Gough-Wilson

When we have a new type of problem to think about, I am learning to have students estimate the answer first.

I asked them to “drop a point” at the centroid of the triangle. We looked at the responses on the graph first and then as a list of ordered pairs.

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What is significant about the coordinates of the centroid?

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Students then interacted with dynamic geometry software.

Centroid_1.gif

What changes? What stays the same?

Do you see a pattern?
What conjecture can you make about the relationship between the coordinates of the vertices of a triangle and the coordinates of its centroid?

Some students needed to interact on a different grid setup to see a relationship.

Centroid_2.gif

After a few minutes, I sent another poll to find out what they figured out.

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And then we confirmed student conjectures as a whole class.

And so the journey to make the Math Practices our habitual practice in learning mathematics continues …

 

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MP7: The Diagonal of an Isosceles Trapezoid

 

SMP7 #LL2LU Gough-Wilson

I’ve written about the diagonals of an isosceles trapezoid before.

When we practice “I can look for and make use of structure”, we practice: “contemplate before you calculate”.

We practice: “look before you leap”.

We ask: “what you can you make visible that isn’t yet pictured?”

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We make mistakes; the first auxiliary line we draw isn’t always helpful.

Or sometimes we see more than is helpful to see all at one time.

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We persevere.

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Even with the same auxiliary lines, we don’t always see the same picture.

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We learn from each other.

And so the journey to make the Math Practices our habitual practice in learning mathematics continues …

 
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Posted by on August 16, 2016 in Angles & Triangles, Geometry, Polygons

 

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MP8: The Medians of a Triangle

How do you give students the opportunity to practice “I can look for and express regularity in repeated reasoning”?

SMP8 #LL2LU Gough-Wilson

When we have a new type of problem to think about, I am learning to have students estimate the answer first.

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I asked for their estimate in two slightly different problems because I wanted them to pay attention to what was given and what was asked for.

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Students then interacted with dynamic geometry software.

Centroid_1.gif

What changes? What stays the same?

Do you see a pattern?
What conjecture can you make about the relationship between a median of a triangle and its segments partitioned by the centroid?

As students moved the vertices of the triangle, the automatic data capture feature of TI-Nspire collected the measurements in a spreadsheet.

Centroid_2.gif

I sent another poll.

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And then we confirmed student conjectures on the spreadsheet.

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And so the journey to make the Math Practices our habitual practice in learning mathematics continues …

 
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Posted by on August 15, 2016 in Angles & Triangles, Geometry

 

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SMP7 – The Triangle Sum Theorem

G-CO.C.10 Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.

How do you provide opportunities for your students to look for and make use of structure? I’m finding that deliberate practice in looking for and making use of structure is making the practice a habit for my students.

SMP7 #LL2LU Gough-Wilson

We ask: “what you can you make visible that isn’t yet pictured?”

We practice: “contemplate before you calculate”.

We practice: “look before you leap”.

We make mistakes; the first auxiliary line we draw isn’t always helpful.

We persevere.

We learn from each other.

 

Months ago, our goal was to prove the Triangle Sum Theorem.

We thought first about what we already knew … what we had already added to our deductive system.

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Then we practiced “I can look for and make use of structure”.

And so the journey to make the Math Practices our habitual practice in learning mathematics continues …

 
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Posted by on July 1, 2016 in Angles & Triangles, Geometry

 

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Structure, Flexibility, and Planning

I set up a recent lesson by asking students to deliberately practice SMP7, look for and make use of structure.

2 SMP7 Number #LL2LU Gough-Wilson

This practice requires us to make visible what isn’t showing. In geometry, that often means drawing auxiliary lines.

We don’t always see structure in the same way or at the same rate, so once you’ve found one way to solve the problem, I want you to also deliberately work on your mathematical flexibility. Find a second way to work the problem.

1 Flexibility #LL2LU Gough

I had 5 questions prepared, the last of which I learned about in Justin’s and Kate’s posts last year about a Five Triangles task. I’ve been thinking a lot this year about not only planning learning episodes but also planning ahead what instructional adjustments I’ll make based on the feedback I get from my students. In my planning, I struggled with which question to use first. Which question would you use first with your students?

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Last year, I had the following results in the following order.

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After this first Quick Poll, I didn’t display the correct answer, asked students to team with someone else in the room, and sent the Poll again.

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After this Quick Poll, we had a student who answered 53˚ share his reasoning with the rest of the class so that we could figure out where the reasoning went wrong.

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The class went fine. But I wondered what would have happened if I had started with a question that required the use of auxiliary lines (even though students struggled with the question that already had them drawn). So I tried that this year.

I could “hear” thinking and I could “see” productive struggle as students started out working the problem individually. Once they started sharing some of the ways that they made visible what wasn’t pictured, I saw evidence of SMP7. Because I had deliberately asked them to work on their math flexibility, they weren’t satisfied with only one way to solve the problem.

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Many wanted to share their way with the whole class.

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They tried another one, and again, you could “hear” thinking. I didn’t even have to suggest individual think time to the class, as they naturally all wanted to try it by themselves first.

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I posed the folded rectangle problem, but the bell rang before students could really dig in to solving it. Maybe next year I’ll be brave enough to start with it, as the journey continues …

p.s. I’m currently reading Ilana Horn’s Strength in Numbers: Collaborative Learning in Secondary Mathematics, and before I was able to publish this post, I happened to read a section entitled “Turning Some Pet Ideas about Mathematics Teaching on Their Heads: Start with Challenging Stuff, Not Easy Stuff”. Her premise is that starting with easy stuff is inequitable, as students who get the mathematics quickly can take over the problem, and those who don’t miss out on the opportunity work with their team. Starting with challenging stuff levels the playing field for all students to contribute and learn.

 
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Posted by on October 26, 2015 in Angles & Triangles, Geometry

 

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Angle Bisection and Midpoints of Line Segments, Take Two

Last year’s lesson using the Illustrative Mathematics task Angle Bisection and Midpoints of Line Segments had plenty of room for improvement. This year, students left with a better understanding of proof and giving feedback on proof.

Our goal? SMP3: I can construct a viable argument and critique the reasoning of others.

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Students started by reading through both parts of the proof, noticing and wondering.

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I’ll admit, I really wanted someone to notice that parts a and b were converses. (I didn’t expect them to use that language … I was just looking for anything about the parts being “opposite”.) I wasn’t ready to tell them, so I specifically asked, “what is the difference between parts a and b”.

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In triangle a thhey already give you the midpoint of line QR and asking you to draw the angle bisector, but in triangle b they are giving you the angle bisector and are asking you to find the midpoint of line QR.        1

In part a, you’re trying to find the angle bisector from the midpoint, but in part b, you’re trying to find the midpoint using the angle bisector. So they’re basically the opposite of each other, but you have the same point and the same line. They were just found in different ways.  1

Part a starts of with finding the midpoint to segment QR and then creates a line from P to go through the midpoint while part b starts with an angle bisector PS then goes to see if it intersects the midpoint to of segment QR.       1

in part a your contructing a midpoint, in part b you are constructing a bisector         1

In part a you are justifying that PM is a bisector of QPR, but in part b you are justifying that PS meets QR at its midpoint.         1

The difference is that part a to show that the bisector will go through the midpoint, while part b is asking to show that the bisector does go through the midpoint rather than just some random point.       1

In part A the midpoint is labeled M and in part B the midpoint is labeled S, but it is the same point. Also part A and part B make the same image, but the just switch the order they made the image. like finding the midpoint first then the bisector, vice versa    1

Students spent a few minutes creating an argument for part a. Then we looked at some of the student work from last year to critique the arguments.

In Embedding Formative Assessment, Dylan Wiliam suggests that students learning how to give feedback should start with anonymous student work … and eventually move towards student work from peers in the same class. This seemed to work well for this task. Additionally, I had the opportunity to purposefully select and sequence the work for giving feedback ahead of time, which gave us more time for learning during class.

My geometry students are 1:1 this year with MacBook Airs, and so I sent a PDF of the student work samples through TI-Nspire Navigator for Networked Computers, which gave them an up-close look at the student work instead of my having to stand at the copy machine for a while or students trying to decipher from it only being displayed on the board at the front of the room.

We looked at one student work sample at a time using Think-Pair-Share to make student thinking visible. What feedback would you give this student?

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M is the same distance from Q and R, but points on the angle bisector are the same distance from the sides of the angle. How do you know M is the same distance from ray PQ as it is from ray PR? We represent distance from a point to a line as the length of the segment perpendicular from the point to the line.

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What is a perpendicular bisector of an angle?

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What is the difference in saying segment QR is a perpendicular bisector of ray PM and saying ray PM is a perpendicular bisector of segment PM?

Before we looked at the next student work sample, I asked students to practice look for and make use of structure, asking what they saw when segment QR was drawn.

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An angle bisector.

A midpoint.

Triangles.

How many triangles?

3 triangles.

What kind of triangles?

The big one is isosceles.

What do you know about isosceles triangles?

They have two congruent angles.

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Eventually we showed that the two triangles were congruent using SAS.

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Then we looked at another student work sample.

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This student noted that the triangle is isosceles, but jumped from one pair of corresponding congruent sides to the angle bisector.

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And one other student work sample, where the student noted that the triangles were congruent, but didn’t give a reason why.

Students looked at part b for a few minutes. Then we looked at one last student work sample. What do you wonder about this argument?

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Does S have to be the midpoint?

After working for a few more minutes, students gave each other feedback and then revised their argument based on the feedback.

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Are we going to look at a correct argument for this?

Will you check mine to be sure that it is right?

Last year, students didn’t care so much whether their argument was correct, nor did they care about seeing a “viable argument”. Somehow, figuring out how to improve some of the arguments for part a got them more interested in their argument for part b.

We plan to look at the following five arguments tomorrow.

With what do you agree?
With what do you disagree?

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And so the journey continues … thankful for do-overs from one year to the next.

 
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Posted by on September 20, 2015 in Angles & Triangles, Geometry, Tools of Geometry

 

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Placing a Fire Hydrant

Placing a Fire Hydrant

We’ve used the Illustrative Mathematics task Placing a Fire Hydrant for several years now. Each year, the task plays out a bit differently because of the questions that the students ask and the mathematics that students notice. Which is, honestly, why I continue to teach.

I set up our work for the day as practicing I can make sense of problems and persevere in solving them and also I can attend to precision. If you don’t know how to start at Level 3, use Levels 1 and 2 to help you get there.

SMP1 #LL2LU

SMP5 #LL2LU

In an effort not to articulate all of the requirements ahead of time, I simply asked: where would you place a fire hydrant to serve buildings A, B, and C. Students dropped a point at the location they thought best.

It was then obvious from the students’ choices that they thought equidistant was important.

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This year I didn’t put out tools that students might choose to use. Instead, I set the timer for them to work alone on paper for a few minutes and told them to ask for what they needed. Before I could get from the front of the room to the back, almost every hand was raised to request either a ruler or a protractor. (No one asked for a compass this year. Last year, when I had them out on the tables, lots of students used them.)

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I gave students a few more minutes to work individually with the option, this time, of working with the TI-Nspire software to show their thinking. And at the end of that, I added a few more minutes, asking students to focus on how they could justify that their solution always works. Then I gave them a few minutes to discuss their thinking with a partner.

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I watched (or monitored, according to Smith & Stein’s 5 Practices) while they worked using the Class Capture feature of TI-Nspire Navigator. During that time I also selected and sequenced for our whole class discussion. I wanted some of the vocabulary associated with special segments in triangles to come out of our discussion, so I didn’t immediately start with the correct solution.

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We started with Autumn, who had constructed the midpoints of the sides and then created both a midsegment of the triangle and some medians of the triangle. She could tell that the intersection of the midsegment and medians was “too high”.

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C chimed in that she had constructed lots of midsegments. In fact, she had created several midsegment triangles, one inside the other.

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Next we went to Addison, who not only had created all three medians of the triangle but had also measured to show that the medians weren’t the answer.

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That led to S, who had been trying to figure out when the intersection of the medians would be a good location for the fire hydrant.

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Arienne told us about her approach next. She had placed a point inside of the buildings, measured from the point to each building, and she was moving the point around to a location that would be equidistant from the buildings.

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Reagan talked with us about her solution next. She had constructed the perpendicular bisectors and measured from their intersection to each vertex to show that it always worked.

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I wonder what that point has to do with the vertices. What do you see in the diagram? (I was expecting students to “see” a circle. But they didn’t. They saw a triangular prism.) I wasn’t ready to show them the circle, though. How could I help make the circle visible without telling them? A new question came to me: What if we had a 4th building? Where could we place the building so that the fire hydrant served it, too?

I quickly collected Reagan’s file and sent it out to all of the students so that they could create a 4th building that was the same distance from the fire hydrant as A, B, and C.

While they were working, Janie said, “I have a 4th building the same distance, but how do I place it so that it always works?” (On the inside, I was thrilled that Janie asked this question. It is exciting for students to realize this early on in the course that we are about generalizing and proving so that something always works and not just for one case.)

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How do you place the 4th building so that it always works? What is significant about the location of the 3 buildings and the fire hydrant?

Sofia volunteered that her 4th building always works. (I have to admit that I was skeptical, but I made her the Live Presenter and asked how she made it.) Sofia had rotated building C about the fire hydrant to get d. (How many degrees? Does the number of degrees matter? Would rotating always work? Why would it work?) She rotated C again to get a 5th building between A and B. What is significant about the location of the 5 buildings and the fire hydrant?

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And then they saw it. It wasn’t yet pictured, but it had become visible. All of the buildings would form a circle around the fire hydrant! The fire hydrant is the circumcenter of ∆ABC. The circle is circumscribed about the triangle.

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And so the journey continues … every once in a while finding a more beautiful question.

 

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Placing a Fire Hydrant (2014)

I gave a talk at ASSM back in April entitled The Slow Math Movement. The following is an excerpt from that talk that describes how the Illustrative Mathematics Placing a Fire Hydrant task played out in my classroom last year:

Towards the beginning of our geometry course, we give students a task from Illustrative Mathematics called Placing a Fire Hydrant. Where would you place a fire hydrant to serve all three buildings?

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Through a Quick Poll, students drop a point at the location they think is best. Then we introduce the requirement that the fire hydrant should be equidistant from all three buildings.

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They started on paper, using rulers, folding, and compasses.

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Several of them realized that if they could find the circle that contained all three locations,

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the center would be equidistant (and thus the location of the fire hydrant).

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However, their methods for finding a circle to contain all three points were not very precise

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(which meant they didn’t already know everything they needed to know about triangle centers).

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Next they moved to technology. I watched while they worked using the Class Capture feature of our technology, and using what I learned from Smith & Stein’s 5 Practices for Orchestrating Productive Mathematics Discussions to monitor, select, and sequence the student work for our whole class discussion.

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My students didn’t come into this lesson knowing the vocabulary associated with special segments in triangles, so I purposefully included some incorrect solutions for placing the fire hydrant equidistant from the buildings to bring out that new vocabulary.

Kolton had constructed the midpoints of the sides of the triangle. I made him the Live Presenter so that he could discuss his solution and so that students could learn what a median of the triangle was. His measurements showed that his solution didn’t always work,

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but the dynamic feature of our software let him move the buildings around and begin to consider when the intersection of the medians would be equidistant from the sides of the building.

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Chaney had constructed a midsegment of the triangle, and so we looked at hers next to learn that new term.

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Jameria had constructed the three midsegments of the triangle, creating a midsegment triangle. She was able to tell from her measurements that her solution didn’t always work, either, but we looked at anyway, and I told students that we would learn more about the midsegment triangle later in the course.

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We moved next to Sawyer, who recognized that the correct placement of the fire hydrant should be the center of a circle that contained all three buildings, but we could see from his work that he hadn’t yet figured out how to get a circle through all three buildings.

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Quinn had fashioned a circle through the three points, but still hadn’t actually constructed it.

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Caroline had constructed the perpendicular bisectors of each side of the triangle. She had measured from their intersection, the circumcenter, to each building to show that they were equidistant.

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As the Live Presenter, she started moving the buildings around to show that her solution always worked.

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Then we asked her to construct the circumscribed circle to emphasize that the intersection of the perpendicular bisectors is the circumcenter.

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As Caroline continued to move around the buildings, Gabe asked, “Why would we put the fire hydrant there?” Caroline stopped, and we took a good look at the setup.

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She moved the buildings again, to exaggerate how ridiculous it would be to place a fire hydrant that far away. Our dynamic technology made the students realize that the circumcenter isn’t always the most efficient place for the fire hydrant, even if it is equidistant from the three buildings. And so we began to explore when it makes sense to put the fire hydrant equidistant from the buildings and when it no longer makes sense.

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Take just a moment to contrast the Fire Hydrant task with how I used to teach special segments in triangles. Which one of these is the “Fast Math” option? Which one furthers the Slow Math Movement?

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In his book The Falconer, Grant Licthman says, Questions are waypoints on the path of wisdom. Each question leads to one or more new questions or answers. Sometimes answers are dead ends; they don’t lead anywhere. Questions are never dead ends. Every question has the inherent potential to lead to a new level of discovery, understanding, or creation, levels that can range from the trivial to the sublime. (Lichtman, 35 pag.)

The technology that we use provide the impetus for students to ask questions, which leads to more questions and some answers, from and by the students. I get to watch and listen and push and probe my students by asking more questions.

What can you do this week to further The Slow Math Movement?

[Cross posted on The Slow Math Movement]

 
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Posted by on September 9, 2015 in Angles & Triangles, Geometry, Tools of Geometry

 

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Reflected Triangles + #ShowYourWork

I’ve used the Illustrative Mathematics task Reflected Triangles for several years now. This year students practiced “Show Your Work” (from Jill Gough) along with coming up with a correct solution.

Level 4: I can show more than one way to find a solution to the problem.

Level 3: I can describe or illustrate how I arrived at a solution in a way that the reader understands without talking to me.

Level 2: I can find a correct solution to the problem.

Level 1: I can ask questions to help me work toward a solution to the problem

Which of the directions can you follow exactly to construct the line of reflection? Are there any that need clarification?

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Students wrote first and then constructed their lines of reflection. I used the Class Capture feature of TI-Nspire Navigator to watch. We’ve spent longer on this task in the past, but this year, I only had about 5 minutes for a whole class discussion. Whose work would you select for the whole class to see?

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We started with Kaelon’s construction.

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How is Holly’s construction different? Can you tell what Holly did?

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How is Phillip’s different? Can you tell what he did?

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We are well on our way towards learning how to “describe or illustrate a solution in a way that the reader understands without talking to me”, as the journey continues …

 
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Posted by on September 8, 2015 in Angles & Triangles, Geometry, Rigid Motions

 

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