# Tag Archives: Standards for Mathematical Practice

## Segments in Circles

Circles: G-C

Understand and apply theorems about circles

2. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.

We used part of the Math Nspired activity Intersecting Lines and Segment Measures for this lesson.

What is the relationship among the four segments created when two chords intersect?

I sent a Quick Poll asking my students to estimate the value of x before we began our exploration. Just over one-third got it correct.

We moved to the technology to see if we could look for regularity in repeated reasoning. The relationship between the segments formed by the chords was not evident to students, even using the technology.

My students are having to learn how to take risks. They enter my classroom afraid of answering incorrectly. When I ask “what do you notice”, they are hesitant to answer because what they notice might not be what they were supposed to notice. We are having to change the mindset of students. It is okay to be wrong; we learn from our mistakes.

One student finally took a risk: I noticed that the segments were proportional.

Okay. Could you give us a proportion? PA:PD=PB:PC.

Let’s look at that for a moment. Look at your screen. Is it true, with the measurements that you have, that PA:PD=PB:PC?

How would we get a proportion like that?

Another student took a risk: We’ve talked about proportions using similar triangles.

Do you see triangles in this picture?

I see ∆PBC and ∆PAD.

(I saw ∆PBA and ∆PCD. It happens so often that my students see differently than I, which always makes me happy that I asked and didn’t draw in my own triangles.)

Could we show that the triangles are similar? How do we show triangles are similar?

Another student took a risk. I can’t remember now what method she named, but it wasn’t a method for proving triangles similar – it was a method that worked to prove triangles congruent. But it is okay that she was wrong. We learned something from that mistake.

We made a list of methods we could use to prove the triangles similar: a sequence of transformations that included a dilation, or SAS~, AA~, SSS~.

What do we know in this diagram? We know PA:PD=PB:PC.

Well, actually, that is what we are trying to prove. So we can’t use that in our proof of why the triangles must be similar. Instead, that will be a result of similarity.

Some students recognized that ∆PDA~∆PCB because of a rotation and dilation. Someone else asked me to number the angles. I see that ∠2 ≅∠5 because vertical angles are congruent.

So are any other angle pairs congruent?

I waited.

And waited longer.

∠3 is inscribed in arc AB.

And so is ∠6.

So they are congruent.

The green marks on the diagram came after the noticing.

So the triangles are also similar by AA~, which makes the corresponding side lengths proportional.

If you look in a textbook, you won’t see PA:PD=PB:PC. Instead, you’ll see PA∙PC=PD∙PB.

A Quick Poll to assess student understanding showed that we were at just over 80% correct.

What is the relationship among the segments formed when two secant segments intersect outside the circle?

We didn’t take the time for a first guess. We went straight to the exploration.

After watching our dynamic geometry software move, I sent a poll.

I’m trying to ease the hurry syndrome, but the seconds are ticking away. We spent a long time making sense of the proof with the chords, and we only have a few minutes left before the bell will ring to end class.

I stopped the poll once half of the class has gotten it correct.

We formalize our conjecture. We look for and make use of structure. What auxiliary lines do we need to show the similar triangles for this diagram?

And then a visual proof of what we mean when we say that the product of one whole secant segment and its outer secant segment is equal to the product of the other whole secant segment and its outer secant segment.

The area of a rectangle with one whole secant segment as base and its outer secant segment as height is equal to the area of a rectangle with the other whole secant segment as base and its outer secant segment as height.

And when one secant segment turns into a tangent segment?

The area of a rectangle with the whole secant segment as base and its outer secant segment as height is equal to the area of square with the tangent segment as its side length.

Students had practice for this last diagram outside of class – and on the bell work at the beginning of the next class.

And so the journey to ease the hurry syndrome continues, sometimes successfully, and sometimes not …

Posted by on February 20, 2014 in Circles, Geometry

## Hopewell Geometry – Misconceptions

Last year I posted student work from the Mathematics Assessment Project task Hopewell Triangles. This year I want to talk about the conversations that we had.

Students worked on the task for 15-20 minutes by themselves. For those who didn’t need that long, I had two other tasks from Illustrative Mathematics on the back of the handout: Seven Circles I and Shortest Line Segment from a Point P to a Line L.

After students worked by themselves, I collected their responses in the TNS document I had sent. They started talking about their results with their groups. Apparently I didn’t make it clear to the whole class that I would be collecting their responses, because everyone had not keyed in their responses, but I looked at the responses that I did get to monitor student progress while they were talking with their groups. I noticed that one student had answered question 4 incorrectly.

So I went to his group and listened to them as they constructed a viable argument and critiqued the reasoning of others. Alex explained how he had arrived at his wrong answer. Can anyone figure out Alex’s misconception?

In our whole class discussion, I made sure that Alex shared his thinking. We all have something to learn from other’s misconceptions and we have created a classroom where it is okay to learn from incorrect thinking. It turns out that Alex used inverse tangent to calculate the acute angle measures. He then concluded that the triangle was right because it had a right angle (since the two acute angles were complementary), not realizing his circular logic in assuming that the triangle was right to use right triangle trigonometry for his acute angle calculations.

Can we show that Alex’s thinking doesn’t work in conjunction with what we know about the other triangles?

Triangle 3 is a 45°-45°-90° triangle, and so its acute angles are 45°. At least half of the students recognized that they had already calculated one of the acute angles for Triangle 1 in problem #2. Then we can show by the Angle Addition Postulate that we have a problem when the shaded triangle is right since 53°, 90°, and 45° don’t sum to 180°.

Most students used the Pythagorean Theorem to show that the shaded triangle was not right, which was our next whole class conversation. One student told us that the hypotenuse of Triangle 3 was 5√2, which he knew because of our work with Special Right Triangles. Another student used the Pythagorean Theorem on Triangle 2 to get its hypotenuse of 7√5. And then another showed us that (5√2)2+(7√5)2≠(15)2. This would have been the end of the conversation in my class when I first started teaching. But because our focus is not only on correct answers but also on logical arguments and understanding, a student stopped us from continuing by asking a question. Why is the hypotenuse of Triangle 2 7√5 instead of 7√3? What a good question. Why is the hypotenuse of Triangle 2 7√5 instead of 7√3?

How did you get 7√3? The triangle has side lengths 7 and 14, so I assumed the triangle was a 30°-60°-90° triangle. Another student cleared up that misconception. Side lengths of x and 2x are for a 30°-60°-90° triangle only when 2x is the length of the hypotenuse.

The last whole class conversation happened because I overhead a group talking about how they knew Triangle 1 was similar to Triangle A. I had asked them to remember that part of the conversation because I was going to ask them to discuss it with the whole class later. One girl said that she knew the triangles were similar because she saw that 3-4-5 was proportional to 9-12-15. But she was having a hard time visualizing how the triangles were similar. How could we show that the two triangles are similar to each other?

How do we define similarity? Two figures are similar if there is a dilation (and if needed, a sequence of rigid motions) that will map one figure onto the other. Can we show that to be true for Triangle A and Triangle 1?

We need several rigid motions along with the dilation to show that Triangle A and Triangle 1 are similar:

Reflect Triangle A about the side that is 3 units.

Rotate Triangle A’ 270° about the vertex of the angle opposite the side that is 3 units. Let’s call that point X’.

Translate Triangle A’’ by a vector that goes from X’’ to the left vertex of the given rectangle.

Dilate Triangle A’’’ about point X’’’ by a scale factor of 3.

I was amazed at the mathematics in this lesson and the opportunity for students to make connections. What a great task! If I had written the task, it wouldn’t have occurred to me to include a triangle that students might mistake for a 30°-60°-90° triangle. Instead, the students were given the opportunity to construct a viable argument and critique the reasoning of others. It might not have occurred to me to include a calculation in an early question (#2) that students could use again in the final question (#4) – I would have likely changed the angle measures to have them perform two different calculations. Instead, the students were given the opportunity to look for regularity in repeated reasoning. If I had written the task and included a triangle similar to one of the others, I doubt it would have occurred to me to make one of them have a different orientation than the other. Instead, the students were given the opportunity to look for and make use of structure.

And so the journey continues, with great tasks like this one to make student misconceptions evident and to correct those misconceptions …

Posted by on February 2, 2014 in Geometry, Right Triangles

## Unit 6: Dilations – Student Reflections

CCSS-M Standards:

Similarity: G-SRT

Understand similarity in terms of similarity transformations

G-SRT 1. Verify experimentally the properties of dilations given by a center and a scale factor:

a. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged.

b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor.

G-SRT 2. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.

G-SRT 3. Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.

Prove theorems involving similarity

G-SRT 4. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.

G-SRT 5. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.

Circles: G-C

Understand and apply theorems about circles

G-C 1. Prove that all circles are similar.

I can statements:

Level 1: I can identify, define, and perform dilations. G-SRT 1

Level 2: I can determine the similarity of two figures using similarity transformations. G-SRT 2, G-SRT 3

Level 3: I can prove theorems about triangles. G-SRT 4, G-C 1

Level 4: I can solve for and prove relationships in geometric figures using similarity criteria. G-SRT 5

Lessons:

6A Dilations

6B Similarity Theorems

6C-6D The Similarity Ratio

6E Pythagorean Relationships

6F Altitude to the Hypotenuse

6G Dilations Performance Assessment – Dilating a line and a circle

6H Dilations Performance Assessment – Bank Shot

6I Mastering

In which Standard for Mathematical Practice did you engage most often during the unit?

Most students chose MP1, make sense of problems and persevere in solving them, and MP 6, attend to precision.

I asked students whether any of the content seemed like repeats of previously learned material.

• Aside from problems that were solved by utilizing the Pythagorean Theorem, all of the content from this unit was new to me.
• I’ve gone over pythagorean theorem multiple times in the past year, however the rest of this is new to me, and frankly quite difficult.
• I can honestly say I learned everything as new in this unit. This was entirely new for me. I feel I had an easier time learning this material than the other material in other units. I caught on very quickly in this unit compared to others
• the only thing I already knew was that for a figure to be similar all the side lengths had to be proportional.
• The only thing i knew about unit 6 before we learned it was that dilations have to do with making an object larger or smaller.
• There was nothing that was repeated in this lesson. I knew what a dilation was, but I’d never done anything with it, so basically I knew the first 15 minutes of this unit, and the rest was completely new to me.

Which lesson helped you the most in this unit?

• There is not one lesson or activity in particular that stands out to me, but I do think the extra homework helped, even though the thought of extra homework is not a very appealing thought.
• 6C-6D was very helpful. If I hadn’t understood similarity ratio, I would have not understood any of the material we learned this unit. I have never gone into so much detail about similarity and ratios. I’m glad I understand them more.
• The day when we used wax paper to decide if objects were similar or not helped me because it let me visually understand it.
• The homework was extremely helpful. It provided practice for me, and showed me what I needed to work on. It deffinitely made the lesson more clear.
• I think that lessons 6A and 6B, dilations and similarity theorems, really prepared me for all the material that was taught after that. Learning these two skills taught me the basics for learning the rest of the unit.

What did you learn during this unit?

• you have to be very precise and make sure you evaluate your answer
• I have learned about the similarity ratio and how to dilate a line and a circle.
• I have learned my favorite transformation: Dilations. I love dilations now and they are my favorite. I figured out how to tell if figures are similar through a rigid motion and dilation or just through a dilation. I learned that the hypotenuse altitude is the geometric mean between the two sides it divides, which helped a ton in figuring out measurements of the sides of a right triangle. I think this was my favorite unit and I learned so many new things.
• I have learned how to use, apply, and perform dilations to geometric figures. I have also learned that listening to the opinions and ideas of others can lead you to find the answer or the reasoning behind certain answers.

And so the journey continues …

1 Comment

Posted by on January 28, 2014 in Dilations, Geometry

## Triangles & Polygons Unit – Student Reflections

CCSS-M Standards:

Congruence G-CO

G-CO 8

Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.

G-CO 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.

G-CO 11

Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.

I can statements:

Level 1: I can solve problems using congruent triangles.

Level 2: I can explain criteria for triangle congruence.

Level 3: I can determine the congruence of two figures using rigid motions.

Level 4: I can prove theorems about angles in triangles.

Level 1: I can recognize properties of special quadrilaterals.

Level 2: I can use properties of special quadrilaterals to solve problems.

Level 3: I can prove theorems about special quadrilaterals.

Level 4: I can determine sufficient conditions for naming special quadrilaterals.

Lessons:

Congruent Triangles

Congruent Triangle Proofs Using Rigid Motions

Interior and Exterior Angles in Polygons

Parallelograms – Proving Properties

Rhombi  Kites

Trapezoids

Performance Assessment – PARCC Angle Bisector Proof  Floor Pattern

Mastering

Which Standard for Mathematical Practice did you engage most often during the unit?

Most students chose MP1, make sense of problems and persevere in solving them, and MP 7, look for and make use of structure.

I asked students whether any of the content seemed like repeats of previously learned material.

• I don’t remember working with this information before now, other than knowing that a square has four right angles and four congruent sides
• Of course I already knew what all of the different shapes were, but I’m not going to count that because I’d never actually used the shapes to solve complex and important mathematical problems. We actually used squares and parallelograms instead of simply knowing what they were.
• In each lesson we learned I learned something new that I had not already known
• I felt that I already knew some things about congruent triangles, but the lesson on them led me to a deeper understanding of them. I also knew things about the interior and exterior angles in polygons, but that lesson helped me to reason abstractly about them. I also knew the different types of quadrilaterals

Which lesson helped you the most in this unit?

• I don’t remember exactly which lessons and activities were which. However, I know the extra practice in the packets and the quick polls helped for me to see what I did or did not know.
• In 5A/5B about Interior and Exterior Angles in Polygons, i had never truly understood the ongoing continuation of degree sized of adding a side to a polygon. That sentence didn’t make much sense, but what i mean is like a triangle is 180 degrees, a rectangle is 360, a hexagon..etc.
• 5G helped because it gave me problems to solve with the entire unit, which gave me a perspective on what I understood and what I didn’t.
• As with every and all lessons, the usage of interactive diagrams and pictures provides a visual representation of word descriptions and reasons for accurately defining special polygons, determining the congruency of triangles, and finding the values of interior or exterior angles of any reqular polygon.
• The floor plan activity was very helpful, it really helped me understand what we were actually learning.

What did you learn during this unit?

• I have learned that I must attend to precision when talking about quadrilaterals because of their special characteristics.
• The one part about this unit that stood out was that any polygon with one pair of parallel sides is a trapezoid. I always thought trapezoids had to be isosceles trapezoids until this unit, so that’s very interesting to me.
• That everything you do you have to have proof for it and that you have to have a reason how you got that.

I have recently read Transformative Assessment by James Popham. http://www.amazon.com/Transformative-Assessment-W-James-Popham/dp/141660667X

Popham discusses levels of formative assessment that shift through the teacher using formative assessment to adjust instruction (level 1), to the student using formative assessment to adjust learning strategies (level 2), to the classroom of students using formative assessment to ensure that all students in the class are meeting the standards and making adjustments to help each other when that is not happening (level 3), to implementation of formative assessment throughout the school (level 4). I have talked about this model with my students, and so I am pleased to hear them using language like “helped for me to see what I did or did not know” and “gave me a perspective on what I understood and what I didn’t”. They are beginning to pay closer attention to what they have learned and what they have not learned. They are beginning to make adjustments when they haven’t yet met the learning targets.

And so the journey continues, as we learn how to learn …

## Student Reflections for Unit on Angles & Triangles

I’ve been asking my students to reflect on each unit as we finish so that I can have some feedback on what changes to make for next year. I collect their responses through an assignment in Canvas.

1. I can use inductive and deductive reasoning to make conclusions about statements, converses, inverses, and contrapositives.
2. I can use and prove theorems about special pairs of angles.
3. I can solve problems using parallel lines.
4. I can prove theorems about parallel lines.

Which Standard of Mathematical Practice did you use most often in this unit?

The top 3 responses:

Make sense of problems and persevere in solving them.

Construct viable arguments and critique the reasoning of others.

Look for and make use of structure.

Think back through the lessons. Did you feel that any were repeats of material that you already knew? If so, which parts?

Many students recognized vertical angles, parallel lines, and the triangle sum theorem as something they had used before. However, they also recognized that we were learning why those relationships worked – and not just that they worked.

One student replied: None of these were repeats so to speak. I already knew what parallel lines and the different types of angles were, but I had no idea why or how or how you use them so much in math. I never sat there and thought, “Oh, we learned this last year, I don’t have to pay attention.”

Think back through the lessons. Was there a lesson or activity that was particularly helpful for you to meet the learning targets for this unit? If so, how?

• I really like 3A Logic for this lesson. It allowed me to think through things and use “logic” to solve problems.
• Interactive activities were particularly helpful as they visually showed the justification of postulates and theorems such as the Angle Addition Postulate, the Definition of Vertical Angles, etc
• Logic in the beginning helped out with the whole unit because that’s mainly what everything linked to.
• All of the lessons were helpful, but I found that the logic lesson was particularly helpful. It helped me to understand what a proof is and get introduced to them. Proofs were very important to meeting the learning targets for this unit.
• The symbolic logic was helpful in the case that if I could figure out one part, it could lead me to unlocking other parts, kind of like a puzzle.

What have you learned in this unit?

• I have learned how to prove things that I already knew were true.
• I have learned all the objectives. I am also better at solving proofs and proving theorems. I learned about triangles and theorems about those as well as the way I can draw in Auxiliary Lines to find angles in problems.
• Do not procrastinate on homework.
• I’ve learned that math isn’t always numbers and equations; it can be proving things through logic and reasoning and postulates and many other things that I really didn’t know existed before this. I learned WHY parallel lines are parallel and WHY the alternate exterior and alternate interior and corresponding angles are the same and WHY all of the angles of a triangle added together equal 180 degrees. I also learned how to apply this knowledge in geometry and how it can help us in real life.
• I have learned that I cannot give up when problems are complicated. I also learned that my way isn’t always the only way.

I think these reflections are good news as the journey continues ….

## Angle Proofs

CCSS-M Congruence G-CO 9

Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints.

How do you go about having students prove theorems about lines and angles in your class? We certainly don’t formally prove every theorem in our geometry course that we use, but we prove a lot of them. What we do that I believe is more important than proving theorems is exploring the hypothesis to make sense of possible conclusions. I remember memorizing a lot of theorems out of our geometry textbook when I took high school geometry. I don’t think that is what my students will remember from our class. In fact, I know it is not.

So bear with me as I share two student reflections about the course:

“I have never really liked math and wasn’t looking forward to this class because of it. Surprisingly, I actually like it much better than I thought I would. Although geometry has confused me sometimes this year, I have actually felt proud when I figure something out that I thought I couldn’t. I think I have been able to get a new mindset and have been able to switch from thinking regularly to analytically when I enter your class. I can now look at a problem, and try to think about it in different ways until I figure out how to solve it.”

“The main memory that I will keep of Geometry class is so complex that everything I say will simply be an understatement.  Although I will attempt to describe it.  The main memory I will keep of this class is the memory of the first math class that I enjoyed.  I liked the fact that in that class I sometimes had the bravery to answer questions.  I was not nervous to have the wrong answer in front of the class or Mrs. Wilson.  I loved the fact that Mrs. Wilson incorporated enjoyable activies throughout her lessons. This was the first math class I really engaged myself in.”

So we make a big deal about accepting postulates and proving theorems in our class deductive system for geometry. We looked at a list of Euclid’s axioms and postulates and noted that Euclid’s 5th postulate took way longer to make sense of than any of the others.

And then we set out to prove theorem #1.

Vertical angles are congruent. Most of my students know that vertical angles are congruent. But they have not thought about why vertical angles are congruent.

What do we know to be true in this diagram? Students noted that the sum of angles 2 and 3 is 180°. The sum the sum of angles 3 and 4 is 180°. Students noted that the sum of angles 1, 2, 3, and 4 is 360°. How do we know this? We can observe it using technology. We used the angle measurement tool to verify the measures of other angles in the diagram. But how do we know it will always be true? So we proceeded to formalize that vertical angles are congruent … making use of the Angle Addition Postulate that we had agreed to accept into our deductive system without proof. And making use of Euclid’s axioms – if equals are subtracted from equals, the remainders are equal – that we now call the Subtraction Property of Equality. Eventually we proved that angles 2 and 4 were congruent.

Next I sent students a Quick Poll: ∠3 and ∠2 are both supplementary to ∠1. If m∠3=50°, what is m∠2?

And asked them to make a conjecture: If two angles are supplementary to the same angle, then …

And then we proved that if two angles are supplementary to the same angle, then they are congruent to each other..

Given: ∠2 is supplementary to ∠1. ∠3 is supplementary to ∠1.

Prove: ∠2≅∠3

What about the converse?

Is it true?
It didn’t take students long to decide that it wasn’t true. Their counterexample was to suppose m∠2=m∠3=180.

So we moved next to parallel lines, remembering Euclid’s 5th postulate as we began to explore what happens when parallel lines are cut by a transversal.

What can we conclude about lines a and b? lines m and n?

We agreed that when corresponding angles are congruent, the lines are parallel. Do we agree enough to let this in to our deductive system as a postulate?

I heard from another geometry teacher that her students thought of corresponding angles as a translation, which goes right along with figures being congruent if there is a rigid motion that maps one onto the other…and a good reason for us to communicate with each other about what we do and say in the classroom. That idea hasn’t come up in my classroom yet, but I will make sure that it does.

And so from there, we proved that alternate interior angles are congruent when parallel lines are cut by a transversal. We didn’t prove that alternate exterior angles are congruent, since the proof is similar. We proved that consecutive interior angles are supplementary when parallel lines are cut by a transversal. We didn’t prove that consecutive exterior angles are supplementary, since the proof is similar.

And then on the summative assessment, I asked the following, which I believe gave evidence that students are beginning to learn to create arguments and they are not just memorizing proofs to recreate on a test.

Suppose you are structuring your deductive system in such a way that you postulate the following about alternate interior angles:

If two parallel lines are cut by a transversal, then alternate interior angles are congruent.

Prove the Corresponding Angles Theorem using the Alternate Interior Angles Postulate and without using results from any parallel line theorems that we proved in class.

Our last part of parallel lines was specifically to think about auxiliary lines. The progression of questions follows.

I am not convinced that I did them in the right order, although maybe it is significant that all but one student got the 3rd problem correct. Should we have started with the 3rd problem instead of the 1st? Since we did start with the 1st, the effect was subtle for problem #3. Many students drew immediately drew auxiliary lines when they saw the 3rd problem. After the 3rd problem, I drew my students’ attention to the math practice look for and make use of structure, which explicitly states drawing auxiliary lines to support an argument. Student shared their thinking, as their arguments for finding the measure of angle 3 were not all the same.

Are we wasting our time proving a few of these theorems in class? We don’t write formal two-column proofs for every theorem that we prove, but we do come up with a logical argument as to why the theorems are true. What we do is reason abstractly and quantitatively. What we do is construct a viable argument and critique the reasoning of others. My students recognize that we are figuring out geometry instead of being told geometry, whether they appreciate it yet or not.

And so the journey continues …

Posted by on October 28, 2013 in Angles & Triangles, Geometry

## The Shipwrecked Surfer

I had saved this task from the August 2006 Mathematics Teacher to use at some point. One of the authors has a link to the article/paper here.

The problem:

A surfer, shipwrecked on an island in the shape of an equilateral triangle, wants to build a hut so that the sum of its distances to the three beaches is minimal. Where should the hut be located? (See Figure 1.)

I like how the problem is worded; students first have to make sense of the problem, and most have to persevere in solving it. I also like that students have to think about what we mean when we say the distance from a point (hut) to a line (beach).

Students started their work on paper. Many students chose the centroid as the point of interest. Others chose randomly. They used rulers and compasses to test their conjectures. One student figured out pretty quickly that it didn’t matter where you put the point. It doesn’t matter where you put the point? No – the sum of the distances from the point to the sides is the same no matter where you put the point.

We moved to our dynamic geometry software. We placed a point inside the triangle. We constructed the perpendicular from the point to each side. We measured the length of each segment on the perpendicular from the point to the side. We calculated the sum of the lengths. We moved the point around inside the triangle and verified that the sum is the same no matter where we put the point.

So we can put the point anywhere. Now what? What’s the significance of placing the point anywhere? What’s the significance of the sum of the distances?

What’s significant about the current location of the point of interest for this student?

Someone figured out that the sum of the distances equals the length of the altitude. Is that important? The altitude is the length of the segment perpendicular from any vertex to the opposite side.

Look what happens if we redefine our point of interest on the altitude. What type of polygon is IBXC? Compare the value of XI to the sum of CI and BI.

Somewhere around here the bell rang.

And that was it. We haven’t revisited the task, although a few students reflected on the task as part of their problem solving points for this quarter:

“The addition of the length to the sides from any random point in a triangle is equal to the altitude because the altitude is exactly that. If you put a point at the top corner of a triangle and add the distance from there to all three sides you get the altitude. The altitude is just adding all the distances to the sides at a certain point.”

“The entire reason the point inside the triangle was equidistance from the sides was because the triangle was equilateral. I’m not sure if this is more complicated than it seems, but if all of the angles are the same, all the sides are the same, and any way you rotate and turn the triangle it still looks the same, then a point inside the triangle is going to be equidistance from the sides because every point on the plane is equidistance. Think about it this way: If I move my point .2 cm to the left, then the distance from the side on its left is shortened by .2 cm, but the distance from the side on right is lengthened by.2 cm. Up, down, left, right, every distance’s total is equal.”

One of the best parts about teaching is that we get “do-overs”. Not always this year and with these students…but definitely next year with new students. How could I make this conversation more meaningful – and get to the mathematics sooner? I could pose the task the class period before. Students can do their paper work on the task outside of class – and come into class ready to share their conjectures. That would get us to the technology sooner – and to the deep exploration for which the technology provides the opportunity.

And so the journey continues…