# 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

## 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”? 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. What is significant about the coordinates of the centroid? Students then interacted with dynamic geometry software. 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. After a few minutes, I sent another poll to find out what they figured out. 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 …

## MP7: The Diagonal of an Isosceles Trapezoid 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?” 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. We persevere. Even with the same auxiliary lines, we don’t always see the same picture.  We learn from each other.

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

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

## 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”? When we have a new type of problem to think about, I am learning to have students estimate the answer first.  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.  Students then interacted with dynamic geometry software. 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. I sent another poll.  And then we confirmed student conjectures on the spreadsheet. 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

## 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. 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. 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 …

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

## Structure, Flexibility, and Planning

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

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

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.

Many wanted to share their way with the whole class. 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. 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.

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

## 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. Students started by reading through both parts of the proof, noticing and wondering.  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”. 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? 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. What is a perpendicular bisector of an angle? 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. 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. Eventually we showed that the two triangles were congruent using SAS.

Then we looked at another student work sample. This student noted that the triangle is isosceles, but jumped from one pair of corresponding congruent sides to the angle bisector. 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? 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.

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?

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