Tag Archives: I have learned

I have learned …

Students aren’t kidding when they ask, “When will we ever need to know this?” In How the Brain Learns Mathematics, David Sousa suggests that students need a reason to move information from short-term memory to long-term memory. What opportunity do we give our students to reflect on what they are learning and why during class?

Many teachers give Exit Tickets, which can give teachers good information about what students have learned. However, I’ve observed many exit tickets that are more useful for teachers than they are for students. If the exit ticket requires a calculation, when do students find out whether what they’ve submitted is correct? Immediately? Or the next time class meets? How many students then complete homework using wrong ideas?

Exit Tickets can be good formative assessment. In fact, Sousa also notes that closure in a lesson shouldn’t be students packing up their backpacks and walking out of the door. Closure needs to be a cognitive process – students need to think about what they have learned and what questions they have, connecting what they have learned in class today with what they have previously learned and maybe even to what they will learn. Exit Tickets can provide students an opportunity to cognitively think about what they are learning.

My question is what types of formative assessment are we using throughout the class period, instead of just at the end of class?

Are you familiar with Dr. Sousa’s brain research on the Primacy/Recency Effect? In essence, it shows that we remember best what we learn first in a learning episode; we remember second best what we learn last in a learning episode; and we remember least what’s in the middle of the learning episode. Think about how the typical math class has been set up. Students come in, and teachers go over homework (prime learning time). At the end of class, students practice (second prime learning time). In the middle of class, teachers teach the new material for the lesson (least prime learning time).

His research shows that 20 minutes is the ideal length for a learning episode. I teach on a block schedule, and so I find that I must be deliberate about planning shorter (20 minute, when possible) learning episodes within the block.

We were finishing up a unit on Angles & Triangles in geometry earlier this week. We begin each class with an opener of questions that students work through with their teams. I collect their responses, show them the solutions, they try to correct misconceptions with their teams, and then we talk all together about any remaining misconceptions. After the opener (first learning episode) each day, students glance through our learning goals for the unit so that they can think about what they know and what they still need to know.

photo 1

Learning goals:

I can use inductive and deductive reasoning to make conclusions about statements, converses, inverses, and contrapositives.

I can use and prove theorems about special pairs of angles. G-CO 9

I can solve problems using triangles. G-CO 10

I can prove theorems about angles in triangles. G-CO 10

I can solve problems using parallel lines. G-CO 9

I can prove theorems about parallel lines. G-CO 9

I can solve problems using congruent triangles. G-CO 8

I can explain criteria for triangle congruence. G-CO 8

Because it was the last day of the unit, I asked students to answer a Quick Poll letting me know what they have learned and what they still need to know. The more I use “I can” statements for learning goals, the more I notice that they give us a common language for talking about what we can already do and what we can’t do yet.

photo 2Screen Shot 2014-10-20 at 10.00.30 AM

i have learned that i can solve problems using triangles, i still need to know how to prove theorems          1

i still need to touch up on the statemfnts and postulates            1

A.) I have learned to make conclusions, find the measures of angles, and etc..

B.) I still need to know the process in constructing parallel lines on the calc. in greater detail           1

about vertical angles            1

i have learned parrell line

i still need to know alot        1

i still have trouble ex triangle congruences           1

1.i learned prove theorems about special angles

2.i need to work on inductive and deductive statements, theorems about angles in triangles           1

i have learned conditional statements. i still need to go back over them                       1

learned how to prove why things are what they are

still need to know how to correctly prove anything from a given           1

i have learned the conditional statements

CO9 G-CO10 G-CO9,8,8

i need work on the true false charts           1

I have learned how to construct parallel lines using a point.      1

L symbolic logic

NTK proofs    1

how to find exterior angles of triangles           how to form theorems   1

i learned how to do ratios in a triangle.

need to know how to prove theroms          1

i have learned more about parallel lines cut by a transversal i still need to know more about constructing my own proofs         1

i learned about the types of hypothesis. i still need to know the different angle terms.          1

i have learned how to work with ratios.

i stll need to know how to form theorms on my own.       1

I have learned converse, inverse, conditional, and contrapositive statements. I need to learn when to use certain postulates in order to complete proofs.            1

I have learned how to prove statements using postulates. I still need to know how to explain criteria for triangle congruence.  1

i can prove theorems about angles in triangles. explain criteria for triangle.   1

how differemt types of angles are equal anb the different type of statements

i need to know the difference between converse inverse and contrapositive statements      1

learned conditional, converse, invese, and contrapositives.         1

I have learned how to solve problems using triangles. I still need to know how to do well on tables.           1

how to construct parrelel lines

how to write a hypothesis and conclusion in its different forms and determine their truth value    1

how to identify logical statements

how to do proofs       1

i need to work on converse inverse conditional contrapositive   1

i have learned how to construct parallel lines.

i still need to know how to prove the truth value of a statement.           1

learned-how to construct parallel lines

need to know-idk      1

i have learned to solve proplems using parallel lines. i still need to learn how to prove problems.   1

what aternate interior angles are;

how to figure out truth talbes.        1

i have learned if p then        1

I took the information about what students still need to know and used it to structure the rest of the class period, instead of just going through review problems in the order I happened to put them together.

A few years ago, Jill Gough and her colleagues experimented with students and faculty taking a brain break every 20 minutes to tweet what they are learning … you can read more about it here.

What will you do to ensure that you are maximizing the learning episodes in your classes?

And so the journey continues, with thanks to @jgough for making me reflect on how often I do formative assessment throughout a class period.


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Unit 11 – Modeling with Geometry

Student Reflections

I can statements:

Level 1: I can create a visual representation of a design problem. 100% Strongly Agree or Agree

Level 2: I can decompose geometric shapes into manageable parts. 94% Strongly Agree or Agree

Level 3: I can estimate and calculate measures as needed to solve problems. 100% Strongly Agree or Agree

Level 4: I can use geometry to solve a design problem and make valid conclusions. 100% Strongly Agree or Agree



Modeling with Geometry G-MG

Apply geometric concepts in modeling situations

1. Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).

2. Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).

3. Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).


Which Standard of Mathematical Practice did you use most often in this unit? In which other Standard of Mathematical Practice did you engage often during the unit?

Students answered that they used make sense of problems and persevere in solving them the most and model with mathematics next.


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

11A Volumes of Compound Objects

11B 2D Representations of 3D Objects

11C A Tank of Water

11D Popcorn Picker

11E Hot Coffee

11F Two Wheels and a Belt

  • I didn’t think any of these were repeated. We really looked at geometry and used it to solve hard real world problems, and I’ve never had to do that before.
  • I do not feel like this was a repeat. Many of us knew about basic areas and volumes of course, but in this unit we went into depth with them and learned how to get the dimensions of more complex shapes. I think we covered new material, because we have mainly been going over coordinates, trig, and triangles previously and this seemed like a new and fun topic to cover for us.
  • I already knew how to find the volumes of compound objects from unit 10.


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?

  • 11F Popcorn Picker was very helpful in helping me learn the targets for this unit. It helped me realize that although two objects may have the same surface area, their volumes may not be the same.
  • Popcorn Picker definitely helped me understand how exactly different dimensions affect the volume of a cylinder even though the dimensions are nearly the same. Using the piece of paper to compose a cylinder using 8.5 and 11 as two different circumferences as well as the height helped me see that the volumes will be different.
  • The coffee one helped me because it made me talk with others at my table and look for ways to solve the problem.
  • Although I enjoyed the popcorn activity, I feel like the most helpful activity was the one where we cut out the cards and thought through the gradual shapes of the water as it emptied the top figure and filled the bottom figure.  It allowed me to work with a team to reason out our arguments and working with my peers in a collective effort enlightened me with thoughts and ideas that I had not previously thought of or would have otherwise ventured to discuss.
  • I really liked the Hot Coffee unit. I understood it well, and it was a good problem to work and figure out. It was also really good for me to make sure to use the right units and convert correctly, which I don’t do sometimes.


What have you learned during this unit?

  • This unit helped me to realize how much I’ve learned this year in geometry and how to do many things like finding volumes and areas of different shapes.
  • How to use the least amount of information to find the need item.
  • I’ve ;earned how to divide a complex geometric objects into parts and calculate It’s volume. I can find out the necessary information needed to solve this kind of problem and how to use them to solve the problem. I can apply math to every day life and model with mathematics. I can also make visual representation of a design problem.
  • I have learned to attend to precision. Throughout not only the homework lessons but also in class, I learned to be careful and slow down. I often hurried over the question and did not take into account the measurements. For example, I hardly ever noticed when the question gave dimensions in feet but asked for the answer in inches. It’s not a hard concept, but it requires patience and effort that I was trying to shortcut on.
  • During this unit, i learned a lot about using what i already knew and combining that to solve difficult problems.
  • I’ve learned how to break down 3D shapes into simple 3D shapes so I can get their volume, and how to do actual geometry problems with all of the things I’ve learned this year. I’ve learned what the different cross sections of objects can look like, and I learned that the world’s largest coffee cup help 2015 gallons of coffee.
  • I learned that I need to model with mathematics more often.


And so the journey continues … figuring out how to provide more opportunities throughout the course for my students to model with mathematics.


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Unit 10 – Geometric Measure and Dimension

Student Reflections

I can statements:

Level 1: I can use formulas to calculate area and volume. 100% Strongly Agree or Agree

Level 2: I can identify cross-sections of 3-D objects, and I can identify the 3-D objects formed by rotating 2-D objects. 90% Strongly Agree or Agree

Level 3: I can explain the formulas for area and volume. 100% Strongly Agree or Agree

Level 4: I can calculate the area and volume of geometric objects to solve problems. 100% Strongly Agree or Agree



Geometric Measurement and Dimension G-GMD

Explain volume formulas and use them to solve problems

1. Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments.

2. (+) Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures.

3. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.

Visualize relationships between two-dimensional and three-dimensional objects

4. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.


Which Standard of Mathematical Practice did you use most often in this unit? In which other Standard of Mathematical Practice did you engage often during the unit?

Students answered that they used look for and make use of structure the most. They used make sense of problems and persevere in solving them and reason abstractly and quantitatively next.


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

10A Length Area

10B Surface Area

10C Volume

10D Enlargements

10E Cross Sections

10F The Best Box

10G Performance Assessment

10H Mastering Arc Length and Sectors

  • I already knew length and area. I had been taught the formulas for surface area, but I didn’t actually understand why those formulas worked until this unit. Cross sections were completely new to me.
  • I felt like I already knew some of the information for calculating volume and area of 3D forms, but in this unit I learned why and how the formulas work.
  • The length and area was a bit of a repeat of what we’ve learned in the past. It was nice to review and be able to bring to memory what we’d already learned as a basis for the new things we were going to learn this unit.
  • This Volume lesson was a tiny bit of something I already knew, but I never got as much into it as we did. Before, I just learned the formula and never understood. I also only ever learned the volume of a rectangular prism.
  • The very first lesson in which we used the basic formulas was a bit of a repeat from past years. However, in the past we had to memorize the formulas, whereas this year we were given the opportunity to understand the why the formula is what it is. I find that understanding it helps more than simply memorizing it.


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?

  • Yes, Length and Area helped me with Surface Area and Volume by making sense of structure within the problem.
  • When you got the orange peels and separated them into four circles to show how the surface area of a sphere worked, it really helped me understand how surface area works. After that I started thinking about how the formulas worked instead of just memorizing them.
  • 10G Performance Assessment helped me out the most. It summed up most of the chapter and helped my apply what I learned to many different problems.
  • I think the lesson of volume helped the most in understanding the formula. When we talked about the different figures fitting into one another it helped me understand why this certain formula was there. It helped show me that all formulas are taken from a common one.


What have you learned during this unit?

  • In this unit I have learned how to use surface area, lateral area, volume, and cross sections. I now have a better understanding from where the formulas come from by looking at the figures and in a way putting together the puzzle pieces to come up with the final formulas.
  • I learned exactly how area, surface area, and volume are all different.
  • I have learned how to calculate lateral area, surface area, and volume of forms and I also learned why the formulas work and not just to plug in numbers to those formulas.
  • I have learned how to identify cross sections and explain the formulas for surface area and volume.
  • I’ve learned how to use formulas to calculate area and volume and why we use those formulas. I’ve learned how to make sure to convert problems to the same dimensions and same ratios, and I’ve learned what cross sections are and the different ones in different 3D figures.
  • I have learned to understand the formulas and not just memorize, but really understand them. I can use these formulas to solve word problems and figure out tough questions. I’ve also learned about cross-sections and how they differ for each 3D shape. I have also understood the surface areas of 3D shapes and how they work with all shapes.
  • During this unit, I have come to understand the geometric formulas, and I believe that understanding will help me later on in tests like the ACT if I were to forget the formula that I memorized in 7th grade, I would remember the concepts and understandings of why the formula is what it is.


And so the journey continues … still struggling to find a balance between teaching conceptual understanding of mathematics and using mathematics to solve problems … although I do think the students’ language when reflecting on their understanding of geometric measure and dimension is a good sign.


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Unit 9 – Coordinate Geometry

Now that school is out, I am going back through our last few student surveys to make notes for next year.

I can statements:

Level 1: I can represent and use vertices of a geometric figure in the coordinate plane. 100% Strongly Agree or Agree

Level 2: I can use the equation of a circle in the coordinate plane to solve problems. 96% Strongly Agree or Agree

Level 3: I can use slope, distance, and midpoint along with properties of geometric objects to verify claims about the objects. 96% Strongly Agree or Agree

Level 4: I can partition a segment in a given ratio. 81% Strongly Agree or Agree



Expressing Geometric Properties with Equations G-GPE

Translate between the geometric description and the equation for a conic section

1. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.

Use coordinates to prove simple geometric theorems algebraically

4. Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2).

5. Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point).

6. Find the point on a directed line segment between two given points that partitions the segment in a given ratio.

7. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.


Which Standard of Mathematical Practice did you use most often in this unit? In which other Standard of Mathematical Practice did you engage often during the unit?

Students answered that they used look for and make use of structure the most. They used make sense of problems and persevere in solving them and model with mathematics next.


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

9A Lines

9B Quadrilaterals

9C Triangles

9D Area & Perimeter

9E Circles

9F Translating Circles

9G Polygons

9H Loci

9I Coordinate Geometry Performance Assessment

9J Mastering Coordinate Geometry – Partitioning a Segment

  • Yes and No, for me most of the material was new and different. I learned how to partition, and use loci. I already knew the equations but in the course of Unit 9 I learned how to use equations like the Pythagorean theorem to find the side area and perimeters of a triangle.
  • I already knew how to find the area and perimeter of simple geometric shapes but not all the shapes.
  • Pretty much none of these were repeats. I’ve heard of bits of these lessons, but I learned so much more during all of the units that I felt like those bits don’t count.
  • I already knew how to find the slope and do other things in 9A. I learned it in 8th grade Algebra and it was a nice break from just learning new things everyday and just having to review that day.
  • The lesson about the equations of lines and the lesson about area and perimeter were things that I had been taught before, but I was taught them in different ways that weren’t as in depth as what we went to in class.
  • Yes, I did feel that content from numerous lessons (namely 9A, 9B, and 9C) was derived from lessons that I have previously covered either in this class or a previous math-related class. For instance, dealing with parallel lines and perpendicular lines, as well as distinguishing between the two by looking at the x-intercept of an equation’s standard form, was a very prominent concept in the algebra I course I took last year. Also, as were the concepts of calculating distance, midpoint, and slope, all of which were key concepts last year. Finally, I felt that observing shapes on a coordinate plane, as was practiced in lessons 9B and 9C with quadrilaterals and triangles, respectively, resembled a lesson that we covered in November/December as part of units 4 and 5, in which we specifically observed each shape and its properties.


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?

  • There was no particular activity that really sticks out in my mind, but simply watching how others worked the problems on the board really helped. Many people brought up ways to do things that I never thought of, and a majority of the time, their way was more effective than mine.
  • Yes, I think in unit 9E when we had to write a definition for circle in quick paw helped me to get a better understanding of what a circle is and to understand why how I defined it was wrong.
  • 9H was one of the hardest ones for me to do but I think it helped me a lot because it made me think harder than the others.
  • Just like in most all the units, the mastering lesson always helps me. This just lets me go back through everything and make sure I understand it.

What have you learned during this unit?

  • I learned how to gather measurements for shapes from their coordinates. Also, though, I learned to really persevere in solving problems. This unit was one of the harder ones for me, and a lot of times I wanted to give up. I kept trying, though, and when I would eventually understand a problem, it was worth it.
  • I have learned how the coordinate plane and geometry are 2 sides of the same coin and that you can use graphs to calculate geometric properties.
  • I learned that you can find the area and perimeter of all shapes not just the simple geometric shapes.
  • I learned a lot in this unit. I learned about all the shapes and how to use equations to solve problems involving them. I learned what loci was which was new and kind of confusing. I also learned how to partition a ratio for a given segment which was cool because it took a lot of different ways to look at the segment that i wouldn’t have thought of before learning this. This unit contained a lot of review from past years, but also a lot of new material that i had never seen before.
  • If this unit has taught me anything, it is that geometry and algebra can be collaborated so that multiple relationships between and details of certain points, lines, and shapes can be proven, all made possible by borrowing the coordinate plane, distance formula, midpoint formula, and slope formula, all in which we became fluent through two years of algebra.


We have struggled to know whether it is best to have a separate coordinate geometry unit or whether we should be using coordinate geometry all year. To me, it is kind of like wondering in a calculus class whether you do early or late transcendentals. I like late transcendentals, because students go back through differentiation and integration towards the end of the course, and they really have to think about which one they are using and why. It turns out to be a nice culminating review that wouldn’t happen if we had been differentiating exponential and logarithmic functions along the way.


And so the journey continues … always trying to figure out what works best for student learning.


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Unit 2 – Tools of Geometry

Last year we started with this unit. This year, we decided to start with our unit on Rigid Motions, since students have some familiarity with transformations from middle school. I noticed that students were much more willing to look for regularity in repeated reasoning and make sense of problems and persevere in solving them during the Tools of Geometry unit because they had some practice with those practices during the last unit.

CCSS-M Standards:

G-CO 1

Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.

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 12

Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment and constructing a line parallel to a given line through a point not on the line.

G-CO 13

Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.

G-C 3

Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.

I can statements:

Level 1: I can use basic terms in geometry: point, line, plane, polygon, inscribed, circumscribed.

Level 2: I can create basic geometric objects based on their definitions: median, angle bisector, altitude, perpendicular bisector.

Level 3: I can use tools and methods to perform formal constructions.

Level 4: I can solve problems using points of concurrency.

We learned about the tools of geometry through a few applications – Placing a Fire Hydrant & Locating a Warehouse, both of which come from Illustrative Mathematics. We used the Special Segments activity from Geometry Nspired, and we used The Shipwrecked Surfer problem published in the August 2006 Mathematics Teacher. I plan to write a separate post about each of the main lessons.

We spent the rest of our time on constructions – really paying attention to what is congruent as we did each construction. I wanted the constructions to come from a context, instead of just passing out a list of steps for a construction and asking students to do the construction because it was on our list of standards.

I have already looked at the student reflections for this unit. While the majority of students still believe that the practice they most used was make sense of problems and persevere in solving them, the second most used practice was use appropriate tools strategically (which, for a unit on constructions, means that students really thought about which practices they were using and didn’t just mark the first one on the list).

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

  • Everything that I learned this unit was completely new to me.
  • I felt that I didn’t already know any of the material covered; it all seemed new to me.
  • I didn’t know any of this but I learned a lot!
  • Absolutely none of them were repeats for me. There were parts of each unit I’d already learned and heard about, but the overall idea of each lesson was new to me.
  • All of the lessons were excellent introductions into learning about and exploring circumcenters, centroids, orthocenters, and incenters. They required the usage of different techniques and critical thinking to achieve the answers. Finding these various properties of triangles required usage of terms and theorems from the past chapter about distance, perpendicular bisectors, justifications by measuring, etc.
  • Most of the information this unit was new. There were terms that I was already familiar with such as perpendicular and bisector, but I didn’t know all of the information I could gain just by saying that a line is a perpendicular bisector, an angle bisector, an altitude, or a median.

Which lesson helped you the most in this unit? Most students said that The Shipwrecked Surfer helped them the most.

  • The shipwrecked surfer really helped understand all of the different segments and points and centers. For us to have to bring together the median, altitude, angle bisector, and perpendicular bisector and their corresponding centers, so to say, showed me how they all work and when they can all be the same. It made more sense to see it like that.
  • I would say that 2A, 2B, and 2D helped the most because they were realistic situations.

What did you learn during this unit?

  • I learned how to use the incenter, circumcenter, and centroid to solve real-world problems.
  • I learned a lot that I didn’t know. I really enjoyed this unit because of this. I can easily find perpendicular bisectors, altitude, medians, and all the points of concurrency and other things we learned in this unit.
  • Other than the various terms and the application of said terms, I learned not to immediately give up when a problem doesn’t practically solve itself.
  • During this unit I learned many new mathematical terms. I also learned how to classify a triangle by using information other than the degree of it’s [sic] angles.
  • I have learned basically everything that the teacher has taught me. I didn’t know much of it, but I knew enough where I didn’t feel completely lost. I learned a lot of vocabulary words.

I think it is significant that the last student said he “learned a lot of vocabulary words”. When I talk about teaching geometry through exploration, I often have teachers ask about the vocabulary. For many, geometry is all about vocabulary – and assessing whether students know definitions and notation. The biggest difference in how I was taught geometry and how I teach geometry is that we don’t start with the vocabulary. We work through meaningful tasks (the students’ words – not just mine) in which the vocabulary will surface. When the vocabulary does surface, I make a big deal about it. I write the word down on the board. I suggest that they write the word down in their notes. I ask what the object will buy us mathematically, and I sometimes find myself saying how you would see the word defined or the object notated if you were to look in a textbook. I have a list of vocabulary in my lesson plans that needs to surface in the lesson, but that list shows up for the students one word at a time, as we get to it in our exploration, instead of as a list of words to define from the glossary before (or even worse, during) class.

And so the journey continues, right or wrong …



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Reflections on Unit 1 – Rigid Motions

I created a survey in our Canvas course for students to reflect on what they had learned in this unit.

CCSS-M Standards:

G-CO 3

Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.

G-CO 2

Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).

G-CO 4

Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.

G-CO 5

Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.

G-CO 6

Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.

G-CO 7

Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.

I Can statements:

Level 1: I can identify and define transformations and composite transformations.

Level 2: I can perform transformations and composite transformations.

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

Level 4: I can apply transformations and composite transformations to figures in the coordinate plane.

Level 4: I can map a figure onto itself using transformations.

Screen Shot 2013-10-13 at 5.00.05 PM

Every student answered “strongly agree” or “agree” to each of the I can statements.

We focused on learning math using the Standards of Mathematical Practice. I asked students which practice they used most often during this unit.

Screen Shot 2013-10-13 at 5.00.34 PM

55% of students said that they most used make sense of problems and persevere in solving them.

The students have some experience with transformations in middle school. Could they tell that we were deepening their understanding of rigid motions? Did any of the lessons feel like repeats of material they had previously learned?

Screen Shot 2013-10-13 at 5.00.46 PM

  • None of them were complete repeats because I learned about them more detailed than before.
  • Before now, I had a very general knowledge of transformations, but actually putting this knowledge to practice was new.
  • I had already learned most of the basics of transformations. I knew reflections were flips, translations were slides, and so on and so on. However, I’d never gone in depth with the mathematical reasoning behind these transformations and their causes and effects, which brought a whole new light on the subject.
  • I believe that the basic ideas of 1A-E I knew but I still had room to grow on the knowledge these subjects.
  • I felt that I already knew how to do translations and reflections, but in this unit, I noticed new things about them such as that a reflection changes the orientation of the pre-image and that a translation keeps the same orientation as the pre-image. I also knew about rotations, but I wasn’t great at actually preforming a rotation.

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Most students answered that What’s My Rule was most helpful. Several others noted that the lesson on Rotations was helpful.

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  • I learned how to really think about math differently. I learned to double check my work- A LOT. I learned how to tell counterclockwise and clockwise apart. I learned translations and how they move, reflections and how they work, and rotations and how they twist. I learned about matrices and how complex they can be, although I’m still a little confused (I think I have it, but I’m not so sure). I learned the first parts of geometry.
  • I have learned how to reflect a figure across the y=x line and the y=-x line.
  • I learned how to map figures onto itself and reflect on lines other the the x and y.
  • I learned some formulas to help with reflections. I also learned what directions a figure should be rotated if I’m only give the degrees to rotate.
  • I learned how to see if shapes on a coordinate plane are congruent, and how to map something onto itself using rigid motions.
  • A lot of things in this unit were repeats of what I have learned in the past, but there was so much more that I didn’t know about all the transformations. For one, I didn’t even think about the distance from each side to the line of reflection in the past. I just knew the simple things like the points were congruent. I’ve learned a lot and couldn’t type it all. People who don’t learn anything within a whole unit are crazy because you can always learn something even if it’s just a small little thing. I would say that I’ve decided not to give up on hard problems like I would in the past. I didn’t really learn that, but I think it’s important to geometry and, really, everyday life.

I wonder (maybe too often) whether what we are doing in class is important. The student reflections give me evidence that my students are recognizing new content and also connecting it to what they knew before our high school geometry class.

And so the journey continues…


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