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

 

Standards:

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

 

Standards:

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.

 

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

 

Standards:

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.

 

Circles – Student Reflections

Unit 8 Student Reflections

I can statements:

1. I can use relationships between angles and arcs in circles to solve for missing measures. (100% strongly agree or agree)
2. I can use relationships between secants, chords, and tangents in circles to solve for missing measures. (100% strongly agree or agree)
3. I can use similarity to calculate arc length and area of a sector. (92% strongly agree or agree)
4. I can prove relationships between secants, chords, and tangent in circles. (96% strongly agree or agree)

Standards:

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.

4. (+) Construct a tangent line from a point outside a given circle to the circle.

Find arc lengths and areas of sectors of circles

5. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.

 

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 look for and make use of structure second most.

 

Think back through the lessons. Did you feel that any were repeats of material that you already knew? If so, which parts?
8A Angles and Arcs
8B Tangents and Chords
8C Angle Measures
8D Segment Lengths
8E Pi
8F Arc Length and Sectors
8G Performance Assessment
8H Mastering

• The only part that was a little familiar was 8E pi.
• I’ve never gone in depth with circles, so no, none of the material was repeats, except maybe a few spots when we learned what pi was.
• 8E was a similar of a repeat of what we learned in Algebra because we had to find pi in multiple situations. It made it much easier for me to understand. Before then, I did have a small misunderstanding of what I needed, but after remembering what I had learned in eighth grade and then I was able to understand more. This was a lesson that showed me that algebra and geometry were very identical.
• The Pi lesson also had some repeated information, but it was nice to learn about the Pi ratio in detail.
• I have had little experience up until this year with circles, other than memorizing the formulas.

 

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?
8A Angles and Arcs
8B Tangents and Chords
8C Angle Measures
8D Segment Lengths
8E Pi
8F Arc Length and Sectors
8G Performance Assessment
8H Mastering

• It helped me understand circles more when we used real life situations.
• Pi was an integral part of the unit and it helped me kind of piece things together. Finding degree measurements in a circle was like a puzzle, when you got one piece you could move on to the next.
• I believe the day where we thought through ways to find the center of the circle gave me insight into the relationship between the circle and the accuracy necessary to correctly talk about it. It also gave me a chance to apply what I had learned previously to find an accurate solution.
• All of the lessons were very helpful for the understanding of the chapter, especially 8A, the beginning lesson, which became the “backbone” for the later lessons.
• I think lesson 8B helped a lot because it introduced tangents and chords, which was a totally new concept for me.

Some students feel like the practice assignments really helped them make sense of the unit as a whole:

• All of the homework activities helped me understand the unit.
• I find it wasn’t the lesson itself, but the homework. The homework challenged me to really think and on a few of them I had to search how to do them. It really helped me make sense of problems on the test and on the lessons after some homework.
• The 8H homework was actually probably the most helpful of any of the things I did. I thought I knew this unit pretty well, until I went and made a 5/10 on my first try. It really helped me go back, review, and relearn the things we’d been doing because I had some very skewed ideas about circles before that lesson.

Some students feel like the Performance Assessment tasks really helped them make sense of the unit as a whole:

• The performance assessment was very helpful to me because in a sense it was a combination of all the previous lessons. This also was a way to make myself quicker at solving the problems and figuring out short-cuts. This was a great review to prepare for my test, which I believe that really was an asset for me. The test was much better with the performance assessment.
• The Performance Assessment was very helpful because it let me see which parts of the unit I didn’t understand as well as the others and showed me what I needed to work on before the test.
• Every lesson this unit was helpful, but the Performance Assessment and the Mastering lessons helped the most. In the lessons before these, we learned a couple things, but in these lessons, we learned how to combine everything that we learned to find the correct answers to challenging problems.

What have you learned during this unit?

• I learned how to calculate the arc length of a circle using the relationship of it to the whole circle.
• In this unit I have deepened my knowledge of Pi, which is one of my favorite numbers and how to recognize tangents. I have also learned how to find arc measures and how to find an angle measure in a circle.
• I have learned the relationships between radii, chords, tangents, and secants in circles. I’ve also learned so many different ways to solve for missing links within circles. I can also find missing lengths outside of the circle. I find that this unit was the toughest to this point.
• … I’ve also learned that’s circles are more complicated than they seem at first.
• During this unit, I learned about the relationships between arc lengths and the angles that intercept them. I also learned how tangents and chords relate to circles, and how to figure out their lengths. Another thing I learned was what Pi was and where it comes from. A fourth thing I learned was how to determine the arc length and area of sectors in circles.

 

And so the journey continues … trying to determine what is best for students. Do I take out the pi lesson next year, since students had some knowledge of pi coming in to high school geometry? Or do I leave it, since several students noted that they enjoyed learning about pi in detail? Eventually, I’ll have students who should have an informal derivation of the relationship between the circumference and area of a circle in grade 7 (7.G.B.4), but until then, I’ll probably keep going over at least part of the lesson.

 

Unit 7: Right Triangles – Student Reflections

I have just read through student reflections for our unit on Right Triangles.

Unit 7 – Right Triangles

Level 1: I can use the Pythagorean Theorem to solve right triangles in applied problems. G-SRT 8

Level 2: I can solve special right triangles. G-SRT 6

Level 3: I can use trigonometric ratios to solve right triangles. G-SRT 6, G-SRT 7

Level 4: I can use trigonometric ratios and special right triangle ratios to solve right triangles in applied problems. G-SRT 8

Similarity: G-SRT

Define trigonometric ratios and solve problems involving right triangles

G-SRT 6. Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.

G-SRT 7. Explain and use the relationship between the sine and cosine of complementary angles.

G-SRT 8. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.

Lessons:

7A – 45-45-90 Triangles

7B – 30-60-90 Triangles

7C – Trigonometric Ratios

7D – Solving Right Triangles

7E – Performance Assessment (Hopewell Triangles)

7F – Mastering Right Triangles

7G – Assessing Right Triangles

Most students felt like the Math Practices they used most in this unit were Construct viable arguments and critique the reasoning of others and Attend to precision.

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

• Trigonometry is a completely new subject for me, so no, none of the lessons were repeats to me. I had no idea about anything except for the Pythagorean Theorem honestly.
• A student from Iran: yes, I leaned part 7A,7B and 7D in middle school, but the difference was that in middle school we just taught how to use Pythagorean theorem and that the side of triangle opposite to 30 angle is half of h, … which were formula but i didn’t know about sin and cos.
• The only part of this lesson that I knew how to do before was the Pythagorean Theorem.

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?

• Trigonometric ratios and solving right triangles were the most helpful lessons because we learned the most new information in these lessons. What we learned in these lessons also applies to all triangles, not just a special type of right triangle.
• The self-check bell ringer that we did a few days ago was very helpful to me. When I know that one of my answers are wrong, I persevere in solving it correctly.
• When we finally put it all together and we were solving right triangles really helped me to finally grasp the whole idea. I had slowly learned and built a solid base and the solving right triangles lesson put it all together for me.
• I believe that the Trigonometric Ratios helped me the most because they allow me to solve a triangle with a different set of information.

Our Trigonometric Ratios lesson came from Geometry Nspired. I’ll write a post about it some day.

What have you learned during this unit?

• I have learned how to use sine, cosine, and tangent, and also the purpose of these formulas. Before I was taught what the formulas were used for, I was always curious as to what they meant on the calculator. Now not only do I know, but I also know how to work problems using sine, cosine, and tangent with accurate results.
• I have learned that there are “special” right triangles, and that there are quicker ways to solve these triangles. I also learned about trigonometry, which is something completely new to me. It was definitely new, and a little hard since it was the first time I saw it.
• I have learned how to solve right triangles, how to do trigonometry, and how to think about the problem before I try to draw it out.

What I have learned during this unit?

I have learned that the “I can” statements are really important for students. We reviewed them daily so that they could see where we were and where we were going. We are slowly making progress towards having students recognize what they are learning (I can …) and how they are learning it (Standards for Mathematical Practice). And so the journey continues …

 

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 …

 

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 …