Visual: SMP-8 Look for and Express Regularity in Repeated Reasoning #LL2LU

Many students would struggle much less in school if, before we presented new material for them to learn, we took the time to help them acquire background knowledge and skills that will help them learn. (Jackson, 18 pag.)

We want every learner in our care to be able to say

I can look for and express regularity in repeated reasoning.
(CCSS.MATH.PRACTICE.MP8)

But…what if I can’t? What if I have no idea what to look for, notice, take note of, or attempt to generalize?

Investing time in teaching students how to learn is never wasted; in doing so, you deepen their understanding of the upcoming content and better equip them for future success. (Jackson, 19 pag.)

Are we teaching for a solution, or are we teaching strategy to express patterns? What if we facilitate experiences where both are considered essential to learn?

We want more students to experience the burst of energy that comes from asking questions that lead to making new connections, feel a greater sense of urgency to seek answers to questions on their own, and reap the satisfaction of actually understanding more deeply the subject matter as a result of the questions they asked.  (Rothstein and Santana, 151 pag.)

What if we collaboratively plan questions that guide learners to think, notice, and question for themselves?

What do you notice? What changes? What stays the same?

Indeed, sharing high-quality questions may be the most significant thing we can do to improve the quality of student learning. (Wiliam, 104 pag.)

How might we design for, expect, and offer feedback on procedural fluency and conceptual understanding?

Level 4
I can attend to precision as I construct a viable argument to express regularity in repeated reasoning.

Level 3
I can look for and express regularity in repeated reasoning.

Level 2
I can identify and describe patterns and regularities, and I can begin to develop generalizations.

Level 1
I can notice and note what changes and what stays the same when performing calculations or interacting with geometric figures.

If we are to harness the power of feedback to increase student learning, then we need to ensure that feedback causes a cognitive rather than an emotional reaction—in other words, feedback should cause thinking. It should be focused; it should relate to the learning goals that have been shared with the students; and it should be more work for the recipient than the donor. (Wiliam, 130 pag.)

[Cross posted on Experiments in Learning by Doing]

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Jackson, Robyn R. (2010-07-27). How to Support Struggling Students (Mastering the Principles of Great Teaching series) (Pages 18-19). Association for Supervision & Curriculum Development. Kindle Edition.

Rothstein, Dan, and Luz Santana. Make Just One Change: Teach Students to Ask Their Own Questions. Cambridge, MA: Harvard Education, 2011. Print.

Wiliam, Dylan (2011-05-01). Embedded Formative Assessment (Kindle Locations 2679-2681). Ingram Distribution. Kindle Edition.

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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 …