Geometry Semester 1                                                            Organized by Lisa Bejarano: @lisabej_manitou

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Unit 1: Constructions & Rigid Transformations

CCSS

1.1 I Know and use precise definitions of geometric terms.

G.CO.1

1.2 I can make formal geometric constructions sited in standards both by hand and using geometry software.

G.CO 12,13

1.3 Given a geometric figure and a rotation, reflection and translation, I can draw the transformed figure.

G.CO.5

1.4 I Understand and can explain the formal definition of rotation, reflection and translation.

G.CO.4

1.5 I can represent transformations in the plane and describe transformations as functions that take points in the plane as inputs and give other points as outputs.

G.CO.2

Enduring Understandings

Essential Questions

·         You can use special geometric tools to make a figure that is congruent to an original figure without measuring.

·         Construction is more accurate than sketching and drawing.

·         Analyzing geometric relationships develops reasoning and justification skills.

·         Objects in space can be transformed in an infinite number of ways and those transformations can be described and analyzed mathematically.

·         Congruence of two objects can be established through a series of rigid motions.

·         Properties of geometric objects can be analyzed and verified through geometric constructions.

·         Judging, constructing, and communicating mathematically appropriate arguments are central to the study of mathematics.

·         What is the relationship between construction and congruency?

·         How do the properties of lines and angles contribute to geometric understanding?

·         How is visualization essential to the study of geometry?

·         How does the rigid motion connect to congruence?

·         How does geometry explain or describe the structure of our world?

·         How do constructions enhance understanding of the geometric properties of objects?

·         How can reasoning be used to establish or refute conjectures?

·         What are the characteristics of a valid argument?

·         What facts need to be verified in order to establish that two figures are congruent?

Learning Target, Instructional task & Standards for Math Practice

1.1

Geometry Vocabulary

High Tech High

 

Students engage with the academic language needed in this unit in a way that taps into their prior knowledge and provides a place to discuss and dialogue with their peers to activate their prior geometry experience.  

SMP.6

1.1

Starting with Definitions

crazymathteacherlady

Using examples & non-examples, students complete Frayer models and collaborate on writing definitions.

SMP.6

SMP.7

1.2

Construction Design Project  crazymathteacherlady

This activity provides an opportunity for students to familiarize themselves with the precision that can be generated using a straightedge and compass

SMP.1

SMP.8

1.2

Construction Castles

Cheesemonkey

“ I created this Constructions Castle project to give students plenty of practice doing constructions while also giving them a chance to develop their understanding of how shapes and angles fit together.

SMP.1

SMP.8

1.2

http://sciencevsmagic.net/geo/

Ancient Greek geometry

A beautifully simple online tool that allows students to experiment with circles & lines to develop understanding of the capabilities of constructions. It's deceptively simple, and quite challenging. Can you construct a regular pentagon in 15 moves or less? And can you construct other regular polygons that are not shown in the challenges, like the 15-gon and 17-gon?

SMP.5

1.2

Euclid, the Game!

Geogebra

This activity is a great introduction to geogebra as well as the basics of constructions while allowing students to develop methods instead of coping steps. Each level introduces a new geometric challenge to construct with only a virtual compass, ruler, and the previous abilities you've discovered.

SMP.5

1.2

Introduction to Constructions

High Tech High

 

The purpose of this activity is to provide students with a teacher guided foundation of construction.  Additionally, students learn to use geometric tools for constructions, helping them to develop and understand how the tools can be used for later geometric work

SMP.5

1.2

Pizza Delivery Region

NCTM Illuminations

The trick to this task is in constructing perpendicular bisectors between locations to ensure customers receive pizzas from the nearest location.  Students must notice this structure and figure out how to apply bisector constructions to increasingly challenging models in to precisely divide the city grid for the pizza company.

SMP.7

SMP.6

1.2

Constructing Shapes

High Tech High

 

 This activity builds on Introduction to Constructions but here students must build on the ideas from the previous activity to make sense of the problem at hand and must attend to precision in explaining why their construction makes the given shape based on how it was constructed and how this connects to the definition of the figure.

SMP.1

SMP.6

1.2

Constructing digitally

High Tech High

Students use their prior knowledge of how to construct the given figures by hand to understand how to use geometric software.

SMP.5

1.3

MATH

High Tech High

 

Activate students’ prior understanding of how to rotate, reflect and translate a given figure. In order to complete the fourth problem, students must look for and make use of structure found in the previous problems.

SMP.7

1.3

Mrs. Pac Man, Robert Kaplinsky

This lesson provides a real-life context for transformations including rotations, reflections, and translations, which are the foundation for students understanding congruence and similarity.  Rather than begin the lesson by defining the terms and identifying them in the game, the goal is to let students initially describe the movements in their own words and then guide them towards a mathematically precise definition.

SMP.6

1.3

Transforming a Pentagon & Answer Key,  High Tech High

 

This activity is useful in case students still struggle with visualizing given transformations and need more practice; this can provide students with an opportunity to demonstrate repeated reasoning and regularity in transformations.

SMP.8

1.3

Reflected Triangles

Illustrative Mathematics

 

This activity has students demonstrate their understanding of the line of reflection as the perpendicular bisector between any two corresponding points as well as a precise and accurate construction of a perpendicular bisector.

SMP.6

1.4

Defining Reflections

Illustrative Mathematics

 

This task helps students transition to a technical mathematical definition of reflection.  This task requires time and patience and is ideally suited for in class group work. If there are mirrors present in the classroom the teacher may wish to have students experiment so that they can see first-hand how the mirror image is similar and how it differs from the original.

SMP.5

SMP.6

1.4

Defining Rotations

Illustrative Mathematics

 

At this point in the unit, students should be very familiar with the visual representation of rotation and should be able to use this task to help transition to the technical mathematical definition.  By providing four possible definitions and having students work through the validity and precision of each.

SMP.6

1.3

1.5

Transforming 2D Figures, MARS

Students describe transformations through writing and images and also see transformations as functions that take points in the plane as inputs and give other points as outputs.  

SMP.1

SMP.7

SMP.6

1.5

Horizontal Stretch of the Plane, Illustrative Mathematics

Students compare a transformation of the plane (translation) which preserves distances and angles to a transformation of the plane (horizontal stretch) which does not preserve either distances or angles.

SMP.2

1.3

1.4

Face Value, Mathalicious

 

Here students are provided with an opportunity to grapple with the idea of something being symmetric and reflecting onto itself and must persevere through the calculations and analysis of the symmetry score.   This activity also provides students with an opportunity to model the ideas facial symmetry through reflection.

SMP.1

SMP.4

Unit 2: Congruence & proof

CCSS

2.1 Specify sequences of rigid motions that will carry a figure onto another

G.CO.5

2.2 Use the definition of congruence in terms of rigid motions to decide if two figures are congruent or not

G.CO.6

2.3 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 (CPCTC)

G.CO.7

2.4 Can explain how the criteria for triangle congruence follow from the definition of congruence in terms of rigid motions

G.CO.8

2.5 Prove theorems about lines and angles

G.CO.9

2.6 Prove theorems about parallelograms

G.CO.11

2.7 Prove base angles of isosceles triangles are congruent

G.CO.10

Enduring Understandings

Essential Questions

Objects in space can be transformed in an infinite number of ways and those transformations can be described and analyzed mathematically.

o The concept of congruence and its connection to rigid motion.

o Congruence of two objects can be established through a series of rigid motions.

Representation of geometric ideas and relationships allow multiple approaches to geometric problems and connect geometric interpretations to other contexts.

o Attributes and relationships of geometric objects can be applied to diverse context.

o Properties of geometric objects can be analyzed and verified through geometric constructions.

·Judging, constructing, and communicating mathematically appropriate arguments are central to the study of mathematics.

o Assumptions about geometric objects must be proven to be true before the assumptions are accepted as facts.

o The truth of a conjecture requires communication of a series of logical steps based on previously proven statements.

o A valid proof contains a sequence of steps based on principles of logic.

 

·         How is visualization essential to the study of geometry?

o    How does the concept of rigid motion connect to the concept of congruence?

·         How does geometry explain or describe the structure of our world?

o        How do geometric constructions enhance understanding of the geometric properties of objects?

·         How can reasoning be used to establish or refute conjectures?

o        What are the characteristics of a valid argument?

o        What is the role of deductive or inductive reasoning in validating a conjecture?

o   What facts need to be verified in order to establish that two figures are congruent?

Learning Target, Instructional task & Standards for Math Practice

2.1[a][b]

Rigid motions & congruence (specifically the activity part of this lesson)

Better lesson

To investigate congruence through rigid motions, students are given a diagram with four congruent triangles. Students will work individually to Identify the rigid motion(s) that can be used to show congruence. They will then write congruency statements for corresponding parts of the triangles and label all congruent parts on the diagram.

SMP 1

SMP 3

SMP 5

SMP 6

2.1

Sequences of Rigid Motions, High Tech High

The purpose of this activity is to allow students to build on their understandings of single rigid motions before moving directly into congruence.  Here, students must attend to precision in their transformations and must also construct viable arguments when explaining how a certain sequence of transformations can be completed in fewer steps.  Additionally, students must use tools appropriately to help them precisely complete the sequence of rigid motions.

SMP 3

SMP 5

SMP 6

2.1

Sequences of Rigid Motions 2,

U of A/High Tech High

Before learning the definition of congruence through rigid motions, it is important that students are familiar with sequences of rigid motions and the idea that there can be multiple ways to carry a given figure onto another.  The activity helps students to look for and express regularity in repeated reasoning in carrying out sequences of rigid transformations.  Additionally, students must attend to precision when completing the transformations to ensure that when comparing different possible sequences the transformed figure is in fact in the correct location.

SMP 6

SMP 8

2.2

Same size, same shape?,

Illustrative mathematics project and description here: Arguing about Shapes Kate Nowak

The purpose of the task is to help students transition from the informal notion of congruence as "same size, same shape" that they learn in elementary school and begin to develop a definition of congruence in terms of rigid transformations. The task can also be used to illustrate the importance of crafting shared mathematical definitions. Note that the term "congruence" is not used in the task; it should be introduced at the end of the discussion as the word we use to capture a more precise meaning of "same size, same shape."

SMP 3

SMP 6

2.3

Properties of Congruent Triangles, Illustrative Mathematics

At this point, students have experience with sequences of rigid motions and what congruence is.  This problem extends those ideas to begin to build the understanding of properties that congruent triangles have.  As explained on the IM website, “The goal of this task is to understand how congruence of triangles, defined in terms of rigid motions, relates to the corresponding sides and angles of these triangles. In particular, there is a sequence of rigid motions mapping one triangle to another if and only if these two triangles have congruent corresponding sides and angles.” Additionally, students will need to reason abstractly and quantitatively while completing this problem; working between the sequence of rigid motions to map one triangle to the other while keeping track of the different parts of the triangles and how they are related.

SMP 2

2.3

2.4

Congruence Theorems, NCTM Illuminations

The purpose of this activity is to provide students an opportunity to experiment with different arrangements of triangle parts (sides and angles) to see what is necessary for triangle congruence.  The goal is that after testing all possible arrangements, students will be able to explain which arrangements provide triangle congruence and which do not.  Through the testing phase, students must persevere through all possible arrangements and should begin to look for repeated reasoning through the different trials.

SMP 1

SMP 8

2.3

2.4

Analyzing Congruency Proofs, MARS

At this point in the unit, students should be familiar with both CPCTC and criteria for triangle congruence from the previous sample activities.  In their explanations about if the given triangles are congruent or not, students must construct viable arguments to support their ideas.  Through this activity and repeated exposure to different triangle criteria, students should be able to look for and make use of structure in identifying whether triangles are congruent or not.

SMP.3

SMP.7

2.4

Why does SAS Work?, Why does ASA work?,  When does SSA work to determine triangle congruence?,  Why does SSS work?, Illustrative Mathematics

These four tasks are grouped together because they all show triangle congruence, each spotlighting one of the criteria for triangle congruence.  In the above activities, students were exposed to the different criteria so some or all of the activities here can be used as needed to ensure that students understand the different criteria.  Because students have seen these ideas in the previous activities, this is a great place for students to look for and express regularity in repeated reasoning through these ideas.

SMP 8

2.4

Angle bisection and midpoints of line segments, Illustrative Mathematics

This problem allows students to use the constructions they know along with criteria for triangle congruence to prove bisectors and midpoints in the given activity.  Students should be able to model with precision the angle and line bisection constructions needed to represent the conditions presented in this task.

SMP 6

2.7

Isosceles Triangle Theorem, Illustrative Mathematics

This task provides students with an opportunity to use their understandings about triangle congruence in a very specific case; the base angles of an isosceles triangle are congruent.

 

2.5

Angles Between Intersecting and Parallel Lines, Khan Academy

The purpose of this section is to provide students with several examples of congruent angles in given contexts.  Depending on where students are and how much practice they need, teachers can either work through all of the provided examples and videos or only those that they see best fit for their students.  Through the various examples and proofs, students should begin to look for and make use of structure relating to a transversal crossed by parallel lines and vertical angles.

SMP 7

2.5

Parallel line & angle mazes Daniel Schneider (Mathy Mcmatherson)

All high school geometry proofs are essentially one big maze. You’re told where to start and where to end – you just gotta connect all the dots and make the jumps in between. So, I experimented with presenting angle relationships with parallel lines that way as well.  Where this leads: I love the fact that there are multiple answers – almost any student can generate their own path and justify their answers. You can also ask some interesting questions – what is the shortest path. Find a path that uses every angle exactly one. Find a path that doesn’t use the same rule twice in a row

SMP 1

SMP 2

SMP 3

2.5

Points Equidistant from a two points in the plane, Illustrative Mathematics

Students should be comfortable with perpendicular bisectors so the open nature of this problem allows students to explore the distance between the endpoints of the line segment to points on the perpendicular line in a familiar context as the build to the understanding of the above proof. This provides a good opportunity for students to engage in “Construct Viable Arguments and Critique the Reasoning of Others.'' Also, students working on this task have multiple opportunities to engage in ''Use Appropriate Tools Strategically'' as the task makes use of geogebra.

SMP 3

SMP 5

2.5

Origami Angles, Tina Cardone

There are 31 angles to find in the diagram (excluding 180 degree angles but including sums of smaller angles. In each case students are asked to find more than the ‘easy’ angles.  They will need to continue working past the point where angle measures are obvious and this is where some students’ ingenuity shines through.

SMP 3

2.6

Quadrilaterals, Khan Academy

This activity explores definitions, conditions and proofs relating to quadrilaterals.  Specifically, there are videos provided to show[c][d][e] that opposite sides and opposite angles of a parallelogram are congruent as well as that diagonals of a parallelogram bisect each other The purpose of this activity is to provide students with an introduction to the congruence properties found within a parallelogram.

SMP 7

2.6

Quadrilateral dots

Fawn Nguyen via Don Steward

“Join the dots to complete these quadrilaterals — where there are options, try to find the one on the grid with the largest possible area.” This task reinforces quadrilateral definitions, properties and area. Leads to student discussions of congruence criteria.

SMP 1

SMP 3

2.6

Congruence of Parallelograms, Illustrative Mathematics

The purpose of this activity is to connect students’ prior learnings about proving congruence in triangles and applying this to parallelograms.  Students can also use their understandings from the previous activity (Quadrilaterals) as they explore different criteria of parallelograms.  As stated on the IM website “This task is ideal for hands-on work or work with a computer to help visualize the possibilities….This task would be ideally suited for group work since it is open ended and calls for experimentation. Thus it provides a good opportunity for students to engage in  “Construct Viable Arguments and Critique the Reasoning of Others.'' Also, students working on this task have multiple opportunities to engage in ''Use Appropriate Tools Strategically'' as they can use manipulatives or computer software to experiment with constructing different parallelograms.”

SMP 3

SMP 5

2.1

-

2.7

Proof Blocks, Proof Blocks

The purpose of this activity is to provide students with a formalized written look at the visual work they have done in this unit[f][g][h].  There are several instructional suggestions for how the proof blocks can be used in the classroom but however they are used, they help students to construct viable arguments and critique the reasoning of others while looking for and making use of the structure necessary for geometric proofs.

SMP 3

SMP 7



Unit 3: Similarity

CCSS

3.1 I can describe the properties of dilations (non-rigid transformation)

G-SRT.1

3.2 I can apply similar triangles to calculate measurements.

G.SRT.2

3.3 I am able to prove that two figures are similar through a series of transformations.

G-SRT.2

3.4 I can use criteria for proving similar triangles

G-SRT.3

3.5 I can prove and use some theorems about triangles

G-SRT.4,

G-SRT.5,

G-CO.10

3.6  I can prove and use slope criteria for parallel & perpendicular lines

G-GPE.5

3.7 I can construct points that divide a given segment into specified ratios.

G-GPE.6,

G-SRT.5

Enduring Understandings

Essential Questions

·         Similar geometric figures have angles that are congruent and segments that are proportional in length.

·         Similar geometric figures can be created by transformations.  All transformations create similar geometric figures.  Dilations, in particular, create figures that are similar, but may not be congruent.

·         Congruence is also similarity. It is just a more specifically defined similarity where the ratio of lengths is 1:1.

·         The processes of proving include a variety of activities, such as developing conjectures, considering the general case, exploring with examples, looking for structural similarities across cases, and searching for counterexamples.

·         Making sense of others’ arguments and determining their validity are proof-related activities.

·         A proof can have many different valid representational forms, including narrative, picture, diagram, two-column presentation, or algebraic form.

 

·         How can transformations help me to understand similarity?

·         What strategy can be used to prove similarity theorems?

·         How do you solve problems using congruent and similar triangles?

·         How do you prove congruent and similarity relationships in geometric figures?

·         How is visualization essential to the study of geometry?

·         How do geometric constructions enhance understanding of the geometric properties of objects?

·         How can reasoning be used to establish or refute conjectures?

·         What are the characteristics of a valid argument?

·         What facts need to be verified in order to establish that two figures are congruent?

·         How can transformations be used to explain similarity?

·         How is similarity of geometric figures applied and verified?

·         What is the relationship between transformations that produce congruent figures and transformations that produce similar figures

Learning Target, Instructional task & Standards for Math Practice

3.1

Photocopy Faux Pas (page 4), Math Vision Project

The purpose of this activity is to provide an open space for the exploration of the properties of dilations without the need for precise definitions just yet.

SMP 1

SMP 2

3.1

How many noses are in your arm? PBS

This activity motivates students to use proportions to determine the scale factor for the statue of liberty without the term being explicitly defined &  apply the scale factor to determine dimensions of other features of the statue.

SMP.1

SMP.5

SMP.6

3.1

Perform Dilations on objects, Engage NY (grade 8 mod 3 lesson 2)

Students learn how to use a compass and a ruler to perform dilations. Students learn that dilations map lines to lines, segments to segments, and rays to rays.  Students know that dilations are degree preserving.

SMP.5

3.1

Dilations using technology, Khan academy

In most assessments, students will be using technology and so, placed here, this task provides them with the opportunity to digest their understanding of dilations and do so in a technology based environment where they are building on previous knowledge built up earlier in the section. Computer based softwares are excellent for problem solving in geometry and this may also provide an opportunity to introduce students to other tools that will be extremely useful as they progress in their geometric thinking.

SMP 5

3.1

Who is the fairest?, NRICH

A more open activity than some of the previous ones, this task could be used in lieu of some of the previous activities to get at some of the same ideas about properties of dilations and what happens when lines and figures are dilated by a point not on themselves.

SMP 7

SMP 8

 

Flip Family (page 13)

The purpose of this activity is to take students’ understandings of dilations and apply it to a situation on the coordinate plane and extend their understanding to the ways that dilations of a specific size can partition a given line segment. Placed here, this activity could serve as a formative assessment to see where students are with dilations, or as another solidifying understanding task to wrap up this section

SMP.5

SMP.7

SMP.8

3.2

3.3

Triangle Dilations (pages 9-11) Math Vision Project

One purpose of this task is to solidify and formalize the definition of dilation. A second purpose of this task is to examine proportionality relationships between sides of similar figures by identifying and writing proportionality statements based on corresponding sides of the similar figures. A third purpose is to examine a similarity theorem that can be proved using dilation: a line parallel to one side of a triangle divides the other two proportionally.

SMP.1

SMP.7

3.1

3.7

Scaling a Triangle in Coordinate Plane, Illustrative Mathematics

The goal of this task is to apply dilations to a triangle in the coordinate plane. Students will need to find the coordinates of a point which cuts a given segment into one third (part a) and into two thirds (part b). Because the centers for the two dilations are vertices of △ABC and the sum of the dilation factors (one third and two thirds) is one, the two dilated triangles share a vertex

 

 

 

 

 

3.3

Defining similarity through angle preserving
transformations
, Khan Academy

 

Placed here, this activity’s purpose is to further interact with the precise nature of rotations, reflections, translations, and dilations with support from khan’s interface. It has the extra motivating factor of seeming like a puzzle where  students’ attention to precision and detail are extremely important as students describe the rotations, reflection, rotations, and dilations that are occurring.

SMP 5

SMP 6

3.5

Golf Ball Problem, Fawn Nguyen

Standard G-SRT.5 calls for students to "use congruence and
 similarity criteria for triangles to solve problems." Both of these task asks students to use similarity to solve a problem in a modeling context that will be familiar to many (Golf and Pool).In order to solve this problem, students must use the fact that when an object bounces off of a wall, the angle of incidence equals the angle of reflection; that is, the angle at which the object hits the wall is equal to the angle at which the object ricochets from the wall forming similar triangles which can help to solve the given problems

SMP 4

3.5

Similar Circles, Illustrative
 Mathematics

This task explores the proof for all circles being similar meaning that there is a sequence of transformations of the plane (reflections, rotations, translations, and dilations) transforming one to the other. If we give the
 plane a coordinate system, then the circles have equations and the transformations can be described explicitly with equations. The goal of this task is to work on showing that all circles are similar using these two different methods, the first visual and the second algebraic. Part (a) of this problem will require students to make sense of a seemingly challenging problem. Part (b) provides an
 opportunity to look for repeated reasoning between the argument for part (a)  and part (b).

SMP 1

SMP 8

3.5

Pigs in a Blanket, Geoff (Emergent
 Math)

In this investigation, students will take a different look at triangles and do some investigating, exploring, and modeling using their knowledge of similar triangles and hopefully come to some previously unknown ideas about midpoints of triangles, namely that connecting them results in 4 congruent triangles that are similar to the larger triangle.

SMP 1

SMP 4

3.3

Similarity transformations, Engage NY HS Geometry, mod 2 lesson 12

Students define a similarity transformation as the composition of basic rigid motions and dilations. Students define two figures to be similar if there is a similarity transformation that takes one to the other. Students can describe a similarity transformation applied to an arbitrary figure (i.e., not just triangles) and can use similarity to distinguish between figures that resemble each other versus those that are actually similar.

SMP 3

SMP 6

3.4

Identifying Similar Triangles

Math Assessment Project

Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle angle criterion for similarity of triangles.

SMP 1

SMP 2

SMP 3

3.5

Rolling Cups,

Math Assessment Project

Read this first by Dan Meyer

This open-ended modeling task has students experiment with rolling cups to develop a formula to determine the radius of the circle formed when rolling a cup given the cup dimensions. Applications of proportions and similar triangles are usually employed. Students are also asked to analyze the responses of others and critique their approaches.

SMP 1

SMP 2

SMP 3

SMP 4


[a]I love your transition from rigid transformations to triangle congruence.  There is clearly a dearth of materials on this sequence and you have unearthed some great ones.

[b]It went well this year. I think that these materials are awesome, but there must also be teacher knowing their students and when to add some notes, journaling, practice, etc. it's also important to know what activities just won't work.

[c]I have difficulty in my class having students see something and retain it.  I like your follow up lessons and have struggled with an initial lesson.  My success with retaining quadrilateral knowledge has been with a 2-3 minute exercise called "What quadrilateral am I?".  Students ask questions and guess.  I restrict them later to types of qualities and over time.

[d]Sounds like a great idea! Have you blogged about this? I can add a link.

[e]Appreciate it.  However, I have not been good about blogging lately and it just another great idea from our a2i team which includes people like David Wees.

[f]I like Proof Blocks.  This row has me wondering if each row is a day.  If that is the case, is this structure introduced at the end of the unit?  Why did you decide that this is the right place?  I have been wrestling with when to introduce proof and would like to hear your thinking.  If each row is not a day, what is the structure.

[g]I think proof is not just a formal list of statements and reasons. Proof

should start early and appear regularly. Asking students to justify their thinking and explain their reasoning is a great introduction to proofs. I consider the proof blocks another tool for student to describe their thinking.

[h]The more I hear your thinking on proof there I agree. I have been taking a similar approach.  However, I once again feel like I jumped into the formal stuff too soon. From what I am piecing together, it seems like you have a good long term path for students to be successful at proof and would like to hear more.