Tag Archives: sbg

Year 12, Day 0

This week teachers headed into school to prepare for next week and students’ return. This will be my twelfth year!

My teaching this year will include more computer science than ever before:

  • AP Computer Science Principles (full year)
  • AP Computer Science A (full year)
  • Foundations of Computer Science  (spring)
  • Computer Integrated Manufacturing (fall)
  • Precalculus independent study (fall, three students, three separate periods)

I’ll also be working with our new engineering teacher and our librarian+new-computer-science-teacher to help them with their lessons, and collaborating with two geometry teachers around standards-based-grading.

Extracurricular activities and competitions:

  • Coding Club (app development, cybersecurity, & more)
  • Women’s Transportation Seminar’s “Transportation You!” Mentoring Program
  • TRAC bridge builder competition
  • CyberPatriot competition
  • STEM Competition
  • possible (in my mind, I want to do each of these this year): Cyber Movie Mondays, Saturday AP & PLTW study groups, Girls Who Code club
  • probably several others…

Ongoing projects that will occupy some of my time this year include:

  • Comp Hydro (teaching hydrology and flooding through computational simulations & modeling, in partnership with the Baltimore Ecosystem Study)
  • MyDesign (engineering design process app and learning management system, in partnership with NSF & the University of Maryland)
  • Internet of Things project to measure air quality and other environmental factors in schools (in partnership with Cool Green Schools, Johns Hopkins University, and Morgan State University)
  • Continuing work toward my Master’s Degree in Computer Science (taking “Artificial Intelligence ” course this semester)
  • Baltimore City Engineering Alliance, a nonprofit 501(c)(3) we created to provide opportunities to Baltimore City students to further their engineering education, and for which I am treasurer

School starts for students on Tuesday, after Labor Day for the first time in my twelve years here teaching in Baltimore. Wish us luck!

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SBG Update

OK. I haven’t talked much about Standards-Based Grading (SBG) since last September. I’ve changed some things about how I implement SBG, which I shall describe here, along with how things went.

Geometry, Fall 2010

Here’s how I set up my Geometry class procedures–SBG-related, anyway–for my fall semester class. We had thirty-two skills, which were mostly assessed in class three times each. I converted students’ SBG grades into a percentage for reporting out by first averaging them and then using a cubic relationship (the third one from this post) to match that average to a traditional percent scale.

The students pretty much did not initiate any re-assessment. Less than five skills were retaken outside of class time. I think for the most part they were confident that, between the next two in-class retakes, they would improve just by being in class and doing the daily assignments and warm-up reviews. But even if they didn’t improve (or even regressed), the students did not seek out help nor did they come in to re-assess. They were content just to let the 1 or 2 be averaged in. And mostly their scores were high, in the 80s [those who attended class anyway].

Algebra 2 with Trigonometry, Spring 2011

For several reasons, including combating this apathy about un-mastered skills, for my spring Algebra 2 with Trigonometry (A2T) class I made a major change in how I converted SBG skills grades into percentages. Instead of averaging grades and then scaling, I set cut-offs of how many skills need to be mastered to earn a particular grade. Rubric here. This was also in part prompted by criticism of averages as being antithetical to the nature of SBG by the Science Goddess and @mthman.

Other than this big change in grading, I mostly kept the same procedures from fall to spring. There were more skills (blog posted here in draft form, final form here), so we only visited each skill twice in class via our weekly quizzes.

Many more students re-assessed this semester. But many waited until it was too late to seriously improve their grade. For this reason, and because we only assessed each skill in class twice instead of three times, grades were generally lower this semester than in my fall geometry class.

And there were still a few students who seemed lackadaisical about their lack of skills mastery and never came in to re-assess even though they were failing or scraping by with a D-. Perhaps I didn’t motivate them well enough by extolling the glories of SBG and how it revolutionizes grading? Can anyone refer me to a motivational-type speech you use with your students about how SBG helps them?

Another new piece is that I had set some prerequisite skills (mostly from Algebra 1 but a few from Geometry) that in my mind are so fundamental and which the main skills of A2T build upon. To promote my students’ review of these prerequisites so that I could help them build new skills upon that foundation, I required that students master these prerequisites before they would score any points for core skills (again see the rubric I used). But many students did not really take the prerequisites seriously. By which I mean, they did not review work from Algebra 1, and make sure all prerequisites were mastered within the first week. Some waited until the middle of the quarter to remediate the prerequisites, others until the last week of the quarter–finally being motivated to fix them then so they didn’t fail.

Is it fair for me to set this separate category of skills without which they cannot earn any points? I thought so, since they should not be in an Algebra 2 class without being able to understand key Algebra 1 material. If it is fair, how can I make the students understand the value of the prerequisites in supporting further learning and make sure to really review and master them the first week? Could I refuse to let them sit for other quizzes (on core A2T skills) until their prerequisites are mastered? Or should I get rid of the distinction between core and prerequisite skills entirely?

I welcome your thoughts and criticisms.

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Principles of Engineering Skills

So I’m thinking of giving a skills list a try in my Principles of Engineering (POE) course this semester too. It wouldn’t be full-on standards-based grading (SBG), since in such a project-driven class I need projects and reports to be the major component to student grades, but I think I can adapt a skills list for quizzes (and occasional outside-of-quiz skills demonstration). I’m tentatively planning 50% projects, 25% skills quizzes, and 25% portfolio–including engineer’s notebook.

Below find my first draft for a skills-based outline of how I intend to teach POE this semester. I will be teaching a mix of ninth and twelfth graders, with backgrounds ranging from Algebra I to Precalculus. This will be tough, as POE is the most math-intensive of the PLTW engineering courses. I must try to teach advanced math applications while not boring my students out of their minds, while at the same time exposing students to the great concepts and societal role of engineering and its subfields.

To all the former, current, and future engineers out there, I welcome your input on the skills listed below. Likewise to engineering high school teachers across the country (for your reference, I’ve mixed up the unit order due to equipment lacks: my order goes Unit 2,4,3,1). Or anyone else with an opinion on engineering education.

While I am constrained somewhat by PLTW’s POE curriculum, I do have some choice in what I emphasize and in which skills I test. Are these skills phrased well? Are they representative of what engineering is all about? Are the major subfields of engineering represented (this is a survey course)? Are there any that are too vague (or too narrow) or seem like they don’t belong?

I’m still welcoming feedback to my Algebra 2 with Trigonometry skills list here for about one more day, so my more mathematically-minded readers may like to head over there to ponder and critique.

POE Skills List

General STEM Skills

  1. Solve equations for a single variable
  2. Substitute numbers for variables in algebraic formulae
  3. Measure lengths and angles to appropriate precision (given the context of the application and the accuracy of the tool)
  4. Use trigonometry to solve for missing sides or angles
  5. Apply the Pythagorean Theorem to find unknown sides in right triangles
  6. Use trigonometry to find unknown sides & angles in right triangles
  7. Use the digital dropbox on TS3/Blackboard to submit work
  8. Identify problems to be solved in an engineering context
  9. List multiple possible solutions to engineering problems
  10. Evaluate each possible solution based on specifications & test results
  11. Show knowledge of, and skillful application of, the engineering design process
  12. Show knowledge of various careers in engineering and other STEM fields

Unit 2 – Materials & Structures

  1. Identify five types of bridges by name, definition, and/or picture
  2. Split a force vector into its x- and y-components
  3. Calculate the centroid of various shapes
  4. Calculate forces and moments acting on various objects
  5. Pick appropriate formulae relating to stress, strain, and material testing
  6. Analyze stress-strain graphs to determine material properties
  7. Calculate bridge efficiency
  8. Analyze a bridge for structural and material strengths and weaknesses

Unit 4 – Statistics & Kinematics

  1. Collect and analyze data using statistical measures of center and variance
  2. Calculate speed and velocity
  3. Calculate the effect of gravity on velocity and position
  4. Analyze horizontal and vertical components of projectile motion

Unit 3 – Control Systems

  1. Create flow charts to represent a process
  2. Identify inputs and outputs in a control system
  3. Identify elements of a flow chart or RoboPro program and their key attributes
  4. Utilize branches in a flow chart or RoboPro program
  5. Utilize variables in a flow chart or RoboPro program
  6. Interpret a flow chart or RoboPro program
  7. Identify open and closed loop systems
  8. Demonstrate an understanding of pneumatic and hydraulic power

Unit 1 – Energy & Power

  1. Calculate ideal mechanical advantage for each simple machine
  2. Calculate actual mechanical advantage and efficiency for each simple machine
  3. Calculate gear ratio
  4. Demonstrate an understanding of electricity and electrical circuits (series/parallel)
  5. Use Ohm’s Law and Kirchhoff’s Laws to calculate resistance, current, and voltage
  6. Calculate work, energy, power, and power efficiency
  7. Demonstrate an understanding of the laws of thermodynamics and thermal energy transfer
  8. Demonstrate knowledge of alternative and renewable energy sources

A total of 40 skills. What do you think?


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Algebra 2 / Trig Skills

Hi! I’m looking for some feedback on what skills I should use in my standards-based grading Algebra 2 with Trigonometry (A2T) course spring semester. I just threw together the following list as a rough draft, and do expect to edit it extensively over the coming week.

A note on organization: I’m planning to focus on algebra third quarter and analysis fourth quarter (did I miscategorize any skills?). Fourth quarter tends to be the shortest quarter, mainly due to our state standardized tests, so it has fewer skills. The first six skills each quarter are meant to be prerequisite skills from Algebra I, Geometry, or even middle-school math; I will hold students accountable for knowing and remembering these skills, but aside from a miniature review I do not intend to teach skills #1-6. I’ve italicized skills that seem weak or a little too easy, and may be combined with others or even eliminated. I’ve bolded skills that seem more advanced or non-essential and possibly not for every student. I’m toying with the idea of requiring students demonstrate mastery of certain numbers of core skills and advanced skills to earn different letter grades.

ALGEBRA (especially of the second degree)!

  1. Graph a line given its equation
  2. Determine the equation of a line from its graph
  3. Solve a linear equation in any form
  4. Solve a system of two linear equations
  5. Manipulate algebraic expressions to simplify or expand
  6. Substitute numbers for variables in an algebraic expression
  7. Convert between scientific and standard notations of numbers
  8. Apply the rules of exponents to simplify an expression
  9. Find the degree and leading coefficient of a polynomial
  10. Determine the end behavior of a function
  11. Combine like terms to add or subtract polynomials
  12. Use the distributive property to multiply polynomials
  13. Use long division to divide polynomials
  14. Factor a monic quadratic expression
  15. Factor a quadratic expression with any leading coefficient
  16. Characterize the shape of a parabola given its equation
  17. Graph a parabola given its equation
  18. Determine the equation of a parabola given its graph
  19. Solve quadratic equations in factored form using the zero product property
  20. Solve quadratic equations in vertex form by isolating the variable
  21. Solve quadratic equations in standard form using the quadratic formula
  22. Solve quadratic equations in standard form by completing the square
  23. Combine like terms to get an equation in standard form
  24. Apply quadratic equations to physics or other real-world scenarios
  25. Plot a complex number in the complex plane
  26. Add and subtract complex numbers
  27. Multiply complex numbers
  28. Determine the magnitude of a complex number
  29. Solve quadratic equations with complex roots
  30. Add and subtract matrices, and multiply a matrix by a scalar
  31. Multiply matrices
  32. Find matrix determinants
  33. Find the inverse of a matrix
  34. Solve matrix equations
  35. Solve systems of linear equations
  36. Graph linear inequalities
  37. Solve systems of linear inequalities by linear programming
  38. Plot the graph of various conic sections based on its equation
  39. Identify key characteristics of each conic section
  40. Explain the relationship among the conic sections


  1. Determine a function’s domain, range, maxima, and minima
  2. Determine a function’s zeros and y-intercepts
  3. Determine the end behavior of a function
  4. Convert between tables of values, equations, and graphs of functions
  5. Apply the Pythagorean Theorem to find unknown sides in right triangles
  6. Use trigonometry to find unknown sides & angles in right triangles
  7. Use laws of sines and cosines to find unknown sides & angles in any triangle
  8. Solve entire triangles given three pieces of information
  9. Apply triangular trigonometry to real-world scenarios
  10. Draw and measure angles in standard position
  11. Convert between degrees and radians
  12. Find coordinates and slope where an angle meets the unit circle
  13. Solve trigonometric equations (including multiple solutions)
  14. Use common trigonometric identities
  15. Sketch the graph of trigonometric functions
  16. Find period and amplitude of periodic functions
  17. Plot points using polar coordinates
  18. Convert between polar and Cartesian coordinates
  19. Plot various polar functions
  20. Evaluate exponential expressions
  21. Apply the rules of exponents to simplify an expression
  22. Identify characteristics of exponential functions
  23. Use exponential functions to model real-world scenarios
  24. Convert between exponential and logarithmic equations
  25. Evaluate logarithmic expressions
  26. Apply the rules of logarithms to simplify an expression
  27. Solve exponential and logarithmic equations
  28. Use logistic growth to model real-world scenarios
  29. Identify the effects of common function transformations
  30. Graph functions and their inverses
  31. Use inverse operations to solve equations
  32. Compose functions
  33. Discuss the effect of transforming, inverting, and composing functions on domain and range

I’m looking for feedback of any kind now, all the way from how to rephrase a skill better and more specifically, to what quintessential A2T skills are missing that I need to add right away. Also, are there some skills here that I can get rid of entirely? Even with the idea of prerequisite/core/advanced skills, 73 seems like way too many (I had only 32 skills this fall for geometry).

[PS I promise a post up soon reflecting back on what worked and what didn’t from semester 1’s courses, and looking ahead to my plans for the new semester’s courses, including how those plans reflect on SBG and PBL.]


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Fractal Difficulties

As you may know from my posts planning out the skills and projects for my fall course on geometry, I recently completed a 2-week project on fractals with my students. I had never taught fractals before, so most of this experience was completely new, and I have a lot that I could have done better.

I plan to follow this post up with discussion of some successes of the fractal project, but today I plan to focus on the negative aspects.

  • Poor motivation. After the first two days introducing the Sierpinski Triangle, the Sierpinski Carpet, and connections between the former and patterns in Pascal’s Triangle, I put together a presentation about how fractals actually represent growth and objects in the real world better than many basic geometrical objects. While there was some pushback about this (stop signs are octagons, buildings are rectangular prisms, the Koch snowflake doesn’t look like real snowflakes), probably stemming from my confusion of ‘natural’ with ‘real-world’, the students were mostly intrigued by this presentation. However, most of the actual pieces of the project, like drawing a Sierpinski carpet and computing its area at each iteration, didn’t seem to connect with the real world or my presentation.
  • “This is too much work.” N0w, I’ve heard that before, but at times during this project students complained that I was asking them to do too much drawing, too much measuring, too much calculating, too much reading, and too many practice questions. Moreso than other projects so far this year in geometry (volume, tessellations, & similarity/scale)
  • Reading = hated with a vengeance. I came across what I felt was an excellent explanation of fractal dimension, at a reading level I believed was appropriate. So after a brief introduction, I asked the students to read, ask questions, and then do some practice with finding dimensions of various objects. This went horribly. I think — to some extent — because they were being asked to read (of all things!) in a math class at all, an idea to which all members of the class were vehemently opposed. Or maybe I misjudged the text’s reading level and accessibility. Perhaps fractal dimension was too difficult a concept to have them read about independently, even if the writing was at the right level.
  • Fractal Dimension. Partly due to the point just discussed, but even with revisiting and reteaching, I did not manage to teach this concept clearly to my students. The logarithm part of the formula was a mystery (not taught until Algebra 2), the idea of any of these fractals being infinitely realized (instead of stopping at a particular iteration on the way) was not accepted by my students, and what to count in terms of number of copies and scale factor (two pieces of the formula) was rarely clear to them. I feel that this part of the project was by far the weakest, and I welcome suggestions for how I could improve it.
  • SBG Skills Quizzes. It started with me formulating an extremely vague skill#12 back on my original skills list. “Students will demonstrate an understanding of fractals and self-similarity.” I mean, come on. In practice, I decided that meant that students would follow an iterative process to draw unfamiliar fractals. This ended up seemingly testing their understanding of the words used to describe the procedure rather than fractals or self-similarity. While the first two quizzes on this skill went poorly, I think I succeeded with the question (and my students succeeded with their answers) on the third. Still, I welcome any advice on what sort of quiz question would better fit my nominal skill, and/or how to phrase the skill better.

OK. Tomorrow: what went right with the Fractals Project. Stay tuned for the exciting details of computer-based exploration and cross-curricular collaboration!


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Why SBG?

Though others have written far more convincingly and elegantly about why they use Standards-Based Grading (see dy/dan’s “How Math Must Assess”, Think Thank Thunk’s SBG ManifestoFAQ, and Parts I & II on this post from Take It to the Limit, among many others), I figured I should at least have some explanation for my halfway jump into it this year.

A discussion over in the comments at Sam’s blog got me thinking, and this post was inspired by writing a response there.

Standards-Based Grading means two things to me:

  1. A way of decluttering grades: simplifying them, while at the same time making them more specific. Usually a student might have a grade for homework, a grade for attendance, a grade for Quiz 1, a grade for chapter4 test, a grade for daily classwork, and several attempted extra credit opportunities, all floating around to give a number like a 78 on the report card. Instead, students are only graded on what they know, via skills or topic assessments. [Or, in my case, half of their grade is what they know and half is how they apply what they know through projects.] Furthermore, rather than grouping an assortment of skills together in quiz/test grades, grades are reported out separately by skill or topic. So instead of seeing a 55% on Quiz 3, a student sees grades reflecting complete mastery on calculating area but only a very limited understanding of identifying lines of symmetry.
  2. A system of re-assessment built in. At our school, we have long had a mastery re-do policy, where students are allowed to re-do work and re-take asssessments to demonstrate mastery of course content. Details of this policy’s implementation are left up to individual teachers, so we may set late penalties for work not turned in on time, while still accepting it for a grade, or say that a student retaking a test failed the first time around may not receive more than a 90, or just allow their new grade to supersede the old. The whole idea of allowing re-assessments and the mastery re-do policy is to recognize that students might not always learn according to our schedule, but if they do learn and successfully master the content in a class, their grade should reflect that. For example, a student may be totally confused by rotation symmetry after I first teach it and give a quiz a couple days later. But then we spend a week creating tilings of the plane and examining the types of symmetry found therein, and the student gains a better understanding of the topic. In a standard class, the quiz has gone by, and the student didn’t learn it in time, so too bad. With SBG, topics/skills re-appear on assessments. A student’s new grade will replace an older grade, so their current grade will better reflect what they currently know about that topic. Additionally, within some reasonable limits, students are allowed to re-assess individual skills after school / before school during coach class.

While meeting a constant flow of deadlines is a key life skill, even more important in my view are the following: independent self-directed learning, setting your own goals on your way to a deadline, and keeping trying until you have reached a goal. A math professor I studied with in college would always say that failure is the only way we truly learn anything. Making a mistake (using what we know and failing) opens our eyes and prompts us to learn a new way that will succeed. And this skill: the skill of persistence in the face of failure, and learning from your mistakes, is perhaps the most important skill I wish to teach my students. Even though it’s not on my SBG skills list 🙂, the very nature of SBG and re-assessment seem to promote this continual striving to get better instead of being satisfied with mediocrity.


I believe this will be my last SBG-details post for a while. I’m hoping to get back to posting more about the content of what I’m teaching in Geometry and Computer Integrated Manufacturing, describing some of the projects we’re doing, and reflecting on how things go. But, for those of you who have liked my last few posts, I do plan on continuing to update my folder over on Scribd with weekly skills quizzes and other relevant documents from my foray into using SBG in Geometry.


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My SBG Geometry Plans

After some dialogue with gasstationwithoutpumps and some additional thought on my part, I made a few changes to the skills list I talked about last post. Thanks for the ideas and discussion, g! My new, updated Geometry Skills List and personal course guide is posted here. I am mostly happy with it, including the fact that its number of skills is a power of 2! However, I am aware that #12 & #23 are both fuzzy, and I have yet to decide exactly how I will assess them. I’d still be happy of additional input for next year into the makeup of the list, even though the list is mostly set for this year. Or also thoughts on how to (more concretely) assess those two fuzzy skills.

A few more details on my SBG (Standards-Based Grading) plan for the semester:

  • I decided to steal Sam’s rubric and test it out, see how it goes. I admire it, as well as his thoughts on SBG at various times including his most recent detailed account of his new system.
  • I’ve borrowed a lot of Dan’s style in the following documents which I hereby share: a blank skills list handout for students to track their own progress, a more-visually-appealing version of the previously-discussed skills list, and the very first quiz I gave last Thursday assessing my skills #1, 2, 3, 4, 5 and 6. Dan has a bunch of good philosophy and practical tips at that same page of his I linked: his “Comprehensive Math Assessment Resource”.
  • Although I didn’t steal handouts from other people, many many other math/science teacher bloggers and tweeps have helped me shape my ideas [and I’m sure will continue to do so as the semester progresses], so a big THANK YOU to y’all as well!
  • I plan to teach and assess 2-3 new skills each week, and assess/re-assess each skill 3-2 times, with the latest grade overwriting earlier ones in the gradebook (and of course the option to reassess more times in after-school or lunchtime coach class). Skills will have a quarter-based deadline to reassess by, since that is a date set by the district. I’m attracted to the idea of two perfect scores allowing you to skip the skill question when it rolls around for a third or more time (perfect scores of level 4, not just mastery at level 3); I may try this too.
  • To convert a student’s grade on skills into a percentage, I’m planning to go with my third conversion formula discussed in this post, though I do wish to revisit the issue when I plan for An SBG version of Algebra 2 with Trigonometry at the start of the spring semester. I shall give cumulative midterm and final exams, which are factored into the grade separately from the quarter grade of 50% skills quizzes and 50% projects.


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Geometry Skills List

Hi all. I am three days into my new year. In Geometry this year, I decided to combine a version of standards-based grading (SBG) with project-based learning (PBL – love those acronyms!).

I shared my syllabus with the class the first day, both as a detail-oriented hard copy and as a more thematic zooming ‘Prezi’ presentation. Now, this will be a very non-traditional Geometry class, idiosyncratic in many ways to me and my mathematical/engineering interests (while still aligning with Maryland state standards). The first half of the course will be focused on patterns and motion, with units on symmetry+tessellations, fractals, and polyhedral patterns; following a one-week fast-paced review of measurement. The second half of the course will look in more detail at measurement and also construction: of triangles, of other polygons built via triangles, of circles, and of some familiar 3D shapes.

But even though some topics are based on my individual passions (although I had considered it before, I was inspired/motivated to actually do a fractals unit after hearing Bob Devaney speak at MathFest this summer), I believe in some ways this represents a truer picture of the real field of geometry than a curriculum driven by axiom/theorem/2-column-proofs where some of the things that go in the reason column are complicated names for something completely obvious (think ‘transitive property of equality’) while others are intuitive leaps that most high school students would never come up with. Some higher-level study of geometries focuses on symmetry and the invariants of a space (e.g. which transformations are isometries and leave an object/shape fundamentally unchanged, and which are not). I took a math course in college that focused on tiling theory (think your bathroom floor tiles or M.C.Escher’s tessellations). Fractals are a new and currently expanding area of study, unlike most of a secondary geometry course that is over 2000 years old. I’m hoping our unit on fractals will change the misperception that all math is static, unchanging, and always has an already-known-by-the-teacher right answer.

Some of my sequence’s focus on measurement and construction is influenced by my recent foray into learning and teaching engineering, while some of it is based on envisioning what real-life skills my students can learn via the course. Additionally, I admit that one personal goal in teaching this course is to convince students not to hate math, so this has also been a factor in deciding how I will teach geometry on this, my fourth go-round.

All of that is a roundabout justification of some of my choices of topics. Grading-wise, I decided to go 50/50 skills quizzes and projects. I am a big proponent of project-based learning, and I’m making some changes this year so that the geometry class will become more project-driven and less of, every week, oh here’s a project that sort of relates to what you’re learning. I believe that projects can assess applications of math, connections to other subjects, and a deeper contextual synthesis of multiple math understandings that is not reflected in discrete standards-based skills assessments.  Perhaps one of the old hands at SBG can tell me how to refine standards based grading to get at these aspects.  But since I don’t know how to incorporate this sort of learning into SBG, for this course I will weight the projects 50% and SBG skills assessments as the other 50%.

Anyway, here’s my Geometry Skills List. I would love feedback as to any topics I excluded that you consider essential to a students’ understanding of geometry, to their knowledge of the world, or to their grasp of future mathematics. I’d also appreciate hearing if any of the skills I listed are less important, or if any should be broken up into multiple skills or combined into a single skill.

  1. Student will measure lengths to the nearest 1/8 in or 0.1cm.
  2. Students will measure and draw angles accurate to the nearest degree.
  3. Students will classify angles by type.
  4. Students will identify the number of dimensions an object has.
  5. Students will use formulas to compute areas of various shapes.
  6. Students will compute perimeters of various shapes.
  7. Students will use formulas to compute volumes of various shapes.
  8. Students will reflect, rotate, and translate shapes with a high degree of accuracy.
  9. Students will identify reflectional, rotational, and translational symmetry.
  10. Students will classify plane tilings by the shapes around a vertex, and by symmetries.
  11. Students will use proportions to determine unknown sides in similar shapes.
  12. Students will demonstrate an understanding of fractals and self-similarity.
  13. Students will measure perimeter, area, and volume of fractals at various iterations.
  14. Students will count the edges, vertices, and faces of polyhedra.
  15. Students will use Euler’s Formula to predict aspects of unfamiliar polyhedra.
  16. Student will determine an appropriate unit of measurement given a situation.
  17. Students will convert between different units of measure by dimensional analysis.
  18. Students will use distance and midpoint formulas for lengths in the coordinate plane.
  19. Students will identify types of triangles, by angle and by side lengths.
  20. Students will construct triangles given side and/or angle measures.
  21. Students will use the congruence theorems to prove triangles congruent.
  22. Students will apply the Pythagorean Theorem to find unknown sides in right triangles.
  23. Students will use trigonometry to find unknown sides and angles in right triangles.
  24. Students will construct circles passing through 1, 2, or 3 points.
  25. Students will identify parts of a circle.
  26. Students will calculate lengths and areas of a circle and parts thereof.
  27. Students will use formulas to compute surface areas of various shapes.
  28. Students will use formulas to compute volumes of various shapes.

I look forward to discussing these with you all. By the way, there’s still time to influence my decision and change the list–any but the first few that are on tomorrow’s quiz :^)


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Standard-Based Grades Into Percentages?

A twitter colleague, @druinok, posed the following challenge, which prompted me a great deal of thought:

How do you convert from standard-based grades (often on a 1-4 or 1-5 system) into a percent score required by a gradebook or school district, while still maintaining a sense of what the 1/2/3/4(/5) system means?

While the formulas involved may be too complicated for a gradebook, I approached this problem mathematically: can I create a map from the SBG 1-4 system to the grades I believe they should represent? My understanding of the SBG system is as follows, which informed my selected points and function choice:

  • 4 = exceeds standard; demonstrates complete mastery and conceptual understanding with no nontrivial errors
  • 3 = meets standard; demonstrates mastery of a skill/topic, perhaps with minor errors
  • 2 = approaching standard; shows some understanding, with major (or many minor) errors
  • 1 = below standard; shows little or no understanding
  • 0 = did not attempt

In my conversation with @druinok, she stated (and I concur) that a student with all 3s has met all standards and should receive a B+/A-. All 4s should clearly represent 100%. In my opinion, all 2s should be enough to scrape by with barely passing the class (60% or D); some may argue that since that student has not shown mastery of the standards he or she should receive a failing grade.  A student with a 2.5 average (half twos and half threes, e.g.) should receive a C. A student with 1s should not pass. In my district, the lowest grade we can assign on a report card is a 50, so a 1 average yields this.

In playing around with the numbers, they seemed to fit naturally into a symmetric pattern around 2.5 = 75%. This in turn prompted me to find differences and look for a cubic equation with inflection point around (2.5, 75).

  • 1 average = 50%
  • 2 average = 60%
  • 2.5 average = 75%
  • 3 average = 90%
  • 4 average = 100%

My first attempt is odd since it reverses the independent and dependent variables (x/y). My second attempt is more understandable and develops the equation -\frac{20}{3} x^3 + 50 x^2 - \frac{280}{3} x +100 . Its graph is shown below.

I also tried a third attempt with different points (3=85% & 2=65%) since I was slightly unhappy that the above function from my second attempt was not monotonically increasing on the interval [1,4]. Still, my second attempt is closer to my understanding of the 4-point scale.

For what it’s worth, I also tried fitting a 5-point scale to percentages.

Anyway, that was a bit of fun. But the formulas aren’t perfect, and are certainly a bit complex for gradebooks (the original challenge).

So my main point in writing this post was to ask: those of you who are standards-based grading aficionados, how do you handle the conversion to a percent or letter score?

Certainly any solution, whether it be a formula like this or a more holistic approach like “a student must meet all standards with a 3 or higher to achieve a grade of A, etc”, must involve clear and open communication about the grading policy with students & families.


Filed under math, teaching