Here is a suggested arrangement of the high school standards into courses, developed with funding from the Bill and Melinda Gates Foundation and the Pearson Foundation, by a group of people including Patrick Callahan and Brad Findell. I haven’t looked at it closely, but it seems to be a solid effort by people familiar with the standards, so I put it up for comment and discussion. There are five files: the first four are graphic displays of the arrangement of the standards into both traditional and integrated sequences, with the standards referred to by their codes. The fifth is a description of the arrangement with the text of the standards and commentary.

9_11 Scope and Sequence_traditional1

9_11 Scope and Sequence_traditional2

9_11 Scope and Sequence_integrated1

9_11 Scope and Sequence_integrated2

High School Units-All-03feb12

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## About Bill McCallum

I was born in Australia and came to the United States to pursue a Ph. D. in mathematics at Harvard University, met my wife, and never went back. I am a professor at the University of Arizona, working in number theory and mathematics education.

This is pretty awesome material, especially the timelines and the commentaries in the course outlines. It occurs to me that many teachers may need actual lesson plans to get started as the course descriptions will not track with current textbooks. Additionally, some teachers will have difficulty with the higher order learning expectations (analysis, evaluation, creating — connecting concepts and big ideas) as they have been entrenched for years teaching content knowledge with only some minor application.

Hopefully, someone will write a CCSS “modeling” handbook for teachers, as well as a book/website with suggested “projects” aimed directly at meeting the standards.

Debbie – Have you seen the Illustrative Mathematics project? http://illustrativemathematics.org/standards/hs

There are already some tasks designed for modeling and I think some tasks could be reworked into projects with a little fine tuning. Maybe it’s a start?

I could not agree more with debbieqd! Current textbooks do not address the 8 Mathematical Practices and coming up with lessons that teach them is a difficult task to complete in a short period of time.

The glaring difference I spy between this and the Appendix A released by CCSS is the location of exponential functions…Algebra 2 or Algebra1…which should it be?!?

A few quick comments about these High School Scope and Sequences. The design challenge was to first organize all of the CCSS standards into “coherent units” of about 2-4 weeks duration. (Phil Daro has said that the “unit” is the optimal grain size for organizing mathematics for learning). Page iv of the “High School Units All” lists them out. There is an Algebra sequence of 10 units that weaves together all the Number and Quantity, Algebra, and Functions standards. They key point here is that these exact same units can be organized into either a Traditional or Integrated sequence. This was done to facilitate collaboration at the unit level without needing to distinguish between traditional and integrated. I do not believe that there is one “correct” or “optimal” sequencing, but we wanted to put out an example that we felt supported the design and spirit of the CCSS. Many states and organizations are dividing the standards up into grade level or courses. These Scope and Sequence documents are to support and encourage people thinking and designing at the unit level.

Many decisions were needed to organize the units into Traditional and Integrated sequences. Those can be compared on pages ii and iii of the “High School Units” document. Since this post is getting lengthy I will respond to some of the rational behind specific choices in other posts.

In New York State, the probability unit has been moved into the Geometry course. Figure that one out.

And that is the rub for me, Appendix A (PARCC) has “Application of Probability” as Unit 6 of Geometry? Is this truly intentional? I’ve seen NYS may give Scope and Sequence in summer 2012, is this related to the above effort?

While this arragement is nice, it doesn’t match Appendix A from the CCSS. I counted 14 standards in the Appendix A version of Alg II that are missing from this version of Alg II. As previously stated, some Appendix A Alg I standards have been shifted to Alg II, including exponential functions.

Looking at it again……perhaps some of the missing standards will be inlcuded in the modeling units…….

One of the worries was that the modeling standards would get lost or neglected. Our approach was to create specific Modeling Units that would foster applying mathematical ideas from across other units. This is not to say there should not be plenty of modeling within the other units (e.g. the first algebra unit is “Modeling with Functions”), rather we thought it would be good to have additional opportunities for students to focus on modeling projects or activities that did not lend themselves to “we are doing a unit on quadratic functions, so I guess we are supposed to be modeling with quadratics” type student responses.

Agreed. The absence of exponential functions is a significant departure from Appendix A as well as the spirit of their inclusion. The standards emphasize the need to compare and contrast linear functions with other functions to more fully develop the peculiarities of each function. In addition, the order of Algebra 2 with polynomial and rational expressions be placed after exponentials, trigonometric, and “functions” situates them after modeling and limits, a bit, the development that could go on in the functions unit. Are this in draft form? Who can we seek more information from on the decisions being made here, etc.? Thank you for posting them here!

A Concern.

In K-8 the CCSS for Math did such a good job of addressing the critical areas of focus for each course. In High School that is not done in the CCSS for Math document. Instead it is noted on page 84 that additional pathways for the courses will be available. I suppose that Appendix A is one such available pathway. I believe that the Appendix A document is very helpful because it again focuses on the critical areas for each course. Although, I do not believe that it is intended as a scope and sequence it seems to be a clear designation of topics/standards that are important within the course. I would like to know the reasoning behind the movement of many of the exponential topics from Algebra 1 to Algebra 2 and why probability is moved from Geometry to Algebra 2. Basically, I want to be convinced. Why the difference between this Scope and Sequence offered here and the Appendix A? The concern for me is not the movement within a course but between them.

Once I feel like I have a good hold on things, additional resources such as this come out and completely knock me back 2 steps (or 10). I am not frustrated by the supporting documents (thank you for your hard work on these). I am frustrated that there are several discrepencies between this document and the Appendix A. I have focused most of my time at this point reviewing the differences in the Algebra I materials. I have noted several areas where this document lists standards that do not appear in the Appendix A Traditional Model. They are N-CN.1, 2, 7, A-SSE.4, F-LE.4, F-TF.1,2,5,8, F-IF.7c, A-APR.2,3,4,6, A-REI.2. Additionally, the whole section on statistics has been removed according to this document.

Although I agree with some of the changes (and have concerns about others), I am more concerned about the work that we are doing with districts who are working on developing their scope and sequences for all grades. I am simply concerned about where to move next. I want to guide our teachers in our district in the right direction. If anyone can be of service, I would greatly appreciate it.

As mentioned above, is there any further discussion of completing a curricular priorities document that breaks high school courses down seperately? New York State has released their version, but in Algebra I, for example, all of the standards are listed as either 70% or 20%. There are no 10% standards. The URL to this document is http://engageny.org/wp-content/uploads/2012/03/nys-math-emphases-k-hs.pdf. I have a hard time believing that what is the emphasis in one course should automatically be considered an emphasis in the next.

Dr. McCallum and team… thank you again for all of your work around the shift to the Common Core. Your materials are invaluable and we have been basing most of our work around what you are producing. We can’t thank you enough. If you have any ideas/suggestions to help lead this work in the right direction, we would appreciate the feedback!

The main difference between this scope and sequence and the Appendix A is that we wanted to organize every HS standard into coherent units (see comments above) that could be sequenced in either a Traditional or Integrated way.

This requires making some estimates about the number of days (which is included in the above documents) to fit them into courses. We wanted to avoid the “mile wide inch deep” syndrome and allow sufficient time for depth. If you compare the K-8 with the HS there are significantly more HS standards, so the units are an approach to get at the critical ideas.

An important point to make about moving exponentials to Algebra 2 is to recall that student have seen them in middle school (in particular Grade 8 Expressions and Equations). So there is every expectation that 9th grade students will compare and analyze situations that involve linear, polynomial, and exponential behavior (e.g. 2x versus x^2 versus 2^x) but the more rigorous work with exponential equations and logarithms is done in Algebra 2.

Also note that many standards are repeated throughout the scope and sequence. So when we cross out part of a standard the intent is to provide focus early on, but that standard will occur later in full (nothing is crossed out in the end).

I agree exponential functions could be split between Algebra 1 and Algebra 2 as Patrick suggests. However, they don’t occur in the Grade 8 standards. Students see integer exponents there, but not functions where the variable is in the exponent.

Bill,

Thanks for the clarification. I was thinking more in terms of what have been called “baby exponentials”. For example a doubling sequence like 2^n (for whole number n) as opposed to analyzing exponentials as continuous functions like 2^x (for x real). These seem like a good and natural source of examples to contrast with linear and polynomial relationships.

I may have overlooked the topic, is there a learning standard for simplifying radicals, I think in a traditional pathway it would be covered in Algebra 1 or Pre-Algebra (grade 8), or even perhaps Geometry. However, I don’t see it anywhere and shall I assume it is just a skill we’re teaching to our own standard?

The standard N-RN.2, Rewrite expressions involving radicals and rational exponents using the properties of exponents, could support some work along these lines. But the standards overall try to get away from demanding that students “simplify” things. For example, they don’t expect students to find the least common denominator when adding fractions, or to reduce fractions to lowest terms. When thinking about radicals, it’s not at all obvious that 3 \sqrt{3} is simpler than \sqrt{27}, and the latter form is more useful for some purposes. For example, you can see that the number is slightly bigger than 5 much more easily from this form.

Thank you so much for your ongoing effort in creating ways to help us organize our transition to the Common Core Standards. My only question regarding the Scope and Sequence timeline is the separation of the Modeling Units (for example 4 days following 45 days A1 – A3 in the traditional Algebra 1 sequence). Perhaps I am misunderstanding the term “modeling” but I thought it would be preferable to incorporate activities and tasks (modeling?) as introductory and as reinforcement elements throughout each unit rather than set time aside after some units have been covered?

I could see doing it either way, but I assume the design in these documents is to allow time for extended modeling tasks that take account of the entire modeling cycle, as described in the original standards document at corestandards.org.

I am just curious where PARCC and the assessment piece comes into play here. I guess a big question I have is will PARCC being including exponential function on the 10th grade assessment, if so won’t we have to include this unit in Algebra 1? Any thoughts on this?

Neither PARCC nor SBAC has issued any guidelines on course boundaries in high school, but I assume they will have to do that as they develop their assessments. You are quite right that it will have an effect.

I have noticed the * next CCSS Domain and Standard Listing on the Scope and Sequence Charts. What is the meaning of the * symbol? Thanks

Dear Mike, that’s a modeling star, indicating a standard that connects with modeling in high school. See the standards document at corestandards.org

Is PARRC using the resource from this page to determine what should be included in each of the high school courses or are they using the Appendix A?

I don’t know, but my guess is that they won’t be tied to either of these documents, but will try to find a solution based on critiques of Appendix A from their member states, which might involve some adjustments to Appendix A.

Any thoughts on the placement of polynomial and rational expressions in the Algebra 2 sequence? Why after the transcendental functions, modeling unit, and unit on functions? Intrigued by this placement. One other surprise was how different this coverage of standards was to the “base” version of the ADP Algebra II course? Does Achieve have a common core “course”? In addition, are you working with any states or advising other states on their HS course construction? Interested to see other solid, well-researched interpretations.

I only found 35 out of 38 pages? Where is the Algebra 2 material?

Bill,

The text file of all of the courses is missing pages 36-38. Can you repost with the complete file?

Sorry for the confusion, the file is complete at 35 pages, Word messed up the count, so there are no missing pages. The modeling units are not described since they are intended to be opportunities for students to apply concepts from multiple units without introducing new standards.

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Almost half of the Algebra 2 progression is Probability and Statistics (42% of non-modeling days) and the single biggest unit in Algebra 1 is Statistics. This makes me wonder… Why not combine Algebra 1 and Algebra 2 into one course and have a new course called Probability and Statistics?

Either way, I am happy to see a reasonable amount of time going to such an essential pair of topics.

Can someone please clarify modeling units versus projects and what that might look like? My undertanding is that modeling units would be similar to the SBAC sample assessments. The projects might be something that students create based on their deep undertanding of the units.

The modeling units would be similar to some of the extended assessments from SBAC. They are meant to connect and draw from multiple units of content.

Projects are where students USE and DO mathematics to create, build, develop something that hopefully necessitates the content. For example, building catapults to analyze projectile motion and quadratic functions.

What about at the elementary level? What does modeling look like there? I realize that there is “modeling to do math” and “modeling to learn math”. I think I’ve got ahold of the second one, but am confused about the first one for elementary grades, say 3 – 5.

Hi Maria, there is some discussion of modeling in the thread on general questions about the standards (the sticky post at the top of the blog). If you search on the term “modeling” you should find it. If this doesn’t answer your question, please feel free to post again.

Hi, again,

I looked for the thread on modeling, but it took me to Task #4 Contest. I am concerned the definition of modeling because it seems that most people I speak to have their own perception.

I can see what modeling looks like at the high school level (there are many examples out there), but I’m still confused as to how it would look at grades 3-5. Is a number line modeling? I can make an argument for it’s use as a model, say for explaining or proving addition of fractions. I guess I need some examples of what modeling would look like at these grades.

Dear Maria, go to the thread on General Questions About the Mathematics Standards, and use your browser’s “Find” or “Search” menu to look for the word “modeling” on that page. Don’t use the search box at the top of the blog.

Bill,

Thank you for the help. I found your answer to Susie H and modeling is stating to get clarified for me. I will study what I’ve found.

In your suggested Scope and Sequence for the high school Common Core State Standards, you refer to 6 Projects. Do you have 6 specific projects in mind? Have they been created or are they just place holders?

No specific projects have been created. Rather the point is to build in days for projects where students use the mathematics.

I am still confused as to why all the standards from the CCSSM are NOT in these documents. Where are the Real Number System standards?

Also, is there any coorelation/cooperation between these and the Dana Center’s scope and sequence?

The Real Number System standards are in there. N-RN 1 and 2 are in Unit A7, and N-RN 3 is in unit A10.

We are currently revising a High School course in Probability and Statistics aligned to CCSSM. I would appreciate any information on current prob/stat curricula already aligned.

Additionally, we are charged with developing a pre-test for the course. I am using information from the Pathway documents as a rationale for which standards will be included in the pre-test.

In addition to a text on modeling it would be a good thing if publishers paid attention and began meaningful work on the development of a textbook aligned to any of the pathways detailed in Appendix A.

Publishers may be waiting to see if people really use one of the sequences outlined in Appendix A. Those two sequences were created to help show that teaching the standards would be possible, not as a finished sequence. The consortia have some different ideas about the sequences. The standards themselves don’t require a certain sequence.

If we wanted to use this resource, who should we reference as the source?

This is a draft circulating for the purpose of soliciting comments and suggestions.

High School Mathematics Scope and Sequence (Draft) 2012.

Made possible by grants from the Pearson Foundation and the Bill and Melinda Gates Foundation.

Publishers may be waiting to see if people really use one of the sequences outlined in Appendix A. Those two sequences were created to help show that teaching the standards would be possible, not as a finished sequence. The consortia have some different ideas about the sequences. The standards themselves don’t require a certain sequence.

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I noticed that most of the course pacing documents include about 150 instructional days. Was this done intentionally with the goal of wrapping up new content prior to the PARCC assessment window? If so, are there suggestions as to what to focus on for the remainder of the year such as preloading content for the next course?

If this was not the intent, is there a concern about spacing out the content a bit more? For example, my district has 180 contact days with students. If we were to spend more time on certain units then an entire unit, such as probability and statistics, could fall after the PARCC assessment window. Is the expectation that all content will be tested or is it expected that certain content at the end of the year would not be tested on the end of course tests.

I appreciate any feedback. We are working hard to develop more in depth documents for our teachers and would like to know our time frame. Thanks!

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