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2010 Alabama Course of Study: Mathematics College- and Career-Ready Standards Welcome to Phase I of Alabama professional development for the implementation.

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Presentation on theme: "2010 Alabama Course of Study: Mathematics College- and Career-Ready Standards Welcome to Phase I of Alabama professional development for the implementation."— Presentation transcript:

1 2010 Alabama Course of Study: Mathematics College- and Career-Ready Standards
Welcome to Phase I of Alabama professional development for the implementation of the 2010 Alabama Course of Study: Mathematics. This session is intended only to be an overview of the new mathematics program and we will address some of the challenges that Alabama educators will face during the implementation of this document. Suggestions for successful transitioning into this program will be offered and explored. Together, we can produce the best mathematics curriculum for all Alabama students. (CLICK) Professional Development: Grades 9 – 12 Phase I Regional Inservice Center Summer PART B

2 Literacy Standards for Grades 6 – 12
2010 Alabama Course of Study: Mathematics College- and Career-Ready Standards This section focuses on understanding the Literacy Standards in the 2010 Alabama Course of Study: Mathematics which includes reading and writing in the mathematics classroom. Literacy will now be included in all courses of study for all subject areas. (CLICK) Literacy Standards for Grades 6 – 12

3 APPENDIX C Literacy Standards for Grades 6 – 12 History/Social Studies, Science and Technical Subjects “These standards are designed to supplement students’ learning of the mathematical standards by helping them meet the challenges of reading, writing, speaking, listening, and language in the field of mathematics.” The 2010 Mathematics Course of Study contains Literacy Standards for grades 6 – 12. They are found in Appendix C but can also be viewed in the English Language Arts Course of Study in Appendix B. The Literacy standards are designed to be a supplement to other Course of Study Standards including mathematics, listed as a ‘Technical Subject’ within the document. Within the Position Statements on p. 4 of the Mathematics Course of Study we see the Literacy Standards specifically mentioned under the section on Curriculum. (READ THE STATEMENT ON SCREEN) (CLICK)

4 It is essential for educators to:
select and develop resources that ensure students can connect their curriculum with the real world. help students recognize and apply math concepts in areas outside of the mathematics classroom. provide students with opportunities to participate in mathematical investigations. Through the Literacy Standards, educators are to facilitate instruction so students make connections between mathematics and other subject areas. It is essential for Educator to: (CLICK AND READ x 4) help students develop problem-solving techniques and skills which enable them to interconnect ideas and build on existing content.

5 Basis of Literacy Standards
The Literacy Standards for Reading and Writing are based on the College and Career Readiness (CCR) anchor standards as outlined in the English Language Arts (ELA) common core. Both of which are outlined in Appendix C. The Literacy Standards are based on the English Language Arts (ELA) ‘College and Career Readiness (CCR)’ Anchor Standards, for both Reading and Writing. They are broken down into two parts: Reading in Technical Subjects Writing in Technical subjects As mentioned earlier ‘Technical Subjects’ includes mathematics. (CLICK)

6 Layout of the Literacy Standards
Appendix C p College and Career Readiness Anchor Standards for Reading p Reading Standards for Literacy in History/Social Studies 6-12 p Reading Standards for Science and Technical Subjects 6-12 p College and Career Readiness Anchor Standards for Writing p Writing Standards for Literacy in History/Social Studies, Science and Technical Subjects grades (through p. 134) In the appendices, here is how the Standards are presented: (CLICK) First, on page 128, there is an outline of the College and Career Readiness Anchor Standards. These are the standards we mentioned in the previous slide; the basis of the Literacy Standards. Next are the Reading Standards for History and Social Studies On Page 130, you will see the specific requirements for mathematics grades 6-12: The Reading Standards for Technical Subjects which includes Mathematics. Then, just like the outline for reading, the Course of Study gives us the ‘Outline of the College and Career Anchor Standards for Writing’. Again, these are taken from the ELA (English Language Arts Common Core) and are the basis of the writing portion of the Literacy Standards. Finally, on page 132, begins the 3 pages that outline the standards for ‘Writing in Technical Subjects’ – mathematics. (CLICK FOR EMPHASIS x 2) Mathematics teachers in grades 6-12 will look directly to both: The Reading and Writing Standards for Technical Subjects for their objectives. Students in Grades K-5 will receive literacy standards in their English language arts standards. (CLICK)

7 Snapshot View Notice that the ten reading standards are in grade spans
Snapshot View Notice that the ten reading standards are in grade spans A great layout to assist teachers in differentiating instruction! We will take a close-up look in a moment. But first, here is an overall ‘snapshot view’ of what the READING Standards look like in the appendices. (If attendees have a copy of the standards, say: ) Please turn to page 130 in your standards to view this more closely. Notice how the Standards are in Grade Spans: (Point to the columns) 6-8 9-10 11-12 This is a very useful layout for teachers as they differentiate instruction to meet the needs of various learners and to show progression in the instruction. Let’s take a closer look at the first standard. (CLICK)

8 A closer look: 1.Cite specific textual evidence to support analysis of science and technical texts. 1. Cite specific textual evidence to support analysis of science and technical texts, attending to the precise details of explanations or descriptions. 1. Cite specific textual evidence to support analysis of science and technical texts, attending to important distinctions the author makes and to any gaps or inconsistencies in the account. Here we isolated the first Reading Standard from page 130. This is the first of the ten reading standards. Notice how the standard is common throughout all grade spans, with the level of difficulty increasing as the student progresses. (Allow time for participants to read and note differences between grade bands – point out as needed.) The next slide gives a fun example of this standard at all levels. (CLICK)

9 Famous Mathematician Cards (aka: ‘The Baseball Card Project’)
This project is just a fun way to have students write a report on a famous mathematician. It is also a great project when you are not wanting to introduce a new math concept, for example, during testing week. Students can format their report just like a baseball card. (CLICK) The front has the name, a picture, and the date the mathematician lived. The back of the card has any factual information: Birthplace Famous quotes Education Field of mathematics Why the mathematician was famous Interesting stories If you notice, this student has simple references listed on the back: The Internet and an Encyclopedia. This is a young student. As the age span progresses, you would expect more specific information. In 9th and 10th grade, I would expect to see the specific website and the date the information was found, or the specific Encyclopedia. An 11th or 12th grade student should cite each entry on the project in proper APA or MLA format. Check with your English Department on which format the students are taught. Lets take a look at another reading standard. FRONT BACK

10 Another Reading Standard:
Grades 6 – 8: Students should be able to read a word problem and create an image of some sort (diagrams, graphs, etc…) Grades 9 – 10: Students should ALSO be able to reverse this skill: translate diagrams and charts into meaningful problems or equations. Grades 11 – 12: Finally, they should expand this skill to other sources (video, data) and use it to address questions and solve problems. This is Reading Standard #7, just as it appears in the Appendices on page 130. Notice again, that the standard progresses by grade. Essentially, it says: (CLICK) In Grade 6-8 (Read) In Grade 9-10 (Read) In Grade (Read) As stated earlier, this is a great layout for differentiating instruction as teachers strive to meet the needs of Tier II students in the classroom, under the guidelines of RtI (Response to Instruction). For example: A teacher teaching Algebra I to 9th graders would expect them to meet the standards in the middle column. But, if they had a student who struggled in reading, they could possibly look to the column for grade span 6-8 for Tier II instruction. Similarly, if a teacher has students who are excelling, the student could be pushed toward the Grade span. Let’s look at a sample problem for this same standard.

11 Grade 8 Mathematics / Standard 22
Course Standard: 22. Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions. ELA Standard: Reading Standard 7: Integrate quantitative or technical information expressed in words in a text with a version of that information expressed visually. Problem: Settlers in the Old West would often fashion tents out of a piece of cloth thrown over tent poles. They would secure it to the to the ground with stakes forming an isosceles triangle for the opening. How long would the cloth have to be so that the opening of the tent was 4 meters high and 6 meters wide? Here is an 8th Grade Geometry problem on Pythagorean Theorem. We are using Course Standard #22. (READ) (CLICK) It uses Reading Standard #7 which we recently reviewed. Here is the problem (READ) Aside from solving the equation, what must the student demonstrate to show mastery of Reading Standard #7? (Wait for responses) Yes, the student must draw and label the diagram correctly. Let’s look at the solution.

12 Teacher/Instructional Leader Notes:
Solution: Original Problem: Settlers in the Old West would often fashion tents out of a piece of cloth thrown over tent poles. They would secure it to the to the ground with stakes forming an isosceles triangle for the opening. How long would the cloth have to be so that the opening of the tent was 4 meters high and 6 meters wide? The vertical pole forms 2 right triangles, so I am using the Pythagorean Theorem. a2 + b2 = c2 = c2 = c2 25 = c2 5 = c But, there are two sides to the tent, so the material needs to be 10 meters long. Answer: The cloth needs to be 10 meters long. 4 6 The 8th grade student is to ‘create an image’ from the word problem. The teacher should continuously challenge the student to include detailed information such as the congruent sides and the right angle notation. And then, (CLICK) the teacher should challenge the student to show all work and (CLICK) explain necessary details through the incorporation of writing. This is an example of how literacy is incorporated into mathematics. Notice that this problem is not only about settlers in the old west, it brings in mathematics through the use of the Pythagorean theorem, translation of verbal expressions into algebraic expressions, and construction of drawings based on written descriptions . Let’s look at an additional sample of a problem where the teacher incorporates a Reading Standard. (CLICK) 3 (1/2 of 6) Teacher/Instructional Leader Notes: Expressing answers as a complete sentence incorporates routine writing. Continually express the importance of accuracy and clarity in diagrams. Assess student ability to translate the word problem into a diagram separately from the ability to solve a problem. It is vital to include word problems in mathematics instruction. It is equally important students be given an opportunity to share idea’s or concerns about their work and to receive timely feedback.

13 Algebra I / Standard 11 Course Standard: 11. Create equations (inequalities) in one variable and use them to solve problems (linear, quadratic, simple rational, exponential) . ELA Standard: Reading #4: Determine the meaning of symbols, key terms, and other domain-specific words and phrases as they are used in a specific scientific or technical context relevant to their grade level. Problem: Identify the word problems below as linear, quadratic, simple rational or exponential stating evidence for your choice. Then, write an equation using variables when appropriate. Solve your equation. A car travels 125 miles in 3 hours. How far would it travel in 5 hours? At a concert, Nabila purchased three t-shirts and a concert program that cost $15. In total, Nabila spent $90. Find the cost of a single t-shirt if they were all the same price. The product of two consecutive positive even integers is 14 more than their sum. Find the integers. Here, the Algebra I students are being asked to create an equation based on the word problem and to identify the type of problem they create (Linear, quadratic, etc…). (CLICK) The ELA Standard has a very similar objective. (READ) Here is the sample problem. (Allow time for audience to READ.) What is the expected outcome of the student? (Give an opportunity for participants to answer/discuss.) Yes, the student should be able to set up an equation and solve the equation. (CLICK)

14 Teacher/Instructional Leader Notes:
Solution: The first problem is a simple rational equation. It compares miles and hours: so, 3x = and x = miles * Answer: The car would travel miles in 5 hours. The second problem is linear because it has one unknown, the price of the t-shirts. Let x be the number of t-shirts 3x+15 = 90 so, x = 25 Answer: The cost of each t-shirt is $25. The final problem is quadratic because it has two unknowns but the second one is related to the first. Let x be the initial even number; (x+2) is the next consecutive even integer x(x+2) = 14 + x + (x+2) x2 + 2x = x x2 = 16, Answer: The first number is 4 and the second number is 6 . x = Proof: 6+4 = 10 and 6(4) = 24 which is 14 more than 10. Here is the solution. Notice the student has their final answer in a complete sentence and offers explanation as needed. (CLICK) It is important that students be able to defend their problem set-up and process toward solving the problem. (READ TEACHER INSTRUCTIONAL NOTES) We’ve been talking a lot about how students write their responses but we haven’t looked at our writing standards. Let’s look at the first one now. Teacher/Instructional Leader Notes: Assess for student logic and ability to pull key vocabulary used in mathematical text. Reading problems provide the teacher with tremendous insight into students understanding.

15 Snapshot View : Writing Standards (page 1 of 3)
Just like the reading standards, we will look at the writing standards more closely in a moment. First, note that the Writing Standards stretch over 3 pages. This is page 132 in your Course of Study. The Writing Standards are also divided by Grade Spans: 6-8, 9-10, and (Point at columns) Unlike the Reading Standards, which are written differently for Social Studies than they are for Science and Mathematics, the Writing Standards are the same for History/Social Studies, Science and Technical Subjects. The teacher needs to consider the specific discipline when applying the standards. This first standard goes into great detail on how a student should construct a formal written argument in regards to discipline specific texts. Take a moment to glance over this first standard and its subparts. (PAUSE) Students need to be taught to do this, step by step. A good place to start might be back at the sample problem we just viewed. (CLICK)

16 REVISITING: Algebra I, Standard 11
Original Standard: 11. Create equations (inequalities in one variable and use them to solve problems (linear, quadratic, simple rational, exponential) . Original Problem: Identify the word problems below as Linear, quadratic, simple rational or exponential stating evidence for your choice. Then, write an equation using variables when appropriate. Solve your equation. 1. A car travels 125 miles in 3 hours. How far would it travel in 5 hours? 2. At a concert, Nabila purchased three t-shirts and a concert program that cost $15. In total, Nabila spent $90. Find the cost of a single t-shirt if they all had the same price. 3. The product of two consecutive positive even integers is 14 more than their sum. Find the integers. NEXT Standard: Writing Standard #1: Write arguments focused on discipline-specific content. Recall in our previous example the teacher had the students identify word problems as being linear, simple rational or quadratic. Then they were to set up the equation using ‘x’ as the variable and finally, solve the equation. (CLICK) We are now going to add a Writing Standard to the same problem. (READ) In this scenario, the teacher presents the problems solved incorrectly either through their initial identification of the problem type and/or their solution logic. The students write an argument, using applicable mathematics vocabulary, that disputes or defends the logic used in a sequential manner. NEW ASSIGNMENT: The teacher presents problems from the original assignment which were labeled incorrectly, had faulty logic and/or incorrect solutions. Randomly distribute and direct students to write a brief argument for each problem to either defend or dispute the logic used.

17 Snapshot View : Writing Standards p. 2
Close-up: 2. Write informative/explanatory texts, including the narration of historical events, scientific procedures/ experiments, or technical processes: Introduction Development of ideas Transitions Vocabulary Style Conclusion Here is a snap-shot view of the Second Writing Standard. Which is also written with subparts. (CLICK) In short, it guides the student through writing informative text (read square). This could be as simple as a two-column proof.

18 Writing Standard #3 In science and technical subjects, students must be able to write precise enough descriptions of the step-by-step procedures they use in their investigations or technical work so others can replicate them and (possibly) reach the same results. Writing standard #3 does not stand alone as a requirement, but rather is part of all the other objectives. (READ SCREEN) (CLICK)

19 Snapshot View : Writing Standards p. 3
This is a snapshot view of the remaining 7 (of 10) writing standards. They cover: The Production and Distribution of Writing Research to build and present knowledge Range of writing (CLICK)

20 *The same for all three levels
A closer look: *The same for all three levels 10. Write routinely over extended time frames (time for reflection and revision) and shorter time frames (a single sitting or a day or two) for range of discipline-specific tasks, purposes, and audiences. Here we isolated the final writing standard. One that makes a very important statement. And, is the same for all three grade spans. (READ) If your school is an AMSTI school, especially at the High School level, this is incorporated in your texts. The AMSTI workbook “Cookies”, for example, takes the students through multiple reading and writing activities as they learn to read and write Linear Inequalities. Students revisit their logic throughout the text, extending it with each activity. If your school is not an AMSTI school, there are numerous ways to incorporate writing into the mathematics classroom and many teachers are already doing this on a daily basis throughout the state. Let’s look at some questioning that encourages thought towards writing in the mathematics classroom. (CLICK)

21 Questioning that Encourages Thought
Does the rule I am using work for all cases? Why, why not? How can I describe what is happening without using specific numbers? How can I predict what’s going to happen without doing all the calculations? Was my prediction correct; how was my logic faulty? What process reverses the one I am using and when is it appropriate? How did two different students reach two different answers? Defend logic. Are there multiple ways to work the same problem? If so, how did you decide which process to use? Can you write a scenario (word problem) for the diagram? Describe your frustrations with the process learned today. Outline everything you understand about the process you used today. (CLICK AND READ x 10) The teacher needs to learn to present prompts that help express their understanding of the content. Some activities that encourage this process are: -Journaling -Exit Slips -Portfolios -Foldables and more. (CLICK) Driscoll, M. Fostering Algebraic Thinking

22 Grade 8 / Geometry Standard 18
Course Standard: 18. Describe the effect of dilations, translations, rotations and reflections on two-dimensional figures using coordinates. [G3] ELA Standard(s): Reading Standard 7: Integrate quantitative or technical information expressed in words in a text with a version of that information expressed visually. Writing Standard 2f: Provide a concluding statement that follows from and supports the information presented. Problem: Mr. Smith asked his students to plot the following points in order, connecting them to form a triangle: (3,0) (7, 1) (4, 5) (3,0). Here are the student responses. Which is correct? Describe the errors in logic made on the remaining three and the result of those errors. A B C D. Here is a final sample problem using both a reading and writing standard. Review the course standard. (CLICK) Note we are using both a reading and writing standard. (READ) Lets review the problem (READ) And the solution.

23 Possible Solution: Triangle A is plotted correctly
A B C D Triangle A is plotted correctly In triangle B, the student switched the (x,y) coordinates. Instead of plotting (3,0) they plotted (0, 3). The student might be confused about which axis is the x and which axis is the y. In Triangle C, the student went in the negative direction for the x coordinate but plotted the y correctly, this caused a reflection around the y-axis. In triangle D the student plotted the x coordinate correctly but went in the negative direction for the y, this caused a rotation and a shift in the graph. Here is the student solution. I’m going to pause to give you time to read through their responses. (PAUSE) As teachers and teacher leaders, what should we be looking for in the response? (CLICK to answer on next slide)

24 Teacher/Teacher Leader Notes
Is the student’s mathematical logic correct? Can they describe errors found in others’ mathematical thinking? Activities like this may be used as a bell ringer (sparking discussion prior to a lesson) or as an exit slip to assess understanding. It is important that the responses are assessed by the teacher through grading or discussion. (CLICK AND READ x 4) After viewing the Reading and Writing Standards, and looking at some simple examples; there is one overall thought we want to leave you with as teachers and administrators in regards to the Literacy Standards. (CLICK)

25 Final Thought: The Literacy Standards allow Flexibility in Reading and Writing
Predictions Proofs Compare/Contrast methods Reflection Journals Word Problems Summarize One thing the inclusion of technical standards in the ELA documents allows for is flexibility. Nowhere is the mathematics teacher told HOW to incorporate these standards into their mathematics curriculum, just that they are to do it often and with rigor. If one method doesn’t work well with one class, try another – try several! The sky is the limit! (CLICK) Writing directions for replication by others Descriptions of Process or change

26 ?? Questions ?? Are there any questions concerning the Literacy Standards in the 2010 Alabama Mathematics Course of Study? If not, we will continue with the Domains of Study/Conceptual Categories and Learning Progressions/Trajectories. (CLICK)

27 2010 Alabama Course of Study: Mathematics College- and Career-Ready Standards
We will now discuss and explore the domains of study, conceptual categories, and learning progressions/trajectories in this section of the presentation. (CLICK) Domains of Study/Conceptual Categories Learning Progressions/Trajectories

28 CHARACTERISTICS Aligned with college and work expectations
Written in a clear, understandable, and consistent format Designed to include rigorous content and application of knowledge through high-order skills Formulated upon strengths and lessons of current state standards Informed by high-performing mathematics curricula in other countries to ensure all students are prepared to succeed in our global economy and society Grounded on sound evidence-based research All standards contained in this document are: (CLICK) Aligned with college and work expectations; (CLICK) Written in a clear, understandable, and consistent format; (CLICK) Designed to include rigorous content and application of knowledge through high-order skills; (CLICK) Formulated upon strengths and lessons of current state standards; (CLICK) Informed by high-performing mathematics curricula in other countries to ensure all students are prepared to succeed in our global economy and society; and (CLICK) Grounded on sound evidence-based research. (CLICK)

29 2010 Alabama Course of Study: Mathematics
Coherent Rigorous Well-Articulated Enables Students to Make Connections All content contained in this document is (CLICK) coherent, (CLICK) rigorous, (CLICK) well-articulated across the grades, (CLICK) and focuses on enabling students to make connections between important mathematical ideas. (CLICK) ACOS, p. 4

30 Coherence Articulated progressions of topics and performances that are developmental and connected to other progressions. Conceptual understanding and procedural skills stressed equally. Real-world/Situational application expected. (CLICK) The mathematics curriculum has been sorted into topics articulated as progressions that are developmental and connected. (CLICK) Both conceptual understandings and procedural skills have equal importance. (CLICK) Application of learning is expected at all levels. (CLICK)

31 FOCUS Key ideas, understandings, and skills are identified.
Deep learning stressed. Key ideas, understandings, and skills are identified in each grade or course, while stressing deep learning. (CLICK)

32 Mathematical Content Format
Grade Domain Cluster Standard Course Conceptual Category Domain Cluster Standard The organizational structure of the 2010 Alabama Course of Study was discussed briefly during the Components of the Course of Study section. Let’s look in more detail at the format. The format is consistent throughout K-8, with an additional dimension in high school. The high school standards are organized into conceptual categories which have been grouped into courses. (CLICK) K-8 9-12

33 What’s the difference? Domain Cluster Standards
Read the top part of page 10. Turn and discuss the question on the slide with your shoulder partner. (Allow a couple participants to report out whole group. Wrap up with the next slide.) (CLICK)

34 What’s the difference? Domain: Overarching “big ideas” that connect content across the grade levels. Cluster: Group of related standards below a domain. Standards: Define what a student should know (understand) and do at the conclusion of a course or grade. (CLICK) Domains are large groups of related standards. (CLICK) Clusters are groups of related standards and appear below domains. (CLICK) Standards are numbered/lettered beneath each cluster. (CLICK)

35 High School Conceptual Categories
Overarching big ideas that connect mathematics across high school Illustrate progression of increasing complexity May appear in all courses Organize high school standards Conceptual Categories are: The “big ideas” that connect mathematics across high school A progression of increasing complexity Descriptions of the WHAT or mathematical content to be learned - elaborated through domains, clusters, and standards. (CLICK)

36 High School Conceptual Categories
Number & Quantity Algebra Functions Modeling Geometry Statistics & Probability The Real Number System Seeing Structure in Expressions Interpreting Functions Modeling is best interpreted not as a collection of isolated topics but rather in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). Congruence Interpreting Categorical and Quantitative Data Quantities Arithmetic with Polynomials & Rational Expressions Building Functions Similarity, Right Triangles, and Trigonometry Making Inferences and Justifying Conclusions The Complex Number System Creating Equations Linear, Quadratic and Exponential Models Circles Conditional Probability and the Rules of Probability Vector and Matrix Quantities Reasoning with Equations and Inequalities Trigonometric Functions Expressing Geometric Properties with Equations Using Probability to Make Decisions Geometric Measurement and Dimension Modeling with Geometry Domains (CLICK) (CLICK) The High School Standards are presented differently; they are presented by conceptual categories for the grade-span 9-12 (not by grade level). (Number and Quantity, Algebra, Functions, Geometry, Statistics and Probability). Modeling standards are integrated within the other five categories, and are indicated with a star. Except for Conceptual Categories, High School has the same K-8 structure of domain, cluster and standard. Standards indicated as (+) are beyond the college and career readiness level but are necessary for advanced mathematics courses, such as calculus, discrete mathematics, and advanced statistics. However, standards with a (+) may still be found in courses expected for all students. (CLICK)

37 High School Clusters Multiple Courses
Illustrate Progression of Increasing Complexity from Grade to Grade Standards that are related are identified by a cluster. The same cluster may appear in multiple courses with their respective standards growing developmentally. Let’s look at an example on the next slide (CLICK)

38 Algebra II with Trigonometry
9-12 Cluster Algebra I Algebra II with Trigonometry Interpret the structure of expressions. (Linear, exponential, quadratic.) 7. Interpret expressions that represent a quantity in terms of its context.* [A-SSE1] Interpret parts of an expression such as terms, factors, and coefficients. [A-SSE1a] Interpret complicated expressions by viewing one or more of their parts as a single entity. [A-SSE1b] 8. Use the structure of an expression to identify ways to rewrite it. [A-SSE2] Interpret the structure of expressions. (Polynomial and rational.) 6. Interpret expressions that represent a quantity in terms of its context.* Interpret parts of an expression such as terms, factors, and coefficients. [A-SSE1a] 7. Use the structure of an expression to An example of a cluster appearing in high school courses is shown on the slide. The cluster is Interpret the structure of expressions. Notice that the standards within the cluster are worded exactly alike, but the Algebra I standards require students to work with linear, exponential and quadratic expressions while the Algebra II with Trigonometry standards require students to work with polynomial and rational expressions. This example demonstrates the progression of mathematical content throughout the high school standards. (CLICK)

39 Content Standards Content standards in this document contain minimum required content. Each content standard completes the phrase “Students will.” Reflect both mathematical understandings and skills, which are equally important. (Read the slide (CLICK) Minimum in quantity, not quality. (CLICK) (CLICK) ) The order in which standards are listed within a course or grade is not intended to convey a sequence for instruction. (CLICK)

40 High School Standards Conceptual Categories Cross course boundaries
Span high school years Standards “Core” for common mathematics curriculum for all students to be college and career ready “College and Career Ready” for entry-level, credit- bearing academic college courses and work-force training programs. “STEM” (+) Additional mathematics that students should learn in order to take courses such as calculus, discrete mathematics, or advanced statistics. As seen in a previous slide- The conceptual categories are: Number and Quantity, Algebra, Functions, Modeling, Geometry, and Statistics and Probability. Although three of the six categories, Algebra, Geometry, and Statistics and Probability, are familiar course names, the conceptual categories are not confined to any one specific course, rather, they cross course boundaries and span high school years. STEM standards are additional standards that take the student above and beyond the core curriculum. As previously mentioned, stem standards are included in some required courses, as well as, electives. (CLICK)

41 High School Conceptual Categories
“The Mathematics Standards for High School” begin on page 66 of the 2010 Alabama Course of Study document with an introduction that describes how the grade nine through twelve content is grouped according to six conceptual categories. These categories provide a coherent view of high school mathematics content. (CLICK)

42 Jigsaw Activity Number Functions & & Quantity Modeling Geometry
Algebra Number & Quantity Functions Statistics & Probability Earlier, we briefly discussed the overviews and narratives of each grade and high school conceptual category. (CLICK X 6) Each Conceptual Category begins with a narrative that clarifies the content meaning and pedagogical focus rather than describing the big ideas as in K-8. (Have participants number 1 – 6.) Read the following Conceptual Category narrative: 1’s - Number & Quantity Conceptual Category, 2’s – Algebra Conceptual Category, 3’s – Functions, 4’s – Modeling, 5’s – Geometry, and 6’s – Statistics & Probability. (Distribute the High School Standards pp. 66 – 122 and the “A Close Look Handout.” Allow participants about 20 minutes to group together to complete their jigsaw parts. Each group should place their information on poster/chart paper and post it on wall. Once a group post their work have them to also complete their portion of the “Final Summation Chart” (record the sum of the total number of Domains, Clusters, Standards, Advanced Standards, and Alabama Standards for your Conceptual Category. When all charts are posted allow each group to do a Gallerywalk. As they do their Gallerywalk ask them to record observations and questions on their foldable. Once groups complete their Gallerywalk have them to return to their table groups and allow a few to share out. This activity will take 30 minutes. (CLICK)

43 How many skills? Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational. [N-RN3] AI.3.1. Explain why the sum of two rational numbers is rational. AI.3.2. Explain why the product of two rational numbers is rational. AI.3.3. Explain that the sum of a rational number and an irrational number is irrational. AI.3.4. Explain that the product of a nonzero rational number and an irrational number is irrational. (When participants return to their tables after Gallerywalk ask the following question in reference to the “Final Summation” chart.) Decide how the number of standards compare to the number of skills? Are they the same? Explain. Turn and talk to your shoulder partner. Identify the number of skills in standard 3 found in Algebra I (p. 81). During your discussion, did you identify these skills? (CLICK) (CLICK) (CLICK) (CLICK) Notice the coding of the standard. The first number represents the grade level, or course for high school. The second number represents the standard and the third is the number of the identified skill(s) contained in the standard. Do not worry if you do not fully understand how to analyze standards; we will be looking at skills and subskills more closely in Phase II professional development. (CLICK)

44 Mathematics Domains of Study by Grade
Let’s look at the mathematical foundation students receive in K-8 prior to entering high school. The standards in Grades K – 8 are focused around eleven domains of study. They are listed in the first column. The predominate grades each domain is studied is noted with a check. The progression of domains demonstrates the mathematical learning progression through grade K-8. All domains are not present in each grade, however a domain that stops or changes names is foundational and serves in the learning of new ideas or applications. An example is Number and Operations in Base Ten which spans Kindergarten through grade 5 and Fractions which spans Grades 3 – 5. These two domains merge into The Number System in 6-8. Geometry spans K-8 (as well as High School). (CLICK)

45 The Big Picture High School Functions, Statistics, Modeling and Proof
K-2 Number and number sense. 3-5 Operations and Properties (Number and Geometry) Fractions 6-8 Algebraic and Geometric Thinking Data Analysis and using Properties Broad topics can be used to organize instruction for the entire year. Let’s look at The Big Picture of the focus of study by grade bands. (CLICK)

46 Learning Progressions/Trajectories
Confrey (2007) “Developing sequenced obstacles and challenges for students…absent the insights about meaning that derive from careful study of learning, would be unfortunate and unwise.” CCSS, p. 4 “… the development of these Standards began with research-based learning progressions detailing what is known today about how students’ mathematical knowledge, skill, and understanding develop over time.” In addition the sequence of topics and performances that is outlined in a body of mathematics standards must also respect what is known about how students learn. As Confrey (2007) points out, “developing sequenced obstacles and challenges for students…absent the insights about meaning that derive from careful study of learning, would be unfortunate and unwise.” (CLICK) “… the development of these Standards began with research-based learning progressions detailing what is known today about how students’ mathematical knowledge, skill, and understanding develop over time.” CCSS 2010, p. 4 Just exactly what are learning progressions/trajectories? Think about this question as you participate in the next activity. (CLICK)

47 What Are Learning Trajectories. And What Are They Good For
What Are Learning Trajectories? And What Are They Good For? —Save the Last Word for Me!!! Read the excerpt from Learning Trajectories in Mathematics: A Foundation for Standards, Curriculum, Assessment, and Instruction Identify 3 ideas that you are willing to talk about with colleagues. Highlight the location in the text where these ideas appear. This activity is: “Save the Last Word for Me.” You will begin reading at the first full paragraph of the article. (Read the directions on the slide. Continued on next slide.) (CLICK) 47

48 Directions for Save the Last Word for Me
Designate a facilitator and timekeeper. A volunteer begins by reading the sentence(s) from the text that embody one of his/her selected ideas. The speaker does not comment on the text at this point. The individual to right of first speaker takes up to one minute to comment on the selected text. The next two individuals also take up to one minute to comment on the initial speaker’s idea. The individual selecting the idea has up to 1 minutes to react to colleagues’ ideas and to talk about why she or he thought this was important. Another group member introduces one idea, and the group follows the same protocol. Continue until all members have shared or until time is called. (Read slide and explain to participants the directions for the activity.) How was this a useful way to explore the ideas in the text and to explore your own thinking? What questions were raised during this protocol? (CLICK)

49 Learning Trajectories – sometimes called learning progressions – are sequences of learning experiences hypothesized and designed to build a deep and increasingly sophisticated understanding of core concepts and practices within various disciplines. The trajectories are based on empirical evidence of how students’ understanding actually develops in response to instruction and where it might break down. Daro, Mosher, & Corcoran, 2011 To better grasp the mathematical understandings, it is important to have a common reference for the meaning of a learning progression.(Read what is on the slide first, then read the next two sentences.) The K- 8 Common Core Math Standards which are incorporated into Alabama’s Mathematics Course of Study are formatted into learning progressions based on the domains. (CLICK)

50 Learning Progression Framework
Ending Point Ending Point Starting Point Starting Point K 1 2 3 4 5 6 7 8 HS Counting and Cardinality Number and Operations in Base Ten Ratios and Proportional Relationships Number and Quantity Number and Operations – Fractions The Number System Operations and Algebraic Thinking Expressions and Equations Algebra Functions Geometry Measurement and Data Statistics and Probability The domains are the learning progressions. Guiding Question (Discussion for Table Groups) What do you notice when looking at the K-8 domains? (Allow participants to talk at their tables as facilitator rotates around the room. Bring participants back together whole group and allow table groups to report out. Before charting responses to show the starting and ending point arrows. (CLICK X 2) Chart the rest of the responses.) (Possible responses are: Starting and ending points – students need to have understanding of standards in the domains to be able to move on to the next domain. Counting and Cardinality only in Kindergarten. If a child does not attend Kindergarten, he is already behind. So how will you help the child learn these missed concepts? Fractions are not taught until Grade 3 Ratio and Proportion taught in 6th and 7th only Functions are not taught until Grade 8 Operations and Algebraic Thinking in K-5 is the foundation for both Expressions and Equations, and Functions in Grades 6-8. K-5 Measurement and Data, which splits into Statistics and Probability and Geometry in Grade 6) What is not easily seen, is how a learning progression may indirectly impact multiple progressions that start later in school years. Math educators are use to thinking about strands in standards (there are 5 broad strands in state standards in Alabama’s current COS) – you can think about the progressions (domains) as small strands – slightly more finely divided with one important difference which is that the progressions have a starting point and an ending point. In particular there is a transition from K-5 to 6-8 which brings in a new set of progressions - the implication is that the O&A, NBT, and Fractions work is meant to be finished at the end of grade 5 so we are meant to use that as foundation to move on to a higher level of understanding –domains have a beginning and an ending point. (CLICK)

51 Investigating the Domains/Conceptual Categories
Domains provide common learning progressions. Curriculum and teaching methods are not dictated. Standards are not presented in a specific instructional order. Standards should be presented in a manner that is consistent with local collaboration. (CLICK) Domains are common learning progressions that can progress across grade levels. (CLICK) Domains do not dictate instructional curriculum or teaching methods. (CLICK) Topics within domains are not meant to be taught in the order presented. (CLICK) Teachers should present the standards in a manner that is consistent with decisions that are made in collaboration with their K-12 mathematics team. (CLICK) 51 51

52 Learning Progressions
K 1 2 3 4 5 6 7 8 HS Counting and Cardinality Number and Operations in Base Ten Ratios and Proportional Relationships Number and Quantity Number and Operations – Fractions The Number System This diagram illustrates how the learning progressions are distributed across the Alabama Course of Study Standards (coherence). Let’s look at the Number Domains and Conceptual Categories as they build across K-12. (CLICK)

53 Table Team Work Beginning at the lowest grade examine the domain and conceptual category, cluster and standards at your grade level - identify how the use of numbers and number systems change from K- 12. Counting & Cardinality (CC)– K only Number and Operations in Base Ten (NBT) – K-5 Number and Operations – Fractions (NF) – 3-5 The Number System (NS) – 6-8 Number and Quantity (N)– 9-12 Look at the grade level above and grade level below (to see the context). Make notes that reflect a logical progression, increasing complexity. As a table group share a vertical progression (bottom–up or top-down) on chart paper. (Group participants in gradebands K-2, 3-5, and 6-8, 9-12) To help you with this activity you might want to look for the Content Standard Identifiers (CC, NBT, NF, NS, and N(9-12)). (Go over directions on slide.) (Possible information for presenter: Numbers and Number Systems. During the years from kindergarten to eighth grade, students must repeatedly extend their conception of number. At first, “number” means “counting number”: 1, 2, 3... Soon after that, 0 is used to represent “none” and the whole numbers are formed by the counting numbers together with zero. The next extension is fractions. At first, fractions are barely numbers and tied strongly to pictorial representations. Yet by the time students understand division of fractions, they have a strong concept of fractions as numbers and have connected them, via their decimal representations, with the base-ten system used to represent the whole numbers. During middle school, fractions are augmented by negative fractions to form the rational numbers. In Grade 8, students extend this system once more, augmenting the rational numbers with the irrational numbers to form the real numbers. In high school, students will be exposed to yet another extension of number, when the real numbers are augmented by the imaginary numbers to form the complex numbers. With each extension of number, the meanings of addition, subtraction, multiplication, and division are extended. In each new number system—integers, rational numbers, real numbers, and complex numbers—the four operations stay the same in two important ways: They have the commutative, associative, and distributive properties and their new meanings are consistent with their previous meanings.) (CLICK)

54 What Goes On The Chart Paper?
Summary and/or representation of how the concept of the use of numbers grows throughout your grade band. Easy for others to interpret or understand. Visual large enough for all to see. More than just the letters and numbers of the standards – include key words or phrases. Summary and/or representation of how the concept of the use of numbers grows throughout your grade band. Easy for others to interpret or understand Visual large enough for all to see More than just the letters and numbers of the standards – include key words or phrases (CLICK)

55 Banding Together for a Common Message
Display posters side-by-side and in order on the wall. Begin at the grade band you studied. Read the posters for your grade band. Discuss similarities and differences between the posters. Establish a clear vision for your grade band. Notice not only what students need to know before they enter your course, but what they need to know for the next course as well. (CLICK)

56 Let’s Talk About It! As a table group, consider your journey through the 2010 ACOS as you studied the concept of the use of numbers K-12. What did you learn? What surprised you? What questions do you still have? Read slide. (CLICK)

57 Value of Learning Progressions/Trajectories to Teachers
Know what to expect about students’ preparation. More readily manage the range of preparation of students in your class. Know what teachers in the next grade expect of your students. Identify clusters of related concepts at grade level. Clarity about the student thinking and discourse to focus on conceptual development. Engage in rich uses of classroom assessment. Here are some ways learning progressions/trajectories can be valuable to teachers. Reflect on the following two questions and then turn and talk to your table partner: How do the progressions apply to your work? Which of the statements on the screen would be most beneficial to you? (CLICK)

58 . Review the Format 2003 ACOS 2010 ACOS Contains bullets
Does not contain bullets Does not contain a glossary Contains a glossary . Let’s wrap up this section of the PD by looking briefly at some of the similarities and especially the differences between the current and new documents. Take a few minutes to discuss and review the similarities and differences between the current Alabama Standards and the 2010 ACOS format. (Allow about 2-3 minutes for participants to compare the current ACOS to the 2010 format for a couple of more statements to add to the chart. ) (Possible Responses: Bullets are no longer included in standards Lettered and numbered items (just as important – both must be mastered). Modeling standards STEM standards Glossary in 2010 but not in 2003 The grade clusters are different; K-8, and 9-12. The Domains progress across grade levels – not all domains are addressed in every grade; The number of standards are different for each grade and course. They range from 22 in Grade K to 28 in Grade 8. ) (CLICK)

59 ?? Questions ?? Are there any questions concerning the Domains of Study, Conceptual Categories, or Learning Progressions/Trajectories? If not, we will continue with identifying content shifts in the 2010 Alabama Mathematics Course of Study. (CLICK)

60 2010 Alabama Course of Study: Mathematics College- and Career-Ready Standards
Each new course of study provides an opportunity to study the standards and adjust teaching plans accordingly. Because several standards have changed grade levels or moved to a different course, there will be some content from the 2010 standards that will have to be taught (along with the 2003 standards) during the school year to prepare students for full implementation of the 2010 ACOS during the school year. However, by studying these new standards with an eye toward developing plans to address this content, teachers and curriculum coordinators can insure that students are prepared for the new standards. Some may choose to develop modules specifically to address this content, while others may simply opt to add this additional content to their classes this year. (CLICK) Content Shifts

61 Content Correlation ALGEBRA I – ALGEBRA – Seeing Structure in Expressions Write expressions in equivalent forms to solve problems. (Quadratic and exponential.) 9. Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.* [A-SSE3] Factor a quadratic expression to reveal the zeros of the function it defines. [A-SSE3a] Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines [A-SSE3b] Determine a quadratic equation when given its graph or roots. (AL) Use the properties of exponents to transform expressions for exponential functions. [A-SSE3c] First, let’s see what is meant by a content shift by working with some Algebra I standards. The conceptual category we are looking at is Algebra, the domain is Seeing Structure in Expressions. The cluster is Write expressions in equivalent forms to solve problems. We are limited to quadratic and exponential functions. We’ll look at a cluster of standards and see what, if any, content has changed courses. (Trainer will read the standard.) What does the * mean at the end of the standard? (This is a modeling standard. Modeling links classroom mathematics to everyday life, work and decision making. It is choosing and using appropriate mathematics to analyze situations and make improved decisions.) Which of these standards are currently in the Alabama mathematics standards? In which course do we find these standards? (CLICK)

62 Correlates with 2010 ACOS – ALGEBRA I
9. Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.* [A-SSE3] a. Factor a quadratic expression to reveal the zeros of the function it defines. [A-SSE3a] d. Use the properties of exponents to transform expressions for exponential functions. [A-SSE3c] Correlates with 2003 ACOS – ALGEBRA I Factor binomials, trinomials, and other polynomials using GCF, difference of squares, perfect square trinomials, and grouping. 9. Solve quadratic equations using the zero product property. 1.1 Applying laws of exponents to simplify expressions, including those containing zero and negative integral exponents. When studying these standards, we find the 9 and 9a are found in Algebra I. (Std. 6 and Std. 9) Also, 9d is found in Algebra I (Std. 1.1) But, the 2003 ACOS for Algebra I does not explicitly mention exponential functions. And, exponential functions must be addressed by the Algebra I teachers. (CLICK)

63 2003 ACOS – ALGEBRA II W/ TRIGONOMETRY
9b. Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines. [A-SSE3b] 9c. Determine a quadratic equation when given its graph or roots. (AL) Correlates with 2003 ACOS – ALGEBRA II W/ TRIGONOMETRY 4. Determine approximate real zeros of functions graphically and numerically and exact real zeros of polynomial functions. 4.1 Using the zero product property, completing the square, and the quadratic formula 5.1 Generating an equation when given its roots or graph Standards 9b and 9c are found in the 2003 Algebra II w/ Trigonometry COS. (CLICK) Standard 9b (2010) correlates with Standards 4 and 4.1 (2003) (CLICK) Standard 9c (2010) correlates with Standards 5.1 (2003) That means the content from 2010 standards 9b and 9c represent a content shift. This was previously taught in Algebra II with Trigonometry but has now been moved back to Algebra I. So, in the school year, the Algebra teacher must address this content that has been moved down from Algebra II w/ Trigonometry into Algebra I in order to prepare her students for Algebra II with Trigonometry in the school year Does everyone understand the procedure for determining content shifts? We have conducted content analysis on each grade level, specifically looking for content that has either moved from one grade to another, is new to the Alabama standards, or is no longer taught in the ACOS. From those efforts, we developed a Content Analysis Document to assist you with planning for the 2011 school year. Let’s take a look at the document. (CLICK)

64 Content Correlation Document
2003 ACOS 2010 ACOS CURRENT ALABAMA CONTENT PLACEMENT 2010 GRADE 1 CONTENT 1.1 Demonstrate concepts of number sense by counting forward and backward by ones, twos, fives, and tens up to 100; counting forward and backward from an initial number other than 1; and using multiple representations for a given number. 1.9. Count to 120, starting at any number less than 120. In this range, read and write numerals and represent a number of objects with a written numeral. [1-NBT1] 1.1.B.1 Identifying position using the ordinal numbers 1st through 10th CONTENT NO LONGER ADDRESSED IN GRADE 1 1.1.B.2 Using vocabulary, including the terms equal, all, and none, to identify sets of objects 1.7. Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false. [1-OA7] 1.1.B.3 Recognizing that the quantity remains the same when the spatial arrangement changes CONTENT NOW ADDRESSED IN KINDERGARTEN: 1.4.b. Understand that the last number name said tells the number of objects counted. The number of objects is the same regardless of their arrangement or the order in which they were counted. [K-CC4b] First, we’ll discuss the different sections of the content correlation document. This document is a content correlation for Grade 1. The first section is a correlation of the 2003 ACOS with the 2010 ACOS. In the left column, (CLICK) you will find all 2003 course of study standards and bullets for the Grade 1. (CLICK) In the right column you find the 2010 standard(s) that have been correlated with the current standard. Any content that is highlighted, indicates additional content that must be introduced along with the 2003 standard. For example, Count to 120 is highlighted in the 2010 content standard that has been correlated with standard 1 of the 2003 document. When you read the 2010 standard and compare it with the 2003 standard, you notice that the 2003 standards required students to count up to 100. The 2010 standard requires that they count to 120. This represents additional content that must be addressed in the upcoming school year. (CLICK) Notice that the 2003 standard. 1.1.B.1 Identifying position using the ordinal numbers 1st through 10th is no longer addressed in Grade 1 in the 2010 standards. In fact, this standard is not addressed at all in the 2010 ACOS. (CLICK) Now, look at standard. 1.1.B.3, Recognizing that the quantity remains the same when the spatial arrangement changes. This content is now addressed in Kindergarten, standard 4b, which states: Understand that the last number name said tells the number of objects counted. The number of objects is the same regardless of their arrangement or the order in which they were counted. [K-CC4b] (CLICK)

65 Content Correlation Document
2003 ACOS 2010 ACOS CONTENT MOVED TO GRADE 1 IN 2010 ACOS 2.6 Solve problems using the associative property of addition. 1.3. Apply properties of operations as strategies to add and subtract. (Students need not use formal terms for these properties.) [1-OA3] (Associative property of addition) 3.1.B.1 Comparing numbers using the symbols >, <, =, and 1.11. Compare two two-digit numbers based on meanings of the tens and ones digits, recording the results of comparisons with the symbols >, =, and <. [1-NBT3] 4.10 Complete addition and subtraction number sentences with a missing addend or subtrahend. 1.8. Determine the unknown whole number in an addition or subtraction equation relating three whole numbers. [1-OA8] The next section in the Content Correlation document reflects content that has moved from another grade level to Grade 1. Notice in the left column, once again you find 2003 standards, in the right column the 2010 standards. In this example, notice that what is currently taught in second grade, (CLICK) Solve problems using the associative property of addition, is now found in the Grade 1 Standard #3 (CLICK), Apply properties of operations as strategies to add and subtract. (Students need not use formal terms for these properties.) [1-OA3] (Associative property of addition). Note the code at the end of the standard from the 2010 ACOS. (CLICK)- [1-OA3] – The number in brackets is the content standard identifier and indicates where this standard can be found in the CCSS document. This number will help you correlate resources that are available from other organizations, such as NCTM or other national organizations. Also, some standards from Grades 3 and 4 have been moved to Grade 1. First grade students are now expected to compare two-digit numbers and record their results of the comparisons with the symbols >, =, and <. This was previously a third grade standard. Another example, what was previously a fourth grade standard, Complete addition and subtraction number sentences with a missing addend or subtrahend, is now a requirement of first grade, standard #8, Determine the unknown whole number in an addition or subtraction equation relating three whole numbers. [1-OA8]. Once again, the highlighted portion indicates additional content that must be addressed in the school year. In order to assist teachers who will be teaching new content from another grade-level, this would be a perfect opportunity to have the teachers collaborate in designing lessons. For example, if a first-grade teacher now must teach content that was taught in Grade 3 in the 2003 document, the third-grade teacher could collaborate with the first-grade teacher. (CLICK)

66 Content Correlation Document
NEW GRADE 1 CONTENT IN 2010 ACOS None The final section of the document (CLICK) contains content that is entirely new to the Alabama Course of Study. In Grade 1, you can see there is no new content. (CLICK)

67 Geometry Content Correlation
Which 2010 standard(s) correlates to standard 5 from the 2003 ACOS? Is there any additional content related to standard 10 that should be addressed in the upcoming school year? What content has been moved from Grade 8 to Geometry? Is there any content that is no longer addressed in Geometry? How many standards and/or bullets? Is there any content that is new to Alabama in the Geometry course in the 2010 ACOS? (The participants will have an opportunity to answer questions from the Content Correlation Document). Use your Geometry document in your handouts to discuss the following questions with your shoulder partners. Answers: Which 2010 standard(s) correlates to standard 5 from the 2003 ACOS? #11, #25, #42 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. [G-CO11] 25. Prove that all circles are similar. [G-C1] 42. Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).* [G-MG3] Is there any additional content related to standard 10 that should be addressed in the upcoming school year? Derive the formula A = (1/2)ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side. [G-SRT9] What content has been moved from Grade 8 to Geometry? (8.8.B.3) Constructing congruent and similar polygons, congruent angles, congruent segments, and parallel and perpendicular lines, (8.9) Determine the measures of special angle pairs, including adjacent, vertical, supplementary, and complementary angles, and angles formed by parallel lines cut by a transversal, (8.11.B.2) Determining the appropriate units of measure to describe surface area and volume, (8.14.B.3) Computing the probability of two independent events and two dependent events, and (8.14.B.4) Determining the probability of an event through simulation Is there any content that is no longer addressed in Geometry? How many standards? Yes, 9 standards and bullets. Is there any content that is new to Alabama in the Geometry course in the 2010 ACOS? Yes. Constructing a tangent line from a point outside a given circle to the circle. Are there any questions on reading this content correlation document?

68 How can I be sure my students are prepared for the implementation of the ACOS in the school year? First of all, you must be aware of these content changes and begin to make plans to address any standards that may constitute a content change for your students. Schools cannot wait until the year of implementation ( ) to make these plans. Teachers will need to address these content shifts during the school year to insure that students are prepared for the full implementation the next year. Let’s see how this Content Correlation document can be used to assist with these plans. CLICK

69 Algebra I Mathematics Curriculum
First Nine Weeks 2003 COS DESCRIPTION NEW CONTENT TO BE ADDED FROM 2010 COS AI.1 Simplify numerical expressions using properties of real numbers and order of operations, including those involving square roots, radical form, or decimal approximations. AI.1.B1 Applying laws of exponents to simplify expressions, including those containing zero and negative integral exponents AI.7 Solve multistep equations and inequalities including linear, radical, absolute value, and literal equations. AI.5 Perform operations of addition, subtraction, and multiplication on polynomial expressions AI.3 Determine characteristics of a relation, including its domain, range, and whether it is a function, when given graphs, tables of values, mappings, or sets of ordered pairs. AI.11 Solve problems algebraically that involve area and perimeter of a polygon, area and circumference of a circle, and volume and surface area of right circular cylinders or right rectangular prisms. AI.2 Analyze linear functions from their equations, slopes, and intercepts. AI.1. Notation for radicals in terms of rational exponents [N-RN1] AI. 2 Rewrite expressions involving radicals and rational exponents [N-RN2] AI.1 Notation for radicals in terms of rational exponents [N-RN1] AI.2. Rewrite expressions involving radicals and rational exponents [N-RN2] AI.16. Solve quadratic equations and inequalities in one variable, including equation with coefficients represented by letters [A-REI3] AI.10. Include quadratics [A-APR1] AI.24. Focus on linear and exponential and on arithmetic and geometric sequences [F-IF1] Here is an example of a curriculum plan for the first nine weeks in an Algebra I classroom. Let’s look at the Algebra I Content Analysis Document and see if any changes should be made in our plans for next year. First find standard 1. Is there any new content required to prepare our students? (CLICK) Yes.. ..(Presenter will read aloud the text that appears on the screen) What about for standard 1.1? Any new content required there? (CLICK) Yes..(Presenter will read aloud the text that appears on the screen) Look at standard 7. Any new content required here? (CLICK) Yes.. (Presenter will read aloud the text that appears on the screen) Does everyone understand how to use the content analysis document to determine if there is new content that must be added to this Algebra I course? (Ask participants to finish standards 5, 3, 11 and 2. Allow them 10 minutes to do this. Then, ask for a volunteer to share out. As you receive feedback from participants, click to make answers appear in table.) (CLICK X 6) (CLICK) NO NEW CONTENT AI.26. Recognize that sequences are functions [F-IF3] AI.27. Include exponential and quadratic functions [F-IF4] AI.46. Interpret slope and intercept of a linear model in the context of the data [S-ID7]

70 Algebra II with Trigonometry Pacing Guide 2nd Nine Weeks
2003 ACOS # TOPICS CONTENT THAT SHOULD BE ADDED AIIT.6 Add, subtract, multiply, and divide functions – including polynomial and rational functions Inverse functions Composition of functions AIIT.3 Graphing rational functions – including horizontal and vertical asymptotes, and holes AIIT.7 Solving rational equations and inequalities AIIT.6.B.3 Extension – constructing graphs NO NEW CONTENT NO NEW CONTENT NO NEW CONTENT AIIT.29. Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. [F-BF3] Now, it’s your turn to determine if any content should be added to the Algebra II w/ Trigonometry curriculum. You will use your Algebra II w/ Trigonometry Content Analysis Document. (Participants will have copy of this handout in their packet) (Each participant will analyze the Algebra II w/ Trigonometry Pacing Guide to determine what content, if any, should be added to adequately prepare their students.) (Allow 5-6 minutes for their analysis. Then, ask for volunteers to share out. As they share out, you may CLICK and the answers will appear on the screen.) (CLICK X 6) Of course, resources that you are currently using may have to be supplemented to insure that all skills in both the 2003 and 2010 are addressed. (CLICK) AIIT.20. Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. [A-REI2] NO NEW CONTENT

71 What about the assessments?
Just a little background on the assessments. The ARMT is based on standards from the 2003 ACOS. In 2008 and 2009, when the 2009 course of study for mathematics was being developed, the course of study committee used a draft of the common core state standards for mathematics as a reference document. So, standards in the 2009 ACOS correlated quite well with the common core standards. After the 2009 ACOS was adopted by the state board, assessment went to work on items based on the 2009 standards. Then, the decision was made not to implement the 2009 document but to wait for the official release of the Common Core State Standards for Mathematics and see if Alabama would decide to adopt these standards. By this time, the contract for the assessment on the 2003 standards had expired and assessment moved to develop items correlated to the 2009 ACOS. These items are more rigorous. The ARMT+ was developed based on the 2009 ACOS. Once the decision was made not to implement the 2009 ACOS, assessment removed any items that correlated with the 2009 ACOS and not the So, on the ARMT+, the items DO correlate with the 2003 document but they are written at a more rigorous level than the previous versions. Further information on the assessments will be provided in the fall by the SDE. (CLICK)

72 2003 ACOS + Identified Content from 2010 ACOS 2010 ACOS + Identified Content from 2003 ACOS So, you can see, it’s essential that you continue to teach the 2003 standards, but at a more rigorous level. Expect more from your students. Also, to insure that all necessary content has been covered and that students are prepared for full implementation of the 2010 ACOS during the school year, teachers must teach the 2003 course of study along with identified content from the 2010 ACOS. This is where the Content Correlation Document will help you with planning your school year. Then, during school year, when full implementation of the 2010 ACOS takes place and teachers must teach the standards from the 2010 COS, you must continue to address some of the standards from the 2003 course of study because the ARMT+ will still be used and it is based on the 2003 course of study standards. There will be an additional correlation document provided during a later phase of the implementation PD to assist with this transition. (CLICK)

73 ?? Questions ?? Are there any questions concerning content shifts? If not, we will continue with Algebra I in Grade 8, Considerations/Consequences in the 2010 Alabama Mathematics Course of Study. (CLICK)

74 Algebra I in the 8th Grade: Considerations and Consequences
2010 Alabama Course of Study: Mathematics College- and Career-Ready Standards Taking Algebra I in the 9th Grade as a “first choice,” and following up with Geometry, Algebra II with Trigonometry, and another mathematics course, will ensure that at the end of high school a student will be college and career ready. The State Department of Education does not advocate for the Algebra I course in Grade 8, but allows for system level decisions to determine whether offering coursework that includes a Grade 8 experience of Algebra I. This session of the Phase I professional development on the 2010 Alabama Course of Study: Mathematics will explore the considerations and consequences of offering an Algebra I course in the Grade 8. (CLICK) Algebra I in the 8th Grade: Considerations and Consequences

75 Algebra I in Grade 8: Considerations and Consequences
Do you have middle school students who should have the option of taking Advanced Placement (AP) Mathematics, or two advanced mathematics courses as part of their high school experience? Some Pathways for Students Who Complete Algebra I in Grade 8 Geometry Geometry Geometry Algebra II W/Trig Algebra II W/Trig Algebra II W/Trig Precalculus Discrete Mathematics Precalculus Analytical Mathematics Precalculus Advanced Placement (AP) Mathematics Course (ACOS: Mathematics, 2010, p. 127) How you answer this question will be one of the considerations in deciding whether to offer Algebra I in Grade 8. If a successful Algebra I experience at Grade 9 is the goal for all students, then an early experience of Algebra I in the Grade 8 is not for all students. And if the purpose is to give a child the opportunity to take college-level AP Mathematics in high school, then the early experience of Algebra I in the Grade 8 may not be the path that needs to be taken by the majority of students or even the average student. (CLICK)

76 Algebra I in Grade 8: Considerations and Consequences
“Systems offering Algebra I in the eighth grade have the responsibility of ensuring that all Algebra I course content standards and Grade 8 course content standards be included in instruction.” (ACOS: Mathematics, 2010, p. 81) The State Department of Education will provide further guidance and training (Phase II) in the fall of relative to issues local education agencies may encounter in providing an Algebra I course in Grade 8. First Bullet – COS guidance for systems that want to offer Algebra I in the Grade 8 is summed up in the first bullet. Providing a Grade 8 Algebra I course requires some serious decision-making. One of the first issues to face is the rigor and the sheer number of skills that must be mastered by students contained in the 2010 Alabama Course of Study: Mathematics for the content in the Grade 8 course and in the Algebra I course. (CLICK) Second Bullet – While Phase I training introduces considerations and consequences, the Phase II training will be able to look in greater depth at the arrangement of standards for course sequences, and discuss other key parts of the 2010 COS like the critical areas of focus, overview of possible teaching units, clusters of standards with instructional notes, and maintaining alignment of learning progressions as it relates to accelerating students through middle school content into an Algebra I experience. (CLICK)

77 Algebra I in Grade 8: Considerations and Consequences
Decisions to accelerate students into a high school Algebra I course before Grade 9 should not be rushed. Placing students into an Algebra I course too early should be avoided at all costs. Local education agency’s decision should: Be Advertised Be Equitable Provide Written Policy Decisions to accelerate students into a high school Algebra I course before Grade 9 should be based on solid evidence of student learning. First and Second Bullets – One of the main goals of the new 2010 COS is to prepare students to successfully complete an Algebra I course in the Grade 9. If there is any question about a student’s ability to master all of the content in Grades 7, 8, and Algebra I courses in middle school, then the student should not be accelerated into Algebra I coursework ahead of time. (CLICK) Third Bullet - If a LEA allows students to take Algebra I in Grade 8, then there should be three items addressed in writing: 1. Registration for this accelerated course should be advertised for all to see. 2. The admission to the early course should be equitable for all students who meet the enrollment requirements. 3. The policy for early admission into the course should be a part of LEA written policy. (CLICK) Fourth Bullet - According to the Brookings Institute's 2009 Brown Center Report on American Education, the NAEP scores of students taking Algebra I in the Grade 8 varied widely. The data showed that of the students who took Algebra I early (in Grade 8), the bottom 10 percent scored far below grade level. This is a reminder that, rather than skipping or rushing through content, students should have appropriate progressions through foundational content to maximize their likelihoods of success in high school mathematics. This also indicates that some students would likely have greater success in mathematics by following the normal progression through Grades 7 and 8 courses and take Algebra I in Grade 9. (CLICK)

78 Algebra I in Grade 8: Considerations and Consequences
Not all students are ready for Algebra I in Grade 8. The 2010 COS Algebra I content is not the same as the Algebra I content in earlier Alabama Courses of Study. Much of what was previously included in Algebra I will now be taught in Grades 6-8 in the 2010 COS. A Southern Regional Education Board (SREB) study found a difference in readiness for higher-level mathematics between students scoring in the bottom quartile and those scoring in the top 3 quartiles. Graduates who completed Geometry or Algebra II as ninth graders earned an average Grade 12 NAEP mathematics at the Proficient level. First Bullet – Only two states have mandated that all students take Algebra I in the 8th Grade (Minnesota and California). Currently, California’s policy that all students take Algebra I in the 8th Grade has gone to court and is still pending. (CLICK) Second Bullet –Given the shifts of content that have occurred through the progression of courses, it would be wise to closely examine who is ready for Algebra I in Grade 8. The mathematics courses for all grades in the 2010 COS are more rigorous than the same grade level courses in previous years. The middle school mathematics courses are more rigorous in order to prepare students for a more rigorous high school Algebra I experience. If there is any doubt concerning whether a child will be able to successfully accelerate through Grades 7, 8, and Algebra I content before the 9th Grade, then the child should take the regular Grade 7 and Grade 8 course sequence in order to have a successful experience of Algebra I in Grade 9. Third Bullet – The SREB study found that students which scored in the lowest quartile (bottom 25%) on achievement tests had slightly higher failure rates when enrolled in higher-level mathematics, than those enrolled in lower-level mathematics (Cooney & Bottoms, 2009, p. 2). Students scoring in the top three quartiles (top 75%) were less likely to fail if they were enrolled in more demanding courses. Students scoring in the top three quartiles may be ready for Algebra I in Grade 8. Fourth Bullet – A transcript study from Grade 12 NAEP testing found that students that took more advanced Grade 9 course work (that is, took Algebra I in Grade 8 to make way for a more advanced course in Grade 9) had higher average NAEP scores than students that took Algebra I in the Grade 9. “Graduates who completed geometry or algebra II as ninth graders earned an average NAEP mathematics score at the Proficient level.” (U.S. Department of Education, Institute of Education Sciences, National Center for Education Statistics, High School Transcript Study (HSTS), Students that show high levels of proficiency may be ready for Algebra I in the Grade 8. (CLICK)

79 Algebra I in Grade 8: Considerations and Consequences
“Mathematics leaders need to ensure equitable access to courses by carefully monitoring barriers to participation.” (A Guide to Mathematics Leadership, 2010, p. 4) There are valid reasons for placing students in certain courses, but mathematics leaders are expected to ensure that there are no systematic barriers that discriminate against specific populations. (A Guide to Mathematics Leadership, 2010, pp ) An accelerated Grade 7 course and a Grade 8 Algebra I course differ from the Grade 7, Grade 8, and Algebra I courses found in the 2010 COS in that they contain additional content by comparison and demand a faster pace for instruction and learning. First Bullet - Local leaders need to ensure equitable access to courses and monitor the process of selection for potential barriers. The mathematics leader must also set children up for success in course selection, and not failure. According to the research, students must be performing in the upper 3 quartiles in mathematics and be able to work and learn at an accelerated pace in order to do well in a Grade 8 Algebra I course. (CLICK) Second Bullet - How are students identified for participation in a course? Leadership must monitor each of these means commonly used to place students into a course: Placement tests Student study skills Teacher/counselor recommendation or selection Parental recommendation Student behavior Scheduling issues There are valid reasons for using the methods mentioned above to place students into the appropriate course. Math leaders are simply expected to ensure that there are no systematic barriers that discriminate in a negative way. Third Bullet – The goal is to equip the student for successful experiences in mathematics based on their learning needs, current abilities, and future aspirations. (CLICK)

80 Algebra I in Grade 8: Considerations and Consequences
Ensure that the Grade 8 Algebra I course is not watered down. Ensure that there is equity in support materials for all courses, that is, that more and better materials and supports are not just in place for the accelerated and advanced courses. Ensure that interventions are in place so that all students are prepared for high school Algebra I coursework. First Bullet – The goal is not whether standards are “covered,” so that the students are exposed to the content, but rather that students are learning and mastering the content. (CLICK) Second Bullet – The old saying, “What is good for the best is good for the rest,” applies to all of the math instruction that takes place in a school. For example, materials such as computer software or graphing calculators or interactive technology that are used to support advanced courses and accelerated instruction should also be provided for regular mathematics instruction. Third Bullet – In determining support for students with a Grade 8 Algebra I experience, leaders should not overlook the goal of moving the majority of students to be ready for the Algebra I course in Grade 9. Supports and interventions should be in place to make sure that all students will have a successful Algebra I experience in Grade 9. (CLICK)

81 Ways to Provide Greater Access to Mathematics
Allow students to take two math electives simultaneously, provided the course progression chart is followed. Use block scheduling to take a math course both semesters. Offer Credit Advancement. Offer Dual Enrollment. Offer summer courses that are designed to provide an equivalent experience of a full course. Some students may not have the preparation to enter Algebra I in the Grade 8, but may still be interested in taking higher level math as part of their high school experience. Here are some possible ways to provide greater access to mathematics for students that may not have had early Algebra I as an option. (CLICK)

82 ?? Questions ?? Are there any questions concerning Algebra I in Grade 8: Considerations and Consequences? (CLICK)

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84 Contact Information ALSDE Office of Student Learning Curriculum and Instruction Cindy Freeman, Mathematics Specialist Phone:


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