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**Fulton County Schools Curriculum and Instruction Division**

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**Fulton County Board of Education Roundtable**

Meeting the Needs of All Students with the Georgia Performance Standards (GPS) Fulton County Board of Education Roundtable February 27, 2007

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**Quality Core Curriculum (QCC) Continuous Achievement **

Fulton County Board of Education Policy IHE: “Each student must be accepted on the level at which he/she is functioning and should be challenged to move through the curriculum at a rate commensurate with the student’s total capabilities.” Quality Core Curriculum (QCC) Continuous Achievement Georgia Performance Standards (GPS) Differentiation Fulton County’s Continuous Achievement Model for English Language Arts and Mathematics Differentiation with acceleration and support (accelerated remediation) as appropriate On this slide, you see a quote from Policy IHE that expresses our philosophy as it relates to meeting the needs of all students. In the table, you see a visual that expresses how this philosophy was met under the Quality Core Curriculum and how we are meeting it now under the Georgia Performance Standards. The Georgia Performance Standards is a far different curriculum than the Quality Core Curriculum, and as a result the manner is which one supports students working below level and accelerates students capable of working above level is quite different.

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**Curriculum Differentiation**

Differentiation is the “fit” or “match” of the curriculum to the needs of the learner. Differentiated instruction promotes high-level and powerful curriculum for all students, but varies the level of teacher support, task complexity, pacing, and avenues to learning based on student readiness, interest, and learning profile. (Tomlinson, 2000)

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QCC – GPS Crosswalk Before any decisions could be made about how we would differentiate for our students, we had to know more about the differences between the GPS and the QCC. The next few slides are intended to give you some insight about the process we used to do this.

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**THREE SLIDES FOLLOW THIS ONE**

GPS 5th FCSS 5th Grade Number and Operations M5N. Students will further develop their understanding of the concept of whole numbers. They will also understand the meanings of multiplication and division of decimal fractions and use decimal fractions and common fractions in computation, as well as in problem solving situations. NOT EXPLICITLY STATED M5N1. Students will further develop their understanding of whole numbers. NOT EXPLICITLY STATED a. Classify the set of counting numbers into subsets with distinguishing characteristics (odd/even, prime/composite). NOT INCLUDED b. Find multiples and factors. c. Analyze and use divisibility rules. Applies skills and concepts for computation of division of whole numbers including estimation and mental computation a) Applies divisibility rules for 2, 3, 5, and 10 b) Divides by up to 2 digit divisors and uses the calculator for larger numbers. M5N2. Students will further develop their understanding of decimal fractions as part of the base-ten number system. Demonstrates and applies concepts of decimals through thousandths. a) Uses concrete and visual models to represent parts of as whole for decimals. b) Compares and order decimals using a number line when appropriate. c) Rounds decimals to the nearest thousandths. a. Understand place value. Applies skills and concepts for addition and subtraction of whole numbers and decimals, including estimation and mental computation a) Identifies the information and steps needed to solve –one, -two, and –3 step word problems. b. Analyze the effect on the product when a number is multiplied by 10, 100, 1000, 0.1, and 0.01. NOT INCLUDED M5N3. Students will further develop their understanding of the meaning of multiplication and division with decimal fractions and use them. a. Model multiplication and division of decimal fractions by another decimal fraction. Applies concepts for multiplication and division of decimals, including estimation and mental computation b. Explain the process of multiplication and division, including situations in which the multiplier and divisor are both whole numbers and decimal fractions. NOT INCLUDED c. Multiply and divide with decimal fractions including decimal fractions less than one and greater than one. NOT INCLUDED d. Understand the relationships and rules for multiplication and division of whole numbers also apply to decimal fractions. M5N4. Students will continue to develop their understanding of the meaning of common fractions and compute with them. Demonstrates and applies concepts of fractions with denominators of 2, 3, 4, 5, 6, 7, 10, 16, or 100. a) Uses concrete and visual models to represent a part of a whole, a part of a set, and a point on a number line. b) Identifies numbers as odd, even, prime, or composite. c) Writes prime factorization of composite numbers and identifies greatest common factor (GCF). d) Writes fractions in simplest form. e) Identifies multiples of a given number and determines least common multiple (LCM). f) Compares and orders fractions with and without models. g) Changes improper fractions to mixed numbers and the reverse. Understand division of whole numbers can be represented as a fraction (a/b = a ÷ b). What you see on this slide are the 5th grade GPS standards/elements on the left and the 5th grade QCC objectives on the right. The GPS standards (identified in the format M6N1, for example) are the broad statements of the knowledge and skills students are to be able to apply and may appear across several grade levels. The elements detail the specifics students are to learn toward that standard for the grade level and rarely repeat from grade level to grade level. The QCC objectives are generally broadly stated and do not provide the level of detailed focus that the GPS elements do. GPS in mathematics increases expectations and academic rigor for all students. Note the standards/elements in the 5th grade QCC-GPS Crosswalk that are in the Georgia Performance Standards that were not in the QCC. These are indicated by the phrases “NOT INCLUDED” or “NOT EXPLICITLY STATED.” Additionally, it must be noted that the QCC was rife with repetition from grade level to grade level, while there is little repetition of elements from one grade to the next in GPS. Greater expectations mean the development of DEEPER mathematical knowledge rather than a broad spectrum of surface level knowledge. The curriculum is no longer “a mile wide and an inch deep.” THREE SLIDES FOLLOW THIS ONE

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GPS 5th FCSS 5th Grade b. Understand the value of a fraction is not changed when both its numerator and denominator are multiplied or divided by the same number because it is the same as multiplying or dividing by one. Applies skills and concepts for computation of fractions and mixed numbers a) Adds and subtracts fractions and mixed numbers. b) Multiplies and divides fractions and mixed numbers. c. Find equivalent fractions and simplify fractions. NOT INCLUDED d. Model the multiplication and division of common fractions. e. Explore finding common denominators using concrete, pictorial, and computational models. f. Use <, >, or = to compare fractions and justify the comparison. g. Add and subtract common fractions and mixed numbers with unlike denominators. h. Use fractions (proper and improper) and decimal fractions interchangeably. i. Estimate products and quotients. M5N5. Students will understand the meaning of percentage. Uses concrete and visual models to represent parts of a whole for percents and uses percents interchangeable with fractions and decimals. Computes the percent of a number a. Model percent on 10 by 10 grids. b. Apply percentage to circle graphs. Measurement M5M. Students will compute the area of geometric plane figures. They will also understand the concept of volume and compute the volume of simple geometric solids and measure capacity. Students will convert from one unit to another within one system of measurement NOT EXPLICITY STATED M5M1. Students will extend their understanding of area of fundamental geometric plane figures. NOT INCLUDED a. Estimate the area of fundamental geometric plane figures. Uses concrete models to develop and apply formulas for area, perimeter, and volume. b. Derive the formula for the area of a parallelogram (e.g., cut the parallelogram apart and rearrange it into a rectangle of the same area). NOT INCLUDED c. Derive the formula for the area of a triangle (e.g. demonstrate and explain its relationship to the area of a rectangle with the same base and height). d. Find the areas of triangles and parallelograms using formulae. e. Estimate the area of a circle through partitioning and tiling and then with formula (let pi = 3.14). (Discuss square units as they apply to circles.) Identifies terms associated with a circle (diameter, radius, and circumference) and finds the circumference using pi f. Find the area of a polygon (regular and irregular) by dividing it into squares, rectangles, and/or triangles and find the sum of the areas of those shapes. This slide and the next two continue to give you a visual look at the differences between GPS and QCC at the fifth grade level in mathematics. We will be giving you at the end of this presentation a notebook that contains this crosswalk for grades K-8.

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GPS 5th FCSS 5th Grade M5M3. Students will measure capacity with appropriately chosen units and tools. Estimates, selects, and applies customary and metric units of measurement a) Length – inch, foot, yard, mile, mm, cm, dm, m, km. Measures length to the nearest quarter inch and mm. b) Capacity/volume – cup, pint, quart, gallon, fluid ounce, teaspoon, tablespoon, milliliter, liter c) Weight/mass – ounce, pound, ton, gram, kilogram d) Elapsed time – hour and minute e) Temperature – Celsius and Fahrenheit/above and below freezing a. Use milliliters, liters, fluid ounces, cups, pints, quarts, and gallons to measure capacity. b. Compare one unit to another within a single system of measurement (e.g., 1 quart = 2 pints). Converts from one customary unit to another customary unit and from one metric unit to another metric unit. M5M4. Students will understand and compute the volume of a simple geometric solid. NOT EXPLICITY STATED a. Understand a cubic unit (u3) is represented by a cube in which each edge has the length of 1 unit. NOT INCLUDED b. Identify the units used in computing volume as cubic centimeters (cm3), cubic meters (m3), cubic inches (in3), cubic feet (ft3), and cubic yards (yd3). NOT INCLUDED c. Derive the formula for finding the volume of a cube and a rectangular prism using manipulatives. d. Compute the volume of a cube and a rectangular prism using formulae. e. Estimate the volume of a simple geometric solid. f. Understand the similarities and differences between volume and capacity. Geometry M5G. Students will further develop their understanding of geometric figures. M5G1. Students will understand congruence of geometric figures and the correspondence of their vertices, sides, and angles. M5G2. Students will understand the relationship of the circumference of a circle to its diameter is pi (π ≈ 3.14). Algebra M5A. Students will represent and investigate mathematical expressions algebraically by using variables. NOT EXPLICITY STATED M5A1. Students will represent and interpret the relationships between quantities algebraically. Explores basic algebraic concepts a) Uses order of operations to simplify numeric expressions that involve addition and subtraction with and without parenthesis. b) Uses a variable to represent an unknown amount in a mathematical expression or equation (number sentence) and evaluates simple algebraic expressions. c) Finds the value of or solves for the variable in a simple algebraic equation such as a + 6 = 10. d) Identifies and uses symbols of equality and inequality a. Use variables, such as n or x, for unknown quantities in algebraic expressions. b. Investigate simple algebraic expressions by substituting numbers for the unknown. c. Determine that a formula will be reliable regardless of the type of number (whole numbers or decimal fractions) substituted for the variable.

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GPS 5th FCSS 5th Grade Data Analysis and Probability M5D. Students will gather, organize, and display data and interpret graphs. Collects and organizes data, determines the appropriate scale and constructs interprets and makes predictions based on data displays such as: tables, charts, pictographs, bar graphs, circle graphs, and simple line graphs using a variety of scales. M5D1. Students will analyze graphs. a. Analyze data presented in a graph. b. Compare and contrast multiple graphic representations (circle graphs, line graphs, bar graphs, etc.) for a single set of data and discuss the advantages/disadvantages of each. M5D2. Students will collect, organize, and display data using the most appropriate graph. NOT EXPLICITY STATED Process Skills EMBEDDED, BUT NOT EXPLICITY STATED M5P. Each topic studied in this course should be developed with careful thought toward helping every student achieve the following process standards. EMBEDDED, BUT NOT EXPLICITY STATED M5P1. Students will solve problems (using appropriate technology). a. Build new mathematical knowledge through problem solving. b. Solve problems that arise in mathematics and in other contexts. c. Apply and adapt a variety of appropriate strategies to solve problems. d. Monitor and reflect on the process of mathematical problem solving. M5P2. Students will reason and evaluate mathematical arguments. a. Recognize reasoning and proof as fundamental aspects of mathematics. b. Make and investigate mathematical conjectures. c. Develop and evaluate mathematical arguments and proofs. d. Select and use various types of reasoning and methods of proof. M5P3. Students will communicate mathematically.

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**Authentic Performance Assessment – Grade 2**

Standards and elements to be assessed: M2N4 a. Model, identify, label, and compare fractions (thirds, sixths, eighths, tenths) as a representation of equal parts of a whole or of a set. b. Know that when all fractional parts are included, such as three thirds, the result is equal to the whole. Description of the Task: You are planning a Pizza Party. Four adults and ten children will eat pizza. You have three pizzas. Adults will each get 2 slices and children will each get 1 slice. All slices must be the same size. You will use models, drawings, and explanations to show what fraction of a pizza each guest will get (use all of the slices of pizza). Be sure to have enough slices of pizza for everyone invited! The difference in GPS and QCC is not just about content. It is also about how students are expected to demonstrate their understanding of that content. This slide is an example of an authentic performance assessment at the end of a grade 2 unit on the comparison of fractional parts and their relationships. Note what the students are asked to do. In the notebooks we provided you on Friday, there is a complete description of this task, with the written materials the students would use to help them solve this problem. It should also be noted that students would have had numerous instructional activities before this assessment was given, all directed at teaching students the knowledge and skills necessary to solve the problem. In contrast, with the Quality Core Curriculum, students would more likely be asked to answer a series of multiple choice questions that would have the student say whether one fraction is greater than, equal to, or less than another fraction. Other questions might deal with whether a combination of fractions equals a whole. This task clearly requires a deeper understanding of the mathematics than would the multiple choice questions described above. Students, however, will be exposed to both selected response (multiple choice) and performance assessments as part of a balanced assessment approach demanded by the Georgia Performance Standards.

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**Approaching the Standard**

Performance Assessment Rubric – 2nd Grade Criteria The student: 1 Below the Standard 2 Approaching the Standard 3 Meets the Standard 4 Exceeds the Standard Score Works cooperatively with the members of his or her group. Does not stay on task or contribute to the group Mostly involved; some off-task behavior Actively involved with the group Displays leadership skills and respects the ideas of others. M2N4. a. Identifies the fractions of pizza served to children and adults at the party. Accurately completes less than two tasks: models, draws, labels, or explains the fractions. Accurately completes two or three of these tasks: models, draws, labels, or explains the fractions. Accurately models, draws, labels, and explains all of the fractions. Accurately models, draws, labels, and explains the fractions. Converts some fractions into reduced form. 2/6 = 1/3 a. Compares the fractions of pizza served at the party. Does not compare the different fractions used. Identifies the fractions used, yet compares only two. Compares each fraction used to each other fraction used, using words or math symbols (<,>,=) Compares each fraction used to each other fraction used, using words and math symbols (<,>,=) b. Identifies the fraction naming each whole pizza. Accurately completes less than two tasks: models, draws, labels, or explains the fraction naming a whole pizza. Accurately completes two or three of these tasks: models, draws, labels, or explains the fraction naming a whole pizza. Accurately models, draws, labels, and explains the fraction naming a whole pizza. Accurately models, draws, labels, and explains the fraction naming a whole pizza and identifies other fractions that would name a whole. a. Creates given fractions Wrong number and/or differently sized fractions. Correct numbers of pieces, but the pieces are not close to the same size. Correct number of pieces and the pieces are close to the size. Correct number of pieces, and the pieces are accurately sized. Correctly answers problem solving ques-tions. Answers are missing or incorrect. Correctly answers two of the three problems. Correctly answers all three problems. Correctly answers all three problems, offering every possible answer. This is the rubric that teachers use to evaluate the task. Notice that a 3 represents meeting the standard and that a 4 represents exceeding the standard. Students who consistently score 4’s would typically be working at an advanced on-grade level.

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**Performance Assessment Sample – 7th Grade**

Standards and elements to be assessed M7D.1 a. Formulate questions and collect data from a census of at least 30 objects and from samples of varying sizes. b. Construct frequency distributions. c. Analyze data using measures of central tendency (mean, median, and mode), including recognition of outliers. d. Analyze data with respect to measures of variation (range, quartiles, inter-quartile range). e. Compare measures of central tendency and variation from samples to those from a census. Observe that sample statistics are more likely to approximate the population parameters as sample size increases. f. Analyze data using appropriate graphs, including pictographs, histograms, bar graphs, line graphs, circle graphs, and line plots introduced earlier, and using box and-whisker plots and scatter plots. g. Analyze and draw conclusions about data, including describing the relationship between two variables. M7P3.a. Organize and consolidate their mathematical thinking through communication. b. Communicate their mathematical thinking coherently and clearly to peers, teachers, and others. c. Analyze and evaluate the mathematical thinking and strategies of others. M7P4.c. Recognize and apply mathematics in contexts outside of mathematics. M7P5.c. Use representations to model and interpret physical, social, and mathematical phenomena. This is an example of a seventh grade performance assessment after a unit on experimentAL design and data analysis. The complete description of the task is in your notebooks. Description of the Task: (GRASPS) Which is faster, man or machine? Your role as a scientist is to create simple experiments involving daily routines at school to determine which is faster, man or machine. Before you conduct the experiments you will survey 30 of your peers to gather their prediction about which is faster man or machine for each experiment. Your audience is your colleagues (other scientists) that are not a part of your study. Each individual group will display results of each experiment in box and whisper plots and discuss if their conclusions upheld their predictions. Your goals are to: design experiments to test man vs. machine survey your peers to find their answer to that question for each experiment test man versus machine 4. analyze the results and check the accuracy of the predictions

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**Performance Assessment Rubric – 7th Grade**

Criteria The student: 4 Exceeds Standards 3 Meets Standards 2 Approaches Standards 1 Below Standards Score Questions M7D1.a Questions demonstrate understanding of quantitative data and relevance to audience. Questions created demonstrate some understanding of quantitative data. Questions suggest little thought of audience and do not measure quantitative data. Questions suggest little or no thought of audience and do not measure quantitative data. Data Collection and Frequency Distribution Table M7D1.a, b Frequency table includes all survey data, as well as appropriate and accurate intervals. Frequency table includes most survey data, as well as appropriate and accurate intervals. Frequency table includes some survey data, as well as appropriate and accurate intervals. Frequency table includes little survey data, as well as appropriate and accurate intervals. Chooses the appropriate methods for displaying their data. M7D1.f Correctly displays data using a double line graph, double bar graph and/or a pictograph. Correctly displays data using a line graph, bar graph, and/or a pictograph independently. Chooses the appropriate graphs but displays data incorrectly. Uses inappropriate graphs to display data (i.e. Circle Graph). Measures of Central Tendency M7D1.c,e Analyzes data using all measures of central tendency, extending interpretation to include other mathematical information. Analyzes data using most measures of central tendency, extending interpretation to include other mathematical information. Analyzes data using some measures of central tendency, extending interpretation to include other mathematical information. Analyzes data using few measures of central tendency, extending interpretation to include other mathematical information. Box-and-Whisker Plot M7D1.d,f Correctly identifies all of the following: Least and greatest values, upper and lower quartiles, and median Correctly identifies most of the following: Correctly identifies some of the following: Correctly identifies few of the following: Conclusion M7P3.a,b,c M7D1.g Includes both individual and group work. Fully justifies conclusions using the data. Includes both individual and group work. Justifies conclusions using the data. Includes both individual and group work. Partially justifies conclusions using the data. Includes both individual and group work. Poorly justifies conclusions using the data. Presentation M7P5.c Demonstrates full knowledge by using accurate statistical information, including all measures of central tendency. Demonstrates knowledge by using accurate statistical information, including most measures of central tendency. Demonstrates partial knowledge by using some accurate statistical information, including the measures of central tendency. Demonstrates little knowledge by using inaccurate statistical information, including few measures of central tendency. This is the rubric the seventh grade teacher uses to evaluate the performance of students on the task.

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**Fulton County Tools That Support Differentiation and Acceleration**

Pre and Post Unit Assessments Differentiation Matrix Acceleration Assessments Placement Assessments The tools listed on this slide support teachers in differentiating effectively the curriculum so that students are placed and move through the curriculum successfully and without gaps in their learning.

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**PRE AND POST UNIT ASSESSMENTS**

A unit pre-assessment is administered at the beginning of each unit to inform the teacher at what level the student should be working—be that advanced, on-level or on-level with support. A unit post-assessment is administered at the end of each unit to determine the level at which the student has attained the standards/elements taught within the unit and whether re-teaching of selected standards/elements is required.

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**DIFFERENTIATION MATRIX SAMPLE**

The Differentiation Matrix provides the teacher a “roadmap” for acceleration or support by standard/element.

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You see on this slide a sample of the differentiation matrix designed for our current implementation of K-2 mathematics. Such a matrix is under design for the 3-5 mathematics to be implemented this fall. Notice the 2nd grade column. StudentS are administered a pre-test before each unit to assure that prerequisite skills are in place and to determine how much of the new information students may already have acquired. If the student knows the prerequisites but none of the new material for the unit, the teacher would find the elements taught in the unit in the second grade column and plan on-level activities for the student to give ample opportunity for the student to acquire the grade level concepts. If the student has few of the prerequisite skills, the teacher would look at the first grade standards/elements on which the second grade curriculum is built and instruction would be planned to provide support to the student by helping them acquire the prerequisites needed to then master the second grade curriculum. If the student already has met the standards/elements to be taught in the unit, the teacher would plan instruction that would build on the second grade standards/elements toward the connected standards/elements at third grade. Students are often inconsistent in their mastery of concepts and may find themselves needing support on some standards/elements and advancement on others. Students who work consistently at the advanced level will have the opportunity to take the acceleration test, in this case, for third grade. If the student performs at the 95% level, he/she will be accelerated to Grade 4 Mathematics in their third grade year.

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**ACCELERATION ASSESSMENTS**

Acceleration assessments are available for each grade level. These assessments are to be administered at the end of the school year to students who have consistently and successfully worked through the grade-level curriculum at the advanced level. If the student scores 95% or better on the acceleration assessment, he/she will be accelerated a grade level.

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**How Do We Place New Students to Fulton County?**

Students new to Fulton County are administered a series of placement assessments that enables their placement in the most appropriate grade level mathematics. Placement assessments are available to ensure teachers have an understanding of either the support or acceleration needs of students new to our system.

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**Typical K-12 Progression (with support and/or advancement as appropriate)**

Kindergarten Grade K Mathematics First Grade Grade 1 Mathematics Second Grade Grade 2 Mathematics Third Grade Grade 3 Mathematics Fourth Grade Grade 4 Mathematics Fifth Grade Grade 5 Mathematics Sixth Grade Grade 6 Mathematics Seventh Grade Grade 7 Mathematics Eighth Grade Grade 8 Mathematics Ninth Grade either Math I or Accelerated Math I Tenth Grade either Math II or Accelerated Math II Eleventh Grade either Math III or Accelerated Math III Twelfth Grade either Math IV or AP Statistics or AP Calculus AB or AP Calculus BC This slide shows the progression through the curriculum for students working either on-level, on-level with support, or advanced in grades K-5. It is important to note that students who complete Grade 5 Mathematics at an advanced level, and who meet the requirements, would place in Grade 6 Mathematics-Advanced. If successful in Grade 8 Mathematics-Advanced, they would qualify for Accelerated Math I in the 9th grade.

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**Accelerated Progression K-12 Sample**

Kindergarten Grade K Advanced First Grade Grade 1 Advanced* Second Grade Grade 3 Advanced Third Grade Grade 4 Advanced Fourth Grade Grade 5 Advanced Fifth Grade Grade 6 Advanced Sixth Grade Grade 7 Advanced Seventh Grade Grade 8 Advanced Eighth Grade Accelerated Math I Ninth Grade Accelerated Math II Tenth Grade Accelerated Math III Eleventh Grade AP Calculus AB or BC Twelfth Grade AP Calculus BC or Calculus II / III (videoconferencing with GA Tech) This slide shows the progression through the curriculum for students working at an advanced level in K-5 as well as accelerating a year in second grade. Acceleration in grades K-5 can take place at any year following a student working consistently and successfully at an advanced level within the grade and achieving a 95% or better on the appropriate acceleration assessment.

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QUESTIONS?

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**Fulton County Schools Curriculum and Instruction Division**

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