Fulton County Schools

Curriculum and Instruction Division

Meeting the Needs of Meeting the Needs of All Students with the All Students with the Georgia Performance Georgia Performance

Standards (GPS)Standards (GPS)Fulton County Board of Education Fulton County Board of Education

RoundtableRoundtable

February 27, 2007February 27, 2007

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 Quality Core Curriculum (QCC)Curriculum (QCC)

Continuous AchievementContinuous Achievement

Georgia Georgia Performance Performance

Standards (GPS)Standards (GPS)DifferentiationDifferentiation

Fulton County’s Fulton County’s Continuous Continuous Achievement Model Achievement Model for English Language for English Language Arts and MathematicsArts and Mathematics

Differentiation with Differentiation with acceleration and acceleration and support (accelerated support (accelerated remediation) as remediation) as appropriateappropriate

Curriculum DifferentiationCurriculum 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)

QCC – GPS QCC – GPS CrosswalkCrosswalk

GPS 5thGPS 5th FCSS 5th GradeFCSS 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. NOT INCLUDED

c. Analyze and use divisibility rules. 5.1.07 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.

5.1.02 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. 5.1.03 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. NOT EXPLICITLY STATED

a. Model multiplication and division of decimal fractions by another decimal fraction. 5.1.11 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. NOT INCLUDED

M5N4. Students will continue to develop their understanding of the meaning of common fractions and compute with them.

5.2.05 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).

GPS 5thGPS 5th FCSS 5th GradeFCSS 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.

5.2.06 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. NOT INCLUDED

e. Explore finding common denominators using concrete, pictorial, and computational models. NOT INCLUDED

f. Use <, >, or = to compare fractions and justify the comparison. NOT INCLUDED

g. Add and subtract common fractions and mixed numbers with unlike denominators. NOT INCLUDED

h. Use fractions (proper and improper) and decimal fractions interchangeably. NOT INCLUDED

i. Estimate products and quotients. NOT INCLUDED

M5N5. Students will understand the meaning of percentage. 5.2.12 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. NOT INCLUDED

b. Apply percentage to circle graphs. NOT INCLUDED

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. 5.1.06 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). NOT INCLUDED

d. Find the areas of triangles and parallelograms using formulae. NOT INCLUDED

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.)

5.2.10 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. NOT INCLUDED

GPS 5thGPS 5th FCSS 5th GradeFCSS 5th Grade

M5M3. Students will measure capacity with appropriately chosen units and tools. 5.2.03 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). 5.2.04 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. NOT INCLUDED

d. Compute the volume of a cube and a rectangular prism using formulae. NOT INCLUDED

e. Estimate the volume of a simple geometric solid. NOT INCLUDED

f. Understand the similarities and differences between volume and capacity. NOT INCLUDED

Geometry M5G. Students will further develop their understanding of geometric figures. NOT INCLUDED

M5G1. Students will understand congruence of geometric figures and the correspondence of their vertices, sides, and angles. NOT INCLUDED

M5G2. Students will understand the relationship of the circumference of a circle to its diameter is pi (π ≈ 3.14). NOT INCLUDED

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.

5.2.01 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.

GPS 5thGPS 5th FCSS 5th GradeFCSS 5th Grade

Data Analysis and Probability M5D. Students will gather, organize, and display data and interpret graphs.

5.1.09 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). EMBEDDED, BUT NOT EXPLICITY STATED

a. Build new mathematical knowledge through problem solving. EMBEDDED, BUT NOT EXPLICITY STATED

b. Solve problems that arise in mathematics and in other contexts. EMBEDDED, BUT NOT EXPLICITY STATED

c. Apply and adapt a variety of appropriate strategies to solve problems. EMBEDDED, BUT NOT EXPLICITY STATED

d. Monitor and reflect on the process of mathematical problem solving. EMBEDDED, BUT NOT EXPLICITY STATED

M5P2. Students will reason and evaluate mathematical arguments. EMBEDDED, BUT NOT EXPLICITY STATED

a. Recognize reasoning and proof as fundamental aspects of mathematics.EMBEDDED, BUT NOT EXPLICITY STATED

b. Make and investigate mathematical conjectures. EMBEDDED, BUT NOT EXPLICITY STATED

c. Develop and evaluate mathematical arguments and proofs. EMBEDDED, BUT NOT EXPLICITY STATED

d. Select and use various types of reasoning and methods of proof. EMBEDDED, BUT NOT EXPLICITY STATED

M5P3. Students will communicate mathematically. EMBEDDED, BUT NOT EXPLICITY STATED

Authentic Performance Authentic Performance Assessment – Grade 2Assessment – Grade 2

Standards and elements to be assessed: M2N4a. 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!

Performance Assessment RubricPerformance Assessment Rubric – 2 – 2ndnd Grade Grade

CriteriaThe 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

M2N4.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 (<,>,=)

M2N4.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.

M2N4.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.

Performance Assessment SamplePerformance Assessment Sample – 7th Grade – 7th Grade

Standards and elements to be assessedM7D.1a. 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.

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: 1.design experiments to test man vs. machine 2.survey your peers to find their answer to that question for each experiment 3.test man versus machine4. analyze the results and check the accuracy of the predictions

Performance Assessment RubricPerformance Assessment Rubric – 7 – 7thth Grade Grade

CriteriaThe student:

4 Exceeds Standards

3 Meets Standards

2 Approaches Standards

1 Below Standards

Score

QuestionsM7D1.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 TendencyM7D1.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 PlotM7D1.d,f

Correctly identifies all of the following:Least and greatest values, upper and lower quartiles, and median

Correctly identifies most of the following:Least and greatest values, upper and lower quartiles, and median

Correctly identifies some of the following:Least and greatest values, upper and lower quartiles, and median

Correctly identifies few of the following:Least and greatest values, upper and lower quartiles, and median

ConclusionM7P3.a,b,cM7D1.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.

PresentationM7P5.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.

Pre and Post Unit AssessmentsPre and Post Unit Assessments Differentiation MatrixDifferentiation Matrix Acceleration AssessmentsAcceleration Assessments Placement AssessmentsPlacement Assessments

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

PRE AND POST UNIT PRE AND POST UNIT ASSESSMENTSASSESSMENTS

DIFFERENTIATION DIFFERENTIATION MATRIX SAMPLEMATRIX SAMPLE

ACCELERATION ACCELERATION ASSESSMENTSASSESSMENTS

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

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

KindergartenKindergarten Grade K Mathematics Grade K Mathematics

First GradeFirst Grade Grade 1 Mathematics Grade 1 Mathematics

Second GradeSecond Grade Grade 2 MathematicsGrade 2 Mathematics

Third GradeThird Grade Grade 3 MathematicsGrade 3 Mathematics

Fourth GradeFourth Grade Grade 4 MathematicsGrade 4 Mathematics

Fifth GradeFifth Grade Grade 5 MathematicsGrade 5 Mathematics

Sixth GradeSixth Grade Grade 6 MathematicsGrade 6 Mathematics

Seventh GradeSeventh Grade Grade 7 MathematicsGrade 7 Mathematics

Eighth GradeEighth Grade Grade 8 Mathematics Grade 8 Mathematics

Ninth GradeNinth Grade either Math I or Accelerated Math I either Math I or Accelerated Math I

Tenth GradeTenth Grade either Math II or Accelerated Math II either Math II or Accelerated Math II

Eleventh GradeEleventh Grade either Math III or Accelerated Math III either Math III or Accelerated Math III

Twelfth GradeTwelfth Grade either Math IV or AP Statistics or AP Calculus AB either Math IV or AP Statistics or AP Calculus AB or AP Calculus BC or AP Calculus BC

Accelerated Progression K-12Accelerated Progression K-12SampleSample

KindergartenKindergarten Grade K Advanced Grade K Advanced

First GradeFirst Grade Grade 1 Advanced* Grade 1 Advanced*

Second GradeSecond Grade Grade 3 Advanced Grade 3 Advanced

Third GradeThird Grade Grade 4 Advanced Grade 4 Advanced

Fourth GradeFourth Grade Grade 5 Advanced Grade 5 Advanced

Fifth GradeFifth Grade Grade 6 Advanced Grade 6 Advanced

Sixth Grade Sixth Grade Grade 7 Advanced Grade 7 Advanced

Seventh GradeSeventh Grade Grade 8 Advanced Grade 8 Advanced

Eighth GradeEighth Grade Accelerated Math I Accelerated Math I

Ninth GradeNinth Grade Accelerated Math II Accelerated Math II

Tenth GradeTenth Grade Accelerated Math III Accelerated Math III

Eleventh GradeEleventh Grade AP Calculus AB or BC AP Calculus AB or BC

Twelfth GradeTwelfth Grade AP Calculus BC or AP Calculus BC or Calculus II / III Calculus II / III (videoconferencing with GA Tech) (videoconferencing with GA Tech)

QUESTIONS?QUESTIONS?

Fulton County Schools

Curriculum and Instruction Division