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Individualized Scope and Sequence
Introduction
Student Profile:
This scope and sequence was written for R.J. Wisenbaker, a young man who will be in 10th grade in the fall of 1998. R.J. tested into the gifted program in elementary school and has been taking gifted classes since that time. When I asked R.J. to describe himself, he quickly responded "smart, musically talented and coordinated". R.J. has a great interest in music; he takes piano and cello lessons, plays cello in the school orchestra, and sings in an after school chorus as well as the youth choir at his church. This academic year, he auditioned and was selected for both All-State Chorus and All-State Orchestra. He feels very confident of his musical abilities and is comfortable performing in front of an audience. This confidence does not carry over to other areas; he does not like to present reports in front of his peers and has no interest in acting. He loves to read and has an appreciation of art and architecture. At this time, he is interested in a career in music, but isn't totally committed to that path. R.J. is outgoing and has a sunny disposition. He enjoys and gets along well with adults but is having some difficulties relating to his peers this year. He sometimes feels that his classmates are ignoring him and has been depressed due to a sense of abandonment and isolation. He has fairly good study habits but occasionally elects not to hand in assignments. He doesn't like the process of writing factual papers and tends to procrastinate when faced with such a task. As an 8th grader, R.J. took Algebra I. He is currently taking Geometry and is averaging close to 90%. His math teacher has recommended that R.J. take Algebra II next year. He likes math, but it is not always his favorite subject. His favorite classes this year are Great Books, Physical Science, and Geometry. His least favorite subject is history. Addressing the Student's Needs through the ISS In order to determine his areas of strength and weakness, RJ took a final exam in Algebra I (Elementary Algebra) in April. Based on his performance, a content standard on scientific notation was added and standard #20 from the QCC standards was omitted. Algebra II QCC content standards have either been included as overarching goals or differentiated to be more appropriate for a gifted student. To augment the QCC standards and to further tailor this course to R.J.'s interests, two additional foci were created: A Geometry/Measurement focus with an architectural slant and an independent study unit focusing on the relationship of mathematics and music. Additional foci and content standards have been woven into the ISS to extend the concepts developed in the QCC and to add an historical as well as an affective dimension. Collaborative objectives are emphasized to help R.J. engage with his peers in a positive and productive way. Opportunities to write and present findings with the support of a group are provided to help him overcome his discomfort with those processes. Objectives dealing with characteristics of different mathematicians and the impact of mathematics on career choices will hopefully help R. J. feel more comfortable about himself and informed about his career options.
Overarching Differentiated Goals
- To provide a meaningful context for gifted students to learn Algebra II and Geometry and Measurement concepts and skills.
- To provide experiences that will encourage the development of mathematical reasoning, problem solving and communication.
- To create an environment that will reflect and reinforce the value of mathematics as a creative human endeavor and as a crucial element in many of the occupations to which gifted student might aspire.
- To engage gifted students in the process of mathematics through the exploration of its history and the people whom have stretched its limits.
- To provide opportunities for gifted students to become proficient with up-to-date technological tools and to use those tools to help develop conceptual understanding.
- To engender in gifted students a sense of confidence and pride in their own mathematical abilities and an appreciation and valuing of the abilities of others.
- To offer collaborative opportunities for students to make sense of mathematics and their relationships with others.
Foci and Associated ContentStandards Over Time--Algebra II
A. August-September 1. Focus: Investigating number sets, their properties and the people who developed them.
a. Compares and contrasts subsets of the real numbers and locates any real number on the number line. b. Evaluates the use of real number properties in proofs and in solving equations. c. Collaboratively researches the evolution of number sets and examines the relation of each new set to the societal needs or historical aspects of the time. d. Performs an in-depth investigation of one of the various people who have been instrumental in the evolution of number sets and compares that personality to himself. Results to be discussed as a group. e. Investigates the mathematics of "secret" codes, the people who develop them, the jobs available, and considers its attraction as a career option. 2. Focus: Investigating linear relationships in one variable. a. Solves problems by writing and solving linear equations and inequalities in one variable, including compound inequalities. b. Solves linear inequalities in one variable and distinguishes between the different representations of solutions sets, including graphs and interval notation. c. In cooperation with other students, prepares a visual aid and lesson that uses distances on the real number line to distinguish the different solution paths in solving absolute value equalities and inequalities in one variable and relates them to algebraic algorithms. B. October - November 3. Focus: The coordinate plane and linear equations a. Graphs and identifies ordered pairs in the coordinate plane and relates it to locating one's position on a map. b. Collaboratively investigates the life and times of Rene Descartes; creates a visual (picture, tableau, puppet show, web page, power point presentations,etc.) of some mathematical aspect of Descartes' life; discusses the apocryphal story of his invention of the coordinate plane; considers what characteristics are necessary for making that kind of intuitive leap; analyses who in the class might have those characteristics. c. Distinguishes the different characteristics (slope, intercepts, equations) of a horizontal line, a vertical line and a line that's neither. d. Compares and contrasts the different forms of equations for a line and writes and graphs equations of lines given various information (such as slope, intercepts, points on the line). e. Uses computers and calculators to investigate linear functions. f. With parental permission, views two or three movies/documentaries that portray mathematicians. Compares those portrayals with that of the stereotypical "nerd". Compares the life of a mathematician in today's world to that of mathematician in Descartes' day. Discusses mathematics as a career option. g. Writes and graphs linear equations from data; considers systems of these equations and the implications of their intersection or lack there of. h. Cooperatively, in a small group, analyzes one method of solving linear systems (algebraically, graphically, etc); communicates findings to class and discusses the appropriateness of each method presented i. Investigates higher order systems and considers their application to real world situations. j. Solves linear programming and decision-making problems by solving linear inequalities k. Collaboratively creates applications problem that could be solved using a system of linear inequalities in two variables, solves them by several methods including calculators and computers, presents problems and solutions to class for discussion. C. December 4. Focus: Investigating Functions & Relations and how we "relate" to each other. a. Formulates relations from data; constructs graphs using several methods (by hand, calculators and computers) and interprets the graphs in relation to the data. Compares and contrasts the domain and range of the relations formulated; distinguishes between meanings of domain and range. b. Formulates relations and functions from data, constructs their graphs using several methods (by hand, calculator and computer). Distinguishes between relations and functions. c. Collaboratively creates written instructions for composing functions; researches the applications of function composition, presents findings to class. d. Creates inverse relations from given functions algebraically and graphically. Distinguishes between inverse relations and inverse functions. Considers the composition of a function and its inverse. e. Discusses group relations and the different roles and responsibilities taken in group endeavors. f. Cooperating in a small group, develops creative definitions that distinguish between "functions" and "relations" among people and compares it to the mathematical definitions; shares definitions with class. g. Discusses reaction to the realization that there are others who function at higher levels--how do you learn to live with it? D. January-February 5. Focus: Investigating Rational Expressions and Equations, and the "rationality"of humans. a. Analyzes the procedures for performing operations with numerical fractions and applies them to the same operations (addition, subtraction, multiplication and division) involving rational expressions. b. Investigates techniques for solving rational equations, including fractional equations and equations with fractional coefficients. c. Investigates classes of problems that are solved using rational equations; creates and solves examples of each. d. Collaboratively, compares the mathematical definitions of rational and irrational with commonly used definitions regarding human behavior. Discusses reasons for irrational and rational behavior. 6. Focus: Investigating Radicals and Exponents
a. Analyzes the laws of exponents and uses them to simplify expressions with integer exponents. b. Investigate the use of scientific notation and procedures for translation from one notation form to another. Apply laws of exponents to operations involving numbers written in scientific notation. c. Collaboratively proposes and applies a strategy for applying the laws of exponents to expressions containing non-integral exponents and analyzes the results for consistency. d. Analyzes the properties of radicals and applies them to operations involving radical expressions, including rationalizing denominators. e. Collaboratively determines the reason(s) denominators are rationalized and judges its necessity today; considers and discusses other mathematical topics that might be discarded as suggested in NCTM's Standards. f. Solves radical equations with one or two radical terms and solves application problems involving such radical equations. g. Collaboratively, compares complex numbers to real numbers; examines procedures for determining the additive inverse, conjugate and absolute value of complex numbers and procedures for operations (addition, subtraction, multiplication and division) with complex numbers.
E. March -April
7. Focus: Topics in Geometry, Measurement and Architecture a. Compares the metric units of measurement with the US customary units. Debates the advantages/disadvantages of each and possible reasons why the US hasn't converted to metric. b. Compares and contrasts the concepts of perimeter, area and volume; analyzes and applies the formulae for determining each for different geometric figures. c. Investigates scaling and its use in architectural drawings and blueprints; creates a drawing or blueprint with a selected scale. d. Collaboratively, constructs selected geometric figures in two and three dimensions and determines the surface area, perimeter, and volume as appropriate; shares and compares findings with other groups. e. Uses measurement concepts to solve application problems. f. Formulates and solves real-life problems utilizing the concepts of perimeter, area and volume. g. Collaboratively, researches the history of measurement and creates a presentation on an aspect of interest. h. Selects an architectural topic, using The Art of Construction (Mario Salvador, 1990) and Builders of the Ancient World (National Geographic Society, 1986) as a springboard for ideas; constructs model on selected topic. i. Considers and discusses architecture as a career option after researching topic and hearing a guest speaker or visiting an architect on the job. F. May - June 8. Focus: Independent Study on Music and Mathematics. Possible Topics of Investigation, Underlying Objectives, and Products:
Topic of Investigation | Underlying Objective | Products |
1 | The Mathematics of Music Theory | Correlating number sets with scales, composition, etc. | Video, power point presentation, etc. |
2 | The Mathematics of Sound | Investigating the mathematical aspects of sound. | Video, power-point presentation, demonstration with cello |
3 | Scientists and Mathematicians Who Play Music and Why! | Combining music with other careers, investigating how they enrich each other. | Book, poster, etc. |
4 | What is the Music of the Spheres? | Investigating Kepler, Pythagoras and others and their musical/mystical slant on mathematics. | Written report, video, power-point presentation, etc. |
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