Lakeside Middle School Mathematics Department

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Goal of the Program

The main goal of our program is to provide a strong foundation in mathematics via challenging courses that are appropriate to the age, abilities, and needs of our students. We hope that, in addition to being well prepared for the mathematics they may encounter in future schooling or careers, our students will emerge from the program impressed with the elegance and scope of the subject, and excited by its vast potential for fun and creativity.

We also aim to equip students with the mathematical skills of a competent citizen in today's world. These skills include the ability to model situations mathematically, to estimate and compare magnitudes, to interpret graphs and statistics, to calculate probabilities, to evaluate numerical and spatial conclusions, to solve problems mentally as well as with paper, calculator, and computer, and to communicate effectively in these areas. Finally, while much of the above is exercised in the context of individual work, we have the further goal of fostering the skills and value of doing mathematics cooperatively with others.

Course Content Overview

We define mathematics very broadly as a systematic approach to studying and representing quantities, forms, patterns, and logic. Thus the content of our courses encompasses the wide range of arithmetic, number theory, measurement, geometry, probability, statistics, graphing, pattern recognition, logic, and problem solving strategy.

As an overall description, grades five and six concentrate on arithmetic skills and the use of numbers and mathematical thinking in a variety of contexts (measurement, data collection, patterns, problem solving, etc). Grade seven consolidates and advances these skills and adds the conceptual and symbolic framework that will later be used extensively in algebra and geometry. Grade eight is a first year algebra (and trigonometry) course. For more details of course topics and the range within grade levels, see below under "Specific Course Content by Grade Level".

Features of the Middle School Math Program

The major strengths of the middle school math program include competent and dedicated teachers, small class sizes, and a department curriculum that is carefully considered yet allows maximum room for a teacher’s initiative and creativity. We also benefit from an ample budget and access to well-supported and well-equipped computer labs.

The Helical Curriculum

At all levels of the math curriculum, considerable reinforcement is achieved in each course by returning at a higher level to concepts and skills introduced in previous courses. In this “helical” curriculum, students periodically revisit certain basic mathematical concepts to study them in ways that make use of their greater capacities for perception and conceptualization. What may have first been understood intuitively through a visual or concrete exercise may be later rediscovered and extended via a more comprehensive and abstract system of symbols and laws. This is in keeping with the departmental view that, while students should be exposed to a wide variety of topics, certain topics must be returned to over time in order to be extended and mastered.

Accounting for a Range of Ability and Experience

In general, as the population of Lakeside student is skewed toward the higher end of the scale of mathematical ability, all our courses aim to be challenging for able students. However, for the rare student who is comparatively weak in math, the Skills Department offers tutoring in math as a further resource to help meet his or her needs.

Decisions regarding placement into the ability-grouped seventh and eighth grade courses are considered carefully, with several factors taken into account, including the student's motivation, teacher recommendations, previous grades, and standardized test scores. One year's placement does not rigidly determine the placement for the following year. A student in Math 7 may very well take Algebra 8a the following year, and a student in Math 7a does not automatically proceed to Algebra 8a. Our goal is to place each student into the course that best serves his or her learning in depth.

Methods of Instruction:

The goals of the Middle school math department are the constants that guide its teachers from year to year in all grade levels. These goals are expressed in the Lakeside Mission Statement, in the Lakeside Teaching Evaluation Criteria, and in this program description. However, as noted above, the program must account for a variety of students' abilities, math experience, and cognitive styles. Thus while the goals remain constant, the teachers' methods of instruction will vary from year to year, grade level to grade level, student to student, topic to topic, and even moment to moment. Our methods are the variables that will change as teachers ask themselves how best to fulfill our goals while taking into consideration the current context in which they are teaching. These methods may include grouping by ability or not, team teaching or teaching individually, project-based instruction or topic-based instruction, using manipulatives or using symbolic constructs, working on paper or working on the computer or calculator, "real-life" applications or imaginary applications, individual problem-solving or group problem-solving, etc. No method that is consistent with the school mission and teaching criteria is considered inherently "right" or inherently "wrong". Rather, each is evaluated according to its effectiveness for reaching our goals in the situation at hand.

However, some generalizations can nonetheless be made:

Middle school math courses may use a textbook as basic outline, but the classes frequently roam into different topics and activities than those found in the textbook. Our intention is to let the teachers decide, within broad outlines, what topics and materials are right for the students in their classes.
The middle school math teachers emphasize problem-solving and physical applications and illustrations as a concrete realization of (and often the precursor to) nearly all theoretical work. Particularly engaging for students are problems and models related to real lives, situations from architecture, government, science, sports, etc. The real-life context can provide useful experience and the student’s intuition of how real things behave may serve as a guide. A student will also see plenty of contrived, humorous, impractical, "unreal" problems, and these too may provide an experience of real learning and a lot of fun.
In teaching mathematics, we try to pose questions and problems that encourage a discovery of mathematical generalizations by the students themselves. The focus is on having students gain confidence in what they know by basing their knowledge on their own experience, not merely the distilled experience of others.
As a part of most mathematical instruction, we try to find something physical for students to see or do while learning. Students run into a lot of Geoboards, Cuisinaire Rods, blocks, Lab Gear, compasses, dice, etc. while at Lakeside. They do a lot of moving around. They use a good many tools. Our goal here, in addition to exploiting the fun and vividness inherent in seeing and doing something, is to have students get a feel for the mathematics, a physical understanding based on seeing and doing.

Use of technology:

The availability of calculators and computers has made it possible to teach certain topics much earlier and in new ways. Particularly useful are general tools such as Excel, Geometer’s SketchPad and Visual Basic. Data collection, functions, variables, geometric construction, sequences and series, and graphing are a few examples of topics explored through the use of computers in our math classes. All 7th and 8th graders use laptop computers. Their use involves weekly projects to explore, exemplify, and extend concepts in problem solving, graphing, modeling, and programming.

5^th Grade Course Content and Features

The 5th grade mathematics course explores the patterns and relationships that lie at the heart of mathematics. In close integration with the science course, the curriculum is organized around interesting mathematical problems derived from real situations (or imaginary extensions), building a conceptual framework as needed. Students solve problems and in so doing they observe patterns and relationships, which can then be formalized and tested. Math concepts are used in the 5th grade science course, and scientific experiments are used as examples of math problems or illustration of math concepts. The environment of the classroom is organized to encourage cooperation, as well as individual questioning, conjecturing, and mathematical reasoning. Stress is given to the advantages of estimation for development of number sense. Exercise in mental computation is a frequent activity. Note: 5th grade mathematics is not organized into ability groups. Variations in ability or background are addressed as necessary via tutoring, remedial work, and enrichment activities in the context of the course itself.

This course, taught within the 5th grade block, includes the following topics: operations on positive numbers; fractions, decimals, percents; proportional reasoning ; rounding and estimating; probability; statistics and data analysis (mean, mode, median, etc.); number theory (gcf, lcm, divisibility, composites, primes); place value; graphing and other visualization of data; geometry (area, perimeter, volume); polygons; measurement of time, mass, length, volume, temperature; the metric system; calculator use; Microsoft Excel for data and sorting.

6^th Grade Course Content and Features

The 6th grade mathematics course extends the 5th grade experience with patterns and relationships by covering a wide range of topics to develop abstract and logical thinking skills, problem-solving proficiency, and mathematical techniques. The main thread of the course might be called "advanced arithmetic". It is a thorough building of the real number system and its associated operations: addition, subtraction, multiplication, division, exponentiation, and roots. It is a framework in which to review basic ideas (fraction arithmetic, ratio and proportion, etc.) and also introduce topics that are new to most sixth graders (operations with negative numbers, conversion of repeating decimals to fractions, scientific notation, countability, etc.). Coursework consists of daily homework, long-term projects, and class presentations, done both as individual work and group tasks. Note: 6th grade mathematics is not organized into ability groups. Variations in ability or background are addressed as necessary via tutoring, remedial work, and extensions of the homework into a wide variety of optional problems.

This course, taught within the 6th grade block, includes the following topics: operations on positive and negative numbers; fractions, decimals, percents; ratios and proportions ; estimation, problem-solving techniques ; probability experiments; statistics experiments; number theory (gcf, lcm, divisibility, composites, primes); prime factorization; exponents; scientific notation; square roots; the Pythagorean Theorem; introduction to algebra (variables to represent unknowns); coordinate graphing ; geometry (area, perimeter, volume); geometric constructions; use of compass and protractor ; three-dimensional visualizations; polyhedra; measurement (time, mass, length, volume, temperature); the metric system and U.S. customary system; scientific calculator use; Microsoft Excel (formulae, coordinate graphing); Geometer's Sketchpad (constructions).

7th Grade Course Content and Features

Math 7: This is a problem-solving and pre-algebra course which includes the following topics: operations on positive and negative numbers; fractions, decimals, percents; percent increase/decrease; ratios and proportions; estimation (divisibility, order of magnitude, reasonableness); probability experiments and theory; statistics - mean, median, mode, quartiles; counting techniques (elementary counting principle); number theory (divisibility, composites, primes); prime factorization; number bases; exponents; scientific notation; square and other roots; the Pythagorean Theorem; pre-algebra topics (linear expressions, linear equations and higher order expressions); properties: commutative, distributive, etc.; coordinate graphing, including y=mx+b; geometry (area, perimeter, volume); polygons; polyhedra; constructions, and use of a compass and protractor; precision, unit conversion; the metric system; calculator use; Geometer's Sketchpad; additional topics include an introduction to matrices, the phenomenon of pi, and logic puzzles.

Math 7a: This course is for students with high achievement in their current class, mastery of basic concepts (fractions, decimals, percents, area, volume, etc.), and strong interest in problem solving. Essentially the same topics are covered as in Math 7, but with greater depth, pace, and expectation of independent work.

8th Grade Course Content and Features

Algebra 8: This is a standard first year algebra course, with additional topics and use of computers and calculators to explore the subject of algebra. Topics include: operations on positive and negative real numbers (integer, rational, radical); classification of the real numbers (rational, radical, transcendental, etc.); absolute value; algebraic ratios and proportions; exponents, exponential growth and decay; scientific notation; radicals, numerical and algebraic; solving radical equations; coordinate graphing, linear and non-linear; statistics; estimation, problem-solving techniques; probability; relative frequency; absolute value; use of calculators, graphing calculators, and spreadsheets; functions, f(x) notation; linear, quadratic, and rational algebraic expressions and equations; polynomials; multivariable systems; sums and products of algebraic and complex fractions; factoring; generalized laws of exponents; fractional, negative, and zero exponents; logarithms; using the Quadratic Formula, the Pythagorean Theorem, the Midpoint Formula, etc.

Algebra 8a: This is a first-year algebra course for highly motivated and talented math students who want a strong focus on mathematics in their eighth grade year. (See note above.) While including the topics mentioned above under Algebra 8, this course covers the concepts of algebra, graphing, and trigonometry with greater depth, pace, and expectation of independent work and with a special emphasis on challenging word problems and formal application of the constructs of algebra, including ventures into limits and derivatives.