Mathematics 7: Pre-Algebra

Units

Unit Essential Questions Content Skills and Processes Assessment Resources Multicultural Dimension
NUMBER THEORY AND THE FUNDAMENTALS

• How can the patterns and relationships around us be expressed mathematically?
• How can these mathematical generalizations be used to derive new relationships or be evaluated in specific situations?
• How can the operations and relations among numbers be characterized symbolically and then built into a theoretical system of study?

*VOCABULARY AND DEFINITIONS
*SET NOTATION AND VENN DIAGRAMS: the basics of set theory
*NUMBER THEORY-REAL NUMBERS AND THEIR SUBSETS: the concept of prime and divisibility, factors and multiples, Fundamental Theorem of Arithmetic, decimal, base-5, and binary, the real number line, absolute value, negative values, counting numbers, zero, integers, rational and irrational numbers, exponents and roots, zero as an exponent, simple logarithms, properties of real numbers, examination of properties using matrices
*COMPUTATION WITH REAL NUMBERS: order of operations, grouping symbols: ( ), [ ], { }, | |, ______; computation with negative integers, computation with exponents, the many ways to express multiplication, absolute value, evaluating arithmetic expressions, matrix addition and subtraction, scientific notation, exponential growth and decay

Factorization

Prime factorization

Use of Venn diagrams to observe the relationship between two numbers in terms of their prime factors, to determine the GCF, to see how other factors are created, and to determine the LCM (least common multiple)

Placement of numbers onto the number line to gain an appreciation for their relative magnitude.

Conversion of numbers from one place-value system to another.

 Items in this concept area are embedded throughout the course and assessed in part throughout the year.

DATA COLLECTION, ANALYSIS, AND GRAPHING
  • What is data?
  • For what purposes is data collected?
  • What knowledge can be summarized and/or derived from data?
  • How do the choices of collection techniques, organization formats, analytical tools, and graphic representation of data influence understanding of the populations that data sets represent?
  • How does this data inform subsequent actions or decisions?

*HYPOTHESIS: stating an initial hypothesis. Choose the type of data to collect based upon a chosen hypothesis or topic of investigation

*DATA CHOICE and COLLECTION STRATEGIES: numerical/ non-numerical data, absolute and relative measure, degree of accuracy, defining the population, random sampling.  Collect data using appropriate measures and being mindful of degree of accuracy and significant figures and note sources of bias in the collection process

*RAW DATA TALLY AND COLLATION:  Organize collected data into data line plots, tally/frequency tables, spread sheets, or stem-leaf plots. 

*STATISTICAL REPRESENTATION: measures of central tendency--mean, median, mode, weighted averages;  measures of spread--range, outliers, 5-number summary, interquartile range, range of normal, outliers.

*GRAPHING:
Bar Graphs, Histograms, Circle Graphs, Box-Whisker Plots, scatter plots, line graphs.

*INTERPRETATION:  Interpret data given in graphical form

*PREDICTION:
trends, interpolation, extrapolation, correlation, probabilities based upon data analysis

*BASIC SKILLS:
Basic operations with rational numbers, conversion between decimals, fractions, and percent, degree measure of angles, rounding, solving proportions, finding a mean, using a formula, and using a protractor and compass.  Students also use a scientific calculator and software (TinkerPlots and Microsoft Excel) to expedite and enhance data collection, collation, graphic representation, and statistical analysis. .

*Project includes a novel hypothesis, data collection and organization, visual display of graphs, and a one-page numerical analysis and conclusion to accompany the visual display
*Written unit exam and quizzes to assess skill development

*TI 30X  multiview scientific calculators
*TinkerPlots software
*Microsoft Excel spreadsheet software
*various internet sites

 

*Dependent upon projects chosen
*Awareness of bias when collecting, analyzing, interpreting, and graphing data
*Care given to the types of sample data given to the students
*Authentic, updated material
 

RATIO, RATE, PROPORTION, PERCENT, AND PROBABILITY

• How does the relative measure of data yield new insights?
• How can ratios be used to describe relationships between correlated data?
• How can ratios and rates be used to calculate proportional change and thus predict future outcomes?

*MEASUREMENT: direct vs relative measure, units and abbreviations, categories, conversions between units/systems of measure
* RATIO: definition, part to part, part to whole, expressed as a fraction in lowest terms, gear ratios, Golden Ratio
* PROPORTION: set up and solving proportions, scale factor, similar figures
* PERCENT: set up and calculation; conversion between %, fraction, and decimal representation; %change; weighted average
*RATE: set up, dimensional analysis, unit rates, RPM and speed
*RATE OF CHANGE: Positive/Negative, linear, exponential, periodic
*PROBABILITY: single and multiple events, odds, possible outcomes vs favorable outcomes, Fundamental Counting Principle, Multiplication Principle, combinatorics, factorial, sample space, dependent and independent events

*Covert between fractions, decimals, and percent
*Set up and solve equations involving proportion
*Use concept of part of to solve problems involving percent, fractions, or decimal fractions
*Solve problems involving percent increase and decrease and percent off or mark-ups
*Calculate rate problems related to space travel and other real situations
*Use scale factor to solve problems
*Calculate the probability of single and multiple, dependent and independent events, weighted averages, and permutations and combinations using formulae and calculator functions
*Basic skills: solve a proportion equations, create equivalent fractions, multiply decimal fractions, multiply fractions

*Problem solving group activities
*Creation of a game, a scale model, an amusement park ride, and a scale factor enlargement or reduction using a grid or projection
*Unit exam

*Math Review: A Quick Reference Guide
* LEGOS
*Roller Coaster software
* Kitchen and shop tools, phonograph turntable, dice, stop watches, rulers, egg timers, bicycles

* Real-world examples and applications from a wide range of experiences and reflecting student solicited interest

MEASUREMENT AND GEOMETRY

By what descriptive and numerical measures can the spatial world be understood?

 

*MEASUREMENT: Know the English standard and metric units of and tools for measuring angles, number, location, length, area, volume, capacity, weight/mass,  and temperature. 

*DIMENSIONS:  0, 1 , 2, and 3 dimensionals.

* CARTESIAN COORDINATE SYSTEM: locating points in a plane and in space.

*IDENTIFY, DESCRIBE, CLASSIFY, AND COMPARE points and 1-3 D structures:
Points, lines, rays, line segments, angles, vertices.
Triangles, quadrilaterals, regular polygons, circles.
Platonic Solids, prisms, cylinders, pyramids, cones, spheres.

*FORMULAE: Surface area and volume of prisms, pyramids, cylinders, cones.

*SPECIAL CHARACTERISTICS AND RELATIONSHIPS:
Regular polygons and regular polyhedra.
Relationships between vertices, edges, angles, and faces.

*TRIANGLES: angle measurement, Pythagorean Theorem, similarity, trigonometric ratios (sin, cos, tan), angle of depression and elevation

* CIRCLES: pi, radius, diameter, and circumference relationships, area--using the formulae and deriving the formula

*STRUCTURAL CONSEQUENCES: hexagons, honeycombs

 

 

* MEASURE: use appropriate tools to directly measure geometric figures, indirectly measure using appropriate formulae
* DESCRIBE: describe using appropriate vocabulary one-, two-, and three-dimensional geometric figures
* DERIVE: through logical steps derive formulae used to calculate the measure of various geometric figures
* GENERALIZE OBSERVED RELATIONSHIPS: observe relationships between various geometric structures and describe these relationships algebraically

*Poster illustrating the affect of a linear change on area and volume
*Tilings--creation of original tiling designs as well as writing a short paragraph on the geometry used in the art of a chosen culture
*Village Project--design, measure, and create a 3 dimensional structure as a cuminating project.
*Quizzes and tests

* art and cultures connections

ALGEBRA: EXPRESSIONS AND EQUATIONS

• How can the patterns and relationships around us be expressed mathematically?
• How can these mathematical generalizations be used to derive new relationships or be evaluated in specific situations?
• How can the operations and relations among numbers be characterized symbolically and then built into a theoretical system of study?

*VOCABULARY OF ALGEBRAIC EXPRESSIONS AND EQUATIONS
*ALGEBRAIC EXPRESSIONS: Translating from words to symbols, describing patterns symbolically, evaluating expressions, naming the parts of an algebraic expression (coefficient, constant, term, variable, factor, exponent)
* OPERATIONS WITH ALGEBRAIC EXPRESSIONS: combining like terms, multiplying monomials, the distributive property, adding and subtracting polynomial expressions, multiplying polynomial expressions, basic factoring, dividing monomial expressions
*ALGEBRAIC EQUATIONS: categorizing (linear, quadratic), using formulae, slope as an expression of the rate of change, slope of a line, solution sets of equations, linear equations
*ALGEBRAIC PROBLEM-SOLVING: equations/inequalities that accurately represent the parameters of a given situation, observed phenomena/relationships
*GRAPHING: solution sets and ordered pairs, data tables from linear equations, Cartesian coordinate system, , describe scenarios that fit a given graph

* SIMPLIFY: simplify algebraic expressions by combining like terms, using the distributive property, multiplication and division of monomial and polynomial expressions
* EVALUATE: find the value of algebraic expressions using appropriate order of operations and properties of real numbers
* SOLVE: be able to solve linear equations and inequalities using simple one and two step methods, be able to solve more complex linear equations and inequalities using simplification
*FACTOR: recognize common monomial factors
*TRANSLATE: write algebraic expressions, equations, and inequalities that describe given data or observed relationships or patterns
*GRAPH: develop data tables from linear equations and then graph the resultant ordered pairs using a Cartesian coordinate system, graph linear equations and inequalities

* group and individual assessment of the basic skills and concepts of algebra as well as projects involving application and algebraic problem-solving
* emphasis on process--using vertical format and logical steps
* assessment of the ability of students to solve algebraically problems from the SAT, AMC 8, and MathCounts

* Math on Call
*Math Review: A Quick Reference Guide

* application questions for this unit are taken from a wide range of experiences and interests