DOWAGIAC UNION HIGH SCHOOL MATH BENCHMARK REVIEW OCT. 2004
I = Introduce

M = Master

R = Review

MATHEMATICS

BENCHMARKS

 ALGEBRA 1A ALGEBRA 1B CP ALGEBRA INF. GEOMETRY CP GEOMETRY CP ADV. ALGEBRA CP F.S.T. CP PRE-CALC

I. Patterns, Relationships and Functions
 
Content Standard 1: Students recognize similarities and generalize patterns, use patterns to create models and make predictions, describe the nature of patterns and relationships, and construct representations of mathematical relationships. (Patterns)
 
1. Analyze and generalize mathematical patterns including sequences, series and recursive patterns.

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2. Analyze, interpret and translate among representations of patterns including tables, charts, graphs, matrices and vectors

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3. Study and employ mathematical models of patterns to make inferences, predictions and decisions

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4. Explore patterns (graphic, numeric, etc.) characteristic of families of functions; explore structural patterns within systems of objects, operations or relations.  

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5. Use patterns and reasoning to solve problems and explore new content

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Content Standard 2: Students describe the relationships among variables, predict what will happen to one variable as another variable is changed, analyze natural variation and sources of variability, and compare patterns of change. (Variability and Change)
 
 

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CP 1

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Adv

FST

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1. Identify and describe the nature of change and begin to use the more formal language such as rate of change, continuity, limit, distribution and deviation.

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2. Develop a mathematical concept of function and recognize that functions display characteristic patterns of change (e.g., linear, quadratic, exponential).

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3. Expand their understanding of function to include non-linear functions, composition of functions, inverses offunctions, and piecewise- and recursively- defined functions.          

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4. Represent functions using symbolism such as matrices, vectors and functional representation (f(x)).

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5. Differentiate and analyze classes of functions including linear, power, quadratic, exponential, circular and trigonometric functions, and realize that many different situations can be modeled by a particular type of function.      

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6. Increase their use of functions and mathematical modelsto solve problems in context.

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II. Geometry and Measurement

 
Content Standard 1: Students develop spatial sense, use shape as an analytic and descriptive tool, identify characteristics and define shapes, identify properties and describe relationships among shapes. (Shape and Shape Relationships)
 
 

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CP 1

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1. Use shape to identify plane and solid figures, graphs, loci, functions and data distributions.

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2. Determine necessary and sufficient conditions for the existence of a particular shape and apply those conditions to analyze shapes.

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3. Use transformational, coordinate or synthetic methods to verify (prove) the generalizations they have made about properties of classes of shapes.

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4. Draw and construct shapes in two and three dimensions and analyze and justify the steps of their constructions.

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5. Study transformations of shapes using isometries, size transformations and coordinate mappings.

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6. Compare and analyze shapes and formally establish the relationships among them, including congruence, similarity, parallelism, perpendicularity and incidence.

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7. Use shape, shape properties and shape relationships to describe the physical world and to solve problems.

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Content Standard 2: Students identify locations of objects, identify location relative to other objects, and describe the effects of transformations (e.g., sliding, flipping, turning, enlarging, reducing) on an object. (Position)
 
 

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1. Locate and describe objects in terms of their position, including polar coordinates, three-dimensional Cartesian coordinates, vectors and limits.

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2. Locate and describe objects in terms of their orientation and relative position, including displacement (vectors), phase shift, maxima, minima and inflection points; give precise mathematical descriptions of symmetries

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3. Give precise mathematical descriptions of transformations and describe the effects of transformations on size, shape, position and orientation.

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4. Describe the locus of a point by a rule or mathematical expression; trace the locus of a moving point.

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5. Use concepts of position, direction and orientation to describe the physical world and to solve problems.

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Content Standard 3: Students compare attributes of two objects, or of one object with a standard (unit), and analyze situations to determine what measurement(s) should be made and to what level of precision. (Measurement)
 
 

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1. Select and use appropriate tools; make accurate measurements using both metric and common units, and measure angles in degrees and radians.

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2. Continue to make and apply measurements of length, mass (weight), time, temperature, area, volume, angle; classify objects according to their dimensions.

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3. Estimate measures with a specified degree of accuracy and evaluate measurements for accuracy, precision and tolerance.
4. Interpret measurements and explain how changes in one measure may affect other measures.

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5. Use proportional reasoning and indirect measurements, including applications of trigonometric ratios, to measure inaccessible distances and to determine derived measures such as density.

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6. Apply measurement to describe the real world and to solve problems.

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III. Data Analysis and Statistics
 
Content Standard 1: Students collect and explore data, organize data into a useful form, and develop skill in representing and reading data displayed in different formats. (Collection, Organization and Presentation of Data)
 
 

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1. Collect and explore data through observation, measurement, surveys, sampling techniques and simulations.

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2. Organize data using tables, charts, graphs, spreadsheets and data bases.

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3. Present data using the most appropriate representation and give a rationale for their choice; show how certain representations may skew the data or bias the presentation.

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4. Identify what data are needed to answer a particular question or solve a given problem and design and implement strategies to obtain, organize and present those data.

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Content Standard 2: Students examine data and describe characteristics of a distribution, relate data to the situation from which they arose, and use data to answer questions convincingly and persuasively. (Description and Interpretation)
 
 

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1. Critically read data from tables, charts or graphs and explain the source of the data and what the data represent.

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2. Describe the shape of a data distribution and determine measures of central tendency, variability and correlation.

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3. Use the data and their characteristics to draw and support conclusions.

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4. Critically question the sources of data; the techniques used to collect, organize and present data; the inferences drawn from the data; and the sources of bias and measures taken to eliminate such bias.

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5. Formulate questions and problems and gather and interpret data to answer those questions.

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Content Standard 3: Students draw defensible inferences about unknown outcomes, make predictions, and identify the degree of confidence they have in their predictions. (Inference and Prediction)
 
 

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1. Make and test hypothesis

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2. Design investigations to model and solve problems; also employ confidence intervals and curve fitting in analyzing the data.

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3. Formulate and communicate arguments and conclusions based on data and evaluate their arguments and those of others.
4. Make predictions and decisions based on data, including interpolations and extrapolations.

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5. Employ investigations, mathematical models, and simulations to make inferences and predictions to answer questions and solve problems.

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IV. Number Sense and Numeration

 
Content Standard 1: Students experience counting and measuring activities to develop intuitive sense about numbers, develop understanding about properties of numbers, understand the need for and existence of different sets of numbers, and investigate properties of special numbers. (Concepts and Properties of Numbers)
 
 

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1. Develop an understanding of irrational, real/complex #s

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2. Use the (ax+bi) and polar forms of complex numbers.

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3. Develop an understanding of the properties of the real and complex number systems and of the properties of special numbers including π, I, e, and conjugates.

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4. Apply their understanding of number systems to model, and solve mathematical and applied problems.

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Content Standard 2: Students recognize that numbers are used in different ways such as counting, measuring, ordering and estimating, understand and produce multiple representations of a number, and translate among equivalent representations. (Representation and Uses of Numbers)
 
 

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1. Give decimal representations of rational and irrational numbers and coordinate and vector representations of complex numbers.

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2. Develop an understanding of more complex representations of numbers, including exponential and logarithmic expressions, and select an appropriate representation to facilitate problem solving.          

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3. Determine when to use rational approximations and the exact values of numbers such as e, π and the irrational.

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4. Apply estimation in increasingly complex situations.

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5. Select appropriate representations for numbers, including representations of rational and irrational numbers and coordinate and vector representations of complex numbers, in order to simplify and solve problems.

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Content Standard 3: Students investigate relationships such as equality, inequality, inverses, factors and multiples, and represent and compare very large and very small numbers. (Number Relationships)
 
 

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1. Compare and order real numbers and compare rational approximations to exact values.

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2. Express numerical comparisons as ratios and rates.

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3. Extend the relationships of primes, factors, multiples and divisibility in an algebraic setting.

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4. Express number relationships using positive and negative rational exponents, logarithms and radicals.

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5. Apply their understanding of number relationships in solving problems.

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V. Numerical and Algebraic Operations and Analytical Thinking

 
Content Standard 1: Students understand and use various types of operations (e.g., addition, subtraction, multiplication, division) to solve problems. (Operations and their Properties).
 
 

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1. Present and explain geometric and symbolic models and operations with real and complex numbers and algebraic expressions.

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2. Compute with real numbers, complex numbers, algebraic expressions, matrices and vectors using technology and for simple instances, with paper-and-pencil algorithms.

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3. Describe the properties of operations with numbers, algebraic expressions, vectors and matrices, and make generalizations about the properties of given mathematical systems.

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4. Efficiently and accurately apply operations with real numbers, complex numbers, algebraic expressions, matrices and vectors in solving problems.

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Content Standard 2: Students analyze problems to determine and appropriate process for solution, and use algebraic notations to model or represent problems. (Algebraic and Analytic Thinking)
 
 

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1. Identify important variables in a context, symbolize them and express their relationships algebraically.

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2. Represent algebraic concepts and relationships with matrices, spreadsheets, diagrams, graphs, tables, physical models, vectors, equations and inequalities; and translate among the various representations.

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3. Solve linear equations and inequalities algebraically and non-linear equations using graphing, symbol-manipulating or spreadsheet technology; and solve linear and non-linear systems using appropriate methods.

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4. Analyze problems that can be modeled by functions, determine strategies for solving the problems and evaluate the adequacy of the solutions in the context of the problems.

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5. Explore problems that reflect the contemporary uses of mathematics in significant contexts and use the power of technology and algebraic and analytic reasoning to experience the ways mathematics is used in society.

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VI. Probability and Discrete Mathematics

 
Content Standard 1: Students develop an understanding of the notion of certainty and of probability as a measure of the degree of likelihood that can be assigned to a given event based on the knowledge available, and make critical judgments about claims that are made in probabilistic situations. (Probability)
 
 

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1. Develop an understanding of randomness and chance variation and describe chance and certainty in the language of probability.

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2. Give a mathematical definition of probability and determine the probabilities of more complex events, and generate and interpret probability distributions.

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3. Analyze events to determine their dependence or independence and calculate probabilities of compound events.

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4. Use sampling and simulations to determine empirical probabilities and, when appropriate, compare them to the corresponding theoretical probabilities; understand and apply the law of large numbers.

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5. Conduct probability experiments and simulations, to model and solve problems, including compound events.

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Content Standard 2: Students investigate practical situations such as scheduling, routing, sequencing, networking, organizing and classifying, and analyze ideas like recurrence relations, induction, iteration, and algorithm design. (Discrete Mathematics)
 
 

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1. Derive and use formulas for calculating permutations and combinations.

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2. Use sets and set relationships to represent algebraic and geometric concepts.

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3. Use vertex-edge graphs to solve network problems such as finding circuits, critical paths, minimum spanning trees and adjacency matrices.
4. Analyze and use discrete ideas, such as induction, iteration and recurrence relations.

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5. Describe and analyze efficient algorithms to accomplish a task or solve a problem in a variety of contexts, including practical, mathematical and computer-related situations.

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6. Use discrete mathematics concepts as described above to model situations and solve problems; and look for whether or not there is a solution (existence problems), determine how many solutions there are (counting problems) and decide upon a best solution (optimization problems).            

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