MICHIGAN FORESTS FOREVER TEACHERS GUIDE
MATHEMATICS |
Michigan
Department of Education - Mathematics
Version on this website as of summer, 2001
STRANDS, STANDARDS, & BENCHMARKS USED IN THIS WEBSITE
Strands are in GREEN and indented.
Standards are in BROWN.
Benchmarks are in RED.
Strand I. Patterns, Relationships, and Functions
Patterns, relationships and functions comprise one of the most important themes in the study of mathematics. Mathematical thinking begins with the recognition of similarities among objects or events, proceeds to generalization and abstraction, and culminates in the ability to understand, explain and make predictions. Contexts that exhibit structure and regularity provide rich opportunities for describing the physical world, studying mathematics and solving problems.
Standard I.1 Patterns
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.
Wherever there is mathematics there are patterns, and wherever there are patterns there is mathematics. Patterns are regularities or similarities that characterize sets of numbers, shapes, graphs, tables or other mathematical objects. Mathematicians look for patterns in everything they do; thus, mathematics is frequently defined as the science of patterns. In studying mathematics, students learn to recognize, describe, analyze and create patterns, to extend and generalize patterns, to create mathematical models based on observed patterns, and to predict the behavior of real-world phenomena based on such observed patterns. They learn to communicate the nature of mathematical patterns and relationships in various ways including words, physical models, diagrams, tables, charts, graphs, and equations. Since each representation highlights different aspects of the patterns and relationships, students must be able to construct multiple representations of mathematical relationships and to translate among them.
M.I.1.ms2 Represent and record patterns in a variety of ways including tables, charts and graphs, and translate between various representations.
M.I.1.ms3 Use patterns and their generalizations to make and justify inferences and predictions.
M.I.1.ms5 Use patterns and generalizations to solve problems and explore new content.
Standard I.2 Variability and
Change
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 are as fundamental to mathematics as they are to the physical world, and an understanding of the concept of a variable is essential to mathematical thinking. Students must be able to describe the relationships among variables, to predict what will happen to one variable as another variable is changed, and to compare different patterns of change. The study of variability and change provides a basis for making sense of the world and of mathematical ideas.
M.I.2.ms1 Identify and describe the nature of change; recognize change in more abstract and complex situations and explore different kinds of change and patterns of variation.
Strand II. Geometry and Measurement
We live in a three-dimensional world. In order to interpret and make sense of that world, students need both analytical and spatial abilities. Geometry and measurement, which involve notions of shape, size, position, and dimension, are used extensively to describe and understand the world around us.
Standard II.1 Shape
and Shape Relationships
Students develop spatial sense, use shape as an analytic and descriptive tool,
identify characteristics and define shapes, identify properties and describe relationships
among shapes.
Spatial sense is developed when students recognize, draw, construct, visualize, compare, classify and transform geometric shapes in both two and three dimensions. They learn to identify those characteristics that are necessary to define a given shape, and they can differentiate one shape from another. Students also develop an awareness of the properties of a shape and of the relationships among shapes. This includes hierarchical classifications of shapes (e.g., all squares are rhombuses), relationships among components of a shape (e.g., opposite sides of a rectangle are parallel), symmetries of a shape, congruence and similarity.
M.II.1.ms7 Use shape, shape properties and shape relationships to describe the physical world and to solve problems.
Standard II.2 Position
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 refers to the location of physical objects or points in space as well as to the relative locations and positions of objects, points, lines, planes and other geometric elements. It includes such notions as betweenness, collinearity and coordinates in two and three dimensions, as well as the locus of a point as it moves through space and the location of special points.
M.II.2.ms1 Locate and describe objects in terms of their position, including compass directions, Cartesian coordinates, latitude and longitude and midpoints.
M.II.2.ms2 Locate and describe objects in terms of their orientation and relative position, including coincident, collinear, parallel, perpendicular; differentiate between fixed (e.g., N- S- E- W) and relative (e.g., right-left) orientations; recognize and describe examples of bilateral and rotational symmetry.
M.II.2.ms5 Use concepts of position, direction and orientation to describe the physical world and to solve problems.
Standard II.3 Measurement
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 reflects the usefulness and practicality of mathematics and puts students in touch with the world around them. Measurement requires the comparison of an attribute (distance, surface, capacity, mass, time, temperature) between two objects or to a known standard, the assignment of a number to represent the comparison, and the interpretation of the results. Measurement also introduces students to the important concepts of precision, approximation, tolerance, error and dimension.
M.II.3.ms1 Select and use appropriate tools; measure objects using standard units in both the metric and common systems and measure angles in degrees.
M.II.3.ms2 Identify the attribute to be measured and select the appropriate unit of measurement for length, mass (weight), time, temperature, perimeter, area, volume and angle.
M.II.3.ms3 Estimate measures with a specified degree of accuracy and decide if an estimate or a measurement is "close enough."
M.II.3.ms5 Use proportional reasoning and indirect measurements to draw inferences.
M.II.3.ms6 Apply measurement to describe the real world and to solve problems.
Strand III. Data Analysis and Statistics
We live in a sea of information. In order not to drown in the data that inundate our lives every day, we must be able to process and transform data into useful knowledge. The ability to interpret data and to make predictions and decisions based on data is an essential basic skill for every individual.
Standard III.1 Collection,
Organization and Presentation of Data
Students collect and explore data, organize data into a useful form, and develop skill
in representing and reading data displayed in different formats.
Knowing what data to collect and where and how to collect them is the starting point of quantitative literacy. The mathematics curriculum should capitalize on students' natural curiosity about themselves and their surroundings to motivate them to collect and explore interesting statistics and measurements derived from both real and simulated situations. Once the data are gathered, they must be organized into a useful form, including tables, graphs, charts and pictorial representations. Since different representations highlight different patterns within the data, students should develop skill in representing and reading data displayed in different formats, and they should discern when one particular representation is more desirable than another.
M.III.1.ms1 Collect and explore data through observation, measurement, surveys, sampling techniques and simulations.
M.III.1.ms2 Organize data using tables, charts, graphs, spreadsheets and data bases.
M.III.1.ms3 Present data using a variety of appropriate representations and explain why one representation is preferred over another or how a particular representation may bias the presentation.
M.III.1.ms4 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.
Standard III.2 Description
and Interpretation
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.
Students must be able to examine data and describe salient characteristics of the distribution. They also must be able to relate the data to the physical situation from which they arose. Students should use the data to answer key questions and to convince and persuade.
M.III.2.ms1 Critically read data from tables, charts or graphs and explain the source of the data and what the data represent.
M.III.2.ms2 Describe the shape of a data distribution and identify the center, the spread, correlations and any outliers.
M.III.2.ms3 Draw, explain and justify conclusions based on data.
M.III.2.ms4 Critically question the sources of data; the techniques used to collect, organize and present data; the inferences drawn from the data; and the possible sources of bias in the data or their presentation.
Standard III.3 Inference and Prediction
Students draw defensible inferences about unknown outcomes, make predictions, and
identify the degree of confidence they have in their predictions.
Based on known data, students should be able to draw defensible inferences about unknown outcomes. They should be able to make predictions and to identify the degree of confidence that they place in their predictions.
M.III.3.ms1 Make and test hypotheses.
M.III.3.ms2 Design experiments to model and solve problems using sampling, simulations and controlled investigations.
M.III.3.ms3 Formulate and communicate arguments and conclusions based on data and evaluate their arguments and those of others.
M.III.3.ms4 Make predictions and decisions based on data, including interpolations and extrapolations.
M.III.3.ms5 Employ investigations, mathematical models and simulations to make inferences and predictions to answer questions and solve problems.
Strand IV. Number Sense and Numeration
Number sense is to mathematics what vocabulary is to language. Students must learn to quantify and measure, concretely at first and increasingly more abstractly as they mature. They also must develop an understanding of numeration systems and of the structure of such systems. They must learn to estimate mathematical quantities and to represent and communicate mathematical ideas in the language of mathematics.
Standard IV.1 Concepts
and Properties of Numbers
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.
Fundamental questions like "What is a number?" or "What is three?" can be deceptively difficult to answer. Students require extensive involvement with concrete experiences of counting and measuring in order to develop an intuitive sense about number. Through both informal and formal means, students develop understanding about important properties of numbers such as even vs. odd, whole number vs. fraction, positive vs. negative. They understand the existence of different sets of numbers (whole numbers, integers, rationals, reals, ...) and the properties of special numbers such as 0, 1, [pi], or the inverse of a number.
M.IV.1.ms1 Develop an understanding of integers and rational numbers and represent rational numbers in both fraction and decimal form.
M.IV.1.ms4 Apply their understanding of number systems to model and solve mathematical and applied problems.
Standard IV.3
Number Relationships
Students investigate relationships such as equality, inequality, inverses, factors and
multiples, and represent and compare very large and very small numbers.
Students develop understanding of important relationships among numbers including the relationships of (=, [not equal]) and (<, >); of opposites (additive inverses) and reciprocals (multiplicative inverses); of factors and multiples; of primes, composites, and relatively prime numbers; of powers and roots. They understand and can represent very large and very small numbers and can compare the orders of magnitude of numbers.
M.IV.3.ms5 Apply their understanding of number relationships in solving problems.
Strand V. Numerical and Algebraic Operations and Analytical Thinking
The ability to represent quantitative situations with algebraic symbolism, numerical operations and algebraic thinking is essential to solving problems in significant contexts and applications. The concepts of number and variable and their symbolic representation and manipulation are central to the understanding of arithmetic and its generalization in algebra. The contemporary applications of mathematics in virtually every field of work and study rely on algebraic and analytic thinking and communication as fundamental tools.
Standard V.1 Operations and their
Properties
Students understand and use various types of operations (e.g., addition, subtraction,
multiplication, division) to solve problems.
The ultimate reason for mastering the operations of arithmetic and algebra is to solve problems. To that end, understanding the basic computational operations and their algorithms is essential for competence in mathematics, but the emphasis must be on understanding and using the operations, not on memorizing algorithms. In computation, understanding and accuracy are always more important than speed. Understanding the operations requires the concomitant understanding and application of the properties of those operations, and it involves knowing what operations to use in a particular situation. There is no one way to perform a calculation. Students must be competent in performing calculations, but they need not have a rigid adherence to one algorithm. Methods of computation include proficiency with mental calculation, paper and pencil, and calculators; the ability to represent computations with manipulatives and geometric models; and the discernment of which computational method to use in a given situation. Computational methods also involve estimating and assessing the reasonableness of the results of a computation.
M.V.1.ms2 Compute with integers, rational numbers and simple algebraic expressions using mental computation, estimation, calculators and paper-and-pencil; explain what they are doing and how they know which operations to perform in a given situation.
M.V.1.ms4 Efficiently and accurately apply operations with integers, rational numbers and simple algebraic expressions in solving problems.
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