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Through the application of mathematics to other disciplines, middle school students should observe the need for the appropriate use of numbers beyond whole numbers. They should begin to recognize the role of number relationships in developing computational techniques. As students should extend their development of number sense to include all real numbers, they investigate the need for numbers as they apply to number theory concepts (e.g. primes, factors, multiples) in real world and mathematical-problem situations. Students begin to develop, analyze, and explain procedures for computation and techniques for estimation.

By the end of the sixth grade, students understand, compute and solve problems with whole numbers, fractions, decimals, percents, ratios and proportions. Relationships among units of measurement and properties of two- and three- dimensional shapes are used to solve problems. Numerical representations and functional relationships use symbolic representations to solve problems. Statistical techniques are introduced as a way to systematically analyze data and probability situations.

By the end of seventh grade, students use equivalent representations of real numbers, including percents, decimals, fractions, powers, roots and absolute value. Applications of proportional reasoning are used to solve numerical and geometric problems. A basic sense of the meaning of variable and algebraic expressions is acquired when using them to model simple situations.

By the end of the eighth grade, students understand, use and connect a variety of techniques for solving linear equations, inequalities and systems of equations in applied contexts. The meaning of variables, expressions, equations and inequalities, are used as models for problem situations. Students evaluate, graph and interpret the graphs of a variety of functions, and connect the behavior of the graphs to their corresponding representations as tables, equations and situations. Students apply proportional reasoning to solve problems involving scale drawings and similar figures, and connect geometric situations to algebraic and numerical situations.

CONTENT STANDARD 1

1. Number Sense and Operations

Students will:

• Understand the relationships between numbers and quantities

• Understand and describe simple addition and subtraction situations

• Use estimation strategies in computation and problem solving

• Understand the use of numbers up to 100

• Demonstrate the meaning of addition and subtraction and use the inverse relationship of addition and subtraction to solve problems

• Make reasonable estimates when comparing numbers

• Understand the relationship among whole numbers up to 1000

• Calculate and solve problems using addition and subtraction

• Model and solve simple problems involving multiplication and division

• Understand that fractions, sets, and subsets refer to parts of a whole

• Model and solve problems by representing, adding and subtracting amounts of money

• Use estimation strategies in computation and problem solving that involve numbers that use ones, tens, hundreds and thousands places

• Understand the place value of whole numbers

• Calculate and solve problems that involve addition, subtraction, multiplication, and division

• Understand the relationship between whole numbers, simple fractions, and decimals

• Understand the place value of whole numbers and decimals, how these relate to addition of simple fractions and decimals, and use the concepts of negative numbers

• Solve problems involving addition, subtraction, multiplication, and division of whole numbers, including the addition and subtraction of negative numbers, and understand the relationship among numbers

• Compute with very large and very small numbers, negative and positive numbers, decimals and fractions and understand the relationship between fractions, decimals, and percents

Performance Indicators

1.MS.1 Use concrete objects to model and order sets of real numbers.

1.MS.2 Investigate the size of large and small numbers in an environmental context.

1.MS.3 Categorize numbers and develop arguments justifying the choice of categories such as prime, composite, square, triangular, odd, even, etc.

1.MS.4 Compute the prime factorization of composite numbers and evaluate powers of numbers.

1.MS.5 Use calculators to generate lists of multiples of given numbers and find common multiples and least common multiples.

1.MS.6 Generate lists of factors of given numbers and find their common factors.

1.MS.7 Solve word problems that involve writing proportions and solving for unknown quantities.

1.MS.8 Concretely and symbolically demonstrate relationships among fractions and their reciprocals, decimals, and percents.

1.MS.9 Use real-world situations to show the relevance of fractions and decimals to daily life.

1.MS.10 Compare, order, and perform arithmetic operations with positive and negative rational numbers.

1.MS.11 Use exponential and scientific notation to solve realistic problems.

1.MS.12 Research ancient number systems, including fractional systems, and compare them to our present system.

1.MS.13 Identify situations in which particular representations of numbers are appropriate.

1.MS.14 Explain the relationship between prime factors of numbers and the permutations possible from the factors.

1.MS.15 Determine whether to use fractions, decimals, or percents in known problem situations and the advantages or disadvantages of each.

1.MS.16 Represent fractions, decimals, and percents using concrete, pictorial, and abstract representations (graphs, models, etc.)

1.MS.17 Use multiple representations of numbers in problem writing.

1.MS.18 Find examples of scientific research in which whole numbers and decimals are expressed in exponential form and using scientific notation.

1.MS.19 Illustrate and use the properties of addition and multiplication to simplify expressions.

1.MS.20 Demonstrate the hierarchy of the real number system with set notation.

1.MS.21 Demonstrate the results of arithmetic operations on positive and negative integers and write rules to govern the operations.

1.MS.22 Compare and order irrational numbers on number lines.

1.MS.23 Represent irrational numbers geometrically, such as square roots of non-perfect squares.

1.MS.24 Explain the attributes of prime and composite numbers.

1.MS.25 Use relationships among ratio, percent, and proportion to solve real-world problems.

1.MS.26 Make generalizations about the size of fractions as their denominators and/or numerators move toward specific limits

1.MS.27 Utilize 24-hour and 12-hour clocks in elapsed time problems.

1.MS.28 Investigate constant and varying rates in relationship to time.

1.MS.29 Solve elapsed time problems.

1.MS.30 Formulate problems involving constant and varying rates in relationship to time.

1.MS.31 Calculate rates of change.

1.MS.32 Formulate and solve non-routine problems involving many denominations of money.

1.MS.33 Use scales in map reading.

CONTENT STANDARD 2

2. Measurement and Geometry

Students will:

• Identify common geometric objects and explore properties such as length, weight, capacity

• Describe, draw and compare the attributes of plane and solid geometric figures and classify them by common attributes and describe their relative location and use their understanding to show relationships and solve problems

• Students explore the concept of time

• Use direct comparison and non-standard units to describe the measurements of objects

• Understand that measurement is accomplished by identifying a unit of measure and comparing it to the item to be measured

• Identify and describe the elements that compose common figures in the plane and common objects in space

• Choose appropriate units and measurement tools to quantify the properties of objects

• Understand perimeter, area, and volume of objects and measure in English and Metric units

Performance Indicators

2.MS.1 Convert units from one system to another within the Metric system.

2.MS.2 Develop, demonstrate, and explain new methods to measure common objects.

2.MS.3 Define and apply precision in measurement to the nearest 1/16 or 1/8 of an inch or to the nearest degree.

2.MS.4 Measure lengths and angles using rulers and protractors.

2.MS.5 Estimate lengths, areas, and volumes using the metric system.

2.MS.6 Estimate the measurements of objects and validate the estimations with actual measurements in metric and English measures.

2.MS.7 Choose appropriate metric or English units to determine length, width, volume, area, or weight of objects.

2.MS.8 Write about problem situations that contain Metric or English measurements.

2.MS.9 Read thermometers in Celsius and Fahrenheit degrees and be able to convert from one unit to another.

2.MS.10 Discuss the amount of error acceptable in a particular measurement.

2.MS.11 Determine the inherent error in measuring instruments by comparing a variety of measurements made with given instruments (i.e. speedometers).

2.MS.12 Demonstrate the proper use of a variety of measuring instruments such as rulers, protractors, compasses, thermometers, scales, balances, etc.

2.MS.13 Construct angle bisectors and perpendicular bisectors using rulers and compasses.

2.MS.14 Compare and contrast lines, segments, rays, points, and planes.

2.MS.15 Discuss similarities and differences of planes, lines, rays, and line segments.

2.MS.16 Estimate the numbers of degrees in given angles, and construct angles using compasses and protractors.

2.MS.17 Construct geometric designs using perpendicular and parallel lines, intersecting lines, right angles, obtuse angles, and acute angles.

2.MS.18 Identify, describe, compare, and classify geometric figures.

2.MS.19 Construct various solid and plane figures.

2.MS.20 Identify mirror images of given figures.

2.MS.21 Measure attributes of geometric figures using different units and express the measurements as fractions, decimals, and natural numbers.

2.MS.22 Create and solve problems using perimeter, area, volume, angle measurement, capacity and weight.

2.MS.23 Use instructional aids (protractors, compasses, etc.) to construct objects, or draw models to scale.

2.MS.24 Identify pairs of angles with special relationships: supplementary, complementary, vertical, etc.

2.MS.25 Classify and construct triangles according to their attributes.

2.MS.26 Identify, describe, compare, and classify geometric figures.

2.MS.27 Draw visual representations of geometric shapes from descriptions.

2.MS.28 Investigate the attributes of equilateral, scalene, and obtuse triangles.

2.MS.29 Use the Pythagorean relationship to determine the unknown side of right triangles.

2.MS.30 Identify objects in the classroom and in the environment, which are proportioned according to the golden ratio.

2.MS.31 Identify, name, and discuss scalene, equilateral, and right triangles.

2.MS.32 Identify and analyze fundamental translations, reflection, and rotations.

2.MS.33 Identify and classify the symmetries of geometric figures.

2.MS.34 Construct regular and semi-regular tessellation and write conjectures about regular polygons, which may or may not tessellate.

2.MS.35 Identify corresponding parts of congruent or similar figures.

2.MS.36 Use compasses and straight edges to construct congruent lines segments, angles, and bisectors.

2.MS.37 Make and display scale drawings of objects.

CONTENT STANDARD 3

3. Algebra

Students will:

• Sort and classify objects

• Use number sentences to solve problems

• Model, represent, and interpret number relationships to create and solve problems involving addition and subtraction

• Select appropriate symbols, operations and properties to represent, describe, simplify, and solve problems involving number relationships

• Use and interpret variables, mathematical symbols and properties to write and simplify expressions and equations, compute the value or values of a specific variable, and plot and interpret the results

Performance Indicators

3.MS.1 Graph ordered pairs in the four quadrants of coordinate planes

3.MS.2 Determine the slopes of lines graphed on coordinate planes.

3.MS.3 Graph and identify sets of ordered pairs on coordinate planes.

3.MS.4 Understand the concepts of variable, expression, and equation.

3.MS.5 Represent situations and numbers patterns with tables, graphs, verbal rules, and equations and explore the interrelationships of these representations.

3.MS.6 Apply algebraic methods to solve a variety of real-world and mathematical problems.

CONTENT STANDARD 4

4. Probability and Statistics

Students will:

• Collect information about objects and events in the environment

• Organize, represent, compare, and interpret categorical data on simple graphs and charts and communicate their findings

• Collect, record, organize, display, and interpret numerical data on bar graphs, and other representations

• Display, analyze, compare and interpret different data sets, including data sets that are not the same size

• Demonstrate an understanding of patterns and how they grow, and describe them in general ways

• Conduct simple probability experiments by determining the number of possible outcomes, and make simple predictions

Performance Indicators

4.MS.1 Read line, bar, and other graphs to elicit important information.

4.MS.2 Solve multi-step problems using a variety of strategies such as writing lists, constructing charts, tables, etc.

4.MS.3 Discuss applications, differences, and similarities of pictographs, bar, line, broken line, and circle graphs.

4.MS.4 Discuss applications, differences, and similarities of pictographs, bar, line, broken line, and circle graphs.

4.MS.5 Select relevant topics and display the collected information in graphs.

4.MS.6 Interpret charts and tables and make inferences based on the information.

4.MS.7 Communicate the results of simple surveys by making use of graphs, charts, and tables.

4.MS.8 Discuss applications, differences, and similarities of pictographs, bar, line, broken line, and circle graphs.

4.MS.9 Choose appropriate scales to construct various charts, tables, or diagrams for data.

4.MS.10 Interpret and analyze data from tables, charts, or diagrams.

4.MS.11 Construct graphs and charts using collected data

4.MS.12 Collect and analyze collected data and organize information in the form of tables, charts, and graphs.

4.MS.13 Find the measures of central tendency of sets of data. 4.MS.14 Predict the theoretical probability of an event occurring after analyzing collected data.

4.MS.15 Justify accepting a specific error of measurement in a particular problem.

4.MS.16 Predict future trends using graphs.

4.MS.17 Discuss the importance of statistics in everyday life.

4.MS.18 Model problems concretely and verify results.

4.MS.19 Utilize technology and apply solution techniques to real-world situations.

4.MS.20 Determine the probability of events occurring and predict the number of times it will occur in given situations.

4.MS.21 Interpret statistics and make predictions in sports, weather, and other world-related situations.

4.MS.22 Simulate real life probability problems using cards, dice, spinners, etc. and predict future outcomes based on these experiments.

4.MS.23 Determine the number of permutations and combinations possible in sets of at least three objects.

4.MS.24 Use computers to determine the probabilities in complex situations.

4.MS.25 Compare probabilities of collected data to computer simulated probabilities.

CONTENT STANDARD 5

5. Problem Solving and Reasoning

Students will:

• Make decisions about how to set up a problem, justify their reasoning, and make connections between problems

• Use strategies, skills and concepts to make decisions about how to approach problems, find solutions, and make generalizations to other situations

Performance Indicators

5.MS.1 Complete sequences of whole numbers in which there are combinations of operations.

5.MS.2 Generalize solutions to problems.

5.MS.3 Determine the rules for generating given patterns of numbers.

5.MS.4 Solve real life problems involving proportions, ratios, percents, probability, statistics and rational numbers.

5.MS.6 Work cooperatively to solve non-routine problems.

5.MS.7 Investigate the possibilities of real-world problems involving multi-step operations that may or may not include extraneous or insufficient information.

5.MS.8 Examine the consequences of choosing a particular problem solving strategy.

5.MS.9 Formulate problems to be solved based on students’ own interests.

5.MS.10 Use proportional reasoning, to solve real life problems involving probability, geometric attributes, and statistics.

5.MS.11 Solve realistic problems having consumer and career applications.

5.MS.12 Compare and contrast different solutions to the same problem.

5.MS.13 Solve multi-step problems using mathematical operations with real numbers.

5.MS.14 Select appropriate problem solving strategies to solve real life problems.

5.MS.15 Evaluate alternative problem solving techniques to choose effective approaches to situations.

5.MS.16 Formulate questions from given sets of graphical, written, or oral information.

5.MS.17 Determine the validity of simple arguments.

5.MS.18 Perform simple deductive and inductive reasoning exercises.

Revised September 2000

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