It is designed to build on a rigorous K–6 experience such as one indicated by the expectations outlined in Achieve's Elementary Mathematics Benchmarks, Grades K–6 or documents developed by the National Council of Teachers of Mathematics. It also builds on the National Assessment of Educational Progress (NAEP) elementary guidelines as well as the expectations set in many states’ elementary grade standards. In particular, it is expected that students will come to this course with a strong conceptual foundation as well as computational facility with whole and rational numbers.

During this middle school course, students will further their understanding of the operations and properties of the rational number system including ratios, rates, proportions and numbers expressed using whole number exponents. The geometry and measurement of lines, angles, triangles, and circles consolidates knowledge gleaned during the elementary years and offers teachers and students opportunities to connect the branches of mathematics through measurement formulas, similarity, and scaling. This course concludes with basic concepts of probability and data. When additional work from Middle School Course 2 is complete, students should be prepared for success on eighth-grade state tests as well as the NAEP grade 8 assessment. They should also be prepared to successfully tackle algebra and geometry taught at the high school level.

Appropriate use of technology is expected in all work. In middle school this includes employing technological tools to assist students in creating graphs and data displays, transforming graphs, conceptualizing and analyzing geometric situations, and solving problems. Testing with and without technological tools is recommended.

How a particular subject is taught influences not only the depth and retention of the content of a course but also the development of skills in inquiry, problem solving, and critical thinking. Every opportunity should be taken to link the concepts of this middle school course to concepts students have encountered in earlier grades as well as to other disciplines. Students should be encouraged to be creative and innovative in their approach to problems, to be productive and persistent in seeking solutions, and to use multiple means to communicate their insights and understanding.

The Major Concepts below provide the focus for Middle School Course 1, which is intended to prepare students for a high school–level course in Algebra I or the first course in an Integrated Mathematics sequence. They should be taught using a variety of methods and applications so that students attain a deep understanding of these concepts.

• Number Representation and Computation
• Ratio, Rates, Scaling, and Similarity
• Measurement Systems
• Angles, Triangles, and Circles
• Three-Dimensional Geometry
• Data Analysis
• Probability

Maintenance Concepts should have been taught previously and are important foundational concepts that will be applied in this course. Continued facility with and understanding of the Maintenance Concepts is essential for success in the Major Concepts defined for Middle School Course 1.

• Positive Rational Number Operations
• Numeric Relationships
• Graphing in the Coordinate Plane
• Geometric Shapes and Measurement
• Data Collection and Organization

A. Number Representation and Computation

Students entering middle school are expected to have computational fluency with whole number operations. This unit extends that fluency to rational numbers, including translating among different rational number representations. Students learn to think flexibly about fractions, decimals, and percents; locate and order rational numbers on the number line; and estimate to predict results and verify calculations. They will not only analyze some algorithms applied to whole numbers in elementary school but also some of those they will learn for operations with the rational numbers. The structure of the multiplicative and additive patterns of numbers studied in elementary school will receive a more formal analysis and be both extended and generalized.

Successful students will:

A1 Extend and apply understanding about rational numbers; translate among different representations of rational numbers.

Rational numbers are those that can be expressed in the form , where p and q are integers and q ≠ 0.

1. Use inequalities to compare rational numbers and locate them on the number line; apply basic rules of inequalities to transform numeric expressions involving rational numbers.

A2 Apply the properties of computation (e.g., commutative property, associative property, distributive property) to positive and negative rational number computation; know and apply effective methods of calculation with rational numbers.

1. Demonstrate understanding of the algorithms for addition, subtraction, multiplication, and division (non-zero divisor) of numbers expressed as fractions, terminating decimals, or repeating decimals by applying the algorithms and explaining why they work.
2. Add, subtract, multiply, and divide (non-zero divisor) rational numbers and explain why these operations always produce another rational number.
3. Interpret parentheses and employ conventional order of operations in a numerical expression, recognizing that conventions are universally agreed upon rules for operating on expressions.
4. Check answers by estimation or by independent calculations, with or without calculators and computers.
5. Solve practical problems involving rational numbers.

   Examples: Calculate markups, discounts, taxes, tips, average speed.

A3 Recognize, describe, extend, and create well-defined numerical patterns.

A pattern is a sequence of numbers or objects constructed using a simple rule. Of special interest are arithmetic sequences, those generated by repeated addition of a fixed number, and geometric sequences, those generated by repeated multiplication by a fixed number.

A4 Know and apply the Fundamental Theorem of Arithmetic.

Every positive integer is either prime itself or can be written as a unique product of primes (ignoring order).

1. Identify prime numbers; describe the difference between prime and composite numbers; determine and divisibility rules (2, 3, 5, 9, 10), explain why they work, and use them to help factor composite numbers.
2. Determine the greatest common divisor and least common multiple of two whole numbers from their prime factorizations; explain the meaning of the greatest common divisor (greatest common factor) and the least common multiple and use them in operations with fractions.
3. Use greatest common divisors to reduce fractions and ratios n:m to an equivalent form in which the gcd (n, m) = 1.

   Fractions in which gcd (n, m) = 1 are said to be in lowest terms.

4. Write equivalent fractions by multiplying both numerator and denominator by the same non-zero whole number or dividing by common factors in the numerator and denominator.
5. Add and subtract fractions by using the least common multiple (or any common multiple) of denominators.

A5 Identify situations where estimates are appropriate and use estimates to predict results and verify the reasonableness of calculated answers.

1. Use rounding, regrouping, percentages, proportionality, and ratios as tools for mental estimation.
2. Develop, apply, and explain different estimation strategies for a variety of common arithmetic problems.

   Examples: Estimating tips, adding columns of figures, estimating interest payments, estimating magnitude.

3. Explain the phenomenon of rounding error, identify examples, and, where possible, compensate for inaccuracies it introduces.

   Examples: Analyzing apportionment in the U.S. House of Representatives; creating data tables that sum properly; analyzing what happens to the sum if you always round down when summing 100 terms.

A6 Use the rules of exponents to simplify and evaluate expressions.

1. Evaluate expressions involving whole number exponents and interpret such exponents in terms of repeated multiplication.

A7 Analyze and apply simple algorithms.

1. Identify and give examples of simple algorithms.

   An algorithm is a procedure (a finite set of well-defined instructions) for accomplishing some task that, given an initial state, will terminate in a well-defined end-state. Recipes and assembly instructions are everyday examples of algorithms.

2. Analyze and compare simple computational algorithms.

   Examples: Write the prime factorization for a large composite number; determine the least common multiple for two positive integers; identify and compare mental strategies for computing the total cost of several objects.

3. Analyze and apply the iterative steps in standard base-10 algorithms for addition and multiplication of numbers.

B. Ratios, Rates, Scaling and Similarity

In conjunction with the study of rational numbers, middle school students examine ratios, rates, and proportionality both procedurally and conceptually. Proportionality concepts connect many areas of the curriculum—number, similarity, scaling, slope, and probability—and serve as a foundation for future mathematics study. Examining proportionality first with numbers then geometrically with similarity concepts and scaling begins to establish important understandings for more formal study of these concepts in high school algebra and geometry courses.

Successful students will:

B1 Use ratios, rates, and derived quantities to solve problems.

1. Interpret and apply measures of change such as percent change and rates of growth.
2. Calculate with quantities that are derived as ratios and products.

   Examples: Interpret and apply ratio quantities including velocity and population density using units such as feet per second and people per square mile; interpret and apply product quantities including area, volume, energy and work using units such as square meters, kilowatt hours, and person days.

3. Solve data problems using ratios, rates, and product quantities.
4. Create and interpret scale drawings as a tool for solving problems.

   A scale drawing is a representation of a figure that multiplies all the distances between corresponding points by a fixed positive number called the scale factor.

B2 Analyze and represent the effects of multiplying the linear dimensions of an object in the plane or in space by a constant scale factor, r.

1. Use ratios and proportional reasoning to apply a scale factor to a geometric object, a drawing, or a model, and analyze the effect.
2. Describe the effect of a scale factor r on length, area, and volume.

B3 Interpret the definition and characteristics of similarity for figures in the plane and apply to problem solving situations.

Informally, two geometric figures in the plane are similar if they have the same shape. More formally, having the same shape means that one figure can be transformed onto the other by applying a scale factor.

1. Apply similarity in practical situations; calculate the measures of corresponding parts of similar figures.
2. Use the concepts of similarity to create and interpret scale drawings.

C. Measurement Systems

This unit extends the measurement of everyday objects in elementary school to an in-depth study of units, approximation, and precision of measurements and measurement conversions within the same system. The geometric measurements of perimeter, area, volume, and surface area are extended to include all shapes and composite shapes. Formulas are examined for relationships, and students apply these formulas in problem-solving situations.

Successful students will:

C1 Make, record, and interpret measurements.

1. Recognize that measurements of physical quantities must include the unit of measurement, that most measurements permit a variety of appropriate units, and that the numerical value of a measurement depends on the choice of unit; apply these ideas when making measurements.
2. Recognize that real-world measurements are approximations; identify appropriate instruments and units for a given measurement situation, taking into account the precision of the measurement desired.
3. Plan and carry out both direct and indirect measurements.

   Indirect measurements are those that are calculated based on actual recorded measurements.

4. Apply units of measure in expressions, equations, and problem situations; when necessary, convert measurements from one unit to another within the same system.
5. Use measures of weight, money, time, information, and temperature; identify the name and definition of common units for each kind of measurement.
6. Record measurements to reasonable degrees of precision, using fractions and decimals as appropriate.

   A measurement context often often defines a reasonable level of precision to which the result should be reported.

   Example: The U.S. Census bureau reported a national population of 299,894,924 on its Population Clock in mid-October of 2006. Saying that the U.S. population is 3 hundred million (3x10⁸) is accurate to the nearest million and exhibits one-digit precision. Although by the end of that month the population had surpassed 3 hundred million, 3x10⁸ remained accurate to one-digit precision.

C2 Identify and distinguish among measures of length, area, surface area, and volume; calculate perimeter, area, surface area, and volume.

1. Calculate the perimeter and area of triangles, quadrilaterals, and shapes that can be decomposed into triangles and quadrilaterals that do not overlap; know and apply formulas for the area and perimeter of triangles and rectangles to derive similar formulas for parallelograms, rhombi, trapezoids, and kites.
2. Given the slant height, determine the surface area of right prisms and pyramids whose base(s) and sides are composed of rectangles and triangles; know and apply formulas for the surface area of right circular cylinders, right circular cones, and spheres; explain why the surface are of a right circular cylinder is a rectangle whose length is the circumference of the base of the cylinder and whose width is the height of the cylinder.
3. Given the slant height, determine the volume of right prisms, right pyramids, right circular cylinders, right circular cones, and spheres.
4. Estimate lengths, areas, surface areas, and volumes of irregular figures and objects.

D. Angles, Triangles, and Circles

Elementary school students should enter middle school with a solid foundation in the basic properties of angles, lines, and triangles in the plane. Middle school students should formalize and use these properties to solve more rigorous problems involving parallel lines cut by a transversal and to identify congruent, supplementary, and complementary angles. A major focus of this unit is on the study of circles, the relationships among their parts, the development of the formulas for the area and circumference, and methods for approximating π.

Successful students will:

D1 Know the definitions and properties of angles and triangles in the plane and use them to solve problems.

1. Know and apply the definitions and properties of complementary, supplementary,interior, and exterior angles.
2. Know and distinguish among the definitions and properties of vertical, adjacent, corresponding, and alternate interior angles; identify pairs of congruent angles and explain why they are congruent.

D2 Identify and explain the relationships among the radius, diameter, circumference, and area of a circle; know and apply formulas for the circumference and area of a circle, semicircle, and quarter-circle

1. Identify the relationship between the circumference of a circle and its radius or diameter as a direct proportion and between the area of a circle and the square of its radius or the square of its diameter as a direct proportion.
2. Demonstrate why the formula for the area of a circle (radius times one-half of its circumference) is plausible and makes geometric sense.
3. Show that for any circle, the ratio of the circumference to the diameter is the same as the ratio of the area to the square of the radius and that these ratios are the same for different circles; identify the constant ratio A/r² = ½Cr/r² = C/2r = C/d as the number π and know that although the rational numbers 3.14, or are often used to approximate π, they are not the actual values of the irrational number π.
4. Identify and describe methods for approximating π.

E. Three-Dimensional Geometry

This unit develops middle school students’ spatial visualization skills. Students have previously worked with simple polyhedrons; these expectations expand middle school students’ repertoire of shapes and more fully develop the characteristics and properties of those shapes. They provide students with experiences translating between two- and three-dimensional representations.

Successful students will:

E1 Visualize solids and surfaces in three-dimensional space.

1. Relate a net, top-view, or side-view to a three-dimensional object that it might represent; visualize and be able to reproduce solids and surfaces in three-dimensional space when given two-dimensional representations (e.g., nets, multiple views).
2. Interpret the relative position and size of objects shown in a perspective drawing.
3. Visualize and describe three-dimensional shapes in different orientations; draw two-dimensional representations of three-dimensional objects by hand and using software; sketch two-dimensional representations of basic three-dimensional objects such as cubes, spheres, pyramids, and cones.
4. Create a net, top-view, or side-view of a three-dimensional object by hand or using software; visualize, describe, or sketch the cross-section of a solid cut by a plane that is parallel or perpendicular to a side or axis of symmetry of the solid.

F. Data Analysis

Students in this course build on the descriptive statistics of elementary school to pose questions of interest, collect relevant data, display data and communicate results. Technology is utilized to both analyze and display data. Students look critically at data studies and reports for possible sources of bias or misrepresentation. Middle school students expand their library of data display tools beyond line, bar, and circle graphs.

Successful students will:

F1 Formulate questions about a phenomenon of interest that can be answered with data; collect and record data; display data using tables, charts, or graphs; evaluate the accuracy of the data.

1. Understand that data are numbers in context and identify appropriate units.
2. Organize written or computerized data records, making use of computerized spreadsheets.
3. Define measurements that are relevant to the questions posed.

F2 Analyze and interpret categorical and quantitative data; judge the accuracy, reasonableness, and potential for misrepresentation.

1. Represent both univariate and bivariate categorical data accurately and effectively; for univariate data, make use of frequency and relative frequency tables and bar graphs; for bivariate data, make use of two-way frequency and relative frequency tables and bar graphs.
2. Identify and explain misleading uses of data by considering the completeness and source of the data, the design of the study, and the way the data are analyzed and displayed.

   Examples: Determine whether the height or area of a bar graph is being used to represent the data; evaluate whether the scales of a graph are consistent and appropriate or whether they are being adjusted to alter the visual information conveyed.

G. Probability

Students have an opportunity in this unit to apply both rational number and proportional reasoning skills to probability situations. Students use theoretical probability and proportions to predict outcomes of simple events. Frequency distributions are examined and created to analyze the likelihood of events. The Law of Large Numbers is used to link experimental and theoretical probabilities.

Successful students will:

G1 Represent probabilities using ratios and percents; use sample spaces to determine the (theoretical) probabilities of events; compare probabilities of two or more events and recognize when certain events are equally likely.

1. Calculate theoretical probabilities in simple models (e.g., number cubes, coins, spinners).
2. Know and use the relationship between probability and odds.

   The odds of an event occurring is the ratio of the number of favorable outcomes to the number of unfavorable outcomes, whereas the probability is the ratio of favorable outcomes to the total number of possible outcomes.

G2 Describe the relationship between probability and relative frequency; use a probability distribution to assess the likelihood of the occurence of an event.

1. Recognize and use relative frequency as an estimate for probability.

   If an action is repeated n times and a certain event occurs b times, the ratio b/n is called the relative frequency of the event occurring.

2. Use theoretical probability, where possible, to determine the most likely result if an experiment is repeated a large number of times.
3. Identify, create, and describe the key characteristics of frequency distributions of discrete and continuous data.

   A frequency distribution shows the number of observations falling into each of several ranges of values; if the percentage of observations is shown, the distributions called a relative frequency distribution. Both frequency and relative frequency distributions are portrayed through tables, histograms, or broken-line graphs.

4. Analyze and interpret actual data to estimate probabilities and predict outcomes.

   Example: In a sample of 100 randomly selected students, 37 of them could identify the difference in two brands of soft drink. Based on these data, what is the best estimate of how many of the 2,352 students in the school could distinguish between the brands of soft drink?

5. Compare theoretical probabilities with the results of simple experiments (e.g., tossing number cubes, flipping coins, spinning spinners).
6. Explain how the Law of Large Numbers explains the relationship between experimental and theoretical probabilities.

   The Law of Large Numbers indicates that if an event of probability p is observed repeatedly during independent repetitions, the ratio of the observed frequency of that event to the total number of repetitions approaches p as the number of repetitions becomes arbitrarily large.

7. Use simulations to estimate probabilities.
8. Compute and graph cumulative frequencies.

About the Benchmarks

Elementary (K–6) Strands and Grade Levels

Secondary (7–12) Strands

Secondary Model Course Sequences

Secondary Assessments and Tasks

Correlations to the Secondary Benchmarks

Supporting Resources

Home

Jump to:

A. Number Representation and Computation

B. Ratios, Rates, Scaling and Similarity

C. Measurement Systems

D. Angles, Triangles and Circles

E. Three-Dimensional Geometry

F. Data Analysis

G. Probability

Tasks Related to This Course

• Autobahn
• Cones Launch
• Congruence Challenge
• Equal Salaries for Equal Work?
• Function Transformations
• How Odd . . .
• King's Deli
• Neighborhood Park
• Regional Triangles
• Rock, Paper, Scissors
• Rolling for the Big One
• A Safe Load
• Satellite
• Talk Is Cheap