Stage H - Mathematics

Descriptors

6A - Students who meet the standard can demonstrate knowledge and use of numbers and their many representations in a broad range of theoretical and practical settings. (Representations)

1. Recognize and use exponential, scientific, and calculator notation. **
2. Represent, order, and compare rational numbers using a variety of methods and materials.
3. Place rational numbers on a number line.

6B - Students who meet the standard can investigate, represent, and solve problems using number facts, operations and their properties, algorithms, and relationships. (Operations and properties)

1. Determine the least common multiple and greatest common factor of a set of numbers using prime factorization containing exponents.
2. Determine and describe the effects of arithmetic operations with decimals and integers (e.g., multiply by a decimal between zero and one, divide by a negative integer).
3. Simplify arithmetic expressions containing integers using the field properties and order of operations.
4. Describe and use the inverse relationships of squaring and finding square roots to simplify computations and solve problems. **
5. Justify divisibility rules for 3, 4, 6, 8, and 9.

6C - Students who meet the standard can compute and estimate using mental mathematics, paper-and-pencil methods, calculators, and computers. (Choice of method)

1. Select, use, and justify appropriate operations, methods, and tools to compute or estimate with real numbers. **
2. Analyze algorithms for computing with real numbers and develop fluency in their use. **

6D - Students who meet the standard can solve problems using comparison of quantities, ratios, proportions, and percents.

1. Develop, use, analyze, and explain methods for solving number sentences or word problems involving proportions with rational numbers. **
2. Solve problems that involve percents, including percent increase and decrease, regardless of the piece of information that is missing.

7A - Students who meet the standard can measure and compare quantities using appropriate units, instruments, and methods. (Performance and conversion of measurements)

1. Solve simple scale conversions, contractions, and dilations (e.g., maps and diagrams).

7B - Students who meet the standard can estimate measurements and determine acceptable levels of accuracy. (Estimation)

1. Measure any quantity to the greatest degree of accuracy determined by the tool.
2. Determine the maximum error in measurements.

7C - Students who meet the standard can select and use appropriate technology, instruments, and formulas to solve problems, interpret results, and communicate findings. (Progression from selection of appropriate tools and methods to application of measurements to solve problems)

1. Solve simple problems involving rates and other derived measurements such as velocity and density. **
2. Solve problems involving angle measurement in polygons and circles.
3. Develop and describe surface area and volume formulas for cones and cylinders by relating pyramids to cones and prisms to cylinders.
4. Solve problems involving time, temperature, mass, speed, distance, density, and monetary values.
5. Solve problems involving scale drawings, models, maps, or blueprints.
6. Determine derived measurements.
7. Determine the surface area of three-dimensional figures.
8. Determine the volume of a sphere.

8A - Students who meet the standard can describe numerical relationships using variables and patterns. (Representations and algebraic manipulations)

1. Investigate and describe linear, quadratic, and exponential patterns recursively. **
2. Investigate and write algebraic expressions to describe the nth term of a simple linear, power, or exponential sequence.
3. Determine a specific term of a pattern of numbers or drawings.
4. Create arithmetic and geometric sequences to fit a given set of conditions.
5. Recognize and generate equivalent forms for linear equations, including transforming linear equations into standard and slope-intercept form. **

8B - Students who meet the standard can interpret and describe numerical relationships using tables, graphs, and symbols. (Connections of representations including the rate of change)

1. Graph linear equations and inequalities on the Cartesian plane.
2. Graph a set of points and describe the relationship as linear or nonlinear.
3. Describe the relationships between symbolic expressions and graphs of lines using the appropriate vocabulary for the intercepts and slope of the line. **
4. Graph absolute values on a number line.
5. Determine the slope of a line from a graph.

8C - Students who meet the standard can solve problems using systems of numbers and their properties. (Problem solving)

1. Solve arithmetic and simple algebraic equations using properties of real numbers, equality and inequality, and justify the procedures.
2. Solve simple algebraic equations for a given variable using inverse operations.

8D - Students who meet the standard can use algebraic concepts and procedures to represent and solve problems. (Connection of 8A, 8B, 8C to solve problems)

1. Solve algebraic equations or word problems that involve linear equations or inequalities using algebraic or graphical representations. **
2. Solve absolute value equations or inequalities in one variable using algebraic or graphical representations.
3. Create word problems that meet given conditions and represent linear relationships.

9A - Students who meet the standard can demonstrate and apply geometric concepts involving points, lines, planes, and space. (Properties of single figures, coordinate geometry and constructions)

1. Represent and analyze the properties of geometric shapes using coordinate geometry. **
2. Draw the image of an object after a combination of transformations.
3. Identify possible types of two- or three-dimensional figures that would match a set of given conditions.
4. Determine if a triangle is possible using side lengths and the triangle inequality.
5. Solve pictorial or word problems that involve geometric relationships within a single geometric shape or figure, including the Pythagorean theorem.
6. Analyze the results of a combination of reflections, rotations, and translations of a figure, and determine alternate motions that could produce the same results.
7. Combine simple construction techniques to construct squares, equilateral triangles, or other simple combinations of equal segments, angles, etc.
8. Analyze properties of a shape that enable it to tessellate the plane.

9B - Students who meet the standard can identify, describe, classify and compare relationships using points, lines, planes, and solids. (Connections between and among multiple geometric figures)

1. Create and analyze scale models using proportional reasoning.
2. Solve problems involving similar figures.
3. Examine the congruence or similarity of objects using transformations. **
4. Analyze properties of a combination of shapes that enable them to tessellate the plane.

9C - Students who meet the standard can construct convincing arguments and proofs to solve problems. (Justifications of conjectures and conclusions)

1. Create and critique arguments concerning geometric ideas and relationships, such as congruence, similarity, the Pythagorean relationship, or formulas for surface areas or volume of simple three-dimensional objects. **
2. Justify the simple construction methods used to produce angle bisectors, perpendicular lines, and equilateral triangles.
3. Represent, solve, and explain numerical and algebraic relationships using geometric concepts.
4. Provide examples or counter-examples to either illustrate or disprove conjectures about geometric characteristics.

9D - Students who meet the standard can use trigonometric ratios and circular functions to solve problems.

1. Recognize Pythagorean Triples.
2. Identify the basic trigonometric ratios in terms of lengths of the sides of a right triangle and an acute angle.
3. Solve for missing side lengths using the trigonometric ratios in right triangles.
4. Determine and justify the side length relationships present in 45o-45o-90o triangles and 30o-60o-90o triangles.
5. Determine the ratio of lengths of sides of a right triangle with given measures for its acute angles using appropriate technologies.

10A - Students who meet the standard can organize, describe and make predictions from existing data. (Data Analysis)

1. Construct, read, interpret, infer, predict, draw conclusions, and evaluate data from various displays, including histograms and scatter plots. **
2. Determine the best measure of central tendency from mean, median, or mode.
3. Discuss how data can be manipulated to represent different points of view based on the use of different measures of central tendency and based on different graphical displays.
4. Discuss biased reporting of data and questions that should be asked when data is viewed.
5. Analyze graphical displays of data for possible misleading characteristics.
6. Make conjectures about the possible correlation between two characteristics of a sample on the basis of scatter plots of the data and approximate lines of fit. *

10B - Students who meet the standard can formulate questions, design data collection methods, gather and analyze data, and communicate findings. (Data collection)

1. Formulate a question, design a study to answer the question, and collect data. **
2. Analyze potential methods of collecting information and decide which methods would produce the most reliable and accurate data.
3. Analyze instruments used for surveys for errors and bias.
4. Analyze potential experiments or simulations for errors and bias.

10C - Students who meet the standard can determine, describe and apply the probabilities of events. (Probability, including counting techniques)

1. Describe and explain complementary and mutually exclusive events using appropriate terminology. **
2. Design and conduct experiments or simulations for probability, including the possible use of technology to simulate events.
3. Discuss the difference in empirical and theoretical probability.
4. Compute probabilities for simple compound events using a variety of methods, including area models.
5. Identify situations where dependent and independent events occur.
6. Determine probabilities using simple counting techniques.
7. Discuss situations where permutations and combinations should be used in counting outcomes.

* National Council of Teachers of Mathematics. Principles and Standards for School Mathematics. Reston, Va: National Council of Teachers of Mathematics, 2000.
** Adapted from: National Council of Teachers of Mathematics. Principles and Standards for School Mathematics. Reston, Va: National Council of Teachers of Mathematics, 2000.

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