Stage F - 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. Represent place values from units through billions using powers of ten.
2. Represent, order, compare, and graph integers.
3. Identify fractional pieces that have the same value but different shapes.
4. Compare and order fractions and decimals efficiently and find their approximate position on a number line. **
5. Represent repeated factors using exponents.

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. Write prime factorizations of numbers.
2. Determine the least common multiple and the greatest common factor of a set of numbers.
3. Demonstrate the meaning of multiplication of fractions (e.g.,1/2 x 3 is 1/2 of a group of three objects).
4. Simplify simple arithmetic expressions with rational numbers using the field properties and the order of operations.
5. Recognize and use the inverse relationships of addition and subtraction, multiplication and division to simplify computations and solve problems. **
6. Solve multiplication number sentences and word problems with whole numbers and familiar fractions.

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 and use appropriate operations, methods, and tools to compute or estimate using whole numbers with natural number exponents. **
2. Analyze algorithms for computing with whole numbers, familiar fractions, and decimals and develop fluency in their use. **

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

1. Solve number sentences and word problems using percents.
2. Demonstrate and explain the meaning of percents, including greater than 100 and less than 1. **
3. Create and explain a pattern that shows a constant ratio.
4. Analyze situations to determine whether ratios are appropriate to solve problems.
5. Determine equivalent ratios.

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

1. Investigate the history of the U.S. customary and metric systems of measurement.
2. Measure, with a greater degree of accuracy, any angle using a protractor or angle ruler.

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

1. Estimate distance, weight, temperature, and elapsed time using reasonable units and with acceptable levels of accuracy.

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. Select and justify an appropriate formula to find the area of triangles, parallelograms, and trapezoids. **
2. Select an appropriate formula or strategy to find the surface area and volume of rectangular and triangular prisms. **
3. Develop and use formulas for determining the area of triangles, parallelograms, and trapezoids.
4. Develop and use the formula for determining the volume of a rectangular and triangular prism.
5. Calculate the surface area of a cube, rectangular prism, and triangular prism.
6. Develop and use formulas for determining the circumference and arc of circles.

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

1. Investigate, extend, and describe arithmetic and geometric sequences of numbers whether presented in numeric or pictorial form. **
2. Evaluate algebraic expressions for given values.
3. Express properties of numbers and operations using variables (e.g., the commutative property is m + n = n + m).
4. Simplify algebraic expressions involving like terms.

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 simple inequalities on a number line.
2. Create a table of values that satisfy a simple linear equation and plot the points on the Cartesian plane.
3. Describe, verbally, symbolically, and graphically, a simple relationship presented by a set of ordered pairs of numbers.

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

1. Identify and explain incorrect uses of the commutative, associative, and distributive properties.
2. Identify and provide examples of the identity property of addition and multiplication.
3. Identify and provide examples of inverse operations.
4. Explain why division by zero is undefined.

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. Create, model, and solve algebraic equations using concrete materials.
2. Solve linear equations, including direct variation, with whole number coefficients and solutions using algebraic or graphical representations.

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. Plot and read ordered pairs of numbers in all four quadrants.
2. Describe sizes, positions, and orientations of shapes under transformations, including dilations.
3. Perform simple constructions (e.g., equal segments, angle and segment bisectors, or perpendicular lines, inscribing a hexagon in a circle) with a compass and straightedge or a mira.
4. Determine and describe the relationship between pi, the diameter, the radius, and the circumference of a circle.
5. Determine unknown angle measures using angle relationships and properties of a triangle or a quadrilateral.

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. Determine the relationships between the number of vertices or sides in a polygon, the number of diagonals, and the sum of its angles.
2. Solve problems that involve vertical, complementary, and supplementary angles.
3. Analyze quadrilaterals for defining characteristics.
4. Create a three-dimensional object from any two-dimensional representation of the object, including multiple views, nets, or technological representations.

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

1. Make, test, and justify conjectures about various quadrilateral and triangle relationships, including the triangle inequality.
2. Justify the relationship between vertical angles.
3. Justify that the sum of the angles of a triangle is 180 degrees.
4. 9D - Students who meet the standard can use trigonometric ratios and circular functions to solve problems.

(9D not applicable for Stage F)

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 circle graphs. **
2. Recognize and explain misleading displays of data due to inappropriate intervals on a scale.

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

1. Gather data by conducting simple simulations.
2. Collect data over time with or without technology.

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

1. Record probabilities as fractions, decimals, or percents.
2. Demonstrate that the sum of all probabilities equals one.
3. Determine empirical probabilities from a set of data provided.
4. Set up a simulation to model the probability of a single event.
5. Discuss the effect of sample size on the empirical probability compared to the theoretical probability.
6. List outcomes by a variety of methods (e.g., tree diagram).
7. Determine theoretical probabilities of simple events.

* 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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