Stage E - 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. Place mixed numbers and decimals on a number line.
2. Show equivalent representations of a number by changing from one form to another form (e.g., standard form to expanded form, fraction to decimal, decimal to percent, improper fraction to mixed number).
3. Differentiate how fractions are used (part of a whole, part of a set, location on a number line, and division of a whole number).
4. Analyze how the size of the whole affects the size of the fraction (e.g., 1/2 of a large pizza is not the same as 1/2 of a small pizza).
5. Describe integers using familiar applications (e.g., a thermometer, above/below sea level).

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 whether a number is prime or composite.
2. Identify all the whole number factors of a composite number.
3. Explore and identify properties of square numbers.
4. Compute with 10, 100, 1000, and other powers of 10.
5. Explore and use divisibility rules.
6. Solve number sentences and word problems using addition and subtraction of fractions with unlike denominators.
7. Solve number sentences and word problems using addition and subtraction of decimals.

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

1. Develop and use strategies to estimate computations involving familiar fractions and decimals in situations relevant to students' experience * (e.g., double a recipe with 3/8 cup sugar, will more than a cup of sugar be needed).
2. Evaluate estimates to judge their reasonableness and degree of accuracy.
3. Select and use appropriate operation(s) and tool(s) (e.g., mental math, pencil-and-paper, estimation, calculator, computer) to perform calculations on whole numbers, fractions, and decimals according to the context and nature of the computation. **
4. Determine and justify whether exact answers or estimates are appropriate.

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

1. Identify and express ratios using appropriate notation (i.e., a/b, a to b, a:b).
2. Model the concept of percent using manipulatives or drawings.

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

1. Convert U.S. customary and metric measurements into larger or smaller units.
   Draw an angle of any given measure using a protractor or angle ruler.

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

1. Explain that all measurements are approximations.
2. Describe how precision is affected by choice of units.
3. Estimate the perimeter, area, and/or volume of regular and irregular shapes and objects.

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 appropriate tools to measure, draw, or construct figures.
2. Develop and discuss strategies for determining area and perimeter of irregular shapes.
3. Develop and use formulas to determine the area of squares, rectangles, and right triangles.
4. Read and interpret a scale on a map or a scale drawing using the idea of a constant ratio (e.g., 1" represents 1 mile), and use it to answer questions about actual measurement.
5. Explain the meaning of a measurement answer in context.

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

1. Describe, extend, and make generalizations about given geometric and numeric patterns. **
2. Describe a pattern, with at least two operations, verbally and symbolically, given a table of input/output numbers.
3. Demonstrate equality of two expressions with variables (e.g., 28 + 35 = 35 + n).
4. Describe situations involving inverse relationships (e.g., the more people, the fewer cookies per person).

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. Model problem situations with objects and equations to draw conclusions. **
2. Represent and analyze patterns and functions using words, tables, and graphs. *
3. Demonstrate how the change in one quantity affects the other in a functional relationship involving whole numbers and unit fractions.
4. Identify, describe, and compare situations with constant and varying rates of change using words, tables, and graphs (e.g., two quantities that vary together are the length of the side of a square and its area). **

8C - Students who meet the standard can solve problems using systems of numbers and their properties. (Problem solving; number systems, systems of equations, inequalities, algebraic functions)

1. Solve problems with whole numbers using order of operations, equality properties, and appropriate field properties.

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

1. Create and solve linear equations involving whole numbers using a variety of methods (e.g., guess and check, bean stick counters).

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. Identify, compare, and analyze attributes of two- and three-dimensional shapes and develop vocabulary to describe the attributes. *
2. Classify two- or three-dimensional shapes according to their properties (e.g., regular and irregular, concave and convex, types of quadrilaterals, pyramids, and prisms). **
3. Investigate and describe the results of subdividing and combining shapes. **
4. Describe paths using coordinate systems. **
5. Determine the distance between points along horizontal and vertical lines of a coordinate system. **
6. Identify and justify rotational symmetry in two- and three-dimensional shapes. **
7. Identify and describe how geometric figures are used in practical settings (e.g., construction, art, advertising, architecture).
8. Identify, sketch, and build two- and three-dimensional shapes given attribute clues.
9. Copy a line segment or an angle using a straightedge and a compass.
10. Construct a perpendicular bisector of a line segment.

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. Demonstrate congruence of plane figures using transformations (translation, rotation, reflection).
2. Determine if two polygons are congruent using measures of angles and sides.
3. Match a front, right side, and top view drawing with a three-dimensional model built with cubes.
4. Identify and describe the five regular polyhedra.
5. Create regular and semi-regular tessellations using pattern blocks, other manipulatives, or technology.

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

1. Make and test conjectures about mathematical properties and relationships and develop logical arguments to justify conclusions. **
2. Make and test conjectures about the results of subdividing and combining shapes. **

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

1. Represent given data using double bar graphs, double line graphs, and stem and leaf plots with and without technology.
2. Select an appropriate graph format to display given data.
3. Read, interpret, infer, predict, draw conclusions, and evaluate data from any graph.
4. Determine mean, median, mode, minimum value, maximum value, and range, and discuss what each does to help interpret a given set of data.

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

1. Design investigations to address a question and consider how data-collection methods affect the nature of a data set.
2. Propose and justify conclusions and predictions that are based on data, and design studies to further investigate the conclusions or predictions. *

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

1. List all possible outcomes of compound, independent events (e.g., toss a coin and spin a spinner).
2. Assign a value of zero to probabilities that are impossible and a value of one to probabilities that are certain.
3. Express simple probabilities as a fraction between zero and one.
4. Predict the probability of outcomes of simple experiments and test the predictions. *

* 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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