Stage C - 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, order, and compare whole numbers to demonstrate an understanding of the base-ten number system.
2. Recognize equivalent representations of whole numbers and generate them by composing and decomposing numbers (e.g., 123 = 100 + 20 + 3). **
3. Judge the size of fractions using models, benchmarks, and equivalent forms. **
4. Represent, order, label, and compare familiar fractions.
   Recognize and generate equivalent forms of familiar fractions. **
5. Explore and discuss uses of decimals.

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. Show and use the relationship between multiplication and division.
2. Demonstrate and describe the effects of multiplying and dividing whole numbers using appropriate mathematical notation and vocabulary.
3. Explore, identify, and use relationships between and among properties of operations (e.g., commutativity applies to addition but not to subtraction).
4. Demonstrate fluency with basic multiplication and division facts.
5. Solve multiplication and division number sentences and word problems.
6. Apply knowledge of basic multiplication facts (factors 0-10) to related facts (e.g., 3 x 4 = 12, 30 x 4 = 120, 300 x 4 = 1200).
7. Select and use one of various algorithms to add and subtract.

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 (i.e. rounding) to estimate the results of whole-number computations and to judge the reasonableness of such results. **
2. Select appropriate methods and tools for computing with whole numbers from mental
3. computation, estimation, calculators, and paper/pencil according to the context and
4. nature of the computation and use of the selected method or tool. *
5. Determine whether exact answers or estimates are appropriate for solutions to problems.

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

1. Describe the relationship between two sets using ">", "<", and "=", "not equal to".

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

1. Explain the need for using standard units for measuring. **
2. Measure objects using standard units in the U.S. customary and metric systems. **
3. Perform simple unit conversions within a system of measurement (e.g., three feet is the same as a
   yard). **
4. Describe multiple measurable attributes (e.g., length, mass/weight, time, temperature, area, volume, capacity) of a single object.
5. Show and explain perimeter of an object by measuring and adding its linear units.
6. Show and explain the area of an object by counting square units.

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

1. Develop and use common referents for linear measures to make comparisons and estimates.
2. Estimate perimeter of simple polygons.

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 apply appropriate standard units and tools to measure length, area, volume, weight, time, and temperature. *
2. Determine elapsed time between events.
3. Solve problems using perimeter and area of simple polygons.
4. Make change from a given amount using bills and coins.

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

1. Extend geometric and simple numeric patterns using concrete objects or paper and pencil.
2. Demonstrate how to create a pattern given a set of directions.
3. Identify errors in a given pattern.
4. Represent the idea of a variable as an unknown quantity using a letter or a symbol in a numerical sentence. **
5. Express mathematical relationships using equations.

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. Represent and analyze simple patterns and operations using words, tables, and graphs. **
2. Describe situations with constant rates of change using words, tables, and graphs (e.g., walking at a constant rate of speed).

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. Apply the relationship of multiplication and division fact families to solve for an unknown quantity.

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. Demonstrate how to select and use an appropriate operation to solve problems involving patterns (e.g., save one penny on day 1, double that amount each day for 10 days).
2. Solve one-step linear equations using concrete materials.

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. Specify locations using a coordinate system. **
2. Predict and describe the results of translations, rotations, and reflections of two-dimensional shapes.
3. Identify, draw, and build polygons.

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. Decompose a three-dimensional object into two-dimensional components.
2. Describe the difference between congruence and similarity. **
3. Describe a motion or a series of motions that will show that two shapes are congruent. *
4. Identify and build a threedimensional object from two-dimensional representations of that object. *
5. Apply geometric ideas and relationships to problems that arise in the classroom or in everyday life. **
6. Apply geometric ideas and relationships to other disciplines. **

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 justify the conclusions. **

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

1. Organize, describe, and make predictions from existing data. *
2. Represent data using tables and graphs such as tallies and bar graphs.
3. Describe the important features of a set of data displayed by a graph.
4. Determine the median of data on a graph.

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

1. Create and administer a survey considering which questions will be asked and how the answers will be recorded.
2. Propose a follow-up survey to investigate questions that arise from the initial survey.

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

1. Describe events as likely or unlikely and discuss the degree of likelihood using such words as certain, equally likely, and impossible. *
2. Explain probability as a fractional part of a group to the whole group (e.g., A tossed coin can land on heads or tails; therefore, it should land on heads 1/2 of the time.)
3. Make predictions based on the results received from a probability experiment.
4. Create and perform a probability experiment (e.g., a penny is flipped 100 times) and record the results.
5. Understand that the measure of the likelihood of an event can be represented by a number from zero to one, inclusive. **

* 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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