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N.1. Number and Operations
1.1. Understand numbers, ways of representing numbers, relationships among numbers, and number systems.
1.1.1. Understand the place-value structure of the base-ten number system and be able to represent and compare whole numbers and decimals.
1.1.3. Develop understanding of fractions as parts of unit wholes, as parts of a collection, as locations on number lines, and as divisions of whole numbers.
1.1.4. Use models, benchmarks, and equivalent forms to judge the size of fractions.
1.1.5. Recognize and generate equivalent forms of commonly used fractions, decimals, and percents.
1.1.6. Explore numbers less than 0 by extending the number line and through familiar applications.
1.2. Understand meanings of operations and how they relate to one another.
1.2.1. Understand various meanings of multiplication and division.
1.2.2. Understand the effects of multiplying and dividing whole numbers.
1.2.3. Identify and use relationships between operations, such as division as the inverse of multiplication, to solve problems.
1.2.4. Understand and use properties of operations, such as the distributivity of multiplication over addition.
1.3. Compute fluently and make reasonable estimates.
1.3.1. Develop fluency with basic number combinations for multiplication and division and use these combinations to mentally compute related problems, such as 30 x 50.
1.3.2. Develop fluency in adding, subtracting, multiplying, and dividing whole numbers.
1.3.3. Develop and use strategies to estimate the results of whole-number computations and to judge the reasonableness of such results.
1.3.4. Develop and use strategies to estimate computations involving fractions and decimals in situations relevant to students' experience.
1.3.5. Use visual models, benchmarks, and equivalent forms to add and subtract commonly used fractions and decimals.
1.3.6. Select appropriate methods and tools for computing with whole numbers from among mental computation, estimation, calculators, and paper and pencil according to the context and nature of the computation and use the selected method or tools.
N.11. Grade 5 Curriculum Focal Points
11.1. Number and Operations and Algebra: Developing an understanding of and fluency with division of whole numbers
11.1.1. Students apply their understanding of models for division, place value, properties, and the relationship of division to multiplication as they develop, discuss, and use efficient, accurate, and generalizable procedures to find quotients involving multi-digit dividends. They select appropriate methods and apply them accurately to estimate quotients or calculate them mentally, depending on the context and numbers involved. They develop fluency with efficient procedures, including the standard algorithm, for dividing whole numbers, understand why the procedures work (on the basis of place value and properties of operations), and use them to solve problems. They consider the context in which a problem is situated to select the most useful form of the quotient for the solution, and they interpret it appropriately.
11.2. Number and Operations: Developing an understanding of and fluency with addition and subtraction of fractions and decimals
11.2.1. Students apply their understandings of fractions and fraction models to represent the addition and subtraction of fractions with unlike denominators as equivalent calculations with like denominators. They apply their understandings of decimal models, place value, and properties to add and subtract decimals. They develop fluency with standard procedures for adding and subtracting fractions and decimals. They make reasonable estimates of fraction and decimal sums and differences. Students add and subtract fractions and decimals to solve problems, including problems involving measurement.
11.3. Geometry and Measurement and Algebra: Describing three-dimensional shapes and analyzing their properties, including volume and surface area
11.3.1. Students relate two-dimensional shapes to three-dimensional shapes and analyze properties of polyhedral solids, describing them by the number of edges, faces, or vertices as well as the types of faces. Students recognize volume as an attribute of three-dimensional space. They understand that they can quantify volume by finding the total number of same-sized units of volume that they need to fill the space without gaps or overlaps. They understand that a cube that is 1 unit on an edge is the standard unit for measuring volume. They select appropriate units, strategies, and tools for solving problems that involve estimating or measuring volume. They decompose three-dimensional shapes and find surface areas and volumes of prisms. As they work with surface area, they find and justify relationships among the formulas for the areas of different polygons. They measure necessary attributes of shapes to use area formulas to solve problems.
N.12. Connections to the Grade 5 Focal Points
12.1. Algebra: Students use patterns, models, and relationships as contexts for writing and solving simple equations and inequalities. They create graphs of simple equations. They explore prime and composite numbers and discover concepts related to the addition and subtraction of fractions as they use factors and multiples, including applications of common factors and common multiples. They develop an understanding of the order of operations and use it for all operations.
12.2. Measurement: Students' experiences connect their work with solids and volume to their earlier work with capacity and weight or mass. They solve problems that require attention to both approximation and precision of measurement.
12.3. Data Analysis: Students apply their understanding of whole numbers, fractions, and decimals as they construct and analyze double-bar and line graphs and use ordered pairs on coordinate grids.
12.4. Number and Operations: Building on their work in grade 4, students extend their understanding of place value to numbers through millions and millionths in various contexts. They apply what they know about multiplication of whole numbers to larger numbers. Students also explore contexts that they can describe with negative numbers (e.g., situations of owing money or measuring elevations above and below sea level).
2.1. Understand patterns, relations, and functions.
2.1.1. Describe, extend, and make generalizations about geometric and numeric patterns.
2.1.2. Represent and analyze patterns and functions, using words, tables, and graphs.
2.2. Represent and analyze mathematical situations and structures using algebraic symbols.
2.2.1. Identify such properties as commutativity, associativity, and distributivity and use them to compute with whole numbers.
2.2.3. Express mathematical relationships using equations.
2.3. Use mathematical models to represent and understand quantitative relationships.
2.3.1. Model problem situations with objects and use representations such as graphs, tables, and equations to draw conclusions.
2.4. Analyze change in various contexts.
2.4.2. Identify and describe situations with constant or varying rates of change and compare them.
3.1. Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships.
3.1.1. Identify, compare, and analyze attributes of two- and three-dimensional shapes and develop vocabulary to describe the attributes.
3.1.2. Classify two- and three-dimensional shapes according to their properties and develop definitions of classes of shapes such as triangles and pyramids.
3.1.4. Explore congruence and similarity.
3.2. Specify locations and describe spatial relationships using coordinate geometry and other representational systems.
3.2.1. Describe location and movement using common language and geometric vocabulary.
3.2.2. Make and use coordinate systems to specify locations and to describe paths.
3.4. Use visualization, spatial reasoning, and geometric modeling to solve problems.
3.4.1. Build and draw geometric objects.
3.4.5. Use geometric models to solve problems in other areas of mathematics, such as number and measurement.
3.4.6. Recognize geometric ideas and relationships and apply them to other disciplines and to problems that arise in the classroom or in everyday life.
4.1. Understand measurable attributes of objects and the units, systems, and processes of measurement.
4.1.1. Understand such attributes as length, area, weight, volume, and size of angle and select the appropriate type of unit for measuring each attribute.
4.1.2. Understand the need for measuring with standard units and become familiar with standard units in the customary and metric systems.
4.1.3. Carry out simple unit conversions, such as from centimeters to meters, within a system of measurement.
4.1.4. Understand that measurements are approximations and how differences in units affect precision.
4.2. Apply appropriate techniques, tools, and formulas to determine measurements.
4.2.2. Select and apply appropriate standard units and tools to measure length, area, volume, weight, time, temperature, and the size of angles.
4.2.4. Develop, understand, and use formulas to find the area of rectangles and related triangles and parallelograms.
4.2.5. Develop strategies to determine the surface areas and volumes of rectangular solids.
N.5. Data Analysis and Probability
5.1. Formulate questions that can be addressed with data and collect, organize, and display relevant data to answer them.
5.1.2. Collect data using observations, surveys, and experiments.
5.2. Select and use appropriate statistical methods to analyze data.
5.2.2. Use measures of center, focusing on the median, and understand what each does and does not indicate about the data set.
5.4. Understand and apply basic concepts of probability.
5.4.1. Describe events as likely or unlikely and discuss the degree of likelihood using such words as certain, equally likely, and impossible.
5.4.3. Understand that the measure of the likelihood of an event can be represented by a number from 0 to 1.
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