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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.
1.1.2. Recognize equivalent representations for the same number and generate them by decomposing and composing numbers.
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.
1.3.5. 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 4 Curriculum Focal Points
11.1. Number and Operations and Algebra: Developing quick recall of multiplication facts and related division facts and fluency with whole number multiplication
11.1.1. Students use understandings of multiplication to develop quick recall of the basic multiplication facts and related division facts. They apply their understanding of models for multiplication (i.e., equal-sized groups, arrays, area models, equal intervals on the number line), place value, and properties of operations (in particular, the distributive property) as they develop, discuss, and use efficient, accurate, and generalizable methods to multiply multi-digit whole numbers. They select appropriate methods and apply them accurately to estimate products or calculate them mentally, depending on the context and numbers involved. They develop fluency with efficient procedures, including the standard algorithm, for multiplying whole numbers, understand why the procedures work (on the basis of place value and properties of operations), and use them to solve problems.
11.2. Number and Operations: Developing an understanding of decimals, including the connections between fractions and decimals
11.2.1. Students understand decimal notation as an extension of the base-ten system of writing whole numbers that is useful for representing more numbers, including numbers between 0 and 1, between 1 and 2, and so on. Students relate their understanding of fractions to reading and writing decimals that are greater than or less than 1, identifying equivalent decimals, comparing and ordering decimals, and estimating decimal or fractional amounts in problem solving. They connect equivalent fractions and decimals by comparing models to symbols and locating equivalent symbols on the number line.
11.3. Measurement: Developing an understanding of area and determining the areas of two-dimensional shapes
11.3.1. Students recognize area as an attribute of two-dimensional regions. They learn that they can quantify area by finding the total number of same-sized units of area that cover the shape without gaps or overlaps. They understand that a square that is 1 unit on a side is the standard unit for measuring area. They select appropriate units, strategies (e.g., decomposing shapes), and tools for solving problems that involve estimating or measuring area. Students connect area measure to the area model that they have used to represent multiplication, and they use this connection to justify the formula for the area of a rectangle.
N.12. Connections to the Grade 4 Focal Points
12.1. Algebra: Students continue identifying, describing, and extending numeric patterns involving all operations and nonnumeric growing or repeating patterns. Through these experiences, they develop an understanding of the use of a rule to describe a sequence of numbers or objects.
12.2. Geometry: Students extend their understanding of properties of two-dimensional shapes as they find the areas of polygons. They build on their earlier work with symmetry and congruence in grade 3 to encompass transformations, including those that produce line and rotational symmetry. By using transformations to design and analyze simple tilings and tessellations, students deepen their understanding of two-dimensional space.
12.3. Measurement: As part of understanding two-dimensional shapes, students measure and classify angles.
12.4. Data Analysis: Students continue to use tools from grade 3, solving problems by making frequency tables, bar graphs, picture graphs, and line plots. They apply their understanding of place value to develop and use stem-and-leaf plots.
12.5. Number and Operations: Building on their work in grade 3, students extend their understanding of place value and ways of representing numbers to 100,000 in various contexts. They use estimation in determining the relative sizes of amounts or distances. Students develop understandings of strategies for multi-digit division by using models that represent division as the inverse of multiplication, as partitioning, or as successive subtraction. By working with decimals, students extend their ability to recognize equivalent fractions. Students' earlier work in grade 3 with models of fractions and multiplication and division facts supports their understanding of techniques for generating equivalent fractions and simplifying fractions.
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.
3.3. Apply transformations and use symmetry to analyze mathematical situations.
3.3.3. Identify and describe line and rotational symmetry in two- and three-dimensional shapes and designs.
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.
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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