## Math

Measurement First Grade Math Ordering and Comparing Numbers Third Grade Math Addition Facts First Grade Math Attributes First Grade Math Ordinals First Grade Math Fractions Greater Than or Less Than 1/2 Second Grade Math Polygon Characteristics Fifth Grade Math ### 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 3 Curriculum Focal Points

#### 11.1. Number and Operations and Algebra: Developing understandings of multiplication and division and strategies for basic multiplication facts and related division facts

##### 11.1.1. Students understand the meanings of multiplication and division of whole numbers through the use of representations (e.g., equal-sized groups, arrays, area models, and equal 'jumps' on number lines for multiplication, and successive subtraction, partitioning, and sharing for division). They use properties of addition and multiplication (e.g., commutativity, associativity, and the distributive property) to multiply whole numbers and apply increasingly sophisticated strategies based on these properties to solve multiplication and division problems involving basic facts. By comparing a variety of solution strategies, students relate multiplication and division as inverse operations.

#### 11.2. Number and Operations: Developing an understanding of fractions and fraction equivalence

##### 11.2.1. Students develop an understanding of the meanings and uses of fractions to represent parts of a whole, parts of a set, or points or distances on a number line. They understand that the size of a fractional part is relative to the size of the whole, and they use fractions to represent numbers that are equal to, less than, or greater than 1. They solve problems that involve comparing and ordering fractions by using models, benchmark fractions, or common numerators or denominators. They understand and use models, including the number line, to identify equivalent fractions.

#### 11.3. Geometry: Describing and analyzing properties of two-dimensional shapes

##### 11.3.1. Students describe, analyze, compare, and classify two-dimensional shapes by their sides and angles and connect these attributes to definitions of shapes. Students investigate, describe, and reason about decomposing, combining, and transforming polygons to make other polygons. Through building, drawing, and analyzing two-dimensional shapes, students understand attributes and properties of two-dimensional space and the use of those attributes and properties in solving problems, including applications involving congruence and symmetry.

### N.12. Connections to the Grade 3 Focal Points

#### 12.1. Algebra: Understanding properties of multiplication and the relationship between multiplication and division is a part of algebra readiness that develops at grade 3. The creation and analysis of patterns and relationships involving multiplication and division should occur at this grade level. Students build a foundation for later understanding of functional relationships by describing relationships in context with such statements as, 'The number of legs is 4 times the number of chairs.'

#### 12.2. Measurement: Students in grade 3 strengthen their understanding of fractions as they confront problems in linear measurement that call for more precision than the whole unit allowed them in their work in grade 2. They develop their facility in measuring with fractional parts of linear units. Students develop measurement concepts and skills through experiences in analyzing attributes and properties of two-dimensional objects. They form an understanding of perimeter as a measurable attribute and select appropriate units, strategies, and tools to solve problems involving perimeter.

#### 12.3. Data Analysis: Addition, subtraction, multiplication, and division of whole numbers come into play as students construct and analyze frequency tables, bar graphs, picture graphs, and line plots and use them to solve problems.

#### 12.4. Number and Operations: Building on their work in grade 2, students extend their understanding of place value to numbers up to 10,000 in various contexts. Students also apply this understanding to the task of representing numbers in different equivalent forms (e.g., expanded notation). They develop their understanding of numbers by building their facility with mental computation (addition and subtraction in special cases, such as 2,500 + 6,000 and 9,000 - 5,000), by using computational estimation, and by performing paper-and-pencil computations.

### N.2. Algebra

#### 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.

### N.3. Geometry

#### 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.

##### 3.4.5. 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.

### N.4. Measurement

#### 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.

##### 4.1.4. 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

#### 5.2. 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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