Math+Standards

=NCTM Math Standards (Grades Pre-K-5)=

Pre-K–2 Expectations: In pre-K through grade 2 all students should–
• count with understanding and recognize "how many" in sets of objects; • use multiple models to develop initial understandings of place value and the base-ten number system; • develop understanding of the relative position and magnitude of whole numbers and of ordinal and cardinal numbers and their connections; • develop a sense of whole numbers and represent and use them in flexible ways, including relating, composing, and decomposing numbers; • connect number words and numerals to the quantities they represent, using various physical models and representations; • understand and represent commonly used fractions, such as 1/4, 1/3, and 1/2.

Grades 3–5 Expectations: In grades 3–5 all students should–
• understand the place-value structure of the base-ten number system and be able to represent and compare whole numbers and decimals; • recognize equivalent representations for the same number and generate them by decomposing and composing numbers; • 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; • use models, benchmarks, and equivalent forms to judge the size of fractions; • recognize and generate equivalent forms of commonly used fractions, decimals, and percents; • explore numbers less than 0 by extending the number line and through familiar applications; • describe classes of numbers according to characteristics such as the nature of their factors.

Pre-K–2 Expectations: In pre-K through grade 2 all students should–
• understand various meanings of addition and subtraction of whole numbers and the relationship between the two operations; • understand the effects of adding and subtracting whole numbers; • understand situations that entail multiplication and division, such as equal groupings of objects and sharing equally.

Grades 3–5 Expectations: In grades 3–5 all students should–
• understand various meanings of multiplication and division; • understand the effects of multiplying and dividing whole numbers; • identify and use relationships between operations, such as division as the inverse of multiplication, to solve problems; • understand and use properties of operations, such as the distributivity of multiplication over addition.

Pre-K–2 Expectations: In pre-K through grade 2 all students should–
• develop and use strategies for whole-number computations, with a focus on addition and subtraction; • develop fluency with basic number combinations for addition and subtraction; • use a variety of methods and tools to compute, including objects, mental computation, estimation, paper and pencil, and calculators.

Grades 3–5 Expectations: In grades 3–5 all students should–
• develop fluency with basic number combinations for multiplication and division and use these combinations to mentally compute related problems, such as 30 × 50; • develop fluency in adding, subtracting, multiplying, and dividing whole numbers; • develop and use strategies to estimate the results of whole-number computations and to judge the reasonableness of such results; • develop and use strategies to estimate computations involving fractions and decimals in situations relevant to students' experience; • use visual models, benchmarks, and equivalent forms to add and subtract commonly used fractions and decimals; • 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.

Pre-K–2 Expectations: In pre-K through grade 2 all students should–
• sort, classify, and order objects by size, number, and other properties; • recognize, describe, and extend patterns such as sequences of sounds and shapes or simple numeric patterns and translate from one representation to another; • analyze how both repeating and growing patterns are generated.

Grades 3–5 Expectations: In grades 3–5 all students should–
• describe, extend, and make generalizations about geometric and numeric patterns; • represent and analyze patterns and functions, using words, tables, and graphs.

Pre-K–2 Expectations: In pre-K through grade 2 all students should–
• illustrate general principles and properties of operations, such as commutativity, using specific numbers; • use concrete, pictorial, and verbal representations to develop an understanding of invented and conventional symbolic notations.

Grades 3–5 Expectations: In grades 3–5 all students should–
• identify such properties as commutativity, associativity, and distributivity and use them to compute with whole numbers; • represent the idea of a variable as an unknown quantity using a letter or a symbol; • express mathematical relationships using equations.

3. Use mathematical models to represent and understand quantitative relationships
Pre-K–2 Expectations: In pre-K through grade 2 all students should–

• model situations that involve the addition and subtraction of whole numbers, using objects, pictures, and symbols.

Grades 3–5 Expectations: In grades 3–5 all students should–
• model problem situations with objects and use representations such as graphs, tables, and equations to draw conclusions.

Pre-K–2 Expectations: In pre-K through grade 2 all students should–
• describe qualitative change, such as a student's growing taller; • describe quantitative change, such as a student's growing two inches in one year.

Grades 3–5 Expectations: In grades 3–5 all students should–
• investigate how a change in one variable relates to a change in a second variable; • identify and describe situations with constant or varying rates of change and compare them.

Pre-K–2 Expectations: In pre-K through grade 2 all students should–
• recognize, name, build, draw, compare, and sort two- and three-dimensional shapes; • describe attributes and parts of two- and three-dimensional shapes; • investigate and predict the results of putting together and taking apart two- and three-dimensional shapes.

Grades 3–5 Expectations: In grades 3–5 all students should–
• identify, compare, and analyze attributes of two- and three-dimensional shapes and develop vocabulary to describe the attributes; • classify two- and three-dimensional shapes according to their properties and develop definitions of classes of shapes such as triangles and pyramids; • investigate, describe, and reason about the results of subdividing, combining, and transforming shapes; • explore congruence and similarity; • make and test conjectures about geometric properties and relationships and develop logical arguments to justify conclusions.

Pre-K–2 Expectations: In pre-K through grade 2 all students should–
• describe, name, and interpret relative positions in space and apply ideas about relative position; • describe, name, and interpret direction and distance in navigating space and apply ideas about direction and distance; • find and name locations with simple relationships such as "near to" and in coordinate systems such as maps.

Grades 3–5 Expectations: In grades 3–5 all students should–
• describe location and movement using common language and geometric vocabulary; • make and use coordinate systems to specify locations and to describe paths; • find the distance between points along horizontal and vertical lines of a coordinate system.

Pre-K–2 Expectations: In pre-K through grade 2 all students should–
• recognize and apply slides, flips, and turns; • recognize and create shapes that have symmetry.

Grades 3–5 Expectations: In grades 3–5 all students should–
• predict and describe the results of sliding, flipping, and turning two-dimensional shapes; • describe a motion or a series of motions that will show that two shapes are congruent; • identify and describe line and rotational symmetry in two- and three-dimensional shapes and designs.

Pre-K–2 Expectations: In pre-K through grade 2 all students should–
• create mental images of geometric shapes using spatial memory and spatial visualization; • recognize and represent shapes from different perspectives; • relate ideas in geometry to ideas in number and measurement; • recognize geometric shapes and structures in the environment and specify their location.

Grades 3–5 Expectations: In grades 3–5 all students should–
• build and draw geometric objects; • create and describe mental images of objects, patterns, and paths; • identify and build a three-dimensional object from two-dimensional representations of that object; • identify and draw a two-dimensional representation of a three-dimensional object; • use geometric models to solve problems in other areas of mathematics, such as number and measurement; • recognize geometric ideas and relationships and apply them to other disciplines and to problems that arise in the classroom or in everyday life.

Pre-K–2 Expectations: In pre-K through grade 2 all students should–
• recognize the attributes of length, volume, weight, area, and time; • compare and order objects according to these attributes; • understand how to measure using nonstandard and standard units; • select an appropriate unit and tool for the attribute being measured.

Grades 3–5 Expectations: In grades 3–5 all students should–
• understand such attributes as length, area, weight, volume, and size of angle and select the appropriate type of unit for measuring each attribute; • understand the need for measuring with standard units and become familiar with standard units in the customary and metric systems; • carry out simple unit conversions, such as from centimeters to meters, within a system of measurement; • understand that measurements are approximations and how differences in units affect precision; • explore what happens to measurements of a two-dimensional shape such as its perimeter and area when the shape is changed in some way.

Pre-K–2 Expectations: In prekindergarten through grade 2 all students should–
• measure with multiple copies of units of the same size, such as paper clips laid end to end; • use repetition of a single unit to measure something larger than the unit, for instance, measuring the length of a room with a single meterstick; • use tools to measure; • develop common referents for measures to make comparisons and estimates.

Grades 3–5 Expectations: In grades 3–5 all students should–
• develop strategies for estimating the perimeters, areas, and volumes of irregular shapes; • select and apply appropriate standard units and tools to measure length, area, volume, weight, time, temperature, and the size of angles; • select and use benchmarks to estimate measurements; • develop, understand, and use formulas to find the area of rectangles and related triangles and parallelograms; • develop strategies to determine the surface areas and volumes of rectangular solids.

Pre-K–2 Expectations: In pre-K through grade 2 all students should–
• pose questions and gather data about themselves and their surroundings; • sort and classify objects according to their attributes and organize data about the objects; • represent data using concrete objects, pictures, and graphs.

Grades 3–5 Expectations: In grades 3–5 all students should–
• design investigations to address a question and consider how data-collection methods affect the nature of the data set; • collect data using observations, surveys, and experiments; • represent data using tables and graphs such as line plots, bar graphs, and line graphs; • recognize the differences in representing categorical and numerical data.

Pre-K–2 Expectations: In pre-K through grade 2 all students should–
• describe parts of the data and the set of data as a whole to determine what the data show.

Grades 3–5 Expectations: In grades 3–5 all students should–
• describe the shape and important features of a set of data and compare related data sets, with an emphasis on how the data are distributed; • use measures of center, focusing on the median, and understand what each does and does not indicate about the data set; • compare different representations of the same data and evaluate how well each representation shows important aspects of the data.

Pre-K–2 Expectations: In pre-K through grade 2 all students should–
• discuss events related to students' experiences as likely or unlikely.

Grades 3–5 Expectations: In grades 3–5 all students should–
• propose and justify conclusions and predictions that are based on data and design studies to further investigate the conclusions or predictions.

Grades 3–5 Expectations: In grades 3–5 all students should–
• describe events as likely or unlikely and discuss the degree of likelihood using such words as certain, equally likely, and impossible; • predict the probability of outcomes of simple experiments and test the predictions; • understand that the measure of the likelihood of an event can be represented by a number from 0 to 1.