Virginia K–6 mathematics standards

Use objects, drawings, words, or numbers to compose and decompose numbers less than or equal to 5 in multiple ways.

Recognize and describe with fluency part-part-whole relationships for numbers up to 5 in a variety of configurations.

Model and identify the number that makes 5 when added to a given number less than or equal to 5.

Use objects, drawings, words, or numbers to compose and decompose numbers less than or equal to 10 in multiple ways.

Model and identify the number that makes 10 when added to a given number less than or equal to 10.

Model and solve single-step contextual problems (join, separate, and part-part-whole) using 10 or fewer concrete objects.

Use direct comparisons to compare, describe, and justify the: lengths of two objects using the terms longer or shorter;

Use direct comparisons to compare, describe, and justify the: heights of two objects using the terms taller or shorter;

Use direct comparisons to compare, describe, and justify the: weights of two objects using the terms heavier or lighter;

Use direct comparisons to compare, describe, and justify the: volumes of two containers using the terms more or less; and

Use direct comparisons to compare, describe, and justify the: amount of time spent on two events using the terms longer or shorter.

Identify and name concrete and pictorial representations of circles, triangles, squares, and rectangles regardless of their orientation in space.

Describe triangles, squares, and rectangles to include the number of sides and number of vertices.

Describe a circle using terms such as round and curved.

Distinguish between examples and nonexamples of identified plane figures (circles, triangles, squares, and rectangles).

Compare and contrast two plane figures using characteristics to describe similarities and differences.

Construct plane figures (circles, triangles, squares, and rectangles) using a variety of materials (e.g., straws, sticks, pipe cleaners).

Identify a calendar as a tool used to measure time.

Name the days of the week and state that there are seven days in one week.

Determine the day before and after a given day (e.g., yesterday, today, tomorrow).

Name the twelve months of the year and state that there are twelve months in one year.

Distinguish between days of the week and months of the year.

Use one-to-one correspondence to determine how many are in a given set containing 30 or fewer concrete objects (e.g., cubes, pennies, balls), and describe the last number named as the total number of objects counted.

Recognize and explain that the number of objects remains the same regardless of the arrangement or the order in which the objects are counted.

Represent forward counting by ones using a variety of tools, including five-frames, tenframes, and number paths (a prelude to number lines).

Count forward orally by ones from 0 to 100.

Count forward orally by ones, within 100, starting at any given number.

Count backward orally by ones when given any number between 1 and 20.

State the number after, without counting, when given any number between 0 and 30.

State the number before, without counting, when given any number between 1 and 20.

Use objects, drawings, words, or numbers to compose and decompose numbers 11-19 into a ten and some ones.

Group a collection of up to 100 objects (e.g., counters, pennies, cubes) into sets of ten and count by tens to determine the total (e.g., there are 3 groups of ten and 6 leftovers, 36 total objects).

Read, write, and identify the numerals 0 through 30.

Construct a set of objects that corresponds to a given numeral within 30, including an empty set.

Determine and write the numeral that corresponds to the total number of objects in a given set of 30 or fewer concrete objects or pictorial models.

Given a set of up to 30 objects, construct another set which has more, fewer, or the same number of objects using concrete or pictorial models.

Given a numeral up to 30, construct a set which has more, fewer, or the same number of objects using concrete or pictorial models.

Compare two sets containing up to 30 concrete objects or pictorial models, using the terms more, fewer, or the same as (equal to).

Compare numbers up to 30, to the benchmarks of 5 and 10 using various models (e.g., five frames, ten frames, number paths [a prelude to number lines], beaded racks, hands) using the terms greater than, less than, or the same as (equal to).

Identify and describe the core found in repeating patterns.

Extend a repeating pattern by adding at least two complete repetitions of the core to the pattern.

Create and describe a repeating pattern using objects, colors, sounds, movements, or pictures.

Sort and classify concrete objects into appropriate subsets (categories) based on one attribute (e.g., size, shape, color, thickness).

Describe and label attributes (e.g., size, color, shape) of a set of objects (e.g., coins, counters, buttons) that has been sorted.

Pose questions, given a predetermined context, that require the collection of data (limited to 25 or fewer data points for no more than four categories).

Determine the data needed to answer a posed question, and collect the data using various methods (e.g., counting objects, drawing pictures).

Organize and represent a data set (vertically or horizontally) by sorting concrete objects into organized groups to form a simple object graph.

Organize and represent a data set (vertically or horizontally) using pictures to form a simple picture graph.

Analyze data represented in object graphs and picture graphs and communicate results: ask and answer questions about the data represented in object graphs and picture graphs (e.g., how many in each category, which categories have the greatest, least, or the same amount of data); and

Analyze data represented in object graphs and picture graphs and communicate results: draw conclusions about the data and make predictions based on the data.

The student will recall with automaticity addition and subtraction facts within 10 and represent, solve, and justify solutions to single-step problems, including those in context, using addition and subtraction with whole numbers within 20.

Recognize and describe with fluency part-part-whole relationships for numbers up to 10 in a variety of configurations.

Demonstrate fluency with addition and subtraction within 10 by applying reasoning strategies (e.g., count on/count back, one more/one less, doubles, make ten).

Recall with automaticity addition and subtraction facts within 10.

Investigate, recognize, and describe part-part-whole relationships for numbers up to 20 in a variety of configurations (e.g., beaded racks, double ten frames).

Solve addition and subtraction problems within 20 using various strategies (e.g., inverse relationships: if 9 + 3 = 12 then 12 - 3 = 9; decomposition using known sums/differences: 9 + 7 can be thought of as 9 decomposed into 2 and 7, then use doubles, 7 + 7 = 14; 14 + 2 = 16 or decompose the 7 into 1 and 6; make a ten: 1 + 9 = 10; 10 + 6 = 16).

Represent, solve, and justify solutions to single-step addition and subtraction problems (join, separate, and part-part-whole) within 20, including those in context, using words, objects, drawings, or numbers.

Determine the unknown whole number that will result in a sum or difference of 10 or 20 (e.g., 14 - __ = 10 or 15 + __ = 20).

Identify and use (+) as a symbol for addition and (-) as a symbol for subtraction.

Describe the equal symbol (=) as a balance representing an equivalent relationship between expressions on either side of the equal symbol (e.g., 6 and 1 is the same as 4 and 3; 6 + 1 is balanced with 4 + 3; 6 + 1 = 4 + 3).

Use concrete materials to model, identify, and justify when two expressions are not equal (e.g., 10 - 3 is not equal to 3 + 5).

Use concrete materials to model an equation that represents the relationship of two expressions of equal value.

Write an equation that could be used to represent the solution to an oral, written, or picture problem.

The student will reason mathematically using nonstandard units to measure and compare objects by length, weight, and volume.

Use nonstandard units to measure the: lengths of two objects (units laid end to end with no gaps or overlaps) and compare the measurements using the terms longer/shorter, taller/shorter, or the same as;

Use nonstandard units to measure the: weights of two objects (using a balance scale or a pan scale) and compare the measurements using the terms lighter, heavier, or the same as; and

Use nonstandard units to measure the: volumes of two containers and compare the measurements using the terms more, less, or the same as.

Measure the length, weight, or volume of the same object or container with two different units and describe how and why the measurements differ.

The student will describe, sort, draw, and name plane figures (circles, triangles, squares, and rectangles), and compose larger plane figures by combining simple plane figures.

Describe triangles, squares, and rectangles using the terms sides, vertices, and angles. Describe a circle using terms such as round and curved.

Sort plane figures based on their characteristics (e.g., number of sides, vertices, angles, curved).

Draw and name the plane figure (circle, square, rectangle, triangle) when given information about the number of sides, vertices, and angles.

Identify, name, and describe representations of circles, squares, rectangles, and triangles, regardless of orientation, in different environments and explain reasoning.

Recognize and name the angles found in rectangles and squares as right angles.

Compose larger plane figures by combining two or three simple plane figures (triangles, squares, and/or rectangles).

The student will demonstrate an understanding of the concept of passage of time (to the nearest hour and half-hour) and the calendar.

Identify different tools to measure time including clocks (analog and digital) and calendar.

Describe the units of time represented on a clock as minutes and hours.

Tell time to the hour and half-hour, using analog and digital clocks.

Describe the location of the hour hand relative to time to the hour and half-hour on an analog clock.

Describe the location of the minute hand relative to time to the hour and half-hour on an analog clock.

Match the time shown on a digital clock to an analog clock to the hour and half-hour.

Identify specific days/dates on a calendar (e.g., What date is Saturday? How many Fridays are in October?).

Use ordinal numbers first through tenth to describe the relative position of specific days/dates (e.g., What is the first Monday in October? What day of the week is May 6th?).

Determine the day/date before and after a given day/date (e.g., Today is the 8 th , so yesterday was the ?), and a date that is a specific number of days/weeks in the past or future (e.g., Tim’s birthday is in 10 days, what will be the date of his birthday?).

The student will utilize flexible counting strategies to determine and describe quantities up to 120.

Count forward orally by ones from 0 to 120 starting at any number between 0 and 120.

Count backward orally by ones when given any number between 1 and 30.

Represent forward counting patterns when counting by groups of 5 and groups of 10 up to 120 using a variety of tools (e.g., objects, coins, 120 chart).

Represent forward counting patterns when counting by groups of 2 up to at least 30 using a variety of tools (e.g., beaded number strings, number paths [a prelude to number lines], 120 chart).

Group a collection of up to 120 objects into tens and ones, and count to determine the total (e.g., 5 groups of ten and 6 ones is equal to 56 total objects).

Identify a penny, nickel, and dime by their attributes and describe the number of pennies equivalent to a nickel and a dime.

Count by ones, fives, or tens to determine the value of a collection of like coins (pennies, nickels, or dimes), whose total value is 100 cents or less.

The student will represent, compare, and order quantities up to 120.

Read and write numerals 0-120 in sequence and out of sequence.

Estimate the number of objects (up to 120) in a given collection and justify the reasonableness of an answer.

Create a concrete or pictorial representation of a number using tens and ones and write the corresponding numeral up to 120 (e.g., 47 can be represented as 47 ones or it can be grouped into 4 tens with 7 ones left over).

Describe the number of groups of tens and ones when given a two-digit number and justify reasoning.

Compare two numbers between 0 and 120 represented pictorially or with concrete objects using the terms greater than, less than, or equal to.

Order three sets, each set containing up to 120 objects, from least to greatest, and greatest to least.

The student will use mathematical reasoning and justification to solve contextual problems that involve partitioning models into two and four equal-sized parts.

Represent equal shares of a whole with two or four sharers, when given a contextual problem.

Represent and name halves and fourths of a whole, using a region/area model (e.g., pie pieces, pattern blocks, paper folding, drawings) and a set model (e.g., eggs, marbles, counters) limited to two or four items.

Describe and justify how shares are equal pieces or equal parts of the whole (limited to halves, fourths) when given a contextual problem.

The student will identify, describe, extend, create, and transfer repeating patterns and increasing patterns using various representations.

Identify and describe repeating and increasing patterns.

Analyze a repeating or increasing pattern and generalize the change to extend the pattern using objects, colors, movements, pictures, or geometric figures.

Create a repeating or increasing pattern using objects, pictures, movements, colors, or geometric figures.

Transfer a repeating or increasing pattern from one form to another.

The student will apply the data cycle (pose questions; collect or acquire data; organize and represent data; and analyze data and communicate results) with a focus on object graphs, picture graphs, and tables.

Sort and classify concrete objects into appropriate subsets (categories) based on one or two attributes, such as size, shape, color, and/or thickness (e.g., sort a set of objects that are both red and thick).

Describe and label attributes of a set of objects that has been sorted.

Pose questions, given a predetermined context, that require the collection of data (limited to 25 or fewer data points for no more than four categories).

Determine the data needed to answer a posed question and collect the data using various methods (e.g., counting objects, drawing pictures, tallying).

Organize and represent a data set by sorting the collected data using various methods (e.g., tallying, T-charts).

Represent a data set (vertically or horizontally) using object graphs, picture graphs, and tables.

Analyze data represented in object graphs, picture graphs, and tables and communicate results: ask and answer questions about the data represented in object graphs, picture graphs, and tables (e.g., total number of data points represented, how many in each category, how many more or less are in one category than another); and

Analyze data represented in object graphs, picture graphs, and tables and communicate results: draw conclusions about the data and make predictions based on the data.

The student will recall with automaticity addition and subtraction facts within 20 and estimate, represent, solve, and justify solutions to single-step and multistep problems, including those in context, using addition and subtraction with whole numbers where addends or minuends do not exceed 100.

Apply strategies (e.g., rounding to the nearest 10, compatible numbers, other number relationships) to estimate a solution for single-step addition or subtraction problems, including those in context, where addends and minuends do not exceed 100.

Apply strategies (e.g., the use of concrete and pictorial models, place value, properties of addition, the relationship between addition and subtraction) to determine the sum or difference of two whole numbers where addends or minuends do not exceed 100.

Represent, solve, and justify solutions to single-step and multistep contextual problems (e.g., join, separate, part-part-whole, comparison) involving addition or subtraction of whole numbers where addends or minuends do not exceed 100.

Demonstrate fluency with addition and subtraction within 20 by applying reasoning strategies (e.g., doubles, near doubles, make-a-ten, compensations, inverse relationships).

Recall with automaticity addition and subtraction facts within 20.

Use patterns, models, and strategies to make generalizations about the algebraic properties for fluency (e.g., 4 + 3 is equal to 3 + 4; 0 + 8 = 8).

Determine the missing number in an equation (number sentence) through modeling and justification with addition and subtraction within 20 (e.g., 3 + = 5 or + 2 = 5; 5 – = 3 or 5 – 2 = ).

Use inverse relationships to write all related facts connected to a given addition or subtraction fact model within 20 (e.g., given a model for 3 + 4 = 7, write 4 + 3 = 7, 7 – 4 = 3, and 7 – 3 = 4).

Describe the not equal symbol (≠) as representing a relationship where expressions on either side of the not equal symbol represent different values and justify reasoning.

Represent and justify the relationship between values and expressions as equal or not equal using appropriate models and/or symbols (e.g., 9 + 24 = 10 + 23; 45 - 9 = 46 - 10; 15 +16 ≠ 31 +15).

The student will reason mathematically using standard units (U.S. Customary) with appropriate tools to estimate, measure, and compare objects by length, weight, and liquid volume to the nearest whole unit.

Explain the purpose of various measurement tools and how to use them appropriately by: identifying a ruler as an instrument to measure length;

Explain the purpose of various measurement tools and how to use them appropriately by: identifying different types of scales as instruments to measure weight; and

Explain the purpose of various measurement tools and how to use them appropriately by: identifying different types of measuring cups as instruments to measure liquid volume.

Use U.S. Customary units to estimate, measure, and compare the two for reasonableness: the length of an object to the nearest inch, using a ruler;

Use U.S. Customary units to estimate, measure, and compare the two for reasonableness: the weight of an object to the nearest pound, using a scale; and

Use U.S. Customary units to estimate, measure, and compare the two for reasonableness: the liquid volume of a container to the nearest cup, using a measuring cup.

The student will demonstrate an understanding of the concept of time to the nearest five minutes, using analog and digital clocks.

Identify the number of minutes in an hour (60 minutes) and the number of hours in a day (24 hours).

Determine the unit of time (minutes, hours, days, or weeks) that is most appropriate when measuring a given activity or context and explain reasoning (e.g., Would you measure the time it takes to brush your teeth in minutes or hours?).

Show, tell, and write time to the nearest five minutes, using analog and digital clocks.

Match a written time (e.g., 1:35, 6:20, 9:05) to the time shown on an analog clock to the nearest five minutes.

The student will identify, describe, and create plane figures (including circles, triangles, squares, and rectangles) that have at least one line of symmetry and explain its relationship with congruency.

Explore a figure using a variety of tools (e.g., paper folding, geoboards, drawings) to show and justify a line of symmetry, if one exists.

Create figures with at least one line of symmetry using various concrete and pictorial representations.

Describe the two resulting figures formed by a line of symmetry as being congruent (having the same shape and size).

The student will describe, name, compare, and contrast plane and solid figures (circles/spheres, squares/cubes, and rectangles/rectangular prisms).

Trace faces of solid figures (cubes and rectangular prisms) to create the set of plane figures related to the solid figure.

Compare and contrast models and nets (cutouts) of cubes and rectangular prisms (e.g., number and shapes of faces, edges, vertices).

Given a concrete or pictorial model, name and describe the solid figure (sphere, cube, and rectangular prism) by its characteristics (e.g., number of edges, number of vertices, shapes of faces).

Compare and contrast plane and solid figures (circles/spheres, squares/cubes, and rectangles/rectangular prisms) according to their characteristics (e.g., number and shapes of their faces, edges, vertices).

The student will utilize flexible counting strategies to determine and describe quantities up to 200.

Represent forward counting patterns when counting by groups of 2 up to at least 50, starting at various multiples of 2 and using a variety of tools (e.g., objects, number lines, hundreds charts).

Represent forward counting patterns created when counting by groups of 5s, 10s, and 25s starting at various multiples up to at least 200 using a variety of tools (e.g., objects, number lines, hundreds charts).

Describe and use patterns in skip counting by multiples of 2 (to at least 50), and multiples of 5, 10, and 25 (to at least 200) to justify the next number in the counting sequence.

Represent forward counting patterns when counting by groups of 100 up to at least 1,000 starting at 0 using a variety of tools (e.g., objects, number lines, calculators, one thousand charts).

Represent backward counting patterns when counting by groups of 10 from 200 or less using a variety of tools including objects, number lines, calculators, and hundreds charts.

Describe and use patterns in skip counting backwards by 10s (from at least 200) to justify the next number in the counting sequence.

Choose a reasonable estimate up to 1,000 when given a contextual problem (e.g., What would be the best estimate for the number of students in our school – 5, 50, or 500?).

Represent even numbers (up to 50) with concrete objects, using two equal groups or two equal addends.

Represent odd numbers (up to 50) with concrete objects, using two equal groups with one leftover or two equal addends plus 1.

Determine whether a number (up to 50) is even or odd using concrete objects and justify reasoning (e.g., dividing collections of objects into two equal groups, pairing objects).

The student will demonstrate an understanding of the ten-to-one relationships of the base 10 number system to represent, compare, and order whole numbers up to 999.

Write the three-digit whole number represented by a given model (e.g., concrete objects, pictures of base 10 blocks).

Read, write, and represent three-digit numbers in standard form, expanded form, and word form, using concrete or pictorial representations.

Apply patterns within the base 10 system to determine and communicate, orally and in written form, the place (ones, tens, hundreds) and value of each digit in a three-digit whole number (e.g., in 352, the 5 represents 5 tens and its value is 50).

Investigate and explain the ten-to-one relationships among ones, tens, and hundreds, using models.

Compose and decompose whole numbers up to 200 by making connections between a variety of models (e.g., base 10 blocks, place value cards, presented orally, in expanded or standard form) and counting strategies (e.g., 156 can be 1 hundred, 5 tens, 6 ones; 1 hundred, 4 tens, 16 ones; 15 tens, 6 ones).

Plot and justify the position of a given number up to 100 on a number line with pre-marked benchmarks of 1s, 2s, 5s, 10s, or 25s.

Compare two whole numbers, each 999 or less, represented concretely, pictorially, or symbolically, using words (greater than, less than, or equal to) and symbols (>, <, or =). Justify reasoning orally, in writing, or with a model.

Order up to three whole numbers, each 999 or less, represented concretely, pictorially, or symbolically from least to greatest and greatest to least.

The student will use mathematical reasoning and justification to solve contextual problems that involve partitioning models into equal-sized parts (halves, fourths, eighths, thirds, and sixths).

Model and describe fractions as representing equal-size parts of a whole.

Describe the relationship between the number of fractional parts needed to make a whole and the size of the parts (i.e., as the whole is divided into more parts, each part becomes smaller).

Compose the whole for a given fractional part and its value (in context) for halves, fourths, eighths, thirds, and sixths (e.g., when given 1 4 , determine how many pieces would be needed to make 4 4 ).

Using same-size fraction pieces, from a region/area model, count by unit fractions up to two wholes (e.g., zero one-fourths, one one-fourth, two one-fourths, three one-fourths, four onefourths, five one-fourths; or zero-fourths, one-fourth, two-fourths, three-fourths, four-fourths, five-fourths).

Given a context, represent, name, and write fractional parts of a whole for halves, fourths, eighths, thirds, and sixths using: region/area models (e.g., pie pieces, pattern blocks, geoboards);

Given a context, represent, name, and write fractional parts of a whole for halves, fourths, eighths, thirds, and sixths using: length models (e.g., paper fraction strips, fraction bars, rods, number lines); and

Given a context, represent, name, and write fractional parts of a whole for halves, fourths, eighths, thirds, and sixths using: set models (e.g., chips, counters, cubes).

Compare unit fractions for halves, fourths, eighths, thirds, and sixths using words (greater than, less than or equal to) and symbols (>, <, =), with region/area and length models.

The student will solve problems that involve counting and representing money amounts up to $2.00.

Identify a quarter and its value and determine multiple ways to represent the value of a quarter using pennies, nickels, and/or dimes.

Count by ones, fives, tens, and twenty-fives to determine the value of a collection of mixed coins and one-dollar bills whose total value is $2.00 or less.

Construct a set of coins and/or bills to total a given amount of money whose value is $2.00 or less.

Represent the value of a collection of coins and one-dollar bills (limited to $2.00 or less) using the cent (¢) and dollar ($) symbols and decimal point (.).

The student will describe, extend, create, and transfer repeating and increasing patterns (limited to addition of whole numbers) using various representations.

Identify and describe repeating and increasing patterns.

Analyze a repeating or increasing pattern and generalize the change to extend the pattern using objects, pictures, and numbers.

Create a repeating or increasing pattern using various representations (e.g., objects, pictures, numbers).

Transfer a given repeating or increasing pattern from one form to another (e.g., objects, pictures, numbers) and explain the connection between the two patterns.

The student will apply the data cycle (pose questions; collect or acquire data; organize and represent data; and analyze data and communicate results) with a focus on pictographs and bar graphs.

Pose questions, given a predetermined context, that require the collection of data (limited to 25 or fewer data points for no more than six categories).

Determine the data needed to answer a posed question and collect the data using various methods (e.g., voting; creating lists, tables, or charts; tallying).

Organize and represent a data set using a pictograph where each symbol represents up to 2 data points. Determine and use a key to assist in the analysis of the data.

Organize and represent a data set using a bar graph with a title and labeled axes (limited to 25 or fewer data points for up to six categories, and limit increments of scale to multiples of 1 or 2).

Analyze data represented in pictographs and bar graphs and communicate results: ask and answer questions about the data represented in pictographs and bar graphs (e.g., total number of data points represented, how many in each category, how many more or less are in one category than another). Pictograph keys will be limited to symbols representing 1, 2, 5, or 10 pieces of data and bar graphs will be limited to scales with increments in multiples of 1, 2, 5, or 10; and

Analyze data represented in pictographs and bar graphs and communicate results: draw conclusions about the data and make predictions based on the data.

The student will estimate, represent, solve, and justify solutions to single-step and multistep problems, including those in context, using addition and subtraction with whole numbers where addends and minuends do not exceed 1,000.

Determine and justify whether an estimate or an exact answer is appropriate when solving single-step and multistep contextual problems involving addition and subtraction, where addends and minuends do not exceed 1,000.

Apply strategies (e.g., rounding to the nearest 10 or 100, using compatible numbers, using other number relationships) to estimate a solution for single-step or multistep addition or subtraction problems, including those in context, where addends or minuends do not exceed 1,000.

Apply strategies (e.g., place value, properties of addition, other number relationships) and algorithms, including the standard algorithm, to determine the sum or difference of two whole numbers where addends and minuends do not exceed 1,000.

Identify and use the appropriate symbol to distinguish between expressions that are equal and expressions that are not equal (e.g., 256 - 13 = 220 + 23; 457 + 100 ≠ 557 + 100).

Represent, solve, and justify solutions to single-step and multistep contextual problems involving addition and subtraction with whole numbers where addends and minuends do not exceed 1,000.

The student will recall with automaticity multiplication and division facts through 10 × 10; and represent, solve, and justify solutions to single-step contextual problems using multiplication and division with whole numbers.

Represent multiplication and division of whole numbers through 10 × 10, including in a contextual situation, using a variety of approaches and models (e.g., repeated addition/subtraction, equal-sized groups/sharing, arrays, equal jumps on a number line, using multiples to skip count).

Use inverse relationships to write the related facts connected to a given model for multiplication and division of whole numbers through 10 × 10.

Apply strategies (e.g., place value, the properties of multiplication and/or addition) when multiplying and dividing whole numbers.

Demonstrate fluency with multiplication facts through 10 × 10 by applying reasoning strategies (e.g., doubling, add-a-group, subtract-a-group, near squares, and inverse relationships).

Represent, solve, and justify solutions to single-step contextual problems that involve multiplication and division of whole numbers through 10 × 10.

Recall with automaticity the multiplication facts through 10 × 10 and the corresponding division facts.

Create an equation to represent the mathematical relationship between equivalent expressions using multiplication and/or division facts through 10 × 10 (e.g., 4 × 3 = 14 - 2, 35 ÷ 5 = 1 × 7).

The student will reason mathematically using standard units (U.S. Customary and metric) with appropriate tools to estimate and measure objects by length, weight/mass, and liquid volume to the nearest half or whole unit.

Justify whether an estimate or an exact measurement is needed for a contextual situation and choose an appropriate unit.

Estimate and measure: length of an object to the nearest U.S. Customary unit (1 2 inch, inch, foot, yard) and metric unit (centimeter, meter);

Estimate and measure: weight/mass of an object to the nearest U.S. Customary unit (pound) and metric unit (kilogram); and

Estimate and measure: liquid volume to the nearest U.S. Customary unit (cup, pint, quart, gallon) and metric unit (liter).

Compare estimates of length, weight/mass, or liquid volume with the actual measurements.

The student will use multiple representations to estimate and solve problems, including those in context, involving area and perimeter (in both U.S. Customary and metric units).

Solve problems, including those in context, involving area: describe and give examples of area as a measurement in contextual situations; and

Solve problems, including those in context, involving area: estimate and determine the area of a given surface by counting the number of square units, describe the measurement (using the number and unit) and justify the measurement.

Solve problems, including those in context, involving perimeter: describe and give examples of perimeter as a measurement in contextual situations;

Solve problems, including those in context, involving perimeter: estimate and measure the distance around a polygon (with no more than six sides) to determine the perimeter and justify the measurement; and

Solve problems, including those in context, involving perimeter: given the lengths of all sides of a polygon (with no more than six sides), determine its perimeter and justify the measurement.

The student will demonstrate an understanding of the concept of time to the nearest minute and solve single-step contextual problems involving elapsed time in one-hour increments within a 12-hour period.

Tell and write time to the nearest minute, using analog and digital clocks.

Match a written time (e.g., 4:38, 7:09, 12:51) to the time shown on analog and digital clocks to the nearest minute.

Solve single-step contextual problems involving elapsed time in one-hour increments, within a 12-hour period (within a.m. or within p.m.) when given: the starting time and the ending time, determine the amount of time that has elapsed;

Solve single-step contextual problems involving elapsed time in one-hour increments, within a 12-hour period (within a.m. or within p.m.) when given: the starting time and amount of elapsed time in one-hour increments, determine the ending time; or

Solve single-step contextual problems involving elapsed time in one-hour increments, within a 12-hour period (within a.m. or within p.m.) when given: the ending time and the amount of elapsed time in one-hour increments, determine the starting time.

The student will identify, describe, classify, compare, combine, and subdivide polygons.

Describe a polygon as a closed plane figure composed of at least three line segments that do not cross.

Classify figures as polygons or not polygons and justify reasoning.

Identify and describe triangles, quadrilaterals, pentagons, hexagons, and octagons in various orientations, with and without contexts.

Identify and name examples of polygons (triangles, quadrilaterals, pentagons, hexagons, octagons) in the environment.

Classify and compare polygons (triangles, quadrilaterals, pentagons, hexagons, octagons).

Combine no more than three polygons, where each has three or four sides, and name the resulting polygon (triangles, quadrilaterals, pentagons, hexagons, octagons).

Subdivide a three-sided or four-sided polygon into no more than three parts and name the resulting polygons.

The student will use place value understanding to read, write, and determine the place and value of each digit in a whole number, up to six digits, with and without models.

Read and write six-digit whole numbers in standard form, expanded form, and word form.

Apply patterns within the base 10 system to determine and communicate, orally and in written form, the place and value of each digit in a six-digit whole number (e.g., in 165,724, the 5 represents 5 thousands and its value is 5,000).

Compose, decompose, and represent numbers up to 9,999 in multiple ways, according to place value (e.g., 256 can be 1 hundred, 14 tens, 16 ones, but also 25 tens, 6 ones), with and without models.

The student will demonstrate an understanding of the base 10 system to compare and order whole numbers up to 9,999.

Compare two whole numbers, each 9,999 or less, using symbols (>, <, =, ≠) and/or words (greater than, less than, equal to, not equal to), with and without models.

Order up to three whole numbers, each 9,999 or less, represented with and without models, from least to greatest and greatest to least.

The student will use mathematical reasoning and justification to represent and compare fractions (proper and improper) and mixed numbers with denominators of 2, 3, 4, 5, 6, 8, and 10), including those in context.

Represent, name, and write a given fraction (proper or improper) or mixed number with denominators of 2, 3, 4, 5, 6, 8, and 10 using: region/area models (e.g., pie pieces, pattern blocks, geoboards);

Represent, name, and write a given fraction (proper or improper) or mixed number with denominators of 2, 3, 4, 5, 6, 8, and 10 using: length models (e.g., paper fraction strips, fraction bars, rods, number lines); and

Represent, name, and write a given fraction (proper or improper) or mixed number with denominators of 2, 3, 4, 5, 6, 8, and 10 using: set models (e.g., chips, counters, cubes).

Identify a fraction represented by a model as the sum of unit fractions.

Use a model of a fraction greater than one to count the fractional parts to name and write it as an improper fraction and as a mixed number (e.g., 1 4 , 2 4 , 3 4 , 4 4 , 5 4 = 1 1 4 ).

Compose and decompose fractions (proper and improper) with denominators of 2, 3, 4, 5, 6, 8, and 10 in multiple ways (e.g., 7 4 = 4 4 + 3 4 or 4 6 = 3 6 + 1 6 = 2 6 + 2 6 ) with models.

Compare a fraction, less than or equal to one, to the benchmarks of 0, 1 2 , and 1 using area/region models, length models, and without models.

Compare two fractions (proper or improper) and/or mixed numbers with like numerators of 2, 3, 4, 5, 6, 8, and 10 (e.g., 2 3 > 2 8 ) using words (greater than, less than, equal to) and/or symbols (>, <, =), using area/region models, length models, and without models.

Compare two fractions (proper or improper) and/or mixed numbers with like denominators of 2, 3, 4, 5, 6, 8, and 10 (e.g., 3 6 < 4 6 ) using words (greater than, less than, equal to) and/or symbols (>, <, =), using area/region models, length models, and without models.

Represent equivalent fractions with denominators of 2, 3, 4, 5, 6, 8, or 10, using region/area models and length models.

The student will solve problems, including those in context, that involve counting, comparing, representing, and making change for money amounts up to $5.00.

Determine the value of a collection of bills and coins whose total is $5.00 or less.

Construct a set of bills and coins to total a given amount of money whose value is $5.00 or less.

Compare the values of two sets of coins or two sets of bills and coins, up to $5.00, with words (greater than, less than, equal to) and/or symbols (>, <, =) using concrete or pictorial models.

Solve contextual problems to make change from $5.00 or less by using counting on or counting back strategies with concrete or pictorial models.

The student will identify, describe, extend, and create increasing and decreasing patterns (limited to addition and subtraction of whole numbers), including those in context, using various representations.

Identify and describe increasing and decreasing patterns using various representations (e.g., objects, pictures, numbers, number lines).

Analyze an increasing or decreasing pattern and generalize the change to extend the pattern or identify missing terms using various representations.

Solve contextual problems that involve identifying, describing, and extending patterns.

Create increasing and decreasing patterns using objects, pictures, numbers, and number lines.

Investigate and explain the connection between two different representations of the same increasing or decreasing pattern.

The student will apply the data cycle (formulate questions; collect or acquire data; organize and represent data; and analyze data and communicate results) with a focus on pictographs and bar graphs.

Formulate questions that require the collection or acquisition of data.

Determine the data needed to answer a formulated question and collect or acquire existing data (limited to 30 or fewer data points for no more than eight categories) using various methods (e.g., polls, observations, tallies).

Organize and represent a data set using pictographs that include an appropriate title, labeled axes, and key. Each pictograph symbol should represent 1, 2, 5 or 10 data points.

Organize and represent a data set using bar graphs with a title and labeled axes, with and without the use of technology tools. Determine and use an appropriate scale (increments limited to multiples of 1, 2, 5 or 10).

Analyze data represented in pictographs and bar graphs, and communicate results orally and in writing: describe the categories of data and the data as a whole (e.g., data were collected on preferred ways to cook or prepare eggs - scrambled, fried, hard boiled, and egg salad);

Analyze data represented in pictographs and bar graphs, and communicate results orally and in writing: identify parts of the data that have special characteristics, including categories with the greatest, the least, or the same (e.g., most students prefer scrambled eggs);

Analyze data represented in pictographs and bar graphs, and communicate results orally and in writing: make inferences about data represented in pictographs and bar graphs;

Analyze data represented in pictographs and bar graphs, and communicate results orally and in writing: use characteristics of the data to draw conclusions about the data and make predictions based on the data (e.g., it is unlikely that a third grader would like hard boiled eggs); and

Analyze data represented in pictographs and bar graphs, and communicate results orally and in writing: solve one- and two-step addition and subtraction problems using data from pictographs and bar graphs.

The student will estimate, represent, solve, and justify solutions to single-step and multistep problems, including those in context, using addition and subtraction with whole numbers.

Determine and justify whether an estimate or an exact answer is appropriate when solving contextual problems involving addition and subtraction with whole numbers. Refine estimates by adjusting the final amount, using terms such as closer to, between, and a little more than.

Apply strategies (e.g., rounding to the nearest 100 or 1,000, using compatible numbers, other number relationships) to estimate a solution for single-step or multistep addition or subtraction problems with whole numbers, where addends or minuends do not exceed 10,000.*

Apply strategies (e.g., place value, properties of addition, other number relationships) and algorithms, including the standard algorithm, to determine the sum or difference of two whole numbers, where addends and minuends do not exceed 10,000.*

Estimate, represent, solve, and justify solutions to single-step and multistep contextual problems involving addition and subtraction with whole numbers where addends and minuends do not exceed 1,000,000.

The student will estimate, represent, solve, and justify solutions to single-step and multistep problems, including those in context, using multiplication with whole numbers, and single-step problems, including those in context, using division with whole numbers; and recall with automaticity the multiplication facts through 12 × 12 and the corresponding division facts.

Determine and justify whether an estimate or an exact answer is appropriate when solving contextual problems involving multiplication and division of whole numbers. Refine estimates by adjusting the final amount, using terms such as closer to, between, and a little more than.

Recall with automaticity the multiplication facts through 12 × 12 and the corresponding division facts.*

Create an equation using addition, subtraction, multiplication, and division to represent the relationship between equivalent mathematical expressions (e.g., 4 × 3 = 2 × 6; 10 + 8 = 36 ÷ 2; 12 × 4 = 60 − 12).

Identify and use the appropriate symbol to distinguish between expressions that are equal and expressions that are not equal, using addition, subtraction, multiplication, and division (e.g., 4 × 12 = 8 × 6 and 64 ÷ 8 ≠ 8 × 8).

Determine all factor pairs for a whole number 1 to 100, using concrete, pictorial, and numerical representations.

Determine common factors and the greatest common factor of no more than three numbers.

Apply strategies (e.g., rounding, place value, properties of multiplication and/or addition) and algorithms, including the standard algorithm, to estimate and determine the product of two whole numbers when given: a two-digit factor and a one-digit factor;*

Apply strategies (e.g., rounding, place value, properties of multiplication and/or addition) and algorithms, including the standard algorithm, to estimate and determine the product of two whole numbers when given: a three-digit factor and a one-digit factor;* or

Apply strategies (e.g., rounding, place value, properties of multiplication and/or addition) and algorithms, including the standard algorithm, to estimate and determine the product of two whole numbers when given: a two-digit factor and a two-digit factor.*

Estimate, represent, solve, and justify solutions to single-step and multistep contextual problems that involve multiplication with whole numbers.

Apply strategies (e.g., rounding, compatible numbers, place value) and algorithms, including the standard algorithm, to estimate and determine the quotient of two whole numbers, given a one-digit divisor and a two- or three-digit dividend, with and without remainders.*

Estimate, represent, solve, and justify solutions to single-step contextual problems involving division with whole numbers.

Interpret the quotient and remainder when solving a contextual problem.

The student will estimate, represent, solve, and justify solutions to single-step problems, including those in context, using addition and subtraction of fractions (proper, improper, and mixed numbers with like denominators of 2, 3, 4, 5, 6, 8, 10, and 12), with and without models; and solve single-step contextual problems involving multiplication of a whole number (12 or less) and a unit fraction, with models.

Estimate and determine the sum or difference of two fractions (proper or improper) and/or mixed numbers, having like denominators limited to 2, 3, 4, 5, 6, 8, 10, and 12 (e.g., 3 8 + 3 8 , 21 5 + 4 5 , 7 4 - 5 4 ) and simplify the resulting fraction. Addition and subtraction with fractions may include regrouping.*

Estimate, represent, solve, and justify solutions to single-step contextual problems using addition and subtraction with fractions (proper or improper) and/or mixed numbers, having like denominators limited to 2, 3, 4, 5, 6, 8, 10, and 12, and simplify the resulting fraction. Addition and subtraction with fractions may include regrouping.

Solve single-step contextual problems involving multiplication of a whole number, limited to 12 or less, and a unit fraction (e.g., 6 × 1 3 , 1 5 × 8, 2 × 1 10 ), with models.*

Apply the inverse property of multiplication in models (e.g., use a visual fraction model to represent 4 4 or 1 as the product of 4 × 1 4 ).

The student will estimate, represent, solve, and justify solutions to single-step and multistep problems, including those in context, using addition and subtraction of decimals through the thousandths, with and without models.

Apply strategies (e.g., rounding to the nearest whole number, using compatible numbers) and algorithms, including the standard algorithm, to estimate and determine the sum or difference of two decimals through the thousandths, with and without models, in which:* decimals do not exceed the thousandths; and

Apply strategies (e.g., rounding to the nearest whole number, using compatible numbers) and algorithms, including the standard algorithm, to estimate and determine the sum or difference of two decimals through the thousandths, with and without models, in which:* addends, subtrahends, and minuends are limited to four digits.

Estimate, represent, solve, and justify solutions to single-step and multistep contextual problems using addition and subtraction of decimals through the thousandths.

The student will reason mathematically to solve problems, including those in context, that involve length, weight/mass, and liquid volume using U.S. Customary and metric units.

Determine an appropriate unit of measure to use when measuring: length in both U.S. Customary (inch, foot, yard, mile) and metric units (millimeter, centimeter, meter);

Determine an appropriate unit of measure to use when measuring: weight/mass in both U.S. Customary (ounce, pound) and metric units (gram, kilogram); and

Determine an appropriate unit of measure to use when measuring: liquid volume in both U.S. Customary (cup, pint, quart, gallon) and metric units (milliliter, liter).

Estimate and measure: length of an object to the nearest U.S. Customary unit (1 2 inch, 1 4 inch, 1 8 inch, foot, yard) and nearest metric unit (millimeter, centimeter, or meter);

Estimate and measure: weight/mass of an object to the nearest U.S. Customary unit (ounce, pound) and nearest metric unit (gram, kilogram); and

Estimate and measure: liquid volume to the nearest U.S. Customary unit (cup, pint, quart, gallon) and nearest metric unit (milliliter, liter).

Compare estimates of length, weight/mass, or liquid volume with the actual measurements.

Given the equivalent measure of one unit, solve problems, including those in context, by determining the equivalent measures within the U.S. Customary system for: length (inches and feet, feet and yards, inches and yards);

Given the equivalent measure of one unit, solve problems, including those in context, by determining the equivalent measures within the U.S. Customary system for: weight/mass (ounces and pounds); and

Given the equivalent measure of one unit, solve problems, including those in context, by determining the equivalent measures within the U.S. Customary system for: liquid volume (cups, pints, quarts, and gallons).

The student will solve single-step and multistep contextual problems involving elapsed time (limited to hours and minutes within a 12-hour period).

Solve single-step and multistep contextual problems involving elapsed time in hours and minutes, within a 12-hour period (within a.m., within p.m., and across a.m. and p.m.) when given: the starting time and the ending time, determine the amount of time that has elapsed in hours and minutes;

Solve single-step and multistep contextual problems involving elapsed time in hours and minutes, within a 12-hour period (within a.m., within p.m., and across a.m. and p.m.) when given: the starting time and amount of elapsed time in hours and minutes, determine the ending time; or

Solve single-step and multistep contextual problems involving elapsed time in hours and minutes, within a 12-hour period (within a.m., within p.m., and across a.m. and p.m.) when given: the ending time and the amount of elapsed time in hours and minutes, determine the starting time.

The student will use multiple representations to develop and use formulas to solve problems, including those in context, involving area and perimeter limited to rectangles and squares (in both U.S. Customary and metric units).

Use concrete materials and pictorial models to develop a formula for the area and perimeter of a rectangle (including a square).

Determine the area and perimeter of a rectangle when given the measure of two adjacent sides (in whole number units), with and without models.

Determine the area and perimeter of a square when given the measure of one side (in whole number units), with and without models.

Use concrete materials and pictorial models to explore the relationship between area and perimeter of rectangles.

Identify and represent rectangles with the same perimeter and different areas or with the same area and different perimeters.

Solve contextual problems involving area and perimeter of rectangles and squares.

The student will identify, describe, and draw points, rays, line segments, angles, and lines, including intersecting, parallel, and perpendicular lines.

Identify and describe points, lines, line segments, rays, and angles, including endpoints and vertices.

Describe endpoints and vertices in relation to lines, line segments, rays, and angles.

Draw representations of points, line segments, rays, angles, and lines, using a ruler or straightedge.

Identify parallel, perpendicular, and intersecting lines and line segments in plane and solid figures, including those in context.