VA Math Preview - Smart Board Learning

Virginia Math Preview

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1st grade Math preview

Number and Number Sense

1.NS.1 The student will utilize flexible counting strategies to determine and describe quantities up to 120.

Students will demonstrate the following Knowledge and Skills:

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

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

c) 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).

d) 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).

e) 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).

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

g) 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.

1.NS.2 The student will represent, compare, and order quantities up to 120.

Students will demonstrate the following Knowledge and Skills:

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

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

c) 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).

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

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

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

1.NS.3 The student will use mathematical reasoning and justification to solve contextual problems that involve partitioning models into two and four equal-sized parts.

Students will demonstrate the following Knowledge and Skills:

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

b) 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.

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

Computation and Estimation

1.CE.1 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.

Students will demonstrate the following Knowledge and Skills:

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

b) 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).

c) Recall with automaticity addition and subtraction facts within 10.

d) 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).

e) 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).

f) 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.

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

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

i) 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).

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

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

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

Measurement and Geometry

1.MG.1 The student will reason mathematically using nonstandard units to measure and compare objects by length, weight, and volume.

Students will demonstrate the following Knowledge and Skills:

a) Use nonstandard units to measure the:

i) 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;

ii) 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

iii) volumes of two containers and compare the measurements using the terms more, less, or the same as.

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

1.MG.2 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.

Students will demonstrate the following Knowledge and Skills:

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

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

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

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

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

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

1.MG.3 The student will demonstrate an understanding of the concept of passage of time (to the nearest hour and half-hour) and the calendar.

Students will demonstrate the following Knowledge and Skills:

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

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

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

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

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

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

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

h) 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?).

i) 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?).

Probability and Statistics

1.PS.1 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.

Students will demonstrate the following Knowledge and Skills:

a) 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).

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

c) 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).

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

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

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

g) Analyze data represented in object graphs, picture graphs, and tables and communicate results:

i) 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

ii) draw conclusions about the data and make predictions based on the data.

Patterns, Functions, and Algebra

1.PFA.1 The student will identify, describe, extend, create, and transfer repeating patterns and increasing patterns using various representations.

Students will demonstrate the following Knowledge and Skills:

a) Identify and describe repeating and increasing patterns.

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

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

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

2nd grade Math preview

Number and Number Sense

2.NS.1 The student will utilize flexible counting strategies to determine and describe quantities up to 200.

Students will demonstrate the following Knowledge and Skills:

a) 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).

b) 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).

c) 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.

d) 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).

e) 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.

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

g) 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?).

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

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

j) 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).

2.NS.2 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.

Students will demonstrate the following Knowledge and Skills:

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

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

c) 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).

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

e) 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).

f) 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.

g) 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.

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

2.NS.3 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).

Students will demonstrate the following Knowledge and Skills:

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

b) 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).

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

d) 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 one-fourths, five one-fourths; or zero-fourths, one-fourth, two-fourths, three-fourths, four-fourths, five-fourths).

e) Given a context, represent, name, and write fractional parts of a whole for halves, fourths, eighths, thirds, and sixths using:

i) region/area models (e.g., pie pieces, pattern blocks, geoboards);

ii) length models (e.g., paper fraction strips, fraction bars, rods, number lines); and

iii) set models (e.g., chips, counters, cubes).

f) 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.

2.NS.4 The student will solve problems that involve counting and representing money amounts up to $2.00.

Students will demonstrate the following Knowledge and Skills:

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

b) 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.

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

d) 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 (.).

Computation and Estimation

2.CE.1 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.

Students will demonstrate the following Knowledge and Skills:

a) 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.

b) 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.

c) 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.

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

e) Recall with automaticity addition and subtraction facts within 20.

f) 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).

g) 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 = ).

h) 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).

i) 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.

j) 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).

Measurement and Geometry

2.MG.1 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.

Students will demonstrate the following Knowledge and Skills:

a) Explain the purpose of various measurement tools and how to use them appropriately by:

i) identifying a ruler as an instrument to measure length;

ii) identifying different types of scales as instruments to measure weight; and

iii) identifying different types of measuring cups as instruments to measure liquid volume.

b) Use U.S. Customary units to estimate, measure, and compare the two for reasonableness:

i) the length of an object to the nearest inch, using a ruler;

ii) the weight of an object to the nearest pound, using a scale; and

iii) the liquid volume of a container to the nearest cup, using a measuring cup.

2.MG.2 The student will demonstrate an understanding of the concept of time to the nearest five minutes, using analog and digital clocks.

Students will demonstrate the following Knowledge and Skills:

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

b) 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?).

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

d) 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.

2.MG.3 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.

Students will demonstrate the following Knowledge and Skills:

a) 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.

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

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

2.MG.4 The student will describe, name, compare, and contrast plane and solid figures (circles/spheres, squares/cubes, and rectangles/rectangular prisms).

Students will demonstrate the following Knowledge and Skills:

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

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

c) 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).

d) 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).

Probability and Statistics

2.PS.1 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.

Students will demonstrate the following Knowledge and Skills:

a) 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).

b) 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).

c) 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.

d) 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).

e) Analyze data represented in pictographs and bar graphs and communicate results:

i) 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

ii) draw conclusions about the data and make predictions based on the data.

Patterns, Functions, and Algebra

2.PFA.1 The student will describe, extend, create, and transfer repeating and increasing patterns (limited to addition of whole numbers) using various representations.

Students will demonstrate the following Knowledge and Skills:

a) Identify and describe repeating and increasing patterns.

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

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

d) 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.

3rd grade Math preview

Number and Number Sense

3.NS.1 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.

Students will demonstrate the following Knowledge and Skills:

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

b) 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).

c) 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.

3.NS.2 The student will demonstrate an understanding of the base 10 system to compare and order whole numbers up to 9,999.

Students will demonstrate the following Knowledge and Skills:

a) 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.

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

3.NS.3 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.

Students will demonstrate the following Knowledge and Skills:

a) 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:

i) region/area models (e.g., pie pieces, pattern blocks, geoboards);

ii) length models (e.g., paper fraction strips, fraction bars, rods, number lines); and

iii) set models (e.g., chips, counters, cubes).

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

c) 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 ).

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

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

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

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

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

3.NS.4 The student will solve problems, including those in context, that involve counting, comparing, representing, and making change for money amounts up to $5.00.

Students will demonstrate the following Knowledge and Skills:

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

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

c) 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.

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

Computation and Estimation

3.CE.1 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.

Students will demonstrate the following Knowledge and Skills:

a) 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.

b) 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.

c) 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.

d) 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).

e) 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.

3.CE.2 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.

Students will demonstrate the following Knowledge and Skills:

a) 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).

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

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

d) 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).

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

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

g) 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).

Measurement and Geometry

3.MG.1 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.

Students will demonstrate the following Knowledge and Skills:

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

b) Estimate and measure:

i) length of an object to the nearest U.S. Customary unit ( inch, inch, foot, yard) and metric unit (centimeter, meter);

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

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

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

3.MG.2 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).

Students will demonstrate the following Knowledge and Skills:

a) Solve problems, including those in context, involving area:

i) describe and give examples of area as a measurement in contextual situations; and

ii) 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.

b) Solve problems, including those in context, involving perimeter:

i) describe and give examples of perimeter as a measurement in contextual situations;

ii) estimate and measure the distance around a polygon (with no more than six sides) to determine the perimeter and justify the measurement; and

iii) given the lengths of all sides of a polygon (with no more than six sides), determine its perimeter and justify the measurement.

3.MG.3 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.

Students will demonstrate the following Knowledge and Skills:

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

b) 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.

c) 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:

i) the starting time and the ending time, determine the amount of time that has elapsed;

ii) the starting time and amount of elapsed time in one-hour increments, determine the ending time; or

iii) the ending time and the amount of elapsed time in one-hour increments, determine the starting time.

3.MG.4 The student will identify, describe, classify, compare, combine, and subdivide polygons.

Students will demonstrate the following Knowledge and Skills:

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

b) Classify figures as polygons or not polygons and justify reasoning.

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

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

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

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

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

Probability and Statistics

3.PS.1 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.

Students will demonstrate the following Knowledge and Skills:

a) Formulate questions that require the collection or acquisition of data.

b) 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).

c) 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.

d) 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).

e) Analyze data represented in pictographs and bar graphs, and communicate results orally and in writing:

i) 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);

ii) 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);

iii) make inferences about data represented in pictographs and bar graphs;

iv) 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

v) solve one- and two-step addition and subtraction problems using data from pictographs and bar graphs.

Patterns, Functions, and Algebra

3.PFA.1 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.

Students will demonstrate the following Knowledge and Skills:

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

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

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

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

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

4th grade Math preview

Number and Number Sense

4.NS.1 The student will use place value understanding to read, write, and identify the place and value of each digit in a nine-digit whole number.

Students will demonstrate the following Knowledge and Skills:

a) Read nine-digit whole numbers, presented in standard form, and represent the same number in written form.

b) Write nine-digit whole numbers in standard form when the numbers are presented orally or in written form.

c) 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 nine-digit whole number (e.g., in 568,165,724, the 8 represents 8 millions and its value is 8,000,000).

4.NS.2 The student will demonstrate an understanding of the base 10 system to compare and order whole numbers up to seven digits.

Students will demonstrate the following Knowledge and Skills:

a) Compare two whole numbers up to seven digits each, using words (greater than, less than, equal to, not equal to) and/or using symbols (>, <, =, ≠).

b) Order up to four whole numbers up to seven digits each, from least to greatest or greatest to least.

4.NS.3 The student will use mathematical reasoning and justification to represent, compare, and order fractions (proper, improper, and mixed numbers with denominators 12 or less), with and without models.

Students will demonstrate the following Knowledge and Skills:

a) Compare and order no more than four fractions (proper or improper), and/or mixed numbers, with like denominators by comparing the number of parts (numerators) using fractions with denominators of 12 or less (e.g., < ). Justify comparisons orally, in writing, or with a model.*

b) Compare and order no more than four fractions (proper or improper), and/or mixed numbers, with like numerators and unlike denominators by comparing the size of the parts using fractions with denominators of 12 or less (e.g., < ). Justify comparisons orally, in writing, or with a model.*

c) Use benchmarks (e.g., 0, , or 1) to compare and order no more than four fractions (proper or improper), and/or mixed numbers, with like and unlike denominators of 12 or less. Justify comparisons orally, in writing, or with a model.*

d) Compare two fractions (proper or improper) and/or mixed numbers using fractions with denominators of 12 or less, using the symbols >, <, and = (e.g., > ). Justify comparisons orally, in writing, or with a model.*

e) Represent equivalent fractions with denominators of 12 or less, with and without models.*

f) Compose and decompose fractions (proper and improper) and/or mixed numbers with denominators of 12 or less, in multiple ways, with and without models.*

g) Represent the division of two whole numbers as a fraction given a contextual situation and a model (e.g., means the same as 3 divided by 5 or represents the amount of muffin each of five children will receive when sharing three muffins equally).

* On the state assessment, items measuring this objective are assessed without the use of a calculator.

4.NS.4 The student will use mathematical reasoning and justification to represent, compare, and order decimals through thousandths, with and without models.

Students will demonstrate the following Knowledge and Skills:

a) Investigate and describe the ten-to-one place value relationship for decimals through thousandths, using concrete models (e.g., place value mats/charts, decimal squares, base 10 blocks).

b) Represent and identify decimals expressed through thousandths, using concrete, pictorial, and numerical representations.

c) Read and write decimals expressed through thousandths, using concrete, pictorial, and numerical representations.

d) Identify and communicate, both orally and in written form, the place and value of each digit in a decimal through thousandths (e.g., given 0.385, the 8 is in the hundredths place and has a value of 0.08).

e) Compare using symbols (<, >, =) and/or words (greater than, less than, equal to) and order (least to greatest and greatest to least), a set of no more than four decimals expressed through thousandths, using multiple strategies (e.g., benchmarks, place value, number lines). Justify comparisons with a model, orally, and in writing.

4.NS.5 The student will reason about the relationship between fractions and decimals (limited to halves, fourths, fifths, tenths, and hundredths) to identify and represent equivalencies.

Students will demonstrate the following Knowledge and Skills:

a) Represent fractions (proper or improper) and/or mixed numbers as decimals through hundredths, using multiple representations, limited to halves, fourths, fifths, tenths, and hundredths.*

b) Identify and model equivalent relationships between fractions (proper or improper) and/or mixed numbers and decimals, using halves, fourths, fifths, tenths, and hundredths.*

c) Write the decimal and fraction equivalent for a given model (e.g., = 0.25 or 0.25 = ; 1.25 = or 1 ; 1.02 = or 1).*

* On the state assessment, items measuring this objective are assessed without the use of a calculator.

Computation and Estimation

4.CE.1 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.

Students will demonstrate the following Knowledge and Skills:

a) 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.

b) 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.*

d) 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.

*On the state assessment, items measuring this objective are assessed without the use of a calculator.

4.CE.2 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.

Students will demonstrate the following Knowledge and Skills:

a) 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.

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

c) 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).

d) 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).

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

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

g) 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:

i) a two-digit factor and a one-digit factor;*

ii) a three-digit factor and a one-digit factor;* or

iii) a two-digit factor and a two-digit factor.*

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

i) 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.*

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

k) Interpret the quotient and remainder when solving a contextual problem.

* On the state assessment, items measuring this objective are assessed without the use of a calculator.

4.CE.3 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.

Students will demonstrate the following Knowledge and Skills:

a) 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., + , 2 + , – ) and simplify the resulting fraction. Addition and subtraction with fractions may include regrouping.*

b) 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.

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

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

* On the state assessment, items measuring this objective are assessed without the use of a calculator.

4.CE.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.

Students will demonstrate the following Knowledge and Skills:

a) 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:*

i) decimals do not exceed the thousandths; and

ii) addends, subtrahends, and minuends are limited to four digits.

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

* On the state assessment, items measuring this objective are assessed without the use of a calculator.

Measurement and Geometry

4.MG.1 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.

Students will demonstrate the following Knowledge and Skills:

a) Determine an appropriate unit of measure to use when measuring:

i) length in both U.S. Customary (inch, foot, yard, mile) and metric units (millimeter, centimeter, meter);

ii) weight/mass in both U.S. Customary (ounce, pound) and metric units (gram, kilogram); and

iii) liquid volume in both U.S. Customary (cup, pint, quart, gallon) and metric units (milliliter, liter).

b) Estimate and measure:

i) length of an object to the nearest U.S. Customary unit ( inch, inch, inch, foot, yard) and nearest metric unit (millimeter, centimeter, or meter);

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

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

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

d) 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:

i) length (inches and feet, feet and yards, inches and yards);

ii) weight/mass (ounces and pounds); and

iii) liquid volume (cups, pints, quarts, and gallons).

4.MG.2 The student will solve single-step and multistep contextual problems involving elapsed time (limited to hours and minutes within a 12-hour period).

Students will demonstrate the following Knowledge and Skills:

a) 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:

i) the starting time and the ending time, determine the amount of time that has elapsed in hours and minutes;

ii) the starting time and amount of elapsed time in hours and minutes, determine the ending time; or

iii) the ending time and the amount of elapsed time in hours and minutes, determine the starting time.

4.MG.3 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).

Students will demonstrate the following Knowledge and Skills:

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

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

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

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

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

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

4.MG.4 The student will identify, describe, and draw points, rays, line segments, angles, and lines, including intersecting, parallel, and perpendicular lines.

Students will demonstrate the following Knowledge and Skills:

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

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

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

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

e) Use symbolic notation to name points, lines, line segments, rays, angles, and to describe parallel and perpendicular lines.

4.MG.5 The student will classify and describe quadrilaterals (parallelograms, rectangles, squares, rhombi, and/or trapezoids) using specific properties and attributes.

Students will demonstrate the following Knowledge and Skills:

a) Develop definitions for parallelograms, rectangles, squares, rhombi, and trapezoids through the exploration of properties and attributes.

b) Identify and describe points, line segments, angles, and vertices in quadrilaterals.

c) Identify and describe parallel, intersecting, perpendicular, and congruent sides in quadrilaterals.

d) Compare, contrast, and classify quadrilaterals (parallelograms, rectangles, squares, rhombi, and/or trapezoids) based on the following properties and attributes:

i) parallel sides;

ii) perpendicular sides;

iii) congruence of sides; and

iv) number of right angles.

e) Denote properties of quadrilaterals and identify parallel sides, congruent sides, and right angles by using geometric markings.

f) Use symbolic notation to name line segments and angles in quadrilaterals.

4.MG.6 The student will identify, describe, compare, and contrast plane and solid figures according to their characteristics (number of angles, vertices, edges, and the number and shape of faces), with and without models.

Students will demonstrate the following Knowledge and Skills:

a) Identify concrete models and pictorial representations of solid figures (cube, rectangular prism, square pyramid, sphere, cone, and cylinder).

b) Identify and describe solid figures (cube, rectangular prism, square pyramid, and sphere) according to their characteristics (number of angles, vertices, edges, and by the number and shape of faces).

c) Compare and contrast plane and solid figures (limited to circles, squares, triangles, rectangles, spheres, cubes, square pyramids, and rectangular prisms) according to their characteristics (number of sides, angles, vertices, edges, and the number and shape of faces).

Probability and Statistics

4.PS.1 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 line graphs.

Students will demonstrate the following Knowledge and Skills:

a) Formulate questions that require the collection or acquisition of data.

b) Determine the data needed to answer a formulated question and collect or acquire existing data (limited to 10 or fewer data points) using various methods (e.g., observations, measurements, experiments).

c) Organize and represent a data set using line graphs with a title and labeled axes with whole number increments, with and without the use of technology tools.

d) Analyze data represented in line graphs and communicate results orally and in writing:

i) describe the characteristics of the data represented in a line graph and the data as a whole (e.g., the time period when the temperature increased the most);

ii) identify parts of the data that have special characteristics and explain the meaning of the greatest, the least, or the same (e.g., the highest temperature shows the warmest day);

iii) make inferences about data represented in line graphs;

iv) draw conclusions about the data and make predictions based on the data to answer questions; and

v) solve single-step and multistep addition and subtraction problems using data from line graphs.

4.PS.2 The student will model and determine the probability of an outcome of a simple event.

Students will demonstrate the following Knowledge and Skills:

a) Describe probability as the degree of likelihood of an outcome occurring using terms such as impossible, unlikely, equally likely, likely, and certain.

b) Model and determine all possible outcomes of a given simple event where there are no more than 24 possible outcomes, using a variety of manipulatives (e.g., coins, two-sided counters, number cubes, spinners).

c) Write the probability of a given simple event as a fraction between 0 and 1, where there are no more than 24 possible outcomes.

d) Determine the likelihood of an event occurring and relate it to its whole number or fractional representation (e.g., impossible or zero; equally likely; certain or one).

e) Create a model or contextual problem to represent a given probability.

Patterns, Functions, and Algebra

4.PFA.1 The student will identify, describe, extend, and create increasing and decreasing patterns (limited to addition, subtraction, and multiplication of whole numbers), including those in context, using various representations.

Students will demonstrate the following Knowledge and Skills:

a) Identify, describe, extend, and create increasing and decreasing patterns using various representations (e.g., objects, pictures, numbers, number lines, input/output tables, and function machines).

b) Analyze an increasing or decreasing single-operation numerical pattern found in lists, input/output tables, or function machines and generalize the change to identify the rule, extend the pattern, or identify missing terms.

c) Given a rule, create increasing and decreasing patterns using numbers and input/output tables (including function machines).

d) Solve contextual problems that involve identifying, describing, and extending increasing and decreasing patterns using single-operation input and output rules.

5th grade Math preview

Number and Number Sense

5.NS.1 The student will use reasoning and justification to identify and represent equivalency between fractions (with denominators that are thirds, eighths, and factors of 100) and decimals; and compare and order sets of fractions (proper, improper, and/or mixed numbers having denominators of 12 or less) and decimals (through thousandths).

Students will demonstrate the following Knowledge and Skills:

a) Use concrete and pictorial models to represent fractions with denominators that are thirds, eighths, and factors of 100 in their equivalent decimal form.*

b) Use concrete and pictorial models to represent decimals in their equivalent fraction form (thirds, eighths, and factors of 100).*

c) Identify equivalent relationships between decimals and fractions with denominators that are thirds, eighths, and factors of 100 in their equivalent decimal form, with and without models.*

d) Compare (using symbols <, >, =) and order (least to greatest and greatest to least) a set of no more than four decimals and fractions (proper, improper) and/or mixed numbers using multiple strategies (e.g., benchmarks, place value, number lines). Justify solutions orally, in writing, or with a model.*

* On the state assessment, items measuring this objective are assessed without the use of a calculator.

5.NS.2 The student will demonstrate an understanding of prime and composite numbers, and determine the prime factorization of a whole number up to 100.

Students will demonstrate the following Knowledge and Skills:

a) Given a whole number up to 100, create a concrete or pictorial representation to demonstrate whether the number is prime or composite, and justify reasoning.

b) Classify, compare, and contrast whole numbers up to 100 using the characteristics prime and composite.

c) Determine the prime factorization for a whole number up to 100.

Computation and Estimation

5.CE.1 The student will estimate, represent, solve, and justify solutions to single-step and multistep contextual problems using addition, subtraction, multiplication, and division with whole numbers.

Students will demonstrate the following Knowledge and Skills:

a) Estimate the sum, difference, product, and quotient of whole numbers in contextual problems.

b) Represent, solve, and justify solutions to single-step and multistep contextual problems by applying strategies (e.g., estimation, properties of addition and multiplication) and algorithms, including the standard algorithm, involving addition, subtraction, multiplication, and division of whole numbers, with and without remainders, in which:

i) sums, differences, and products do not exceed five digits;

ii) factors do not exceed two digits by three digits;

iii) divisors do not exceed two digits; or

iv) dividends do not exceed four digits.

c) Interpret the quotient and remainder when solving a contextual problem.

5.CE.2 The student will estimate, represent, solve, and justify solutions to single-step and multistep problems, including those in context, using addition and subtraction of fractions with like and unlike denominators (with and without models), and solve single-step contextual problems involving multiplication of a whole number and a proper fraction, with models.

Students will demonstrate the following Knowledge and Skills:

a) Determine the least common multiple of two numbers to find the least common denominator for two fractions.

b) Estimate and determine the sum or difference of two fractions (proper or improper) and/or mixed numbers, having like and unlike denominators limited to 2, 3, 4, 5, 6, 8, 10, and 12 (e.g., + , − , 3 + 2 ), and simplify the resulting fraction.*

c) Estimate and solve single-step and multistep contextual problems involving addition and subtraction with fractions (proper or improper) and/or mixed numbers having like and unlike denominators, with and without models. Denominators should be limited to 2, 3, 4, 5, 6, 8, 10, and 12. Answers should be expressed in simplest form.

d) Solve single-step contextual problems involving multiplication of a whole number, limited to 12 or less, and a proper fraction (e.g., 9 × , 8 × ), with models. The denominator will be a factor of the whole number and answers should be expressed in simplest form.*

* On the state assessment, items measuring this objective are assessed without the use of a calculator.

5.CE.3 The student will estimate, represent, solve, and justify solutions to single-step and multistep problems, including those in context, using addition, subtraction, multiplication, and division with decimal numbers.

Students will demonstrate the following Knowledge and Skills:

a) Apply estimation strategies (e.g., rounding to the nearest whole number, tenth or hundredth; compatible numbers, place value) to determine a reasonable solution for single-step and multistep contextual problems involving addition, subtraction, and multiplication of decimals, and single-step contextual problems involving division of decimals.

b) Estimate and determine the product of two numbers using strategies and algorithms, including the standard algorithm, when given:

i) a two-digit factor and a one-digit factor (e.g., 2.3 × 4; 0.08 × 0.9; .16 × 5);*

ii) a three-digit factor and a one-digit factor (e.g., 0.156 × 4, 3.28 × 7, 8.09 × 0.2);* and

iii) a two-digit factor and a two-digit factor (e.g., 0.85 × 3.7, 14 × 1.6, 9.2 × 3.5).*

(Products will not exceed the thousandths place, and leading zeroes will not be considered when counting digits.)

c) Estimate and determine the quotient of two numbers using strategies and algorithms, including the standard algorithm, in which:*

i) quotients do not exceed four digits with or without a decimal point;

ii) quotients may include whole numbers, tenths, hundredths, or thousandths;

iii) divisors are limited to a single digit whole number or a decimal expressed as tenths; and

iv) no more than one additional zero will need to be annexed.

d) Solve single-step and multistep contextual problems involving addition, subtraction, and multiplication of decimals by applying strategies (e.g., estimation, modeling) and algorithms, including the standard algorithm.

e) Solve single-step contextual problems involving division with decimals by applying strategies (e.g., estimation, modeling) and algorithms, including the standard algorithm.

* On the state assessment, items measuring this objective are assessed without the use of a calculator.

5.CE.4 The student will simplify numerical expressions with whole numbers using the order of operations.

Students will demonstrate the following Knowledge and Skills:

a) Use order of operations to simplify numerical expressions with whole numbers, limited to addition, subtraction, multiplication, and division in which:*

i) expressions may contain no more than one set of parentheses;

ii) simplification will be limited to five whole numbers and four operations in any combination of addition, subtraction, multiplication, or division;

iii) whole numbers will be limited to two digits or less; and

iv) expressions should not include braces, brackets, or fraction bars.

b) Given a whole number numerical expression involving more than one operation, describe which operation is completed first, which is second, and which is third.*

* On the state assessment, items measuring this objective are assessed without the use of a calculator.

Measurement and Geometry

5.MG.1 The student will reason mathematically to solve problems, including those in context, that involve length, mass, and liquid volume using metric units.

Students will demonstrate the following Knowledge and Skills:

a) Determine the most appropriate unit of measure to use in a contextual problem that involves metric units:

i) length (millimeters, centimeters, meters, and kilometers);

ii) mass (grams and kilograms); and

iii) liquid volume (milliliters and liters).

b) Estimate and measure to solve contextual problems that involve metric units:

i) length (millimeters, centimeters, and meters);

ii) mass (grams and kilograms); and

iii) liquid volume (milliliters and liters).

c) Given the equivalent metric measure of one unit, in a contextual problem, determine the equivalent measurement within the metric system:

i) length (millimeters, centimeters, meters, and kilometers);

ii) mass (grams and kilograms); and

iii) liquid volume (milliliters and liters).

5.MG.2 The student will use multiple representations to solve problems, including those in context, involving perimeter, area, and volume.

Students will demonstrate the following Knowledge and Skills:

a) Investigate and develop a formula for determining the area of a right triangle.

b) Estimate and determine the area of a right triangle, with diagrams, when the base and the height are given in whole number units, in metric or U.S. Customary units, and record the solution with the appropriate unit of measure (e.g., 16 square inches).

c) Describe volume as a measure of capacity and give examples of volume as a measurement in contextual situations.

d) Investigate and develop a formula for determining the volume of rectangular prisms using concrete objects.

e) Solve problems, including those in context, to estimate and determine the volume of a rectangular prism using concrete objects, diagrams, and formulas when the length, width, and height are given in whole number units. Record the solution with the appropriate unit of measure (e.g., 12 cubic inches).

f) Identify whether the application of the concept of perimeter, area, or volume is appropriate for a given situation.

g) Solve contextual problems that involve perimeter, area, and volume in standard units of measure.

5.MG.3 The student will classify and measure angles and triangles, and solve problems, including those in context.

Students will demonstrate the following Knowledge and Skills:

a) Classify angles as right, acute, obtuse, or straight and justify reasoning.

b) Classify triangles as right, acute, or obtuse and equilateral, scalene, or isosceles and justify reasoning.

c) Identify congruent sides and right angles using geometric markings to denote properties of triangles.

d) Compare and contrast the properties of triangles.

e) Identify the appropriate tools (e.g., protractor, straightedge, angle ruler, available technology) to measure and draw angles.

f) Measure right, acute, obtuse, and straight angles, using appropriate tools, and identify measures in degrees.

g) Use models to prove that the sum of the interior angles of a triangle is 180 degrees and use the relationship to determine an unknown angle measure in a triangle.

h) Solve addition and subtraction contextual problems to determine unknown angle measures on a diagram.

Probability and Statistics

5.PS.1 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 line plots (dot plots) and stem-and-leaf plots.

Students will demonstrate the following Knowledge and Skills:

a) Formulate questions that require the collection or acquisition of data.

b) Determine the data needed to answer a formulated question and collect or acquire existing data (limited to 30 or fewer data points) using various methods (e.g., polls, observations, measurements, experiments).

c) Organize and represent a data set using a line plot (dot plot) with a title, labeled axes, and a key, with and without the use of technology tools. Line plots (dot plots) may contain whole numbers, fractions, or decimals.

d) Organize and represent numerical data using a stem-and-leaf plot with a title and key, where the stems are listed in ascending order and the leaves are in ascending order, with or without commas between the leaves.

e) Analyze data represented in line plots (dot plots) and stem-and-leaf plots and communicate results orally and in writing:

i) describe the characteristics of the data represented in a line plot (dot plot) and stem-and-leaf plot as a whole (e.g., the shape and spread of the data);

ii) make inferences about data represented in line plots (dot plots) and stem-and-leaf plots (e.g., based on a line plot (dot plot) of the number of books students in a bus line have in their backpack, every student will have from two to four books in their backpack);

iii) identify parts of the data that have special characteristics and explain the meaning of the greatest, the least, or the same (e.g., the stem-and-leaf plot shows that the same number of students scored in the 90s as scored in the 70s);

iv) draw conclusions about the data and make predictions based on the data to answer questions; and

v) solve single-step and multistep addition and subtraction problems using data from line plots (dot plots) and stem-and-leaf plots.

5.PS.2 The student will solve contextual problems using measures of center and the range.

Students will demonstrate the following Knowledge and Skills:

a) Describe mean as fair share.

b) Describe and determine the mean of a set of data values representing data from a given context as a measure of center.

c) Describe and determine the median of a set of data values representing data from a given context as a measure of center.

d) Describe and determine the mode of a set of data values representing data from a given context as a measure of center.

e) Describe and determine the range of a set of data values representing data from a given context as a measure of spread.

5.PS.3 The student will determine the probability of an outcome by constructing a model of a sample space and using the Fundamental (Basic) Counting Principle.

Students will demonstrate the following Knowledge and Skills:

a) Determine the probability of an outcome by constructing a sample space (with a total of 24 or fewer equally likely possible outcomes), using a tree diagram, list, or chart to represent and determine all possible outcomes.

b) Determine the number of possible outcomes by using the Fundamental (Basic) Counting Principle.

Patterns, Functions, and Algebra

5.PFA.1 The student will identify, describe, extend, and create increasing and decreasing patterns with whole numbers, fractions, and decimals, including those in context, using various representations.

Students will demonstrate the following Knowledge and Skills:

a) Identify, describe, extend, and create increasing and decreasing patterns using various representations (e.g., objects, pictures, numbers, number lines, input/output tables, function machines).

b) Analyze an increasing or decreasing single-operation numerical pattern found in lists, input/output tables, and function machines, and generalize the change to identify the rule, extend the pattern, or identify missing terms. (Patterns will be limited to addition, subtraction, multiplication, and division of whole numbers; addition and subtraction of fractions with like denominators of 12 or less; and addition and subtraction of decimals expressed in tenths or hundredths).

c) Solve contextual problems that involve identifying, describing, and extending increasing and decreasing patterns using single-operation input and output rules (limited to addition, subtraction, multiplication, and division of whole numbers; addition and subtraction of fractions with like denominators of 12 or less; and addition and subtraction of decimals expressed in tenths or hundredths).

5.PFA.2 The student will investigate and use variables in contextual problems.

Students will demonstrate the following Knowledge and Skills:

a) Describe the concept of a variable (presented as a box, letter, or other symbol) as a representation of an unknown quantity.

b) Write an equation (with a single variable that represents an unknown quantity and one operation) from a contextual situation, using addition, subtraction, multiplication, or division.

c) Use an expression with a variable to represent a given verbal expression involving one operation (e.g., “5 more than a number” can be represented by y + 5).

d) Create and write a word problem to match a given equation with a single variable and one operation.