Courses of Study : Mathematics (Grade 3)

Students:
Given any multiplication problem in the form a x b = c,

Interpret the equation as a groups of b objects equals the product c, the total number of items.

Example: Given 5 x 7 = 35, students explain that 35 represents the total, 5 is the number of groups and 7 is the number in each group.

Use concrete materials/pictorial representations to model multiplication situations.

Write expressions and equations illustrated by models and drawings.

Write word problems to represent a multiplication situation.

Students know:

• that in multiplication, one factor represents the number of groups and the other factor represents the number of items in each group, and the product represents the total number of items in all of the groups.

Students are able to:

• Use a model or drawing to illustrate the product of two whole numbers.
• Write an expression or equation to represent the product of two whole numbers identifying the number of equal groups and the group size.

Students understand that:

• a multiplication problem can be interpreted as x groups of y objects.

Learning Objectives:
M.3.1.1: Identify and define the parts of a multiplication problem including factors, multiplier, multiplicand and product.
M.3.1.2: Use multiplication to find the total number of objects arranged in rectangular arrays based on columns and rows.
M.3.1.3: Write an equation to express the product of the multipliers (factors).
M.3.1.4: Relate multiplication to repeated addition and skip counting.
M.3.1.5: Apply concepts of multiplication through the use of manipulatives, number stories, skip counting arrays, area of a rectangle, or repeated addition.
M.3.1.6: Apply basic multiplication facts through 9 x 9 using manipulatives, solving problems, and writing number stories.
M.3.1.7: Solve addition problems with multiple addends.
M.3.1.8: Represent addition using manipulatives.

Students:
Given any division problem (including word situations) in the form a ÷ b = c,

Use concrete materials/pictorial representations to model various division situations.

Identify and explain the meanings of the quantities given as well as the meaning and quantity of the missing information.

Example: Given 35 ÷ 5, depending on context, students explain that 35 is the number of objects partitioned into 5 equal shares, and 7 is the size of each share, or that 35 is the number of objects partitioned into groups of 5 objects each, and 7 is the number of groups shared.

Explain the strategy or reasoning used to find a quotient (or missing factor).

Write expressions and equations illustrated by models and drawings.

Write word problems to represent a situation involving division.

Students know:

• that division is related to multiplication in terms of finding a missing factor. The missing factor being either the number of groups or the number of items in each group.

Students are able to:

• Interpret quantities in a division situation as the number of objects in each group or the number of equal groups.
• Use a model or drawing to illustrate a quotient.
• Write word problems for division context involving equal groups and fair shares.

Students understand that:

• a division expression represents either the number of objects in each group when the total number is partitioned evenly into a given number of groups or the number of groups when the total number is partitioned into groups that each contain a given number.

Learning Objectives:
M.3.2.1: Define the parts of a division problem including divisor, dividend, and quotient.
M.3.2.2: Write a division equation.
M.3.2.3: Apply the signs ÷ and = to the action of separating sets.
M.3.2.4: Recognize division as either repeated subtraction, parts of a set, parts of a whole, or the inverse of multiplication.
M.3.2.5: Model grouping with basic division facts partitioned equally (e.g. 8/2).
M.3.2.6: Apply properties of operations as strategies to subtract.
M.3.2.7: Subtract within 20.
M.3.2.8: Represent equal groups using manipulatives.

Students:
When given a variety of word problems involving multiplication and division within 100,

• Write and evaluate multiplication and division expressions to represent the word situation.
• Explain and justify solutions using a variety of representations (equal groups, arrays, area models, number lines, tape diagrams).
• Use the relationship between multiplication and division to write equations with an unknown factor.

Students know:

• Multiplication situations can be related to division contexts by identifying the total number of groups and the number of items in a group.
• Strategies to solve problems involving multiplication and division.

Students are able to:

• Use models, drawings, and equations to represent a multiplication or division situation.
• Use symbols to represent unknown quantities in equations.
• Solve word situations with multiplication and division within 100 involving equal groups, arrays, and measurement quantities.

Students understand that:

• a word problem with an unknown product is a multiplication problem, and a word problem with an unknown number of groups or an unknown group size can be thought of as a division problem or a multiplication problem with an unknown factor.

Learning Objectives:
M.3.3.1: Demonstrate computational understanding of multiplication and division by solving authentic problems with multiple representations using drawings, words, and/or numbers.
M.3.3.2: Identify key vocabulary words to solve multiplication and division word problems.
M.3.3.3: Solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.
M.3.3.4: Recall basic multiplication facts.
M.3.3.5: Add and subtract within 20.
M.3.3.6: Represent repeated addition, subtraction, and equal groups using manipulatives.

Students:

• Relate three whole numbers to determine an unknown number in a multiplication equation.
• 
  Example: 8 x ? = 48 or ? x 8 = 48.
• Relate three whole numbers to determine an unknown whole number in a division equation.
• 
  Example: 5 = ? ÷ 3 or ? ÷ 3 = 5.
• Relate missing factor multiplication equations to division equations using both symbols ? or ÷ for division.

Students know:

• how to use the meaning of multiplication and division and the relationship between the two operations to determine an unknown number in a given equation.

Students are able to:

• Relate three whole numbers to determine the unknown factor in a multiplication equation.
• Relate three whole numbers to determine the unknown whole number in a division equation.

Students understand that:

• the unknown number in a multiplication or division equation is the number that makes the equation true.

Learning Objectives:
M.3.4.1: Use arrays to show equal groups in multiplication and division.
M.3.4.2: Recall basic multiplication facts.
M.3.4.3: Determine the unknown whole number in an addition or subtraction equation relating three whole numbers.
M.3.4.4: Represent repeated addition, repeated subtraction, and equal groups using manipulatives.

Students:

• Use their understanding of multiplication and division to develop and apply a variety of properties to various situations.

Examples: given 4 x 9 = 36 is known, then 9 x 4 = 36 is also known. (commutative property). If 3 x 5 x 2 can be found by 3 x 5 =15, then 15 x 2 = 30 or 5 x 2 = 10, then 3 x 10 = 30. (associative property). If 8 x 5 = 40 and 8 x 2 = 16 are known, then 8 x 7 can be found by 8 x (5 + 2) = (8 x 5) + (8 x 2) = 40 + 16 = 56. (distributive property.)
• Identify an equivalent expression where the properties of operations has been applied.
• Describe the properties of multiplication related to visual models.
• Write multiplication equations to represent visual models.
• Apply properties of operations as strategies to multiply and divide.

Note: Students need not use formal terms for the properties of operations.

Students know:

• When any factor, x, is multiplied by a factor of 1, the product is the value of x.
• If one factor is zero, then there are zero groups or zero items in a group and the product is zero.
• The commutative property of multiplication shows a x b = c and b x a = c.
• The associative property of multiplication shows that when multiplying three or more numbers, the product is always the same regardless of the grouping.
• The distributive property will help in finding products of more difficult multiplication facts.

Students are able to:

• Develop properties as strategies for multiplication and division.
• Apply properties of operations as strategies to multiply and divide.

Students understand that:

• applying properties of operations can help develop strategies to find solutions to multiplication and division problems.

Learning Objectives:
M.3.5.1: Define properties of operations.
M.3.5.2: Apply basic multiplication facts.
M.3.5.3: Apply properties of operations as strategies to add and subtract.
M.3.5.4: Count to answer "how many?" questions about as many as 30 things arranged in a rectangular array.

Students:
When given a division problem with an unknown quotient,

• Use a variety of strategies to solve the division problem and justify the solution.
• Use the relationship between multiplication and division to write an equation with an unknown factor to represent a division problem.
• Use symbols to represent the unknown quantities in equations.
• Use the inverse relationship between multiplication and division to find quotients.

Students know:

• Multiplication and division are related operations.
• Using known multiplication facts and the relationship between multiplication and division, will help build fluency with division facts.

Students are able to:

• Use the relationship between multiplication and division to find quotients.
• Write a multiplication equation with a missing factor to represent a division situation.
• Use symbols to represent an unknown quantity in equations.

Students understand that:

• Multiplication and division are related operations.
• The dividend in a division equation is the same as the product in a related multiplication equation.

Learning Objectives:
M.3.6.1: Apply divisibility rules for 2, 5, and 10.
M.3.6.2: Apply basic multiplication facts.
M.3.6.3: Understand subtraction as an unknown-addend problem.
M.3.6.4: Recognize division as repeated subtraction, parts of a set, parts of a whole, or the inverse of multiplication.

Students:

• When given any single digit multiplication problem, use an efficient strategy (recall, inverse operations, arrays, derived facts, properties of operations, doubling, skip counting, square numbers) to name the product.
• When given a division problem with a single digit divisor and an unknown single digit quotient, use an efficient strategy (recall, inverse operations, arrays, derived facts, properties of operations, doubling, skip counting, square numbers) to name the quotient.

Students know:

• Strategies for finding products and quotients.
• How to use multiplication facts in terms of a missing factor to learn division facts.

Students are able to:

• Use strategies based on properties of operations and patterns of multiplication to find products and quotients.
• Use efficient multiplication and division strategies based on the numbers in the problems. -Use multiplication facts in terms of a missing factor to learn division facts.

Students understand that:

• they can use the meaning of the numbers in multiplication and division situations to determine strategies to become fluent with multiplication and division facts.

Learning Objectives:
M.3.7.1: Name the first 10 multiples of each one-digit natural number.
M.3.7.2: Recognize multiplication as repeated addition, and division as repeated subtraction.
M.3.7.3: Apply properties of operations as strategies to add and subtract.
M.3.7.4: Recall basic addition and subtraction facts.

Students:
When given a variety of two-step word problems involving all four operations,

• Apply understanding of operations to find solutions.
• Use a model to represent the problem situation.
• Write an equation to represent the problem using a symbol for the unknown quantity.
• Explain and justify strategies and solutions.
• Apply understanding of operations and estimation strategies including rounding to evaluate reasonableness of the solution.

Students know:

• Characteristics of addition, subtraction, multiplication, and division.
• Strategies for addition, subtraction, multiplication, and division.
• Strategies for mental computation and estimating sums, differences, products, and quotients.

Students are able to:

• Use a variety of strategies to solve two-step word problems involving all four operations.
• Write an equation to represent the problem context, and use a symbol for the unknown quantity.
• Justify strategy and solutions using mathematical vocabulary.
• Determine and justify reasonableness of solutions using mental computation strategies and estimation strategies.

Students understand that:

• Mathematical problems can be solved using a variety of strategies, models, and representations.
• Contextual situations represented by multiplication and division.
• Reasonableness of solutions can be evaluated by using estimation strategies.

Learning Objectives:
M.3.8.1: Define the identity property of addition and multiplication.
M.3.8.2: Estimating sums and differences using multiple methods, including compatible numbers and rounding, to judge the reasonableness of an answer.
M.3.8.3: Apply commutative, associative, and identity properties for all operations to solve problems.
M.3.8.4: Identify a rule when given a pattern.
M.3.8.5: Solve addition and subtraction problems, including word problems, involving one-and two digit numbers with and without regrouping, using multiple strategies. M 3.8.6: Solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.
M.3.8.7: Represent multiplication and division with manipulatives.
M.3.8.8: Recall basic addition and subtraction facts.

Students:

• Recognize, describe, and explain arithmetic patterns.
• Given a number pattern, find the next number or numbers in the pattern.
• Given a number pattern, find a characteristic of the next number or numbers in the pattern.
• Given an addition table or multiplication table, find the missing values in the table.
• Given an addition table or multiplication table, find a characteristic of a row or column of that table.

Students know:

• that mathematical ideas and concepts build on patterns and recognize and identify those patterns to make sense of math, and the ability to make generalizations is the foundation for algebraic reasoning.

Students are able to:

• Identify arithmetic patterns in number sequences, in the addition table, or multiplication table.
• Use logical reasoning and properties of numbers and operations to explain characteristics of arithmetic patterns.

Students understand that:

• Mathematical concepts build on patterns.
• When consecutive terms always differ by the same amount, an arithmetic pattern is formed.
• Visual patterns can be found in the multiplication table.

Learning Objectives:
M.3.9.1: Define arithmetic patterns: geometric or numeric.
M.3.9.2: Explain arithmetic patterns using properties of operations.
M.3.9.3: Recognize arithmetic patterns (including geometric patterns or patterns in the addition table or multiplication table).
M.3.9.4: Construct repeating and growing patterns with a variety of representations.
M.3.9.5: Demonstrate computational fluency, including quick recall, of addition and multiplication facts.
M.3.9.6: Duplicate an existing pattern.
M.3.9.7: Skip count.
M.3.9.8: Represent addition and multiplication with manipulatives.

Students:
When given two-digit or three-digit number to round to the nearest 10 or 100,

• Identify the ten or hundreds that the number falls between.
• Plot the number on a number line between the tens or hundreds.
• Identify the nearest ten or hundred and justify the answer.
• Identify a possible value of the unknown number when instructed that an unknown number will round to a given number when rounding to the nearest 10 or 100.

Example: An unknown number will round to 340 when rounded to the nearest 10. Identify a possible value for the unknown number.

Students know:

• Values of the digits in the ones, tens, and hundreds places.
• How to determine what is halfway between two multiples of 10 or 100.
• Strategies for rounding to the nearest 10 or 100.
• Use place value vocabulary and logical reasoning to justify solutions when rounding.

Students are able to:

• Round whole numbers to the nearest 10 or 100.
• Identify a possible value for a number which will result in a given number rounded to the nearest 10 or 100.
  Example: What value will result in 270 when rounded to the nearest 10? Identify the possible values.

Students understand that:

• rounding is determining which ten or hundred a number is closer to.

Learning Objectives:
M.3.10.1: Define rounding.
M.3.10.2: Round whole numbers from 100 to 999 using whole numbers from 10 to 99.
M.3.10.3: Model rounding whole numbers to the nearest 100.
M.3.10.4: Round whole numbers from 10 to 99 using whole numbers from 1 to 9.
M.3.10.5: Model rounding whole numbers to the nearest 10.
M.3.10.6: Identify the steps in rounding two- and three-digit numbers.
Example: Identify the digit that may change and the number to the right.
M.3.10.7: Round whole numbers from 1 to 9 and model to show proficiency.
M.3.10.8: Understand that the two digits of a two-digit number represent amounts of tens and ones.
M.3.10.9: Match the number in the ones, tens, and hundreds position to a pictorial representation or manipulative of the value.

Students:
When given problems of addition and subtraction within 1000,

• Fluently find sums and differences using strategies based on place value, properties of operations, and the relationship between addition and subtraction.
• Justify strategies and by relating the strategy to a written method and explain reasoning used.
• Use estimation strategies to check for reasonableness and justify solutions.

Note: Standard algorithm for addition and subtraction is not a grade 3 expectation.

Students know:

• The relationship between addition and subtraction operations.
• How conceptual models support and give understanding to procedures for addition and subtraction.

Students are able to:

• Use a variety of strategies to solve addition and subtraction problems within 1000.

Students understand that:

• Strategies for addition and subtraction will vary depending on the problem.
• Strategies can include place value, properties of operations, and the relationship between addition and subtraction.

Learning Objectives:
M.3.11.1: Define the commutative and associative properties of addition and subtraction.
M.3.11.2: Subtract within 100 using strategies and algorithms based on the relationship between addition and subtraction.
M.3.11.3: Subtract within 100 using strategies and algorithms based on properties of operations.
M.3.11.4: Subtract within 100 using strategies and algorithms based on place value.
M.3.11.5: Add within 100 using strategies and algorithms based on the relationship between addition and subtraction.
M.3.11.6: Add within 100 using strategies and algorithms based on properties of operations.
M.3.11.7: Add within 100 using strategies and algorithms based on place value.
M.3.11.8: Recall basic addition and subtraction facts.

Students:

• Efficiently use strategies based on place value and properties of operations to multiply one-digit numbers by multiples of 10 (from 10-90) and justify their answers.

Students know:

• a variety of strategies or tools to find products (skip counting, properties of operations, concrete materials, number lines, arrays, etc.).

Students are able to:

• Find the product of a 1-digit factor and multiple of 10 (from 10 to 90).
• Use concrete materials and pictorial models to find the product.
• Use properties of operations to find the product.
• Justify products.

Students understand that:

• A one-digit number multiplied by ten gives a multiple of ten. Ex: 6 x 10 = 60 is the same as that number of ones (60) or that number of tens (6 tens).
• Adding a zero to the product of two non-zero whole numbers does not demonstrate the relationship between the product and its place value.

Learning Objectives:
M.3.12.1: Model place value by multiplying vertically.
M.3.12.2: Model properties of operations by multiplying horizontally.
M.3.12.3: Recall basic multiplication facts.
M.3.12.4: Recall multiplication as repeated addition.
M.3.12.5: Apply properties of operations as strategies to add.

Students:
When given any fraction in form a/b,

• Create an area model to represent the fraction.
• Use a number line to represent the fraction.
• Explain the relationship between the fraction and the model including the corresponding number of unit fractions.
  Example: 3/4 is composed of 3 units of 1/4 or 3/4 is the same as 1/4 + 1/4 + 1/4.
• Identify a point to represent the fraction when given on a number line labeled with multiple points.

Note: Set models (parts of a group) are not models used in grade 3.

Students know:

• Fractional parts of a whole must be of equal size but not necessarily equal shape.
• Denominators represent the number of equal size parts that make a whole.
• The more equal pieces in the whole, the smaller the size of the pieces.
• The numerator represents the number of equal pieces in the whole that are being counted or considered.

Students are able to:

• Use an area model and length model to show a unit fraction as one part of an equally partitioned whole.
• Explain that given a fraction with a numerator greater than one, the numerator indicates the number of unit fraction pieces represented by the fraction.
  Example: 3/4 is the same as 3 units of 1/4 size, or three 1/4 pieces, 3 copies of 1/4, or 3 iterations of 1/4.
• Identify and describe the fractional name given a visual fraction model.
• Identify and demonstrate fractional parts of a whole that are the same size but not the same shape using concrete materials.

Students understand that:

• Given the same size whole, the larger the denominator, indicating the number of equal parts in the whole, the smaller the size of the pieces because there are more pieces in the whole.
• Fractions are numbers that represent a quantity less than, equal to, or greater than 1.
• Fractions represent equal partitions of a whole.

Learning Objectives:
M.3.13.1: Define fraction, numerator, and denominator.
M.3.13.2: Identify the parts of a fraction.
M.3.13.3: Label numerator, denominator, and fraction bar.
M.3.13.4: Identify parts of a whole with two, three, or four equal parts.
M.3.13.5: Distinguish between equal and non-equal parts.
M.3.13.6: Partition circles and rectangles into two and four equal shares; describe the shares using the words halves, fourths, and quarters; and use the phrases half of, fourth of, and quarter of.

Students:
When given a fraction a/b (with denominators of 2, 3, 4, 6, 8),

• Use a number line and partition the interval between 0 and 1 into b equal parts, specified by the denominator.
• Use a number line and partition the interval between 0 and 1 into b equal parts and mark off a lengths of 1/b unit fractions.
• Model a fraction with a point on a number line and recognize the length of the fraction as the distance from the fraction point to 0.
• Extend the number to include fractions greater than one as a continuation of counting unit fractions.
• Given a fraction, draw a model to represent the fraction using a number line.
• Given a fraction and a number line with labeled points, identify the labeled point that represents the fraction.
• Given a point on a number line, identify the fraction modeled by the point.

Students know:

• How to use fraction strips as a model to connect to finding fractional parts on a number line.
• Fractions are numbers that can be represented on a number line.
• Fractions can be placed on the number line by marking off equal parts between two whole numbers.
• Fractions equal to 1 have the same numerator and same denominator.
• Fractions greater than 1 have a numerator that will be greater than the denominator.

Students are able to:

• Represent fractions on a number line.
• Locate fractions on a number line.
• Use a number line and partition an interval from 0 to 1 into equal parts as specified by the denominator of a fraction.
• Represent a non unit fraction on a number line by marking off unit fraction lengths as specified by the numerator from zero.
• Extend the number line to include fractions greater than one as a continuation of counting unit fractions.

Students understand that:

• A number line is a length model.
• Fractions are numbers that represent a quantity less than, equal to, or greater than 1 and can be placed on a number line.
• A number line can be partitioned to represent equal parts of a whole.

Learning Objectives:
M.3.14.1: Recognize fractions as lengths from zero to one.
M.3.14.2: Represent whole numbers as lengths from 0 on a number line diagram with equally spaced points corresponding to the numbers 0, 1, 2…, and represent whole-number sums and differences within 100 on a number diagram.
M.3.14.3: Identify a number line.
M.3.14.4: Recognize whole numbers as lengths from zero to one.
M.3.14.5: Represent whole numbers as lengths from 0 on a number line diagram with equally spaced points corresponding to the numbers 0, 1, 2…, and represent whole-number sums and differences within 100 on a number diagram.
M.3.14.6: Identify a number line.
M.3.14.7: Label the fractions on a pre-made number line diagram.
M.3.14.8: Represent whole numbers as lengths from 0 on a number line diagram with equally spaced points corresponding to the numbers 0, 1, 2…, and represent whole-number sums and differences within 100 on a number diagram.
M.3.14.9: Recognize a number line diagram with equally spaced points.

Students:

• Use a variety of area models and length models to identify equivalent fractions.
• Use a variety of area models and length models to illustrate equivalent fractions.
• Use visual representations to find fractions equal to 1.
• Illustrate and explain fractions equivalent to whole numbers (limited to 0 through 5).
• Compare two fractions by reasoning about their size and use <, >, or = to record the comparison.
• Compare two fractions using visual fraction models.
• Use symbols <, >, or = to record the comparison between two fractions.

Note: Tasks in grade 3 are limited to fractions with denominators 2, 3, 4, 6, or 8.

Students know:

• Fractions with different names can be equal.
• Two fractions are equivalent if they are the same size, cover the same area, or are at the same point on a number line.
• Unit fraction counting continues beyond 1 and whole numbers can be written as fractions.
• Use a variety of area models and length models to show that a whole number can be expressed as a fraction and to show that fractions can be equivalent to whole numbers.
• Comparing two fractions is only reasonable if they refer to the same whole.
• The meaning of comparison symbols <, >, = .
• Reason about the size of a fraction to help compare fractions.
• Use a variety of area and length models to represent two fractions that are the same size but have different names.
• Use a fraction model to explain how equivalent fractions can be found.
• Use a variety of area models and length models to demonstrate that any fraction that has the same nonzero numerator and denominator is equivalent to 1.
• Use models to show that the numerator of a fraction indicates the number of parts, so if the denominators of two fractions are the same, the fraction with the greater numerator is the greater fraction.
• Use models to show that the denominator of a fraction indicates the size of equal parts a whole is partitioned into, and that the greater the denominator, the smaller the parts. -Determine when two fractions can not be compared because they do not refer to the same size whole.

Students are able to:

• Explain equivalence of two fractions using visual models and reasoning about their size.
• Compare two fractions with same numerators or with same denominators using visual models and reasoning about their size.
• Express whole numbers as fractions.
• Identify fractions equivalent to whole numbers.
• Record comparisons of two fractions using <, >, or = and justify conclusion.
• Explain that the whole must be the same for the comparing of fractions to be valid.

Students understand that:

• A fraction is a quantity which can be illustrated with a length model or an area model.
• Two fractions can be the same size but have different fraction names.
• A fraction can be equivalent to a whole number.
• Any fraction that has the same nonzero numerator and denominator is equivalent to 1.
• The numerator of a fraction indicates the number of parts, so if the denominators of two fractions are the same, the fraction with the greater number of parts is the greater fraction.
• The denominator of a fraction indicates the size of equal parts in a whole, so the greater the denominator, the smaller the size of the parts in a whole.

Learning Objectives:
M.3.15.1: Define equivalent.
M.3.15.2: Recognize pictorial representations of equivalent fractions.
M.3.15.3: Recognize different interpretations of fractions, including parts of a set or a collection, points on a number line, numbers that lie between two consecutive whole numbers, and lengths of segments on a ruler.
M.3.15.4: Recognize that equal shares of identical wholes need not have the same shape.
M.3.15.5: Compose two-dimensional shapes (rectangles, squares, trapezoids, triangles, half-circles, and quarter-circles) or three-dimensional shapes (cubes, right rectangular prisms, right circular cones, and right circular cylinders) to create a composite shape, and compose new shapes from the composite shape.
M.3.15.6: Label a fraction with multiple representations.
M.3.15.7: Recognize that a whole can be partitioned into differing equal parts (halves, fourths, eighths, etc.).
M.3.15.8: Partition circles and rectangles into two and four equal shares; and describe the shares using the words halves, fourths, and quarters; and use the phrases half of, fourth of, and quarter of.
M.3.15.9: Recognize different interpretations of fractions, including parts of a set or a collection, points on a number line, numbers that lie between two consecutive whole numbers, and lengths of segments on a ruler.
M.3.15.10: Label a pictorial representation.
M.3.15.11: Recognize that a fraction is a part of a whole.
M.3.15.12: Partition circles and rectangles into two and four equal shares; describe the shares using the words halves, fourths, and quarters; and use the phrases half of, fourth of, and quarter of.

Students

Organize data and draw a scaled picture graph (with scales other than 1) to represent a data set with several categories.

Organize data and draw a scaled bar graph (with scales other than 1) to represent a data set with several categories.

Given a scaled picture graph or bar graph, solve one-And two-step problems using information presented in the graphs.

Determine simple probability from a context that includes a picture or information displayed in a graph.

Example: A picture graph displays data to represent the type of transportation for students traveling to school as 10 students walk, 8 students ride bikes, 38 ride the bus, and 12 ride in cars. Another student enrolls in school. What is the least likely way they will travel to school? Why?
Note: Students are expected to reason about probability, not calculate a probability.

Students know:

• Strategies for collecting, organizing, and recording data in picture graphs and bar graphs.
• Describe and interpret data on picture and bar graphs.
• Strategies for solving addition and subtraction one-And two-step problems.

Students are able to:

• Collect and categorize data to display graphically.
• Draw a scaled picture graph (with scales other than 1) to represent a data set with several categories.
• Draw a scaled bar graph (with scales other than 1) to represent a data set with several categories.
• Determine simple probability from a context that includes a picture.
  Example: A bar graph displays data to represent students' favorite colors with data showing 4 students choose red, 11 students choose blue, 2 students choose green, and 4 students choose purple. If Jamal is a student in the class, what do you think his favorite color might be? Why?
• Solve one-And two-step "how many more" and "how many less" problems using information presented in scaled graphs.

Students understand that:

• Questions concerning mathematical contexts can be answered by collecting and organizing data scaled pictographs and bar graphs.
• Understand that logical reasoning and connections between representations provide justifications for solutions.

Learning Objectives:
M.3.16.1: Define picture graph, bar graph, and data.
M.3.16.2: Interpret the data to solve problems.
M.3.16.3: Identify the parts of a graph (x-axis, y-axis, title, key, equal intervals, labels).
M.3.16.4: Locate the data on a picture graph and a bar graph.
M.3.16.5: Organize, represent, and interpret data with up to three categories; ask and answer questions about the total number of data points, how many in each category, and how many more or less are in one category than in another.
M.3.16.6: Directly compare two objects, with a measurable attribute in common, to see which object has "more of" or "less of" the attribute, and describe the difference.

Students:

• Measure objects to the nearest 1/2 inch.
• Measure objects to the nearest 1/4 (quarter) inch.
• Create a line plot marked off in appropriate units (whole numbers, halves, or quarters) to represent data of several objects.
• Create a line plot marked off in appropriate units (whole numbers, halves, or quarters) to represent data of repeated measurements.

Example: Measuring how far a marble rolls under certain conditions.

Students know:

• Nearest half and nearest quarter inch on a ruler.
• A ruler is a type of number line and shows fraction of 1/2 and 1/4.

Students are able to:

• Measure objects to the nearest half and fourth of an inch.
• Create a line plot to display the data of the objects measured.

Students understand that:

• A line plot is a graph that displays a distribution of data values, including whole numbers, halves and quarters, such that each data value is marked above a horizontal line with an X or dot.
• A ruler is a type of number line partitioned equally and shows halves and fourths.

Learning Objectives:
M.3.17.1: Define line plot.
M.3.17.2: Identify the parts of a line plot.
M.3.17.3: Measure objects to the nearest inch.
M.3.17.4: Identify one-inch units on a ruler starting with 0.
M.3.17.5: Express the length of an object as a whole number of length units by laying multiple copies of a shorter object (the length unit) end to end; understand that the length measurement of an object is the number of same-size length units that span it with no gaps or overlaps.
M.3.17.6: Directly compare two objects, with a measurable attribute in common, to see which object has "more of" or "less of" the attribute, and describe the difference.

Students:

• Tell and record time to the nearest minute using an analog clock.
• Determine elapsed time using a number line.
• Solve simple word problems using elapsed time in minutes (within 90 minutes) or hours.

Students know:

• Conventions for time notation.
• Time sequence patterns.
• Strategies to determine elapsed time.

Students are able to:

• Accurately read and write time to the nearest minute from analog and digital clocks.
• Measure time intervals in minutes.
• Illustrate elapsed time using a number line.
• Solve problems involving elapsed time in minutes (with 90 minutes) or hours.

Students understand that:

• An analog clock is a whole partitioned into 60 parts and each part is one minute.
• A number line can be partitioned to show time intervals in minutes.
• A number line can be used to solve word problems that involve time intervals.

Learning Objectives:
M.3.18.1: Compare equivalent units of time using hours and minutes.
Examples: 60 minutes = one hour, 30 minutes = one half of an hour.
M.3.18.2: Recognize key vocabulary and/or phrases associated with time.
M.3.18.3: Compare the lengths of time to complete everyday activities.
M.3.18.4: Tell and write time in hours and half-hours using analog and digital clocks.
M.3.18.5: Recognize hour, minute, and second hands on an analog clock.
M.3.18.6: Count by 5's to 60.

Students:

• Accurately measure the liquid volume and mass of objects by selecting and using appropriate tools (such as balance and spring scales, graduated cylinders, beakers, and measuring cups) to determine measures to the nearest whole unit.
• Given an image of a measurement device, determine the volume or mass shown in the image.
• Use the four operations to solve one-step word problems involving liquid volume or mass measurements.
• Given two measurement quantities or two images of a measuring device, determine the total volume/mass, or find the difference between the two volumes/masses.
• Given the volume or mass of an object, determine the volume/mass of more than one object using multiplication.
• Given the total volume or mass of multiple identical objects, determine the volume/mass of a single object using division.
• Explain and justify solutions using a variety of representations.

Students know:

• Personal benchmarks for metric standard units of measure, mass (gram & kilogram) and liquid volume (liter), and the use of related tools (such as balance, spring scales, graduated cylinders, beakers, measuring cups) for measurement to those units.
• Characteristics of addition, subtraction, multiplication, and division contexts that involve measurements.
• How to represent quantities and operations physically, pictorially, or symbolically.
• Strategies to solve one-step word problems that involve measurement.

Students are able to:

• Measure liquid volume and mass in metric standard units.
• Choose appropriate measurement tools and units of measure.
• Represent quantities and operations physically, pictorially, or symbolically,
• Use a variety of strategies to solve one-step word problems that involve measurement.

Students understand that:

• Capacity indicates the measure of the volume (dry or liquid) in a container.
• Mass indicates the amount of matter in an object and can be represented with different sized units.

Learning Objectives:
M.3.19.1: Define liquid volume, mass, grams, kilograms, and liters.
M.3.19.2: Recognize how the standard units of measure compare to one another.
M.3.19.3: Identify key terms for word problems.
M.3.19.4: Express the length of an object as a whole number of length units by laying multiple copies of a shorter object (the length unit) end to end; understand that the length measurement of an object is the number of same-size length units that span it with no gaps or overlaps.
M.3.19.5: Recall basic addition, subtraction, multiplication, and division facts.
M.3.19.6: Describe measurable attributes of objects such as length or weight. Describe several measurable attributes of a single object.

Students:

• Use manipulatives to tile a rectangle with unit squares to find the area.
• Given a rectangle drawn on a coordinate grid, determine the area of the rectangle.
• Given a rectangle tiled with unit squares, determine the area of the rectangle.

Students know:

• area is a measurable attribute of two-dimensional figures.

Students are able to:

• Find the area of a rectangle by tiling it without gaps or overlaps.
• Measure the area of a rectangle by counting the number of unit squares needed to cover the shape.

Students understand that:

• Area is the number of unit squares needed to cover a surface.
• Multiple unit squares can be combined to measure the area of rectangles so long as the unit squares completely cover the figure without overlapping each other or extending beyond the edge of the figure.

Learning Objectives:
M.3.20.1: Define length.
M.3.20.2: Recognize that units of measure must be equal.
M.3.20.3: Express the length of an object as a whole number of length units by laying multiple copies of a shorter object (the length unit) end to end.
M.3.20.4: Recognize that the length measurement of an object is the number of same-size length units that span it with no gaps or overlaps.

Students:

• Accurately measure area by counting standard unit squares (square cm, square m, square in, and square ft) and non-standard unit squares (e.g., orange pattern blocks or floor tiles).

Students know:

• area is a measurable attribute of two-dimensional figures.

Students are able to:

• Determine area of a rectangle by counting unit squares.

Students understand that:

• A unit square is a square with a side length of 1 unit, and that such a square represents a unit of measurement.
• The area of a plane figure is measured by counting the number of same-size squares (unit squares) that exactly cover the interior space of the figure.

Learning Objectives:
M.3.21.1: Recognize that unit squares are equal.
M.3.21.2: Define the units of measurement (cm, m, in, ft).
M.3.21.3: Express the length of an object as a whole number of length units by laying multiple copies of a shorter object (the length unit) end to end; understand that the length measurement of an object is the number of same-size length units that span it with no gaps or overlaps.

Students:

• Use arrays and area models to represent the distributive property in mathematical reasoning.
• Given the side lengths of a rectangle, use a multiplication expression to determine the area of the rectangle.
• Solve real-world problems involving areas of rectangles using concrete materials, reasoning, multiplication, and distributive property.

Students know:

• The area measurement of rectangular regions has a multiplicative relationship of the number of square units in a row and the number of rows.
• The area of a rectangle with whole number side lengths is the same as the product of multiplying side lengths.
• How to use an area model to illustrate the distributive property.

Students are able to:

• Relate the area of a rectangle with whole number side lengths and show that the area is the same as would be found by multiplying side lengths.
• Use concrete materials, arrays and area models to illustrate the distributive property.
• Solve real-world problems involving areas of rectangles using concrete materials, reasoning, multiplication, and distributive property.

Students understand that:

• side lengths with unit squares produces rows and columns and that multiplying the number of rows by the number of columns is equivalent to the total number of squares just like arrays.
• The side length of a rectangle can be rewritten as the sum of two numbers and that when the other side is multiplied by each of those two numbers, then the sum of the products is equal to the area of the rectangle.

Learning Objectives:
M.3.22.1: Recognize arrays as multiplication or repeated addition.
M.3.22.2: Recall basic addition and multiplication facts.
M.3.22.3: Build and draw shapes to possess defining attributes.
M.3.22.4: Compose simple shapes to form larger shapes.

Students:
Will use concrete materials to find areas of rectilinear figures by

• Decomposing the figure into non overlapping rectangles, finding the area of each, and then finding the sum of the areas.
• Composing rectilinear figures by joining two rectangles (without overlapping) and determine the area of the composed rectilinear figure is the sum of the areas of the two joined rectangles.

Students know:

• Area is a measurable attribute of two-dimensional figures.
• The area measurement of rectangular regions has a multiplicative relationship of the number of square units in a row and the number of rows.

Students are able to:

• Decompose rectilinear figures as non-overlapping rectangles using concrete materials.
• Find the area of two rectangles, and create a rectilinear figure by joining the two rectangles (without overlapping), and determine the area of the created rectilinear figure as the sum of the two rectangles.

Students understand that:

• rectilinear shapes can be decomposed into non overlapping rectangles, and the sum of the areas of the nonverlapping rectangles is equivalent to the area of the original rectilinear shape.

Learning Objectives:
M.3.23.1: Label pre-made arrays.
M.3.23.2: Partition a rectangle into rows and columns of same-size squares, and count to find the total number of them.
M.3.23.3: Recall basic addition and multiplication facts.
M.3.23.4: Compose two-dimensional shapes (rectangles, squares, trapezoids, triangles, half-circles, and quarter-circles).
M.3.23.5: Identify a rectangle.

Students:

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

Students know:

• Perimeter is a measurable attribute of rectangles.
• Area is a measurable attribute of rectangles.

Students are able to:

• Construct rectangles with a given perimeter.
• Construct rectangles with a given area.
• Construct rectangles with the same perimeters but differing areas.
• Construct rectangles with the same areas but differing perimeters.

Students understand that:

• Perimeter and area are measurable attributes of rectangles.
• Perimeter is the distance around a figure found by adding side lengths.
• The area of a plane figure is measured by the number of square units that cover the interior space of the rectangle.

Learning Objectives:
M.3.24.1: Define perimeter.
M.3.24.2: Recall the formula for perimeter (P= L+L+W+W or P=2L + 2W).
M.3.24.3: Recall basic addition and multiplication facts.
M.3.24.4: Build and draw shapes to possess defining attributes.
M.3.24.5: Express the length of an object as a whole number of length units by laying multiple copies of a shorter object (the length unit) end to end; understand that the length measurement of an object is the number of same-size length units that span it with no gaps or overlaps.
M.3.24.6: Describe measurable attributes of objects such as length or weight.

Students:

• Measure to find the perimeter of shapes.
• Given a figure, determine the perimeter of the figure.
• Given a figure with a missing side length and a given perimeter, determine the missing side length.
• Solve real-world problems involving perimeters of polygons.
• Use a multiplication expression to find perimeter of a polygon when all side measures of the polygon are equal.

Students know:

• Measurable attributes of objects, specifically perimeter.
• Strategies for modeling measurement problems involving perimeter.
• Strategies for representing and computing perimeter.

The Students are able to:

• Solve real-world and mathematical problems involving perimeters of polygons.
• Find the perimeter of a figure given the side lengths.
• Find an unknown side length of a polygon given the perimeter and one missing side length.

Students understand that:

• Perimeter is measured in length units and is the distance around a two-dimensional figure.
• If all the sides of a polygon are equal, then the perimeter can be determined by multiplying one side length by the total number of sides.

Learning Objectives:
M.3.25.1: Define perimeter.
M.3.25.2: Recall the formula for perimeter (P= L+L+W+W or P=2L + 2W).
M.3.25.3: Recall basic addition and multiplication facts.
M. 3.23.4: Build and draw shapes to possess defining attributes.
M.3.25.4: Express the length of an object as a whole number of length units by laying multiple copies of a shorter object (the length unit) end to end; understand that the length measurement of an object is the number of same-size length units that span it with no gaps or overlaps.
M.3.25.5: Describe measurable attributes of objects such as length or weight.

Students:

• Describe, analyze, and compare properties of two-dimensional shapes.
• Identify shapes that are and are not quadrilaterals by examining the properties of geometric shapes.
• Draw rhombuses, rectangles, and squares as examples of quadrilaterals and draw examples of quadrilaterals that do not belong to any of these subcategories.
• Use geometric terms when describing quadrilaterals.
• Identify attributes that are needed to belong to the subcategories of rhombuses, rectangles, and squares, and recognize when a shape does not have those attributes.

Example: A quadrilateral with all four sides of different lengths will not be a rhombus, rectangle, or square.

Students know:

• that shapes in different categories may share attributes and that the shared attributes can define a larger category.

Students are able to:

• Identify two-dimensional shapes.
• Sort shapes according to number of sides.
• Sort quadrilaterals based on the presence or absence of square corners.
• Draw examples of squares, rectangles, and rhombuses.
• Draw quadrilaterals that are not rhombuses, rectangles, and squares.

Students understand that:

• Attributes of a shape help make decisions about how to categorize the shape.
• Certain attributes are needed to belong to the subcategories of rhombuses, rectangles, and squares.
• Sometimes a shape does not have the attributes needed to belong to the subcategories of rhombuses, rectangles, and squares.

Learning Objectives:
M.3.26.1: Recall the vocabulary of shapes (labels, sides, faces, vertices, etc.).
M.3.26.2: Recognize and draw shapes having specified attributes such as a given number of angles.
M.3.26.3: Build and draw shapes to possess defining attributes.
M.3.26.4: Sort shapes into categories.