Courses of Study : Mathematics (Grade 5)

Students:
When given a mathematical expression in words, will

• Write numerical expressions to represent the context and evaluate the expression.
• Explain their thinking as they use the order of operations to evaluate a variety of problems.

Given a numerical expression involving multiple operations and up to two sets of grouping symbols, will

• Evaluate the expression.
• Explain the meaning of the expression without evaluating the expression.

Note: Expressions should not contain nested grouping symbols, should be limited to expressions found in application of associative or distributive properties, and not always limited to whole numbers.

Students know:

• Vocabulary associated with the four operations to write the symbolic notation of the mathematical expression.
  Example: The phrase, "the product of 4 and 3" is written as "4 x 3."
• Strategies for evaluating a numerical expression and replace it with an equivalent form.
  Example: Given (22 + 16) + 43 can be replaced with 38 + 43 and then further simplified.

Students are able to:

• Write, explain, and evaluate numerical expressions representing two-step problems in context.
• Evaluate numerical expressions with grouping symbols.
• Translate a numerical expression into words.
• Write a numerical expression given a mathematical expression in words.

Students understand that:

• multi-step word problems can be represented by numerical expressions using operations and grouping symbols to indicate order of evaluating them.

Learning Objectives:
M.5.1.1: Define parentheses, braces, and brackets.
M.5.1.2: Distinguish between non-numerical and numerical expression.
M.5.1.3: Recognize expressions.
M.5.1.4: Apply properties of operations as strategies to add and subtract.
M.5.1.5: Represent addition and subtraction with objects, mental images, drawings, expressions, or equations.

Students:
Use two related rules to

• Generate numerical patterns and record as ordered pairs.
• Graph ordered pairs consisting of corresponding terms.
• Identify and explain relationships between corresponding terms.

Example: Given the rule "add 1 starting at 0", and "add 2 starting at 0," explain the terms in one sequence as 1/2 the corresponding terms in the other sequence, or the terms in one sequence are twice the corresponding terms in the other sequence.

Students know:

• Strategies to identify numerical patterns and recognize the relationship between the terms in the pattern.
• Reasoning strategies to generate a numerical pattern which follow a given rule.

Students are able to:

• Generate two numerical patterns using two given rules.
• Complete an input/output table for data.
• Identify relationship between terms in an input/output table.
• Form ordered pairs from an input/output table.
• Graph ordered pairs on a coordinate plane.

Students understand that:

• relationships between two numerical patterns can be represented by ordered pairs and graphed in the first quadrant of the coordinate plane.

Learning Objectives:
M.5.2.1: Construct repeating and growing patterns with a variety of representations.
M.5.2.2: Continue an existing pattern.
M.5.2.3: Identify arithmetic patterns (including patterns in the addition table or multiplication table).

Students:

• Use models to illustrate the relationship between two successive place values in whole numbers and decimals.
• Explain that a digit in one place represents 1/10 of what it represents to its left or the place value is 10 times the place value on the right.
• Use strategies to find products and explain patterns when multiplying by powers of 10.
  Example: The product of 420 x 200 = 42 x 10 x 2 x 100 is the same as (42 x 2) x (10 x 100) = 84 x 1000 = 84,000 shows multiplying by three powers of 10 shifts the digits in the product three place values greater (to the left.)
• Use strategies to find products and quotients and use place value understanding to explain patterns in the placement of the decimal point when involving a power of 10.
• Write powers of 10 in standard form and using exponential notation.

Students know:

• Each place value position represents 10 times what it represents in the place to its right.
  Example: In 433, the underlined 3 represents 3 tens and has a value of 30 which is ten times the value of the 3 ones to its right.
• Place value understanding is extended to apply reasoning that a place value position represents 1/10 of what it represents in the place value to its left.
  Example: In 433, the underlined 3 represents 3 ones and has a value of 3 which is one-tenth of the value of the 3 tens or 30 to its left.
• A given number multiplied by a power of 10 shifts the digits in the given number one place value greater (to the left) for each factor of 10. -A given number divided by a power of 10 shifts the digits in the given number one-tenth of the value (to the right) for each factor of 10.

Students are able to:

• Reason and explain the relationship between two successive place values.
• Explain patterns of zeros of the product when multiplying by powers of 10.
• Explain patterns in placement of decimals when multiplying or dividing by power of 10.
• Write powers of 10 using exponential notation.

Students understand that:

• The relationship of adjacent places values in the base ten system extend beyond whole numbers to decimal values.
• Multiplying or dividing by a power of 10 shifts the digits in a whole number or decimal that many places to the left or right respectively.

Learning Objectives:
M.5.3.1: Use place value understanding to round whole numbers to the nearest 10 or 100.
M.5.3.2: Compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits using >, =, and < symbols to record the results of comparisons.
M.5.3.3: Identify that the three digits of a three-digit number represent amounts of hundreds, tens, and ones.

Students:

• Given a decimal number in one form (words, base-ten numerals, expanded), identify the number in another form.
• Read decimals with number names.

Example: Read 4.023 as "four and 23 thousandths."
• Write decimals using base-ten numerals and expanded form.

Example: 4.023 as 4 x 1 + 2 x 1/100 + 3 x 1/1000 or 4 x 1 + 2 x 0.01 + 3 x 0.001.
• Use place value understanding to compare two decimals.

Students know:

• How to read and write whole numbers in standard form, word form, and expanded form.
• How to compare two whole numbers using place value understanding.
• Prior place value understanding with whole numbers is extended and applied to decimal values.
• Recognize and model decimal place value using visual representations to compare.

Students are able to:

• Read and write decimal values in word form, standard form, and expanded form.
• Compare decimals to thousandths using <, >, or = .

Students understand that:

• the adjacent place value relationship in the base ten system extends to decimals and is used to write decimals in expanded form and compare decimals.

Learning Objectives:
M.5.4.1: Recognize decimals as parts of a whole.
M.5.4.2: Compare whole numbers.
M.5.4.3: Write whole numbers in words and expanded form.
M.5.4.4: Read whole numbers.
M.5.4.5: Define expanded notation and standard form.
M.5.4.6: Convert a number written in expanded to standard form.
M.5.4.7: Define hundredths and thousandths.
M.5.4.8: Compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits using >, =, and < symbols to record the results of comparisons.
M.5.4.9: Identify comparison symbols.
Examples: >, =, and <.

Students:
When given a decimal number,

• Use place value vocabulary and models to justify the rounding of the number to a specified place value.
• Rounded to a specified place value, identify a number that could have resulted in that rounding.

Students know:

• how to use place value understanding to round multi-digit whole numbers to any place.

Students are able to:

• Round decimals using place value understanding.

Students understand that:

• in the base ten system, the adjacent place value relationship extends to decimals and is used to round decimals.

Learning Objectives:
M.5.5.1: Round multi-digit whole numbers to any place.
M.5.5.2: Round whole numbers to the nearest 10 or 100.

Students:
When given a context for multiplication of two whole numbers,

• Choose the most appropriate strategy to find the product.
• Accurately use standard algorithm when appropriate.

Students know:

• Strategies based on place value and properties of operations for finding products of two factors including a one-digit and up to a four-digit factor and two two-digit factors.
• Decomposition of a given number into base ten units.
• How to illustrate a product of two factors using an area model.
• Connections between an area model and finding partial products when multiplying.

Students are able to:

• Use the standard algorithm to find a product.

Students understand that:

• properties of operations and the base ten system are foundational to the computation of products using the standard algorithm.

Learning Objectives:
M.5.6.1: Demonstrate steps in setting up a long multiplication problem.
M.5.6.2: Multiply 2-digit numbers by 1-digit multiplier.
M.5.6.3: Multiply 1-digit numbers by 1-digit multiplier.
M.5.6.4: Recall basic multiplication facts.
M.5.6.5: Recall repeated addition facts.

Students:

• Choose strategies based on place value, partial quotients, properties of operations, or the relationship between multiplication and division to find whole number quotients and remainders.
• Solve word problem situations involving division.
• Justify solution path for quotients using equations, arrays or area models.
Note: Standard algorithm for division is not an expectation at grade 5.

Students know:

• Efficient strategies to find a whole number quotient when a multi-digit number (up to 4-digit dividend) is divided by a single-digit divisor.
• How to justify quotients using an illustration or the relationship between multiplication and division.

Students are able to:

• Find whole number quotients and remainders using a variety of strategies based on place value and properties of operations.
• Illustrate and explain the calculation using equations, arrays, and area models.

Students understand that:

• Strategies for division by a one-digit divisor are extended to two-digit divisors.
• Visual models are used to illustrate division.
• Remainders may be written as a fraction or decimal and interpreted based on context of the problem situation.

Learning Objectives:
M.5.7.1: Construct a division equation with an example of the division algorithm.
M.5.7.2: Illustrate the division algorithm using a one-digit divisor and a 2-digit dividend.
M.5.7.3: Identify the place value of a division problem.
M.5.7.4: Restate the inverse process of division as multiplication.
M.5.7.5: Recall basic multiplication facts.

Students:

• Use strategies based on place value, properties of operations, and relationship between addition and subtraction to find sums and differences of decimals.
• Use strategies based on place value, properties of operations, and relationship between multiplication and division to find products and quotients of decimals.
• Use models to justify the sum, difference, product or quotient of decimals.
• Solve real-world problems with decimals to hundredths.

Note: Products are limited to thousandths and quotients are either whole numbers or decimals terminating at the tenths or hundredths place.

Students know:

• Strategies based on place value understanding, properties, and relationship between operations to find the sum, difference, product, and quotient of whole numbers.
• How to write decimal notation for fractions with denominators of 10 or 100.
• Use estimation strategies to assess reasonableness of answers.

Students are able to:

• Use concrete models, drawings, and strategies to add, subtract, multiply, and divide decimals.
• Relate strategies for operations with decimals to a written method and explain reasoning used.
• Solve real-world context problems involving decimals.

Students understand that:
Problems involving operations with decimals

• Can be solved using a variety of strategies based on place value, properties of operations, or the relationship between the operations.
• Can be illustrated using concrete models or drawings.

Learning Objectives:
M.5.8.1: Use decimal notation for fractions with denominators 10 or 100.
M.5.8.2: Multiply and divide within 100, using strategies such as the relationship between multiplication and division or properties of operations.
M.5.8.3: Add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction.
M.5.8.4: Apply properties of operations as strategies to multiply and divide.
M.5.8.5: Identify that 100 can be thought of as a bundle of ten tens, called a "hundred".
M.5.8.6: Identify that the numbers 100, 200, 300, 400, 500, 600, 700, 800, 900 refer to one, two, three, four, five, six, seven, eight, or nine hundreds (and 0 tens and 0 ones).
M.5.8.7: Recall basic addition, subtraction, multiplication, and division facts.

Students:

• Model and solve real-world problems involving sums and differences of fractions (including mixed numbers) with unlike denominators.
• Use visual models to illustrate the problem situation involving fractions.
• Use fraction understanding and estimation strategies to assess the reasonableness of answers.

Students know:

• The meaning and magnitude of fractions expressed in units of halves, fourths, eighths, thirds, sixths, twelfths, fifths, tenths, and hundredths.
• Strategies to find sums of two or more fractions with like denominators.
• Strategies to find the difference of two fractions with like denominators.
• How to decompose a fraction greater than 1 and express as a mixed number.
  Example: 7/3 = 3/3 + 3/3 + 1/3 = 2 1/3.

Students are able to:

• Solve real-word problems involving addition and subtraction of fractions with unlike denominators.
• Represent problems using fraction models or equations.
• Assess reasonableness of answers using estimation and benchmark fractions.

Students understand that:

• solving word problems involving addition and subtraction of fractions with unlike units
• Require strategies to find equivalent fractions in a common unit, and the sum or difference will be expressed in the common unit.
• Can be assessed for reasonableness of answers using estimation strategies.

Learning Objectives:
M.5.9.1: Add and subtract mixed numbers with like denominators.
M.5.9.2: Recognize that comparisons are valid only when the two fractions refer to the same whole.
M.5.9.3: Record the results of comparisons with symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.
M.5.9.4: Recognize a fraction as a number on the number line; represent fractions on a number line diagram.
M.5.9.5: Recognize key terms to solve word problems.
M.5.9.6: Apply properties of operations for addition and subtraction.
M.5.9.7: Recall basic addition and subtraction facts.

Students:

• Use a variety of strategies and fraction equivalence to find sums and differences of fractions and mixed numbers with unlike denominators.

Students know:

• Strategies to determine if two given fractions are equivalent.
• How to use a visual model to illustrate fraction equivalency.
• Contextual situations for addition and subtraction.

Students are able to:

• Use fraction equivalence to add and subtract fractions and mixed numbers with unlike denominators.

Students understand that:
Addition and subtraction of fractions and mixed numbers with unlike units,

• Require strategies to find equivalent fractions in a common unit, and the sum or difference will be expressed in the common unit.
• Can be assessed for reasonableness of answers using estimation strategies.

Learning Objectives:
M.5.10.1: Add and subtract mixed numbers with like denominators, e.g., by replacing each mixed number with an equivalent fraction, and/or by using properties of operations and the relationship between addition and subtraction.
M.5.10.2: Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size.
M.5.10.3: Identify two fractions as equivalent (equal) if they are the same size or the same point on a number line.
M.5.10.4: Recognize and generate simple equivalent fractions.
M.5.10.5: Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers.
M.5.10.6: Compare two fractions with the same numerator or the same denominator by reasoning about their size.
M.5.10.7: Recall basic addition, subtraction, multiplication, and division facts.

Students:

• Solve problems involving division of whole numbers leading to quotients of a fraction or mixed number.

Example: Given that 3 cookies are shared equally with 6 people, find what fraction of the cookies each person receives. Each person receives 3/6 of a cookie or 1/2 of a cookie.
Example: Given that 3 cookies are shared equally with 2 people, find what fraction of the cookies each person receives. Each person receives 3/2 cookies or 1 1/2 cookies.
• Model and interpret a fraction as division.
• Use models, drawings, or equations to represent word problems.

Students know:

• Contextual situations for division.
• Strategies to equipartition.

Students are able to:

• Solve word problems involving division of whole numbers leading to quotients with fractions.
• Use fraction models, drawings, equations to represent word problems.
• Model and interpret a fraction as division.

Students understand that:

• a ÷ b is a division expression and can be written as a/b showing division of the numerator by the denominator (including cases where the value of a < b).

Learning Objectives:
M.5.11.1: Define a mixed number.
M.5.11.2: Generate equivalent fractions.
M.5.11.3: Recognize a fraction as a number on the number line; represent fractions on a number line diagram.

Students:
Given a fraction times a whole number,

• use visual models to illustrate the product to develop the procedure (a/b) × q.
• Create a story context for the equation (a/b) × q.

Given a fraction times a fraction,

• Use visual models to illustrate the product to develop the procedure (a/b) × (c/d).
• Create a story context for the equation (a/b) × (c/d).

Given a rectangle with two fractional side lengths,

• Use an area model to illustrate and find the rectangular area.
• Find the area by tiling it with unit squares of the appropriate unit fraction.

Given a rectangle with fractional side lengths including mixed numbers,

• Use an area model to illustrate and find the rectangular area to lead to answers in the form of whole numbers or mixed numbers.

Example: Using an area model, a rectangle with dimensions of 1 1/2 x 1 2/3 will have partial products of 1, 1/2, 1/3, and 1/6 and the sum of the partial products will give an area of 2 sq units.
• Use an area model to find the area of a rectangle by tiling the rectangle with unit squares.

Example: Using an area model, a rectangle with dimensions 1 1/2 x 1 1/3 will be tiled with unit squares of 1/6 size showing the tiled partial products as 6/6, 3/6, 2/6, and 1/6 for a total area of 12/6 sq units, so it would take 12 tiles of size 1/6 units to cover the area of the rectangle.

Students know:

• How to write an equation involving repeated addition with fractions as a multiplication equation of a whole number times the fraction.
  Example: 2/9 + 2/9 + 2/9 + 2/9 = 4 x 2/9 = 8/9.
• The relationship of partial products to an area model when multiplying by two whole numbers.
• Area of a rectangle is determined by multiplying side lengths and is found in square units.

Students are able to:

• Use previous understandings of multiplication to
• Find products of a fraction times a whole number and products of a fraction times a fraction.
• Use area models, linear models or set models to represent products.
• Create a story context to represent equations (a/b) × q and (a/b) × (c/d) to interpret products.
• Find area of rectangles with fractional side lengths and represent products as rectangular areas.
• Find the area of a rectangle by tiling the area of a rectangle with unit squares.

Students understand that:

• Any whole number can be written as a fraction.
• The general rule for multiplication involving fractions can be justified through visual models.
• A variety of contextual situations can be represented by multiplication involving fractions.
• Tiling with unit squares can be used to find the area of a rectangle with fractional side lengths.

Learning Objectives:
M.5.12.1: Define proper fraction.
M.5.12.2: Multiply fractions using denominators between 2 and 5.
M.5.12.3: Identify proper and improper fractions.
M.5.12.4: Recall basic multiplication facts.
M.5.12.5: Model changing a whole number to a fraction.
M.5.12.6: Partition a rectangle into rows and columns of same-size squares, and count to find the total number of them.
M.5.12.7: Label the numerator and denominator of a fraction.
M.5.12.8: Count the area squares for the length and width.
M.5.12.9: Identify the width and length of a rectangle.

Students:

• Reason about the impact of scaling one or both factors on the size of the product before multiplying and justify their thinking.

Example: Which is greater? 3/5 x 13 or 13 x 3/4? 13 x 3/4 is greater than 3/5 x 13 because both expressions contain a factor of 13, but the scale factor of 3/4 will result in a greater product than a scale factor of 3/5 because 3/4 > 3/5.
• Explain the size of the product when multiplying a number by a fraction greater than 1 and when multiplying a number by a fraction less than 1.

Students know:

• How to interpret multiplicative comparisons.
• Strategies to compare products with whole numbers using reasoning and justification.
  Example: Which is greater? 5 x 2 x 13 or 13 x 9? 10 x 13 is greater than 9 x 13 because both expressions contain a factor of 13, but the scale factor of 10 will result in a greater product than a scale factor of 9.
• Fraction meaning and magnitude of fractions less than and greater than 1.

Students are able to:

• Interpret multiplication as scaling.
• Use reasoning to compare products of multiplication expressions.
• Reason and explain when multiplying a given number by a fraction why the product will be greater than or less than the original number.

Students understand that:

• a product reflects the size of its factors.

Learning Objectives:
M.5.13.1: Define scaling.
M.5.13.2: Define principle of fraction equivalence.
M.5.13.3: Multiply a fraction by a whole number.
M.5.13.4: Compare two fractions with the same numerator or the same denominator by reasoning about their size.
M.5.13.5: Recognize that comparisons are valid only when the two fractions refer to the same whole.
M.5.13.6: Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.
M.5.13.7: Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers.
M.5.13.8: Identify factor and product.
M.5.13.9: Use comparison symbols.
Examples: >, =, or <.

Students:

• Use a variety of strategies, including models, pictures, tables, and patterns to solve problems that provide a context for multiplying fractions and mixed numbers.

Students know:

• Contextual situations for multiplication.
• How to use an area model to illustrate the product of two whole numbers and its relationship to partial products and extend this knowledge to illustrate products involving fractions and mixed numbers.

Students are able to:

• Solve real-word problems involving multiplication of fractions and mixed numbers.
• Write equations to represent the word situation.
• Use visual fraction models to represent the problem.

Students understand that:

• A variety of strategies are used to model and solve problems that provide a context for multiplying fractions and mixed numbers.
• Solutions are interpreted based on the meaning of the quantities and the context of the problem situation.

Learning Objectives:
M.5.14.1: Define improper fraction, mixed number, fraction, equations, numerator, denominator.
M.5.14.2: Multiply proper fractions with common denominators 2-10.
M.5.14.3: Solve problems using whole numbers.
M.5.14.4: Write number sentences for word problems.
M.5.14.5: Identify key terms to solve multiplication word problems.
Examples: times, every, at this rate, each, per, equal/equally, in all, total.
M.5.14.6: Recall basic multiplication facts.

Students:

• Solve real-world problems involving division of a unit fraction by a non-zero whole number, or division of a whole number by a unit fraction.
• Justify solutions using visual models, drawings, and equations to represent the problem context.
• Explain quotients using the relationship between multiplication and division.
• Create a story context for a unit fraction divided by a whole number and use models to illustrate the quotient.
• Create a story context for a whole number divided by a unit fraction and use models to illustrate the quotient.

Students know:

• Contextual situations involving division with whole numbers and unit fractions.
• Strategies for representing a division problem with a visual model.

Students are able to:

• Use previous understandings of operations to
• Divide unit fractions by a whole number and whole numbers by unit fractions.
• Use visual models to illustrate quotients.
• Create story contexts for division.
• Use the relationship between multiplication and division to explain quotients.

Students understand that:

• A variety of contextual situations are represented with division of a whole number by a fraction or a fraction by a whole number.
• Quotients resulting from division of a whole number by a fraction or a fraction by a whole number can be illustrated and justified with a visual model.
• The relationship between multiplication and division can be used to justify quotients resulting from division of a whole number by a fraction or a fraction by a whole number.

Learning Objectives:
M.5.15.1: Define quotient.
M.5.15.2: Multiply a fraction by a whole number.
M.5.15.3: Recognize key terms to solve word problems.
Examples: times, every, at this rate, each, per, equal/equally, in all, total.
M.5.15.4: Recall basic multiplication and division facts.
M.5.15.5: Express whole numbers as fractions.
M.5.15.6: Recognize fractions that are equivalent to whole numbers.
M.5.15.7: Recall basic multiplication and division facts.
M.5.15.8: Solve word problems involving multiplication of a fraction by a whole number.
M.5.15.9: Recognize key terms to solve word problems.
M.5.15.10: Recall basic multiplication and division facts.

Students:
Produce a data set by measuring objects to the nearest one-eighth unit and

• Construct a line plot to display, analyze, and interpret data set.
• Use the four operations to solve problems involving the data set presented in the line plot.
• Use the data set to create problems (involving the four operations) and solve them.

Note: Division is limited to unit fractions by whole numbers and whole numbers by a unit fractions.

Students know:

• Strategies to equipartition a length model.
• Measurement in units of halves, fourths, and eighths using a tool for standard units of measure.
• Strategies to solve problems using the four operations with fractions.

Students are able to:

• Create a line plot with appropriate intervals.
• Represent data on a line plot.
• Apply strategies for solving problems involving all four operations with the fractional data.

Students understand that:

• mathematical data can be collected, analyzed, and organized in a data display to solve problems involving the data.

Learning Objectives:
M.5.16.1: Make a line plot to display a data set of measurements in fractions of a unit.
M.5.16.2: Solve problems involving addition and subtraction of fractions by using information presented in line plots.
M.5.16.3: Draw a scaled picture graph and a scaled bar graph to represent a data set with several categories.
M.5.16.4: Solve one- and two-step "how many more" and "how many less" problems using information presented in scaled bar graphs.
M.5.16.5: Draw a picture graph and a bar graph (with single-unit scale) to represent a data set with up to four categories.
M.5.16.6: Solve simple put-together, take-apart, and compare problems using information presented in a bar graph.

Students:

• Convert different-sized measurement units within the same system.
• Solve multi-step word problems involving conversion of metric or customary units.

Students know:

• Strategies for converting a larger unit of measure to a smaller unit in the same system.
• Relative size of customary and metric units of measure.
• Strategies for converting between units of measure in the same system.

Students are able to:

• Convert measurement units.
• Solve multi-step word problems involving measurement conversions.

Students understand that:

• the multiplicative relationship between units of measures given in the same measurement system is essential when converting units to a larger or smaller unit.

Learning Objectives:
M.5.17.1: Identify relative sizes of measurement units within one system of units, including km, m, cm; kg, g; lb, oz; l, ml; and hr, min, sec.
M.5.17.2: Express measurements in a larger unit in terms of a smaller unit.
M.5.17.3: Solve two-step word problems.
M.5.17.4: Measure and estimate liquid volumes and masses of objects using standard units of grams (g), kilograms (kg), and liters (l).
M.5.17.5: Add, subtract, multiply, or divide to solve one-step word problems involving masses or volumes that are given in the same units, e.g., by using drawings (such as a beaker with a measurement scale) to represent the problem.
M.5.17.6: Recall basic addition, subtraction, multiplication, and division facts.

Students:

• Measure volume of rectangular solids by packing the figure and counting the number of same-sized unit cubes needed to completely fill the figure.

Students know:

• strategies or the formula to find the area of a rectangle.

Students are able to:

• Count unit cubes to find volume.
• Demonstrate volume by packing a solid figure with unit cubes.

Students understand that:

• volume represents the amount of space enclosed in a three-dimensional figure and is measured by the number of same-size cubes that exactly fill the interior space of the object.

Learning Objectives:
M.5.18.1: Define volume including the formulas V = L × W x h, and V = B x h.
M.5.18.2: Define solid figures.
M.5.18.3: Define unit cube.
M.5.18.4: Recognize that shapes in different categories (e.g., rhombuses, rectangles, and others) may share attributes (e.g., having four sides), and that the shared attributes can define a larger category (e.g., quadrilaterals).
M.5.18.5: Describe attributes of three-dimensional figures.
M.5.18.6: Describe attributes of two-dimensional figures.
M.5.18.7: Compare the unit size of volume/capacity in the metric system including milliliters and liters.
M.5.18.8: Define cubic inches, cubic centimeters, and cubic feet.
M.5.18.9: Compare the unit size of volume/capacity in the customary system including fluid ounces, cups, pints, quarts, gallons.
M.5.18.10: Measure areas by counting unit squares (square cm, square m, square in, square ft, and improvised units).
M.5.18.11: Measure and estimate liquid volumes and masses of objects using standard units of grams (g), kilograms (kg), and liters (l).
M.5.18.12: Add, subtract, multiply, or divide to solve one-step word problems involving masses or volumes that are given in the same units, e.g., by using drawings (such as a beaker with a measurement scale) to represent the problem.
M.5.18.13: Recall basic multiplication facts.
M.5.18.14: Fluently add.

Students:
Given right rectangular prisms with whole number edge lengths,

• Use associative property of multiplication to find volume and relate it to packing a solid with unit cubes.
• Apply formula V = l × w × h, where V represents volume and l, w, and h represent the three dimensions of the prism (length, width, height) and relate the formula to a unit cube filled model.
• Apply formula V = B × h, where V represents volume, B is the base-area, and h represents the height (number of layers of the base-area) and relate the formula to a unit cube filled model.

• Given a solid figure composed of two or more right rectangular prisms in real-world or mathematical contexts, find the total volume by decomposing the figure into non-overlapping rectangular prisms and find the sum of the volumes.

Students know:

• Measurable attributes of area and how it relates to finding the volume of objects.
• Units of measurement for volume, specifically unit cubes.

Students are able to:

• Solve word problems involving volume.
• Use associative property of multiplication to find volume.
• Relate operations of multiplication and addition to finding volume.
• Apply formulas to find volume of right rectangular prisms.
• Find volume of solid figures composed of two rectangular prisms.

Students understand that:

• Volume is a derived attribute based on a length unit and can be computed as the product of three length measurements or as the product of one base area and one length measurement.
• Volume is an extension of area and can be found as the area of the base being repeated for a given number of layers.

Learning Objectives:
M.5.19.1: Define volume.
M.5.19.2: Recognize angle measure as additive.
M.5.19.3: Apply the area and perimeter formulas for rectangles in real-world and mathematical problems.
M.5.19.4: Solve real-world and mathematical problems involving perimeters of polygons, including finding the perimeter given the side lengths, finding an unknown side length, and exhibiting rectangles with the same perimeter and different areas or with the same area and different perimeters.
M.5.19.5: Recognize the formula for volume.
M.5.19.6: Recall the attributes of three-dimensional solids.
M.5.19.7: Recall basic multiplication facts.
M.5.19.8: Fluently add.
M.5.19.9: Compare the unit size of volume/capacity in the metric system including milliliters and liters.
M.5.19.10: Measure and estimate liquid volumes.
M.5.19.11: Recall basic multiplication facts.
M.5.19.12: Compare the unit size of volume/capacity in the metric system including milliliters and liters.
M.5.19.13: Recognize the formula for volume.
M.5.19.14: Recall basic multiplication facts.
M.5.19.15: Describe attributes of three-dimensional figures.
M.5.19.16: Describe attributes of two-dimensional figures.
M.5.19.17: Identify solid figures.

Students:

• Use the first quadrant in a coordinate plane to identify coordinates of a given point.
• Use the first quadrant in a coordinate plane to explain how the location of an ordered pair is determined.
• Given a real-world situation involving a relationship between two variables, graph a representation of the situation and interpret coordinate values of the points in the context of the problem.
• Given a graph representing a real-world situation, interpret the coordinate values of the points in the context of the situation.

Students know:

• Specific directions and vocabulary to explain ordered pair location.
• The first number of an ordered pair indicates how far to travel from the origin in the direction of one axis and the second number indicates how far to travel in the direction of the second axis.

Students are able to:

• Graph points in the first quadrant.
• Interpret coordinate values in context of the problem.

Students understand that:

• graphing points on a coordinate plane provides a representation of a mathematical context which aids in visualizing situations and solving problems.

Learning Objectives:
M.5.20.1: Define ordered pair of numbers, quadrant one, coordinate plane, and plot points.
M.5.20.2: Label the horizontal axis (x).
M.5.20.3: Label the vertical axis (y).
M.5.20.4: Identify the x- and y- values in ordered pairs.
M.5.20.5: Model writing ordered pairs.

Students:
When given a variety of triangles,

• Measure sides and angles to classify triangles based on side length and angle measure.

Students know:

• Measurable attributes of triangles include length of side and angle measures.
• Appropriate tools and units of measure for length of side and angle measures.

Students are able to:

• Classify triangles according to side measures and angle measures.

Students understand that:

• triangles can be described and classified by their properties of side length, angle size, or cross-classify to include both side length and angle size.

Learning Objectives:
M.5.21.1: Define isosceles, equilateral, scalene, right and equiangular triangles; obtuse, acute, and right angle; vertex/vertices.
M.5.21.2: Identify a right triangle.
M.5.21.3: Sort and categorize shapes.
M.5.21.4: Recognize and draw shapes having specified attributes.

Students:
When given variety of two-dimensional figures,

• Use attributes of shapes to explain their classification in as many categories and subcategories as possible.
• Distinguish properties that are more general from those that are more specific and make connections between and within categories of figures.

Example: A quadrilateral is a figure with a general property of 4-sides, while a parallelogram is a specific type of quadrilateral with two pairs of opposite sides which are both parallel and congruent. Based on this hierarchy, all parallelograms are quadrilaterals, but not all quadrilaterals are parallelograms.

Students know:

• properties or attributes of two-dimensional shapes.

Students are able to:

• Classify quadrilaterals based on properties.

Students understand that:

• Quadrilaterals can be identified by general properties to more specific properties.
• Properties belonging to a category of two-dimensional figures also belong to all subcategories of that category.

Learning Objectives:
M.5.22.1: Define vertex/vertices and angle.
M.5.22.2: Identify that shapes in different categories (e.g., rhombuses, rectangles, and others) may share attributes (e.g., having four sides), and that the shared attributes can define a larger category (e.g., quadrilaterals).
M.5.22.3: Recognize and draw shapes having specified attributes such as a given number of angles or a given number of equal faces.
M.5.22.4: Identify triangles, quadrilaterals, pentagons, hexagons, heptagons, and octagons based on the number of sides, angles, and vertices.

Students:
Given a variety of two-dimensional figures,

• Use attributes to classify a shape into categories and subcategories.
• Use mathematical vocabulary to explain that attributes belonging to one category are also attributes of the subcategory.

Students know:

• vocabulary associated with the properties of shapes.

Students are able to:

• Explain the relationship between shapes in categories and subcategories.

Students understand that:

• Quadrilaterals can be identified by general properties to more specific properties.
• Properties belonging to a category of two-dimensional figures also belong to all subcategories of that category.

Learning Objectives:
M.5.23.1: Understand that shapes in different categories (e.g., rhombuses, rectangles, and others) may share attributes (e.g., having four sides), and that the shared attributes can define a larger category (e.g., quadrilaterals).
M.5.23.2: Recognize rhombuses, rectangles, and squares as examples of quadrilaterals, and draw examples of quadrilaterals that do not belong to any of these subcategories.
M.5.23.3: Recognize attributes of shapes.
M.5.23.4: Recall the vocabulary of shapes (labels, sides, faces, vertices, etc.).
M.5.23.5: Sort shapes into categories.