Category: Grade 5 Math

Algebra: Patterns and Graphing

Unit description: Students will begin to learn about the coordinate system.  Students will be able to graph an ordered pair in the first quadrant and consider what that ordered pair might represent.  Students will extend their understanding of patterns to interpret two connected numerical patterns with two rules. 

Essential Outcomes of the Unit  

Geometry- Graph points on the coordinate plane to solve real-world and mathematical problems

5.G.2 Represent real world and mathematical problems by graphing points in the first quadrant of the coordinate plane, and interpret coordinate values of points in the context of the situation.

Other Standards Addressed in the Unit

Operations and Algebraic Thinking- Analyze patterns and relationships.

5.OA.3 Generate two numerical patterns using two given rules. Identify apparent relationships between corresponding terms. Form ordered pairs consisting of corresponding terms from the two patterns, and graph the ordered pairs on a coordinate plane.

Geometry- Graph points on the coordinate plane to solve real-world and mathematical problems

5.G.1 Use a pair of perpendicular number lines, called axes, to define a coordinate system, with the intersection of the lines (the origin) arranged to coincide with the 0 on each line and a given point in the plane located by using an ordered pair of numbers, called its coordinates. Coherence: NY-5.G.1 → NY-6.NS.6 Understand that the first number 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, with the convention that the names of the two axes and the coordinates correspond.

Essential Questions and Big Ideas

What is a coordinate system?

• A coordinate system is created with a pair of perpendicular lines called axes with the intersection of the lines arranged to coincide with 0 on each line (the origin).  
• A given point in the plane can be located by using an ordered pair of numbers called coordinates.  

How can I use the coordinate plane to represent and solve problems? 

• Points can be graphed on a coordinate plane to represent patterns and relationships.  
• The relationship between points on a coordinate plane can be used to solve problems.  

How can I interpret patterns?

• Patterns can be found by interpreting changes in values over time.  

Download the complete Grade 5 Math Unit 6 framework to customize for your own planning.

Geometry and Volume

Unit description: Students will build on their understanding of area from fourth grade to consider 3-D shapes and their volumes.  Students will develop methods for finding the volume of rectangular prisms that include multiplying the area of the base times the height, multiplying length by the width by the height, and counting unit cubes. 

Essential Outcomes of the Unit

Measurement and Data- Geometric measurement: understand concepts of volume and relate volume to multiplication and to addition.

• 5.MD.5 Relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume.
• 5.MD.5a Find the volume of a right rectangular prism with whole-number side lengths by packing it with unit cubes, and show that the volume is the same as would be found by multiplying the edge lengths, equivalently by multiplying the height by the area of the base.
• 5.MD.5b Apply the formulas V = l × w × h and V = B × h for rectangular prisms to find volumes of right rectangular prisms with whole-number edge lengths in the context of solving real world and mathematical problems.
• 5.MD.5c Recognize volume as additive. Find volumes of solid figures composed of two nonoverlapping right rectangular prisms by adding the volumes of the non-overlapping parts, applying this technique to solve real world problems.

Other Standards Addressed in the Unit

Measurement and Data- Geometric measurement: understand concepts of volume and relate volume to multiplication and to addition.

• 5.MD.3 Recognize volume as an attribute of solid figures and understand concepts of volume measurement.
• 5.MD.3a Recognize that a cube with side length 1 unit, called a “unit cube,” is said to have “one cubic unit” of volume, and can be used to measure volume
• 5.MD.3b Recognize that a solid figure which can be packed without gaps or overlaps using n unit cubes is said to have a volume of n cubic units.
• 5.MD.4 Measure volumes by counting unit cubes, using cubic cm, cubic in., cubic ft., and improvised units.

Geometry- Classify two-dimensional figures into categories based on their properties.

• 5.G.3 Understand that attributes belonging to a category of two-dimensional figures also belong to all subcategories of that category.
• 5.G.4 Classify two-dimensional figures in a hierarchy based on properties.

Essential Questions and Big Ideas

• How do I classify quadrilaterals?
  • Quadrilaterals can be classified in multiple ways. 
  • Trapezoids and parallelograms are classified based on parallel sides.
  • Rectangles and squares are classified based on right angles. 
  • Squares and rhombi are classified based on equal side lengths. 
• What is volume?
  • Volume represents the amount of 3-D space a 3-D shape takes up. 
  • Volume is measured in cubic units. 
• What is a “unit cube” and how do I use it to measure volume?
  • A “unit cube” represents one cubic unit. 
  • “Unit cubes” can be combined together to represent a volume. 
• How is volume related to area?
  • Rectangular prisms are built of layers of areas of unit cubes. 
  • The area of a base can be multiplied by a height to find a volume.

Download the complete Grade 5 Math Unit 5 framework to customize for your own planning.

Convert Units of Measure

Unit description: Students will extend their knowledge of metric and customary conversions from fourth grade to work with metric units of more sizes, as well as to convert customary volume and mass units.  

Essential Outcomes of the Unit

Measurement and Data- Convert like measurement units within a given measurement system.

• 5.MD.1 Convert among different-sized standard measurement units within a given measurement system when the conversion factor is given. Use these conversions in solving multi-step, real world problems.

Essential Questions and Big Ideas

How do I convert metric units? 

• The metric system is based on powers of 10.  
• Conversions in the metric system can be completed by multiplying or dividing by powers of 10.  

How do I convert customary length units?

• Conversion factors can be used to convert length units.  
• When converting units, you must consider the relationship between units and their sizes, which is represented as a conversion factor.

How do I convert customary volume units? 

• Conversion factors can be used to convert length units.  
• When converting units, you must consider the relationship between units and their sizes, which is represented as a conversion factor.

How do I convert customary mass units?  

• Conversion factors can be used to convert length units.  
• When converting units, you must consider the relationship between units and their sizes, which is represented as a conversion factor.

Download the complete Grade 5 Math Unit 4 framework to customize for your own planning.

Fractions: Add and Subtract, Multiply and Divide, Line Plots with Fractions

Unit description:  Students will extend their knowledge of adding and subtracting fractions with like denominators from fourth grade to fractions with unlike denominators.  Students will extend their knowledge of multiplying fractions by whole numbers to multiply fractions by other fractions and mixed numbers.  Students will consider the contexts for dividing unit fractions by whole numbers and dividing whole numbers by unit fractions and the relationship of the quotient to the dividend and divisor.  Students will also use their knowledge of fractions to create and interpret line plots with fractional measures on the scale.  

Essential Outcomes of the Unit 

Number and Operations—Fractions- Use equivalent fractions as a strategy to add and subtract fractions.

• 5.NF.1 Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators.
• 5.NF.2 Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators. Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers

Number and Operations—Fractions- Apply and extend previous understandings of multiplication and division to multiply and divide fractions.

• 5.NF.3 Interpret a fraction as division of the numerator by the denominator (𝑎𝑎 𝑏𝑏 = a ÷ b).  Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers.
• 5.NF.4 Apply and extend previous understandings of multiplication to multiply a fraction by a whole number or a fraction.
• 5.NF.4a Interpret the product 𝑎𝑎 𝑏𝑏 × q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a × q ÷ b.
• 5.NF.4b Find the area of a rectangle with fractional side lengths by tiling it with rectangles of the appropriate unit fraction side lengths, and show that the area is the same as would be found by multiplying the side lengths. Multiply fractional side lengths to find areas of rectangles, and represent fraction products as rectangular areas.
• 5.NF.6 Solve real world problems involving multiplication of fractions and mixed numbers.

Measurement and Data- Convert like measurement units within a given measurement system.

• 5.MD.2 Convert among different-sized standard measurement units within a given measurement system when the conversion factor is given. Use these conversions in solving multi-step, real world problems.

Other Standards Addressed in the Unit

Operations and Algebraic Thinking- Operations and Algebraic Thinking

• 5.OA.2 Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them.

Number and Operations—Fractions- Apply and extend previous understandings of multiplication and division to multiply and divide fractions.

• 5.NF.5 Interpret multiplication as scaling (resizing).
• 5.NF.5a Compare the size of a product to the size of one factor on the basis of the size of the other factor, without performing the indicated multiplication.
• 5.NF.5b Explain why multiplying a given number by a fraction greater than 1 results in a product greater than the given number (recognizing multiplication by whole numbers greater than 1 as a familiar case). Explain why multiplying a given number by a fraction less than 1 results in a product smaller than the given number. Relate the principle of fraction equivalence 𝑎𝑎 𝑏𝑏 = 𝑎𝑎 𝑏𝑏 × 𝑛𝑛 𝑛𝑛 to the effect of multiplying 𝑎𝑎 𝑏𝑏 by 1.
• 5.NF.7 Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions. Note: Students able to multiply fractions in general can develop strategies to divide fractions in general, by reasoning about the relationship between multiplication and division. But division of a fraction by a fraction is not a requirement until grade 6 (NY-6.NS.1).
• 5.NF.7a Interpret division of a unit fraction by a non-zero whole number, and compute such quotients.
• 5.NF.7b Interpret division of a whole number by a unit fraction, and compute such quotients.
• 5.NF.7c Solve real-world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions.

Essential Questions and Big Ideas

How do operations with fractions relate to operations with whole numbers?

• Addition, subtraction, multiplication, and division can be completed with fractions.  
• Addition and subtraction still represent putting together or taking apart and part and whole relationships.  
• Multiplication can represent finding a total made from equal groups or scaling. 
• Division represents splitting an amount into equal groups.  

What do equivalent fractions represent and why are they useful when solving equations with fractions?

• Equivalent fractions are fractions that are equal or take up the same amount of space.  
• To add and subtract fractions, they must have the same denominator or unit size.  

How can I find the area of a rectangle with fractional side lengths?

• Length times width equals area.  
• Side lengths can be broken up and multiplied using the distributive property.  

What are the effects of multiplying by quantities greater than 1 compared to the effects of multiplying by quantities less than 1?

• Multiplying by an amount greater than one creates a product that is greater than the other factor. 
• Multiplying by an amount less than one creates a product that is less than the other factor.  

What does it mean to divide by a fraction?  Or to divide a fraction by a whole number?

• Dividing by a fraction represents splitting a fraction into equal parts.  
• Dividing a whole number by a fraction represents splitting a whole into fractional parts.    

Download the complete Fractions: Add and Subtract, Multiply and Divide framework to customize for your own planning.

Four Operations with Decimals

Students will deepen their understanding of decimals as they work with decimals to the thousandths place. Students will extend their strategies for computing with whole numbers to add, subtract, multiply, and divide decimals.

Note: Lessons will vary in length, depending on the amount of time you have with students, the resources that you choose to accompany the unit, the level of rigor within each learning target, and any other factors that may contribute to the pacing of your learning progressions. It is recommended that you adjust the pace and length of each learning progression(s) accordingly in response to these factors.

These learning progressions were developed using Next Generation Learning Standards and were crosswalked with the Common Core Standards.

Download the complete Grade 5 Math Unit 2 framework to customize for your own planning.

Essential Outcomes

Numbers in Base Ten

• NY-5.NBT.7 – Using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between operations:
  • add and subtract decimals to hundredths;
  • multiply and divide decimals to hundredths.
  • Relate the strategy to a written method and explain the reasoning used.
• NY-5.NBT.3 Read, write, and compare decimals to thousandths.
• NY-5.NBT.3b Compare two decimals to thousandths based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons.
• NY-5.NBT.4 – Use place value understanding to round decimals to any place.

Other Standards Addressed in this Unit

Numbers in Base Ten

• NY-5.NBT.1 – Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left.
• NY-5.NBT.2 – Use whole-number exponents to denote powers of 10. Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10.

Essential Questions and Big Ideas

• How do numbers with decimals relate to the base ten number system?
  • Decimals maintain the relationships between digits that are based on powers of ten.
  • A digit in one place is ten times the value of the same digit in the place to the right.
  • A digit in one place is 1/10 the value of the same digit in the place to the left.
  • The patterns in place values continue in the decimal places (tenths, hundredths, thousandths).
  • Numbers with decimals can be written in expanded form and word form.
• How do I add and subtract with decimals?
  • Addition and subtraction is based on the base ten number system.
  • Like place values are added together.
  • If there is not a digit in a certain place value, a zero can be used as a placeholder.
• How do I multiply with decimals?
  • Numbers with decimals can be multiplied in a similar way to whole numbers.
  • When multiplying tenths by tenths, hundredths are created.
  • Multiplying decimal places create smaller decimal places.
• How do I divide with decimals?
  • Numbers with decimals can be divided in a similar way to whole numbers.
  • When dividing by an amount with a decimal, multiply by a power of ten to create a whole number. Multiply the dividend by the same power of ten to keep the relationship between amounts the same.
  • When dividing a decimal by a whole number, maintain the location of the decimal place.
• How do I solve multi-step problems?
  • When solving multi-step problems, identify the unknowns.
  • After identifying the unknowns, identify the actions and relationships within the problem.
  • Write number sentences to represent the actions and relationships within a problem.

Download the complete Grade 5 Math Unit 2 framework to customize for your own planning.

Place Value, Multiplication and Division with Whole Numbers, and Expressions

Students will build on their place value understandings from fourth grade, and begin to compare digits that are to the left of other digits, in addition to the right, multiply three and four digit numbers, and divide with two digit divisors.

Essential Outcomes

Number in Operations in Base Ten

• 5.NBT.1: Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left.
• 5.NBT.2: Use whole-number exponents to denote powers of 10. Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10.
• 5.NBT.5: Fluently multiply multi-digit whole numbers using a standard algorithm.
• 5.NBT.6: Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

Operations and Algebraic Thinking

• 5.OA.1: post-test – Apply the order of operations to evaluate numerical expressions, e.g.:
  • 6+8÷2
  • (6 + 8) ÷ 2
  • Note: Exponents and nested grouping symbols are not included.
• 5.OA.2: post-test – Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. e.g., Express the calculation “add 8 and 7, then multiply by 2” as (8 + 7) × 2. Recognize that 3 × (18,932 + 921) is three times as large as 18,932 + 921, without having to calculate the indicated sum or product

Essential Questions and Big Ideas

• What is the base ten number system and how can I use it to represent numbers?
  • Numbers are based on powers of 10.
  • A digit in one place represents 10 times as much as it represents in the place to its right.
  • A digit in one place represents 1/10 of what it represents in the place to its left.
  • Exponents can be used to represent powers of ten.
  • An exponent is used to indicate how many times to multiply a number (base) by itself. Ex: a^3 = a x a x a
  • Powers of 10 are the values of 10 with different exponents.
  • Powers of 10 represent different place values.
  • Numbers can be written in numeral form, word form, and expanded form.
  • Numbers can be written in expanded form with powers of ten.
• How can I fluently multiply whole numbers?
  • Multiplication represents repeated addition.
  • Multiplication represents finding a total made from equal groups.
  • The distributive property can be used to multiply larger numbers by breaking them up based on place value and multiplying each part.
• How can I fluently divide with whole numbers?
  • Division represents breaking a total into equal groups.
  • When dividing you take away multiples of the divisor until you’ve completed the dividend.
  • When dividing by two-digit divisors, it can help to write out multiples of the divisor.
• What are expressions, and how do I solve them?
  • Expressions are number sentences without an equal sign.
  • Expressions can be written with words or with numbers.
  • The order of operations represents the sequence to complete to solve an expression.
  • Parentheses in an expression can note which steps to complete first in an expression.

Download the complete Grade 5 Math Unit 1 framework to customize for your own planning.