Math - Fifth Grade

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Intro

Resources by Strand

Resources by Strand

Mathematical Practices
(5.MP) Operations and Algebraic Thinking
(5.OA) Number and Operations in Base Ten
(5.NBT) Number and Operations - Fractions
(5.NF) Measurement and Data
(5.MD) Geometry
(5.G)

Strand: MATHEMATICAL PRACTICES (5.MP)

The Standards for Mathematical Practice in Fifth Grade describe mathematical habits of mind that teachers should seek to develop in their students. Students become mathematically proficient in engaging with mathematical content and concepts as they learn, experience, and apply these skills and attitudes.

Standard 5.MP.1
Make sense of problems and persevere in solving them. Explain the meaning of a problem, look for entry points to begin work on the problem, and plan and choose a solution pathway. When a solution pathway does not make sense, look for another pathway that does. Explain connections between various solution strategies and representations. Upon finding a solution, look back at the problem to determine whether the solution is reasonable and accurate, often checking answers to problems using a different method or approach.
Standard 5.MP.2
Reason abstractly and quantitatively. Make sense of quantities and their relationships in problem situations. Contextualize quantities and operations by using images or stories. Decontextualize a given situation and represent it symbolically. Interpret symbols as having meaning, not just as directions to carry out a procedure. Know and flexibly use different properties of operations, numbers, and geometric objects.
Standard 5.MP.3
Construct viable arguments and critique the reasoning of others. Use stated assumptions, definitions, and previously established results to construct arguments. Explain and justify the mathematical reasoning underlying a strategy, solution, or conjecture by using concrete referents such as objects, drawings, diagrams, and actions. Listen to or read the arguments of others, decide whether they make sense, ask useful questions to clarify or improve the arguments, and build on those arguments.
Standard 5.MP.4
Model with mathematics. Identify the mathematical elements of a situation and create a mathematical model that shows the relationships among them. Identify important quantities in a contextual situation, use mathematical models to show the relationships of those quantities, analyze the relationships, and draw conclusions. Models may be verbal, contextual, visual, symbolic, or physical.
Standard 5.MP.5
Use appropriate tools strategically. Consider the tools that are available when solving a mathematical problem, whether in a real-world or mathematical context. Choose tools that are relevant and useful to the problem at hand, such as drawings, diagrams, technologies, and physical objects and tools, as well as mathematical tools such as estimation or a particular strategy or algorithm.
Standard 5.MP.6
Attend to precision. Communicate precisely to others by crafting careful explanations that communicate mathematical reasoning by referring specifically to each important mathematical element, describing the relationships among them, and connecting their words clearly to representations. Calculate accurately and efficiently, and use clear and concise notation to record work.
Standard 5.MP.7
Look for and make use of structure. Recognize and apply the structures of mathematics such as patterns, place value, the properties of operations, or the flexibility of numbers. See complicated things as single objects or as being composed of several objects.
Standard 5.MP.8
Look for and express regularity in repeated reasoning. Notice repetitions in mathematics when solving multiple related problems. Use observations and reasoning to find shortcuts or generalizations. Evaluate the reasonableness of intermediate results.

Strand: OPERATIONS AND ALGEBRAIC THINKING (5.OA)

Write and interpret numerical expressions (Standards 5.OA.1–2)

Core Guide

OER Curriculum

Engage NY

Grade 5 Math Module 2: Multi-Digit Whole Number and Decimal Fraction Operations
In Module 2 students apply patterns of the base ten system to mental strategies and a sequential study of multiplication via area diagrams and the distributive property leading to fluency with the standard algorithm. Students move from whole numbers to multiplication with decimals, again using place value as a guide to reason and make estimations about products. Multiplication is explored as a method for expressing equivalent measures in both whole number and decimal forms. A similar sequence for division begins concretely with number disks as an introduction to division with multi-digit divisors and leads student to divide multi-digit whole number and decimal dividends by two-digit divisors using a vertical written method. In addition, students evaluate and write expressions, recording their calculations using the associative property and parentheses. Students apply the work of the module to solve multi-step word problems using multi-digit multiplication and division with unknowns representing either the group size or number of groups. An emphasis on the reasonableness of both products and quotients, interpretation of remainders and reasoning about the placement of decimals draws on skills learned throughout the module, including refining knowledge of place value, rounding, and estimation.
Grade 5 Math Module 4: Multiplication and Division of Fractions and Decimal Fractions (EngageNY)
Grade 5's Module 4 extends student understanding of fraction operations to multiplication and division of both fractions and decimal fractions. Work proceeds from interpretation of line plots which include fractional measurements to interpreting fractions as division and reasoning about finding fractions of sets through fraction by whole number multiplication. The module proceeds to fraction by fraction multiplication in both fraction and decimal forms. An understanding of multiplication as scaling and multiplication by n/n as multiplication by 1 allows students to reason about products and convert fractions to decimals and vice versa. Students are introduced to the work of division with fractions and decimal fractions. Division cases are limited to division of whole numbers by unit fractions and unit fractions by whole numbers. Decimal fraction divisors are introduced and equivalent fraction and place value thinking allow student to reason about the size of quotients, calculate quotients and sensibly place decimals in quotients. Throughout the module students are asked to reason about these important concepts by interpreting numerical expressions which include fraction and decimal operations and by persevering in solving real-world, multistep problems which include all fraction operations supported by the use of tape diagrams.
Grade 5 Math Module 6: Problem Solving with the Coordinate Plane (EngageNY)
In this 40-day module, students develop a coordinate system for the first quadrant of the coordinate plane and use it to solve problems. Students use the familiar number line as an introduction to the idea of a coordinate, and they construct two perpendicular number lines to create a coordinate system on the plane. Students see that just as points on the line can be located by their distance from 0, the planes coordinate system can be used to locate and plot points using two coordinates. They then use the coordinate system to explore relationships between points, ordered pairs, patterns, lines and, more abstractly, the rules that generate them. This study culminates in an exploration of the coordinate plane in real world applications.

Georgia Standards of Excellence Curriculum Frameworks

Grade 5 Unit 1: Order of Operations & Whole Numbers (Georgia Standards)
In this unit students will solve problems by representing mathematical relationships between quantities using mathematical expressions and equations. Use the four whole number operations efficiently, including the application of order of operations. Write, evaluate, and interpret mathematical expressions with and without using symbols. Apply strategies for multiplying a 2- or 3-digit number by a 2-digit number. Develop paper-and-pencil multiplication algorithms (the U.S. traditional algorithm is not an expectation) for 3- or 4-digit number multiplied by a 2- or 3-digit number.

Instructional Tasks

Bowling for Numbers
The purpose of this game is to help students think flexibly about numbers and operations and to record multiple operations using proper notation.
Comparing Products
The purpose of this task is to generate a classroom discussion that helps students synthesize what they have learned about multiplication in previous grades.
Seeing is Believing
The purpose of this task is to help students see that 4Ã(9+2) is four times as big as (9+2).
Using Operations and Parentheses
The purpose of this task is to give students a chance to work creatively with three of the four fundamental arithmetic operations (addition, subtraction, and multiplication). It is well suited for helping students develop fluency with addition, subtraction, and multiplication of single digit numbers.
Video Game Scores
This task asks students to exercise two complementary skills, writing an expression in part and interpreting a given expression.
Watch Out for Parentheses 1
This problem asks the student to evaluate six numerical expressions that contain the same integers and operations yet have differing results due to placement of parentheses. It helps students see the purpose of using parentheses.
Words to Expressions 1
This problem allows student to see words that can describe the expression from part (c) of "5.OA Watch out for Parentheses." Additionally , the words (add, sum) and (product, multiply) are all strategically used so that the student can see that these words have related meanings.

Review Activitiy

Standard 5.OA.1
Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols.
Standard 5.OA.2
Write and interpret simple numerical expressions.
Write simple expressions that record calculations with numbers. For example, use 2 x (8+7) to express the calculation "add 8 and 7, then multiply by 2."
Interpret numerical expressions without evaluating them. For example, use conceptual understanding of multiplication to interpret 3 x (18939 + 921) as being three times as large as 18932 + 921 without calculating the indicated sum or product.

Analyze patterns and relationships (Standard 5.OA.3)

Standard 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. For example, given the rule "add 3" and the starting number 0, and given the rule "add 6" and the starting number 0, generate terms in the resulting sequences, and observe that the terms in one sequence are twice the corresponding terms in the other sequence. Explain informally why this is so.

Strand: NUMBER AND OPERATIONS IN BASE TEN (5.NBT)

Understand the place value system (Standards 5.NBT.1–4)

Core Guide

OER Curriculum

Engage NY

Grade 5 Math Module 1: Place Value and Decimal Fractions
In Module 1, students understanding of the patterns in the base ten system are extended from Grade 4s work with place value of multi-digit whole numbers and decimals to hundredths to the thousandths place. In Grade 5, students deepen their knowledge through a more generalized understanding of the relationships between and among adjacent places on the place value chart, e.g., 1 tenth times any digit on the place value chart moves it one place value to the right. Toward the modules end students apply these new understandings as they reason about and perform decimal operations through the hundredths place.
Grade 5 Math Module 2: Multi-Digit Whole Number and Decimal Fraction Operations
In Module 2 students apply patterns of the base ten system to mental strategies and a sequential study of multiplication via area diagrams and the distributive property leading to fluency with the standard algorithm. Students move from whole numbers to multiplication with decimals, again using place value as a guide to reason and make estimations about products. Multiplication is explored as a method for expressing equivalent measures in both whole number and decimal forms. A similar sequence for division begins concretely with number disks as an introduction to division with multi-digit divisors and leads student to divide multi-digit whole number and decimal dividends by two-digit divisors using a vertical written method. In addition, students evaluate and write expressions, recording their calculations using the associative property and parentheses. Students apply the work of the module to solve multi-step word problems using multi-digit multiplication and division with unknowns representing either the group size or number of groups. An emphasis on the reasonableness of both products and quotients, interpretation of remainders and reasoning about the placement of decimals draws on skills learned throughout the module, including refining knowledge of place value, rounding, and estimation.

Georgia Standards of Excellence Curriculum Frameworks

Grade 5 Unit 1: Order of Operations & Whole Numbers (Georgia Standards)
In this unit students will solve problems by representing mathematical relationships between quantities using mathematical expressions and equations. Use the four whole number operations efficiently, including the application of order of operations. Write, evaluate, and interpret mathematical expressions with and without using symbols. Apply strategies for multiplying a 2- or 3-digit number by a 2-digit number. Develop paper-and-pencil multiplication algorithms (the U.S. traditional algorithm is not an expectation) for 3- or 4-digit number multiplied by a 2- or 3-digit number.
Grade 5 Unit 2: Adding and Subtracting with Decimals (Georgia Standards)
In this unit students will solve problems by understanding that like whole numbers, the location of a digit in a decimal number determines the value of the digit. Understand that rounding decimals should be sensible for the context of the problem. Understand that decimal numbers can be represented with models. Understand that addition and subtraction with decimals are based on the fundamental concept of adding and subtracting the numbers in like position values.
Grade 5 Unit 3: Multiplying and Dividing with Decimals (Georgia Standards)
General methods used for computing products of whole numbers extend to products of decimals. Because the expectations for decimals are limited to thousandths and expectations for factors are limited to hundredths at this grade level, students will multiply tenths with tenths and tenths with hundredths, but they need not multiply hundredths with hundredths.

Instructional Tasks

Are these equivalent to 9.52?
The purpose of this task is to help students develop the understanding that a single base-ten number can be represented in many different ways.
Comparing Decimals on the Number Line
This task involves using number lines to compare decimal numbers. The numbers selected in this task are purposefully chosen to target student misconceptions.
Drawing Pictures to Illustrate Decimal Comparisons
The purpose of this task is for students to compare decimal numbers using pictures or diagrams. Using such visual representations helps develop a deep understanding of the base-ten system and underscores that the relative place value of the digits can be more important than the value of the digits as numbers between 0 and 9.
Kipton's Scale
The task allows students to explore both the structure of our place value system and how we use that to efficiently multiply and divide powers of 10.
Marta's Multiplication Error
This task highlights a common misconception among students deriving the rules for multiplying a number by a power of 10. It could be used to ground a classroom discussion during the first day of multiplying decimals by powers of 10 or would also be appropriate for a formative assessment to check for student understanding of this pivotal transition from whole number reasoning to decimal reasoning.
Millions and Billions of People
The purpose of this task is to help students understand the multiplicative relationship between commonly used large numbers (millions and billions) by using their understanding of place value.
Multiplying Decimals by 10
The purpose of this task is to help students understand and explain why multiplying a decimal number by 10 shifts all the digits one place to the left.
Placing Thousandths on the Number Line
Though this task primarily deals with comparing decimal numbers on a number line, it also requires students to draw upon what they know about the base ten system.
Rounding to Tenths and Hundredths
The purpose of this task is for students to use the position of a number on the number line to round the number without knowing its exact value. Though this task deals most directly with rounding, it also requires students to understand or figure out that one tenth of 0.1 is 0.01.
Tenths and Hundredths
The purpose of this task is for students to explore the relationship between tenths and hundredths (as well as the relationship between tens and hundreds).
Which number is it?
The purpose of this task is to help students understand the fact that the value of a digit in one place is ten times the value of the same digit in the place to the right.

Review Activitiy

Standard 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.
Standard 5.NBT.2
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. Use whole-number exponents to denote powers of 10.
Standard 5.NBT.3
Read, write, and compare decimals to thousandths.
Read and write decimals to thousandths using base-ten numerals, number names, and expanded form. For example, 347.392 = 3 x 100 + 4 x 10 + 7 x 1 + 3 x (1/10) + 9 x (1/100) + 2 x (1/1000).
Compare two decimals to thousandths based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons.
Standard 5.NBT.4
Use place value understanding to round decimals to any place.

Perform operations with multidigit whole numbers and with decimals to hundredths (Standards 5.NBT.5–7)

Core Guide

OER Curriculum

Engage NY

Grade 5 Math Module 1: Place Value and Decimal Fractions
In Module 1, students understanding of the patterns in the base ten system are extended from Grade 4s work with place value of multi-digit whole numbers and decimals to hundredths to the thousandths place. In Grade 5, students deepen their knowledge through a more generalized understanding of the relationships between and among adjacent places on the place value chart, e.g., 1 tenth times any digit on the place value chart moves it one place value to the right. Toward the modules end students apply these new understandings as they reason about and perform decimal operations through the hundredths place.
Grade 5 Math Module 2: Multi-Digit Whole Number and Decimal Fraction Operations
In Module 2 students apply patterns of the base ten system to mental strategies and a sequential study of multiplication via area diagrams and the distributive property leading to fluency with the standard algorithm. Students move from whole numbers to multiplication with decimals, again using place value as a guide to reason and make estimations about products. Multiplication is explored as a method for expressing equivalent measures in both whole number and decimal forms. A similar sequence for division begins concretely with number disks as an introduction to division with multi-digit divisors and leads student to divide multi-digit whole number and decimal dividends by two-digit divisors using a vertical written method. In addition, students evaluate and write expressions, recording their calculations using the associative property and parentheses. Students apply the work of the module to solve multi-step word problems using multi-digit multiplication and division with unknowns representing either the group size or number of groups. An emphasis on the reasonableness of both products and quotients, interpretation of remainders and reasoning about the placement of decimals draws on skills learned throughout the module, including refining knowledge of place value, rounding, and estimation.
Grade 5 Math Module 4: Multiplication and Division of Fractions and Decimal Fractions (EngageNY)
Grade 5's Module 4 extends student understanding of fraction operations to multiplication and division of both fractions and decimal fractions. Work proceeds from interpretation of line plots which include fractional measurements to interpreting fractions as division and reasoning about finding fractions of sets through fraction by whole number multiplication. The module proceeds to fraction by fraction multiplication in both fraction and decimal forms. An understanding of multiplication as scaling and multiplication by n/n as multiplication by 1 allows students to reason about products and convert fractions to decimals and vice versa. Students are introduced to the work of division with fractions and decimal fractions. Division cases are limited to division of whole numbers by unit fractions and unit fractions by whole numbers. Decimal fraction divisors are introduced and equivalent fraction and place value thinking allow student to reason about the size of quotients, calculate quotients and sensibly place decimals in quotients. Throughout the module students are asked to reason about these important concepts by interpreting numerical expressions which include fraction and decimal operations and by persevering in solving real-world, multistep problems which include all fraction operations supported by the use of tape diagrams.

Georgia Standards of Excellence Curriculum Frameworks

Grade 5 Unit 1: Order of Operations & Whole Numbers (Georgia Standards)
In this unit students will solve problems by representing mathematical relationships between quantities using mathematical expressions and equations. Use the four whole number operations efficiently, including the application of order of operations. Write, evaluate, and interpret mathematical expressions with and without using symbols. Apply strategies for multiplying a 2- or 3-digit number by a 2-digit number. Develop paper-and-pencil multiplication algorithms (the U.S. traditional algorithm is not an expectation) for 3- or 4-digit number multiplied by a 2- or 3-digit number.
Grade 5 Unit 2: Adding and Subtracting with Decimals (Georgia Standards)
In this unit students will solve problems by understanding that like whole numbers, the location of a digit in a decimal number determines the value of the digit. Understand that rounding decimals should be sensible for the context of the problem. Understand that decimal numbers can be represented with models. Understand that addition and subtraction with decimals are based on the fundamental concept of adding and subtracting the numbers in like position values.
Grade 5 Unit 3: Multiplying and Dividing with Decimals (Georgia Standards)
General methods used for computing products of whole numbers extend to products of decimals. Because the expectations for decimals are limited to thousandths and expectations for factors are limited to hundredths at this grade level, students will multiply tenths with tenths and tenths with hundredths, but they need not multiply hundredths with hundredths.

Instructional Tasks

Review Activitiy

Standard 5.NBT.5
Fluently multiply multi-digit whole numbers using the standard algorithm.
Standard 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.
Standard 5.NBT.7
Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used. In this standard, dividing decimals is limited to a whole number dividend with a decimal divisor or a decimal dividend with a whole number divisor. Compare the value of the quotient on the basis of the values of the dividend and divisor.

Strand: NUMBER AND OPERATIONS - FRACTIONS (5.NF)

Use equivalent fractions as a strategy to add and subtract fractions (Standards 5.NF.1–2)

Core Guide

OER Curriculum

Instructional Tasks

Addition of Fractions Using a Visual Model
Adding two fractions with unlike denominators is the focus of this video lesson. Students will learn how to use a visual model to work with these fractions. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort.
Commutative and Associative Equations
This lesson focuses on how to rearrange and combine parts of algebraic expressions by using the commutative and associative properties of addition. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort.
Do These Add Up?
This set of questions is designed to enhance a student's understanding of when it is and is not appropriate to add fractions.
Egyptian Fractions
Because the Egyptians represented fractions differently than we do, this task can help students understand that there can be many ways of representing the same number. This helps prepare them for writing algebraic expressions in 6th grade.
Finding Common Denominators to Add
This task asks students to find and use two different common denominators to add two given fractions. It also ask students to draw pictures to help them to see why finding a common denominator is an important part of solving the given addition problems.
Finding Common Denominators to Subtract
Part of this task asks students to use two different denominators to subtract fractions. The purpose of this is to help students realize that any common denominator will work, not just the least common denominator. They are also asked to draw pictures to help them see why finding a common denominator is important.
Fractions on a Line Plot
The purpose of this task is for students to add unit fractions with unlike denominators and solve addition and subtraction problems involving fractions that have more than one possible solution.
Fractions with Borrowing
This Teaching Channel video shows how students use decomposition to subtract fractions and mixed numbers. (14 min.)
Jog-a-Thon
The purpose of this task is to present students with a situation where it is natural to add fraction with unlike denominators; it can be used for either assessment or instructional purposes.
Making S'Mores
The purpose of this instructional task is to motivate a discussion about adding fractions and the meaning of the common denominator. The different parts of the task have students moving back and forth between the abstract representation of the fractions and the meaning of the fractions in the context.
Mixed Numbers with Unlike Denominators
The purpose of this task is to help students realize there are different ways to add mixed numbers and is most appropriate for use in an instructional setting.
Salad Dressing
The purpose of this task is to have students add fractions with unlike denominators and divide a unit fraction by a whole number. This accessible real-life context provides students with an opportunity to apply their understanding of addition as joining two separate quantities.
Sharing Chocolate
This task was designed to include specific features that support access for all students and align to best practice for English Language Learner (ELL) instruction.
Sharing Lunches
This task requires students to think about how a single situation involving fractions can be accurately represented using addition or multiplication.
To Multiply Or Not to Multiply, Variation 2
This task was written as part of a collaborative project between Illustrative Mathematics, the Smarter Balanced Digital Library, and the Teaching Channel.

Review Activitiy

Standard 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. For example, 2/3 + 5/4 = 8/12 + 15/12 = 23/12. (In general, a/b + c/d = (ad + bc)/bd.)
Standard 5.NF.2
Solve real-world problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators by, for example, using visual fraction models or equations to represent the problem. Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers. For example, recognize 2/5 + 1/2 = 3/7 as an incorrect result, by observing that 3/7 < 1/2.

Apply and extend previous understandings of multiplication and division to multiply and divide fractions (Standards 5.NF.3–7)

Core Guide

OER Curriculum

Engage NY

Grade 5 Math Module 4: Multiplication and Division of Fractions and Decimal Fractions (EngageNY)
Grade 5's Module 4 extends student understanding of fraction operations to multiplication and division of both fractions and decimal fractions. Work proceeds from interpretation of line plots which include fractional measurements to interpreting fractions as division and reasoning about finding fractions of sets through fraction by whole number multiplication. The module proceeds to fraction by fraction multiplication in both fraction and decimal forms. An understanding of multiplication as scaling and multiplication by n/n as multiplication by 1 allows students to reason about products and convert fractions to decimals and vice versa. Students are introduced to the work of division with fractions and decimal fractions. Division cases are limited to division of whole numbers by unit fractions and unit fractions by whole numbers. Decimal fraction divisors are introduced and equivalent fraction and place value thinking allow student to reason about the size of quotients, calculate quotients and sensibly place decimals in quotients. Throughout the module students are asked to reason about these important concepts by interpreting numerical expressions which include fraction and decimal operations and by persevering in solving real-world, multistep problems which include all fraction operations supported by the use of tape diagrams.
Grade 5 Math Module 5: Addition and Multiplication with Volume and Area (EngageNY)
In this 25-day module, students work with two- and three-dimensional figures. Volume is introduced to students through concrete exploration of cubic units and culminates with the development of the volume formula for right rectangular prisms. The second half of the module turns to extending students understanding of two-dimensional figures. Students combine prior knowledge of area with newly acquired knowledge of fraction multiplication to determine the area of rectangular figures with fractional side lengths. They then engage in hands-on construction of two-dimensional shapes, developing a foundation for classifying the shapes by reasoning about their attributes. This module fills a gap between Grade 4's work with two-dimensional figures and Grade 6's work with volume and area.

Georgia Standards of Excellence Curriculum Frameworks

Grade 5 Unit 4: Adding, Subtracting, Multiplying, and Dividing Fractions (Georgia Standards)
Students apply their understanding of fractions and fraction models to represent the addition and subtraction of fractions with unlike denominators as equivalent calculations with like denominators.

Instructional Tasks

Area Model for Multiplication of Fractions
In this lesson and activity students will use area models of fractions to understand how to multiply them. They will also make predictions about results, reduce answers to their simplest forms, and note any patterns they observe. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort.
Banana Pudding
The purpose of this task is to provide students with a concrete situation involving a recipe that they can model by dividing a whole number by a unit fraction.
Calculator Trouble
In this task students are told "Luke had a calculator that will only display numbers less than or equal to 999,999,999." They are then given a list of multiplication problems and asked which of them his calculator would display and they must explain their answer.
Chavone's Bathroom Tiles
This task helps students link the concepts of multiplication and area.
Comparing Heights of Buildings
The goal of this task is to compare three quantities using the notion of multiplication as scaling. Students will recognize (5.NF.B.5) that the Burj Khalifa is taller than the Eiffel tower and that the Eiffel Tower is shorter than the Willis Tower using the size of the given multiplicative scalars.
Comparing a Number and a Product
The purpose of this task is for students to compare a number and its product with other numbers that are greater than and less than one. As written, this task could be used in a summative assessment context, but it might be more useful in an instructional setting where students are asked to explain their answers either to a partner or in a whole class discussion.
Connecting the Area Model to Context
This task is designed to assess students conceptual understanding of the area model (one of the visual fraction models referred to in the standard for multiplying fractions).
Connor and Makayla Discuss Multiplication
The purpose of this task is to have students think about the meaning of multiplying a number by a fraction and use this burgeoning understanding of fraction multiplication to make sense of the commutative property of multiplication in the case of fractions.
Converting Fractions of a Unit into a Smaller Unit
In this task each of these problems students are given a set of a specified size and a specified number of subsets into which it is to be divided. The questions ask the student to find out the size of each of the subsets.
Cornbread Fundraiser
This task is designed to introduce students to an area representation for multiplying a fraction by a fraction.
Cross Country Training
This task was designed to provide students with opportunities to extend their understanding of whole number multiplication to multiplication with fractions.
Dividing a Whole Number by a Unit Fraction
In this lesson a visual model is used to help students learn how a fraction can be used to divide a whole number. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort.
Dividing by One-Half
This task requires students to recognize both "number of groups unknown" and "group size unknown" division problems in the context of a whole number divided by a unit fraction.
Drinking Juice
This is the question for this task: "Alisa had 1/2 a liter of juice in a bottle. She drank 3/4 of the juice that was in the bottle. How many liters of juice did she drink?"
Folding Strips of Paper
The purpose of this task is to provide students with a concrete experience they can relate to fraction multiplication. The task also purposefully relates length and locations of points on a number line, a common trouble spot for students. This task is meant for instruction and would be a useful as part of an introductory unit on fraction multiplication.
Fundraising
This task reads "Cai, Mark, and Jen were raising money for a school trip. Cai collected 2/12 times as much as Mark. Mark collected 2/3 as much as Jen. Who collected the most? Who collected the least? Explain."
Grass Seedlings
This task reads "The students in Raulâs class were growing grass seedlings in different conditions for a science project. He noticed that Pabloâs seedlings were 1 1/2 times a tall as his own seedlings. He also saw that Celinaâs seedlings were 3/4 as tall as his own. Which of the seedlings shown below must belong to which student? Explain your reasoning."
Half of a Recipe
Here is the question for this task: "Kendra is making 1/2 of a recipe. The full recipe calls for 3 1/4 cup of flour. How many cups of flour should Kendra use?"
How Many Marbles?
This task presents this problem: "Julius has 4 blue marbles. If one third of Julius' marbles are blue, how many marbles does Julius have? Draw a diagram and explain."
How Many Servings of Oatmeal?
This task provides a context for performing division of a whole number by a unit fraction.
How Much Pie?
The purpose of this task is to help students see the connection between aÃ·b and a/b in a particular concrete example.
Making Cookies
This task provides an opportunity to discuss unit conversion and rounding in a very realistic context.
Modeling Fraction and Mixed Number Division Using Arrays
Students will learn how to solve word problems that involve dividing fractions and mixed numbers by using a visual model. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort.
Models for the Multiplication and Division of Fraction
This lesson plan shows students what happens when they multiply and divide fractions by using visual area models. The students can also create their own models based on problems they solve.
Mrs. Gray's Homework Assignment
This task is intended to assess students ability to 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.
New Park
Part 1 of this task is designed to elicit student thinking about multiplication of fractions and the commutative property. Part 2 of the task uses the area of a rectangle to help students understand why the commutative property always holds.
Origami Stars
The purpose of this task is to present students with a situation in which they need to divide a whole number by a unit fraction in order to find a solution. Calculating the number of origami stars that Avery and Megan can make illustrates students' understanding of the process of dividing a whole number by a unit fraction.
Painting a Room
The purpose of this task is to provide students with a situation in which it is natural for them to divide a unit fraction by a non-zero whole number.
Painting a Wall
The purpose of this task is for students to find the answer to a question in context that can be represented by fraction multiplication. This task is appropriate for either instruction or assessment depending on how it is used and where students are in their understanding of fraction multiplication.
Reasoning about Multiplication
In this task a rule is posited "When you multiply by a number, you will always get a bigger answer." Students are asked for what numbers will the rule work? For what numbers will the rule not work? Explain and give examples.
Running a Mile
Students are given this statement "Curt and Ian both ran a mile. Curt's time was 8/9 Ian's time." They must determine who ran faster, explain their answer and draw a picture.
Running to School
This task asks for the solution to this problem: "The distance between Rosa's house and her school is 3/4 mile. She ran 1/3 of the way to school. How many miles did she run?"
Salad Dressing
The purpose of this task is to have students add fractions with unlike denominators and divide a unit fraction by a whole number. This accessible real-life context provides students with an opportunity to apply their understanding of addition as joining two separate quantities.
Scaling Up and Down
This task is structured so that a teacher can diagnose the depth of a students understanding of what it means to multiply a number by a fraction, specifically in terms of scaling (as such it can be used formatively or summatively).
Sharing Chocolate
This task was designed to include specific features that support access for all students and align to best practice for English Language Learner (ELL) instruction.
Sharing Lunches
This task requires students to think about how a single situation involving fractions can be accurately represented using addition or multiplication.
Standing in Line
The purpose of this task is for students to solve a problem in context that can be solved in different ways, but in particular by dividing a whole number by a unit fraction.
To Multiply Or Not to Multiply, Variation 2
This task was written as part of a collaborative project between Illustrative Mathematics, the Smarter Balanced Digital Library, and the Teaching Channel.
To Multiply or Not to Multiply?
In this task students are given some problems which can be solved by multiplying 1/82/5, while others need a different operation. Students are to select the ones that can be solved by multiplying these two numbers and for the remaining, tell what operation is appropriate. In all cases they must solve the problem (if possible) and include appropriate units in their answer.
What is 23 divided by 5?
This task involves whole number division problems which do not result in a whole number quotient. It is important for students to be able to decide whether the context requires the result to be reported as a whole number with remainder or a mixed number/decimal.

Review Activitiy

Standard 5.NF.3
Interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b). Solve real-world problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, through the use of visual fraction models or equations to represent the problem. For example, interpret 3/4 as the result of dividing three by four, noting that 3/4 multiplied by four equals three, and that when three wholes are shared equally among four people each person has a share of size 3/4. If nine people want to share a 50-pound sack of rice equally by weight, how many pounds of rice should each person get? Between what two whole numbers does your answer lie?
Standard 5.NF.4
Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.
Interpret the product (a/b) x q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a x q ÷ b using a visual fraction model. For example, use a fraction model to show (2/3) x 4 = 8/3, and create a story context for this equation. Do the same with (2/3) x (4/5) = 8/15. (In general, (a/b) x (c/d) = ac/bd.)
Find the area of a rectangle with fractional side lengths by tiling it with unit squares 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.
Standard 5.NF.5
Interpret multiplication as scaling.
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. For example, the products of expressions such as 5 x 3 or ½ x 3 can be interpreted in terms of a quantity, three, and a scaling factor, five or ½. Thus in addition to knowing that 5 x 3 = 15, they can also say that 5 x 3 is five times as big as three, without evaluating the product. Likewise they see ½ x 3 as half the size of three.
Explain why multiplying a given number by a fraction greater than one results in a product greater than the given number (recognizing multiplication by whole numbers greater than one as a familiar case); explain why multiplying a given number by a fraction less than one results in a product smaller than the given number; and relate the principle of fraction equivalence. For example, 6/10 = (2x3)/(2x5). In general, a/b = (n x a)/(n x b) has the effect of multiplying a/b by one.
Standard 5.NF.6
Solve real-world problems involving multiplication of fractions and mixed numbers, for example, by using visual fraction models or equations to represent the problem.
Standard 5.NF.7
Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions. Use strategies to divide fractions by reasoning about the relationship between multiplication and division. Division of a fraction by a fraction is not a requirement at this grade.
Interpret division of a unit fraction by a non-zero whole number, and compute such quotients. For example, create a story context for (1/3) ÷ 4, and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that (1/3) ÷ 4 = 1/12 because (1/12) x 4 = 1/3.
Interpret division of a whole number by a unit fraction, and compute such quotients. For example, create a story context for 4 ÷ (1/5), and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that 4 ÷ (1/5) = 20 because 20 x (1/5) = 4.
Solve real-world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions, for example, by using visual fraction models and equations to represent the problem. For example, how much chocolate will each person get if three people share 1/2 lb. of chocolate equally? How many 1/3-cup servings are in two cups of raisins?

Strand: MEASUREMENT AND DATA (5.MD)

Convert like measurement units within a given measurement system (Standard 5.MD.1)

Standard 5.MD.1
Convert among different-sized standard measurement units within a given measurement system (for example, convert 5 cm to 0.05 m); use these conversions in solving multi-step, real-world problems.

Represent and interpret data (Standard 5.MD.2)

Standard 5.MD.2
Make a line plot to display a data set of measurements in fractions of a unit (halves, quarters, eighths). Use operations on fractions for this grade to solve problems involving information presented in line plots. For example, given graduated cylinders with different measures of liquid in each, find the amount of liquid each cylinder would contain if the total amount in all the cylinders were redistributed equally.

Understand concepts of geometric measurement and volume, as well as how multiplication and addition relate to volume (Standard 5.MD.3-5)

Standard 5.MD.3
Recognize volume as an attribute of solid figures and understand concepts of volume measurement.
A cube with side length one unit, called a "unit cube," is said to have "one cubic unit" of volume, and can be used to measure volume.
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.
Standard 5.MD.4
Measure volumes by counting unit cubes, using cubic cm, cubic in., cubic ft., and improvised units.
Standard 5.MD.5
Relate volume to the operations of multiplication and addition and solve real-world and mathematical problems involving volume.
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. Represent threefold whole-number products as volumes, for example, to represent the associative property of multiplication.
Apply the formulas V = l x w x h and V = b x 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.
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.

Strand: GEOMETRY (5.G)

Graph points on the coordinate plane to solve real-world and mathematical problems in quadrant one (Standards 5.G.1–2)

Battleship Using Grid Paper
The purpose of this task is to give students practice plotting points in the first quadrant of the coordinate plane and naming coordinates of points. It could be easily adapted to plotting points with negative coordinates. It also provides teachers with a good opportunity to assess how well their students understand how to plot ordered pairs and identify the coordinates of points.
Battleships - Example
Use this link as an example for "Battleship Using Grid Paper" activity.
Cartesian Coordinate System
This lesson is designed to help students understand the Cartesian plane, specifically how to plot points, read coordinates and find the ratio of the rise over run for slope.
Graphing Points on a Coordinate Plane: Harry's Feet Compete
This Cyberchase video follows Harry as he decides to get fit and establish a workout program. He charts his progress on a graph. The classroom activity consists of a ball-balancing game in which students graph their attempts. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort.
Hoop Dream: Use a Diagram to Solve Real-World Math Problems
Students learn about creating diagrams and plotting points on a plane by watching these short videos about kids playing a game involving making baskets. They then play a game where they plot points on a plane. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort.
Horizontal and Vertical Distances on the Cartesian Graph
In this activity students place marine animals on a Cartesian graph and then determine the horizontal and vertical distance between them. The classroom activity builds on the student's understanding of distances between points on a Cartesian graph. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort.
IXL Game: Graph points on a coordinate plane
This game is designed to help fifth graders learn to graph points on a coordinate plane. This is just one of many online games that supports the Utah Math core. Note: The IXL site requires subscription for unlimited use.
Locating Points on the Cartesian Graph
In this activity students use logic and clues to plot the location of marine animals on a Cartesian graph. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort.
Maze Game
In this applet students move a robot through a mine field by using coordinates on the Cartesian Plane in order to remove the land mines.

Standard 5.G.1
Compose and understand the coordinate plane.
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 zero on each line and a given point in the plane located by using an ordered pair of numbers, called its coordinates.
Using quadrant one on the coordinate plane, understand that the first number in a coordinate pair indicates how far to travel from the origin in the direction of the horizontal axis, and the second number indicates how far to travel in the direction of the vertical axis, with the convention that the names of the two axes and the coordinates correspond (x-axis and x-coordinate, y-axis and y-coordinate).
Standard 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.

Classify two-dimensional figures into categories based on their properties. (Standards 5.G.3–4)

Standard 5.G.3
Understand that attributes belonging to a category of two-dimensional figures also belong to all subcategories of that category. For example, all rectangles have four right angles and all squares are rectangles, so all squares have four right angles.
Standard 5.G.4
Classify two-dimensional figures in a hierarchy based on properties.

The Online Core Resource pages are a collaborative project between the Utah State Board of Education and the Utah Education Network. If you would like to recommend a high quality resource, contact Trish French (Elementary) or Lindsey Henderson (Secondary). If you find inaccuracies or broken links contact resources@uen.org.

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Math - Fifth Grade