For the MD College and Career Ready Standards

Fifth Grade Mathematics Curriculum Guide for the MD College and Career Ready Standards

2014-2015

Fifth Grade Overview Fifth Grade Mathematics Curriculum for MD College and Career Ready Standards

Operations and Algebraic Thinking (OA)  Write and interpret numerical expressions.  Analyze patterns and relationships.

Number and Operations in Base Ten (NBT)  Understand the place value system.  Perform operations with multi-digit whole numbers and with decimals to hundredths.

Number and Operations—Fractions (NF)  Use equivalent fractions as a strategy to add and subtract fractions.  Apply and extend previous understandings of multiplication and division to multiply and divide fractions.

Measurement and Data (MD)  Convert like measurement units within a given measurement system.  Represent and interpret data.  Geometric measurement: understand concepts of volume and relate volume to multiplication and to addition.

Geometry (G)  Graph points on the coordinate plane to solve real-world and mathematical problems.  Classify two-dimensional figures into categories based on their properties.

Major Cluster Supporting Cluster Additional Cluster Standards for Mathematical Practice Standards Explanations and Examples 1. Make sense of Students solve problems by applying their understanding of operations with whole numbers, decimals, and fractions including

Grade 5 Wicomico County Public Schools 2014-2015 2 Fifth Grade Mathematics Curriculum for MD College and Career Ready Standards problems and mixed numbers. They solve problems related to volume and measurement conversions. Students seek the meaning of a persevere in solving problem and look for efficient ways to represent and solve it. They may check their thinking by asking themselves, ―What is them. the most efficient way to solve the problem? ―”Does this make sense?”, and ―”Can I solve the problem in a different way?”

2. Reason abstractly Fifth graders should recognize that a number represents a specific quantity. They connect quantities to written symbols and and quantitatively. create a logical representation of the problem at hand, considering both the appropriate units involved and the meaning of quantities. They extend this understanding from whole numbers to their work with fractions and decimals. Students write simple expressions that record calculations with numbers and represent or round numbers using place value concepts. 3. Construct viable In fifth grade, students may construct arguments using concrete referents, such as objects, pictures, and drawings. They arguments and explain calculations based upon models and properties of operations and rules that generate patterns. They demonstrate and critique the reasoning explain the relationship between volume and multiplication. They refine their mathematical communication skills as they of others. participate in mathematical discussions involving questions like ―”How did you get that?” and ―”Why is that true?” They

explain their thinking to others and respond to others’ thinking. 4. Model with Students experiment with representing problem situations in multiple ways including numbers, words (mathematical mathematics. language), drawing pictures, using objects, making a chart, list, or graph, creating equations, etc. Students need opportunities to connect the different representations and explain the connections. They should be able to use all of these representations as needed. Fifth graders should evaluate their results in the context of the situation and whether the results make sense. They also evaluate the utility of models to determine which models are most useful and efficient to solve problems. 5. Use appropriate Fifth graders consider the available tools (including estimation) when solving a mathematical problem and decide when tools strategically. certain tools might be helpful. For instance, they may use unit cubes to fill a rectangular prism and then use a ruler to measure the dimensions. They use graph paper to accurately create graphs and solve problems or make predictions from real world data. 6. Attend to Students continue to refine their mathematical communication skills by using clear and precise language in their discussions precision. with others and in their own reasoning. Students use appropriate terminology when referring to expressions, fractions, geometric figures, and coordinate grids. They are careful about specifying units of measure and state the meaning of the symbols they choose. For instance, when figuring out the volume of a rectangular prism they record their answers in cubic units. 7. Look for and In fifth grade, students look closely to discover a pattern or structure. For instance, students use properties of operations as make use of strategies to add, subtract, multiply and divide with whole numbers, fractions, and decimals. They examine numerical patterns structure. and relate them to a rule or a graphical representation. 8. Look for and Fifth graders use repeated reasoning to understand algorithms and make generalizations about patterns. Students connect express regularity in place value and their prior work with operations to understand algorithms to fluently multiply multi-digit numbers and perform

Grade 5 Wicomico County Public Schools 2014-2015 3 Fifth Grade Mathematics Curriculum for MD College and Career Ready Standards repeated reasoning. all operations with decimals to hundredths. Students explore operations with fractions with visual models and begin to formulate generalizations.

GRADE 5 MD COLLEGE AND CAREER READY STANDARDS INTRODUCTION

In Grade 5, instructional time should focus on three critical areas: (1) developing fluency with addition and subtraction of fractions, and developing understanding of the multiplication of fractions and of division of fractions in limited cases (unit fractions divided by whole numbers and whole numbers divided by unit fractions); (2) extending division to 2-digit divisors, integrating decimal fractions into the place value system and developing understanding of operations with decimals to hundredths, and developing fluency with whole number and decimal operations; and (3) developing understanding of volume.

1. 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. They develop fluency in calculating sums and differences of fractions, and make reasonable estimates of them. Students also use the meaning of fractions, of multiplication and division, and the relationship between multiplication and division to understand and explain why the procedures for multiplying and dividing fractions make sense. (Note: this is limited to the case of dividing unit fractions by whole numbers and whole numbers by unit fractions.)

2. Students develop understanding of why division procedures work based on the meaning of base-ten numerals and properties of operations. They finalize fluency with multi-digit addition, subtraction, multiplication, and division. They apply their understandings of models for decimals, decimal notation, and properties of operations to add and subtract decimals to hundredths. They develop fluency in these computations, and make reasonable estimates of their results. Students use the relationship between decimals and fractions, as well as the relationship between finite decimals and whole numbers (i.e., a finite decimal multiplied by an appropriate power of 10 is a whole number), to understand and explain why the procedures for multiplying and dividing finite decimals make sense. They compute products and quotients of decimals to hundredths efficiently and accurately.

3. Students recognize volume as an attribute of three-dimensional space. They understand that volume can be measured by finding the total number of same-size units of volume required to fill the space without gaps or overlaps. They understand that a 1-unit by 1-unit by 1-unit cube is the standard unit for measuring volume. They select appropriate units, strategies, and tools for solving problems that involve estimating and measuring volume. They decompose three-dimensional shapes and find volumes of right rectangular prisms by

Grade 5 Wicomico County Public Schools 2014-2015 4 Fifth Grade Mathematics Curriculum for MD College and Career Ready Standards viewing them as decomposed into layers of arrays of cubes. They measure necessary attributes of shapes in order to determine volumes to solve real world and mathematical problems.

Fifth Grade Math At-A-Glance 2014 – 2015

Units *You can quickly jump to a unit Important Dates by selecting it. Unit 1: Developing Multiplication and Division Strategies September 1 Labor Day through Volume (23 days) September 26 Professional Day October 17 MSEA Convention Unit 2: Extend Place Value to Decimals (21 days) Mathematics Intermediate Interim Assessment 1 - October 21 October 28 End term 1 November 3 Professional Day November 4 Election Day Unit 3: Fraction Operations (33 days) November 17–21 American Education Week November 26–28 Thanksgiving Holiday Mathematics Intermediate Interim Assessment 2 – December 11 December 22 January 2 Winter Break January 20 Martin Luther King’s birthday Unit 4: Decimal Operations (15 days) January 23 End term 2 January 26 Professional Day February 13 Professional Day Unit 5: Solving Problems with Fractional Quantities (21 days) February 16 Presidents’ Day

PARCC PBA 3/2 – 3/27 (window)

Unit 6: Classifying 2D Figures (15 days)

March 27 – End of term 3

Grade 5 Wicomico County Public Schools 2014-2015 5 Fifth Grade Mathematics Curriculum for MD College and Career Ready Standards

Unit 7: Representing Algebraic Thinking ( 12 days) April 1-6 Spring Break April 7 Professional Day PARCC EOY 4/20 – 5/15 (window)

Unit 8: Coordinate Plane ( 15 days)

Unit 9: Working Towards Fluency with Fractions (13 days) Mathematics Placement Assessment GR 5 – May 22 Unit 10: Multiplication & Division of Whole Nos. – Fluency (12 May 25 – Memorial Day days)

Unit One - Developing Multiplication and Division Strategies through Volume (23 days) The Grade 5 year begins with developing conceptual understanding of volume as an attribute of solid figures. This is a new concept for grade 5 and provides an engaging context that supports problem solving throughout the year. Students practice and refine their multiplication and division strategies, attaining fluency in multiplication with whole numbers by the end of the year. Students expand their understandings of geometric measurement and spatial structuring to include volume as an attribute of three-dimensional space. In this unit, students develop this understanding using concrete models to discover strategies for finding volume, and then go on to use this understanding while solving real-world problems and apply strategies and formulas. Volume is a major emphasis in Grade 5. The connection to multiplication and addition provides an opportunity for students to start the year off by applying the multiplication and addition strategies they learned in previous grades in a new interesting context. Also in this unit, students build on their work from previous grade levels to refine their strategies for multiplication and division in order to reach fluency in multiplication of multi-digit whole numbers by the end of the year. Students continue to develop more sophisticated strategies for division (including quotients with remainders 4.NBT.B.6) to become flexible and efficient with the standard algorithm in Grade 6. Students begin to find quotients with two digit divisors early in the year to build strategies for accurate computations. Since students have solved two step word problems using the four operations in third grade and multi-step equations in 4th grade, the understanding of order or operations within the four operations should have been mastered.

It is important that students continue to multiply or divide to solve word problems involving multiplicative comparison (4.OA.A.2.), e.g., by using drawings/bar models and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison. Multiplicative comparison problems involve a comparison of two quantities in which one is described as a multiple of the other. The relation between quantities is described in terms of how many times larger one is than the other. The number quantifying this relationship is not an identifiable quantity. Table 2 illustrates all of the story structures that should be used for multiplication and division word problems.

Multiplicativ Smaller Unknown: Larger Unknown: Multiplier Unknown: e Comparison The giraffe is 18 feet tall. She is 3 times The giraffe in the zoo is 3 times as tall as the kangaroo. The The giraffe is 18 feet tall. The th 4 Grade as tall as the kangaroo. How tall is the kangaroo is 6 feet tall. How tall is the giraffe? kangaroo is 6 feet tall. The giraffe is kangaroo? how many times taller than the

4.OA.2 Measurement Example: kangaroo? Measurement Example: A rubber band is 6 centimeters long. How long will the A rubber band is stretched to be 18 rubber band be when it is stretched to be 3 times as long? Measurement Example: centimeters long and that is 3 times as A rubber band was 6 centimeters long long as it was at first. How long was the at first. Now it is stretched to be 18 rubber band at first? centimeters long. How many times as long is the rubber band now as it was at first? The giraffe is 18 feet tall. The kangaroo The kangaroo is 6 feet tall. She is 1/3 as tall as the giraffe. The giraffe is 18 feet tall. The is 1/3 as tall as the giraffe. How tall is How tall is the giraffe? kangaroo is 6 feet tall. What fraction 5th Grade the kangaroo? of the height of the giraffe is the height of the kangaroo?

Unit One Standards 5.MD.C.3 Recognize volume as an attribute of solid figures and understand concepts of volume measurement. a. 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. b. 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.C.4 Measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and improvised units

5.MD.C.5 Relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume. a. 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, e.g., to represent the associative property of multiplication. b. 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. c. Recognize volume as additive. Find volumes of solid figures composed of two non-overlapping right rectangular prisms by adding the volumes of the non- overlapping parts, applying this technique to solve real world problems. 5.NBT.B.5 Fluently multiply multi-digit whole numbers using the standard algorithm. 5.NBT.B.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.

Math Language

The terms below are for teacher reference. Although these do not need to be memorized by the students, students should be encouraged to use these math terms in explanations or justifications. algorithm multiplicand remainder height associative property multiplier layers decompose order of operations (conventional Volume measurement distributive property order) area of base nonstandard cubic units dividend partition division (fair-sharing) attribute overlap divisor partial product capacity right rectangular prism equation partial quotient cubic units solid figure exponents product cubic cm, cubic in, cubic ft unit cube expression properties of operations edge lengths variable measurement division quotient gap

Essential Questions How do we measure volume? How are area and volume alike and different? How can you find the volume of cubes and rectangular prisms? How do you convert volume between units of measure? What is the relationship between the volumes of geometric solids? Why are some tools better to use than others when measuring volume? Why is volume represented with cubic units and area represented with square units? How can estimating help us when solving multiplication problems? What strategies can we use to efficiently solve multiplication problems? How can I use what I know about multiplying multiples of ten to multiply two whole numbers? How can estimating help us when solving division problems? What strategies can we use to efficiently solve division problems?

Common Misconceptions 1. When solving problems that require regrouping units, students use their knowledge of regrouping the numbers as with whole numbers. Students need to pay attention to the unit of measurement which dictates the regrouping and the number to use. The same procedures used in regrouping whole numbers should not be taught when solving problems involving measurement conversions.

2. Some students may think the term “square” refers only to the geometric figure with equal length sides. They will need to understand that area of any rectangle is measured in square units. The same idea may be present in “cubic units”. Students may think it only has to do with the geometric solid “cube”. They need to understand that “cubic units” are used to measure any rectangular prism.

Grade 5 Unit One – Developing Multiplication and Division Strategies through Volume (23 days) Back to “Year at a Glance”

Standard Connections/Notes Resources 5.NBT.B.5 Fluently multiply In prior grades, students used various strategies to multiply a whole Teaching Student Centered Mathematics multi-digit whole numbers number of up to four digits by a one-digit whole number, and multiply Invented Strategies for Multiplication pgs. 113-115 using the standard algorithm. two two-digit numbers. Strategies were based on place value and the Area Model Development pgs. 116-117 properties of operations and were illustrated and explained by using Traditional Algorithm for Multiplication pgs. 118-120 equations, rectangular arrays, and/or area models. Expanded Lesson Area Model Multiplication pgs. 129- 130 Students can continue to use these different strategies as long as they are efficient, but must also understand and be able to use the Lessons standard algorithm. In applying the standard algorithm, students Using Partial Products and Visual Models for recognize the importance of place value. Multiplication The area model for multiplication is a pictorial way of representing Multiply by Multiples of Ten Using Patterns and multiplication. In the area model, the length and width of a rectangle

represent factors, and the area of the rectangle represents their Properties product. An area model not only helps to explain why the standard algorithm Activities and Tasks commonly taught in the United States for multiplication works, it is an Toss a Rectangle efficient recording alternative. Elmer’s Multiplication Error (IM)

Some students (especially visual learners and those who have Videos difficulty keeping numbers lined up in multiplication problems) may Multiply multi-digit numbers using an area model (LZ) prefer it. Furthermore, this method has certain benefits. It illuminates Area Models for Multiplication using Kidspiration important mathematical concepts (such as the distributive property), allows for computational flexibility (expanded notations allow students Online to use derived facts), and reinforces the concept of area. Rectangle Multiplication (common) NLVM Templates and Visuals

Standard Connections/Notes Resources Van de Walle Template Partial Products - #36 Supplemental Unit on Area Models Partial Product Template Area Model and Partial Products Example

Examples: 123 x 34. When students apply the standard algorithm, they, decompose 34 into 30 + 4. Then they multiply 123 by 4, the value of the number in the ones place, and then multiply 123 by 30, the value of the 3 in the tens place, and add the two products.

5.NBT.B.6. Find whole-number In fourth grade, students’ experiences with division were limited to Teaching Student Centered Mathematics quotients of whole numbers dividing by one-digit divisors. This standard extends students’ prior Invented Strategies for Division pgs. 121-123 with up to four-digit dividends experiences with strategies, illustrations, and explanations. When the and two-digit divisors, using two-digit divisor is a ―familiar number, a student might decompose Activities and Tasks strategies based on place the dividend using place value. Minutes and Days (IM)

Standard Connections/Notes Resources value, the properties of Partition the Dividend (K5) operations, and/or the Example: Division Strategy Multiplying Up (K5) relationship between  Using expanded notation ~ 2682 ÷ 25 = Soda Task multiplication and division. (2000 + 600 + 80 + 2) ÷ 25 Illustrate and explain the Using his or her understanding of the relationship between 100 and Videos calculation by using equations, 25, a student might think ~ Using Base 10 Blocks to Model Long Division (UT) rectangular arrays, and/or area I know that 100 divided by 25 is 4 so 200 divided by 25 is 8 and Division (chunking) (UT) models. 2000 divided by 25 is 80. 600 divided by 25 has to be 24. Since 3 x 25 is 75, I know that 80 Divide-4digit-dividends-by-2-digit-divisors-by-setting- divided by 25 is 3 with a reminder of 5. (Note that a student might up-an-equation (LZ) divide into 82 and not 80) I can’t divide 2 by 25 so 2 plus the 5 leaves a remainder of 7. 80 + 24 + 3 = 107. So, the answer is 107 Online with a remainder of 7. Rectangle Division NLVM The Quotient Café - Illuminations Using an equation that relates division to multiplication, 25 x n = 2682, a student might estimate the answer to be slightly larger than 100 Templates and Visuals because s/he recognizes that 25 x 100 = 2500. Division Using the Area Model

Examples: 968 ÷ 21  Using base ten models, a student can represent 962 and use the models to make an array with one dimension of 21. The student continues to make the array until no more groups of 21 can be made. Remainders are not part of the array

Standard Connections/Notes Resources

9984 ÷ 64  An area model for division is shown below. As the student uses the area model, s/he keeps track of how much of the 9984 is left to divide.

5. MD.C.3 Recognize volume Students’ prior experiences with volume were restricted to liquid Teaching Student Centered Mathematics as an attribute of solid figures volume. As students develop their understanding of volume, they Activity 9.10 “Box Comparison – Cubic Units” pg. 267 and understand concepts of understand that a 1-unit by 1-unit by 1-unit cube is the standard unit volume measurement. for measuring volume. This cube has a length of 1 unit, a width of 1 Lessons unit and a height of 1 unit and is called a cubic unit. This cubic unit is Estimating Volume - Counting on Frank - Illuminations

a. A cube with side length 1 written with an exponent of 3 (e.g., in³, m³). Students connect this Collecting the Rays unit, called a “unit cube,” is notation to their understanding of powers of 10 in our place value Finding Volume by Packing with Cubic Units and said to have “one cubic unit” of system. Models of cubic inches, centimeters, cubic feet, etc. are Counting volume, and can be used to helpful in developing an image of a cubic unit. Student’s estimate how measure volume. many cubic yards would be needed to fill the classroom or how many Activities and Tasks b. A solid figure which can be cubic centimeters would be needed to fill a pencil box. Build a Cubic Meter (K5) packed without gaps or Rectangular Prism Nets for Packing overlaps using n unit cubes is Insides and Outsides - Utah Education Network (Do not said to have a volume of n extend to surface area.)

Standard Connections/Notes Resources cubic units. Videos Understanding Volume LZ

Online Cubes Illuminations 5.MD.C.4 Measure volumes by Students understand that same sized cubic units are used to measure Teaching Student Centered Mathematics counting unit cubes, using volume. They select appropriate units to measure volume. For Activity 9.10 “Box Comparison – Cubic Units” pg. 267 cubic cm, cubic in, cubic ft, and example, they make a distinction between which units are more improvised units. appropriate for measuring the volume of a gym and the volume of a Lessons box of books. They can also improvise a cubic unit using any unit as a Estimating Volume - Counting on Frank - Illuminations length (e.g., the length of their pencil). Students can apply these ideas Collecting the Rays - Illuminations by filling containers with cubic units (wooden cubes) to find the Fill ‘Er Up - Illuminations volume. They may also use drawings or interactive computer software Insides and Outsides (Do not extend to surface area.) to simulate the same filling process. Explore Volume by Counting Unit Cubes

Activities and Tasks What’s the Volume (K5) Ordering Rectangular Prisms by Volume (K5) Building Rectangular Prisms with a Given Volume (K5)

Videos Identify and label three dimensional figures (LZ) Find-volume-by-counting-cubes (LZ)

Online Cubes - Illuminations

Templates and Visuals Rectangular Prism Nets for Packing (cm³) 5.MD.C.5 Relate volume to the Students need multiple opportunities to measure volume by filling Teaching Student Centered Mathematics operations of multiplication and rectangular prisms with cubes and looking at the relationship between Activity 9.10 “Box Comparison – Cubic Units” pg. 267 addition and solve real world the total volume and the area of the base. They derive the volume

Standard Connections/Notes Resources and mathematical problems formula (volume equals the area of the base times the height) and Lessons involving volume. explore how this idea would apply to other prisms. Students use the Cutting Corners (AIMS) associative property of multiplication and decomposition of numbers Estimating Volume - Counting on Frank - Illuminations a. Find the volume of a right using factors to investigate rectangular prisms with a given number of Collecting the Rays - Illuminations rectangular prism with whole- cubic units. They have to be able to decompose a prism, understanding that Fill ‘Er Up – Illuminations number side lengths by it can be partitioned into layers, and each layer partitioned into rows, and each Compose and Decompose Rectangular Prisms Using Layers packing it with unit cubes, and row into cubes. They also have to be able to compose such a structure, Use Multiplication to Calculate Volume show that the volume is the multiplicatively, back into higher units. That is, they eventually learn to Find Volume of Solid Figures Composed of Two same as would be found by conceptualize a layer as a unit that itself is composed of units of units—rows, Rectangular Prisms multiplying the edge lengths, each row composed of individual cubes—and they iterate that structure. Design a Sculpture equivalently by multiplying the Design a Sculpture 2 height by the area of the base. Examples: Paper Volume Cubes Represent threefold whole- number products as volumes, Activities and Tasks e.g., to represent the PARCC Sample Task associative property of Designing a Cereal Box (K5) multiplication. Designing a Toy Box (K5) Students are given a net and asked to predict the number of Joe’s Buildings (K5) b. Apply the formulas V = l ´ w cubes required to fill the container formed by the net. In such Roll a Rectangular Prism ´ h and V = b ´ h for tasks, students may initially count single cubes or repeatedly Candy Boxes rectangular prisms to find add the number of cubes in a row to determine the number in Filling Boxes volumes of right rectangular each layer, and repeatedly add the number in each layer to find Packing Blocks prisms with whole-number the total number of unit cubes. In folding the net to make the Box of Clay (IM) edge lengths in the context of shape, students can see how the side rectangles fit together and Using Volume to Understand the Assoc.Property (IM) solving real world and determine the number of layers. You Can Multiply Three Numbers in Any Order (IM) mathematical problems. Insides and Outsides - (Do not extend to surface area.) When given 24 cubes, students make as many rectangular prisms as The Sky’s the Limit Calculating Volume c. Recognize volume as possible with a volume of 24 cubic units. Students build the prisms and Volume of Solid Figures additive. Find volumes of solid record possible dimensions. figures composed of two non- Videos overlapping right rectangular Length Width Height Find-the-volume-of-a-solid-figure-by-multiplying (LZ) prisms by adding the volumes 1 2 12 Recognize that volume is additive (LZ) of the non-overlapping parts, 2 2 6 applying this technique to 4 2 3 Online

Standard Connections/Notes Resources solve real world problems. 8 3 1 Cubes – Illuminations Space Blocks NLVM Isometric Drawing Tool (cube) Illuminations A homeowner is building a swimming pool and needs to calculate the volume of water needed to fill the pool. The design of the pools are Templates and Visuals shown in the illustrations below. Power Point Volume

Unit Two - Extend Place Value to Decimals (21 days) In Grade 5, students extend their understanding of the base-ten system to decimals to the thousandths place, building on their Grade 4 work with tenths and hundredths. Students extend their understanding of the base-ten system to the relationship between adjacent places, how numbers compare, and how numbers round for decimals to thousandths. Students use their understanding of structure of whole numbers to generalize this understanding to decimals (MP.7) and explain the relationship between the numerals (MP.6). Grade 5 is the last grade in which the NBT domain appears in MDCCRS. Later work in the base ten system relies on the meanings and properties of operations. This also contributes to deepening students’ understanding of computation and algorithms in

Grade 5 Wicomico County Public Schools 2014-2015 15 Fifth Grade Mathematics Curriculum for MD College and Career Ready Standards the new domains that start in Grade 6. The extension of the place value system from whole numbers to decimals is a major intellectual accomplishment involving understanding and skill with base-ten units and fractions. In this unit students also apply both their understanding of comparing fractions and their understanding of place value to compare decimals.

Unit Two Standards 5.NBT.A.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.A.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.

5.NBT.A.3. Read, write, and compare decimals to thousandths. a. Read and write decimals to thousandths using base-ten numerals, number names, and expanded form, e.g., 347.392 = 3 × 100 + 4 × 10 + 7 × 1 + 3 × (1/10) + 9 × (1/100) + 2 × (1/1000). b. Compare two decimals to thousandths based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons

5.NBT.A.4. Use place value understanding to round decimals to any place.

Math Language The terms below are for teacher reference. Although these do not need to be memorized by the students, students should be encouraged to use these math terms in explanations or justifications. associative property distributive property place value thousandths exponents fraction powers of 10 value decimal hundredths rounding decimal point ones tenths

Essential Questions What is the relationship between decimals and fractions? How can we read, write, and represent decimal values? How are decimal numbers placed on a number line? How can rounding decimal numbers be helpful? How can you decide if your answer is reasonable? How do we compare decimals? How do you round decimals? How does context help me round decimals? How can we use exponents to represent powers of 10?

How does multiplying or dividing by a power of ten affect the product?

Common Misconceptions 1. A common misconception that students have when trying to extend their understanding of whole number place value to decimal place value is that as you move to the left of the decimal point, the number increases in value. Reinforcing the concept of powers of ten is essential for addressing this issue.

2. A second misconception that is directly related to comparing whole numbers is the idea that the longer the number the greater the number. With whole numbers, a 5-digit number is always greater that a 1-, 2-, 3-, or 4-digit number. However, with decimals a number with one decimal place may be greater than a number with two or three decimal places. For example, 0.5 is greater than 0.12, 0.009 or 0.499. One method for comparing decimals it to make all numbers have the same number of digits to the right of the decimal point by adding zeros to the number, such as 0.500, 0.120, 0.009 and 0.499. A second method is to use a place-value chart to place the numerals for comparison.

Grade 5 Unit Two - Extend Place Value to Decimals (21 days) Back to “Year at a Glance”

Standard Connections/Notes Resources 5.NBT.A.1 Recognize that in a In fourth grade, students examined the relationships of the digits in Teaching Student Centered Mathematics

Standard Connections/Notes Resources multi-digit number, a digit in numbers for whole numbers. This standard extends this understanding A Two-Way Relationship pg. 184 one place represents 10 times to the relationship of decimal fractions to thousandths. In grade 4, as much as it represents in the students recognize that in a multi-digit whole number, a digit in one Lessons place to its right and 1/10 of place represents ten times what it represents in the place to its right Changing Places (AIMS) what it represents in the place (4.NBT.A.1). Students use base ten blocks, pictures of base ten Relate Base Ten Units from Millions to Thousandths to its left. blocks, and interactive images of base ten blocks to manipulate and Patterns R Us investigate the place value relationships. Meaning of Place Value Connections: 5.NBT.A.2 5.MD.A.1 They use their understanding of unit fractions to compare decimal Activities and Tasks places and fractional language to describe those comparisons. Before Kipton’s Scale (IM) considering the relationship of decimal fractions, students express their Tenths and Hundredths (IM) understanding that in multi-digit whole numbers, a digit in one place Which Number Is It? (IM) represents 10 times what it represents in the place to its right and 1/10 Comparing Digits (K5) of what it represents in the place to its left. Place Value Tasks Word Problems with Template A student thinks ―I know that in the number 5555, the 5 in the tens place (5555) represents 50 and the 5 in the hundreds place (5555) Videos represents 500. So a 5 in the hundreds place is ten times as much as a Multiply and divide with multiples of ten 5 in the tens place or a 5 in the tens place is 1/10 of the value of a 5 in Determine-the-value-of-a-digit-in-the-thousandths- the hundreds place. place (LZ)

To extend this understanding of place value to their work with decimals, Online students use a model of one unit; they cut it into 10 equal pieces, shade Base Blocks Decimals - NLVM in, or describe 1/10 of that model using fractional language. (This is 1 out of 10 equal parts, so it is 1/10. I can write this using 1/10 or 0.1). Templates and Visuals They repeat the process by finding 1/10 of a 1/10 (e.g., dividing 1/10 Decimal of the Day into 10 equal parts to arrive at 1/100 or 0.01) and can explain their Number Connections reasoning, 0.01 is 1/10 of 1/10 thus is 1/100 of the whole unit. Place Value Template for Sheet Protector

Examples: 1(10,000) + 2(1,000) + 4(100) + 3(10) + 2(1) + 5(1/10) + 3(1/100). What number is one-tenth of the expanded form above? 5.NBT.A.2 Explain patterns in Powers of 10 is a fundamental aspect of the base ten system, thus Lessons the number of zeros of the 5.NBT.A.2 can help students extend their understanding of place Use Exponents to Name Place Value Units product when multiplying a value to incorporate decimals to hundredths. When students explain

Standard Connections/Notes Resources number by powers of 10, and patterns in the number of zeroes of the product when multiplying a explain patterns in the number by powers of 10 (5.NBT.2), they have an opportunity to look for Activities and Tasks placement of the decimal point and express regularity in repeated reasoning. New at Grade 5 is the Multiplying Whole Numbers by a Power of Ten (K5) when a decimal is multiplied or use of whole number exponents to denote powers of 10. Students Multiplying a Decimal by a Power of 10 divided by a power of 10. Use understand why multiplying by a power of 10 shifts the digits of a whole Dividing a Decimal by a Power of 10 (K5) whole-number exponents to number or decimal that many places to the left. Patterns in the number Dividing a Whole Number by a Power of 10 (K5) denote powers of 10. of 0s in products of a whole numbers and a power of 10 and the Marta’s Multiplication Error (IM) location of the decimal point in products of decimals with Multiplying Decimals by 10 (IM) powers of 10 can be explained in terms of place value. Patterns R Us Word Problems with Template Examples: Students might write: Videos  36 x 10 = 36 x = 360 Multiply-decimal-numbers-by-a-power-of-ten (LZ)  36 x 10 x 10 = 36 x = 3600 Templates and Visuals  36 x 10 x 10 x 10 = 36 x = 36,000 Decimal of the Day  36 x 10 x 10 x 10 x 10 = 36 x = 360,000 Number Connections Place Value Template for Sheet Protector Students might think and/or say:  I noticed that every time, I multiplied by 10 I added a zero to the end of the number. That makes sense because each digit’s value became 10 times larger. To make a digit 10 times larger, I have to move it one place value to the left.  When I multiplied 36 by 10, the 30 became 300. The 6 became 60 or the 36 became 360. So I had to add a zero at the end to have the 3 represent 3 one-hundreds (instead of 3 tens) and the 6 represents 6 tens (instead of 6 ones).

Students should be able to use the same type of reasoning as above to explain why the following multiplication and division problem by powers of 10 make sense.

 523 x = 523,000 The place value of 523 is increased by three places.

Standard Connections/Notes Resources  52.3 ÷ = 5.23 The place value of 52.3 is decreased by one place.  2.5 × 103 = 2.5 × (10 × 10 × 10) = 2.5 × 1,000 = 2,500 Students should reason that the exponent above the 10 indicates how many places the decimal point is moving (not just that the decimal point is moving but that you are multiplying or making the number 10 times greater three times) when you multiply by a power of 10. Since we are multiplying by a power of 10 the decimal point moves to the right.

Standard Connections/Notes Resources 5.NBT.A.3 Read, write, and Teaching Student Centered Mathematics compare decimals to Students build on the understanding they developed in fourth grade to Activity 7.9 “Line ‘Em Up” pg. 191 thousandths. read, write, and compare decimals to thousandths. They connect their Activity 7.10 “Close Nice Numbers” pg. 192 prior experiences with using decimal notation for fractions and addition a. Read and write decimals to of fractions with denominators of 10 and 100. They use concrete Lessons thousandths using base-ten models and number lines to extend this understanding to decimals to Name Decimal Fractions in Various Forms numerals, number names, and the thousandths. Models may include base ten blocks, place value Compare Decimal Fractions to the Thousandths expanded form, e.g., 347.392 charts, grids, pictures, drawings, manipulatives, technology-based, etc. Show Me the Money (AIMS) = 3 × 100 + 4 × 10 + 7 × 1 + 3 They read decimals using fractional language and write decimals in Dueling Decimals (AIMS) × (1/10) + 9 × (1/100) + 2 × fractional form, as well as in expanded notation as shown in the Decimal Detectives (AIMS) (1/1000). standard 3a. This investigation leads them to understanding Decimal Designs (AIMS) b. Compare two decimals to equivalence of decimals (0.8 = 0.80 = 0.800). Dealing with Decimals (AIMS) thousandths based on Clued into Decimals (AIMS) meanings of the digits in each Students need to understand the size of decimal numbers and relate place, using >, =, and < them to common benchmarks such as 0, 0.5 (0.50 and 0.500), and 1. Activities and Tasks symbols to record the results Comparing tenths to tenths, hundredths to hundredths, and Decimal of the Week of comparisons thousandths to thousandths is simplified if students use their Decimal Numbers Exit Slips understanding of fractions to compare decimals. Decimal Assessment Connections: Comparing Decimals QR Code Activity 5.NBT.A.4 Example: Comparing Decimals (K5) Some equivalent forms of Representing Decimals with Base Ten Blocks (K5) 0.72 are: 70/100 + 2/100 Representing Decimals in Different Ways (K5) 72/100 Are These Equivalent to 9.52? (IM) 0.720 Comparing Decimals on the Number Line (IM) 7/10 + 2/100 Placing Thousandths on the Number Line (IM) 7 x (1/10) + 2 x (1/100) Drawing Pictures to Illustrate Decimal Comparisons (IM) 7 x (1/10) + 2 x (1/100) + 0 x (1/1000) Base Ten Pictures for Decimals Decimal Number of the Day Prompts 720/1000 0.70 + 0.02 Decimal War Decimal Review Game Cards Comparing 0.25 and 0.17, a student might think, 25 hundredths is more Place Value Path Game than 17 hundredths. They may also think that it is 8 hundredths more. Decimal Representations on a Number Line They may write this comparison as 0.25 > 0.17 and recognize that 0.17 Spin and Compare Decimals < 0.25 is another way to express this comparison. Writing Decimals in Expanded Form Comparing 0.207 to 0.26, a student might think, both numbers have 2 Corn Shucks

Standard Connections/Notes Resources tenths, so I need to compare the hundredths. The second number has Race to a Meter 6 hundredths and the first number has no hundredths so the second number must be larger. Another student might think while writing Videos fractions, I know that 0.207 is 207 thousandths (and may write Write-decimals-in-expanded-form (LZ) 207/1000). 0.26 is 26 hundredths (and may write 26/100) but I can also think of it as 260 thousandths (260/1000). So, 260 thousandths is more Online than 207 thousandths. Number Line Worksheets with Decimals Math Aids Base Blocks Decimals - NLVM

Templates and Visuals Base Ten Blocks as Decimals Decimal Place Value Cards Decimal Square Sets Large Thousandths Decimal Square Place Value Mat Decimal Arrow Cards 5.NBT.A.4 Use place value Teaching Student Centered Mathematics understanding to round When rounding a decimal to a given place, students may identify the Close “Nice” Numbers pg. 192 decimals to any place. two possible answers, and use their understanding of place value to compare the given number to the possible answers. Lessons Connections: Decimal Round-Up/Round-Down 5.NBT.A.3 Example: Rounding Decimals Using the Vertical Number Line Round 14.235 to the nearest tenth. Students recognize that the possible answer must be in tenths thus, it is Activities and Tasks either 14.2 or 14.3. They then identify that 14.235 is closer to 14.2 Rounding to Tenths and Hundredths (IM) (14.20) than to 14.3 (14.30). Batter Up Smarter Balance Sample Question

Videos Round-decimals-to-the-nearest-tenth-number-line (LZ) Decompose 1.57 to round to the nearest tenth. Online Base Blocks Decimals - NLVM

Standard Connections/Notes Resources Templates and Visuals Decimal Number Lines Template

Unit Three - Fraction Operations (33 days) In this unit students use what they’ve learned in Grades 3 and 4 about fraction equivalency in terms of visual models and benchmarks to extend understanding of adding and subtracting fractions, including mixed numbers. They reason about size of fractions to make sense of their answers—e.g. they understand that the sum of 1/2 and 2/3 will be greater than 1. It is important to note that in some cases it may not be necessary to find least common denominator to add fractions with unlike denominators. Students should be encouraged to use their conceptual understanding of fractions rather than just using the algorithm for adding fractions. In addition, there is no mathematical reason for students to write fractions in simplest form.

In this unit students extend their understanding of multiplying a fraction by a whole number to multiplying fractions by fractions. In previous grades, students have developed understanding of fractions as numbers. In this grade level, students develop an understanding of the connection between fractions and division. They will use this understanding to explore the relationship of multiplication and division when multiplying fractions. Students build on their work with “compare” problems in Grade 4 (4.OA.A.1) to develop a foundational understanding of multiplication as scaling. They interpret, represent, and explain situations involving multiplication of fractions. Students apply their whole number work with multiplication to develop conceptual understanding of multiplying a fraction by a fraction. Scaling is foundational for developing an understanding of ratios and proportion in future grade levels.

Students will also use their understanding of the relationship between multiplication and division to develop a conceptual understanding of division with fractions (division of a whole number by a unit fraction or a unit fraction by a whole number).

Unit Three Standards 5.NF.A.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.)

5.NF.A.2 Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators, e.g., by 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.

5.NF.B.3 Interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or equations to represent the problem.

5.NF.B.4 Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction. a. Interpret the product (a/b) × 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. b. 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

5.NF.B.5 Interpret multiplication as scaling (resizing), by: a. Comparing 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. b. Explaining 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); explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the given number; and relating the principle of fraction equivalence a/b = (n × a)/(n × b) to the effect of multiplying a/b by 1

5.NF.B.6 Solve real world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem

5.NF.B.7 Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions a. Interpret division of a unit fraction by a non-zero whole number, and compute such quotients. b. Interpret division of a whole number by a unit fraction, and compute such quotients. c. Solve real world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions, e.g., by using visual fraction models and equations to represent the problem. For example, how much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 1/3-cup servings are in 2 cups of raisins?

Math Language The terms below are for teacher reference. Although these do not need to be memorized by the students, students should be encouraged to use these math terms in explanations or justifications. addition/add decompose improper fraction subtraction/subtract associative property distributive property mixed number unit fraction benchmark fraction equation numerator unlike denominator common denominator equivalent reasonableness variable difference estimate scaling

Essential Questions How are equivalent fractions helpful when solving problems? How can a fraction be greater than 1? How can a model help us make sense of a problem? How can comparing factor size to 1 help us predict what will happen to the product? How can decomposing fractions or mixed numbers help us model fraction multiplication? How can decomposing fractions or mixed numbers help us multiply fractions? How can fractions be used to describe fair shares? How can fractions with different denominators be added together? How can looking at patterns help us find equivalent fractions? How can making equivalent fractions and using models help us solve problems? How can modeling an area help us with multiplying fractions? How can we describe how much someone gets in a fair-share situation if the fair share is less than 1? How can we describe how much someone gets in a fair-share situation if the fair share is between two whole numbers? How can we model an area with fractional pieces? How can we model dividing a unit fraction by a whole number with manipulatives and diagrams? How can we tell if a fraction is greater than, less than, or equal to one whole?

How does the size of the whole determine the size of the fraction? What connections can we make between the models and equations with fractions? What do equivalent fractions have to do with adding and subtracting fractions? What does dividing a unit fraction by a whole number look like? What does dividing a whole number by a unit fraction look like? What does it mean to decompose fractions or mixed numbers? What models can we use to help us add and subtract fractions with different denominators? What strategies can we use for adding and subtracting fractions with different denominators? When should we use models to solve problems with fractions? How can I use a number line to compare relative sizes of fractions?

Grade 5 Unit Three - Fraction Operations (33 days) Back to “Year at a Glance”

Standard Connections/Notes Resources Teaching Student Centered Mathematics 5.NF.A.1 Add and subtract Students in Grade 4 added and subtracted fractions with like Addition and Subtraction pgs. 162-166 fractions with unlike denominators. Fifth grade students should apply their understanding of An Area Model Approach pgs. 155-156 denominators (including equivalent fractions developed in fourth grade and their ability to rewrite mixed numbers) by replacing fractions in an equivalent form to find common denominators. They Lessons given fractions with equivalent should know that multiplying the denominators will always give a fractions in such a way as to common denominator but may not result in the smallest denominator. Use Benchmark Numbers to Assess produce an equivalent sum or Simplifying answers is not part of this standard, yet conversations Reasonableness of Addition and Subtraction difference of fractions with like should take place when an answer is 6/4 cup of flour, for example. Equations denominators. For example, 2/3 + 5/4 = 8/12 + 15/12 = Suggested manipulatives to develop this concept: fraction strips, Add Fractions Using Visual Models to Create 23/12. (In general, a/b + c/d = Cuisenaire rods, fraction towers, fraction fringes (AIMS), fraction Equivalent Fractions (ad + bc)/bd.) squares, fraction circles, two color counters, and pattern blocks. Subtract Fractions Using Visual Models to Create Visual fraction models include tape diagrams, area models, Equivalent Fractions number line diagrams, set diagrams (See Van de Walle pgs.162- 166). Add and Subtract Whole and Mixed Numbers Using the Number Line Examples: Royal Rugs (AIMS) Fraction Time (AIMS) Fractions with Pattern Blocks (AIMS) Estimating Sums and Differences Delightfully Different Fractions Utah Ed Network Area Model Investigating Equivalent Fractions with Relationship Rods - Illuminations

Activities and Tasks Adding Mixed Numbers (K5) Equivalent Fractions to Add Unlike Fractions (K5) Equivalent Fractions to Sub Unlike Fractions (K5) Subtracting Mixed Numbers with Unlike Denominators (K5) Closest to 25 (K5) Linear Model ( + ) Common Assessment on Adding and Subtracting

Standard Connections/Notes Resources Fractions and Mixed Numbers

Egyptian Fractions (IM) Finding Common Denominators to Add (IM) Jog a Thon (IM) Making S’more (IM)

Videos Using Pattern Blocks to Add Fractions Using Pattern Blocks to Subtract Fractions Using Pattern Blocks to Multiply Fractions Find-common-denom.-by-creating-area-models(LZ)

Online Equivalent Fractions (like denominators) Fractions – Adding NLVM

Templates and Visuals Fraction Spinners to Eighths Fraction Spinners to Sixteenths Fraction Spinners 5.NF.A.2 Solve word Some problems should appear as multistep. Lessons problems involving addition Solve Two-Step Word Problems Involving Fractions and subtraction of fractions Examples: referring to the same whole, Jerry was making two different types of cookies. One recipe needed 3/4 Solve Multi-Step Word Problems with Mixed including cases of unlike cup of sugar and the other needed 2/3 cup of sugar. How much sugar Numbers denominators, e.g., by using did he need to make both recipes? visual fraction models or  Mental estimation: Activities and Tasks equations to represent the A student may say that Jerry needs more than 1 cup of sugar but less PARCC Sample Question problem. Use benchmark than 2 cups. An explanation may compare both fractions to ½ and state Measuring Cups (IM) fractions and number sense that both are larger than ½ so the total must be more than 1. In Do These Add Up? (IM) of fractions to estimate addition, both fractions are slightly less than 1 so the sum cannot be Salad Dressing (IM) mentally and assess the more than 2. Fraction Word Problems Unlike Denominators(K5) reasonableness of answers. Opus Math – Word problems For example, recognize an Example: Using a bar diagram

Standard Connections/Notes Resources incorrect result 2/5 + 1/2 =  Sonia had 2 candy bars. She promised her brother that she Videos 3/7, by observing that 3/7 < would give him ½ of a candy bar. How much will she have left Estimate-answers-to-problems-involving-fractions- 1/2. after she gives her brother the amount she promised? using-area-models (LZ)

 If Mary ran 3 miles every week for 4 weeks, she would reach her goal for the month. The first day of the first week she ran

1 ¾ miles. How many miles does she still need to run the first week?  Using addition to find the answer: 1 ¾ + n = 3  A student might add 1 ¼ to 1 ¾ to get to 3 miles. Then he or she would add 1/6 more. Thus 1 ¼ miles + 1/6 of a mile is what Mary needs to run during that week.

Example: Using an area model to subtract  This model shows 1 ¾ subtracted from 3 leaving 1 + ¼ + which a student can then change to 1 + + = 1 .

 This diagram models a way to show how 3 and 1 can be expressed with a denominator of 12. Once this is accomplished, a student can complete the problem, 2 – 1 = 1 .

5.NF.B.3 Interpret a fraction Students are expected to demonstrate their understanding using Lessons as division of the numerator concrete materials, drawing models, and explaining their thinking when Interpret a Fraction as Division by the denominator (a/b = a ÷ working with fractions in multiple contexts. They read 3/5 as three fifths b). Solve word problems and after many experiences with sharing problems, learn that 3/5 can Use Tape Diagrams to Model Fractions as Division involving division of whole also be interpreted as 3 divided by 5. numbers leading to answers Solve Word Problems Involving Division of Whole in the form of fractions or Examples: Numbers Leading to Answers in the Form of mixed numbers, e.g., by using  Ten team members are sharing 3 boxes of cookies. How much Fractions visual fraction models or of a box will each student get? When working this problem a equations to represent the student should recognize that the 3 boxes are being divided Activities and Tasks problem. into 10 groups, so s/he is seeing the solution to the following Converting Fractions of a Unit to a Smaller Unit (IM) For example, interpret 3/4 as equation, 10 x n = 3 (10 groups of some amount is 3 boxes) How Much Pie? (IM) the result of dividing 3 by 4, which can also be written as n = 3 ÷ 10. Using models or Relating Fractions to Division (K5) noting that 3/4 multiplied by 4 diagram, they divide each box into 10 groups, resulting in each equals 3, and that when 3 team member getting 3/10 of a box. wholes are shared equally Videos  Two afterschool clubs are having pizza parties. For the Math among 4 people each person Use-picture-models-to-divide-whole-numbers-leading-to- Club, the teacher will order 3 pizzas for every 5 students. For has a share of size 3/4. If 9 answers-in-the-form-of-fractions (LZ) the student council, the teacher will order 5 pizzas for every 8 people want to share a 50- students. Since you are in both groups, you need to decide pound sack of rice equally by which party to attend. How much pizza would you get at each weight, how many pounds of party? If you want to have the most pizza, which party should rice should each person get? you attend? Between what two whole numbers does your answer  The six fifth grade classrooms have a total of 27 boxes of lie? pencils. How many boxes will each classroom receive? Students may recognize this as a whole number division

Standard Connections/Notes Resources Connections: problem but should also express this equal sharing problem as 5.NBT.B.6 . They explain that each classroom gets boxes of pencils and can further determine that each classroom gets 4 or 4 boxes of pencils.

(cont.)

Example:

5.NF.B.4 Apply and extend Representing multiplication of fractions with visual and concrete Teaching Student Centered Mathematics

Standard Connections/Notes Resources previous understandings of models is fundamental to this unit in order for students to make Multiplication pgs. 167-172 multiplication to multiply a sense of multiplying fractions by fractions (MP.1, MP.4). fraction or whole number by a Lessons fraction. Students are expected to multiply fractions including proper fractions, Cuisenaire Rods Fractions of a Number improper fractions, and mixed numbers. (Mixed numbers will be Relate Fractions as Division to Fraction of a Set

a. Interpret the product (a/b) addressed in Unit 5.) They multiply fractions efficiently and accurately Multiply any Whole Number by a Fraction Using × q as a parts of a partition of as well as solve problems in both contextual and non-contextual Tape Diagrams q into b equal parts; situations. equivalently, as the result of a As they multiply fractions such as 3/5 x 6, they can think of the Find a Fraction of a Measurement and Solve Word sequence of operations a × q operation in more than one way. Problems ÷ b. For example, use a visual fraction model to show  3 x (6 ÷ 5) or (3 x 6/5) (2/3) × 4 = 8/3, and create a  (3 x 6) ÷ 5 or 18 ÷ 5 (18/5) story context for this Activities and Tasks Suggested manipulatives/visual models to develop this concept: equation. Do the same with Connor and Mikayla Discuss Multiplication (IM) area model templates (adapted from AIMS), pattern blocks, paper (2/3) × (4/5) = 8/15. (In Folding Strips of Paper (IM) for folding, Cuisenaire rods, fraction squares, graph paper general, (a/b) × (c/d) = ac/bd.) Parts of a Whole The Whole Matters b. Find the area of a Students create a story problem for 3/5 x 6 such as, rectangle with fractional side Greatest Product lengths by tiling it with unit  Isabel had 6 feet of wrapping paper. She used 3/5 of the paper Area Word Problems with Fractional Side Lengths (K5) squares of the appropriate to wrap some presents. How much does she have left? Using Models to Multiply Fractions Common Assessment unit fraction side lengths, and  Every day Tim ran 3/5 of mile. How far did he run after 6 days? show that the area is the (Interpreting this as 6 x 3/5) Videos same as would be found by Cuisenaire Rods to Model Multiplication of Whole multiplying the side lengths. Examples: Numbers by Fractions, Fractions by Fractions, and Multiply fractional side lengths Building on previous understandings of multiplication Mixed Numbers by Fractions to find areas of rectangles, Using Pattern Blocks to Multiply Fractions and represent fraction  Rectangle with dimensions of products as rectangular 2 and 3 showing that 2 x 3 = 6. areas. Find-the-Area-of-a-Rectangle-with-Fractional-Side- Lengths-by-Tiling (LZ)

Multiplying a Fraction by a Fraction using an area model (LZ)

Standard Connections/Notes Resources Online Fractions – Rectangle Multiplication NLVM

Templates and Visuals Fraction Spinners to Eighths Fraction Spinners to Sixteenths Fraction Spinners Area Model Template adapted from AIMS Rectangle with dimensions of 2 and showing that 2 x =

5.NF.B.5 Interpret Multiplication as scaling In preparation for Grade 6 work in ratios and Lessons multiplication as scaling proportional reasoning, students learn to see products such as 5 x 3 or Compare the Size of the Product to the Size of the (resizing), by: x 3 as expressions that can be interpreted in terms of a quantity, 3, and Factors a scaling factor, 5 or . Thus, in addition to knowing that 5 x 3 = 15, Compare the Size of the Product to the Size of the a. Comparing the size of a they can also say that 5 x 3 is 5 times as big as 3, without evaluating Factors Lesson 2 product to the size of one the product. Likewise, they see x 3 as half the size of 3. Students factor on the basis of the size reason abstractly and practice communicating their thinking in real Activities and Tasks of the other factor, without world situations (MP.2, MP.6). They use number lines and other visual Multiplication and Scale Problems (K5) performing the indicated models to interpret situations involving multiplication by numbers larger Comparing a Number and a Product (IM)

Standard Connections/Notes Resources multiplication. than one (when the result will be larger than the original quantity) and Fundraising (IM) involving multiplication by a fraction smaller than 1 (when the result will Reasoning about Multiplication b. Explaining why be smaller than the original quantity) Running a Mile multiplying a given number by (MP.4). a fraction greater than 1 Videos results in a product greater Understand-the-effect-of-multiplying-by-a-fraction- than the given number greater-than-one (LZ) (recognizing multiplication by whole numbers greater than 1 as a familiar case); explaining why multiplying a given (cont.) number by a fraction less than 1 results in a product smaller than the given number; and Examples: relating the principle of x 7 is less than 7 because 7 is multiplied by a factor less than 1 fraction equivalence a/b = (n so the product must be less than 7. × a)/(n × b) to the effect of multiplying a/b by 1

 2 x 8 must be more than 8 because 2 groups of 8 is 16 and 2 is almost 3 groups of 8. So the answer must be close to, but less than 24.

 = because multiplying by is the same as multiplying by 1.

5.NF.B.6 Solve real world Some problems should appear as multistep. Lessons problems involving Solve Word Problems Using Tape Diagrams multiplication of fractions and Multiplication of mixed numbers will appear in Unit 5. mixed numbers, e.g., by using Activities and Tasks visual fraction models or Example: Sue ran of a mile. Alex ran 3 times as far as Sue. How far Using Area Models to Multiply Fractions Common equations to represent the did Alex run? (multiplicative comparison GR4) Assessment problem Multiplying Fractions by Dividing Rectangles (K5) Multiplying a Fraction by a Fraction Word Problems (K5) Running to School (IM) To Multiply or Not to Multiply (IM) Drinking Juice (IM) Painting a Wall (IM) Fraction by Fraction Word Problems (K5)

PARCC Sample Question

Videos Area Model to Multiply Mixed Numbers. Multiply-a-fraction-by-a-whole-number-using-visual- representations (LZ)

Online Fractions – Rectangle Multiplication NLVM

Standard Connections/Notes Resources 5.NF.B.7 Apply and extend In this unit it is critical for students to use concrete objects or pictures Teaching Student Centered Mathematics previous understandings of to help conceptualize, create, and solve problems. In fifth grade, Whole number and unit fraction only: division to divide unit fractions students experience division problems with whole number divisors and Partition Concept pgs. 172-173 by whole numbers and whole unit fraction dividends (fractions with a numerator of 1) or with unit Measurement Concept pgs.174-175 numbers by unit fractions fraction divisors and whole number dividends. Students extend their a. Interpret division of a unit understanding of the meaning of fractions, how many unit fractions are Lessons fraction by a non-zero whole in a whole, and their understanding of multiplication and division as Divide a Whole Number by a Unit Fraction number, and compute such involving equal groups or shares and the number of objects in each Divide a Unit Fraction by a Whole Number Lesson quotients. For example, group/share. In sixth grade, they will use this foundational create a story context for understanding to divide into and by more complex fractions and Activities and Tasks (1/3) ÷ 4, and use a visual develop abstract methods of dividing by fractions. Divide a Unit Fraction by a Whole Number (K5) fraction model to show the Divide a Whole Number by a Unit Fraction (K5) quotient. Use the relationship Relating Fractions to Division (K5) between multiplication and Example: Knowing the number of groups/shares and finding how Dividing by One Half (IM) division to explain that (1/3) ÷ many/much in each group/share How Many Servings of Oatmeal (IM) 4 = 1/12 because (1/12) × 4 = How Many Marbles (IM) 1/3.  Four students sitting at a table were given 1/3 of a pan of Origami Stars (IM) b. Interpret division of a brownies to share. How much of a pan will each student get if whole number by a unit they share the pan of brownies equally? Videos fraction, and compute such Divide-unit-fractions-by-whole-numbers-by- quotients. For example, drawing-a-model (LZ) create a story context for 4 ÷ Divide-whole-numbers-by-unit-fractions-using-a- (1/5), and use a visual number-line (LZ) fraction model to show the quotient. Use the relationship Online between multiplication and Number Line Bars – Fractions NLVM division to explain that 4 ÷ (1/5) = 20 because 20 × (1/5) = c. Solve real world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions, e.g., by using visual fraction models and equations

Unit Four – Decimal Operations (15 days) Measurement is used in this unit as a context for operations with decimals. Students’ previous experiences with decimal fractions and fraction computations are applied here to provide multiple ways of thinking about operations with decimals. Students can use their understanding of a decimal fraction equivalencies, concrete or visual models, and place value to reason about decimal quantities and operations.

Unit Four Standards 5.NBT.B.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.

5.NBT.A.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

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

Math Language The terms below are for teacher reference. Although these do not need to be memorized by the students, students should be encouraged to use these math terms in explanations or justifications. array hundredths partial products associative property of multiplication identity property of multiplication partition/partitive division (or fair-sharing) centimeter kilometer place value commutative property of multiplication measurement division (or repeated subtraction) product distributive property meter quotient dividend metric system remainder division millimeter ten thousands divisor millions tens factor multiple tenths hundred thousands multiplier thousands hundreds ones

Essential Questions Why is place value important when adding whole numbers and decimal numbers? How does the placement of a digit affect the value of a decimal number? Why is place value important when subtracting whole numbers and decimal numbers? What strategies can I use to add and subtract decimals? How can we use models to help us multiply and divide decimals? How do the rules of multiplying whole numbers relate to multiplying decimals? How are multiplication and division related? How are factors and multiples related to multiplication and division? What are some patterns that occur when multiplying and dividing by decimals? How can we efficiently solve multiplication and division problems with decimals? What strategies are effective for finding a missing factor or divisor? How can we check for errors in multiplication or division of decimals?

Common Misconceptions 1. Students might compute the sum or difference of decimals by lining up the right-hand digits as they would whole number. For example, in computing the sum of 15.34 + 12.9, students will write the problem in this manner:

15.34 + 12.9 16.63 To help students add and subtract decimals correctly, have them first estimate the sum or difference. Providing students with a decimal-place value chart will enable them to place the digits in the proper place.

2. Multiplication can increase or decrease a number. From previous work with computing whole numbers, students understand that the product of multiplication is greater than the factors. However, multiplication can have a reducing effect when multiplying a positive number by a decimal less than one or multiplying two decimal numbers together. We need to put the term multiplying into a context with which we can identify and which will then make the situation meaningful. Also using the terms times and of interchangeably can assist with the contextual understanding.

Grade 5 Unit Four – Decimal Operations (15 days) Back to “Year at a Glance”

Standard Connections/Notes Resources 5.NBT.B.7 Add, subtract, Addition and Subtraction: Addition and Subtraction of Decimals: multiply, and divide decimals to Because of the uniformity of the structure of the base-ten system, hundredths, using concrete students use the same place value understanding for adding Teaching Student Centered Mathematics models or drawings and and subtracting decimals that they used for adding and subtracting Computation with Decimals pgs.196-198 strategies based on place whole numbers. Instead of just computing answers, students reason Activity 7.11 “Exact Sums and Differences” pg. 198 value, properties of operations, about both the relationship between fraction and decimal and/or the relationship operations and the relationship between whole number Lessons between addition and computation and fractional/decimal computation (MP.2, MP.3). Add Decimals Using Place Value Strategies subtraction; relate the strategy This standard requires students to extend the models and strategies Subtracting Decimals Using Place Value Strategies to a written method and they developed for whole numbers in grades 1-4 to decimal values. explain the reasoning used. Before students are asked to give exact answers, they should Activities and Tasks estimate answers based on their understanding of operations and the The Value of Education (IM) value of the numbers (Rounding 5.NBT.A.4). Some problems Base Ten Decimal Bag Addition should appear as multistep. Base Ten Decimal Bag Subtraction (K5) Total Ten (K5) Examples: Decimal Addition to 500 (K5)  3.6 + 1.7 Decimal Cross Number Puzzles (K5) A student might estimate the sum to be larger than 5 because 3.6 is Decimal race to Zero (K5) more than 3 ½ and 1.7 is more than 1 ½. Magic Squares Decimal Addition (K5)  5.4 – 0.8 Decimal Addition Bingo A student might estimate the answer to be a little more than 4.4 Decimal Addition Spin because a number less than 1 is being subtracted. Decimal Subtraction Spin Decimal Dynamo Students should be able to express that when they add decimals they Race to the Finish Line add tenths to tenths and hundredths to hundredths. So, when they Race to 10 or Bust are adding in a vertical format (numbers beneath each other), it is Race to 1 or Bust important that they write numbers with the same place value beneath Sum with Decimals each other. This understanding can be reinforced by connecting Sum with Decimals 2 addition of decimals to their understanding of addition of fractions. Shopping Spree Adding fractions with denominators of 10 and 100 is a standard in fourth grade. Videos Use addition and subtraction to solve real-world Example: 4 - 0.3 problems involving decimals

Standard Connections/Notes Resources 3 tenths subtracted from 4 wholes. The wholes must be divided into Online tenths. The answer is 3 and 7/10 or 3.7. Base Blocks Decimals - NLVM . Diffy (Subtraction) NLVM

Templates and Visuals Addition of Decimals Example: Decimal Squares Sets 1.7 – 0.8 = 17 tenths – 8 tenths = 9 tenths = 0.9

Multiplication and Division: Multiplication and Division Resources: General methods used for computing products of whole numbers extend to products of decimals. Because the expectations for Teaching Student Centered Mathematics decimals are limited to thousandths and expectations for factors are Computation with Decimals pgs. 198-201 limited to hundredths at this grade level, students will multiply tenths Activity 7.12 “Where Does the Decimal Go? pg. 199 with tenths and tenths with hundredths, but they need not multiply Activity 7.13 “Where Does the Decimal Go? pg. 200 hundredths with hundredths. Before students are asked to give exact answers, they should estimate answers based on their understanding Lessons of operations and the value of the numbers. Some problems should Multiplying Decimals to Hundredths (MSDE) appear as multistep. Approximate Decimal Quotients through Reasoning about Decimal Placement Example: An area model can be useful for illustrating products. Relate Decimal and Fraction Multiplication Operation Decimals (AIMS)

Activities and Tasks What is 23 Divided by 5? (IM) Decimal Division Model Lab Sheet Multiplication and Division of Decimals Multiplication of Decimals with an Area Model Target One Students should be able to describe the partial products displayed by the area model. For example, Videos Base ten blocks to multiply 1.3 x 2.2 Visual model of $3.71 7.

Standard Connections/Notes Resources Division of Decimal Numbers. Multiply-decimals-to-the-hundredths-by-using-fractions Multiplying Decimals using an area model (whole number by decimal to the hundredths) LZ Example: 6 x 2.4 Multiplying Decimals using an area model LZ A student might estimate an answer between 12 and 18 since 6 x 2 is 12 and 6 x 3 is 18. Templates and Visuals To divide to find the number of groups, a student might: Multiplying Decimals Poster  Draw a segment to represent 1.6 meters. In doing so, s/he Multiply and Divide Decimals Using Models (PPT) would count in tenths to identify the 6 tenths, and be able to Modeling Decimal Multiplication identify the number of 2 tenths within 6 tenths. The student Area Model Multiplying Decimals can then extend the idea of counting by tenths to divide the Decimal Squares Tenths for Multiplication one meter into tenths and determine that there are 5 more Decimal Squares for Division groups of 2 tenths. Decimal Squares for Multiplication  Use their understanding of multiplication and think, “8 groups of 2 is 16, so 8 groups of 2/10 is 16/10 or 1 6/10.”

Two-Step Example: A gardener installed 42.6 meters of fencing in a week. He installed 13.45 meters on Monday and 9.5 meters on Tuesday. He installed the rest of the fence in equal lengths on Wednesday through Friday. How many meters of fencing did he install on each of the last three days?

Example using a Bar Model: The top surface of a desk has a length of 5.6 feet. The length is 4

Standard Connections/Notes Resources times its width. What is the width of the desk?

5.NBT.A.2 Explain patterns in NBT.A.2. was introduced in Unit One. In Unit Four students will apply Lessons the number of zeros of the what they know to help them multiply and divide decimals. Use Exponents with Application to Metric Conversions product when multiplying a number by powers of 10, and Before students consider decimal multiplication more generally, they explain patterns in the can study the effect of multiplying 0.1 and by 0.01 to explain why the Videos placement of the decimal point product is ten or a hundred times as small as the multiplicand (moves Explain patterns in zeros when multiplying by a power when a decimal is multiplied or one or two places to the right). They can then extend their reasoning of ten divided by a power of 10. Use to multipliers that are single-digit multiples of 0.1 and 0.01 (e.g., 0.2 whole-number exponents to and 0.02, etc.). There are several lines of reasoning that students can denote powers of 10. use to explain the placement of the decimal point in other products of decimals. Students can think about the product of the smallest base- ten units of each factor. For example, a tenth times a tenth is a hundredth, so 3.2 x 7.1 will have an entry in the hundredth place.

As with decimal multiplication, students can first examine the cases of dividing by 0.1 and 0.01 to see that the quotient becomes 10 times or 100 times as large as the dividend.

Example: 40.25 x = x 754.2 ÷ 10³ = x

Standard Connections/Notes Resources 5.MD.A.1 Convert among This standard provides measurement conversion as a context Teaching Student Centered Mathematics different-sized standard for not only working with decimals but a deeper understanding Figure 7.6 pg. 186 measurement units within a for place value and the connection to the metric system. given measurement system Lessons (e.g., convert 5 cm to 0.05 m), In fifth grade, students build on their prior knowledge of related Decimal Multiplication to Express Equivalent and use these conversions in measurement units to determine equivalent measurements. Prior to Measurements solving multi-step, real world making actual conversions, they examine the units to be converted, problems. determine if the converted amount will be more or less units than the Find a Fraction of a Measurement original unit, and explain their reasoning. They use several strategies to convert measurements. When converting metric measurement, Activities and Tasks students apply their understanding of place value and decimals. This PARCC Sample Question is an excellent opportunity to reinforce notions of place value for Comparing Units of Metric Linear Measurement whole numbers and decimals, and connection between fractions and Measurement Conversion Word Problems decimals (e.g., 2 ½ meters can be expressed as 2.5 meters or 250 centimeters). Videos Select Appropriate Measurement-conversions (LZ) Examples: 764 mL ÷ 10³ = 0.764 liters Templates and Visuals Converting Metric Units of Measure The length of the bar for a high jump competition must always be 4.75 m. Express this measurement in millimeters. Explain your thinking using an equation that includes an exponent. 4.75 x 10³ = 4,750 mm

Unit Five – Solving Problems with Fractional Quantities (21 days) In this unit students use data and other contexts to solve real world problems involving fractional computations. This unit provides an opportunity for students to extend their understanding of multiplication of fractions to mixed numbers. All of the different problem types in Table 2 in the Common Core State Standards for Mathematics should be addressed in this unit.

Unit Five Standards 5.NF.B.4 Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction. a. Interpret the product (a/b) × 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. b. 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

5.NF.B.5 Interpret multiplication as scaling (resizing), by: a. Comparing 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. b. Explaining 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); explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the given number; and relating the principle of fraction equivalence a/b = (n × a)/(n × b) to the effect of multiplying a/b by 1

5.NF.B.6 Solve real world problems involving multiplication of fractions and mixed numbers

5.NF.B.7 Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions c. Solve real world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions.

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

5.MD.B.2 Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). Use operations on fractions for this grade to solve problems involving information presented in line plots.

Math Language The terms below are for teacher reference. Although these do not need to be memorized by the students, students should be encouraged to use these math terms in explanations or justifications. addition/add difference interval unit fraction benchmark fraction equation mixed number unlike denominator variable common denominator equivalent numerator data estimate reasonableness decompose improper fraction subtraction/subtract

Essential Questions How are equivalent fractions helpful when solving problems? How can a fraction be greater than 1? How can a model help us make sense of a problem? How can comparing factor size to 1 help us predict what will happen to the product? How can decomposing fractions or mixed numbers help us model fraction multiplication? How can decomposing fractions or mixed numbers help us multiply fractions? How can fractions be used to describe fair shares? How can fractions with different denominators be added together? How can looking at patterns help us find equivalent fractions? How can making equivalent fractions and using models help us solve problems? How can modeling an area help us with multiplying fractions? How can we describe how much someone gets in a fair-share situation if the fair share is less than 1? How can we describe how much someone gets in a fair-share situation if the fair share is between two whole numbers? How can we model an area with fractional pieces? How can we model dividing a unit fraction by a whole number with manipulatives and diagrams? How can we tell if a fraction is greater than, less than, or equal to one whole? How does the size of the whole determine the size of the fraction? What connections can we make between the models and equations with fractions? What do equivalent fractions have to do with adding and subtracting fractions?

What does dividing a unit fraction by a whole number look like? What does dividing a whole number by a unit fraction look like? What does it mean to decompose fractions or mixed numbers? What models can we use to help us add and subtract fractions with different denominators? What strategies can we use for adding and subtracting fractions with different denominators? When should we use models to solve problems with fractions? How can I use a number line to compare relative sizes of fractions? How can I use a line plot to compare fractions?

Grade 5 Unit Five – Solving Problems with Fractional Quantities (21 days) Back to “Year at a Glance”

Standard Connections/Notes Resources 5.NF.B.4 Apply and extend In Unit 3, students multiplied fractions using visual fraction models or Teaching Student Centered Mathematics previous understandings of equations. Unit 5 will extend this understanding to mixed numbers. Factors Greater than One pg.171 multiplication to multiply a fraction or whole number by a Examples: Lessons fraction. Area of Rectangles a. Interpret the product (a/b) × New Floor Lesson q as a parts of a partition of q Real World Problems into b equal parts; equivalently, Real World Problems 2 as the result of a sequence of operations Find the Area of Rectangles by Tiling and Relate to b. Find the area of a rectangle Multiplication with fractional side lengths by tiling it with unit squares of the Measure to Find the Area of Rectangles with Fractional appropriate unit fraction side Side Lengths lengths, and show that the area is the same as would be Multiply Mixed Number Factors Relate to the found by multiplying the side Distributive Property and the Area Model lengths. Multiply fractional side lengths to find areas of rectangles, and represent Multiply mixed numbers by mixed numbers using visual fraction products as representations rectangular areas. Activities and Tasks 5.NF.B.5 Interpret Making Cookies (IM) multiplication as scaling Half of a Recipe (IM) (resizing), by: Fractions Multiplied by Mixed Numbers a. Comparing the size of a Mixed Numbers by Fractions product to the size of one Whole Number by Mixed Number with Cuisinaire Rods factor on the basis of the size Common Assessment Multiplying Fractions of the other factor, without Closest to 25 performing the indicated multiplication. Once understanding has been developed, move students to the b. Explaining why multiplying a algorithm: Videos given number by a fraction 2 x 1 = x = = 3 sq ft Area Model to Multiply Mixed Numbers. greater than 1 results in a

Standard Connections/Notes Resources product greater than the given number (recognizing Online multiplication by whole Multiplication of Mixed Numbers using the area model: Fractions – Rectangle Multiplication NLVM numbers greater than 1 as a 3 x 3 = familiar case); explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the given number; and relating the principle of fraction equivalence a/b = (n × a)/(n × b) to the effect of multiplying a/b by 1.

5.NF.B.6 Solve real world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem. c. Solve real world problems Parts a. and b. of this standard were taught in Unit 3. Teaching Student Centered Mathematics involving division of unit Division pg. 172-173 fractions by non-zero whole In fifth grade, students experience division problems with whole numbers and division of whole number divisors and unit fraction dividends (fractions with a numerator numbers by unit fractions, e.g., of 1) or with unit fraction divisors and whole number dividends. Lessons by using visual fraction models Students extend their understanding of the meaning of fractions, how Solve Problems Involving Fraction Division and equations to represent the many unit fractions are in a whole, and their understanding of A Fifth-Grade Fractions Lesson – Math Solutions problem. For example, how multiplication and division as involving equal groups or shares and the much chocolate will each number of objects in each group/share. In sixth grade, they will use Activities and Tasks person get if 3 people share this foundational understanding to divide into and by more complex Division Fraction Word Problems (K5) 1/2 lb of chocolate equally? fractions and develop abstract methods of dividing by fractions. Smarter Balanced Sample Question How many 1/3-cup servings are in 2 cups of raisins? Division Example: Knowing the number of groups/shares and finding how many/much in each group/share

 Four students sitting at a table were given 1/3 of a pan of brownies to share. How much of a pan will each student get if they share the pan of brownies equally?

Examples: Knowing how many in each group/share and finding how many groups/shares  Angelo has 4 lbs of peanuts. He wants to give each of his friends 1/5 lb. How many friends can receive 1/5 lb of peanuts?

A diagram for 4 ÷ is shown below. Students explain that since there are five fifths in one whole, there must be 20 fifths in 4 lbs.

How much rice will each person get if 3 people share lb of rice equally?  A student may think or draw and cut it into 3 equal groups then determine that each of those parts is .  A student may think of as equivalent to . divided by 3 is . 5.MD.A.1 Convert among Lessons different-sized standard Example: Sam made 2/3 liter of punch and 3/4 liter of tea to take to a Fraction Multiplication with Measurement Conversions measurement units within a party. Is that more or less than 1.5 liters? Explain. given measurement system + = = 1 is less than 0.5, so it is less. Find a Fraction of a Measurement and Solve Word (e.g., convert 5 cm to 0.05 m), Problems and use these conversions in solving multi-step, real world Activities and Tasks problems. Comparing Measurements Task Punch Task Check Up Task PARCC Sample Question Converting Unit Tasks Converting Units of Time Converting Standard Units of Length Converting Standard Units of Capacity

Videos Solve real world distance problems with unit conversions 5.MD.B.2 Make a line plot to This standard is included here so measurement line plots can be display a data set of used as a context for students to apply fraction computation Lessons measurements in fractions of a strategies. Students use line plots and other tools/technology to Measure Pencil Lengths and Construct Line Plots unit (1/2, 1/4, 1/8). Use reason about problem situations. Students attend to the underlying operations on fractions for this meaning of the quantities and operations when solving problems Activities and Tasks

Standard Connections/Notes Resources grade to solve problems rather than just how to compute answers. Students should work Fractions on a Line Plot (IM) involving information presented with data in science and other subjects. Sacks of Flour (K5) in line plots. For example, Dog treats Line Plot Activity given different measurements Students in grade 4 solved problems involving addition and subtraction of liquid in identical beakers, of fractions by using information presented in line plots. The standard Videos find the amount of liquid each in this cluster provides an opportunity for solving real-world problems Solve-multistep-problems-using-information-in-a-line- beaker would contain if the with operations on fractions, connecting directly to both number and plot (LZ) total amount in all the beakers operations. were redistributed equally Templates and Visuals Line Plot Template

(cont.)

Examples: Ten beakers, measured in liters, are filled with a liquid.

The line plot above shows the amount of liquid in liters in 10 beakers. If the liquid is redistributed equally, how much liquid would each beaker have? (This amount is the mean.) Students apply their understanding of operations with fractions. They use either addition and/or multiplication to determine the total number of liters in the beakers. Then the sum of the liters is shared evenly

Standard Connections/Notes Resources among the ten beakers.

Unit Six – Classifying 2D Figures (15 days) In this unit the emphasis is on the hierarchical relationship among 2-dimensional geometric figures. Students have had previous experience classifying shapes using defining attributes.

* Clarification: A trapezoid is a quadrilateral with at least one pair of parallel sides. Unit Six Standards 5.G.B.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 squares are rectangles, so all squares have four right angles.

5.G.B.4 Classify two-dimensional figures in a hierarchy based on properties.

Math Language The terms below are for teacher reference. Although these do not need to be memorized by the students, students should be encouraged to use these math terms in explanations or justifications. acute angle isosceles triangle plane figure right triangle acute triangle kite polygon scalene triangle circle obtuse angle protractor symmetry/symmetrical congruence/congruent obtuse triangle quadrilateral square equilateral triangle parallel lines rectangle triangle half circle/semicircle parallelogram regular polygon trapezoid hexagon pentagon rhombus/rhombi two-dimensional figure irregular polygon perpendicular lines right angle vertex

Essential Questions How can plane figures be categorized and classified? What is a quadrilateral? What are the properties of quadrilaterals? How can you classify different types of quadrilaterals? How are quadrilaterals alike and different? How can angle and side measures help us to create and classify triangles? Where is geometry found in your everyday world? What careers involve the use of geometry? Why are some quadrilaterals classified as parallelograms? Why are kites not classified as parallelograms? Why is a square always a rectangle? What are ways to classify triangles?

Grade 5 Unit Six – Classifying 2D Figures (15 days) Back to “Year at a Glance”

Standard Connections/Notes Resources 5.G.B.3 Understand that Geometric properties include properties of sides (parallel, Teaching Student Centered Mathematics attributes belonging to a perpendicular, congruent), properties of angles (type, measurement, Activity 8.5 “Can You Make It?” pg. 217 category of two-dimensional congruent), and properties of symmetry (point and line). Two-Dimensional Shapes pgs. 220-222 figures also belong to all Activity 8.8 “Property Lists for Quadrilaterals pg. 226 subcategories of that category. If the opposite sides on a parallelogram are parallel and congruent, Property Lists for Quadrilaterals (Omit rotation) pg.226 For example, all rectangles then rectangles are parallelograms have four right angles and Lessons squares are rectangles, so all Examples: Quadrilateral Characteristics Utah Education Network. squares have four right angles.  A parallelogram has 4 sides with both sets of opposite sides Constructing Quadrilaterals parallel. What types of quadrilaterals are parallelograms? Quadrilateral Tangram Challenge Parallelogram's Attributes  Regular polygons have all of their sides and angles Rectangles and Rhombuses congruent. Name or draw some regular polygons. Draw and Define Trapezoids Based on Attributes  All rectangles have 4 right angles. Squares have 4 right angles so they are also rectangles. True or False? Activities and Tasks  A trapezoid has 2 sides parallel so it must be a Sometimes Always Never parallelogram. True or False? Quadrilateral Word Problems PARCC Sample Question Types of Triangles a QR Activity

Videos

Standard Connections/Notes Resources Analyze-and-compare-triangles-by-angles (LZ)

Online Geoboard NLVM Shape Sorter Illuminations

Templates and Visuals Frayer Model Example/Non-Example

Standard Connections/Notes Resources 5.G.B.4 Classify two- IRC has die cuts for: circle, oval, square, rectangles, octagon, Teaching Student Centered Mathematics dimensional figures in a hexagon, pentagon, trapezoid, triangle, and rhombus. One piece of Activity 8.12 True or False pg.231 hierarchy based on properties. construction paper will produce 4 of each 2D figure for students to use in sorting activities. Ordering multiple figures will allow you to Lessons modify the 2D shapes. Example: Cut one trapezoid to make it a right Classify 2D Figures trapezoid. Draw 2D Figures from Given Attributes Shape-Up from NCTM Illuminations Sorting Polygons - Illuminations

Activities and Tasks What Is a Trapezoid? (IM) Identifying Quadrilaterals (K5) Properties of figures may include: Quadrilateral Criteria (K5)  Properties of sides – parallel, perpendicular, congruent, Quadrilateral Hierarchy Diagram (K5) Regular/Irregular Hierarchy Diagram (K5) number of sides Triangle Hierarchy (K5)  Properties of angles – types of angles, congruent Triangle Hierarchy 2 (K5) Examples: PARCC Sample Question  A scalene triangle can be right, acute, and obtuse.  A right triangle can be both scalene and isocoles, but not Videos equilateral. Classify-quadrilaterals-in-a-hierarchy Online Shape Sorter Illuminations

Unit Seven – Representing Algebraic Thinking (12 days) In this unit students explore algebraic expressions more formally to represent and interpret calculations involving whole numbers, fractions, and decimals. They apply their understanding of the different algebraic properties of operations and explain the relationships between the quantities with the written expressions. This unit includes opportunities to both evaluate expressions and reason about expressions without calculating a solution. This is foundational for further work with numbers in later grades.

Unit Seven Standards 5.OA.A.1 Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols to represent the problem.

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

Math Language The terms below are for teacher reference. Although these do not need to be memorized by the students, students should be encouraged to use these math terms in explanations or justifications. braces evaluate properties of operations brackets expression parentheses conventional order interpret evaluate distributive property numerical expression equation order of operations Essential Questions Why is it important to follow an order of operations? How can I write an expression that demonstrates a situation or context? How can an expression be written given a set value? What is the difference between an equation and an expression? In what kinds of real world situations might we use equations and expressions? How can we evaluate expressions?

Grade 5 Unit Seven – Representing Algebraic Thinking (12 days) Back to “Year at a Glance”

Standard Connections/Notes Resources 5.OA.A.1 Use parentheses, This standard builds on the expectations of third grade where students Lessons brackets, or braces in are expected to start learning the conventional order (3.OA.D.8.). Exploring Krypto numerical expressions, and Students need experiences with multiple expressions that use Following the Order evaluate expressions with grouping symbols throughout the year to develop understanding of Evaluating Expressions these symbols to represent the when and how to use parentheses, brackets, and braces. First, problem. students use these symbols with whole numbers. Then the symbols Activities and Tasks can be used as students add, subtract, multiply and divide decimals A Sequence of Events and fractions. Expressions with Parentheses Numerical Expressions Wall Clock Examples: Bowling for Numbers (IM)  (26 + 18) ÷ 4 Answer: 11 Watch Out for Parentheses (IM) Same or Different  {[2 x (3+5)] – 9} + [5 x (23-18)] Answer: 32 Trick Answers  12 – (0.4 x 2) Answer: 11.2 Solving Numerical Expressions  (2 + 3) x (1.5 – 0.5) Answer: 5 Target Number Dash  { 80 [ 2 x (3 ½ + 1 ½ ) ] } + 100 Answer: 900 Operation Target Picturing Factors in Different Orders (IM) Using Operations and Parentheses (IM) To further develop students’ understanding of grouping symbols and You Can Multiply Three Numbers in Any Order (IM) facility with operations, students place grouping symbols in equations Order of Operations Bingo to make the equations true or they compare expressions that are grouped differently. Videos Examples: Interpret-parentheses-as-do-this-first (LZ)  15 - 7 – 2 = 10 → 15 - (7 – 2) = 10  3 x 125 ÷ 25 + 7 = 22 → [3 x (125 ÷ 25)] + 7 = 22 Online  Compare 3 x 2 + 5 and 3 x (2 + 5) Order of Operations Activities Pan Balance Numbers Compare 15 – 6 + 7 and 15 – (6 + 7)  Primary Krypto

Grade 5 Unit Seven – Representing Algebraic Thinking (12 days) Back to “Year at a Glance”

Standard Connections/Notes Resources 5.OA.A.2 Write simple Students use their understanding of operations and grouping Lessons expressions that record symbols to write expressions and interpret the meaning of a Interpret and Evaluate Expressions with Fractions calculations with numbers, and numerical expression. The expressions described in 5.OA.A.2 interpret numerical Create Story Contexts for Numerical Expressions should be no more complex than the expressions one finds in an expressions without evaluating application of the associative or distributive property. Include Including Fractions them. For example, express fraction and decimal expressions. the calculation “add 8 and 7, Compare and Evaluate Expressions with Fractions and then multiply by 2” as 2 × (8 + Parentheses 7). Recognize that 3 × (18932 Examples: + 921) is three times as large Students write an expression for calculations given in words such as as 18932 + 921, without divide 144 by 12, and then subtract 7/8. They write (144 ÷ 12) – 7/8. having to calculate the Activities and Tasks indicated sum or product. Verbal and Algebraic Expressions Students recognize that 0.5 x (300 ÷ 15) is ½ of (300 ÷ 15) without Operation Bingo Connections: calculating the quotient. Comparing Products (IM) 5.OA.A.1 Seeing is Believing (IM) Describe how the expression 5(10 x 10) relates to 10 x 10. Video Game Scores (IM) The expression 5 (10 x 10) is 5 times larger than the expression Word to Expressions 1 (IM) 10 x 10 since I know that 5 (10 x 10) means that I have 5 groups of (10 x 10). Videos Write-expressions-to-represent-numerical-situations (LZ) Students write an expression for calculations given in words such as divide 144 by 12, and then subtract 7/8. They write (144 ÷ 12) – 7/8.

Students recognize that 0.5 x (300 ÷ 15) is ½ of (300 ÷ 15) without calculating the quotient.

Unit Eight – Coordinate Plane (15 days)

In this unit students are introduced to the coordinate plane, applying their knowledge of the number line to understand the relationship of the two dimensions of a point in the coordinate plane. Students connect their work with numerical patterns to form ordered pairs and graph these ordered pairs in the first quadrant of a coordinate plane. Students use this model to make sense of and explain the relationships within the numerical patterns they generate. This prepares students for future work with functions and proportional relationships in the middle grades.

Unit Eight Standards 5.OA.B.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.

5.G.A.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. 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 (e.g., x-axis and x-coordinate, y-axis and y-coordinate).

5.G.A.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.

Math Language The terms below are for teacher reference. Although these do not need to be memorized by the students, students should be encouraged to use these math terms in explanations or justifications. axis/axes first quadrant origin vertical coordinates horizontal parallel x-axis coordinate plane intersection of lines perpendicular x-coordinate coordinate system line point y-axis distance ordered pairs rule y-coordinate

Essential Questions How does the coordinate system work? How do coordinate grids help you organize information? What relationships can be determined by analyzing two sets of given rules? How might a coordinate grid help me understand a relationship between two numbers? How can we represent numerical patterns on a coordinate grid? How can a line graph help us determine relationships between two numerical patterns?

How can the coordinate system help you better understand other map systems?

Common Misconceptions Students reverse the points when plotting them on a coordinate plane. They count up first on the y-axis and then count over on the x-axis. The location of every point in the plane has a specific place.

Grade 5 Unit Eight – Coordinate Plane (15 days) Back to “Year at a Glance”

Standard Connections/Notes Resources 5.OA.B.3 Generate two Students should have experienced generating and analyzing Teaching Student Centered Mathematics numerical patterns using two numerical patterns using a given rule in Grade 4. Students precisely Activity 10.6 “What’s Next and Why?” pg. 299 given rules. Identify apparent describe the coordinates of points and the relationship of the Graphing Patterns pgs. 297-298 relationships between coordinate plane to the number line (MP.6). Students both Activity 10.7 “Start and Jump Numbers” pgs. 299-300 corresponding terms. Form generate and identify relationships in numerical patterns, using ordered pairs consisting of the coordinate plane as a way of representing these relationships Lessons corresponding terms from the and patterns (MP.4). Analyzing Patterns and Relationships two patterns, and graph the Willie the Wheel Man (AIMS) ordered pairs on a coordinate Generate Two Number Patterns from Given Rules, Plot plane. For example, given the Use the rule add 3 to write a sequence of numbers. Starting with a 0, the Points, and Analyze the Patterns rule “Add 3” and the starting students write 0, 3, 6, 9, 12, . . . Investigate Patterns on a Coordinate Plane number 0, and given the rule Use the rule add to write a sequence of numbers. Starting with 0, Plot Points and Describe Patterns with Coordinate “Add 6” and the starting number students write 0, 6, 12, 18, 24, . . . Pairs 0, generate terms in the Generate a Number Pattern from a Rule and Plot the resulting sequences, and After comparing these two sequences, the students notice that each Points observe that the terms in one term in the second sequence is twice the corresponding terms of the sequence are twice the first sequence. One way they justify this is by describing the patterns Activities and Tasks corresponding terms in the of the terms. Their justification may include some mathematical PARCC Sample Question other sequence. Explain notation (See example below). A student may explain that both informally why this is so. sequences start with zero and to generate each term of the second Videos Connections: 5.G.A.1 sequence he/she added 6, which is twice as much as was added to Identify-the-relationship-between-two-numerical-patterns (LZ) 5.G.A.2 produce the terms in the first sequence. Students may also use the distributive property to describe the relationship between the two numerical patterns by reasoning that 6 + 6 + 6 = 2 (3 + 3 + 3).

Standard Connections/Notes Resources 0, +3 3, +3 6, +3 9, +312, . . . 0, +6 6, +6 12, +618, +6 24, . . .

Once students can describe that the second sequence of numbers is twice the corresponding terms of the first sequence, the terms can be written in ordered pairs and then graphed on a coordinate grid. They should recognize that each point on the graph represents two quantities in which the second quantity is twice the first quantity.

Standard Connections/Notes Resources 5.G.A.1 Use a pair of Students can use a classroom size coordinate system to physically Teaching Student Centered Mathematics perpendicular number lines, locate the coordinate point (5,3) by starting at the origin point (0,0) by Location Activities pg. 239 called axes, to define a walking 5 units along the x axis to find the first number in the pair and coordinate system, with the then walking up 3 units for the second number in the pair. Lessons intersection of the lines (the Fly on the Ceiling origin) arranged to coincide with Name Points Using Coordinate Pairs the 0 on each line and a given Battleship point in the plane located by Draw Symmetric Figures on Coordinate Plane (challenging) using an ordered pair of Construct a Coordinate System on a Plane numbers, called its coordinates. Understand that the first Activities and Tasks number indicates how far to Opus Math sample questions travel from the origin in the Pieces of Eight direction of one axis, and the Blackbeard’s Treasure Box second number indicates how Coordinate Grid Geoboards (K5) far to travel in the direction of Coordinate Grid Tangram (K5) the second axis, with the Coordinate Shapes convention that the names of Geometric Shapes on a Coordinate Grid (K5) the two axes and the Coordinate Grid Swap (K5) coordinates correspond (e.g., x- Battleship Using Grid Paper (IM) axis and x-coordinate, y-axis PARCC Sample Question and y-coordinate). Videos Use-a-model-to-identify-parts-of-a-coordinate-plane (LZ)

Online Geoboard – Coordinate NLVM (quadrant one only) Hidden Message Coordinate Graphing Worksheet Works Printable Grid Sheets Math Aids

Templates and Visuals First Quadrant Grids

Standard Connections/Notes Resources 5.G.A.2 Represent real world Although students can often “locate a point,” these understandings Teaching Student Centered Mathematics and mathematical problems by are beyond simple skills. For example, initially, students often fail to Graphing Patterns pgs. 297-298 graphing points in the first distinguish between two different ways of viewing the point (2, 3), quadrant of the coordinate say, as instructions: “right 2, up 3”; and as the point defined by being Lessons plane, and interpret coordinate a distance 2 from the x-axis and a distance 3 from the y-axis. In these Describe the Graph Illuminations values of points in the context of two descriptions, the 2 is first associated with the x-axis, then with the Use Coordinate Systems to Solve Real World Problems the situation. y-axis. .

Examples: Activities and Tasks Sara has saved $20. She earns $8 for each hour she works. Opus Math Sample problems  If Sara saves all of her money, how much will she have after Meerkat Coordinate Plane Task (IM) working 3 hours? 5 hours? 10 hours?  Create a graph that shows the relationship between the Videos hours Sara worked and the amount of money she has saved. Find-distance-between-points-on-a-coordinate-plane-by- Use the graph below to determine how much money Jack makes counting (LZ) after working exactly 9 hours. Templates and Visuals Describe the Graph Activity recording Sheet for NCTM Lesson

Unit Nine – Working Towards Fluency with Fractions (+, – , x) (13 days)

Addition, subtraction, and multiplication of fractions is taught in Grade 5 and not again in Grade 6. These standards are appearing again at the end of the year so that students can continue to use the different strategies they have learned this year and move towards fluency. They may continue to use strategies that are efficient, but must also understand and be able to use some kind of algorithm.

Unit Nine Standards 5.NF.A.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.)

5.NF.A.2 Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators, e.g., by 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.

5.NF.B.4 Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction. a. Interpret the product (a/b) × 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. b. 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.

5.NF.B.6 Solve real world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem.

Grade 5 Unit Nine – Working Towards Fluency with Fractions (13 days) Back to “Year at a Glance”

Connections/Notes Resources Lessons (For more description go back to Unit 3.) Add Fractions Making Like Units Numerically Subtract Fractions Making Like Units Numerically Examples: Strategize to Solve Multi-Term Problems Solve and Create Fraction Word Problems Involving  Addition, Subtraction, and Multiplication

Grade 5 Unit Nine – Working Towards Fluency with Fractions (13 days) Back to “Year at a Glance”

Connections/Notes Resources Time for Recess Unit (NYCDOE)

 Alexis spent of her money. She gave of the remainder to her brother. She had $120 left. How Activities and Tasks much money did she have at the beginning? Cindy’s Cats PARCC Sample Question

Bar Model

$120 ÷ 3 = $40, so each seventh represents $40. 7 x $40 = $280

 PARCC Sample Question

Unit Ten – Multiplication & Division of Whole Numbers – Fluency (12 days) These standards were introduced in Unit 1 to provide opportunities throughout the year for students to work towards fluency. In this unit students demonstrate fluency in multiplication with whole numbers and continue to practice division with whole numbers using various strategies. Students need to solve multistep word problems incorporating the story structures of Table 2.

Unit Ten Standards 5.NBT.B.5 Fluently multiply multi-digit whole numbers using the standard algorithm.

5.NBT.B.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.

Grade 5 Unit Ten – Multiplication & Division of Whole Numbers – Fluency (12 days) Back to “Year at a Glance”

Standard Connections/Notes Resources 5.NBT.B.5 Fluently multiply Grade 5 students need to become fluent with the standard Lessons multi-digit whole numbers multiplication algorithm with multi-digit whole numbers. In prior Use Your Head (AIMS) using the standard algorithm. grades, students used various strategies to multiply. Students can Fence IT (Extend task to larger nos.) continue to use these different strategies as long as they are efficient, Solve Two-Step Word Problems with Multi-Digit Mul but must also understand and be able to use the standard algorithm. Fluently Multiply Using the Standard Algorithm In applying the standard algorithm, students recognize the importance Design and Construct Boxes of place value. Finding Products of a Whole Number and a Decimal Gifts Galore Xmas in June (AIMS) Examples: 371 x 2,784 Activities and Tasks PARCC Toy Animals Task Batter-Up Multiplication Game Multiplication Mix PARCC Sample Question

5.NBT.B.6 Find whole-number 5.NBT.B.6 is a milestone along the way to reaching fluency with Lessons quotients of whole numbers the standard algorithm in Grade 6 (6.NS.B.2). Students should Counting the Miles (AIMS) with up to four-digit dividends continue to interpret remainders. Three Digit Divisibility Dilemma (AIMS) and two-digit divisors, using How Thick Is It (AIMS) strategies based on place value, the properties of operations, and/or the Activities and Tasks relationship between Double Dutch Treat multiplication and division. End of Year Practice Problems (AIMS) Illustrate and explain the calculation by using equations. rectangular arrays, and/or area models.

Table 2

The following resources have been used to create these units for WCBOE: Building Conceptual Understanding and Fluency through Games (Public Schools of North Carolina) Engage NY Arizona Academic Content Standards Progressions for the Common Core Standards Georgia DOE North Carolina DOE Illustrative Math Howard County, MD PARCC NYCDOE AIMS Learn Zillion Gizmos

K to 5 Math Resources NCTM Illuminations

Grade 5 Wicomico County Public Schools 2014-2015 69