math4u.wicomico.wikispaces.netmath4u.wicomico.wikispaces.net/file/view/Grade 5... · Web viewFifth Grade. Mathematics Curriculum. Guide. f. or . ... (12 days) Mathematics Placement

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

Grade 5 Wicomico County Public Schools 2014-2015 2


GRADE 5 MD COLLEGE AND CAREER READY STANDARDS INTRODUCTION


Standards for Mathematical PracticeStandards Explanations and Examples1. Make sense of problems and persevere in solving them.

Students solve problems by applying their understanding of operations with whole numbers, decimals, and fractions including mixed numbers. They solve problems related to volume and measurement conversions. Students seek the meaning of a problem and look for efficient ways to represent and solve it. They may check their thinking by asking themselves, ―What is 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 and quantitatively.

Fifth graders should recognize that a number represents a specific quantity. They connect quantities to written symbols and 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 arguments and critique the reasoning of others.

In fifth grade, students may construct arguments using concrete referents, such as objects, pictures, and drawings. They explain calculations based upon models and properties of operations and rules that generate patterns. They demonstrate and explain the relationship between volume and multiplication. They refine their mathematical communication skills as they 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 mathematics.

Students experiment with representing problem situations in multiple ways including numbers, words (mathematical 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 tools strategically.

Fifth graders consider the available tools (including estimation) when solving a mathematical problem and decide when 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 precision.

Students continue to refine their mathematical communication skills by using clear and precise language in their discussions 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 make use of structure.

In fifth grade, students look closely to discover a pattern or structure. For instance, students use properties of operations as strategies to add, subtract, multiply and divide with whole numbers, fractions, and decimals. They examine numerical patterns and relate them to a rule or a graphical representation.

8. Look for and express regularity in repeated reasoning.

Fifth graders use repeated reasoning to understand algorithms and make generalizations about patterns. Students connect place value and their prior work with operations to understand algorithms to fluently multiply multi-digit numbers and perform all operations with decimals to hundredths. Students explore operations with fractions with visual models and begin to formulate generalizations.


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 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 by selecting it. Important Dates

Unit 1: Developing Multiplication and Division Strategiesthrough Volume (23 days)

September 1 Labor DaySeptember 26 Professional Day

Unit 2: Extend Place Value to Decimals (21 days)October 17 MSEA Convention

Mathematics Intermediate Interim Assessment 1 - October 21October 28 End term 1

Unit 3: Fraction Operations (33 days)

November 3 Professional DayNovember 4 Election Day

November 17–21 American Education WeekNovember 26–28 Thanksgiving Holiday

Mathematics Intermediate Interim Assessment 2 – December 11December 22 January 2 Winter Break

Unit 4: Decimal Operations (15 days)January 20 Martin Luther King’s birthday

January 23 End term 2January 26 Professional Day

Unit 5: Solving Problems with Fractional Quantities (21 days)February 13 Professional DayFebruary 16 Presidents’ Day

PARCC PBA 3/2 – 3/27 (window)

Unit 6: Classifying 2D Figures (15 days)

Unit 7: Representing Algebraic Thinking ( 12 days)March 27 – End of term 3

April 1-6 Spring BreakApril 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)

Unit 10: Multiplication & Division of Whole Nos. – Fluency (12 days)Mathematics Placement Assessment GR 5 – May 22

May 25 – Memorial Day

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.

Multiplicative Comparison

Smaller Unknown: Larger Unknown: Multiplier Unknown:

4th Grade

4.OA.2

The giraffe is 18 feet tall. She is 3 times as tall as the kangaroo. How tall is the kangaroo?

Measurement Example:A rubber band is stretched to be 18 centimeters long and that is 3 times as long as it was at first. How long was the rubber band at first?

The giraffe in the zoo is 3 times as tall as the kangaroo. The kangaroo is 6 feet tall. How tall is the giraffe?

Measurement Example:A rubber band is 6 centimeters long. How long will the rubber band be when it is stretched to be 3 times as long?

The giraffe is 18 feet tall. The kangaroo is 6 feet tall. The giraffe is how many times taller than the kangaroo?

Measurement Example:A rubber band was 6 centimeters long at first. Now it is stretched to be 18 centimeters long. How many times as long is the rubber band now as it was at first?

5th Grade

The giraffe is 18 feet tall. The kangaroo is 1/3 as tall as the giraffe. How tall is the kangaroo?

The kangaroo is 6 feet tall. She is 1/3 as tall as the giraffe. How tall is the giraffe?

The giraffe is 18 feet tall. The kangaroo is 6 feet tall. What fraction of the height of the giraffe is the height of the kangaroo?

Unit One Standards5.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 LanguageThe 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.algorithmassociative property decomposedistributive property dividend divisor equation exponents expression measurement division

multiplicand multiplier order of operations (conventional order) partition division (fair-sharing) partial product partial quotient product properties of operations quotient

remainder

Volumearea of baseattribute capacitycubic units cubic cm, cubic in, cubic ftedge lengths gap

height layersmeasurement nonstandard cubic units overlap right rectangular prism solid figure unit cube variable

Essential QuestionsHow 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 Misconceptions1. 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 Resources5.NBT.B.5 Fluently multiply multi-digit whole numbers using the standard algorithm.

In prior grades, students used various strategies to multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers. Strategies were based on place value and the properties of operations and were illustrated and explained by using equations, rectangular arrays, and/or area models.

Students can continue to use these different strategies as long as they are efficient, but must also understand and be able to use the standard algorithm. In applying the standard algorithm, students recognize the importance of place value.

The area model for multiplication is a pictorial way of representing multiplication. In the area model, the length and width of a rectangle represent factors, and the area of the rectangle represents their product.An area model not only helps to explain why the standard algorithm commonly taught in the United States for multiplication works, it is an efficient recording alternative.

Some students (especially visual learners and those who have difficulty keeping numbers lined up in multiplication problems) may prefer it. Furthermore, this method has certain benefits. It illuminates important mathematical concepts (such as the distributive property), allows for computational flexibility (expanded notations allow students to use derived facts), and reinforces the concept of area.

Teaching Student Centered MathematicsInvented Strategies for Multiplication pgs. 113-115Area Model Development pgs. 116-117Traditional Algorithm for Multiplication pgs. 118-120Expanded Lesson Area Model Multiplication pgs. 129-130

LessonsUsing Partial Products and Visual Models forMultiplication

Multiply by Multiples of Ten Using Patterns and Properties

Activities and TasksToss a RectangleElmer’s Multiplication Error (IM)

VideosMultiply multi-digit numbers using an area model (LZ)Area Models for Multiplication using Kidspiration

OnlineRectangle Multiplication (common) NLVM

Templates and VisualsVan de Walle Template Partial Products - #36Supplemental Unit on Area ModelsPartial Product TemplateArea Model and Partial Products Example


http://nlvm.usu.edu/en/nav/frames_asid_192_g_2_t_1.html?from=category_g_2_t_1.html

http://teachertube.com/viewVideo.php?video_id=104876

https://learnzillion.com/lessons/528-multiply-multidigit-numbers-using-an-area-model



Standard Connections/Notes ResourcesExamples: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 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.

In fourth grade, students’ experiences with division were limited to dividing by one-digit divisors. This standard extends students’ prior experiences with strategies, illustrations, and explanations. When the two-digit divisor is a ―familiar number, a student might decompose the dividend using place value.

Example: Using expanded notation ~ 2682 ÷ 25 =

(2000 + 600 + 80 + 2) ÷ 25Using his or her understanding of the relationship between 100 and 25, a student might think ~

I know that 100 divided by 25 is 4 so 200 divided by 25 is 8 and 2000 divided by 25 is 80.600 divided by 25 has to be 24. Since 3 x 25 is 75, I know that 80 divided by 25 is 3 with a reminder of 5. (Note that a student might divide into 82 and not 80) I can’t divide 2 by 25 so 2 plus the 5leaves a remainder of 7. 80 + 24 + 3 = 107. So, the answer is 107 with a remainder of 7.

Teaching Student Centered MathematicsInvented Strategies for Division pgs. 121-123

Activities and TasksMinutes and Days (IM)Partition the Dividend (K5)Division Strategy Multiplying Up (K5)Soda Task

VideosUsing Base 10 Blocks to Model Long Division (UT)Division (chunking) (UT)

Divide-4digit-dividends-by-2-digit-divisors-by-setting-up-an-equation (LZ)

Online


http://learnzillion.com/lessons/550-divide-4digit-dividends-by-2-digit-divisors-by-setting-up-an-equation

http://learnzillion.com/lessons/550-divide-4digit-dividends-by-2-digit-divisors-by-setting-up-an-equation

http://www.youtube.com/watch?v=JA1dazUC3sM&feature=share

http://www.youtube.com/watch?v=8IXAqXGDMXw&feature=share



Standard Connections/Notes Resources

Using an equation that relates division to multiplication, 25 x n = 2682, a student might estimate the answer to be slightly larger than 100 because s/he recognizes that 25 x 100 = 2500.

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

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.

Rectangle Division NLVMThe Quotient Café - Illuminations

Templates and VisualsDivision Using the Area Model


http://illuminations.nctm.org/Activity.aspx?ID=4197




Standard Connections/Notes Resources5. 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.

Students’ prior experiences with volume were restricted to liquid volume. As students develop their understanding of volume, they understand that a 1-unit by 1-unit by 1-unit cube is the standard unit for measuring volume. This cube has a length of 1 unit, a width of 1 unit and a height of 1 unit and is called a cubic unit. This cubic unit is written with an exponent of 3 (e.g., in³, m³). Students connect this notation to their understanding of powers of 10 in our place value system. Models of cubic inches, centimeters, cubic feet, etc. are helpful in developing an image of a cubic unit. Student’s estimate how many cubic yards would be needed to fill the classroom or how many cubic centimeters would be needed to fill a pencil box.

Teaching Student Centered MathematicsActivity 9.10 “Box Comparison – Cubic Units” pg. 267

LessonsEstimating Volume - Counting on Frank - Illuminations Collecting the RaysFinding Volume by Packing with Cubic Units and Counting

Activities and TasksBuild a Cubic Meter (K5)Rectangular Prism Nets for PackingInsides and Outsides - Utah Education Network (Do not extend to surface area.)

VideosUnderstanding Volume LZ

OnlineCubes Illuminations

5.MD.C.4 Measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and improvised units.

Students understand that same sized cubic units are used to measure volume. They select appropriate units to measure volume. For example, they make a distinction between which units are more appropriate for measuring the volume of a gym and the volume of a box of books. They can also improvise a cubic unit using any unit as a length (e.g., the length of their pencil). Students can apply these ideas by filling containers with cubic units (wooden cubes) to find the volume. They may also use drawings or interactive computer software to simulate the same filling process.


LessonsEstimating Volume - Counting on Frank - IlluminationsCollecting the Rays - IlluminationsFill ‘Er Up - Illuminations Insides and Outsides (Do not extend to surface area.)Explore Volume by Counting Unit Cubes

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


http://illuminations.nctm.org/Lesson.aspx?id=3085



https://learnzillion.com/lessons/1796-understanding-volume





Standard Connections/Notes ResourcesVideosIdentify and label three dimensional figures (LZ)Find-volume-by-counting-cubes (LZ)

OnlineCubes - Illuminations

Templates and VisualsRectangular Prism Nets for Packing (cm³)


http://learnzillion.com/lessons/1264-find-volume-by-counting-cubes

http://learnzillion.com/lessons/1485-identify-and-label-threedimension-figures



Standard Connections/Notes Resources5.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

Students need multiple opportunities to measure volume by filling rectangular prisms with cubes and looking at the relationship between the total volume and the area of the base. They derive the volume formula (volume equals the area of the base times the height) and explore how this idea would apply to other prisms. Students use the associative property of multiplication and decomposition of numbers using factors to investigate rectangular prisms with a given number of cubic units. They have to be able to decompose a prism, understanding that it can be partitioned into layers, and each layer partitioned into rows, and each row into cubes. They also have to be able to compose such a structure, multiplicatively, back into higher units. That is, they eventually learn to conceptualize a layer as a unit that itself is composed of units of units—rows, each row composed of individual cubes—and they iterate that structure.

Examples:

Students are given a net and asked to predict the number ofcubes required to fill the container formed by the net. In suchtasks, students may initially count single cubes or repeatedlyadd the number of cubes in a row to determine the number ineach layer, and repeatedly add the number in each layer to findthe total number of unit cubes. In folding the net to make theshape, students can see how the side rectangles fit together anddetermine the number of layers.

When given 24 cubes, students make as many rectangular prisms as possible with a volume of 24 cubic units. Students build the prisms and record possible dimensions.

Length Width Height


LessonsCutting Corners (AIMS)Estimating Volume - Counting on Frank - IlluminationsCollecting the Rays - IlluminationsFill ‘Er Up – Illuminations Compose and Decompose Rectangular Prisms Using LayersUse Multiplication to Calculate VolumeFind Volume of Solid Figures Composed of Two Rectangular PrismsDesign a SculptureDesign a Sculpture 2Paper Volume Cubes

Activities and TasksPARCC Sample TaskDesigning a Cereal Box (K5)Designing a Toy Box (K5)Joe’s Buildings (K5)Roll a Rectangular PrismCandy BoxesFilling BoxesPacking BlocksBox of Clay (IM)Using Volume to Understand the Assoc.Property (IM)You Can Multiply Three Numbers in Any Order (IM)Insides and Outsides - (Do not extend to surface area.)The Sky’s the Limit Calculating VolumeVolume of Solid Figures

VideosFind-the-volume-of-a-solid-figure-by-multiplying (LZ)Recognize that volume is additive (LZ)

Online


https://learnzillion.com/lessons/1592-recognize-that-volume-is-additive-by-finding-the-volume-of-a-3d-figure-composed-to-two-rectangular-prisms

http://learnzillion.com/lessons/1803-find-the-volume-of-a-solid-figure-by-multiplying-v-l-x-w-x-h

http://www.illustrativemathematics.org/illustrations/1631


http://www.parcconline.org/sites/parcc/files/Grade5-TwoAquariumTanks.pdf






Standard Connections/Notes Resourcessolve real world problems. 1 2 12

2 2 64 2 38 3 1

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 shown in the illustrations below.

Cubes – IlluminationsSpace Blocks NLVMIsometric Drawing Tool ( cube) Illuminations

Templates and VisualsPower 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


http://illuminations.nctm.org/Activity.aspx?id=4182

http://nlvm.usu.edu/en/nav/frames_asid_195_g_2_t_2.html?open=activities&from=category_g_2_t_2.html


system relies on the meanings and properties of operations. This also contributes to deepening students’ understanding of computation and algorithms in 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 Standards5.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 LanguageThe 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 propertyexponentsdecimal decimal point

distributive propertyfraction hundredths ones

place value powers of 10rounding tenths

thousandthsvalue

Essential QuestionsWhat 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 Misconceptions1. 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 Resources5.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

In fourth grade, students examined the relationships of the digits in numbers for whole numbers. This standard extends this understanding to the relationship of decimal fractions to thousandths. In grade 4, students recognize that in a multi-digit whole number, a digit in one

Teaching Student Centered MathematicsA Two-Way Relationship pg. 184

Lessons




Standard Connections/Notes Resourcesplace to its right and 1/10 of what it represents in the place to its left.

Connections: 5.NBT.A.25.MD.A.1

place represents ten times what it represents in the place to its right (4.NBT.A.1). Students use base ten blocks, pictures of base ten blocks, and interactive images of base ten blocks to manipulate and investigate the place value relationships.

They use their understanding of unit fractions to compare decimal places and fractional language to describe those comparisons. Before considering the relationship of decimal fractions, students express their understanding that in multi-digit whole numbers, a digit in one place represents 10 times what it represents in the place to its right and 1/10 of what it represents in the place to its left.

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) represents 500. So a 5 in the hundreds place is ten times as much as a 5 in the tens place or a 5 in the tens place is 1/10 of the value of a 5 in the hundreds place.

To extend this understanding of place value to their work with decimals, students use a model of one unit; they cut it into 10 equal pieces, shade 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). They repeat the process by finding 1/10 of a 1/10 (e.g., dividing 1/10 into 10 equal parts to arrive at 1/100 or 0.01) and can explain their reasoning, 0.01 is 1/10 of 1/10 thus is 1/100 of the whole unit.

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?

Changing Places (AIMS)Relate Base Ten Units from Millions to ThousandthsPatterns R UsMeaning of Place Value

Activities and TasksKipton’s Scale (IM)Tenths and Hundredths (IM)Which Number Is It? (IM)Comparing Digits (K5)Place Value TasksWord Problems with Template

VideosMultiply and divide with multiples of tenDetermine-the-value-of-a-digit-in-the-thousandths-place (LZ)

OnlineBase Blocks Decimals - NLVM

Templates and VisualsDecimal of the DayNumber ConnectionsPlace Value Template for Sheet Protector

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

Powers of 10 is a fundamental aspect of the base ten system, thus 5.NBT.A.2 can help students extend their understanding of place value to incorporate decimals to hundredths. When students explain patterns in the number of zeroes of the product when multiplying a number by powers of 10 (5.NBT.2), they have an opportunity to look for and express regularity in repeated reasoning. New at Grade 5 is the use of whole number exponents to denote powers of 10. Students

LessonsUse Exponents to Name Place Value Units

Activities and TasksMultiplying Whole Numbers by a Power of Ten (K5)Multiplying a Decimal by a Power of 10



http://learnzillion.com/lessons/2519-determine-the-value-of-a-digit-in-the-thousandths-place

http://learnzillion.com/lessons/2519-determine-the-value-of-a-digit-in-the-thousandths-place

https://learnzillion.com/lessons/3013-multiply-and-divide-with-multiples-of-ten



Standard Connections/Notes Resourcesdivided by a power of 10. Use whole-number exponents to denote powers of 10.

understand why multiplying by a power of 10 shifts the digits of a whole number or decimal that many places to the left. Patterns in the number of 0s in products of a whole numbers and a power of 10 and the location of the decimal point in products of decimals withpowers of 10 can be explained in terms of place value.

Examples:Students might write:

36 x 10 = 36 x 101 = 360 36 x 10 x 10 = 36 x 102 = 3600 36 x 10 x 10 x 10 = 36 x 103 = 36,000 36 x 10 x 10 x 10 x 10 = 36 x 104 = 360,000

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 103= 523,000 The place value of 523 is increased by three places.

52.3 ÷ 101 = 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 = 2,500Students should reason that the exponent above the 10 indicates how many places the decimal point is moving (not just that the decimal point

Dividing a Decimal by a Power of 10 (K5)Dividing a Whole Number by a Power of 10 (K5)Marta’s Multiplication Error (IM)Multiplying Decimals by 10 (IM)Patterns R UsWord Problems with Template

VideosMultiply-decimal-numbers-by-a-power-of-ten (LZ)

Templates and VisualsDecimal of the DayNumber ConnectionsPlace Value Template for Sheet Protector


http://learnzillion.com/lessons/435-multiply-decimal-numbers-by-a-power-of-ten



Standard Connections/Notes Resourcesis 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 Resources5.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

Connections:5.NBT.A.4

Students build on the understanding they developed in fourth grade to read, write, and compare decimals to thousandths. They connect their prior experiences with using decimal notation for fractions and addition of fractions with denominators of 10 and 100. They use concrete models and number lines to extend this understanding to decimals to the thousandths. Models may include base ten blocks, place value charts, grids, pictures, drawings, manipulatives, technology-based, etc.They read decimals using fractional language and write decimals in fractional form, as well as in expanded notation as shown in the standard 3a. This investigation leads them to understanding equivalence of decimals (0.8 = 0.80 = 0.800).

Students need to understand the size of decimal numbers and relate them to common benchmarks such as 0, 0.5 (0.50 and 0.500), and 1. Comparing tenths to tenths, hundredths to hundredths, and thousandths to thousandths is simplified if students use their understanding of fractions to compare decimals.

Example:Some equivalent forms of 0.72 are:72/100

7/10 + 2/100

7 x (1/10) + 2 x (1/100)

0.70 + 0.02

70/100 + 2/100

0.720

7 x (1/10) + 2 x (1/100) + 0 x (1/1000)

720/1000

Comparing 0.25 and 0.17, a student might think, 25 hundredths is more than 17 hundredths. They may also think that it is 8 hundredths more. They may write this comparison as 0.25 > 0.17 and recognize that 0.17 < 0.25 is another way to express this comparison.Comparing 0.207 to 0.26, a student might think, both numbers have 2 tenths, so I need to compare the hundredths. The second number has

Teaching Student Centered MathematicsActivity 7.9 “Line ‘Em Up” pg. 191Activity 7.10 “Close Nice Numbers” pg. 192

LessonsName Decimal Fractions in Various FormsCompare Decimal Fractions to the ThousandthsShow Me the Money (AIMS)Dueling Decimals (AIMS)Decimal Detectives (AIMS)Decimal Designs (AIMS)Dealing with Decimals (AIMS)Clued into Decimals (AIMS)

Activities and TasksDecimal of the WeekDecimal Numbers Exit SlipsDecimal AssessmentComparing Decimals QR Code ActivityComparing Decimals (K5)Representing Decimals with Base Ten Blocks (K5)Representing Decimals in Different Ways (K5)Are These Equivalent to 9.52? (IM)Comparing Decimals on the Number Line (IM)Placing Thousandths on the Number Line (IM)Drawing Pictures to Illustrate Decimal Comparisons (IM)Base Ten Pictures for DecimalsDecimal Number of the Day PromptsDecimal WarDecimal Review Game CardsPlace Value Path GameDecimal Representations on a Number LineSpin and Compare DecimalsWriting Decimals in Expanded FormCorn ShucksRace to a Meter




Standard Connections/Notes Resources6 hundredths and the first number has no hundredths so the second number must be larger. Another student might think while writing fractions, I know that 0.207 is 207 thousandths (and may write 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 than 207 thousandths.

VideosWrite-decimals-in-expanded-form (LZ)

OnlineNumber Line Worksheets with Decimals Math AidsBase Blocks Decimals - NLVM

Templates and VisualsBase Ten Blocks as DecimalsDecimal Place Value CardsDecimal Square SetsLarge Thousandths Decimal SquarePlace Value MatDecimal Arrow Cards

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

Connections:

5.NBT.A.3

When rounding a decimal to a given place, students may identify the two possible answers, and use their understanding of place value to compare the given number to the possible answers.

Example:Round 14.235 to the nearest tenth.Students recognize that the possible answer must be in tenths thus, it is either 14.2 or 14.3. They then identify that 14.235 is closer to 14.2 (14.20) than to 14.3 (14.30).

Decompose 1.57 to round to the nearest tenth.

Teaching Student Centered MathematicsClose “Nice” Numbers pg. 192

LessonsDecimal Round-Up/Round-DownRounding Decimals Using the Vertical Number Line

Activities and TasksRounding to Tenths and Hundredths (IM)Batter UpSmarter Balance Sample Question

VideosRound-decimals-to-the-nearest-tenth-number-line (LZ)

OnlineBase Blocks Decimals - NLVM

Templates and VisualsDecimal Number Lines Template



http://learnzillion.com/lessons/432-round-decimals-to-the-nearest-tenth-using-a-number-line

http://sampleitems.smarterbalanced.org/itempreview/sbac/index.htm


http://learnzillion.com/lessons/3285-write-decimals-in-expanded-form






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 Standards5.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?


addition/add associative propertybenchmark fraction common denominator difference

decomposedistributive propertyequationequivalent estimate

improper fraction mixed number numerator reasonableness scaling

subtraction/subtract unit fraction unlike denominator variable

Essential QuestionsHow 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”


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.)

Students in Grade 4 added and subtracted fractions with like denominators. Fifth grade students should apply their understanding of equivalent fractions developed in fourth grade and their ability to rewrite fractions in an equivalent form to find common denominators. They should know that multiplying the denominators will always give a common denominator but may not result in the smallest denominator. Simplifying answers is not part of this standard, yet conversations should take place when an answer is 6/4 cup of flour, for example.

Suggested manipulatives to develop this concept: fraction strips, Cuisenaire rods, fraction towers, fraction fringes (AIMS), fraction squares, fraction circles, two color counters, and pattern blocks.

Visual fraction models include tape diagrams, area models, number line diagrams, set diagrams (See Van de Walle pgs.162-166).

Examples:

Area Model

Linear Model ( 34

+ 23

)

Teaching Student Centered MathematicsAddition and Subtraction pgs. 162-166An Area Model Approach pgs. 155-156

Lessons

Use Benchmark Numbers to Assess Reasonableness of Addition and Subtraction Equations

Add Fractions Using Visual Models to Create Equivalent Fractions

Subtract Fractions Using Visual Models to Create Equivalent Fractions

Add and Subtract Whole and Mixed Numbers Using the Number Line

Royal Rugs (AIMS)Fraction Time (AIMS)Fractions with Pattern Blocks (AIMS)Estimating Sums and DifferencesDelightfully Different Fractions Utah Ed NetworkInvestigating Equivalent Fractions with Relationship Rods - Illuminations

Activities and TasksAdding 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)Common Assessment on Adding and Subtracting Fractions and Mixed Numbers




http://www.uen.org/Lessonplan/preview.cgi?LPid=21526



Standard Connections/Notes ResourcesEgyptian Fractions (IM)Finding Common Denominators to Add (IM)Jog a Thon (IM)Making S’more (IM)

VideosUsing Pattern Blocks to Add FractionsUsing Pattern Blocks to Subtract FractionsUsing Pattern Blocks to Multiply FractionsFind-common-denom.-by-creating-area-models(LZ)

OnlineEquivalent Fractions (like denominators)Fractions – Adding NLVM

Templates and VisualsFraction Spinners to EighthsFraction Spinners to SixteenthsFraction Spinners

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. For example, recognize an incorrect result 2/5 + 1/2 = 3/7, by observing that 3/7 < 1/2.

Some problems should appear as multistep.

Examples:Jerry was making two different types of cookies. One recipe needed 3/4 cup of sugar and the other needed 2/3 cup of sugar. How much sugar did he need to make both recipes?

Mental estimation:A student may say that Jerry needs more than 1 cup of sugar but less than 2 cups. An explanation may compare both fractions to ½ and state that both are larger than ½ so the total must be more than 1. In addition, both fractions are slightly less than 1 so the sum cannot be more than 2.

Example: Using a bar diagram

Sonia had 2 13

candy bars. She promised her brother that she

would give him ½ of a candy bar. How much will she have left

LessonsSolve Two-Step Word Problems Involving Fractions

Solve Multi-Step Word Problems with Mixed Numbers

Activities and TasksPARCC Sample QuestionMeasuring Cups (IM)Do These Add Up? (IM)Salad Dressing (IM)Fraction Word Problems Unlike Denominators(K5)Opus Math – Word problems

VideosEstimate-answers-to-problems-involving-fractions-using-area-models (LZ)


http://learnzillion.com/lessons/3181-estimate-answers-to-problems-involving-fractions-using-area-models

http://learnzillion.com/lessons/3181-estimate-answers-to-problems-involving-fractions-using-area-models

http://www.opusmath.com/common-core-clusters/5.nf.a-add-and-subtract-fractions

http://www.parcconline.org/sites/parcc/files/PARCC_SampleItems_Mathematics_G5PencilBox_082713_Final.pdf

http://nlvm.usu.edu/en/nav/frames_asid_106_g_2_t_1.html


http://learnzillion.com/lessons/2638-find-common-denominators-by-creating-area-models

http://www.youtube.com/watch?v=UllVhLmSRXc

http://www.youtube.com/watch?v=kmsd1HoYoug

http://www.youtube.com/watch?v=0l1jVDQe1a4



Standard Connections/Notes Resourcesafter she gives her brother the amount she promised?

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 16

leaving 1 + ¼ + 16

which a student can then change to 1 + 312

+ 2

12 = 1

512

.

This diagram models a way to show how 316

and 1 34

can be

expressed with a denominator of 12.

Once this is accomplished, a student can complete

the problem,

2 1412

– 1 9

12 = 1

512

.





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.For example, interpret 3/4 as the result of dividing 3 by 4, noting that 3/4 multiplied by 4 equals 3, and that when 3 wholes are shared equally among 4 people each person has a share of size 3/4. If 9 people want to share a 50-pound sack of rice equally by weight, how many pounds of rice should each person get? Between what two whole numbers does your answer lie?

Connections:5.NBT.B.6

Students are expected to demonstrate their understanding using concrete materials, drawing models, and explaining their thinking when working with fractions in multiple contexts. They read 3/5 as three fifths and after many experiences with sharing problems, learn that 3/5 can also be interpreted as 3 divided by 5.

Examples: Ten team members are sharing 3 boxes of cookies. How much

of a box will each student get? When working this problem a student should recognize that the 3 boxes are being divided into 10 groups, so s/he is seeing the solution to the following equation, 10 x n = 3 (10 groups of some amount is 3 boxes) which can also be written as n = 3 ÷ 10. Using models or diagram, they divide each box into 10 groups, resulting in each team member getting 3/10 of a box.

Two afterschool clubs are having pizza parties. For the Math Club, the teacher will order 3 pizzas for every 5 students. For the student council, the teacher will order 5 pizzas for every 8 students. Since you are in both groups, you need to decide which party to attend. How much pizza would you get at each party? If you want to have the most pizza, which party should you attend?

The six fifth grade classrooms have a total of 27 boxes of pencils. How many boxes will each classroom receive? Students may recognize this as a whole number division problem but should also express this equal sharing problem as 276 . They explain that each classroom gets

276 boxes of

LessonsInterpret a Fraction as Division

Use Tape Diagrams to Model Fractions as Division

Solve Word Problems Involving Division of Whole Numbers Leading to Answers in the Form of Fractions

Activities and TasksConverting Fractions of a Unit to a Smaller Unit (IM)How Much Pie? (IM)Relating Fractions to Division (K5)

VideosUse-picture-models-to-divide-whole-numbers-leading-to-answers-in-the-form-of-fractions (LZ)


http://learnzillion.com/lessons/3588-use-picture-models-to-divide-whole-numbers-leading-to-answers-in-the-form-of-fractions

http://learnzillion.com/lessons/3588-use-picture-models-to-divide-whole-numbers-leading-to-answers-in-the-form-of-fractions




Standard Connections/Notes Resourcespencils and can further determine that each classroom gets 4 36 or 4

12 boxes of pencils.

(cont.)

Example:

5.NF.B.4 Apply and extend previous understandings of multiplication to multiply a

Representing multiplication of fractions with visual and concrete models is fundamental to this unit in order for students to make sense of multiplying fractions by fractions (MP.1, MP.4).

Teaching Student Centered MathematicsMultiplication pgs. 167-172




Standard Connections/Notes Resourcesfraction 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. For example, use a visual fraction model to show (2/3) × 4 = 8/3, and create a story context for this equation. Do the same with (2/3) × (4/5) = 8/15. (In general, (a/b) × (c/d) = ac/bd.)

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.

Students are expected to multiply fractions including proper fractions, improper fractions, and mixed numbers. (Mixed numbers will be addressed in Unit 5.) They multiply fractions efficiently and accurately as well as solve problems in both contextual and non-contextual situations.As they multiply fractions such as 3/5 x 6, they can think of the operation in more than one way.

3 x (6 ÷ 5) or (3 x 6/5) (3 x 6) ÷ 5 or 18 ÷ 5 (18/5)

Suggested manipulatives/visual models to develop this concept: area model templates (adapted from AIMS), pattern blocks, paper for folding, Cuisenaire rods, fraction squares, graph paper

Students create a story problem for 3/5 x 6 such as, Isabel had 6 feet of wrapping paper. She used 3/5 of the paper

to wrap some presents. How much does she have left? Every day Tim ran 3/5 of mile. How far did he run after 6 days?

(Interpreting this as 6 x 3/5)

Examples:Building on previous understandings of multiplication

Rectangle with dimensions of2 and 3 showing that 2 x 3 = 6.

LessonsCuisenaire Rods Fractions of a NumberRelate Fractions as Division to Fraction of a SetMultiply any Whole Number by a Fraction Using Tape Diagrams

Find a Fraction of a Measurement and Solve Word Problems

Activities and TasksConnor and Mikayla Discuss Multiplication (IM)Folding Strips of Paper (IM)Parts of a WholeThe Whole MattersGreatest ProductArea Word Problems with Fractional Side Lengths (K5)Using Models to Multiply Fractions Common Assessment

VideosCuisenaire Rods to Model Multiplication of Whole Numbers by Fractions, Fractions by Fractions, and Mixed Numbers by Fractions

Using Pattern Blocks to Multiply Fractions

Find-the-Area-of-a-Rectangle-with-Fractional-Side- Lengths-by-Tiling (LZ)

Multiplying a Fraction by a Fraction using an areamodel (LZ)

OnlineFractions – Rectangle Multiplication NLVM

Templates and Visuals


The area model and line segments show the area is the same quantity as the product of the side lengths.


http://www.youtube.com/watch?v=x8ys1JRdlKg

http://learnzillion.com/lessons/1542-find-the-area-of-a-rectangle-with-fractional-side-lengths-by-tiling

http://learnzillion.com/lessons/1542-find-the-area-of-a-rectangle-with-fractional-side-lengths-by-tiling

http://www.youtube.com/watch?v=UllVhLmSRXc

http://www.youtube.com/watch?v=I78nSG4IqCc








Rectangle with dimensions of 2 and 23

showing that 2 x 23 =

43

Fraction Spinners to EighthsFraction Spinners to SixteenthsFraction SpinnersArea Model Template adapted from AIMS

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 as scaling In preparation for Grade 6 work in ratios and proportional reasoning, students learn to see products such as 5 x 3 or 12 x 3 as expressions that can be interpreted in terms of a quantity, 3,

and a scaling factor, 5 or 12 . Thus, in addition to knowing that 5 x 3 =

15, they can also say that 5 x 3 is 5 times as big as 3, without

evaluating the product. Likewise, they see 12 x 3 as half the size of 3.

Students reason abstractly and practice communicating their thinking in real world situations (MP.2, MP.6). They use number lines and other visual models to interpret situations involving multiplication by numbers larger than one (when the result will be larger than the original quantity) and involving multiplication by a fraction smaller than 1 (when the result

LessonsCompare the Size of the Product to the Size of the FactorsCompare the Size of the Product to the Size of the Factors Lesson 2

Activities and TasksMultiplication and Scale Problems (K5)Comparing a Number and a Product (IM)Fundraising (IM)Reasoning about MultiplicationRunning a Mile

VideosUnderstand-the-effect-of-multiplying-by-a-fraction-greater-than-one (LZ)


http://learnzillion.com/lessons/3312-understand-the-effect-of-multiplying-by-a-fraction-greater-than-one

http://learnzillion.com/lessons/3312-understand-the-effect-of-multiplying-by-a-fraction-greater-than-one







Standard Connections/Notes Resourcesmultiplication 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

will be smaller than the original quantity)(MP.4).

(cont.)

Examples:34 x 7 is less than 7 because 7 is multiplied by a factor less than 1

so the product must be less than 7.

2 23

x 8 must be more than 8 because 2 groups of 8 is 16 and 2 23

is

almost 3 groups of 8. So the answer must be close to, but less than 24.

34 =

5 x35x 4 because multiplying

34 by

55 is the same as

multiplying by 1.


x 7




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

Some problems should appear as multistep.

Multiplication of mixed numbers will appear in Unit 5.

Example: Sue ran 23of a mile. Alex ran 3 times as far as Sue. How

far did Alex run? (multiplicative comparison GR4)

LessonsSolve Word Problems Using Tape Diagrams

Activities and TasksUsing Area Models to Multiply Fractions Common AssessmentMultiplying 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

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

OnlineFractions – Rectangle Multiplication NLVM

b. Interpret division of a whole number by a unit fraction, and compute such quotients. For example, create a story context for 4 ÷ (1/5), and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that 4 ÷ (1/5) = 20 because 20 × (1/5) = c. Solve real world problems

In this unit it is critical for students to use concrete objects or pictures to help conceptualize, create, and solve problems. In fifth grade, students experience division problems with whole number divisors and unit fraction dividends (fractions with a numerator of 1) or with unit fraction divisors and whole number dividends. Students extend their understanding of the meaning of fractions, how many unit fractions are in a whole, and their understanding of multiplication and division as involving equal groups or shares and the number of objects in each group/share. In sixth grade, they will use this foundational understanding to divide into and by more complex fractions and develop abstract methods of dividing by fractions.

Teaching Student Centered Mathematics Whole number and unit fraction only:Partition Concept pgs. 172-173Measurement Concept pgs.174-175

LessonsDivide a Whole Number by a Unit FractionDivide a Unit Fraction by a Whole Number Lesson

Activities and TasksDivide a Unit Fraction by a Whole Number (K5)Divide a Whole Number by a Unit Fraction (K5)


23

43

63


http://learnzillion.com/lessons/3421-multiply-a-fraction-by-a-whole-number-using-visual-representations

http://learnzillion.com/lessons/3421-multiply-a-fraction-by-a-whole-number-using-visual-representations

http://www.youtube.com/watch?v=lCbswBRlGyY

http://www.parcconline.org/sites/parcc/files/PARCC_SampleItems_Mathematics_G5AreaofBoard_081913_Final.pdf



Standard Connections/Notes Resourcesinvolving 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?

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 ÷ 15 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 12 lb of rice

Relating Fractions to Division (K5)Dividing by One Half (IM)How Many Servings of Oatmeal (IM)How Many Marbles (IM)Origami Stars (IM)

VideosDivide-unit-fractions-by-whole-numbers-by-drawing-a-model (LZ)Divide-whole-numbers-by-unit-fractions-using-a-number-line (LZ)

OnlineNumber Line Bars – Fractions NLVM


1 lb peanuts

15 lb of peanuts

The diagram shows the 1/3 pan divided into 4 equal shares with each share equaling 1/12 of the pan.


http://learnzillion.com/lessons/1041-divide-whole-numbers-by-unit-fractions-using-a-number-line

http://learnzillion.com/lessons/1041-divide-whole-numbers-by-unit-fractions-using-a-number-line

http://learnzillion.com/lessons/1037-divide-unit-fractions-by-whole-numbers-by-drawing-a-model

http://learnzillion.com/lessons/1037-divide-unit-fractions-by-whole-numbers-by-drawing-a-model



Standard Connections/Notes Resourcesequally?

A student may think or draw 12

and cut it into 3 equal groups

then determine that each of those parts is 16

.

A student may think of 12 as equivalent to

36 .

36 divided by 3 is

16 .

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 Standards5.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.


array associative property of multiplicationcentimeter commutative property of multiplication distributive property dividend division divisor factor hundred thousands hundreds

hundredths identity property of multiplication kilometermeasurement division (or repeated subtraction)metermetric systemmillimeter millions multiple multiplier ones

partial products partition/partitive division (or fair-sharing) place value product quotient remainder ten thousands tens tenths thousands

Essential QuestionsWhy 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 Misconceptions1. 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 Resources5.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.

Addition and Subtraction:Because of the uniformity of the structure of the base-ten system,students use the same place value understanding for addingand subtracting decimals that they used for adding and subtractingwhole numbers. Instead of just computing answers, students reason about both the relationship between fraction and decimal operations and the relationship between whole number computation and fractional/decimal computation (MP.2, MP.3).This standard requires students to extend the models and strategies

Addition and Subtraction of Decimals:

Teaching Student Centered MathematicsComputation with Decimals pgs.196-198Activity 7.11 “Exact Sums and Differences” pg. 198

LessonsAdd Decimals Using Place Value StrategiesSubtracting Decimals Using Place Value Strategies

Activities and Tasks





.

they developed for whole numbers in grades 1-4 to decimal values. Before students are asked to give exact answers, they should estimate answers based on their understanding of operations and the value of the numbers (Rounding 5.NBT.A.4). Some problems should appear as multistep.

Examples: 3.6 + 1.7

A student might estimate the sum to be larger than 5 because 3.6 is more than 3 ½ and 1.7 is more than 1 ½.

5.4 – 0.8A student might estimate the answer to be a little more than 4.4 because a number less than 1 is being subtracted.

Students should be able to express that when they add decimals they add tenths to tenths and hundredths to hundredths. So, when they are adding in a vertical format (numbers beneath each other), it is important that they write numbers with the same place value beneath each other. This understanding can be reinforced by connecting addition of decimals to their understanding of addition of fractions. Adding fractions with denominators of 10 and 100 is a standard in fourth grade.

Example: 4 - 0.3

3 tenths subtracted from 4 wholes. The wholes must be divided into tenths. The answer is 3 and 7/10 or 3.7.

Example: 1.7 – 0.8 = 17 tenths – 8 tenths = 9 tenths = 0.9

Multiplication and Division:

The Value of Education (IM)Base Ten Decimal Bag AdditionBase Ten Decimal Bag Subtraction (K5)Total Ten (K5)Decimal Addition to 500 (K5)Decimal Cross Number Puzzles (K5)Decimal race to Zero (K5)Magic Squares Decimal Addition (K5)Decimal Addition BingoDecimal Addition SpinDecimal Subtraction SpinDecimal DynamoRace to the Finish LineRace to 10 or BustRace to 1 or BustSum with DecimalsSum with Decimals 2Shopping Spree

VideosUse addition and subtraction to solve real-world problems involving decimals

OnlineBase Blocks Decimals - NLVMDiffy (Subtraction) NLVM

Templates and VisualsAddition of DecimalsDecimal Squares Sets

Multiplication and Division Resources:

Teaching Student Centered MathematicsComputation with Decimals pgs. 198-201




https://learnzillion.com/lessons/1150-use-addition-and-subtraction-to-solve-realworld-problems-involving-decimals

https://learnzillion.com/lessons/1150-use-addition-and-subtraction-to-solve-realworld-problems-involving-decimals



Standard Connections/Notes ResourcesGeneral methods used for computing products of whole numbersextend to products of decimals. Because the expectations for decimals are limited to thousandths and expectations for factors are limited to hundredths at this grade level, students will multiply tenthswith tenths and tenths with hundredths, but they need not multiplyhundredths with hundredths. Before students are asked to give exact answers, they should estimate answers based on their understanding of operations and the value of the numbers. Some problems should appear as multistep.

Example: An area model can be useful for illustrating products.

Students should be able to describe the partial products displayed by the area model. For example,

Example: 6 x 2.4A student might estimate an answer between 12 and 18 since 6 x 2 is 12 and 6 x 3 is 18.

To divide to find the number of groups, a student might: Draw a segment to represent 1.6 meters. In doing so, s/he

would count in tenths to identify the 6 tenths, and be able to identify the number of 2 tenths within 6 tenths. The student can then extend the idea of counting by tenths to divide the one meter into tenths and determine that there are 5 more

Activity 7.12 “Where Does the Decimal Go? pg. 199Activity 7.13 “Where Does the Decimal Go? pg. 200

LessonsMultiplying Decimals to Hundredths (MSDE)Approximate Decimal Quotients through Reasoning about Decimal PlacementRelate Decimal and Fraction MultiplicationOperation Decimals (AIMS)

Activities and TasksWhat is 23 Divided by 5? (IM)Decimal Division Model Lab SheetMultiplication and Division of DecimalsMultiplication of Decimals with an Area ModelTarget One

VideosBase ten blocks to multiply 1.3 x 2.2Visual model of $3.71 ÷ 7.Division of Decimal Numbers.Multiply-decimals-to-the-hundredths-by-using-fractionsMultiplying Decimals using an area model (whole number by decimal to the hundredths) LZMultiplying Decimals using an area model LZ

Templates and VisualsMultiplying Decimals PosterMultiply and Divide Decimals Using Models (PPT)Modeling Decimal MultiplicationArea Model Multiplying DecimalsDecimal Squares Tenths for MultiplicationDecimal Squares for DivisionDecimal Squares for Multiplication


http://www.youtube.com/watch?v=nY8IEwClNxg

http://www.youtube.com/watch?v=3SmzSEqsyUQ

http://learnzillion.com/lessons/555-multiply-decimals-to-the-hundredths-by-using-fractions

http://www.youtube.com/watch?v=21rhQWKUlAk

http://www.youtube.com/watch?v=QtxjPbCORrk

http://www.youtube.com/watch?v=zLOxYYdZAWE



Standard Connections/Notes Resourcesgroups of 2 tenths.

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 times its width. What is the width of the desk?

5.NBT.A.2 Explain patterns in the number of zeros of the product when multiplying a

NBT.A.2. was introduced in Unit One. In Unit Four students will apply what they know to help them multiply and divide decimals.

LessonsUse Exponents with Application to Metric Conversions




Standard Connections/Notes Resourcesnumber 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.

Before students consider decimal multiplication more generally, they can study the effect of multiplying 0.1 and by 0.01 to explain why the product is ten or a hundred times as small as the multiplicand (moves one or two places to the right). They can then extend their reasoning to multipliers that are single-digit multiples of 0.1 and 0.01 (e.g., 0.2 and 0.02, etc.). There are several lines of reasoning that students can 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 104 = x 754.2 ÷ 10³ = x

VideosExplain patterns in zeros when multiplying by a power of ten


https://learnzillion.com/lessons/2471-explain-patterns-in-zeros-when-multiplying-by-a-power-of-ten-

https://learnzillion.com/lessons/2471-explain-patterns-in-zeros-when-multiplying-by-a-power-of-ten-



Standard Connections/Notes Resources5.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.

This standard provides measurement conversion as a context for not only working with decimals but a deeper understanding for place value and the connection to the metric system.

In fifth grade, students build on their prior knowledge of related measurement units to determine equivalent measurements. Prior to making actual conversions, they examine the units to be converted, determine if the converted amount will be more or less units than the original unit, and explain their reasoning. They use several strategies to convert measurements. When converting metric measurement, students apply their understanding of place value and decimals. This is an excellent opportunity to reinforce notions of place value for whole numbers and decimals, and connection between fractions and decimals (e.g., 2 ½ meters can be expressed as 2.5 meters or 250 centimeters).

Examples: 764 mL ÷ 10³ = 0.764 liters

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

Teaching Student Centered MathematicsFigure 7.6 pg. 186

LessonsDecimal Multiplication to Express Equivalent Measurements

Find a Fraction of a Measurement

Activities and TasksPARCC Sample QuestionComparing Units of Metric Linear MeasurementMeasurement Conversion Word Problems

VideosSelect Appropriate Measurement-conversions (LZ)

Templates and VisualsConverting Metric Units of Measure


http://learnzillion.com/lessons/3527-select-appropriate-measurement-conversions


U nit 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 Standards5.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.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.




addition/add benchmark fraction common denominator datadecompose

differenceequationequivalent estimateimproper fraction

intervalmixed number numerator reasonableness subtraction/subtract

unit fraction unlike denominator variable

Essential QuestionsHow 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 Resources5.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 operationsb. 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

In Unit 3, students multiplied fractions using visual fraction models or equations. Unit 5 will extend this understanding to mixed numbers.

Examples:

Once understanding has been developed, move students to the algorithm:

2 14

x 113

= 94

x 43

= 3612

= 3 sq ft

Teaching Student Centered MathematicsFactors Greater than One pg.171

LessonsArea of RectanglesNew Floor LessonReal World ProblemsReal World Problems 2

Find the Area of Rectangles by Tiling and Relate to Multiplication

Measure to Find the Area of Rectangles with Fractional Side Lengths

Multiply Mixed Number Factors Relate to the Distributive Property and the Area Model

Multiply mixed numbers by mixed numbers using visual representations

Activities and TasksMaking Cookies (IM)Half of a Recipe (IM)Fractions Multiplied by Mixed NumbersMixed Numbers by FractionsWhole Number by Mixed Number with Cuisinaire RodsCommon Assessment Multiplying FractionsClosest to 25

VideosArea Model to Multiply Mixed Numbers.

Online


http://www.youtube.com/watch?v=lCbswBRlGyY

https://learnzillion.com/lesson_plans/33

https://learnzillion.com/lesson_plans/33



Standard Connections/Notes Resourcesmultiplication 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.

Multiplication of Mixed Numbers using the area model:

3 23

x 3 23

=

Fractions – Rectangle Multiplication NLVM

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, 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

Parts a. and b. of this standard were taught in Unit 3.

In fifth grade, students experience division problems with whole number divisors and unit fraction dividends (fractions with a numerator of 1) or with unit fraction divisors and whole number dividends. Students extend their understanding of the meaning of fractions, how many unit fractions are in a whole, and their understanding of multiplication and division as involving equal groups or shares and the number of objects in each group/share. In sixth grade, they will use this foundational understanding to divide into and by more complex fractions and develop abstract methods of dividing by fractions.

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?

Teaching Student Centered MathematicsDivision pg. 172-173

LessonsSolve Problems Involving Fraction DivisionA Fifth-Grade Fractions Lesson – Math Solutions

Activities and TasksDivision Fraction Word Problems (K5)Smarter Balanced Sample Question


http://sampleitems.smarterbalanced.org/itempreview/sbac/index.htm




Standard Connections/Notes Resourcesare in 2 cups of raisins?

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 ÷ 15 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 12 lb of rice

equally?

A student may think or draw 12

and cut it into 3 equal groups


1 lb peanuts

15 lb of peanuts

The diagram shows the 1/3 pan divided into 4 equal shares with each share equaling 1/12 of the pan.




then determine that each of those parts is 16

.

A student may think of 12 as equivalent to

36 .

36 divided by 3

is 16

.


Example: Sam made 2/3 liter of punch and 3/4 liter of tea to take to a party. Is that more or less than 1.5 liters? Explain.23 +

34 =

1712 = 1

512

512 is less than 0.5, so it is less.

LessonsFraction Multiplication with Measurement Conversions

Find a Fraction of a Measurement and Solve Word Problems

Activities and TasksComparing Measurements TaskPunch TaskCheck Up TaskPARCC Sample QuestionConverting Unit TasksConverting Units of TimeConverting Standard Units of LengthConverting Standard Units of Capacity

VideosSolve real world distance problems with unit conversions


https://learnzillion.com/lessons/3501-solve-real-world-distance-problems-with-unit-conversions

https://learnzillion.com/lessons/3501-solve-real-world-distance-problems-with-unit-conversions



Standard Connections/Notes Resources5.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. For example, given different measurements of liquid in identical beakers, find the amount of liquid each beaker would contain if the total amount in all the beakers were redistributed equally

This standard is included here so measurement line plots can be used as a context for students to apply fraction computation strategies. Students use line plots and other tools/technology to reason about problem situations. Students attend to the underlying meaning of the quantities and operations when solving problems rather than just how to compute answers. Students should work with data in science and other subjects.

Students in grade 4 solved problems involving addition and subtraction of fractions by using information presented in line plots. The standard in this cluster provides an opportunity for solving real-world problems with operations on fractions, connecting directly to both number and operations.

(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

LessonsMeasure Pencil Lengths and Construct Line Plots

Activities and TasksFractions on a Line Plot (IM)Sacks of Flour (K5)Dog treats Line Plot Activity

VideosSolve-multistep-problems-using-information-in-a-line-plot (LZ)

Templates and VisualsLine Plot Template


http://learnzillion.com/lessons/3455-solve-multistep-problems-using-information-in-a-line-plot

http://learnzillion.com/lessons/3455-solve-multistep-problems-using-information-in-a-line-plot



Standard Connections/Notes Resourcesbeaker 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 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 Standards5.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 acute trianglecircle congruence/congruent equilateral triangle half circle/semicircle hexagon irregular polygon

isosceles triangle kite obtuse angleobtuse triangle parallel lines parallelogram pentagon perpendicular lines

plane figure polygon protractorquadrilateral rectangle regular polygon rhombus/rhombi right angle

right triangle scalene triangle symmetry/symmetricalsquare triangle trapezoid two-dimensional figure vertex

Essential QuestionsHow 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 Resources5.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.

Geometric properties include properties of sides (parallel, perpendicular, congruent), properties of angles (type, measurement, congruent), and properties of symmetry (point and line).

If the opposite sides on a parallelogram are parallel and congruent, then rectangles are parallelograms

Examples: A parallelogram has 4 sides with both sets of opposite sides

parallel. What types of quadrilaterals are parallelograms? Regular polygons have all of their sides and angles

Teaching Student Centered MathematicsActivity 8.5 “Can You Make It?” pg. 217Two-Dimensional Shapes pgs. 220-222Activity 8.8 “Property Lists for Quadrilaterals pg. 226Property Lists for Quadrilaterals (Omit rotation) pg.226

LessonsQuadrilateral Characteristics Utah Education Network.Constructing QuadrilateralsQuadrilateral Tangram ChallengeParallelogram's Attributes




Standard Connections/Notes Resourcescongruent. Name or draw some regular polygons.

All rectangles have 4 right angles. Squares have 4 right angles so they are also rectangles. True or False?

A trapezoid has 2 sides parallel so it must be a parallelogram. True or False?

Rectangles and RhombusesDraw and Define Trapezoids Based on Attributes

Activities and TasksSometimes Always NeverQuadrilateral Word ProblemsPARCC Sample QuestionTypes of Triangles a QR Activity

VideosAnalyze-and-compare-triangles-by-angles (LZ)

OnlineGeoboard NLVMShape Sorter Illuminations

Templates and VisualsFrayer ModelExample/Non-Example

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

IRC has die cuts for: circle, oval, square, rectangles, octagon, hexagon, pentagon, trapezoid, triangle, and rhombus. One piece of construction paper will produce 4 of each 2D figure for students to use in sorting activities. Ordering multiple figures will allow you to modify the 2D shapes. Example: Cut one trapezoid to make it a right trapezoid.

Properties of figures may include: Properties of sides – parallel, perpendicular, congruent,

number of sides Properties of angles – types of angles, congruent

Examples: A scalene triangle can be right, acute, and obtuse.

Teaching Student Centered MathematicsActivity 8.12 True or False pg.231

LessonsClassify 2D FiguresDraw 2D Figures from Given AttributesShape-Up from NCTM IlluminationsSorting Polygons - Illuminations

Activities and TasksWhat Is a Trapezoid? (IM)Identifying Quadrilaterals (K5)Quadrilateral Criteria (K5)Quadrilateral Hierarchy Diagram (K5)Regular/Irregular Hierarchy Diagram (K5)Triangle Hierarchy (K5)Triangle Hierarchy 2 (K5)PARCC Sample Question


http://illuminations.nctm.org/Lesson.aspx?ID=1072



http://learnzillion.com/lessons/3297-analyze-and-compare-triangles-by-angles

http://www.google.com/url?sa=i&rct=j&q=right%20trapezoid&source=images&cd=&cad=rja&uact=8&docid=oGz0LTrPPTpStM&tbnid=oA3xoyjVh57_5M:&ved=0CAUQjRw&url=http://www.imomath.com/index.php?options%3D543%26&ei=RFWYU4nTDq7JsQTYooHADA&bvm=bv.68693194,d.cWc&psig=AFQjCNGMO9ydIUTiPqXryMUsNDJzZsoetA&ust=1402578582349150



Standard Connections/Notes Resources A right triangle can be both scalene and isocoles, but not

equilateral. VideosClassify-quadrilaterals-in-a-hierarchy

OnlineShape Sorter Illuminations

Un it 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 foundationalfor further work with numbers in later grades.

Unit Seven Standards



http://learnzillion.com/lessons/3513-classify-quadrilaterals-in-a-hierarchy


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 LanguageThe 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.bracesbracketsconventional orderdistributive property equation

evaluateexpressioninterpretnumerical expression order of operations

properties of operationsparenthesesevaluate

Essential QuestionsWhy 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 Resources5.OA.A.1 Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols to represent the problem.

This standard builds on the expectations of third grade where students are expected to start learning the conventional order (3.OA.D.8.). Students need experiences with multiple expressions that use grouping symbols throughout the year to develop understanding of when and how to use parentheses, brackets, and braces. First, students use these symbols with whole numbers. Then the symbols can be used as students add, subtract, multiply and divide decimals

LessonsExploring KryptoFollowing the OrderEvaluating Expressions

Activities and TasksA Sequence of Events


http://www.uen.org/Lessonplan/preview.cgi?LPid=16331

http://illuminations.nctm.org/LessonDetail.aspx?id=L803



Standard Connections/Notes Resourcesand fractions.

Examples: (26 + 18) ÷ 4 Answer: 11 {[2 x (3+5)] – 9} + [5 x (23-18)] Answer: 32 12 – (0.4 x 2) Answer: 11.2 (2 + 3) x (1.5 – 0.5) Answer: 5 { 80 [ 2 x (3 ½ + 1 ½ ) ] } + 100 Answer: 900

To further develop students’ understanding of grouping symbols and facility with operations, students place grouping symbols in equations to make the equations true or they compare expressions that are grouped differently.Examples:

15 - 7 – 2 = 10 → 15 - (7 – 2) = 10 3 x 125 ÷ 25 + 7 = 22 → [3 x (125 ÷ 25)] + 7 = 22 Compare 3 x 2 + 5 and 3 x (2 + 5) Compare 15 – 6 + 7 and 15 – (6 + 7)

Expressions with ParenthesesNumerical Expressions Wall ClockBowling for Numbers (IM)Watch Out for Parentheses (IM)Same or DifferentTrick AnswersSolving Numerical ExpressionsTarget Number DashOperation TargetPicturing Factors in Different Orders (IM)Using Operations and Parentheses (IM)You Can Multiply Three Numbers in Any Order (IM)Order of Operations Bingo

VideosInterpret-parentheses-as-do-this-first (LZ)

OnlineOrder of Operations ActivitiesPan Balance NumbersPrimary Krypto

5.OA.A.2 Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. For example, express the calculation “add 8 and 7, then multiply by 2” as 2 × (8 + 7). Recognize that 3 × (18932 + 921) is three times as large as 18932 + 921, without having to calculate the indicated sum or product.

Students use their understanding of operations and grouping symbols to write expressions and interpret the meaning of a numerical expression. The expressions described in 5.OA.A.2 should be no more complex than the expressions one finds in an application of the associative or distributive property. Include fraction and decimal expressions.

Examples: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.

LessonsInterpret and Evaluate Expressions with Fractions

Create Story Contexts for Numerical Expressions Including Fractions

Compare and Evaluate Expressions with Fractions and Parentheses

Activities and TasksVerbal and Algebraic ExpressionsOperation Bingo




http://www.math-aids.com/Order_of_Operations/

http://learnzillion.com/lessons/2409-interpret-parentheses-as-do-this-first

http://illuminations.nctm.org/lesson.aspx?id=2583





Standard Connections/Notes ResourcesConnections:5.OA.A.1 Describe how the expression 5(10 x 10) relates to 10 x 10.

The expression 5 (10 x 10) is 5 times larger than the expression 10 x 10 since I know that 5 (10 x 10) means that I have 5 groups of (10 x 10).

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.

Comparing Products (IM)Seeing is Believing (IM)Video Game Scores (IM)Word to Expressions 1 (IM)

VideosWrite-expressions-to-represent-numerical-situations (LZ)

Unit Eight – Coordinate Plane (15 days)


http://learnzillion.com/lessons/2737-write-expressions-to-represent-numerical-situations






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 Standards5.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.


axis/axes coordinates coordinate plane coordinate system distance

first quadrant horizontal intersection of lines line ordered pairs

origin parallelperpendicularpoint rule

vertical x-axis x-coordinate y-axis y-coordinate

Essential QuestionsHow 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 MisconceptionsStudents 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 Resources5.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. For example, given the rule “Add 3” and the starting number 0, and given the rule “Add 6” and the starting number 0, generate terms in the resulting sequences, and observe that the terms in one sequence are twice the corresponding terms in the other sequence. Explain informally why this is so.Connections: 5.G.A.15.G.A.2

Students should have experienced generating and analyzing numerical patterns using a given rule in Grade 4. Students precisely describe the coordinates of points and the relationship of the coordinate plane to the number line (MP.6). Students both generate and identify relationships in numerical patterns, using the coordinate plane as a way of representing these relationships and patterns (MP.4).

Use the rule add 3 to write a sequence of numbers. Starting with a 0, students write 0, 3, 6, 9, 12, . . .Use the rule add to write a sequence of numbers. Starting with 0, students write 0, 6, 12, 18, 24, . . .

After comparing these two sequences, the students notice that each term in the second sequence is twice the corresponding terms of the first sequence. One way they justify this is by describing the patterns of the terms. Their justification may include some mathematical notation (See example below). A student may explain that both sequences start with zero and to generate each term of the second sequence he/she added 6, which is twice as much as was added to 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).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.

Teaching Student Centered MathematicsActivity 10.6 “What’s Next and Why?” pg. 299Graphing Patterns pgs. 297-298Activity 10.7 “Start and Jump Numbers” pgs. 299-300

LessonsAnalyzing Patterns and RelationshipsWillie the Wheel Man (AIMS)Generate Two Number Patterns from Given Rules, Plot the Points, and Analyze the PatternsInvestigate Patterns on a Coordinate PlanePlot Points and Describe Patterns with Coordinate PairsGenerate a Number Pattern from a Rule and Plot the Points

Activities and TasksPARCC Sample Question

VideosIdentify-the-relationship-between-two-numerical-patterns (LZ)


http://learnzillion.com/lessons/3194-identify-the-relationship-between-two-numerical-patterns




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).

Students can use a classroom size coordinate system to physically locate the coordinate point (5,3) by starting at the origin point (0,0) by walking 5 units along the x axis to find the first number in the pair and then walking up 3 units for the second number in the pair.

Teaching Student Centered MathematicsLocation Activities pg. 239

LessonsFly on the CeilingName Points Using Coordinate PairsBattleshipDraw Symmetric Figures on Coordinate Plane (challenging)Construct a Coordinate System on a Plane

Activities and TasksOpus Math sample questionsPieces of EightBlackbeard’s Treasure BoxCoordinate Grid Geoboards (K5)Coordinate Grid Tangram (K5)Coordinate ShapesGeometric Shapes on a Coordinate Grid (K5)Coordinate Grid Swap (K5)Battleship Using Grid Paper (IM)PARCC Sample Question

VideosUse-a-model-to-identify-parts-of-a-coordinate-plane (LZ)

OnlineGeoboard – Coordinate NLVM (quadrant one only)



http://learnzillion.com/lessons/1700-use-a-model-to-identify-parts-of-a-coordinate-plane

http://www.opusmath.com/common-core-standards/5.g.1-use-a-pair-of-perpendicular-number-lines-called-axes-to-define-a



Standard Connections/Notes ResourcesHidden Message Coordinate Graphing Worksheet WorksPrintable Grid Sheets Math Aids

Templates and VisualsFirst Quadrant Grids

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.

Although students can often “locate a point,” these understandings are beyond simple skills. For example, initially, students often fail to distinguish between two different ways of viewing the point (2, 3), say, as instructions: “right 2, up 3”; and as the point defined by being a distance 2 from the x-axis and a distance 3 from the y-axis. In these two descriptions, the 2 is first associated with the x-axis, then with the y-axis.

Examples: Sara has saved $20. She earns $8 for each hour she works.

If Sara saves all of her money, how much will she have after working 3 hours? 5 hours? 10 hours?

Create a graph that shows the relationship between the hours Sara worked and the amount of money she has saved.

Use the graph below to determine how much money Jack makes after working exactly 9 hours.

Teaching Student Centered MathematicsGraphing Patterns pgs. 297-298

LessonsDescribe the Graph IlluminationsUse Coordinate Systems to Solve Real World Problems.

Activities and TasksOpus Math Sample problemsMeerkat Coordinate Plane Task (IM)

VideosFind-distance-between-points-on-a-coordinate-plane-by-counting (LZ)

Templates and VisualsDescribe the Graph Activity recording Sheet for NCTM Lesson

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


http://learnzillion.com/lessons/1703-find-distance-between-points-on-a-coordinate-plane-by-counting

http://learnzillion.com/lessons/1703-find-distance-between-points-on-a-coordinate-plane-by-counting

http://s3.amazonaws.com/illustrativemathematics/illustration_pdfs/000/001/516/original/illustrative_mathematics_1516.pdf?1393462359

http://www.opusmath.com/common-core-standards/5.g.1-use-a-pair-of-perpendicular-number-lines-called-axes-to-define-a


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 Standards5.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

(For more description go back to Unit 3.)

Examples:

Alexis spent 37

of her money. She gave 14

of the remainder to her brother. She had $120 left.

How much money did she have at the beginning?

Bar Model

$120 ÷ 3 = $40, so each seventh represents $40.7 x $40 = $280

PARCC Sample Question

LessonsAdd Fractions Making Like Units NumericallySubtract Fractions Making Like Units NumericallyStrategize to Solve Multi-Term ProblemsSolve and Create Fraction Word Problems Involving Addition, Subtraction, and MultiplicationTime for Recess Unit (NYCDOE)

Activities and TasksCindy’s CatsPARCC Sample Question

Unit Ten – Multiplication & Division of Whole Numbers – Fluency (12 days)


37spent

$120 left

14 of remainder


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 Standards5.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 Resources5.NBT.B.5 Fluently multiply multi-digit whole numbers using the standard algorithm.

Grade 5 students need to become fluent with the standardmultiplication algorithm with multi-digit whole numbers. In prior grades, students used various strategies to multiply. Students can continue to use these different strategies as long as they are efficient, but must also understand and be able to use the standard algorithm. In applying the standard algorithm, students recognize the importance of place value.

Examples: 371 x 2,784

LessonsUse Your Head (AIMS)Fence IT (Extend task to larger nos.)Solve Two-Step Word Problems with Multi-Digit MulFluently Multiply Using the Standard AlgorithmDesign and Construct BoxesFinding Products of a Whole Number and a DecimalGifts Galore Xmas in June (AIMS)

Activities and TasksPARCC Toy Animals TaskBatter-Up Multiplication GameMultiplication MixPARCC Sample Question

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.

5.NBT.B.6 is a milestone along the way to reaching fluency with the standard algorithm in Grade 6 (6.NS.B.2). Students should continue to interpret remainders.

LessonsCounting the Miles (AIMS)Three Digit Divisibility Dilemma (AIMS)How Thick Is It (AIMS)

Activities and TasksDouble Dutch TreatEnd of Year Practice Problems (AIMS)


http://nrich.maths.org/2663


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


Documents

math4u.wicomico.wikispaces.netmath4u.wicomico.wikispaces.net/file/view/Grade 5... · Web viewFifth Grade. Mathematics Curriculum. Guide. f. or . ... (12 days) Mathematics Placement