Fifth Grade Unit 1: Order of Operations and Whole Numbers 3 WeeksIn this unit students will: Solve problems by representing mathematical relationships between quantities using mathematical expressions and equations. Use the four whole number operations efficiently, including the application of order of operations. Write, evaluate, and interpret mathematical expressions with and without using symbols. Apply strategies for multiplying a 2- or 3-digit number by a 2-digit number. Develop paper-and-pencil multiplication algorithms (not limited to the traditional algorithm) for 3- or 4-digit number multiplied by a 2- or 3-digit number. Apply paper-and-pencil strategies for division (not the standard algorithm) Solve problems involving multiplication and division. Investigate the effects of multiplying whole numbers by powers of 10. Fluent use of standard division algorithm is a 6th grade standardUnit Resources:Unit 1 Overview Video Parent Letter (Spanish) Parent Standards Clarification Number Talks Vocabulary Cards Prerequisite Skills Assessment Sample Post Assessment Student Friendly Standards Concept Map

Topic 1: Order of Operations and Whole Numbers Big Ideas/Enduring Understandings: Engage in mathematical discourse using precise math vocabulary Write, evaluate, and interpret mathematical expressions with and without using symbols Solve problems by representing mathematical relationships between quantities using mathematical expressions and equations. Use the four whole number operations efficiently, including the application of order of operations.Essential Questions: Why is it important to follow an order of operations? How can I effectively critique the reasoning of others? 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? How can an expression be written?Content StandardsContent standards are interwoven and should be addressed throughout the year in as many different units and activities as possible in order to emphasize the natural connections that exist among mathematical topics.Write and Interpret Numerical ExpressionsMGSE5.OA.1 Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols.

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MGSE5.OA.2 Write simple expressions that record calculations with number, and interpret numerical expressions without evaluating them. For example, express the calculation “add 8 and 7, then multiply by 2” as 2 x(8 + 7). Recognize that 3 x (18932 + 921) is three times as large as 18932 + 921, without having to calculate the indicted sum or product.

Vertical Articulation Fourth Grade Number and Operations in Base TenUse the four operations with whole numbers to solve problems.MGSE4.OA.1 Understand that a multiplicative comparison is a situation in which one quantity is multiplied by a specified number to get another quantity.

a. Interpret a multiplication equation as a comparison e.g., interpret 35 = 5 × 7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5.

b. Represent verbal statements of multiplicative comparisons as multiplication equations.

MGSE4.OA.2 Multiply or divide to solve word problems involving multiplicative comparison. Use drawings and equations with a symbol or letter for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison.

Sixth Grade Algebraic Expressions MGSE6.EE.1 Write and evaluate numerical expressions involving whole-number exponents. MGSE6.EE.2 Write, read, and evaluate expressions in which letters stand for numbers.MGSE6.EE.2a Write expressions that record operations with numbers and with letters standing for numbers. For example, express the calculation “Subtract y from 5” as 5–𝑦. MGSE6.EE.2b Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity. For example, describe the expression 2(8+7) as a product of two factors; view (8+7) as both a single entity and a sum of two terms. MGSE6.EE.2c Evaluate expressions at specific values for their variables. Include expressions that arise from formulas in real-world problems. Perform arithmetic operations, including those involving whole-number exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations). For example, use the formulas 𝑉=𝑠3 and 𝐴=6𝑠2 to find the volume and surface area of a cube with sides of length 𝑠=1/2.

Instructional StrategiesWrite and Interpret Numerical ExpressionsOA.1This standard builds on the expectations of third grade where students are expected to start learning the conventional order. 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 and fractions.Examples:

To further develop students’ understanding of grouping symbols and facility with operations, students place grouping symbols in equations to make the equations 2

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true or they compare expressions that are grouped differently.

Examples:

OA.2This standard refers to expressions. Expressions are a series of numbers and symbols (+, −,×,÷) without an equals sign. Equations result when two expressions are set equal to each other (2 + 3 = 4 + 1).Example:4(5 + 3) is an expression.When we compute 4(5 + 3) we are evaluating the expression. The expression equals 32. 4(5 + 3) = 32 is an equation.

This standard also calls for students to verbally describe the relationship between expressions without actually calculating them. This standard calls for students to apply their reasoning of the four operations as well as place value while describing the relationship between numbers. The standard does not include the use of variables, only numbers and signs for operations.

Example:Teacher: Write an expression for the steps “double five and then add 26.”Student: (2 × 5) + 26Teacher: Describe how the expression 5(10 × 10) relates to (10 × 10)Student: The expression 5(10 × 10) is 5 times larger than the expression (10 × 10) since I know that 5(10 × 10) means that I have 5 groups of (10 × 10).

Students use their understanding of operations and grouping symbols to write expressions and interpret the meaning of a numerical expression.

Other Examples:

Students write an expression for calculations given in words such as “divide 144 by 12 and then subtract78 .” They write (144 ÷ 12) −78 .

Students recognize that 0.5 × (300 ÷ 15) is 12 of (300 ÷ 15) without calculating the quotient.

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PEMDAS

There are always exceptions to solving problems with PEMDAS Multiplication and Division are interchangeable (it depends on the order the operations occur in the number sentence) Addition and Subtraction are interchangeable (it depends on the order the operations occur in the number sentence)

Use discovery approaches when teaching PEMDAS Example: Allow students to explore solving problems with operations in the order they occur in the number sentence. Then, have students compare their

solutions to the response given by a scientific calculator. Use the difference in solutions to discuss the order of operations. Explore the exceptions to PEMDAS. For example, in this problem:

Students should actually add the numbers in the numerator and subtract the numbers in the denominator prior to dividing. In a different example, completing the operations inside the parentheses is not always a priority with the distributive property. For example, I get the same

answer whether I complete:

OR

OA.1 & OA.2Students should be given ample opportunities to explore and evaluate numerical expressions with mixed operations. Eventually this should include real-world contexts that would require the use of grouping symbols in order to describe the context as a single expression. This is the foundation for evaluating algebraic expressions that will include whole-number exponents in Grade 6.

There are conventions (rules) determined by mathematicians that must be learned with no conceptual basis. For example, multiplication and division are always done before addition and subtraction.

Begin with expressions that have two operations without any grouping symbols (multiplication or division combined with addition or subtraction) before introducing expressions with multiple operations. Using the same digits, with the operations in a different order, have students evaluate the expressions and discuss why the value of the expression is different. For example, have students evaluate 5 × 3 + 6 and 5 + 3 × 6.

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Discuss the rules that must be followed. Have students insert parentheses around the multiplication or division part in an expression. A discussion should focus on the similarities and differences in the problems and the results. This leads to students being able to solve problem situations which require that they know the order in which operations should take place.

After students have evaluated expressions without grouping symbols, present problems with one grouping symbol, beginning with parentheses, then adding expressions that have brackets and/or braces. Have students write numerical expressions in words without calculating the value. This is the foundation for writing algebraic expressions. Then, have students write numerical expressions from phrases without calculating them.

Using both brackets and braces (nesting symbols) isn’t a fifth grade expectation but it can be taught given an explanation. However, the main emphasis in fifth grade is the use of the parenthesis.

Engage NY Lessons are included in the activity file. Coming Soon…Common MisconceptionsStudents may believe the order in which a problem with mixed operations is written is the order to solve the problem. The use of mnemonic phase “Please Excuse My Dear Aunt Sally” to remember the order of operations (Parentheses, Exponents, Multiplication, Division, Addition, Subtraction) can also mislead students to always perform multiplication before division and addition before subtraction. To correct this thinking, students need to understand that addition and subtraction are inverse operations and multiplication and division are inverse operations, as in they have the same “impact”. At this level, students need opportunities to explore the “impact” of the various operations on numbers and solve equations starting with the operation of greatest “impact”.

Example:3 + 2 = 5, 5 − 2 = 3 (generalize subtraction “undoes” addition – inverse operation)3 × 2 = 6, 6 ÷ 2 = 3 (generalize division “undoes” multiplication – inverse operation and multiplication and division have a greater “impact” on a number than addition and subtraction)32 = 9 (generalize, exponents have a greater “impact” on a number)

Allow students to use calculators to determine the value of the expression, and then discuss the order the calculator used to evaluate the expression. Do this with four-function and scientific calculators.

Students need lots of experience with writing multiplication in different ways. Multiplication can be indicated with a raised dot, such as 4 ⋅ 5, with a raised cross symbol, such as 4 × 5, or with parentheses, such as 4(5) 𝑜𝑟 (4)(5). Note that the raised cross symbol is not the same as the letter “x”, and so care should be taken when writing or typing it. Students need to be exposed to all three notations and should be challenged to understand that all are useful. In instruction, teachers are encouraged to use a notation and stay consistent. Students also need help and practice remembering the convention that we write a rather than 1 · a or 1a, especially in expressions such as a + 3a.DifferentiationIncrease the RigorOA.1

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Ty says that 16 - (4 - 4) = 8. Jade says it equals 12. Who do you agree with? Do these two equations have the same solution? Why or why not? (18 + 5) x 2 AND 18 + (5 x 2) Given 12 ÷ 4 + 2, where should the parentheses be in this equation to make it equal 2? Write a numerical expression with at least two operations so that when evaluated it equals 18. Why do we need order of operations? What might happen if we didn't have order of operations?OA.2 Jackie has 34 marbles and Natalie has 123 marbles. Sarah has four times the amount of marbles as Jackie and Natalie combined. Write the number sentence

that shows the number of marbles that Sarah has. Do you need to do the calculations to know which answer is larger? 6 x (184 + 948) OR (948 + 184) x 3? How much larger will the answer be? What is a way to write an expression that is five times as much as 96 ÷ 3? How are the expressions 8 - a and a - 8 different? Explain your thinking. Write this statement as expression: “Twenty-three less than the sum of forty-nine and thirty-seven.” Write an expression for twice a number, decreased by twenty-nine, is seven. How would you write the following expression in words? 32 = 2x + 7. Write a story problem that would go along with this expression.

Acceleration Intervention Coming Soon…Evidence of LearningBy completion of this lesson, students will be able to: Write and solve expressions including parentheses and brackets Interpret numerical expressions without evaluating them. Apply the rules for order of operations to solve problems.Additional Assessments: Shared Assessments: See formative assessment folder for Topic 1.Adopted ResourcesMy MathChapter 7

7.1 Hands On: Numerical Expressions

*These lessons are not to be completed consecutively as it is way too much material. They are designed to help support you as you teach your standards.

Adopted Online ResourcesMy Mathhttp://connected.mcgraw-hill.com/connected/login.doTeacher User ID: ccsde0(enumber)Password: cobbmath1Student User ID: ccsd(student ID)Password: cobbmath1Exemplarshttp://www.exemplarslibrary.com/User: Cobb EmailPassword: cobbmath

Think Math:

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Additional Web ResourcesK-5 Math teaching Resources http://www.k-5mathteachingresources.com/5th-grade-number-activities.htmlOA.1Target Number DashNumerical Expressions ClockOA.2Equivalent Expressions MatchIllustrative Mathematics provides instructional and assessment tasks, lesson plans, and other resources https://www.illustrativemathematics.org/OA.1Watch Out for Parentheses 1Bowling for NumbersUsing Operations and Parentheses OA.2Comparing ProductsWords to Expressions 1Video Game ScoresSeeing is BelievingLearn Zillion https://learnzillion.com/OA.1Determine if Parentheses Change the Value of an ExpressionMake Equations True by Adding ParenthesesOA.2Write a Numerical Expression to Represent a Verbal Description of a CalculationRepresent a Real World Situation as a Numerical ExpressionRecord a Numerical Expression as a Written DescriptionDetermine Whether a Description of a Numerical Expression is AccurateCompose a Real World Problem by Interpreting a Given Numerical ExpressionReason about Expressions to Compare Their Values Without Evaluating ThemOrder of Operations Bingo http://illuminations.nctm.org/Lesson.aspx?id=2583Inside Mathematics - Professional Resource for Educators http://www.insidemathematics.orgSuggested Manipulativesbase ten blockscalculatorsplace value chartnumber linegrid paper

Vocabulary exponentexpressionbracesbracketsparenthesis

Suggested Literature

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digit cards VideosSEDL for NBT.6

Task DescriptionsScaffolding Task Task that build up to the learning task.Constructing Task Task in which students are constructing understanding through deep/rich contextualized problem solving Practice Task Task that provide students opportunities to practice skills and concepts.Culminating Task Task designed to require students to use several concepts learned during the unit to answer a new or unique situation. Formative Assessment Lesson (FAL)

Lessons that support teachers in formative assessment which both reveal and develop students’ understanding of key mathematical ideas and applications.

3-Act Task Whole-group mathematical task consisting of 3 distinct parts: an engaging and perplexing Act One, an information and solution seeking Act Two, and a solution discussion and solution revealing Act Three.

State TasksTask Name Task Type/

Grouping StrategyContent Addressed Standard(s) Task Description

Order of Operations Scaffolding TaskSmall Group/Individual Task

Deriving the rules of order of operations MGSE.OA.1

Evaluate expressions with symbols using 1-inch square tiles

Trick Answers Constructing TaskIndividual/Partner Task Order of operations MGSE.OA.1 Discovering rules of

order of operations

Money for Chores Constructing TaskIndividual/Partner Task Write and evaluating expressions. MGSE.OA.1

Write expressions to solve equations for chores

Hogwarts House Cup Constructing TasksIndividual/Partner Task

Evaluate expressions with parentheses ( ), brackets [ ] and

braces { }.

MGSE.OA.1MGSE.OA.2

Writing expressions to earn points for Hogwarts

Hogwarts House Cup Part 2 Practice TaskIndividual/Partner Task

Evaluate expressions with parentheses ( ), brackets [ ] and

braces {}.

MGSE.OA.1MGSE.OA.2

Writing expressions to earn points for Hogwarts

The Beanbag Dartboard Performance TaskPartner/Small Group Task

Write and evaluate expressions and equations to represent a real

life situation

MGSE.OA.1MGSE.OA.2

Wonderings about throwing a beanbag on a dartboard

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Expression Puzzle Practice TaskIndividual/Partner Task Interpret Numerical Expressions MGSE.OA.2

Matching numeric expressions to its meaning in written words

Fifth Grade Unit 1: Order of Operations and Whole NumbersTopic 2: Place ValueBig Ideas/Enduring Understandings: Investigate the effects of multiplying and dividing whole numbers by powers of 10Essential Questions: How does multiplying a whole number by a power of ten affect the product?Content Standards

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Content standards are interwoven and should be addressed throughout the year in as many different units and activities as possible in order to emphasize the natural connections that exist among mathematical topics.Understand the Place Value System MGSE5.NBT.1 Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left. MGSE5.NBT.2 Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10.

Vertical ArticulationFourth Grade Place Value StandardsGeneralize place value understanding for multi-digit whole numbersMGSE4.NBT.1 Recognize that in a multi-digit whole number, a digit in any one place represents ten times what it represents in the place to its right. For example, recognize that 700 ÷ 70 = 10 by applying concepts of place value and division.

Sixth Grade Algebraic Expressions StandardsApply and evaluate numerical expressions involving whole-number exponents. MGSE6.EE.1 Write and evaluate numerical expressions involving whole-number exponents.

Instructional StrategiesUnderstand the Place Value SystemNBT.1This standard calls for students to reason about the magnitude of numbers. Students should work with the idea that the tens place is ten times as much as the ones place, and the ones place is 1/10th the size of the tens place.

In 4th grade, students examined the relationships of the digits in numbers for whole numbers only. This standard extends this understanding to the relationship of decimal fractions, however, that will be addressed in a later unit. Review to the grade level overview for more information. 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.Example:The 2 in the number 542 is different from the value of the 2 in 324. The 2 in 542 represents 2 ones or 2, while the 2 in 324 represents 2 tens or 20. Since the 2 in 324 is one place to the left of the 2 in 542 the value of the 2 is 10 times greater. Meanwhile, the 4 in 542 represents 4 tens or 40 and the 4 in 324 represents

4 ones or 4. Since the 4 in 324 is one place to the right of the 4 in 542 the value of the 4 in the number 324 is 110𝑡ℎ of its value in the number 542.

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 110 of the value of a 5 in the hundreds place”.

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Based on the base-10 number system digits to the left are times as great as digits to the right; likewise, digits to the right are 110𝑡ℎ of digits to the left. For

example, the 8 in 845 has a value of 800 which is ten times as much as the 8 in the number 782. In the same spirit, the 8 in 782 is 110𝑡ℎ the value of the 8 in

845.Example: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 110”. I can write this using

110 or 0.1”). the repeat the process by

finding 110 of a

110 (e.g., dividing

110 into 10 equal parts to arrive at

1100 or 0.01) and can explain their reasoning, “0.01 is

110 of

110 thus is

1100 of the whole

unit.”

In the number 55.55, each digit is 5, but the value of the digits is different because of the placement.

The 5 that the arrow points to is 110 of the 5 to the left and 10 times the 5 to the right. The 5 in the ones place is

110 of 50 and 10 times five tenths.

The 5 that the arrow points to is 110 of the 5 to the left and 10 times the 5 to the right. The 5 in the tenths place is 10 times five hundredths.

In Grade 5, the concept of place value is extended to include decimal values to thousandths. The strategies for Grades 3 and 4 should be drawn upon and extended for whole numbers and decimal numbers. For example, students need to continue to represent, write and state the value of numbers including decimal numbers. For students who are not able to read, write and represent multi-digit numbers, working with decimals will be challenging.

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When converting in the metric system, have students extend their prior knowledge of the base-ten system as they multiply or divide by powers of ten (as referenced in Units 1 and 2). Teaching conversions should focus on the relationship of the measurements, not merely rote memorization. The questions ask the student to find out the size of each of the subsets. Students are not expected to know e.g. that there are 5280 feet in a mile. If this is to be used as an assessment task, the conversion factors should be given to the students. However, in a teaching situation it is worth having them realize that they need that information rather than giving it to them upfront; having students identify what information they need to have to solve the problem and knowing where to go to find it allows them to engage in Standard for Mathematical Practice 5, Use appropriate tools strategically.Retrieved from Illustrative Mathematicshttp://www.illustrativemathematics.org/standards/k8NBT.2This standard includes multiplying by multiples of 10 and powers of 10, including 102 which is 10 10=100, and 103 which is 10 10 10 =1,000. Students should have experiences working with connecting the pattern of the number of zeros in the product when you multiply by powers of 10. Students should notice the shift of the digits/decimal point when multiplying by a power of 10. Examples: 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 shifting (not just that the decimal point is shifting 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 digits shifts to the right.

350 ÷ 103 = 350 ÷ 1,000 = 0.350 = 0.35350 ÷ 10 = 35, 35 ÷ 10 = 3.53.5 ÷ 10 = 0.35, 𝑜𝑟 350 × 110 , 35 x 110 , 3.5 x 110This will relate well to subsequent work with operating with fractions. This example shows that when we divide by powers of 10, the exponent above the 10 indicates how many places the digits are shifting (how many times we are dividing by 10 , the number becomes ten times smaller). Since we are dividing by powers of 10, the digits will shift to the left.

Students need to be provided with opportunities to explore this concept and come to this understanding; this should not just be taught procedurally.

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.

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523 × 103 = 523,000 The place value of 523 is increased by 3 places. 5.223 × 102 = 522.3 The place value of 5.223 is increased by 2 places. 52.3 × 101 = 5.23 The place value of 52.3 is decreased by one place.

Engage NY Lessons are included in the activity file. Coming Soon…Common MisconceptionsA common misconception that students have when trying to extend their understanding of whole number place value to decimal place value is that as you shift 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. DifferentiationIncrease the RigorNBT.1 How is the value of 6 different in 496 and 9.64? How are .7, 7, 70 related? Why is 35 x 10 = 350? Draw pictures and/or use number sentences to illustrate your explanation. Explain why 6 ÷ 10 = .6 Draw pictures and or use number sentences to illustrate your explanation. Jesse puts 10 jellybeans on a scale and the scale reads 12.0 grams. How much would you expect 1 jellybean to weigh? Why? How are 24 and .24 alike and different?NBT.2 Why does the product of 5 x 80 have two zeros? When solving 382 x 10, Mary knows to take 382 and add a zero to make the answer 3,820. Why does this work? Use what you know about place value to

explain. John says if you divide 5,624 by 100, you just move the decimal point to the left two places to get the answer of 56.24. Is John right? Use your place value

understanding to explain why this works. The approximate height (in feet) of the Statue of Liberty can be expressed as 3 x 102. Using what you know about exponents and place value, what is the

height of the statue in whole numbers? Jack is multiplying 64.15 x 10 so he put a zero at the end of the number to get his answer. 64.15 x 10 = 64.150. Explain why you agree or disagree with the

Jack's thinking.

Acceleration Intervention Coming Soon…Evidence of LearningBy completion of this lesson, students will be able to: Solve word problems involving the multiplication of 3- or 4- digit multiplicand by a 2- or 3- digit multiplier. Use exponents to represent powers of ten. Solve problems involving the division of 3- or 4- digit dividends by 2-digit divisors.Additional Assessment

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Shared Assessments: See formative assessment folder for Topic 2.Adopted ResourcesMy Math:1.1 Place Value Through Millions

Adopted Online ResourcesMy Mathhttp://connected.mcgraw-hill.com/connected/login.doTeacher User ID: ccsde0(enumber)Password: cobbmath1Student User ID: ccsd(student ID)Password: cobbmath1

Exemplarshttp://www.exemplarslibrary.com/User: Cobb EmailPassword: cobbmath “Planely” a Problem (NBT.1)

Think MathChapter 8 8.1 Using Place Value 8.2 Introducing Decimals 8.3 Zooming in on the Number Line 8.4 Decimals on the Number Line

Additional Web ResourcesK-5 Math teaching Resources http://www.k-5mathteachingresources.com/5th-grade-number-activities.htmlNBT.1Place Value ConcentrationNBT.2Multiplying a Whole Number by a Power of 10Multiplying a Decimal by a Power of 10Illustrative Mathematics provides instructional and assessment tasks, lesson plans, and other resources https://www.illustrativemathematics.org/NBT.1Kipton’s ScaleWhich Number Is It?Tenths and HundredthsMillions and Billions of PeopleNBT.2Marta’s Multiplication ErrorMultiplying Decimals by 10Learn Zillion https://learnzillion.com/NBT.1Determine the Value of a Digit in the Thousandths PlaceUnderstand that Place Value Increases Ten Times with each Shift to the Left in a Multi-Digit NumberUse Number Lines to Show Increases in Place Value with each Shift to the Left in a Multi-Digit Number

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Use Base-Ten Blocks to Understand how Place Value Decrease with each Shift to the Right in a Multi-Digit NumberUse Number Lines to Show Decreases in Place Value with every Shift to the Right in a Multi-Digit NumberRecognize Place Value Relationships by Multiplying and Dividing by TenNBT.2Explain Patterns in Zeros when Multiplying by a Power of TenRepresent Powers of 10 Using Whole Number ExponentsExplain Patterns in the Placement of the Decimal Point when Dividing a Decimal by Powers of 10Suggested Manipulativesbase ten blockscalculatorsplace value chartnumber linegrid paperdigit cards

Vocabularyexponentpower of 10place valuetenthshundredthsthousandths

Suggested LiteratureOne Grain of RiceToothpaste MillionaireCount to a MillionHow Much is a MillionG is for GoogleEarthquakes The Kings ChessboardIf you Made a Million

Task DescriptionsScaffolding Task Task that build up to the learning task.Constructing Task Task in which students are constructing understanding through deep/rich contextualized problem solving Practice Task Task that provide students opportunities to practice skills and concepts.Culminating Task Task designed to require students to use several concepts learned during the unit to answer a new or unique situation. Formative Assessment Lesson (FAL)

Lessons that support teachers in formative assessment which both reveal and develop students’ understanding of key mathematical ideas and applications.

3-Act Task Whole-group mathematical task consisting of 3 distinct parts: an engaging and perplexing Act One, an information and solution seeking Act Two, and a solution discussion and solution revealing Act Three.

State TasksTask Name Task Type/

Grouping StrategyContent Addressed Standard(s) Task Description

Patterns R Us Constructing TaskPartner/Small Group Task

Exploring powers of ten with exponents

MGSE.NBT.1MGSE.NBT.2

Power of ten and patterns

Fifth Grade Unit 1: Order of Operations and Whole Numbers

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Topic 3: Multiplication & DivisionBig Ideas/Enduring Understandings: Multiplication may be used to find the total number of objects when objects are arranged in equal groups, rectangular arrays/area models. One of the factors in multiplication indicates the number of objects in a group and the other factor indicates the number of groups. Unfamiliar multiplication problems may be solved by using, invented strategies or known multiplication facts and properties of multiplication and division.

For example, 8 x 7 = (8 x 2) + (8 x 5) and 18 x 7 = (10 x 7) + (8 x 7). There are two common situations where division may be used: fair sharing (given the total amount and the number of equal groups, determine how

many/much in each group) and measurement (given the total amount and the amount in a group, determine how many groups of the same size can be created).

The dividend, divisor, quotient, and remainder are related in the following manner: dividend = divisor x quotient + remainder. Some division situations will produce a remainder, but the remainder will always be less than the divisor. If the remainder is greater than the divisor, that

means at least one more can be given to each group (fair sharing) or at least one more group of the given size (the dividend) may be created.Essential Questions: 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? How can I use the situation in a story problem to determine the best operation to use? How can I effectively explain my mathematical thinking and reasoning to others?Content StandardsContent standards are interwoven and should be addressed throughout the year in as many different units and activities as possible in order to emphasize the natural connections that exist among mathematical topics.Perform Operations with Multi-Digit Whole NumbersMGSE5.NBT.5 Fluently multiply multi-digit whole numbers using the standard algorithm. MGSE5.NBT.6 Find whole-number quotients of whole number 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.

Vertical ArticulationFourth Grade Multiplication & Division StandardsUse place value understanding and properties of operations to perform multi-digit arithmetic. MGSE4.NBT.5 Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the

Sixth Grade Multiplication & Division StandardsCompute fluently with multi-digit numbers and find common factors and multiples. MGSE6.NS.2 Fluently divide multi-digit numbers using the standard algorithm

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calculation by using equations, rectangular arrays and/or area models. MGSE4.NBT.6 Find whole-number quotients and remainders with up to four-digit dividends and one-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. Instructional StrategiesPerform operations with multi-digit whole numbers NBT.5This standard refers to fluency which means students select and use a variety of methods and tools to compute, including objects, mental computation, estimation, paper and pencil, and calculators. They work flexibly with basic number combinations and use visual models, benchmarks, and equivalent forms. They are accurate and efficient (use a reasonable amount of steps), and flexible (use strategies such as the distributive property or breaking numbers

apart (decomposing and recomposing) also using strategies according to the numbers in the problem, 26 × 4 may lend itself to (25 × 4) + 4 where as another problem might lend itself to making an equivalent problem 32 × 4 = 64 × 2.

This standard builds upon students’ work with multiplying numbers in third and fourth grade. In fourth grade, students developed understanding of multiplication through using various strategies. While the standard algorithm is mentioned, alternative strategies are also appropriate to help students develop conceptual understanding.

The size of the numbers should NOT exceed a three-digit factor by a two-digit factor.

In previous grade levels, students have used various models and strategies to solve problems involving multiplication with whole numbers, so they should be able to transition to using standard algorithms effectively. With guidance from the teacher, they should understand the connection between the standard algorithm and their strategies. 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.Example:Find the product of 123 × 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. The ways in which students are taught to record this method may vary, but ALL should emphasize the place-value nature of the algorithm, For example, one might write

Note that a further decomposition of 123 into 100 + 20 + 30 and recording of the partial products would also be acceptable.

Examples of alternative strategies:

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There are 225 dozen cookies in the bakery. How many cookies are there?

Draw an array model for 225 × 12 → 200 × 10, 200 × 2, 20 × 10, 20 × 2, 5 × 10, 5 × 2.

Because students have used various models and strategies to solve problems involving multiplication with whole numbers, they should be able to transition to using standard algorithms effectively. With guidance from the teacher, they should understand the connection between the standard algorithm and their strategies. Connections between the algorithm for multiplying multi-digit whole numbers and strategies such as partial products or lattice multiplication are necessary for students’ understanding.

You can multiply by listing all the partial products. For example:

The multiplication can also be done without listing the partial products by multiplying the value of each digit from one factor by the value of each digit from the other factor. Understanding of place value is vital in using the standard algorithm.

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In using the standard algorithm for multiplication, when multiplying the ones, 32 ones is 3 tens and 2 ones. The 2 is written in the ones place. When multiplying the tens, the 24 tens is 2 hundreds and 4 tens. But, the 3 tens from the 32 ones need to be added to these 4 tens, for 7 tens. Multiplying the hundreds, the 16 hundreds is 1 thousand and 6 hundreds. But, the 2 hundreds from the 24 tens need to be added to these 6 hundreds, for 8 hundreds.

NBT.6This standard references various strategies for division. Division problems can include remainders. Even though this standard leads more towards computation, the connection to story contexts is critical.

Make sure students are exposed to problems where the divisor is the number of groups and where the divisor is the size of the groups. 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:There are 1,716 students participating in Field Day. They are put into teams of 16 for the competition. How many teams get created? If you have left over students, what do you do with them?

Example:968 ÷ 21 =?Using base ten models, a student can represent 968 and use the models to make an array with one dimension of 21. The student continues to make the array

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until no more groups of 21 can be made. Remainders are not part of the array.

Example:9,984 ÷ 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.

Example: Using expanded notation: 2682 ÷ 25 = (2000 + 600 + 80 + 2) ÷ 25 Using 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 × 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.)

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I can’t divide 2 by 25 so 2 plus the 5 leaves a remainder of 7. 80 + 24 + 3 = 107. So, the answer is 107 with a remainder of 7.

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

By fifth grade, students should understand that division can mean equal sharing or partitioning of equal groups or arrays. They should also understand that it is the same as repeated subtraction, and since it’s the inverse of multiplication, the quotient can be thought of as a missing factor. In fourth grade, students divided 4-digit dividends by 1-digit divisors. They also used contexts to interpret the meaning of remainders. Division is extended to 2-digit divisors in fifth grade, but fluency of the traditional algorithm is not expected until sixth grade. Division models and strategies that have been used in previous grade levels, such as arrays, number lines, and partial quotients, should continue to be used in fifth grade as students deepen their conceptual understanding of this division.

Engage NY Lessons are included in the activity file. Coming Soon…Common MisconceptionsDifferentiationIncrease the RigorNBT.5 Write a 3-digit by 1-digit multiplication problem with a product close to 2,000? List two numbers whose product is between 400 and 600. Is 4,000 a reasonable estimate for 312 x 9? Explain your reasoning. How does the order of the digits in the factors impact the product? (e.g. 452 x 7 compared to 425 x 7) In a multiplication problem, what happens to the product if each factor is multiplied by 2? What happens to the product if one factor is doubled and the

other factor is divided by 2? Is the product of 42 x 63 over or under 2,400? Use what you know about place value in the standard algorithm to explain your answer. You multiply two two-digit numbers and the product is close to 2,600.What two numbers did you multiply? Use 1,2,3,4,5 to make the a multiplication problem with the greatest product? How does the placement of the numbers impact the product? NBT.6 If the quotient is 34, what could a possible dividend and divisor be? (use either a three- or four-digit dividend) How do you know that 34 is not the quotient of 1,216 ÷ 4? Find a number that when divided by either 2, 3, or 5 has a remainder of 1. List three numbers that when divided by 5 each have a remainder of 1. Using the digits 4, 9, 7, 1 and 5, create a division sentence with a two-digit divisor and the greatest possible quotient. Write a division problem that has a four-digit dividend and a one-digit divisor with a quotient that is even. What is the relationship between multiplication and division? Provide examples to show your thinking. Use two different division strategies to solve 9,754 ÷ 5.

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How does understanding place value help when solving 439 ÷14 with the partial quotient strategy? Explain by using examples.

Acceleration Intervention Coming Soon…Evidence of LearningBy completion of this lesson, students will be able to: Apply strategies for multiplying a 2- or 3-digit number by a 2-digit number. Develop paper-and-pencil multiplication algorithms (not limited to the traditional algorithm) for 3- or 4-digit number multiplied by a 2- or 3-digit number. Apply paper-and-pencil strategies for division (not the standard algorithm) Solve problems involving multiplication and division.Additional Assessment:Formative Assessment Lesson - Division and Interpreting Remainders MGSE5.NBT.6 Shared Assessments: See formative assessment folder for Topic 3.Adopted ResourcesMy Math:Chapter 22.4 Multiplication Patterns2.6 Use Partial Products and the Distributive Property2.7 The Distributive Property2.8 Estimate Products

2.9 Multiply by One-Digit Numbers 2.10 Multiply by Two-Digit NumbersChapter 3

3.1 Relate Division to Multiplication 3.2 Hands On: Division Models 3.3 Two-digit Dividends 3.4 Division Patterns 3.5 Estimate Quotients 3.6 Hands on: Division Models with Greater Numbers 3.7 Hands On: Distributive Property and Partial Products 3.8 Divide Three-and-Four Digit Dividends 3.9 Place the First Digit 3.10 Quotients with Zeros 3.11 Hands On: Use models to Interpret the Remainder 3.12 Interpret the Remainder 3.13 Problem-Solving Investigation Chapter 4

Adopted Online ResourcesMy Mathhttp://connected.mcgraw-hill.com/connected/login.doTeacher User ID: ccsde0(enumber)Password: cobbmath1Student User ID: ccsd(student ID)Password: cobbmath1

Exemplarshttp://www.exemplarslibrary.com/User: Cobb EmailPassword: cobbmath

Hot Dogs for a Picnic (NBT.5) Apple Pies (NBT.5 & NBT.6) Soup, Fruit, and Juice Cans (NBT.5) There’s always Room for Jell-O (NBT.5) Camping (NBT.6) Equal Snacks (NBT.6) Great Pizza Dilemma (NBT.6) Mrs. Hasson’s Decorating Dilemma (NBT.6) M&M Cookie Combos (NBT.6) Hanging Airplanes (NBT.6)

Think MathChapter 2 2.4 Multiplying by Multiples of 10 2.5 Working with Large Numbers 2.6 Connecting Multiplication and Division 2.7 Arrays with Leftovers 2.11 Finding Products of Large FactorsChapter 5 5.1 Multiplying Multi-Digit Numbers 5.2 Writing Vertical Records 5.3 Writing Shorter Records 5.5 Multiplying Large Numbers

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4.1 Estimate Quotients 4.2 Hands On: Divide Using Base-Ten Blocks 4.3 Divide by a Two-Digit 4.4 Adjust Quotients 4.5 Divide Greater Numbers 4.6 Problem-Solving Investigation: Solve a Simpler ProblemAdditional Web ResourcesK-5 Math teaching Resources http://www.k-5mathteachingresources.com/5th-grade-number-activities.htmlNBT.5Multiplication Race (2 x 3 digit)NBT.6Division Strategy: Partition the Dividend (ver. 2)Estimate the Quotient (ver. 2)Write It, Solve It, Check It! (ver. 3)Illustrative Mathematics provides instructional and assessment tasks, lesson plans, and other resources https://www.illustrativemathematics.org/NBT.5Elmer’s Multiplication ErrorNBT.6Minutes and DaysLearn Zillion https://learnzillion.com/NBT.5Use Area Models for MultiplicationUse Partial Products for MultiplicationUse the Standard Algorithm for MultiplicationNBT.6Divide 4-Digit Dividends by 2-Digit Divisors by Setting up an EquationDivide 4-Digit Dividends by 2-Digit Divisors by Estimating and Adjusting the QuotientUse an Area Model for Division of 4-Digit Dividends by 2-Digit DivisorsDivide 4-Digit Dividends by 2-Digit Divisors by Using a Rectangular ArrayDivide 4-Digit Dividends by 2-Digit Divisors by Using Expanded NotationEstimation 180 is a website of 180 days of estimation ideas that build number sense http://www.estimation180.com/days.htmlSuggested Manipulativesbase-ten blocksgrid paperplace value chartnumber line

Vocabularyproductquotientdividenddivisor

Suggested LiteratureA Remainder of OneHershey’s Kisses: Multiplication and DivisionAmanda Bean’s Amazing DreamThe Best of Times

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digit cards factors The Doorbell RangDivide and Ride7 x 9 = Trouble!Each Orange Had 8 Slices

Task DescriptionsScaffolding Task Task that build up to the learning task.Constructing Task Task in which students are constructing understanding through deep/rich contextualized problem solving Practice Task Task that provide students opportunities to practice skills and concepts.Culminating Task Task designed to require students to use several concepts learned during the unit to answer a new or unique situation. Formative Assessment Lesson (FAL)

Lessons that support teachers in formative assessment which both reveal and develop students’ understanding of key mathematical ideas and applications.

3-Act Task Whole-group mathematical task consisting of 3 distinct parts: an engaging and perplexing Act One, an information and solution seeking Act Two, and a solution discussion and solution revealing Act Three.

State TasksTask Name Task Type/

Grouping StrategyContent Addressed Standard(s) Task Description

Multiplication Three in a Row Practice TaskSmall Group/Partner Task Multiply multi-digit numbers MGSE.NBT.5

Multiplication game 2-digit by 2 or 3-digit multiplication

Preparing a Prescription CTE Task Individual/Partner Task

Single and double digit multiplication in a real-world

contextMGSE.NBT.5

Using multiplication to determine cost of patient’s illness, and creating an invoice

The Grass is Always Greener Constructing TaskSmall Group/ Individual Task

Applying multiplication to problem solving situations

MGSE.NBT.5MGSE.NBT.6

Determining which turf is a better deal

Division Four in a Row Practice TaskPartner/Small Group Task

Divide four-digit dividends by one and two-digit divisors MGSE.NBT.6 Dividing numbers up to 4-digits by 1

and 2-digit divisors

Are These All 365 ÷ 15? Constructing TaskIndividual/Partner Task

Conceptual Understanding of Division Problem Types MGSE.NBT.6 Analyzing three different division

situations

Start of the Year Celebration Culminating TaskIndividual Task

Write expressions which involve multiplication and division of

whole numbers

MGSE.OA.1MGSE.OA.2MGSE.NBT.5MGSE.NBT.6

Determining expressions for amount of tables and chairs needed for a party

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