y 308: a a fourth grade

COMMUNITY UNIT DISTRICT 308: MATHEMATICS

CURRICULUM GUIDE

This document serves as the curriculum guide for District 308 mathematics. All components are fully

aligned to the Illinois Early Learning and Developmental Standards and Common Core State Standards.

Fourth Grade

2014-2015

EC-5 Math Curriculum Team Members: Brokaw Early Childhood Center:

Susan Craig (EC)

Darlene Howell (EC)

East View Kindergarten Center:

Jean Rampala (K)

Jennifer Friel (K)

Boulder Hill Elementary:

Alexandra Wooden (3d)

Brittany Morelli (2nd

)

Churchill Elementary:

Christine Gamlin (2nd

)

Kelly Fleagle (5th

)

Fox Chase Elementary:

Amber Denbo (1st

)

Amy Huffman (5th)

Grande Park Elementary:

Tamara Allen (2nd

)

Jeffrey Rainaldi (4th

)

Homestead Elementary

Jennifer Rusin (2nd

)

Julie Wilson (4th

)

Hunt Club Elementary:

Rosa Brolley (3rd

-Dual)

Toni Morgan (1st

-Dual)

Lakewood Creek Elementary:

Ann Lutz (5th

)

Long Beach Elementary:

Kirsten Brandwein (3rd

)

Andrea Daleiden (5th

)

Jessica Richter (2nd

)

Old Post Elementary:

Dana Miles (2nd

)

Deanne Todd (4th

)

Prairie Point Elementary:

Lisa Paluch (2nd

)

Sara Studer (3rd

)

Southbury Elementary:

Anne Marie Simmons (1st

)

Brendan Stephens (4th

)

Jennifer Weaver (K)

The Wheatlands Elementary:

Tamarac Maddox (2nd

)

Susan Rost (4th

)

Wolf’s Crossing Elementary:

Amanda Armitage (1st

)

Joy Varney (5th

)

Administrators:

Jodi Ancel

Tammie Harmon

Melissa McDowell

Lisa Smith

Katelyn Hutchison (1st

)

Community Unit District 308 Mathematics Units Fourth Grade

Priority Standards Supporting Standards Additional Standards 1

Unit One: Number and Operations

Approximate Time Frame: 4-5 Weeks Connections to Previous Learning: In Grade 3, students can fluently add and subtract within 1,000 and round numbers to the nearest 10 and 100. Focus of the Unit: In this unit, students will use a standard algorithm to fluently add and subtract numbers up to a million. They will learn to round multi-digit whole numbers and use this skill when assessing the reasonableness of answers to word problems. Students will generate shape and number patterns by following a given rule and identify other features in the pattern that were not given in the rule itself. Connections to Subsequent Learning: In fifth grade, students will extend their understanding of addition and subtraction of multi-digit whole numbers by adding and subtracting decimals. They will continue their fourth grade work on number patterns by generating two numerical patterns using two given rules, forming ordered pairs from the two patterns, and graphing the ordered pairs on a coordinate plane.

Progression Citation: From the K-5, Number and Operations in Base Ten progression document, p.13 Use place value understanding and properties of operations to perform multi-digit arithmetic. At Grade 4, students become fluent with the standard addition

and subtraction algorithms.4.NBT.4 As at the beginning of this progression, these algorithms rely on adding or subtracting like base-ten units (ones with ones, tens

with tens, hundreds with hundreds, and so on) and composing or decomposing base-ten units as needed (such as composing 10 ones to make 1 ten or decomposing 1 hundred to make 10 tens). In mathematics, an algorithm is defined by its steps and not by the way those steps are recorded in writing. With this in mind, minor variations in methods of recording standard algorithms are acceptable.

Transfer: Students will apply…

Students will apply concepts and procedures for adding, subtracting, multiplying and dividing multi-digit whole numbers to solve real-world and mathematical problems.

Ex. Based on census data from 2010, the population of Springfield, Illinois was 116,250. The population of the second largest city in Illinois, Aurora, was 197,899. How many more people were living in Aurora than Springfield in 2010?

Understandings: Students will understand that …

Patterns are generated by following a specific rule.

Rounding numbers can be used when estimating answers to real-world problems.

The standard algorithm for addition and subtraction relies on adding or subtracting like base-ten units.



Essential Questions:

How does place value help represent the value of numbers?

What strategies can be used to find rules for patterns and what predictions can the pattern support?

What strategies can I use to add or subtract?

How does understanding place value help you solve multi-digit addition and subtraction problems and how can rounding be used to estimate answers to problems?

How does the position of a digit in a number affect its value, and how can the value of digits be used to compare two numbers?

Prerequisite Skills/Concepts: Students should already be able to…

Identify arithmetic patterns and explain using properties.

Fluently add and subtract within 1000.

Advanced Skills/Concepts: Some students may be ready to…

Generate a pattern that follows a two-step rule.

Identify the rule and extend the pattern for a two-step function.

Knowledge: Students will know…

Patterns are generated by following a specific rule. (4.OA.5)

Rounding can be used to estimate reasonable answers for word problems. (4.NBT.3)

Skills: Students will be able to do…

Generate a pattern that follows a rule. (4.OA.5)

Given a pattern, identify the rule and extend the pattern and also identify apparent features of a pattern that follows a given rule, which are not explicit in the rule itself. (4.OA.5)

Round multi-digit whole numbers to a given place. (4.NBT.3)

Explain the rounding process using visuals and/or language. (4.NBT.3)

Add and subtract multi-digit whole numbers up to 1,000,000. (4.NBT.4)

WIDA Standard: English language learners communicate information, ideas and concepts necessary for academic success in the content area of Mathematics. English language learners will benefit from:

A preview of critical vocabulary terms before instruction.

The use of visuals to make explicit connections between the vocabulary and the content being learned.

Desired Outcomes:

Standards: Generate and analyze patterns. 4.OA.5 Generate a number or shape pattern that follows a given rule. Identify apparent features of the pattern that were not explicit in the rule itself. For example, given the rule “Add 3” and the starting number 1, generate terms in the resulting sequence and observe that the terms appear to alternate between odd and even numbers. Explain informally why the numbers will continue to alternate in this way. Generalize place value understanding for multi-digit whole numbers. 4.NBT.3 Use place value understanding to round multi-digit whole numbers to any place. Use place value understanding and properties of operations to perform multi-digit arithmetic. 4.NBT.4 Fluently add and subtract multi-digit whole numbers using the standard algorithm.



Highlighted Mathematical Practices: (Practices to be explicitly emphasized are indicated with an *.)

*1. Make sense of problems and persevere in solving them. Students will solve word problems using all operations. Students will represent word problems using various modalities. Students will demonstrate their perseverance by selecting an effective modality to solve problems. *2. Reason abstractly and quantitatively. Students determine what operations to use to solve word problems and think about the reasonableness of their solutions. Students use reasonable estimates based upon the value of the numbers. 3. Construct viable arguments and critique the reasoning of others. Students demonstrate their ability to construct viable arguments when they talk and write about the steps they take to solve problems. Students restate and respond to each other about their mathematical thought processes. 4. Model with mathematics. Students write equations when recording all steps used to solve multiplication and division problems. They use all modalities (picture, manipulatives, written symbols, real world situations, oral/written language) when representing a situation. *5. Use appropriate tools strategically. Students choose and use the appropriate tools to solve problems. Tools could include: color tiles, base ten blocks, counters, math lines, geoboards, unifix cubes, websites, formulas, numerical equations, numerical expressions, diagrams, graph paper, shape diagrams, songs, and white boards. 6. Attend to precision. Students demonstrate precision in calculation by using their inverse operations to check their work and by using precise vocabulary in their oral and written explanations. *7. Look for and make use of structure. Students recognize and identify patterns existing between addition and multiplication along with division and subtraction. Students use this knowledge when applying strategies to evaluate real-world problems such as area and perimeter. They also use problem solving structures to aid in solving real world problems. *8. Look for and express regularity in repeated reasoning. Students identify and explain patterns within multiplication and division arrays (prime numbers, square numbers, and composite numbers, factors, multiples), determine and communicate whether identified patterns always work, determine and communicate when to apply specific patterns, describe the relationship between multiplication, addition, subtraction, and division, and justify that solutions make sense (i.e., using mental computation and/or estimation).



Academic Vocabulary:

Critical Terms Supplemental Terms

English:

Rounding

Estimation Digit

Place value

Expanded form

Period

Standard form

Word form

Is Equal to

Is greater than

Is less than

Number line

Minuend

Subtrahend

Associative property of addition

Commutative property of addition

Equation

Identity property of addition

Unknown

Variable

Spanish:

Estimación

Digito de estimación

Valor posicional

Forma desarrollada

Punto

Forma estándar

Forma verbal

Es igual a

Es mayor que

Es menor que

Línea numérica

Minuendo

Sustraendo

Propiedad asociativa de adición

Propiedad conmutativa de adición

Ecuación

Propiedad de identidad de adición

Incógnita

Variable

English:

Patterns

Rules

Ten thousands

Round

Spanish:

Patrones

Reglas

Decenas de millar

Redondear

Assessments: Utilize Pre-Assessments, Post-Assessments, Formative Assessments via My Math and Mastery Connect. Utilize District 308 Unit Summative Assessments via Mastery Connect.



Unit Two: Multiplication and Division Concepts Approximate Time Frame: 6-7 Weeks

Connections to Previous Learning:

In Grade 3, students fluently add and subtract within 1,000, and they also fluently multiply and divide within 100. They develop an understanding of the meanings of multiplication and division of whole numbers through activities and problems involving equal-sized groups, arrays, and area models. 4th grade students will extend these skills to develop fluency with efficient procedures for multiplying whole numbers and apply their understanding of models for multiplication and division. Focus of the Unit: Much of this unit focuses on extending previous skills to multi-digit whole numbers. Students will read, write, and compare multi-digit whole numbers up to a million, conceptually understanding that a digit in one place represents 10 times what it represents in the place to its right. Students will also fluently multiply (4-digit by 1-digit, 2-digit by 2-digit) and divide (4-digit by 1-digit) using strategies based on place-value and the properties of operations. They will illustrate and explain their calculations by using equations, rectangular arrays, and/or area models. Students will develop an understanding of multiplication as multiplicative comparisons (35 is 5 times as much as 7.) They will apply their understanding of multiplication comparison, multiplication skills, and division skills while estimating and problem-solving. Students extend the idea of decomposition to multiplication and learn to use the terms multiple and factor. Connections to Subsequent Learning: In fifth grade, students will extend their understanding of the base-ten system for decimals to thousandths. They will fluently compute products of whole numbers using the standard algorithm. They will continue their fourth grade work on division, extending it to computation of whole number quotients with dividends of up to four digits and two-digit divisors.

Progression Citation: From the K-5, Number and Operations in Base Ten, pp. 12-16, K-5 Operations and Algebraic Thinking, pp. 29-31 At Grade 4, students extend their work in the base-ten system. They use standard algorithms to fluently add and subtract. They use methods based on place value and properties of operations supported by suitable representations to multiply and divide with multi-digit numbers. Generalize place value understanding for multi-digit whole numbers. In the base-ten system, the value of each place is 10 times the value of the place to the immediate right.4.NBT.1 Because of this, multiplying by 10 yields a product in which each digit of the multiplicand is shifted one place to the left. To read numerals between 1,000 and 1,000,000, students need to understand the role of commas. Each sequence of three digits made by commas is read as hundreds, tens, and ones, followed by the name of the appropriate base-thousand unit (thousand, million, billion,



trillion, etc.). Thus, 457,000 is read “four hundred fifty seven thousand.”4.NBT.2 The same

methods students used for numbers called one tenth and one hundredth, work with these fractions can be used as preparation to extend the base-ten system to non-whole numbers. Using the unit fractions 1/10 and 1/100, non-whole numbers like 23 7/10 can be written in comparing and rounding numbers in previous grades apply to these numbers, because of the uniformity of the base-ten system. Decimal notation and fractions - students in Grade 4 work with fractions having denominators 10 and 100.4.NF.5 Because it involves partitioning into 10 equal parts and treating the parts as an expanded

form that extends the form used with whole numbers: 2 10 3 1 7 1 10 .4.NF.4b As with whole-number

expansions in the base-ten system, each unit in this decomposition is ten times the unit to its right. This can be connected with the use of base-ten notation to represent 2 x 10 + 3 x 1+7 x 1/10 as 23.7.

Using decimals allows students to apply familiar place value reasoning to fractional quantities.4.NF.6

The Number and Operations—Fractions Progression discusses decimals to hundredths and

comparison of decimals4.NF.7

in more detail. The decimal point is used to signify the location of the ones place, but its location may suggest there should be a “oneths" place to its right in order to create symmetry with respect to the decimal point. However, because one is the basic unit from which the other base ten units are derived, the symmetry occurs instead with respect to the ones place. Ways of reading decimals aloud vary. Mathematicians and scientists often read 0.15 aloud as “zero point one five" or “point one five." (Decimals smaller than one may be written with or without a zero before the decimal point.) Decimals with many non-zero digits are more easily read aloud in this manner. (For example, the number π, which has infinitely many non-zero digits, begins 3.1415…) Other ways to read 0.15 aloud are “1 tenth and 5 hundredths” and “15 hundredths,” just as 1,500 is sometimes read “15 hundred” or “1 thousand, 5 hundred.” Similarly, 150 is read “one hundred and fifty” or “a hundred fifty” and understood as 15 tens, as 10 tens and 5 tens, and as 100+50. Just as 15 is understood as 15 ones and as 1 ten and 5 ones in computations with whole numbers, 0.15 is viewed as 15 hundredths and as 1 tenth and 5 hundredths in computations with decimals. It takes time to develop understanding and fluency with the different forms. Layered cards for decimals can help students become fluent with decimal equivalencies such as three tenths is thirty hundredths. Use place value understanding and properties of operations to perform multi-digit arithmetic. At Grade 4, students become fluent with the standard addition



and subtraction algorithms.4.NBT.4 As at the beginning of this progression, these algorithms rely on adding or subtracting like base-ten units (ones with ones, tens with tens, hundreds with hundreds, and so on) and composing or decomposing base-ten units as needed (such as composing 10 ones to make 1 ten or decomposing 1 hundred to make 10 tens). In mathematics, an algorithm is defined by its steps and not by the way those steps are recorded in writing. With this in mind, minor variations in methods of recording standard algorithms are acceptable. In fourth grade, students compute products of one-digit numbers and multi-digit numbers (up to four digits) and products of two two-digit numbers. They divide multi-digit numbers (up to four digits) by one-digit numbers. As with addition and subtraction, students should use methods they understand and can explain. Visual representations such as area and array diagrams that students draw and connect to equations and other written numerical work are useful for this purpose. By reasoning repeatedly about the connection between math drawings and written numerical work, students can come to see multiplication and division algorithms as abbreviations or summaries of their reasoning about quantities. Students can invent and use fast special strategies while also working towards understanding general methods and the standard algorithm. One component of understanding general methods for multiplication is understanding how to compute products of one-digit numbers and multiples of 10, 100, and 1000. This extends work in Grade 3 on products of one-digit numbers and multiples of 10. We can calculate 6 x 700 by calculating 6 x 7 and then shifting the

result to the left two places (by placing two zeros at the end to show that these are hundreds) because 6 groups of 7 hundred is hundreds, which is 42 hundreds, or 4,200. Students can use this place value reasoning, which can also be supported with diagrams of arrays or areas, as they develop and practice using the patterns in relationships among products such as 6 x 7, 6 x 70, 6 x 700, and 6 x 7000. Products of 5 and even numbers, such as 5 x 4, 5 x 40, 5 x 400, 5 x 4000 and 4 x 5, 4 x 50, 4 x 5000 might be discussed and practiced separately afterwards because they may seem at first to violate the patterns by having an “extra” 0 that comes from the one-digit product. Another part of understanding general base-ten methods for multi-digit multiplication is understanding the role played by the distributive property. This allows numbers to be decomposed into base-ten units, products of the units to be computed, and then combined. By decomposing the factors into like base-ten units and applying the distributive property, multiplication computations are reduced to single-digit multiplications and products of numbers with multiples of 10, of 100, and of 1000. Students can connect diagrams of areas or arrays to numerical work to develop understanding of general base-ten multiplication methods. Computing products of two two-digit numbers requires using the distributive property several times when the factors are decomposed into base-ten units.



For example, 36 x 94 = (30 + 6) x (90 + 4) = (30 + 6)90 + (30 + 6) x 4: General methods for computing quotients of multi-digit numbers and one-digit numbers rely on

the same understandings as for multiplication, but cast in terms of division.4.NBT.6 One component

is quotients of multiples of 10, 100, or 1000 and one-digit numbers. For example, 42 ÷ 6 is related to 420 ÷ 6and 4200 ÷ 6. Students can draw on their work with multiplication and they can also reason that 4200 6 means partitioning 42 hundreds into 6 equal groups, so there are 7 hundreds in each group. Another component of understanding general methods for multi-digit division computation is the idea of decomposing the dividend into like base-ten units and finding the quotient unit by unit, starting with the largest unit and continuing on to smaller units. As with multiplication, this relies on the distributive property. This can be viewed as finding the side length of a rectangle (the divisor is the length of the other side) or as allocating objects (the divisor is the number of groups). See the figures on the next page for examples. Multi-digit division requires working with remainders. In preparation for working with remainders, students can compute sums of a product and a number, such as 4 x 8 + 3. In multi-digit division, students will need to find the greatest multiple less than a given number. For example, when dividing by 6, the greatest multiple of 6 less than 50 is 6 x 8 = 48. Students can think of these “greatest multiples” in terms of putting objects into groups. For example, when 50 objects are shared among 6 groups, the largest whole number of objects that can be put in each group is 8, and 2 objects are left over. (Or when 50 objects are allocated into groups of 6, the largest whole number of groups that can be made is 8, and 2 objects are left over.) The equation 6 x 8 + 2 = 50 (or 8 x 6 + 2 = 50) corresponds with this situation. Cases involving 0 in division may require special attention.




Students will apply concepts and procedures for adding, subtracting, multiplying and dividing multi-digit whole numbers to solve real-world and mathematical problems. Ex: A basketball costs 3 times as much as a tennis ball. If the tennis ball costs $4, how much does a basketball cost? Ex: The dimensions of a small-size soccer field are 65 yards by 45 yards. What is the area of the soccer field?




Place value is based on groups of ten and the value of a number is determined by the place of its digits.

Whole numbers are read from left to right using the name of the period; commas are used to separate periods.

A number can be written using its name, standard, or expanded form.

Flexible methods of computation involve grouping numbers in strategic ways.

The distributive property is connected to the area model and/or partial products method of multiplication.

Multiplication and division are inverse operations.

There are three different structures for multiplication and division problems: Area/Arrays, Equal Groups, and Comparison , and the unknown quantity in multiplication and division situations is represented in three ways: Unknown Product, Group Size Unknown, and Number of Groups Unknown.


In what ways can numbers be composed and decomposed?

How are multiplication and division related?

How can I communicate multiplication?

How can I multiple by a two-digit number?

What are different models for multiplication and division?

How does division affect numbers?

What are efficient methods for finding products and quotients, and how can place value properties aid computation?

How are dividends, divisors, quotients, and remainders related?

What real-life situations require the use of multiplication or division?

How can a remainder affect the answer in a division word problem?

How are patterns used in mathematics?


Add and subtract fluently within 1000.

Read and write numbers to 1000 using base- ten numerals, number names, and expanded form.

Compare two three-digit numbers.

Apply properties to solve problems.

Interpret products and quotients of whole numbers.

Solve word problems involving x and ÷ with equal groups, arrays, and measurement.

Determine unknowns in x and ÷ equations.

Understand division as unknown factor problems.

Fluently x and ÷ within 100.

Multiply 1-digit whole number by multiples of 10.


Multiply larger than 2 digit by 2 digit numbers.

Divide by multiples of ten.

Solve problems involving multi-step equations with two or more operation.

Create multi-step problems using multiple structures.

Use parentheses to represent and solve multi-step problems.



Solve two-step word problems involving +, -, x and ÷


Multiplication equations can show comparisons. (4.OA.1)

When to apply single equations or more than one equation using manipulatives, and/or diagrams to represent multiplicative comparison. (4.OA.1)

Verbal statements of multiplicative comparisons can be written as equations with and without variables. (i.e., Sally is five years old. Her mom is eight times older. How old is Sally’s Mom? 5 x 8 = 40) (4.OA.1)

A digit in one place represents ten times what it represents in the place to its right, by using manipulatives, pictures, language, and/or equations to explain their reasoning. (4.NBT.1)

Strategies for multiplying and dividing based on place value, the properties of operations, and/or the relationship between multiplication and division. (4.NBT.5) (4.NBT.6)


Translate comparative situations into drawings and equations with a symbol for the unknown and unknowns in all 3 locations. (4.OA.2)

Solve word problems involving multiplicative comparison using drawings and (multiplication or division) equations with a symbol for the unknown number and unknowns in all 3 locations. (4.OA.2)

Explain the difference between additive comparison and multiplicative comparison using visuals and words. (4.OA.2)

Read and write whole numbers up to a million using standard, word, and expanded form. (4.NBT.2)

Compare two multi-digit (up to a million) numbers. (4.NBT.2)

Use manipulatives, pictures, and language to show the relationship between the numerals and their place value representations in multiple ways. (4.NBT.2)

Identify all factor pairs for any given number 1-100. Recognize that a whole number is a multiple of each of its factors. (4.OA.4)

Determine whether a given whole number in the range 1-100 is a multiple of a given one-digit number. (4.OA.4)

Determine whether a given whole number in the range 1-100 is prime or composite. (4.OA.4)

Use visuals, symbols and/or language to explain their reasoning. (4.OA.4)

Multiply up to 4-digit by 1-digit numbers and 2-digit by 2-digit numbers. (4.NBT.5)

Use place value manipulatives to represent multiplication calculations. Illustrate and explain the calculation by using written equations, rectangular arrays, and area models. (4.NBT.5)






Desired Outcomes:

Standards: Use the four operations with whole numbers to solve problems. 4.OA.1 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. Represent verbal statements of multiplicative comparisons as multiplication equations. 4.OA.2 Multiply or divide to solve word problems involving multiplicative comparison, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison. Generalize place value understanding for multi-digit whole numbers. 4.NBT.1 Recognize that in a multi-digit whole number, a digit in 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. 4.NBT.2 Read and write multi-digit whole numbers using base-ten numerals, number names, and expanded form. Compare two multi-digit numbers based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons. Gain familiarity with factors and multiples. 4.OA.4 Find all factor pairs for a whole number in the range 1–100. Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range 1–100 is a multiple of a given one-digit number. Determine whether a given whole number in the range 1–100 is prime or composite. 4.OA.5 Generate a number or shape pattern that follows a given rule. Identify apparent features of the pattern that were not explicit in the rule itself. For example, given the rule “Add 3” and the starting number 1, generate terms in the resulting sequence and observe that the terms appear to alternate between odd and even numbers. Explain informally why the numbers will continue to alternate in this way. Use place value understanding and properties of operations to perform multi-digit arithmetic. 4.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 calculation by using equations, rectangular arrays, and/or area models. 4.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. Use the four operations with whole numbers to solve problems. 4.OA.3 Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.


1. Make sense of problems and persevere in solving them. Students will solve word problems using all operations. Students will represent word problems using various modalities. Students will demonstrate their perseverance by selecting an effective modality to solve problems. *2. Reason abstractly and quantitatively. Students determine what operations to use to solve word problems and think about the reasonableness of their solutions. Students use reasonable estimates based upon the value of the numbers. 3. Construct viable arguments and critique the reasoning of others. Students demonstrate their ability to construct viable arguments when they talk and



write about the steps they take to solve problems. Students restate and respond to each other about their mathematical thought processes. *4. Model with mathematics. Students write equations when recording all steps used to solve multiplication and division problems. They use all modalities (picture, manipulatives, written symbols, real world situations, oral/written language) when representing a situation. 5. Use appropriate tools strategically. Students choose and use the appropriate tools to solve problems. Tools could include: color tiles, base ten blocks, counters, math lines, geoboards, unifix cubes, websites, formulas, numerical equations, numerical expressions, diagrams, graph paper, shape diagrams, songs, and white boards. 6. Attend to precision. Students demonstrate precision in calculation by using their inverse operations to check their work and by using precise vocabulary in their oral and written explanations. *7. Look for and make use of structure. Students recognize and identify patterns existing between addition and multiplication along with division and subtraction. Students use this knowledge when applying strategies to evaluate real-world problems such as area and perimeter. They also use problem solving structures to aid in solving real world problems. 8. Look for and express regularity in repeated reasoning. Students identify and explain patterns within multiplication and division arrays (prime numbers, square numbers, and composite numbers, factors, multiples), determine and communicate whether identified patterns always work, determine and communicate when to apply specific patterns, describe the relationship between multiplication, addition, subtraction, and division, and justify that solutions make sense (i.e., using mental computation and/or estimation).



English:

Multiplicative comparison

Standard form

Written form

Expanded form

Area model

Factor

Multiple

Prime

Composite

Divisor

Dividend

Remainder

Associative property of multiplication

Commutative property of multiplication

Decompose

Spanish:

Comparación multiplicativa

Forma estándar

Forma verbal

Forma desarrollada

Modelo de área

Factor

Múltiplo

Número primo

Número compuesto

Divisor

Dividendo

Residuo

Propiedad asociativa de multiplicación

Propiedad conmutativa de multiplicación

Descomponer

English:

Array

Equation

Product

Quotient

Multiply

Divide

Equation

Spanish:

Arreglo

Ecuación

Producto

Cociente

Multiplicar

Divide

ecuación



Dividend

Divisor

Facts family

Factor

Identity property of multiplication

Multiple

Product

Quotient

Repeated subtraction

Zero property of multiplication

Distributive property

Partial products

Regroup

Operation

Compatible numbers

Partial quotients

Remainders

Input

Nonnumeric pattern

Numeric pattern

Output

Pattern

Rule

Sequence

Term

Dividendo

Divisor

Familia de operaciones

Factor

Propiedad de la identidad de la multiplicación

Múltiplo

Producto

Cociente

Resta repetida

Propiedad del cero de la multiplicación

Propiedad distributiva

Productos parciales

Reagrupar

Operación

Números compatibles

Cocientes parciales

Resto

Entrada

Patrón no numérico

Patrón numérico

Salida

Patrón

Regla

Secuencia

Término




Unit Three: Multiplication and Division Application Approximate Time Frame: 5-6 Weeks

Connections to Previous Learning:

In Grade 3, students fluently add and subtract within 1,000, and they also fluently multiply and divide within 100. They develop an understanding of the

meanings of multiplication and division of whole numbers through activities and problems involving equal-sized groups, arrays, and area models. 4th grade students will extend these skills to develop fluency with efficient procedures for multiplying whole numbers and apply their understanding of models for multiplication and division. Focus of the Unit: Much of this unit focuses on extending previous skills to multi-digit whole numbers. Students will read, write, and compare multi-digit whole numbers up to a million, conceptually understanding that a digit in one place represents 10 times what it represents in the place to its right. Students will also fluently multiply (4-digit by 1-digit, 2-digit by 2-digit) and divide (4-digit by 1-digit) using strategies based on place-value and the properties of operations. They will illustrate and explain their calculations by using equations, rectangular arrays, and/or area models. Students will develop an understanding of multiplication as multiplicative comparisons (35 is 5 times as much as 7.) They will apply their understanding of multiplication comparison, multiplication skills, and division skills while estimating and problem-solving. Students extend the idea of decomposition to multiplication and learn to use the terms multiple and factor. Connections to Subsequent Learning: In fifth grade, students will extend their understanding of the base-ten system for decimals to thousandths. They will fluently compute products of whole numbers using the standard algorithm. They will continue their fourth grade work on division, extending it to computation of whole number quotients with dividends of up to four digits and two-digit divisors.

Progression Citation: From the K-5, Number and Operations in Base Ten, pp. 12-16, K-5 Operations and Algebraic Thinking, pp. 29-31 At Grade 4, students extend their work in the base-ten system. They use standard algorithms to fluently add and subtract. They use methods based on place value and properties of operations supported by suitable representations to multiply and divide with multi-digit numbers. Generalize place value understanding for multi-digit whole numbers. In the base-ten system, the value of each place is 10 times the value of the place to the immediate right.4.NBT.1 Because of this, multiplying by 10 yields a product in which each digit of the multiplicand is shifted one place to the left. To read numerals between 1,000 and 1,000,000, students need to understand the role of commas. Each sequence of three digits made by commas is read as hundreds, tens, and



ones, followed by the name of the appropriate base-thousand unit (thousand, million, billion,

trillion, etc.). Thus, 457,000 is read “four hundred fifty seven thousand.” 4.NBT.2 The same

methods students used for comparing and rounding numbers in previous grades apply to these numbers, because of the uniformity of the base-ten system. Decimal notation and fractions - students in Grade 4 work with fractions having denominators 10 and 100.4.NF.5 Because it involves partitioning into 10 equal parts and treating the parts as numbers called one tenth and one hundredth, work with these fractions can be used as preparation to extend the base-ten system to non-whole numbers. Using the unit fractions 1/10 and 1/100, non-whole numbers like 23 7/10 can be written in an

expanded form that extends the form used with whole numbers: 2 10 3 1 7 1 10 .4.NF.4b As with

whole-number expansions in the base-ten system, each unit in this decomposition is ten times the

unit to its right. This can be connected with the use of base-ten notation to represent 2 x 10 + 3 x 1+7 x 1/10 as 23.7. Using decimals allows students to apply familiar place value reasoning to

fractional quantities.4.NF.6 The Number and Operations—Fractions Progression discusses decimals to

hundredths and comparison of decimals4.NF.7

in more detail. The decimal point is used to signify the location of the ones place, but its location may suggest there should be a “oneths" place to its right in order to create symmetry with respect to the decimal point. However, because one is the basic unit from which the other base ten units are derived, the symmetry occurs instead with respect to the ones place. Ways of reading decimals aloud vary. Mathematicians and scientists often read 0.15 aloud as “zero point one five" or “point one five." (Decimals smaller than one may be written with or without a zero before the decimal point.) Decimals with many non-zero digits are more easily read aloud in this manner. (For example, the number π, which has infinitely many non-zero digits, begins 3.1415…) Other ways to read 0.15 aloud are “1 tenth and 5 hundredths” and “15 hundredths,” just as 1,500 is sometimes read “15 hundred” or “1 thousand, 5 hundred.” Similarly, 150 is read “one hundred and fifty” or “a hundred fifty” and understood as 15 tens, as 10 tens and 5 tens, and as 100+50. Just as 15 is understood as 15 ones and as 1 ten and 5 ones in computations with whole numbers, 0.15 is viewed as 15 hundredths and as 1 tenth and 5 hundredths in computations with decimals. It takes time to develop understanding and fluency with the different forms. Layered cards for decimals can help students become fluent with decimal equivalencies such as three tenths is thirty hundredths.



Use place value understanding and properties of operations to perform multi-digit arithmetic. At Grade 4, students become fluent with the standard addition and subtraction algorithms.4.NBT.4 As at the beginning of this progression, these algorithms rely on adding or subtracting like base-ten units (ones with ones, tens with tens, hundreds with hundreds, and so on) and composing or decomposing base-ten units as needed (such as composing 10 ones to make 1 ten or decomposing 1 hundred to make 10 tens). In mathematics, an algorithm is defined by its steps and not by the way those steps are recorded in writing. With this in mind, minor variations in methods of recording standard algorithms are acceptable. In fourth grade, students compute products of one-digit numbers and multi-digit numbers (up to four digits) and products of two two-digit numbers. They divide multi-digit numbers (up to four digits) by one-digit numbers. As with addition and subtraction, students should use methods they understand and can explain. Visual representations such as area and array diagrams that students draw and connect to equations and other written numerical work are useful for this purpose. By reasoning repeatedly about the connection between math drawings and written numerical work, students can come to see multiplication and division algorithms as abbreviations or summaries of their reasoning about quantities. Students can invent and use fast special strategies while also working towards understanding general methods and the standard algorithm. One component of understanding general methods for multiplication is understanding how to compute products of one-digit numbers and multiples of 10, 100, and 1000. This extends work in Grade 3 on products of one-digit numbers and multiples of 10. We can calculate 6 x 700 by calculating 6 x 7 and then shifting the result to the left two places (by placing two zeros at the end to show that these are hundreds) because 6 groups of 7 hundred is 6 x 7 hundreds, which is 42 hundreds, or 4,200. Students can use this place value reasoning, which can also be supported with diagrams of arrays or areas, as they develop and practice using the patterns in relationships among products. Products of 5 and even numbers, such as 5 x 4, 5 x 40, 5 x 400, 5 x 4000 and 4 x 5, 4 x 50, 4 x 5000 might be discussed and practiced separately afterwards because they may seem at first to violate the patterns by having an “extra” 0 that comes from the one-digit product. Another part of understanding general base-ten methods for multi-digit multiplication is understanding the role played by the distributive property. This allows numbers to be decomposed into base-ten units, products of the units to be computed, and then combined. By decomposing the factors into like base-ten units and applying the distributive property, multiplication computations are reduced to single-digit multiplications and products of numbers with multiples of 10, of 100, and of 1000. Students can connect diagrams of areas or arrays to numerical work to develop understanding of general base-ten multiplication methods.



Computing products of two two-digit numbers requires using the distributive property several times when the factors are decomposed into base-ten units. For example, 36 x 94 = (30 + 6) x (90 + 4) = (30 + 6)90 + (30 + 6) x 4: General methods for computing quotients of multi-digit numbers and one-digit numbers rely on the

same understandings as for multiplication, but cast in terms of division.4.NBT.6 One component is

quotients of multiples of 10, 100, or 1000 and one-digit numbers. For example, 42 ÷ 6 is related to 420 ÷ 6and 4200 ÷ 6. Students can draw on their work with multiplication and they can also reason that 4200 6 means partitioning 42 hundreds into 6 equal groups, so there are 7 hundreds in each group. Another component of understanding general methods for multi-digit division computation is the idea of decomposing the dividend into like base-ten units and finding the quotient unit by unit, starting with the largest unit and continuing on to smaller units. As with multiplication, this relies on the distributive property. This can be viewed as finding the side length of a rectangle (the divisor is the length of the other side) or as allocating objects (the divisor is the number of groups). See the figures on the next page for examples. Multi-digit division requires working with remainders. In preparation for working with remainders, students can compute sums of a product and a number, such as 4 x 8 + 3. In multi-digit division, students will need to find the greatest multiple less than a given number. For example, when dividing by 6, the greatest multiple of 6 less than 50 is 6 x 8 = 48. Students can think of these “greatest multiples” in terms of putting objects into groups. For example, when 50 objects are shared among 6 groups, the largest whole number of objects that can be put in each group is 8, and 2 objects are left over. (Or when 50 objects are allocated into groups of 6, the largest whole number of groups that can be made is 8, and 2 objects are left over.) The equation 6 x 8 + 2 = 50 (or 8 x 6 + 2 = 50) corresponds with this situation. Cases involving 0 in division may require special attention.




Students will apply concepts and procedures for adding, subtracting, multiplying and dividing multi-digit whole numbers to solve real-world and mathematical problems. Ex: A basketball costs 3 times as much as a tennis ball. If the tennis ball costs $4, how much does a basketball cost? Ex: The dimensions of a small-size soccer field are 65 yards by 45 yards. What is the area of the soccer field?


Place value is based on groups of ten and the value of a number is determined by the place of its digits.

Whole numbers are read from left to right using the name of the period; commas are used to separate periods.



A number can be written using its name, standard, or expanded form.

Flexible methods of computation involve grouping numbers in strategic ways.

The distributive property is connected to the area model and/or partial products method of multiplication.

Multiplication and division are inverse operations.

There are three different structures for multiplication and division problems: Area/Arrays, Equal Groups, and Comparison , and the unknown quantity in multiplication and division situations is represented in three ways: Unknown Product, Group Size Unknown, and Number of Groups Unknown.

Some division situations will produce a remainder, but the remainder should 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 giv en size (the dividend) may be created. When using division to solve word problems, how the remainder is interpreted depends on the problem situation.


In what ways can numbers be composed and decomposed?

How are multiplication and division related?

How can I communicate multiplication?

How can I multiple by a two-digit number?

What are different models for multiplication and division?

How does division affect numbers?

What are efficient methods for finding products and quotients, and how can place value properties aid computation?

How are dividends, divisors, quotients, and remainders related?

What real-life situations require the use of multiplication or division?

How can a remainder affect the answer in a division word problem?

How are patterns used in mathematics?


Add and subtract fluently within 1000.

Read and write numbers to 1000 using base- ten numerals, number names, and expanded form.

Compare two three-digit numbers.

Apply properties to solve problems.

Interpret products and quotients of whole numbers.

Solve word problems involving x and ÷ with equal groups, arrays, and measurement.

Determine unknowns in x and ÷ equations.

Understand division as unknown factor problems.



Multiply larger than 2 digit by 2 digit numbers.

Divide by multiples of ten.

Solve problems involving multi-step equations with two or more operation.

Create multi-step problems using multiple structures.

Use parentheses to represent and solve multi-step problems.




Solve two-step word problems involving +, -, x and ÷.


Multiplication equations can show comparisons. (4.OA.1)

When to apply single equations or more than one equation using manipulatives, and/or diagrams to represent multiplicative comparison. (4.OA.1)

Verbal statements of multiplicative comparisons can be written as equations with and without variables. (i.e., Sally is five years old. Her mom is eight times older. How old is Sally’s Mom? 5 x 8 = 40) (4.OA.1)

A digit in one place represents ten times what it represents in the place to its right, by using manipulatives, pictures, language, and/or equations to explain their reasoning. (4.NBT.1)

Strategies for multiplying and dividing based on place value, the properties of operations, and/or the relationship between multiplication and division. (4.NBT.5) (4.NBT.6)


Translate comparative situations into drawings and equations with a symbol for the unknown and unknowns in all 3 locations. (4.OA.2)

Solve word problems involving multiplicative comparison using drawings and (multiplication or division) equations with a symbol for the unknown number and unknowns in all 3 locations. (4.OA.2)

Explain the difference between additive comparison and multiplicative comparison using visuals and words. (4.OA.2)

Read and write whole numbers up to a million using standard, word, and expanded form. (4.NBT.2)

Compare two multi-digit (up to a million) numbers. (4.NBT.2)

Use manipulatives, pictures, and language to show the relationship between the numerals and their place value representations in multiple ways. (4.NBT.2)

Identify all factor pairs for any given number 1-100. Recognize that a whole number is a multiple of each of its factors. (4.OA.4)

Determine whether a given whole number in the range 1-100 is a multiple of a given one-digit number. (4.OA.4)

Determine whether a given whole number in the range 1-100 is prime or composite. (4.OA.4)

Use visuals, symbols and/or language to explain their reasoning. (4.OA.4)

Multiply up to 4-digit by 1-digit numbers and 2-digit by 2-digit numbers. (4.NBT.5)

Use place value manipulatives to represent multiplication calculations. Illustrate and explain the calculation by using written equations, rectangular arrays, and area models. (4.NBT.5)






Desired Outcomes:

Standards: Use the four operations with whole numbers to solve problems. 4.OA.1 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. Represent verbal statements of multiplicative comparisons as multiplication equations. 4.OA.2 Multiply or divide to solve word problems involving multiplicative comparison, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison. Generalize place value understanding for multi-digit whole numbers. 4.NBT.1 Recognize that in a multi-digit whole number, a digit in 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. 4.NBT.2 Read and write multi-digit whole numbers using base-ten numerals, number names, and expanded form. Compare two multi-digit numbers based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons. Gain familiarity with factors and multiples. 4.OA.4 Find all factor pairs for a whole number in the range 1–100. Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range 1–100 is a multiple of a given one-digit number. Determine whether a given whole number in the range 1–100 is prime or composite. 4.OA.5 Generate a number or shape pattern that follows a given rule. Identify apparent features of the pattern that were not explicit in the rule itself. For example, given the rule “Add 3” and the starting number 1, generate terms in the resulting sequence and observe that the terms appear to alternate between odd and even numbers. Explain informally why the numbers will continue to alternate in this way. Use place value understanding and properties of operations to perform multi-digit arithmetic. 4.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 calculation by using equations, rectangular arrays, and/or area models. 4.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. Use the four operations with whole numbers to solve problems. 4.OA.3 Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.




1. Make sense of problems and persevere in solving them. Students will solve word problems using all operations. Students will represent word problems using various modalities. Students will demonstrate their perseverance by selecting an effective modality to solve problems. *2. Reason abstractly and quantitatively. Students determine what operations to use to solve word problems and think about the reasonableness of their solutions. Students use reasonable estimates based upon the value of the numbers. 3. Construct viable arguments and critique the reasoning of others. Students demonstrate their ability to construct viable arguments when they talk and write about the steps they take to solve problems. Students restate and respond to each other about their mathematical thought processes. *4. Model with mathematics. Students write equations when recording all steps used to solve multiplication and division problems. They use all modalities (picture, manipulatives, written symbols, real world situations, oral/written language) when representing a situation. 5. Use appropriate tools strategically. Students choose and use the appropriate tools to solve problems. Tools could include: color tiles, base ten blocks, counters, math lines, geoboards, unifix cubes, websites, formulas, numerical equations, numerical expressions, diagrams, graph paper, shape diagrams, songs, and white boards. 6. Attend to precision. Students demonstrate precision in calculation by using their inverse operations to check their work and by using precise vocabulary in their oral and written explanations. *7. Look for and make use of structure. Students recognize and identify patterns existing between addition and multiplication along with division and subtraction. Students use this knowledge when applying strategies to evaluate real-world problems such as area and perimeter. They also use problem solving structures to aid in solving real world problems. 8. Look for and express regularity in repeated reasoning. Students identify and explain patterns within multiplication and division arrays (prime numbers, square numbers, and composite numbers, factors, multiples), determine and communicate whether identified patterns always work, determine and communicate when to apply specific patterns, describe the relationship between multiplication, addition, subtraction, and division, and justify that solutions make sense (i.e., using mental computation and/or estimation).



English:

Multiplicative comparison

Standard form

Written form

Expanded form

Area model

Factor

Multiple

Prime

Composite

Divisor

Spanish:

Comparación multiplicativa

Forma estándar

Forma verbal

Forma desarrollada

Modelo de área

Factor

Múltiple

Número primo

Compuesto

Divisor

English:

Array

Equation

Product

Quotient

Fourths

Halves

Thirds

Fractions

Place Value

Spanish:

Arreglo

Ecuación

Producto

Cociente

Cuartos

Mitades

Tercios

Fracciones

Valor posicional



Dividend

Remainder

Benchmark fractions

Composite number

Denominator

Equivalent fractions

Factor pairs

Greatest common factor

Improper fraction

Least common multiple

Mixed number

Numerator

Prime number

Simplest form

Like fractions

Decimal

Hundredth

Tenth

Dividendo

Residuo

Fracciones de referencia

Número compuesto

Denominador

Fracciones equivalentes

Pares de factores

Máximo común factor

Fracción impropia

Mínimo común múltiplo

Número mixto

Numerador

Número Primo

Forma reducida

Fracciones semejantes

Decimal

Centenas

Decenas




Unit Four: Fractions and Decimals Approximate Time Frame: 7-8 Weeks

Connections to Previous Learning: In third grade, students developed an understanding of fractions as numbers, including whole numbers written as fractions, by representing them on a number line and with visual fraction models. They began building the foundation of equivalence and comparison by exploring through the use of manipulatives. They apply their knowledge of halves and fourths by measuring with rulers and displaying the data on a line plot. Fourth graders extend their understanding of fraction equivalence and comparison by using symbols and visual models, and they apply previous understanding of operations on whole numbers to operations with fractions. Focus of the Unit: This unit focuses on building the fundamental understanding of fraction concepts and applications. Students in fourth grade must use concrete representations to develop a strong understanding of the concepts (fraction tiles, pattern blocks, visual fraction models, number lines, etc.) so they can later apply to more complex work with fractions. In this unit, the expectations are limited to fractions with denominators, 2, 3, 4, 5, 6, 8, 10, 12, and 100. Students will use visual fraction models to explore the idea of creating equivalent fractions by multiplying both the numerator and denominator by the same number or by dividing a shaded region into various parts. They see that the numerical process corresponds physically to partitioning each unit fraction piece into smaller equal pieces. Students will explore a variety of strategies for comparing fractions. They will use their knowledge of equivalent fractions to compare fractions with unlike denominators, by finding equivalent fractions with common numerators or common denominators. They will also compare fractions using benchmark fractions, such as ½ and 1.

In fourth grade, the focus is on building on the work of equivalent fractions by having students change fractions with a denominator of 10 into equivalent fractions with a denominator of 100 and then adding these fractions(e.g., write 7/10 as 70/100 and add 7/10 + 3/100 = 73/100). Students will learn to use decimal notation for expressing fractions and locating decimals on a number line. Students will compare two decimal fractions to hundredths by reasoning about their size, referring to the same whole. Students will use symbols to record the results of these comparisons and be able to justify their conclusions. Students explore the idea that all fractions are a sum of unit fractions. They use this knowledge to compose and decompose fractions with the same denominator, including converting between mixed numbers and improper fractions. Using their understanding of composing and decomposing fractions, they will add and subtract fractions and mixed numbers with the same denominator. Students will also apply their knowledge of unit fractions to multiplying a fraction times a whole number, by using number lines, visual fraction models, and equations. In this unit, students will connect their understanding of fractions to solve word problems using the customary measurement system. They will convert from larger units to smaller units and record measurement equivalents in a two-column table. They will also display a data set of measurements in a line plot, and use the information to solve problems involving addition and subtraction of fractions. Connections to Subsequent Learning:



In fifth grade, students will relate their understanding of equivalent fractions and addition/subtraction of fractions to add and subtract fractions and mixed numbers with like and unlike denominators. Fifth graders will extend previous understandings of multiplication and division to multiply (fraction by fraction) and divide fractions (unit fraction by whole number and whole number by unit fraction). In fifth grade, students will read, write and compare decimals to thousandths using base-ten numerals, number names, and expanded form. Place value understandings will be used to round decimals to any place. Students will add, subtract, multiply and divide decimals to hundredths using various strategies.

From the 3-5 Number and Operations – Fractions progression document, pp. 5-8 Grade 4 students learn a fundamental property of equivalent fractions: multiplying the numerator and denominator of a fraction by the same non-zero whole number results in a fraction that represents the same number as the original fraction. This property forms the basis for much of their other work in Grade 4, including the comparison, addition, and subtraction of fractions and the introduction of finite decimals.

Equivalent fractions-students can use area models and number line diagrams to reason about equivalence.4.NF.1 They see that the numerical process of

multiplying the numerator and denominator of a fraction by the same number, n, corresponds physically to partitioning each unit fraction piece into n smaller equal pieces. The whole is then partitioned into n times as many pieces, and there are n times as many smaller unit fraction pieces as in the original fraction. This argument, once understood for a range of examples, can be seen as a general argument, working directly from the Grade 3 understanding of a fraction as a point on the number line. The fundamental property can be presented in terms of division, as in, e.g.


Students will apply concepts and procedures to determine fraction equivalence and compare fractions. EX: If Sheila ate 4/8 of a candy bar and Robert ate ½ of an equal sized candy bar. Did one person eat more than the other? Justify your reasoning.

They will use their understanding of unit fractions to add and subtract fractions and mixed numbers with like denominators in real world and mathematical problems. EX: A cookie recipe calls for ¾ cup of sugar, ¼ cup of brown sugar, and 2/4 cup of chocolate chips. How many cups of dry ingredients are need to make the cookies?

Solving real-world problems involving decimal conversions in metric measurement (e.g. the length of a board being 0.74 meters).

Use the four operations to solve real world problems involving money. (e.g. finding the cost of a group of items priced at $0.65, $4.89, $0.59 and $12.08).


Fractions can be represented visually and in written form.

Comparisons are valid only when the two fractions refer to the same whole.

Fractions and Mixed Numbers are composed of unit fractions and can be decomposed as a sum of unit fractions.

Improper Fractions and Mixed Numbers represent the same value.

Addition and subtraction of fractions involves joining and separating parts referring to the same whole.

A product of a fraction times a whole number can be written as a multiple of a unit fraction.



When converting measurements within one system, the size, length, mass, volume of the object remains the same.

Fractions with denominators of 10 can be expressed as an equivalent fraction with a denominator of 100.

Fractions with denominators of 10 and 100 may be expressed using decimal notation.

When comparing two decimals to hundredths, the comparisons are valid only if they refer to the same whole.


What is the difference between a prime and composite number?

How do I determine the factors of a number?

How can different fractions name the same amount?

How can I use operations to model real-world fractions?

How are fractions and decimals related?

How are fractions used in problem-solving situations?

How are fractions composed, decomposed, compared and represented?

Why is it important to identify, label, and compare fractions as representations of equal parts of a whole or of a set?

How can multiplying a whole number by a fraction be displayed as repeated addition (as a multiple of a unit fraction)?

How can visual models be used to help with understanding decimals?

How can visual models be used to determine and compare equivalent fractions and decimals?

How would you compare and order decimals through hundredths?


Understand fractions as numbers.

Represent fractions on a number line.

Recognize, generate, explain equivalence and compare fractions using visual fraction models.

Fluently add and subtract whole numbers.

Use a ruler to measure to the nearest ¼, ½, and whole of an inch.

Display data on a line plot.


Add fractions with same denominators.

Add and subtract fractions and mixed numbers with unlike denominators using visual fraction models.

Solve word problems requiring the use of equivalent fractions.


A fraction a/b is equivalent to a fraction (n x a)/(n x b). (4.NF.1)

Fractions with different denominators can be compared by using visual fraction models, benchmark fractions, finding common denominators, and finding common numerators. (4.NF.2)

Addition and subtraction of fractions as joining and separating parts referring to the same whole using manipulatives, pictures, symbols, language, and real-life examples. (4.NF.3)

A fraction with a denominator of 10 can also be expressed as an equivalent fraction with a denominator of 100. (4.NF.5)


Recognize and generate equivalent fractions .(4.NF.1)

Compare 2 fractions with different denominators and different numerators by representing the fractions with symbols, visual models and words and by comparing to a benchmark fraction using symbols, visual models and words. (4.NF.2)

Identify if comparisons are valid or invalid and explain why. (4.NF.2)

Represent unit fractions as a fraction with a numerator of 1 with manipulatives, pictures, symbols, language, and real-life examples. (4.NF.3)



A number can be represented as both a fraction and a decimal. (4.NF.6)

Decimal comparisons are only valid when the two decimals refer to the same whole. (4.NF.7)

Decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation. Justify decompositions, e.g., by using a visual fraction model. (4.NF.3)

Add and subtract mixed numbers with like denominators and model the decomposition of the mixed numbers into unit fractions using manipulatives, pictures, symbols, language, and real-life examples. (4.NF.3)

Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators using visual models and/or equations. (4.NF.3)

Represent multiplication of a fraction by a whole number as repeated addition using area or linear models. (4.NF.4)

Represent that a fraction, such as ¾, is made up of 3 unit fractions of ¼ using a multiplication equation, such as 3 x ¼ = ¾(4.NF.4).

Multiply a fraction by a whole number by decomposing the fraction into a multiple of a unit fraction such as ¾ x 2 = 3 x 2 x ¼ which equals 6/4, using manipulatives, pictures, symbols, language, and real-life examples. (4.NF.4)

Represent improper fractions with visual models to demonstrate their relationship to the two closest whole numbers. (4.NF.4)

Solve word problems involving multiplication of any fraction by a whole number by using visual models and/or equations. (4.NF.4)

Identify relative sizes of measurement units within one system (customary) of units including lb., oz.; hr., min, sec (4.MD.1)

Represent the larger unit of measure in terms of the smaller unit of measure within the same measurement system (customary), including lb., oz.; hr., min, sec. using manipulatives, pictures, language and/or equations. (4.MD.1)

Record customary measurement equivalents in a two column table. (4.MD.1)

Use pictures and equations to represent and solve addition, subtraction, multiplication and division word problems involving distance, elapsed time, liquid volumes, and masses of objects (customary system). (4.MD.2)

Measure objects to the nearest ½, ¼ or 1/8 of a unit. (4.MD.4)



Make a line plot to display a set of measurements to the nearest ½, ¼ or 1/8 of a unit. (4.MD.4)

Solve problems involving addition and subtraction of fractions by using information presented in line plots. (i.e. range) (4.MD.4)

Represent a fraction with denominator 10 as an equivalent fraction with a denominator of 100. (4.NF.5)

Add two fractions with denominators of 10 and 100 using manipulatives, pictures, written symbols, and language to explain the process. (4.NF.5)




Desired Outcomes:

Standards: Extend understanding of fraction equivalence and ordering. 4.NF.1 Explain why a fraction a/b is equivalent to a fraction (n x a)/(n x b) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions. 4.NF.2 Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as ½. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model. Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers. 4.NF.3 Understand a fraction a/b with a > 1 as a sum of fractions 1/b. a. Understand addition and subtraction of fractions as joining and separating parts referring to the same whole. b. Decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation. Justify decompositions, e.g., by using a visual fraction model. Examples: 3/8 = 1/8 + 1/8 + 1/8; 3/8 = 1/8 + 2/8; 2 1/8 = 1 + 1 +1/8 = 8/8 + 8/8 + 1/8. c. Add and subtract mixed numbers with like denominators, e.g., by replacing each mixed number with an equivalent fraction, and/or by using properties of operations and the relationship between addition and subtraction. d. Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators, e.g., by using visual fraction models and equations to represent the problem. 4.NF.4 Apply and extend previous understandings of multiplication to multiply a fraction by a whole number. a. Understand a fraction a/b as a multiple of 1/b. For example, use a visual fraction model to represent 5/4 and the product 5 x (1/4), recording the conclusion by the equation 5/4 = 5 x (1/4). b. Understand a multiple of a/b as a multiple of 1/b, and use this understanding to multiply a fraction by a whole number. For example, use a visual fraction



model to express 3 x (2/5) as 6 x (1/5), recognizing this product as 6/5. (In general, n x (a/b) = (n x a)/b.) c. Solve word problems involving multiplication of a fraction by a whole number, e.g., by using visual fraction models and equations to represent the problem. For example, if each person at a party will eat 3/8 of a pound of roast beef, and there will be 5 people at the party, how many pounds of roast beef will be needed? Between what two whole numbers does your answer lie? 4.NF.5 Express a fraction with denominator 10 as an equivalent fraction with denominator 100, and use this technique to add two fractions with respective denominators 10 and 100. For example, express 3/10 as 30/100, and add 3/10 + 4/100 = 34/100. 4.NF.6 Use decimal notation for fractions with denominators 10 or 100. For example, rewrite 0.62 as 62/100; describe a length of 0.62 meters; locate 0.62 on a number line diagram. 4.NF.7 Compare two decimals to hundredths by reasoning about their size. Recognize that comparisons are valid only when the two decimals refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual model. Solve problems involving measurement and conversion of measurement from a larger unit to a smaller unit. 4.MD.1 Know relative sizes of measurement units within one system of units including km, m, cm; kg, g; lb., oz.; l. ml; hr, min, sec. Within a single system of measurement, express measurements in a larger unit in terms of a smaller unit. Record measurement equivalents in a two-column table. For example, know that if 1 ft is 12 times as long as 1 in. Express the length of a 4 ft snake as 48 in. Generate conversion table for feet and inches listing the number pairs (1, 12), (2,24), (3, 36), … 4.MD.2 Use the four operations to solve word problems involving distances, intervals of time, liquid volumes, masses of objects, and money, including problems involving simple fractions or decimals, and problems that require expressing measurement quantities using diagrams such as number line diagrams that feature a measurement scale. Represent and interpret data. 4.MD.4 Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). Solve problems involving addition and subtraction of fractions by using information presented in line plots. For example, from a line plot find and interpret the difference in length between the longest and shortest specimens in an insect collection.


1. Make sense of problems and persevere in solving them. Students know that doing mathematics involves solving problems and discussing how they solved them. Students explain to themselves the meaning of a problem and look for ways to solve it. Fourth graders may use concrete objects or pictures to help them conceptualize and solve problems. *2. Reason abstractly and quantitatively. Fourth graders should recognize that a number represents a specific quantity. They connect the quantity to written symbols and create a logical representation of the problem at hand, considering both the appropriate unit s involved and the meaning of quantities. They extend this understanding from whole numbers to their work with fractions. *3. Construct viable arguments and critique the reasoning of others. In fourth grade, students may construct arguments using concrete referents, such as objects, pictures, and drawings. They explain their thinking and make connections between models and equations. 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. 5. Use appropriate tools strategically. Fourth 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 fractions tiles, visual fraction models, equations, or a number line to add and subtract fractions. 6. Attend to precision. As fourth graders develop their mathematical communication skills, they try to use clear and precise language in their discussions with others and in their own reasoning. They state the meaning of the symbols they choose. For instance, they use appropriate labels when creating a line plot. 7. Look for and make use of structure. *8. Look for and express regularity in repeated reasoning. Students use models to examine patterns and generate their own algorithms. For example, students use visual fraction models to write equivalent fractions.



English:

Benchmark fractions

Common denominators

Improper fraction

Mixed numbers

Visual fraction model

Range

Decimal

Tenths

Hundredths

Decimal grids

Spanish:

Fracciones de referencia

Denominadores comunes

Fracción impropia

Número mixto

Modelo visual de la fracción

Rango

Decimal

Décimas

Centésimas

Cuadricula de decimales

English:

Unit fractions

Decompose

Compose

Equivalent

Numerator

Denominator

Symbols

Number line

Line plot

Distances (inches and feet)

Intervals (of time)

Elapsed time (seconds, minutes, hours, days, etc.)

Liquid volume (fluid ounce, cup, pint, quart, gallon)

Weight (ounce, pound, ton)

Quarters

Halves

Fourths

Halves

Thirds

Spanish:

Fraccione unitaria

Descomponer

Componer

Equivalente

Numerador

Denominador

Símbolo

Línea numérica

Grafica lineal

Distancias (pulgadas y pies)

Intervalos (del tiempo)

Tiempo transcurrido (segundos, minutos, horas, días, etc.)

Capacidad (onzas liquidas, tasa, pinta, cuarto, galón)

Peso (onza, libra, tonelada)

Cuartos

Mitades

Cuartos

Tercios



Fraction

Place value

Fracción

Valor posicional




Unit Five: Computation and Applications Approximate Time Frame: 6-7 Weeks

Connections to Previous Learning: In third grade, students use their problem-solving skills to solve skills to solve two-step problems using all four operations. Third graders work with the measurement systems by estimating and measuring length, liquid volume, and masses of objects. They also explore the concepts of perimeter of shapes and area of rectangles. Focus of the Unit: They will use all four operations to solve real-world problems using multi-digit whole numbers and represent problems as equations with a variable in place of the unknown quantity. Students will work with the metric system of measurement to convert from a larger unit of measure to a smaller unit of measure and record these measurement equivalents in a two-column chart. Students will solve problems involving distances, liquid volumes, and masses of objects. They will apply the formulas for perimeter and area of rectangles when solving real world and mathematical problems. Connections to Subsequent Learning: After converting from larger units of measurement to smaller units of measurement in fourth grade, students in fifth grade will convert measurements from smaller units to larger units.



From the K-5, Number and Operations in Base Ten progression document, p.13

In fourth grade, students compute products of one-digit numbers and multi-digit numbers (up to four digits) and products of two two-digit numbers. They divide multi-digit numbers (up to four digits) by one-digit numbers. As with addition and subtraction, students should use methods they understand and can explain. Visual representations such as area and array diagrams that students draw and connect to equations and other written numerical work are useful for this purpose. By reasoning repeatedly about the connection between math drawings and written numerical work, students can come to see multiplication and division algorithms as abbreviations or summaries of their reasoning about quantities. Students can invent and use fast special strategies while also working towards understanding

general methods and the standard algorithm. One component of understanding general methods

for multiplication is understanding how to compute products of one-digit numbers and multiples of 10, 100, and 1000. This extends work in Grade 3 on products of one-digit numbers and multiples of 10. We can calculate 6 x 700 by calculating 6 x 7 and then shifting the result to the left two places (by placing two zeros at the end to show that these are hundreds) because 6 groups of 7 hundred is 6 7 hundreds, which is 42 hundreds, or 4,200. Students can use this place value reasoning, which can also be supported with diagrams of arrays or areas, as they develop and practice using the patterns in relationships among products such as 6 x 7, 6 x 70, 6 x 700, and 6 x 7000. Products of 5 and even numbers, such as 5 x 4, 5 x 40, 5 x 400, 5 x 4000 and 4x 5, 4 x 50, 4 x 500, 4 x 5000 might be discussed and practiced separately afterwards because they may seem at first to violate the patterns by having an “extra” 0 that comes from the one-digit product. Another part of understanding general base-ten methods for multi-digit multiplication is understanding the role played by the distributive property. This allows numbers to be decomposed into base-ten units, products of the units to be computed, and then combined. By decomposing the factors into like base-ten units and applying the distributive property, multiplication computations are reduced to single-digit multiplications and products of numbers with multiples of 10, of 100, and of 1000. Students can connect diagrams of areas or arrays to numerical work to develop understanding of general base-ten multiplication methods.


Students will apply concepts and procedures for adding, subtracting, multiplying and dividing multi-digit whole numbers to solve real-world and mathematical problems. Ex. Based on census data from 2010, the population of Springfield, Illinois was 116,250. The population of the second largest city in Illinois, Aurora, was 197,899. How many more people were living in Aurora than Springfield in 2010?

Students will apply the area and perimeter formulas for rectangles when solving real-world problems. Ex. Shana wants to plant a vegetable garden in her



backyard. She has a rectangular area set aside that is 36 square feet. If the length of her garden is 9 feet, what would be the width of her garden?


The four operations are interconnected.

Converting from larger to smaller units of measurement in the metric system is done by multiplying by powers of ten.

Perimeter is a real life application of addition and subtraction.

Area is a real life application of multiplication and division.


How are the four basic operations related to one another?

Why does the size, length, mass, volume of an object remain the same when converted to another unit of measurement?

How are the units of measure within the metric system related?

How do you find the area and perimeter of geometric figures and how can using the formulas for perimeter and area help you solve real- world problems?

Why do we convert measurements?

How can conversions of measurements help me solve real-world problems?



Solve two-step word problems involving +, -, x and ÷.

Measure length, liquid volume, and mass using metric units.


Find the perimeter of geometric figures by using tiling and addition equations.

Find the area of rectangles by using tiling and multiplication equations.

Relate area to addition & multiplication (Arrays).


Convert measurements in the metric system and solve word problems that require the answer to be converted within the metric system.


How the four operations can be used to solve real-world and mathematical problems. (4.OA.3)

The relative size of measurement units within the metric system. (4.MD.1)

The formula for perimeter of geometric figures. (4.MD.3)

The formula for area of rectangles. (4.MD.3)


Solve multistep word problems posed with whole numbers and having whole- number answers using the four operations. (4.OA.3)

Represent multi-step word problems using equations with a variable standing for the unknown quantity. (4.OA.3)

Assess the reasonableness of answers using mental computation and estimation strategies, including rounding. (4.OA.3)

Represent the larger unit of measure in terms of the smaller unit of measure within the metric system, using manipulatives, pictures, language and/or equations. (4.MD.1)



Record measurement equivalents in a two-column table. (4.MD.1)

Use pictures and equations to represent and solve addition, subtraction, multiplication and division word problems involving measurement, distance, liquid volumes and masses of objects.(4.MD.2)

Solve problems involving area and perimeter of rectangles using visuals and equations that represent the formulas for area and perimeter of rectangles. (4.MD.3)




Desired Outcomes:

Standards: Use the four operations with whole numbers to solve problems. 4.OA.3 Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding. Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit. 4.MD.1 Know relative sizes of measurement units within one system (metric) of units including km, m, cm; kg, g. Within a single system of measurement, express measurements in a larger unit in terms of a smaller unit. Record measurement equivalents in a two-column table. For example, know that 1ft is 12 times as long as 1in. Express the length of a 4ft snake as 48 in. Generate a conversion table for feet and inches listing the number pairs (1, 12), (2, 24), (3, 36), … 4.MD.2 Use the four operations to solve word problems involving distances, intervals of time, liquid volumes, masses of objects, and money, including problems involving simple fractions or decimals, and problems that require expressing measurement quantities using diagrams such as number line diagrams that feature a measurement scale. 4.MD.3 Apply the area and perimeter formulas for rectangles in real world and mathematical problems. For example, find the width of a rectangular room given the area of the flooring and the length, by viewing the area formula as a multiplication equation with an unknown factor. 4.OA.5 Generate a number or shape pattern that follows a given rule. Identify apparent features of the pattern that were not explicit in the rule itself. For example, given the rule “Add 3” and the starting number 1, generate terms in the resulting sequence and observe that the terms appear to alternate between odd and even numbers. Explain informally why the numbers will continue to alternate in this way.




*1. Make sense of problems and persevere in solving them. Students will solve word problems using all operations. Students will represent word problems using various modalities. Students will demonstrate their perseverance by selecting an effective modality to solve problems. 2. Reason abstractly and quantitatively. Students determine what operations to use to solve word problems and think about the reasonableness of their solutions. Students use reasonable estimates based upon the value of the numbers. 3. Construct viable arguments and critique the reasoning of others. Students demonstrate their ability to construct viable arguments when they talk and write about the steps they take to solve problems. Students restate and respond to each other about their mathematical thought processes. 4. Model with mathematics. Students write equations when recording all steps used to solve multiplication and division problems. They use all modalities (picture, manipulatives, written symbols, real world situations, oral/written language) when representing a situation. *5. Use appropriate tools strategically. Students choose and use the appropriate tools to solve problems. Tools could include: color tiles, base ten blocks, counters, math lines, geoboards, unifix cubes, websites, formulas, numerical equations, numerical expressions, diagrams, graph paper, shape diagrams, songs, and whiteboards. *6. Attend to precision. Students demonstrate precision in calculation by using their inverse operations to check their work and by using precise vocabulary in their oral and written explanations. 7. Look for and make use of structure. Students recognize and identify patterns existing between addition and multiplication along with division and subtraction. Students use this knowledge when applying strategies to evaluate real-world problems such as area and perimeter. They also use problem solving structures to aid in solving real world problems. *8. Look for and express regularity in repeated reasoning. Students identify and explain patterns within multiplication and division arrays (prime numbers, square numbers, and composite numbers, factors, multiples), determine and communicate whether identified patterns always work, determine and communicate when to apply specific patterns, describe the relationship between multiplication, addition, subtra ction, and division, and justify that solutions make sense (i.e., using mental computation and/or estimation).



English:

Rule

Variable

Formula

Capacity

Convert

Cup

Customary System

Fluid ounce

Foot

Gallon

Line plot

Spanish:

Regla

Variable

Formula

Capacidad

Convertir

Tasa

Sistema inglesas

Onza líquidas

Pie

Galón

Grafica lineal

English:

Metric units of measurement

Distance

Liquid volume

Mass

Length

Spanish:

Unidades métricas de medir

Distancia

Capacidad

Masa

Longitud



Ounce

Pint

Pound

Mile

Quart

Second

Ton

Weight

Yard

Centimeter

Gram

Kilogram

Kilometer

Liter

Mass

Meter

Metric system

Milliliter

Millimeter

Perimeter

Area

Square unit

Unit square

Onza

Pinta

Libra

Milla

Cuarto

Segundo

Tonelada

Peso

Yarda

Centímetro

Gramo

Kilogramo

Kilometro

Litro

Masa

Metro

Sistema métrico

Mililitro

Milímetro

Perímetro

Área

Cuadrado unitario

Unidad cuadrada




Unit Six: Geometry Approximate Time Frame: 2-3 Weeks

Connections to Previous Learning: Students in third grade analyze, compare, and classify two-dimensional shapes based on their properties. They understand that shapes in different categories may share attributes (e.g., having four sides), and that shared attributes can define a larger category (e.g., quadrilaterals). In first through third grade, students have had ample opportunity to investigate, describe, and reason about decomposing and composing polygons to make other polygons. In fourth grade, students will expand their knowledge of attributes of polygons to include parallel and perpendicular lines and angle measurements. They also explore the concepts of perimeter of shapes and area of rectangles. Focus of the Unit: In this unit, students expand their vocabulary of geometric objects and develop a connection of many of the concepts they have been developing, including points, lines, line segments, rays, angles, parallel and perpendicular lines to classify two-dimensional figures. Students begin to use protractors to measure and draw acute, right and obtuse angles. Once they understand the types of angles, they can apply their knowledge to the classification of triangles. In addition, they learn how to cross-classify figures, for example, naming a shape as a right isosceles triangle. Students also explore line segments, rays, angles, parallelism, etc. in varied contexts to connect meaning to what normally are isolated concepts. For example, an angle is measured with reference to a circle with its center at the common endpoint of the rays. When working with angles, students need to be able to solve addition and subtraction problems to find unknown angles on a diagram in real world and mathematical problems recognizing the whole angle is a sum of smaller angles composing it. Students will apply the formulas for perimeter and area of rectangles when solving real world and mathematical problems. They also connect lines to lines of symmetry in two-dimensional figures. Connections to Subsequent Learning: Fifth grade students continue to build upon their prior knowledge of classification of two-dimensional shapes to be able to classify polygons into a hierarchy based on their properties. They use the concept of perpendicular lines to define a coordinate system.

From the K-6 progression document, Geometry pp. 10-17 Grade 2, Students learn to name and describe the defining attributes of categories of two- dimensional shapes, including circles, triangles, squares, rectangles, rhombuses, trapezoids, and the general category of quadrilateral. They describe pentagons, hexagons, septagons, octagons, and other polygons by the number of

sides, for example, describing a septagon as either a “seven-gon” or simply “seven-sided shape” (MP2).2.G.1

Because they have developed both verbal descriptions of these categories and their defining attributes and a rich store of associated mental images, they are able to draw shapes with specified attributes, such as a shape with five sides or a shape with six angles.2.G.1 They can represent these shapes’ attributes accurately (within the constraints of fine motor skills). They use length to identify the properties of shapes (e.g., a specific figure is a rhombus because all four of its sides have equal length). They recognize right angles, and can explain the distinction between a rectangle and a parallelogram without right angles and with sides of different lengths (sometimes called a “rhomboid”).



Students learn to combine their composition and decomposition competencies to build and operate on composite units (units of units), intentionally substituting arrangements or composites of smaller shapes or substituting several larger shapes for many smaller shapes, using geometric knowledge and spatial reasoning to develop foundations for area, fraction, and proportion. For example, they build the same shape from different parts, e.g., making with pattern blocks, a regular hexagon from two trapezoids, three rhombuses, or six equilateral triangles. They recognize that the hexagonal faces of these constructions have equal area that each trapezoid has half of

that area, and each rhombus has a third of that area.2.G.3

This example emphasizes the fraction concepts that are developed; students can build and recognize more difficult composite shapes and solve puzzles with numerous pieces. For example, a tangram is a special set of 7 shapes which compose an isosceles right triangle. The tangram pieces can be used to make many different configurations and tangram puzzles are often posed by showing pictures of these configurations as silhouettes or outlines. These pictures often are made more difficult by orienting the shapes so that the sides of right angles are not parallel to the edges of the page on which they are displayed. Such pictures often do not show a grid that shows the composing shapes and are generally not preceded by analysis of the composing shapes. Students also explore decompostitions of shapes into regions that are congruent or have equal area. 2.G.3 For example, two squares can be portioned into fourths in different ways. Any of these fourths represents an equal share of the shapes (e.g., “the same amount of cake”) even though they have different shapes. Foundational activities, such as forming arrays by tiling a rectangle with identical squares (as in building a floor or wall from blocks) should have developed students’ basic spatial structuring competencies before second grade—if not, teachers should ensure that their students learn these skills. Spatial structuring can be further developed with several activities with grids. Games such as “battleship” can be useful in this regard. Another useful type of instructional activity is copying and creating designs on grids. Students can copy designs drawn on grid paper by placing manipulative squares and right triangles onto other copies of the grid. They can also create their own designs, draw their creations on grid paper, and exchange them, copying each other’s designs. Another, more complex, activity designing tessellations by iterating a “core square.” Students’ design a unit composed of smaller units: a core square composed of a 2 by 2 array of squares filled



with square or right triangular regions. They then create the tessellation (“quilt”) by iterating that core in the plane. This builds spatial structuring because students are iterating “units of units” and reflecting on the resulting structures. Computer software can aid in this iteration. These various types of composition and decomposition experiences simultaneously develop students’ visualization skills, including recognizing, applying, and anticipating (MP1) the effects of applying rigid motions (slides, flips, and turns) to two-dimensional shapes. Grade 3, Students analyze, compare, and classify two-dimensional shapes by their properties (see the footnote on p. 3).3.G.1 They explicitly relate and combine these classifications. Because they have built a firm foundation of several shape categories, these categories can be the raw material for thinking about the relationships between classes. For example, students can form larger, superordinate, categories, such as the class of all shapes with four sides, or quadrilaterals, and recognize that it

includes other categories, such as squares, rectangles, rhombuses, parallelograms, and trapezoids. They also recognize that there are quadrilaterals that are not in any of those subcategories. A description of these categories of quadrilaterals is illustrated in the margin. The Standards do not require that such representations be constructed by Grade 3 students, but they should be able to draw examples of quadrilaterals that are not in the subcategories. Similarly, students learn to draw shapes with prespecified attributes, without making a prior assumptions regarding their classification.MP1 For example, they could solve the problem of making a shape with two long sides of the same length and two short sides of the same length that is not a rectangle. Students investigate, describe, and reason about decomposing and composing polygons to make other polygons. Problems such as finding all the possible different compositions of a set of shapes involve geometric problem solving and notions of congruence and symmetry (MP7). They also involve the practices of making and testing conjectures (MP1), and convincing others that conjectures are correct (or not) (MP3). Such problems can be posed even forsets of simple shapes such as tetrominoes, four squares arranged to form a shape so that every square shares at least one side and sides coincide or share only a vertex. More advanced paper-folding (origami) tasks afford the same mathematical practices of seeing and using structure, conjecturing, and justifying conjectures. Paper folding can also illustrate many geometric concepts. For example, folding a piece of paper creates a line segment. Folding a square of paper twice, horizontal edge to horizontal edge, then vertical edge to vertical edge, creates a right angle, which can be



unfolded to show four right angles. Students can be challenged to find ways to fold paper into rectangles or squares and to explain why the shapes belong in those categories.

Students also develop more competence in the composition and decomposition of rectangular regions, that is, spatially structuring rectangular arrays. They learn to partition a rectangle into identical squares3.G.2 by anticipating the final structure and thus forming the array by drawing rows and columns (see the bottom right example on p. 11; some students may still need work building or drawing squares inside the rectangle first). They count by the number of columns or rows, or use multiplication to determine the number of squares in the array. They also learn to rotate these arrays physically and mentally to view them as composed of smaller arrays, allowing illustrations of properties of multiplication (e.g., the commutative property and the distributive property). Grade 4, Students describe, analyze, compare, and classify two-dimensional shapes by their properties (see the footnote on p. 3), including explicit use of angle sizes4.G.1 and the related geometric properties of perpendicularity and parallelism.4.G.2 They can identify these properties in two-dimensional figures. They can use side length to classify triangles as equilateral, equiangular, isosceles, or scalene; and can use angle size to classify them as acute, right, or obtuse. They then learn to cross-classify, for example, naming a shape as a right isosceles triangle. Thus, students develop explicit awareness of and vocabulary for many concepts they have been developing, including points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines. Such mathematical terms are useful in communicating geometric ideas, but more important is that constructing examples of these concepts, such as drawing angles and triangles that are acute, obtuse, and right, 4.G.1 help students form richer concept images connected to verbal definitions. That is, students have more complete and accurate mental images and associated vocabulary for geometric ideas (e.g., they understand that angles can be larger than 90 and their concept images for angles include many images of such obtuse angles). Similarly, students see points and lines as abstract objects: Lines are infinite in extent and points have location but no dimension. Grids are made of points and lines and do not end at the edge of the paper. Students also learn to apply these concepts in varied contexts (MP4). For example, they learn to represent angles that occur in various contexts as two rays, explicitly including the reference line, e.g., a horizontal or vertical line when considering slope or a “line of sight” in turn contexts. They understand the size of the angle as a rotation of a ray on the reference line to a line depicting

slope or as the “line of sight” in computer environments. Students might solve problems of

drawing shapes with turtle geometry. Analyzing the shapes in order to construct them (MP1) requires students to explicitly formulate their ideas about the shapes (MP4, MP6). For instance, what series of commands would produce a square? How many degrees would the turtle turn? What is the measure of the resulting angle? What would be the commands for an equilateral triangle? How many degrees would the turtle turn? What is the measure of the resulting angle? Such experiences help students connect what are often initially isolated ideas about the concept of angle. Students might explore line segments, lengths, perpendicularity, and parallelism on different types of grids, such as rectangular and triangular (isometric) grids (MP1, MP2).4.G.2, 4.G.3 Can you find a non-



rectangular parallelogram on a rectangular grid? Can you find a rectangle on a triangular grid? Given a segment on a rectangular grid that is not parallel to a grid line, draw a parallel segment of the same length with a given endpoint. Given a half of a figure and a line of symmetry, can you accurately draw the other half to create a symmetric figure? Students also learn to reason about these concepts. For example, in “guess my rule” activities, they may be shown two sets of shapes and asked where a new shape belongs (MP1, MP2).4.G.2 In an interdisciplinary lesson (that includes science and engineering ideas as well as items from mathematics), students might encounter another property that all triangles have: rigidity. If four fingers (both thumbs and index fingers) form a shape (keeping the fingers all straight), the shape of that quadrilateral can be easily changed by changing the angles. However, using three fingers (e.g., a thumb on one hand and the index and third finger of the other hand), students can see that the shape is fixed by the side lengths. Triangle rigidity explains why this shape is found so frequently in bridge, high-wire towers, amusement park rides, and other constructions where stability is sought. Grade 5, By the end of Grade 5, competencies in shape composition and decomposition, and especially the special case of spatial structuring of rectangular arrays (recall p. 11), should be highly developed (MP7). Students need to develop these competencies because they form a foundation for understanding multiplication, area, volume, and the coordinate plane. To solve area problems, for example, the ability to decompose and compose shapes plays multiple roles. First, students understand that the area of a shape (in square units) is the number of unit squares it takes to cover the shape without gaps or overlaps. They also use decomposition in other ways. For example, to calculate the area of an “L-shaped” region, students might decompose the region into rectangular regions, and then decompose each region into an array of unit squares, spatially structuring each array into rows or columns. Students extend their spatial structuring in two ways. They learn to spatially structure in three dimensions; for example, they can decompose a right rectangular prism built from cubes into layers, seeing each layer as an array of cubes. They use this understanding to find the volumes of right rectangular prisms with edges whose lengths are whole numbers as the number of unit cubes that pack the prisms (see the Geometric Measurement Progression). Second, students extend their knowledge of the coordinate plane, understanding the continuous nature of two-dimensional space and the role of fractions in specifying locations in that space. Thus, spatial structuring underlies coordinates for the plane as well, and students learn both to apply it and to distinguish the objects that are structured. For example, they learn to interpret the components of a rectangular grid structure as line segments or lines (rather than regions) and understand the precision of location that these lines require, rather than treating them as fuzzy boundaries or indicators of intervals. Students learn to reconstruct the levels of counting and quantification that they had already constructed in the domain of discrete objects to the coordination of (at first) two continuous linear measures. That is, they learn to apply their knowledge of number and length to the order and distance relationships of a coordinate grid and to coordinate this across two dimensions.5.G.1 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 y-axis and a distance 3 from the x-axis. In these two descriptions the 2 is first associated with the x-axis, then with the y-axis.



They connect ordered pairs of (whole number) coordinates to points on the grid, so that these coordinate pairs constitute numerical objects and ultimately can be operated upon as single mathematical entities. Students solve mathematical and real-world problems using coordinates. For example, they plan to draw a symmetric figure using computer software in which students’ input coordinates that are then connected by line segments.5.G.2 Students learn to analyze and relate categories of two-dimensional and three-dimensional shapes explicitly based on their properties.5.G.4 Based on analysis of properties; they classify two-dimensional figures in hierarchies. For example, they conclude that all rectangles are parallelograms, because they are all quadrilaterals with two pairs of opposite, parallel, equal-length sides (MP3). In this way, they relate certain categories of shapes as subclasses of other

categories.5.G.3 This leads to understanding propagation of properties; for example, students understand that squares possess all properties of rhombuses and of

rectangles. Therefore, if they then show that rhombuses’ diagonals are perpendicular bisectors of one another, they infer that squares’ diagonals are.


Students will apply concepts and procedures of classifying two-dimensional figures based on the presence of parallel or perpendicular lines. Example: How many acute, obtuse, or right angles are in this shape?

Students will apply concepts and procedures of decomposing angles into smaller parts. Example: A windshield wiper rotates 65 degrees and then pauses. It must rotate a total of 150 degrees to clear the windshield. What is the remaining amount of degrees the windshield wiper must rotate to complete its rotation to clear the windshield?

Students will apply the area and perimeter formulas for rectangles when solving real-world problems. Example: Shana wants to plant a vegetable garden in her backyard. She has a rectangular area set aside that is 36 square feet. If the length of her garden is 9 feet, what would be the width of her garden?


Shapes can be classified by properties of their lines and angles.

Angles are measured in the context of a central angle of a circle

Angles are composed of smaller angles.

Perimeter is a real life application of addition and subtraction.

Area is a real life application of multiplication and division.


How are different ideas about geometry connected?

What are the types of angles and the relationships?

What are the types of triangles and the relationships?

How are angles applied in the context of a circle?

How are parallel lines and perpendicular lines used in classifying two-dimensional shapes?

How are protractors used to measure and aid in drawing angles and triangles?

How can an addition or subtraction equation be used to solve a missing angle measure when the whole angle has been divided into two angles and



only one measurement is given?

How do you find the area and perimeter of geometric figures and how can using the formulas help you solve real-world problems?


Classify shapes based on the number and length of sides and number of angles.

Compose and decompose polygons to make other polygons.

Find the perimeter of geometric figures by using tiling and addition equations.

Find the area of rectangles by using tiling and multiplication equations.

Relate area to addition and multiplication (arrays).


Apply their knowledge of geometric attributes to sort and classify two- dimensional and three-dimensional shapes.

Measure angles greater than 180 degrees and relate them to the fractional part of a circle.


Points, lines, line segments, rays, right angles, acute angles, obtuse angles, perpendicular lines, parallel lines can be identified within 2-dimensional figures. (4.G.1)

Angles are formed wherever two rays share a common endpoint. (4.MD.5)

An angle measure is a fraction of circular arc between the points where the two rays intersect the circle. (4.MD.5)

Benchmark angles and transfer their understanding that a

3600 rotation about a point makes a complete circle to

recognize and sketch angles that measure approximately

900 and 180

0

. (4.MD.5) An angle that turns through 1/360 of a circle is called a “one-degree

angle,” and can be used to measure angles. (4.MD.5)

Angle measure is additive. (4.MD.7)

A line of symmetry for a two-dimensional figure is a line across the figure such that the figure can be folded along the line into matching parts. (4.G.3)

The formula for perimeter of geometric figures.

The formula for area of rectangles.


Draw points, lines, line segments, rays, right angles, acute angles, obtuse angles, perpendicular lines, and parallel lines. (4.G.1)

Classify 2-dimensional figures based on the presence or absence of parallel or perpendicular lines and right, acute or obtuse angles. (4.G.2)

Identify and classify triangles. Label the categories of triangles (right triangles, scalene, isosceles). (4.G.2)

Recognize a line of symmetry for a two-dimensional figure as a fold-line, where the figure can be folded into matching parts. (4.G.3)

Determine whether a figure has one or more lines of symmetry and draw lines of symmetry. (4.G.3)

Identify the components of an angle and the number of degrees in a circle. (4.MD.5)

Use visuals and language to show the relationship between the components of an angle to a circle. (I.e. the center of the circle is the endpoint of the rays of the angle). (4.MD.5)

Measure angles in whole-number degrees using a protractor. (4.MD.6)

Sketch angles of a specified measure. (4.MD.6)

Use diagrams, manipulatives and equations to show that angle measure is additive. (4.MD.7)

Solve addition and subtraction problems to find unknown angles on a diagram of adjacent angles. (non-overlapping angles) (4.MD.7)

Solve problems involving area and perimeter of rectangles using visuals and equations that represent the formula for area and



perimeter of rectangles.




Desired Outcomes:

Standards: Draw and identify lines and angles, and classify shapes by properties of their lines and angles. 4.G.1 Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines. Identify these in two-dimensional figures. 4.G.2 Classify two-dimensional figures based on the presence or absence of parallel or perpendicular lines, or the presence or absence of angles of a specified size. Recognize right triangles as a category, and identify right triangles. 4.G.3 Recognize a line of symmetry for a two-dimensional figure as a line across the figure such that the figure can be folded along the line into matching parts. Identify line-symmetric figures and draw lines of symmetry. 4.MD.3 Apply the area and perimeter formulas for rectangles in real world and mathematical problems. For example, find the width of a rectangular room given the area of the flooring and the length, by viewing the area formulas as a multiplication equation with an unknown factor. 4.MD.5 Recognize angles as geometric shapes that are formed wherever two rays share a common endpoint, and understand concepts of angle measurement: a. An angle is measured with reference to a circle with its center at the common endpoint of the rays, by considering the fraction of the circular arc between the points where the two rays intersect the circle. An angle that turns through 1/360 of a circle is called a “one-degree angle,” and can be used to measure angles. b. An angle that turns through n one-degree angles is said to have an angle measure of n degrees. 4.MD.6 Measure angles in whole-number degrees using a protractor. Sketch angles of specified measure. 4.MD.7 Recognize angle measure as additive. When an angle is decomposed into non-overlapping parts, the angle measure of the whole is the sum of the angle measures of the parts. Solve addition and subtraction problems to find unknown angles on a diagram in real world and mathematical problems, e.g., by using an equation with a symbol for the unknown angle measure.


*1. Make sense of problems and persevere in solving them. Students use a drawing to conceptualize the given angle measures when solving for unknown angle measurements. 2. Reason abstractly and quantitatively. Before students begin measuring angles with protractors, they need to have some experiences with benchmark angles. They transfer their understanding that a 360º rotation about a point makes a complete circle to recognize and sketch angles that measure approximately 90º and 180º. *3. Construct viable arguments and critique the reasoning of others. Students will justify their reasoning when classifying objects. 4. Model with mathematics. Students may use transparencies with lines to arrange two lines in different ways to determine that the 2 lines might intersect



in one point or may never intersect. Further investigations may be initiated using geometry software. These types of explorations may lead to a discussion on angles. *5. Use appropriate tools strategically. Students may use straight edges, rulers, and geometry software to create and analyze geometric objects. 6. Attend to precision. Students will measure angles to the nearest degree. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning.



English:

Points

End points

Lines

Line segments

Rays

Angles (right, acute, obtuse)

Central

Adjacent angles

Perpendicular lines

Parallel lines

Protractor

Degrees

Symmetry

Right triangle

Scalene triangle

Isosceles triangle

Area

Perimeter

Formula

Acute Angle

Acute triangle

Angle

Degree

Endpoint

Intersecting

Spanish:

Puntos Extremos Rectas Segmento de recta Semirrecta Ángulos (recto, agudo,

obtuso) Central Ángulos adyacentes Rectas perpendiculares Rectas paralelas Transportador Grados Simetría Triángulo rectángulo Triángulo escaleno Triangulo isósceles Área Perímetro Formula Angulo agudo Triángulo acutángulo Angulo Grado

Extremo Secantes

English:

Plane (two-dimensional) figures

Quadrilaterals

Square

Rhombus

Rectangle

Circle

Triangle

Additive

Spanish:

Plano (bidimensional) figuras

Cuadriláteros Cuadrado Rombo Rectángulo Circulo Triángulo

Aditivo



Line

Line of symmetry

Obtuse angle

Obtuse triangle

One-degree angle

Parallel

Parallelogram

Perpendicular

Point

Ray

Rhombus

Right angle

Right triangle

Trapezoid

línea Eje de simetría Angulo obtuso Triángulo obtusángulo Angulo de un grado Paralelas Paralelogramo Perpendiculares Punto Semirrecta Rombo Ángulo recto Triangulo rectángulo

Trapecio


References:

Common Core State Standards for Mathematics. (2010). : Common Core State Standards Initiative.

Connell, S., & SanGiovanni, J. (2013). Putting the Practices Into Action: Implementing the Common Core

Standards for Mathematical Practice, K-8. : Heinemann.

Illinois Early Learning Developmental Standards (2013). Springfield: Illinois State Board of Education.

Illinois Mathematics Curriculum Model Units. (2013). Springfied: Illinois State Board of Education .

K-8 Publishers' Criteria for the Common Core State Standards for Mathematics. (2013, April 9). . Retrieved

November 1, 2014, from http://www.corestandards.org/assets/Math_Publishers_Criteria_K-

8_Summer%202012_FINAL.pdf

Marzano, R. J., & Simms, J. A. (2013). Vocabulary for the Common Core. : Marzano Research Laboratory.

My Math. (2013). Columbus: McGraw-Hill Education.

PARCC Model Content Frameworks Mathematics . (2012). Washington: Partnership for Assessment of

Readiness for College and Careers .

Tomlinson, C. A., & McTighe, J. (2006). Integrating Differentiated Instruction & Understanding by Design

Connecting Content and Kids. Alexandria, Va.: Association for Supervision and Curriculum Development.

http://www.corestandards.org/assets/Math_Publishers_Criteria_K-8_Summer%202012_FINAL.pdf

http://www.corestandards.org/assets/Math_Publishers_Criteria_K-8_Summer%202012_FINAL.pdf

y 308: a a fourth grade

Documents