Grade 4 Math

Course Description

In this course, students will deepen their understanding of mathematical concepts through hands-on activities and real-world applications. Emphasizing problem-solving and critical thinking, the course covers:

Place Value and Whole Number Operations: Students will build a strong understanding of place value by expressing numbers in various forms and using visual models to compare and compute with whole numbers. They will fluently recall addition and subtraction facts to solve problems with the standard algorithm, use keywords and relationships to choose operations, and apply formulas to solve real-world problems. Estimation through rounding and understanding number meaning will help students make sense of computations and their practical use.
Multiplication and Division Problems: Students will solve one-step multiplication and division word problems involving measurement conversions and time. They will also interpret line plots and measure length, mass, and liquid volume using different units. These skills help them apply math to real-world measurement situations.
Extend and Apply Multiplication: Students will multiply multiples of 10, 100, and 1,000 by two-digit factors using patterns, partial products, and area models. They will understand how factors represent rectangle side lengths and use area formulas to find and combine areas in real-world problems. Students will also choose estimation strategies and explain how different multiplication methods relate to each other.
Fractions and Decimals: Students understand factors, multiples, and divisibility to identify prime and composite numbers, generate number patterns, and write fractions in simplest form. They use fraction equivalence, benchmark fractions, and common numerators or denominators to compare and relate fractions and decimals, especially in money contexts involving tenths and hundredths. Additionally, students draw and identify geometric elements like points, lines, rays, angles, and use protractors to measure and solve problems involving angles and their properties in two-dimensional figures.
Operations with Fractions: Students will understand that unit fractions are the building blocks of all fractions and that fractions represent equal parts of a whole. They will learn to add, subtract, and multiply fractions and mixed numbers by combining numerators, finding common denominators, and using visual models to connect to real-life situations. Students will also apply the Distributive Property and recognize how fractions greater than one can be expressed as mixed numbers through regrouping and renaming.
Two-Dimensional Figures and Symmetry:Students will learn to identify and draw parallel and perpendicular lines using tools, and classify triangles by their angles (acute, right, obtuse) and sides (scalene, isosceles, equilateral). They will distinguish different quadrilaterals by their angle properties and understand how symmetry divides shapes into equal parts without measuring. Students will also describe patterns created by repeating shapes and visualize lines of symmetry.
Measurement, Data, and Time: Students will understand customary and metric units of length, weight, and liquid volume, using everyday objects and measurement tools like rulers and measuring cups. They will learn how units relate through multiplication and division, create and interpret line plots with fractions, and distinguish clock hands to calculate elapsed time with appropriate regrouping. Using visual models and strategies, students will solve real-world problems involving both mixed and standard measures of time and measurement.

Course Big Ideas

Numbers have different forms, and each digit’s value depends on its place, which is ten times the value to its right. This understanding helps compare numbers, use > and <, round correctly, and estimate reasonable answers.
Adding and subtracting multi-digit numbers requires aligning digits by place value and regrouping as needed. Identifying unknowns guides operation choice, and concepts like perimeter connect math to real-world problems.
Students fluently use multiplication and division to combine and separate equal groups, apply place value concepts—including rounding and multiplying/dividing by 10, 100, and 1,000—and use properties of operations to solve problems.
Students connect visual models to math processes, interpret keywords in word problems, and understand digit values to accurately solve multi-step problems involving whole numbers and operations.
Students use patterns and basic facts to multiply multiples of 10, 100, and 1,000 by two-digit factors, applying strategies like partial products and the area model to understand and connect different multiplication methods.
Understanding area as the product of side lengths helps students use formulas, break shapes into smaller parts, and combine areas to solve real-world problems, translating known values to find unknowns.
Estimation and analyzing the relationship between numbers guide students in choosing the best strategies and operations to solve problems involving multiplication and area.
Understanding factors, multiples, and divisibility helps students simplify and compare fractions by identifying equivalent fractions, common denominators, and using benchmarks like 0, ½, and 1 whole. They also learn to distinguish prime and composite numbers to apply divisibility rules effectively.
Students recognize patterns in fractions by applying multiplication to numerators and denominators, creating equivalent fractions, and choosing efficient strategies for comparing fractions with unlike denominators using common numerators or denominators in real-world contexts.
Connecting fractions, decimals, and money, students relate fractions like 1/10 and 1/100 to coins, understand decimal notation, and use this knowledge to compare decimals and solve money-related problems in everyday life.
Students explore geometric concepts by drawing and identifying points, lines, angles, and shapes, using tools like protractors to measure angles, understanding angle degrees as fractions of a circle, and applying benchmark angles to solve real-world angle problems.
Fractions are made of unit fractions representing equal parts, where adding and subtracting combine numerators using common denominators, including mixed numbers with regrouping, supported by visual models.
Multiplying fractions by whole numbers multiplies the numerator only, and the Distributive Property connects multiplication of whole and mixed numbers, with visual models linking these ideas to real-world problems.
Students identify and draw parallel and perpendicular lines, classify triangles and quadrilaterals by their sides and angles, and understand symmetry as dividing shapes into equal parts.
They visualize lines of symmetry, describe repeating shape patterns, and apply geometric reasoning to analyze and classify shapes in real-world contexts.
Students measure length, mass, and liquid volume using appropriate units, and represent data clearly using line plots to analyze and interpret measurement information.
Students solve one-step word problems involving measurement conversions within the same unit and apply addition and subtraction to calculate time intervals using both visual and digital time formats.

Course Essential Questions

How does understanding place value and number forms help us read, write, compare, round, and estimate large and multi-digit numbers accurately?
In what ways can we use operations, properties, and visual models—such as area models, bar models, and line plots—to solve real-world problems involving whole numbers, fractions, measurement, and time?
How do patterns, factors, multiples, and divisibility support us in simplifying, comparing, and operating with fractions, decimals, and whole numbers across various contexts?
How can geometric concepts like lines, angles, symmetry, and shapes be identified, classified, and measured to solve problems and recognize patterns in our surroundings?
How do estimation strategies and understanding relationships between numbers guide us in choosing efficient methods to solve multi-step problems involving operations, measurement, and geometry?

Course Competencies

Students will understand place value and number forms to read, write, compare, round, and estimate multi-digit numbers accurately, using visual models and applying these skills to solve problems involving addition, subtraction, multiplication, and division with whole numbers.
They will use factors, multiples, and patterns to simplify, compare, and operate with fractions and decimals, including generating equivalent fractions, relating fractions to decimals and money, and applying the Distributive Property for multiplication of whole and mixed numbers.
Students will identify, classify, and measure geometric figures by understanding lines, angles, symmetry, triangles, and quadrilaterals, using tools like rulers and protractors, and apply formulas for area and perimeter to solve real-world problems involving decomposed shapes.
They will measure length, mass, and liquid volume using customary and metric units, create and interpret line plots, solve problems involving measurement conversions and elapsed time, and connect math operations to real-life contexts through estimation and strategic problem solving.

Course Assessments

Growth Measure Benchmarks
Acadience Math Assessments
IXL Math Diagnostic
Are You Ready?
Module 1 Assessment
Module 2 Assessment
Module 3 Assessment
Module 4 Assessment
Module 5 Assessment
Module 8 Assessment
Module 9 Assessment
Module 6 Assessment
Module 7 Assessment
Module 10 Assessment
Module 11 Assessment
Module 12 Assessment
Module 13 Assessment
Module 17 Assessment
Module 18 Assessment
Module 14 Assessment
Module 15 Assessment
Module 16 Assessment
Module 19 Assessment
Module 20 Assessment
Module 21 Assessment

Course Units

Unit 1: Place Value & Whole Number Operations

Standards

PA Core Mathematics Standards
Module 1

CC.2.1.4.B.1 Students acquire the knowledge and skills needed to: Apply place-value concepts to show an understanding of multi-digit whole numbers.
- M04.A-T.1.1.1 Demonstrate an understanding that in a multi-digit whole number (through 1,000,000), a digit in one place represents ten times what it represents in the place to its right.
- M04.A-T.1.1.2 Read and write whole numbers in expanded, standard, and word form through 1,000,000.
- M04.A-T.1.1.3 Compare two multi-digit numbers through 1,000,000 based on meanings of the digits in each place, using >, =, and < symbols.
- M04.A-T.1.1.4 Round multi-digit whole numbers (through 1,000,000) to any place.

Module 2

CC.2.1.4.B.2 Students acquire the knowledge and skills needed to: Use place-value understanding and properties of operations to perform multi-digit arithmetic.
- M04.A-T.2.1.1 Add and subtract multi-digit whole numbers (limit sums and subtrahends up to and including 1,000,000).
- M04.A-T.2.1.4 Estimate the answer to addition, subtraction, and multiplication problems using whole numbers through six digits (for multiplication, no more than 2 digits × 1 digit, excluding powers of 10).
CC.2.4.4.A.1 Students acquire the knowledge and skills needed to: Solve problems involving measurement and conversions from a larger unit to a smaller unit.
- M04.D-M.1.1.3 Apply the area and perimeter formulas for rectangles in real-world and mathematical problems (may include finding a missing side length). Whole numbers only. The formulas will be provided.

Know

Numbers can be written in different forms for different purposes.
The value of a digit is determined by its position in a number.
In a multi-digit whole number, a digit in one place represents ten times the value it would have in the place to its right.
When comparing numbers, distinguish the meaning between the > and < symbols.
When adding larger numbers, addends need to be aligned by place value, and then adding begins with the digits in the ones place, moving left one place at a time, regrouping when necessary.
When subtracting, align the two numbers by place value and begin subtracting starting in the ones place, moving left one place at a time, regrouping when necessary.
Apply the rules of rounding (number-line, place value understanding) in multi-digit numbers.
An estimate can help determine if an answer is reasonable.
Identifying the unknown value in a problem helps determine which operation (addition & subtraction) to use to find the solution.
Perimeter is the total length of all sides of a shape.

Understanding/Key Learning

Understanding place value relationships is a foundational skill in identifying numbers and performing estimation and computation with whole numbers.
Expressing numbers in different ways helps students develop an understanding of place-value meaning.
Being able to switch fluently between the standard form and expanded form of numbers, as well as using visual models, sets up students to be able to quickly compare numbers of any size, as well as perform computations using these numbers and understand what the numbers represent in real-world situations.
Comparing numbers involves comparing digits with the greatest place value possible and moving to digits with decreasing place values until one number is shown to have a greater value than another
Recognize rounding as a strategy for estimation and understand the rules for rounding numbers based on their digits.
Fluent recall of addition and subtraction facts aids in solving computational problems using the standard algorithm with larger numbers.
Keywords and relationships between values in a problem help identify which operation to use.
Formulas can be used to solve problems.

Do

Module 1

Read and write multi-digit numbers in different ways (standard form, word form, expanded form) to 1,000,000.
Describe the values of digits in a number to 1,000,000.
Use visual representations (base ten blocks, place value charts) to group and name multi-digit whole numbers to 1,000,000.
Use place-value charts and number lines to compare (>, <, =) and order whole numbers.
Round whole numbers and recognize rounding as an estimation strategy.

Module 2

Use the standard algorithm to add whole numbers and assess reasonableness using mental math and estimates.
Use the standard algorithm to find the difference between two whole numbers and assess the reasonableness using mental math and estimates.
Interpret and solve comparison problems using addition and subtraction by drawing bar models.
Use a formula to find the perimeter of a rectangle or find an unknown side given the perimeter and the length of one side of the rectangle.
Determine the reasonableness of sums and differences.

Unit Essential Questions

How does understanding the value of a digit help us read and write large numbers?
In what ways can we group and compare multi-digit whole numbers to make sense of their value and order?
Why is rounding whole numbers important, and how does it help us in real-life situations?
How can we use mental math and estimates to check if our sums and differences are reasonable when using the standard algorithm?
In what ways do bar models help us visualize and solve comparison problems involving addition and subtraction?
How can understanding the formula for the perimeter of a rectangle help us solve real-world problems involving measurements?

Lesson Essential Questions

L1.1 In what ways can you visually represent numbers to show the value of each digit?
L1.2 Why do we write numbers in different forms?
L1.3 How can I regroup and rename numbers to show the same value?
L1.4 How can I use my knowledge of place value to compare and order numbers?
L1.5 Can you round any number to a given place and explain how you did it?
L2.1 How do I add whole numbers greater than 1000 and make sure my answer is reasonable?
L2.2 How do I subtract whole numbers greater than 1000 and make sure my answer is reasonable?
L2.3 How do I draw a visual model to identify the missing value and solve comparison problems?
L2.4 How do I find the perimeter of an object?

Materials/Resources

Vocabulary

Assessments

Unit 2: Multiplication and Division Problems

Standards

PA Core Mathematics Standards
Modules 3-7

CC.2.2.4.A.1 Students acquire the knowledge and skills needed to: Represent and solve problems involving the four operations.
- M04.B-O.1.1.1 Interpret a multiplication equation as a comparison. Represent verbal statements of multiplicative comparisons as multiplication equations.
- M04.B-O.1.1.2 Multiply or divide to solve word problems involving multiplicative comparison, distinguishing multiplicative comparison from additive comparison.
- M04.B-O.1.1.4 Identify the missing symbol (+, –, ×, ÷, =, <, and >) that makes a number sentence true (single-digit divisor only).
- M04.B-O.1.1.3 Solve multi-step word problems posed with whole numbers using the four operations. Answers will be either whole numbers or have remainders that must be interpreted, yielding a final answer that is a whole number. Represent these problems using equations with a symbol or letter standing for the unknown quantity.
CC.2.1.4.B.2 Students acquire the knowledge and skills needed to: Use place-value understanding and properties of operations to perform multi-digit arithmetic.
- M04.A-T.2.1.2 Multiply a whole number of up to four digits by a one-digit whole number and multiply 2 two-digit numbers.
- M04.A-T.2.1.3 Divide up to four-digit dividends by one-digit divisors with answers written as whole-number quotients and remainders.
- M04.A-T.1.1.4 Round multi-digit whole numbers (through 1,000,000) to any place.
- M04.A-T.2.1.4 Estimate the answer to addition, subtraction, and multiplication problems using whole numbers through six digits (for multiplication, no more than 2 digits × 1 digit, excluding powers of 10).
- M04.B-O.1.1.3 Solve multi-step word problems posed with whole numbers using the four operations. Answers will be either whole numbers or have remainders that must be interpreted, yielding a final answer that is a whole number. Represent these problems using equations with a symbol or letter standing for the unknown quantity.

Modules 8- 9

CC.2.1.4.B.2 Students acquire the knowledge and skills needed to: Use place-value understanding and properties of operations to perform multi-digit arithmetic.
- M04.A-T.2.1.2 Multiply a whole number of up to four digits by a one-digit whole number and multiply 2 two-digit numbers.
- M04.A-T.1.1.4 Round multi-digit whole numbers (through 1,000,000) to any place.
- M04.A-T.2.1.4 Estimate the answer to addition, subtraction, and multiplication problems using whole numbers through six digits (for multiplication, no more than 2 digits × 1 digit, excluding powers of 10).
CC.2.2.4.A.1 Students acquire the knowledge and skills needed to: Represent and solve problems involving the four operations.
- M04.B-O.1.1.3 Solve multi-step word problems posed with whole numbers using the four operations. Answers will be either whole numbers or have remainders that must be interpreted yielding a final answer that is a whole number. Represent these problems using equations with a symbol or letter standing for the unknown quantity.

Know

Multiplication is combining equal groups, and addition is combining unequal groups.
Fluent recall of basic multiplication and division facts (0-12).
The relationship between multiplication and division as combining and separating equal groups.
Understand keywords/phrases used in word problems.
Understand the relationship between visual models and the mathematical processes.
In a multi-digit whole number, a digit in one place represents ten times the value it would have in the place to its right.
The process of rounding in multi-digit numbers.
Multiplication and division of multiples of 10, 100, and 1,000.
The value of a digit is determined by its position in a number.
Properties of multiplication and division can be used to solve problems.
Multiplication of one-digit numbers and multiples of 10.
The process of rounding whole numbers to the nearest 10 and 100.
The steps to solving two-step word problems using the four operations.
The formula to solve for the perimeter of polygons, as well as to identify an unknown side length.
Understand the difference between the area and perimeter of a rectangle.

Understanding/Key Learning

Understanding the meaning of multiplication and division and their representation within a visual model, as well as when written as an equation.
Understanding that the phrase “times as many” refers to multiplying the quantity of the group by the number of groups.
Distinguish between multiplicative comparisons (equal groups) and additive comparisons (unequal groups) in visual models and equations.
Understand that division and multiplication are inverse operations.
Demonstrate math fact fluency by replacing missing values in a multiplicative comparison.
Determine a solution path, including the operations to use, based on the relationship between the numbers in the problem.
Apply basic math fact knowledge and patterns to multiply and divide multiples of 10, 100, and 1,000.
Understanding that estimation is used to determine whether an answer is a reasonable solution to the problem.
Understand how the quotient changes when a dividend is increased or decreased, or how the quotient changes when a divisor is increased or decreased.
Mental math skills allow you to calculate quickly and fluently, without getting bogged down with individual calculations.
Understand how showing multiplication methods visually prepares students for regrouping in multi-digit multiplication problems.
Understand that a factor can be broken down into smaller parts to be multiplied by another factor, and it does not change the final product.
Relate various multiplication strategies to others to explain their process in multiplying multi-digit numbers (i.e, how are the partial products and area model related to the standard algorithm for multiplication).
Understand that all dividends do not divide equally by all divisors and that the remainder is included as a leftover portion of the quotient, which can be used or discarded, depending on the situation in the problem.
Relate various division strategies to others to explain their process in dividing multi-digit numbers (i.e, how are the partial quotients, area model, and repeated subtraction related to the standard algorithm for division).
Understand how showing division methods visually prepares students for regrouping in multi-digit division problems.
Apply basic math fact knowledge and patterns to multiply multiples of 10, 100, and 1,000 when multiplying by 2-digit factors.
Determine the best estimation strategy for a given situation when an exact value is not necessary.
Understand that factors can be written as the side lengths of a rectangle and that the area of the shape is the product of the factors.
Understand that the partial products of the area model can be used to find the product without drawing the rectangle.
Relate various multiplication strategies to others to explain their process in multiplying multi-digit numbers (i.e, how are the partial products and area model related to the standard algorithm for multiplication).
Analyze a situation and its relationship to the numbers in the problem in order to solve the problem.
Understand that the formula for area is the relationship between the figure’s side length and width, which determines its interior space.
Understand that a figure can be combined into two smaller shapes and that the total area can be found by combining the areas of the shapes together.
Translate information given in a formula by substituting the known values into the appropriate spaces to determine the unknown values.
Determine when to remove or combine areas to connect real-world problems to find the total area.

Do

Module 3

Solve multiplicative and additive comparison problems using visual models and equations.
Explain how to decide whether a comparison problem should use multiplication or addition.
Understand and apply inverse operations.
Determine the appropriate operation to solve multi-step word problems with whole numbers.

Module 4

Explain how using patterns and place value can help find a product of multiples of 10, 100, and 1,000.
Explain how basic facts can be used to divide multiples of 10, 100, and 1,000.
Use estimation to determine the reasonableness of answers to multiplication and division problems.
Use compatible numbers to estimate quotients.
Use mental math to solve multiplication and division problems, including the Commutative Property of Multiplication and the Distributive Property.

Module 5

Use base-ten blocks to represent multiplication of 2- and 3- digit numbers by 1-digit numbers.
Use area models, the Distributive Property, and Partial Products as strategies to multiply whole numbers.
Use place value to multiply up to 4-digit by 1-digit numbers.
Solve multi-step word problems using multiplication of whole numbers.
Using mental math and estimation, assess the reasonableness of the multiplication of whole numbers.

Module 6

Use visual representations to develop a conceptual understanding of division with and without remainders.
Find whole number quotients and remainders with up to 4-digit dividends and 1-digit divisors.
Illustrate and explain the calculations by using equations, rectangular arrays, partial quotients and area models.
Interpret remainders in real-world contexts.
Solve multi-step word problems using division of whole numbers.
Using mental math and estimation, assess the reasonableness of the division of whole numbers.

Module 7

Connect concrete and visual models to using place value to solve division problems with regrouping.
Divide up to 4-digit numbers by 1-digit numbers using the standard algorithm.
Solve multi-step word problems using multiplication and division of whole numbers and assess the reasonableness of answers.
Interpret remainders in real-world contexts.

Module 8

Use area models, place value, and partial products to multiply 2-digit whole numbers.
Round multi-digit whole numbers to any place.
Solve multi-step word problems using the four operations.

Module 9

Apply the area formula to solve real-world and mathematical problems.
Apply the area formula for rectangles to find the area of the combined rectangles.
Find the unknown measure of a rectangle given the length of one side and the area or perimeter.
Find the area of figures that consist of a rectangle with one or more smaller rectangles removed.

Unit Essential Questions

How can using visual models and equations help me to solve problems?
How do I know which steps and mathematical operations to use when solving a problem?
How does understanding inverse operations help me to solve problems and check my work?
How can patterns in multiplication and division help me to solve problems?
How does place value help me multiply and divide multi-digit numbers by 1-digit numbers?
How does estimation help me decide if my answer makes sense?
How can I use mental math and properties of multiplication to solve problems more efficiently?
How do remainders affect my answer, especially in real-world situations?
How can I use what I know about place value to multiply one-digit numbers and multiples of 10?
What steps can I take to solve multi-step word problems, and how do I know which operations to use?
How can I find the perimeter of a polygon and use that to figure out a missing side length?
How is perimeter different from area?
How can I use the area formula to solve real-world and math problems involving rectangles?
How do I find the area of a shape when part of a rectangle is removed?

Lesson Essential Questions

L.3.1 How can you use visual models and equations to represent and interpret multiplicative comparisons?
L.3.2 How do you determine whether a comparison is multiplicative or additive?
L.3.3 How can you use division to show a multiplicative comparison?
L.3.4 How can you represent unknown values using letters in multiplication and addition equations?
L.3.5 How can you represent unknown values using letters in division equations?
L4.1 How do I use patterns to multiply by multiples of 10, 100, and 1,000?
L4.2 How do I use patterns to divide by multiples of 10, 100, and 1,000?
L4.3 How can you determine whether your product is reasonable?
L4.4 How do compatible numbers help me to estimate a quotient?
L4.5 How can I use mental math strategies to help me multiply and divide numbers?
L5.1 How can I use arrays to represent multiplication?
L5.2 How do you use the array model and distributive property to multiply a 2-digit factor by a 1-digit
factor?
L5.3 What does the partial products method look like when solving 3-digit by 1-digit numbers?
L5.4 How does the partial products method change when you’re solving up to 4-digit by 1-digit numbers?
L5.5 What are the steps of the standard algorithm for multiplication of a 2-digit by 1-digit number?
L5.6 How does the standard algorithm method change when you’re solving up to 4-digit by 1-digit numbers?
L5.7 What problem-solving steps can be used to solve multi-step word problems?
L8.1 What strategies can you use to multiply multiples of 10?
L8.2 How can you predict if the estimated product is related to the actual product?
L8.3 How can you determine which factors to multiply in order to solve a multiplication problem using an
area model?
L8.4 Why is it necessary to align partial products when using the partial products method to multiply?
L8.5 What are the steps of the standard algorithm for multiplication of two 2-digit numbers using
regrouping?
L8.6 What method of multiplication works best for you when solving multiplication problems using two
2-digit numbers?
L8.7 What problem-solving steps can be used to solve multi-step word problems?
L9.1 What is the formula for finding the area of a rectangle?
L9.2 How can you find the area of a figure of 2 combined rectangles?
L9.3 How can you find the measure of an unknown side when you are given the length of one side, or the
area or perimeter of the shape?
L9.4 How can you find the area of a shape by using the subtraction of an interior shape?
L6.1 How can you use visual models and equations to represent division?
L6.2 How does your visual model show the quotient and remainder of a division problem?
L6.3 What does the remainder represent in the context of a division problem?
L6.4 How can the Distributive Property and the area model help you solve multi-digit division problems?
L6.5 How is repeated subtraction related to division?
L6.6 What does the partial quotients method look like when solving multi-digit by 1-digit numbers?
L7.1 How can you use visual models and equations to represent the division of a multi-digit by 1-digit
number divisor?
L7.2 How can you determine the number of digits that will be in your quotient?
L7.3 What are the steps of the standard algorithm for division of a multi-digit by 1-digit number divisor?
L7.4 How can you determine if the answer to a multiplication or division word problem is reasonable?

Materials/Resources

Vocabulary

Assessments

Unit 3: Fractions and Decimals

Standards

PA Core Mathematics Standards
Module 10

CC.2.2.4.A.2 Students acquire the knowledge and skills needed to: Develop and/or apply number theory concepts to find factors and multiples.
- M04.B-O.2.1.1 Find all factor pairs for a whole number in the interval 1 through 100. Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the interval 1 through 100 is a multiple of a given one-digit number. Determine whether a given whole number in the interval 1 through 100 is prime or composite.
CC.2.2.4.A.4 Students acquire the knowledge and skills needed to: Generate and analyze patterns using one rule.
- M04.B-O.3.1.1 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.
- M04.B-O.3.1.2 Determine the missing elements in a function table (limit to +, –, or × and to whole numbers or money).
- M04.B-O.3.1.3 Determine the rule for a function given a table (limit to +, –, or × and to whole numbers).

Module 11

CC.2.1.4.C.1 Students acquire the knowledge and skills needed to: Extend the understanding of fractions to show equivalence and ordering.
- M04.A-F.1.1.2 Compare two fractions with different numerators and different denominators (denominators limited to 2, 3, 4, 5, 6, 8, 10, 12, and 100) using the symbols >, =, or < and justify the conclusions.
- M04.A-F.1.1.1 Recognize and generate equivalent fractions.

Module 12

CC.2.1.4.C.3 Students acquire the knowledge and skills needed to: Connect decimal notation to fractions, and compare decimal fractions (base 10 denominator, e.g., 19/100).
- M04.A-F.3.1.2 Use decimal notation for fractions with denominators 10 or 100.
- M04.A-F.3.1.1 Add two fractions with respective denominators 10 and 100.
- M04.A-F.3.1.3 Compare two decimals to hundredths using the symbols >, =, or <, and justify the conclusions.
CC.2.4.4.A.1 Students acquire the knowledge and skills needed to: Solve problems involving measurement and conversions from a larger unit to a smaller unit.
- M04.D-M.1.1.2 Use the four operations to solve word problems involving distances, intervals of time (such as elapsed time), liquid volumes, masses of objects; money, including problems involving simple fractions or decimals; and problems that require expressing measurements given in a larger unit in terms of a smaller unit.

Module 13

CC.2.3.4.A.1 Students acquire the knowledge and skills needed to: Draw lines and angles and identify these in two-dimensional figures.
- M04.C-G.1.1.1 Draw points, lines, line segments, rays, angles (right, acute, and obtuse), and perpendicular and parallel lines. Identify these in two-dimensional figures.
- M04.D-M.3.1.1 Measure angles in whole-number degrees using a protractor. With the aid of a protractor, sketch angles of specified measure.
CC.2.4.4.A.6 Students acquire the knowledge and skills needed to: Measure angles and use properties of adjacent angles to solve problems.
- M04.D-M.3.1.2 Solve addition and subtraction problems to find unknown angles on a diagram in real-world and mathematical problems. (Angles must be adjacent and non-overlapping.)

Know

Fluent recall of basic multiplication and division facts (0-12).
Division is a process used to find an unknown factor.
Arithmetic patterns are used to explain patterns.
Fractions are equivalent when their shaded parts are equal.
Fractions can be compared using visual models.
Values of coins and their relationship to a dollar.
Examples of various quadrilaterals.
Unit fractions are 1 part of a whole partitioned into equal parts.
Formulas of area and perimeter to solve real-world problems.

Understanding/Key Learning

Identifying all of the factor pairs of a number helps to write fractions in simplest form.
Applying knowledge of divisibility, factors, and multiples helps to generate equivalent fractions, as well as compare fractions.
Understand the relationship between factors and multiples (Factor x Factor = Multiple).
Understand the difference between numbers that are prime and composite and how that determines divisibility in other numbers.
Understand that patterns begin with a starting number and follow a rule to generate the next terms in the pattern.
When comparing fractions, comparisons are only valid when two fractions refer to the same whole.
Benchmarks such as 0, ½, and 1 whole can be used to compare fractions that have different numerators and denominators.
Fractional parts can be subdivided into the same number of equal parts without changing the value of the whole.
When mathematical patterns are applied, numerators and denominators are multiplied by another number, and therefore, an equivalent fraction is formed.
Identifying common denominators in fractions can be used to compare fractions with unlike denominators.
Determine when using a common numerator or denominator is more efficient to compare fractions with unlike denominators.
Determine the best strategy for creating common numerators and denominators in real-world situations.
Understand the relationship between the fraction 1/10 as it relates to money (10 cents).
Understand the relationship between the fraction 1/100 as it relates to money (1 cent).
Understand the relationship between the fraction, decimal, and money values for fractions of tenths and hundredths (0.2, 0.20, 2/10, 20/100, 20 cents).
Use knowledge of decimals to compare decimals in tenths and hundredths.
Relate the value of combinations of coins to decimals and fractions.
Relate decimal amounts to money values to connect money related problems into everyday life scenarios.
Drawing examples of points, lines, line segments, rays, angles, perpendicular and parallel lines helps to identify them in various two-dimensional shapes.
Understand that the distance between two rays determines the angle’s measurement.
Understand that fractions of a circle are based on how many times the angle turns within the circle (1/360), and are identified as degrees.
Use knowledge of unit fractions to name the degree measurement of an angle.
Understand that a protractor is used to measure the degree of angles and identify the steps used in accurately measuring an angle.
Recognize that angles can be joined and separated without changing the measurement of the total angle.
Identify benchmark angles (45, 60, 90, and 180 degrees) to help solve real-world problems involving angles.

Do

Module 10

Find all factor pairs of a whole number.
Determine whether a number is a multiple of a given number.
Determine whether a whole number is prime or composite.
Generate a number pattern that follows a given rule.

Module 11

Use a benchmark fraction to compare two fractions.
Use visual models to explain fraction equivalence and solve a numerical process.
Use fraction equivalence to rewrite fractions with the same numerator and/or denominator in order. to compare the two.

Module 12

Create equivalent decimals and fractions with denominators of 10 and 100.
Using models, solve word problems involving money.
Use visual models and place value to compare decimals.
Represent the relationship between fractions, decimals, and money.

Module 13

Draw points, lines, rays, and angles, as well as identify them in two-dimensional figures.
Identify angles and use a protractor to measure and draw them.
Recognize angles as additive.
Solve problems involving angles.

Unit Essential Questions

How can we use fluent recall of multiplication and division facts to solve problems and find unknown factors?
In what ways do visual models and arithmetic patterns help us compare and understand fractions, including unit fractions and equivalence?
How can we decide whether a whole number is prime or composite, and why is this distinction important?
How can recognizing equivalent fractions assist in comparing fractions with different numerators and denominators?
How do we generate number patterns that follow a specific rule, and how can these patterns help us predict future numbers?
How do the values of coins relate to a dollar, and how can we use this knowledge in real-world math situations?
How can we identify different quadrilaterals and use formulas for area and perimeter to solve real-world problems?
What steps do we follow to measure and draw angles using a protractor?
How can understanding that angles are additive help us solve problems involving multiple angles?

Lesson Essential Questions

L10.1 How can I show the factor pairs of a number using a visual model?
L10.2 How can I use division to determine whether a number is a factor of another number?
L10.3 How can you determine whether a number is a multiple of another number and how can you create a list of multiples of a number?
L10.4 How can you determine whether a number is prime or composite and how does knowing whether a number is prime or composite help you to solve a problem?
L10.5 How can you use rules to continue a pattern and identify features of a pattern that are not directly Stated?
L11.1 How can models help you to compare two fractions with different numerators and denominators?
L11.2 How can a benchmark fraction (½) help you compare two fractions with unlike numerators and denominators?
L11.3 How can models help you to explain why two fractions are equivalent?
L11.4 Why are multiplication and division used to generate equivalent fractions?
L11.5 How does knowing the common multiples of two denominators help you to find equivalent fractions?
L11.6 How can you use common numerators and denominators to compare two fractions?
L11.7 What are some strategies you can use to compare and order fractions with unlike numerators or denominators?
L12.1 How can you use visual models and decimal notation to represent fractions, mixed numbers, and decimals in tenths?
L12.2 How can you use visual models and decimal notation to represent fractions, mixed numbers, and decimals in hundredths?
L12.3 How can you convert a fraction using tenths and hundredths into a decimal number in tenths and hundredths?
L12.4 How can a model, number line, and place value chart help you to compare two decimal numbers?
L12.5 How can you use models of hundredths, fractions, and decimals to represent money amounts?
L12.6 What steps are needed to represent money amounts as fractions, decimals, and mixed numbers in real-world situations?
L13.1 How can you identify, draw, and name angles?
L13.2 How can you compare angle sizes and determine how many unit angles there are in a given circle?
L13.3 How do angles represent fractional parts of a circle?
L13.4 How can you determine the measurement of an angle by using its fractional part of a circle?
L13.5 How can you use a protractor to measure and draw angles?
L13.6 What operations can you use to join and separate given angles?
L13.7 How can you use the measurement of a known angle to determine the measurement of an unknown angle?

Materials/Resources

Vocabulary

Assessments

Unit 4: Operations and Fractions

Standards

PA Core Mathematics Standards
Modules 14-16

CC.2.1.4.C.2 Students acquire the knowledge and skills needed to: Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers.
- M04.A-F.2.1.2 Decompose a fraction or a mixed number into a sum of fractions with the same denominator (denominators limited to 2, 3, 4, 5, 6, 8, 10, 12, and 100), recording the decomposition by an equation. Justify decompositions (e.g., by using a visual fraction model).
- M04.A-F.2.1.1 Add and subtract fractions with a common denominator (denominators limited to 2, 3, 4, 5, 6, 8, 10, 12, and 100; answers do not need to be simplified; and no improper fractions as the final answer).
- M04.A-F.2.1.4 Solve word problems involving addition and subtraction of fractions referring to the same whole or set and having like denominators(denominators limited to 2, 3, 4, 5, 6, 8, 10, 12, and 100).
- M04.A-F.2.1.5 Multiply a whole number by a unit fraction (denominators limited to 2, 3, 4, 5, 6, 8, 10, 12, and 100 and final answers do not need to be simplified or written as a mixed number).
- M04.A-F.2.1.6 Multiply a whole number by a non-unit fraction (denominators limited to 2, 3, 4, 5, 6, 8, 10, 12, and 100 and final answers do not need to be simplified or written as a mixed number).
- M04.A-F.2.1.7 Solve word problems involving multiplication of a whole number by a fraction (denominators limited to 2, 3, 4, 5, 6, 8, 10, 12, and 100).
CC.2.1.4.C.3 Students acquire the knowledge and skills needed to: Connect decimal notation to fractions, and compare decimal fractions (base 10 denominator, e.g., 19/100).
- M04.A-F.3.1.1 Add two fractions with respective denominators 10 and 100.
- M04.A-F.2.1.3 Add and subtract mixed numbers with a common denominator(denominators limited to 2, 3, 4, 5, 6, 8, 10, 12, and 100; no regrouping with subtraction; fractions do not need to be simplified; and no improper fractions as the final answers).

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Know

Understanding/Key Learning

Understand that unit fractions are building blocks of all fractions.
Recognize that fractions are a combination of same-sized parts within the same whole.
Understand that when adding fractions, only the numerators are combined.
Understand that when subtracting fractions, only the numerators are removed.
Connect visual models of addition and subtraction of fractions to real-life scenarios.
Understand the process of common denominators by changing denominators of tenths into hundredths to add and subtract fractions.
Understand that if a numerator is greater than the denominator, the value of the fraction is greater than 1.
Understand that when adding and subtracting fractions greater than 1, only the numerators are combined or removed.
Understand that fractions greater than 1 can be named as whole numbers and fractions (mixed numbers).
Understand that when adding and subtracting mixed numbers, the mixed numbers are combined, and the fractions are combined.
Understand that when adding and subtracting mixed numbers, regrouping of whole numbers and fractions greater than 1 will need to be renamed as mixed numbers.
Connect visual models of addition and subtraction of mixed numbers to real-life scenarios.
Understand that fractions are multiples of unit fractions.
Understand that multiplying a fraction by a whole number is the same as multiplying the whole number and the numerator, while keeping the denominator the same.
Connect visual models of multiplication of whole numbers and fractions to real-life scenarios.
Understand that the Distributive Property can be used to multiply whole numbers and mixed numbers.

Do

Module 14

Decompose a fraction into a sum of fractions.
Use models and equations to represent and solve real-world problems involving the addition and subtraction of fractions.
Add fractions with denominators of 10 and 100.

Module 15

Use visual, concrete models, and equations to represent and solve real-world problems involving the addition and subtraction of mixed numbers.
Use visual models to decompose a sum of fractions.
Rename mixed numbers as equivalent fractions to add and subtract.
Apply properties of addition to add mixed numbers.

Module 16

Multiply fractions by whole numbers.
Write a fraction as a multiple of a whole number and a unit fraction.
Use visual models and equations to represent and solve real-world problems involving products of a fraction and a whole number.

Unit Essential Questions

How does a unit fraction help us understand other fractions?
How does knowing multiplication and division facts help us understand and work with fractions?
How can we use pictures, models, or equations to solve real-world problems with fractions?
What strategies help us add and subtract fractions in everyday situations?
How can we break apart (decompose) fractions to make them easier to work with?
Why might we rename a mixed number as a fraction, and how does that help us add or subtract?
How can visual models and equations help us solve real-world problems that involve multiplying fractions and whole numbers?
When might we need to multiply a fraction by a whole number in real life?

Lesson Essential Questions

L14.1 How can you model subtracting fractions? How can you use addition of unit fractions to create fractions?
L14.2 How do models help you solve problems with addition of fractions?
L14.3 How can number lines help you solve real-world problems with addition of fractions?
L14.4 How do models help you solve problems with subtractions of fractions?
L14.5 How can you express subtracting fractions using equations?
L14.6 How can you rename fractions of tenths as hundredths in order to add fractions?
L15.1 How can you add and subtract fractions that are greater than one with like denominators to solve real-world problems?
L15.2 How can you rename mixed numbers as a sum of fractions with like denominators?
L15.3 How do visual models and equations help you to add and subtract mixed numbers with like denominators?
L15.4 How can you rename mixed numbers to fractions greater than one with like denominators to solve real-world problems?
L15.5 How do the properties of addition help you add fractions and mixed numbers?
L15.6 How can you add and subtract fractions and mixed numbers to solve real-world problems?
L16.1 How can you use repeated addition to show multiplication of a whole number and a fraction?
L16.2 How can you show multiplication of a whole number and a fraction as a product of a whole number and a unit fraction?
L16.3 How can models and equations help you find the product of a whole number and a fraction?
L16.4 How can you multiply mixed numbers and whole numbers to solve real-world problems?

Materials/Resources

Vocabulary

Assessments

Unit 5: Two-Dimensional Figures and Symmetry

Standards

PA Core Mathematics Standards
Module 17

CC.2.3.4.A.1 Students acquire the knowledge and skills needed to: Draw lines and angles and identify these in two-dimensional figures.
- M04.C-G.1.1.1 Draw points, lines, line segments, rays, angles (right, acute, and obtuse), and perpendicular and parallel lines. Identify these in two-dimensional figures.
CC.2.3.4.A.2 Students acquire the knowledge and skills needed to: Classify two-dimensional figures by properties of their lines and angles.
- M04.C-G.1.1.2 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.
- M04.D-M.3.1.1 Measure angles in whole-number degrees using a protractor. With the aid of a protractor, sketch angles of specified measure.

Module 18

CC.2.3.4.A.3 Students acquire the knowledge and skills needed to: Recognize symmetric shapes and draw lines of symmetry.
- M04.C-G.1.1.3 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 mirroring parts. Identify line-symmetric figures and draw lines of symmetry (up to two lines of symmetry).
- M04.B-O.3.1.1 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.

Know

Understand features of quadrilaterals.
Understand that rectangles can have the same area and different perimeter, and same perimeter and different area.
Perimeter is the distance around a closed plane figure.
Understand two-dimensional shapes by their attributes.
Understand the features of two dimensional shapes.
Understand how to distinguish patterns in addition and multiplication tables.
When given a shape pattern, describe and extend the pattern.

Understanding/Key Learning

Understand the difference between parallel and perpendicular lines and use tools to accurately draw them.
Demonstrate the ability to classify any triangle as acute, right, or obtuse.
Demonstrate the ability to classify any triangle as scalene, isosceles, or equilateral.
Distinguish between the different types of quadrilaterals.
Understand that quadrilaterals are made up of varying angles that combine to create a shape.
Understand that symmetry is dividing a shape into two equal parts.
Visualize lines of symmetry without measuring, cutting, or folding.
Describe patterns formed by repeating shapes.

Do

Module 17

Describe, and draw two-dimensional shapes including their sides and angles.
When given the perimeter of a rectangle calculate the unknown measure of a side.
Identify and draw perpendicular and parallel lines.
Identify and classify triangles by their sides and angles.
Identify and classify quadrilaterals.
Accurately measure and draw angles of two-dimensional figures.

Module 18

Identify and draw lines of symmetry in various two-dimensional shapes by visually inspecting or folding, explaining how the shape is divided into two mirror-image halves.
Describe, extend, and generate shape patterns.

Unit Essential Questions

How can we use the properties of sides, angles, and lines—including parallel, perpendicular, and lines of symmetry—to understand, describe, and accurately draw two-dimensional shapes?
How can we identify and classify triangles by their sides and angles, and use lines, angles, and symmetry to describe and draw these two-dimensional shapes accurately?
How can we identify and classify quadrilaterals by their sides and angles, and use lines—such as parallel, perpendicular, and lines of symmetry—to describe, draw, and understand their unique properties?
How can we use a protractor to accurately measure and draw angles in two-dimensional shapes?

Lesson Essential Questions

L17.1 How can you identify parallel, perpendicular, and intersecting lines?
L17.2 How can you classify triangles by the measure of their angles?
L17.3 How can you classify triangles by the length of their sides?
L17.4 How can you identify different types of quadrilaterals?
L17.5 How can you use a protractor to measure and draw angles?
L18.1 How can you identify a line of symmetry on a two-dimensional figure?
L18.2 How can you identify two-dimensional figures that have line symmetry?
L18.3 How can you generate and identify shape patterns?

Materials/Resources

Vocabulary

Assessments

Unit 6: Measurement , Data, and Time

Standards

PA Core Mathematics Standards
Module 19-21

CC.2.4.4.A.1 Students acquire the knowledge and skills needed to: Solve problems involving measurement and conversions from a larger unit to a smaller unit.
- M04.D-M.1.1.1 Know relative sizes of measurement units within one system of units including standard units (in., ft, yd, mi; oz., lb; and c, pt, qt, gal), metric units (cm, m, km; g, kg; and mL, L), and time (sec, min, hr, day, wk, mo, and yr). Within a single system of measurement, express measurements in a larger unit in terms of a smaller unit. A table of equivalencies will be provided.
- M04.D-M.1.1.2 Use the four operations to solve word problems involving distances, intervals of time (such as elapsed time), liquid volumes, masses of objects; money, including problems involving simple fractions or decimals; and problems that require expressing measurements given in a larger unit in terms of a smaller unit.
- M04.D-M.1.1.4 Identify time (analog or digital) as the amount of minutes before or after the hour.
CC.2.4.4.A.2 Students acquire the knowledge and skills needed to: Translate information from one type of data display to another.
- M04.D-M.2.1.3 Translate information from one type of display to another (table, chart, bar graph, or pictograph).
CC.2.4.4.A.4 Students acquire the knowledge and skills needed to: Represent and interpret data involving fractions using information provided in a line plot.
- M04.D-M.2.1.1 Make a line plot to display a data set of measurements in fractions of a unit (e.g., intervals of 1/2, 1/4, or 1/8).
- M04.D-M.2.1.2 Solve problems involving addition and subtraction of fractions by using information presented in line plots (line plots must be labeled with common denominators, such as 1/4, 2/4, 3/4).

Know

The process of measuring the length, mass, and liquid volume of an object.
Line plots display data using various units of measurement.
The steps to solving one-step word problems involving measurement conversions within the same unit.
Time expressed in visual and digit formats.
The steps to solving one-step word problems involving addition and subtraction of time intervals.

Understanding/Key Learning

Understand the relative size of different customary units of length, weight, and liquid volume using ordinary objects.
Understand tools of measurement including: rulers, meter sticks, measuring cups, etc.
Understand that the metric system is based on powers of 10.
Understand that units within a system can be multiplied and divided to show equivalence between units.
Understand that fractions and measurement data can be used to create and use line plots to solve problems.
Understand the difference between the minutes, seconds, and hour hand on a clock.
Understand the difference in time can be expressed as elapsed time.
When adding and subtracting amounts of time, different regrouping rules must be applied.
Apply a variety of strategies, like visual models, counting up or down, or adding and subtracting to solve real-world problems involving time.
Understand the difference between mixed measures of time and measurement and that operations are not applied in the same way.

Do

Modules 19-20

Use nonstandard units to measure lengths, weights, and volumes of familiar objects.
Compare the size of measurements using customary units to familiar objects.
Compare lengths, weights, and volumes given in different customary units.
Convert measurements from larger to smaller units.
Organize and compare measures using a table.
Organize data to make a line plot.
Use data from a line plot to answer questions involving addition and subtraction.
Compare data presented on a line plot to answer questions.
Compare size of measurements using metric units to familiar objects.
Identify common objects with a given mass, length, or liquid volume.
Compare relative sizes of metric units of length using fractions and decimals.
Compare relative sizes of metric units of mass and liquid volume using fractions and decimals.
Solve real-world problems involving measurement.

Module 21

Compare lengths of time given in different units.
Find the elapsed time based on a start and end time.
Use number lines, visual models, and equations to model elapsed time.
Find the start or end time based on the elapsed time.
Solve multistep word problems involving distances and times given in different units.

Unit Essential Questions

How do we measure the length, mass, and liquid volume of different objects accurately?
What can line plots tell us about data when we use different units of measurement?
What steps do we follow to solve one-step word problems that involve converting measurements within the same unit?
How can we read and represent time in both visual and digital formats, and solve word problems that add or subtract time intervals?

Lesson Essential Questions

L19.1 How can you select and use nonstandard units to measure lengths, weights, and liquid volume?
L19.2 How can you represent and compare customary lengths of measures?
L19.3 How can you represent and compare customary units of weight?
L19.4 How can you represent and compare customary units of liquid volume?
L19.5 How do you construct and analyze a line plot?
L20.1 How can you choose an appropriate metric unit to estimate the measure of the length, mass, or liquid volume?
L20.2 How can you represent and compare metric lengths of measure?
L20.3 How can you represent and compare metric units of mass and liquid volume?
L20.4 How does understanding the relative size of measurement units help you solve real world problems involving metric units and customary units of measurement?
L21.1 How can you represent and compare units of time in hours, minutes, and seconds?
L21.2 How can you use visual modules to solve elapsed time problems?
L21.3 How can you use visual modules to add and subtract times when given a start and end times?
L21.4 How can you use the understanding of time to solve problems using more than one operation to solve?