Mathematics Curriculum Maps

Kindergarten

Summary of Year:

Kindergarten mathematics is about (1) representing, relating, and operating on whole numbers, initially with sets of objects; and (2) describing shapes and space. More learning time in Kindergarten should be devoted to number than to other topics.

Key Areas of Focus for K-2: Addition and subtraction—concepts, skills, and problem solving

Required Fluency: K.OA.5 Add and subtract within 5.

Module 1: Numbers to 10

Module 2: Two-Dimensional and Three-Dimensional Shapes

Module 3: Comparison of Length, Weight, Capacity, and Numbers to 10

Module 4: Number Pairs, Addition and Subtraction to 10

Module 5: Numbers 10–20 and Counting to 100

Module 6: Analyzing, Comparing, and Composing Shapes

CCLS Major Emphasis Clusters

Counting and Cardinality

• Know number names and count sequence.

• Count to tell the number of objects.

• Compare numbers.

Operations and Algebraic Thinking

• Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from.

Number and Operations in Base Ten

• Work with numbers 11-19 to gain foundations for place value.

Quarter 1 Modules & Themes:

Module 1: Numbers to 10

In Module 1, Kindergarten starts out with solidifying the meaning of numbers to 10 with a focus on embedded numbers and relationships to 5 using fingers, cubes, drawings, 5 groups and the Rekenrek. Students then investigate patterns of “1 more” and “1 less” using models such as the number stairs. Because fluency with addition and subtraction within 5 is a Kindergarten goal, addition within 5 is begun in Module 1 as another representation of the decomposition of numbers.

Quarter 2 Modules & Themes:

Module 2 Two-Dimensional and Three-Dimensional Shapes

Module 3 Comparison of Length, Weight, Capacity, and Numbers to 10

In Module 2, Students learn to identify and describe squares, circles, triangles, rectangles, hexagons, cubes, cones,

cylinders and spheres. During this module students also practice their fluency with numbers to 10.

In Module 3, students begin to experiment with comparison of length, weight and capacity. Students first learn to identify the attribute being

compared, moving away from non-specific language such as “bigger” to “longer than,” “heavier than,” or “more than.” Comparison begins with developing the meaning of the word “than” in the context of “taller than,” “shorter than,” “heavier than,” “longer than,” etc. The terms “more” and

“less” become increasingly abstract later in Kindergarten. “7 is 2 more than 5” is more abstract than “Jim is taller than John.”

Quarter 3 Modules & Themes:

Module 4

In Module 4, number comparison leads to a further study of embedded numbers (e.g., “3 is less than 7” leads to, “3 and 4 make 7,” and 3 + 4 = 7,). “1

more, 2 more, 3 more” lead into addition (+1, +2, +3). Students now represent stories with blocks, drawings, and equations.

Quarter 4 Modules & Themes:

Module 5 and 6

After Module 5, after students have a meaningful experience of addition and subtraction within 10 in Module 4, they progress to exploration of

numbers 10-20. They apply their skill with and understanding of numbers within 10 to teen numbers, which are decomposed as “10 ones and some

ones.” For example, “12 is 2 more than 10.” The number 10 is special; it is the anchor that will eventually become the “ten” unit in the place value system in Grade 1.

Module 6 rounds out the year with an exploration of shapes. Students build shapes from components, analyze and compare them, and discover that

they can be composed of smaller shapes, just as larger numbers are composed of smaller numbers.

1st Grade

Summary of Year:

First Grade mathematics is about (1) developing understanding of addition, subtraction, and strategies for addion and subtraction within 20; (2) developing understanding of whole number relationships and place value, including grouping in tens and ones; (3) developing understanding of linear measurement and measuring lengths as iterating length units; and (4) reasoning about attributes of, and composing and decomposing geometric shapes.

Key Areas of Focus for K-2: Addition and subtraction—concepts, skills, and problem solving

Required Fluency: 1.OA.6 Add and subtract within 10.

Module 1: Sums and Differences to 10

Module 2: Introduction to Place Value Through Addition and Subtraction Within 20

Module 3: Ordering and Comparing Length Measurements as Numbers

Module 4: Place Value, Comparison, Addition and Subtraction to 40

Module 5: Identifying, Composing, and Partitioning Shapes

Module 6: Place Value, Comparison, Addition and Subtraction to 100

CCLS Major Emphasis Cluster:

Operations and Algebraic Thinking

• Represent and solve problems involving addition and subtraction.

• Understand and apply properties of operations and the relationship between addition and subtraction.

• Add and subtract within 20.

• Work with addition and subtraction equations.

Number and Operations in Base Ten

• Extend the counting sequence.

• Understand place value.

• Use place value understanding and properties of operations to add and subtract.

Measurement and Data

• Measure lengths indirectly and by iterating length units.

Quarter 1 Modules & Themes:

Module 1: Sums and Differences to 10

In Grade 1, work with numbers to 10 continues to be a major stepping-stone in learning the place value system. In Module 1, students work to

further understand the meaning of addition and subtraction begun in Kindergarten, largely within the context of the Grade 1 word problem types.

They begin intentionally and energetically building fluency with addition and subtraction facts—a major gateway to later grades.

Quarter 2 Modules & Themes:

Module 2: Introduction to Place Value Through Addition and Subtraction Within 20

In Module 2, students add and subtract within 20. Work begins by modeling “adding and subtracting

across ten” in word problems and with equations. Solutions involving decomposition and

composition like that shown to the right for 8 + 5 reinforce the need to “make 10.” In Module 1,

students loosely grouped 10 objects to make a ten. They now transition to conceptualizing that ten as

a single unit (using 10 linking cubes stuck together, for example). This is the next major stepping-stone

in understanding place value, learning to group “10 ones” as a single unit: 1 ten. Learning to

“complete a unit” empowers students in later grades to understand “renaming” in the addition

algorithm, to add 298 and 35 mentally (i.e., 298 + 2 + 33), and to add measurements like 4 m, 80 cm,

and 50 cm (i.e., 4 m + 80 cm + 20 cm + 30 cm = 4 m + 1 m + 30 cm = 5 m 30 cm).

Module 3: Ordering and Comparing Length Measurements as Numbers

Module 3, which focuses on measuring and comparing lengths indirectly and by iterating length units, gives students a few weeks to practice and

internalize “making a 10” during daily fluency activities.

Quarter 3 Modules & Themes:

Module 4: Place Value, Comparison, Addition and Subtraction to 40

Module 4 returns to understanding place value. Addition and subtraction within 40 rest on firmly establishing a “ten” as a unit that can be counted,

first introduced at the close of Module 2. Students begin to see a problem like 23 + 6 as an opportunity separate the “2 tens” in 23 and concentrate

on the familiar addition problem 3 + 6. Adding 8 + 5 is related to solving 28 + 5; complete a unit of ten and add 3 more.

Module 5: Identifying, Composing, and Partitioning Shapes

In Module 5, students think about attributes of shapes and practice composing and decomposing geometric shapes. They also practice work with

addition and subtraction within 40 during daily fluency activities (from Module 4). Thus, this module provides important “internalization time” for

students between two intense number-based modules. The module placement also gives more spatially-oriented students the opportunity to build

their confidence before they return to arithmetic.

Quarter 4 Modules & Themes:

Module 6: Place Value, Comparison, Addition and Subtraction to 100

Although Module 6 focuses on “adding and subtracting within 100,” the learning goal differs from the “within 40” module. Here, the new level of

complexity is to build off the place value understanding and mental math strategies that were introduced in earlier modules. Students explore by

using simple examples and the familiar units of 10 made out of linking cubes, bundles, and drawings. Students also count to 120 and represent any

number within that range with a numeral.

2nd Grade

Summary of Year:

Module 1: Sums and Differences to 20

Module 2: Addition and Subtraction of Length Units

Module 3: Place Value, Counting, and Comparison of Numbers to 1000

Module 4: Addition and Subtraction Within 200 with Word Problems to 100

Module 5: Addition and Subtraction Within 1000 with Word Problems to 100

Module 6: Foundations of Multiplication and Division

Module 7: Problem Solving with Length, Money, and Data

Module 8: Time, Shapes, and Fractions as Equal Parts of Shapes

Second Grade mathematics is about:

(1) extending understanding of base-ten notation;

(2) building fluency with addition and subtraction;

(3) using standard units of measure; and

(4) describing and analyzing shapes.

Key Areas of Focus for K-2:

Addition and subtraction—concepts, skills, and problem solving

Required Fluency:

2.OA.2 Add and subtract within 20.

2.NBT.5 Add and subtract within 100.

Quarter 1 Modules & Themes:

Module 1 establishes a motivating, differentiated fluency program in the first few weeks that will provide each student with enough practice to achieve mastery of the new required fluencies (i.e., adding and subtracting within 20 and within 100) by the end of the year. Students learn to represent and solve word problems using addition and subtraction: a practice that will also continue throughout the year.

Module 2, students learn to measure and estimate using standard units for length and solve measurement word problems involving addition and

subtraction of length. A major objective is for students to use measurement tools with the understanding that linear measure involves an iteration of

units and that the smaller a unit, the more iterations are necessary to cover a given length.

Module 3, students work exclusively with metric units, i.e.

centimeters and meters, in this module to support upcoming work with place value concepts.

Quarter 2 Modules & Themes:

Module 3 (Topics F & G) Place Value, Counting, and Comparison of Numbers to 1,000

In Module 3, students extend their understanding of base-ten notation and apply their understanding of place value to count and compare numbers to 1000. In Grade 2 the place value units move from a proportional model to a non-proportional number disk model (see picture). The place value table with number disks can be used through Grade 5 for modeling very large numbers and decimals, thus providing students greater facility with and understanding of mental math and algorithms.

Module 4 (Topics A-E) Addition and Subtraction Within 200 with Word Problems to 100

In Module 4, students apply their work with place value units to add and subtract within 200 moving from concrete to pictorial to abstract. This work deepens their understanding of base-ten, place value, and the properties of operations. It also challenges them to apply their knowledge to one-step and two-step word problems. During this module, students also continue to develop one of the required fluencies of the grade: addition and subtraction within 100.

Quarter 3 Modules & Themes:

Module 5 Addition and Subtraction Within 1,000 with Word Problems to 100

Module 5 builds upon the work of Module 4. Students again use place value strategies, manipulatives, and math drawings to extend their conceptual understanding of the addition and subtraction algorithms to numbers within 1000. They maintain addition and subtraction fluency within 100 through daily application work to solve one- and two-step word problems of all types. A key component of Modules 4 and 5 is that students use place value reasoning to explain why their addition and subtraction strategies work.

Module 6 Foundations of Multiplication and Division

In Module 6, students extend their understanding of a unit to build the foundation for multiplication and division wherein any number, not just powers of ten, can be a unit. Making equal groups of “four apples each” establishes the unit “four apples” (or just four) that can then be counted: 1 four, 2 fours, 3 fours, etc. Relating the new unit to the one used to create it lays the foundation for multiplication: 3 groups of 4 apples equal 12 apples (or 3 fours is 12).

Quarter 4 Modules & Themes:

Module 7 Problem Solving with Length, Money, and Data

Module 7 provides another opportunity for students to practice their algorithms and problem-solving skills with perhaps the most well-known, interesting units of all: dollars, dimes, and pennies. Measuring and estimating length is revisited in this module in the context of units from both the customary system (e.g., inches and feet) and the metric system (e.g., centimeters and meters). As they study money and length, students represent data given by measurement and money data using picture graphs, bar graphs, and line plots.

Module 8 Time, Shapes, and Fractions as Equal Parts of Shapes

In Module 8, students investigate, describe, and reason about the composition and decomposition of shapes to form other shapes. Through building, drawing, and analyzing two- and three-dimensional shapes, students develop a foundation for understanding area, volume, congruence, similarity, and symmetry in later grades.

### 3rd Grade

Summary of Year:

Third Grade mathematics is about (1) developing understanding of multiplication and division and strategies for multiplication and division within 100; (2) developing understanding of fractions, especially unit fractions (fractions with numerator 1); (3) developing understanding of the structure of rectangular arrays and of area; and (4) describing and analyzing two-dimensional shapes.

Module 1: Properties of Multiplication and Division and Solving Problems with Units of 2–5 and 10 Module 2: Place Value and Problem Solving with Units of Measure

Module 3: Multiplication and Division with Units of 0, 1, 6–9, and Multiples of 10

Module 4: Multiplication and Area

Module 5: Fractions as Numbers on the Number Line Module 6: Collecting and Displaying Data

Module 7: Geometry and Measurement Word Problems

Key Areas of Focus for 3-5: Multiplication and division of whole numbers and fractions—concepts, skills, and problem solving

Required Fluency: 3.OA.7 Multiply and divide within 100. 3.NBT.2 Add and subtract within 1000.

CCLS Major Emphasis Clusters:

￼Operations and Algebraic Thinking

• Represent and solve problems involving multiplication and division.

• Understand the properties of multiplication and the relationship between multiplication and division.

• Multiply and divide within 100.

• Solve problems involving the four operations and identify and explain patterns in arithmetic. Number and Operations – Fractions

• Develop understanding of fractions as numbers. Measurement and Data

• Solve problems involving measurement and estimation of intervals of time, liquid volumes, and masses of objects.

• Geometric measurement: understand concepts of area and relate area to multiplication and to addition.

Quarter 1 Modules & Themes:

Properties of Multiplication and Division and Solving Problems with Units of 2–5 and 10

The first module builds upon the foundation of multiplicative thinking with units started in Grade 2. First, students concentrate on the meaning of multiplication and division and begin developing fluency for learning products involving factors of 2, 3, 4, 5, and 10. The restricted set of facts keeps learning manageable and also provides enough examples for one- and two-step word problems and for starting measurement problems involving weight, capacity, and time in the second module.

Module 2 is separated over A1 and A2 for pacing considerations. A1 assesses Topics A through C, which focus on measurement of time, metric weight and capacity, place value, and rounding. Module 2 provides students with internalization time for learning the 2, 3, 4, 5, and 10 facts as part of their fluency activities. Students then apply their new understanding of numbers to use place value understandings and the number line as tools to round two-, three-, and four-digit measurements to the nearest ten or hundred. Weight and capacity are assessed in A2, since the standard encompasses multiplication and division, which are further developed through Module 3.

Quarter 2 Modules & Themes:

Module 2 D through E, Module 3, and Module 4 Topics A and B

Place Value and Problem Solving with Units of Measure (continued)

In A2, students continue to move through Module 2, applying their new understanding of numbers to compose larger units when adding and decompose into smaller units when subtracting. Students will also solve word problems using number lines and proportional tape diagrams. Drawing the relative sizes of the lengths involved in the model prepares students to locate fractions on a number line in Module 5 (where they learn to locate points on the number line relative to each other and relative to the whole unit).

Multiplication and Division with Units of 0, 1, 6–9, and Multiples of 10

Students learn the remaining multiplication and division facts in Module 3 as they continue to develop their understanding of multiplication and division strategies within 100 and use those strategies to solve two-step word problems. The “2, 3, 4, 5 and 10 facts” module (Module 1) and the “0, 1, 6, 7, 8, 9 and multiples of 10 facts” module (Module 3) both provide important, sustained time for work in understanding the structure of rectangular arrays to prepare students for area in Module 4. This work is necessary because students may initially find it difficult to distinguish the different units in a grid, count them, and recognize that the count is related to multiplication. Tiling also supports a correct interpretation of the grid. Modules 1 and 3 slowly build up to the area model rectangular arrays in the context of learning multiplication and division.

Multiplication and Area

By Module 4, students are ready to investigate area. This module is separated over A2 and A3 for pacing consideration. In Topics A and B, students begin to conceptualize area as the amount of two dimensional surface that is contained within a plane figure. They also measure the area of a shape by finding the total number of same- size units of area, e.g. tiles, required to cover the shape without gaps or overlaps. Students connect their extensive work with rectangular arrays and multiplication to eventually discover the area formula for a rectangle, which is formally introduced in Grade 4.

Quarter 3 Modules & Themes:

Module 4 Topics C and D and Module 5 Topics A through D

Multiplication and Area (continued)

Students continue to progress with Module 4. In Topic C, students manipulate rectangular arrays to concretely demonstrate the arithmetic properties. They apply tiling and multiplication skills to determine all whole number possibilities for the side lengths of rectangles given their areas. Topic D creates an opportunity for students to solve problems involving area. Students decompose or compose composite regions into non-overlapping rectangles, find the area of each region, and then add or subtract to determine the total area of the original shape.

Fractions as Numbers on the Number Line

In Module 5, students deepen Grade 2 practice with equal shares to understand fractions as equal partitions of a whole. A3 covers Topics A through D, where student’s knowledge of fractions becomes more formal as they work with area models and the number line. One goal of Module 5 is for students to transition from thinking of fractions as area or parts of a figure to points on a number line. To make that jump, students think of fractions as being constructed out of unit fractions. Once the unit fraction has been established, counting them is as easy as counting whole numbers: 1 fourth, 2 fourths, 3 fourths, 4 fourths, 5 fourths, etc.

Quarter 4 Modules & Themes:

Module 5 Topic E and F, Module 6, and Module 7

Fractions as Numbers on the Number Line (continued)

In A4 students continue to progress through Module 5. In Topic E, they should notice that some fractions with different units are placed at the exact same point on the number line, and therefore, are equal. Students should recognize that whole numbers can be written as fractions. Topic F concludes the module with comparing fractions that have the same numerator. As students compare fractions by reasoning about their size, they understand that fractions with the same numerator and a larger denominator are actually smaller pieces of the whole. Topic F leaves students with a new method for precisely partitioning a number line into unit fractions of any size without using a ruler.

Collecting and Displaying Data

In Module 6, students leave the world of exact measurements behind. By applying their knowledge of fractions from Module 5, they estimate lengths to the nearest halves and fourths of an inch and record that information in bar graphs and line plots. This module also prepares students for the multiplicative comparison problems of Grade 4 by asking students “how many more” and “how many less” questions about scaled bar graphs. The year rounds out with plenty of time to solve two-step word problems involving the four operations, and to improve fluency for concepts and skills initiated earlier in the year.

Geometry and Measurement Word Problems

In Module 7, students also describe, analyze, and compare properties of two-dimensional shapes. By now, students have done enough work with both linear and area measurement models to understand that there is no relationship in general between the area of a figure and perimeter, which is a concept taught in the last module.

4th Grade

Summary of Year:

Fourth grade mathematics is about (1) developing understanding and fluency with multi-digit multiplication, and developing understanding of dividing to find quotients involving multi-digit dividends; (2) developing an understanding of fraction equivalence, addition and subtraction of fractions with like denominators, and multiplication of fractions by whole numbers; and (3) understanding that geometric figures can be analyzed and classified based on their properties, such as having parallel sides, perpendicular sides, particular angle measures, and symmetry.

Module 1: Place Value, Rounding, and Algorithms for Addition and Subtraction Module 2: Unit Conversions and Problem Solving with Metric Measurement Module 3: Multi-Digit Multiplication and Division

Module 4: Angle Measure and Plane Figures

Module 5: Fraction Equivalence, Ordering, and Operations Module 6: Decimal Fractions

Module 7: Exploring Multiplication

Key Areas of Focus for 3-5: Multiplication and division of whole numbers and fractions—concepts, skills, and problem solving

Required Fluency: 4.NBT.4 Add and subtract within 1,000,000.

CCLS Major Emphasis Clusters:

Operations and Algebraic Thinking

• Use the four operations with whole numbers to solve problems.

Number and Operations in Base Ten

• Generalize place value understanding for multi-digit whole numbers.

• Use place value understanding and properties of operations to perform multi-digit arithmetic.

Number and Operations – Fractions

• Extend understanding of fraction equivalence and ordering.

• Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers.

• Understand decimal notation for fractions, and compare decimal fractions.

Quarter 1 Modules & Themes:

Module 1, Module 2, and Module 3 Topics A through C

Place Value, Rounding, and Algorithms for Addition and Subtraction

In Grade 4, students extend their work with whole numbers. They begin with large numbers using familiar units (tens and hundreds) and develop their understanding of thousands by building knowledge of the pattern of times ten in the base ten system on the place value chart. In Grades 2 and 3 students focused on developing the concept of composing and decomposing place value units within the addition and subtraction algorithms. Now, in Grade 4, those (de)compositions and are seen through the lens of multiplicative comparison, e.g. 1 thousand is 10 times as much as 1 hundred. They next apply their broadened understanding of patterns on the place value chart to compare, round, add and subtract. The module culminates with solving multi-step word problems involving addition and subtraction modeled with tape diagrams that focus on numerical relationships.

Unit Conversions and Problem Solving with Metric Measurement

The algorithms continue to play a part in Module 2 as students relate place value to metric units. Students work with metric measurement in the context of the addition and subtraction algorithms, mental math, place value, and word problems. Customary units are used as a context for fractions in Module 5.

Multi-Digit Multiplication

Module 3 is separated over A1 and A2 for pacing considerations. Students investigate the formulas for area and perimeter (which will be assessed in A2, using the full magnitude of numbers expected for multiplication and division in Grade 4). They then solve multiplicative comparison problems including the language of “times as much as” with a focus on problems using area and perimeter as a context. This is foundational for understanding multiplication as scaling in Grade 5 and sets the stage for proportional reasoning in Grade 6. This Grade 4 module, beginning with area and perimeter, allows for new and interesting word problems as students learn to calculate with larger numbers and interpret more complex problems. Students decompose numbers into base ten units in order to find products of single-digit by multi-digit numbers. Students bridge partial products to the recording of multiplication via the standard algorithm. Finally, the partial products method, the standard algorithm, and the area model are compared and connected by the distributive property.

Quarter 2 Modules & Themes:

Module 3 Topics D through H

Multi-Digit Multiplication (continued) and Division

In A2, students continue to progress with Module 3. Students apply their new multiplication skills to solve multi- step word problems and multiplicative comparison problems. Students write equations from statements within the problems and use a combination of addition, subtraction, and multiplication to solve. Students focus on interpreting the remainder within division problems, both in word problems and long division. Students 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. Students then practice using the standard algorithm to record long division, and they solve word problems. Students explore factors, multiples, and prime and composite numbers within 100, gaining valuable insights into patterns of divisibility as they test for primes and find factors and multiples. The module closes as students multiply two-digit by two-digit numbers.

Quarter 3 Modules & Themes:

Module 5: Fraction Equivalence, Ordering, and Operations

Students build on their Grade 3 work with unit fractions as they explore fraction equivalence and extend this understanding to mixed numbers. This leads to the comparison of fractions and mixed numbers and the representation of both in a variety of models. Benchmark fractions play an important part in students’ ability to generalize and reason about relative fraction and mixed number sizes. Students then have the opportunity to apply what they know to be true for whole number operations to the new concepts of fraction and mixed number operations.

Quarter 4 Modules & Themes:

Module 4, Module 6, and Module 7 Topics A through C

*Module 7 Topic D “Year in Review” is not included

Angle Measure and Plane Figures

Module 4 focuses as much on solving unknown angle problems using letters and equations as it does on building, drawing, and analyzing two-dimensional shapes in geometry. Students have already used letters and equations to solve word problems in earlier grades. They continue to do so in Grade 4, and now they also learn to solve unknown angle problems: work that challenges students to build and solve equations to find unknown angle measures. Students learn the definition of a degree and learn how to measure angles in degrees using a protractor. They also apply their understanding that angle measures are additive. Unknown angle problems help to unlock algebraic concepts for students because such problems are visual.

Decimal Fractions

Module 6 starts with the realization that decimal place value units are simply special fractional units. Students explore decimal numbers via their relationship to decimal fractions, expressing a given quantity in both fraction and decimal forms. Fluency plays an important role in this topic as students learn to recognize that 3/10 = 0.3 = 3 tenths. They also recognize that 3 tenths is equal to 30 hundredths and subsequently have their first experience adding and subtracting fractions with unlike denominators. Utilizing the understanding of fractions developed throughout Module 5, students apply the same reasoning to decimal numbers, building a solid foundation for Grade 5 work with decimal operations.

Exploring Measurement with Multiplication

Module 7 is focused on multiplication and measurement as students solve multi-step word problems. Students build their competencies in measurement as they relate multiplication to the conversion of measurement units. Throughout the module, students will explore multiple strategies for solving measurement problems involving unit conversion.

5th Grade

Summary of Year:

Fifth grade mathematics is about (1) developing fluency with addition and subtraction of fractions, and developing understanding of the multiplication of fractions and of division of fractions in limited cases (unit fractions divided by whole numbers and whole numbers divided by unit fractions); (2) extending division to two-digit divisors, integrating decimal fractions into the place value system and developing understanding of operations with decimals to hundredths, and developing fluency with whole number and decimal operations; and (3) developing understanding of volume.

Module 1: Place Value and Decimal Fractions

Module 2: Multi-Digit Whole Number and Decimal Fraction Operations Module 3: Addition and Subtraction of Fractions

Module 4: Multiplication and Division of Fractions and Decimal Fractions Module 5: Addition and Multiplication with Volume and Area

Module 6: Problem Solving with the Coordinate Plane

Key Areas of Focus for 3-5: Multiplication and division of whole numbers and fractions—concepts, skills, and problem solving

Required Fluency: 5.NBT.5 Multi-digit multiplication.

CCLS Major Emphasis Clusters:

Number and Operations in Base Ten

• Understand the place value system.

• Perform operations with multi-digit whole numbers and with decimals to hundredths. Number and Operations – Fractions

• Use equivalent fractions as a strategy to add and subtract fractions.

• Apply and extend previous understandings of multiplication and division to multiply and divide fractions.

Measurement and Data

• Geometric measurement: understand concepts of volume and relate volume to multiplication and to addition.

Quarter 1 Modules & Themes:

Module 1 and Module 2 Topics A and C

Place Value and Decimal Fractions

Students’ experiences with the algorithms as ways to manipulate place value units in Grades 2-4 really begin to pay dividends in Grade 5. In Module 1, whole number patterns with number disks on the place value table are easily generalized to decimal numbers. As students work word problems with measurements in the metric system, where the same patterns occur, they begin to appreciate the value and the meaning of decimals. Students begin to apply their work with place value to adding, subtracting, multiplying and dividing decimal numbers with tenths and hundredths (continued in Module 2).

Multi-Digit Whole Number and Decimal Fraction Operations

Module 2 begins by using place value patterns and the distributive and associative properties to multiply multi- digit numbers by multiples of 10 and leads to fluency with multi-digit whole number multiplication. (Multi-digit decimal multiplication such as 4.1 × 3.4 and division such as 4.5 ÷ 1.5 are studied in Module 4.) Students evaluate and write simple expressions to record their calculations using the associative property and parentheses to record the relevant order of calculations. In this Module, place value understanding moves toward understanding the distributive property via area models which are used to generate and record the partial products of the standard algorithm. For multiplication, students must grapple with and fully understand the distributive property (one of the key reasons for teaching the multi-digit algorithm). While the multi-digit multiplication algorithm is a straightforward generalization of the one-digit multiplication algorithm, the division algorithm with two-digit divisors requires far more care to teach because students have to also learn estimation strategies, error correction strategies, and the idea of successive approximation (all of which are central concepts in math, science, and engineering).

Quarter 2 Modules & Themes:

Modules 2 Topics C through H and Module 3

Multi-Digit Whole Number and Decimal Fraction Operations (continued)

In A2 students continue work with Module 2. Students move from whole numbers to multiplication with decimals, again using place value as a guide to reason and make estimations about products. Students explore multiplication as a method for expressing equivalent measures. For example, they multiply to convert between meters and centimeters or ounces and cups with measurements in both whole number and decimal form. Topics E through H provide a similar sequence for division, beginning with an introduction to division with multi-digit whole numbers. Students use properties of operations to interpret 420 ÷ 60 as 420 ÷ 10 ÷ 6. This Module leads students to divide multi-digit dividends by two-digit divisors using the written vertical method. Students use their understanding to divide decimals by two-digit divisors in a sequence similar to that of Topic F with whole numbers. Students apply the work of the module to solve multi-step word problems using multi-digit division with unknowns representing either the group size or number of groups. There is an emphasis on checking the reasonableness of their answers that draws on skills learned throughout the module, including refining their knowledge of place value, rounding, and estimation.

Addition and Subtraction of Fractions

Work with place value units paves the path toward fraction arithmetic in Module 3 as elementary math’s place value emphasis shifts to the larger set of fractional units for algebra. Like units are added to and subtracted from like units. The new complexity is that when units are not equivalent, they must be changed for equal units so that they can be added or subtracted. The equivalence is represented symbolically as students engage in active meaning-making rather than obeying the perhaps mysterious command to “multiply the top and bottom by the same number.” The full breadth of 5.OA.1 and 5.OA.2 will be assessed, including the use of brackets, in A2. Though introduced in Module 2, 5.OA.1 and 5.OA.2 are not fully covered by any Modules to include using brackets to write and evaluate numerical expressions. Please make adjustments as needed.

Quarter 3 Modules & Themes:

Module 4

Multiplication and Division of Fractions and Decimal Fractions

In Module 4, equal sharing with area models (both concrete and pictorial) provides students with an opportunity to understand division of whole numbers with answers in the form of fractions or mixed numbers. Discussion also includes an interpretation of remainders as a fraction. Students interpret finding a fraction of a set (3/4 of 24) as multiplication of a whole number by a fraction (3/4 × 24). This, in turn, leads students to see division by a whole number as being equivalent to multiplication by its reciprocal. Students apply their knowledge of a fraction of a set and previous conversion experiences (with scaffolding from a conversion chart, if necessary) to find a fraction of a measurement, thus converting a larger unit to an equivalent smaller unit. Students start with multiplying a unit fraction by a unit fraction and progress to multiplying two non-unit fractions. This intensive work with fractions positions students to extend their previous work with decimal-by-whole number multiplication to decimal-by-decimal multiplication. Students extend their understanding of multiplication to include scaling. Students compare the product to the size of one factor, given the size of the other factor without calculation. This reasoning, along with the other work of this module, sets the stage for reasoning about the size of products when quantities are multiplied by numbers larger than 1 and smaller than 1. Students relate their previous work with equivalent fractions to interpreting multiplication by 𝑛𝑛/𝑛𝑛 as multiplication by 1. Students also begin the work of division with both fractions and decimal fractions. Using the same thinking developed in Module 2 to divide whole numbers, students reason about how many fourths are in 5 when considering such cases as 5 ÷ 1/4 . They also reason about the size of the unit when 1/4 is partitioned into 5 equal parts: 1/4 ÷ 5. Using this thinking as a backdrop, students are introduced to decimal fraction divisors and use equivalent fraction and place value thinking to reason about the size of quotients, calculate quotients, and sensibly place the decimal in quotients.

Quarter 4 Modules & Themes:

Module 5 and Module 6

*not including Module 6 Topic E and F “Multi-Step Word Problems” and “The Years in Review: A Reflection on A Story of Units”

Addition and Multiplication with Volume and Area

Frequent use of the area model in Modules 3 and 4 prepares students for an in-depth discussion of area and volume in Module 5. But the module on area and volume also reinforces work done in the fraction module. Now, questions about how the area changes when a rectangle is scaled by a whole or fractional scale factor may be asked and missing fractional sides may be found. Measuring volume once again highlights the unit theme, as a unit cube is chosen to represent a volume unit and used to measure the volume of simple shapes composed out of rectangular prisms.

Problem Solving with the Coordinate Plane

Scaling is revisited in the last module on the coordinate plane. Since Kindergarten, where growth and shrinking patterns were first introduced, students have been using bar graphs to display data and patterns. Extensive bar- graph work has set the stage for line plots, which are both the natural extension of bar graphs and the precursor to linear functions. It is in this final module of K-5 that a simple line plot of a straight line is presented on a coordinate plane and students are asked about the scaling relationship between the increase in the units of the vertical axis for 1 unit of increase in the horizontal axis. This is the first hint of slope and marks the beginning of the major theme of middle school: ratios and proportions.

6th Grade

Summary of Year:

Sixth grade mathematics is about (1) connecting ratio and rate to whole number multiplication and division and using concepts of ratio and rate to solve problems; (2) completing understanding of division of fractions and extending the notion of number to the system of rational numbers, which includes negative numbers; (3) writing, interpreting, and using expressions and equations; and (4) developing understanding of statistical thinking.

Module 1: Ratios and Unit Rates

Module 2: Arithmetic Operations Including Dividing by a Fraction

Module 3: Rational Numbers

Module 4: Expressions and Equations

Module 5: Area, Surface Area, and Volume Problems

Module 6: Statistics

Key Areas of Focus: Ratios and proportional reasoning; early expressions and equations

Required Fluency:

6.NS.2 Multi-digit division

6.NS.3 Multi-digit decimal operations

CCLS Major Emphasis Clusters:

Ratios and Proportional Relationships

• Understand ratio concepts and use ratio reasoning to solve problems. The Number System

• Apply and extend previous understandings of multiplication and division to divide fractions by fractions.

• Apply and extend previous understandings of numbers to the system of rational numbers. Expressions and Equations

• Apply and extend previous understandings of arithmetic to algebraic expressions.

• Reason about and solve one-variable equations and inequalities.

• Represent and analyze quantitative relationships between dependent and independent variables.

Quarter 1 Modules & Themes:

PModule 1 and 2 Topic A

Ratios and Unit Rates

In Module 1, students build on their prior work in measurement and multiplication and division as they study the concepts and language of ratios and unit rates. Students use proportional reasoning to solve problems. In particular, students solve ratio and rate problems using tape diagrams, tables of equivalent ratios, double number line diagrams, and equations. They plot pairs of values generated from a ratio or rate on the first quadrant of the coordinate plane.

Dividing Fractions by Fractions

Students expand their understanding of the number system and build their fluency in arithmetic operations in Module 2 (separated over A1 and A2 for pacing considerations). Students learned in Grade 5 to divide whole numbers by unit fractions and unit fractions by whole numbers. Now, they apply and extend their understanding of multiplication and division to divide fractions by fractions. The meaning of this operation is connected to real- world problems as students are asked to create and solve fraction division word problems.

Quarter 2 Modules & Themes:

Module 2 Topics B, C, D and Module 3

Arithmetic Operations

Students continue to expand their understanding of the number system and build their fluency in arithmetic operations as they complete Module 2. Students continue (from Fifth Grade) to build fluency with dividing multi-digit whole numbers, and adding, subtracting, multiplying, and dividing multi-digit decimal numbers using the standard algorithms.

Rational Numbers

Major themes of Module 3 are to understand rational numbers as points on the number line and to extend previous understandings of numbers to the system of rational numbers, which now include negative numbers. Students extend coordinate axes, from Quadrant 1 in Fifth Grade, to represent points in the plane with negative number coordinates and, as part of doing so, see that negative numbers can represent quantities in real-world contexts. They use the number line to order numbers and to understand the absolute value of a number. They begin to solve real-world and mathematical problems by graphing points in all four quadrants, a concept that continues throughout to be used into high school and beyond.

Quarter 3 Modules & Themes:

Module 3: Rational Numbers

Major themes of Module 3 are to understand rational numbers as points on the number line and to extend previous understandings of numbers to the system of rational numbers, which now include negative numbers. Students extend coordinate axes, from Quadrant 1 in Fifth Grade, to represent points in the plane with negative number coordinates and, as part of doing so, see that negative numbers can represent quantities in real-world contexts. They use the number line to order numbers and to understand the absolute value of a number. They begin to solve real-world and mathematical problems by graphing points in all four quadrants, a concept that continues throughout to be used into high school and beyond.

Module 4: Expressions and Equations

With their sense of numbers expanded to include negative numbers, in Module 4 students begin a formal study of algebraic expressions and equations. Students learn equivalent expressions by continuously relating algebraic expressions back to arithmetic and the properties of arithmetic (commutative, associative, and distributive). They write, interpret, and use expressions and equations as they reason about and solve one-variable equations and inequalities and analyze quantitative relationships between two variables.

Quarter 4 Modules & Themes:

Module 5 and 6

Area, Surface Area, and Volume Problems

Module 5 is an opportunity to practice the material learned in Module 4 in the context of geometry; students apply their newly acquired capabilities with expressions and equations to solve for unknowns in area, surface area, and volume problems. They find the area of triangles and other two dimensional figures and use the formulas to find the volumes of right rectangular prisms with fractional edge lengths. Students use negative numbers in coordinates as they draw lines and polygons in the coordinate plane. They also find the lengths of sides of figures, joining points with the same first coordinate or the same second coordinate and apply these techniques to solve real-world and mathematical problems.

Statistics

In Module 6, students develop an understanding of statistical variability and apply that understanding as they summarize, describe, and display distributions. In particular, careful attention is given to measures of center and variability.

7th Grade

Summary of Year:

Seventh grade mathematics is about (1) developing understanding of and applying proportional relationships; (2) developing understanding of operations with rational numbers and working with expressions and linear equations; (3) solving problems involving scale drawings and informal geometric constructions, and working with two- and three-dimensional shapes to solve problems involving area, surface area, and volume; and (4) drawing inferences about populations based on samples.

Module 1: Ratios and Proportional Relationships Module 2: Rational Numbers

Module 3: Expressions and Equations

Module 4: Percent and Proportional Relationships Module 5: Statistics and Probability

Module 6: Geometry

Key Areas of Focus for Grade 7: Ratios and proportional reasoning; arithmetic of rational numbers

CCLS Major Emphasis Clusters:

Ratios and Proportional Relationships

• Analyze proportional relationships and use them to solve real-world and mathematical problems. The Number System

• Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.

Expressions and Equations

• Use properties of operations to generate equivalent expressions.

• Solve real-life and mathematical problems using numerical and algebraic expressions and equations.

Quarter 1 Modules & Themes:

Module 1 and 2 Topic A

Ratios and Proportional Relationship

In Module 1, students build on their Grade 6 experiences with ratios, unit rates, and fraction division to analyze proportional relationships. They decide whether two quantities are in a proportional relationship, identify constants of proportionality, and represent the relationship by equations. These skills are then applied to real- world problems, including scale drawings.

Rational Numbers

In Grade 6, students formed a conceptual understanding of integers through the use of the number line, absolute value, and opposites and extended their understanding to include the ordering and comparing of rational numbers. In Module 2 (separated over A1 and A2 for pacing considerations), students continue to build an understanding of the number line from their work in Grade 6. They learn to add, subtract rational numbers by A1. Previous work in computing the sums, differences, products, and quotients of fractions and decimals serves as a significant foundation as well.

Quarter 2 Modules & Themes:

Module 2 Topics B and C and Module 3 Topic A

Rational Numbers (continued)

Students continue with Module 2 to multiply and divide rational numbers. Students develop the rules for multiplying and dividing signed numbers. They use the properties of operations and their previous understanding of multiplication as repeated addition to represent the multiplication of a negative number as repeated subtraction. Students represent the division of two integers as a fraction, extending product and quotient rules to all rational numbers. They realize that any rational number in fractional form can be represented as a decimal that either terminates in 0s or repeats. Students recognize that the context of a situation often determines the most appropriate form of a rational number, and they use long division, place value, and equivalent fractions to fluently convert between these fractions and decimal forms. Module 2 includes rational numbers as they appear in expressions and equations—work that is continued in Module 3.

Expressions and Equations

In Grade 6, students interpreted expressions and equations as they reasoned about one-variable equations. Module 3 (separated over A2 and A3 for pacing considerations) consolidates and expands upon students’ understanding of equivalent expressions as they apply the properties of operations (associative, commutative, and distributive) to write expressions in both standard form (by expanding products into sums) and in factored form (by expanding sums into products). To begin this module, students will generate equivalent expressions using the fact that addition and multiplication can be done in any order with any grouping and will extend this understanding to subtraction (adding the inverse) and division (multiplying by the multiplicative inverse, also known as the reciprocal). They extend the properties of operations with numbers (learned in earlier grades) and recognize how the same properties hold true for letters that represent numbers. Knowledge of rational number operations from Module 2 is demonstrated as students collect like terms containing both positive and negative integers.

Quarter 3 Modules & Themes:

Module 3 Topics B and C, Module 4, and Module 4 Topics A and B

Expressions and Equations (continued)

Students continue with Module 3. Students solve real-life and mathematical problems using numerical and algebraic expressions and equations. Their work with expressions and equations is applied to finding unknown angles and problems involving area, volume, and surface area, which continues in Module 6.

Percent and Proportional Relationships

Module 4 parallels Module 1’s coverage of ratio and proportion, but this time with a concentration on percent. Problems in this module include simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, and percent error. Additionally, this module includes percent problems about populations, which prepare students for probability models about populations covered in the next module.

Statistics and Probability

Module 5 is separated over A3 and A4 for pacing considerations. In Module 5, students learn to interpret the probability of an event as the proportion of the time that the event will occur when a chance experiment is repeated many times. They learn to compute or estimate probabilities using a variety of methods, including collecting data, using tree diagrams, and using simulations. Students move to comparing probabilities from simulations to computed probabilities that are based on theoretical models. They calculate probabilities of compound events using lists, tables, tree diagrams, and simulations. They learn to use probabilities to make decisions and to determine whether or not a given probability model is plausible.

Quarter 4 Modules & Themes:

Module 5 Topics C and D and Module 6

Statistics and Probability (continued)

In Module 5 Topics C and D, students focus on using random sampling to draw informal inferences about a population. In Topic C, they investigate sampling from a population. They learn to estimate a population mean using numerical data from a random sample. They also learn how to estimate a population proportion using categorical data from a random sample. In Topic D, students learn to compare two populations with similar variability. They learn to consider sampling variability when deciding if there is evidence that the means or the proportions of two populations are actually different

Geometry

The year concludes with students drawing and constructing geometrical figures in Module 6. They also revisit unknown angle, area, volume, and surface area problems, which now include problems involving percentages of areas or volumes.

8th Grade

Summary of Year:

Eighth grade mathematics is about (1) formulating and reasoning about expressions and equations, including modeling an association in bivariate data with a linear equation, and solving linear equations and systems of linear equations; (2) grasping the concept of a function and using functions to describe quantitative relationships; (3) analyzing two- and three-dimensional space and figures using distance, angle, similarity, and congruence, and understanding and applying the Pythagorean Theorem.

Module 1: Integer Exponents and Scientific Notation Module 2: The Concept of Congruence

Module 3: Similarity

Module 4: Linear Equations

Module 5: Examples of Functions from Geometry

Module 6: Linear Functions

Module 7: Introduction to Irrational Numbers Using Geometry

Key Areas of Focus for Grade 8: Linear algebra

CCLS Major Emphasis Clusters:

• Expressions and Equations

• Work with radicals and integer exponents.

• Understand the connections between proportional relationships, lines, and linear equations.

• Analyze and solve linear equations and pairs of simultaneous linear equations. Functions

• Define, evaluate, and compare functions.

Geometry

• Understand congruence and similarity using physical models, transparencies, or geometry software.

• Understand and apply the Pythagorean Theorem.

Quarter 1 Modules & Themes:

Module 1, Module 2, and Module 3 Topic A

Integer Exponents and Scientific Notation

In Module 1, students’ knowledge of operations on numbers will be expanded to include operations on numbers in integer exponents. Module 1 also builds on students’ understanding from previous grades with regard to transforming expressions. Students build upon their foundation with exponents as they make conjectures about how zero and negative exponents of a number should be defined and prove the properties of integer exponents. They make sense out of very large and very small numbers. Having established the properties of integer exponents, students learn to express the magnitude of a positive number through the use of scientific notation and to compare the relative size of two numbers written in scientific notation. Students explore use of scientific notation and choose appropriately sized units as they represent, compare, and make calculations with very large quantities and very small quantities.

The Concept of Congruence

In Module 2, students learn about rigid motions, i.e. translations, reflections, and rotations, in the plane and, more importantly, how to use them to precisely define the concept of congruence. Students verify experimentally the basic properties of rotations, reflections, and translations and deepen their understanding of these properties using reasoning. They describe the sequence of various combinations of rigid motions while maintaining the basic properties of individual rigid motions. Students learn that congruence is just a sequence of basic rigid motions. ￼The module ends by introducing the Pythagorean Theorem

Similarity

Module 3 is separated over A1 and A2 for pacing considerations. The experimental study of rotations, reflections, and translations in Module 2 prepares students for the more complex work of understanding the effects of dilations on geometrical figures in their study of similarity in Module 3. Students learn about dilation and similarity. They describe the effect of dilations on two dimensional figures in general and using coordinates. Students learn that dilations are angle-preserving transformations. Students apply this knowledge of proportional relationships and rates to determine if two figures are similar, and if so, by what scale factor one can be obtained from the other. By looking at the effect of a scale factor on the length of a segment of a given figure, students will write proportions to find missing lengths of similar figures.

Quarter 2 Modules & Themes:

Module 3 Topics B and C and Module 4 Topics A through C

Similarity (continued)

In A2, students continue to progress through Module 3. Students demonstrate that a two-dimensional figure is similar to another if the second can be obtained from a dilation followed by congruence. Knowledge of basic rigid motions is reinforced throughout the module, specifically when students describe the sequence that exhibits a similarity between two given figures.

Linear Equations

Module 4 is separated over A2 and A3 for pacing considerations. In Module 4, students use similar triangles learned in Module 3 to explain why the slope of a line is well-defined. Students learn the connection between proportional relationships, lines, and linear equations as they develop ways to represent a line by different equations (y = mx + b, y – y1 = m (x – x1), etc.). Students learn that not every linear equation has a solution and learn how to transform given equations into simpler forms until an equivalent equation results in a unique solution, no solution, or infinitely many solutions. Students must write and solve linear equations in real-world and mathematical situations. Students know that the slope of a line describes the rate of change of a line. Students first encounter slope by interpreting the unit rate of a graph. In general, students learn that slope can be determined using any two distinct points on a line by relying on their understanding of properties of similar triangles from Module 3. Students derive 𝑦𝑦 = 𝑚𝑚𝑥𝑥 and 𝑦𝑦 = 𝑚𝑚𝑥𝑥 + 𝑏𝑏 for linear equations by examining similar triangles.

Quarter 3 Modules & Themes:

Module 4 Topics D and E, Module 5, and Module 6 Topics A through C

System of Equations

In A3 students continue to progress with linear equations through systems of equations. They analyze and solve linear equations and pairs of simultaneous linear equations. Students graph simultaneous linear equations to find the point of intersection and then verify that the point of intersection is in fact a solution to each equation in the system. To motivate the need to solve systems algebraically, students graph systems of linear equations whose solutions do not have integer coordinates. Students learn to solve systems of linear equations by substitution and elimination. Students understand that a system can have a unique solution, no solution, or infinitely many solutions, as they did with linear equations in one variable. Finally, students apply their knowledge of systems to solve problems in real-world contexts.

Examples of Functions from Geometry

Students are introduced to functions in the context of linear equations and area/volume formulas in Module 5. Students learn that the definition of a graph of a function is the set of ordered pairs consisting of an input and the corresponding output. Students inspect the rate of change of linear functions and conclude that the rate of change is the slope of the graph of a line. They learn to interpret the equation 𝑦𝑦 = 𝑚𝑚𝑥𝑥 + 𝑏𝑏 as defining a linear function whose graph is a line. Once students understand the graph of a function, they begin comparing two functions represented in different ways. They define, evaluate, and compare functions using equations of lines as a source of linear functions and area and volume formulas as a source of non-linear functions.

Linear Functions

Module 6 is separated over A3 and A4 for pacing considerations. In Grades 6 and 7, students worked with data involving a single variable. In this module, students use their understanding of functions to model the relationships of bivariate data. Students examine the relationship between two variables using linear functions. Linear functions are connected to a context using the initial value and slope as a rate of change to interpret the context. Slope is also interpreted as an indication of whether the function is increasing or decreasing and as an indication of the steepness of the graph of the linear function. Nonlinear functions are explored by examining nonlinear graphs and verbal descriptions of nonlinear behavior. Students use linear functions to model the relationship between two quantitative variables as students move to the domain of statistics and probability. Students make scatter plots based on data. They also examine the patterns of their scatter plots or given scatter plots. Students assess the fit of a linear model by judging the closeness of the data points to the line.

Quarter 4 Modules & Themes:

A4 includes Module 6 Topic D and Module 7

Linear Functions (continued)

In A3, students continue to progress through Module 6. Students use linear and nonlinear models to interpret the rate of change and the initial value in context. They use the equation of a linear function and its graph to make predictions. Students also examine graphs of nonlinear functions and use nonlinear functions to model relationships that are nonlinear. Students gain experience with the mathematical practice of “modeling with mathematics” (MP.4). Students examine bivariate categorical data by using two-way tables to determine relative frequencies. They use the relative frequencies calculated from tables to informally assess possible associations between two categorical variables.

Introduction to Irrational Numbers Using Geometry

By Module 7 students have been using the Pythagorean Theorem for several months. They are sufficiently prepared to learn and explain a proof of the theorem on their own. The Pythagorean Theorem is also used to motivate a discussion of irrational square roots (irrational cube roots are introduced via volume of a sphere). Thus, as the year began with looking at the number system, so it concludes with students understanding irrational numbers and ways to represent them (radicals, non-repeating decimal expansions) on the real number line.