math schedule of assessed standards (sas) development and...

EngageNY Aligned Math SAS Supporting Document 2016-2017

© 2016 Achievement Network. 1

Math Schedule of Assessed Standards (SAS) Development and RationaleThis document includes information about the purpose, development, and content of the math SAS, as well as detailed grade-level sequencing comments.

What is the purpose of the SAS?

ANet interim assessments are formative assessments designed to provide teachers with actionable information about what students know and can do, relative to what has been taught. The assessments are carefully crafted to meet the demands of the Common Core and to provide teachers with timely, actionable information to inform instruction. The SAS is an overview of what will be assessed across interims throughout the school year. The purpose of the SAS is

● to identify the standards assessed on each interim● to supplement a pacing guide or other curriculum materials● to aid in the vertical planning process● to support how you approach prioritizing standards and to support in making connections across

standards, clusters, and domains The SAS does not

● tell you when to begin teaching a standard, the order in which you should teach the standards, or how much instructional time should be spent on any given standard

● act as a stand-alone planning document (e.g. unit plan) or provide full standard descriptions● unpack standards to a “teachable” level (tell you how to teach the standards)

Please Note: Teachers are encouraged to plan ahead and teach some standards before the cycle in which they are first assessed on the SAS, especially fluency standards, culminating standards, and opportunities for in-depth focus.

How was the 16-17 SAS developed?

The EngageNY Aligned Math SAS was developed to follow the sequencing of the EngageNY Math Modules, which is a high-quality curriculum resource that aligns to the Common Core standards and is provided for free by the New York State Education Department. This SAS also aligns to the sequence of standards in Eureka Math. Assessments will be available in paper-based and online formats. The 16-17 Math SAS development was a collaborative process, incorporating

● input and feedback from teachers, instructional leaders, and partners● recommendations from ANet coaches and the Assessment Team● information published by trusted resources (e.g. PARCC, Smarter Balanced, the Charles A. Dana Center,

and EngageNY)● feedback and recommendations from Student Achievement Partners (SAP), a trusted partner

organization that includes authors of the Common Core standards



High-level standards-placement rationaleIn general, ANet's approach to standards placement is as follows, but varies based on partner input:

● Standards are arranged in coherent and connected groups within and across interims.● All major cluster standards are assessed to support coherence across domain progressions and grade-

level expectations.● Supporting and additional clusters are usually placed where their content will enhance the major cluster

standards. This reinforces the coherence of the standards and the connections expected in instruction.● Review standards represent areas for in-depth focus, fluency standards, and the major work of the grade.

Progression across interimsThere are multiple ways to effectively sequence the standards in instruction and assessment. ANet aims for a sequence that

● developmentally makes sense, so that skills necessary to master a standard are assessed on the current or previous interims or in a previous grade level

● can be broken into cohesive units for instruction● promotes a balance in conceptual, application, and procedural learning as called for by the Common Core

standards

Focus on the major workNot all content in a given grade is emphasized equally in the standards. The point distribution on assessments across the year reflects a focus on the major work of the grade.

● Grade 2: at least 85% of points assess the major work of the grade● Grades 3–5: at least 75% of points assess the major work of the grade● Grades 6–8: at least 65% of points assess the major work of the grade

Assessment of secondary and additional standards may be pared down in consideration of assessment length and focus on major cluster standards.

What are the key shifts and notes for 16-17?

Updates to Constructed-Response Analysis Guides

The exemplar student work included on the analysis guides will, where possible, be typed rather than handwritten in order to reflect student responses seen either on paper or in the online platform. The analysis guides for 2-point constructed-response items will also include common misconceptions, new for 16-17.

Consistency in number of innovative items The percentage of innovative items (multiple-select and technology-enhanced items for online) incorporated across the year will remain at approximately 15% for paper assessments and between 20-30% for online assessments.

Consistency in cluster-level items The number of cluster-level items will remain approximately the same for 16-17 and will continue to align to teacher-scored items only.



Assessment Time Estimates

● Assessment time estimates are for students' completion of content and have been informed by test

design guidance as well as feedback from school partners. Estimates do not include time for administration steps.

● Actual times may vary based on how long it takes students to answer the questions and/or respond to the constructed-response prompts. Schools can use these for guidance to support scheduling but ultimately understand their students and classrooms best and should make decisions that are right for them.

Paper Estimated Range in Minutes

Grade A1 A2 A3 A4

2nd grade 55–75 55–75 55–75 60–80

3rd grade 85–105 100–120 115–135 105–125

4th grade 85–105 110–130 105–125 110–130

5th grade 95–115 110–130 110–130 110–130

6th grade 95–115 115–135 115–135 110–130

7th grade 100–120 100–120 115–135 105–125

8th grade 85–105 100–120 115–135 115–135

Online Estimated Range in Minutes

Grade A1 A2 A3 A4

2nd grade 55–75 55–75 55–75 60–80

3rd grade 90–110 110–130 120–140 110–130

4th grade 90–110 115–135 110–130 115–135

5th grade 100–120 120–140 120–140 115–135

6th grade 100–120 120–140 120–140 115–135

7th grade 110–130 105–125 120–140 115–135

8th grade 90–110 105–125 120–140 120–140



What does the SAS include?

For each of the interims in each grade, the SAS includes the following:

● Assessed standards organized by: Major Clusters, Supporting Clusters, Additional Clusters, and Review Standards

● The total number of items organized by: Machine Scored (selected-response, multiple-select, and short-answer items) and Teacher Scored (2-point and 4-point constructed-response items)

o Item counts by standard, including constructed-response standards, will be finalized and displayed on the myANet standards tab at least five weeks before administration for each interim.

o Based on feedback from partners and to ensure greater consistency in assessment length, each interim will contain no more than 35 items.

● Required materials

o Reference sheets are not allowed in grades 2–4, but are ANet-provided in grades 5–8 for all interims.

o A protractor is ANet-provided when needed for the interim (4th grade only).o Calculator-eligible standards are underlined in interims where there is a calculator section in

grades 6–8.o Rulers are not required in any grade.o For online grades 5–8, graph paper is suggested in some interims based on assessed standards.o For online grade 8, tracing paper is suggested in some interims based on assessed standards.

● Brief descriptions of the standards, which do not include or represent the full extent of the Common

Core standardso The brief descriptions are not intended to indicate what will or will not be assessed.o Teachers should refer to the full standard language on myANet or at corestandards.org in order to

know the full content of the standard and to provide instruction that addresses the full breadth of the standard. You can also download a free Common Core standards app.1

● Additional notes for some standardso Notes are included if:

▪ There is a restriction or limitation on how a standard will be assessed in that interim▪ A restriction or limitation on a standard from a previous interim has been lifted in a later

interim (content restrictions are lifted by the end of the year to ensure full coverage of the standards)

▪ Additional rationale for how a standard will be assessed is neededo On the paper SAS, notes are indicated with an asterisk (*) next to the standard code and are shown

at the bottom of the page or on the next page. o On the myANet standards tab, notes can be seen by hovering the cursor over the standard code;

this will show the entire standard language along with any notes for that standard in bold.

1 https://www.masteryconnect.com/state-apps.html



Sample 2016-2017 Math SAS



Grade-Level Sequencing Comments

Language for the Grade-Level Sequencing for the SAS was compiled from EngageNY Common Core Curriculum2 Module Overviews and Curriculum Maps for grades PK–53 and 6–84. Grade 2: Overview of the Grade: “In Grade 2, instructional time should focus on four critical areas: (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.” 5

A1 Themes Sums and Differences to 20 Module 1 sets the foundation for students to master the sums and differences to 20 and to subsequently apply these skills to fluently add one-digit to two-digit numbers at least through 100 using place value understandings, properties of operations, and the relationship between addition and subtraction. Addition and Subtraction of Length Units In Module 2, students work to deepen their conceptual understanding of measurement and to relate addition and subtraction to length. Their work in Module 2 is exclusively with metric units in order to support place value concepts. Customary units will be introduced in Module 7 and assessed in A4. Place Value, Counting, and Comparison of Numbers to 1,000 Module 3, which is separated over A1 and A2 for pacing consideration, has students expand their skill with, and understanding of, units by bundling ones, tens, and hundreds up to 1 thousand. Students begin to see the efficiency of place value and base-ten numbers as they repeatedly bundle 10 ones to make 1 ten and subsequently bundle 10 tens to make 1 hundred. Students work with base ten numerals representing modeled numbers as they practice moving fluidly between word form, unit form, standard form, and expanded form.

A1 includes Module 1, Module 2, and Module 3 Topics A through E for a total of 31 instructional days. A2 Themes Place Value, Counting, and Comparison of Numbers to 1,000 (continued) In A2, students continue to progress through Module 3. Students compare numbers using the symbols <, >, and =. They apply their comparison and place value skills to order more than two numbers in different forms, and to skip-count to find 1, 10, and 100 more or less than a number.

2 https://www.engageny.org/common-core-curriculum 3 https://www.engageny.org/resource/pre-kindergarten-grade-5-mathematics-curriculum-map-and-guiding-documents 4 https://www.engageny.org/resource/grades-6-8-mathematics-curriculum-map 5 Read the full Common Core Standard language here: http://www.corestandards.org/Math/. The requirements of the grade-level standards inform all standards placement decisions on the math SAS.



Addition and Subtraction within 200 with Word Problems to 100 In Module 4, students develop place value strategies to fluently add and subtract within 100; they represent and solve one- and two-step word problems of varying types within 100; and they develop conceptual understanding of addition and subtraction of multi-digit numbers within 200. Students develop an understanding of the composition and decomposition of units, and they relate these representations to the standard algorithm for addition and subtraction.

A2 includes Module 3 Topics F and G and Module 4 for a total of 37 instructional days.

A3 Themes

Addition and Subtraction within 1,000 with Word Problems to 100 In Module 5, students build upon their mastery of renaming place value units and extend their work with conceptual understanding of the addition and subtraction algorithms to numbers within 1,000. Throughout the module, students continue to focus on strengthening and deepening conceptual understanding and fluency.

Foundations of Multiplication and Division Module 6 lays the conceptual foundation for multiplication and division in Grade 3 and for the idea that numbers other than 1, 10, and 100 can serve as units. Topics in this module include: Formation of Equal Groups, Arrays and Equal Groups, Rectangular Arrays as a Foundation for Multiplication and Division, and The Meaning of Even and Odd Numbers.

A3 includes Module 5 and Module 6 for a total of 40 instructional days.

A4 Themes

Problem Solving with Length, Money, and Data Module 7 presents an opportunity for students to practice addition and subtraction strategies within 100 and problem-solving skills as they learn to work with various types of units within the contexts of length, money, and data. Students represent categorical and measurement data using picture graphs, bar graphs, and line plots. They revisit measuring and estimating length from Module 2, though now using both metric and customary units.

Time, Shapes, and Fractions as Equal Parts of Shapes In Module 8, the final module of the year, students extend their understanding of part-whole relationships through the lens of geometry. As students compose and decompose shapes, they begin to develop an understanding of unit fractions as equal parts of a whole.

A4 includes Module 7 and Module 8 for a total of 42 instructional days. Grade 3: Overview of the Grade: “In Grade 3, instructional time should focus on four critical areas: (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.”



A1 Themes

Properties of Multiplication and Division and Solving Problems with Units of 2–5 and 10 Module 1 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.

Place Value and Problem Solving with Units of Measure Module 2 uses place value to unify measurement, rounding skills, and the standard algorithms for addition and subtraction. Module 2 is separated over A1 and A2 for pacing considerations. A1 includes Topic A, which focuses on measurement of time and solving problems involving addition and subtraction of time intervals, and Topic B, which focuses on metric weight and capacity. Module 2 also provides students with internalization time for learning the 2, 3, 4, 5, and 10 facts as part of their fluency activities. Since 3.MD.A.2 encompasses multiplication and division, which are further developed through Module 3, weight and capacity are assessed in A2.

A1 includes Module 1 and Module 2 Topics A and B for a total of 32 instructional days.

A2 Themes

Place Value and Problem Solving with Units of Measure (continued) In A2, students continue to move through Module 2. Now more experienced with measurement and estimation, students further develop their skills by learning to round in Topic C. They use their understanding of place value and the number line to round two-, three-, and four-digit measurements to the nearest ten or hundred. Students apply their new understanding of numbers to compose larger units when adding and decompose into smaller units when subtracting. Students 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 in order 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.

A2 includes Module 2 Topics C through E and Module 3 for a total of 31 instructional days. A3 Themes

Multiplication and Area By Module 4, students are ready to investigate area. 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. 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 apply practice with equal shares (from Grade 2) to understanding fractions as equal parts of a whole. A3 covers Topics A through D, where students’ 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.

A3 includes Module 4 and Module 5 Topics A through D for a total of 35 instructional days.

A4 Themes

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 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 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 the area and perimeter of a figure are distinct measurements, which is a concept taught in the last module.

A4 includes Module 5 Topic E and F, Module 6, and Module 7 for a total of 50 instructional days.

Grade 4: Overview of the Grade: “In Grade 4, instructional time should focus on three critical areas: (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; (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.” A1 Themes Place Value, Rounding, and Algorithms for Addition and Subtraction In Module1, 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 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 progress from finding partial products to recording 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.

A1 includes Module 1, Module 2, and Module 3 Topics A through C for a total of 35 instructional days. A2 Themes 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 them. 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.

A2 includes Module 3 Topics D through H for a total of 27 instructional days.

A3 Themes

Fraction Equivalence, Ordering, and Operations Module 5 centers on equivalent fractions and operations with fractions. 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. Students add and subtract fractions with like units using the area model and the number line. They multiply a fraction by a whole number where the interpretation is as repeated addition. Through this introduction to fraction arithmetic, they gradually come to understand fractions as units they can manipulate, just like whole numbers. Throughout the module, customary units of measurement provide a relevant context for the arithmetic.

A3 includes Module 5 for a total of 41 instructional days. Module 4 will be assessed in A4.

A4 Themes

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 find unknown angle measures by building and solving equations. 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.

A4 includes Module 4, Module 6, and Module 7 Topics A through C for a total of 46 instructional days (Module 7 Topic D “Year in Review” is not included).



Grade 5:Overview of the Grade: “In Grade 5, instructional time should focus on three critical areas: (1) developing fluency with addition and subtraction of fractions, and developing understanding of the multiplication of fractions and of division of fractions in limited cases (unit fractions divided by whole numbers and whole numbers divided by unit fractions); (2) extending division to 2-digit divisors, integrating decimal fractions into the place value system and developing understanding of operations with decimals to hundredths, and developing fluency with whole number and decimal operations; and (3) developing understanding of volume.” A1 Themes Place Value and Decimal Fractions Students’ experiences with the algorithms as ways to manipulate place value units in Grades 2–4 are referenced throughout Grade 5 modules. In Module 1, whole number patterns with number disks on the place value table are easily generalized to decimal numbers. As students work on word problems with measurements in the metric system, where the same patterns occur, they begin to recognize the application 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. This 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 simple expressions. They also write simple expressions, recording their calculations using the associative property and parentheses to record the relevant order of operations. In this module, students understand the distributive property via area models which are used to generate and record the partial products of the standard algorithm. For multiplication, students must fully understand the distributive property (one of the key reasons for teaching the multi-digit algorithm). The multi-digit multiplication algorithm is a straightforward generalization of the one-digit multiplication algorithm. However, when learning the division algorithm with two-digit divisors, students must also learn estimation strategies, error correction strategies, and the idea of successive approximation (all of which are central concepts in math, science, and engineering).

A1 includes Module 1 and Module 2 Topics A and B for a total of 25 instructional days. A2 Themes 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 place value 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 In Module 3, students' understanding of whole-number operations is expanded to fractional units. Like units are added to, and subtracted from, like units; when units are not equivalent, they must be changed to equal units. The equivalence is represented symbolically as students engage in active meaning-making, rather than simply following an algorithm.

A2 includes Module 2 Topics C through H and Module 3 for a total of 36 instructional days. A3 Themes 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 fractions. 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 / 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.

A3 includes Module 4 for a total of 33 instructional days. A4 Themes 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. However, 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.

A4 includes Module 5 and Module 6 for a total of 41 instructional days (not including Module 6 Topics E and F “Multi-Step Word Problems” and “The Years in Review: A Reflection on A Story of Units”).

Grade 6:Overview of the Grade: “In Grade 6, instructional time should focus on four critical areas: (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 numbers 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.”A1 Themes

Ratios and Unit Rates In Module 1, students build on their prior work in measurement, 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.

Arithmetic Operations 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.

A1 includes Module 1 and Module 2 Topic A for a total of 36 instructional days. A2 Themes

Arithmetic Operations (continued) Students continue to expand their understanding of the number system and build their fluency in arithmetic operations as they complete Module 2. Students continue 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 includes negative numbers. Students extend coordinate axes, from Quadrant 1 in Grade 5, 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 to be used into high school and beyond.

A2 includes Module 2 Topics B through D and Module 3 for a total of 30 instructional days. A3 Themes 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 about 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.

A3 includes Module 4 for a total of 34 instructional days. A4 Themes 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 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.

A4 includes Module 5 and Module 6 for a total of 42 instructional days.



Grade 7: Overview of the Grade: “In Grade 7, instructional time should focus on four critical areas: (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.” A1 Themes Ratios and Proportional Relationships 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 and subtract rational numbers in A1. Previous work in computing the sums, differences, products, and quotients of fractions and decimals serves as a significant foundation as well.

A1 includes Module 1 and Module 2 Topic A for a total of 31 instructional days. A2 Themes

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.

A2 includes Module 2 Topics B and C and Module 3 Topic A for a total of 20* instructional days. (*Upon completion of the 20 instructional days included in A2, teachers are encouraged to begin A3 content/modules. Some of the instructional days for Module 3 Topics B and C in A3 could fall into A2, but the standards in these topics are not assessed in A2 because in order to assess some of these standards at their full breadth requires knowledge of percentages, which is not covered until Module 4.)

A3 Themes

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 from comparing simulated probabilities to computing 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.

A3 includes Module 3 Topics B and C, Module 4, and Module 5 Topics A and B for a total of 50* instructional days. (*Upon completion of the 20 instructional days included in A2, teachers are encouraged to begin A3 content/modules. Some of the instructional days for Module 3 Topics B and C in A3 could fall into A2, but the standards in these topics are not assessed in A2 because in order to assess some of these standards at their full breadth requires knowledge of percentages, which is not covered until Module 4.)

A4 Themes

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.

A4 includes Module 5 Topics C and D and Module 6 for a total of 38 instructional days.

Grade 8:

Overview of the Grade: “In Grade 8, instructional time should focus on three critical areas: (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.”

A1 Themes

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, e.g. rotations, reflections, and translations, 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.

A1 includes Module 1, Module 2, and Module 3 Topic A for a total of 36 instructional days. A2 Themes 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 is congruent to the first after a dilation. 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 ( , – – ,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 and 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 and for linear equations by examining similar triangles.

A2 includes Module 3 Topics B and C and Module 4 Topics A through C for a total of 30 instructional days. A3 Themes 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 = + 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.

A3 includes Module 4 Topics D and E, Module 5, and Module 6 Topics A through C for a total of 31 instructional days.

A4 Themes Linear Functions (continued) In A4, 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.

A4 includes Module 6 Topic D and Module 7 for a total of 25 instructional days.

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