Shelby County Schools’ mathematics instructional maps are ...
IntroductionIn 2014, the Shelby County Schools Board of Education adopted a set of ambitious, yet attainable goals for school and student performance. The District is committed to these goals, as further described in our strategic plan, Destination2025. By 2025,80% of our students will graduate from high school college or career ready90% of students will graduate on time100% of our students who graduate college or career ready will enroll in a post-secondary opportunityIn order to achieve these ambitious goals, we must collectively work to provide our students with high quality, college and career ready aligned instruction. The Tennessee State Standards provide a common set of expectations for what students will know and be able to do at the end of a grade. College and career readiness is rooted in the knowledge and skills students need to succeed in post-secondary study or careers. The TN State Standards represent three fundamental shifts in mathematics instruction: focus, coherence and rigor. 4171952286635The Standards for Mathematical Practice describe varieties of expertise, habits of minds and productive dispositions that mathematics educators at all levels should seek to develop in their students. These practices rest on important National Council of Teachers of Mathematics (NCTM) “processes and proficiencies” with longstanding importance in mathematics education. -571500457200Throughout the year, students should continue to develop proficiency with the eight Standards for Mathematical Practice.This curriculum map is designed to help teachers make effective decisions about what mathematical content to teach so that, ultimately our students, can reach Destination 2025. To reach our collective student achievement goals, we know that teachers must change their practice so that it is in alignment with the three mathematics instructional shifts. Throughout this curriculum map, you will see resources as well as links to tasks that will support you in ensuring that students are able to reach the demands of the standards in your classroom. In addition to the resources embedded in the map, there are some high-leverage resources around the content standards and mathematical practice standards that teachers should consistently access:The TN Mathematics StandardsThe Tennessee Mathematics Standards: can access the Tennessee State standards, which are featured throughout this curriculum map and represent college and career ready learning at reach respective grade level.Standards for Mathematical Practice Mathematical Practice Standards can access the Mathematical Practice Standards, which are featured throughout this curriculum map. This link contains more a more detailed explanation of each practice along with implications for instructions.Purpose of the Mathematics Curriculum MapsThis curriculum framework or map is meant to help teachers and their support providers (e.g., coaches, leaders) on their path to effective, college and career ready (CCR) aligned instruction and our pursuit of Destination 2025. It is a resource for organizing instruction around the TN State Standards, which define what to teach and what students need to learn at each grade level. The framework is designed to reinforce the grade/course-specific standards and content—the major work of the grade (scope)—and provides a suggested sequencing and pacing and time frames, aligned resources—including sample questions, tasks and other planning tools. Our hope is that by curating and organizing a variety of standards-aligned resources, teachers will be able to spend less time wondering what to teach and searching for quality materials (though they may both select from and/or supplement those included here) and have more time to plan, teach, assess, and reflect with colleagues to continuously improve practice and best meet the needs of their students.The map is meant to support effective planning and instruction to rigorous standards; it is not meant to replace teacher planning or prescribe pacing or instructional practice. In fact, our goal is not to merely “cover the curriculum,” but rather to “uncover” it by developing students’ deep understanding of the content and mastery of the standards. Teachers who are knowledgeable about and intentionally align the learning target (standards and objectives), topic, task, and needs (and assessment) of the learners are best-positioned to make decisions about how to support student learning toward such mastery. Teachers are therefore expected--with the support of their colleagues, coaches, leaders, and other support providers--to exercise their professional judgement aligned to our shared vision of effective instruction, the Teacher Effectiveness Measure (TEM) and related best practices. However, while the framework allows for flexibility and encourages each teacher/teacher team to make it their own, our expectations for student learning are non-negotiable. We must ensure all of our children have access to rigor—high-quality teaching and learning to grade-level specific standards, including purposeful support of literacy and language learning across the content areas. Additional Instructional SupportShelby County Schools adopted our current math textbooks for grades 6-8 in 2010-2011. ?The textbook adoption process at that time followed the requirements set forth by the Tennessee Department of Education and took into consideration all texts approved by the TDOE as appropriate. ?We now have new standards; therefore, the textbook(s) have been vetted using the Instructional Materials Evaluation Tool (IMET). This tool was developed in partnership with Achieve, the Council of Chief State Officers (CCSSO) and the Council of Great City Schools. The review revealed some gaps in the content, scope, sequencing, and rigor (including the balance of conceptual knowledge development and application of these concepts), of our current materials.?The additional materials purposefully address the identified gaps in alignment to meet the expectations of the CCR standards and related instructional shifts while still incorporating the current materials to which schools have access. ?Materials selected for inclusion in the Curriculum Maps, both those from the textbooks and external/supplemental resources (e.g., EngageNY), have been evaluated by district staff to ensure that they meet the IMET criteria.How to Use the Mathematics Curriculum MapsOverviewAn overview is provided for each quarter. The information given is intended to aid teachers, coaches and administrators develop an understanding of the content the students will learn in the quarter, how the content addresses prior knowledge and future learning, and may provide some non-summative assessment items.Tennessee State StandardsThe TN State Standards are located in the left column. Each content standard is identified as the following: Major Work, Supporting Content or Additional Content.; a key can be found at the bottom of the map. The major work of the grade should comprise 65-85% of your instructional time. Supporting Content are standards that supports student’s learning of the major work. Therefore, you will see supporting and additional standards taught in conjunction with major work. It is the teacher’s responsibility to examine the standards and skills needed in order to ensure student mastery of the indicated standard. ContentTeachers are expected to carefully craft weekly and daily learning objectives/ based on their knowledge of TEM Teach 1. In addition, teachers should include related best practices based upon the TN State Standards, related shifts, and knowledge of students from a variety of sources (e.g., student work samples, MAP, etc.). Support for the development of these lesson objectives can be found under the column titled ‘Content’. The enduring understandings will help clarify the “big picture” of the standard. The essential questions break that picture down into smaller questions and the objectives provide specific outcomes for that standard(s). Best practices tell us that clearly communicating and making objectives measureable leads to greater student mastery.Instructional Support and ResourcesDistrict and web-based resources have been provided in the Instructional Resources column. Throughout the map you will find instructional/performance tasks, i-Ready lessons and additional resources that align with the standards in that module. The additional resources provided are supplementary and should be used as needed for content support and differentiation. Topics Addressed in QuarterUse Positive & Negative Numbers to Represent Quantities in Real-World ContextRational Numbers and the Number LineRational Numbers and the Coordinate PlaneAbsolute Value of Rational NumbersOrdering Rational NumbersGraphing Points Draw Polygons in Coordinate PlaneOverview By quarter 2 students are familiar with the number line and determining the location of positive fractions, decimals, and whole numbers from previous grades. Students will extend the number line (both horizontally and vertically) to include the opposites of whole numbers (6.NS.C.6a, 6.NS.C.6c). The number line model is extended to two-dimensions, as students use the coordinate plane to model and solve real-world problems involving rational numbers (6.NS.C.6b, 6.NS.8, 6.G.3). Next, students will extend their arithmetic work to include using letters to represent numbers in order to understand that letters are simply "stand-ins" for numbers and that arithmetic is carried out exactly as it is with numbers (6.EE.A.2c). Students explore operations in terms of verbal expressions and determine that arithmetic properties hold true with expressions (6.EE.A.3). They understand the relationships of operations and use them to generate equivalent expressions (6.EE.A.4, 6.G.A.1), ultimately extending arithmetic properties from manipulating numbers to manipulating expressions and to evaluating expressions in order to develop and evaluate formulas.Grade Level StandardType of RigorFoundational StandardsSample Assessment Items6.NS.5Conceptual UnderstandingLearn Zillion: 6.NS.5-76.NS.6Conceptual Understanding6.NS.5, 5.G.1Learn Zillion: 6.NS.6 & 86.NS.7Conceptual UnderstandingTNCore Assessment Task: 6.NS.76.NS.8Procedural Skill and Fluency5.G.2, 6.NS.6 6.EE.1Procedural Skill and Fluency5.NBT.2Achieve the Core: 6.EE6.EE.2Procedural Skill and Fluency5.OA.2, 5.OA.3Learn Zillion: 6.EE.1-26.EE.3Application5.OA.2, 6.NS.4TNCore Assessment Task: 6.EE.3-46.EE.4Conceptual Understanding5.OA.2, 6.NS.46.G.1Application5.NF.46.G.3Application5.G.2Engage NY Assessment: Select Grade 6Fluency NCTM PositionProcedural fluency is a critical component of mathematical proficiency. Procedural fluency is the ability to apply procedures accurately, efficiently, and flexibly; to transfer procedures to different problems and contexts; to build or modify procedures from other procedures; and to recognize when one strategy or procedure is more appropriate to apply than another. To develop procedural fluency, students need experience in integrating concepts and procedures and building on familiar procedures as they create their own informal strategies and procedures. Students need opportunities to justify both informal strategies and commonly used procedures mathematically, to support and justify their choices of appropriate procedures, and to strengthen their understanding and skill through distributed practice.The fluency standards for 6th grade listed below should be incorporated throughout your instruction over the course of the school year. Click Engage NY Fluency Support to access exercises that can be used as a supplement in conjunction with building conceptual understanding. 6.NS.2 Fluently divide multi-digit numbers using standard algorithms6.NS.3 Fluently add, subtract, multiply and divide multi-digit decimalsReferences: STATE STANDARDSCONTENTINSTRUCTIONAL SUPPORT & RESOURCESRational Numbers, Number Lines and Absolute Value (Allow approximately 3 weeks for instruction, review and assessment)Domain: The Number SystemCluster: Apply and extend previous understandings of numbers to the system of rational numbers. 6.NS.5 Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation.Enduring Understanding(s):The system of rational numbers includes negative numbers as well as positive ones.Essential Question(s):How are positive and/or negative numbers used in real-life situations?How is absolute value used in real-world situations?Objective(s):Students will use a number line to explore the absolute value of an integer.Students use positive integers to locate negative integers by moving in the opposite direction from zero.Students use positive and negative numbers to indicate a change (gain or loss) in elevation with a fixed reference point, temperature, and the balance in a bank account. Additional Information:Example(s):a. Use an integer to represent 25 feet below sea levelb. Use an integer to represent 25 feet above sea level.c. What would 0 (zero) represent in the scenario above?Solution:a. -25b. +25c. 0 would represent sea levelThe Great Barrier Reef is the world’s largest reef system and is located off the coast ofAustralia. It reaches from the surface of the ocean to a depth of 150 meters. Students could represent this value as less than -150 meters or a depth no greater than 150 meters below sea level.Recognize that an account balance less than – 30 dollars represents a debt greater than 30 dollars.Glencoe7-3B Integers and Absolute Value (p. 419 only and Problem Solving and H.O.T. problems)Holt11-1 Integers and Absolute ValueBuilding Conceptual Understanding:TN Task Arc: Locating, Ordering and Finding... 6.NS.C.5 (Tasks 1, 3 5-7)Engage NY: Real-World Positive & Negative Numbers & Zero Lesson 2Utah Education Network Lesson Plans: Combining IntegersChoose from the following resources and use them to ensure that the intended outcome and level of rigor (mainly conceptual understanding and application) of the standards are met.Recommended Additional Lesson(s):Engage NY: Positive & Negative Numbers on a Number Line-Opposite Direction & Value Lesson 1Engage NY: Real-World Positive & Negative Numbers & Zero Lesson 3Task(s): You must establish a free account to access resources.TN Task: Fun at the Ocean 6.NS.C.5, 6.NS.C.6, 6.NS.C.7TN Assessment Task: Ordering TaskAdditional Resource(s):Contest Winner & Positive and Negative Events Tasks (Scroll to 6.NS.C.5 section)Math Station Activities pp. 22 & 29Correlated iReady Lesson(s):Four-Digit DividendsVocabulary: positive number, negative number, integer, opposite, quadrantWriting in Math:Students will write down 5 real-world situations that can be represented by integers (e.g., a gain or loss of points at a soccer game, elevation of a mountain). After explaining why the situation is positive or negative, they will draw a number line and graph the points.Integers on a Number Line/Real-World SituationsDomain: The Number SystemCluster: Apply and extend previous understandings of numbers to the system of rational numbers. 6.NS.C.6: Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates.6.NS.C.6a Recognize opposite signs of numbers as indicating locations on opposite sides of 0 on the number line; recognize that the opposite of the opposite of a number is the number itself, e.g., – (–3) = 3, and that 0 is its own opposite.6.NS.C.6b Understand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane; recognize that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes.6.NS.C.6c Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane. Enduring Understanding(s):Points can be graphed in all four quadrants of a coordinate grid by using ordered pairs to determine location.A rational number can be represented as a point on a number line and the number line can be used as a tool to order rational numbers.Essential Question(s):How do you locate points in the coordinate plane?Objective(s):Students will construct the coordinate plane’s vertical and horizontal axes and discover the relationship between the four quadrants and the signs of the coordinates of points that lie in each quadrantStudents will use reflections to change the positions of figures in the coordinate plane.Students will recognize that finding the opposite of any rational number is the same as finding an integer’s opposite.Students will recognize that two rational numbers that lie on the same side of zero will have the same sign, while those that lie on opposites sides of zero will have opposite signs.Additional Information:Number lines can be used to show numbers and their opposites. Both 3 and -3 are 3 units from zero on the number line. Graphing points and reflecting across zero on a number line extends to graphing and reflecting points across axes on a coordinate grid. The use of both horizontal and vertical number line models facilitates the movement from number lines to coordinate grids.Students recognize the point where the x-axis and y-axis intersect as the origin. Students identify the four quadrants and are able to identify the quadrant for an ordered pair based on the signs of the coordinates. For example, students recognize that in Quadrant II, the signs of all ordered pairs would be (–, +).Glencoe7-3C The Coordinate Plane (page 423-427)Holt11-3 Coordinate Plane (p. 604-607)11–4 Transformations in the Coordinate Plane (pg. 608-611) (Only cover the examples and exercises that deal with Reflections because 6.NS.C.6b only refers to reflections and the other transformations will be addressed in 8th grade.)Building Conceptual Understanding:CMP CCSS Investigation 3: Integers and the Coordinate Plane (p. 23 if printed; p. 67 if viewing from computer and student pages began on p.79)TN Task Arc: Locating, Ordering & Finding Distance Between Integers (Task 2)Choose from the following resources and use them to ensure that the intended outcome and level of rigor (mainly conceptual understanding and application) of the standards are met.Recommended Additional Lesson(s):Engage NY: 6.NS.6 (Choose lessons from Module 3 Topics A-C)Engage NY: Lesson 4 The Opposite of a Number (Problem Set Lesson)Engage NY: Lesson 5 The Opposite of a Number's Opposite (Problem Set Lesson)Engage NY: Lesson 6 Rational Numbers on a Number LineTask(s):Illustrative Math: Reflecting Points over Coordinate axesIllustrative Math: Plotting Points in the Coordinate Plane Illustrative Math: Integers on a Number lineIllustrative Math: Extending the Number lineAmusement Park Task. (Scroll to 6.NS C.6 section)Additional Resources: Math Station Activities: Refer to p. 15-28Learn : Opposites and the Meaning of ZeroLearn : The Opposite of an OppositeCorrelated iReady Lesson(s):Rational Numbers and Absolute ValuePlotting Ordered PairsReview Plotting Ordered PairsCoordinate Plane and Absolute ValueRational Numbers and Absolute ValueVocabulary: coordinate plane, axes, x-axis, y-axis, quadrants, origin, x-coordinate, y-coordinate, polygon, reflectionWriting in Math: Ask students to define “opposite” as it relates to integers on a number line.Graphic Organizer:Vocabulary Building: Students are to make flash cards and/or Frayer Model for the vocabulary words. Students are to write the definition of the word on the front of the card and draw an illustration on the back of the card. Frayer Model ExampleCreate a two-tab foldable to help students understand the concepts of distance on the coordinate plane. Label one Flap “Reflection across the x-axis.” Label the other flap “Reflection across the y-axis.” Write important ideas about each type of reflection under the appropriate flap. Foldable LinkCoordinate Plane Guided Notes and Foldable ActivityDomain: The Number SystemCluster: Apply and extend previous understandings of numbers to the system of rational numbers. 6.NS.C.7 Understand ordering and absolute value of rational numbers.6.NS.C.7a Interpret statements of inequality as statements about the relative position of two numbers on a number line. 6.NS.C.7b Write, interpret, and explain statements of order for rational numbers in real-world contexts.6.NS.C.7c Understand the absolute value of a rational number as its distance from 0 on the number line; interpret absolute as magnitude for a positive or negative quantity in a real-world situation.6.NS.C.7d Distinguish comparisons of absolute value from statements about order. For example, recognize that an account balance less than –30 dollars represent a debt greater than 30 dollars.Enduring Understanding(s):Absolute value can be described in more than one way, depending upon the real-world context. It can be distance, or it can be size (magnitude)Essential Question(s):How do you use positive and negative numbers to describe quantities having opposite values?What is absolute value?Objective(s):Students will order rational numbers on a number line.Students will interpret statements of inequality as statements about relative position of two numbers on a number line diagram. Students will write, interpret, and explain statements of order for rational numbers in real-world contexts.Students will identify the absolute value of a rational number.Students will use models to determine the truth of inequalities.Additional Information:Students recognize the distance from zero as the absolute value or magnitude of a rational number. Students need multiple experiences to understand the relationships between numbers, absolute value, and statements about order. Students will interpret statements of inequality as statements about the relative position of two numbers on a number line diagram.Students will write, interpret, and explain statements of order for rational numbers in real-world contexts. By placing two numbers on the same number line, they are able to write inequalities and make statements about the relationships between the numbers. Case 1: Two positive numbers 5 > 3 5 is greater than 3 Case 2: One positive and one negative number 3 > -3positive 3 is greater than negative 3negative 3 is less than positive 3Case 3: Two negative numbers -3 > -5negative 3 is greater than negative 5 negative 5 is less than negative 3Glencoe7-2A Solve Inequalities Using Models7-2B Inequalities7-2D Write and Graph Inequalities7-3A Explore Absolute Value7-3B Integers & Absolute ValueAdditional Lesson 5 - Compare and Order Integers (page 795-798)Holt11-1 Integers and Absolute Value (page 594-597)11-2 Comparing and Ordering Integers(page 598-601)Building Conceptual Understanding: TN Task Arc: Locating, Ordering & Finding Distance Between Integers (Tasks 2,3,4 & 7)Engage NY Lessons: Order and Absolute Value Choose from the following resources and use them to ensure that the intended outcome and level of rigor (mainly conceptual understanding and application) of the standards are met.Recommended Additional Lesson(s):Learn Zillion: Relationship Between Rational NumbersLearn Zillion: Understand Relationship Between Rational NumbersTask(s)Illustrative Math: Comparing Temperatures TaskIllustrative Math: Jumping Flea TaskIllustrative Math: Fractions on the Number Line TaskIllustrative Math: Integers on the Number Line 2 TaskTN Assessment Tasks: Absolute Value, Comparing on a Number Line & TemperatureAdditional Resources:Better Lesson: 6.NS.7Compare and Order Integers Lesson Plan and AttachmentsMath Station Activities pp. 22 & 29Correlated iReady Lesson(s):Rational Numbers and Absolute ValueVocabulary: rational number, absolute value, magnitude, greater than ( >), less than ( < ), greater than or equal to (≥), less than or equal to ( ≤)Writing in Math:Students will analyze pre-cut number line strips that have two numbers on them, and write inequalities as well as make statements about the relationships between the numbers. To illustrate, if students are given a number line strip with a 5 and 3 on it, they should be able to write the inequality 5>3 or 3<5. In addition, they should write statements of order for that same example (5 is greater than 3).Number Line Generator for Math Journals/ActivitiesDomain: The Number SystemCluster: Apply and extend previous understandings of multiplication and division to divide fractions by fractions. 6.NS.8 Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate.6.G.3 Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. Apply these techniques in the context of solving real-world and mathematical problems. (This standard supports and extends 6.NS.8 because it asks students to apply graphing points in a coordinate plane to solve real-world problems. It will also be covered more in depth in Q3.)Enduring Understanding(s):Integers have magnitude and direction.On the coordinate plane, a point represents the two facets of information associated with an ordered pair. Essential Question(s):How can you find distances between points on the same vertical or horizontal line?Objective(s):Students will calculate absolute value. Students will graph points in all four quadrants of the coordinate plane.Students will solve real-world problems by graphing points in all four quadrants of a coordinate plane.Students will calculate the distances between two points with the same first coordinate or the same second coordinate using absolute value, given only coordinates. Students will describe how to use the absolute value to find the distance between two points on the coordinate plane.Additional Information:Students understand the relationship between two ordered pairs differing only by signs as reflections across one or both axes. For example, in the ordered pairs (-2, 4) and (-2, -4), the y-coordinates differ only by signs, which represents a reflection across the x-axis. A change is the x-coordinates from (-2, 4) to (2, 4), represents a reflection across the y-axis. When the signs of both coordinates change, [(2, -4) changes to (-2, 4)], the ordered pair has been reflected across both axes.Also, students will locate numbers on a number line according to their placement in order of value.Example(s): Graph the following points in the correct quadrant of the coordinate plane. If you reflected each point across the x-axis, what are the coordinates of the reflected points? What similarities do you notice between coordinates of the original point and the reflected point? 12 , -3 12 , -12 , -3 , (0.25, -0.75)Note that the y-coordinates are opposites.Students will interpret and solve mathematical problems by graphing points in all four quadrants of the coordinate plane including the use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate.Students will find the distance between points when ordered pairs have the same x-coordinate (vertical) or same y-coordinate (horizontal)Example(s):What is the distance between (–5, 2) and (–9, 2)?Solution: The distance would be 4 units. This would be a horizontal line since the y-coordinates are the same. In this scenario, both coordinates are in the same quadrant. The distance can be found by using a number line to find the distance between –5 and –9. Students could also recognize that –5 is 5 units from 0 (absolute value) and that –9 is 9 units from 0 (absolute value). Since both of these are in the same quadrant, the distance can be found by finding the difference between the distances 9 and 5. (| 9 | - | 5 |).If the points on the coordinate plane below are the three vertices of a rectangle, what are the coordinates of the fourth vertex? How do you know? What are the length and width of the rectangle? To determine the distance along the x-axis between the point (-4, 2) and (2, 2) a student must recognize that -4 is -4or 4 units to the left of 0 and 2 is 2or 2 units to the right of zero, so the two points are total of 6 units apart along the x-axis.GlencoeAdditional Lesson 6: Distance on the Coordinate Plane (p. 799-800)Lesson 7: Polygons on the Coordinate Plane (page 801-806)Holt11-3 Coordinate Plane Choose from the following resources and use them to ensure that the intended outcome and level of rigor (mainly conceptual understanding and application) of the standards are met.Recommended Additional Lesson(s):Engage NY Lesson 18: 6.NS.6 & 8Connected Math Investigation: Integers and the Coordinate Plane (6.NS.6.B & 6.NS.8)Learn Zillion Coordinate PlaneTask(s):Illustrative Math: Distances Between Points TaskIllustrative Math: Walking the Block 6.G.3Additional Resources:PBS Learning: Coordinate Grid VideoVirtual Nerd: Integers and the Coordinate PlaneCorrelated iReady Lesson(s):Plotting Ordered PairsReview Plotting Ordered PairsVocabulary: coordinate plane, absolute value, x-coordinate, y-coordinate Writing in Math:Have students explain in writing how they describe the process of locating points in a coordinate plane.Write and Evaluate Expressions( Allow approximately 3 weeks for instruction, review and assessment )Domain: Expressions and EquationsCluster: Apply and extend previous understandings of arithmetic to algebraic expressions. 6.EE.A.1: Write and evaluate numerical expressions involving whole-number exponents. 6.EE.A.2c: Evaluate expressions at specific values of their variables. Include expressions that arise from formulas used in real-world problems. Perform arithmetic operations, including those involving whole number exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations).6.G.A.1 Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems. (This standard supports 6.EE.a and 6.EE.2c. because it shows the usefulness of exponential notation in a geometric context and it gives students an opportunity to see that sometimes it is easier to write a number as a numeric expression rather than evaluating the expression. This standard will be covered in Q3) Enduring Understanding(s):Exponential notation is a way to express repeated products of the same number.Algebraic expressions may be used to represent and generalize mathematical problems and real life situations.Essential Question(s):How do arithmetic properties contribute to algebraic understanding?How do the order of operations and properties help simplify and evaluate algebraic expressions?How does the result change when the value of the variable is changed?Objective(s):Students will evaluate expressions that include whole-number exponents.Students will substitute values for variables and evaluate expressions.Students will use the order of operations to simplify expressions and solve problemsAdditional Information:Order of operations is introduced throughout elementary grades, including the use of grouping symbols, ( ), { }, and [ ] in 5th grade. Order of operations with exponents is the focus in 6th grade.Students demonstrate the meaning of exponents to write and evaluate numerical expressions with whole number exponents.Problems involve expressing b-fold products a?a?…?a in the form ab, where a and b are non-zero whole numbers. Students are not required to use of the laws of exponents. Numerical values in these expressions may include whole numbers, fractions, and decimals.Students will evaluate expressions with the understanding that a variable is a letter used to represent a number. Example(s):Evaluate: 43 (Solution: 4 x 4 x 4 = 64) 5 + 24 ● 6 (Solution: 5+ 2x2x2x2 x 6= 101) 72 – 24 ÷ 3 + 25 (Solution: 7x7-24÷3+25= 67)Evaluate 5(n + 3) – 7n, when n = 12Solution:5(1/2 + 3) - 7(1/2)5(3 1/2) - 3 1/2 note: 7(1/2)=7/2=3 1/217 1/2 - 3 1/2 14Students may also reason that 5 groups of 3 1/2 take away 1 group of 3 1/2 would give 4 groups of 3 1/2. Multiply 4 times 3 1/2 to get 14.Use the formulas V=s3 and A=6s2 to find the volume and surface area of a cube with sides of length s= 1/2.Glencoe1-3A Exponents (p. 62-65)5-1A Numerical Expressions: Order Of Operations (p. 270-273)5-1B Algebra: Variables and Expressions (p. 274-278)Holt1–2 Exponents (p. 10-13)1-3 Order Of Operations (p. 18-21)2-1 Variables and Expressions (p. 50-53)Building Conceptual Understanding:EngageNY Lessons: 6.EE.1 Lessons 5 & 6Learn Zillion: Understanding Algebraic ExpressionsLearn Zillion: Evaluate Algebraic ExpressionsChoose from the following resources and use them to ensure that the intended outcome and level of rigor (mainly conceptual understanding and application) of the standards are met.Recommended Additional Lesson(s):Learn Zillion: Write and Evaluate Algebraic ExpressionsTask(s):Illustrative Math: Djinni's Offer Exponent Illustrative Math: Sierpinski's CarpetIllustrative Math: Exponent Experimentation 1Illustrative Math: Exponent Experimentation 2TN Assessment Task: Expressions 6.EE.A.1Additional Resources:Math Station Activities p. 60Exponents Interactive Jeopardy (Click on ‘The Game’ near center of page)Correlated iReady Lesson(s):Numerical Expressions and Order of OperationsAlgebraic ExpressionsVocabulary: base, exponent, powers, exponential form, expressions, numerical expressions, variable, algebraic expressionGraphic Organizer:Students will create information frames on the following vocabulary words include: Information frames can be used to help students organize and remember concepts. Students write the topic in the middle rectangle. Then students write related concepts in the spaces around the rectangle. Related concepts can include words, numbers, example, definition, non-example, procedure, or details. Information Frame Graphic Organizer6.EE.A.2: Write, read, and evaluate expressions in which letters stand for numbers.6.EE.A.2.a Write expressions that record operations with numbers and with letters standing for numbers. “Subtract y from 5” as 5-y 6.EE.B.6: Use variables to represent numbers and write expression when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.6.EE.A.2b: Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity. Enduring Understanding(s):Expressions can be written from verbal descriptions using letters and numbersEssential Question(s):How do we generalize numerical relationships and express mathematical ideas using expressions and equations?Objective(s):Students will translate word phrases to algebraic expressionsStudents will evaluate expressions.Students will identify parts of an expression using precise languageAdditional Information:When students write expressions from verbal descriptions, they must understand that the order is important when writing subtraction and division problems. Students will understand that expressions, such as 5n, means to multiply 5 to the value of "n".Examples: 7 more than 3 times a number (Solution: 3x + 7) 3 times the sum of a number and 5 (Solution: 3(x + 5))7 less than the product of 2 and a number (Solution: 2x – 7)Twice the difference between a number and 5 (Solution: 2(z – 5))Students should identify parts of an expression. Consider the following expression:x2 + 5y + 3x + 6The variables are x and y.There are 4 terms, x2, 5y, 3x, and 6.There are 3 variable terms, x2, 5y, 3x. They have coefficients of 1, 5, and 3 respectively. The coefficient of x2 is 1, since x2 = 1x2. The term 5y represent 5y’s or 5 * y.There is one constant term, 6.The expression represents a sum of all four terms.Connecting writing expressions with story problems and/or drawing pictures will give students a context for this work. It is important for students to read algebraic expressions in a manner that reinforces that the variable represents a number. Maria has three more than twice as many crayons as Elizabeth. Write an algebraic expression to represent the number of crayons that Maria has.(Solution: 2c + 3 where c represents the number of crayons that Elizabeth has.)(Note that 6.EE.B.6 does not expect students to solve the expressions; however, 6.EE.2c does have student evaluating expressions.)Glencoe5-1B Algebra: Variables and Expressions (p. 274-278)5-1C Explore Write Expressions (p. 279-281) 5-1D Algebra: Write Expressions (p. 282-285)Holt2 – 1 Variables and Expressions (p. 50-53)2 – 2 Translating Between Words and Math (p. 54-57)Building Conceptual Understanding:EngageNY: 6.EE.2 & 6 Lessons 18-22, 28-29Connected Math Investigation: Number Properties and Algebraic Expressions p. 41 (Teacher Notes) & 53 (Student Pages)Choose from the following resources and use them to ensure that the intended outcome and level of rigor (mainly conceptual understanding and application) of the standards are met.Recommended Additional Lesson(s):Math Shell Lesson: Evaluating Statements About Number OperationsLearn Zillion: 6.EE.2a & bTasks:Illustrative Math Tasks: 6.EE.2 Illustrative Math Tasks: 6.EE.6TN Instructional Task: Math CompetitionAdditional Resources:Khan Academy- Variables and ExpressionsExpressions and VariablesKhan Academy-Writing Expressionswrite-variable-expressions-to-represent-word-problemsMath Station Activities p. 68Correlated iReady Lesson(s):Algebraic ExpressionsVocabulary: expression, numerical expression, algebraic expression, factor, coefficient, equivalent expression, product, term, quotient, sumWriting in Math:Graphic Organizer:Have students write as many key words for the four basic math functions as they can think of. Translating Words to Math Graphic WheelHave students create Frayer Model of the vocabulary words in this lesson which include: variable, algebraic expression, evaluate, term, coefficient, constant, product, factor, quotientFrayer ModelEquivalent Expressions ( Allow approximately 3 weeks for instruction, review and assessment )Domain: Expressions and EquationsCluster: Apply and extend previous understandings of arithmetic to algebraic expressions.6.EE.A.3: Apply the properties of operations to generate equivalent expressions.6.NS.4 Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1–100 with a common factor as a multiple of a sum of two whole numbers with no common factor. (This standard supports the work of generating equivalent expressions. Students must understand that GCF, LCM and properties of operations, mainly the distributive property, can be used to easily generate equivalent expressions.)Enduring Understanding(s):Algebraic expressions are used to represent quantitative relationships.Properties of operations can be used to generate equivalent expressions. Essential Question(s):How can order of operations be applied to a mathematical expression?Objective(s):Students will describe how to use the properties of operations to generate equivalent expressions. Students will use the order of operations to simplify expressions and solve problemsAdditional Information:Students will use the distributive property to write equivalent expressions. They will also have to find common factors of terms in an expression.Example(s):Apply the distributive property to generate the equivalent expression of 5(m + 3).Solution: 5m + 15Write the equivalent expression for 24a - 12.Solution: 12(2a - 1) Students interpret y as referring to one y. Thus, they can reason that one y plus one y plus one y must be 3y. They also use the distributive property, the multiplicative identity property of 1, and the commutative property for multiplication to prove that y + y + y = 3y: y + y + y = y x 1 + y x 1 + y x 1 = y x (1 + 1 + 1) = y x 3 = 3yGlencoe5-2A Algebra: Properties (p. 289-293)5-2-B Explore The Distributive Property (p. 294-295) 5-2C The Distributive Property (p. 296-299)Holt1 – 4 Properties and Mental MathA-1 Model Arithmetic PropertiesBuilding Conceptual Understanding:TN Equivalent Expression Task Arc (Tasks 5-8)Choose from the following resources and use them to ensure that the intended outcome and level of rigor (mainly conceptual understanding and application) of the standards are met.Recommended Additional Lesson(s):Engage NY: 6.EE.3 Lessons Math Shell Concept Development Lesson: Representing the Laws of ArithmeticLearn Zillion: 6.EE.3CCSS Investigation 2: Number Properties and Algebraic EquationsTask: Illustrative Math Task: 6.EE.3Additional Resources:Math Station Activities p. 75Correlated iReady Lesson(s):Equivalent ExpressionsVocabulary: order of operations, properties of operationsGraphic Organizer:Students will create a 4-tab foldable and label with each of the four properties and come up with their own example illustrating each of the properties. Students must have one example involving numbers and one example in which they must use variables. Properties Foldable Example: Copy and Paste the link directly. of Numbers Foldable Activity,practice,&quiz.pdfDomain: Expressions and EquationsCluster: Apply and extend previous understandings of arithmetic to algebraic expressions. 6.EE.A.4: Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them). 6.NS.4 Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1–100 with a common factor as a multiple of a sum of two whole numbers with no common factor.(This standard supports the work of identifying equivalent expressions. Students must understand that GCF, LCM and properties of operations, mainly the distributive property, can be used to easily identify when two expressions are equivalent.)Enduring Understanding(s):Algebraic expressions are used to represent quantitative relationships.Properties of operations can be used to generate equivalent expressions. Essential Question(s):What strategies can be used to determine if two expressions are equivalent?Objective(s):Students will extend their knowledge of GCF and the distributive property to determine if two expressions are equivalent.Additional Information:Students connect their experiences with finding and identifying equivalent forms of whole numbers and can write expressions in various forms. Students generate equivalent expressions using the associative, commutative, and distributive properties. They can prove that the expressions are equivalent by simplifying each expression into the same form. Glencoe(This link will take you to the Holt Middle School Math Material on-line. Click on the section that says additional Common Core Material to access the Curriculum Companion.)HoltCurriculum Companion 4 – 3A Equivalent Expressions Building Conceptual Understanding:TN Equivalent Expression Task Arc (Tasks 1-4)Engage NY: Module 4 Lessons (These lessons lay the foundation for mastery of 6.EE.4)Choose from the following resources and use them to ensure that the intended outcome and level of rigor (mainly conceptual understanding and application) of the standards are met.Recommended Additional Lesson(s):Learn Zillion: Equivalent ExpressionsCCSS Investigation 2: Number Properties and Algebraic EquationsTasks:TN Assessment Task: RectangleIllustrative Math: Equivalent Expressions Illustrative math: Watch out for Parenthesis Illustrative Math: Rectangle PerimeterCorrelated iReady Lesson(s):Equivalent ExpressionsVocabulary: order of operations, properties of operationsWriting in Math:Graphic Organizer(s):Students will create a property folder to distinguish between the associative, distributive, and commutative properties. They can glue these folders in their interactive math notebooks or math journals.Property Foldable Examples with Simplifying Expressions HYPERLINK "" Dinah Zike's Book of FoldablesStudents will write the characteristics of the math properties and equivalent expression examples in foldable Frayer models.Frayer Model FoldableRESOURCE TOOLBOXNWEA MAP Resources: - Sign in and Click the Learning Continuum Tab – this resources will help as you plan for intervention, and differentiating small group instruction on the skill you are currently teaching. (Four Ways to Impact Teaching with the Learning Continuum) - These Khan Academy lessons are aligned to RIT scores.Textbook Resourcesconnected.mcgraw-TN Core/CCSSTNReady Math StandardsTNCoreAchieve the CoreEdutoolboxVideosKhan AcademyWatch Know LearnLearn ZillionVirtual NerdMath PlaygroundStudyJamsCalculator ActivitiesGreatest Common Factor CalculatorTI-73 ActivitiesCASIO ActivitiesInteractive ManipulativesNational Library of Virtual Manipulatives - NLVMNumber & Operations Virtual ManipulativesInteractive Cartesian Plane Math Is FunAdditional SitesPBS GCF GameLCM GameGCF Game AAA MathFrayer Model TemplateGrade 6 Flip Book(This book contains valuable resources that help develop the intent, the understanding and the implementation of the state standards) ................
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