Unit 4 Integers and the Coordinate Plane

Transcription

1 Number of Days: 28 days 1/8/18 2/16/18 Unit Goals Stage 1 Unit Description: Students will extend the number line to include negative numbers using both horizontal and vertical number lines. Students will understand the absolute value of a number and will graph ordered pairs in all four quadrants as well as drawing polygons given coordinates for the vertices. In addition, students will also learn seventh grade standards of operations with rational numbers. They learn to add, subtract, multiply, and divide integers, while maintaining the properties of operations and the relationships between addition and subtraction, and multiplication and division. Students will then apply their understanding to perform the four operations on positive and negative rational numbers in the context of real-world situations. Materials: coordinate grid cling, number line, graph paper, two-color counters, Desmos Standards for Mathematical Practice SMP 1 SMP 2 SMP 3 SMP 4 SMP 5 SMP 6 SMP 7 SMP 8 Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. Standards for Mathematical Content Clusters Addressed [m] 6.NS.C Apply and extend previous understandings of numbers to the system of rational numbers. [s] 6.G.A Solve real-world and mathematical problems involving Transfer Goals Students will be able to independently use their learning to Make sense of never-before-seen problems and persevere in solving them. Construct viable arguments and critique the reasoning of others. Making Meaning UNDERSTANDINGS Students will understand that The number system includes negative numbers. Positive and negative numbers are used together to describe quantities having opposite directions or values. Each positive number has an opposite, a corresponding negative number that is the same distance away from zero on the left side of zero. Understand the absolute value of a rational number as its distance from 0 on the number line. Integer operations can be represented in multiple ways, such as using manipulatives, number lines, and real-world situations. Ordered pairs can be used to identify points on a coordinate plane. KNOWLEDGE Students will know The definition of academic vocabulary words, such as absolute value, additive inverse, axis, coordinate plane, integers, opposites, origin, and quadrants. Opposite signs of numbers indicate locations on opposite sides of 0 on the number line. The sum of opposite numbers is zero. ESSENTIAL QUESTIONS Students will keep considering What real world situations can be represented with negative and positive numbers? How are adding and subtracting integers related? How are multiplying and dividing integers related? Acquisition SKILLS Students will be skilled at and/or be able to Locate and position integers and other rational numbers on a vertical and horizontal number line. Use rational numbers to represent quantities in realworld contexts. Write and interpret statements about the order of rational numbers in real-world contexts. Find the absolute value of a rational number. LONG BEACH UNIFIED SCHOOL DISTRICT 1 Posted 12/21/17

2 Unit 4 Integers and the Coordinate Plane area, surface area, and volume. [m] 7.NS.A Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers. Unit Goals Stage 1 Any subtraction problem can be written as an addition problem by adding the additive inverse, p q = p + ( q). A coordinate plane has an x-axis and a y-axis. Add, subtract, multiply, and divide rational numbers to solve real-world and mathematical problems. Apply properties of operations as strategies to add, subtract, multiply, and divide rational numbers. Plot ordered pairs of integers and other rational numbers on a coordinate plane. Solve real-world and mathematical problems by using coordinates and absolute value to find the distance between points with the same first coordinate or same second coordinate. Recognize that if the coordinates only differ by the signs, the points are reflections across one or both axes. Draw polygons in the coordinate plane, given coordinates for the vertices in the context of solving real-world and mathematical problems. LONG BEACH UNIFIED SCHOOL DISTRICT 2 Posted 12/21/17

3 Standards for Mathematical Practice SMP 1 Make sense of problems and persevere in solving them. SMP 2 Reason abstractly and quantitatively. SMP 3 Construct viable arguments and critique the reasoning of others. SMP 4 Model with mathematics. SMP 5 Use appropriate tools strategically. SMP 6 Attend to precision. SMP 7 Look for and make use of structure. SMP 8 Look for and express regularity in repeated reasoning. Assessed Grade Level Standards Standards for Mathematical Content [m] 6.NS.C 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. 6.NS.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. a. 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. b. 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. c. 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. 6.NS.7 Understand ordering and absolute value of rational numbers. a. Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram. For example, interpret 3 > 7 as a statement that 3 is located to the right of 7 on a number line oriented from left to right. b. Write, interpret, and explain statements of order for rational numbers in real-world contexts. For example, write 3 C > 7 C to express the fact that 3 C is warmer than 7 C. c. Understand the absolute value of a rational number as its distance from 0 on the number line; interpret absolute value as magnitude for a positive or negative quantity in a real-world situation. For example, for an account balance of 30 dollars, write 30 = 30 to describe the size of the debt in dollars. d. Distinguish comparisons of absolute value from statements about order. For example, recognize that an account balance less than 30 dollars represents a debt greater than 30 dollars. 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. [s] 6.G.A Solve real-world and mathematical problems involving area, surface area, and volume. 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. LONG BEACH UNIFIED SCHOOL DISTRICT 3 Posted 12/21/17

4 Assessed Grade Level Standards [m] 7.NS.A Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers. 7.NS.1 Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram. a. Describe situations in which opposite quantities combine to make 0. For example, a hydrogen atom has 0 charge because its two constituents are oppositely charged. b. Understand p + q as the number located a distance q from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing real world contexts. c. Understand subtraction of rational numbers as adding the additive inverse, p q = p + ( q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real world contexts. d. Apply properties of operations as strategies to add and subtract rational numbers. 7.NS.2 Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers. a. Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as ( 1)( 1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts. b. Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with non-zero divisor) is a rational number. If p and q are integers, then (p/q) = ( p)/q = p/( q). Interpret quotients of rational numbers by describing realworld contexts. c. Apply properties of operations as strategies to multiply and divide rational numbers. d. Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats. 7.NS.3 Solve real-world and mathematical problems involving the four operations with rational numbers. Key: [m] = major clusters; [s] = supporting clusters; [a] = additional clusters LONG BEACH UNIFIED SCHOOL DISTRICT 4 Posted 12/21/17

5 Assessment Evidence Unit Assessment Evidence of Learning Stage 2 Claim 1: Students can explain and apply mathematical concepts and carry out mathematical procedures with precision and fluency. Concepts and skills that may be assessed in Claim1: 6.NS.C The student uses positive and negative numbers to represent quantities in real-world contexts. The student can identify the location of ordered pairs on the coordinate plane based on the signs of the numbers in an ordered pair. The student locates and positions integers and other rational numbers on a number line. The student positions ordered pairs of integers and other rational numbers on a coordinate plane. The student writes and interprets statements about the order of rational numbers in real-world contexts. The student represents the absolute value of a rational number as the distance from zero on a number line. The student can make comparisons of absolute value from statements about order. The student solves real-world and mathematical problems by graphing ordered pairs on a coordinate plane and using coordinates and absolute value to find the distance between points with the same first coordinate or same second coordinate. 6.G.A The student draws polygons in the coordinate plane, given coordinates for the vertices in the context of solving real-world and mathematical problems. 7.NS.A The student interprets rational number values on a number line, including modeling addition and subtraction expressions. The student applies properties of operations as strategies to add and subtract rational numbers. The student applies properties of operations as strategies to multiply and divide rational numbers. The student converts from a fractional form of rational numbers to a decimal form of rational numbers. The student solves real-world and mathematical problems involving the four operations with rational numbers. Claim 2: Students can solve a range of wellposed problems in pure and applied mathematics, making productive use of knowledge and problem-solving strategies. Standard clusters that may be assessed in Claim 2: 6.NS.C 6.G.A 7.NS.A Claim 3: The student can clearly and precisely construct viable arguments to support their own reasoning and critique the reasoning of others. Standard clusters that may be assessed in Claim 3: 6.NS.C 7.NS.A Claim 4: The student can analyze complex, real-world scenarios and can construct and use mathematical models to interpret and solve problems. Standard clusters that may be assessed in Claim 4: 6.NS.C 6.G.A 7.NS.A LONG BEACH UNIFIED SCHOOL DISTRICT 5 Posted 12/21/17

6 Other Evidence Formative Assessment Opportunities Informal teacher observations Checking for understanding using active participation strategies Exit slips/summaries s Modeling Lessons (SMP 4) Formative Assessment Lessons (FAL) Quizzes/Chapter Tests SBAC Interim Assessment Blocks Access Using Formative Assessment for Differentiation for suggestions. Located on the LBUSD website M Mathematics Curriculum Documents LONG BEACH UNIFIED SCHOOL DISTRICT 6 Posted 12/21/17

7 I will explore integers and the coordinate plane by participating in the Opening. OPENING TASK (choose one) Extending the Number Line is an activity in which students explore negative numbers on a number line. Teachers: Prepare the transparencies and number lines. (SMP 7, 8) Extending the Number Line Exploration Horizontal Number Lines Vertical Number Lines 1 day Using a Coordinate Grid is an online video with a corresponding activity. Have students work in pairs on the discussion questions. Share as a whole group. (SMP 3) Using a Coordinate Grid Video Questions to Using a Coordinate Grid Video A Fly on the Ceiling is a children s book in which students explore the coordinate plane. A Fly on the Ceiling 2 days I will explore numbers that are less than zero by 6.NS.5, 6.NS.6, 6.NS.6a Extending vertical and horizontal number lines to include negative numbers. Explaining that a rational number is a point on a number line. (SMP 3) Using positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of zero. (SMP 3) Recognizing opposite signs of numbers as indicating locations on opposite sides of zero. Recognizing that the opposite of the opposite of a number is the number itself. o How can you represent numbers that are less than zero? o Describe real-life examples that use numbers that are less than zero. o Is zero considered positive or negative? Explain your reasoning. o Synergy Item Bank: Item ID 53035, 53204, 55355, 56104, 51268, 56090, Section 6.1 (Activities 1, 2, and 3; Examples 1, 2, and 3) Positive and Negative Numbers in Real Life Integers on the Number Line Opposite Directions Khan Academy: Intro to negative numbers Khan Academy: Number opposites Mile High Lakes of North America LONG BEACH UNIFIED SCHOOL DISTRICT 7 Posted 12/21/17

8 1-2 days I will compare and order integers by Finding and positioning integers and other rational numbers (fractions and decimals) on horizontal and vertical number lines. Interpreting statements of inequality as statements about the relative position of two numbers on a number line. Writing, interpreting, and explaining statements of order for rational numbers in real-world contexts. (SMP 2, 3) o Explain how to use a number line to compare two integers. (SMP 3, 5) o How can you use a number line to compare positive and negative fractions and decimals? o Does every number have an opposite? Give examples to justify your reasoning. o The farther right a number is on the number line, the greater it is. Do you agree or disagree with this statement? Explain your reasoning. o The farther left a number is on the number line, the less it is. Do you agree or disagree with this statement? Explain your reasoning. o Synergy Item Bank: Item ID 53290, 55348, 55349, 55361, 55363, 55571, Section 6.2 (Activities 1 and 2; Examples 1, 2, 3, 4, and 5) Section 6.3 (Activities 2 and 3; Examples 2 and 3) STEM Video: Tuning a Guitar Guess My Integer Plotting Rational Numbers Khan Academy: Negative decimals & fractions on the number line Khan Academy: Comparing negative numbers Desmos: Order Rational Numbers Desmos: Ordering Integers Fractions on the Number Line STEM Video Performance : Tuning Guitars 6.NS.5, 6.NS.6c, 6.NS.7a, 6.NS.7b LONG BEACH UNIFIED SCHOOL DISTRICT 8 Posted 12/21/17

9 I will explore Explaining that the absolute value of a rational Section 6.4 absolute value number is its distance from 0 on the number line. (Activities 1 and 2; by (SMP 3) Examples 1, 2, and 3) Interpreting absolute value as magnitude for a positive or negative quantity in a real-world situation. Comparing and ordering absolute value expressions. 2 days o What does the absolute value of a number represent? o Absolute values are always positive. Do you agree or disagree with this statement? Explain your reasoning. o Synergy Item Bank: Item ID 52921, 53042, 6.NS.7c, 6.NS.7d 53222, days I will add integers by Using manipulatives to create conjectures about the rules for adding integers. (SMP 5, 7, 8) Using a horizontal and vertical number line diagram. Showing that a number and its opposite have a sum of zero (called additive inverses). Explaining p + q as the number located a distance q from p, in the positive or negative direction depending on whether q is positive or negative. (SMP 3) Interpreting sums of integers by describing realworld contexts. (SMP 2) o Why do additive inverses have a sum of zero? o How can you add integers using multiple methods? o How is adding integers similar to adding whole numbers? How is it different? o Synergy Item Bank: Item ID 53043, 53059, 53334, Section 11.2 (Activities 1, 2, 3, and 4; Examples 1, 2, and 3) STEM Video: Freezing Solid Absolute Value Khan Academy: Absolute value Above and Below Sea Level Adding Integers with Number Lines Adding Two-Digit Numbers Additive Inverse Video mypd Course #2572: Adding Integers With Two Color Counters Integer Addition Beat the Computer Drill Adding Integers Tic-Tac-Toe Khan Academy: Intro to adding negative numbers STEM Video Performance : Melting Matters 7.NS.1a-d LONG BEACH UNIFIED SCHOOL DISTRICT 9 Posted 12/21/17

10 3 days I will subtract integers by Using manipulatives to create conjectures about the rules for subtracting integers. (SMP 5, 7, 8) Showing that the distance between two integers on a number line is the absolute value of their difference. Explaining subtraction of integers as adding the additive inverse, p q p ( q). (SMP 3) Interpreting differences of integers by describing real-world contexts. (SMP 2) o Why do you add the inverse when subtracting? o How can you subtract integers using multiple methods? o How is subtracting integers similar to subtracting whole numbers? How is it different? (SMP 7) o Synergy Item Bank: Item ID 53316, 57151, 64844, Section 11.3 (Activities 1, 2, 3, and 4; Examples 1, 2, and 3) Subtracting Integers Using Two-Color Counters PowerPoint Subtracting Two-Digit Numbers Integer Operations on the Number Line Which One Doesn t Belong? Adding and Subtracting Integers Comic Strip Activity Top This Game Adding and Subtracting Integers Math-O Adding and Subtracting Integers Tic-Tac-Toe Khan Academy: Intro to subtracting negative numbers Khan Academy: Adding & subtracting with negatives on the number line Cliff Diving Differences of Integers Modeling: How Much Did the Temperature Drop? (SMP 4) 7.NS.1a-d LONG BEACH UNIFIED SCHOOL DISTRICT 10 Posted 12/21/17

11 2-3 days I will check my understanding of adding and subtracting of integers by participating in the FAL. OPTION 1: FORMATIVE ASSESSMENT LESSON Adding and Subtracting Directed Numbers (SMP 1, 2, 3, 5, 6, 7, 8) 2 days I will multiply integers by Using manipulatives to create conjectures about the rules for multiplying integers. (SMP 5, 7, 8) Explaining why the product of two negative numbers must be positive using properties, i.e. ( 1)( 1) = 1. (SMP 3) Interpreting products of integers in real-world contexts. (SMP 2) o How can you multiply integers using multiple methods? o How is multiplying integers similar to multiplying whole numbers? How is it different? o Synergy Item Bank: Item ID 53058, 53339, 53354, 53444, 53449, Section 11.4 (Activities 1, 2, 3, and 4; Examples 1, 2, 3, and 4) mypd Course #2574: Multiplying Integers with Two Color Counters Multiplying Two-Digit Numbers Why is a Negative Times a Negative Always Positive? Integer Multiplication Beat the Computer Drill Right on Target Game Add, Subtract, Multiply Integer Operations How Do They Fit? Integers Product Game Khan Academy: Multiply & divide negative numbers Khan Academy: Exponents with negative bases 7.NS.2a-d Multiply to Make 64 The Running Videos LONG BEACH UNIFIED SCHOOL DISTRICT 11 Posted 12/21/17

12 2 days I will divide integers by Using manipulatives to create conjectures about the rules for dividing integers. (SMP 5, 7, 8) Explaining that integers can be divided as long as the divisor is not zero and know that every quotient of integers is a rational number. (SMP 3) Interpreting quotients of integers in real-world contexts. o Compare and contrast multiplying and dividing integers. o How can you divide integers using multiple methods? o How is dividing integers similar to dividing whole numbers? How is it different? (SMP 7) o Synergy Item Bank: Item ID 53315, 53442, 53446, 53448, Section 11.5 (Activities 1, 2, 3, and 4; Examples 1, 2, 3, and 4) Dividing Positive and Negative Numbers Video Dividing Two-Digit Numbers Missing Factor Model for Dividing Integers mypd Course #2467: Dividing Integers with Two Color Counters Integer Division Beat the Computer Drill Multiplying and Dividing Integers Connect 4 Mixed Operations with Integers Math-O Substituting Negative Values into Algebraic Expressions Khan Academy: Multiply & divide negative numbers Khan Academy: Negative signs in fractions 7.NS.2a-d LONG BEACH UNIFIED SCHOOL DISTRICT 12 Posted 12/21/17

13 2 days I will apply operations of rational numbers by Converting a fraction into a decimal. Adding, subtracting, multiplying, and dividing positive and negative decimals, fractions, and mixed numbers. Solving real-world and mathematical problems involving the four operations with positive and negative rational numbers. Applying properties and order of operations as strategies to add, subtract, multiply, and divide positive and negative rational numbers. o When would you need to add, subtract, multiply, or divide rational numbers in a real-world context? o How is adding rational numbers similar to adding integers? How is it different? o Synergy Item Bank: Item ID 70785, 53053, 53268, 53332, 53475, 67819, Section 12.1 (Example 1) Section 12.2 (Activities 1 and 2; Examples 1, 2, 3, and 4) Section 12.3 (Activities 1 and 2; Examples 1, 2, 3, and 4) Section 12.4 (Examples 1, 2, and 4) STEM Video: Carpenter or Joiner Converting Rational Numbers Video Decimal Expansions of Fractions Operations on the Number Line Rational Numbers Which One Doesn t Belong? Multiplying Fractions 3 Converting Between Fractions and Decimals Decimal Maze Multiplying Rational Numbers Product Game Khan Academy: Adding & subtracting negative fractions Khan Academy: Multiply & divide negative fractions Differences and Distances Who is the Better Batter? STEM Video Performance : Precisely Perfect 7.NS.3 LONG BEACH UNIFIED SCHOOL DISTRICT 13 Posted 12/21/17

14 I will explore the coordinate plane by 3 days Identifying the components of the coordinate plane, such as the axes, origin, and quadrants. Plotting ordered pairs in all quadrants, including ordered pairs with rational numbers. Identifying the coordinates of an ordered pair in a graph. Identifying the location of an ordered pair based on the signs of the numbers in the ordered pair. Recognizing that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one of both axes. Describing how the signs of ordered pairs are affected by reflections across the x-axis and/or y-axis. o Explain how to locate points on the coordinate plane. (SMP 3) o How do you know if a point is to the right or the left of the y-axis? How do you know if a point is above or below the x-axis? o Describe the location of an ordered pair with zero as its x or y coordinate. o What conjectures can you make about the signs of coordinates in each quadrant? o Describe how to reflect a point over the x-axis, y-axis, or both axes. o Synergy Item Bank: Item ID 52423, 52427, 53232, 53233, 53273, 53282, Section 6.5 (Activities 1 and 2; Examples 1, 2, and 4) Extension 6.5 (Examples 1 and 2) Plotting Points in the Coordinate Plane Graphing Points on a Coordinate Plane Desmos: Coordinate Plane Activity Desmos: The (Awesome) Coordinate Plane Activity Coordinate Plane: Reflecting Points Video Reflecting Points on the Coordinate Plane Technology Activity Khan Academy: Coordinate plane Desmos: Battle Boats Port Royal A Sunken City 6.NS.6b, 6.NS.6c, 6.NS.8 LONG BEACH UNIFIED SCHOOL DISTRICT 14 Posted 12/21/17

15 1 day I will check my understanding of graphing on the coordinate plane by participating in the FAL. OPTION 2: FORMATIVE ASSESSMENT LESSON Reflecting Points Over Coordinate Axes (SMP 1, 2, 3, 5, 6, 7, 8) 2 days I will solve realworld and mathematical problems involving the coordinate plane by Drawing polygons in the coordinate plane given coordinates for the vertices. Finding distance between ordered pairs on a graph with the same first coordinate or the same second coordinate. Using of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate. o Why is the length of a segment always positive? o Explain how you find the distance between two points. o When finding distance between ordered pairs on a graph with the same first coordinate or the same second coordinate, which do you prefer: counting the distance or adding the absolute values? Explain your reasoning. o Synergy Item Bank: Item ID 52408, 52470, 53106, Section 4.4 (Activities 1, 2, and 3; Examples 2 and 3) Section 6.5 (Example 3; Exercises 25-34, 53) Desmos: Zombie Apocalypse Area of a Quadrilateral on a Coordinate Plane Distance Between Points Khan Academy: Polygons on the coordinate plane Desmos: Rectangle Ordered Pairs Card Sort Facial Recognition 6.G.3 LONG BEACH UNIFIED SCHOOL DISTRICT 15 Posted 12/21/17

16 I will prepare for Incorporating the Standards for Mathematical Ch. 6 Study Help (p. 266) the unit Practice (SMPs) along with the content standards to Ch. 6 Review assessment on review the unit. (p ) 1 day integers and the Ch. 11 Study Help (p. 494) coordinate plane Ch. 11 Review by (p ) Temperature Hotel Elevator Bottling 1 day Unit Assessment Students will take the Synergy Online Unit Assessment. Unit Assessment Resources (Word or PDF) can be used throughout the unit. At this point, all standards addressed in the Grade 6 SBAC Interim Assessment Block The Number System and Grade 6 SBAC Interim Assessment Block Geometry have been covered. This block may now be administered. LONG BEACH UNIFIED SCHOOL DISTRICT 16 Posted 12/21/17