6thGradeMathematicsCurriculumGuide
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Unit 3: Rational Numbers – No Calculators Time Frame: Quarter 2 – about 23 days 7
Connections to Previous Learning Students extend their previous understandings of number and the ordering of numbers to the full system of rational numbers, which includes negative rational numbers, and in particular negative integers. Examples that use positive and negative numbers to describe nature, financial credits and debits, or electricity help build a context for learning about integers and the meaning of “0”. Focus within the Grade Level: Much of the learning in this unit is related to distances on a number line. Students learn that between two whole numbers on a number line, there are points that are described by rational numbers. Students compare and order rational numbers on the number line using statements about the relative position of the numbers on the line and record these comparisons using inequalities. For instance, ‐5 > ‐8 is described as ‐5 is located to the right of ‐8 on a number line oriented from left to right. Nature, finances or temperatures might be used as contexts to describe the numbers. For instance, ‐3° Centigrade is warmer than ‐7°. Students’ experiences placing rational numbers on vertical and horizontal number lines prepare them to plot points in all 4 quadrants of the coordinate plane. They see the sign of the number as an indicator of directionality and the number, itself as the distance a point is from zero, or the origin. They reason about the order and absolute value of rational numbers, and learn to interpret absolute value l5l as the magnitude for a negative or positive number. For example, for a money account of – 5 dollars, the l5l means the quantity of money owed or debited. Through experiences with number lines and other contexts, students learn that the opposite of the opposite of a number is the number itself, e.g., ‐ (‐6) = 6 and they learn that 0 is its own opposite. Connections to Subsequent Learning: Students in Grade 6 also build on their work with distance in elementary school by reasoning about relationships among shapes to determine distances. Students will apply what they learn about integers to their work on expressions in Unit 4. They prepare for work on scale drawings and constructions in Grade 7 by drawing polygons in the coordinate plane. Mathematical Practices 1. Make Sense of Problems and Persevere in Solving Them. 5. Use Appropriate Tools Strategically. 2. Reason Abstractly and Quantitatively. 6. Attend to Precision. 3. Construct Viable Arguments and Critique the Reasoning of Others. 7. Look for and Make Use of Structure. 4. Model with Mathematics. 8. Look for and Express Regularity in Repeated Reasoning. 1 6thGradeMathematicsCurriculumGuide
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Stage 1 Desired Results
Transfer Goals Students will be able to independently use their learning to…
 Apply a sense of number size to their lives. Meaning Goals UNDERSTANDINGS Students will understand that…  Quantities having more or less than zero are described using positive and negative numbers.  Number lines are visual models used to represent the density principle: between any two whole numbers are many rational numbers, including decimals and fractions.  The rational numbers can extend to the left or to the right on the number line, with negative numbers going to the left of zero, and positive numbers going to the right of zero.  The coordinate plane is a tool for modeling real‐world and mathematical situations and for solving problems. ESSENTIAL QUESTIONS
 How are positive and negative numbers used?  How do rational numbers relate to integers?  What is modeled on the coordinate plane? Acquisition Goals Students will know…  The meaning of a negative number.  The meaning of absolute value.  Parts of a coordinate graph.  Parts of a coordinate pair.  Place value determines the significance of a number.  Visual fraction model.  Positive and negative numbers.  Absolute value as the distance from zero on the number line.  The coordinate plane: ordered pairs, quadrants, axes, reflections.  Integers.  Rational numbers.  Rational numbers have an order and an absolute value. Students will be skilled at…
 Identify an integer and its opposite and the directions they represent in real‐world contexts. (6.NS.5)  Use integers to represent quantities in real‐world situations (above/ below sea level) (6.NS.5)  Understand the meaning of 0 and where it fits into a situation(6.NS.5)  Represent and explain the value of a rational number as a point on a number line (6.NS.6)  Recognize that a number line can be both vertical and horizontal (6.NS.6)  Represent a number and its opposite equidistant from zero on a number line. (6.NS.6)  Identify that the opposite of the opposite of the number is itself. (6.NS.6)  Incorporate opposites on the number line or plot opposite points on a coordinate grid where x and y intersect at zero. (6.NS.6)  Represent signs of numbers in ordered pairs as locations in quadrants on the coordinate plane and explain the relationship between the location and the signs. (6.NS.6)  Represent and explain reflections of ordered pairs on a coordinate plane (6.NS.6) 2 6thGradeMathematicsCurriculumGuide
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Locate and position integers and other rational numbers on horizontal or vertical number lines (6.NS.6) Locate and position integers and other rational numbers on a coordinate plane. (6.NS.6) Identify the absolute value of a number as the distance from zero (6.NS.7) Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram. (6.NS.7) Use inequalities to order integers relative to their position on the number line(6.NS.7) Write statements of order for rational numbers in real‐world contexts. (6.NS.7) Interpret & explain statements of order for rational numbers in real‐world contexts. (6.NS.7) Represent the absolute value of a rational number as the distance from zero and recognize the symbol │ x │. (6.NS.7) Interpret absolute value as magnitude for a positive or negative quantity in a real‐world situation. (6.NS.7) Distinguish comparisons of absolute value from statements about order. (Compare rational numbers using absolute value in real‐world situations. For negative numbers, as the absolute values increases, the value of the number decreases.) (6.NS.7) Solve real‐world problems by graphing points in all four quadrants of the coordinate plane (6.NS.8) Use coordinates to find distances between points with the same first coordinate or the same second coordinate. (6.NS.8) Use absolute value to find distances between points with the same first coordinate or the same second coordinate. (6.NS.8) 3 6thGradeMathematicsCurriculumGuide
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Calculator 6.NS.5 6.NS.6 no 6.NS.6a no 6.NS.6b no 6.NS.6c no 6.NS.7 6.NS.7a no 6.NS.7b no 6.NS.7c no 6.NS.7d no 6.NS.8 Materials Needed:
Engage NY Module 3 Additional – if needed Holt Course 1 Holt Course 2 no Stage 1 Established Goals: Common Core State Standards for Mathematics Number System 6.NS.C Apply and extend previous understandings of numbers to the system of rational numbers. 6.NS.C.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.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. 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 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. 6.NS.C.7b Write, interpret, and explain statements of order for rational numbers in real‐world contexts. For example, write –3 oC > –7 oC to express the fact that –3 oC is warmer than –7 oC. 6.NS.C.7c 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. 6.NS.C.7d Distinguish comparisons of absolute value from statements about order. For example, recognize that an account balance less than –30 dollars represents a debt 4 6thGradeMathematicsCurriculumGuide
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greater than 30 dollars. 6.NS.C.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.  Major Clusters  Supporting Clusters  Additional Clusters Suggested Assessments Fluency Activities
Vocabulary absolute value (6) The absolute value of a number is the distance between the number and zero on the number line.; shown by l l. axes (3) The horizontal number line (x‐axis) and the vertical number line (y‐axis) on the coordinate plane coordinate (5) one of the numbers of an ordered pair that locate a point on a coordinate graph. coordinate grid/plane (5) A plane formed by the intersection of a horizontal number line called the x‐axis and a vertical number line called the y‐axis. Credit (6) To add an amount of something debit (6) To withdraw an amount of something distance (4) A numerical description of how far away objects or numbers are away from each other. greater than or equal to (6) ≥ greater than (K) (>) is symbol used to compare two numbers, with the greater number given first. inequality (5) A mathematical sentence that shows the relationship between quantities that are not equal. Integer (6) A member of the set of whole numbers and their opposites. less than or equal to (6) ≤ less than (K) (<) is a symbol used to compare two numbers, with the lesser number given magnitude (6) how far away the math term is from zero. 5 6thGradeMathematicsCurriculumGuide
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negative (6) A number less than zero number line diagram (1) a line on which numbers are marked at intervals, used to illustrate simple numerical operations. opposite (5) Two numbers that are an equal distance from zero on a number line. ordered pair (5) A pair of numbers that can be used to locate a point on a coordinate plane. positive (6) A number greater than zero quadrant (6) The x‐ and y‐ axes divide the coordinate plane into four regions. Each region is called a quadrant. rational numbers (6) Any number that can be expressed as a ratio of two integers. reflect (8) A transformation of a figure that flips the figure across a line. relative position (6) x‐coordinate (6) The first number in an ordered pair; it tells the distance to move right or left from the origin, (0,0). y‐coordinate (6) The second number in an ordered pair; it tells the distance to move up or down from the origin, (0,0). > (1) The symbol for “greater than” < (1) The symbol for “less than” ≥ () The symbol for “greater than or equal to” ≤ () The symbol for “less than or equal to” 6 6thGradeMathematicsCurriculumGuide
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6.NS.C.5 Vocab – integers, rational numbers, positive, negative, opposite, Students use rational numbers (fractions, decimals, and integers) to represent real‐world contexts and understand the meaning of 0 in each situation. Example 1: a. Use an integer to represent 25 feet below sea level b. Use an integer to represent 25 feet above sea level. c. What would 0 (zero) represent in the scenario above? Solution: a. ‐25 b. +25 c. 0 would represent sea level 6.NS.C.6 Vocab – quadrants, line diagrams, coordinate, x‐coordinate, y‐coordinate, ordered pair, axis, reflect 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. Example:  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? 7 6thGradeMathematicsCurriculumGuide
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6.NS.C.7 Vocab ‐ absolute value, distance, inequality, greater than, >, less than, <, greater than or equal to, ≥, less than or equal to, ≤, credit/debit. Common models to represent and compare integers include number line models, temperature models and the profit‐loss model. On a number line model, the number is represented by an arrow drawn from zero to the location of the number on the number line; the absolute value is the length of this arrow. The number line can also be viewed as a thermometer where each point of on the number line is a specific temperature. In the profit‐loss model, a positive number corresponds to profit and the negative number corresponds to a loss. Each of these models is useful for examining values but can also be used in later grades when students begin to perform operations on integers. In working with number line models, students internalize the order of the numbers; larger numbers on the right or top of the number line and smaller numbers to the left or bottom of the number line. They use the order to correctly locate integers and other rational numbers on the number line. By placing two numbers on the same number line, they are able to write inequalities and make statements about the relationships between the numbers. Comparative statements generate informal experience with operations and lay the foundation for formal work with operations on integers in grade 7. Example:  One of the thermometers shows ‐3°C and the other shows ‐7°C. Which thermometer shows which temperature? Which is the colder temperature? How much colder? Write an inequality to show the relationship between the temperatures and explain how the model shows this relationship. 8 6thGradeMathematicsCurriculumGuide
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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. Example:  The Great Barrier Reef is the world’s largest reef system and is located off the coast of Australia. 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. 6.NS.C.8 Vocabulary: X‐coordinate, y‐coordinate, Students find the distance between points when ordered pairs have the same x‐coordinate (vertical) or same y‐coordinate (horizontal). Example:  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? 9 6thGradeMathematicsCurriculumGuide
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Stage 2 ‐ Evidence
Evaluative Criteria/Assessment Level Descriptors (ALDs): 6.NS.C – (SBAC Target D) Level 4 – no descriptor Level 3 students should be able to apply and extend previous understandings of numbers to relate statements of inequality to relative positions on a number line, place points with rational coordinates on a coordinate plane, and solve problems involving the distance between points when they share a coordinate. They should be able to understand absolute value and ordering by using number lines and models and relate reflection across axes to changes in sign. Level 2 students should be able to apply and extend previous understandings of whole numbers to order rational numbers and interpret statements of their order in the context of a situation. They should be able to place all rational numbers on a number line and integer pairs on a coordinate plane with various axis increments. They should be able to relate changes in sign to placements on opposite sides of the number line and understand the absolute value of a number as its distance from zero on a number line. Level 1 students should be able to place all integers on a number line and integer pairs on a coordinate plane with one‐unit increments on both axes. SBA Examples Claim 1 Item Specs 10 6thGradeMathematicsCurriculumGuide
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Stage 3 – Learning Plan Sample Summary of Key Learning Events and Instruction that serves as a guide to a detailed lesson planning
NOTES:
LEARNING ACTIVITIES: **Days may change depending on any tasks or assessing you choose to do. Each lesson is ONE day, and ONE day is considered a 45‐minute period.
6.NS.5, 6.NS.6, 6.NS.7, 6.NS.8
IXL Ex. Lessons 1‐2: Ratios is 2 days (one per lesson) unless otherwise noted in margins. F.9, M.1, M.2, M.3, M.4, P.1, P.2, P.3, P.6, S.9, W.1, W.2, W.3, W.4, W.5, Z.2, BB.17, BB.18 Topic A: Understanding Positive and Negative Numbers on the Number Line (6.NS.C.5, 6.NS.C.6a, 6.NS.C.6c) Intervention Holt Course 1 Lesson 11‐1 Integers in Real‐World Situations Lesson 1: Positive and Negative Numbers on the Number Line—Opposite Direction and Value Holt Course 1 Lesson 11‐2 Comparing and Ordering Integers Lessons 2–3: Real‐World Positive and Negative Numbers and Zero Holt Course 2 Lesson 2‐11 Comparing and Ordering Rational Lesson 4: The Opposite of a Number Numbers Lesson 5: The Opposite of a Number’s Opposite Lesson 6: Rational Numbers on the Number Line Performance Task Positive and Negative Events – 6.NS.5 Topic B: Order and Absolute Value (6.NS.C.6c, 6.NS.C.7) Mathtastic Amusement Park – 6.NS.6bc Lessons 7–8: Ordering Integers and Other Rational Numbers If behind, consider omitting Lesson 8, as it Percent Cards 6.NS.6 shares the same Student Outcomes with Lesson 7. Consider reserving the Opening Exercise and Exercise 1 from Lesson 8 for fluency review in later lessons. TI Activities Verbal to Visual – 6.NS.6 Lesson 9: Comparing Integers and Other Rational Numbers Lesson 10: Writing and Interpreting Inequality Statements Involving Rational Numbers 6.NS.C.7 Lesson 11: Absolute Value—Magnitude and Distance Intervention Lesson 12: The Relationship Between Absolute Value and Order Holt Course 2 Lesson 2‐1 Integers (absolute value) Lesson 13: Statements of Order in the Real World Performance Task Mid‐Module Assessment Symbols of Inequalities and the Coordinate System 6.NS.7 Topics A through B (assessment 1 day, return 1 day, remediation or further applications 1 day) Representing Rational Numbers on the Number Line 6.NS.7 What’s Your Sign? 6.NS.6, 6.NS.7 Topic C: Rational Numbers and the Coordinate Plane (6.NS.C.6b, 6.NS.C.6c, 6.NS.C.8) Lesson 14: Ordered Pairs Lesson 15: Locating Ordered Pairs on the Coordinate Plane Lesson 16: Symmetry in the Coordinate Plane Lesson 17: Drawing the Coordinate Plane and Points on the Plane Lesson 18: Distance on the Coordinate Plane Lesson 19: Problem Solving and the Coordinate Plane 11 6thGradeMathematicsCurriculumGuide
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Stage 3 – Learning Plan Sample End‐of‐Module Assessment Topics A through C (assessment 1 day, return 1 day, remediation or further applications 1 day) Daily Lesson Plan Learning Target: Opening Activity: Activities:  Whole Group:  Small Group/Guided/Collaborative/Independent:  Whole Group: Checking for Understanding (before, during and after): Assessments: 6.NS.C.8 Intervention Holt Course 1 Lesson 11‐3 The Coordinate Plane Performance Task Graphing on the Coordinate Plane 6.NS.6, 6.NS.8 12