6.NS.6 - About the Math, Learning Targets, and Increasing Rigor

Full Standard

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.

1. 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.
2. 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.
3. 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.

Learning Targets

• Plot integers and rational numbers on number lines.
• Graph ordered pairs in all four quadrants using integers and other rational numbers.
• Describe the meaning of opposites using number lines and physical models (i.e. two-color counters or algebra tiles).
• Explain how the meaning of the signs of ordered pairs influences their location on a coordinate plane.
• Recognize that locations of the points, that differ only by signs, are related by reflections across one or both axes.
• Explain that if the only difference between coordinates is their signs they are the same distance from the origin and one or both axes.

About the Math

• Plot integers and rational numbers on number lines.
• Graph ordered pairs in all four quadrants using integers and other rational numbers.
• Explain how the meaning of the signs of ordered pairs influences their location on a coordinate plane.
• Recognize that locations of the points, that differ only by signs, are related by reflections across one or both axes.

In earlier grades, students worked with positive fractions, decimals and whole numbers on the number line and in quadrant 1 of the coordinate plane. In 6th grade, students extend the number line to represent all rational numbers and recognize that number lines may be either horizontal or vertical (e.g., thermometer) which facilitates the movement from number lines to coordinate grids. Students continue to place fractions, decimals and whole numbers on the number line as well as recognize that a number and its opposite are equidistance from zero (reflections about the zero). The opposite sign (–) shifts the number to the opposite side of 0. For example, – 4 could be read as “the opposite of 4” which would be negative 4. In the example, – (–6.4) would be read as “the opposite of the opposite of 6.4” which would be 6.4.

Students worked with Quadrant I in earlier grades. As the x-axis and y-axis are extending to include negatives, students begin to work with the Cartesian Coordinate system. 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 (–, +). 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.

• Describe the meaning of opposites using number lines and physical models (i.e. two-color counters or algebra tiles).
• Explain that if the only difference between coordinates is their signs they are the same distance from the origin and one or both axes.

Integers are opposites if they are the same distance from 0. 4 and -4 are both 4 from 0 and thus they are opposite. Students in 6th grade need to understand opposites in this way. Connecting this understanding to algebra tiles or two sided counters will help reinforce this foundation concept for work with computing integers later. 

Essential vocabulary for this standard includes: rational numbers, opposites, absolute value, origin, quadrants, coordinate plane, ordered pairs, x-axis, y-axis, and coordinates. 

Progression of Standard within Grade 6 (5 AGL)

This progression informs how to develop the standard within the grade level. This progression is provided by HCPSS Elementary Mathematics.

Progression of this Standard Across Grades

This progression is informed by the Achieve the Core Coherence Map Links to an external site.. Information is not the complete standard.

TASKS

These tasks can be used with small group or whole group instruction.

• Look at point (2, 4), if you change the y-coordinate to be the opposite of the existing integer, then you will be reflecting over which axis? Plot both points to support your answer. (over the x-axis) What if you change the x-coordinate to the opposite, which axis will you reflect over? (y-axis)
• Between which two integers will -2.7 fall? Which whole number integer is it closest to?
• Create a number line and plot the following rational numbers on the line: -3.5, 4, 1.5, -3, and -.5.
• The opposite of 4 is -4 and the opposite of -9 is 9. What is the opposite of zero? Explain your reasoning.

SLIDE-BASED TASKS 

  These links are HCPSS created instructional tasks. These tasks are provided in Google slides. These tasks should be used for inspiration and resources, but instruction should start with students having the opportunity to engage with the math first (often involving physical and/or visual models) followed by discussion and explicit instruction to ensure student understanding.

Additional Tasks 

  These links provide instructional ideas connected to this standard. [NOTE: NCTM membership required for access to Illuminations lessons.] 

• Points on a Number Line Links to an external site.(Illustrative Math Lesson 7.2)
• Comparing Distances from Zero Links to an external site.(Illustrative Math Lesson 7.7)
• Points on a Coordinate Plane Links to an external site.(Illustrative Math Lesson 7.11)
• Integers on the Number Line 2 Links to an external site.  (Links to an external site.)  (Illustrative Math)
• Reflecting points over coordinate axes  (Links to an external site.) Links to an external site. (Illustrative Math)
• Plotting points in the coordinate plane  (Links to an external site.) Links to an external site. (Illustrative Math)
• Deflategate Links to an external site. (YummyMath) 
• Amusement Park Download Amusement Park (Lesson Collection)
• Line 'em Up Links to an external site. (Illuminations, NCTM)
• Coordinate Plane Download Coordinate Plane
  (PPT)
• Opposite Numbers on a Number Line Download Opposite Numbers on a Number Line
  (HCPSS Lesson)
• Graphing Coordinates Links to an external site. (MSDE Lesson Resource)

Tasks From Print Resources

  These publications have been provided for each school. They are typically stored in team closets or the media center. Check with your team leader if you cannot find them. 

Print Resources

Book Thumbnail	Book Title	Grade	Pages
	Teaching Student-Centered Mathematics	6-8	Coordinate Slides, Activity 13.15, Page 288 Coordinate Reflections, Activity 13.16, Page 289
	Groundworks, Algebraic Thinking	5	1-7 ('Get the Point')
	Hands-On Standards Common Core	6	30 (Lesson 3)

Centers

  These print resources can be used during independent or center time. These resources could also be used as lesson seeds.

INDEPENDENT PRACTICE/HOMEWORK/ASSESSMENT

  These resource sheets can be used for independent practice, homework, or assessment. They are intended to reinforce procedures and concepts. They should not be used as a source of direct instruction or whole-group practice.

• Integers Using Number Lines and Ordered Pairs a Links to an external site.
• Integers Using Number Lines and Ordered Pairs b Links to an external site.

• Plot Integers on a Number Line Links to an external site.
• Plot Integers and Identify Opposites on a Number Line Links to an external site.
• Identify Opposites on a Number Line Links to an external site.
• Plot Integers, Rational Numbers, and Their Opposites on a Number Line Links to an external site.
• Plot and Reason About Opposites on a Number Line Links to an external site.
• Plot and Reason About Points on a Number Line and Coordinate Grid Links to an external site.
• Plot and Identify Points and Their Reflections on a Coordinate Grid Links to an external site.
• Reason About Reflections on a Coordinate Grid Links to an external site.
• Reason About Reflections on a Coordinate Grid_2 Links to an external site.
• Identify Reflected Points on a Coordinate Grid Links to an external site.

Learning Targets

• Plot integers and rational numbers on number lines.
• Graph ordered pairs in all four quadrants using integers and other rational numbers.
• Describe the meaning of opposites using number lines and physical models (i.e. two-color counters or algebra tiles).
• Explain how the meaning of the signs of ordered pairs influences their location on a coordinate plane.
• Recognize that locations of the points, that differ only by signs, are related by reflections across one or both axes.
• Explain that if the only difference between coordinates is their signs they are the same distance from the origin and one or both axes.

Learning targets identify what students should be able to do. This rubric can be applied to tasks and observations for assessment and/or grading. 

Rubric for Tasks Links to an external site.