Unit 3

Rational Numbers

Grade 6

Math

Unit Length and Description:

 

25 days

In this unit students begin the study of negative numbers, their relationship to positive numbers, and the meaning and uses of absolute value. Students understand that all numbers have an opposite, and that a number and its opposite are equidistant from zero. Students will use the number line to order rational numbers and understand the absolute value of a number. Students will also begin work with the Cartesian coordinate system. They identify the four quadrants and can identify the quadrant for an ordered pair based on the signs of the coordinates. Students will also solve real-world and mathematical problems using all four quadrants of a coordinate plane.

 

Standards:

 

NS The Number System

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.

a.   Recognize opposite signs of numbers as indicating locations on opposite sides of 0 on the number line;

b.   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.

c.    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.

d.   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.

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 d dollars represents a debt 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.

 

Foundational Standards

Develop understanding of fractions as numbers.

3.NF.A.2

 

Understand a fraction as a number on the number line; represent fractions on a number line diagram.

a.   Represent a fraction 1/b on a number line diagram by defining the interval from 0 to 1 as the whole and partitioning it into b equal parts. Recognize that each part has size 1/b and that the endpoint of the part based at 0 locates the number 1/b on the number line.

b.   Represent a fraction a/b on a number line diagram by marking off a lengths 1/b from 0. Recognize that the resulting interval has size a/b and that its endpoint locates the number a/b on the number line.

 

Draw and identify lines and angles, and classify shapes by properties of their lines and angles.

4.G.A.3

Recognize a line of symmetry for a two-dimensional figure as a line across the figure such that the figure can be folded along the line into matching parts. Identify line-symmetric figures and draw lines of symmetry.

Graph points on the coordinate plane to solve real-world and mathematical problems.

5.G.A.1

Use a pair of perpendicular number lines, called axes, to define a coordinate system, with the intersection of the lines (the origin) arranged to coincide with the 0 on each line and a given point in the plane located by using an ordered pair of numbers, called its coordinates. Understand that the first number indicates how far to travel from the origin in the direction of one axis, and the second number indicates how far to travel in the direction of the second axis, with the convention that the names of the two axes and the coordinates correspond (e.g., -axis and -coordinate, -axis and -coordinate).

5.G.A.2

Represent real-world and mathematical problems by graphing points in the first quadrant of the coordinate plane, and interpret coordinate values of points in the context of the situation.

Standards for Mathematical Practices

1.   Make sense of problems and persevere in solving them.

2.   Reason abstractly and quantitatively.

3.   Construct viable arguments and critique the reasoning of others.

4.   Model with mathematics.

5.   Use appropriate tools strategically.

6.   Attend to precision.

7.   Look for and make use of structure.

8.   Look for and express regularity in repeated reasoning.

Instructional Outcomes

 

        6.NS.C.5:

o   I can indentify an integer and its opposite.

o   I can identify a negative number as being to the left of 0 on a number line and a positive number as being to the right of 0 on a number line.

o   I can use integers to represent quantities in real world situations (for example, above/below sea level, temperature above/below zero, debit/credit.)

o   I can explain the meaning of 0 in real-world contexts.

        6.NS.C.6a 6c:

o   I can identify a rational number as a point on the number line.

        6.NS.C.6a

o   I can identify the location of zero on a number line in relation to positive and negative numbers.

o   I can recognize opposite signs of numbers as locations on opposite sides of 0 on the number line.

o   I can recognize that the opposite of the opposite of a number is the number itself.

o   I can recognize that 0 is its own opposite.

        6.NS.C.6b

o   I can identify the quadrant of the coordinate plane a point is in based on the sign of the numbers in an ordered pair.

o   I can reason that when only the x value in a set of ordered pairs are opposites, it creates a relfection over the y axis, for example, (x,y) and (x,-y).

o   I can recognize that when only the y value in a set of ordered pairs are opposites, it creates a reflection over the x axis, for example, (x,y) and (x, -y).

o   I can reason that when two ordered pairs differ only by signs, the locations of the points are related by reflections across both axes, for example, (-x,-y) and (x,y).

        6.NS.6c

o   I can find and position integers and other rational numbers on a horizontal or vertical number line diagram.

o   I can find and position pairs of integers and other rational numbers on a coordinate plane.

        6.NS.C.7a 7d

o   I can order rational numbers on a number line.

        6.NS.C.7a

o   I can interpret statements of inequality as statements about relative position of two numbers on a number line diagram.

        6.NS.C.7b

o   I can write, interpret, and explain statements of order for rational numbers in real-world contexts.

        6.NS.C.7c

o   I can identify absolute value of rational numbers.

o   I can interpret absolute value of a rational number as distance from 0 on a number line.

o   I can apply the ideas of absolute value to real-world quantities, when appropriate.

        6.NS.C.7d

o   I can distinguish comparsons of absolute value from statements about order and apply to real world context.

        6.NS.C.8

o   I can calculate absolute value.

o   I can graph points in all four quadrants of the coordinate plane.

o   I can solve real-world problems by graphing points in all four quadrants of a coordinate plane.

o   I can calculate the distances between two points with the same first coordinate or the same second coordinate using absolute value, given only coordiantes.

 

 

 

Enduring Understandings:

 

        Numeric fluency includes both the understanding of and the ability to appropriately use numbers.

        A quantity can be represented numerically in various ways.

        The symbolic language of algebra is used to communicate and generalize the patterns in mathematics.

        Coordinate geometry can be used to represent and verify geometric/ algebraic relationships.

 

Essential Questions:

 

        What is the meaning of positive and negative numbers and zero in real-life situations?

        How can I compare rational numbers on a number line?

        How do you find value of an integer on the number line?

        How can you write and graph positive and negative integers?

        How does absolute value relate to distance on a number line?

        How can you plot points on a coordinate plane?