Rational Numbers
Unit description: In this unit the students will learn to develop the concept of opposite numbers and absolute values, and that zero is its own opposite. They will use positive and negative numbers to represent real-world quantities and compare and order integers and rational numbers with and without number lines. The students will describe the relationship between rational numbers in real-world contexts through comparison and using understanding of absolute value. The students will learn to plot points in all four quadrants, find the distance between points, identify reflections across both axes, and create polygons.
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Essential Outcomes of the Unit
The Number System
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. 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. Use number lines and coordinate axes to represent points on a number line and in the coordinate plane with negative number coordinates.
- 6.NS.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, and that 0 is its own opposite.
- 6.NS.6c Find and position integers and other rational numbers on a horizontal or vertical number line. 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.
Other Standards Addressed in the Unit
The Number System
Apply and extend previous understandings of numbers to the system of rational numbers
- 6.NS.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.7a Interpret statements of inequality as statements about the relative position of two numbers on a number line.
- 6.NS.7b Write, interpret, and explain statements of order for rational numbers in real-world contexts
- 6.NS.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.
- 6.NS.7d Distinguish comparisons of absolute value from statements about order.
- 6.NS.8 Solve real-world and mathematical problems by graphing points on a coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate.
Geometry
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.
Essential Questions and Big Ideas
How are positive and negative numbers used?
- Quantities having more or less than zero are described using positive and negative numbers.
How do rational numbers relate to integers?
- 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.
What is modeled on the coordinate plane?
- The coordinate plane is a tool for modeling real-world and mathematical situations and for solving problems.