Download Unit 1C - Rational Numbers

Unit 1c – The Number System
Class Notes
Date
Locating a Point and Its Opposite on a Number Line
Common Core Standard: 6.NS.C.6 Understand a rational number as a point on the number line. Recognize opposite
signs of numbers as indicating locations on opposite sides of 0 on the number line.
Learning Target: I can locate a point and its opposite on a number line.
Important Terms:
 Positive Number: Any number greater than 0. They may be written with a positive sign (+) but they are usually
without it. Ex 8 or +8
 Negative Number: Any number less than 0. They are always written with a negative sign (-). Ex. -8
 Opposite: Numbers that are the same distance from 0 on a number line but on different sides of 0. Ex. 3 and -3
 Integer: A set of all whole numbers and their opposites.
o Whole Numbers: Zero and the counting numbers. Ex. 0, 1, 2, 3, 4, …
Examples:
Try This:
Graph each integer and its opposite on the number line.
1)
7
2)
-6
3) 0
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Page 1
4) 3.25
5) -2.6
6) 4
5
8
1
7) −9 2
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Page 2
Date
Describe Quantities Using Positive and Negative Numbers
Common Core Standard: 6.NS.C.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.
Learning Target: I can describe quantities using positive and negative numbers.
Important Information:

Positive Number: Any number greater than 0. They may be written with a positive sign (+) but they are usually
without it. Ex 8 or +8
 Negative Number: Any number less than 0. They are always written with a negative sign (-). Ex. -8
Examples:
Name a positive or negative number to represent each situation.
Write a situation that each integer could represent.
A. 15
saving $15.00 dollars
B. -9
a loss of 9 yards in football
C. 18
an increase of 18 points
D. 0
zero degrees outside
Try This:
Name a positive or negative number for each situation.
1) Earning $50
3) 7 feet above sea level
2) 20° below zero
4) a decrease of 39 points
5) a house sinking 5 inches
6) elevator going up 5 floors
Write a situation that each integer could represent.
7) +49
9) -83
8) -7
10) +15
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Page 3
Date
Absolute Value
Common Core Standard: 6.NS.C.7 Understand ordering and absolute value of rational numbers. Understand the
absolute value of a rational number as its distance from zero on the number line.
Learning Target: I can identify and explain the absolute value of a number.
Important Terms:
 Absolute Value: The distance a number is from 0 on a number line. The symbol for absolute value is ││.
Ex. │-5│ = 5 and │3│ = 3
Important Information:
 Absolute values are never negative.
 Opposite integers have the same absolute value. Ex. │-5│ = 5 and │5│ = 5
 The absolute value of zero is zero. Ex. │0│ = 0
Examples:
Try This:
Use the number line to find the absolute value of each integer.
1) │-9│
3) │6│
2) │3│
4) │-2│
Find each absolute value.
5) │419│
8) │96│
6) │-295│
9) │723│
7) │-806│
10) │-150│
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Page 4
Date
Converting Between Decimals and Fractions
Common Core Standard:
4.NF.C.6 Use decimal notations for fractions.
5.NF.B.3 Interpret a fraction as a division of the numerator by the denominator.
Learning Target: I can convert between decimals and fractions.
Important Terms:
 Mixed Number: A number that contains both a whole number greater than zero and a fractions, such as 1 ½.
 Terminating Decimal: A decimal number that has a finite number of decimal places, a decimal that ends or
terminates, such as 0.75.
 Repeating Decimal: A decimal number that has a block of one or more digits that repeats continuously, such
as 0.6666…
Important Information:
Examples:
Part A: Writing Fractions as Decimals
Part B: Writing Decimals as Fractions
Try This:
Write each fraction or mixed number as a decimal.
2
7
1)
2) 2
3) 2
Write each decimal as a fraction or mixed number.
5) 0.15
6) 2.6
7) 0.625
5
8
5
6
4)
17
20
8) 3.375
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Page 5
Date
Compare Rational Numbers
Common Core Standard: 6.NS.C.7 Understand ordering and absolute value of rational numbers. Interpret statements
of inequality as statements about the relative position of two numbers on a number line diagram. Write, interpret and
explain statements of order for rational numbers in real-world contexts.
Learning Target: I can compare rational numbers, including integers, fractions and decimals.
Important Information:
 >, is the symbol for greater than.
 <, is the symbol for less than.
 =, is the symbol for equal to.
Examples:
Comparing Decimals
A. 8.04
8.403
<
When comparing the tenths place, the 0 in 8.04
is smaller than the 4 in 8.403. As a result, 8.04 is
less than 8.403.
B. 0.0906
-0.6801
>
When comparing a positive and a negative number
the positive number is always the greater one,
regardless of the numbers. As a result, 0.0906 is
greater than -0.6801.
Comparing Fractions
(you can also change all fractions to decimals and then follow the steps above)
Comparing Integers
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Page 6
Try This:
Compare. Write <, > or =.
1) 3.578
4) 0.78
3.758
7
9
2)
3
5
5) 0.65
2
21
2
−3
3) -8
-9
6) -21
-1
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Page 7
Date
Order Rational Numbers
Common Core Standard: 6.NS.C.7 Understand ordering and absolute value of rational numbers. Interpret statements
of inequality as statements about the relative position of two numbers on a number line diagram. Write, interpret and
explain statements of order for rational numbers in real-world contexts.
Learning Target: I can order rational numbers, including integers, fractions and decimals.
Examples:
Ordering Decimals
Ordering Fractions
(you can also change all fractions to decimals and then follow the steps above)
Ordering Integers
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Page 8
Try This:
Order the fractions, decimals and integers from least to greatest.
2 4 11
1) 32.525, 32.52524. 31.6257
2) 5 , 9 , 15
Order the fractions, decimals and integers from greatest to least.
3
1
2
2
4) − 0.38, 4 , − 5
5) 5 , 0.7, − 3
3) -18, -9, -31, 15, 0
6) -67, 82, -73, -10, 20
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Page 9
Date
Graph/Identify Points
Common Core Standard: 6.NS.C.8 Solve real-world and mathematical problem 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.
Learning Target: I can graph and identify point on the coordinate plane in all four quadrants.
Important Terms:
 Coordinate grid (plane): A plane formed by the intersection of a horizontal line called the x-axis and a vertical
line called the y-axis.
 Ordered pair: A pair of numbers that can be used to locate a point on a coordinate plane.
 Origin: The point where the x-axis and y-axis intersect on the coordinate plane. (0,0)
 Quadrant: The x-axis and y-axis divide the coordinate plane into four regions. Each region is called a quadrant.
 x-coordinate: The first number in an ordered pair. It tells the distance to move right or left from the origin.
 y-coordinate: The second number in an ordered pair. It tells the distance to move up or down from the origin.
Important Information:
Examples:
Part A: Identifying Quadrants
Part B: Locating Points on a Coordinate Plane
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Page 10
Part C: Graphing Points on a Coordinate Plane
Try This:
Name the quadrant where each point is located.
1. Q
2. X
3. Y
4. Z
Give the coordinates of each point.
5. P
6. R
7. T
8. H
Graph each point on a coordinate plane.
9. L (0, 3)
10. M (3, -3)
11. V (-4, 3)
12. N (-2, -1)
13. T (-5, 0)
14. S (1, -4)
Name the quadrant or axis each ordered pair is located.
15. (3, -1)
16. (0, 5)
17. (-4, -4)
17. (-2, 3)
18. (-12, 0)
19. (21, 13)
20. (0, 0)
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