# 6.3 and *6.3 and *6.6 ab - Integers (No Calc)

6.3  The student will

a)  identify and represent integers;

b)  order and compare integers; and

c)  identify and describe absolute value of integers.

6.3  The student will

a)  identify and represent integers;

b)  compare and order integers; and

c)  identify and describe absolute value of integers.

6.6  The student will

a)  add, subtract, multiply, and divide integers;*  (with NO CALCULATOR)

b)  solve practical problems involving operations with integers; and

Bloom's Level:  Remember, Understand, Analyze

### BIG IDEAS

• I can determine yard line after each play of a football game, calculate amount of force required to stop a drag racer, figure golf score in relation to par, and track my finances over time.
• I will understand the role of negative values in practical situations.

• I can determine yard line after each play of a football game, calculate amount of force required to stop a drag racer, figure golf score in relation to par, and track my finances over time.
• I will understand numbers and their opposites are equal distance from zero on a number line and the role of negative values in practical situations.

### UNDERSTANDING THE STANDARD

• Integers are the set of whole numbers, their opposites, and zero.
• Positive integers are greater than zero.
• Negative integers are less than zero.
• Zero is an integer that is neither positive nor negative.
• A negative integer is always less than a positive integer.
• When comparing two negative integers, the negative integer that is closer to zero is greater.
• An integer and its opposite are the same distance from zero on a number line. For example, the opposite of 3 is -3.
• The absolute value of a number is the distance of a number from zero on the number line regardless of direction. Absolute value is represented |-6| = 6.
• On a conventional number line, a smaller number is always located to the left of a larger number (e.g.,  –7 lies to the left of –3, thus –7 < –3; 5 lies to the left of 8 thus 5 is less than 8).

UNDERSTAND INTEGERS

2016 VDOE Curriculum Framework - 6.3 Understanding

·  The set of integers includes the set of whole numbers and their opposites {…-2, -1, 0, 1, 2, …}. Zero has no opposite and is an integer that is neither positive nor negative.

·  Integers are used in practical situations, such as temperature (above/below zero), deposits/withdrawals in a checking account, golf (above/below par), time lines, football yardage, positive and negative electrical charges, and altitude (above/below sea level).

·  Integers should be explored by modeling on a number line and using manipulatives, such as two-color counters, drawings, or algebra tiles.

·  The opposite of a positive number is negative and the opposite of a negative number is positive.

·  Positive integers are greater than zero.

·  Negative integers are less than zero.

·  A negative integer is always less than a positive integer.

·  When comparing two negative integers, the negative integer that is closer to zero is greater.

·  An integer and its opposite are the same distance from zero on a number line.

-  Example: the opposite of 3 is −3 and the opposite of −10 is 10.

·  On a conventional number line, a smaller number is always located to the left of a larger number (e.g.,–7 lies to the left of –3, thus –7 < –3; 5 lies to the left of 8 thus 5 is less than 8)

·  The absolute value of a number is the distance of a number from zero on the number line regardless of direction. Absolute value is represented using the symbol (e.g.,  and).

·  The absolute value of zero is zero.

INTEGER OPERATIONS

2016 VDOE Curriculum Framework - 6.6 Understanding

·  The set of integers is the set of whole numbers and their opposites (e.g., …-3, -2, -1, 0, 1, 2, 3…). Zero has no opposite and is neither positive nor negative.

·  Integers are used in practical situations, such as temperature changes (above/below zero), balance in a checking account (deposits/withdrawals), golf, time lines, football yardage, and changes in altitude (above/below sea level).

·  Concrete experiences in formulating rules for adding, subtracting, multiplying, and dividing integers should be explored by examining patterns using calculators, using a number line, and using manipulatives, such as two-color counters, drawings, or by using algebra tiles.

·  Sums, differences, products and quotients of integers are either positive, negative, undefined or zero. This may be demonstrated through the use of patterns and models.

·  The order of operations is a convention that defines the computation order to follow in simplifying an expression. Having an established convention ensures that there is only one correct result when simplifying an expression.

·  The order of operations is as follows:

–  First, complete all operations within grouping symbols.1 If there are grouping symbols within other grouping symbols, do the innermost operation first.

–  Second, evaluate all exponential expressions.

–  Third, multiply and/or divide in order from left to right.

–  Fourth, add and/or subtract in order from left to right.

1Parentheses , absolute value  (e.g.,   ), and the division bar (e.g.,  ) should be treated as grouping symbols.

·  Expressions are simplified using the order of operations and applying the properties of real numbers. Students should use the following properties, where appropriate, to further develop flexibility and fluency in problem solving (limitations may exist for the values of a, b, or c in this standard):

-  Commutative property of multiplication:

-  Associative property of multiplication:

-  Subtraction and division are neither commutative nor associative.

-  Distributive property (over addition/subtraction):  and

-  Identity property of multiplication (multiplicative identity property):

-  The additive identity is zero (0) because any number added to zero is the number. The multiplicative identity is one (1) because any number multiplied by one is the number. There are no identity elements for subtraction and division.

-  Multiplicative property of zero:

-  Substitution property: If  then b can be substituted for a in any expression, equation or inequality.

·  The power of a number represents repeated multiplication of the number (e.g., 83 = 8 · 8 · 8). The base is the number that is multiplied, and the exponent represents the number of times the base is used as a factor. In the example, 8 is the base, and 3 is the exponent.

Any number, except zero, raised to the zero power is 1. Zero to the zero power (is undefined.

### ESSENTIALS

• What role do negative integers play in practical situations?
Some examples of the use of negative integers are found in temperature (below 0), finance (owing money), below sea level. There are many other examples.
• How does the absolute value of an integer compare to the absolute value of its opposite?
They are the same because an integer and its opposite are the same distance from zero on a number line.

The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to

6.3a1  Identify an integer represented by a point on a number line.

·  6.3a2  Identify an integer represented by a point on a number line.

6.3a2  Represent integers on a number line.

·  6.3a1  Model integers, including models derived from practical situations.

6.3b1  Order and compare integers using a number line

·  6.3b1  Compare and order integers using a number line.

6.3b2  Compare integers, using mathematical symbols (<, >, =).

·  6.3b2  Compare integers, using mathematical symbols ().

6.3c1  Identify and describe the absolute value of an integer.

·  6.3c1  Identify and describe the absolute value of an integer.

INTEGER OPERATIONS

·  6.6a1  Model addition, subtraction, multiplication and division of integers using pictorial representations or concrete manipulatives.

·  6.6a2  Add, subtract, multiply, and divide two integers.

·  6.6b1  Solve practical problems involving addition, subtraction, multiplication, and division with integers.

### KEY VOCABULARY

integer, absolute value, positive integers, negative integers, whole numbers, number line, set, opposite

Updated: Oct 27, 2017