Topic 12: Number Properties
    for use after
    Bits and Pieces III
    Investigation 4
    As you work with operations on rational numbers, you may notice certain
    characteristics that hold true. For example, you know that 7 ? 3
    10 and
    3 ? 7
    10. The Commutative Property of Addition says that you can add
    two numbers in any order and the result will be the same.
    The table below shows some important number properties that are true for
    all rational numbers.
    Number properties can help you perform mental calculations quickly. For
    example, to simplify 13 ? 19 ? 7 mentally, you might use the Commutative
    Property of Addition to add 13 and 7 before adding 19.
    Use the properties of numbers to find the value of the variable in each
    equation.
    A. 3 ? 9 ? 12 =
    a
    ? 3 ? 12
    Use the Commutative Property of Addition to reorder the first two
    numbers, so 3 ? 9 ? 12
    9 ? 3 ? 12. Therefore,
    a
    9.
    B. 25 3 (4 3 12) ? 3
    (4 3
    k
    ) 3 12 ? 3
    Use the Associative Property of Multiplication to regroup the
    multiplication, so 25 3 (4 3 12) ? 3
    (25 3 4) 3 12 ? 3. Use the
    Commutative Property of Multiplication to reorder the multiplication,
    so (25 3 4) 3 12 ? 3 5 (4 3 25) 3 12 ? 3.So,
    k
    5 25.
    5
    5
    55
    Property
    Description
    Commutative
    Property of
    Addition
    Changing the order of
    the numbers you are
    adding does not
    change the sum.
    Examples
    3 ? 7 ? 10 and 7 ? 3 ? 10,
    so 3 ? 7 ? 7 ? 3
    Commutative
    Property of
    Multiplication
    Changing the order of
    the numbers you are
    multiplying does not
    change the product.
    4 ? 5 ? 20 and 5 ? 4 ? 20,
    so 4 ? 5 ? 5 ? 4
    Associative
    Property of
    Addition
    Changing the grouping
    of the numbers you are
    adding does not
    change the sum.
    2 ? (6 ? 1) ? 9 and
    (2 ? 6) ? 1 ? 9, so
    2 ? (6 ? 1) ? (2 ? 6) ? 1
    Associative
    Property of
    Multiplication
    Changing the grouping
    of the numbers you are
    multiplying does not
    change the product.
    3 ? (2 ? 2) ? 12 and
    (3 ? 2) ? 2 ? 12, so
    3 ? (2 ? 2) ? (3 ? 2) ? 2
    5
    5
    Problem
    12.1
    Problem
    12.2
    Consider the two rectangles at the right. You can
    find the total area of the rectangles by writing
    (3 3 4) ?
    (3 3 6) = 12 ? 18,or 30.
    You can also find the total area by combining the
    smaller rectangles into one large rectangle as shown.
    The rectangle area is 3(4 ? 6) = 3(10), or 30.
    This model illustrates another property of
    numbers. The Distributive Property combines
    multiplication with addition and subtraction. To
    multiply a sum or difference, you multiply each number within the
    parentheses by the number outside the parentheses. For example, you can
    write 3(4 ? 6) ? 3(4) ? 3(6).
    Use the Distributive Property to find the value of x in the equation
    16(97)
    ?
    16(100)
    ?
    16(x).
    You can write 97 as a difference of two integers, so 16(97) ? 16(100
    3).
    Using the Distributive Property, 16(100 3) ? 16(100) ? 16( 3).
    Since 16(100) ? 16( 3) ? 16(100)
    16(3), the complete equation is
    16(97) ? 16(100)
    16(3).
    Therefore,
    x
    ? 3.
    Exercises
    Name the property illustrated in each equation.
    1. 15 ? 27 ? 25 = 27 ? 15 ? 25
    2. 14 3 20 ? 14 3 5 = 14 3 15
    3. (5 3 7) ? (3 3 2) = (7 3 5) ? (3 3 2)
    4. 9 ? (11 ? 6) = (9 ? 11) ? 6
    Use mental math and number properties to simplify each expression.
    5. 25 3 102
    6. (51 ? 13) ? (9 ? 7)
    7. 4 3 (17 3 25)
    8. (20 3 19) 3 5
    9. Given that 212 3 4 ? 7 = 855, simplify 4 3 212 ? 7.
    10. Draw a model to show that 6(7) = 6(5) ? 6(2).
    11. Find values of
    x
    ,
    y
    , and
    z
    that satisfy the equation
    14(9 ? 11) 3 25 ? 18 3 13 = 13 3
    x
    ? 14 3
    y
    (11 ?
    z
    )
    2
    22
    22
    2
    3
    4
    3
    6
    3
    46
    Topic 10: Number Properties
    Teaching Guide
    Mathematical Goals
    • Use a factor tree to find a prime factorization
    • Write a prime factorization using exponents
    Vocabulary
    •
    Associative Property
    of Addition
    •
    Associative Property
    of Multiplication
    •
    Commutative
    Property of Addition
    •
    Commutative
    Property of
    Multiplication
    •
    Distributive Property
    At a Glance
    As students work with rational numbers, they may notice that addition,
    subtraction, multiplication, and division follow certain rules. These rules are
    described by the properties of rational numbers. Make sure that students
    understand that these properties hold true for all rational numbers. Some
    students may be confused by the use of the word “property” rather than
    “rule.” Students should recognize that these properties are characteristics of
    operations with rational numbers. Students can then use the properties to
    simplify mental calculations.
    At this point, students are not expected to prove that these properties
    are true, but they should be able to use examples to demonstrate the
    properties. For example, you may want students to evaluate several
    expressions involving the Associative Property of Addition to convince
    them that the property is true.
    Before Example 10.1, make sure that students are comfortable
    recognizing the properties in simple equations like 3 1 5 5 5 1 3 and
    (7 3 6) 3 4 5 7 3 (6 3 4). Students may be confused when variables or
    extra terms, like 1 12, are added to the equations.
    During Example 10.1, ask:
    •
    Is the order of the terms different on the two sides of the equation?
    •
    Is the grouping of the terms different on the two sides of the equation?
    For Exercises 5–8, students will use the properties of numbers to simplify
    mental calculations. Explain to students that rearranging the order of
    calculations or breaking a large number into two smaller numbers can help
    them do calculations more quickly. By using the properties of numbers,
    students can be sure that they are not changing the value of the expression
    when they simplify it.
    Homework Check
    When reviewing Exercise 5, ask:
    •
    What two numbers add up to 102 and are easy to multiply 25 by?
    When reviewing Exercise 6, ask:
    •
    What pairs of digits add up to 10?
    •
    Which pairs of numbers in the expression add up to a multiple of 10?
    •
    Which property can you use to change the grouping of the numbers in
    the expression?
    •
    Which property can you use to change the order of the numbers in the
    expression?
    PACING
    1 day
    Assignment Guide for Topic 10
    Core
    1–11
    Answers to Topic 10
    Exercises
    1.
    Commutative Property of Addition
    2.
    Distributive Property
    3.
    Commutative Property of Multiplication
    4.
    Associative Property of Addition
    5.
    2,550
    6.
    80
    7.
    1,700
    8.
    1,900
    9.
    855
    10.
    Answers may vary. Sample:
    11.
    x
    5 18,
    y
    5 25,
    z
    5 9
    52
    6

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