Problem
    7.2
    Problem
    7.1
    Topic 7: Multiplying and Dividing Rational Numbers
    for use after
    Accentuate the Negative
    Investigation 3
    A. Do the following calculations and use the results to help you answer
    the questions that follow.
    Multiplication
    Division
    12 ? 3 ?
    12 ? 3 ?
    18 ??
    18 ??
    12 ??
    12 ??
    18 ??
    18 ??
    1. When you multiply a number by a value
    greater than 1
    , how can you
    describe the size of the product as compared to the number? Give an
    example from the table.
    2. When you multiply a number by a value
    between 0 and 1,
    how can you
    describe the size of the product as compared to the number? Give an
    example from the table.
    3. When you divide a number by a value
    greater than 1,
    how can you
    describe the size of the quotient as compared to the number? Give an
    example from the table.
    4. When you divide a number by a value
    between 0 and 1,
    how can you
    describe the size of the quotient compared to the number? Give an
    example from the table.
    A. Dana used rounding to get 180 as an estimated product for 59 ? 3.
    How did she round the factors?
    B. Natalya used rounding to get 400 as an estimated product for
    41.7 ? 9.6. How did she round the factors?
    C. In order to estimate the quotient 573 ? 28.2, Sam rounded the
    numbers to the nearest whole numbers, 573 and 28.
    1. Do you agree with Sam’s rounding? Explain.
    2. Are there better numbers to use when rounding? If so, what is your
    suggestion?
    3. What estimated quotient do you get with your suggested numbers?
    1
    9
    1
    9
    1
    3
    1
    3
    1
    1
    1
    1
    3
    3
    Problem
    7.3
    A. A benchmark fraction is a fraction that is easy to calculate with.
    1. To estimate the product
    what benchmark fraction could you
    use instead of ?
    2. Estimate the product using the benchmark fraction.
    B. Rafiq wants to estimate 23.4 ? .
    1. What benchmark fraction is close to ?
    2. If Rafiq uses that benchmark fraction, what is the best way to round
    23.4 to make it a compatible number?
    3. Using the benchmark fraction and the compatible number, what
    estimated product will Rafiq get?
    Exercises
    1. Predict whether each product or quotient is greater or less than the
    first factor in the problem.
    a. 129 ? 0.57
    b. ?
    c. 0.8 ? 0.64
    d. 306 ?
    e. 3 ? 4
    f. ?
    g. 1.2 ? 63.9
    h. 72 ? 21.9
    2. Estimate each product or quotient.
    a. 24.9 ? 9.3
    b. 11.4 ? 6.6
    c. 24 ?
    d. 80 ?
    e.
    ? 33
    f. 0.85 ? 7.3
    g. 81 ?
    h. 72.2 ? 25.3
    3. Estimate 21.8 ? 8 twice, using a different set of compatible numbers
    each time. Determine which is the more accurate estimate by
    calculating the actual quotient and comparing that with your estimated
    quotients.
    4. Hector says to estimate the quotient for 43 ? 13 by using 40 ? 10.
    Rachel says to use 45 ? 15 and that her estimate will be more accurate
    than Hector’s.
    a. Estimate using Hector’s method of rounding.
    b. Estimate using Rachel’s method of rounding.
    c. Which estimate will be more accurate? Give your reasoning.
    d. Check your prediction by calculating the exact value of the
    quotient.
    89
    9
    43
    119
    15
    7
    1
    13
    2
    5
    9
    4
    3
    8
    4
    5
    12
    7
    1
    8
    1
    3
    9
    30
    9
    30
    5
    9
    60 3
    5
    9
    Topic 7: Multiplying and Dividing
    Rational Numbers
    Guided Instruction
    Mathematical Goals
    • Relate the size of a number used for multiplying or dividing to the size of the
    product or quotient, respectively.
    • Estimate the results of multiplying and dividing rational numbers.
    Vocabulary
    •
    benchmark fraction
    At a Glance
    Having some sense of the dimensions of a product or quotient enables a
    student to judge the appropriateness of a proposed calculation for a word
    problem. Additionally, comparing a completed calculation with an estimate
    provides a check on the reasonableness of the results of the calculation.
    One way to anticipate the result of a multiplication operation involving
    positive numbers is to determine whether one of the factors is less than or
    greater than 1. For a division operation, the question is whether the divisor
    is less than or greater than 1. This type of analysis will reveal whether the
    product or quotient is greater or less than the other number in the
    calculation.
    Some students will always tend to round to a multiple of ten. However, a
    student who knows the perfect squares to 225 may use 22.5 ? 1.5 ? 15 as
    an estimate for 22.6 ? 1.48, and should be encouraged to do so.
    After Problem 7.2, ask:
    •
    How can choosing which way to round the numbers in a multiplication
    problem affect the estimated product as compared to the actual product?
    (If you round both numbers up, the estimated product will be higher
    than the actual product. If you round both numbers down, the
    estimated product will be less than the actual product.)
    •
    Does it make a difference whether you round the dividend and the
    divisor in the same direction?
    (Yes. Rounding the dividend up and the
    divisor down gives an estimated quotient that is higher than the actual
    quotient. Rounding the dividend down and the divisor up gives an
    estimated quotient that is lower than the actual quotient.)
    After Problem 7.3, ask:
    •
    How is substituting a benchmark fraction for an actual fraction similar
    to rounding the actual fraction?
    (They both involve choosing a
    convenient value that is close to an actual value.)
    You will find additional work on rational number operations in the
    grade 6 unit
    Bits and Pieces II.
    PACING
    1 day
    Assignment Guide for Topic 7
    Core
    1–4
    Answers to Topic 7
    Problem 7.1
    A.
    Multiplication
    Division
    12 ? 3 ? 36
    12 ? 3 ? 4
    18 ?? 24
    18 ??
    12 ?? 4
    12 ?? 36
    18 ?? 2
    18 ?? 162
    1.
    The product is greater than the number;
    12 ? 3 ? 36
    2.
    The product is less than the number;
    12 ?? 4
    3.
    The quotient is less than the number;
    12 ? 3 ? 4
    4.
    The quotient is greater than the number;
    12 ?? 36
    Problem 7.2
    A.
    Dana rounded 59 to 60.
    B.
    Natalya rounded 41.7 to 40, and 9.6 to 10.
    C. 1.
    No, 573 and 28 are not compatible.
    2.
    Use 600 and 30.
    3.
    20
    Problem 7.3
    A. 1.
    2.
    30
    B. 1.
    2.
    Round to 24.
    3.
    8
    Exercises
    1. a.
    greater than
    b.
    less than
    c.
    less than
    d.
    greater than
    e.
    less than
    f.
    greater than
    g.
    greater than
    h.
    less than
    2.
    Answers may vary. Samples:
    a.
    250
    b.
    2
    c.
    2
    d.
    40
    e.
    11
    f.
    7
    g.
    8
    h.
    3
    3.
    Answers may vary. Sample: Estimates 20 ? 10
    ? 2 and 21 ? 7 ? 3. The actual quotient is
    2.73. The second estimate is more accurate
    than the first.
    4. a.
    4
    b.
    3
    c.
    Answers may vary. Sample: Rachel’s,
    because her rounded numbers are just 2
    away from the actual numbers, while
    Hector’s are 3 away.
    d.
    The actual quotient is 3.31.
    1
    3
    1
    2
    1
    3
    1
    3
    1
    9
    1
    9
    1
    3
    1
    3
    27
    1
    1
    2
    1
    1
    3
    3

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