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Title: Bonnie Vondracek Susan Pittman


1
Bonnie Vondracek Susan Pittman
August 2224, 2006 Washington, DC
2
GED 2002 Series Tests
  • Math Experiences
  • One picture tells a thousand words
  • one experience tells a thousand pictures.

3
Who are GED Candidates?
  • Average Age 24.7 years
  • Gender 55.1 male 44.9 female
  • Ethnicity
  • 52.3 White
  • 18.1 Hispanic Origin
  • 21.5 African American
  • 2.7 American Indian or Alaska Native
  • 1.7 Asian
  • 0.6 Pacific Islander/Hawaiian
  • Average Grade Completed 10.0

4
Statistics from GEDTS
  • Standard Score Statistics for
    Mathematics

Mathematics continues to be the most difficult
content area for GED candidates.
5
Statistics from GEDTS
  • GED Standard Score and Estimated National Class
    Rank of
    Graduating U.S. High School Seniors, 2001

Source 2001 GED Testing Service Data
6
Statistical Study
  • There is a story often told about the writer
    Gertrude Stein. As she lay on her deathbed, a
    brave friend leaned over and whispered to her,
    Gertrude, what is the answer? With all her
    strength, Stein lifted her head from the pillow
    and replied, What is the question?Then she
    died.

7
The Question Is . . .
  • GEDTS Statistical Study for Mathematics
  • Results were obtained from three operational test
    forms.
  • Used the top 40 of the most frequently missed
    test items.
  • These items represented 40 of the total items on
    the test forms.
  • Study focused on those candidates who passed (410
    standard score) /- 1 SEM called the NEAR group
    (N107,163), and those candidates whose standard
    scores were /- 2 SEMs below passing called the
    BELOW group (N10,003).
  • GEDTS Conference, July 2005

8
Most Missed Questions
  • How are the questions distributed between the two
    halves of the test?
  • Total number of questions examined 48
  • Total from Part I (calculator) 24
  • Total from Part II (no calculator) 24

9
Math Themes Applying Basic Math Principles to
Calculation
  • Because mathematics is so often conveyed in
    symbols, oral and written communication about
    mathematical ideas is not always recognized as an
    important part of mathematics education. Students
    do not necessarily talk about mathematics
    naturally teachers need to help them to do so.
  • (NCTM 1996)

10
Math Themes Most Missed Questions
  • Theme 1 Geometry and Measurement
  • Theme 2 Applying Basic Math Principles to
    Calculation
  • Theme 3 Reading and Interpreting Graphs and
    Tables

11
An Unusual Phenomenon
  • Select a four-digit number (except one that has
    all digits the same).
  • Rearrange the digits of the number so they form
    the largest number possible.
  • Now rearrange the digits of the number so that
    they form the smallest number possible.
  • Subtract the smaller of the two numbers from the
    larger.
  • Take the difference and continue the process over
    and over until something unusual happens.

12
Most Missed Questions Applying Basic Math
Principles to Calculation
Summarizing Comparison of Most Commonly Selected
Incorrect Responses
  • Its clear that both groups find the same
    questions to be most difficult and both groups
    are also prone to make the same primary errors.

13
Most Missed Questions Applying Basic Math
Principles to Calculation
  • Visualizing reasonable answers, including those
    with fractional parts
  • Determining reasonable answers with percentages
  • Calculating with square roots
  • Interpreting exponent as a multiplier
  • Selecting the correct equation to answer a
    conceptual problem

14
Most Missed Questions Applying Basic Math
Principles to Calculation
  • When Harold began his word-processing job, he
    could type only 40 words per minute. After he had
    been on the job for one month, his typing speed
    had increased to 50 words per minute.


  • By what percent did Harolds typing speed
    increase?
  • (1) 10 (2) 15 (3) 20 (4) 25
    (5) 50

15
Most Missed Questions Applying Basic Math
Principles to Calculation
  • Harolds typing speed, in words per minute,
    increased from 40 to 50.
  • Increase of 10 4 words per minute 40 4
    44 not enough (50).
  • Increase of 20 (10 10) 40 4 4 48
    not enough.
  • Increase of 30 (10 10 10) 40 4 4 4
    52 too much.
  • Answer is more than 20, but less than 50
    answer is (4) 25.

16
Most Missed Questions Applying Basic Math
Principles to Calculation
  • A positive number less than or equal to ½ is
    represented
  • by x. Three expressions involving x are given
  • (A) x 1 (B) 1/x (C) 1 x2
  • Which of the following series lists the
    expressions from
  • least to greatest?
  • A, B, C
  • B, A, C
  • B, C, A
  • C, A, B
  • C, B, A

17
Most Missed Questions Applying Basic Math
Principles to Calculation
  • A positive number less than or equal to ½ is
    represented by x. Three expressions involving x
    are given
  • (A) x 1 (B) 1/x (C) 1 x2
  • Which of the following series lists the
    expressions from least to greatest?
  • A, B, C
  • B, A, C
  • B, C, A
  • C, A, B
  • C, B, A

Select a fraction and decimal and try each.
½ 0.1 Evaluate A, B, and C using ½
and then 0.1. A 1 ½ A 1.1 B 2 B 10 C 1 ¼
C 1.01 Arrange (Least Greatest) 1 ¼, 1 ½,
2 (C, A, B) 1.01, 1.1, 10 (C, A, B)
18
Most Missed Questions Applying Basic Math
Principles to Calculation
  • A survey asked 300 people which of the three
    primary colors, red, yellow, or blue was their
    favorite. Blue was selected by 1/2 of the people,
    red by 1/3 of the people, and the remainder
    selected yellow. How many of the 300 people
    selected YELLOW?
  • (1) 50
  • (2) 100
  • (3) 150
  • (4) 200
  • (5) 250

19
Most Missed Questions Applying Basic Math
Principles to Calculation
Visualizing a Reasonable Answer When Calculating
With Fractions
  • Of all the items produced at a manufacturing
    plant on Tuesday, 5/6 passed inspection. If 360
    items passed inspection on Tuesday, how many were
    PRODUCED that day?
  • Which of the following diagrams correctly
    represents the relationship between items
    produced and those that passed inspection?

20
Most Missed Questions Applying Basic Math
Principles to Calculation
  • Of all the items produced at a manufacturing
    plant on Tuesday, 5/6 passed inspection. If 360
    items passed inspection on Tuesday, how many were
    PRODUCED that day?
  • 300
  • 432
  • 492
  • 504
  • (5) 3000
  • Hint The items produced must be greater than
    the number passing inspection.

21
Most Missed Questions Applying Basic Math
Principles to Calculation
Checking Your Visualization Skills
22
Most Missed Questions Applying Basic Math
Principles to Calculation
  • A cross-section of a uniformly thick piece of
  • tubing is shown at the right. The width of
  • the tubing is represented by x. What is the
  • measure, in inches, of x?
  • 0.032
  • 0.064
  • 0.718
  • 0.750
  • 2.936

23
Most Missed Questions Applying Basic Math
Principles to Calculation
  • Exponents
  • The most common calculation error appears to be
    interpreting the exponent as a multiplier rather
    than a power.
  • On Part I, students should be able to use the
    calculator to raise numbers to a power several
    ways.
  • On Part II, exponents are found in two
    situations simple calculations and scientific
    notation.

24
Most Missed Questions Applying Basic Math
Principles to Calculation
  • If a 2 and b -3, what is the value of 4a ?
    ab?
  • -96
  • -64
  • -48
  • 2
  • (5) 1

25
Most Missed Questions Applying Basic Math
Principles to Calculation
  • Calculation with Square Roots
  • Any question for which the candidate must find a
    decimal approximation of the square root of a
    non-perfect square will only be found on Part I.
  • Questions involving the Pythagorean Theorem may
    require the candidate to find a square root.
    Other questions also contain square roots.

26
Most Missed Questions Applying Basic Math
Principles to Calculation
  • The Golden Rectangle discovered by the ancient
    Greeks is thought to have an especially pleasing
    shape. The length (L) of this rectangle in terms
    of its width (W) is given by the following
    formula.
  • L W ? (1 ?5)
  • 2
  • If the width of a Golden Rectangle is 10 meters,
    what is its approximate length in meters?
  • (1) 6.1 (2) 6.6 (3) 11.2 (4) 12.2
    (5) 16.2

27
Most Missed Questions Applying Basic Math
Principles to Calculation
  • L W ? (1 ?5)
  • 2
  • The width (W) is known to be 10.
  • L is more than W ? (1
    ?4)
  • 2
  • L is more than 10 ? (1
    2)
  • 2
  • L is more than 10 ? 3
  • 2
  • L is more than 15.
  • Only one alternative fits the conditions set.
  • (1) 6.1 (2) 6.6 (3) 11.2 (4) 12.2 (5)
    16.2

28
Final Tips
  • Candidates do not all learn in the same manner.
    Presenting alternate ways of approaching the
    solution to questions during instruction will tap
    more of the abilities that the candidates possess
    and provide increased opportunities for the
    candidates to be successful.
  • After the full range of instruction has been
    covered, consider revisiting the following areas
    once again before the candidates take the test.

29
Tips from GEDTS Applying Basic Math Principles
to Calculation
  • Replace a variable with a REASONABLE number, then
    test the alternatives.
  • Be able to find 10 of ANY number.
  • Try to think of reasonable (or unreasonable)
    answers for questions, particularly those
    involving fractions.
  • Try alternate means of calculation, particularly
    testing the alternatives.
  • Remember that exponents are powers, and that a
    negative exponent in scientific notation
    indicates a small decimal number.
  • Be able to access the square root on the
    calculator alternately, have a sense of the size
    of the answer.
  • Kenn Pendleton, GEDTS Math Specialist

30
Reflections
  • What are the mathematical concepts that you feel
    are necessary in order to provide a full range of
    math instruction in the GED classroom?
  • What naturally occurring classroom activities
    could serve as a context for teaching these
    skills?
  • How do students representations help them
    communicate their mathematical understandings?
  • How can teachers use these various
    representations and the resulting conversations
    to assess students understanding and plan
    worthwhile instructional tasks?
  • How will you incorporate the area of applying
    basic math principles to calculation, as
    identified by GEDTS as a problem area, into the
    math curriculum?
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