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.

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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.

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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

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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

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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)

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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

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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.

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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.

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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

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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

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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.

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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

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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)
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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

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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?

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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.

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Most Missed Questions Applying Basic Math
Principles to Calculation
Checking Your Visualization Skills
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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

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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.

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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

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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.

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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

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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

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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.

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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

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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?