```GED Test Mathematics
• New information from
GEDTS
• Most frequently missed
math test items
• Students need both
content and strategies
• Tips for success
• Reflections
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
Statistics from GEDTS
Standard Score Statistics for Mathematics
Median
Mean
Mathematics Score for All U.S.
GED Completers
460
469
Mathematics Score for All U.S.
GED Passers
490
501
Mathematics continues to be the most
difficult content area for GED candidates.
GEDTS Statistical Study
• Studied three
operational test forms
• Analyzed the 40 most
frequently missed
items
• These were 40% of the
total items
• 2003-04 data; released
July 2005
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
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
Puzzler: Exploring Patterns
What curious property do each of the following figures
share?
10
8
3
6
6
4
4
2
Most Missed Questions: Geometry
and Measurement
• Pythagorean Theorem
• Area, perimeter, volume
–
–
–
–
Visualizing type of formula to be used
Comparing area, perimeter, and volume of figures
Partitioning of figures
Using variables in a formula
• Parallel lines and angles
Most
Missed
Questions:
Geometry
.
and Measurement
One end of a 50-ft cable is attached to the top of a 48-ft
tower. The other end of the cable is attached to the
ground perpendicular to the base of the
tower at a distance x feet from
cable
tower
50 ft
the base. What is the measure,
48 ft
in feet, of x?
(1)
(2)
(3)
(4)
(5)
2
4
7
12
14
Which incorrect alternative
would these candidates
most likely have chosen?
(1) 2
Why?
The correct answer is (5): 14

x

Most Missed Questions: Geometry
and Measurement
The height of an A-frame storage
shed is 12 ft. The distance from the
center of the floor to a side of the
shed is 5 ft. What is the measure,
in feet, of x?
(1) 13
(2) 14
(3) 15
(4) 16
(5) 17
side x
height
12 ft

Which incorrect alternative
would these candidates
most likely have chosen?
(5) 17
Why?
The correct answer is (1): 13
5 ft 
Most Missed Questions: Geometry
and Measurement
• Were either of the incorrect alternatives in the last two
questions even possible if triangles were formed?
• Theorem: The measure of any side of a triangle must
be LESS THAN the sum of the measures of the other
two sides. (This same concept forms the basis for other
questions in the domain of Geometry.)
Most Missed Questions: Geometry
and Measurement
Below are rectangles A and B with no text. For each,
or perimeter?
A
B
A: Area Perimeter
Either/both
Perimeter
B: Area Perimeter
Either/both
Area
Most Missed Questions: Geometry
and Measurement
Area by Partitioning
• An L-shaped flower garden is shown by the shaded area in the
diagram. All intersecting segments are perpendicular.
32 ft
6 ft
20 ft
house
6 ft
Most Missed Questions: Geometry
and Measurement
32 ft
32 ft
6 ft
6 ft
20 ft
32 × 6 = 192
+ 14 × 6 = 84
14 ft
house
6 ft
6 ft
26 ft
6 ft
20 ft
26 × 6 = 156
+ 20 × 6 = 120
6 ft
276 ft2
6 ft
276
ft2
6 ft
26 ft
6 ft
14 ft
6 ft
26 × 6 = 156
+ 14 × 6 = 84
+ 6 × 6 = 36
276 ft2
Most Missed Questions: Geometry
and Measurement
x+2
x–2
Which expression represents the area of the rectangle?
(1)
2x
(2)
x2
(3)
x2 – 4
(4)
x2 + 4
(5)
x2 – 4x – 4
Most Missed Questions: Geometry
and Measurement
x+2
x–2
Choose a number for x.
I choose 8. Do you see any
restrictions? Determine
(8 + 2 = 10; 8 – 2 = 6; 10  6 = 60)
Which alternative yields that value?
(1) 2x
(1) (2)
(3)
(4)
(5)
x2
x2 – 4
x2 + 4
x2 – 4x – 4
2  8 = 16; not correct (60).
82 = 64; not correct.
82 – 4 = 64 – 4 = 60; correct!
82 + 4 = 64 + 4 = 68.
82 – 4(8) – 4 = 64 – 32 – 4 = 28
Most Missed Questions: Geometry
and Measurement
1
3
5
7
2
a
4
6
8
b
Parallel Lines
•
If a || b, ANY pair of angles above will satisfy one of these two equations:
x = y
x + y = 180
Which one would you pick?
If the angles look equal (and the lines are parallel), they are!
If they don’t appear to be equal, they’re not!
Most Missed Questions: Geometry
and Measurement
These are
not parallel.
1
4
parallelograms
2
3
5
6
8
7
trapezoids
Where else are candidates likely to use the relationships
among angles related to parallel lines?
Most Missed Questions: Geometry
and Measurement
•
Comparing Areas/Perimeters/Volumes
A rectangular garden had a length of 20 feet and a
width of 10 feet. The length was increased by 50%,
and the width was decreased by 50% to form a new
garden. How does the area of the new garden compare
to the area of the original garden?
The area of the new garden is
(1) 50% less
(2) 25% less
(3) the same
(4) 25% greater
(5) 50% greater
Most Missed Questions: Geometry
and Measurement
20 ft (length)
10 ft
(width)
Area:
20 x 10 = 200 ft2
original garden
30 ft
Area:
30 x 5 = 150 ft2
5 ft
new garden
The new area is 50 ft2 less; 50/200 = 1/4 = 25% less.
Most Missed Questions: Geometry
and Measurement
20 ft (length)
10 ft
(width)
Area:
20 x 10 = 200 ft2
original garden
30 ft
Area:
30 x 5 = 150 ft2
5 ft
new garden
How do the perimeters of the above two figures compare?
What would happen if you decreased the length by 50% and
increased the width by 50%
Tips from GEDTS: Geometry and
Measurement
• Any side of a triangle CANNOT be the sum or difference of the
other two sides (Pythagorean Theorem).
• If a geometric figure is shaded, the question will ask for area; if
only the outline is shown, the question will ask for perimeter
(circumference).
• To find the area of a shape that is not a common geometric figure,
partition the area into non-overlapping areas that are common
geometric figures.
• If lines are parallel, any pair of angles will either be equal or have
a sum of 180°.
• The interior angles within all triangles have a sum of 180°.
• The interior angles within a square or rectangle have a sum of
360°.
Kenn Pendleton, GEDTS Math Specialist
Reflections
• What are the geometric concepts that you feel are
necessary in order to provide a full range of math
instruction in the GED classroom?
• How will you incorporate the areas of geometry
identified by GEDTS as most problematic into the
math curriculum?
• If your students have little background knowledge in
geometry, how could you help them develop and use
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
Investigate 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.
Most Missed Questions: Applying Basic
Math Principles to Calculation
including those with fractional parts
percentages
• Calculating with square roots
• Interpreting exponent as a multiplier
• Selecting the correct equation to answer
a conceptual problem
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 Harold’s typing speed
increase?
(1) 10% (2) 15% (3) 20% (4) 25% (5) 50%
Most Missed Questions: Applying Basic
Math Principles to Calculation
• Harold’s 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%;
Most Missed Questions: Applying Basic
Math Principles to Calculation
A positive number less than or equal to 1/2 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?
(1) A, B, C
(2) B, A, C
(3) B, C, A
(4) C, A, B
(5) C, B, A
Most Missed Questions: Applying Basic
Math Principles to Calculation
A positive number less than or
equal to 1/2 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?
(1) A, B, C
(2) B, A, C
(3) B, C, A
(4) C, A, B
(5) 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)
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)
(2)
(3)
(4)
(5)
50
100
150
200
250
Most Missed Questions: Applying Basic
Math Principles to Calculation
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?
A
produced
passed
B
produced
passed
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?
(1) 300
(2) 432
(3) 492
(4) 504
(5) 3000
Hint: The items produced must be greater than the number passing
inspection.
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?
(1) 0.032
x
(2) 0.064
inside
diameter
1.436 in
(3) 0.718
(4) 0.750
(5) 2.936
outside diameter
1.500 in
+ 1.436 +
= 1.500
x
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.
Most Missed Questions: Applying Basic
Math Principles to Calculation
If a = 2 and b = -3, what is the value of 4a  ab?
(1) -96
(2) -64
(3) -48
(4) 2
(5) 1
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.
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
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
• 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?
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
Time Out for a Math Starter!
Let’s get started problem solving with
graphics by looking at the following graph.
Who is represented by each point?
Interpreting Graphs and Tables
•
•
•
•
•
Comparing graphs
Transitioning between text and graphics
Interpreting values on a graph
Interpreting table data for computation
Selecting table data for computation
Interpreting Graphs and Tables
Increasing House Value
\$200,000
House A
Initial \$100,000
Cost
0
\$0
4
Time (years)
8
House A cost \$100,000 and increased in value as shown in
the graph.
House B cost less than house A and increased in value at a
greater rate. Sketch a graph that might show the changing
value of house B.
Interpreting Graphs and Tables
\$200,000
A (1)
B
\$100,000
\$0
4
8
Time (years)
0
\$200,000
A (3)
B
B
A (4)
\$100,000
0
\$0
4
8
Time (years)
\$200,000
\$0
4
8
Time (years)
\$200,000
A (5)
B
\$100,000
\$0
0
Which
One?
0
\$100,000
0
\$100,000
\$0
A (2)
B
\$200,000
4
8
Time (years)
4
8
Time (years)
Interpreting Graphs and Tables
The changing values of two investments are shown in
the graph below.
Investment A
\$2000
Investment B
Amount of
Investment
\$1000
0
\$0
4
8
Time (years)
12
Interpreting Graphs and Tables
How does the amount initially invested and the rate
of increase for investment A compare with those of
investment B?
Investment A
\$2000
Investment B
Amount of
Investment
\$1000
0
\$0
4
8
Time (years)
12
Interpreting Graphs and Tables
Investment A
\$2000
Investment B
Amount of
Investment
\$1000
0
\$0
4
8
Time (years)
12
Compared to investment B, investment A had a
(1) lesser initial investment and a lesser rate of increase.
(2) lesser initial investment and the same rate of increase.
(3) lesser initial investment and a greater rate of increase.
(4) greater initial investment and a lesser rate of increase.
(5) greater initial investment and a greater rate of increase.
Interpreting Graphs and Tables
\$400
\$200
\$0
0
Profit/Loss in
Thousands of
Dollars
4,000
8,000
12,000
-\$200
Video Games Sold
The profit, in thousands of dollars, that a company expects to
make from the sale of a new video game is shown in the graph.
Interpreting Graphs and Tables
\$400
\$200
\$0
0
Profit/Loss in
Thousands of
Dollars
4,000
8,000
12,000
-\$200
Video Games Sold
What is the expected profit/loss before any video games are sold?
(1) \$0 (2) -\$150 (3) -\$250 (4) -\$150,000 (5) -\$250,000
Interpreting Graphs and Tables
Results of Internet Purchase Survey
Number of Purchases
Number of Respondents
0
14
1
22
2
39
3
25
What was the total number of Internet purchases made by
the survey respondents?
(1) 86
(2) 100
(3) 106
(4) 175
(5) 189
(0  14) + 1  22 + 2  39 + 3  25 = 22 + 78 + 75 = 175
Interpreting Graphs and Tables
Claude is sewing 3 dresses in style B using fabric that is 54
inches wide. The table below contains information for
determining the yards of fabric needed.
Dress Size
Yardage
Information
10
Style A
Fabric
Width
14
16
Yards of Fabric Needed
35 in
45 in
54 in
60 in
Style B
Fabric
Width
12
3.25
3.875
3.875
3.875
3
3
3.25
3.25
2.375
2.5
2.75
2.75
2.25
2.25
2.25
2.5
Yards of Fabric Needed
35 in
45 in
54 in
60 in
3.875
4
4.125
4.625
3.125
3.25
3.25
3.625
2.5
2.875
3
3
2.25
2.375
2.5
2.75
Interpreting Graphs and Tables
What is the minimum number of yards of fabric
recommended for one dress each of size 10, 12, and 14?
Dress Size+
Yardage
Information
10
Style A
Fabric
Width
14
16
Yards of Fabric Needed
35 in
45 in
54 in
60 in
Style B
Fabric
Width
12
3.25
3.875
3.875
3.875
3
3
3.25
3.25
2.375
2.5
2.75
2.75
2.25
2.25
2.25
2.5
Yards of Fabric Needed
35 in
45 in
54 in
60 in
3.875
4
4.125
4.625
3.125
3.25
3.25
3.625
2.5
2.875
3
3
2.25
2.375
2.5
2.75
Interpreting Graphs and Tables
What is the minimum number of yards of fabric
recommended for one dress each of size 10, 12, and 14?
Dress Size
Yardage
Information
10
Style A
Fabric
Width
14
16
Yards of Fabric Needed
35 in
45 in
54 in
60 in
Style B
Fabric
Width
12
3.25
3.875
3.875
3.875
3
3
3.25
3.25
2.375
2.5
2.75
2.75
2.25
2.25
2.25
2.5
Yards of Fabric Needed
35 in
45 in
54 in
60 in
3.875
4
4.125
4.625
3.125
3.25
3.25
3.625
2.5
2.875
3
3
2.25
2.375
2.5
2.75
Interpreting Graphs and Tables
• Have candidates find examples of different
types of graphs.
• Have candidates create questions for their
graphics and/or those of others.
• Develop the capacity to translate from graphics
to text as well as text to graphics.
• Develop the capacity to select pertinent
information from the information presented.
• Reinforce the need to read and interpret scales,
present graphs without scales or without units.
Kenn Pendleton, GEDTS Math Specialist
Reflections
• What are the major concepts that you feel are
necessary in order to provide a full range of
graphic literacy instruction in the GED
classroom?
• How will you incorporate the areas of graphic
literacy identified by GEDTS as most
problematic into the math curriculum?
• If your students have difficulty in interpreting
graphics, how could you help them develop
and use such skills in your classroom?
```