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Improving Candidate Performance

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Improving Candidate Performance

1. Compare sets of difficult questions for candidates scoring near the cut score with those scoring 1 – 2 SEM below the cut score.

2. Using the top 40% of MC questions

most frequently missed, identify themes.

3. Develop sample questions related to

identified themes.

- Are the two groups most frequently missing the same or different questions?

- Are the two groups making the same or different errors?

- Are the two groups more likely to find the challenging questions on the same or different halves (Part I – calculator; Part II – no calculator) of the test?

A comparison of the 16 questions most frequently missed by candidates near the cut score with those missed by candidates 1 – 2 SEM below the cut score: Form #1

Near

1. A

2. B

3. C

4. D

5. E

Below

1. A

2. C

3. B

4. D

5.G

Near

6. F

7. G

8. H

9. I

10. J

11. K

Below

6. I

7. H

8.F

9.*

10.K

11.E

Near

12. L

13. M

14. N

15. O

16. P

Below

12.N

13. M

14. *

15. O

16. L

A comparison of the 16 questions most frequently missed by candidates near the cut score with those missed by candidates 1 – 2 SEM below the cut score: Form #2

Near

1. A

2. B

3. C

4. D

5. E

Below

1. A

2. D

3. B

4. E

5. C

Near

6. F

7. G

8. H

9. I

10. J

11. K

Below

6. F

7. G

8. M

9. K

10. L

11. H

Near

12. L

13. M

14. N

15. O

16. P

Below

12. J

13. N

14. I

15. O

16. P

A comparison of the 16 questions most frequently missed by candidates near the cut score with those missed by candidates 1 – 2 SEM below the cut score: Form #3

Near

1. A

2. B

3. C

4. D

5. E

Below

1. A

2. B

3. C

4. D

5.H

Near

6. F

7. G

8. H

9. I

10. J

11. K

Below

6. I

7. G

8.F

9. K

10.E

11.M

Near

12. L

13. M

14. N

15. O

16. P

Below

12.L

13.J

14. *

15. P

16.*

Comparison Summary

Form 1

14 of 16 questions

are the same

Form 2

all 16 questions

are the same

Form 3

14 of 16 questions

are the same

Generally speaking, the questions on which we will focus are the same for both groups. From this point on, the analysis will be done using questions for candidates near the cut score. When appropriate, comments will be made to reference those below the cut score.

Identifying Common Themes Among the Questions

more difficult (total of 48 questions) less difficult

Theme 1

Theme 2

Theme 3

Other

Theme 1:

Geometry

Theme 2:

Calculation

Theme 3:

Graphs and tables

(Remember those four questions for candidates 1-2 SEM below the cut score that did not match questions for candidates near the cut score? Three related to graphs/tables, and one related to calculation.)

Summarizing Comparison of Most Commonly Selected Incorrect Responses

It should now be clear that both groups find the same questions to be most difficult and both groups are also prone to make the same primary errors.

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

Name the type of Geometry question that is most likely to be challenging for the candidates.

My answer:

The Pythagorean theorem

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 the base.

cable 50 ft

tower48 ft

x

What is the measure, in feet, of x?

(1) 2 (2) 4 (3) 7 (4) 12 (5) 14

The correct answer is (5): 14

Which incorrect alternative would these candidates most likely have chosen?

(1) 2

Why?

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.

side x

height 12 ft

5 ft

What is the measure, in feet, of x?

The correct answer is (1): 13

(1) 13 (2) 14 (3) 15 (4) 16 (5) 17

Which incorrect alternative would these candidates most likely have chosen?

(5) 17

Why?

Below are rectangles A and B with no text. For each, do you think that a question would be asked about area or perimeter?

A

B

A: Area Perimeter Either/both

Perimeter

B: Area Perimeter Either/both

Area

This visual distinction is always used on the GED Mathematics Test.

Area by Partitioning

32 ft

An L-shaped flower garden is shown by the shaded area in the diagram. All intersecting segments are perpendicular.

6 ft

20 ft

house

6 ft

Partition the L-shaped area into shapes whose areas GED candidates could likely find. Label the dimensions appropriate for finding area.

Compare your partitioning with someone near you. Are they exactly the same? Will the total areas be the same?

Three Possible Solutions

32 ft

6 ft

6 ft

20 ft

32 × 6 = 192 + 14 × 6 = 84 276 ft2

14 ft

house

6 ft

6 ft

6 ft

26 ft

26 ft

6 ft

6 ft

6 ft

26 × 6 = 156 + 14 × 6 = 84 + 6 × 6 = 36 276 ft2

20 ft

14 ft

26 × 6 = 156 + 20 × 6 = 120 276 ft2

6 ft

6 ft

Area/Perimeter When Variables are Involved

x + 2

x – 2

Is this an area or a perimeter problem?

Which expression represents the area of the rectangle?

(1) 2x (2) x2 (3) x2 – 4 (4) x2 + 4 (5) x2 – 4x - 4

How can you approach this question if your algebra skills are not strong?

Pretend that x is a number!

x + 2

This is a strategy that will be repeated when we look at calculation items.

x – 2

Choose a number for x. Do you see any restrictions?

(I choose 8.)

Determine the area numerically.

(8 + 2 = 10; 8 – 2 = 6; 10 6 = 60)

Which alternative yields that value?

(1) 2x (2) x2 (3) x2 – 4 (4) x2 + 4 (5) x2 – 4x – 4

2 8 = 16; not correct (60).

82 = 64; not correct.

82 – 4 = 64 – 4 = 60; correct!

82 + 4 = 64 + 4 = 68; not correct

82 – 4(8) – 4 = 64 – 32 – 4 = 28; not correct

Parallel Lines

1

2

a

3

4

5

6

b

7

8

If a || b, ANY pair of angles above will satisfy one of these two equations:

x = y

x + y = 180

Which one should you use?

If the angles look equal (and the lines are parallel), they are!If they don’t appear to be equal, they’re not!

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

20 ft (length)

10 ft (width)

Area: 20 × 10 = 200 ft2

original garden

30 ft

Area: 30 × 5 = 150 ft2

5 ft

new garden

The new area is 50 ft2 less; 50/200 = 1/4 = 25% less.

How do the perimeters compare?

What would be the case if the WIDTH was increased by 50% and the LENGTH was decreased by 50%?

Calculation

Eleven of the 20 calculation questions appeared on Part I where the calculator is available.

The calculator can provide an alternate means of determining the correct response for certain questions. Candidates should have practice with this strategy so that they can use the technique on the test.

For both halves of the test, having a sense of what is reasonable will go a long way towards selecting the appropriate alternative.

Here are questions similar to two items found on the same form of the test. Which was easier for the Near candidates? The Below candidates?

(Part I) A game show contestant can win a maximum of $480. For each question answered correctly, 1/8 of the money will be awarded. What will the contestant win if 2 questions are answered correctly?

(Part II) An ad agency needed 75 people in a survey group and estimated that 9 of the invited would not come. Each of 6 agency employees invited the same number of people to attend. If x is the number of people each called, 6x – 9 = 75. What is the value of x?

Near Below

+ 23.3%+15.6%

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%

This question was intended for Part II. Any percentages found on Part II will involve only simple calculation. Candidates who can estimate/calculate 10% of any number and 25% of a whole number will have an advantage on problems of this type.

Harold’s typing speed, in words per minute, increased from 40 to 50: an increase of 10. If a candidate could find 25% of 40, on to the next problem!

Suppose a candidate can only find 10% of any number.

10% of 40 is 4.An increase of 10% would mean an increase of 4 words per minute.

(1) 10% (2) 15% (3) 20% (4) 25% (5) 50%

X

Increase of 10%: 40 + 4 = 44; not enough (50).

Increase of 20% (10% + 10%); 40 + 4 + 4 = 48; not enough.

X

X

Increase of 30% (10% + 10%+ 10%); 40 + 4 + 4 + 4= 52; too much.

X

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

Chose a number for x that agrees with the information in the first sentence.

1/2

0.1

Evaluate A, B, and C.

A: 1.5 B: 2 C: 1.25

A: 1.1 B: 10 C: 1.01

Arrange (Least Greatest)

1.25, 1.5, 2 (C,A,B)

1.01, 1.1, 10 (C,A,B)

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

This question was designed for Part II. As was true with percent, any calculation with fractions on Part II is relatively easy.

Blue: 1/2 of 300 = 150

Red: 1/3 of 300 = 100

Blue + Red = 250; the remainder (300 – 250 = 50) selected yellow.

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?

B

A

produced

passed

produced

passed

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?

The items produced must be greater than the number passing inspection. Here are the alternatives for this question.

(1) 300 (2) 432 (3) 492 (4) 504 (5) 3000

Which incorrect alternative do you think was selected most often?

300!

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?

inside diameter 1.436 in

x

x

outside diameter 1.500 in

(1) 0.032 (2) 0.064 (3) 0.718 (4) 0.750 (5) 2.936

This question can be answered by subtracting and dividing. It can also be answered by only adding.

= 1.500

+ 1.436 +

This question was designed for Part I.

The most common calculation error seems to be interpreting the exponent as a multiplier rather than a power.

On Part I of the test, remember that the calculator can raise numbers to a power several ways.

Exponents on Part II of the test would be found in two situations: simple calculations or scientific notation. When numbers are written in scientific notation, candidates should recognize that positive exponents represent large numbers and negative exponents represent small decimal numbers; they must be able to convert from one expression to the other.

If a = 2 and b = -3, what is the value of 4a ab?

(1) -96 (2) -64 (3) -48 (4) 2 (5) 1

This question was designed for Part I, so the calculator could be used to find the correct answer.

Knowing that 2-3 represents a small decimal number and not a negative number would have enabled them to eliminate three of the five alternatives.

It hopefully goes without saying that 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, which were discussed earlier, certainly may require the candidate to find a square root. Other questions also contain square roots. This example is from one of the official Practice Tests.

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

This question may be difficult even with the calculator. Is there another way to get an idea of what the correct answer may be?

L = W (1 + 5) 2

The width (W) is known to be 10.

Suppose a candidate is reluctant to use/trust the calculator but recognizes that 5 is slightly more than 4, and also knows that 4 is 2.

Alternatives

L is more than W (1 + 4) 2

(1) 6.1 (2) 6.6 (3) 11.2 (4) 12.2 (5) 16.2

L is more than 10 (1 + 2) 2

L is more than 10 3 2

L is more than 15.

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 areas of Geometry and calculation once again before the candidates take the test.

- 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 an 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°.

- 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 (best on Part I).

- 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.