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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 I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
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1. GED Test Mathematics • New information from GEDTS • Most frequently missed math test items • Students need both content and strategies • Tips for success • Reflections

2. 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

3. Statistics from GEDTS Standard Score Statistics for Mathematics Mathematics continues to be the most difficult content area for GED candidates.

4. 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

5. 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

6. 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

7. 4 2 3 4 6 10 8 6 Puzzler: Exploring Patterns What curious property do each of the following figures share?

8. 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

9. cable 50 ft tower48 ft  x  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 the base. What is the measure, in feet, of x? (1) 2 (2) 4 (3) 7 (4) 12 (5) 14 Which incorrect alternative would these candidates most likely have chosen? (1) 2 Why? The correct answer is (5): 14

10. side x height 12 ft  5 ft  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 Which incorrect alternative would these candidates most likely have chosen? (5) 17 Why? The correct answer is (1): 13

11. 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.)

12. A B Most Missed Questions: Geometry and Measurement 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: Area Perimeter Either/both Perimeter B: Area Perimeter Either/both Area

13. 32 ft 6 ft 20 ft house 6 ft 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.

14. 32 ft 32 ft 6 ft 6 ft 32 × 6 = 192 + 14 × 6 = 84 20 ft 14 ft house 276 ft2 6 ft 6 ft 6 ft 26 ft 26 ft 6 ft 6 ft 6 ft 20 ft 26 × 6 = 156 + 14 × 6 = 84 + 6 × 6 = 36 14 ft 26 × 6 = 156 + 20 × 6 = 120 6 ft 6 ft 276 ft2 276 ft2 Most Missed Questions: Geometry and Measurement

15. x + 2 x – 2 Most Missed Questions: Geometry and Measurement Which expression represents the area of the rectangle? (1) 2x (2) x2 (3) x2 – 4 (4) x2 + 4 (5) x2 – 4x – 4

16. Most Missed Questions: Geometry and Measurement x + 2 Choose a number for x. I choose 8. Do you see any restrictions? Determine the answer numerically. x – 2 (8 + 2 = 10; 8 – 2 = 6; 10  6 = 60) Which alternative yields that value? 2  8 = 16; not correct (60). • 2x • (2) x2 (3) x2 – 4 (4) x2 + 4 (5) x2 – 4x – 4 82 = 64; not correct. 82 – 4 = 64 – 4 = 60; correct! 82 + 4 = 64 + 4 = 68. 82 – 4(8) – 4 = 64 – 32 – 4 = 28

17. 1 2 a 3 4 5 6 b 7 8 Most Missed Questions: Geometry and Measurement 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!

18. 1 2 4 3 These are not parallel. parallelograms 5 6 8 7 trapezoids Most Missed Questions: Geometry and Measurement Where else are candidates likely to use the relationships among angles related to parallel lines?

19. 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 • 50% less • 25% less • the same • 25% greater • 50% greater

20. 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 Most Missed Questions: Geometry and Measurement The new area is 50 ft2 less; 50/200 = 1/4 = 25% less.

21. 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 Most Missed Questions: Geometry and Measurement 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%

22. 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

23. 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 such skills in your classroom?

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

25. Investigate an Unusual Phenomenon 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. • 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.

26. 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

27. 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%

28. 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%; answer is (4) 25%.

29. 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? • A, B, C • B, A, C • B, C, A • C, A, B • C, B, A

30. 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? • 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)

31. 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

32. A B produced passed produced passed 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?

33. 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.

34. inside diameter 1.436 in x x outside diameter 1.500 in + 1.436 + = 1.500 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

35. 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.

36. 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

37. 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.

38. 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

39. 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?

40. 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

41. 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?

42. Most Missed Questions: Reading and 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

43. Increasing House Value \$200,000 House A Initial Cost \$100,000 \$0 0 4 8 Time (years) Most Missed Questions: Reading and Interpreting Graphs and Tables 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.

44. \$200,000 A A \$200,000 B B \$100,000 \$100,000 \$0 \$0 0 4 8 0 4 8 Time (years) Time (years) \$200,000 \$200,000 \$200,000 A B \$100,000 \$100,000 \$100,000 \$0 \$0 \$0 4 8 0 Time (years) Most Missed Questions: Reading and Interpreting Graphs and Tables (1) (2) B A (3) (4) 4 8 0 Time (years) A (5) B Which One? 4 8 0 Time (years)

45. Investment A \$2000 Investment B Amount of Investment \$1000 \$0 0 4 8 12 Time (years) Most Missed Questions: Reading and Interpreting Graphs and Tables The changing values of two investments are shown in the graph below.

46. Investment A \$2000 Investment B Amount of Investment \$1000 \$0 0 4 8 12 Time (years) Most Missed Questions: Reading and Interpreting Graphs and Tables How does the amount initially invested and the rate of increase for investment A compare with those of investment B?

47. Investment A \$2000 Investment B Amount of Investment \$1000 \$0 0 4 8 12 Time (years) Most Missed Questions: Reading and Interpreting Graphs and Tables • Compared to investment B, investment A had a • lesser initial investment and a lesser rate of increase. • lesser initial investment and the same rate of increase. • lesser initial investment and a greater rate of increase. • greater initial investment and a lesser rate of increase. • greater initial investment and a greater rate of increase.

48. \$400 \$200 Profit/Loss in Thousands of Dollars \$0 0 4,000 8,000 12,000 -\$200 Video Games Sold Most Missed Questions: Reading and Interpreting Graphs and Tables 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.

49. \$400 \$200 Profit/Loss in Thousands of Dollars \$0 0 4,000 8,000 12,000 -\$200 Video Games Sold Most Missed Questions: Reading and Interpreting Graphs and Tables What is the expected profit/loss before any video games are sold? (1) \$0 (2) -\$150 (3) -\$250 (4) -\$150,000 (5) -\$250,000

50. Most Missed Questions: Reading and Interpreting Graphs and Tables Results of Internet Purchase Survey 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