Bonnie Vondracek Susan Pittman

2. GED 2002 Series Tests Math = Experiences One picture tells a thousand words; one experience tells a thousand pictures. We�ve all heard such phrases as �the face that launched a thousand ships� or �picture is worth a thousand words.� However, we should not forget that math is also a very experiential subject. We do not learn mathematical concepts merely through rote memorization or reading a textbook. We need to help our students to access real-life experiences or provide those experiences if we want them to be true problem solvers rather than having them capable of merely parroting facts that we have provided. Throughout this workshop, you will experience and discover connections within those areas that GED candidates exhibit the most difficulty and will hopefully share your own personal experiences and expertise with others. But before we begin our exploration of those specific areas that provide students with the most difficulty, let�s take a few minutes to look at who our GED students are.We�ve all heard such phrases as �the face that launched a thousand ships� or �picture is worth a thousand words.� However, we should not forget that math is also a very experiential subject. We do not learn mathematical concepts merely through rote memorization or reading a textbook. We need to help our students to access real-life experiences or provide those experiences if we want them to be true problem solvers rather than having them capable of merely parroting facts that we have provided. Throughout this workshop, you will experience and discover connections within those areas that GED candidates exhibit the most difficulty and will hopefully share your own personal experiences and expertise with others. But before we begin our exploration of those specific areas that provide students with the most difficulty, let�s take a few minutes to look at who our GED students are.

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 Who are our GED students? Have they changed over the years? According to the annual statistical report, there have been some changes. It is important in the teaching of mathematics to know who our students are. [Note: Information is obtained from Who Passed the GED Tests? 2004 an annual statistical report produced by the General Educational Development Testing Service of the American Council on Education.]Who are our GED students? Have they changed over the years? According to the annual statistical report, there have been some changes. It is important in the teaching of mathematics to know who our students are. [Note: Information is obtained from Who Passed the GED Tests? 2004 an annual statistical report produced by the General Educational Development Testing Service of the American Council on Education.]

4. Statistics from GEDTS Standard Score Statistics for Mathematics Each year, GEDTS analyzes the statistical data for the three operational versions of the GED Test. In 2004, the most recent year for statistics, the average score in mathematics for all GED completers in the United States was a 469. The score for those GED candidates who passed the GED was a 501. The minimal score for passing each GED subtest is a 410 with an overall average requirement for all five subtests at a 450. Although the 469 and 501 appear to be adequate scores, mathematics continues to be the lowest average score among the five subtests. As in the past, mathematics continues to be the most difficult content area for GED candidates. [Note: The first set of mathematics scores are based on each candidate�s best score earned in 2004 and is based on all U.S. completers. The GED standard score for all GED passers in 2004 in the area of mathematics was an average of 501.] However, a mean or median score does not provide the type of information that is most helpful to an instructor who wishes to assist students in attaining better math skills and ultimately a passing score on the GED Mathematics Test. This requires a more intensive study of question types and those which are missed most often by students who do not pass the test.Each year, GEDTS analyzes the statistical data for the three operational versions of the GED Test. In 2004, the most recent year for statistics, the average score in mathematics for all GED completers in the United States was a 469. The score for those GED candidates who passed the GED was a 501. The minimal score for passing each GED subtest is a 410 with an overall average requirement for all five subtests at a 450. Although the 469 and 501 appear to be adequate scores, mathematics continues to be the lowest average score among the five subtests. As in the past, mathematics continues to be the most difficult content area for GED candidates. [Note: The first set of mathematics scores are based on each candidate�s best score earned in 2004 and is based on all U.S. completers. The GED standard score for all GED passers in 2004 in the area of mathematics was an average of 501.] However, a mean or median score does not provide the type of information that is most helpful to an instructor who wishes to assist students in attaining better math skills and ultimately a passing score on the GED Mathematics Test. This requires a more intensive study of question types and those which are missed most often by students who do not pass the test.

5. Statistics from GEDTS GED Standard Score and Estimated National Class Rank of Graduating U.S. High School Seniors, 2001 Remember the 469. Take a look at the chart that correlates a GED Standard Score to an Estimated National Class Ranking. Would you feel comfortable that students who graduated in the bottom half of the class would possess higher order mathematics skills? Probably not. Students who function within the range of 1-2 SEMs below the passing score or who pass with a minimal score need to develop improved mathematical thinking skills in order to be successful in both postsecondary education and in the workforce. Remember the 469. Take a look at the chart that correlates a GED Standard Score to an Estimated National Class Ranking. Would you feel comfortable that students who graduated in the bottom half of the class would possess higher order mathematics skills? Probably not. Students who function within the range of 1-2 SEMs below the passing score or who pass with a minimal score need to develop improved mathematical thinking skills in order to be successful in both postsecondary education and in the workforce.

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. [Note: You may wish to use this short story as a lead into an overview on the statistical study.][Note: You may wish to use this short story as a lead into an overview on the statistical study.]

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 (N=107,163), and those candidates whose standard scores were +/- 2 SEMs below passing called the BELOW group (N=10,003). GEDTS Conference, July 2005 Review with the participants how the information for the statistical study was obtained. GED candidates who were NEAR passing and those with scores BELOW passing became the target groups for the study. By focusing on these groups, the most troubling/difficult items in the area of the GED Mathematics Test were identified. One SEM equates to approximately 50 points. Review with the participants how the information for the statistical study was obtained. GED candidates who were NEAR passing and those with scores BELOW passing became the target groups for the study. By focusing on these groups, the most troubling/difficult items in the area of the GED Mathematics Test were identified. One SEM equates to approximately 50 points.

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 Discuss that when the GED 2002 Series Mathematics Test was first developed, many people were concerned that the calculator would create a less challenging test. The analysis supports that this is an untrue statement. Of the items most often missed, an equal number were located on each part of the test. The use of the calculator made no difference in a student providing a correct answer for the most frequently missed questions.Discuss that when the GED 2002 Series Mathematics Test was first developed, many people were concerned that the calculator would create a less challenging test. The analysis supports that this is an untrue statement. Of the items most often missed, an equal number were located on each part of the test. The use of the calculator made no difference in a student providing a correct answer for the most frequently missed questions.

9. Math Themes: Geometry and Measurement �The notion of building understanding in geometry across the grades, from informal to formal thinking, is consistent with the thinking of theorists and researchers.� (NCTM 2000, p. 41) Do you sleep on a rectangle, drink out of a cylinder, eat ice cream from a cone, or have meals at a square table? Then you have experienced geometry. Geometry touches on every aspect of our lives. It is important to explore the shapes, lines, angles, and space that are woven into our students� daily lives as well as our own. In fact �geo� means earth and �metry� to measure. So it�s not unusual to think of geometry as real-life types of measurement. Geometry is the development of spatial sense and the actual measuring and the concepts related to units of measure. As with all areas of mathematics, instructors should actively involve students in activities in order to build their understanding of geometric ideas, to see the power and usefulness of geometry in their lives, and to feel confident in their own capabilities as problem solvers. When students can be engaged in using and applying geometric knowledge to investigate and/or think about situations that relate to geometry, true problem solving occurs. [Note: If using this PowerPoint as part of a workshop, you may wish to begin the presentation here.]Do you sleep on a rectangle, drink out of a cylinder, eat ice cream from a cone, or have meals at a square table? Then you have experienced geometry. Geometry touches on every aspect of our lives. It is important to explore the shapes, lines, angles, and space that are woven into our students� daily lives as well as our own. In fact �geo� means earth and �metry� to measure. So it�s not unusual to think of geometry as real-life types of measurement. Geometry is the development of spatial sense and the actual measuring and the concepts related to units of measure. As with all areas of mathematics, instructors should actively involve students in activities in order to build their understanding of geometric ideas, to see the power and usefulness of geometry in their lives, and to feel confident in their own capabilities as problem solvers. When students can be engaged in using and applying geometric knowledge to investigate and/or think about situations that relate to geometry, true problem solving occurs. [Note: If using this PowerPoint as part of a workshop, you may wish to begin the presentation here.]

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 Although geometry is indeed everywhere around us, it is also one of the math themes of the most missed questions. Discuss that three primary themes were identified by the study as being the areas in which the GED candidates had the most difficulty. This section of the workshop will deal specifically with the area of geometry. [Note: For each section of the workshop, you may wish to begin with a math starter/math bender. If a workshop is being conducted only in the area of Geometry, you will also want to include a basic icebreaker in order to allow instructors time to introduce themselves, as well as setting the stage for the workshop.] Although geometry is indeed everywhere around us, it is also one of the math themes of the most missed questions. Discuss that three primary themes were identified by the study as being the areas in which the GED candidates had the most difficulty. This section of the workshop will deal specifically with the area of geometry. [Note: For each section of the workshop, you may wish to begin with a math starter/math bender. If a workshop is being conducted only in the area of Geometry, you will also want to include a basic icebreaker in order to allow instructors time to introduce themselves, as well as setting the stage for the workshop.]

11. Puzzler: Exploring Patterns What curious property do each of the following figures share? What curious property do each of the following figures share? Debrief the activity by having instructors discuss what pattern(s) they discovered in the figures. Follow-up the activity by asking if this property is true of all rectangles, squares, circles, and triangles? Select one shape, the rectangle, and have instructors explore what other numerical (integer) values create this same curious property. Debrief the activity by having instructors share what they have discovered about different geometric figures. Discuss that finding patterns is an important skill for students to develop in mathematics, including the area of geometry. This type of activity provides instructors with the opportunity to use their problem-solving skills in the area of geometry. [Note: This is a sample activity. You may wish to include a different problem-solving activity to open the workshop.] What curious property do each of the following figures share? Debrief the activity by having instructors discuss what pattern(s) they discovered in the figures. Follow-up the activity by asking if this property is true of all rectangles, squares, circles, and triangles? Select one shape, the rectangle, and have instructors explore what other numerical (integer) values create this same curious property. Debrief the activity by having instructors share what they have discovered about different geometric figures. Discuss that finding patterns is an important skill for students to develop in mathematics, including the area of geometry. This type of activity provides instructors with the opportunity to use their problem-solving skills in the area of geometry. [Note: This is a sample activity. You may wish to include a different problem-solving activity to open the workshop.]

12. Most Missed Questions: Geometry and Measurement Do the two groups most commonly select the same or different incorrect responses? It�s clear that both groups find the same questions to be most difficult and both groups are also prone to make the same primary errors. As you know, the most frequently missed items occurred equally on both parts of the GED Mathematics Test. But how did each group, Near and Below, perform on the test items. Did these two groups miss the same types of items or with the difference in SEM, did they miss different types of items? It�s clear that both groups found the same types of questions to be most difficult. Also, as we look at the different types of questions that were missed, you will notice that similar error patterns also occurred. GED candidates not only missed similar questions, but they also selected the same incorrect answer, known as a distracter.As you know, the most frequently missed items occurred equally on both parts of the GED Mathematics Test. But how did each group, Near and Below, perform on the test items. Did these two groups miss the same types of items or with the difference in SEM, did they miss different types of items? It�s clear that both groups found the same types of questions to be most difficult. Also, as we look at the different types of questions that were missed, you will notice that similar error patterns also occurred. GED candidates not only missed similar questions, but they also selected the same incorrect answer, known as a distracter.

13. Most Missed Questions: Geometry and Measurement Name the type of Geometry question that is most likely to be challenging for the candidates What type of geometry question do you think was most challenging for the GED candidates? Did you say the Pythagorean Theorem? If you did, you were correct. Questions regarding the Pythagorean Theorem were found on each of the three operational forms of the GED Mathematics Test and all GED candidates found them to be difficult.What type of geometry question do you think was most challenging for the GED candidates? Did you say the Pythagorean Theorem? If you did, you were correct. Questions regarding the Pythagorean Theorem were found on each of the three operational forms of the GED Mathematics Test and all GED candidates found them to be difficult.

14. 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 Use of variables in a formula Parallel lines and angles The Pythagorean Theorem was not the only area of difficulty for students on the GED Mathematics Test. Students also found area, perimeter, and volume questions to be difficult, as well as questions that dealt with parallel lines and angles. [Note: The following slides will provide instructors with sample questions similar to those that were missed by GED candidates. These questions have been provided by GEDTS. They mirror exactly the question types missed, as well as the distracters that were selected most often by the candidates.]The Pythagorean Theorem was not the only area of difficulty for students on the GED Mathematics Test. Students also found area, perimeter, and volume questions to be difficult, as well as questions that dealt with parallel lines and angles. [Note: The following slides will provide instructors with sample questions similar to those that were missed by GED candidates. These questions have been provided by GEDTS. They mirror exactly the question types missed, as well as the distracters that were selected most often by the candidates.]

15. Getting Started with Geometry and Measurement! In the following diagram of the front view of the Great Pyramid, the measure of the angle PRQ is 120 degrees, the measure of the angle PQR is 24 degrees, and the measure of the angle PST is 110 degrees. What is the measure of the angle RPS in degrees? Read the question on the slide and have instructors identify the types of knowledge that students must possess in order to answer the question.Read the question on the slide and have instructors identify the types of knowledge that students must possess in order to answer the question.

16. Getting Started with Geometry and Measurement! Hint: How many degrees are there in a triangle or a straight line? There are 180 degrees in a triangle. That is, the sum of the angles in a triangle is 180 degrees. A straight line is 180 degrees. The concept that both a triangle and a straight line are 180 degrees can be difficult for some students to comprehend. Have instructors share sample activities that they have used to assist students in better understanding this concept. [Note: One activity to assist students in �seeing� that the sum of the angles of a triangle and a straight line both equal 180 degrees is to have them cut out a triangle and then tear the triangle apart into its three angles. Have students arrange the angles to form a straight line showing that the interior angles correlate to a straight line.] There are 180 degrees in a triangle. That is, the sum of the angles in a triangle is 180 degrees. A straight line is 180 degrees. The concept that both a triangle and a straight line are 180 degrees can be difficult for some students to comprehend. Have instructors share sample activities that they have used to assist students in better understanding this concept. [Note: One activity to assist students in �seeing� that the sum of the angles of a triangle and a straight line both equal 180 degrees is to have them cut out a triangle and then tear the triangle apart into its three angles. Have students arrange the angles to form a straight line showing that the interior angles correlate to a straight line.]

17. Answer 180 degrees � 120 degrees = 60 degrees 180 degrees � 110 degrees = 70 degrees 60 degrees + 70 degrees = 130 degrees 180 degrees � 130 degrees = 50 degrees In words, the problem would be as follows: Angle PRQ = 120 degrees so Angle PRS has 60 degrees. Angle PST has 110 degrees so Angle PSR has 70 degrees. We know that the triangle PRS has 60 + 70 degrees in two of its angles to equal 130 degrees, therefore the third angle RPS is 180 � 130 degrees or 50 degrees. Review with instructors the different steps that students must take in order to solve this most missed question.Review with instructors the different steps that students must take in order to solve this most missed question.

18. 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? Ask instructors what type of skills this question assesses. Instructors will share that the question assesses a student�s knowledge of the Pythagorean Theorem. Walk instructors through which of the incorrect alternatives that GED candidates were most likely to have selected. [Note: Although instructors and texts teach Pythagorean Theorem, it appears that students have difficulty in applying the formula to different types of situations. Students who missed this question selected the distracter #1. From the analysis, it was noted that students generally use addition or subtraction as their first method of solving a problem. Because the answer for subtracting 48 from 50 was one of the options, students automatically selected this as the correct answer.]Ask instructors what type of skills this question assesses. Instructors will share that the question assesses a student�s knowledge of the Pythagorean Theorem. Walk instructors through which of the incorrect alternatives that GED candidates were most likely to have selected. [Note: Although instructors and texts teach Pythagorean Theorem, it appears that students have difficulty in applying the formula to different types of situations. Students who missed this question selected the distracter #1. From the analysis, it was noted that students generally use addition or subtraction as their first method of solving a problem. Because the answer for subtracting 48 from 50 was one of the options, students automatically selected this as the correct answer.]

19. 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 Most Missed Questions: Geometry and Measurement Although this is also a question regarding the Pythagorean Theorem, the height is indicated by a dotted line. Again, students seem to select addition or subtraction as their computation of choice. In this problem, students selected the distracter that resulted in adding the two numbers indicated on the graphic.Although this is also a question regarding the Pythagorean Theorem, the height is indicated by a dotted line. Again, students seem to select addition or subtraction as their computation of choice. In this problem, students selected the distracter that resulted in adding the two numbers indicated on the graphic.

20. 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.) Comprehending whether or not a triangle is possible is an important skill for students to internalize. Although many students may know the rule that the measure of any side of a triangle must be less than the sum of the measures of the other two sides, they need experiences with creating possible triangles and analyzing why other triangles are impossible. Comprehending whether or not a triangle is possible is an important skill for students to internalize. Although many students may know the rule that the measure of any side of a triangle must be less than the sum of the measures of the other two sides, they need experiences with creating possible triangles and analyzing why other triangles are impossible.

21. 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? Visualizing what math terminology means is important in order for students to identify the correct formula to use. Discuss with instructors the need for students to have real-life experiences with area and perimeter in order to understand what the formulas really mean. One cue for students when taking the test is to identify which figures indicate area versus perimeter. On the GED Mathematics Test, area is always represented by a shaded figure; whereas, perimeter figures are not. [Note: This visual distinction is always used on the GED Mathematics Test. However, students must understand that in real life this distinction is not generally available.]Visualizing what math terminology means is important in order for students to identify the correct formula to use. Discuss with instructors the need for students to have real-life experiences with area and perimeter in order to understand what the formulas really mean. One cue for students when taking the test is to identify which figures indicate area versus perimeter. On the GED Mathematics Test, area is always represented by a shaded figure; whereas, perimeter figures are not. [Note: This visual distinction is always used on the GED Mathematics Test. However, students must understand that in real life this distinction is not generally available.]

22. 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. Have instructors partition (�cut�) the L-shaped area into shapes whose areas GED candidates could likely find. Have them label the dimensions appropriate for finding area and compare their partitioning with someone near them. Many students look at this type of question and give up. They don�t believe that they have enough information because they don�t know the dimensions of the house. Instructors should have students actually �cut� the figure in order to understand the concept of partitioning. Also, using a �hands-on� approach is excellent for students whose learning strength is not visual, but rather kinesthetic. Remind instructors that the shaded area would indicate that, on the GED Mathematics Test, students would be calculating the area of the shaded portion. Have instructors partition (�cut�) the L-shaped area into shapes whose areas GED candidates could likely find. Have them label the dimensions appropriate for finding area and compare their partitioning with someone near them. Many students look at this type of question and give up. They don�t believe that they have enough information because they don�t know the dimensions of the house. Instructors should have students actually �cut� the figure in order to understand the concept of partitioning. Also, using a �hands-on� approach is excellent for students whose learning strength is not visual, but rather kinesthetic. Remind instructors that the shaded area would indicate that, on the GED Mathematics Test, students would be calculating the area of the shaded portion.

23. Most Missed Questions: Geometry and Measurement [Note: Have instructors share with the group what types of partitioning they used in order to solve the problem. See whether or not different methods were used from the above possibilities. Ask whether or not there are other possible ways to solve the problem. What would those methods be?] Have instructors brainstorm different types of lessons that they could use to reinforce the concept of partitioning.[Note: Have instructors share with the group what types of partitioning they used in order to solve the problem. See whether or not different methods were used from the above possibilities. Ask whether or not there are other possible ways to solve the problem. What would those methods be?] Have instructors brainstorm different types of lessons that they could use to reinforce the concept of partitioning.

24. 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 Is this an area or perimeter problem? How would you teach students to solve this problem if their algebra skills are not strong? Have you ever used substitution? For some candidates, the presence of variables in a question can cause significant concern. A test-taker with algebra skills will be able to answer some questions more quickly than someone who does not have or cannot recall these concepts. However, there are other ways to determine the correct solution for a multiple-choice question. Substitution is one method. [Note: When any number can be chosen, avoid selecting 0 or 1. Each of these numbers can lead to a solution that appears to be correct but may not be. Also, remind instructors that because the figure is shaded, on the GED Mathematics Test, students would be asked to find the area.] Is this an area or perimeter problem? How would you teach students to solve this problem if their algebra skills are not strong? Have you ever used substitution? For some candidates, the presence of variables in a question can cause significant concern. A test-taker with algebra skills will be able to answer some questions more quickly than someone who does not have or cannot recall these concepts. However, there are other ways to determine the correct solution for a multiple-choice question. Substitution is one method. [Note: When any number can be chosen, avoid selecting 0 or 1. Each of these numbers can lead to a solution that appears to be correct but may not be. Also, remind instructors that because the figure is shaded, on the GED Mathematics Test, students would be asked to find the area.]

25. Most Missed Questions: Geometry and Measurement Discuss that substitution is a strategy that can be used in calculation problems as well. Provide instructors with different examples of how students can use substitution to solve a problem. Identify different conditions that should exist when identifying a number to substitute for x, such as it should be larger than 2, easy to calculate, such as a single digit number, and one that is not a fraction or decimal. The example uses the number 8 to substitute for x. The process of substituting values for variables is not the most time-efficient way to find the correct answer. However, it is an approach that should be considered if the GED candidate cannot recall necessary algebra skills. Candidates should consider working on these problems last so that they will have enough time to also work on other questions. [Note: Have instructors urge students to consider checking their work by selecting another value for the variable and evaluating that the alternative is the same.] Discuss that substitution is a strategy that can be used in calculation problems as well. Provide instructors with different examples of how students can use substitution to solve a problem. Identify different conditions that should exist when identifying a number to substitute for x, such as it should be larger than 2, easy to calculate, such as a single digit number, and one that is not a fraction or decimal. The example uses the number 8 to substitute for x. The process of substituting values for variables is not the most time-efficient way to find the correct answer. However, it is an approach that should be considered if the GED candidate cannot recall necessary algebra skills. Candidates should consider working on these problems last so that they will have enough time to also work on other questions. [Note: Have instructors urge students to consider checking their work by selecting another value for the variable and evaluating that the alternative is the same.]

26. 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! Have instructors identify which expression they would select and why. [Note: Reinforce with instructors that if the angles look equal and the lines look parallel, they are. If they don�t appear to be equal and the lines don�t look to be parallel, they are not. The GED Mathematics Test makes a clear distinction with equal versus non-equal angles and lines.]Have instructors identify which expression they would select and why. [Note: Reinforce with instructors that if the angles look equal and the lines look parallel, they are. If they don�t appear to be equal and the lines don�t look to be parallel, they are not. The GED Mathematics Test makes a clear distinction with equal versus non-equal angles and lines.]

27. Most Missed Questions: Geometry and Measurement Where else are students likely to use the relationships among angles related to parallel lines? Have instructors brainstorm different types of scenarios where their students would use relationships regarding angles related to parallel lines. Where else are students likely to use the relationships among angles related to parallel lines? Have instructors brainstorm different types of scenarios where their students would use relationships regarding angles related to parallel lines.

28. 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? Another most missed question deals with comparing areas, perimeters, and volumes. Which distracter do you think students selected most often? What strategy would you teach so that students would more likely select the correct answer?Another most missed question deals with comparing areas, perimeters, and volumes. Which distracter do you think students selected most often? What strategy would you teach so that students would more likely select the correct answer?

29. Most Missed Questions: Geometry and Measurement Many students are not visual learners. By drawing a picture of what the question is asking, students are more likely to set up the equation correctly. Many students are not visual learners. By drawing a picture of what the question is asking, students are more likely to set up the equation correctly.

30. Most Missed Questions: Geometry and Measurement Assess what would occur if the length of the figure was decreased by 50% and the width increased by 50%. Compare this answer to the original. Have instructors brainstorm how they could use this activity in class to help students develop a deeper understanding of this concept. Assess what would occur if the length of the figure was decreased by 50% and the width increased by 50%. Compare this answer to the original. Have instructors brainstorm how they could use this activity in class to help students develop a deeper understanding of this concept.

31. 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 Geometry is the development of spatial sense and the actual measuring and the concepts related to units of measure. As with all areas of mathematics, instructors should actively involve students in activities in order to build their understanding of geometric ideas, to see the power and usefulness of geometry in their lives, and to feel confident in their own capabilities as problem solvers. When students can be engaged in using and applying geometric knowledge to investigate and/or think about situations that relate to geometry, true problem solving occurs. Geometry is the development of spatial sense and the actual measuring and the concepts related to units of measure. As with all areas of mathematics, instructors should actively involve students in activities in order to build their understanding of geometric ideas, to see the power and usefulness of geometry in their lives, and to feel confident in their own capabilities as problem solvers. When students can be engaged in using and applying geometric knowledge to investigate and/or think about situations that relate to geometry, true problem solving occurs.

32. 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 area of geometry once again before the candidates take the test. Review the ideas on the slide. [Note: GEDTS recommends that after the full range of instruction has been covered, that these specific areas of learning be reviewed prior to the test.] Review the ideas on the slide. [Note: GEDTS recommends that after the full range of instruction has been covered, that these specific areas of learning be reviewed prior to the test.]

33. 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? So, how can you help students better understand geometric concepts? become better problem solvers? Take a few minutes to reflect on the following questions. Share your ideas with your group. 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? So, how can you help students better understand geometric concepts? become better problem solvers? Take a few minutes to reflect on the following questions. Share your ideas with your group. 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?