1. ©2014 IPDAE. All rights reserved. All content in this presentation is the proprietary property of The Institute for the Professional Development of Adult Educators A Look at the Most-Missed Items on the GED ® Mathematical Reasoning Test Bonnie Goonen – bv73008@aol.com

2. ©2014 IPDAE. All rights reserved. All content in this presentation is the proprietary property of The Institute for the Professional Development of Adult Educators Algebraic Sugar Cane 10 factories produce sugar cane. The second produced twice as much as the first. The third and fourth each produced 80 more than the first. The fifth produced twice as much as the second. The sixth produced 40 more than the fifth. The seventh and eighth each produced 40 less than the fifth. The ninth produced 80 more than the second. The tenth produced nothing due to drought in Australia. If the sum of the production equaled 11,700, how much sugar cane did the first factory produce?

3. ©2014 IPDAE. All rights reserved. All content in this presentation is the proprietary property of The Institute for the Professional Development of Adult Educators The Answer 500. First write down what each factory produced in relation to the first factory (whose production is x): 1. x 2. 2x 3. x + 80 4. x + 80 5. 4x 6. 4x + 40 7. 4x - 40 8. 4x - 40 9. 2x + 80 10. 0x Add them all up and set equal to 11,700 to get the equation: 23x + 200 = 11,700 Solution is: x = 500

4. “The essence of mathematics is not to make simple things complicated, but to make complicated things simple.” ©2014 IPDAE. All rights reserved. All content in this presentation is the proprietary property of The Institute for the Professional Development of Adult Educators

5. 2014 GED ® Test GED ® Mathematical Reasoning Test One test with calculator allowed (except for the first five items) 115 minutes in length Content 45% - Quantitative Problem Solving Number operations Geometric thinking 55% - Algebraic Problem Solving Texas Instruments - TI 30XS Multiview™ Integration of mathematical practices

6. ©2014 IPDAE. All rights reserved. All content in this presentation is the proprietary property of The Institute for the Professional Development of Adult Educators 2014 GED ® Test StatusMathRLAScienceSocial Studies Passed56%77%73%68% National Passing56%77%73%69% Passed with Honors 3%6%3%6% GEDTS Data Retrieved 11/2014

7. ©2014 IPDAE. All rights reserved. All content in this presentation is the proprietary property of The Institute for the Professional Development of Adult Educators 2014 GED ® Test GEDTS Data Retrieved 11/2014

8. ©2014 IPDAE. All rights reserved. All content in this presentation is the proprietary property of The Institute for the Professional Development of Adult Educators 2014 GED ® Test Area of Concern Math continues to be most difficult Different test takers have gaps Test takers are not accessing support available (materials/ tutorials/practice tests)

9. MOST MISSED ITEMS ANALYSIS Quantitative Reasoning

10. ©2014 IPDAE. All rights reserved. All content in this presentation is the proprietary property of The Institute for the Professional Development of Adult Educators Most-Missed Items Quantitative Reasoning Compute area/circumference of circles o Find radius or diameter when given area or circumference

11. ©2014 IPDAE. All rights reserved. All content in this presentation is the proprietary property of The Institute for the Professional Development of Adult Educators Most-Missed Items Quantitative Reasoning Compute perimeter/area of polygons o Find side length when given perimeter or area

12. ©2014 IPDAE. All rights reserved. All content in this presentation is the proprietary property of The Institute for the Professional Development of Adult Educators Most-Missed Items Quantitative Reasoning Compute perimeter/area of 2-dimensional composite shapes

13. ©2014 IPDAE. All rights reserved. All content in this presentation is the proprietary property of The Institute for the Professional Development of Adult Educators Most-Missed Items Quantitative Reasoning Use scale factors to determine magnitude of a size change

14. ©2014 IPDAE. All rights reserved. All content in this presentation is the proprietary property of The Institute for the Professional Development of Adult Educators Most-Missed Items Quantitative Reasoning Solve two-step real world problems involving percent, such as simple interest, tax, markups/markdowns, gratuities, commissions, percent increase/decrease

15. QUANTITATIVE PROBLEM SOLVING SKILLS

16. ©2014 IPDAE. All rights reserved. All content in this presentation is the proprietary property of The Institute for the Professional Development of Adult Educators Geometric Reasoning Seeking relationships Checking effects of transformations Generalizing geometric ideas o Conjecturing about the “always” & “every” o Testing the conjecture o Drawing a conclusion about the conjecture o Making a convincing argument Balancing exploration with deduction o Exploring structured by one or more explicit limitation/restriction o Taking stock of what is being learned through the exploration

17. Triangle Quadrilateral Rotation Reflection Congruent Right Angle Polynomial Parallel

18. ©2014 IPDAE. All rights reserved. All content in this presentation is the proprietary property of The Institute for the Professional Development of Adult Educators Tangrams Originated in China Its invention is unrecorded in history The word “tangram” may have come from the Tanka people who were traders who played this puzzle and past it on via sailors. The word might come from an obscure English word “trangram” which means puzzle or trinket. Became popular during the 19th century in Europe and America.

19. ©2014 IPDAE. All rights reserved. All content in this presentation is the proprietary property of The Institute for the Professional Development of Adult Educators Tangrams Tangram puzzles are made of seven shapes: 5 triangles 1 square 1 parallelogram

20. ©2014 IPDAE. All rights reserved. All content in this presentation is the proprietary property of The Institute for the Professional Development of Adult Educators What can I do with Tangrams? Teach math in an interesting manner Recreate given shapes Create new shapes Integrate mathematical skills using manipulatives (e.g., What fraction of the original shape are each of the 7 pieces?)

21. ©2014 IPDAE. All rights reserved. All content in this presentation is the proprietary property of The Institute for the Professional Development of Adult Educators What can I do with Tangrams? Reinforce geometric principles and ratios Given a specified side length, find the area of each of the smaller shapes making up the tangram. Determine different ways all 7 shapes be used to make a square. Given a specified side length, determine individual side lengths (and/or perimeters) of each of the 7 pieces. Find the smallest perimeter using all 7 shapes. The maximum perimeter.

22. ©2014 IPDAE. All rights reserved. All content in this presentation is the proprietary property of The Institute for the Professional Development of Adult Educators What can I do with Tangrams? Can you make a square using all of your tangram pieces?

23. ©2014 IPDAE. All rights reserved. All content in this presentation is the proprietary property of The Institute for the Professional Development of Adult Educators What can I do with Tangrams? Can you make a square using all of your tangram pieces?

24. ©2014 IPDAE. All rights reserved. All content in this presentation is the proprietary property of The Institute for the Professional Development of Adult Educators Create the Angel fish

25. ©2014 IPDAE. All rights reserved. All content in this presentation is the proprietary property of The Institute for the Professional Development of Adult Educators Create the Angel fish

26. ©2014 IPDAE. All rights reserved. All content in this presentation is the proprietary property of The Institute for the Professional Development of Adult Educators Create the sitting cat

27. ©2014 IPDAE. All rights reserved. All content in this presentation is the proprietary property of The Institute for the Professional Development of Adult Educators Create the sitting cat

28. ©2014 IPDAE. All rights reserved. All content in this presentation is the proprietary property of The Institute for the Professional Development of Adult Educators It’s All About Geometric Reasoning Sample problem If the entire tangram is worth $120, how much is each piece worth? What skills will a student use with this problem?

29. ©2014 IPDAE. All rights reserved. All content in this presentation is the proprietary property of The Institute for the Professional Development of Adult Educators It’s All About Geometric Reasoning What fractional part of the original shape are each of the 7 pieces? Can you determine the decimal equivalency?

30. ©2014 IPDAE. All rights reserved. All content in this presentation is the proprietary property of The Institute for the Professional Development of Adult Educators It’s All About Geometric Reasoning Activity 1 How many different lengths are involved? Number the lengths in order from smallest to largest. Start by labeling the smallest side 1, the next largest side 2, etc.

31. ©2014 IPDAE. All rights reserved. All content in this presentation is the proprietary property of The Institute for the Professional Development of Adult Educators It’s All About Geometric Reasoning Answer to Activity 1 How many different lengths are involved? There are four different measurements. Number the lengths in order from smallest to largest. Start by labeling the smallest side 1, the next largest side 2, etc. 1 1 1 1 1 1 1 1 1 1 2 3 2 2 2 2 3 3 3 4 4

32. ©2014 IPDAE. All rights reserved. All content in this presentation is the proprietary property of The Institute for the Professional Development of Adult Educators It’s All About Geometric Reasoning Activity 2 If the area of the parallelogram is equal to 4 square units find the area of each piece. 4

33. ©2014 IPDAE. All rights reserved. All content in this presentation is the proprietary property of The Institute for the Professional Development of Adult Educators It’s All About Geometric Reasoning Answer to Activity 2 If the area of the parallelogram is equal to 4 square units find the area of each piece. 4 4 4 2 2 8 8

34. ©2014 IPDAE. All rights reserved. All content in this presentation is the proprietary property of The Institute for the Professional Development of Adult Educators It’s All About Geometric Reasoning Activity 3 If the area of the small square is equal to 4, find the length of each side of each piece. 4

35. ©2014 IPDAE. All rights reserved. All content in this presentation is the proprietary property of The Institute for the Professional Development of Adult Educators It’s All About Geometric Reasoning Answer to Activity 3 If the area of the small square is equal to 4, find the length of each side of each piece.

36. ©2014 IPDAE. All rights reserved. All content in this presentation is the proprietary property of The Institute for the Professional Development of Adult Educators It’s All About Geometric Reasoning Activity 4 Place the three isosceles right triangles on top of each other. Compare the corresponding angles. What is the relationship between the three triangles? Suppose the length of the leg on the smallest triangle is equal to 2 units. How long is each of the nine sides of the three triangles?

37. MOST MISSED ITEMS ANALYSIS Algebraic Reasoning

38. ©2014 IPDAE. All rights reserved. All content in this presentation is the proprietary property of The Institute for the Professional Development of Adult Educators Most-Missed Items Algebraic Reasoning Locate points in coordinate plane Determine slope of a line from graph, equation, or table

39. ©2014 IPDAE. All rights reserved. All content in this presentation is the proprietary property of The Institute for the Professional Development of Adult Educators Most-Missed Items Algebraic Reasoning Graph two-variable linear equations

40. ©2014 IPDAE. All rights reserved. All content in this presentation is the proprietary property of The Institute for the Professional Development of Adult Educators Most-Missed Items Algebraic Reasoning Solve One-variable linear equations, and formulas with multiple variables Solve linear inequalities with one variable Solve one-variable quadratic equations with real solutions

41. ©2014 IPDAE. All rights reserved. All content in this presentation is the proprietary property of The Institute for the Professional Development of Adult Educators Most-Missed Items Algebraic Reasoning Create linear expressions as part of word-to- symbol translations On an algebra test, the highest grade was 42 points higher than the lowest grade. The sum of the two grades was 138. Find the lowest grade.

42. ©2014 IPDAE. All rights reserved. All content in this presentation is the proprietary property of The Institute for the Professional Development of Adult Educators Most-Missed Items Algebraic Reasoning Create one- or two-variable linear equations to represent situations given The cost of admission to a popular music concert was $162 for 12 children and 3 adults. The admission was $122 for 8 children and 3 adults in another music concert. How much was the admission for each child and adult?

43. ©2014 IPDAE. All rights reserved. All content in this presentation is the proprietary property of The Institute for the Professional Development of Adult Educators Most-Missed Items Algebraic Reasoning Create one-variable linear inequalities to represent situations you have been given The perimeter of a square must be less than 160 feet. What is the maximum length of a side in feet? x = length of a side 4x < 160 x < 40 Each side must be less than 40 feet.

44. ©2014 IPDAE. All rights reserved. All content in this presentation is the proprietary property of The Institute for the Professional Development of Adult Educators Algebra Students need to be exposed to the underlying structure of algebra while working with arithmetic. Students need to recognize that the total number of items in two sets can be written as 6 + 9, rather than only 15. Algebra should begin at the ABE level

45. ©2014 IPDAE. All rights reserved. All content in this presentation is the proprietary property of The Institute for the Professional Development of Adult Educators Where do we begin? Create opportunities for algebraic thinking as a part of regular instruction Integrate elements of algebraic thinking into arithmetic instruction o Acquiring symbolic language o Recognizing patterns and making generalizations Reorganize formal algebra instruction to emphasize its applications Adapted from National Institute for Literacy, Algebraic Thinking in Adult Education, Washington, DC 20006

46. ©2014 IPDAE. All rights reserved. All content in this presentation is the proprietary property of The Institute for the Professional Development of Adult Educators Some Big Ideas of Algebra Variable Symbolic Notation Equality Ratio and Proportion Pattern Generalization Equations and Inequalities Multiple Representations of Functions

47. Two Fundamental Problems

48. ©2014 IPDAE. All rights reserved. All content in this presentation is the proprietary property of The Institute for the Professional Development of Adult Educators The Problem with Equivalence Students believe In reality The = sign is the announcement of the result or answer of an arithmetic operation. 3 + 4 = (ta da) 7 The = sign is a symbol of equivalence (a symbol that denotes a relationship between two quantities).

49. ©2014 IPDAE. All rights reserved. All content in this presentation is the proprietary property of The Institute for the Professional Development of Adult Educators The Problem with Equivalence Teach students to think of a balance scale when they are performing basic arithmetic operations Avoid the words makes, equals, and totals Use the phrase “is equivalent to” or “is the same as”

50. ©2014 IPDAE. All rights reserved. All content in this presentation is the proprietary property of The Institute for the Professional Development of Adult Educators The Problem with Equivalence Flip number sentences so the numbers and operations are on the right rather than the left side Insert a letter instead of a blank or box

51. ©2014 IPDAE. All rights reserved. All content in this presentation is the proprietary property of The Institute for the Professional Development of Adult Educators The Problem with Variables Students have difficulty understanding variables. Variables play a central role in understanding algebra are hard to define vary dependent on the context in which they are used

52. ©2014 IPDAE. All rights reserved. All content in this presentation is the proprietary property of The Institute for the Professional Development of Adult Educators The Problem with Variables Misconception 1: Variables can be combined

53. ©2014 IPDAE. All rights reserved. All content in this presentation is the proprietary property of The Institute for the Professional Development of Adult Educators The Problem with Variables Misconception 2: Variables have Defined Values

54. ©2014 IPDAE. All rights reserved. All content in this presentation is the proprietary property of The Institute for the Professional Development of Adult Educators The Problem with Variables Misconception 3: Variables can be any number

55. ©2014 IPDAE. All rights reserved. All content in this presentation is the proprietary property of The Institute for the Professional Development of Adult Educators The Problem with Variables Misconception 4: Variables = Objects

56. ©2014 IPDAE. All rights reserved. All content in this presentation is the proprietary property of The Institute for the Professional Development of Adult Educators How can the problem be solved? Make sure students understand the difference between abbreviations and variables. “6 meters” instead of “6m” Have students write out the literal translation of the equation/expression. 6P = S Six times the number of professors equals the number of students.

57. ©2014 IPDAE. All rights reserved. All content in this presentation is the proprietary property of The Institute for the Professional Development of Adult Educators How can the problem be solved? Use a Math Translation Guide

58. ©2014 IPDAE. All rights reserved. All content in this presentation is the proprietary property of The Institute for the Professional Development of Adult Educators How can the problem be solved? Practice Translating Jennifer has 10 fewer DVDs than Brad. j – 10 = b (common answer, but incorrect) Insert the words and see the difference in the equation. j (has) = b (fewer) – 10 so j = b – 10

59. ©2014 IPDAE. All rights reserved. All content in this presentation is the proprietary property of The Institute for the Professional Development of Adult Educators How can the problem be solved? Use Multiple Representations Represent problems using symbols, expressions, and equations, tables, and graphs Model real-world situations Complete problems different ways (flexibility in problem solving)

60. ©2014 IPDAE. All rights reserved. All content in this presentation is the proprietary property of The Institute for the Professional Development of Adult Educators How can the problem be solved? Which would you rather do? 5861 x 12465861 x 1246

61. ©2014 IPDAE. All rights reserved. All content in this presentation is the proprietary property of The Institute for the Professional Development of Adult Educators How can the problem be solved? Use Vertical Multiplication of Polynomials

63. PROBLEM SOLVING Developing Quantitative and Algebraic Reasoning Skills

64. “Mathematical problem solving skills are critical to successfully function in today’s technologically advanced society. Solving word problems requires understanding the relationships and outcomes of problems. You must make connections between the different meanings, interpretations, and relationships to mathematical operations.” Van de Walle, 2004 ©2014 IPDAE. All rights reserved. All content in this presentation is the proprietary property of The Institute for the Professional Development of Adult Educators

65. It’s All About Problem Solving Example of “Student Problem Solving” without guidance

66. ©2014 IPDAE. All rights reserved. All content in this presentation is the proprietary property of The Institute for the Professional Development of Adult Educators It’s All About Problem Solving Problem Solving Strategies Look for patterns Consider all possibilities Make an organized list Draw a picture Guess and check Write an equation Construct a table or graph Act it out Use objects Work backward Solve a simpler (or similar) problem

67. ©2014 IPDAE. All rights reserved. All content in this presentation is the proprietary property of The Institute for the Professional Development of Adult Educators It’s All About Problem Solving Polya’s Four Steps to Problem Solving Understand the problem Devise a plan Carry out the plan Look back (reflect) Polya, George. How To Solve It, 2nd ed. (1957). Princeton University Press.

68. ©2014 IPDAE. All rights reserved. All content in this presentation is the proprietary property of The Institute for the Professional Development of Adult Educators It’s All About Problem Solving Explicit Instruction Matters Shows consistent positive effects on performance Places students’ attention on mathematical ideas Develops “mathematical power” Develops students’ beliefs that they are capable of doing mathematics Provides ongoing assessment data

69. ©2014 IPDAE. All rights reserved. All content in this presentation is the proprietary property of The Institute for the Professional Development of Adult Educators It’s All About Problem Solving “Anyone who has never made a mistake has never tried anything new.” Albert Einstein Remember, once is not enough when teaching problem solving.

70. Meaning Making The single greatest tool/instructional method we can use in our classrooms

71. ©2014 IPDAE. All rights reserved. All content in this presentation is the proprietary property of The Institute for the Professional Development of Adult Educators Resources

72. ©2014 IPDAE. All rights reserved. All content in this presentation is the proprietary property of The Institute for the Professional Development of Adult Educators Thank You By Educators For Educators www.floridaipdae.org Thank You! Bonnie Goonen bv73008@aol.com