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CASE STUDY UNIT

Mathematics: Identifying and Addressing

Student Errors

Created by Janice Brown, PhD, Vanderbilt UniversityKim Skow, MEd, Vanderbilt University

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The contents of this resource were developed under a grant from the U.S. Department of Education, #H325E120002. However, those contents do not necessarily represent the policy of the U.S. Department of Education, and you should not assume endorse- ment by the Federal Government. Project Officer, Sarah Allen

Mathematics: Identifying and Addressing Student Errors

Contents: Page

Credits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ii Standards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv STAR Sheets

Collecting Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Identifying Error Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Word Problems: Additional Error Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Determining Reasons for Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Addressing Error Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

Case Studies Level A, Case 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 Level A, Case 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 Level B, Case 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 Level B, Case 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 Level C, Case 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

Answer Key . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

TABLE OF CONTENTS

* For an Answer Key to this case study, please email your full name, title, and institutional affiliation to the IRIS Center at iris@vanderbilt .edu .

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To Cite This Case Study Unit

Brown J ., Skow K ., & the IRIS Center . (2016) . Mathematics: Identifying and addressing student errors. Retrieved from http:// iris .peabody .vanderbilt .edu/case_studies/ics_matherr .pdf

Content Contributors

Janice Brown Kim Skow

Case Study Developers

Janice Brown Kim Skow

Editor Jason Miller

Reviewers

Diane Pedrotty Bryant David Chard Kimberly Paulsen Sarah Powell Paul Riccomini

Graphics Brenda KnightPage 27- Geoboard Credit: Kyle Trevethan

Mathematics: Identifying and Addressing Student Errors

CREDITS

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Mathematics: Identifying and Addressing Student Errors

STANDARDS

Licensure and Content Standards This IRIS Case Study aligns with the following licensure and program standards and topic areas .

Council for the Accreditation of Educator Preparation (CAEP) CAEP standards for the accreditation of educators are designed to improve the quality and effectiveness not only of new instructional practitioners but also the evidence-base used to assess those qualities in the classroom .

• Standard 1: Content and Pedagogical Knowledge

Council for Exceptional Children (CEC) CEC standards encompass a wide range of ethics, standards, and practices created to help guide those who have taken on the crucial role of educating students with disabilities .

• Standard 1: Learner Development and Individual Learning Differences

Interstate Teacher Assessment and Support Consortium (InTASC) InTASC Model Core Teaching Standards are designed to help teachers of all grade levels and content areas to prepare their students either for college or for employment following graduation .

• Standard 6: Assessment • Standard 7: Planning for Instruction

National Council for Accreditation of Teacher Education (NCATE) NCATE standards are intended to serve as professional guidelines for educators . They also overview the “organizational structures, policies, and procedures” necessary to support them

• Standard 1: Candidate Knowledge, Skills, and Professional Dispositions

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Error analysis is a type of diagnostic assessment that can help a teacher determine what types of errors a student is making and why . More specifically, it is the process of identifying and reviewing a student’s errors to determine whether an error pattern exists—that is, whether a student is making the same type of error consistently . If a pattern does exist, the teacher can identify a student’s misconceptions or skill deficits and subsequently design and implement instruction to address that student’s specific needs . Research on error analysis is not new: Researchers around the world have been conducting studies on this topic for decades . Error analysis has been shown to be an effective method for identifying patterns of mathematical errors for any student, with or without disabilities, who is struggling in mathematics .

Steps for Conducting an Error Analysis An error analysis consists of the following steps: Step 1. Collect data: Ask the student to complete at least 3 to 5 problems of the same type (e .g .,

multi-digit multiplication) . Step 2. Identify error patterns: Review the student’s solutions, looking for consistent error patterns

(e .g ., errors involving regrouping) . Step 3. Determine reasons for errors: Find out why the student is making these errors . Step 4. Use the data to address error patterns: Decide what type of instructional strategy will best

address a student’s skill deficits or misunderstandings .

Benefits of Error AnalysisBenefits of Error Analysis An error analysis can help a teacher to:

• Identify which steps the student is able to perform correctly (as opposed to simply marking answers either correct or incorrect, something that might mask what it is that the student is doing right)

• Determine what type(s) of errors a student is making • Determine whether an error is a one-time miscalculation or a persistent issue that

indicates an important misunderstanding of a mathematic concept or procedure • Select an effective instructional approach to address the student’s misconceptions and

to teach the correct concept, strategy, or procedure

Mathematics: Identifying and Addressing Student Errors

INTRODUCTION

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References Ashlock, R . B . (2010) . Error patterns in computation (10th ed .) . Boston: Allyn & Bacon . Ben-Zeev, T . (1998) . Rational errors and the mathematical mind . Review of General Psychology,

2(4), 366–383 . Cox, L . S . (1975) . Systematic errors in the four vertical algorithms in normal and handicapped

populations . Journal for Research in Mathematics Education, 6(4), 202–220 . Idris, S . (2011) . Error patterns in addition and subtraction for fractions among form two students .

Journal of Mathematics Education, 4(2), 35–54 . Kingsdorf, S ., & Krawec, J . (2014) . Error analysis of mathematical word problem solving across

students with and without learning disabilities . Learning Disabilities Research & Practice, 29(2), 66–74 .

Radatz, H . (1979) . Error analysis in mathematics education . Journal for Research in Mathematics Education, 10(3), 163–172 .

Riccomini, P . J . (2014) . Identifying and using error patterns to inform instruction for students struggling in mathematics. Webinar slideshow .

Yetkin, E . (2003) . Student difficulties in learning elementary mathematics . ERIC Clearinghouse for Science, Mathematics, and Environmental Education. Retrieved from http://www .ericdigests . org/2004-3/learning .html

References for the Following Cases Ashlock, R . B . (2010) . Error patterns in computation (10th ed .) . Boston: Allyn & Bacon . Sherman, H . J ., Richardson, L . I ., & Yard, G . J . (2009) . Teaching learners who struggle with

mathematics: Systematic invervention and remediation (2nd ed .) . Upper Saddle River, NJ: Merrill/Pearson .

Chapin, S . H . (1999) . Middle grades math: Tools for success (course 2): Practice workbook. New Jersey: Prentice-Hall .

☆ What a STAR Sheet isWhat a STAR Sheet is A STAR (STrategies And Resources) Sheet provides you with a description of a well- researched strategy that can help you solve the case studies in this unit .

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Mathematics: Identifying and Addressing Student Errors Collecting Data

STAR SHEET

About the Strategy Collecting data involves asking a student to complete a worksheet, test, or progress monitoring measure containing a number of problems of the same type .

What the Research and Resources Say • Error analysis data can be collected using formal (e .g ., chapter test, standardized test) or

informal (e .g ., homework, in-class worksheet) measures (Riccomini, 2014) . • Error analysis is one form of diagnostic assessment . The data collected can help teachers

understand why students are struggling to make progress on certain tasks and align instruction with the student’s specific needs (National Center on Intensive Intervention, n .d .; Kingsdorf & Krawec, 2014) .

• To help determine an error pattern, the data collection measure must contain at a minimum three to five problems of the same type (Special Connections, n .d .) .

Identifying Data Sources To conduct an error analysis for mathematics, the teacher must first collect data . She can do so by using a number of materials completed by the student (i .e ., student product) . These include worksheets, progress monitoring measures, assignments, quizzes, and chapter tests . Homework can also be used, assuming the teacher is confident that the student completed the assignment independently . Regardless of the type of student product used, it should contain at a minimum three to five problems of the same type . This allows a sufficient number of items with which to determine error patterns .

Scoring To better understand why students are struggling, the teacher should mark each incorrect digit in a student’s answer, as opposed to simply marking the entire answer incorrect . Evaluating each digit in the answer allows the teacher to more quickly and clearly identify the student’s error and to determine whether the student is consistently making this error across a number of problems . For example, take a moment to examine the worksheet below . By marking the incorrect digits, the teacher can determine that, although the student seems to understand basic math facts, he is not regrouping the “1” to the ten’s column in his addition problems . Note: Marking each incorrect digit might not always reveal the error pattern . Review the STAR Sheets Identifying Error Patterns, Word Problems: Additional Error Patterns, and Determining Reasons for Errors to learn about identifying the different types of errors students make .

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TipsTips • Typically, addition, subtraction, and multiplication problems should be

scored from RIGHT to LEFT . By scoring from right to left, the teacher will be sure to note incorrect digits in the place value columns . However, division problems should be scored LEFT to RIGHT .

• If the student is not using a traditional algorithm to arrive at a solution, but instead using a partial algorithm (e .g ., partial sums, partial products) then addition, subtraction, multiplication, and division problems should be scored from LEFT to RIGHT .

References Kingsdorf, S ., & Krawec, J . (2014) . Error analysis of mathematical word problem solving across

students with and without learning disabilities . Learning Disabilities Research and Practice, 29(2), 66–74 .

National Center on Intensive Intervention . (n .d .) . Informal academic diagnostic assessment: Using data to guide intensive instruction. Part 3: Miscue and skills analysis . PowerPoint slides . Retrieved from http://www .intensiveintervention .org/resource/informal-academic-diagnostic- assessment-using-data-guide-intensive-instruction-part-3

Riccomini, P . J . (2014) . Identifying and using error patterns to inform instruction for students struggling in mathematics . Webinar series, Region 14 State Support Team .

Special Connections . (n .d .) . Error pattern analysis . Retrieved from http://www .specialconnections . ku .edu/~specconn/page/instruction/math/pdf/patternanalysis .pdf

The University of Chicago School Mathematics Project . (n .d .) . Learning multiple methods for any mathematical operation: Algorithms. Retrieved from http://everydaymath .uchicago .edu/about/ why-it-works/multiple-methods/

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STAR SHEETSTAR SHEET Mathematics: Identifying and Addressing Student Errors

Identifying Error Patterns

About the Strategy Identifying error patterns refers to determining the type(s) of errors made by a student when he or she is solving mathematical problems .

What the Research and Resources Say Three to five errors on a particular type of problem constitute an error pattern (Howell, Fox, & Morehead, 1993; Radatz, 1979) . Typically, student mathematical errors fall into three broad categories: factual, procedural, and conceptual . Each of these errors is related either to a student’s lack of knowledge or a misunderstanding (Fisher & Frey, 2012; Riccomini, 2014) . Not every error is the result of a lack of knowledge or skill . Sometimes, a student will make a mistake simply because he was fatigued or distracted (i .e ., careless errors) (Fisher & Frey, 2012) . Procedural errors are the most common type of error (Riccomini, 2014) . Because conceptual and procedural knowledge often overlap, it is difficult to distinguish conceptual errors from procedural errors (Rittle-Johnson, Siegler, & Alibali, 2001; Riccomini, 2014) .

Types of Errors 1. Factual errors are errors due to a lack of factual information (e .g ., vocabulary, digit identification,

place value identification) . 2. Procedural errors are errors due to the incorrect performance of steps in a mathematical process

(e .g ., regrouping, decimal placement) . 3. Conceptual errors are errors due to misconceptions or a faulty understanding of the underlying

principles and ideas connected to the mathematical problem (e .g ., relationship among numbers, characteristics, and properties of shapes) .

FYI FYI Another type of error that a student might make is a careless error . The student fails to correctly solve a given mathematical problem despite having the necessary skills or knowledge . This might happen because the student is tired or distracted by activity elsewhere in the classroom . Although teachers can note the occurrence of such errors, doing so will do nothing to identify a student’s skill deficits . For many students, simply pointing out the error is all that is needed to correct it . However, it is important to note that students with learning disabilities often make careless errors .

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Common Factual Errors Factual errors occur when students lack factual information (e .g ., vocabulary, digit identification, place value identification) . Review the table below to learn about some of the common factual errors committed by students .

Factual Error Examples

Has not mastered basic number facts: The student does not know basic mathematics facts and makes errors when adding, subtracting, multiplying, or dividing single-digit numbers .

3 + 2 = 7 7 − 4 = 2 2 × 3 = 7 8 ÷ 4 = 3

Misidentifies signs 2 × 3 = 5 (The student identifies the multiplication sign as an addition sign .) 8 ÷ 4 = 4 (The student identifies the division sign as a minus sign .)

Misidentifies digits The student identifies a 5 as a 2 .

Makes counting errors 1, 2, 3, 4, 5, 7, 8, 9 (The student skips 6 .)

Does not know mathematical terms (vocabulary)

The student does not understand the meaning of terms such as numerator, denominator, greatest common factor, least common multiple, or circumference .

Does not know mathematical formulas The student does not know the formula for calculating the area of a circle .

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Procedural Error Examples Regrouping Errors

Forgetting to regroup: The student forgets to regroup (carry) when adding, multiplying, or subtracting .

77 + 54

121

The student added 7 + 4 correctly but didn’t regroup one group of 10 to the tens column .

123 − 76

53

The student does not regroup one group of 10 from the tens column, but instead subtracted the number that is less (3) from the greater number (6) in the ones column .

56 × 2 102

After multiplying 2 × 6, the student fails to regroup one group of 10 from the tens column .

Regrouping across a zero: When a problem contains one or more 0’s in the minuend (top number), the student is unsure of what to do .

304 − 21

323

The student subtracted the 0 from the 2 instead of regrouping .

Performing incorrect operation: Although able to correctly identify the signs (e .g ., addition, minus), students often subtract when they are suppose to add, or vice versa . However, students might also perform other incorrect operations, such as multiplying instead of adding .

234 − 45

279

The student added instead of subtracting .

3 + 2

6

The student multiplied instead of adding .

Fraction Errors Failure to find common denominator when adding and subtracting fractions

3 1 4 — + — = — 4 3 7

The student added the numerators and then the denominators without finding the common denominator .

Failure to invert and then multiply when dividing fractions 1 1 2 2

— ÷ 2 = — × — = — = 1 2 2 1 2

The student did not invert the 2 to before multiplying to get the correct answer of .

Failure to change the denominator in multiplying fractions 2 5 10 — × — = — 8 8 8

The student did not multiply the denominators to get the correct answer .

Incorrectly converting a mixed number to an improper fraction

1 4 1— = — 2 2

To find the numerator, the student added 2 + 1 + 1 to get 4, instead of following the correct procedure ( 2 × 1 + 1 = 3 ) .

Common Procedural Errors Procedural knowledge is an understanding of what steps or procedures are required to solve a problem . Procedural errors occur when a student incorrectly applies a rule or an algorithm (i .e ., the formula or step-by-step procedure for solving a problem) . Review the table below to learn more about some common procedural errors .

1 4

1 2

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Common Conceptual Errors Conceptual knowledge is an understanding of underlying ideas and principles and a recognition of when to apply them . It also involves understanding the relationships among ideas and principles . Conceptual errors occur when a student holds misconceptions or lacks understanding of the underlying principles and ideas related to a given mathematical problem (e .g ., the relationship between numbers, the characteristics and properties of shapes) . Examine the table below to learn more about some common conceptual errors .

Conceptual Error Examples Misunderstanding of place value: The student doesn’t understand place value and records the answer so that the numbers are not in the appropriate place value position .

67 + 4

17

The student added all the numbers together ( 6 + 7 + 4 = 17 ), not understanding the values of the ones and tens columns .

10 + 9

91

The student recorded the answer with the numbers reversed, disregarding the appropriate place value position of the numbers or digits .

Write the following as a number:

When expressing a number beyond two digits, the student does not have a conceptual understanding of the place value position .

a) seventy-six b) nine hundred seventy-

four c) six thousand, six

hundred twenty-four

Student answer: a) 76 b) 90074 c) 600060024

Procedural Error cont Examples cont Decimal Errors

Not aligning decimal points when adding or subtracting: The student aligns the numbers without regard to where the decimal is located .

120 .4 +

63 .21 75 .25

The student did not align the decimal points to show digits in like places . In this case, .4 and .2 are in the tenths place and should be aligned .

Not placing decimal in appropriate place when multiplying or dividing: The student does not count and add the number of decimal places in each factor to determine the number of decimal places in the product . Note: This could also be a conceptual error related to place value.

3 .4 × .2

6 .8

As with adding or subtracting, the student aligns the decimal point in the product with the decimal points in the factors . The student did not count and add the number of decimal places in each factor to determine the number of decimal places in the product

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Conceptual Error cont . Examples cont .

Overgeneralization: Because of lack of conceptual understanding, the student incorrectly applies rules or knowledge to novel situations .

321 −

245 124

Regardless of whether the greater number is in the minuend (top number) or subtrahend (bottom number), the student always subtracts the number that is less from the greater number, as is done with single-digit subtraction .

Put the following fractions in order from smallest to largest .

The student puts fractions in the order , , , because he doesn’t understand the relation between the numerator and its denominator; that is, larger denominators mean smaller fractional parts .

Overspecialization: Because of lack of conceptual understanding, the student develops an overly narrow definition of a given concept or of when to apply a rule or algorithm .

Which of the triangles below are right triangles?

The student chooses a because she only associates a right triangle with those with the same orientation as a .

a)

b)

c) both

Student answer: a

90˚

12 200

1 351

77 486

12 200

1 351

77 486

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References Ashlock, R . B . (2010) . Error patterns in computation (10th ed .) . Boston: Allyn & Bacon . Ben-Hur, M . (2006) . Concept-rich mathematics instruction . Alexandria, VA: ASCD . Cohen, L . G ., & Spenciner, L . J . (2007) . Assessment of children and youth with special needs (3rd

ed .) . Upper Saddle River, NJ: Pearson . Educational Research Newsletter and Webinars . (n .d .) . Students’ common errors in working with

fractions . Retrieved from http://www .ernweb .com/educational-research-articles/students- common-errors-misconceptions-about-fractions/

El Paso Community College . (2009) . Common mistakes: Decimals. Retrieved from http://www . epcc .edu/CollegeReadiness/Documents/Decimals_0-40 .pdf

El Paso Community College . (2009) . Common mistakes: Fractions . Retrieved from http://www . epcc .edu/CollegeReadiness/Documents/Fractions_0-40 .pdf

Fisher, D ., & Frey, N . (2012) . Making time for feedback . Feedback for Learning, 70(1), 42–46 . Howell, K . W ., Fox, S ., & Morehead, M . K . (1993) . Curriculum-based evaluation: Teaching and

decision-making. Pacific Grove, CA: Brooks/Cole . National Council of Teachers of Mathematics . (2000) . Principles and standards for school

mathematics . Reston, VA: Author . Riccomini, P . J . (2014) . Identifying and using error patterns to inform instruction for students

struggling in mathematics . Webinar series, Region 14 State Support Team . Radatz, H . (1979) . Error analysis in mathematics education . Journal for Research in Mathematics

Education, 10(3), 163–172 . Rittle-Johnson, B ., Siegler, R . S ., & Alibali, M . W . ( 2001) . Developing conceptual understanding

and procedural skill in mathematics: An iterative process . Journal of Educational Psychology, 93(2), 346–362 .

Sherman, H . J ., Richardson, L . I ., & Yard, G . J . (2009) . Teaching learners who struggle with mathematics: Systematic intervention and remediation (2nd ed .) . Upper Saddle River, NJ: Merrill/Pearson .

Siegler, R ., Carpenter, T ., Fennell, F ., Geary, D ., Lewis, J ., Okamoto, Y ., Thompson, L ., & Wray, J . (2010) . Developing effective fractions instruction for kindergarten through 8th grade: A practice guide (NCEE #2010-4039) . Washington, DC: National Center for Education Evaluation and Regional Assistance, Institute of Education Sciences, U .S . Department of Education . Retrieved from http://ies .ed .gov/ncee/wwc/pdf/practice_guides/fractions_pg_093010 .pdf

Special Connections . (n .d .) . Error pattern analysis. Retrieved from http://www .specialconnections . ku .edu/~specconn/page/instruction/math/pdf/patternanalysis .pdf

Yetkin, E . (2003) . Student difficulties in learning elementary mathematics. ERIC Clearinghouse for Science, Mathematics, and Environmental Education . Retrieved from http://www .ericdigests . org/2004-3/learning .html

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STAR SHEETSTAR SHEET Mathematics: Identifying and Addressing Student Errors

Word Problems: Additional Error Patterns

About the Strategy A word problem presents a hypothetical real-world scenario that requires a student to apply mathematical knowledge and reasoning to reach a solution .

What the Research and Resources Say • Students consider computational exercises more difficult when they are expressed as word

problems rather than as number sentences (e .g ., 3 + 2 =) (Sherman, Richardson, & Yard, 2009) .

• When they solve word problems, students struggle most with understanding what the problem is asking them to do . More specifically, students might not recognize the problem type and therefore do not know what strategy to use to solve it (Jitendra et al ., 2007; Sherman, Richardson, & Yard, 2009; Powell, 2011; Shin & Bryant, 2015) .

• Word problems require a number of skills to solve (e .g ., reading text, comprehending text, translating the text into a number sentence, determining the correct algorithm to use) . As a result, many students, especially those with math and/or reading difficulties, find word problems challenging (Powell, Fuchs, Fuchs, Cirino, & Fletcher, 2009; Reys, Lindquist, Lambdin, & Smith, 2015) .

• Word problems are especially difficult for students with learning disabilities (Krawec, 2014; Shin & Bryant, 2015) .

Common Difficulties Associated with Solving Word Problems A student might solve word problems incorrectly due to factual, procedural, or conceptual errors . However, a student might encounter additional difficulties when trying to solve word problems, many of which are associated with reading skill deficits, such as those described below . Poor vocabulary knowledge: The student does not understand many mathematics terms (e .g ., difference, factor, denominator) . Limited reading skills: The student has difficulty reading text with vocabulary and complex sentence structure . Because of this, the student struggles to understand what is being asked . Inability to identify relevant information: The student has difficulty determining which pieces of information are relevant and which are irrelevant to solving the problem . Lack of prior knowledge: The student has limited experience with the context in which the problem is embedded . For example, a student unfamiliar with cooking might have difficulty solving a fraction problem presented within the context of baking a pie . Inability to translate the information into a mathematical equation: The student has difficulty translating the information in the word problem into a mathematical equation that they can solve . More specifically, the student might not be able to put the numbers in the correct order in the equation or determine the correct operation to use .

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Example The word problem below illustrates why students might have difficulty solving this type of problem .

Jonathan would like to buy a new 21-speed bicycle. The bike costs $119.76. Jonathan received $25 for his birthday. He also worked for 3 months last summer and earned $59.50. Find the difference between what the bike costs and the amount of money Jonathan has.

In addition to solving this word problem incorrectly due to factual, procedural, or conceptual errors, the student might struggle for reasons related to reading skill deficits .

• Poor vocabulary knowledge—The student might be unfamiliar with the term difference . • Limited reading skills—The student might struggle with the problem’s final sentence because of

its complex structure . If the student doesn’t understand some of the vocabulary (e .g ., received, earned), it might impede his or her ability to solve the problem .

• Inability to identify relevant information—The student might attend to irrelevant information, such as the type of bicycle or the number of months Jonathan worked, and therefore solve the problem incorrectly .

• Lack of prior knowledge—The student might have limited knowledge about the process of making purchases .

• Inability to translate information into a mathematical equation—The student might have difficulty determining which operations to perform with which numbers . This situation might be made worse in cases involving problems with multiple steps .

References Jitendra, A . K ., Griffin, C . C ., Haria, P ., Leh, J ., Adams, A ., & Kaduvettoor, A . (2007) . A

comparison of single and multiple strategy instruction on third-grade students’ mathematical problem solving . Journal of Educational Psychology, 99(1), 115–127 .

Krawec, J . L . (2014) . Problem representation and mathematical problem solving of students of varying math ability . Journal of Learning Disabilities, 47(2), 103–115 .

Powell, S . R . (2011) . Solving word problems using schemas: A review of the literature . Learning Disabilities Research & Practice, 26(2), 94–108 .

Powell, S . R ., Fuchs, L . S ., Fuchs, D ., Cirino, P . T ., & Fletcher, J . M . (2009) . Do word-problem features differentially affect problem difficulty as a function of students’ mathematics difficulty with and without reading difficulty? Journal of Learning Disabilities 20(10), 1–12

Reys, R ., Lindquist, M . M ., Lambdin, D . V ., & Smith, N . L . (2015) . Helping children learn mathematics (11th ed .) . Hoboken, NJ: John Wiley & Sons .

Sherman, H . J ., Richardson, L . I ., & Yard, G . J . (2009) . Teaching learners who struggle with mathematics: Systematic intervention and remediation (2nd ed .) . Upper Saddle River, NJ: Merrill/Pearson .

Shin, M ., & Bryant, D . P . (2015) . A synthesis of mathematical and cognitive performances of students with mathematics learning disabilities . Journal of Learning Disabilities, 48(1), 96–112 .

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Mathematics: Identifying and Addressing Student Errors Determining Reasons for Errors

CASE STUDY

About the Strategy Determining the reason for errors is the process through which teachers determine why the student is making a particular type of error .

What the Research and Resources Say • To help them to improve their mathematical performance, teachers must first identify and

understand why students make particular errors (Radatz, 1979; Yetkin, 2003) . • Typically, a student’s errors are not random; instead, they are often based on incorrect

algorithms or procedures applied systematically (Cox, 1975; Ben-Zeev, 1998) . • Knowing what a student is thinking when she is solving a problem can be a rich source of

information about what she does and does not understand (Hunt & Little, 2014; Baldwin & Yun, 2012) .

Helpful Strategies Determining exactly why a student is making a particular error is important in that it informs the teacher’s instructional response . Though it is sometimes obvious why a student is making a certain type of errors, at other times determining a reason proves more difficult . In these latter instances, the teacher can use one or more of the following strategies . Interview the student—It is sometimes unclear why a student is making a particular type of error . For example, it can be difficult for a teacher to distinguish between procedural or conceptual errors . For this reason, it can be beneficial to ask a student to talk through his or her process for solving the problem . Teachers can ask general questions such as “How did you come up with that answer?” or prompt the student with statements such as “Show me how you got that answer .” Another reason teachers might want to interview the student is to make sure the student has the prerequisite skills to solve the problem . Observe the student—A student might also reveal information through nonverbal means . This can include gestures, pauses, signs of frustration, and self-talk . The teacher can use information of this type to identify at what point in the problem-solving task that the student experiences difficulty or frustration . It can also help the teacher determine which procedure or set of rules a student is applying and why . Look for exceptions to an error pattern—In addition to looking for error patterns, a teacher should note instances when the student does not make the same error on the same type of problem . This, too, can be informative because it might indicate that the student has partial or basic understanding of the concept in question . For example, Cammy completed a worksheet on multiplying whole numbers by fractions . She seemed to get most of them wrong; however, she correctly answered the problems in which the fraction was . This seems to indicate that, though Cammy conceptually understands what of a whole is, she most likely does not know the process for multiplying whole numbers by fractions .

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Considerations for Students with Learning Disabilities Approximately 5–8% of students exhibit mathematics learning disabilities . Therefore, it is important to understand that their unique learning differences might impact their ability to learn and correctly choose and apply solution strategies to solve mathematics problems . A few characteristics that teachers might notice with students with learning disabilities is that these students often:

• Have difficulty mastering basic number facts • Make computational errors even though they might have a strong conceptual understanding • Have difficulty making the connection between concrete objects and semiabstract (visual

representations) or abstract knowledge or mathematical symbols • Struggle with mathematical terminology and written language • Have visual-spatial deficits, which result in difficulty visualizing mathematical concepts (although

this is quite rare)

References Baldwin, E . E ., & Yun, J . T . (2012) . Mathematics curricula and formative assessments: Toward an

error-based approach to formative data use in mathematics. Santa Barbara, CA: University of California Educational Evaluation Center .

Ben-Zeev, T . (1998) . Rational errors and the mathematical mind . Review of General Psychology, 2(4), 366–383 .

Cox, L . S . (1975) . Systematic errors in the four vertical algorithms in normal and handicapped populations . Journal for Research in Mathematics Education, 6(4), 202–220 .

Garnett, K . (n .d .) . Math learning disabilities . Retrieved from http://www .ldonline .org article/ Math_Learning_Disabilities

Hunt, H . H ., & Little, M . E . (2014) . Intensifying interventions for students by identifying and remediating conceptual understandings in mathematics . Teaching Exceptional Children, 46(6), 187–196 .

PBS, & the WGBH Educational Foundation . (2002) . Difficulties with mathematics. Retrieved from http://www .pbs .org/wgbh/misunderstoodminds/mathdiffs .html

Radatz, H . (1979) . Error analysis in mathematics education . Journal for Research in Mathematics Education, 10(3), 163–172 .

Sherman, H . J ., Richardson, L . I ., & Yard, G . J . (2009) . Teaching learners who struggle with mathematics: Systematic intervention and remediation. Upper Saddle River, NJ: Pearson .

Shin, M ., & Bryant, D . P . (2015) . A synthesis of mathematical and cognitive performances of students with mathematics learning disabilities . Journal of Learning Disabilities, 48(1), 96–112 .

Special Connections . (n .d .) . Error pattern analysis. Retrieved from http://specialconnections . ku .edu/~specconn/page/instruction/math/pdf/patternanalysis .pdf

Yetkin, E . (2003) . Student difficulties in learning elementary mathematics . ERIC Clearinghouse for Science, Mathematics, and Environmental Education . Retrieved from http://www .ericdigests . org/2004-3/learning .html

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STAR SHEET Mathematics: Identifying and Addressing Student Errors

Addressing Error Patterns

About the Strategy Addressing error patterns is the process of providing instruction that focuses on a student’s specific error .

What the Research and Resources Say • Students will continue to make procedural errors if they do not receive targeted instruction to

addresses those errors . Simply providing more opportunities to practice working a given problem is typically not effective (Riccomini, 2014) .

• By conducting an error analysis, the teacher can target specific misunderstandings or missteps, rather than re-teaching the entire skill or concept (Fisher & Frey, 2012) .

• Without intervention, students have been shown to continue to apply the same error patterns one year later (Cox, 1975) .

• Addressing a student’s conceptual errors might require the use of concrete or visual representations, as well as a great deal of re-teaching . Students can often use concrete objects to solve problems that they initially answered incorrectly (Riccomini, 2014; Yetkin, 2003) .

• Simply teaching the formula or the steps to solve a mathematics problem is typically not sufficient to help students gain conceptual understanding (Sweetland & Fogarty, 2008) .

How To Address Student Errors After the teacher has determined what types of error(s) a student is making, he or she can address the error in the following way . Discuss the error with the student: After the teacher has interviewed the student and examined work products, the teacher should briefly describe the student’s error and explain that they will work together to correct it . Provide effective instruction to address the student’s specific error: The teacher should target the student’s specific error instead of re-teaching how to work this type of problem in general . For example, if a student’s error is related to not regrouping during addition, the teacher should focus on where exactly in the process the student makes the error . The teacher must pinpoint the instruction to focus on the error and help the student to understand what he is doing incorrectly . Simply re-teaching the lesson will not ensure that the student understands the error and how to correctly solve the problem . Use effective strategies: With the type of error in mind, the teacher should select an effective strategy that will help to correct the student’s misunderstandings or missteps . Below are two effective strategies that teachers might find helpful to address some—if not all—error patterns .

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Manipulatives Manipulatives are concrete objects—for example, base-ten blocks, a geoboard, or integer chips—that a student can use to develop a conceptual understanding of mathematic topics . These objects help a student to represent the mathematical idea she is trying to learn or the problem she is trying to solve . For example, the teacher might demonstrate the idea of fractions by using fraction blocks or fraction strips . It is important that the teacher make explicit the connection between the concrete object and the abstract or the symbolic concept being taught . After a student has gained a basic understanding of the mathematical concept, the concrete objects should be replaced by visual representations such as images of a number line or geoboard (a small board with nails on which students stretch rubber bands to explore a variety of basic geometry concepts) . The goal is for the student to eventually understand and apply the concept with numerals and symbols . It is important that the teacher’s instruction match the needs of the student . Teachers should keep in mind that some students will need concrete objects to understand a concept, whereas others will be able to understand the concept using visual representations . Additionally, some students will require the support of concrete objects longer than will other students .

FYIFYI Recall that students with learning disabilities sometimes have visual-spatial deficits, which makes it difficult for them to learn concepts using visual representations . For these students, teachers should teach concepts using concrete materials accompanied by strong, precise verbal descriptions or explanations .

Keep in MindKeep in Mind The type of instruction a teacher uses to correct conceptual errors will likely differ from that used to address factual or procedural errors . Simply teaching a student the formula or the steps to solve a mathematics problem will not help the student gain conceptual understanding .

Geoboard Credit: Kyle Trevethan

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Explicit, systematic instruction Explicit, systematic instruction involves teaching a specific skill or concept in a highly structured environment using clear, direct language and incorporating the components listed in the table below .

Components of Explicit Instruction Modeling • The teacher models thinking aloud to demonstrate the completion of

a few sample problems . • The teacher leads the student through more sample problems . • The teacher points out difficult aspects of the problems .

Guided Practice • The student completes problems with the help of either teacher or peer guidance .

• The teacher monitors the student’s work . • The teacher offers positive corrective feedback .

Independent Practice

• The student completes the problems independently . • The teacher checks the student’s performance on independent

work . Adapted from Bender (2009), pp. 31–32

Reassess student skills: After providing instruction to correct the student’s error(s), the teacher should conduct a formal or informal assessment to make sure that the student has mastered the skill or concept in question .

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Instructional Tips • Check for prerequisite skills: Make sure the student has the prerequisite skills needed to solve

the problem with which he has been struggling . For example, if the student is making errors while adding two-digit numbers, the teacher needs to make sure the student knows basic math facts . If the student lacks the necessary pre-skills, the teacher should begin instruction at that point .

• Model examples and nonexamples: Be sure to model the completion of a minimum of three to five problems of the kind the student is struggling with . Add at least one nonexample of the error pattern to prevent overgeneralization (incorrectly applying the rule or knowledge to novel situations) and overspecialization (developing an overly narrow definition of the concept of or when to apply a rule or procedure) . For example, in the case of a student who does not regroup when subtracting, a teacher modeling how to solve this type of problem should also include problems that do not require regrouping .

• Pinpoint error: During modeling and guided practice, focus only on the place in the problem where the student makes an error . It is not necessary to work through the entire problem . For example, if the student’s error pattern is that she fails to find the common denominator when adding and subtracting fractions, the teacher would only model the process and explain the underlying conceptual knowledge of finding the common denominator . She would stop at that point, as opposed to completing the problem because the student knows the process from that point forward . The teacher should then continue in same manner for the remaining problems .

• Provide ample opportunities for practice: As with modeling, provide a minimum of three to five problems for guided practice, making sure to include a nonexample .

• Start with simple problems: During modeling and guided practice, begin with simple problems and gradually progress to more difficult ones as the student gains an understanding of the error and how to correctly complete the problem .

• Move the error around: When possible, move the error around so that it does not always occur in the same place . For example, if the student’s error is not regrouping when multiplying, the teacher should include examples that require regrouping in the ones and tens column, instead of always requiring the regrouping to occur in the ones column .

1 1 — + — 4 2

1 2 — + — 4 4

[Stop at this point because you have addressed the error pattern; the student knows how to add fractions.]

Problems 1 and 3 are examples that require regrouping, whereas problem 2, which does not require regrouping, is a nonexample . 121 231 376 − 17 − 120 − 229

1 . 2 . 3 .

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References Colarussso, R ., & O’Rourke, C . (2004) . Special education for all teachers (3rd ed .) . Dubuque, IA:

Kendall Hunt . Cox, L . S . (1975) . Systematic errors in the four vertical algorithms in normal and handicapped

populations . Journal for Research in Mathematics Education, 6(4), 202–220 . Fisher, D ., & Frey, N . (2012) . Making time for feedback . Feedback for Learning, 70(1), 42–46 . Riccomini, P . J . (2014) . Identifying and using error patterns to inform instruction for students

struggling in mathematics. Webinar series, Region 14 State Support Team . Retrieved from http://www .ohioregion14 .org/perspectives/?p=1005

Sweetland, J ., & Fogarty, M . (2008) . Prove it! Engaging teachers as learners to enhance conceptual understanding . Teaching Children Mathematics, 68–73 . Retrieved from http://www . uen .org/utahstandardsacademy/math/downloads/level-2/5-2-ProveIt .pdf

Yetkin, E . (2003) . Student difficulties in learning elementary mathematics. ERIC Clearinghouse for Science, Mathematics, and Environmental Education . Retrieved from http://www .ericdigests . org/2004-3/learning .html

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Background Student: Dalton Age: 12 Grade: 7th

Scenario Mrs . Moreno, a seventh-grade math teacher, is concerned about Dalton’s performance . Because Dalton has done well in her class up to this point, she believes that he has strong foundational mathematics skills . However, since beginning the lessons on multiplying decimals, Dalton has performed poorly on his independent classroom assignments . Mrs . Moreno decides to conduct an error analysis on his last homework assignment to determine what type of error he is making .

Possible Strategies • Collecting Data • Identifying Error Patterns

! ! AssignmentAssignment 1 . Read the Introduction. 2 . Read the STAR Sheets for the possible strategies listed above . 3 . Score Dalton’s classroom assignment below . For ease of scoring, an answer key has been provided . 4 . Examine the scored worksheet and determine Dalton’s error pattern .

Answer Key 1) 7 .488 2) 3 .065 3) 0 .5976 4) .00084 5) .5040 6) 2 .6724 7) .006084 8) 7 .602 9) .00183 10) 4 .6098 11) $39 .00 12) 732 .48 cm

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Mathematics: Identifying and Addressing Student Errors Level A • Case 1

CASE STUDY

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Background Student: Madison Age: 8 Grade: 2nd

Scenario Madison is a bright and energetic third-grader with a specific learning disability in math . Her class just finished a chapter on money, and her teacher, Ms . Brooks, was pleased with Madison’s performance . Ms . Brooks believes that Madison’s success was largely due to the fact that she used play money to teach concepts related to money . As is noted in Madison’s individualized education program (IEP), she more easily grasps concepts when using concrete objects (i .e ., manipulatives such as play coins and dollar bills) . In an attempt to build on this success, Ms . Brooks again used concrete objects—in this case, cardboard clocks with movable hands—to teach the chapter on telling time . The class is now halfway through that chapter, and to Ms . Brooks’ disappointment, Madison seems to be struggling with this concept . Consequently, Ms . Brooks decides to conduct an error analysis on Madison’s most recent quiz .

Possible Stragegies • Collecting Data • Identifying Error Patterns

! ! AssignmentAssignment 1 . Read the Introduction . 2 . Read the STAR Sheets for the possible strategies listed above . 3 . Score Madison’s quiz below by marking each incorrect response . 4 . Examine the scored quiz and determine Madison’s error pattern .

Answer Key

1) 3:00 2) 9:25 3) 7:15 4) 5)

6) 7) 8) 9)

10)

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Mathematics: Identifying and Addressing Student Errors Level A • Case 2

CASE STUDY

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Mathematics: Identifying and Addressing Student Errors Level B • Case 1

CASE STUDY

Background Student: Shayla Age: 10 Grade: 5th

Scenario Shayla and her family just moved to a new school district . Her math class is currently learning how to add and subtract fractions with unlike denominators . Shayla’s math teacher, Mr . Holden, is concerned because Shayla is performing poorly on assignments and quizzes . Before he can provide instruction to target Shayla’s skill deficits or conceptual misunderstandings, he needs to determine why she is having difficulty . For this reason, he decides to conduct an error analysis to discover what type of errors she is making .

Possible Strategies • Collecting Data • Identifying Error Patterns • Word Problems: Additional Error Patterns

! ! AssignmentAssignment 1 . Read Introduction . 2 . Read the STAR Sheets for the possible strategies listed above . 3 . Score Shayla’s assignment below by marking each incorrect digit . 4 . Examine the scored assignment and discuss at least three possible reasons for Shayla’s error pattern .

4 8

3 18

6 12

1 10

5 6

7 8

3 4

1 4

7 16

2 6

5 8

3 6

Answer Key

1) 2) 3) 4) 5)

6) 7) 8) 0 9) 10)

11) 12) 13)

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Mathematics: Identifying and Addressing Student Errors Level B • Case 2

CASE STUDY

Background Student: Elías Age: 7 Grade: 2nd

Scenario A special education teacher at Bordeaux Elementary School, Mrs . Gustafson has been providing intensive intervention to Elías, who has a learning disability, and collecting progress monitoring data for the past six weeks . His data indicate that he is not making adequate progress to meet his end-of- year goals . Mrs . Gustafson has decided that she needs to conduct a diagnostic assessment to identify areas of difficulty and to determine specific instructional needs . As part of the diagnostic assessment, Mrs . Gustafson conducts an error analysis using Elías’ progress monitoring data .

Possible Activities • Collecting Data • Identifying Error Patterns • Determining Reasons for Errors

! ! AssignmentAssignment 1 . Read the Introduction . 2 . Read the STAR Sheets for the possible strategies listed above . 3 . Score Elías’ progress monitoring probe below by marking each incorrect digit . 4 . When Mrs . Gustafson scores the probe, she finds two possible explanations . One is that Elías is

making a conceptual error, and the other is that he doesn’t understand or is not applying the correct procedure .

a . Assume that his error pattern is procedural . Describe Elías’ possible procedural error pattern .

b . Assume that his error pattern is conceptual . Describe Elías’ possible conceptual error pattern .

5 . Because the instructional adaptations Mrs . Gustafson will make will depend on Elías’ error pattern, she must be sure of the reasons for his errors . Explain at least one strategy Mrs . Gustafson could use to determine Elías’ error type .

Answer Key

1) 40 2) 87 3) 45 4) 22 5) 42

6) 34 7) 5 8) 122 9) 5 10) 80

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For illustrative purposes, only 10 of the 25 problems are shown .

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Mathematics: Identifying and Addressing Student Errors Level C • Case 1

CASE STUDY

Background Student: Wyatt Age: 12 Grade: 6th

Scenario Mr . Goldberg has been teaching a unit on fractions . He was pleased that all of his students seemed to quickly master adding and subtracting two fractions . However, when he began teaching the students how to multiply fractions, a small number of them did not readily learn the content . But after a quick mini-lesson, it appears that all but three students seem to understand how to solve the problems . One of these students, Wyatt, seems to be really struggling . Mr . Goldberg determines that he needs to collect some data to help him decide what type of error Wyatt is making so that he can provide appropriate instruction to help Wyatt be successful . To do so, he decides to evaluate Wyatt’s most recent independent classroom assignment .

! ! AssignmentAssignment 1 . Read the Introduction. 2 . Read the STAR Sheets . 3 . Score Wyatt’s classroom assignment below by marking each incorrect digit . 4 . Review Wyatt’s scored assignment sheet .

a . Describe Wyatt’s error pattern . b . Discuss any exceptions to this error pattern . What might these indicate?

5 . Based on Wyatt’s error pattern, which of the two strategies described in the Addressing Error Patterns STAR Sheet would you recommend that Mr . Goldberg use to remediate this error? Explain your response .

1 8

2 9

14 48

12 25

21 56

12 121

24 108

48 48

2 6

1 3

1 4

2 12

6 12

Answer Key

1) 2) 3) 4) 5)

6) 7) 8) or 1 9) or 10)

11) 12)

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Dealer Satisfaction

Dealer Satisfaction by Region and Year

0 2010 2011 2012 2013 2014 South America 2010 2011 2012 2013 2014 Europe 2010 2011 2012 2013 2014 Pacific Rim 2010 2011 2012 2013 2014 China 2012 2013 2014 1 0 1 1 2 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 2010 2011 2012 2013 2014 South America 2010 2011 2012 2013 2014 Europe 2010 2011 2012 2013 2014 Pacific Rim 2010 2011 2012 2013 2014 China 2012 2013 2014 0 0 1 2 3 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 2 2010 2011 2012 2013 2014 South America 2010 2011 2012 2013 2014 Europe 2010 2011 2012 2013 2014 Pacific Rim 2010 2011 2012 2013 2014 China 2012 2013 2014 2 2 1 6 5 0 0 1 1 2 1 1 1 1 1 1 1 1 0 1 0 1 1 3 2010 2011 2012 2013 2014 South America 2010 2011 2012 2013 2014 Europe 2010 2011 2012 2013 2014 Pacific Rim 2010 2011 2012 2013 2014 China 2012 2013 2014 14 14 8 12 15 2 2 4 3 4 3 2 2 2 4 2 1 1 2 2 1 4 5 4 2010 2011 2012 2013 2014 South America 2010 2011 2012 2013 2014 Europe 2010 2011 2012 2013 2014 Pacific Rim 2010 2011 2012 2013 2014 China 2012 2013 2014 22 20 34 34 44 6 6 11 12 22 7 8 15 21 17 2 3 3 5 7 0 2 8 5 2010 2011 2012 2013 2014 South America 2010 2011 2012 2013 2014 Europe 2010 2011 2012 2013 2014 Pacific Rim 2010 2011 2012 2013 2014 China 2012 2013 2014 11 14 15 45 56 2 2 14 33 60 4 4 7 6 8 0 0 1 3 2 0 0 2

This chart is showing Dealer Satisfaction between North America, South America, Europe, Pacific Rim and China. The data that was selected was rated on a a survery scale from 0-5 and between the the years of 2010-2014, except for China who started later in 2012. North America was leading in sample size and "in 5s" dealer satisfacion for "excelltence". Although North America recieved the highest total numbers in dealer satisfactions for excellent rankings, in 2014, South America recieved 60 surverys and North America recieved 56 within the level 5 category.

End-User Satisfaction

End-User Satisfaction by Region and Year

0 2010 2011 2012 2013 2014 South America 2010 2011 2012 2013 2014 Europe 2010 2011 2012 2013 2014 Pacific Rim 2010 2011 2012 2013 2014 China 2012 2013 2014 1 1 1 0 0 1 1 0 0 0 1 1 1 1 0 2 1 1 0 0 0 1 0 1 2010 2011 2012 2013 2014 South America 2010 2011 2012 2013 2014 Europe 2010 2011 2012 2013 2014 Pacific Rim 2010 2011 2012 2013 2014 China 2012 2013 2014 3 2 2 2 2 2 3 2 2 2 2 2 1 1 1 3 2 2 2 1 3 2 1 2 2010 2011 2012 2013 2014 South America 2010 2011 2012 2013 2014 Europe 2010 2011 2012 2013 2014 Pacific Rim 2010 2011 2012 2013 2014 China 2012 2013 2014 6 4 5 4 3 5 6 6 5 5 4 5 4 3 2 5 7 5 4 3 3 2 1 3 2010 2011 2012 2013 2014 South America 2010 2011 2012 2013 2014 Europe 2010 2011 2012 2013 2014 Pacific Rim 2010 2011 2012 2013 2014 China 2012 2013 2014 15 18 17 15 15 18 17 19 20 19 21 21 26 17 19 15 15 16 17 19 6 4 3 4 2010 2011 2012 2013 2014 South America 2010 2011 2012 2013 2014 Europe 2010 2011 2012 2013 2014 Pacific Rim 2010 2011 2012 2013 2014 China 2012 2013 2014 37 35 34 33 31 36 36 37 37 37 36 34 37 41 45 41 41 40 40 42 28 30 31 5 2010 2011 2012 2013 2014 South America 2010 2011 2012 2013 2014 Europe 2010 2011 2012 2013 2014 Pacific Rim 2010 2011 2012 2013 2014 China 2012 2013 2014 38 40 41 46 49 38 37 36 36 37 36 37 31 37 33 34 34 36 37 35 10 11 14

This chart is showing End-User Satisfaction between North America, South America, Europe, Pacific Rim and China. The data that was selected was rated on a a survery scale from 0-5 and between the the years of 2010-2014, except for China who started later in 2012. North America, South America, Europe, and the Pacific Rim all have the same sample size of 100 for each year between 2010 through 2014. China has a smaller sample size of 50 between the years of 2012 through 2014. You can see that the ratings of 5's, 4's, and 3's are the highest ratings. North America's rating of 4 decreases every year starting with 2010 while the 5 ratings increase through the years. The Pacfic Rim's 4 ratings are highest rated and is basically constant throughout the years while the 5 ratings are lower then 4 ratings the 5's are constant throughout the years.

Complaints

Complaints by Month and Region

World 40179 40210 40238 40269 40299 40330 40360 40391 40422 40452 40483 40513 40544 40575 40603 40634 40664 40695 40725 40756 40787 40817 40848 40878 40909 40940 40969 41000 41030 41061 41091 41122 41153 41183 41214 41244 41275 41306 41334 41365 41395 41426 41456 41487 41518 41548 41579 41609 41640 41671 41699 41730 41760 41791 41821 41852 41883 41913 41944 41974 169 187 210 226 232 261 245 223 195 174 154 163 195 221 240 264 283 296 269 256 231 214 201 171 200 216 234 253 282 305 296 279 266 243 232 203 216 239 266 284 315 340 319 304 277 250 228 213 240 251 281 298 322 350 330 311 289 265 239 219 NA 40179 40210 40238 40269 40299 40330 40360 40391 40422 40452 40483 40513 40544 40575 40603 40634 40664 40695 40725 40756 40787 40817 40848 40878 40909 40940 40969 41000 41030 41061 41091 41122 41153 41183 41214 41244 41275 41306 41334 41365 41395 41426 41456 41487 41518 41548 41579 41609 41640 41671 41699 41730 41760 41791 41821 41852 41883 41913 41944 41974 102 115 128 136 137 151 140 128 103 96 84 99 123 141 152 163 178 170 153 146 131 125 118 96 112 117 126 138 152 163 156 148 143 131 128 107 110 123 138 150 169 181 169 160 141 123 112 105 121 126 148 155 168 183 170 158 149 136 121 108 SA 40179 40210 40238 40269 40299 40330 40360 40391 40422 40452 40483 40513 40544 40575 40603 40634 40664 40695 40725 40756 40787 40817 40848 40878 40909 40940 40969 41000 41030 41061 41091 41122 41153 41183 41214 41244 41275 41306 41334 41365 41395 41426 41456 41487 41518 41548 41579 41609 41640 41671 41699 41730 41760 41791 41821 41852 41883 41913 41944 41974 12 13 15 16 17 19 18 16 15 14 11 9 10 13 16 20 22 28 25 23 20 16 13 11 15 18 20 23 26 30 28 26 24 21 18 15 19 23 26 30 33 37 34 32 29 26 24 23 26 28 31 35 39 43 41 38 33 30 26 23 Eur 40179 40210 40238 40269 40299 40330 40360 40391 40422 40452 40483 40513 40544 40575 40603 40634 40664 40695 40725 40756 40787 40817 40848 40878 40909 40940 40969 41000 41030 41061 41091 41122 41153 41183 41214 41244 41275 41306 41334 41365 41395 41426 41456 41487 41518 41548 41579 41609 41640 41671 41699 41730 41760 41791 41821 4185 2 41883 41913 41944 41974 52 55 61 67 73 82 80 76 73 62 59 54 59 62 66 70 75 86 81 79 73 68 66 61 66 71 76 79 85 91 89 86 82 76 73 70 74 79 83 88 91 95 92 90 87 83 77 74 80 82 85 89 95 98 95 93 89 85 80 76 Pac 40179 40210 40238 40269 40299 40330 40360 40391 40422 40452 40483 40513 40544 40575 40603 40634 40664 40695 40725 40756 40787 40817 40848 40878 40909 40940 40969 41000 41030 41061 41091 41122 41153 41183 41214 41244 41275 41306 41334 41365 41395 41426 41456 41487 41518 41548 41579 41609 41640 41671 41699 41730 41760 41791 41821 41852 41883 41913 41944 41974 3 4 6 7 5 9 7 3 4 2 0 1 3 5 6 11 8 12 10 8 7 5 4 3 4 6 9 11 14 15 18 15 13 12 10 7 8 10 13 11 15 19 17 15 14 12 10 7 8 10 12 13 12 15 14 13 11 8 7 7 China 40179 40210 40238 40269 40299 40330 40360 40391 40422 40452 40483 40513 40544 40575 40603 40634 40664 40695 40725 40756 40787 40817 40848 40878 40909 40940 40969 41000 41030 41061 41091 41122 41153 41183 41214 41244 41275 41306 41334 41365 41395 41426 41456 41487 41518 41548 41579 41609 41640 41671 41699 41730 41760 41791 41821 41852 41883 41913 41944 41974 3 4 3 2 5 6 5 4 4 3 3 4 5 4 6 5 7 8 7 7 6 6 5 4 5 5 5 6 8 11 10 9 7 6 5 5

This chart is showing PLE's Complaints from registered customers each month within PLE's 5 regions. From this data we can conclude that there is more use of the equipment in the summer months because of the higher number of complaints recieved. China has the fewest number of compaints, this is due to the less customer usage. Based off the data, the Pacific Rim and South America do not have as many complaints as North America does due to less people using or purchasing PLE's equipment. .

Mower Unit Sales

Mower Unit Sales by Month and Region

NA 40179 40210 40238 40269 40299 40330 40360 40391 40422 40452 40483 40513 40544 40575 40603 40634 40664 40695 40725 40756 40787 40817 40848 40878 40909 40940 40969 41000 41030 41061 41091 41122 41153 41183 41214 41244 41275 41306 41334 41365 41395 41426 41456 41487 41518 41548 41579 41609 41640 41671 41699 41730 41760 41791 41821 41852 41883 41913 41944 41974 6000 7950 8100 9050 9900 10200 8730 8140 6480 5990 5320 4640 5980 7620 8370 8830 9310 10230 8720 7710 6320 5840 4960 4350 6020 7920 8430 9040 9820 10370 9050 7620 6420 5890 5340 4430 6100 8010 8430 9110 9730 10120 9080 7820 6540 6010 5270 5380 6210 8030 8540 9120 9570 10230 9580 7680 6870 5930 5260 4830 SA 40179 40210 40238 40269 40299 40330 40360 40391 40422 40452 40483 40513 40544 40575 40603 40634 40664 40695 40725 40756 40787 40817 40848 40878 40909 40940 40969 41000 41030 41061 41091 41122 41153 41183 41214 41244 41275 41306 41334 41365 41395 41426 41456 41487 41518 41548 41579 41609 41640 41671 41699 41730 41760 41791 41821 41852 41883 41913 41944 41974 200 220 250 280 310 300 280 250 230 220 210 180 210 240 250 290 330 310 290 270 250 250 240 210 220 250 270 310 360 330 310 300 280 270 260 230 250 270 280 320 380 360 320 310 300 290 270 260 270 280 300 340 390 380 350 34 0 320 310 300 290 Europe 40179 40210 40238 40269 40299 40330 40360 40391 40422 40452 40483 40513 40544 40575 40603 40634 40664 40695 40725 40756 40787 40817 40848 40878 40909 40940 40969 41000 41030 41061 41091 41122 41153 41183 41214 41244 41275 41306 41334 41365 41395 41426 41456 41487 41518 41548 41579 41609 41640 41671 41699 41730 41760 41791 41821 41852 41883 41913 41944 41974 720 990 1320 1650 1590 1620 1590 1560 1590 1320 990 660 690 1020 1290 1620 1650 1590 1560 1530 1590 1260 900 660 570 840 1110 1500 1440 1410 1440 1410 1350 1080 840 510 480 750 1140 1410 1340 1360 1410 1490 1310 980 770 430 400 750 970 1310 1260 1240 1300 1250 1210 970 650 300 Pacific 40179 40210 40238 40269 40299 40330 40360 40391 40422 40452 40483 40513 40544 40575 40603 40634 40664 40695 40725 40756 40787 40817 40848 40878 40909 40940 40969 41000 41030 41061 41091 41122 41153 41183 41214 41244 41275 41306 41334 41365 41395 41426 41456 41487 41518 41548 41579 41609 41640 41671 41699 41730 41760 41791 41821 41852 41883 41913 41944 41974 100 120 110 120 130 120 140 130 130 120 130 140 140 150 140 150 130 140 150 140 150 160 150 150 160 150 160 170 160 170 160 170 180 180 190 180 200 190 200 210 190 200 200 210 220 210 220 230 200 190 210 220 200 210 230 220 220 230 240 230 China 40179 40210 40238 40269 40299 40330 40360 40391 40422 40452 40483 40513 40544 40575 40603 40634 40664 40695 40725 40756 40787 40817 40848 40878 40909 40940 40969 41000 41030 41061 41091 41122 41153 41183 41214 41244 41275 41306 41334 41365 41395 41426 41456 41487 41518 41548 41579 41609 41640 41671 41699 41730 41760 41791 41821 41852 41883 41913 41944 41974 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 16 22 26 14 15 11 3 1 World 40179 40210 40238 40269 40299 40330 40360 40391 40422 40452 40483 40513 40544 40575 40603 40634 40664 40695 40725 40756 40787 40817 40848 40878 40909 40940 40969 41000 41030 41061 41091 41122 41153 41183 41214 41244 41275 41306 41334 41365 41395 41426 41456 41487 41518 41548 41579 41609 41640 41671 41699 41730 41760 41791 41821 41852 41883 41913 41944 41974 7020 9280 9780 11100 11930 12240 10740 10080 8430 7650 6650 5620 7020 9030 10050 10890 11420 12270 10720 9650 8310 7510 6250 5370 6970 9160 9970 11020 11780 12280 10960 9500 8230 7420 6630 5350 7030 9220 10050 11050 11640 12040 11010 9830 8370 7490 6530 6300 7080 9250 10020 10995 11436 12082 11486 9504 8635 7451 6453 5651

The chart identifies the unit sales on PLE's mower equipment. We can see that the highest peak for mower sales is in the summer months and then a decline in sales starting in early fall months. Looking at the chart, North America is the region with the highest unit sales for PLE's mowers.

Tractor Unit Sales

Tractor Unit Sales by Month and Region

NA 40179 40210 40238 40269 40299 40330 40360 40391 40422 40452 40483 40513 40544 40575 40603 40634 40664 40695 40725 40756 40787 40817 40848 40878 40909 40940 40969 41000 41030 41061 41091 41122 41153 41183 41214 41244 41275 41306 41334 41365 41395 41426 41456 41487 41518 41548 41579 41609 41640 41671 41699 41730 41760 41791 41821 41852 41883 41913 41944 41974 570 611 630 684 650 600 512 500 478 455 407 360 571 650 740 840 830 760 681 670 640 620 570 533 620 792 890 960 1040 1032 1006 910 803 730 699 647 730 930 1160 1510 1650 1490 1460 1390 1360 1340 1240 1103 1250 1550 1820 2010 2230 2490 2440 2334 2190 2080 2050 2004 SA 40179 40210 40238 40269 40299 40330 40360 40391 40422 40452 40483 40513 40544 40575 40603 40634 40664 40695 40725 40756 40787 40817 40848 40878 40909 40940 40969 41000 41030 41061 41091 41122 41153 41183 41214 41244 41275 41306 41334 41365 41395 41426 41456 41487 41518 41548 41579 41609 41640 41671 41699 41730 41760 41791 41821 41852 41883 41913 41944 41974 250 270 260 270 280 270 264 280 290 280 290 280 320 350 390 440 470 490 481 460 460 440 436 420 510 590 610 600 620 640 590 600 670 630 710 570 650 680 724 730 760 800 840 830 820 810 827 750 780 805 830 890 930 980 1002 970 960 930 920 902 Eur 40179 40210 40238 40269 40299 40330 40360 40391 40422 40452 40483 40513 40544 40575 40603 40634 40664 40695 40725 40756 40787 40817 40848 40878 40909 40940 40969 41000 41030 41061 41091 41122 41153 41183 41214 41244 41275 41306 41334 41365 41395 41426 41456 41487 41518 41548 41579 41609 41640 41671 41699 41730 41760 41791 41821 41852 41883 41913 41944 41974 560 600 680 650 580 590 760 645 650 670 888 850 620 760 742 780 690 721 680 711 695 650 680 657 610 680 730 820 810 807 760 720 660 630 603 570 500 590 620 730 740 720 670 610 599 560 550 520 480 523 560 570 590 600 580 570 550 530 517 490 Pac 40179 40210 40238 40269 40299 40330 40360 40391 40422 40452 40483 40513 40544 40575 40603 40634 40664 40695 40725 40756 40787 40817 40848 40878 40909 40940 40969 41000 41030 41061 41091 41122 41153 41183 41214 41244 41275 41306 41334 41365 41395 41426 41456 41487 41518 41548 41579 41609 41640 41671 41699 41730 41760 41791 41821 41852 41883 41913 41944 41974 212 230 240 263 269 280 290 270 263 258 240 230 250 275 270 280 290 300 312 305 290 260 250 240 250 250 260 270 290 310 340 320 313 290 280 260 287 290 300 310 330 340 350 341 330 320 300 290 200 210 220 230 253 270 280 250 230 220 190 190 China 40179 40210 40238 40269 40299 40330 40360 40391 40422 40452 40483 40513 40544 40575 40603 40634 40664 40695 40725 40756 40787 40817 40848 40878 40909 40940 40969 41000 41030 41061 41091 41122 41153 41183 41214 41244 41275 41306 41334 41365 41395 41426 41456 41487 41518 41548 41579 41609 41640 41671 41699 41730 41760 41791 41821 41852 41883 41913 41944 41974 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 10 12 20 22 20 24 20 31 30 37 32 33 35 50 63 68 70 82 80 90 100 102 110 114 111 121 123 120 130 136 134 132 137 130 139 131 World 40179 40210 40238 40269 40299 40330 40360 40391 40422 40452 40483 40513 40544 40575 40603 40634 40664 40695 40725 40756 40787 40817 40848 40878 40909 40940 40969 41000 41030 41061 41091 41122 41153 41183 41214 41244 41275 41306 41334 41365 41395 41426 41456 41487 41518 41548 41579 41609 41640 41671 41699 41730 41760 41791 41821 41852 41883 41913 41944 41974 1592 1711 1810 1867 1779 1740 1826 1695 1681 1663 1825 1720 1761 2035 2142 2340 2280 2271 2154 2146 2085 1970 1936 1850 2000 2324 2510 2672 2780 2813 2716 2581 2476 2317 2324 2080 2202 2540 2867 3348 3550 3432 3400 3261 3209 3132 3027 2777 2821 3209 3553 3820 4133 4476 4436 4256 4067 3890 3816 3717

The chart identifies the unit sales for PLE's tractor equipment. We can see that throughout the years with the World orange line shown in the graph increases total sales between the years of 2010 to 2014. The line is basically increase in a positive direction on this graph. And the increase in tractor sales increase in each region throughout the years as well. Overall there is a positive correlations between time and tractor unit sales over all of the country regions.

Q2

On-Time Delivery

On Time Delivery by Month

Number of deliveries 40179 40210 40238 40269 40299 40330 40360 40391 40422 40452 40483 40513 40544 40575 40603 40634 40664 40695 40725 40756 40787 40817 40848 40878 40909 40940 40969 41000 41030 41061 41091 41122 41153 41183 41214 41244 41275 41306 41334 41365 41395 41426 41456 41487 41518 41548 41579 41609 41640 41671 41699 41730 41760 41791 41821 41852 41883 41913 41944 41974 1086 1101 1116 1216 1183 1176 1198 1205 1223 1209 1198 1243 1220 1241 1237 1258 1262 1227 1243 1281 1272 1295 1298 1318 1281 1320 1352 1336 1291 1342 1352 1377 1385 1356 1362 1349 1386 1358 1371 1362 1350 1381 1392 1371 1402 1384 1399 1369 1401 1388 1395 1412 1403 1415 1426 1431 1445 1425 1413 1456 Number On Time 40179 40210 40238 40269 40299 40330 40360 40391 40422 40452 40483 40513 40544 40575 40603 40634 40664 40695 40725 40756 40787 40817 40848 40878 40909 40940 40969 41000 41030 41061 41091 41122 41153 41183 41214 41244 41275 41306 41334 41365 41395 41426 41456 41487 41518 41548 41579 41609 41640 41671 41699 41730 41760 41791 41821 41852 41883 41913 41944 41974 1069 1080 1089 1199 1168 1160 1181 1189 1210 1194 1180 1223 1201 1224 1217 1242 1246 1212 1227 1264 1254 1278 1281 1296 1264 1304 1334 1320 1276 1326 1337 1360 1368 1338 1346 1333 1371 1342 1356 1348 1338 1366 1378 1359 1387 1370 1377 1357 1390 1376 1385 1401 1392 1402 1415 1420 1426 1414 1403 1427 Percent 40179 40210 40238 40269 40299 40330 40360 40391 40422 40452 40483 40513 40544 40575 40603 40634 40664 40695 40725 40756 40787 40817 40848 40878 40909 40940 40969 41000 41030 41061 41091 41122 41153 41183 41214 41244 41275 41306 41334 41365 41395 41426 41456 41487 41518 41548 41579 41609 41640 41671 41699 41730 41760 41791 41821 41852 41883 41913 41944 41974 0.98434622467771637 0.98092643051771122 0.97580645161290325 0.98601973684210531 0.9873203719357565 0.98639455782312924 0.9858096828046744 0.98672199170124486 0.98937040065412918 0.98759305210918114 0.9849749582637729 0.98390989541432017 0.98442622950819669 0.98630136986301364 0.98383185125303152 0.9872813990461049 0.98732171156893822 0.98777506112469438 0.98712791633145613 0.98672911787665885 0.98584905660377353 0.98687258687258683 0.98690292758089371 0.98330804248861914 0.98672911787665885 0.98787878787878791 0.98668639053254437 0.9880239520958084 0.98838109992254064 0.98807749627421759 0.98890532544378695 0.98765432098765427 0.98772563176895312 0.98672566371681414 0.98825256975036713 0.98813936249073386 0.98917748917748916 0.98821796759941094 0.98905908096280093 0.98972099853157125 0.99111111111111116 0.98913830557566984 0.98994252873563215 0.99124726477024072 0.98930099857346643 0.98988439306358378 0.98427448177269483 0.99123447772096418 0.99214846538187007 0.99135446685878958 0.99283154121863804 0.99220963172804533 0.99215965787598004 0.99081272084805649 0.99228611500701258 0.99231306778476591 0.98685121107266438 0.99228070175438599 0.99292285916489742 0.98008241758241754

We decided to use a clustered column chart to represent the On-Time deliveries for PLE's unit deliveries. The darker backgorund makes it easier to see the difference in the deliveries and the ones that were delivered on time to the customer. For example, for the month of January of 2010, PLE's had a total of 1086 deliveries but out of that number, 98.4% when delivered on-time. This chart makes is easy to compare those deliveries.

Response Time

Response Time by Quarter and Year

Q1 2013 4.356805690747569 5.415645561640849 5.50147957886802 2.7866492627596018 5.5495684291032372 3.6535666521900567 8.0191382648423311 4.0045367922517467 3.3431904438999482 4.9159115332600773 3.5546503494857462 3.5231651208392578 1.2533953549223953 2.1813659868144897 4.3525112841394726 2.4588828336505686 2.0693403411656619 2.9026272313218215 2.5783995324105491 5.4993536350026258 2.4736523454863346 4.2446331617044049 1.8764321948197904 4.2502707783001821 5.0840524335741062 4.4030024509425854 1.6400465637503658 6.4004832592559975 3.6791089013946476 3.9198121311870637 4.1274743279587707 3.3353070575118 182 3.2786815763189225 3.2441311231537839 3.2535645158874105 5.199402282357914 5.281745886293356 4.3296535222340022 4.6425480076664822 2.6515938470198308 3.4188237959257095 3.9721818592966884 1.2641333041188774 6.1579749098542376 6.4025937417114616 1 3.6338166336805444 5.3400354017299829 3.7376013478366077 5.6347801245807201 Q2 2013 4.3325643203628719 4.7253575742855904 1.6261836647812742 4.205002231471008 6.8870843718526888 0.92273817092645904 5.2676703929377258 0.9 3.8496963027922901 5.0034296676371017 3.5156336692365584 5.1965592759428549 5.1282537227292782 5.2852813935955059 1 2.1758940859639551 4.554598807159346 2.1334770720626692 5.241364395557321 4.0773214535205629 4.0392099875374701 5.0861743587360255 7.6592344597214836 4.6470289347111251 0.9 2.0076011863478924 1.3415140968631021 8.0482562664896253 4.913553401207901 5.0573001756914895 3.2576159340591402 4.263339950126829 1.6992101776180788 2.2969732966215815 5.3534252841258425 2.3312703418254386 3.6666470790136372 4.7275287655123979 1.0453071339055895 2.6700355177366872 4.1573383426351942 0.9 3.5076733168592908 5.9505744942056484 2.0504684001265558 8.2124891817569736 2.5168079431081423 3.9860188720253062 2.5933316904469392 1.3390093484544194 Q3 2013 3.7146412572171541 2.5241054166387769 2.6896680131601172 3.4734687281586232 5.121887857355178 1 3.4443303369032221 6.0388986233435578 2.5292204148415478 2.3882014423422517 3.2575328580848875 4.6841771612223244 3.5920977600896733 1.0686919770948591 2.8610331858787688 4.4406181180663413 4.8667564036138362 6.7562134566530592 2.8361203070078047 1.2506345731951298 3.4268334778305145 2.9840077834948899 4.6549896572530276 2.658026692485437 4.9887814887613064 3.7590027707908304 3.1200700098695235 2.1182925186865034 4.3161646820651374 3.6110861904732885 4.020589817925357 2.6307855071779342 4.4749861038569367 4.1842934072762734 4.729422703646124 2.646999978721142 2.3632449077256026 3.6397843862930315 5.6180936147272593 0.9 6.4001208150573081 3.2102573234867307 3.5474379322538154 5.9302431103121496 5.5190132619161165 4.9623297448549426 4.8508693501632667 5.5698431018088019 4.817243512049318 3.1770789567660542 Q4 2013 4.4392094297145377 4.0731587306290749 5.112268023462093 3.4856877947313478 4.6882091838633642 6.3605414298799587 8.2577867134241387 1.9114045345340855 8.9296140787191689 6.8537110665638465 5.687837084318744 3.0470982993429061 5.9130352484353352 1 1.8187038323085289 3.7439606431726133 6.1054524950159248 4.7754579200991429 4.1273587031391799 7.174651283188723 5.7005295376293361 1 3.3979271266653086 2.0414006586215692 4.3706494453581399 2.4660232712485595 3.2023929280549055 5.833204123613541 3.9361662048613653 2.4685073286527768 3.8865800989733543 6.875510290323291 1.7119800860236865 6.3871489247540012 6.5707099666760769 4.1814614734030329 8.8249639803543687 3.3480947750867927 5.499761538070743 6.5071526579267811 0.9 2.8718966505985009 7.4505069379520137 3.4878651250473922 3.0321399536696845 7.4588620110298507 4.844769601826556 2.8833146744582336 0.95167707614018582 3.0501850106738857 Q1 2014 2.7456040207704064 3.2393556203765912 4.3539226190710902 5.5837254386511628 2.894123937135737 5.0948083718190897 2.3263553849625169 1.6863519214035478 3.8792584710841767 3.3915317054430489 5.1440984371816736 0.98274408274446623 2.3405503235204379 2.8036798049521168 3.0573333298030776 2.4015251220640494 1.5885425874381327 3.0502597347600386 1.5024861987563782 5.5816790755721737 3.1106598463389674 1.0826270646299236 3.6316638862495894 1.8572607551555849 1.8951628099835944 6.0711554816458371 1 1 1.1885672812291888 3.7861455403850415 5.8584701456362378 0.9 2.2395776532954188 0.9 3.8749611086182996 2.464285372394079 3.8408806368403021 2.429744468923309 1.5390717600035715 0.9 3.6867980235052529 1.7277737207274186 3.5219481297695894 2.2330224702323904 5.3514018382935316 5.1112406673433721 6.4554624678799879 5.6095641831285317 3.6320509899320315 3.8695416570641101 Q2 2014 3.4465603756718339 1.95467528909212 2.7691193817037858 1.830401933041867 3.7153588062967176 4.588204054819653 1.1652720867306927 1.4585909492627254 1.8973007253254766 2.954022155684652 4.6879442460369321 3.3438613708160121 3.5946013293898433 4.0304668881464751 2.3857898749003654 1.6263281476160047 2.3982745086716024 4.4406580935930835 4.9579172890691554 4.4146033441240435 3.3970261109818241 3.1488661615032472 4.8728326954762453 3.969714915804798 3.8509883405669827 2.8099522832082586 1.7614722390891986 5.5786442397977227 4.9162933545478156 2.6285494722134901 3.2720810930943118 2.8562667092803169 3.8348668648570312 1.7931613082357218 2.7003026924678126 3. 6135908966418357 0.9 3.3844030066422421 4.3807401278929321 2.872878402634524 2.1136076692375356 2.8578058016893921 3.1247515916067643 1.8599295880296269 2.4143211784423331 2.9756362972722856 0.9 1.0139794620801696 4.5589501577371268 5.6660748749738561 Q3 2014 1.6701319585336023 2.5849427136818122 3.4712812824436696 3.1168675112239725 1 5.3960551516211126 3.895330913408543 4.4883640915286378 2.0577209700859385 4.4860002011118922 3.5669281790687819 3.4085343334736535 3.3083657134084206 2.7882290472261957 2.0893796280033712 4.2785482113031321 4.4665714616057812 1.9354151921361336 3.8966397899712319 3.3183290004926675 2.1960299894344644 3.5221082233219931 2.3136046896324842 1 5.8955778361705597 1.0873686808990897 4.5958403309923597 3.5192415528654237 4.1415744438636466 4.1337970136082731 2.4295045553371892 2.3373820643682848 2.5318425476398261 4.1416370853112312 2.6456999724614434 3.211152780593693 3.85011697592563 2.202989783952944 4.573015765643504 2.9913637225290586 4.1850706869154237 3.0259632315646741 1.9018393762307824 2.0914913041706313 1.0339421199460048 2.9528837406614912 7.4192420318722725 3.7933836059237365 2.4752080851867504 2.7128647919453215 Q4 2014 2.5510757682699476 2.3031384176196297 1.0432483764365315 1.5865764185495208 3.1144282689187093 4.0469112450868128 3.3778203219757414 1.2557568157266359 0.9 2.3109832641697721 2.7098836613280581 1.6538044479151721 3.5820508815508219 2.9565219124837312 3.7752575695325503 2.8747584524811827 0.90147952555562361 4.8724379853869326 3.1082047103613148 0.9 3.5162579211377305 3.1823331897161551 0.9 1.3526853040733839 1.6183518896927125 1.8669454407703596 1.0325304361234884 2.31182863949507 1.9896637882542563 3.9689445844036526 1 3.5086081612011184 2.410366592403443 2.4695753796098869 4.0189783890586117 2.0281505344886681 3.6200026175269158 4.1219250038469912 1.4048089001793413 2.4852340362034737 2.6676015937031479 4.3273157376010207 1.9502917626145062 2.7026329421918489 1.758633944109897 2.6436946159723447 4.4879045349720403 1.6248547768103889 1.1000000000000001 4.4970204003679104

From the data in this line graph, on response time between quarters, we are able to determine that there is no correlation between response times and quarters from how the lines on the graph are random.

Part 2 - Shipping Cost

You can see in the table of quartiles with Mowers and Tactors in Existing and Proposed shipping cost locations that Mowers have a slight increase in shipping costs in the proposed then the existing. There is also an increase in shipping cost in Tactors in Proposed locations compared to Existing locations.

Fixed Cost

Part 3 - Regions and Averages

part 3

Average of Price China Eur NA Pac SA 3 3.9 3.71 4.0999999999999996 3.5 Average of Service China Eur NA Pac SA 2.6 3.8666666666666667 4.3099999999999996 4.3 4.24 Average of Ease of Use China Eur NA Pac SA 4.0999999999999996 4.333333333333333 4.2699999999999996 3.9 3.92 Average of Quality China Eur NA Pac SA 3.8 4.0999999999999996 4.5999999999999996 4.4000000000000004 4.28

Q1

Part 3 - 2014 Customer Survey

North America

1 Quality Ease of Use Price Service 1 2 5 0 2 Quality Ease of Use Price Service 0 2 10 3 3 Quality Ease of Use Price Service 3 6 19 8 4 Quality Ease of Use Price Service 30 47 41 44 5 Quality Ease of Use Price Service 66 43 25 45

South America

1 Quality Ease of Use Price Service 1 1 2 1 2 Quality Ease of Use Price Service 0 1 8 0 3 Quality Ease of Use Price Service 4 6 10 6 4 Quality Ease of Use Price Service 24 35 23 22 5 Quality Ease of Use Price Service 21 7 7 21

Europe

1 Quality Ease of Use Price Service 0 0 2 1 2 Quality Ease of Use Price Service 1 0 1 2 3 Quality Ease of Use Price Service 6 3 4 5 4 Quality Ease of Use Price Service 12 14 14 14 5 Quality Ease of Use Price Service 11 13 9 8

Pacific Rim

1 Quality Ease of Use Price Service 0 0 0 0 2 Quality Ease of Use Price Service 0 1 0 0 3 Quality Ease of Use Price Service 1 1 1 1 4 Quality Ease of Use Price Service 4 6 7 5 5 Quality Ease of Use Price Service 5 2 2 4

China

1 Quality Ease of Use Price Service 0 0 0 1 2 Quality Ease of Use Pric e Service 1 0 2 3 3 Quality Ease of Use Price Service 2 1 6 5 4 Quality Ease of Use Price Service 5 7 2 1 5 Quality Ease of Use Price Service 2 2 0 0

In this chart with the frequency distribution for North America, you can see that the quality, ease of use, and service production areas don't need to really change anything. Those areas can do the same thing they are doing. The price section in this chart needs improvment in their pricing, by the wide variation in the distribution, you can reduce costs or use different materials.

In this chart with the frequency distribution for South America, you can see that quality and service areas don't need to change anything they can keep on doing what they are doing. The ease of use can improve in turing all of those 4's into 5's for better ratings. Price again can change by reducing costs or changing materials to reduce the pricing.

In this chart with the frequency distribution shown in a historgram for Europe region, you can see all areas; quality, ease of use, price, and service all need improvments to get higher ratings from consumers. Price can reduce costs. Service can train their service workers to help customers better. Ease of use can improve the design of the product. Quality can improve on the procurment side to making better products.

In this chart with the frequency distribution shown in a histogram for Pacific Rim region, you can see most of the areas most rated number is 4's. So, service, price, and ease of use can improve a little bit to make some of those 4's into 5's. Quality can improve the overall quality in products from the procurment side.

In this chart showning the China regions distribution between areas and ratings. All areas need improvment to make the customers want to get these products again. Quality needs to improve the quality of the product by changing the procument side of things. Ease of use comes from that if the quality is good and making it easy to use will follow a little. We need to train or hire more people to help with the companies customer service so our customers have a good experience with our company. Overall everything is connected so if you focus on some areas the others will some what follow.

Unit Production Costs

Operating & Interest Expenses

Industry Mower Total Sales

Industry Tractor Total Sales

Q3

Defects After Delivery