1. Rosalind Duplechain, PhD University of West Georgia College of Education Fractions: Part 3 Module 7 Diagnosing and Correcting Mathematical Errors: ECED 4251

2. Basic Structure of PPt Questions about Homework Sheets (slide 3) Questions about Homework Sheets Lecture (slides 4-12) Lecture How the D&C Process works with Fraction Concepts and OperationsD&C Process Fraction concepts, number sense, and equivalent fractions Fractionnumber senseequivalent fractions Application (slide 13-14) Application See textbook for error patterns associated with fraction concepts and operations. Homework - (See Course Calendar). Homework

3. QuestionsQuestions about Homework Sheets Homework Sheets #6 and #7.

4. Fraction Concepts Fractional parts are equal shares or equal-sized portions of a whole or unit. A unit can be an object or a collection of things. A unit is counted as 1. On a number line, the distance form 0 to 1 is the unit. The denominator of a fraction tells how many parts of that size are needed to make the whole. For example: thirds require three parts to make a whole. The denominator is the divisor. Larger denominators = smaller parts; Smaller denominators = larger parts The numerator of a fraction tells how many of the fractional parts are under consideration.

5. Number Sense… An intuition about numbers: their size AND how reasonable a quantity is once a number operation has occurred. Use estimation as a strategy for determining whether the answer is reasonable. “understand and represent commonly used fractions, such as ¼, 1/3, and ½ (Ashlock, 2006, p. 46) less than a 0.5 difference between the estimate and answer when operating with fractions (Van de Walle)

6. Equivalent Fractions… Two equivalent fractions are two ways of describing … the same amount by using different-sized fractional parts (Van de Walle, 2004, p. 242) the same point on a number line by using different fractions (Ashlock, 2010, p. 68) To create equivalent fractions with larger denominators, we MULTIPLY both the numerator and the denominator by a common whole number factor. Activity Question: Can we use smaller parts (larger denominators) to cover exactly what we have? To create equivalent fractions in the simplest terms (lowest terms), we DIVIDE both the numerator and the denominator by a common whole number factor. Activity Question: What are the largest parts (smaller denominators) we can use to cover exactly what we have (Ashlock, 2010, p. 65)? Simplest terms means that the numerator and denominator have no common whole number factors (Van de Walle, 2004, p. 261) “Reduce” is no longer used because it implies that we are making a fraction smaller when in fact we are only renaming the fraction, not changing its size (Van de Walle, 2004, p. 261) The concept of equivalent fractions is based upon the multiplicative identity property that says that any number multiplied by 1, or divided by 1, remains unchanged (Van de Walle, 2004, p. 261): ¾ x 1 = ¾ x 3/3 = 9/12 ¾ x 1 = ¾ x 5/5 = 15/20

7. The D&C Process Diagnose Correct Evaluate Reflect

8. Process #1: Diagnose…Diagnose… Basic Fact Errors and Algorithm Errors Collect Data Analyze Data for Errors Pre-diagnose Data Interview Student Final Diagnosis of Data

9. Process #2: Correct…Correct… Basic Facts Errors Teach meaning of operation Teach and practice number relationship strategies Work on automaticity Whole Number Operations/Algorithm Errors Conceptual Only Intermediate Procedural Only Independent Practice

10. Process #3: Evaluate…Evaluate… Did diagnosis and correction work? Collect Student’s Work Sample (post-test) Analyze Work Sample for Errors Diagnose Incorrect Responses Determine Effectiveness of Correction Strategy (based on post- test’s score)

11. Process #4: Reflect…Reflect… Is student’s error fixed? If yes, move on to another area of concern and begin diagnosing and correcting process. If no, return to steps in diagnosing and correcting process? Ask yourself: DIAGNOSING: Did I miss a step in the diagnosing process? Did I miss an error? CORRECTING: Did I miss a step in the correcting process? Did I rush my student by either… blending correction steps? not spending enough time on each step?

12. Final Notes Final Notes About Fraction Concepts and Equivalent Fractions Developing Number Sense Fractional-Parts Counting (Van de Walle, 2004, pp. 246-247) Activity 15.3 p. 248 Fraction Number Sense (Van de Walle, 2004, pp. 251-257 Activity 15.6: Zero, One-half, or One Activity 15.7: Close Fractions Activity 15.8: About How Much? Activity 15.9: Ordering Unit Fractions Activity 15.10: Choose, Explain, Test Activity 15.11: Line ‘Em Up

13. Application Let’s apply what we’ve learned about the D&C Process to violations of algorithms, and in particular to fractions. Dan Linda

14. Application: Guidance Using work samples, diagnose and plan correction. Refer to previous knowledge, textbook, and other resources as needed. Prepare to justify responses. Ask yourself: Which problems are wrong? What exactly is this student doing to get each problem wrong (i.e., skills and/or steps for solving problem)? Refer to diagnosing checklist for fractions (right side of this slide) What mathematical misunderstandings might cause a student to make this error? Refer to diagnosing checklist for fractions (right side of this slide) Diagnosing Checklist for Fractions The procedural error(s): Ask yourself: What exactly is this student doing to get this problem wrong? Basic Facts Violations of Algorithm The conceptual error(s): Ask yourself: What mathematical misunderstandings might cause a student to make this procedural error? Fraction Concepts Part-Whole Relationship Equal Parts/Fair Shares Equivalent Fractions Multiplicative Identity Property Meaning of Operations in general Meaning of Operations when Fractions are involved Properties Commutative Property Associative Property Zero Property Multiplicative Identity Property Number Sense

15. Homework See Course Calendar on CourseDen. Use Instructional Resource Item #28. Complete Homework Sheet #8. Read textbook chapter 6 (fractions only). Use overview sheet for Module 7 as checklist. Complete all items on this checklist.