1. Diagnosing Mathematical Errors: Fractions and Decimals: Addition and Subtraction Dr. Jill Drake College of Education

2. Today’s Topics… Quiz Vocabulary Review Chapter 5: Ashlock (2010) ◦ Diagnosing Errors: Group Work ◦ Correcting Errors: Whole Group Homework

3. Fraction Concepts See Van de Walle (2004), p. 242 ◦ 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. ◦ The numerator of a fraction tells how many of the fractional parts are under consideration.

4. 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). ◦ To create equivalent fractions with larger denominators, we multiply both the numerator and the denominator by a common whole number factor.  Question: Can we use smaller parts (larger denominators) to cover exactly what we have?  (Activity 15.17 – Van de Walle, p. 260). ◦ To create equivalent fractions in the simplest terms (lowest terms), we divide both the numerator and the denominator by a common whole number factor.  Question: What are the largest parts we can use to cover exactly what we have (Ashlock, 2006, p. 146)?  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 property that says that nay number multiplied by, or divided by, 1 remains unchanged (Van de Walle, 2004, p. 261).  ¾ x 1 = ¾ x 3/3 = 9/12

5. Where might a student error in learning fraction and decimal operations? Basic Facts (not known) Procedural ◦ Algorithm difficulties Conceptual ◦ Fraction/Decimal Concepts  Part-Whole Relationship  Equal Parts/Fair Shares  Place Value ◦ Equivalent Fractions/Decimals ◦ Meaning of Operations in general ◦ Meaning of Operations when fractions or decimals are involved ◦ Properties  Commutative Property  Associative Property  Zero Property  Multiplicative Identity Property ◦ Number Sense

6. Demonstrations

7. Diagnosing Errors Work with a group of your peers to reach a consensus about… ◦ Error Type: Conceptual, Procedural or Both? ◦ The procedural error(s)  Ask yourselves: What exactly is this student doing to get this problem wrong? ◦ The conceptual error(s)  Ask yourselves: What mathematical misunderstandings might cause a student to make this procedural error?  Fraction Concepts  Part-Whole Relationship  Equal Parts/Fair Shares  Number Sense

9. Robbie’s Case (A-F-1) Describe Robbie’s error pattern. 1. Procedural Error: Robbie adds the two numerators as the new numerator. Robbie adds the two denominators as the new denominator. 2. Conceptual Error Robbie may not understand the algorithm of fraction addition.

10. Fraction Addition: Correction strategies for Robbie: 1. Conceptual Strategy Estimation: Estimate answers before computing by using benchmark numbers such as ½ and 1. To get students understand the algorithm of fraction addition, consider a simper task such as 5/8 + 2/8 where no fraction needs to be changed. Let’s students to explain why 5/8 + 2/8 make 7/8 and relate the reason the algorithm. To get students to move to common denominator, consider a simpler task such as 3/8 + 2/4 where only one fraction needs to be changed. Let’s students to explain why 5/8 + 2/4 make one whole and there is 1/8 extra. Relate the reason to the need of common denominator for unlike fractions.

11. Fraction Addition: Correction strategies for Robbie 2. Intermediate Strategy Use papers or pictures to represent each addend as fractional parts of a unit region and then divide these fractional parts into the same size (same denominator) of parts. Relate the step-by-step representation procedure to the written algorithm. 3. Procedural Strategy Find the common denominator for addends. Change each addend into an equivalent fraction which has this common denominator. Add the numerators as the numerator for the sum.

12. Diagnosing Errors Work with a group of your peers to reach a consensus about… ◦ Error Type: Conceptual, Procedural or Both? ◦ The procedural error(s)  Ask yourselves: What exactly is this student doing to get this problem wrong? ◦ The conceptual error(s)  Ask yourselves: What mathematical misunderstandings might cause a student to make this procedural error?  Fraction Concepts  Part-Whole Relationship  Equal Parts/Fair Shares  Number Sense

13. Dave

14. Dave’s Case (A-F-2) Describe Dave’s error pattern. 1.Procedural Error: 2.Conceptual Error

15. Fraction Addition: Correction strategies for Dave: 1. Conceptual Strategy Estimation: Estimate answers before computing by using benchmark numbers such as ½ and 1. To get students understand the algorithm of fraction addition, consider a simper task such as 5/8 + 2/8 where no fraction needs to be changed. Let’s students to explain why 5/8 + 2/8 make 7/8 and relate the reason the algorithm. To get students to move to common denominator, consider a simpler task such as 3/8 + 2/4 where only one fraction needs to be changed. Let’s students to explain why 5/8 + 2/4 make one whole and there is 1/8 extra. Relate the reason to the need of common denominator for unlike fractions.

16. Fraction Addition: Correction strategies for Dave Intermediate Strategy Use papers or pictures to represent each addend as fractional parts of a unit region and then divide these fractional parts into the same size (same denominator) of parts. Relate the step-by-step representation procedure to the written algorithm. 3. Procedural Strategy Find the common denominator for addends. Change each addend into an equivalent fraction which has this common denominator. Add the numerators as the numerator for the sum.

17. Diagnosing Errors Work with a group of your peers to reach a consensus about… ◦ Error Type: Conceptual, Procedural or Both? ◦ The procedural error(s)  Ask yourselves: What exactly is this student doing to get this problem wrong? ◦ The conceptual error(s)  Ask yourselves: What mathematical misunderstandings might cause a student to make this procedural error?  Fraction Concepts  Part-Whole Relationship  Equal Parts/Fair Shares  Number Sense

18. Robin

19. Robin’s Case (A-F-3) Describe Robin’s error pattern. 1.Procedural Error: 2.Conceptual Error

20. Fraction Addition: Correction strategies for Robin: 1. Conceptual Strategy Estimation: Estimate answers before computing by using benchmark numbers such as ½ and 1. To get students understand the algorithm of fraction addition, consider a simper task such as 5/8 + 2/8 where no fraction needs to be changed. Let’s students to explain why 5/8 + 2/8 make 7/8 and relate the reason the algorithm. To get students to move to common denominator, consider a simpler task such as 3/8 + 2/4 where only one fraction needs to be changed. Let’s students to explain why 5/8 + 2/4 make one whole and there is 1/8 extra. Relate the reason to the need of common denominator for unlike fractions.

21. Fraction Addition: Correction strategies for Robin Intermediate Strategy Use papers or pictures to represent each addend as fractional parts of a unit region and then divide these fractional parts into the same size (same denominator) of parts. Relate the step-by-step representation procedure to the written algorithm. 3. Procedural Strategy Find the common denominator for addends. Change each addend into an equivalent fraction which has this common denominator. Add the numerators as the numerator for the sum.

22. Diagnosing Errors Work with a group of your peers to reach a consensus about… ◦ Error Type: Conceptual, Procedural or Both? ◦ The procedural error(s)  Ask yourselves: What exactly is this student doing to get this problem wrong? ◦ The conceptual error(s)  Ask yourselves: What mathematical misunderstandings might cause a student to make this procedural error?  Fraction Concepts  Part-Whole Relationship  Equal Parts/Fair Shares  Number Sense

23. Andrew

24. Andrew’s Case (S-F-1) Describe Andrew’s error pattern. 1.Procedural Error: 2.Conceptual Error:

25. Fraction Subtraction: Correction strategies for Andrew  Conceptual  Intermediate  Procedural

26. Correction Strategies… Correctional Strategies for Subtraction of Fractions ◦ See Ashlock’s (2010) text,…  Andrew’s Correction Strategy pages 82. ◦ See Van de Walle’s (2004) activities…  Activity 15.4: Mixed-Number Names (p. 249)  See also pages 257 – 260  Activity 15.13: Different Fillers  Activity 15.14: Dot Paper Equivalencies  Activity 15.15: Group the Counters, Find the Names  Activity 15.16: Missing-Number Equivalencies  Activity 15.17: Slicing Squares

27. Diagnosing Errors Work with a group of your peers to reach a consensus about… ◦ Error Type: Conceptual, Procedural or Both? ◦ The procedural error(s)  Ask yourselves: What exactly is this student doing to get this problem wrong? ◦ The conceptual error(s)  Ask yourselves: What mathematical misunderstandings might cause a student to make this procedural error?  Fraction Concepts  Part-Whole Relationship  Equal Parts/Fair Shares  Number Sense

28. Chuck

29. Chuck’s Case (S-F-2) Describe Chuck’s error pattern. 1. Procedural Error: Chuck records the difference between the two denominators as the new denominator. 2. Conceptual Error Chuck may not understand the algorithm of fraction subtraction.

30. Fraction Subtraction: Correction strategies for Chuck  Conceptual  Intermediate  Procedural

31. Correction Strategies… Correctional Strategies for Subtraction of Fractions ◦ See Ashlock’s (2010) text,…  Chuck’s Correction Strategy pages 83. ◦ See Van de Walle’s (2004) activities…  Activity 15.4: Mixed-Number Names (p. 249)  See also pages 257 – 260  Activity 15.13: Different Fillers  Activity 15.14: Dot Paper Equivalencies  Activity 15.15: Group the Counters, Find the Names  Activity 15.16: Missing-Number Equivalencies  Activity 15.17: Slicing Squares

32. Case Study Questions

33. Non-Basic Facts Correcting ErrorsNon-Basic Facts Correcting Errors… Non-Basic Facts Correcting Errors Conceptual Only – using manipulatives only, emphasize the concepts being taught Teacher Guided Experiences Intermediate – identify the error; re-teach procedures for solving problem using the written symbols; use manipulatives (and/or drawings) to support the symbols (the operation and the answer). Teacher Guided Experiences Procedural Only – identify error (if not already done); re-teach procedures for solving problem using the written symbols; no use of manipulatives. Teacher Guided Experiences Independent Practice (procedural) – allow student to practice procedures away from teacher; once practice is completed, check and give student feedback and decide whether student needs more intermediate work, more procedural only work, or more independent practice. Student-only practice Teacher feedback

34. Questions… Have a blessed week!