Diagnosing Mathematical Errors: Fractions and Decimals (Concepts, Equivalence, and Operations) College of Education Chapter 4 Ashlock (2010)

Today’s Topics… Review Chapter 3 Quiz Review of concepts and vocabulary Demonstrations Chapter 4 - Ashlock (2010) Diagnosing Errors: Group Work Correcting Errors: Whole Group Chapter 4 Ashlock (2010)

Review: Diagnosing and Correcting: Case Study Part II Diagnosing and Correcting Mathematical Errors involves four processes/steps: Diagnosing: collect data, analyze data, pre-diagnose, interview, and make final diagnosis Correcting: conceptual, intermediate, procedural only, and independent practice Evaluating: collect data, analyze data, diagnose, and determine effectiveness of correction strategy Reflecting: instructor will provide this upon your return to class Chapter 4 Ashlock (2010)

Vocabulary Procedural Errors – involve errors in skills and/or step-by-step procedures (algorithms) needed to solve mathematical problems Conceptual Errors – errors that are caused by the misunderstanding of mathematical ideas such as place value, meaning of operations, and number sense. Both Procedural and Conceptual Errors – errors that involve violations of an algorithm AND a misunderstanding of a mathematical idea. Chapter 4 Ashlock (2010)

Vocabulary Algorithm – “step-by-step procedures for accomplishing a task, such as solving a problem” (Ashlock, 2002, p. 256). “a predetermined set of instructions for solving a specific problem in a limited number of steps” (Webster, 1996, p. 34). Meaning of Operations - Addition – Joining two or more addends together resulting in a number that is larger than all addends. Subtraction – Separating a smaller quantity from a larger quantity resulting in a quantity that is smaller than the minuend. Multiplication – repeated addition of a specified number or quantity Division - the process of finding out the number of times a number (the divisor) is contained in another number ( the dividend) Place Value - The understanding that the place of a digit tells its value AND the understanding that a number can be combined and taken apart in different ways (i.e., regrouping). Chapter 4 Ashlock (2010)

Vocabulary 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 fractions, such as ¼, 1/3, and ½ (Ashlock, 2010). Properties of Operations Commutative Property – the order in which two numbers are added (or multiplied) does not change the result Associative Property – the order in which three numbers are added (or multiplied) does hot change the result Distributive Property – adding, or multiplying, in parts Zero Property – Addition and Subtraction – the result is the non-zero number we started with Multiplication – the result is zero Multiplicative Identity Property – any number multiplied by, or divided by, one remains unchanged Chapter 4 Ashlock (2010)

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. The numerator of a fraction tells how many of the fractional parts are under consideration. Chapter 4 Ashlock (2010)

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 Chapter 4 Ashlock (2010)

Decimal Concepts A decimal is a number with a decimal point in it. (6.72, 0.54, 0.019) The value of digits in a decimal is based on the our base-ten system or powers of 10. The digits to the left of the decimal point represent whole numbers whose values increase by powers of ten for each successive position. The digits to the right of the decimal point represent parts of a whole whose values decrease by powers of ten for each successive position. As with whole numbers, the place value positions to the right of the decimal have specific names and values. Chapter 4 Ashlock (2010)

Decimals Concepts Equivalent decimals describe the same amount by using a different number of zeros (.040 or .04 or .0400). Adding zeros to the right of the last digit in a decimal does not change the value of the number. A repeating decimal is one in which at some point in the notation, the sequence of digits repeats indefinitely. When comparing decimals, you progress from the largest place value to the smallest place value consider the digits in each position until a determination can be made. Chapter 4 Ashlock (2010)

Demonstrations Chapter 4 Ashlock (2010)

Diagnosing Errors Work with a group of your peers to reach a consensus about… The procedural error(s) Ask yourselves: What exactly is this student doing to get this problem wrong? Basic Facts Violations of Algorithm 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 Chapter 4 Ashlock (2010)

Gretchen Chapter 4 Ashlock (2010)

Our Conclusions: Chapter 4 Ashlock (2010)

Correction Strategies… Correctional Strategies for Fraction Concepts… See Ashlock’s (2010) text… Gretchen’s Correction Strategies See Van de Walle (2004) text. Fraction Concepts Understanding fractions build on an understanding of fair shares – See Van de Walle (2004), pages 243-244. See also Activity 15.1 and 15.2 (pp. 246-247). Models for fractions – See Van de Walle (2004), pages 244-246 Fraction circles, geoboard, graphing paper, pattern blocks, paper folding, fraction tiles, number lines, folded paper strips, and set models Chapter 4 Ashlock (2010)

Carlos Chapter 4 Ashlock (2010)

Our Conclusions: Chapter 4 Ashlock (2010)

Correction Strategies… Correctional Strategies for Fraction Concepts… See Ashlock’s (2006) text,… Carlos – pp. 143 -144. See Van de Walle (2004) text. Fraction Concepts Understanding fractions build on an understanding of fair shares – See Van de Walle (2004), pages 243-244. See also Activity 15.1 and 15.2 (pp. 246-247). Models for fractions – See Van de Walle (2004), pages 244-246 Fraction circles, geoboard, graphing paper, pattern blocks, paper folding, fraction tiles, number lines, folded paper strips, and set models Chapter 4 Ashlock (2010)

Jill Chapter 4 Ashlock (2010)

Our Conclusions: Chapter 4 Ashlock (2010)

Correction Strategies… Correctional Strategies for Equivalent Fractions See Ashlock’s (2010) text… 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 Chapter 4 Ashlock (2010)

Sue Chapter 4 Ashlock (2010)

Our Conclusions: Chapter 4 Ashlock (2010)

Correction Strategies… Correctional Strategies for Equivalent Fractions See Ashlock’s (2010) text,… 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 Chapter 4 Ashlock (2010)

Tonya Chapter 4 Ashlock (2010)

Our Conclusions: Chapter 4 Ashlock (2010)

Other Correction Strategies… Correctional Strategies for Equivalent Fractions See Ashlock’s (2010) text,… 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 Chapter 4 Ashlock (2010)

Further Information 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 Chapter 4 Ashlock (2010)

Questions… Chapter 4 Ashlock (2010)