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## Fractions

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**Proportional Reasoning**• Number Sense • Are 1.7 and 1/7 the same or are they different? • Are 0.5 and 6/12 the same or different? • Order the following numbers from largest to smallest: 0.48, 5/8, 14/13, 0.99. • What is 5 + ½ +0.5 = ? • Are there any fractions between 2/5 and 3/5? • Are there any decimals between 2/5 and 3/5? • Are there any decimals between 0.46 and 0.47? • Are there any fractions between 0.46 and 0.47?**Fractions Chapter**• Fractions Analysis • Rewriting Fractions • Operations—Adding and Subtracting • Operations—Multiplying • Operations—Dividing**Fractions Chapter**Terminology (page 241) Numerator Denominator Proper fraction Mixed number GCF (Greatest Common Factor) LCD (Least Common Denominator)**Fractions Chapter**Why are fractions so difficult for students?**Today**• Jigsaw activity • 4 groups • Fraction analysis • Rewriting fractions • Adding/fubtracting fractions • Multiplying/dividing fractions**Today**• Jigsaw cont. • 30 minutes to identify • Preskills • General teaching procedures • Diagnosis and remediation • Corrections • Example selection • Develop a 5 min. activity for the group**Today**• Present to group • 10 minutes to present • 5 minutes for activity**Fractions Analysis**• Instruction begins around second grade and includes: • Part/whole • Writing a faction for a diagram • Reading fractions • Determining if fraction is >, <, = to one whole • Reading mixed fractions**Fractions Analysis**• Important features: • Proper and improper introduced concurrently • Initial instruction is to interpret a fraction • Initially fraction instruction is figures divided into parts**Fractions AnalysisPart-Whole**• Number line introduction—Format 12.1 • Diagram introduction—Format 12.2**Fractions AnalysisWriting Fractions**• Format 12.3 • Part A: Students learn parts of fraction • B & C: Daily practice for several weeks • Model Format 12.3 • Example selection guidelines • Vary the number of parts in each whole, number of wholes, and number of parts shaded • Include proper and improper fractions**Fractions AnalysisDrawing Diagrams**Drawing diagrams to represent fractions • Teacher models how to divide circles into equal parts • Worksheet, dividing circles and shading parts used**Fractions AnalysisDecoding Fractions**Decoding fractions (traditional) Format 12. 4 Teacher models reading fractions and tests students**Fractions AnalysisMore, Less, Equal to One**Fractions that are more than one, less than one and equal to one Format 12. 5 Part A: Pictures Part B: Rules Part C: Structured Worksheet Model 12.5, B**Fractions AnalysisMixed Numbers**Reading and writing mixed numbers Format 12. 6 Part A: Picture demonstration Part B: Teacher models and tests reading mixed fractions Part C: Writing numbers**Rewriting Fractions**Identify missing numerator in an equivalent fraction What are/is • Equivalent fractions • Reducing fractions • Converting mixed fractions to improper and vice versa**Rewriting Fractions**What are the general preskills?**Rewriting FractionsEquivalent Fractions**What is the basic strategy? What are the specific preskills? 3 = 4 12 33 = 9 4 3 12 x**Rewriting FractionsEquivalent Fractions**Format 12.7 teaches the rule for factions equal to 1: When the top number is the same as the bottom number, the fraction equals 1.**Rewriting FractionsEquivalent Fractions**Format 12.8 teaches: • Part A shows the concept of equivalent fractions • Part B teaches the rule—When you multiply by a fraction that equals 1, the answer equals the number that you start with. • Part C is the structured presentation of the strategy**Rewriting FractionsEquivalent Fractions**What are the example selection guidelines for 12.8?**Rewriting FractionsReducing Fractions**Two stages: 1. Introducing greatest-common-factor (What does GCF mean?) • Reducing fractions when GCF is difficult to determine**Rewriting FractionsReducing Fractions**What are the preskills?**Rewriting FractionsReducing Fractions**What are the factors of: 15 12 9 36**Rewriting FractionsReducing Fractions**Greatest-common-factor Format 12.10—define GCF and lead students in finding**Rewriting FractionsReducing Fractions**Format 12.11 • Part A, teacher presents the strategy (model) • Part B, the structured worksheet • What are the example selection guidelines for Format 12.11?**Rewriting FractionsReducing Fractions**What strategy do we teach for reducing fractions with big numbers (those with difficult GCF)? 45 = (5) 9 = (3) 3 = 3 75 (5) 15 (3) 5 5**Rewriting Fractions: Converting Mixed Numbers and Improper**Fractions What is a mixed number? What is an improper fraction?**Rewriting Fractions: Converting Mixed Numbers and Improper**Fractions What is the procedure for converting an improper fraction to a mixed number? Format 12.12: Part A shows the concept in pictures Part B teaches the strategy Part C is a worksheet**Rewriting Fractions: Converting Mixed Numbers and Improper**Fractions Format 12.12: What are the example selection guidelines?**Rewriting Fractions: Converting Mixed Numbers and Improper**Fractions What is the procedure for converting mixed numbers to improper fractions? Format 12.13 Part A is converting a whole number Part B is converting a mixed number**Operations—Adding and Subtracting**Three basic problem types of addition/subtraction problems • With like denominators • With unlike denominators with easy lowest-common denominators • With unlike denominators and difficult lowest-common denominators**Operations—Adding and Subtracting**Like denominators: Format 12.14 teaches students (at this point) only to add or subtract fractions in which the whole has the same number of parts. Worksheets should include problems with unlike denominators—why?**Operations—Adding and Subtracting**Fractions with unlike denominators: What are the preskills?**Operations—Adding and Subtracting**Fractions with unlike denominators: Format 12.15 teaches students to find the least common multiple by skip counting for each denominator and selecting the smallest common number. 3 + 1 4 5**Operations—Adding and Subtracting**Adding and subtracting fractions with unlike denominators, Format 12.16: Part A, teacher demonstrates finding the LCM, multiplying both fractions by a fraction of 1, and then adding. Part B and C are worksheets.**Operations—Adding and Subtracting**Adding and subtracting fractions with unlike denominators, Format 12.16: What are the example selection guidelines?**Operations--Multiplication**Three problem types: 1. Multiplying proper fractions 2. Multiplying a fraction and a whole number 3. Multiplying one or more mixed numbers**Operations--Multiplication**Multiplying proper fractions: Students are taught the simple rule: Work the top times the top and the bottom times the bottom. Include multiplying proper fractions with addition and subtraction of fractions on worksheets.**Operations--Multiplication**Multiplying fractions and whole numbers: Format 12.18: Part A teaches changing a whole number to a fraction: 5 = 5/1 Part B and C worksheets, students change whole number to a fraction, multiply, and covert the answer to a mixed number.**Operations--Dividing**Model of the strategy that illustrates the rationale as well as the procedures (not included in the book)**Operations--Dividing**Preskills: Identity element: 1 x a = a Fraction of one has same number top and bottom: a/a Any number divided by 1 equals that number Multiplying fractions Reciprocals: a x b = ab = 1 b a ab**Diagnosis and Remediation**What are the following patterns of errors? What are examples? (Page 268) • Computational • Component-skill • Strategy**Diagnosis and Remediation**Expand the remediation for Summary Box 12.4 • What skills would you teach (often an isolated skill) • What types of problems would you include in your remediation practice problems? • What types of problems would you include in the final problem set?**Decimals**7 areas: • Reading and writing decimals and mixed decimals • Converting decimals to equivalent decimals • Adding and subtracting decimals • Multiplying decimals • Rounding off decimals • Dividing decimals • Converting between decimals and fractions**Decimals**• What are the preskills for all decimal areas? • What decimals skills are preskills for all other decimal areas?**Reading and Writing Decimals:Reading Tenths and Hundredths**Format 13.1: • Structured board: Teaches the rule that one digit after the decimal tells about tenths and two digits after the decimal tell about hundredths • Structured work sheet: Given a decimal students identify the fraction and visa versa**Reading and Writing Decimals:Reading Tenths and Hundredths**Format 13.1: What example sets should be used for this format? What critical behavior must the teacher emphasize?