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?