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Fractions

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
- OperationsAdding and Subtracting
- OperationsMultiplying
- OperationsDividing

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 ,
- 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 introductionFormat 12.1
- Diagram introductionFormat 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

3 3 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 ruleWhen 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 12.10define 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

OperationsAdding 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

OperationsAdding 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

denominatorswhy?

OperationsAdding and Subtracting

- Fractions with unlike denominators
- What are the preskills?

OperationsAdding 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

OperationsAdding 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.

OperationsAdding 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

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 DecimalsReading 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 DecimalsReading Tenths and

Hundredths

- Format 13.1
- What example sets should be used for this format?
- What critical behavior must the teacher

emphasize?

Reading and Writing DecimalsWriting Tenths and

Hundredths

- Format 13.2
- Part A Teacher demonstrates how to write a

fraction as a decimal - Part B Less structured worksheet, given a

fraction, writing a decimal - What examples should we include?

Reading and Writing Mixed Decimals Tenths and

Hundredths

- What must the students be able to do (preskills)?
- Format 13.3
- Part A Teacher demonstrates reading mixed

numbers emphasizing and between whole number and

decimal - Part B Given a mixed fraction student writes

decimal and/or given words students write the

mixed decimal

Reading and Writing DecimalsRepresenting

Thousandths

- Same procedures as Formats 13.1 and 13.2
- What should the initial examples be like?
- After several lessons what examples should be

added?

Equivalent Decimals

- What are equivalent decimals?
- What are equivalent decimal skills needed for?
- Format 13.4
- Part A Rationale for adding zeros (.3 .30)
- Part B Structured board for rewriting decimals
- Part C Worksheet with column chart

Adding and Subtracting Decimals and Mixed Decimals

- Types of problems
- Both numbers have the same number of decimal

places - Numbers have different numbers of digits after

the decimal point

Adding and Subtracting Decimals and Mixed Decimals

- Both numbers have the same number of decimal

places - Teach students to align problems vertically
- Students write the decimal place directly below

other decimal points - Problems oriented horizontally are rewritten

vertically and worked as above

Adding and Subtracting Decimals and Mixed Decimals

- Numbers have different numbers of digits after

the decimal point - What does this require?

Adding and Subtracting Decimals and Mixed Decimals

- Numbers have different numbers of digits after

the decimal point - Format 13.5
- What examples should we include?

Rounding Off Decimals

- What is rounding off a preskill for?
- Format 13.6
- 1. Determine how many digits will be after the

decimal place - Count the number of digits and draw a line
- If the number after the line is 5 or more, round

up

Rounding Off Decimals

- What are the 3 example selection guidelines?
- What type of numbers are particularly hard to

round off?

Multiplying Decimals

- What is the tricky part of multiplying

decimals? - Format 13.7
- Part A Teaching students the rule for

determining where the decimal point goes in the

answer - Part B Structured worksheet, students determine

where to put the decimal point in completed

multiplication problems - What examples could be confusing?

Dividing Decimals

- Complex skill
- What are the preskills? What must be emphasized

in the preskills?

Dividing Decimals

- 4 categories of problems, what are examples of
- No remainder no conversion
- Conversion required to solve without remainder
- A remainder that requires rounding
- Divisor is a decimal

Dividing Decimals

- No remainder no conversion
- Rule Decimal in answer goes directly above the

decimal in the number being divided - Teacher leads students
- What are the example selection guidelines?

Dividing Decimals

- Conversion required to solve without

remainderstudents must rewrite the dividend as

an equivalent decimal to avoid a remainder. - Tell students will work until no remainder
- Model
- What are the example selection guidelines?

Dividing Decimals

- A remainder that requires rounding
- Format 13.8
- Part A Students read the directions, determine

how many digits in the answer, work the problem

until they have that many digits, draw a line,

work one more digit so that they can round off - Special consideration is given to what type of

problems?

Dividing Decimals

- Divisor is a decimal
- What must be done to solve these problems?
- What is the preskill?

Dividing Decimals

- Divisor is a decimal
- Preskill, 13.9multiplying decimals by multiples

of 10 - Part A Rule about moving the decimal
- Part B Structured worksheet
- What types of examples should be included?

Dividing Decimals

- Divisor is a decimal
- Format 13.10
- Part A Teaches that you cant divide by a

decimal, then shows how to divide - Part B Structured worksheet
- What are the two example selection guidelines?

Converting Fractions and Decimals

- Why teach this?
- What are the two preskills?
- What is the rule for converting fractions into

decimals? - What examples should be included?

Diagnosis and Remediation

- Review Summary Box 13.1

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