fractions. proportional reasoning number sense are 1.7 and 1/7 the same or are they different? ...

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 Operations—Adding and Subtracting Operations—Multiplying Operations—Dividing

Fractions Chapter

Terminology (page 241)NumeratorDenominatorProper fractionMixed numberGCF (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 OneFractions that are more than one, less

than one and equal to oneFormat 12. 5

Part A: PicturesPart B: Rules Part C: Structured Worksheet

Model 12.5, B

Fractions AnalysisMixed Numbers

Reading and writing mixed numbersFormat 12. 6

Part A: Picture demonstrationPart 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


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.


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


What are the example selection guidelines for 12.8?

Rewriting FractionsReducing Fractions

Two stages:1. Introducing greatest-common-factor

(What does GCF mean?)2. Reducing fractions when GCF is

difficult to determine


What are the preskills?


What are the factors of:1512936


Greatest-common-factorFormat 12.10—define GCF and lead students in finding


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?


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?

What is the procedure for converting an improper fraction to a mixed number?

Format 12.12:Part A shows the concept in picturesPart B teaches the strategyPart C is a worksheet


Format 12.12:What are the example selection guidelines?


What is the procedure for converting mixed numbers to improper fractions?

Format 12.13Part A is converting a whole numberPart B is converting a mixed number


Operations—Adding and SubtractingThree basic problem types of

addition/subtraction problems1. With like denominators2. With unlike denominators with easy

lowest-common denominators3. With unlike denominators and difficult

lowest-common denominators

Operations—Adding and SubtractingLike 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 SubtractingFractions with unlike denominators:

What are the preskills?

Operations—Adding and SubtractingFractions 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 + 14 5

Operations—Adding and SubtractingAdding 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 SubtractingAdding and subtracting fractions with

unlike denominators, Format 12.16:What are the example selection

guidelines?

Operations--Multiplication

Three problem types:1. Multiplying proper fractions2. Multiplying a fraction and a whole number3. Multiplying one or more mixed numbers


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.


Multiplying fractions and whole numbers:Format 12.18:

Part A teaches changing a whole number to a fraction: 5 = 5/1Part 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 = aFraction of one has same number top and bottom: a/aAny number divided by 1 equals that numberMultiplying fractionsReciprocals: 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


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:1. Reading and writing decimals and mixed

decimals2. Converting decimals to equivalent

decimals3. Adding and subtracting decimals4. Multiplying decimals5. Rounding off decimals6. Dividing decimals7. 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?

Reading and Writing Decimals:Writing Tenths and Hundredths

Format 13.2Part A: Teacher demonstrates how to

write a fraction as a decimal Part B: Less structured worksheet, given a

fraction, writing a decimalWhat 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 Decimals:Representing ThousandthsSame 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 DecimalsTypes of problems:1. Both numbers have the same number

of decimal places2. Numbers have different numbers of

digits after the decimal point

Adding and Subtracting Decimals and Mixed DecimalsBoth 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 DecimalsNumbers have different numbers of digits

after the decimal point:What does this require?

Adding and Subtracting Decimals and Mixed DecimalsNumbers have different numbers of digits

after the decimal point:Format 13.5What examples should we include?

Rounding Off Decimals

What is rounding off a preskill for?Format 13.61. Determine how many digits will be

after the decimal place2. Count the number of digits and draw a

line3. 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:

1. No remainder & no conversion2. Conversion required to solve without

remainder3. A remainder that requires rounding4. 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 remainder—students 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 decimalWhat must be done to solve these

problems?What is the preskill?

Dividing Decimals

Divisor is a decimal Preskill, 13.9—multiplying

decimals by multiples of 10 Part A: Rule about moving the

decimal Part B: Structured worksheetWhat types of examples should be

included?

Dividing Decimals

Divisor is a decimal Format 13.10

Part A: Teaches that you can’t divide by a decimal, then shows how to divide

Part B: Structured worksheetWhat are the two example selection

guidelines?

Converting Fractions and DecimalsWhy teach this?What are the two preskills?What is the rule for converting fractions

into decimals?What examples should be included?


Review Summary Box 13.1

fractions. proportional reasoning number sense are 1.7 and 1/7 the same or are they different? ...

Documents

fractions analysis instruction

activity slide

diagram reading fractions

group slide

common denominator slide

jigsaw activity

following numbers