Download Contents - Math Solutions

Contents
Foreword by Steve Leinwand
xxv
Acknowledgments
xxix
How to Use This Resource
xxxi
Introduction
xxxi
Reference Tables
Reproducibles
xxxvi
Video Clips by Chapter
xxxvii
Video Clips by Grade,
Including Demographics
xli
INTRODUCTION: Why Fractions, Decimals, and Percentages?
1
Fractions Are a Critical Foundation in Mathematics
2
Evidence of Our Struggles (the Data from Research)
2
Why Do Students Struggle with Fractions, Decimals, and
Percentages?
4
Mathematical Areas Linked to Students’
Difficulties with Fractions
4
Procedural Knowledge Linked to Students’
Difficulties with Fractions
7
Looking Ahead
Connecting the Chapter to Your Practice
SECTION I: Understanding Number Talks
CHAPTER 1: What Is a Number Talk?
Classroom Clip 1.1
16 × 2 : Why Number Talks?
8
Why Number Talks?
9
10
11
13
14
14
xi
Understanding Number Talks: Four Foundational
Principles
Establish a Safe Learning Community
20
Classroom Clip 1.2
4 × 1 : Establishing a Safe Learning Community
22
Use Purposeful Problems
23
Record with Purpose
26
Know When to Ask and When to Tell
28
2
Number Talks and the Standards for
Mathematical Practice
Classroom Clip 1.3
Placing 0.89 on the Number Line: Number Talks
and the Standards for Mathematical Practice
Looking Ahead
Connecting the Chapter to Your Practice
CHAPTER 2: How Do I Get Started with Number Talks?
Getting Started: Ten Essentials
xii
19
33
36
37
37
38
39
Essential 1: Organize Your Classroom Space to
Foster Student Engagement
40
Essential 2: Start with Problems That Give All
Students Access and Elicit Thinking from
Multiple Perspectives
41
Essential 3: Insist on Students Solving
Problems Mentally
41
Essential 4: Provide Appropriate Wait Time
42
Classroom Clip 2.1
Using Hand Signals and Wait Time
42
Essential 5: Accept, Respect, and Consider
All Answers
43
Essential 6: Encourage Student Communication
Throughout the Number Talk
44
Classroom Clip 2.2
1 1×2
2 5
: Using Turn and Talk with a Number Talk
45
Essential 7: Record Strategies to Make Student
Thinking Public
49
Essential 8: Be Prepared When the Sharing
of Strategies Doesn’t Go as Anticipated
52
Essential 9: Limit Your Number Talks to Five to
Fifteen Minutes
53
Essential 10: Be Patient with Yourself and Your
Students
53
Five Ways to Collect Evidence of Students’
Understanding
Ask Students to Indicate Which Strategies
Are Most Efficient for Them
Classroom Clip 2.3
1 7 : Thinking About Efficient Strategies
+
2 8
55
55
56
Save Recordings of the Problems Posed and
Corresponding Student Strategies
57
Hold Small-Group Number Talks Throughout
the Week
57
Create and Post Student-Generated Strategy
Charts
58
Give an Occasional Exit Problem at the End of the
Number Talk
59
Looking Ahead
Connecting the Chapter to Your Practice
59
60
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SECTION II: Number Talks to Help Students Build
Fractional Reasoning
CHAPTER 3: What Are the Big Ideas with Rational Numbers?
63
Big Idea 1: Fractions Are Distinct Numbers
64
Big Idea 2: A Fraction Is Part of a Whole
66
Big Idea 3: The Denominator Is a Divisor and Tells
How Many Equal Parts the Whole Is Divided Into
67
Big Idea 4: The Numerator Counts How Many Equal
Parts There Are in the Fraction
68
Big Idea 5: There Is an Infinite Number of Fractions
Between Any Two Given Fractions
68
Big Idea 6: Equivalence Means That a Fraction
Can Be Repartitioned and Renamed in an Infinite
Number of Ways
69
Looking Ahead
71
Connecting the Chapter to Your Practice
71
CHAPTER 4: Number Talks to Help Students Build Fractional
Reasoning
72
Number Talks That Use Story Problems to Build
Fractional Reasoning
73
Classroom Clip 4.1
What is 3 of 12? Using a Story Context
74
Number Talks That Use Area Models to Build
Fractional Reasoning
75
4
xiv
61
Category 1: Reasoning About Equal Parts of
a Fraction
76
Category 2: Reasoning About Equivalence
Among Fractions
82
Classroom Clip 4.2
The Fraction Kit: A Conversation About Equivalence
84
Number Talks That Use Set Models to Build Fractional
Reasoning
89
Number Talks That Use Linear Models to Build
Fractional Reasoning
93
Category 1: Using Integers as Benchmarks
94
Category 2: Using Integers and Fractions as
Benchmarks
96
Number Talks That Use Comparing and Ordering
to Build Fractional Reasoning
99
Category 1: Is the Fraction Closer to 0, 21 ,
or 1 Whole?
100
Category 2: Is the Fraction Closer to 1, 1 21 , or
2 Wholes?
102
Number Talks That Use Strategies for Comparing
and Ordering Fractions
104
Category 1: Using Proximity to 21
106
Classroom Clip 4.3
Comparing 24 and 21
107
Category 2: Using Distance from the Whole
108
Category 3: Using Common Numerators
109
Category 4: Using Proportional Reasoning
110
50
40
Looking Ahead
Connecting the Chapter to Your Practice
CHAPTER 5: Number Talks to Help Students Connect
Fractions, Decimals, and Percentages
111
111
113
The Importance of Focusing on Fractions Before
Decimals and Percentages
114
Number Talks That Connect Fractions to Percentages
114
Classroom Clip 5.1
: Connecting Fractions to Percentages
1 1
×
4 3
119
xv
Category 1: Using Benchmark Fractions to
Find Percentages
120
Category 2: Finding Percentages of
Whole Numbers
122
Category 3: Comparing and Ordering
Fractions and Percentages
124
Number Talks That Connect Fractions to Decimals
125
Classroom Clip 5.2
Placing 0.9, 0.13, 0.255 on the Number Line:
Connecting Fractions to Decimals
127
Category 1: Comparing and Ordering Fractions
and Decimals
128
Category 2: Comparing Decimals to Decimals
129
Number Talks That Connect Fractions, Decimals,
and Percentages
130
Looking Ahead
131
Connecting the Chapter to Your Practice
SECTION III: Number Talks to Help Students Operate
with Fractions and Decimals
CHAPTER 6: Number Talks for Addition with Fractions
How Number Talks Help the Development of
Addition Strategies with Fractions
Making Connections to Strategies for Adding
Whole Numbers
132
133
135
136
137
Classroom Clip 6.1
1
5
+ : Developing Addition Strategies
2 6
xvi
with Fractions
138
Making Connections to the Arithmetic
Properties
139
Using Story Problems
141
Using Models
142
Connecting Students’ Strategies
146
Number Talks That Use Area Models for Addition
with Fractions
147
Classroom Clip 6.2
Are the Addends Equal to 1? Using Area Models
with Addition of Fractions
149
Category 1: Using Area Models Partitioned into
Common Denominators
150
Category 2: Using Area Models Requiring
Partitioning
153
Number Talks That Use Set Models for Addition with
Fractions
157
Category 1: Using Set Models with Eight Dots
159
Category 2: Using Set Models with Twelve Dots
161
Number Talks That Use Linear Models for Addition
with Fractions
163
Category 1: Using a Number Line Partitioned
into Halves and Fourths
164
Category 2: Using a Number Line Partitioned
into Halves and Thirds
166
Number Talks That Highlight the Addition
Strategy “Decomposing to Make a Benchmark
Number”
168
Classroom Clip 6.3
1
1 1 3
+ , + : Developing Addition Strategies
2 2 2 4
with Fractions
169
Category 1: Using Halves as Benchmarks
170
Category 2: Using Wholes as Benchmarks
172
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Number Talks That Highlight the Addition Strategy
“Adjusting an Addend”
174
Looking Ahead
177
Connecting the Chapter to Your Practice
CHAPTER 7: Number Talks for Subtraction with
Fractions
How Number Talks Help the Development
of Subtraction Strategies with Fractions
178
179
180
Making Connections to Strategies for
Subtracting Whole Numbers
182
Using Story Problems
183
Using Models
184
Connecting Students’ Strategies
186
Classroom Clip 7.1
3 3
- : Developing Subtraction Strategies
4 8
with Fractions
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189
Number Talks That Use Area Models for Subtraction
with Fractions
190
Using Area Models Partitioned into Eighths
and Twelfths
192
Number Talks That Use Set Models for Subtraction
with Fractions
195
Number Talks That Use Linear Models for Subtraction
with Fractions
200
Category 1: Using a Number Line Partitioned into
Halves and Fourths
201
Category 2: Using a Number Line Partitioned
into Halves and Thirds
203
Number Talks That Highlight the Subtraction Strategy
“Adding Up”
205
Category 1: Using Like Denominators
207
Category 2: Using Unlike Denominators
209
Number Talks That Highlight the Subtraction Strategy
“Removing the Subtrahend in Parts”
211
Number Talks That Highlight the Subtraction Strategy
“Partial Differences”
214
Looking Ahead
217
Connecting the Chapter to Your Practice
CHAPTER 8: Number Talks for Multiplication with Fractions
How Number Talks Help the Development of
Multiplication Strategies with Fractions
218
219
220
Making Connections to Strategies for Multiplying
Whole Numbers
221
Making Connections to Arithmetic Properties
224
Highlighting the Shifting Whole
225
Classroom Clip 8.1
1 1
× : Developing Multiplication Strategies
4 3
with Fractions
227
Using Story Problems
227
Using Models
229
Classroom Clip 8.2
of 12, 41 of 12: Developing Multiplication Strategies
with Fractions
231
1
2
Connecting Students’ Strategies
232
Number Talks That Use Unit Fractions and Whole
Numbers with Multiplication
236
Number Talks That Use Fractions and Whole
Numbers with Multiplication
238
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Number Talks That Use Models to Focus on the
Shifting Whole
242
Number Talks That Use Unit Fractions with Multiplication 244
Number Talks That Highlight the Multiplication
Strategy “Partial Products”
Classroom Clip 8.3
1 31 × 43 : Developing Multiplication Strategies
with Fractions
247
Category 1: Multiplying a Fraction by a Whole
Number
248
Category 2: Multiplying a Mixed Number by
a Fraction
250
Category 3: Multiplying a Fraction by a Fraction
252
Number Talks That Highlight the Multiplication
Strategy “Switching Numerators or Denominators”
254
Number Talks That Highlight the Multiplication
Strategy “Decomposing Factors into Factors”
258
Category 1: Multiplying a Fraction by
a Whole Number
259
Category 2: Multiplying a Fraction by a Fraction
261
Number Talks That Highlight the Multiplication
Strategy “Doubling and Halving”
Classroom Clip 8.4
Doubling and Halving to Solve
1 2
×
4 3
263
265
Category 1: Multiplying Whole Numbers and
Fractions
266
Category 2: Multiplying a Fraction by
a Fraction
268
Looking Ahead
xx
246
Connecting the Chapter to Your Practice
270
271
CHAPTER 9: Number Talks for Division with Fractions
How Number Talks Help the Development of
Division Strategies with Fractions
273
274
Making Connections to Strategies for Dividing
Whole Numbers
275
Highlighting the Shifting Whole
278
Classroom Clip 9.1
1 ÷ 3 : Developing Division Strategies with Fractions 281
8
Using Story Problems
282
Using Models
284
Connecting Students’ Strategies
287
Number Talks That Use Whole Numbers and
Unit Fractions with Division
290
Category 1: Using Whole Number Divisors with
Mixed Number Quotients
291
Category 2: Using Whole Number Divisors with
Quotients Less Than 1
293
Category 3: Using a Unit Fraction as the Divisor
295
Number Talks That Highlight the Division Strategy
“Proportional Reasoning”
297
Classroom Clip 9.2
6 ÷ 3, 3 ÷ 1 1 , 1 1 ÷ 3 : Proportional
2
2
Reasoning String
4
Number Talks That Highlight the Division Strategy
“Common Denominators”
299
302
Category 1: Using the Same Divisors
303
Category 2: Using Equivalent Divisors
305
Number Talks That Highlight the Division Strategy
“Partial Quotients”
307
xxi
Number Talks That Highlight the Division Strategy
“Multiplying Up” 310
Looking Ahead
314
Connecting the Chapter to Your Practice
315
CHAPTER 10: Number Talks for Operating with Decimals
316
How Number Talks Help the Development of
Computation Strategies with Decimals
318
Making Connections to Strategies for Computing
with Whole Numbers
319
Making Connections to Arithmetic Properties 323
Using Story Problems
324
Using Models 325
Connecting Students’ Strategies
330
Section 1: Number Talks for Addition with
Decimals333
Number Talks That Highlight a Place to Begin with
Addition of Decimals
333
Number Talks That Highlight the Addition Strategy
“Adding by Place Value”
336
Category 1: Adding Tenths and Hundredths
337
Category 2: Adding Hundredths and
Thousandths339
Number Talks That Highlight the Addition Strategy
“Adding Up by Chunks”
Category 1: Adding Hundredths
341
342
Category 2: Adding Hundredths and
Thousandths344
xxii
Number Talks That Highlight the Addition Strategy
“Making Benchmark Numbers”
346
Category 1: Adding Hundredths
347
Category 2: Adding Mixed Decimals
348
Section 2: Number Talks for Subtraction with Decimals
349
Number Talks That Highlight the Subtraction Strategy
“Adding Up”
349
Category 1: Subtracting Hundredths
351
Category 2: Subtracting Multiple Place Values
353
Number Talks That Highlight the Subtraction Strategy
“Adjusting the Minuend or Subtrahend”
354
Category 1: Adjusting the Subtrahend 356
Category 2: Adjusting the Minuend
357
Number Talks That Highlight the Subtraction Strategy
“Constant Difference”
358
Category 1: Subtracting Hundredths
359
Category 2: Subtracting Mixed Decimals
361
Number Talks That Highlight the Subtraction Strategy
“Subtracting by Place Value”
362
Category 1: Using Place Values
364
Category 2: Using Negative Numbers
366
Section 3: Number Talks for Multiplication
with Decimals
368
Number Talks That Highlight a Place to Begin with
Multiplication of Decimals
368
Category 1: Ten Times Smaller
369
Category 2: Ten Times Larger
371
xxiii
Number Talks That Highlight the Multiplication
Strategy “Partial Products”
373
Category 1: Using Whole Numbers and Decimals
375
Category 2: Using Mixed Decimals
377
Number Talks That Highlight the Multiplication
Strategy “Breaking Factors into Factors”
379
Category 1: Using Tenths as a Multiplier
380
Category 2: Using Hundredths as a Multiplier
382
Number Talks That Highlight the Multiplication
Strategy “Doubling and Halving”
Using Number Talk Strings
384
385
Section 4: Number Talks for Division with Decimals
387
Number Talks That Highlight the Division Strategy
“Proportional Reasoning”
387
Category 1: Using Whole Numbers and Decimals 389
Category 2: Using Decimals
391
Number Talks That Highlight the Division Strategy
“Multiplying Up”
393
Number Talks That Highlight the Division Strategy
“Partial Quotients”
396
Looking Ahead
398
Connecting the Chapter to Your Practice 398
References399
To purchase the reproducible versions of the area, set, and linear
models in this resource, visit www.mathsolutions.com/
numbertalksfdp_reproducibles.
xxiv