A
- Accuracy
- in communication, 150, 226
- conceptual engagement and, 112–113
- in discussions, 110
- with factors, 228
- feedback and, 102
- for learners, 112, 124, 198
- observations for, 206
- terminology and, 170
- Activities
- for building norms, 19–23
- for coordinate planes, 109–113
- for data, 169–172, 180–184, 187–190
- discussions for, 29–30, 41–42, 62
- exploration for, 29
- for generalizing, 225–228, 232–236, 240–244
- launches for, 28–29, 40, 47–48
- for number line, 96–100, 103–105
- for objects, 71–75, 87–90
- for play, 40–41, 58
- for proportions, 145–147, 150–153, 158–161
- for symbols, 197–200, 206–209, 216–219
- for unit rates, 121–125, 129–132, 138–141
- for visualization, 143, 165
- Adding machine tape, 18
- Agendas
- for boxes, 85–86
- for coordinate planes, 107–108
- for data, 167–168, 178–179, 185–186
- for generalizing, 223–224, 230–231, 239–240
- for number line, 94–95, 101–102
- for objects, 60, 70–71
- for proportions, 144, 149, 156–157
- for shapes, 26, 38–39, 46–47
- for symbols, 195–196, 204–205, 215
- for unit rates, 118–119, 128, 136–137
- Algebra, 192–194, 196
- Analysis, 172
- Animals jump
- for data, 144–153
- task cards for, 154–155
- Annotated graphs, 150
- Areas, 26–37, 38–45
- Atlantic Hurricanes of 1977, 177
- Atlantic Hurricanes of 1987, 176
- Atlantic Hurricanes of 1997, 175
- Atlantic Hurricanes of 2007, 174
- Atlantic Hurricanes of 2017, 173
- Averages. See Data; Means; Medians; Modes
B
- Balance, 204–205, 208–214
- Base unit, 119–125
- Best deal, 128–135
- Big ideas
- data as, 164–166
- generalizing as, 221–222
- methods compared to, 9–10
- number line as, 92–93
- objects as, 57–59
- polygons as, 24–25
- proportions as, 142–143
- symbols as, 192–194
- unit rates as, 116–117
- visualization of, 216
- Body length, 152–153
- Boxes, 85–87, 89–90, 225
- Brains, 10–13, 166
- Building footprints, 48, 52–56
- Building norms, 19–23
- Burton, Leone, 8
C
- Calculations, 12–13
- Calculators, 18
- Calder, Alexander, 205
- Centimeter cubes, 17
- Centimeter grid paper, 247
- Challenges
- in counting, 99
- discussions of, 73
- in exploration, 97–98
- fractional units as, 183–184
- investigation and, 186
- in knowledge, 182–183
- for learners, 139, 141, 151–152
- motor challenges, 113
- observations and, 146
- for predictions, 102
- psychology of, 124, 146–147, 222
- strategies for, 208
- task cards for, 191
- in tasks, 21
- Charts
- for data, 151
- discussions and, 139, 217–218
- for economics, 133–135
- as evidence, 131
- graphing and, 148
- for groupwork, 242
- organization for, 99
- in play, 130
- for relationships, 217
- for teachers, 111
- Classes, 88
- Collaboration, 241
- colloquial terminology, 171
- Color coding, 6–7, 18, 181, 222, 225
- Common Core curriculum standards, 9–10
- Communication
- accuracy in, 150, 226
- of analysis, 172
- collaboration and, 241
- comparison in, 160
- of data, 88
- descriptions and, 170
- investigation and, 156
- language in, 63
- for learners, 199, 243–244
- materials and, 87
- of observations, 225
- predictions and, 72, 121–122
- in reflections, 200
- of relationships, 198
- of visualization, 123–124, 181
- Comparative language, 99–100
- Comparison
- in communication, 160
- for learners, 131–132
- observations from, 129–130, 160
- in reflections, 190
- of shapes, 171
- strategies for, 99–100
- for utility, 139
- Complex polygons, 38–45
- Comprehension, 89, 160–161, 199
- Conceptual engagement
- accuracy and, 112–113
- for algebra, 192–194
- color coding for, 222
- for Cuisenaire rods, 201–203
- for data, 142, 145
- decisions for, 129
- evidence for, 234, 238
- exploration for, 121
- graphing for, 223
- hurricanes for, 167–177
- for learners, 240
- lesson plans for, 19–23
- memorization compared to, 4–5
- movement words for, 109
- observations and, 123–124
- play for, 165
- puzzles for, 208
- radial patterns for, 220
- restaurant tables in, 157
- for symmetry, 96–97
- technology for, 121–122, 147, 150, 225, 232
- tools for, 137–138
- for variables, 198
- visualization in, 110
- Conceptualization, 24–25
- Confusion, 116
- Conjectures, 88–89, 225–226, 234, 240–241
- Context, 179
- Contrasts, 146–147
- Coordinate planes
- activities for, 109–113
- agendas for, 107–108
- in investigation, 93
- for teachers, 108–109
- templates for, 114–115
- Counterexamples, 234–236
- Counting, 99
- Creativity, 208, 217, 241–242
- Cubes
- dimensions of, 60
- in discussions, 181
- for learners, 180
- lesson plans for, 60–64
- in play, 182
- Cuisenaire rods, 17
- conceptual engagement for, 201–203
- for exploration, 197
- extensions for, 198
- relationships from, 195
- for visualization, 193–194
- writing and, 199
D
- Data
- activities for, 169–172, 180–184, 187–190
- agendas for, 167–168, 178–179, 185–186
- animals jump, 144–153
- as big ideas, 164–166
- charts for, 151
- communication of, 88
- conceptual engagement for, 142, 145
- data points, 172, 179
- data sets, 168, 170, 185, 189–190
- hurricane, 173–177
- in investigation, 117
- methods for, 140
- organization for, 90
- patterns for, 221–222, 226
- in play, 143
- rates and, 157
- for ratios, 136
- Speed, 136–141
- for teachers, 168, 179, 186
- templates for, 191
- unit data, 158–163
- values from, 122, 164–165
- Decisions, 129, 140, 151
- Descriptions, 170–171
- Design, 151–152
- Dimensions, 60
- Discovery, 157, 231–232
- Discussions
- accuracy in, 110
- for activities, 29–30, 41–42, 62
- of challenges, 73
- charts and, 139, 217–218
- conjectures in, 225–226
- counterexamples in, 234–235
- cubes in, 181
- factors in, 226
- for groupwork, 49, 97, 169–170
- knowledge in, 187–188
- launches for, 145
- for learners, 49, 72, 88, 109, 123, 170–171, 234
- for methods, 132
- of mistakes, 62
- of motion, 146
- objects in, 98–99
- observations in, 98, 242
- for partnerships, 73, 111, 151–152, 182
- of patterns, 159
- precision in, 197–198
- radial patterns in, 217
- ratios in, 121, 157, 160
- of relationships, 207–208
- for strategies, 207
- of visualization, 131
- Dot paper, 250–251
- Driscoll, Mark, 8
E
- Economics, 116, 119, 128–135, 222
- Education, 9–10, 57
- Einstein, Albert, 14
- Engagement. See Conceptual engagement
- Equality, 194
- Equivalence, 204, 208–209
- Evidence
- charts as, 131
- for conceptual engagement, 234, 238
- for learners, 73, 235–236
- in reflections, 236
- as strategies, 111–112
- visualization as, 132, 149
- Exploration
- for activities, 29
- challenges in, 97–98
- for conceptual engagement, 121
- creativity in, 217, 241–242
- Cuisenaire rods for, 197
- for data sets, 170
- for groupwork, 48, 130, 188
- journaling and, 98
- in launches, 232
- for learners, 30, 42, 49, 94, 129, 167
- methods of, 138–139
- networking for, 239
- of observations, 110
- for partnerships, 62, 87–88, 110–111, 123, 145–146, 158–159, 169, 180–181, 187, 198, 216–217, 225
- record keeping for, 122
- of shapes, 168
- of strategies, 233–234
- of terminology, 178
- Extensions
- creativity in, 208
- for Cuisenaire rods, 198
- for graphing, 88–89
- grid paper for, 227
- for growth patterns, 242–243
- investigation in, 111–112, 139–140, 188
- for learners, 62, 99, 104
- predictions and, 73, 124–125, 160
- for radial patterns, 216
- as reflections, 141
- “Fluency without Fear,” 2
- Folding
- nets, 71–75, 91
- around zero, 94–100
- Folding objects, 21–23, 57–59, 70, 90
- for lesson plans, 65–69, 76–84, 91
- technology for, 61–62
- Formulas, 179
- Fostering Algebraic Thinking (Driscoll), 8
- Fractional units, 86, 183–184, 196
- Frustration, 187
- Fundamentals, of mathematics, 221
G
- Games, 17, 93
- instructions for, 103–104, 232–233
- number line bounce, 101–106
- with patterns, 231–232
- pennies and paper clips, 230–238
- reflections for, 105
- in research, 2
- shopping, 128–135
- for strategies, 230
- See also Play
- Generalizing
- activities for, 225–228, 232–236, 240–244
- agendas for, 223–224, 230–231, 239–240
- as big ideas, 221–222
- for teachers, 224–225, 231–232, 240
- templates for, 229, 237–238, 245
- Geometry, 17, 72, 117, 160–161
- Graphing
- annotated graphs, 150
- charts and, 148
- for conceptual engagement, 223
- extensions for, 88–89
- groupwork for, 165
- instructions for, 145
- for learners, 143, 190
- lesson plans for, 150
- look‐fors for, 227–228
- observations from, 145–146
- tools for, 247–251
- Grid paper, 227, 247–249
- Groupwork
- charts for, 242
- discussions for, 49, 97, 169–170
- exploration for, 48, 130, 188
- for graphing, 165
- for learners, 19–20
- paper folding as, 21–23
- in play, 130
- record keeping from, 181
- Growth patterns, 192–194
- extensions for, 242–243
- radial patterns and, 224–225
- ratios and, 219
- representation of, 218
H
- Halving, 205
- High‐ceilings, 2–3, 14–15
- Humphreys, Cathy, 8
- Hurricanes, 167–177
I
- Ideas, 2–3, 5–6, 9–10, 21–22
- Imagination, 152
- Implicit connections, 105
- In the Balance (Kroner), 205, 210–214
- Innovation, 153
- Inspiration. See Motivation
- Instructions
- for games, 103–104, 232–233
- for graphing, 145
- for learners, 96
- materials and, 86–87
- for predictions, 61–62
- strategies for, 142
- for teachers, 27–28, 39, 47, 61, 71
- integers. See Number line
- Interpretation, 183–184
- Investigation
- challenges and, 186
- communication and, 156
- coordinate planes in, 93
- data in, 117
- of data sets, 185
- in extensions, 111–112, 139–140, 188
- growth patterns in, 194
- for learners, 14–15, 25, 58–59, 137–138, 235
- for measuring, 166
- in partnerships, 87–88
- patterns in, 143
- of predictions, 241–242
- puzzles for, 222
- for teachers, 108–109, 137
- visualization and, 193
- Irregular polygons, 26–37
- Isometric dot paper, 250
- Iterating units, 152
K
- Knowledge
- challenges in, 182–183
- counterexamples and, 236
- in discussions, 187–188
- implicit connections and, 105
- iterating units in, 152
- methods for, 7–8, 26
- processes for, 189
- quantitative literacy, 164
- in reflections, 147, 153
- science of, 5, 157
- toothpick squares for, 239–245
- Kroner, Lou, 205, 210–214
L
- Language
- in communication, 63
- comparative language, 99–100
- geometric language, 72, 117
- for interpretation, 183
- for learners, 179
- for measuring, 168
- for observations, 227
- visualization and, 74, 108–109
- vocabulary and, 71
- Launches
- for annotated graphs, 150
- for boxes, 87
- conjectures in, 240–241
- for discussions, 145
- exploration in, 232
- for mental strategies, 129–130
- for objects, 61–62, 71–72
- for observations, 109
- observations from, 169, 187
- for partnerships, 121–122, 138
- play and, 180
- for ratios, 216
- for relationships, 197
- for shapes, 28–29, 40, 47–48, 120–121
- for symmetry, 96–97
- technology for, 206
- for visualization, 103
- Learners
- accuracy for, 112, 124, 198
- challenges for, 139, 141, 151–152
- communication for, 199, 243–244
- comparison for, 131–132
- comprehension for, 160–161
- conceptual engagement for, 240
- conjectures for, 88–89
- contrasts for, 146–147
- cubes for, 180
- data points for, 172
- decisions for, 140
- discussions for, 49, 72, 88, 109, 123, 170–171, 234
- economics for, 119
- evidence for, 73, 235–236
- exploration for, 30, 42, 49, 94, 129, 167
- extensions for, 62, 99, 104
- feedback for, 5–6
- fractional units for, 86
- graphing for, 143, 190
- groupwork for, 19–20
- halving for, 205
- high‐ceilings for, 14–15
- imagination for, 152
- innovation for, 153
- instructions for, 96
- interpretation by, 183–184
- investigation for, 14–15, 25, 58–59, 137–138, 235
- journaling for, 6
- language for, 179
- line plots for, 224–229
- measuring devices for, 94–100
- methods for, 104–105, 189–190
- motor challenges for, 113
- patterns for, 108–109, 111, 217–218
- physical tasks for, 74–75
- pictures for, 5–6
- planning for, 140
- play for, 13–14, 24–25, 72
- predictions by, 74, 218
- psychology of, 189
- puzzles for, 210–214
- record keeping for, 88, 90
- reflections for, 125
- relationships for, 207–209
- representation by, 182–183
- sketches for, 159
- strategies for, 131
- symbols for, 9–11
- tasks for, 196
- technology for, 16
- tools for, 164–165
- variables for, 199–200
- visualization for, 10–13, 57–58, 63, 102, 118, 129–130, 221–222, 243
- Lesson plans
- for conceptual engagement, 19–23
- for cubes, 60–64
- folding objects for, 65–69, 76–84, 91
- for graphing, 150
- In the Balance (Kroner), 205, 210–214
- for math mobiles, 204–214
- for means, 178–184
- for medians, 185–191
- for modes, 185–191
- for polygons, 26, 38–39, 46–47
- reflections for, 64
- for shapes, 38–43
- for volume, 85–90
- Line plots, 224–229
- Lockhart, Paul, 8
- Logic, 89–90
- Look‐fors
- for body length, 152–153
- for boxes, 89–90
- for economics, 131–132
- for equivalence, 208–209
- for geometry, 160–161
- for graphing, 227–228
- for motion, 146–147
- for predictions, 74–75
- for radial patterns, 218–219
- for relationships, 199–200, 243–244
- for speed, 140–141
- strategies and, 104–105, 235–236
- for teachers, 31–32, 42–43, 63–64, 99–100, 112–113
- for values, 189–190
- in visualization, 49–50
- Low‐floor tasks, 2–3
M
- Manipulatives, 16–18
- Mason, John, 8
- Materials, 16–18, 86–87
- Math mobiles, 204–209
- Mathematics, 19–23
- algebra, 192–194
- convincing in, 5–9
- of design, 151–152
- Discovering the Art of Mathematics, 231–232
- “Fluency without Fear” for, 2
- fundamentals of, 221
- geometric language for, 72
- means, 164–166, 178–184
- medians, 164–166, 185–191
- modes, 164–166, 185–191
- objects in, 57–59
- open mathematics, 13–14
- predictions in, 63–64
- reason in, 5–9
- speed in, 136–141
- tables for, 122, 143
- thinking in, 5–9
- 2‐D concepts in, 107–115
- WIM, 4
- Youcubed for, 1
- Means, 164–166, 178–184
- Measuring
- devices, 17, 94–100
- investigation for, 166
- language for, 168
- tasks for, 164
- testing and, 189
- tools for, 138–139
- Medians, 164–166, 185–191
- Memorization, 2, 4–5
- Menon, Vinod, 10
- Mental strategies, 102, 129–130
- Methods
- big ideas compared to, 9–10
- for data, 140
- discussions for, 132
- for engagement, 13–14
- of exploration, 138–139
- for feedback, 61–62
- for knowledge, 7–8, 26
- for learners, 104–105, 189–190
- for relationships, 192–194
- for visualization, 58
- Mistakes, 14–15, 61–62
- Modes, 164–166, 185–191
- Motion, 146–147
- Motivation, 4, 208
- Motor challenges, 113
- Movement words, 109
N
- Nets
- filling, 85–90
- folding, 71–75
- templates, 76–84, 91
- Networking, 13, 239
- Number line bounce
- activities for, 96–100, 103–105
- agendas for, 94–95, 101–102
- as big ideas, 92–93
- coordinate planes and, 107–113
- for teachers, 95–96, 102
- templates for, 106, 229
- Number mystery line plot, 229
- Numbers
- number patterns, 223–229
- properties for, 227
- rounding of, 152–153
- visualization of, 243
- Numerical relationships, 116
O
- Objects
- activities for, 71–75, 87–90
- agendas for, 60, 70–71
- as big ideas, 57–59
- in discussions, 98–99
- launches for, 61–62, 71–72
- in mathematics, 57–59
- nets, 76–84
- templates for, 65–70
- See also Folding objects
- Observations
- for accuracy, 206
- challenges and, 146
- communication of, 225
- from comparison, 129–130, 160
- conceptual engagement and, 123–124
- in discussions, 98, 242
- exploration of, 110
- from graphing, 145–146
- language for, 227
- from launches, 169, 187
- launches for, 109
- patterns and, 112, 122, 188
- from play, 207
- predictions for, 158
- in reflections, 100, 228
- of symmetry, 99
- OECD. See Organisation for Economic Co‐operation and Development
- Office supplies, 18
- Open mathematics, 13–14
- Organisation for Economic Co‐operation and Development (OECD), 4
- Organization, 90, 99
P
- Paper clips and pennies (game), 230–238
- Paper folding, 21–23
- Parents, 12
- Partnerships
- discussions for, 73, 111, 151–152, 182
- exploration for, 62, 87–88, 110–111, 123, 145–146, 158–159, 169, 180–181, 187, 198, 216–217, 225
- investigation in, 87–88
- launches for, 121–122, 138
- play for, 73, 103, 150, 206, 232
- psychology of, 62, 72
- strategies in, 104
- technology for, 197
- Patterns
- blocks, 17
- for data, 221–222, 226
- discussions of, 159
- games with, 231–232
- in investigation, 143
- journaling for, 110
- for learners, 108–109, 111, 217–218
- number patterns, 223–229
- observations and, 112, 122, 188
- pattern blocks, 121
- pattern sheets, 217
- in reflections, 219
- symbols and, 243
- tables for, 124, 158–159
- for teachers, 86–87
- tiling patterns, 119–125
- visualization of, 239
- See also Growth patterns; Radial patterns
- Patty paper, 18
- Pennies and paper clips (game), 230–238
- Perimeters, 160–161
- Physical tasks, 74–75
- Pictures, 5–6
- PISA. See Program for International Student Assessment
- Planes. See Coordinate planes
- Planning, 140
- Play, 17, 93
- activities for, 40–41, 58
- charts in, 130
- for conceptual engagement, 165
- for conjectures, 234
- cubes in, 182
- data in, 143
- economics in, 116, 222
- equality in, 194
- geometric language in, 117
- groupwork in, 130
- launches and, 180
- for learners, 13–14, 24–25, 72
- observations from, 207
- for partnerships, 73, 103, 150, 206, 232
- Polygons
- as big ideas, 24–25
- complex polygons, 38–45
- conceptualization of, 24–25
- irregular polygons, 26–32
- lesson plans for, 26, 38–39, 46–47
- templates for, 33–37
- Precision, 99, 151, 197–198
- Predictions
- challenges for, 102
- communication and, 72, 121–122
- extensions and, 73, 124–125, 160
- folding objects and, 70
- instructions for, 61–62
- investigation of, 241–242
- by learners, 74, 218
- look‐fors for, 74–75
- in mathematics, 63–64
- for observations, 158
- reflections and, 113
- Prisms. See Polygons
- Processes, 189
- Program for International Student Assessment (PISA), 4
- Properties, for numbers, 227
- Proportions
- activities for, 145–147, 150–153, 158–161
- agendas for, 144, 149, 156–157
- as big ideas, 142–143
- for teachers, 145, 150, 157
- templates for, 148, 154–155, 162–163
- Psychology
- of calculations, 12–13
- of challenges, 124, 146–147, 222
- of confusion, 116
- of discovery, 157
- of failure, 57–58
- of frustration, 187
- of learners, 189
- of mistakes, 14–15
- of partnerships, 62, 72
- in reflections, 132
- of testing, 3–4
- Puzzles, 208, 210–214, 222
Q
- Quadrilateral in Question, 26, 28–29, 32
- Quantitative literacy, 164
- Questions, 88–89
R
- Radial patterns
- for conceptual engagement, 220
- in discussions, 217
- extensions for, 216
- growth patterns and, 224–225
- look‐fors for, 218–219
- relationships from, 215
- Rates, 138, 157
- Ratios
- Reason
- ideas and, 21–22
- in mathematics, 5–9
- reasoning, 142–143, 227
- spatial reasoning, 232–233
- Record keeping
- for exploration, 122
- from groupwork, 181
- for learners, 88, 90
- Reflections
- communication in, 200
- comparison in, 190
- evidence in, 236
- extensions as, 141
- for folding objects, 90
- for games, 105
- knowledge in, 147, 153
- for learners, 125
- for lesson plans, 64
- observations in, 100, 228
- patterns in, 219
- predictions and, 113
- psychology in, 132
- for shapes, 172
- sketches for, 161
- for solids, 75
- strategies in, 43, 209
- on terminology, 184
- for volume, 50
- Relationships
- charts for, 217
- communication of, 198
- from Cuisenaire rods, 195
- discussions of, 207–208
- launches for, 197
- for learners, 207–209
- look‐fors for, 199–200, 243–244
- methods for, 192–194
- numerical relationships, 116
- from radial patterns, 215
- Representation, 208–209
- of growth patterns, 218
- by learners, 182–183
- Research
- games in, 2
- from PISA, 4
- on symmetry, 92, 95–96
- for teachers, 1
- Restaurant tables, 157–163
- Rods. See Cuisenaire rods
- Rounding, of numbers, 152–153
S
- Science, 5, 10–11, 157
- Self‐monitoring, 4
- Shanghai skyline, 47–48, 51
- Shapes
- agendas for, 26, 38–39, 46–47
- color coding for, 181
- comparison of, 171
- exploration of, 168
- launches for, 28–29, 40, 47–48, 120–121
- lesson plan for, 38–43
- reflections for, 172
- templates for, 44–45, 52–56
- 3‐D shapes, 58
- Shards of a Shape, 38–45
- Shopping, 128–135
- Skepticism, 8, 21–23
- Sketches
- for comprehension, 199
- for learners, 159
- for reflections, 161
- Snap cubes, 17
- Solids, 46–56, 75
- Spatial reasoning, 232–233
- Speed, 136–141
- Square tiles, 17
- STEM fields, 57
- Strategies
- for balance, 208–209
- for challenges, 208
- color coding, 6–7
- for comparison, 99–100
- counterexamples, 234–235
- discussions for, 207
- evidence as, 111–112
- exploration of, 233–234
- games for, 230
- for instructions, 142
- for learners, 131
- look‐fors and, 104–105, 235–236
- mental strategies, 102
- in partnerships, 104
- for rates, 138
- in reflections, 43, 209
- self‐monitoring, 4
- for symmetry, 105
- for teachers, 14–15
- verbal explanations, 61
- Stress, 2
- Students. See Learners
- Summer camp, 3–4, 12, 24–25
- Symbols
- activities for, 197–200, 206–209, 216–219
- agendas for, 195–196, 204–205, 215
- as big ideas, 192–194
- for learners, 9–11
- patterns and, 243
- for teachers, 196, 205, 216
- templates for, 201–203, 210–214, 220
- Symmetry
- conceptual engagement for, 96–97
- launches for, 96–97
- observations of, 99
- research on, 92, 95–96
- strategies for, 105
T
- Tables
- Task cards, 150, 154–155, 191
- Tasks
- challenges in, 21
- comprehension of, 89
- ideas and, 2–3
- for learners, 196
- for measuring, 164
- motivation for, 208
- physical tasks, 74–75
- precision in, 99
- for testing, 2–3
- Teachers
- brains and, 166
- charts for, 111
- colloquial terminology for, 171
- coordinate planes for, 108–109
- data for, 168, 179, 186
- generalizing for, 224–225, 231–232, 240
- instructions for, 27–28, 39, 47, 61, 71
- investigation for, 108–109, 137
- look‐fors for, 31–32, 42–43, 63–64, 99–100, 112–113
- mistakes and, 61
- number line for, 95–96, 102
- patterns for, 86–87
- proportions for, 145, 150, 157
- questions for, 88–89
- research for, 1
- strategies for, 14–15
- symbols for, 196, 205, 216
- technology for, 3
- unit rates for, 119–121, 129, 137–138
- vocabulary for, 73
- Technology
- calculators, 18
- for conceptual engagement, 121–122, 147, 150, 225, 232
- for folding objects, 61–62
- for launches, 206
- for learners, 16
- for partnerships, 197
- for teachers, 3
- for visualization, 71–72, 92
- Templates
- for coordinate planes, 114–115
- for data, 191
- for dot paper, 250–251
- for generalizing, 229, 237–238, 245
- for grid paper, 247–249
- for nets, 76–84, 91
- for number line, 106, 229
- for objects, 65–70
- for polygons, 33–37
- for proportions, 148, 154–155, 162–163
- for quadrilaterals, 33
- for shapes, 44–45, 52–56
- for shopping, 133–135
- for symbols, 201–203, 210–214, 220
- tiling patterns, 119–127
- for unit data, 162–163
- for unit rates, 126–127, 133–135
- Terminology
- accuracy and, 170
- colloquial terminology, 171
- exploration of, 178
- reflections on, 184
- Testing
- measuring and, 189
- psychology of, 3–4
- tasks for, 2–3
- Theories, 8
- Thinking, 5–9, 179
- Thurston, William, 5
- Tiling patterns, 119–127
- Timers, 18
- Tools
- for conceptual engagement, 137–138
- for graphing, 247–251
- for learners, 164–165
- for measuring, 138–139
- Toothpick squares, 239–245
- Transformation, 171–172
U
- Unfolding objects. See Folding objects
- Unit data, 158–163
- Unit rates
- activities for, 121–125, 129–132, 138–141
- agendas for, 118–119, 128, 136–137
- base unit, 119–125
- as big ideas, 116–117
- for teachers, 119–121, 129, 137–138
- templates for, 126–127, 133–135
- Utility, 139
V
- Values
- from data, 122, 164–165
- in feedback, 182
- look‐fors for, 189–190
- tables for, 117
- Variables
- conceptual engagement for, 198
- for learners, 199–200
- in visualization, 209
- Verbal explanations, 61
- Visualization
- activities for, 143, 165
- of big ideas, 216
- communication of, 123–124, 181
- in conceptual engagement, 110
- Cuisenaire rods for, 193–194
- of data sets, 189–190
- discussions of, 131
- as evidence, 132, 149
- investigation and, 193
- language and, 74, 108–109
- launches for, 103
- for learners, 10–13, 24, 57–58, 63, 102, 118, 129–130, 221–222, 243
- look‐fors in, 49–50
- methods for, 58
- of numbers, 243
- for numerical relationships, 116
- of patterns, 239
- technology for, 71–72, 92
- transformation and, 171–172
- for 2‐D concepts, 117
- variables in, 209
- Vocabulary
- for descriptions, 171
- language and, 71
- for teachers, 73
- Volume
- lesson plans for, 85–90
- logic for, 89–90
- reflections for, 50
- of solids, 46–56