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# Personalized Learning Bridges Middle School Math with a Geometric Approach

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• Right-handed user.
• Hold pencil 2-3cm (1 in.)from tip.
• Parallel lines.
• Triangle must hug the t-square.
• This doesn’t work!
• Perpendicular lines.
• Perpendicular lines.
• Still more parallel lines.
• Intersecting lines.
• Intersecting lines.
• Maintain a margin.
• Move the t-square down.
• Draw a side.
• Flip the triangle and repeat.
• Flip the triangle and repeat.
• A bit of erasing.
• The two halves are congruent.
• Divide it in half another way.
• Divide it in half another way.
• Divide it in half another way.
• Divide it in half another way.
• A bit of erasing.
• Flip the triangle and repeat.
• Can you figure out how to do it?
• Finding the center of the base line.
• Be sure the tick mark intersects the line.
• Flip the triangle over and repeat.
• How do you draw the last line?
• The smaller triangles are similar to the larger triangle.
• Name the rhombuses:
• Name the rhombuses: ABDF,
• Name the rhombuses. ABDF, BCDF,
• Name the rhombuses: ABDF, BCDF, BDEF.
• Name the trapezoids:
• Name the trapezoids: BCEF,
• Name the trapezoids: BCEF, ACDF,
• Name the trapezoids: BCEF, ACDF, ABDE.
• Cut it out….
• and fold on the lines.
• and fold on the lines.
• and fold on the dotted lines.
• and fold on the dotted lines.
• and fold on the dotted lines.
• How could you draw eighths?
• Not very esthetic.
• Can you divide it into sixths?
• Name an acute triangle:
• Name an acute triangle: ACE.
• Name a right triangle: ADE, ABE, ABG.
• Draw the base line.
• Move the t-square down.
• Draw the sides.
• Draw the diagonal…
• or just a tick mark.
• Draw the top.
• A bit of erasing.
• Outside square is twice the size of the inscribed square.
• No tracing or measuring.
• No tracing or measuring.
• No tracing or measuring.
• No tracing or measuring.
• No tracing or measuring.
• No tracing or measuring.
• No tracing or measuring.
• No tracing or measuring.
• No tracing or measuring.
• More examples to construct.
• ### Personalized Learning Bridges Middle School Math with a Geometric Approach

1. 1. Personalized Learning Bridges Middle-School Math with a Geometric Approach Presented by Tracy Mittleider, MSEd and Kathleen Lawler Based on work of Joan A. Cotter, Ph.D. Aplus+ Conference October 24, 2012© Activities for Learning, Inc., 2012
2. 2. Why a Geometric Approach?© Activities for Learning, Inc., 2012
3. 3. Why a Geometric Approach? Most students in “middle school” are visual learners.© Activities for Learning, Inc., 2012
4. 4. Why a Geometric Approach? Most students in “middle school” are visual learners. 90% of the math topics can be explored geometrically (visually).© Activities for Learning, Inc., 2012
5. 5. Why a Geometric Approach? Most students in “middle school” are visual learners. 90% of the math topics can be explored geometrically (visually). Therefore, it makes sense to teach them geometrically.© Activities for Learning, Inc., 2012
6. 6. Drawing Tools Paper 11 in. by 8.5 in.© Activities for Learning, Inc., 2012
7. 7. Drawing Tools Drawing board with paper© Activities for Learning, Inc., 2012
8. 8. Drawing Tools Drawing board with paper© Activities for Learning, Inc., 2012
9. 9. Drawing Tools T-square T-square© Activities for Learning, Inc., 2012
10. 10. Drawing Tools T-square T-square© Activities for Learning, Inc., 2012
11. 11. Drawing Tools T-square© Activities for Learning, Inc., 2012
12. 12. Drawing Tools T-square© Activities for Learning, Inc., 2012
13. 13. Drawing Tools T-square© Activities for Learning, Inc., 2012
14. 14. Drawing Tools T-square© Activities for Learning, Inc., 2012
15. 15. Drawing Tools T-square© Activities for Learning, Inc., 2012
16. 16. Drawing Tools T-square© Activities for Learning, Inc., 2012
17. 17. Drawing Tools T-square© Activities for Learning, Inc., 2012
18. 18. Drawing Tools T-square© Activities for Learning, Inc., 2012
19. 19. Drawing Tools T-square© Activities for Learning, Inc., 2012
20. 20. Drawing Tools Pencil (mechanical) and eraser© Activities for Learning, Inc., 2012
21. 21. Drawing Tools Pencil (mechanical) and eraser© Activities for Learning, Inc., 2012
22. 22. Drawing Tools Pencil (mechanical) and eraser© Activities for Learning, Inc., 2012
23. 23. Drawing Tools Pencil (mechanical) and eraser© Activities for Learning, Inc., 2012
24. 24. Drawing Tools Pencil (mechanical) and eraser© Activities for Learning, Inc., 2012
25. 25. Drawing Tools Pencil (mechanical) and eraser© Activities for Learning, Inc., 2012
26. 26. Drawing Tools The 45 triangle 45° 45° 90°© Activities for Learning, Inc., 2012
27. 27. Drawing Tools The 30-60 triangle (a set square) 30° 60° 90°© Activities for Learning, Inc., 2012
28. 28. Drawing Tools The 30-60 triangle© Activities for Learning, Inc., 2012
29. 29. Drawing Tools The 30-60 triangle© Activities for Learning, Inc., 2012
30. 30. Drawing Tools The 30-60 triangle© Activities for Learning, Inc., 2012
31. 31. Drawing Tools The 30-60 triangle© Activities for Learning, Inc., 2012
32. 32. Drawing Tools The 30-60 triangle© Activities for Learning, Inc., 2012
33. 33. Drawing Tools The 30-60 triangle© Activities for Learning, Inc., 2012
34. 34. Drawing Tools The 30-60 triangle© Activities for Learning, Inc., 2012
35. 35. Drawing Tools The 30-60 triangle© Activities for Learning, Inc., 2012
36. 36. Drawing Tools The 30-60 triangle© Activities for Learning, Inc., 2012
37. 37. Drawing Tools The 30-60 triangle© Activities for Learning, Inc., 2012
38. 38. Drawing Tools The 30-60 triangle© Activities for Learning, Inc., 2012
39. 39. Drawing Tools The 30-60 triangle© Activities for Learning, Inc., 2012
40. 40. © Activities for Learning, Inc., 2012
41. 41. Equilateral Triangles Draw the base line© Activities for Learning, Inc., 2012
42. 42. Equilateral Triangles Draw the base line© Activities for Learning, Inc., 2012
43. 43. Equilateral Triangles Draw the sides© Activities for Learning, Inc., 2012
44. 44. Equilateral Triangles Draw the sides© Activities for Learning, Inc., 2012
45. 45. Equilateral Triangles Draw the sides© Activities for Learning, Inc., 2012
46. 46. Equilateral Triangles Draw the sides© Activities for Learning, Inc., 2012
47. 47. Equilateral Triangles Divide it into halves© Activities for Learning, Inc., 2012
48. 48. Equilateral Triangles Divide it into halves© Activities for Learning, Inc., 2012
49. 49. Equilateral Triangles Divide it into halves© Activities for Learning, Inc., 2012
50. 50. Equilateral Triangles Divide it into halves 1 1 2 2© Activities for Learning, Inc., 2012
51. 51. Equilateral Triangles Divide it into halves© Activities for Learning, Inc., 2012
52. 52. Equilateral Triangles Divide it into halves© Activities for Learning, Inc., 2012
53. 53. Equilateral Triangles Divide it into halves© Activities for Learning, Inc., 2012
54. 54. Equilateral Triangles Divide it into halves© Activities for Learning, Inc., 2012
55. 55. Equilateral Triangles Divide it into halves© Activities for Learning, Inc., 2012
56. 56. © Activities for Learning, Inc., 2012
57. 57. Equilateral Triangles Divide it into thirds© Activities for Learning, Inc., 2012
58. 58. Equilateral Triangles Divide it into thirds© Activities for Learning, Inc., 2012
59. 59. Equilateral Triangles Divide it into thirds© Activities for Learning, Inc., 2012
60. 60. Equilateral Triangles Divide it into thirds© Activities for Learning, Inc., 2012
61. 61. Equilateral Triangles Divide it into thirds© Activities for Learning, Inc., 2012
62. 62. Equilateral Triangles Divide it into thirds© Activities for Learning, Inc., 2012
63. 63. Equilateral Triangles Divide it into thirds© Activities for Learning, Inc., 2012
64. 64. Equilateral Triangles Divide it into thirds 1 1 3 3 1 3© Activities for Learning, Inc., 2012
65. 65. Equilateral Triangles Divide it into thirds 1 3 1 1 3 3© Activities for Learning, Inc., 2012
66. 66. © Activities for Learning, Inc., 2012
67. 67. Equilateral Triangles Divide it into fourths© Activities for Learning, Inc., 2012
68. 68. Equilateral Triangles Divide it into fourths Find the center of the base line.© Activities for Learning, Inc., 2012
69. 69. Equilateral Triangles Divide it into fourths tick mark© Activities for Learning, Inc., 2012
70. 70. Equilateral Triangles Divide it into fourths© Activities for Learning, Inc., 2012
71. 71. Equilateral Triangles Divide it into fourths© Activities for Learning, Inc., 2012
72. 72. Equilateral Triangles Divide it into fourths© Activities for Learning, Inc., 2012
73. 73. Equilateral Triangles Divide it into fourths© Activities for Learning, Inc., 2012
74. 74. Equilateral Triangles Divide it into fourths© Activities for Learning, Inc., 2012
75. 75. Equilateral Triangles Divide it into fourths 1 4 1 1 4 1 4 4© Activities for Learning, Inc., 2012
76. 76. Equilateral Triangles Divide it into fourths A B F C E D Name the rhombuses:© Activities for Learning, Inc., 2012
77. 77. Equilateral Triangles Divide it into fourths A B F C E D Name the rhombuses: ABDF© Activities for Learning, Inc., 2012
78. 78. Equilateral Triangles Divide it into fourths A B F C E D Name the rhombuses: ABDF, BCDF© Activities for Learning, Inc., 2012
79. 79. Equilateral Triangles Divide it into fourths A B F C E D Name the rhombuses: ABDF, BCDF,© Activities for Learning, Inc., 2012 BDEF
80. 80. Equilateral Triangles Divide it into fourths A B F C E D Name the trapezoids:© Activities for Learning, Inc., 2012
81. 81. Equilateral Triangles Divide it into fourths A B F C E D Name the trapezoids: BCEF© Activities for Learning, Inc., 2012
82. 82. Equilateral Triangles Divide it into fourths A B F C E D Name the trapezoids: BCEF, ACDF© Activities for Learning, Inc., 2012
83. 83. Equilateral Triangles Divide it into fourths A B F C E D Name the trapezoids: BCEF, ACDF,© Activities for Learning, Inc., 2012 ABDE
84. 84. Equilateral Triangles What if we cut this shape out….© Activities for Learning, Inc., 2012
85. 85. Equilateral Triangles …and folded it on the lines….© Activities for Learning, Inc., 2012
86. 86. Equilateral Triangles© Activities for Learning, Inc., 2012
87. 87. Equilateral Triangles© Activities for Learning, Inc., 2012
88. 88. Equilateral Triangles© Activities for Learning, Inc., 2012
89. 89. Equilateral Triangles© Activities for Learning, Inc., 2012
90. 90. Equilateral Triangles© Activities for Learning, Inc., 2012
91. 91. Equilateral Triangles It’s a tetrahedron!© Activities for Learning, Inc., 2012
92. 92. Equilateral Triangles Divide it into eighths© Activities for Learning, Inc., 2012
93. 93. Equilateral Triangles Divide it into eighths© Activities for Learning, Inc., 2012
94. 94. Equilateral Triangles Divide it into eighths© Activities for Learning, Inc., 2012
95. 95. Equilateral Triangles Divide it into eighths© Activities for Learning, Inc., 2012
96. 96. Equilateral Triangles Divide it into eighths© Activities for Learning, Inc., 2012
97. 97. Equilateral Triangles Divide it into sixths© Activities for Learning, Inc., 2012
98. 98. Equilateral Triangles Divide it into sixths© Activities for Learning, Inc., 2012
99. 99. Equilateral Triangles Divide it into sixths© Activities for Learning, Inc., 2012
100. 100. Equilateral Triangles Divide it into sixths A B F G C E D Name an acute triangle:© Activities for Learning, Inc., 2012
101. 101. Equilateral Triangles Divide it into sixths A B F G C E D Name an acute triangle: ACE© Activities for Learning, Inc., 2012
102. 102. Equilateral Triangles Divide it into sixths A B F G C E D Name a right triangle:© Activities for Learning, Inc., 2012
103. 103. Equilateral Triangles Divide it into sixths A B F G C E D Name a right triangle: ADE© Activities for Learning, Inc., 2012
104. 104. Equilateral Triangles Divide it into sixths A B F G C E D Name a right triangle: ADE, ABE© Activities for Learning, Inc., 2012
105. 105. Equilateral Triangles Divide it into sixths A B F G C E D Name a right triangle: ADE, ABE, ABG© Activities for Learning, Inc., 2012
106. 106. Equilateral Triangles Divide it into sixths A B F G C E D Name an obtuse triangle:© Activities for Learning, Inc., 2012
107. 107. Equilateral Triangles Divide it into sixths A B F G C E D Name an obtuse triangle: GCE© Activities for Learning, Inc., 2012
108. 108. Equilateral Triangles Divide it into twelfths© Activities for Learning, Inc., 2012
109. 109. Equilateral Triangles Divide it into twelfths© Activities for Learning, Inc., 2012
110. 110. Equilateral Triangles Divide it into twelfths© Activities for Learning, Inc., 2012
111. 111. Equilateral Triangles Divide it into twelfths© Activities for Learning, Inc., 2012
112. 112. Equilateral Triangles Divide it into twelfths© Activities for Learning, Inc., 2012
113. 113. Equilateral Triangles Divide it into twelfths© Activities for Learning, Inc., 2012
114. 114. Equilateral Triangles Divide it into twelfths Can you see the cube?© Activities for Learning, Inc., 2012
115. 115. Equilateral Triangles Divide it into twelfths Can you see the cube?© Activities for Learning, Inc., 2012
116. 116. Equilateral Triangles Divide it into twelfths Do you see the hexagon?© Activities for Learning, Inc., 2012
117. 117. Equilateral Triangles Divide it into twelfths Do you see the hexagon?© Activities for Learning, Inc., 2012
118. 118. Equilateral Triangles Ninths© Activities for Learning, Inc., 2012
119. 119. Equilateral Triangles Twenty-sevenths© Activities for Learning, Inc., 2012
120. 120. Equilateral Triangles Twenty-sevenths© Activities for Learning, Inc., 2012
121. 121. Equilateral Triangles© Activities for Learning, Inc., 2012
122. 122. Equilateral Triangles© Activities for Learning, Inc., 2012
123. 123. Equilateral Triangles Thirty-seconds© Activities for Learning, Inc., 2012
124. 124. Equilateral Triangles Stephanie, 6, did 256!© Activities for Learning, Inc., 2012
125. 125. Equilateral Triangles© Activities for Learning, Inc., 2012
126. 126. Equilateral Triangles© Activities for Learning, Inc., 2012
127. 127. Equilateral Triangles© Activities for Learning, Inc., 2012
128. 128. Equilateral Triangles© Activities for Learning, Inc., 2012
129. 129. © Activities for Learning, Inc., 2012
130. 130. Equilateral Triangles Spiral© Activities for Learning, Inc., 2012
131. 131. Equilateral Triangles Spiral© Activities for Learning, Inc., 2012
132. 132. Equilateral Triangles Spiral© Activities for Learning, Inc., 2012
133. 133. Equilateral Triangles Spiral© Activities for Learning, Inc., 2012
134. 134. Equilateral Triangles Spiral© Activities for Learning, Inc., 2012
135. 135. Equilateral Triangles Spiral© Activities for Learning, Inc., 2012
136. 136. Equilateral Triangles Spiral© Activities for Learning, Inc., 2012
137. 137. Equilateral Triangles Spiral© Activities for Learning, Inc., 2012
138. 138. Equilateral Triangles Spiral© Activities for Learning, Inc., 2012
139. 139. Equilateral Triangles Spiral© Activities for Learning, Inc., 2012
140. 140. Equilateral Triangles Spiral© Activities for Learning, Inc., 2012
141. 141. Geometric Approach© Activities for Learning, Inc., 2012
142. 142. Geometric Approach • Basis of CAD (computer aided design).© Activities for Learning, Inc., 2012
143. 143. Geometric Approach • Basis of CAD (computer aided design). • Integrates many concepts.© Activities for Learning, Inc., 2012
144. 144. Geometric Approach • Basis of CAD (computer aided design). • Integrates many concepts. • Incorporates measuring.© Activities for Learning, Inc., 2012
145. 145. Geometric Approach • Basis of CAD (computer aided design). • Integrates many concepts. • Incorporates measuring. • Allows independent work.© Activities for Learning, Inc., 2012
146. 146. Geometric Approach • Basis of CAD (computer aided design). • Integrates many concepts. • Incorporates measuring. • Allows independent work. • Excellent preparation for high school mathematics.© Activities for Learning, Inc., 2012
147. 147. Geometric Approach Goals© Activities for Learning, Inc., 2012
148. 148. Geometric Approach Goals • To use elementary math.© Activities for Learning, Inc., 2012
149. 149. Geometric Approach Goals • To use elementary math. • To learn to read math texts.© Activities for Learning, Inc., 2012
150. 150. Geometric Approach Goals • To use elementary math. • To learn to read math texts. • To prepare for more advanced mathematics.© Activities for Learning, Inc., 2012
151. 151. Geometric Approach Goals • To use elementary math. • To learn to read math texts. • To prepare for more advanced mathematics. • To discover mathematics in the everyday world.© Activities for Learning, Inc., 2012
152. 152. Geometric Approach Goals • To use elementary math. • To learn to read math texts. • To prepare for more advanced mathematics. • To discover mathematics in the everyday world. • To enjoy mathematics.© Activities for Learning, Inc., 2012
153. 153. Why a Geometric Approach?© Activities for Learning, Inc., 2012
154. 154. Why a Geometric Approach? Most students in “middle school” are visual learners.© Activities for Learning, Inc., 2012
155. 155. Why a Geometric Approach? Most students in “middle school” are visual learners. 90% of the math topics can be explored geometrically (visually).© Activities for Learning, Inc., 2012
156. 156. Why a Geometric Approach? Most students in “middle school” are visual learners. 90% of the math topics can be explored geometrically (visually). Therefore, it makes sense to teach them geometrically.© Activities for Learning, Inc., 2012
157. 157. © Activities for Learning, Inc., 2012
158. 158. Squares The 45 triangle.© Activities for Learning, Inc., 2012
159. 159. Squares The 45 triangle is half of a square.© Activities for Learning, Inc., 2012
160. 160. Squares© Activities for Learning, Inc., 2012
161. 161. Squares© Activities for Learning, Inc., 2012
162. 162. Squares© Activities for Learning, Inc., 2012
163. 163. Squares© Activities for Learning, Inc., 2012
164. 164. Squares© Activities for Learning, Inc., 2012
165. 165. Squares© Activities for Learning, Inc., 2012
166. 166. Squares© Activities for Learning, Inc., 2012
167. 167. Squares© Activities for Learning, Inc., 2012
168. 168. Squares© Activities for Learning, Inc., 2012
169. 169. Squares Halves© Activities for Learning, Inc., 2012
170. 170. Squares Fourths© Activities for Learning, Inc., 2012
171. 171. Squares Eighths© Activities for Learning, Inc., 2012
172. 172. Squares Sixteenths© Activities for Learning, Inc., 2012
173. 173. © Activities for Learning, Inc., 2012
174. 174. Squares Halves We need to find the center.© Activities for Learning, Inc., 2012
175. 175. Squares Halves We need to find the center.© Activities for Learning, Inc., 2012
176. 176. Squares Halves We need to find the center.© Activities for Learning, Inc., 2012
177. 177. Squares Halves© Activities for Learning, Inc., 2012
178. 178. Squares Halves© Activities for Learning, Inc., 2012
179. 179. Squares Fourths© Activities for Learning, Inc., 2012
180. 180. Squares Fourths© Activities for Learning, Inc., 2012
181. 181. Squares Eighths Find the center of a small square.© Activities for Learning, Inc., 2012
182. 182. Squares Eighths Find the center of a small square.© Activities for Learning, Inc., 2012
183. 183. Squares Eighths Find the center of a small square.© Activities for Learning, Inc., 2012
184. 184. Squares Eighths Find the center of a small square.© Activities for Learning, Inc., 2012
185. 185. Squares Eighths© Activities for Learning, Inc., 2012
186. 186. Squares Eighths© Activities for Learning, Inc., 2012
187. 187. Squares Sixteenths© Activities for Learning, Inc., 2012
188. 188. Squares Sixteenths© Activities for Learning, Inc., 2012
189. 189. Squares Sixteenths© Activities for Learning, Inc., 2012
190. 190. © Activities for Learning, Inc., 2012
191. 191. Squares Circumscribed squares© Activities for Learning, Inc., 2012
192. 192. Squares Circumscribed squares© Activities for Learning, Inc., 2012
193. 193. Squares Circumscribed squares© Activities for Learning, Inc., 2012
194. 194. Squares Circumscribed squares© Activities for Learning, Inc., 2012
195. 195. Squares Circumscribed squares© Activities for Learning, Inc., 2012
196. 196. Squares Circumscribed squares© Activities for Learning, Inc., 2012
197. 197. Squares Circumscribed squares© Activities for Learning, Inc., 2012
198. 198. Squares Circumscribed squares© Activities for Learning, Inc., 2012
199. 199. Squares Circumscribed squares© Activities for Learning, Inc., 2012
200. 200. Squares Circumscribed squares© Activities for Learning, Inc., 2012
201. 201. Squares Circumscribed squares© Activities for Learning, Inc., 2012
202. 202. © Activities for Learning, Inc., 2012
203. 203. Squares Inscribed squares© Activities for Learning, Inc., 2012
204. 204. Squares Inscribed squares© Activities for Learning, Inc., 2012
205. 205. Squares Inscribed squares© Activities for Learning, Inc., 2012
206. 206. Squares Inscribed squares Compare the area of the squares.© Activities for Learning, Inc., 2012
207. 207. Squares Inscribed squares Compare the area of the squares.© Activities for Learning, Inc., 2012
208. 208. Squares Inscribed squares© Activities for Learning, Inc., 2012
209. 209. Squares Inscribed squares© Activities for Learning, Inc., 2012
210. 210. Squares Inscribed squares© Activities for Learning, Inc., 2012
211. 211. Squares Inscribed squares© Activities for Learning, Inc., 2012
212. 212. Squares Inscribed squares© Activities for Learning, Inc., 2012
213. 213. Squares Inscribed squares© Activities for Learning, Inc., 2012
214. 214. Squares Inscribed squares© Activities for Learning, Inc., 2012
215. 215. Squares Inscribed squares© Activities for Learning, Inc., 2012
216. 216. Squares Right triangle spiral© Activities for Learning, Inc., 2012
217. 217. Squares Right triangle spiral© Activities for Learning, Inc., 2012
218. 218. Squares Right triangle spiral© Activities for Learning, Inc., 2012
219. 219. Squares Right triangle spiral© Activities for Learning, Inc., 2012
220. 220. Squares Right triangle spiral© Activities for Learning, Inc., 2012
221. 221. Squares Right triangle spiral© Activities for Learning, Inc., 2012
222. 222. Squares Right triangle spiral© Activities for Learning, Inc., 2012
223. 223. Squares Right triangle spiral© Activities for Learning, Inc., 2012
224. 224. Squares Right triangle spiral© Activities for Learning, Inc., 2012
225. 225. Squares Right triangle spiral© Activities for Learning, Inc., 2012
226. 226. © Activities for Learning, Inc., 2012
228. 228. Hexagons© Activities for Learning, Inc., 2012
229. 229. Hexagons© Activities for Learning, Inc., 2012
230. 230. Hexagons© Activities for Learning, Inc., 2012
231. 231. Hexagons© Activities for Learning, Inc., 2012
232. 232. Hexagons© Activities for Learning, Inc., 2012
233. 233. Hexagons© Activities for Learning, Inc., 2012
234. 234. Hexagons© Activities for Learning, Inc., 2012
235. 235. Hexagons© Activities for Learning, Inc., 2012
236. 236. Hexagons Draw all the diagonals.© Activities for Learning, Inc., 2012
237. 237. Hexagons Draw all the diagonals.© Activities for Learning, Inc., 2012
238. 238. Hexagons Draw all the diagonals.© Activities for Learning, Inc., 2012
239. 239. Hexagons Draw all the diagonals.© Activities for Learning, Inc., 2012
240. 240. Hexagons Draw all the diagonals.© Activities for Learning, Inc., 2012
241. 241. Hexagons Draw all the diagonals.© Activities for Learning, Inc., 2012
242. 242. Hexagons Do you see another hexagon?© Activities for Learning, Inc., 2012
243. 243. Hexagons Find the figures Do you see another hexagon?© Activities for Learning, Inc., 2012
244. 244. Hexagons Find the figures 2 non-congruent rectangles?© Activities for Learning, Inc., 2012
245. 245. Hexagons Find the figures 2 non-congruent rectangles?© Activities for Learning, Inc., 2012
246. 246. Hexagons Find the figures 2 non-congruent rectangles?© Activities for Learning, Inc., 2012
247. 247. Hexagons Find the figures 3 non-congruent right triangles?© Activities for Learning, Inc., 2012
248. 248. Hexagons Find the figures 3 non-congruent right triangles?© Activities for Learning, Inc., 2012
249. 249. Hexagons Find the figures 3 non-congruent right triangles?© Activities for Learning, Inc., 2012
250. 250. Hexagons Find the figures 3 non-congruent right triangles?© Activities for Learning, Inc., 2012
251. 251. Hexagons Find the figures 2 non-congruent rhombuses?© Activities for Learning, Inc., 2012
252. 252. Hexagons Find the figures 2 non-congruent rhombuses?© Activities for Learning, Inc., 2012
253. 253. Hexagons Find the figures 2 non-congruent rhombuses?© Activities for Learning, Inc., 2012
254. 254. Hexagons Find the figures 3 similar equilateral triangles?© Activities for Learning, Inc., 2012
255. 255. Hexagons Find the figures 3 similar equilateral triangles?© Activities for Learning, Inc., 2012
256. 256. Hexagons Find the figures 3 similar equilateral triangles?© Activities for Learning, Inc., 2012
257. 257. Hexagons Find the figures 3 similar obtuse triangles?© Activities for Learning, Inc., 2012
258. 258. Hexagons Find the figures 3 similar obtuse triangles?© Activities for Learning, Inc., 2012
259. 259. Hexagons Find the figures 3 similar obtuse triangles?© Activities for Learning, Inc., 2012
260. 260. Hexagons Find the figures 3 similar obtuse triangles?© Activities for Learning, Inc., 2012
261. 261. Hexagons Find the figures 3 non-congruent trapezoids?© Activities for Learning, Inc., 2012
262. 262. Hexagons Find the figures 3 non-congruent trapezoids?© Activities for Learning, Inc., 2012
263. 263. Hexagons Find the figures 3 non-congruent trapezoids?© Activities for Learning, Inc., 2012
264. 264. Hexagons Find the figures 3 non-congruent trapezoids?© Activities for Learning, Inc., 2012
265. 265. Hexagons Find the figures 2 similar kites?© Activities for Learning, Inc., 2012
266. 266. Hexagons Find the figures 2 similar kites?© Activities for Learning, Inc., 2012
267. 267. Hexagons Find the figures 2 similar kites?© Activities for Learning, Inc., 2012
268. 268. © Activities for Learning, Inc., 2012
269. 269. Hexagons Constructing the small star Apothem: perpendicular line from side to center.© Activities for Learning, Inc., 2012
270. 270. Hexagons Constructing the small star Apothem: perpendicular line from side to center.© Activities for Learning, Inc., 2012
271. 271. Hexagons Constructing the small star Apothem: perpendicular line from side to center.© Activities for Learning, Inc., 2012
272. 272. Hexagons Constructing the small star© Activities for Learning, Inc., 2012
273. 273. Hexagons Constructing the small star© Activities for Learning, Inc., 2012
274. 274. Hexagons Constructing the small star© Activities for Learning, Inc., 2012
275. 275. Hexagons Constructing the small star© Activities for Learning, Inc., 2012
276. 276. Hexagons Constructing the small star Ratio of the areas of star and hexagon?© Activities for Learning, Inc., 2012
277. 277. Hexagons Constructing the small star Ratio of the areas of star and hexagon?© Activities for Learning, Inc., 2012
278. 278. Hexagons Constructing the large star© Activities for Learning, Inc., 2012
279. 279. Hexagons Constructing the large star© Activities for Learning, Inc., 2012
280. 280. Hexagons Constructing the large star© Activities for Learning, Inc., 2012
281. 281. Hexagons Constructing the large star© Activities for Learning, Inc., 2012
282. 282. Hexagons Constructing the large star© Activities for Learning, Inc., 2012
283. 283. Hexagons Constructing the large star© Activities for Learning, Inc., 2012
284. 284. Hexagons Constructing the large star© Activities for Learning, Inc., 2012
285. 285. Hexagons Constructing the large star© Activities for Learning, Inc., 2012
286. 286. The Clock 11 12 1 10 2 9 3 8 4 7 6 5© Activities for Learning, Inc., 2012
287. 287. The Clock 11 12 1 10 2 9 3 8 4 7 6 5© Activities for Learning, Inc., 2012
288. 288. The Clock 11 12 1 10 2 9 3 8 4 7 6 5© Activities for Learning, Inc., 2012
289. 289. The Clock 11 12 1 10 2 9 3 8 4 7 6 5© Activities for Learning, Inc., 2012
290. 290. The Clock 11 12 1 10 2 9 3 8 4 7 6 5© Activities for Learning, Inc., 2012
291. 291. The Clock 11 12 1 10 2 9 3 8 4 7 6 5 11 12 1 10 2 9 3 8 4 7 6 5© Activities for Learning, Inc., 2012
292. 292. Right Triangle Design© Activities for Learning, Inc., 2012
293. 293. Right Triangle Design© Activities for Learning, Inc., 2012
294. 294. Right Triangle Design© Activities for Learning, Inc., 2012
295. 295. Right Triangle Design© Activities for Learning, Inc., 2012
296. 296. Right Triangle Design© Activities for Learning, Inc., 2012
297. 297. Right Triangle Design© Activities for Learning, Inc., 2012
298. 298. Right Triangle Design© Activities for Learning, Inc., 2012
299. 299. Right Triangle Design© Activities for Learning, Inc., 2012
300. 300. Right Triangle Design© Activities for Learning, Inc., 2012
301. 301. Right Triangle Design© Activities for Learning, Inc., 2012
302. 302. © Activities for Learning, Inc., 2012
303. 303. Tangrams© Activities for Learning, Inc., 2012
304. 304. Tangrams© Activities for Learning, Inc., 2012
305. 305. Tangrams© Activities for Learning, Inc., 2012
306. 306. Tangrams© Activities for Learning, Inc., 2012
307. 307. Tangrams© Activities for Learning, Inc., 2012
308. 308. Tangrams© Activities for Learning, Inc., 2012
309. 309. Tangrams© Activities for Learning, Inc., 2012
310. 310. Tangrams Are the 3 green figures are equal in© Activities for Learning, Inc., 2012 area?
311. 311. Tangrams© Activities for Learning, Inc., 2012
312. 312. Tangrams The 3 green figures ARE equal in area.© Activities for Learning, Inc., 2012
313. 313. Tangrams Constructing a tangram arrangement.© Activities for Learning, Inc., 2012
314. 314. Tangrams Constructing a tangram arrangement.© Activities for Learning, Inc., 2012
315. 315. Tangrams Constructing a tangram arrangement.© Activities for Learning, Inc., 2012
316. 316. Tangrams Constructing a tangram arrangement.© Activities for Learning, Inc., 2012
317. 317. Tangrams Constructing a tangram arrangement.© Activities for Learning, Inc., 2012
318. 318. Tangrams Constructing a tangram arrangement.© Activities for Learning, Inc., 2012
319. 319. Tangrams Constructing a tangram arrangement.© Activities for Learning, Inc., 2012
320. 320. Tangrams Constructing a tangram arrangement.© Activities for Learning, Inc., 2012
321. 321. Tangrams Constructing a tangram arrangement.© Activities for Learning, Inc., 2012
322. 322. Tangrams© Activities for Learning, Inc., 2012
323. 323. Tangram Design Symmetry Start at the left side of the paper.© Activities for Learning, Inc., 2012
324. 324. Tangram Design Symmetry Start at the left side of the paper.© Activities for Learning, Inc., 2012
325. 325. Tangram Design Symmetry Reflect the image vertically across the© Activities for Learning, Inc., 2012 line.
326. 326. Tangram Design Symmetry Reflect the image vertically.© Activities for Learning, Inc., 2012
327. 327. Tangram Design Symmetry© Activities for Learning, Inc., 2012
328. 328. Tangram Design Symmetry© Activities for Learning, Inc., 2012
329. 329. Tangram Design Symmetry© Activities for Learning, Inc., 2012
330. 330. Tangram Design Symmetry© Activities for Learning, Inc., 2012
331. 331. Tangram Design Symmetry© Activities for Learning, Inc., 2012
332. 332. Tangram Design Symmetry© Activities for Learning, Inc., 2012
333. 333. Tangram Design Symmetry Reflect the image horizontally.© Activities for Learning, Inc., 2012
334. 334. Tangram Design Symmetry Reflect the image horizontally.© Activities for Learning, Inc., 2012
335. 335. Tangram Design Symmetry The blue rotated 180° is the yellow.© Activities for Learning, Inc., 2012
336. 336. Tangram Design Symmetry Reflect the image vertically.© Activities for Learning, Inc., 2012
337. 337. Tangram Design Symmetry© Activities for Learning, Inc., 2012
338. 338. Tangram Design Symmetry© Activities for Learning, Inc., 2012
339. 339. Tangram Design Symmetry© Activities for Learning, Inc., 2012
340. 340. Tangram Design Symmetry© Activities for Learning, Inc., 2012
341. 341. Tangram Design Symmetry© Activities for Learning, Inc., 2012
342. 342. Tangram Design Symmetry© Activities for Learning, Inc., 2012
343. 343. Tangram Design Symmetry© Activities for Learning, Inc., 2012
344. 344. Tangram Design Symmetry© Activities for Learning, Inc., 2012
345. 345. Tangram Design Symmetry Two horizontal reflections is the original.© Activities for Learning, Inc., 2012
346. 346. Tangram Design Symmetry • Such constructions require thinking ahead.© Activities for Learning, Inc., 2012
347. 347. Tangram Design Symmetry • Such constructions require thinking ahead. • Each step must be justified; no guessing.© Activities for Learning, Inc., 2012
348. 348. Tangram Design Symmetry • Such constructions require thinking ahead. • Each step must be justified; no guessing. • Symmetry is in everyday life.© Activities for Learning, Inc., 2012
349. 349. Tangram Design Symmetry • Such constructions require thinking ahead. • Each step must be justified; no guessing. • Symmetry is in everyday life. • Symmetry is often on tests.© Activities for Learning, Inc., 2012
350. 350. © Activities for Learning, Inc., 2012
351. 351. Pythagorean Theorem First construct a right triangle.© Activities for Learning, Inc., 2012
352. 352. Pythagorean Theorem© Activities for Learning, Inc., 2012
353. 353. Pythagorean Theorem© Activities for Learning, Inc., 2012
354. 354. Pythagorean Theorem© Activities for Learning, Inc., 2012
355. 355. Pythagorean Theorem© Activities for Learning, Inc., 2012
356. 356. Pythagorean Theorem© Activities for Learning, Inc., 2012
357. 357. Pythagorean Theorem© Activities for Learning, Inc., 2012
358. 358. Pythagorean Theorem© Activities for Learning, Inc., 2012
359. 359. Pythagorean Theorem© Activities for Learning, Inc., 2012
360. 360. Pythagorean Theorem Area of blue squares = yellow square.© Activities for Learning, Inc., 2012
361. 361. Pythagorean Theorem Area of blue squares = yellow square.© Activities for Learning, Inc., 2012
362. 362. Pythagorean Theorem Draw a right triangle.© Activities for Learning, Inc., 2012
363. 363. Pythagorean Theorem© Activities for Learning, Inc., 2012
364. 364. Pythagorean Theorem© Activities for Learning, Inc., 2012
365. 365. Pythagorean Theorem© Activities for Learning, Inc., 2012
366. 366. Pythagorean Theorem© Activities for Learning, Inc., 2012
367. 367. Pythagorean Theorem© Activities for Learning, Inc., 2012
368. 368. Pythagorean Theorem© Activities for Learning, Inc., 2012
369. 369. Pythagorean Theorem Turn the t-square upside down.© Activities for Learning, Inc., 2012
370. 370. Pythagorean Theorem© Activities for Learning, Inc., 2012
371. 371. Pythagorean Theorem© Activities for Learning, Inc., 2012
372. 372. Pythagorean Theorem© Activities for Learning, Inc., 2012
373. 373. Pythagorean Theorem Do not move the t-square.© Activities for Learning, Inc., 2012
374. 374. Pythagorean Theorem Do not move the t-square.© Activities for Learning, Inc., 2012
375. 375. Pythagorean Theorem© Activities for Learning, Inc., 2012
376. 376. Pythagorean Theorem© Activities for Learning, Inc., 2012
377. 377. Pythagorean Theorem© Activities for Learning, Inc., 2012
378. 378. Pythagorean Theorem Area of blue squares = yellow square.© Activities for Learning, Inc., 2012
379. 379. Triangle Area A B C© Activities for Learning, Inc., 2012
380. 380. Triangle Area A B C Draw the altitude.© Activities for Learning, Inc., 2012
381. 381. Triangle Area A B C Area is half of enclosing rectangle.© Activities for Learning, Inc., 2012
382. 382. Triangle Area A B C Another altitude.© Activities for Learning, Inc., 2012
383. 383. Triangle Area A B C Find the area.© Activities for Learning, Inc., 2012
384. 384. Triangle Area A B C Another altitude.© Activities for Learning, Inc., 2012
385. 385. Triangle Area A B C Find the area.© Activities for Learning, Inc., 2012
386. 386. Triangle Area A B C Altitudes intersect at the orthocenter.© Activities for Learning, Inc., 2012
387. 387. Triangle Area A B C 3 areas slightly different. Why?© Activities for Learning, Inc., 2012
388. 388. Geometric Approach© Activities for Learning, Inc., 2012
389. 389. Geometric Approach • Basis of CAD (computer aided design).© Activities for Learning, Inc., 2012
390. 390. Geometric Approach • Basis of CAD (computer aided design). • Integrates many concepts.© Activities for Learning, Inc., 2012
391. 391. Geometric Approach • Basis of CAD (computer aided design). • Integrates many concepts. • Incorporates measuring.© Activities for Learning, Inc., 2012
392. 392. Geometric Approach • Basis of CAD (computer aided design). • Integrates many concepts. • Incorporates measuring. • Allows independent work.© Activities for Learning, Inc., 2012
393. 393. Geometric Approach • Basis of CAD (computer aided design). • Integrates many concepts. • Incorporates measuring. • Allows independent work. • Excellent preparation for high school mathematics.© Activities for Learning, Inc., 2012
394. 394. Geometric Approach Goals© Activities for Learning, Inc., 2012
395. 395. Geometric Approach Goals • To use elementary math.© Activities for Learning, Inc., 2012
396. 396. Geometric Approach Goals • To use elementary math. • To learn to read math texts.© Activities for Learning, Inc., 2012
397. 397. Geometric Approach Goals • To use elementary math. • To learn to read math texts. • To prepare for more advanced mathematics.© Activities for Learning, Inc., 2012
398. 398. Geometric Approach Goals • To use elementary math. • To learn to read math texts. • To prepare for more advanced mathematics. • To discover mathematics in the everyday world.© Activities for Learning, Inc., 2012
399. 399. Geometric Approach Goals • To use elementary math. • To learn to read math texts. • To prepare for more advanced mathematics. • To discover mathematics in the everyday world. • To enjoy mathematics.© Activities for Learning, Inc., 2012
400. 400. Personalized Learning Bridges Middle-School Math with a Geometric Approach – Part 2 Presented by Tracy Mittleider, MSEd and Kathleen Lawler Based on work of Joan A. Cotter, Ph.D. Aplus+ Conference October 24, 2012© Activities for Learning, Inc., 2012
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