1. Counting: Necessary or Detrimental? AMS Conference March 25, 2011 Chicago, Illinois by Joan A. Cotter, Ph.D. [email_address] 7 5 2 Presentation available: ALabacus.com 7 x 7 VII

2. National Math Crisis

9. Math Education is Changing

15. Counting Model

17. Counting Model From the perspective of most teachers

26. Counting Model From a child's perspective Because we’re so familiar with 1, 2, 3, we’ll use letters. A = 1 B = 2 C = 3 D = 4 E = 5, and so forth

27. Counting Model From a child's perspective F + E

28. Counting Model From a child's perspective F + E A

29. Counting Model From a child's perspective F + E A B

30. Counting Model From a child's perspective F + E A C B

31. Counting Model From a child's perspective F + E A F C D E B

32. Counting Model From a child's perspective F + E A A F C D E B

33. Counting Model From a child's perspective F + E A B A F C D E B

34. Counting Model From a child's perspective F + E A C D E B A F C D E B

35. Counting Model From a child's perspective F + E What is the sum? (It must be a letter.) A C D E B A F C D E B

36. Counting Model From a child's perspective K G I J K H A F C D E B F + E

37. Counting Model From a child's perspective Now memorize the facts!! G + D

38. Counting Model From a child's perspective Now memorize the facts!! G + D H + F

39. Counting Model From a child's perspective Now memorize the facts!! G + D H + F D + C

40. Counting Model From a child's perspective Now memorize the facts!! G + D H + F C + G D + C

41. Counting Model From a child's perspective Now memorize the facts!! E + I G + D H + F C + G D + C

42. Counting Model From a child's perspective Try subtracting by “taking away” H – E

43. Counting Model From a child's perspective Try skip counting by B’s to T : B , D , . . . T .

44. Counting Model From a child's perspective Try skip counting by B’s to T : B , D , . . . T . What is D  E ?

45. Counting Model From a child's perspective L is written AB because it is A J and B A’s

46. Counting Model From a child's perspective L is written AB because it is A J and B A’s huh?

47. Counting Model From a child's perspective L is written AB because it is A J and B A’s (twelve)

48. Counting Model From a child's perspective L is written AB because it is A J and B A’s (12) (twelve)

49. Counting Model From a child's perspective L is written AB because it is A J and B A’s (12) (one 10) (twelve)

50. Counting Model From a child's perspective L is written AB because it is A J and B A’s (12) (one 10) (two 1s). (twelve)

51. Counting Model Summary

58. Calendar Math August 29 22 15 8 1 30 23 16 9 2 24 17 10 3 25 18 11 4 26 19 12 5 27 20 13 6 28 21 14 7 31 Sometimes calendars are used for counting.

59. Calendar Math August 29 22 15 8 1 30 23 16 9 2 24 17 10 3 25 18 11 4 26 19 12 5 27 20 13 6 28 21 14 7 31 Sometimes calendars are used for counting.

60. Calendar Math August 29 22 15 8 1 30 23 16 9 2 24 17 10 3 25 18 11 4 26 19 12 5 27 20 13 6 28 21 14 7 31

61. Calendar Math August 29 22 15 8 1 30 23 16 9 2 24 17 10 3 25 18 11 4 26 19 12 5 27 20 13 6 28 21 14 7 31 This is ordinal, not cardinal counting. The 3 doesn’t include the 1 and the 2.

62. Calendar Math August 29 22 15 8 1 30 23 16 9 2 24 17 10 3 25 18 11 4 26 19 12 5 27 20 13 6 28 21 14 7 31 This is ordinal, not cardinal counting. The 4 doesn’t include 1, 2 and 3.

63. Calendar Math August 29 22 15 8 1 30 23 16 9 2 24 17 10 3 25 18 11 4 26 19 12 5 27 20 13 6 28 21 14 7 31 A calendar is NOT like a ruler. On a ruler the numbers are not in the spaces.

64. Calendar Math August 8 1 9 2 10 3 4 5 6 7 Always show the whole calendar. A child needs to see the whole before the parts. Children also need to learn to plan ahead.

68. Research on Counting Karen Wynn’s research Show the baby two teddy bears.

69. Research on Counting Karen Wynn’s research Then hide them with a screen.

70. Research on Counting Karen Wynn’s research Show the baby a third teddy bear and put it behind the screen.

71. Research on Counting Karen Wynn’s research Show the baby a third teddy bear and put it behind the screen.

72. Research on Counting Karen Wynn’s research Raise screen. Baby seeing 3 won’t look long because it is expected.

73. Research on Counting Karen Wynn’s research Researcher can change the number of teddy bears behind the screen.

74. Research on Counting Karen Wynn’s research A baby seeing 1 teddy bear will look much longer, because it’s unexpected.

90. Visualizing Mathematics

91. Visualizing Mathematics “ In our concern about the memorization of math facts or solving problems, we must not forget that the root of mathematical study is the creation of mental pictures in the imagination and manipulating those images and relationships using the power of reason and logic.” Mindy Holte (E I)

92. Visualizing Mathematics “ Think in pictures, because the brain remembers images better than it does anything else.” Ben Pridmore, World Memory Champion, 2009

93. Visualizing Mathematics “ Mathematics is the activity of creating relationships, many of which are based in visual imagery. ” Wheatley and Cobb

94. Visualizing Mathematics “ The process of connecting symbols to imagery is at the heart of mathematics learning.” Dienes

95. Visualizing Mathematics “ The role of physical manipulatives was to help the child form those visual images and thus to eliminate the need for the physical manipulatives.” Ginsberg and others

99. Visualizing Mathematics Ready: How many?

100. Visualizing Mathematics Ready: How many?

101. Visualizing Mathematics Try again: How many?

102. Visualizing Mathematics Try again: How many?

103. Visualizing Mathematics Try again: How many?

104. Visualizing Mathematics Ready: How many?

105. Visualizing Mathematics Try again: How many?

106. Visualizing Mathematics Try to visualize 8 identical apples without grouping.

107. Visualizing Mathematics Try to visualize 8 identical apples without grouping.

108. Visualizing Mathematics Now try to visualize 5 as red and 3 as green.

109. Visualizing Mathematics Now try to visualize 5 as red and 3 as green.

110. Visualizing Mathematics Early Roman numerals I II III IIII V VIII 1 2 3 4 5 8 Romans grouped in fives. Notice 8 is 5 and 3.

111. Visualizing Mathematics Who could read the music? : Music needs 10 lines, two groups of five.

113. Naming Quantities Using fingers

114. Naming Quantities Using fingers Naming quantities is a three-period lesson.

115. Naming Quantities Using fingers Use left hand for 1-5 because we read from left to right.

116. Naming Quantities Using fingers

117. Naming Quantities Using fingers

118. Naming Quantities Using fingers Always show 7 as 5 and 2, not for example, as 4 and 3.

119. Naming Quantities Using fingers

120. Naming Quantities Yellow is the Sun Yellow is the sun. Six is five and one. Why is the sky so blue? Seven is five and two. Salty is the sea. Eight is five and three. Hear the thunder roar. Nine is five and four. Ducks will swim and dive. Ten is five and five. – Joan A. Cotter Also set to music. Listen and download sheet music from Web site.

121. Naming Quantities Recognizing 5

122. Naming Quantities Recognizing 5

123. Naming Quantities Recognizing 5 5 has a middle; 4 does not. Look at your hand; your middle finger is longer to remind you 5 has a middle.

124. Naming Quantities Tally sticks Lay the sticks flat on a surface, about 1 inch (2.5 cm) apart.

125. Naming Quantities Tally sticks

126. Naming Quantities Tally sticks

127. Naming Quantities Tally sticks Stick is horizontal, because it won’t fit diagonally and young children have problems with diagonals.

128. Naming Quantities Tally sticks

129. Naming Quantities Tally sticks Start a new row for every ten.

130. Naming Quantities What is 4 apples plus 3 more apples? Solving a problem without counting How would you find the answer without counting?

131. Naming Quantities What is 4 apples plus 3 more apples? Solving a problem without counting To remember 4 + 3, the Japanese child is taught to visualize 4 and 3. Then take 1 from the 3 and give it to the 4 to make 5 and 2.

132. Naming Quantities Number Chart 1 2 3 4 5

133. Naming Quantities Number Chart 1 2 3 4 5 To help the child learn the symbols

134. Naming Quantities Number Chart 6 1 7 2 8 3 9 4 10 5 To help the child learn the symbols

135. Naming Quantities Pairing Finger Cards Use two sets of finger cards and match them.

136. Naming Quantities Ordering Finger Cards Putting the finger cards in order.

137. Naming Quantities 10 Matching Numbers to Finger Cards Match the number to the finger card. 5 1

138. Naming Quantities Matching Fingers to Number Cards Match the finger card to the number. 9 4 1 6 10 2 8 3 5 7

139. Naming Quantities Finger Card Memory game Use two sets of finger cards and play Memory.

140. Naming Quantities “ Grouped in fives so the child does not need to count.” Black and White Bead Stairs A. M. Joosten This was the inspiration to group in 5s.

141. Naming Quantities Number Rods

142. Naming Quantities Number Rods

143. Naming Quantities Number Rods Using different colors.

144. Naming Quantities Spindle Box 45 dark-colored and 10 light-colored spindles. Could be in separate containers.

145. Naming Quantities Spindle Box 45 dark-colored and 10 light-colored spindles in two containers.

146. Naming Quantities Spindle Box The child takes blue spindles with left hand and yellow with right. 1 2 3 0 4

147. Naming Quantities Spindle Box The child takes blue spindles with left hand and yellow with right. 6 7 8 5 9

148. Naming Quantities Spindle Box The child takes blue spindles with left hand and yellow with right. 6 7 8 5 9

149. Naming Quantities Spindle Box The child takes blue spindles with left hand and yellow with right. 6 7 8 5 9

150. Naming Quantities Spindle Box The child takes blue spindles with left hand and yellow with right. 6 7 8 5 9

151. Naming Quantities Spindle Box The child takes blue spindles with left hand and yellow with right. 6 7 8 5 9

152. Naming Quantities Spindle Box The child takes blue spindles with left hand and yellow with right. 6 7 8 5 9

153. Naming Quantities Black and White Bead Stairs This was the inspiration to group in 5s.

154. AL Abacus Double-sided AL abacus. Side 1 is grouped in 5s. Trading Side introduces algorithms with trading. 1000 10 1 100

155. AL Abacus Cleared

156. 3 Entering quantities AL Abacus Quantities are entered all at once, not counted.

157. 5 AL Abacus Entering quantities Relate quantities to hands.

158. 7 AL Abacus Entering quantities

159. AL Abacus 10 Entering quantities

160. AL Abacus The stairs Can use to “count” 1 to 10. Also read quantities on the right side.

161. AL Abacus Adding

162. AL Abacus Adding 4 + 3 =

163. AL Abacus Adding 4 + 3 =

164. AL Abacus Adding 4 + 3 =

165. AL Abacus Adding 4 + 3 =

166. AL Abacus Adding 4 + 3 = 7 Answer is seen immediately, no counting needed.

167. Problem Solving

171. Problem Solving Part-Whole Circles Part-whole circles children see relationships and solve problems.

172. Problem Solving Part-Whole Circles Whole Part-whole circles children see relationships and solve problems.

173. Problem Solving Part-Whole Circles Whole Part-whole circles children see relationships and solve problems. Part Part

174. Problem Solving Part-Whole Circles 10 If 10 is the whole

175. Problem Solving Part-Whole Circles 10 4 and 4 is one part,

176. Problem Solving Part-Whole Circles 10 4 What is the other part?

177. Problem Solving Part-Whole Circles 10 4 6 What is the other part?

178. Problem Solving Solving a problem Lee received 3 goldfish as a gift. Now Lee has 5. How many did Lee have to start with? A missing addend problem, considered very difficult for first graders. They can do it with Part-Whole Circles.

179. Problem Solving Solving a problem Lee received 3 goldfish as a gift. Now Lee has 5. How many did Lee have to start with? Is 3 a part or whole?

180. Problem Solving Solving a problem Lee received 3 goldfish as a gift. Now Lee has 5. How many did Lee have to start with? Is 3 a part or whole? 3

181. Problem Solving Solving a problem 3 Lee received 3 goldfish as a gift. Now Lee has 5. How many did Lee have to start with? Is 5 a part or whole?

182. Problem Solving Solving a problem Lee received 3 goldfish as a gift. Now Lee has 5. How many did Lee have to start with? Is 5 a part or whole ? 5 3

183. Problem Solving Solving a problem Lee received 3 goldfish as a gift. Now Lee has 5. How many did Lee have to start with? 5 3 What is the missing part?

184. Problem Solving Solving a problem Lee received 3 goldfish as a gift. Now Lee has 5. How many did Lee have to start with? What is the missing part? 5 3 2

185. Problem Solving Solving a problem 5 3 2 Lee received 3 goldfish as a gift. Now Lee has 5. How many did Lee have to start with? Write the equation.

186. Problem Solving Solving a problem Lee received 3 goldfish as a gift. Now Lee has 5. How many did Lee have to start with? 2 + 3 = 5 5 3 2 Write the equation.

187. Problem Solving Solving a problem Lee received 3 goldfish as a gift. Now Lee has 5. How many did Lee have to start with? 2 + 3 = 5 5 3 2 3 + 2 = 5 Write the equation.

188. Problem Solving Solving a problem Lee received 3 goldfish as a gift. Now Lee has 5. How many did Lee have to start with? 2 + 3 = 5 5 3 2 3 + 2 = 5 5 – 3 = 2 Write the equation. Is this an addition or subtraction problem?

189. Problem Solving Part-whole circles young children solve problems. Writing equations do not.

190. Go to the Dump Game Objective: To learn the facts that total 10: 1 + 9 2 + 8 3 + 7 4 + 6 5 + 5 Children use the abacus while playing this “Go Fish” type game.

191. Go to the Dump Game Objective: To learn the ways to partition 10: 1 + 9 2 + 8 3 + 7 4 + 6 5 + 5 Object of the game: To collect the most pairs that equal ten. It is similar to “Go Fish.” Children use the abacus while playing this “Go Fish” type game.

192. Go to the Dump Game The ways to partition 10.

193. “ Math” Way of Naming Numbers

194. “ Math” Way of Naming Numbers 11 = ten 1

195. “ Math” Way of Naming Numbers 11 = ten 1 12 = ten 2

196. “ Math” Way of Naming Numbers 11 = ten 1 12 = ten 2 13 = ten 3

197. “ Math” Way of Naming Numbers 11 = ten 1 12 = ten 2 13 = ten 3 14 = ten 4

198. “ Math” Way of Naming Numbers 11 = ten 1 12 = ten 2 13 = ten 3 14 = ten 4 . . . . 19 = ten 9

199. “ Math” Way of Naming Numbers 11 = ten 1 12 = ten 2 13 = ten 3 14 = ten 4 . . . . 19 = ten 9 20 = 2-ten Don’t say “2-ten s .” We don’t say 3 hundred s eleven for 311.

200. “ Math” Way of Naming Numbers 11 = ten 1 12 = ten 2 13 = ten 3 14 = ten 4 . . . . 19 = ten 9 20 = 2-ten 21 = 2-ten 1 Don’t say “2-ten s .” We don’t say 3 hundred s eleven for 311.

201. “ Math” Way of Naming Numbers 11 = ten 1 12 = ten 2 13 = ten 3 14 = ten 4 . . . . 19 = ten 9 20 = 2-ten 21 = 2-ten 1 22 = 2-ten 2 Don’t say “2-ten s .” We don’t say 3 hundred s eleven for 311.

202. “ Math” Way of Naming Numbers 11 = ten 1 12 = ten 2 13 = ten 3 14 = ten 4 . . . . 19 = ten 9 20 = 2-ten 21 = 2-ten 1 22 = 2-ten 2 23 = 2-ten 3 Don’t say “2-ten s .” We don’t say 3 hundred s eleven for 311.

203. “ Math” Way of Naming Numbers 11 = ten 1 12 = ten 2 13 = ten 3 14 = ten 4 . . . . 19 = ten 9 20 = 2-ten 21 = 2-ten 1 22 = 2-ten 2 23 = 2-ten 3 . . . . . . . . 99 = 9-ten 9

204. “ Math” Way of Naming Numbers 137 = 1 hundred 3-ten 7 Only numbers under 100 need to be said the “math” way.

205. “ Math” Way of Naming Numbers 137 = 1 hundred 3-ten 7 or 137 = 1 hundred and 3-ten 7 Only numbers under 100 need to be said the “math” way.

206. “ Math” Way of Naming Numbers 0 10 20 30 40 50 60 70 80 90 100 4 5 6 Ages (yrs.) Song, M., & Ginsburg, H. (1988). p. 326. The effect of the Korean number system on young children's counting: A natural experiment in numerical bilingualism. International Journal of Psychology, 23, 319-332. Korean formal [math way] Korean informal [not explicit] Chinese U.S. Average Highest Number Counted Shows how far children from 3 countries can count at ages 4, 5, and 6.

207. “ Math” Way of Naming Numbers 0 10 20 30 40 50 60 70 80 90 100 4 5 6 Ages (yrs.) Song, M., & Ginsburg, H. (1988). p. 326. The effect of the Korean number system on young children's counting: A natural experiment in numerical bilingualism. International Journal of Psychology, 23, 319-332. Korean formal [math way] Korean informal [not explicit] Chinese U.S. Average Highest Number Counted Purple is Chinese. Note jump between ages 5 and 6.

208. “ Math” Way of Naming Numbers 0 10 20 30 40 50 60 70 80 90 100 4 5 6 Ages (yrs.) Song, M., & Ginsburg, H. (1988). p. 326. The effect of the Korean number system on young children's counting: A natural experiment in numerical bilingualism. International Journal of Psychology, 23, 319-332. Korean formal [math way] Korean informal [not explicit] Chinese U.S. Average Highest Number Counted Dark green is Korean “math” way.

209. “ Math” Way of Naming Numbers 0 10 20 30 40 50 60 70 80 90 100 4 5 6 Ages (yrs.) Average Highest Number Counted Song, M., & Ginsburg, H. (1988). p. 326. The effect of the Korean number system on young children's counting: A natural experiment in numerical bilingualism. International Journal of Psychology, 23, 319-332. Korean formal [math way] Korean informal [not explicit] Chinese U.S. Dotted green is everyday Korean; notice smaller jump between ages 5 and 6.

210. “ Math” Way of Naming Numbers 0 10 20 30 40 50 60 70 80 90 100 4 5 6 Ages (yrs.) Song, M., & Ginsburg, H. (1988). p. 326. The effect of the Korean number system on young children's counting: A natural experiment in numerical bilingualism. International Journal of Psychology, 23, 319-332. Korean formal [math way] Korean informal [not explicit] Chinese U.S. Average Highest Number Counted Red is English speakers. They learn same amount between ages 4-5 and 5-6.

215. Math Way of Naming Numbers Compared to reading:

219. Math Way of Naming Numbers “ Rather, the increased gap between Chinese and U.S. students and that of Chinese Americans and Caucasian Americans may be due primarily to the nature of their initial gap prior to formal schooling, such as counting efficiency and base-ten number sense.” Jian Wang and Emily Lin, 2005 Researchers

220. Math Way of Naming Numbers Using 10s and 1s, ask the child to construct 48. Research task:

221. Math Way of Naming Numbers Using 10s and 1s, ask the child to construct 48. Research task: Then ask the child to subtract 14.

222. Math Way of Naming Numbers Using 10s and 1s, ask the child to construct 48. Research task: Then ask the child to subtract 14. Children thinking of 14 as 14 ones count 14.

223. Math Way of Naming Numbers Using 10s and 1s, ask the child to construct 48. Research task: Then ask the child to subtract 14. Children thinking of 14 as 14 ones counted 14.

224. Math Way of Naming Numbers Using 10s and 1s, ask the child to construct 48. Research task: Then ask the child to subtract 14. Children thinking of 14 as 14 ones counted 14.

225. Math Way of Naming Numbers Using 10s and 1s, ask the child to construct 48. Research task: Then ask the child to subtract 14. Children thinking of 14 as 14 ones counted 14.

226. Math Way of Naming Numbers Using 10s and 1s, ask the child to construct 48. Research task: Then ask the child to subtract 14. Children thinking of 14 as 14 ones counted 14.

227. Math Way of Naming Numbers Using 10s and 1s, ask the child to construct 48. Research task: Then ask the child to subtract 14. Children thinking of 14 as 14 ones counted 14.

228. Math Way of Naming Numbers Using 10s and 1s, ask the child to construct 48. Research task: Then ask the child to subtract 14. Children thinking of 14 as 14 ones counted 14.

229. Math Way of Naming Numbers Using 10s and 1s, ask the child to construct 48. Research task: Then ask the child to subtract 14. Children thinking of 14 as 14 ones counted 14.

230. Math Way of Naming Numbers Using 10s and 1s, ask the child to construct 48. Research task: Then ask the child to subtract 14. Children who understand tens remove a ten and 4 ones.

231. Math Way of Naming Numbers Using 10s and 1s, ask the child to construct 48. Research task: Then ask the child to subtract 14. Children who understand tens remove a ten and 4 ones.

232. Math Way of Naming Numbers Using 10s and 1s, ask the child to construct 48. Research task: Then ask the child to subtract 14. Children who understand tens remove a ten and 4 ones.

233. Math Way of Naming Numbers Traditional names 4-ten = forty The “ty” means tens.

234. Math Way of Naming Numbers Traditional names 4-ten = forty The “ty” means tens. The traditional names for 40, 60, 70, 80, and 90 follow a pattern.

235. Math Way of Naming Numbers Traditional names 6-ten = sixty The “ty” means tens.

236. Math Way of Naming Numbers Traditional names 3-ten = thirty “ Thir” also used in 1/3, 13 and 30.

237. Math Way of Naming Numbers Traditional names 5-ten = fifty “ Fif” also used in 1/5, 15 and 50.

238. Math Way of Naming Numbers Traditional names 2-ten = twenty Two used to be pronounced “twoo.”

239. Math Way of Naming Numbers Traditional names A word game fireplace place-fire Say the syllables backward. This is how we say the teen numbers.

240. Math Way of Naming Numbers Traditional names A word game fireplace place-fire paper-news newspaper Say the syllables backward. This is how we say the teen numbers.

241. Math Way of Naming Numbers Traditional names A word game fireplace place-fire paper-news box-mail mailbox newspaper Say the syllables backward. This is how we say the teen numbers.

242. Math Way of Naming Numbers Traditional names ten 4 “ Teen” also means ten.

243. Math Way of Naming Numbers Traditional names ten 4 teen 4 “ Teen” also means ten.

244. Math Way of Naming Numbers Traditional names ten 4 teen 4 fourteen “ Teen” also means ten.

245. Math Way of Naming Numbers Traditional names a one left

246. Math Way of Naming Numbers Traditional names a one left a left-one

247. Math Way of Naming Numbers Traditional names a one left a left-one eleven

248. Math Way of Naming Numbers Traditional names two left Two pronounced “twoo.”

249. Math Way of Naming Numbers Traditional names two left twelve Two pronounced “twoo.”

250. Composing Numbers 3-ten

251. Composing Numbers 3-ten

252. Composing Numbers 3-ten 3 0

253. Composing Numbers 3-ten 3 0 Point to the 3 and say 3.

254. Composing Numbers 3-ten 3 0 Point to 0 and say 10. The 0 makes 3 a ten.

255. Composing Numbers 3-ten 7 3 0

256. Composing Numbers 3-ten 7 3 0

257. Composing Numbers 3-ten 7 3 0 7

258. Composing Numbers 3-ten 7 3 0 7 Place the 7 on top of the 0 of the 30.

259. Composing Numbers 3-ten 7 Notice the way we say the number, represent the number, and write the number all correspond. 3 0 7

260. Composing Numbers 7-ten 8 7 8 8 Another example.

261. Composing Numbers 10-ten

262. Composing Numbers 10-ten 1 0 0

263. Composing Numbers 10-ten 1 0 0

264. Composing Numbers 10-ten 1 0 0

265. Composing Numbers 1 hundred

266. Composing Numbers 1 hundred 1 0 0

267. Composing Numbers 1 hundred 1 0 0 Of course, we can also read it as one hun-dred.

268. Composing Numbers 1 hundred 1 0 0 Of course, we can also read it as one hun-dred. 1 0 1 0

269. Composing Numbers 1 hundred 1 0 0 Of course, we can also read it as one hun-dred.

270. Composing Numbers Reading numbers backward 2 5 8 4 8 To read a number, students are often instructed to start at the right (ones column), contrary to normal reading of numbers and text:

271. Composing Numbers 2 5 8 4 5 8 To read a number, students are often instructed to start at the right (ones column), contrary to normal reading of numbers and text: Reading numbers backward

272. Composing Numbers 2 5 8 4 2 5 8 To read a number, students are often instructed to start at the right (ones column), contrary to normal reading of numbers and text: Reading numbers backward

273. Composing Numbers 2 5 8 4 2 5 8 4 To read a number, students are often instructed to start at the right (ones column), contrary to normal reading of numbers and text: Reading numbers backward

274. Composing Numbers 2 5 8 4 2 5 8 4 To read a number, students are often instructed to start at the right (ones column), contrary to normal reading of numbers and text: Reading numbers backward The Decimal Cards encourage reading numbers in the normal order.

275. Fact Strategies

278. Fact Strategies Complete the Ten 9 + 5 =

279. Fact Strategies Complete the Ten 9 + 5 =

280. Fact Strategies Complete the Ten 9 + 5 =

281. Fact Strategies Complete the Ten 9 + 5 = Take 1 from the 5 and give it to the 9.

282. Fact Strategies Complete the Ten 9 + 5 = Take 1 from the 5 and give it to the 9. Use two hands and move the bead simultaneously.

283. Fact Strategies Complete the Ten 9 + 5 = Take 1 from the 5 and give it to the 9.

284. Fact Strategies Complete the Ten 9 + 5 = 14 Take 1 from the 5 and give it to the 9.

285. Fact Strategies Two Fives 8 + 6 =

286. Fact Strategies Two Fives 8 + 6 =

287. Fact Strategies Two Fives 8 + 6 = Two fives make 10.

288. Fact Strategies Two Fives 8 + 6 = Just add the “leftovers.”

289. Fact Strategies Two Fives 8 + 6 = 10 + 4 = 14 Just add the “leftovers.”

290. Fact Strategies Two Fives 7 + 5 = Another example.

291. Fact Strategies Two Fives 7 + 5 =

292. Fact Strategies Two Fives 7 + 5 = 12

293. Fact Strategies Going Down 15 – 9 =

294. Fact Strategies Going Down 15 – 9 =

295. Fact Strategies Going Down 15 – 9 = Subtract 5; then 4.

296. Fact Strategies Going Down 15 – 9 = Subtract 5; then 4.

297. Fact Strategies Going Down 15 – 9 = Subtract 5; then 4.

298. Fact Strategies Going Down 15 – 9 = 6 Subtract 5; then 4.

299. Fact Strategies Subtract from 10 15 – 9 =

300. Fact Strategies Subtract from 10 15 – 9 = Subtract 9 from 10.

301. Fact Strategies Subtract from 10 15 – 9 = Subtract 9 from 10.

302. Fact Strategies Subtract from 10 15 – 9 = Subtract 9 from 10.

303. Fact Strategies Subtract from 10 15 – 9 = 6 Subtract 9 from 10.

304. Fact Strategies Going Up 13 – 9 =

305. Fact Strategies Going Up 13 – 9 = Start with 9; go up to 13.

306. Fact Strategies Going Up 13 – 9 = Start with 9; go up to 13.

307. Fact Strategies Going Up 13 – 9 = Start with 9; go up to 13.

308. Fact Strategies Going Up 13 – 9 = Start with 9; go up to 13.

309. Fact Strategies Going Up 13 – 9 = 1 + 3 = 4 Start with 9; go up to 13.

310. Money Penny

311. Money Nickel

312. Money Dime

313. Money Quarter

314. Money Quarter

315. Money Quarter

316. Money Quarter

317. Bead Frame 1 10 100 1000

318. Bead Frame Montessori’s Patent

319. Bead Frame Montessori’s Patent

320. Bead Frame Montessori’s Patent

321. Bead Frame Montessori’s Patent

322. Bead Frame Montessori’s Patent

323. Bead Frame 8 + 6 1 10 100 1000

324. Bead Frame 8 + 6 1 10 100 1000

325. Bead Frame 8 + 6 1 10 100 1000

326. Bead Frame 8 + 6 1 10 100 1000

327. Bead Frame 8 + 6 1 10 100 1000

328. Bead Frame 8 + 6 1 10 100 1000

329. Bead Frame 8 + 6 1 10 100 1000

330. Bead Frame 8 + 6 1 10 100 1000

331. Bead Frame 8 + 6 1 10 100 1000

332. Bead Frame 8 + 6 14 1 10 100 1000

333. Bead Frame Difficulties for the child

340. AL Abacus Double-sided AL abacus. Side 1 is grouped in 5s. Trading Side introduces algorithms with trading. 1000 10 1 100

341. Trading Side Cleared 1000 10 1 100

342. Trading Side Thousands 1000 10 1 100

343. Trading Side Hundreds 1000 10 1 100 The third wire from each end is not used.

344. Trading Side Tens 1000 10 1 100 The third wire from each end is not used.

345. Trading Side Ones 1000 10 1 100 The third wire from each end is not used.

346. Trading Side Adding 8 + 6 1000 10 1 100

347. Trading Side Adding 8 + 6 1000 10 1 100

348. Trading Side Adding 8 + 6 1000 10 1 100

349. Trading Side Adding 8 + 6 1000 10 1 100

350. Trading Side Adding 8 + 6 14 1000 10 1 100

351. Trading Side Adding 8 + 6 14 Too many ones; trade 10 ones for 1 ten. You can see the 10 ones (yellow). 1000 10 1 100

352. Trading Side Adding 8 + 6 14 Too many ones; trade 10 ones for 1 ten. 1000 10 1 100

353. Trading Side Adding 8 + 6 14 Too many ones; trade 10 ones for 1 ten. 1000 10 1 100

354. Trading Side Adding 8 + 6 14 Same answer before and after trading. 1000 10 1 100

355. Trading Side Bead Trading game Object: To get a high score by adding numbers on the green cards. 1000 10 1 100

356. Trading Side Bead Trading game Object: To get a high score by adding numbers on the green cards. 7 1000 10 1 100

357. Trading Side Bead Trading game Object: To get a high score by adding numbers on the green cards. 7 1000 10 1 100

358. Trading Side Bead Trading game 6 Turn over another card. Enter 6 beads. Do we need to trade? 1000 10 1 100

359. Trading Side Bead Trading game 6 Turn over another card. Enter 6 beads. Do we need to trade? 1000 10 1 100

360. Trading Side Bead Trading game 6 Turn over another card. Enter 6 beads. Do we need to trade? 1000 10 1 100

361. Trading Side Bead Trading game 6 Trade 10 ones for 1 ten. 1000 10 1 100

362. Trading Side Bead Trading game 6 1000 10 1 100

363. Trading Side Bead Trading game 6 1000 10 1 100

364. Trading Side Bead Trading game 9 1000 10 1 100

365. Trading Side Bead Trading game 9 1000 10 1 100

366. Trading Side Bead Trading game 9 Another trade. 1000 10 1 100

367. Trading Side Bead Trading game 9 Another trade. 1000 10 1 100

368. Trading Side Bead Trading game 3 1000 10 1 100

369. Trading Side Bead Trading game 3 1000 10 1 100

375. Trading Side Adding 4-digit numbers 3658 + 2738 1000 10 1 100

376. Trading Side Adding 4-digit numbers 3658 + 2738 Enter the first number from left to right. 1000 10 1 100

377. Trading Side Adding 4-digit numbers 3 658 + 2738 Enter the first number from left to right. 1000 10 1 100

378. Trading Side Adding 4-digit numbers 3 658 + 2738 Enter the first number from left to right. 1000 10 1 100

379. Trading Side Adding 4-digit numbers 3 6 58 + 2738 Enter the first number from left to right. 1000 10 1 100

380. Trading Side Adding 4-digit numbers 36 5 8 + 2738 Enter the first number from left to right. 1000 10 1 100

381. Trading Side Adding 4-digit numbers 365 8 + 2738 Enter the first number from left to right. 1000 10 1 100

382. Trading Side Adding 4-digit numbers 3658 + 273 8 Add starting at the right. Write results after each step. 1000 10 1 100

383. Trading Side Adding 4-digit numbers 3658 + 273 8 Add starting at the right. Write results after each step. 1000 10 1 100

384. Trading Side Adding 4-digit numbers 3658 + 273 8 Add starting at the right. Write results after each step. 1000 10 1 100

385. Trading Side Adding 4-digit numbers 3658 + 273 8 Add starting at the right. Write results after each step. 1000 10 1 100

386. Trading Side Adding 4-digit numbers 3658 + 2738 6 Add starting at the right. Write results after each step. . . . 6 ones. Did anything else happen? 1000 10 1 100

387. Trading Side Adding 4-digit numbers 3658 + 2738 6 Add starting at the right. Write results after each step. 1 Is it okay to show the extra ten by writing a 1 above the tens column? 1000 10 1 100

388. Trading Side Adding 4-digit numbers 3658 + 27 3 8 6 Add starting at the right. Write results after each step. 1 1000 10 1 100

389. Trading Side Adding 4-digit numbers 3658 + 27 3 8 6 Add starting at the right. Write results after each step. 1 Do we need to trade? [no] 1000 10 1 100

390. Trading Side Adding 4-digit numbers 3658 + 2738 9 6 Add starting at the right. Write results after each step. 1 1000 10 1 100

391. Trading Side Adding 4-digit numbers 3658 + 2 7 38 96 Add starting at the right. Write results after each step. 1 1000 10 1 100

392. Trading Side Adding 4-digit numbers 3658 + 2 7 38 96 Add starting at the right. Write results after each step. 1 Do we need to trade? [yes] 1000 10 1 100

393. Trading Side Adding 4-digit numbers 3658 + 2 7 38 96 Add starting at the right. Write results after each step. 1 Notice the number of yellow beads. [3] Notice the number of blue beads left. [3] Coincidence? No, because 13 – 10 = 3. 1000 10 1 100

394. Trading Side Adding 4-digit numbers 3658 + 2 7 38 96 Add starting at the right. Write results after each step. 1 1000 10 1 100

395. Trading Side Adding 4-digit numbers 3658 + 2738 3 96 Add starting at the right. Write results after each step. 1 1000 10 1 100

396. Trading Side Adding 4-digit numbers 3658 + 2738 3 96 Add starting at the right. Write results after each step. 1 1 1000 10 1 100

397. Trading Side Adding 4-digit numbers 3658 + 2 738 396 Add starting at the right. Write results after each step. 1 1 1000 10 1 100

398. Trading Side Adding 4-digit numbers 3658 + 2 738 396 Add starting at the right. Write results after each step. 1 1 1000 10 1 100

399. Trading Side Adding 4-digit numbers 3658 + 2738 6 396 Add starting at the right. Write results after each step. 1 1 1000 10 1 100

400. Trading Side Adding 4-digit numbers 3658 + 2738 6396 Add starting at the right. Write results after each step. 1 1 1000 10 1 100

401. Multiplication on the AL Abacus Basic facts 6  4 = (6 taken 4 times)

402. Multiplication on the AL Abacus Basic facts 6  4 = (6 taken 4 times)

403. Multiplication on the AL Abacus Basic facts 6  4 = (6 taken 4 times)

404. Multiplication on the AL Abacus Basic facts 6  4 = (6 taken 4 times)

405. Multiplication on the AL Abacus Basic facts 6  4 = (6 taken 4 times)

406. Multiplication on the AL Abacus Basic facts 9  3 =

407. Multiplication on the AL Abacus Basic facts 9  3 =

408. Multiplication on the AL Abacus Basic facts 9  3 = 30

409. Multiplication on the AL Abacus Basic facts 9  3 = 30 – 3 = 27

410. Multiplication on the AL Abacus Basic facts 4  8 =

411. Multiplication on the AL Abacus Basic facts 4  8 =

412. Multiplication on the AL Abacus Basic facts 4  8 =

413. Multiplication on the AL Abacus Basic facts 4  8 = 20 + 12 = 32

414. Multiplication on the AL Abacus Basic facts 7  7 =

415. Multiplication on the AL Abacus Basic facts 7  7 =

416. Multiplication on the AL Abacus Basic facts 7  7 = 25 + 10 + 10 + 4 = 49

417. Multiplication on the AL Abacus Commutative property 5  6 =

418. Multiplication on the AL Abacus Commutative property 5  6 =

419. Multiplication on the AL Abacus Commutative property 5  6 =

420. Multiplication on the AL Abacus Commutative property 5  6 = 6  5

421. Multiplication on the AL Abacus 7  8 = This method was used in the Middle Ages, rather than memorize the facts > 5  5. For facts > 5  5

422. Multiplication on the AL Abacus 7  8 = This method was used in the Middle Ages, rather than memorize the facts > 5  5. For facts > 5  5

423. Multiplication on the AL Abacus 7  8 = This method was used in the Middle Ages, rather than memorize the facts > 5  5. For facts > 5  5 Tens:

424. Multiplication on the AL Abacus 7  8 = This method was used in the Middle Ages, rather than memorize the facts > 5  5. For facts > 5  5 Tens:

425. Multiplication on the AL Abacus 7  8 = This method was used in the Middle Ages, rather than memorize the facts > 5  5. For facts > 5  5 Tens: 20 + 30

426. Multiplication on the AL Abacus 7  8 = 50 + This method was used in the Middle Ages, rather than memorize the facts > 5  5. For facts > 5  5 Tens: 20 + 30 50

427. Multiplication on the AL Abacus 7  8 = 50 + This method was used in the Middle Ages, rather than memorize the facts > 5  5. For facts > 5  5 Tens: Ones: 20 + 30 50

428. Multiplication on the AL Abacus 7  8 = 50 + This method was used in the Middle Ages, rather than memorize the facts > 5  5. For facts > 5  5 Tens: Ones: 20 + 30 50

429. Multiplication on the AL Abacus 7  8 = 50 + This method was used in the Middle Ages, rather than memorize the facts > 5  5. For facts > 5  5 Tens: Ones: 3  2 20 + 30 50

432. Multiplication on the AL Abacus 9  7 = For facts > 5  5

433. Multiplication on the AL Abacus 9  7 = For facts > 5  5

434. Multiplication on the AL Abacus 9  7 = For facts > 5  5 Tens:

435. Multiplication on the AL Abacus 9  7 = For facts > 5  5 Tens:

436. Multiplication on the AL Abacus 9  7 = For facts > 5  5 Tens: 40 + 20

437. Multiplication on the AL Abacus 9  7 = 60 + For facts > 5  5 Tens: 40 + 20 60

438. Multiplication on the AL Abacus 9  7 = 60 + For facts > 5  5 Tens: Ones: 40 + 20 60

439. Multiplication on the AL Abacus 9  7 = 60 + For facts > 5  5 Tens: Ones: 40 + 20 60

440. Multiplication on the AL Abacus 9  7 = 60 + For facts > 5  5 Tens: Ones: 1  3 40 + 20 60

443. The Multiplication Board 6 6  4 7 x 7 on original multiplication board. 1 2 3 4 5 6 7 8 9 10

444. The Multiplication Board 6  4 Using two colors. 1 2 3 4 5 6 7 8 9 10 6

445. The Multiplication Board 7  7 7 x 7 on original multiplication board. 1 2 3 4 5 6 7 8 9 10 7

446. The Multiplication Board 7  7 Upper left square is 25, yellow rectangles are 10. So, 25, 35, 45, 49. 1 2 3 4 5 6 7 8 9 10 7

447. The Multiplication Board 7  7 Less clutter.

448. Multiples Patterns Twos 2 4 6 8 10 12 14 16 18 20 Recognizing multiples needed for fractions and algebra.

449. Multiples Patterns Twos 2 4 6 8 10 12 14 16 18 20 Recognizing multiples needed for fractions and algebra.

450. Multiples Patterns Twos 2 4 6 8 10 1 2 14 16 18 20 The ones repeat in the second row. Recognizing multiples needed for fractions and algebra.

451. Multiples Patterns Twos 2 4 6 8 10 12 1 4 16 18 20 The ones repeat in the second row. Recognizing multiples needed for fractions and algebra.

452. Multiples Patterns Twos 2 4 6 8 10 12 14 1 6 18 20 The ones repeat in the second row. Recognizing multiples needed for fractions and algebra.

453. Multiples Patterns Twos 2 4 6 8 10 12 14 16 1 8 20 The ones repeat in the second row. Recognizing multiples needed for fractions and algebra.

454. Multiples Patterns Twos 2 4 6 8 1 0 12 14 16 18 2 0 The ones repeat in the second row. Recognizing multiples needed for fractions and algebra.

455. Multiples Patterns Fours 4 8 12 16 20 24 28 32 36 40

456. Multiples Patterns Fours 4 8 1 2 1 6 2 0 2 4 2 8 3 2 3 6 4 0 The ones repeat in the second row.

457. Multiples Patterns Sixes and Eights 6 12 18 24 30 36 42 48 54 60 8 16 24 32 40 48 56 64 72 80

458. Multiples Patterns Sixes and Eights 6 1 2 1 8 2 4 3 0 3 6 4 2 4 8 5 4 6 0 8 16 24 32 40 48 56 64 72 80

459. Multiples Patterns Sixes and Eights 6 1 2 1 8 2 4 3 0 3 6 4 2 4 8 5 4 6 0 8 1 6 2 4 3 2 4 0 4 8 5 6 6 4 7 2 8 0 Again the ones repeat in the second row.

460. Multiples Patterns Sixes and Eights 6 12 18 24 30 36 42 48 54 60 8 16 24 32 4 0 48 56 64 72 8 0 The ones in the 8s show the multiples of 2.

461. Multiples Patterns Sixes and Eights 6 12 18 24 30 36 42 48 54 60 8 16 24 3 2 4 0 48 56 64 7 2 8 0 The ones in the 8s show the multiples of 2.

462. Multiples Patterns Sixes and Eights 6 12 18 24 30 36 42 48 54 60 8 16 2 4 3 2 4 0 48 56 6 4 7 2 8 0 The ones in the 8s show the multiples of 2.

463. Multiples Patterns Sixes and Eights 6 12 18 24 30 36 42 48 54 60 8 1 6 2 4 3 2 4 0 48 5 6 6 4 7 2 8 0 The ones in the 8s show the multiples of 2.

464. Multiples Patterns Sixes and Eights 6 12 18 24 30 36 42 48 54 60 8 1 6 2 4 3 2 4 0 4 8 5 6 6 4 7 2 8 0 The ones in the 8s show the multiples of 2.

465. Multiples Patterns Sixes and Eights 6 12 18 24 30 36 42 48 54 60 8 16 24 32 40 48 56 64 72 80 6  4 6  4 is the fourth number (multiple).

466. Multiples Patterns Sixes and Eights 6 12 18 24 30 36 42 48 54 60 8 16 24 32 40 48 56 64 72 80 8  7 8  7 is the seventh number (multiple).

467. Multiples Patterns Nines 9 18 27 36 45 90 81 72 63 54 The second row is written in reverse order. Also the digits in each number add to 9.

471. Multiples Patterns Threes 3 6 9 1 2 15 18 2 1 24 27 3 0 The 3s have several patterns: Observe the ones.

472. Multiples Patterns Threes 3 6 9 1 2 15 18 2 1 24 27 3 0 The 3s have several patterns: Observe the ones.

473. Multiples Patterns Threes 3 6 9 1 2 15 18 2 1 2 4 27 3 0 The 3s have several patterns: Observe the ones.

474. Multiples Patterns Threes 3 6 9 1 2 1 5 18 2 1 2 4 27 3 0 The 3s have several patterns: Observe the ones.

475. Multiples Patterns Threes 3 6 9 1 2 1 5 18 2 1 2 4 27 3 0 The 3s have several patterns: Observe the ones.

476. Multiples Patterns Threes 3 6 9 1 2 1 5 18 2 1 2 4 2 7 3 0 The 3s have several patterns: Observe the ones.

477. Multiples Patterns Threes 3 6 9 1 2 1 5 1 8 2 1 2 4 2 7 3 0 The 3s have several patterns: Observe the ones.

478. Multiples Patterns Threes 3 6 9 1 2 1 5 1 8 2 1 2 4 2 7 3 0 The 3s have several patterns: Observe the ones.

479. Multiples Patterns Threes 3 6 9 1 2 1 5 1 8 2 1 2 4 2 7 30 The 3s have several patterns: The tens are the same in each row.

480. Multiples Patterns Threes 3 6 9 12 15 18 21 24 27 30 The 3s have several patterns: Add the digits in the columns.

481. Multiples Patterns Threes 3 6 9 12 15 18 21 24 27 30 The 3s have several patterns: Add the digits in the columns.

482. Multiples Patterns Threes 3 6 9 12 15 18 21 24 27 30 The 3s have several patterns: Add the digits in the columns.

483. Multiples Patterns Sevens 7 14 21 28 35 42 49 56 63 70 The 7s have the 1, 2, 3… pattern.

484. Multiples Patterns Sevens 7 14 2 1 28 35 4 2 49 56 6 3 70 The 7s have the 1, 2, 3… pattern.

485. Multiples Patterns Sevens 7 1 4 2 1 28 3 5 4 2 49 5 6 6 3 70 The 7s have the 1, 2, 3… pattern.

486. Multiples Patterns Sevens 7 1 4 2 1 2 8 3 5 4 2 4 9 5 6 6 3 7 0 The 7s have the 1, 2, 3… pattern.

487. Multiples Patterns Sevens 7 14 21 28 35 42 49 56 63 70 The 7s have the 1, 2, 3… pattern.

488. Multiples Patterns Sevens 7 14 21 2 8 35 42 4 9 56 63 7 0 The 7s have the 1, 2, 3… pattern.

489. Fraction Chart 1 Giving the child the big picture, a Montessori principle. 1 2 1 2 1 3 1 4 1 5 1 6 1 7 1 8 1 9 1 10 1 3 1 3 1 4 1 5 1 6 1 7 1 8 1 9 1 4 1 5 1 6 1 7 1 8 1 4 1 5 1 6 1 7 1 8 1 9 1 5 1 6 1 6 1 7 1 7 1 7 1 8 1 8 1 8 1 8 1 9 1 9 1 9 1 9 1 9 1 9 1 10 1 10 1 10 1 10 1 10 1 10 1 10 1 10 1 10

490. Fraction Chart 1 How many fourths in a whole? Giving the child the big picture, a Montessori principle. 1 2 1 2 1 3 1 4 1 5 1 6 1 7 1 8 1 9 1 10 1 3 1 3 1 4 1 5 1 6 1 7 1 8 1 9 1 4 1 5 1 6 1 7 1 8 1 4 1 5 1 6 1 7 1 8 1 9 1 5 1 6 1 6 1 7 1 7 1 7 1 8 1 8 1 8 1 8 1 9 1 9 1 9 1 9 1 9 1 9 1 10 1 10 1 10 1 10 1 10 1 10 1 10 1 10 1 10

491. Fraction Chart 1 How many fourths in a whole? Giving the child the big picture, a Montessori principle. 1 2 1 2 1 3 1 4 1 5 1 6 1 7 1 8 1 9 1 10 1 3 1 3 1 4 1 5 1 6 1 7 1 8 1 9 1 4 1 5 1 6 1 7 1 8 1 4 1 5 1 6 1 7 1 8 1 9 1 5 1 6 1 6 1 7 1 7 1 7 1 8 1 8 1 8 1 8 1 9 1 9 1 9 1 9 1 9 1 9 1 10 1 10 1 10 1 10 1 10 1 10 1 10 1 10 1 10

492. Fraction Chart 1 How many fourths in a whole? Giving the child the big picture, a Montessori principle. 1 2 1 2 1 3 1 4 1 5 1 6 1 7 1 8 1 9 1 10 1 3 1 3 1 4 1 5 1 6 1 7 1 8 1 9 1 4 1 5 1 6 1 7 1 8 1 4 1 5 1 6 1 7 1 8 1 9 1 5 1 6 1 6 1 7 1 7 1 7 1 8 1 8 1 8 1 8 1 9 1 9 1 9 1 9 1 9 1 9 1 10 1 10 1 10 1 10 1 10 1 10 1 10 1 10 1 10

493. Fraction Chart 1 How many fourths in a whole? Giving the child the big picture, a Montessori principle. 1 2 1 2 1 3 1 4 1 5 1 6 1 7 1 8 1 9 1 10 1 3 1 3 1 4 1 5 1 6 1 7 1 8 1 9 1 4 1 5 1 6 1 7 1 8 1 4 1 5 1 6 1 7 1 8 1 9 1 5 1 6 1 6 1 7 1 7 1 7 1 8 1 8 1 8 1 8 1 9 1 9 1 9 1 9 1 9 1 9 1 10 1 10 1 10 1 10 1 10 1 10 1 10 1 10 1 10

494. Fraction Chart 1 How many fourths in a whole? Giving the child the big picture, a Montessori principle. 1 2 1 2 1 3 1 4 1 5 1 6 1 7 1 8 1 9 1 10 1 3 1 3 1 4 1 5 1 6 1 7 1 8 1 9 1 4 1 5 1 6 1 7 1 8 1 4 1 5 1 6 1 7 1 8 1 9 1 5 1 6 1 6 1 7 1 7 1 7 1 8 1 8 1 8 1 8 1 9 1 9 1 9 1 9 1 9 1 9 1 10 1 10 1 10 1 10 1 10 1 10 1 10 1 10 1 10

495. Fraction Chart 1 How many eighths in a whole? 1 2 1 2 1 3 1 4 1 5 1 6 1 7 1 8 1 9 1 10 1 3 1 3 1 4 1 5 1 6 1 7 1 8 1 9 1 4 1 5 1 6 1 7 1 8 1 4 1 5 1 6 1 7 1 8 1 9 1 5 1 6 1 6 1 7 1 7 1 7 1 8 1 8 1 8 1 8 1 9 1 9 1 9 1 9 1 9 1 9 1 10 1 10 1 10 1 10 1 10 1 10 1 10 1 10 1 10

496. Fraction Chart 1 Which is more, 3/4 or 4/5? 1 2 1 2 1 3 1 4 1 5 1 6 1 7 1 8 1 9 1 10 1 3 1 3 1 4 1 5 1 6 1 7 1 8 1 9 1 4 1 5 1 6 1 7 1 8 1 4 1 5 1 6 1 7 1 8 1 9 1 5 1 6 1 6 1 7 1 7 1 7 1 8 1 8 1 8 1 8 1 9 1 9 1 9 1 9 1 9 1 9 1 10 1 10 1 10 1 10 1 10 1 10 1 10 1 10 1 10

497. Fraction Chart 1 Which is more, 3/4 or 4/5? 1 2 1 2 1 3 1 4 1 5 1 6 1 7 1 8 1 9 1 10 1 3 1 3 1 4 1 5 1 6 1 7 1 8 1 9 1 4 1 5 1 6 1 7 1 8 1 4 1 5 1 6 1 7 1 8 1 9 1 5 1 6 1 6 1 7 1 7 1 7 1 8 1 8 1 8 1 8 1 9 1 9 1 9 1 9 1 9 1 9 1 10 1 10 1 10 1 10 1 10 1 10 1 10 1 10 1 10

498. Fraction Chart 1 Which is more, 3/4 or 4/5? 1 2 1 2 1 3 1 4 1 5 1 6 1 7 1 8 1 9 1 10 1 3 1 3 1 4 1 5 1 6 1 7 1 8 1 9 1 4 1 5 1 6 1 7 1 8 1 4 1 5 1 6 1 7 1 8 1 9 1 5 1 6 1 6 1 7 1 7 1 7 1 8 1 8 1 8 1 8 1 9 1 9 1 9 1 9 1 9 1 9 1 10 1 10 1 10 1 10 1 10 1 10 1 10 1 10 1 10

499. Fraction Chart 1 Which is more, 3/4 or 4/5? 1 2 1 2 1 3 1 4 1 5 1 6 1 7 1 8 1 9 1 10 1 3 1 3 1 4 1 5 1 6 1 7 1 8 1 9 1 4 1 5 1 6 1 7 1 8 1 4 1 5 1 6 1 7 1 8 1 9 1 5 1 6 1 6 1 7 1 7 1 7 1 8 1 8 1 8 1 8 1 9 1 9 1 9 1 9 1 9 1 9 1 10 1 10 1 10 1 10 1 10 1 10 1 10 1 10 1 10

500. Fraction Chart Stairs (Unit fractions) 1 1 2 1 3 1 4 1 5 1 7 1 8 1 10 1 6 1 9

501. Fraction Chart A hyperbola. Stairs (Unit fractions) 1 1 2 1 3 1 4 1 5 1 7 1 8 1 10 1 6 1 9

502. Fraction Chart 1 9/8 is 1 and 1/8. 1 2 1 2 1 3 1 4 1 5 1 6 1 7 1 8 1 9 1 10 1 3 1 3 1 4 1 5 1 6 1 7 1 8 1 9 1 4 1 5 1 6 1 7 1 8 1 4 1 5 1 6 1 7 1 8 1 9 1 5 1 6 1 6 1 7 1 7 1 7 1 8 1 8 1 8 1 8 1 9 1 9 1 9 1 9 1 9 1 9 1 10 1 10 1 10 1 10 1 10 1 10 1 10 1 10 1 10 1 8

503. “ Pie” Model Are we comparing angles, arcs, or area?

504. “ Pie” Model Try to compare 4/5 and 5/6 with this model. 6 1 6 1 6 1 6 1 6 1 6 1 5 1 4 1 2 1 3 1 5 1 5 1 5 1 5 1 4 1 4 1 4 1 3 1 3 1 2 1

505. “ Pie” Model Experts in visual literacy say that comparing quantities in pie charts is difficult because most people think linearly. It is easier to compare along a straight line than compare pie slices. askoxford.com

506. “ Pie” Model Experts in visual literacy say that comparing quantities in pie charts is difficult because most people think linearly. It is easier to compare along a straight line than compare pie slices. askoxford.com Specialists also suggest refraining from using more than one pie chart for comparison. statcan.ca

507. “ Pie” Model Difficulties

514. Fraction War 1 1 2 1 2 1 4 1 4 1 4 1 4 1 8 1 8 1 8 1 8 1 8 1 8 1 8 1 8

515. Fraction War 1 1 2 1 2 1 4 1 4 1 4 1 4 1 8 1 8 1 8 1 8 1 8 1 8 1 8 1 8

516. Fraction War

517. Fraction War Especially useful for learning to read a ruler with inches.

518. Fraction War 1 1 2 1 2 1 4 1 4 1 4 1 8 1 8 1 8 1 8 1 8 1 8 1 8 1 4 1 8

519. Simplifying Fractions

520. Simplifying Fractions

521. Simplifying Fractions The fraction 4/8 can be reduced on the multiplication table as 1/2.

522. Simplifying Fractions The fraction 4/8 can be reduced on the multiplication table as 1/2.

523. Simplifying Fractions In what column would you put 21/28? 21 28

524. Simplifying Fractions In what column would you put 21/28? 21 28

525. Simplifying Fractions In what column would you put 21/28? 21 28

526. Simplifying Fractions 21 28 45 72

527. Simplifying Fractions 21 28 45 72

528. Simplifying Fractions 21 28 45 72

529. Simplifying Fractions 12 16

530. Simplifying Fractions 12 16

531. Simplifying Fractions 6/8 needs further simplifying. 12 16

532. Simplifying Fractions 6/8 needs further simplifying. 12 16

533. Simplifying Fractions 6/8 needs further simplifying. 12 16

534. Simplifying Fractions 12/16 could have put here originally. 12 16

535. In Conclusion

541. Counting: Necessary or Detrimental? AMS Conference March 25, 2011 Chicago, Illinois by Joan A. Cotter, Ph.D. [email_address] 7 5 2 Presentation available: ALabacus.com 7 x 7 VII