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# Fraction Mass HOPE April 2013

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• Notice what happens as the fractions get smaller and smaller. The curve is a hyperbola.
• Notice what happens as the fractions get smaller and smaller. The curve is a hyperbola.
• Notice what happens as the fractions get smaller and smaller. The curve is a hyperbola.
• Notice what happens as the fractions get smaller and smaller. The curve is a hyperbola.
• This is from a second grade textbook that was very popular in the 1980s.
• With this model, could you compare 2/5 and 1/4? Also, children will think fractions are two numbers, but they are one number just as 37 is one number.
• The 1993 textbook using this model does not say the figures represent one. So fourths look like four. How do you compare fourths and thirds?
• The 1993 textbook using this model does not say the figures represent one. So fourths look like four. How do you compare fourths and thirds?
• A study showed that many students and adults thought this was impossible.
• Writing the common multiple in a circle, 3, in this example, helps students remember what they’re dividing by.
• The fraction 4/8 can be simplified by using the multiplication table.
• The fraction 4/8 can be simplified by using the multiplication table.
• In what column could you put 21/28?
• Where can you put 45/72? The rows need not be contiguous.
• Oops, 6/8 is not the simplest form.
• (We could’ve arrived there sooner if 12/16 had been put in the 4s column.)
• 4 thousand minus 2 thousand is 2 thousand. . . .
• 4 thousand minus 2 thousand is 2 thousand.
• 4 hundred minus 3 hundred is 3 hundred.
• Add up the partial subtractions.
• 4 thousand minus 2 thousand is 2 thousand.
• 6 hundred minus 8 hundred is –2 hundred.
• 4 minus 9 is –5.
• Again, add up the partial subtractions.
• 1/7 minus 5/7 is –4/7.
• ### Fraction Mass HOPE April 2013

1. 1. FractionsMassHOPE-TEACHWorcester, MASaturday, April 27, 20132:45pm - 3:45pmJoan A. Cotter, Ph.D.JoanCotter@RightStartMath.com
2. 2. Why Learn Fractions• Sharing pizza
3. 3. Why Learn Fractions• Sharing pizza• Cooking and baking
4. 4. Why Learn Fractions• Sharing pizza• Cooking and baking• Reading rulers
5. 5. Why Learn Fractions• Sharing pizza• Cooking and baking• Reading rulers• Easing into decimals• Learning algebra
6. 6. Fractions in the Comics
7. 7. Fractions in the Comics
8. 8. Fraction Chart11212131415161718191101313141516171819141516171814151617181915161617171718181818191919191919110110110110110110110110110
9. 9. Fraction Chart
10. 10. Fraction Stairs1
11. 11. Fraction Stairs112
12. 12. Fraction Stairs11213
13. 13. Fraction Stairs1121314
14. 14. Fraction Stairs112131415
15. 15. Fraction Stairs11213141516
16. 16. Fraction Stairs1121314151716
17. 17. Fraction Stairs112131415171816
18. 18. Fraction Stairs11213141517181619
19. 19. Fraction Stairs11213141517181101619
20. 20. Fraction Stairs
21. 21. Fraction Stairs
22. 22. Fraction StairsA hyperbola.
23. 23. Fraction Names• In English, except for half, we useordinal numbers to name fractions.
24. 24. Sometimes called quarter.Fourths
25. 25. Sometimes called quarter.• A quarter of a hour (15 min.)Fourths
26. 26. Sometimes called quarter.• A quarter of a hour (15 min.)• A quarter of a dollar (25¢)Fourths
27. 27. Sometimes called quarter.• A quarter of a hour (15 min.)• A quarter of a dollar (25¢)• A quarter of a gallon (quart)Fourths
28. 28. Sometimes called quarter.• A quarter of a hour (15 min.)• A quarter of a dollar (25¢)• A quarter of a gallon (quart)• A Quarter Pounder (4 oz.)Fourths
29. 29. Writing Fractions1 one
30. 30. Writing Fractions1 onedivided by
31. 31. Writing Fractions13onedivided bythree
32. 32. Writing Fractions13onedivided bythree1131313
33. 33. Writing Fractions13onedivided bythreeAvoid saying “over” as in 1 over 3.1131313
34. 34. Unit Fraction WarObjective:To help the children realize a unitfraction decreases as the denominatorincreases.
35. 35. Unit Fraction WarObject of the game:To collect all, or most, of the cardswith the greater unit fraction.Objective:To help the children realize a unitfraction decreases as the denominatorincreases.
36. 36. Unit Fraction War
37. 37. 1514Unit Fraction War
38. 38. 1514Unit Fraction War
39. 39. Unit Fraction War
40. 40. Unit Fraction War118
41. 41. Unit Fraction War118
42. 42. Unit Fraction War
43. 43. Unit Fraction War1616
44. 44. Unit Fraction War1616
45. 45. Unit Fraction War16161314
46. 46. Unit Fraction War
47. 47. Fraction ChartHow many fourths in awhole?11212131415161718191101313141516171819141516171814151617181915161617171718181818191919191919110110110110110110110110110
48. 48. Fraction ChartHow many fourths in awhole?12131516171819110131516171819141516171814151617181915161617171718181818191919191919110110110110110110110110110141412113
49. 49. 12131516171819110131516171819141516171814151617181915161617171718181818191919191919110110110110110110110110110Fraction ChartHow many fourths in awhole?141412113
50. 50. Fraction ChartHow many fourths in awhole?13151617181911013151617181915161718141516171819151616171717181818181919191919191101101101101101101101101101414121131412
51. 51. Fraction ChartHow many fourths in awhole?13151617181911015161718191516171815161718191516161717171818181819191919191911011011011011011011011011014141211312131414
52. 52. Fraction ChartHow many sixths in awhole?11212131415161718191101313141516171819141516171814151617181915161617171718181818191919191919110110110110110110110110110
53. 53. Fraction ChartHow many eighths in awhole?11212131415161718191101313141516171819141516171814151617181915161617171718181818191919191919110110110110110110110110110
54. 54. Concentrating on One GameObjective:To help the children realize that 5 fifths, 8eighths, and so forth, make a whole.
55. 55. Concentrating on One GameObject of the game:To find the pairs that make a whole.Objective:To help the children realize that 5 fifths, 8eighths, and so forth, make a whole.
56. 56. Concentrating on One
57. 57. Concentrating on One53
58. 58. Concentrating on One53
59. 59. Concentrating on One53
60. 60. Concentrating on One5325
61. 61. Concentrating on One
62. 62. Concentrating on One38
63. 63. Concentrating on One38
64. 64. Concentrating on One38
65. 65. Concentrating on One3878
66. 66. Concentrating on One
67. 67. Concentrating on One58
68. 68. Concentrating on One58
69. 69. Concentrating on One58
70. 70. Concentrating on One3858
71. 71. Concentrating on One
72. 72. Concentrating on One
73. 73. Fraction ChartWhich is more, 3/4 or4/5?11212131415161718191101313141516171819141516171814151617181915161617171718181818191919191919110110110110110110110110110
74. 74. 11212131415161718191101313141516171819141516171814151617181915161617171718181818191919191919110110110110110110110110110Fraction ChartWhich is more, 3/4 or4/5?
75. 75. 11212131415161718191101313141516171819141516171814151617181915161617171718181818191919191919110110110110110110110110110Fraction ChartWhich is more, 3/4 or4/5?
76. 76. Fraction ChartWhich is more, 7/8 or8/9?11212131415161718191101313141516171819141516171814151617181915161617171718181818191919191919110110110110110110110110110
77. 77. 11212131415161718191101313141516171819141516171814151617181915161617171718181818191919191919110110110110110110110110110Fraction ChartWhich is more, 7/8 or8/9?
78. 78. 11212131415161718191101313141516171819141516171814151617181915161617171718181818191919191919110110110110110110110110110Fraction ChartAn interesting pattern.
79. 79. Partial Chart11414141412121818181818181818
80. 80. Partial Chart11414141412121818181818181818
81. 81. Partial Chart
82. 82. Partial Chart1 2 3 4 5 6
83. 83. Fraction War
84. 84. Fraction WarObjective:To practice comparing ones, halves,fourths, and eighths in preparation forreading a ruler.
85. 85. Fraction WarObject of the game:To capture all the cards.Objective:To practice comparing ones, halves,fourths, and eighths in preparation forreading a ruler.
86. 86. Fraction War11414141412121818181818181818
87. 87. Fraction War141811414141412121818181818181818
88. 88. Fraction War141811414141412121818181818181818
89. 89. 11414141412121818181818181818Fraction War1418
90. 90. Fraction War11414141412121818181818181818
91. 91. 11414141412121818181818181818Fraction War3458
92. 92. 11414141412121818181818181818Fraction War3458
93. 93. 11414141412121818181818181818Fraction War3458
94. 94. Fraction War11414141412121818181818181818
95. 95. Fraction War343411414141412121818181818181818
96. 96. Fraction War343411414141412121818181818181818
97. 97. Fraction War3434381411414141412121818181818181818
98. 98. Fraction War11414141412121818181818181818
99. 99. Fraction War11414141412121818181818181818
100. 100. 11212131415161718191101313141516171819141516171814151617181915161617171718181818191919191919110110110110110110110110110Fraction ChartHow many fourths equal a half?
101. 101. 11212141516171819110141516171819141516171814151617181915161617171718181818191919191919110110110110110110110110110Fraction ChartHow many fourths equal a half?
102. 102. 11212131415161718191101313141516171819141516171814151617181915161617171718181818191919191919110110110110110110110110110Fraction ChartHow many fourths equal a half?
103. 103. 11212131415161718191101313141516171819141516171814151617181915161617171718181818191919191919110110110110110110110110110Fraction ChartHow many fourths equal a half? Eighths?
104. 104. 11212131415161718191101313141516171819141516171814151617181915161617171718181818191919191919110110110110110110110110110Fraction ChartHow many fourths equal a half? Eighths?
105. 105. 11212131415161718191101313141516171819141516171814151617181915161617171718181818191919191919110110110110110110110110110Fraction ChartHow many fourths equal a half? Eighths?Sevenths?
106. 106. 11212131415161718191101313141516171819141516171814151617181915161617171718181818191919191919110110110110110110110110110Fraction ChartHow many fourths equal a half? Eighths?Sevenths?
107. 107. 11212131415161718191101313141516171819141516171814151617181915161617171718181818191919191919110110110110110110110110110Fraction ChartWhat is 1/4 plus 1/8?
108. 108. 11212131415161718191101313141516171819141516171814151617181915161617171718181818191919191919110110110110110110110110110Fraction ChartWhat is 1/4 plus 1/8?
109. 109. 11212131415161718191101313141516171819141516171814151617181915161617171718181818191919191919110110110110110110110110110Fraction ChartWhat is 1/4 plus 1/8?
110. 110. 11212131415161718191101313141516171819141516171814151617181915161617171718181818191919191919110110110110110110110110110Fraction ChartWhat is 1/4 plus 1/8? [3/8]
111. 111. 11212131415161718191101313141516171819141516171814151617181915161617171718181818191919191919110110110110110110110110110Fraction ChartIf the chance of rain is 3/4, the chance it won’t rain is1/4.
112. 112. 11212131415161718191101313141516171819141516171814151617181915161617171718181818191919191919110110110110110110110110110Fraction ChartIf the chance of rain is 3/4, the chance it won’t rain is1/4.
113. 113. Faulty Fractions
114. 114. Faulty Fractions“Fish Tank” model
115. 115. Faulty Fractions2“Fish Tank” model
116. 116. Faulty Fractions25“Fish Tank” model
117. 117. Faulty Fractions= partwholeCRA model
118. 118. Faulty Fractions= partwhole“Goal: To develop the spatial organization, visuallyand kinesthetically, to read and write fractionscorrectly.CRA model
119. 119. Faulty Fractions= partwhole“Goal: To develop the spatial organization, visuallyand kinesthetically, to read and write fractionscorrectly.“Materials: Red squares and larger black squares aredisplayed to help with sequencing and numberplacement.”CRA model
120. 120. Faulty FractionsThis is fourths.“Words” model
121. 121. Faulty FractionsThis is fourths. This is thirds.“Words” model
122. 122. Faulty Fractions13131314141414“Rounded corners”
123. 123. Faulty FractionsThe middle fractions are greaterthan the fractions at the ends!13131314141414“Rounded corners”
124. 124. Faulty Fractions121121313136 6 6 6 6 617171717171717191101101101101101101101101101101818181818181818141414191919191919191915151515151 1 1 1 1 114“Color” model
125. 125. 114141414121211011011011011011011011011011018181818181818181313131515151515161616161616Faulty Fractions112112112112112112112112112112112112Missing 7ths & 9ths
126. 126. 112Faulty FractionsMissing 7ths & 9ths1121314151811016
127. 127. Faulty FractionsAre we comparing angles, arcs, or area?Circles
128. 128. Faulty Fractions6161616161615141213151515151414141313121Try to compare 4/5 and 5/6 with this model.Circles
129. 129. Faulty FractionsExperts in visual literacy say thatcomparing quantities in pie charts isdifficult because most people thinklinearly. It is easier to compare along astraight line than compare pie slices.askoxford.comCircles
130. 130. Faulty FractionsExperts in visual literacy say thatcomparing quantities in pie charts isdifficult because most people thinklinearly. It is easier to compare along astraight line than compare pie slices.askoxford.comSpecialists also suggest refraining fromusing more than one pie chart forcomparison.www.statcan.caCircles
131. 131. Definition of a FractionWhat is the definition of a fraction?
132. 132. Definition of a FractionWhat is the definition of a fraction?A part of a set or part of a whole, a small part.
133. 133. Definition of a FractionWhat is the definition of a fraction?A part of a set or part of a whole, a small part.This is the everyday meaning of fraction.
134. 134. Definition of a Fraction32What about ?What is the definition of a fraction?A part of a set or part of a whole, a small part.This is the everyday meaning of fraction.
135. 135. Definition of a FractionAn expression that indicatesthequotient of two quantities.American Heritage Dictionary:
136. 136. Definition of a FractionAn expression that indicatesthequotient of two quantities.This is the mathematical meaning of fraction.American Heritage Dictionary:
137. 137. Definition of a FractionThis is the mathematical meaning of fraction.32An expression that indicatesthequotient of two quantities.American Heritage Dictionary:
138. 138. Fractions > 111212131415161718191101313141516171819141516171814151617181915161617171718181818191919191919110110110110110110110110110
139. 139. Fractions > 11121213141516171819110131314151617181914151617181415161718191516161717171818181819191919191911011011011011011011011011018
140. 140. Mixed to Improper FractionsEach row of connected rectangles represents 1.Write each quantity as a mixed numberand as an improper fraction.
141. 141. Mixed to Improper FractionsEach row of connected rectangles represents 1.Write each quantity as a mixed numberand as an improper fraction.
142. 142. Mixed to Improper FractionsEach row of connected rectangles represents 1.2 =34114Write each quantity as a mixed numberand as an improper fraction.
143. 143. Write each quantity as a mixed numberand as an improper fraction.Mixed to Improper Fractions2 =34114Each row of connected rectangles represents 1.two 4s
144. 144. Write each quantity as a mixed numberand as an improper fraction.Mixed to Improper Fractions2 =3411424two 4sEach row of connected rectangles represents 1.
145. 145. Write each quantity as a mixed numberand as an improper fraction.Mixed to Improper Fractions2 =3411424Each row of connected rectangles represents 1.two 4s + 3
146. 146. Write each quantity as a mixed numberand as an improper fraction.Mixed to Improper Fractions2 =3411443Each row of connected rectangles represents 1.two 4s + 32
147. 147. Write each quantity as a mixed numberand as an improper fraction.Mixed to Improper Fractions2 =3411443Each row of connected rectangles represents 1.two 4s + 3 = 112
148. 148. Write each quantity as a mixed numberand as an improper fraction.Mixed to Improper Fractions2 =3411443 11Each row of connected rectangles represents 1.two 4s + 3 = 112
149. 149. Write each quantity as a mixed numberand as an improper fraction.Mixed to Improper Fractions2 =341144 =231434113Each row of connected rectangles represents 1.two 4s + 3 = 112
150. 150. Write each quantity as a mixed numberand as an improper fraction.Mixed to Improper Fractions2 =341144 =231434113Each row of connected rectangles represents 1.two 4s + 3 = 11four 3s + 2 = 142
151. 151. Write each quantity as a mixed numberand as an improper fraction.Mixed to Improper Fractions2 =341144 =231434 =35235Each row of connected rectangles represents 1.4113two 4s + 3 = 11four 3s + 2 = 142
152. 152. Write each quantity as a mixed numberand as an improper fraction.Mixed to Improper Fractions2 =341144 =231434 =35235Each row of connected rectangles represents 1.4113four 3s + 2 = 14two 4s + 3 = 112
153. 153. Write each quantity as a mixed numberand as an improper fraction.Mixed to Improper Fractions2 =341144 =231434 =35235Each row of connected rectangles represents 1.24113four 3s + 2 = 14two 4s + 3 = 11four 5s + 3 = 23
154. 154. Improper to Mixed FractionsCircle the wholes and write each quantity asan improper fraction and as a mixed number.
155. 155. Improper to Mixed FractionsCircle the wholes and write each quantity asan improper fraction and as a mixed number.= 211515
156. 156. Improper to Mixed FractionsCircle the wholes and write each quantity asan improper fraction and as a mixed number.= 211515
157. 157. Improper to Mixed FractionsCircle the wholes and write each quantity asan improper fraction and as a mixed number.= 211515
158. 158. Improper to Mixed FractionsCircle the wholes and write each quantity asan improper fraction and as a mixed number.= 211515= 15323
159. 159. Improper to Mixed FractionsCircle the wholes and write each quantity asan improper fraction and as a mixed number.= 211515= 15323
160. 160. Fraction of Geometric Figures12Shade
161. 161. Fraction of Geometric Figures12Shade
166. 166. Making the WholeDraw the whole.13
167. 167. Making the WholeDraw the whole.13131313
168. 168. Making the WholeDraw the whole.1313131323
169. 169. Making the WholeDraw the whole.13131313232313
170. 170. 112131415161718191101313141516171819141516171814151617181915161617171718181818191919191919110110110110110110110110110Fraction ChartWhat is 1/2 of 1/2?12
171. 171. 1121315161718191101313141516171819141516171814151617181915161617171718181818191919191919110110110110110110110110110Fraction ChartWhat is 1/2 of 1/2?1214
172. 172. 112131415161718191101313141516171819141516171814151617181915161617171718181818191919191919110110110110110110110110110Fraction ChartWhat is 1/3 of 1/2?12
173. 173. Fraction ChartWhat is 1/3 of 1/2?11213141516171819110131314151718191415161718141516171819151616171717181818181919191919191101101101101101101101101101216
174. 174. Simplifying Fractions12
175. 175. Simplifying Fractions36= 12
176. 176. Simplifying Fractions48= 12
177. 177. Simplifying Fractions
178. 178. Simplifying Fractions912
179. 179. Simplifying Fractions912 3
180. 180. Simplifying Fractions912 3
181. 181. Simplifying Fractions912= 343
182. 182. Simplifying Fractions
183. 183. Simplifying Fractions
184. 184. Simplifying Fractions
185. 185. Simplifying Fractions
186. 186. Simplifying Fractions2128
187. 187. Simplifying Fractions2128
188. 188. Simplifying Fractions2128
189. 189. Simplifying Fractions4572
190. 190. Simplifying Fractions4572
191. 191. Simplifying Fractions4572
192. 192. Simplifying Fractions1216
193. 193. Simplifying Fractions1216
194. 194. Simplifying Fractions1216
195. 195. Simplifying Fractions1216
196. 196. Simplifying Fractions1216
197. 197. Simplifying Fractions1216
198. 198. © Joan A. Cotter, Ph.D., 2013Multiples PatternsTwos2 4 6 8 1012 14 16 18 20
199. 199. © Joan A. Cotter, Ph.D., 2013Multiples PatternsTwos2 4 6 8 1012 14 16 18 20The ones repeat in the secondrow.
200. 200. © Joan A. Cotter, Ph.D., 2013Multiples PatternsFours4 8 12 16 2024 28 32 36 40The ones repeat in the secondrow.
201. 201. © Joan A. Cotter, Ph.D., 2013Multiples PatternsSixes and Eights6 12 18 24 3036 42 48 54 608 16 24 32 4048 56 64 72 80
202. 202. © Joan A. Cotter, Ph.D., 2013Multiples PatternsSixes and Eights6 12 18 24 3036 42 48 54 608 16 24 32 4048 56 64 72 80
203. 203. © Joan A. Cotter, Ph.D., 2013Multiples PatternsSixes and Eights6 12 18 24 3036 42 48 54 608 16 24 32 4048 56 64 72 80
204. 204. © Joan A. Cotter, Ph.D., 2013Multiples PatternsSixes and Eights6 12 18 24 3036 42 48 54 608 16 24 32 4048 56 64 72 80The ones in the 8s show the multiples
205. 205. © Joan A. Cotter, Ph.D., 2013Multiples PatternsSixes and Eights6 12 18 24 3036 42 48 54 608 16 24 32 4048 56 64 72 80The ones in the 8s show the multiples
206. 206. © Joan A. Cotter, Ph.D., 2013Multiples PatternsSixes and Eights6 12 18 24 3036 42 48 54 608 16 24 32 4048 56 64 72 80The ones in the 8s show the multiples
207. 207. © Joan A. Cotter, Ph.D., 2013Multiples PatternsSixes and Eights6 12 18 24 3036 42 48 54 608 16 24 32 4048 56 64 72 80The ones in the 8s show the multiples
208. 208. © Joan A. Cotter, Ph.D., 2013Multiples PatternsSixes and Eights6 12 18 24 3036 42 48 54 608 16 24 32 4048 56 64 72 80The ones in the 8s show the multiples
209. 209. © Joan A. Cotter, Ph.D., 2013Multiples PatternsSixes and Eights6 12 18 24 3036 42 48 54 608 16 24 32 4048 56 64 72 806 x 46 x 4 is the fourth number
210. 210. © Joan A. Cotter, Ph.D., 2013Multiples PatternsSixes and Eights6 12 18 24 3036 42 48 54 608 16 24 32 4048 56 64 72 80 8 x 78 x 7 is the seventh number
211. 211. © Joan A. Cotter, Ph.D., 2013Multiples PatternsNines9 18 27 36 4590 81 72 63 54The second row is written in reverseorder.Also the digits in each number add to9.
212. 212. © Joan A. Cotter, Ph.D., 2013Multiples PatternsThrees3 6 912 15 1821 24 2730The 3s have several patterns:Observe the ones.
213. 213. © Joan A. Cotter, Ph.D., 2013Multiples PatternsThrees3 6 912 15 1821 24 2730The 3s have several patterns:Observe the ones.
214. 214. © Joan A. Cotter, Ph.D., 2013Multiples PatternsThrees3 6 912 15 1821 24 2730The 3s have several patterns:Observe the ones.
215. 215. © Joan A. Cotter, Ph.D., 2013Multiples PatternsThrees3 6 912 15 1821 24 2730The 3s have several patterns:Observe the ones.
216. 216. © Joan A. Cotter, Ph.D., 2013Multiples PatternsThrees3 6 912 15 1821 24 2730The 3s have several patterns:Observe the ones.
217. 217. © Joan A. Cotter, Ph.D., 2013Multiples PatternsThrees3 6 912 15 1821 24 2730The 3s have several patterns:Observe the ones.
218. 218. © Joan A. Cotter, Ph.D., 2013Multiples PatternsThrees3 6 912 15 1821 24 2730The 3s have several patterns:Observe the ones.
219. 219. © Joan A. Cotter, Ph.D., 2013Multiples PatternsThrees3 6 912 15 1821 24 2730The 3s have several patterns:Observe the ones.
220. 220. © Joan A. Cotter, Ph.D., 2013Multiples PatternsThrees3 6 912 15 1821 24 2730The 3s have several patterns:Observe the ones.
221. 221. © Joan A. Cotter, Ph.D., 2013Multiples PatternsThrees3 6 912 15 1821 24 2730The 3s have several patterns:Observe the ones.
222. 222. © Joan A. Cotter, Ph.D., 2013Multiples PatternsThrees3 6 912 15 1821 24 2730The 3s have several patterns:Observe the ones.
223. 223. © Joan A. Cotter, Ph.D., 2013Multiples PatternsThrees3 6 912 15 1821 24 2730The 3s have several patterns:The tens are the same in each row.
224. 224. © Joan A. Cotter, Ph.D., 2013Multiples PatternsThrees3 6 912 15 1821 24 2730The 3s have several patterns:Add the digits in the columns.
225. 225. © Joan A. Cotter, Ph.D., 2013Multiples PatternsThrees3 6 912 15 1821 24 2730The 3s have several patterns:Add the digits in the columns.
226. 226. © Joan A. Cotter, Ph.D., 2013Multiples PatternsThrees3 6 912 15 1821 24 2730The 3s have several patterns:Add the digits in the columns.
227. 227. © Joan A. Cotter, Ph.D., 2013Multiples PatternsThrees3 6 912 15 1821 24 2730The 3s have several patterns:Add the “opposites.”
228. 228. © Joan A. Cotter, Ph.D., 2013Multiples PatternsThrees3 6 912 15 1821 24 2730The 3s have several patterns:Add the “opposites.”
229. 229. © Joan A. Cotter, Ph.D., 2013Multiples PatternsThrees3 6 912 15 1821 24 2730The 3s have several patterns:Add the “opposites.”
230. 230. © Joan A. Cotter, Ph.D., 2013Multiples PatternsThrees3 6 912 15 1821 24 2730The 3s have several patterns:Add the “opposites.”
231. 231. © Joan A. Cotter, Ph.D., 2013Multiples PatternsSevens7 14 2128 35 4249 56 6370The 7s have the 1, 2, 3… pattern.
232. 232. © Joan A. Cotter, Ph.D., 2013Multiples PatternsSevens7 14 2128 35 4249 56 6370The 7s have the 1, 2, 3… pattern.
233. 233. © Joan A. Cotter, Ph.D., 2013Multiples PatternsSevens7 14 2128 35 4249 56 6370The 7s have the 1, 2, 3… pattern.
234. 234. © Joan A. Cotter, Ph.D., 2013Multiples PatternsSevens7 14 2128 35 4249 56 6370The 7s have the 1, 2, 3… pattern.
235. 235. © Joan A. Cotter, Ph.D., 2013Multiples PatternsSevens7 14 2128 35 4249 56 6370Look at the tens.
236. 236. © Joan A. Cotter, Ph.D., 2013Multiples PatternsSevens7 14 2128 35 4249 56 6370Look at the tens.
237. 237. © Joan A. Cotter, Ph.D., 2013Multiples PatternsSevens7 14 2128 35 4249 56 6370Look at the tens.
238. 238. Subtracting Fractions4684–2372Preliminary understanding
239. 239. Subtracting Fractions4684–23722000Preliminary understanding
240. 240. Subtracting Fractions4684–23722000300Preliminary understanding
241. 241. Subtracting Fractions4684–2372200030010Preliminary understanding
242. 242. Subtracting Fractions4684–23722000300102Preliminary understanding
243. 243. Subtracting Fractions4684–237220003001022312Preliminary understanding
244. 244. Subtracting Fractions4684–23724684–287920003001022312Preliminary understanding
245. 245. Subtracting Fractions4684–23724684–2879200030010223122000Preliminary understanding
246. 246. Subtracting Fractions4684–23724684–2879200030010223122000–200Preliminary understanding
247. 247. Subtracting Fractions4684–23724684–2879200030010223122000–20010Preliminary understanding
248. 248. Subtracting Fractions4684–23724684–2879200030010223122000–20010–5Preliminary understanding
249. 249. Subtracting Fractions4684–23724684–2879200030010223122000–20010–51805Preliminary understanding
250. 250. Subtracting Fractions354–
251. 251. Subtracting Fractions354–325
252. 252. Subtracting Fractions354–325575– 217
253. 253. Subtracting Fractions354–3253575– 217
254. 254. Subtracting Fractions354–3253– 47575– 217
255. 255. Subtracting Fractions354–3253– 47237575– 217
256. 256. Multiplying Fractions
257. 257. Multiplying Fractions• Multiplication is more than repeatedaddition.
258. 258. Multiplying Fractions4 x 4 = 4 + 4 + 4 + 4• Multiplication is more than repeatedaddition.
259. 259. Multiplying Fractions4 x 4 = 4 + 4 + 4 + 4• Repeated addition doesn’t work well withfractions.• Multiplication is more than repeatedaddition.
260. 260. Multiplying Fractions• Repeated addition doesn’t work well withfractions.12x = + ?12124 x 4 = 4 + 4 + 4 + 4• Multiplication is more than repeatedaddition.
261. 261. Multiplying FractionsArea is a bettermodel.4 x 4 =
262. 262. Multiplying Fractions12x =12The square represents 1.
263. 263. Multiplying Fractions12x =12
264. 264. Multiplying Fractions12x =1214The solution is the double-crosshatchedarea.
265. 265. Multiplying Fractions23x =34
266. 266. Multiplying Fractions23x =34
267. 267. Multiplying Fractions23x =34
268. 268. Multiplying Fractions23x =34612
269. 269. Multiplying Fractionsx =3412=23612
270. 270. Multiplying Fractions23x =34The total number of rectangles is 3 x 4.
271. 271. Multiplying Fractions2334The number of double-crosshatched rectangles is 2The total number of rectangles is 3 x 4.x =
272. 272. Multiplying Fractions2334This is why we multiply fractions bymultiplying numerators and denominators.x =
273. 273. Dividing Fractions
274. 274. Dividing Fractions÷ =121
275. 275. Dividing Fractions÷ =1211 ÷ 1/2 means how many 1/2s in 1.
276. 276. Dividing Fractions÷ =12111212141414141313131 ÷ 1/2 means how many 1/2s in 1.
277. 277. Dividing Fractions÷ =12111212141414141313131 ÷ 1/2 means how many 1/2s in 1.
278. 278. Dividing Fractions÷ =12111212141414141313131 ÷ 1/2 means how many 1/2s in 1.
279. 279. Dividing Fractions÷ =121 211212141414141313131 ÷ 1/2 means how many 1/2s in 1.
280. 280. Dividing Fractions÷ =121 2÷ =13111212141414141313131 ÷ 1/3 means how many 1/3s in 1.
281. 281. Dividing Fractions÷ =121 2÷ =13111212141414141313131 ÷ 1/3 means how many 1/3s in 1.
282. 282. Dividing Fractions÷ =121 2÷ =13111212141414141313131 ÷ 1/3 means how many 1/3s in 1.
283. 283. Dividing Fractions÷ =121 2÷ =13111212141414141313131 ÷ 1/3 means how many 1/3s in 1.
284. 284. Dividing Fractions÷ =121 2÷ =131 311212141414141313131 ÷ 1/3 means how many 1/3s in 1.
285. 285. Dividing Fractions÷ =121 2÷ =131 3÷ =2311121214141414131313
286. 286. Dividing Fractions÷ =121 2÷ =131 3÷ =23111212141414141313131 ÷ 2/3 means how many 2/3s in 1.
287. 287. Dividing Fractions÷ =121÷ =131÷ =231112121414141423231 ÷ 2/3 means how many 2/3s in 1.
288. 288. Dividing Fractions÷ =121 2÷ =131 3÷ =231112121414141423231 ÷ 2/3 means how many 2/3s in 1.
289. 289. Dividing Fractions÷ =121÷ =131÷ =2311121214141414232332231 ÷ 2/3 means how many 2/3s in 1.
290. 290. Dividing Fractions÷ =121÷ =131÷ =2311121214141414232332231 ÷ 2/3 also must be half of 1 ÷ 1/3.
291. 291. Dividing Fractions÷ =121÷ =131÷ =1 3÷ =231 32112121414141413131323
292. 292. Dividing Fractions÷ =121÷ =131÷ =1 3÷ =231 321121214141414131313231 ÷ 3 is simply the definition of a fraction.
293. 293. 1121214141414131313Dividing Fractions÷ =121÷ =13113÷ =1 3÷ =231 32231 ÷ 3 is simply the definition of a fraction.
294. 294. Dividing Fractions÷ =121÷ =131÷ =1 3÷ =1 4÷ =2311121214141414131313322313
295. 295. Dividing Fractions÷ =121÷ =131 14÷ =1 3÷ =1 4÷ =2311121214141414131313322313
296. 296. Dividing Fractions1314÷ =1 3÷ =1 4÷ =1 43121131÷ =231÷ =÷ =11212141414141313133223
297. 297. Dividing Fractions÷ =1 3÷ =1 41121213131314141414÷ =1 43121131÷ =231÷ =÷ =11212141414141313131313143223
298. 298. Dividing Fractions÷ =1 3÷ =1 41121213131314141414÷ =1 43121131÷ =231÷ =÷ =11212141414141313131313143223Only 3/4 of the 4/3 fits into the 1.
299. 299. Dividing Fractions÷ =1 3÷ =1 41121213131314141414÷ =1 43121131÷ =231÷ =÷ =1121214141414131313131314322334Only 3/4 of the 4/3 fits into the 1.
300. 300. Dividing Fractions÷ =1 3÷ =1 4÷ =1 43121131÷ =231÷ =÷ =1314322334
301. 301. Dividing Fractions÷ =121 2÷ =131 31314÷ =1 3÷ =1 434÷ =1 43÷ =231 32
302. 302. Dividing Fractions÷ =121 2÷ =131 31314÷ =1 3÷ =1 434÷ =1 43÷ =231 32The colored pairs are reciprocals of eachother.A reciprocal may be called a multiplicativeinverse.
303. 303. Dividing Fractions÷ =121 2÷ =131 31314÷ =1 3÷ =1 434÷ =1 43÷ =231 32The colored pairs are reciprocals of eachother.A reciprocal may be called a multiplicativeinverse. When multiplied together, theyequal 1.
304. 304. Dividing Fractions÷ =121 2÷ =131 31314÷ =1 3÷ =1 434÷ =1 43÷ =231 32The colored pairs are reciprocals of eachother.A reciprocal may be called a multiplicativeinverse. When multiplied together, theyequal 1.In the equation 6 ÷ 2 = 3, 6 = 2 x 3.
305. 305. Dividing Fractions÷ =121 2÷ =131 31314÷ =1 31÷ =1 4134÷ =1 43÷ =231 32
306. 306. Dividing FractionsSometimes textbooks put a 1 under awhole number to make it look like afraction, but it is not necessary.÷ =121 2÷ =131 31314÷ =1 31÷ =1 4134÷ =1 43÷ =231 32
307. 307. Dividing Fractions÷ = __235To find
308. 308. Dividing Fractions÷ = __235To findFirst think about finding 1 ÷ 2/3.
309. 309. Dividing Fractions÷ = __235÷ =231First findTo find
310. 310. Dividing Fractions÷ = __235÷ =231 32First findTo find
311. 311. Dividing Fractions÷ =235÷ = __235÷ =231 32First findTo findThen
312. 312. Dividing Fractions÷ = __235÷ =231 32First findTo findThen ÷ =235 5 (1 )x ÷23
313. 313. Dividing Fractions= x =325÷ = __235÷ =231 32First findTo findThen ÷ =235 5 (1 )x ÷23
314. 314. Dividing Fractions= x =321525÷ = __235÷ =231 32First findTo findThen ÷ =235 5 (1 )x ÷23
315. 315. Dividing Fractions= x = =32125 7÷ = __235÷ =231 32First findTo findThen ÷ =235 5 (1 )x ÷23152
316. 316. Dividing Fractions= x = =32125 7÷ = __235÷ =231 32First findTo findThen ÷ =235 5 (1 )x ÷23152Does the answer make sense?About how many 2/3s are in 5?
317. 317. 23Dividing Fractions÷ = __Find34
318. 318. 23Dividing Fractions÷ = __Find34(Is the answer more or less than 1?)
319. 319. 114141414121213131323Dividing Fractions÷ = __Find34(Is the answer more or less than 1?)
320. 320. 114141414121213131323Dividing Fractions÷ = __Find34(Is the answer more or less than 1?)
321. 321. 114141414121213131323Dividing Fractions÷ = __Find34(The answer must be less than 1.)
322. 322. Dividing FractionsTo find23÷ = __34÷ =341First find
323. 323. Dividing FractionsTo find23÷ = __34÷ =341 43First find
324. 324. Dividing Fractions÷ =341 43First findTo findThen÷ = __3423÷ =3423
325. 325. Dividing Fractions÷ =341 43First findTo findThen÷ = __342334233423÷ = x (1 ÷ )
326. 326. Dividing Fractions÷ =341 43First findTo find ÷ = __3423Then34233423÷ = x (1 ÷ )= x =4323
327. 327. = x =Dividing Fractions÷ =341 43First findTo find ÷ = __3423Then34233423÷ = x (1 ÷ )894323
328. 328. = x =Dividing Fractions÷ =341 43First findTo find ÷ = __3423Then34233423÷ = x (1 ÷ )894323The answer should be < 1 and it is.
329. 329. = x =Dividing Fractions÷ =341 43First findTo find ÷ = __3423Then34233423÷ = x (1 ÷ )894323The extra step of dividing by 1 can be omitted.
330. 330. ÷ = x (1 ÷ )= x =Dividing Fractions÷ =341 43First findTo find ÷ = __3423Then34233423894323The extra step of dividing by 1 can be omitted.
331. 331. Dividing FractionsIt’s ours to reason whyWe invert and multiply.
332. 332. Fraction Meanings
333. 333. Fraction Meanings• One or more equal parts of a whole.
334. 334. Fraction Meanings• One or more equal parts of a whole.• One or more equal parts of a collection.
335. 335. Fraction Meanings• One or more equal parts of a whole.• One or more equal parts of a collection.• Division of two whole numbers.
336. 336. Fraction Meanings• One or more equal parts of a whole.• One or more equal parts of a collection.• Location on a number line.• Division of two whole numbers.
337. 337. Fraction Meanings• One or more equal parts of a whole.• Ratio of two numbers.• One or more equal parts of a collection.• Location on a number line.• Division of two whole numbers.
338. 338. FractionsMassHOPE-TEACHWorcester, MASaturday, April 27, 20132:45pm - 3:45pmJoan A. Cotter, Ph.D.JoanCotter@RightStartMath.com