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.

Transcript

  • 1. FractionsMassHOPE-TEACHWorcester, MASaturday, April 27, 20132:45pm - 3:45pmJoan A. Cotter, Ph.D.JoanCotter@RightStartMath.com
  • 2. Why Learn Fractions• Sharing pizza
  • 3. Why Learn Fractions• Sharing pizza• Cooking and baking
  • 4. Why Learn Fractions• Sharing pizza• Cooking and baking• Reading rulers
  • 5. Why Learn Fractions• Sharing pizza• Cooking and baking• Reading rulers• Easing into decimals• Learning algebra
  • 6. Fractions in the Comics
  • 7. Fractions in the Comics
  • 8. Fraction Chart11212131415161718191101313141516171819141516171814151617181915161617171718181818191919191919110110110110110110110110110
  • 9. Fraction Chart
  • 10. Fraction Stairs1
  • 11. Fraction Stairs112
  • 12. Fraction Stairs11213
  • 13. Fraction Stairs1121314
  • 14. Fraction Stairs112131415
  • 15. Fraction Stairs11213141516
  • 16. Fraction Stairs1121314151716
  • 17. Fraction Stairs112131415171816
  • 18. Fraction Stairs11213141517181619
  • 19. Fraction Stairs11213141517181101619
  • 20. Fraction Stairs
  • 21. Fraction Stairs
  • 22. Fraction StairsA hyperbola.
  • 23. Fraction Names• In English, except for half, we useordinal numbers to name fractions.
  • 24. Sometimes called quarter.Fourths
  • 25. Sometimes called quarter.• A quarter of a hour (15 min.)Fourths
  • 26. Sometimes called quarter.• A quarter of a hour (15 min.)• A quarter of a dollar (25¢)Fourths
  • 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. 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. Writing Fractions1 one
  • 30. Writing Fractions1 onedivided by
  • 31. Writing Fractions13onedivided bythree
  • 32. Writing Fractions13onedivided bythree1131313
  • 33. Writing Fractions13onedivided bythreeAvoid saying “over” as in 1 over 3.1131313
  • 34. Unit Fraction WarObjective:To help the children realize a unitfraction decreases as the denominatorincreases.
  • 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. Unit Fraction War
  • 37. 1514Unit Fraction War
  • 38. 1514Unit Fraction War
  • 39. Unit Fraction War
  • 40. Unit Fraction War118
  • 41. Unit Fraction War118
  • 42. Unit Fraction War
  • 43. Unit Fraction War1616
  • 44. Unit Fraction War1616
  • 45. Unit Fraction War16161314
  • 46. Unit Fraction War
  • 47. Fraction ChartHow many fourths in awhole?11212131415161718191101313141516171819141516171814151617181915161617171718181818191919191919110110110110110110110110110
  • 48. Fraction ChartHow many fourths in awhole?12131516171819110131516171819141516171814151617181915161617171718181818191919191919110110110110110110110110110141412113
  • 49. 12131516171819110131516171819141516171814151617181915161617171718181818191919191919110110110110110110110110110Fraction ChartHow many fourths in awhole?141412113
  • 50. Fraction ChartHow many fourths in awhole?13151617181911013151617181915161718141516171819151616171717181818181919191919191101101101101101101101101101414121131412
  • 51. Fraction ChartHow many fourths in awhole?13151617181911015161718191516171815161718191516161717171818181819191919191911011011011011011011011011014141211312131414
  • 52. Fraction ChartHow many sixths in awhole?11212131415161718191101313141516171819141516171814151617181915161617171718181818191919191919110110110110110110110110110
  • 53. Fraction ChartHow many eighths in awhole?11212131415161718191101313141516171819141516171814151617181915161617171718181818191919191919110110110110110110110110110
  • 54. Concentrating on One GameObjective:To help the children realize that 5 fifths, 8eighths, and so forth, make a whole.
  • 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. Concentrating on One
  • 57. Concentrating on One53
  • 58. Concentrating on One53
  • 59. Concentrating on One53
  • 60. Concentrating on One5325
  • 61. Concentrating on One
  • 62. Concentrating on One38
  • 63. Concentrating on One38
  • 64. Concentrating on One38
  • 65. Concentrating on One3878
  • 66. Concentrating on One
  • 67. Concentrating on One58
  • 68. Concentrating on One58
  • 69. Concentrating on One58
  • 70. Concentrating on One3858
  • 71. Concentrating on One
  • 72. Concentrating on One
  • 73. Fraction ChartWhich is more, 3/4 or4/5?11212131415161718191101313141516171819141516171814151617181915161617171718181818191919191919110110110110110110110110110
  • 74. 11212131415161718191101313141516171819141516171814151617181915161617171718181818191919191919110110110110110110110110110Fraction ChartWhich is more, 3/4 or4/5?
  • 75. 11212131415161718191101313141516171819141516171814151617181915161617171718181818191919191919110110110110110110110110110Fraction ChartWhich is more, 3/4 or4/5?
  • 76. Fraction ChartWhich is more, 7/8 or8/9?11212131415161718191101313141516171819141516171814151617181915161617171718181818191919191919110110110110110110110110110
  • 77. 11212131415161718191101313141516171819141516171814151617181915161617171718181818191919191919110110110110110110110110110Fraction ChartWhich is more, 7/8 or8/9?
  • 78. 11212131415161718191101313141516171819141516171814151617181915161617171718181818191919191919110110110110110110110110110Fraction ChartAn interesting pattern.
  • 79. Partial Chart11414141412121818181818181818
  • 80. Partial Chart11414141412121818181818181818
  • 81. Partial Chart
  • 82. Partial Chart1 2 3 4 5 6
  • 83. Fraction War
  • 84. Fraction WarObjective:To practice comparing ones, halves,fourths, and eighths in preparation forreading a ruler.
  • 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. Fraction War11414141412121818181818181818
  • 87. Fraction War141811414141412121818181818181818
  • 88. Fraction War141811414141412121818181818181818
  • 89. 11414141412121818181818181818Fraction War1418
  • 90. Fraction War11414141412121818181818181818
  • 91. 11414141412121818181818181818Fraction War3458
  • 92. 11414141412121818181818181818Fraction War3458
  • 93. 11414141412121818181818181818Fraction War3458
  • 94. Fraction War11414141412121818181818181818
  • 95. Fraction War343411414141412121818181818181818
  • 96. Fraction War343411414141412121818181818181818
  • 97. Fraction War3434381411414141412121818181818181818
  • 98. Fraction War11414141412121818181818181818
  • 99. Fraction War11414141412121818181818181818
  • 100. 11212131415161718191101313141516171819141516171814151617181915161617171718181818191919191919110110110110110110110110110Fraction ChartHow many fourths equal a half?
  • 101. 11212141516171819110141516171819141516171814151617181915161617171718181818191919191919110110110110110110110110110Fraction ChartHow many fourths equal a half?
  • 102. 11212131415161718191101313141516171819141516171814151617181915161617171718181818191919191919110110110110110110110110110Fraction ChartHow many fourths equal a half?
  • 103. 11212131415161718191101313141516171819141516171814151617181915161617171718181818191919191919110110110110110110110110110Fraction ChartHow many fourths equal a half? Eighths?
  • 104. 11212131415161718191101313141516171819141516171814151617181915161617171718181818191919191919110110110110110110110110110Fraction ChartHow many fourths equal a half? Eighths?
  • 105. 11212131415161718191101313141516171819141516171814151617181915161617171718181818191919191919110110110110110110110110110Fraction ChartHow many fourths equal a half? Eighths?Sevenths?
  • 106. 11212131415161718191101313141516171819141516171814151617181915161617171718181818191919191919110110110110110110110110110Fraction ChartHow many fourths equal a half? Eighths?Sevenths?
  • 107. 11212131415161718191101313141516171819141516171814151617181915161617171718181818191919191919110110110110110110110110110Fraction ChartWhat is 1/4 plus 1/8?
  • 108. 11212131415161718191101313141516171819141516171814151617181915161617171718181818191919191919110110110110110110110110110Fraction ChartWhat is 1/4 plus 1/8?
  • 109. 11212131415161718191101313141516171819141516171814151617181915161617171718181818191919191919110110110110110110110110110Fraction ChartWhat is 1/4 plus 1/8?
  • 110. 11212131415161718191101313141516171819141516171814151617181915161617171718181818191919191919110110110110110110110110110Fraction ChartWhat is 1/4 plus 1/8? [3/8]
  • 111. 11212131415161718191101313141516171819141516171814151617181915161617171718181818191919191919110110110110110110110110110Fraction ChartIf the chance of rain is 3/4, the chance it won’t rain is1/4.
  • 112. 11212131415161718191101313141516171819141516171814151617181915161617171718181818191919191919110110110110110110110110110Fraction ChartIf the chance of rain is 3/4, the chance it won’t rain is1/4.
  • 113. Faulty Fractions
  • 114. Faulty Fractions“Fish Tank” model
  • 115. Faulty Fractions2“Fish Tank” model
  • 116. Faulty Fractions25“Fish Tank” model
  • 117. Faulty Fractions= partwholeCRA model
  • 118. Faulty Fractions= partwhole“Goal: To develop the spatial organization, visuallyand kinesthetically, to read and write fractionscorrectly.CRA model
  • 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. Faulty FractionsThis is fourths.“Words” model
  • 121. Faulty FractionsThis is fourths. This is thirds.“Words” model
  • 122. Faulty Fractions13131314141414“Rounded corners”
  • 123. Faulty FractionsThe middle fractions are greaterthan the fractions at the ends!13131314141414“Rounded corners”
  • 124. Faulty Fractions121121313136 6 6 6 6 617171717171717191101101101101101101101101101101818181818181818141414191919191919191915151515151 1 1 1 1 114“Color” model
  • 125. 114141414121211011011011011011011011011011018181818181818181313131515151515161616161616Faulty Fractions112112112112112112112112112112112112Missing 7ths & 9ths
  • 126. 112Faulty FractionsMissing 7ths & 9ths1121314151811016
  • 127. Faulty FractionsAre we comparing angles, arcs, or area?Circles
  • 128. Faulty Fractions6161616161615141213151515151414141313121Try to compare 4/5 and 5/6 with this model.Circles
  • 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. 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. Definition of a FractionWhat is the definition of a fraction?
  • 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. 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. 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. Definition of a FractionAn expression that indicatesthequotient of two quantities.American Heritage Dictionary:
  • 136. Definition of a FractionAn expression that indicatesthequotient of two quantities.This is the mathematical meaning of fraction.American Heritage Dictionary:
  • 137. Definition of a FractionThis is the mathematical meaning of fraction.32An expression that indicatesthequotient of two quantities.American Heritage Dictionary:
  • 138. Fractions > 111212131415161718191101313141516171819141516171814151617181915161617171718181818191919191919110110110110110110110110110
  • 139. Fractions > 11121213141516171819110131314151617181914151617181415161718191516161717171818181819191919191911011011011011011011011011018
  • 140. Mixed to Improper FractionsEach row of connected rectangles represents 1.Write each quantity as a mixed numberand as an improper fraction.
  • 141. Mixed to Improper FractionsEach row of connected rectangles represents 1.Write each quantity as a mixed numberand as an improper fraction.
  • 142. Mixed to Improper FractionsEach row of connected rectangles represents 1.2 =34114Write each quantity as a mixed numberand as an improper fraction.
  • 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. Write each quantity as a mixed numberand as an improper fraction.Mixed to Improper Fractions2 =3411424two 4sEach row of connected rectangles represents 1.
  • 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. 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. 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. 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. 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. 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. 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. 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. 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. Improper to Mixed FractionsCircle the wholes and write each quantity asan improper fraction and as a mixed number.
  • 155. Improper to Mixed FractionsCircle the wholes and write each quantity asan improper fraction and as a mixed number.= 211515
  • 156. Improper to Mixed FractionsCircle the wholes and write each quantity asan improper fraction and as a mixed number.= 211515
  • 157. Improper to Mixed FractionsCircle the wholes and write each quantity asan improper fraction and as a mixed number.= 211515
  • 158. Improper to Mixed FractionsCircle the wholes and write each quantity asan improper fraction and as a mixed number.= 211515= 15323
  • 159. Improper to Mixed FractionsCircle the wholes and write each quantity asan improper fraction and as a mixed number.= 211515= 15323
  • 160. Fraction of Geometric Figures12Shade
  • 161. Fraction of Geometric Figures12Shade
  • 162. Fraction of Geometric Figures1223Shade Shade
  • 163. Fraction of Geometric Figures1223Shade Shade
  • 164. Fraction of Geometric Figures122314Shade Shade Shade
  • 165. Fraction of Geometric Figures122314Shade Shade Shade
  • 166. Making the WholeDraw the whole.13
  • 167. Making the WholeDraw the whole.13131313
  • 168. Making the WholeDraw the whole.1313131323
  • 169. Making the WholeDraw the whole.13131313232313
  • 170. 112131415161718191101313141516171819141516171814151617181915161617171718181818191919191919110110110110110110110110110Fraction ChartWhat is 1/2 of 1/2?12
  • 171. 1121315161718191101313141516171819141516171814151617181915161617171718181818191919191919110110110110110110110110110Fraction ChartWhat is 1/2 of 1/2?1214
  • 172. 112131415161718191101313141516171819141516171814151617181915161617171718181818191919191919110110110110110110110110110Fraction ChartWhat is 1/3 of 1/2?12
  • 173. Fraction ChartWhat is 1/3 of 1/2?11213141516171819110131314151718191415161718141516171819151616171717181818181919191919191101101101101101101101101101216
  • 174. Simplifying Fractions12
  • 175. Simplifying Fractions36= 12
  • 176. Simplifying Fractions48= 12
  • 177. Simplifying Fractions
  • 178. Simplifying Fractions912
  • 179. Simplifying Fractions912 3
  • 180. Simplifying Fractions912 3
  • 181. Simplifying Fractions912= 343
  • 182. Simplifying Fractions
  • 183. Simplifying Fractions
  • 184. Simplifying Fractions
  • 185. Simplifying Fractions
  • 186. Simplifying Fractions2128
  • 187. Simplifying Fractions2128
  • 188. Simplifying Fractions2128
  • 189. Simplifying Fractions4572
  • 190. Simplifying Fractions4572
  • 191. Simplifying Fractions4572
  • 192. Simplifying Fractions1216
  • 193. Simplifying Fractions1216
  • 194. Simplifying Fractions1216
  • 195. Simplifying Fractions1216
  • 196. Simplifying Fractions1216
  • 197. Simplifying Fractions1216
  • 198. © Joan A. Cotter, Ph.D., 2013Multiples PatternsTwos2 4 6 8 1012 14 16 18 20
  • 199. © Joan A. Cotter, Ph.D., 2013Multiples PatternsTwos2 4 6 8 1012 14 16 18 20The ones repeat in the secondrow.
  • 200. © Joan A. Cotter, Ph.D., 2013Multiples PatternsFours4 8 12 16 2024 28 32 36 40The ones repeat in the secondrow.
  • 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. © 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. © 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. © 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. © 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. © 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. © 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. © 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. © 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. © 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. © 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. © Joan A. Cotter, Ph.D., 2013Multiples PatternsThrees3 6 912 15 1821 24 2730The 3s have several patterns:Observe the ones.
  • 213. © Joan A. Cotter, Ph.D., 2013Multiples PatternsThrees3 6 912 15 1821 24 2730The 3s have several patterns:Observe the ones.
  • 214. © Joan A. Cotter, Ph.D., 2013Multiples PatternsThrees3 6 912 15 1821 24 2730The 3s have several patterns:Observe the ones.
  • 215. © Joan A. Cotter, Ph.D., 2013Multiples PatternsThrees3 6 912 15 1821 24 2730The 3s have several patterns:Observe the ones.
  • 216. © Joan A. Cotter, Ph.D., 2013Multiples PatternsThrees3 6 912 15 1821 24 2730The 3s have several patterns:Observe the ones.
  • 217. © Joan A. Cotter, Ph.D., 2013Multiples PatternsThrees3 6 912 15 1821 24 2730The 3s have several patterns:Observe the ones.
  • 218. © Joan A. Cotter, Ph.D., 2013Multiples PatternsThrees3 6 912 15 1821 24 2730The 3s have several patterns:Observe the ones.
  • 219. © Joan A. Cotter, Ph.D., 2013Multiples PatternsThrees3 6 912 15 1821 24 2730The 3s have several patterns:Observe the ones.
  • 220. © Joan A. Cotter, Ph.D., 2013Multiples PatternsThrees3 6 912 15 1821 24 2730The 3s have several patterns:Observe the ones.
  • 221. © Joan A. Cotter, Ph.D., 2013Multiples PatternsThrees3 6 912 15 1821 24 2730The 3s have several patterns:Observe the ones.
  • 222. © Joan A. Cotter, Ph.D., 2013Multiples PatternsThrees3 6 912 15 1821 24 2730The 3s have several patterns:Observe the ones.
  • 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. © Joan A. Cotter, Ph.D., 2013Multiples PatternsThrees3 6 912 15 1821 24 2730The 3s have several patterns:Add the digits in the columns.
  • 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. © Joan A. Cotter, Ph.D., 2013Multiples PatternsThrees3 6 912 15 1821 24 2730The 3s have several patterns:Add the digits in the columns.
  • 227. © Joan A. Cotter, Ph.D., 2013Multiples PatternsThrees3 6 912 15 1821 24 2730The 3s have several patterns:Add the “opposites.”
  • 228. © Joan A. Cotter, Ph.D., 2013Multiples PatternsThrees3 6 912 15 1821 24 2730The 3s have several patterns:Add the “opposites.”
  • 229. © Joan A. Cotter, Ph.D., 2013Multiples PatternsThrees3 6 912 15 1821 24 2730The 3s have several patterns:Add the “opposites.”
  • 230. © Joan A. Cotter, Ph.D., 2013Multiples PatternsThrees3 6 912 15 1821 24 2730The 3s have several patterns:Add the “opposites.”
  • 231. © Joan A. Cotter, Ph.D., 2013Multiples PatternsSevens7 14 2128 35 4249 56 6370The 7s have the 1, 2, 3… pattern.
  • 232. © Joan A. Cotter, Ph.D., 2013Multiples PatternsSevens7 14 2128 35 4249 56 6370The 7s have the 1, 2, 3… pattern.
  • 233. © Joan A. Cotter, Ph.D., 2013Multiples PatternsSevens7 14 2128 35 4249 56 6370The 7s have the 1, 2, 3… pattern.
  • 234. © Joan A. Cotter, Ph.D., 2013Multiples PatternsSevens7 14 2128 35 4249 56 6370The 7s have the 1, 2, 3… pattern.
  • 235. © Joan A. Cotter, Ph.D., 2013Multiples PatternsSevens7 14 2128 35 4249 56 6370Look at the tens.
  • 236. © Joan A. Cotter, Ph.D., 2013Multiples PatternsSevens7 14 2128 35 4249 56 6370Look at the tens.
  • 237. © Joan A. Cotter, Ph.D., 2013Multiples PatternsSevens7 14 2128 35 4249 56 6370Look at the tens.
  • 238. Subtracting Fractions4684–2372Preliminary understanding
  • 239. Subtracting Fractions4684–23722000Preliminary understanding
  • 240. Subtracting Fractions4684–23722000300Preliminary understanding
  • 241. Subtracting Fractions4684–2372200030010Preliminary understanding
  • 242. Subtracting Fractions4684–23722000300102Preliminary understanding
  • 243. Subtracting Fractions4684–237220003001022312Preliminary understanding
  • 244. Subtracting Fractions4684–23724684–287920003001022312Preliminary understanding
  • 245. Subtracting Fractions4684–23724684–2879200030010223122000Preliminary understanding
  • 246. Subtracting Fractions4684–23724684–2879200030010223122000–200Preliminary understanding
  • 247. Subtracting Fractions4684–23724684–2879200030010223122000–20010Preliminary understanding
  • 248. Subtracting Fractions4684–23724684–2879200030010223122000–20010–5Preliminary understanding
  • 249. Subtracting Fractions4684–23724684–2879200030010223122000–20010–51805Preliminary understanding
  • 250. Subtracting Fractions354–
  • 251. Subtracting Fractions354–325
  • 252. Subtracting Fractions354–325575– 217
  • 253. Subtracting Fractions354–3253575– 217
  • 254. Subtracting Fractions354–3253– 47575– 217
  • 255. Subtracting Fractions354–3253– 47237575– 217
  • 256. Multiplying Fractions
  • 257. Multiplying Fractions• Multiplication is more than repeatedaddition.
  • 258. Multiplying Fractions4 x 4 = 4 + 4 + 4 + 4• Multiplication is more than repeatedaddition.
  • 259. Multiplying Fractions4 x 4 = 4 + 4 + 4 + 4• Repeated addition doesn’t work well withfractions.• Multiplication is more than repeatedaddition.
  • 260. Multiplying Fractions• Repeated addition doesn’t work well withfractions.12x = + ?12124 x 4 = 4 + 4 + 4 + 4• Multiplication is more than repeatedaddition.
  • 261. Multiplying FractionsArea is a bettermodel.4 x 4 =
  • 262. Multiplying Fractions12x =12The square represents 1.
  • 263. Multiplying Fractions12x =12
  • 264. Multiplying Fractions12x =1214The solution is the double-crosshatchedarea.
  • 265. Multiplying Fractions23x =34
  • 266. Multiplying Fractions23x =34
  • 267. Multiplying Fractions23x =34
  • 268. Multiplying Fractions23x =34612
  • 269. Multiplying Fractionsx =3412=23612
  • 270. Multiplying Fractions23x =34The total number of rectangles is 3 x 4.
  • 271. Multiplying Fractions2334The number of double-crosshatched rectangles is 2The total number of rectangles is 3 x 4.x =
  • 272. Multiplying Fractions2334This is why we multiply fractions bymultiplying numerators and denominators.x =
  • 273. Dividing Fractions
  • 274. Dividing Fractions÷ =121
  • 275. Dividing Fractions÷ =1211 ÷ 1/2 means how many 1/2s in 1.
  • 276. Dividing Fractions÷ =12111212141414141313131 ÷ 1/2 means how many 1/2s in 1.
  • 277. Dividing Fractions÷ =12111212141414141313131 ÷ 1/2 means how many 1/2s in 1.
  • 278. Dividing Fractions÷ =12111212141414141313131 ÷ 1/2 means how many 1/2s in 1.
  • 279. Dividing Fractions÷ =121 211212141414141313131 ÷ 1/2 means how many 1/2s in 1.
  • 280. Dividing Fractions÷ =121 2÷ =13111212141414141313131 ÷ 1/3 means how many 1/3s in 1.
  • 281. Dividing Fractions÷ =121 2÷ =13111212141414141313131 ÷ 1/3 means how many 1/3s in 1.
  • 282. Dividing Fractions÷ =121 2÷ =13111212141414141313131 ÷ 1/3 means how many 1/3s in 1.
  • 283. Dividing Fractions÷ =121 2÷ =13111212141414141313131 ÷ 1/3 means how many 1/3s in 1.
  • 284. Dividing Fractions÷ =121 2÷ =131 311212141414141313131 ÷ 1/3 means how many 1/3s in 1.
  • 285. Dividing Fractions÷ =121 2÷ =131 3÷ =2311121214141414131313
  • 286. Dividing Fractions÷ =121 2÷ =131 3÷ =23111212141414141313131 ÷ 2/3 means how many 2/3s in 1.
  • 287. Dividing Fractions÷ =121÷ =131÷ =231112121414141423231 ÷ 2/3 means how many 2/3s in 1.
  • 288. Dividing Fractions÷ =121 2÷ =131 3÷ =231112121414141423231 ÷ 2/3 means how many 2/3s in 1.
  • 289. Dividing Fractions÷ =121÷ =131÷ =2311121214141414232332231 ÷ 2/3 means how many 2/3s in 1.
  • 290. Dividing Fractions÷ =121÷ =131÷ =2311121214141414232332231 ÷ 2/3 also must be half of 1 ÷ 1/3.
  • 291. Dividing Fractions÷ =121÷ =131÷ =1 3÷ =231 32112121414141413131323
  • 292. Dividing Fractions÷ =121÷ =131÷ =1 3÷ =231 321121214141414131313231 ÷ 3 is simply the definition of a fraction.
  • 293. 1121214141414131313Dividing Fractions÷ =121÷ =13113÷ =1 3÷ =231 32231 ÷ 3 is simply the definition of a fraction.
  • 294. Dividing Fractions÷ =121÷ =131÷ =1 3÷ =1 4÷ =2311121214141414131313322313
  • 295. Dividing Fractions÷ =121÷ =131 14÷ =1 3÷ =1 4÷ =2311121214141414131313322313
  • 296. Dividing Fractions1314÷ =1 3÷ =1 4÷ =1 43121131÷ =231÷ =÷ =11212141414141313133223
  • 297. Dividing Fractions÷ =1 3÷ =1 41121213131314141414÷ =1 43121131÷ =231÷ =÷ =11212141414141313131313143223
  • 298. Dividing Fractions÷ =1 3÷ =1 41121213131314141414÷ =1 43121131÷ =231÷ =÷ =11212141414141313131313143223Only 3/4 of the 4/3 fits into the 1.
  • 299. Dividing Fractions÷ =1 3÷ =1 41121213131314141414÷ =1 43121131÷ =231÷ =÷ =1121214141414131313131314322334Only 3/4 of the 4/3 fits into the 1.
  • 300. Dividing Fractions÷ =1 3÷ =1 4÷ =1 43121131÷ =231÷ =÷ =1314322334
  • 301. Dividing Fractions÷ =121 2÷ =131 31314÷ =1 3÷ =1 434÷ =1 43÷ =231 32
  • 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. 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. 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. Dividing Fractions÷ =121 2÷ =131 31314÷ =1 31÷ =1 4134÷ =1 43÷ =231 32
  • 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. Dividing Fractions÷ = __235To find
  • 308. Dividing Fractions÷ = __235To findFirst think about finding 1 ÷ 2/3.
  • 309. Dividing Fractions÷ = __235÷ =231First findTo find
  • 310. Dividing Fractions÷ = __235÷ =231 32First findTo find
  • 311. Dividing Fractions÷ =235÷ = __235÷ =231 32First findTo findThen
  • 312. Dividing Fractions÷ = __235÷ =231 32First findTo findThen ÷ =235 5 (1 )x ÷23
  • 313. Dividing Fractions= x =325÷ = __235÷ =231 32First findTo findThen ÷ =235 5 (1 )x ÷23
  • 314. Dividing Fractions= x =321525÷ = __235÷ =231 32First findTo findThen ÷ =235 5 (1 )x ÷23
  • 315. Dividing Fractions= x = =32125 7÷ = __235÷ =231 32First findTo findThen ÷ =235 5 (1 )x ÷23152
  • 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. 23Dividing Fractions÷ = __Find34
  • 318. 23Dividing Fractions÷ = __Find34(Is the answer more or less than 1?)
  • 319. 114141414121213131323Dividing Fractions÷ = __Find34(Is the answer more or less than 1?)
  • 320. 114141414121213131323Dividing Fractions÷ = __Find34(Is the answer more or less than 1?)
  • 321. 114141414121213131323Dividing Fractions÷ = __Find34(The answer must be less than 1.)
  • 322. Dividing FractionsTo find23÷ = __34÷ =341First find
  • 323. Dividing FractionsTo find23÷ = __34÷ =341 43First find
  • 324. Dividing Fractions÷ =341 43First findTo findThen÷ = __3423÷ =3423
  • 325. Dividing Fractions÷ =341 43First findTo findThen÷ = __342334233423÷ = x (1 ÷ )
  • 326. Dividing Fractions÷ =341 43First findTo find ÷ = __3423Then34233423÷ = x (1 ÷ )= x =4323
  • 327. = x =Dividing Fractions÷ =341 43First findTo find ÷ = __3423Then34233423÷ = x (1 ÷ )894323
  • 328. = x =Dividing Fractions÷ =341 43First findTo find ÷ = __3423Then34233423÷ = x (1 ÷ )894323The answer should be < 1 and it is.
  • 329. = x =Dividing Fractions÷ =341 43First findTo find ÷ = __3423Then34233423÷ = x (1 ÷ )894323The extra step of dividing by 1 can be omitted.
  • 330. ÷ = x (1 ÷ )= x =Dividing Fractions÷ =341 43First findTo find ÷ = __3423Then34233423894323The extra step of dividing by 1 can be omitted.
  • 331. Dividing FractionsIt’s ours to reason whyWe invert and multiply.
  • 332. Fraction Meanings
  • 333. Fraction Meanings• One or more equal parts of a whole.
  • 334. Fraction Meanings• One or more equal parts of a whole.• One or more equal parts of a collection.
  • 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. 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. 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. FractionsMassHOPE-TEACHWorcester, MASaturday, April 27, 20132:45pm - 3:45pmJoan A. Cotter, Ph.D.JoanCotter@RightStartMath.com