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
    162. 162. Fraction of Geometric Figures1223Shade Shade
    163. 163. Fraction of Geometric Figures1223Shade Shade
    164. 164. Fraction of Geometric Figures122314Shade Shade Shade
    165. 165. Fraction of Geometric Figures122314Shade Shade Shade
    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

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