Expressions and equations with cryptarithmetic
- 2. CRYPTARITHMETIC RULES Find which digits should replace each of the letters in each of the following word sums. For each problem, each of the letters stands for a different digit. The first digit in a number is never zero. Replace the letters so that each sum is correct. Are these possible? If so, can you find more than one solution? Are there restrictions on which digits are possible for any of the letters?
- 3. CCSS CONTENT STANDARDS ADDRESSED Students should : • Understand the difference between face value and place values of digits in multi-digit numbers (2.NBT.1, 5.NBT.1) • Use their place value understanding and properties of operations to perform multi-digit arithmetic (3.NBT.2, 4.NBT.4) • Evaluate expressions in which letters stand for numbers (6.EE.2) • Determine whether a given number makes an equation true (6.EE.5) • Understand that a variable can represent an unknown number or any number in a specified
- 4. CCSS STANDARDS FOR MATHEMATICAL PRACTICE ADDRESSED Make sense of problems and persevere in solving them Understand what the problem is asking – what is the nature of the solutions which exist? Construct viable arguments & critique the reasoning of others Describe the constraints on variables. Listen to others’ solution methods Evaluate efficiencies of different solution methods. Attend to precision Make sure the numbers add correctly. Make sure that each different letter is represented by a different digit. Keep track of the digits they have already used & the digits that are remaining. Find ways to record their valid & invalid solutions. Look for and express regularity in repeated reasoning Look for patterns. Reason abstractly and quantitatively Substitute letters for digits. Understand that the digit chosen for each letter is a face value. Understand the place value of the digit & its corresponding letter. Use appropriate tools strategically Decide what tools to use Look for & make use of structure Create a system to record addends which result in a carry. Determine when a carry to the next higher place value will be used. Model with mathematics Letter-number substitution problems are used in cryptography, a real-world application.
- 5. W I N + T H E R A C E Find which digits should replace each of the letters in each of the following word sums. For each problem, each of the letters stands for a different digit. The first digit in a number is never zero. Replace the letters so that each sum is correct. Can you find more than one solution? Are there restrictions on which digits are possible for any of the letters? Explain how you know.
- 6. Find which digits should replace each of the letters in each of the following word sums. For each problem, each of the letters stands for a different digit. The first digit in a number is never zero. Replace the letters so that each sum is correct. Can you find more than one solution? Are there restrictions on which digits are possible for any of the letters? Explain how you know. +
- 7. REDUCING THE PROBLEM F O U R + O N E F I V E • R + E = E, so R = 0 (identity property of addition) • O + O = I with no carry, so O = 1, 2, 3, 4
- 8. ASSISTING STUDENTS’ WORK ON PROBLEM
- 9. SCAFFOLDING SPREADSHEET #1 Blackboard > Content > FOUR ONE FIVE > Spreadsheet
- 11. EXTENSION QUESTIONS Question 1: Did you notice any shortcuts to finding solutions, and if so, why are these valid shortcuts? Question 2: Given a solution to the FOUR+ONE=FIVE problem, 5130 + 167 = 5297, find as many different solutions as you can using the same digits. How will you know when you have found all of the possible solutions using these digits? Question 3: Find the total number of solutions there are to this word sum. Describe how you found this number of solutions. Question 4: Answer the question, “If I find all of the solutions to this problem for F = 1, can I multiply that number of solutions by 9 to find the total number of solutions to this problem? Why or why not?” Question 5: Create your own letter-number substitution
- 12. SOLUTION METHOD #1: PLACE VALUES F1O1UR = F*1000+O1*100+U*10+R O2NE = O*100+N*10+E F O1 U R 1000 100 10 0 2000 200 20 3000 300 30 4000 400 40 5000 50 6000 60 7000 70 8000 80 9000 90 O2 N E 100 10 1 200 20 2 300 30 3 400 40 4 50 5 60 6 70 7 80 8 90 9
- 13. DISTINCT DIGITS FOR F1, O1, U, R (219 SETS) F1 = 1 F1 = 2 F1= 3 F1 = 4 F1 = 5 F1 = 6 F1 = 7 F1 = 8 F1 = 9 1230 2130 3120 4120 5120 6120 7120 8120 9120 1240 2140 3140 4130 5130 6130 7130 8130 9130 1250 2150 3150 4150 5140 6140 7140 8140 9140 1260 2160 3160 4160 5160 6150 7150 8150 9150 1270 2170 3170 4170 5170 6170 7160 8160 9160 1280 2180 3180 4180 5180 6180 7180 8170 9170 1290 2190 3190 4190 5190 6190 7190 8190 9180 1320 2310 3210 4210 5210 6210 7210 8210 9210 1340 2340 3240 4230 5230 6230 7230 8230 9230 1350 2350 3250 4250 5240 6240 7240 8240 9240 1360 2360 3260 4260 5260 6250 7250 8250 9250 1370 2370 3270 4270 5270 6270 7260 8260 9260 1380 2380 3280 4280 5280 6280 7280 8270 9270 1390 2390 3290 4290 5290 6290 7290 8290 9280 1420 2410 3410 4310 5310 6310 7310 8310 9310 1430 2430 3420 4320 5320 6320 7320 8320 9320 1450 2450 3450 5340 6340 7340 8340 9340 1460 2460 3460 5360 6350 7350 8350 9350 1470 2470 3470 5370 6370 7360 8360 9360 1480 2480 3480 5380 6380 7380 8370 9370 1490 2490 3490 5390 6390 7390 8390 9380 5410 6410 7410 8410 9410 5420 6420 7420 8420 9420 5430 6430 7430 8430 9430 5460 6450 7450 8450 9450 5470 6470 7460 8460 9460 5480 6480 7480 8470 9470 5490 6490 7490 8490 9480
- 14. O2 = 1 O2= 2 O2= 3 O2= 4 120 165 201 256 301 356 401 456 123 167 203 257 302 357 402 457 124 168 204 258 304 358 403 458 125 169 205 259 305 359 405 459 126 170 206 260 306 360 406 460 127 172 207 261 307 361 407 461 128 173 208 263 308 362 408 462 129 174 209 264 309 364 409 463 130 175 210 265 310 365 410 465 132 176 213 267 312 367 412 467 134 178 214 268 314 368 413 468 135 179 215 269 315 369 415 469 136 180 216 270 316 370 416 470 137 182 217 271 317 371 417 471 138 183 218 273 318 372 418 472 139 184 219 274 319 374 419 473 140 185 230 275 320 375 420 475 142 186 231 276 321 376 421 476 143 187 234 278 324 378 423 478 145 189 235 279 325 379 425 479 146 190 236 280 326 380 426 480 147 192 237 281 327 381 427 481 148 193 238 283 328 382 428 482 149 194 239 284 329 384 429 483 150 195 240 285 340 385 430 485 152 196 241 286 341 386 431 486 153 197 243 287 342 387 432 487 154 198 245 289 345 389 435 489 156 246 290 346 390 436 490 157 247 291 347 391 437 491 158 248 293 348 392 438 492 159 249 294 349 394 439 493 160 250 295 350 395 450 495 162 251 296 351 396 451 496 163 253 297 352 397 452 497 164 254 298 354 398 453 498 Distinct digits for O2, N, E (280 sets)
- 15. SUMS FOR EACH COMBINATION OF F1O1UR AND O2NE (219 * 280 = 61,320 SETS) O2NE 120 123 124 125 126 F1O1UR 1230 1230+120=1350 1230+123=1353 1230+124=1354 1230+125=1355 1230+126=1356 1240 1240+120=1360 1240+123=1363 1240+124=1364 1240+125=1365 1240+126=1366 1250 1250+120=1370 1250+123=1373 1250+124=1374 1250+125=1375 1250+126=1376 1260 1260+120=1380 1260+123=1383 1260+124=1384 1260+125=1385 1260+126=1386 1270 1270+120=1390 1270+123=1393 1270+124=1394 1270+125=1395 1270+126=1396 1280 1280+120=1400 1280+123=1403 1280+124=1404 1280+125=1405 1280+126=1406 1290 1290+120=1410 1290+123=1413 1290+124=1414 1290+125=1415 1290+126=1416 1320 1320+120=1440 1320+123=1443 1320+124=1444 1320+125=1445 1320+126=1446 1340 1340+120=1460 1340+123=1463 1340+124=1464 1340+125=1465 1340+126=1466 1350 1350+120=1470 1350+123=1473 1350+124=1474 1350+125=1475 1350+126=1476 1360 1360+120=1480 1360+123=1483 1360+124=1484 1360+125=1485 1360+126=1486 1370 1370+120=1490 1370+123=1493 1370+124=1494 1370+125=1495 1370+126=1496 1380 1380+120=1500 1380+123=1503 1380+124=1504 1380+125=1505 1380+126=1506 1390 1390+120=1510 1390+123=1513 1390+124=1514 1390+125=1515 1390+126=1516
- 16. LOGICAL STATEMENT CHECKS For each of 61,320 possibilities for “F1O1UR + O2NE = F2IVE,” check 10 logic statements: Whether F1 is the same as F2 Whether O1 is the same as O2 Whether F1 is distinct from N, E, I, V Whether O1 is distinct from N, E, I, V Whether U is distinct from N, E, I, V Whether R is distinct from N, E, I, V Whether N is distinct from I, V Whether E is distinct from I, V Whether I is distinct from V If and only if all of the above are true, then the solution is valid for “FOUR + ONE = FIVE.”
- 17. LOGICAL STATEMENT CHECKS – SCREEN SHOT
- 19. Observation #1 F and E do not depend on the other letters. There are 36 pairs of 2-digit combinations: 1,2 2,3 3,4 4,5 5,6 6,7 7,8 8,9 1,3 2,4 3,5 4,6 5,7 6,8 7,9 1,4 2,5 3,6 4,7 5,8 6,9 1,5 2,6 3,7 4,8 5,9 1,6 2,7 3,8 4,9 1,7 2,8 3,9 1,8 2,9 1,9 The possibilities for F and E are interchangeable. METHOD 2: COMBINATORICS #1 F O U R + O N E F I V E
- 20. Observation #2 O + O = I with no carry. Thus, O can only be 1, 2, 3, or 4. Observation #3 U + N = V can, but does not need to, have a carry. Observation #4 The possibilities for U and N are interchangeable (commutative property of addition). * For every combination of F, E, U, N, we can wrote 4 possible arrangements: FEUN, EFUN, FENU, EFNU. Thus the total # of possible solutions is a multiple of four. F O U R + O N E F I V E
- 21. METHOD FOR GENERATING SOLUTIONS – DISTINCT DIGITS For each of the 36 pairs of (F,E) possibilities, find O, I, U, N, V possibilities: O + O = I No Carry from U + N = V 1+1=2 1+2=3 2+3=5 3+4=7 4+5=9 2+2=4 1+3=4 2+4=6 3+5=8 3+3=6 1+4=5 2+5=7 3+6=9 4+4=8 1+5=6 2+6=8 1+6=7 2+7=9 1+7=8 1+8=9 O + O = I Carry from U + N = V 1+1 3 2+9 1 4+8 2 5+8 3 6+9 5 2+2 5 3+8 1 4+9 3 5+9 4 7+8 5 3+3 7 3+9 2 5+6 1 6+7 3 7+9 6 4+4 9 4+7 1 5+7 2 6+8 4 8+9 7 There are 300 solutions for (F, E, O, I, U, N, V) with F<E and U<V, so there are 300* 4 = 1200 total solutions
- 22. METHOD 3: COMBINATORICS #2 • For every combination of (O, I, V), there are 2 combinations of (U,N). • For every combination of (O, I, V, U, N, R=0), there are 12 combinations of (F, E): * Since O, I, V, U, N, R represent 6 distinct digits, there are 4 remaining digits for F and E * For each of the 4 digits that are possible for the value of F, there are 3 remaining digits possible for the value of E. (4 * 3 = 12) If we find the number of combinations of (O, I, U, N, V, R) where U < N, we have 2*12 = 24 times more total solutions to (F, E, O, I, U, N, V, R) F O U R + O N E F I V E
- 23. 50 COMBINATIONS OF (O, U, N, I, V) O U N I V O U N I V O U N I V O U N I V 1 3 4 2 7 2 1 5 4 6 3 1 4 6 5 4 1 2 8 3 1 3 5 2 8 2 1 6 4 7 3 1 7 6 8 4 1 5 8 6 1 3 6 2 9 2 1 7 4 8 3 1 8 6 9 4 1 6 8 7 1 4 5 2 9 2 1 8 4 9 3 2 5 6 7 4 2 3 8 5 1 4 8 3 2 2 3 5 4 8 3 2 7 6 9 4 2 5 8 9 1 5 7 3 2 2 3 6 4 9 3 2 9 7 1 4 2 7 8 9 1 5 9 3 4 2 3 8 5 1 3 4 5 6 9 4 3 6 8 9 1 6 8 3 4 2 4 7 5 1 3 4 8 7 2 4 3 8 9 1 1 6 9 3 5 2 4 9 5 3 3 5 6 7 1 4 5 6 9 1 1 7 8 3 5 2 6 7 4 3 3 5 9 7 4 4 5 7 9 2 1 7 9 3 6 2 6 8 5 4 3 6 8 7 4 4 5 8 9 3 1 8 9 3 7 2 7 9 5 6 3 6 9 7 5 4 6 7 9 3 2 8 9 5 7 4 7 8 9 5 F O U R + O N E F I V E 50 * 24 = 1200 TOTAL SOLUTIONS
- 24. MORE WORD MATH PROBLEMS! HERE + SHE = COMES DAYS + TOO = SHORT SATURN + URANUS = PLANETS HEAD + HAND = KNEE ORANGE + GREEN + RED = YELLOW