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Number Sense Mental Mathematics TH FOR ADDITION Parameters for Assessment • May include up to 3 digit + 3 digit questions. • Questions may only involve carrying numbers in the TENS place value, Strategies for Teaching 1. Traditional (pencil and paper) 156 + 395 551 2. Adding Left to Right (starting with the hundreds) Start on the left and add across each place value. Example: 125 +243 Add 200 + 100 = 300 Add 40 + 20 = 60 Add 5+ 3 = 8 And then add them all together = 368. 3, Compensation This is where we change one number to make the addition easier by "compensating". Example: 99 + 172 Here we would take the 99 and turn it into 100 and then do the addition. That would now make the question 100 + 172 = 272 and then we subtract the I we compensated to get 271 for our answer. 4, Associative Property This is where we are adding more than 2 numbers and we just re-arrange them to pair easier numbers to add together. Example: 27 + 45 + 33 So, we re-arrange the numbers so that we can make pairs of numbers that are easier to add, It would look like this (27 + 33) + 45 and we can make (60) + 45 = 105. This is much easier than starting with 27 + 45 and then adding 33 to that. 5. Commutative Property This is addition with only two numbers. It is much like Associative property except you are re-arranging the numbers so they 'appear easier to add. Example: 17 + 22 = 22 + 17 It is just another way of looking at it and starting with a different number to do the math, 6. Balancing by Zeros -This is a strategy for subtraction but thought it would appropriate to include here. Basically, we are using 10 as a base number in which to subtract other numbers from. Here is an example; 15 -' We add one to this top number because that is what we did on the bottom to get to 10 9-. --> We change this to a 10 to make it easier to subtract -4 6 E- We have the same difference on each side 4 16 10 6 FOR MULTIPLICATION Parameters for Assessment • • May include up to two digit x two digit questions. Must be able to explain TWO different strategies for arriving at correct solution, Strategies for Teaching 1. Traditional (pencil and paper) 23 x14 92 +230 322 2. Skip Counting or Repeated Addition This is simply adding the given number to the previous answer as many times as needed. Example: 2 x 5 is the same as 2 + 2 + 2+2+2= 10or by counting 2, 4, 6, 8, 10 3, Halving and Doubling ("Friendly Numbers") This is creating friendly numbers to multiply with. Example: 16 x 52 Half of 16 is 8 and double 52 is 104. Now we have 8 x 104. Half of 8 is 4 and double 104 is 208. Now we have 4 x 208. Half of 4 is 2 and double 208 is 416 Now we have 2 x 416 (instead of 16 x 52) = 832l 4. Associative Property This is where we are multiplying more than 2 numbers and we just re-arrange them to pair easier numbers to add together. Example: 15 x 9 x 2 can be re-arranged so we have (15 x 2) x 9. Then we can do (30) x 9 = 270. 5. Commutative Property This is multiplication with only two numbers. It is much like Associative property except you are re-arranging the numbers so they "appear" easier to multiply. Example: 15 x 3 is the same as 3 x 15. This way we can do 3x 10(=30)and3x5(=15) and them together quickly to get 45. 6. Distributive Property This is like working through the brackets. The most common sample of this is the Egyptian Box. (See attached sheet.) Example: 23 x 41 is the same as (20 + 3) x (40 + 7) Then you do 20x 40=800,20x7=140,3x40=120,and 3x7=21 and add the answers all together. (Total = 1081) In paper-and-pencil computation, we usually start at the right and work toward the left. To add in your head, start at the left. It also works with decimals. 1.7+3.6 THINK . . 1 + 3 = 4 7 tenths + 6 tenths = 1 and 3 tenths I ©. 4 + 1 and 3 tenths = 5.3 TRY THESE I N YOUR HEAD. Add from the left. 1. 22 + 39 4. 526 + 48 7. 4.5+2.5 2.45+38 5. 329 + 36 8.8.4+1.8 3. 56 + 37 6. 236 + 120 9. 26.5+2.7 10. 43.8 + 10.8 MENTAL MATH IN JUNIOR HIGH Copyright a 1988 by Dais Seymour Publications All MENTAL MATH IN JUNIO. i tHiGH LESSON 1 ADDING FROM THE.LE POWER BUILDER A 1.38+46= 11. 4.7+2.8= 2. 57+25- 12. 3.8 + 1.5 = 3. 44+39= 13. 5.7+2.5= 4.64+18= 14. 8.3 + 4.7 = 1'. 68+35= 15. 5:4+3.8 16. 12.6 + 6.7 = "'68+35= 7. 417+58= 17. 23.6 + 5.9 8.545+228= 18. 45.8 + 3.8 = 9.624+239= 19. 37.8 + 11.2 = 10. 356 + 517 = 2©.45.9+12.8= THINK IT THROUGH December 22 is the first day of winter. March 21 is the last day of winter. How many days does winter offically have? MENTAL MATH IN JUNIOR HIGH LESSON 1 ADDING FROM THE _...,--I POWER BUILDER B 1.46+28= 11. 5.2+3.9= 2. 47 + 25 = 12. 5.8 + 1.5 = 3. 66+19= 13. 3.7 + 4.6 = 4. 24 + 58 = 14. 2.9 + 5.3 = 5.48+35= 15. 2.8 + 6.4 = 6.437+55= 16. 14.6 + 4.9 = 7. 638+29- 17. 43.4 + 4.6 = 8. 345 + 227 = 18. 53.7 + 5.4 = 9.462+219= 19. 33.6 + 12.5 = 10. 456 + 338 = 20.65.8+12.2= THINK IT THROUGH June 22 is the first day of summer. September 21 is the last ; °y of summer. How many days does summer o;-:icially have? 42 <=, Copyngh€ c 1988 by Cale Seymour Publications Mental Math Addition Practice (Carrying from ones into the tens) 243 637 +649 627 + 224 +317 227 333 172 +146 +157 +519 454 639 714 +117 +241 +168 Adding in your head is easier when you make your own compatible pairs, then adjust, Like this .. . Make your own compatibles. Adjust the answer. 1. 75 + 28 4. 427 + 75 7. 795 + 206 2. 69+35 5. 450+65 8. 253+752 3. 188+2 1 3 6. 580 + 423 9. 11 50+356 10. 1250 + 757 Copyright (D 1988 by Dale Seymour Publications MENTAL MATH IN JUNIOR HIGH LESSON 7 MAKE YOUR OWN COMPATIB POWER BUILDER A 1. 25+79= 11. 435 + 568 = 2. 45+57= 12. 295 + 706 = 3. 18+85= 13. 455 + 456 = 4. 75+28= 14. 263 + 738 5.68+33= 15. 375 + 526 6.159+42= 16. 276 + 727 = 7. 125 + 277 17. 459 + 544 = 8. 468 + 35 = 18. 2500 + 501 = 9. 109 + 393 19. 425 + 176 10. 254 + 349 = 20. 725 + 277 = THINK IT THROUGH If 867 + 133 = 1000, what is 867 + 135? 868 + 132? 8.67 + 1.33? Copyright@ 1988 by Dale Seymour Publications MENTAL MATH IN JUNIOR HIGH LESSON 7 MAKE YOUR OWN COMPAT POWER BUILDER B 1.75+26= 11. 345 + 659 = 2. 35+67- 12.307+695= 3. 19+82= 13. 285 + 717 = 4. 27+75= 14. 155 + 846 = 5. 65+38= 15.518+485= 6. 143+58= 16. 475 + 426 = 7. 275+ 127= 17. 365 + 337 = 8. 235+67= 18. 4246 + 555 = 9. 362 + 139 = 19. 425 + 376 = 10.155+249= 20. 525 + 478 = If 655 + 1345 = 2000, what is 655 + 1355? 645 + 1355? 6.55 + 13.45? Copyright © 1988 by Date Seymour Publications In mental math, when a subtraction problem needs regrouping .. . DON'T DO THIS... DO THIS! Use a finger to cover the parts as you think it through. TRY THESE IN YOUR HEAD Subtract i n parts. 4. 800-53 7. 1.35-0.65 2. 62-23 5. 1000-475 8. 6.25 - 1.45 3. 120 -57 6. 500-125 9. 8 - 0.53 75-36 10. $10,00 - $3.50 MENTAL MATH IN JW Copyright 0 1988 by Ups: 1GH dymour Publications 49 MENTAL MATH IN JUNIOR HIGH LESSON 5 SUBTRACTING IN PAF POWER BUILDER A 1. 56 - 38 = 11. 400 - 125 = 2. 80 -44 = 12. 534 - 225 = 3. 65-36= 13.800-275= 4. 50--29= 14. 775 - 485 = 5. 83-35= 15.900--355= 6. 90 - 36 = 16. 1000 - 825 = 7. 9.0 - 3.6 = 17. 6.35-2.55 8. 8.2 - 1.9 = 18. 8.37 - 4.38 = 9. 5.4 - 2.6 = 19. $20 .00 - $3.75 = 10.9-7.8= 20. $10.00 -$8.63= The difference between two numbers is 25. If the numbers are tripled, what is the difference between the numbers? Copyright CD 1988 by pale Seymour Publications MENTAL MATH IN JUNIOR HIGH LESSON 5 SUBTRACTING IN P POW ER BUILDER B 1. 45-27= 11.400-150= 2. 60 - 33 = 12. 627-418= 3. 84 - 55 = 13. 543 - 244 = 4. 70-38= 14. 1000 - 650 = 5. 93 - 46 = 15. 800 - 450 = 6.80-49= 16. 1000 - 735 = 7. 8.0 - 2.5 = 17.8.25 -3.45= 8. 7.8--2.9 = 18. 9.45 - 3.46 = 9. 7.5 - 2.6 = 19. $10.00 - $2.25 = 10. 6 - 3.7 = 20. $20.00 - $6.55 = THINK IT THROUGH The difference between two numbers is 19. If the numbers are doubled, what is the difference between the numbers? 50 Copyright s 1988 by pale Seymour Publications Two numbers that total a nice "tidy sum (like 10, or 100, or 1000) are called compatible numbers. 45 and 55 are compatible. So are 360 and 640. Compatible numbers make mental math easy! Learn to recogniz& them. Find compatible pairs. Find compatible pairs. 4 0 40 71 400 300 550 56 75 29 30 510 620 250 100 44 33 12 67 630 900 700 380 96 70 25 450 750 490 370 88 600 TRY THESE. USE YOUR HEAD. Think about compatible numbers. On scrap paper, list number pairs that total 100. Write as many as you can in one minute. CO! MENTAL MATH IN Copyright ; 1987 b•, MIDDLE GRADES Seymour Publications 2. How many different pairs of numbers total 1000? 61 MENTAL MATH IN THE MIDDLE GRADES LESSON 15 SEARCHING FOR COMPATIBLES POWER BUILDER A 100 11. 400 + = 1000 2. 94+ = 100 12. 250 + = 1000 3. 31+ _ =100 13. 950 + = 1000 4. 46+ = 100 14. 899 + = 1000 5. 25+ _ = 100 15. 375 + = 1000 1. 35+_ 6. 100-17= 16. 1000-501 = 7. 100-53= 17. 1000 - 695 = 8.100--62= 18.1000-99= 9. 100-95= 19.1000-725= 10.100-39= 20.1000-645= How many different pairs of whole numbers add to 100? THINK IT THROUGH Copyright C 1987 By Dale Seymour Publications MENTAL MATH IN THE MIDDLE GRADES LESSON 15 SEARCHING FOR COMPATIBLES POWER BUILDER B 1. 50 + = 100 11. 700+ = 1000 2. 93 + = 100 12. 975 + = 1000 3. 49 + = 100 13. 499 + = 1000 4. 15+ = 100 14. 450 + = 1000 5. 33 + = 100 15. 95 + = 1000 6.100-75= 16. 1000 - 125 = 7. 100-8= 17. 1000 - 901 = 8.100--29= 18.1000-255= 9.100--80= 19. 1000 - 650 = 10. 100 - 42 = 20.1000-575= THINK IT THROUGH 62 Copyright Z 1987 By Dale Seymour Publications g 7 86 4t0 -3 9 Which problem in each pair is easier? Whys "Making tens" can help you subtract in your head. Adding 2 to 28 makes 30. That's easier to subtract. Then I'll adjust 55, too, to balance. r-- 4 is 57 55 ++2 2 --> _2S 30 27 NTAL MATH IN THE MIDDLE GRADES Copyright i 1987 By Date Seymour Publications Remember: Adding the same amount to both numbers leaves the difference unchanged! MENTAL MATH IN THE MIDDLE GRADES LESSON 14 BALANCING IN SUBTRACTION PO "IER BUILDER A 1.53-28= 11.83-25 2.44-19= 12. 46 - 29 = 3. 71 -35= 13. 71 -38 = 4.85-29= 14.82-26= 5. 50 - 28 = 15. 66-18= 6. 45- 17 = 16.80-29= 17. 46 = 7. 81 -39 - 18 = 8. 56 - 37 = 18.94-49= 9. 37 - 16 = 19.90-65= 10. 42 - 28 = 20.73-56= THINK IT THROUGH Subtract the largest two-digit even number from the largest three-digit even number. Copyright MENTAL MATH IN THE MIDDLE GRADES LESSON 14 1987 By Dale Seymour Publications BALANCING IN SUBTRACTION POWER BUILDER B 1. 52 -- 19 = 11.93-15= 2. 83-28= 12. 66-39= 3. 44 - 26 = 13. 81 -48= 4. 55-17= 14.92-35= 5.70-27= 15.76-47- 6. 51 -29= 16.70-28= 7. 62 - 38 = 17. 36-19= 8. 71 -19= 18. 84-36= 9. 65 - 28 19. 80 - 45 = 10.82-66= 20.83-49- THINK IT THROUGH 60 Subtract the smallest three-digit odd number from the smallest four-digit odd number. When you add the same amount to each number A a subtraction problem, the answer does not change. Adding to both numbers balances the problem. Balancing can sometimes make subtraction easier to do in your head. MENTAL MATH TIP Add whatever you need to change the subtrahend (second number) into an easily subtracted number. TRY THESE IN YOUR HEAD. Use balancing to make it easier. 96-59 4. 132 - 88 7. 583 - 298 2. 65-19 5. 151 - 97 8. 846-399 3. 76 - 27 6. 233 - 95 9. 2100 -- 1998 10. 4363 -- 3999 MENTAL MATH IN JUNIOR HIGH Copyright C' 1988 by Dale Seymour Publications 67 LESSON 12 MENTAL MATH IN JUNIOR HIGH SUBTRACTING BY BALANCING POWER BUILDER A 1. 85-49- 11. 469 - 198 = 2. 73 - 59 = 12. 753 - 187 = 3. 84-37= 13. 641 - 285 = 4. 62-28- 14. 704 - 475 = 5. 126-89= 15.333-189= 6.253-78= 16. 4874 -- 596 = 7. 461-95- 17. 8343-997- 8. 282 - 99 = 18. 6454 - 2198 = 9. 544-77- 19. 7826-1997- 10. 632 - 88 = 20. 9544 - 7985 = Subtract the largest 3-digit odd number from the largest 4-digit even number. POWER BUILDER B 1. 76 - 39 = 11. 457 - 199 _ 2. 84-48- 12. 845 - 1888 = 3. 92-67= 13. 832 - 395 = 4. 65--38= 14. 803 - 565 = 5. 146 - 79 = 15. 666 - 178 = 6. 273 - 85 = 16.6752-375= 7.372-96= 17. 9254 - 1999 = 8. 233 - 99 = 18. 7243 - 4998 = 9. 444 - 77 = 19. 8435 - 2997 = 10. 745 -- 78 7 20. 9635 - 8988 = THINK IT THROUGH Subtract the largest 4-digit odd number from the smallest 5-digit even number. Copynght 0 1988 by Dale Seymour Publications Mental Math - 2 Digit Multiplying Teacher's Guide In addition to the Egyptian box method worksheet for students to try multiplying with, check out this website with alternate methods for 2 to 5 different ways to add, subtract, multiply and divide. The Many Ways of Arithmetic in UCSMP Everyday Mathematics http:'/r ,.math.nvu.edu/-braams/links/e -arith.html Check out the site below to see a neat way the Egyptians actually did multiply by using doubling: http:tmembers.cox.net/eIessons2/Egypt/E3 '2MathMulti loin Measurin theE v tianWav. df For your reference, here are some properties of math that may help the students' understanding with these mental math strategies: Commutative Property of Addition: 5+6=6+5 Associative Property of Addition 5 + 6 + 4 can be thought of as either (5+6)+4 or 5+(6+4) Commutative property of Multiplication 5x6=6x5 Associative Property of Multiplication 5 x 6 x 4 can be thought of as either (5x6)x4 or 5x(6x4) Finally there is the Distributive Property of Multiplication over Addition which is our Egyptian box method on the student worksheet: 5 x (3 + 7) = 5 x 10 = 50 orwemaydo 5x(3+7)= (5x3)+(5x7)=15+35=50 When we multiply 2 x 23, we can: 2x(20+3)=(2x20)+(2x3)=40+6=46 This may make a simple problem more complicated, but if we work backwards, it also simplifies. Ex: (9x6)+(9x4)=9x(6+4)=9x I0=90 The above examples were taken from Stein's Refresher Mathematics 7 t' Ed., Allyn & Bacon, Inc. 1980 2 Digit by 2 Digit Multiplication Explaining in different ways Activity Name: Rather than try to multiply numbers by using a memorized method below: Ex. 52 x 16 312 520 832 Let's try to explain how we can do multiplying by breaking up the numbers into tens and ones. 52 = 50 + 2 and 16 = 10 + 6 a)Then let's multiply each of the 4 numbers and write the answers inside the spaces in the box. b) Lastly, let's add up the four numbers from the boxes. 50 2 10 6 If we add up the 4 multiplications that we did in the box, we get This question can also be written like this: (54+2)x(10±6)= (50 x 10) + (50 x 6) + (2 x 10) + (2 x 6) = Let's practice! 1) 24x22 Add the four boxes totals up = 2) 25 x 34 42 x 16 2) Answer: 4) 13 x 24 3) Answer: 5) 51 x 37 4) Answer: 6) 19 x 27 5) Answer: 6) Answer: Multiplying in your head is easier if you break a number into parts and multiply the front-end numbers first. • Break up 524. Multiply from the front to the back .. . • Add as you go along. Focus on the left (front-end) digits by covering the others. 5 3 1. 4 x 55 4. 8 x 25 7. 2 x 545 2. 4 x 76 5. 4 x 625 8. 8 x 625 3.45x6 6.465x3 9.456x5 10. 3 x 235 MENTAL MATH IN JUNIOR HIGH Copyright 0 1988 by Dale Seymour Publications 73 MENTAL MATH IN JUNIOR HIGH LESSON 15 FRONT-END MULTIPLICATION POWER BUILDER A 11. 5x218= 12. 2x849= 1.6x28= 2. 5x82= 3. 7x36= 13. 6x55= 4. 5x66= 5.4x84= 14. 3 x 428 = 15.7x450= 6.6x45= 7.8x53= 16.4x825= 17. 5x315= 18. 3 x 675 = 19. 4x925= 8.9x72= 9.4x126= 10. 4 x 325 = 20.6x215= THINK IT THROUGH 3x37=111 6x37=222 9 x 3 == 333 Look at the number sentences in the box. Find a pattern and use it to mentally calculate 15 x 37 and 21 x 37. 12 x 3=444 Copyright 4 1988 by Date Seymour Publications MENTAL MATH IN JUNIOR HIGH LESSON 15 FRONT-END MUL71PLIC POWE R BUILDER B 1. 7x27= 11. 5x219= 12.2x849= 2. 5x62= 3. 8x46= 4. 5x66= 5.4x84= 6. 6x45= 7. 8x53= 16.4x825= 17. 5x315= 8. 9x72= 18.3x675= 9. 3x126= 19. 8 x 525 10.4x625= 13. 4x65= 14. 3x428= 15.7x450= 20. 5x319= THINK IT THROUGH Look at the number sentences in the box. Find a pattern and use it to mentally calculate 28 x 15,873 and 42 x 15,873. 7 x 15,873 = 111,111 14 x 15,873 = 222,222 L 21 x 15,873 = 333,333 Copyright 1988 by Dale Seymour Publications /At Doubling numbers is something we do every day. Here's an easy way to do it in your head: Double a number by doubling each of its parts. Then add. PouFLF, 126 TRY THb5E IN YUCK HLAD. Double each number by parts 1. Double 34 2. Double 8 1 3. Double 912 4. Double 47 7. Double 64 8. Double 75 5. Double 29 9. Double 54 6. Double 430 10. Double 720 MENTAL MATH IN THE MIDDLE GRADES Copyright 1987 by Dale Seymour Publications LESSON 28 MENTAL MATH IN THE MIDDLE GRADES DOUBLING POWER BU I LDER A 1. Double 23 = 11. Double 42 2. Double 62 = 12. Double 91 = 3. Double 210 13. Double 325 4. Double 207 = 14. Double 36 = 5. Double 45 = 15. Double 55 = 6. Double 508 = 16. )ruble 86 7. Double 57 = 17. 8. Double 98 = 18. Double 128 = 9. Double 250 = 19. Double 256 = ouble 64 Double 512 10. Double 900 Think of a number. Double it. Add 6. Divide by 2. Subtract the number you thought of first. Now do the same thing with a new starting number. Can you explain why your answer is always 3? THINK IT THROUGH opyrig' °', 1987 By D^. MENTAL MATH IN THE MIDDLE GRADES Seymour Publication( LESSON 28 DOUBLING POWER BUILDER B 1. Double 43 = 11. Double 34 2. Double 74 12. Double 83 3. Double 113 13 4. Double 16 14. ,double 409 5. Double 85 15. Double 75 = 6. Double 700 = 16. Double 27 = 7. Dou" 17. Double 54 87 = )ouble 424 8. Double 97 = 18. Double 108 9. Double 65 = 19. Double 216 = 10. Double 840 = 20. Double 432 = THINK IT THROUGH 96 Think of a number. Multiply it by 4. Subtract 8. Divide by 4. Add 2. Now do the same thing with a new starting number. Can you explain your answer? Copyright 4 1987 By Dale Seymour Publications Here's a trick to make mental multiplication easier. x IS If one number is even, you can cut it in half and double the other number. HALF OF 4 ... DOUBLE 15 2 X 30 C THAT'S EASY! This sometimes gives you an easier problem. 60 18x8 You can even keep on halving and doubling, if it helps. . . 18 X 8 C 35X4 4, r 72X2 1-` 144 TRY THESE IN YOUR HEAD. n Halve one, double the other. 1. 4X17 3. 5X88 7. 8X13 2. 6 X 45 4. 35 X 4 8. 12 X 150 5. 25X 15 9. 8X45 6. 125X 12 10. 55X8 MENTAL MATH IN THE MIDDLE GRADES Copyright 1987 by pate Seymour Publications 97 MENTAL MATH IN THE MIDDLE GRADES LESSON 29 HALVING AND DOUBLING POWER BUILDER A 1.4x13= 11. 14 x 15 2. 6x 15= 12. 15x32= 3. 8x35= 13. 14 x 25 = 4.23x4= 14. 18x25= 5. 35 x6= 15. 250 x 16= 6.4x55= 16.150x6=^ 7.6x65= 17. 150 x 14 = 8.8x15= 18.125x8= 9.37x4=_ 19.14x35= 10. 25x6= THINK IT THROUGH 20. 12x 150= Use mental math to decide which of the following equals 64 x 32: 64x16 128x16 32 x 128 Copyright C 1987 By Dale Seymour Publications MENTAL MATH IN TF- - :iDDLE GRADES LESSON 29 HALVING AND DOUBLING POWER BUILDER B 1.4x14= 11. 18 x 15 = 2.6x25= , f 45 3. 12. 16x25= 13.15x64= 4.24x4 14. 24 x 15= 5.45x6= 15.225x8= 6.4x65= 16. 150x8 7.6x55= 17. 16 x 12 8.8x55= 18.125x6= 9. 47x4= 19.18x35= 10.75x6= THINK IT THROUGH 98 20. 15 x 120 Use mental math to decide which of the following equals 48 x 144: 24 x 96 96 x 288 24 x 72 96 x 72 Copyright © 1987 By Dale Seymour Publications LESSON 36 COMPATIBLE FACTORS How would you do this in your head? Multiplying the numbers in order, step by step, is NOT the answer To make multiplication easier, search for compatible factors. Then rearrange the factors to simplify your figuring. T AY THESE IN YUUK HEAL). Search for compatible factors. 2x8x5 2.2x7x 15 3.4x11 x59 4.4x13x25 5. 6 x 9 x 500 MENTAL Copyright N JUNIOR HIGH 988 by Date Seymour Publications 5.2x 19x5 7.25x5x4x2 8. 2 x 13 x 5 x 5 x 2 9.4x7x3x250 10. 15 x 3 x 2 x 2 x 15 123 MENTAL MATH IN JUNIOR HIGH LESSON 36 COMPATIBLE FA POW ER BUILDER A 1. 5x7x2= 11. 15x3x4x2= 2. 2x13x5= 12.4x4x15x5= 3. 2x6x15= 4. 15x4x5= 13. 5x5x6x2x2= 14.5x7x5x4= 15.9x3x4x5= 5.20x7x5= 6.2x7x5x6= 7. 15x7x2x3= 16. 13x2x3x5= 17. 5x7x7x2= 18.5x5x8x2x4= 19. 11 x2x6x25= 8. 6x4x5x25= 9. 11 x4x2x25= 10. 25x5x4x8= 20.9x8x50x2= THINK IT THROUGH The dimensions of a large tank are 25mby 25mby 8m. What is the volume of water it can hold? Copyright © 1988 by Date Seymour pi 3E -itic MENTAL MATH IN JUNIOR HIGH LESSON 36 COMPATIBLt FA POWER BUILDER B 11. 2x3x5x 13= 12. 9x8x5x2= 1. 4x6x25= 2. 2x29x5= 14 o JO 3. 7x 15x2= 13. 5x3x9x2= 14. 7x5x3x4= 15. 15x4x5x5= 4. 4x 15x5= 5. 5x3x12= 6. 6x4x5x2x5= 0 7. 11 x5x2x6= 8. 3x4x25x13=^Q 9. 12x3x4x25= 10. 4x13x25x2= 3 7C __t) t^C3 16. 11 x5x5x8= 17.7x5x20x8= 18. 25x9x5x4= 19. 50x3x8x3= 20. 125x11 x2x4 = 0 0 //cC THINK IT THROUGH Fifteen workers each worked 40 hours a week for 5 weeks at a rate of $8.00 an hour. Calculate the cost of the payroll. 124: Copyright C9) 1988 by Dale Seymour Public 214 x25 Here's a trick that can simplify mental multiplication. . . a Rearrange one or both of the numbers. 24 X 25 Your aim i s to find compatible pairs. A 6 X 4 X 25 cop 6 X 100 = 600 Can you find a different way to rearrange 24 X 25? TRY THESE IN YOUR HEAD. Rearrange to find compatible pairs 8X15 2. 15 X 24 MENTAL MATH IN THE MIDDLE GRADES Copyright . 1987 by Dare Seymour Publications 3. 15X16 7. 12X15 4. 36 X 50 8. 18 X 500 5. 48X 15 9. 12X35 5.24X500 10.15X26 111 MENTAL MATH IN THE MIDDLE GRADES LESSON 3 MAKE-YOUR-OWN COMPATIBLE FACTORS POWER BUILDER A 1.4x35= 11. 22 x 15 = 2. 4 x 45 = 12.25x18= 3. 15x14= 13. 45 x 16 4. 24x 15= 14. 15x38 5.15x18= 15. 35 x 12 = 6. 12x25= 16.60x25- 7. 5x24= 17.55x40 8. 8 x 25 18. 45 x 8C 9.5x32= 19. 25 x 180 10. 25 x 16 = 20. 450 x 8 = 00 THINK IT THROUGH Knowing that 25 x 25 = 625, mentally calculate 24 x 25, 25 x 26, 25 x 27, and 25 x 23. Copyright 0 1987 By Dare Seymour Publications MENTAL MATH IN THE MIDDLE GRADES LESSON 36 MAKE-YOUR-OWN COMPATIBLE FACTORS POWER BU I LDER B 1.6x25= 11. 25 x 18 = 2. 35x6 = 12.25x28= 3. 55x4= 13. 45 x 12 = 4.6x45= 14. 15x26= 5.45x8= 15. 35x14= 6.6x55= 16. 40x35= 7.45x8= 17.50x24= 8. 8x25=- 18. 250 x 16= 9. 15x22= 19. 40 x 450 = 10. 25 x 14 = 20. 15 x 180 THINK IT THROUGH 112 Copyright C 1987 By Dale Seymour Puh1Etrzrinn. LESSON 37 MAKING COMPATIBLE FACTORS simplify this multiplication, rearrange one or both of the numbers. The trick is to look for pairs of factors that are compatible. Then complete the multiplication in steps. Can you find another pair of compatible factors to check your calculation? TRY THESE IN YOUR HEAD. Make your own compatible factors. 8 x 15 4. 12 x 25 7. 15 x 36 2.4x45 5.18x15 8.36x25 3. 15 x 14 6. 28 x 50 9. 32 x 500 10. 12 x 150 A MENTAL MATH IN JUNIOR HIGH Copyright 0 1988 by Date Seymour Publications 125 MENTAL MATH IN JUNIOR HIGH LESSON 37 MAKING C©MPATISLI POWER BUILDER A 1. 35x4= It 22 x 15 2. 4x45= 12.25x18= 3. 15x14= 13. 45 x 16 4. 24 x 15= 14. 15x36= 5. 15x18= 15.35x12= 6. 12x25= 16.60x25= 7.5x24= 17.55x40= 8. 18x50= 18.45x80= 9.25x16= 19.25x180= 20. 450 x 8 10. 5x32= THINK IT THROUGH Rearrange the factors in these problems and calculate the products mentally: 25x1.2 2.5x48 1.5x8 CopyrightC 1988 by Dale Seymour Publi LESSON 37 MENTAL MATH IN JUNIOR HIGH MAKING COMPA, _C- T POWER BUILDER B 1. 6x25= 11. 25x18= 2. 35x6= 12. 25x28= 3. 55x" _ 13. 45x 12= 4. 6x45= 14. 15x26= 5. 45x8= 15. 35x14= 6. 6x55= 16.40x35= 35x8= 17.50x24= 8. 8x25= 18.250x16= 9. 15x22= 19. 40 x 450 10. 25x14= 20. 15x 180= The dimensions of a box are 15 by 10 by 24. If one dimension is doubled, what is the new volume? What happens to the volume if one dimension is halved and another doubled? What if all the dimensions are doubled? Copyright! 1988 by Dare Seymour Pubtica: 50 what happens when one -factor is multiplied by 10 .. . The product is also multiplied by 10. a Iotice 150 You can use that idea to multiply numbers with trailing zeros. For each time that a factor is multiplied by 10, tack another trailing zero onto the product. 5 5 X3 x3 15 15 5[0 x301 5C1 X-3 IM Temember these steps: 60 x 300 • Remove the trailing zeros. Multiply the remaining numbers. • Tack on ALL the zeros. 6M x3 = ► 8 18 o00 TRY THESE IN YOUR HEAD. Tack on trailing zeros. 1. 4 x 20 4. 50 x 50 7. 00 x 30 2. 4 x 50 5. 300 x 9 8. 5 x 8000 3. 50 x 20 6. 7 x 800 9. 30 x 500 10. 200 x 300 MENTAL MATH IN JUNIOR HIGH Copyright 0 1988 by Dale Seymour Publications 71 MENTAL MATH IN JUNIOR HIGH LESSON 14 TACK ON TRAIUNG ZE POWER BUILDER A 1. 7x30- 11. 50x600= 2. 8x60= 12. 300 x 50 = 3.9x20= 13. 90 x 200 = 4. 5x40= 14. 7x8000= 5.500x9= 15. 50x6000= 6.300x8= 16. 800x700= 7.5x800= 17. 900 x 500 = 8. 30 x 200 18. 7000 x 60 = 9. 400 x 60 = 19. 300 x 700 = 10. 70 x 500 THINK IT THROUGH 20. 50 x 8000 = i List all the different products that can be formed by multiplying any two numbers on this card. 40 60 30 20 Copynghtcg) 1988 by Dale Seymour Public= ` is LESSON 14 MENTAL MATH IN JUNIOR HIGH TACK ON TRAILING POWER BUILDER B 1. 6x40= 11.50x200= 2. 7x50= 12. 500 x 50 = 3. 8x30= 13. 70 x 400 = 4. 5x60= 14.3x600= 5. 500x7= 15. 50 x 8000 = 6. 300x6= 16. 600 x 300 = 7.5x400= 17. 800 x 500 = 8.20x400= 18. 8000 x 60 9. 600 x 40 = 19. 200 x 600 = 10. 500 x 90 = 20. 50 x 4000 = THINK IT THROUGH List all the different products that can be formed by multiplying any two numbers on this card. 72 Copyright Q 1988 by Dale Seymour Publications I LESSON 17 TACK ON TRAILING ZEROS DIVIDE IN YOUR HEAD 1200 : 4 N Numbers with trailing zeros are easy to divide in your head. Fo1bo nese steps. Remove the trailing zeros. • Divide the remaining numbers. • Tack the trailing zeros onto your answer. V4x3 • Check by multiplying. TRY THESE IN YOUR HEAD. Cut off and tack on the trailing zeros. 1. 1200 - 2 4. 7 28 00 7. 9)27,000 2. 2400-8 5. 4 38{ 8. 3600-.- 6 3. 1000 - 5 6. 12 2400 9. 3500 35 10. 15 3000 MENTAL MATH IN JUNIOR HIGH Copyright 15 1988 by Dale Seymour Publications LESSON 17 MENTAL MATH IN JUNIOR HIGH TACK ON TRAILING ZE POWER BUILDER A 1. 2400 -,- 6 = 11. 1800 -1- 2 2. 320-+-8- 12.9 18,000 = 3. 7 420 = 13. 42,000 + 6 4. 7 3508 = 14.6 120,000 5. 4)286. 15. 2500 + 25 = 6. 7 16. 48,000 + 6 4900 = 7. 540+6 = 17. 72 ,000 + 8 8. 5600- 8 = 18.7 21,004= 9. 2400 + 3 = 19.5 250,000 = 10. 5 3500 = 20.3 21,000 _ THINK IT THROUGH How many 5-cent stamps can you buy for $25? MENTAL MATH IN JUMOR HIGH LESSON 17 POWER BUILDER B 1. 3200 + 4 11.1600.2= 2. 400+8= 12. S 16,000- 3. 4 280 = 13. 42,000 -- 6 4. 7 4200 14.4 160,000 = 5. 9 270= 15. 1500 + 15 = 6. 8 4800 16. 36,000 + 6 = 7. 540+6= 17. 56,000 ^ 8 = 8. 6300 a- 7 18.7 21,000 9. 2400+3- 19.5 45,000 10. 5 2500 -= 20.3 27,000 = - How many 5-cent stamps can you buy for $100? TACK ON TRAILING t - You can divide both numbers in a division problem by the same amount without changing the answer. Using this idea, it's easy to simplify a problem when both numbers have trailing zeros. 8000 : 400 80 -:- 4 Cancel the common trailing zeros. 80 -4 -2o TRY THESE IN YOUR HEAD. Cancel the common trailing zeros. 1. 9000.30 4. 800 - - 20 7. 5000 ^ 50 2. 900 5. 1000 ^ 50 8. 3600 -= 900 6. 2000 ^ 50 9. 10,000 -- 1 00 3. 9000 300 ^ 3000 10. 1,000, 000 = 2000 MENTAL MATH IN JUNIOR HIGH Copyright 0 1988 by Dale Seymour Publications 79 MENTAL MATH fN JUNIOR HIGH LESSON 18 CANCEL COMMON TRAILING ZER( POWER BUILDER A 1. 800+0 11. 600 1200 = 2. 12,000 - 600 12. 50 3. 15,000. 30 = 13. 72,000 + 900 4. 2400 + 80 = 14. 800 J 3200= 5. 60 15. 30,000 + 60 = 3600 40,000 = 6. 90 72,000 = 16. 45,000+90= 7. 400 32,000 -- 17. 500 20,000 - 8. 50 350 18. 70 9. 800 19. 81,000+900= 4800 = 10. 4900 + 70 = 4200 20. 45,000 + 50 = f THINK IT / THROUGH The state gets a tax of 100 for every dollar of gasoline sold. How many dollars does the state get for gasoline sales of $400,000? Copynght ® 1988 by Dale Seymour Pubticatir- c LESSON 18 M E r 1L MATH IN JUNIOR HIGH CANCEL COMMON TRAIUNG t. POW ER BUILDER B 1. 600 + 30 11. 300 j 2003 2. 16,000 -+- 400 = 12. 50 30,000. _ 3. 18,000 + 60 13. 56 ,000 -:- 700 4. 3200 + 80 = 14. 400 2800 = 5. 50) 2500 = 15.40,000+80= 6. 80 6 000 = 16. 54,000 + 90 = 7. 300 27,000 17. 500 30,000 = 8. 50 18. 80 9. 600 450 = 4 000 = 10. 8100-90 = 7200 = 19. 63 ,L00 + 900 = 20. 35 ,000 - 50 = NK IT 4UGH The state gets a tax of 15c for every dollar of gasoline sold. How much money does the state get on gasoline sales of $600 ,000? 80 Copynght U 1988 by Dale Seymour Pubfocations w LESSON 13 BALANCING WITH DECIMALS Which problem would you rather do in your head? They look different, but they are really the same problem. Balance by adding 0.04.. . 3.42 + 0.04 3.46 -- .96 + 0.04 -^-- 2. That's how balancing can make a problem easier. How could you make these problems easier by balancing? 7.84 1 .95 TRY THESE IN YOUR HEAD. Use balancing to make them easier. 1. 4.15-1.9 4. 8.1-0.7 7. 3.53 -- 0.88 2. 6.4-3.8 5. 4.23 - 1.98 8. 5.75 -- 0.96 3. 9.3-6.9 6. 7.45 - 4.98 9. 8.22 - 1.94 10. 15.362- 4.989 MENTAL MATH IN JUNIOR HIGH Copyright 0 1988 by Dale Seymour Publications 69 MENTAL MATH IN JUNIOR HIGH LESSON 13 BALANCING WITH DECII POWER BUILDER A 1. 4.7.2.9 = 11.6.24.3.88= 2. 7.1-3.8 - 12. 9.23 - 4.96 = 3. 9.2 - 4.7 = 13. 14.52 -- 3.99 = 4. 6.3---2.8- 14.22.62-15.89= 5. 5.14-0.98= 15. 36.03 -- 25.95 = 6. 6.33-0.87= 16. 82.32 - 19.96 7. 8.21 -0.95= 17. 5.276 - 1.999 = 8. 7.42 - 0.97 = 18. 15.825 - 7.998 = 9. 9.32 -- 2."' _ 19. 23.543 - 13.985 = 10. 8.15 --5.79= 20. 45.00' - 19.998 = / THINK IT THROUGH f Take the largest 3-digit decimal less than one and double it. What do you need to add to get a sum of 4? Copyright 0 1988 by Dale Seymour Publi MENTAL MATH IN JUNIOR HIGH LESSON 13 POW `R BUILDER B 1. 5.6 - 3.9 = 11. 5.21 -1.89= 2. 8.2 - 4.7 = 12.8.34--2.87= 3. 7.5.5.8= 13. 15.41 - 4.99 = 4. 7.2 - 3.9 = 14. 21.43 - 20.99 = 5. 6.15 - 4.99 = 18. 23.05 - 19.98 = 6. 7.33 --0.88= 16. 75.34 - 29.97 7. 7.22 - 0.96 = 17. 41.85 - 1.999 = 8. 8.31 - 0.97 = 18. 12.940 - 6.998 = 9. 8.25 - 4.96 = 19. 42.342-20.987= 10. 9.17 - 4.88 20. 50.002 - 30.999 = Take the largest 2-digit decimal less than one and triple it. What do you need to subtract to have a difference of 1 ? BALANCING WITH DE 'ns 4 When pairs of decimal numbers add to a whole number, we can say they are compatible. It works with money amounts, and it works with plain decimals. Co M PnfIBLE P,aiRS ^ I.io+ *0.90 1.74^ 0.26 x8.45 + S I.55 3.7^1.3 Find compatible pairs. Find compatible pairs. $0.52 12.8 $6.90 $9.60 $9.33 $5.40 $0.67 $2.50 $0.48 4.15 5.37 $3.10 9.15 0.85 2.4 3.85 94.63 2.6 TRY THESE IN YOUR HEAD. Find compatible pairs that add to $1.00. $0.85 $0.71 $0.29 $0.15 $0.34 MENTAL MATH IN JUNIOR HIGH Copyright a 1988 by Dale Seymour Publications $0.35 $0.41 $0. 65 $0.66 2. Find compatible pairs that add to 10. 0.85 2.75 6.20 4.55 3.80 5.10 9.15 4.90 7.25 55 MENTAL MATH IN JUNIOR HIGH LESSON 8 SEARCHING FOR COMPATIBLE DECIM. POW ER BUILDER A = $1.00 1. $0.52+ 2. $0.69 + _ $1.00 + 0.36 = 1 +0.88= 1 4. 5. 6. 7. 8. 9. 0.11 + $2.45 + $4.51 + _ $10.00 =$10.00 = 10 9.38+ +3.69= 10 + 5.74 = 10 10. 11. $4.95 + 12. $3.69 + 13. 14. 1.7+ 15. 8.2+ 16. 17. 18.0.74+ = $5.00 = $5.00 + 1.63 = 5 =5 = 10 +4.4=10 19. 9.345+ +17.64=20 =10 = 10 20. +4.745=5 THINK IT THROUGH Megan has only dimes and quarters. She has the same number of quarters as dimes. If she has $3.85, how many quarters does she have? Copyright© 1988 by Dale Seymour Pubf ao," LESSON 8 MENTAL MATH IN JUNIOR HIGH SEARCHING FOR COMPATIBLE DECI, POWER BUILDER B 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. $0.64+ $0,73+ 0.39+ $3.35 + $6.52+ 8.28+ = $1.00 = $1.00 +$0.44=$1.00 + 0.77 = 1 =1 = $10.00 = $10.00 = 10 + 4.59 = 10 + 6.68 = 10 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. $3.72 + $3.57 + 2.6+ 7.3+ 0.74+ 9.125+ = $5.00 = $5.00 + 1.59 = 5 =5 = 10 +5.5=10 + 18.38 = 20 =10 = 10 + 4.085 = 5 Josh has only dimes and quarters. He has the same number of quarters as dimes. The total value of the quarters is 7500 more than the total value of the dimes. How much money does he have?