number sense - wikispacessense+&+mental+math.pdf · number sense mental mathematics. mental...

Number Sense

Mental Mathematics

MENTAL MATH

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

. Traditional (pencil and paper)

156+ 395551

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 = 300Add 40 + 20 = 60Add 5+ 3 = 8

And then add them all together = 368.

3. CompensationThis is where we change one number to make the addition easier by "compensating'.

Example: 99 + 172Here 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 wesubtract the I we compensated to get 271 for our answer.

4. Associative PropertyThis is where we are adding more than 2 numbers and we just re-arrange them to paireasier numbers to add together.

Example : 27 + 45 + 33So, we re-arrange the numbers so that we can make pairs of numbers that are easier toadd. It would look like this (27 + 33) + 45 and we can make (60) + 45 =105. This ismuch easier than starting with 27 + 45 and then adding 33 to that.

5. Commutative PropertyThis is addition with only two numbers. It is much like Associative property except you arere-arranging the numbers so they "appear" easier to add,

Example : 17 + 22 = 22 + 17It 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 toinclude 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 -4 16- 9 4 We change this to a 10 to make it easier to subtract 4 - 106 E- We have the same difference on each side 4 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)

23x14

92+230

322

2. Skip Counting or Repeated AdditionThis is simply adding the given number to the previous answer as many times as needed.

Example : 2x5 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 52Half of 16 is 8 and double 52 is 104.Now we have 8x 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 416Now we have 2 x 416 (instead of 16 x 52) = 832!

4. Associative PropertyThis is where we are multiplying more than 2 numbers and we just re-arrange them to paireasier 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)x9=270.

5. Commutative PropertyThis is multiplication with only two numbers. It is much dike Associative property exceptyou 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) and 3 x 5 (= 15)and them together quickly to get 45.

6. Distributive PropertyThis is like working through the brackets. The most common sample of this is the EgyptianBox. (See attached sheet,)

Example : 23 x 41 is the same as (20 + 3) x (40 + 7)Then you do 20x40= 800,20x7 = 140,3x40 = 120, and 3 x 7 = 21 and add theanswers all together. (Total = 1081)

In paper-and-pencil computation,we usually start at the right andwork toward the left.

To add in your head, start at the left.

1 + 3 = 4

4 + 1 and 3 tenths = 5.3

1.7+3.6

7 tenths + 6 tenths = 1 and 3 tenths ) o o a

TRY THESE IN YOUR HEAD.Add from the left.

1. 22+39 4. 526+48

2.46+38 5. 329+36

3. 56 + 37 6. 236 + 120

7. 4.5+2.5

8. 6.4+1.8

9. 26.5+2.7

10. 43.8 + 10.8MENTAL MATH IN JUNIOR HIGHCopyright 0 1988 by Bale Seymour Pubfications 41

MENTAL MATH IN JUNIOR HIGH

POWER BUILDER A1. 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 =

5.68+35= 15. 5:4 + 3.8 =

6. 268 + 35 = 16. 12.6 + 6.7 =

7. 417+58= 17.23.6+5.9=

8. 545 + 228 = 18. 45.8 + 3.8 =

9. 624 + 239 = 19. 37.8 + 11.2 =

13.356+517= 20. 45.9 + 12.8 =

THINK IT

MENTAL MATH IN .Jr1IOR HIGH

December 22 is the first day of winter. March 21is the fast day of winter. How many days doeswinter offcally have?

LESSON 1 ADDING FROM THE U

Copyright ® 1988 by D_ w Seymour Publics

LESSON 1 Af)DfNG FROM THE ...r"

POWER BUILDER B1.46+28=

2. 47+25=

3. 66+ 19=

4. 24 + 58 =

5. 48+35-

6. 437 + 55 =

7.638+29=

8. 345+227.

9. 462+219=

10.456+338=

11.5.2+3.9=

12. 5.8 + 1.5 =

13. 3.7 + 4.6 =

14.2.9+5.3=

15. 2.8 + 6.4

16.14.6+4.9=

17. 43.4 + 4.6 =

18.53.7+5.4=

19. 33.6 + 12.5

2©.65.8+12.2=

June 22 is the first day of summer. September 21is the last day of summer . How many days doessummer officially have?

Copyright 4 1988 by Dale Seymour Publications

Mental Math Addition Practice (Carrying from ones into the tens)

243637

627+649

+ 224+ 317

227 333 172

+146 +157 + 519

454 639 714

+117 +241 +168

Like this .. .

Make your own compatibles.

Adding in your head is easierwhen you make your owncompatible pairs, then adjust,

Adjust the answer.

1. 75 + 28 4. 427 + 75 7. 795 + 206

2. 69 + 35 5. 450 + 65 8. 253 + 752

3. 188 + 213 6. 580 + 423 9. 1150 + 356

10. 1250 + 757

Copyright c f 988 by pate Seymour Pubhcatians

TRY THESE IN YOUR HEAD.Make compatibles and adjust.


POWER BUILDER A1, 25+79=

2. 45+57=

3. 18+85=

4. 75+28-

5. 68+33=

6. 159+42=

7. 125 + 277 =

8. 468 + 35 =

9. 109 + 393

10. 254 + 349 =THINK IT

THROUGH

1'; )


POWER BUILDER B

Copyright ® 1988 by Dale Seymour Publications

LESSON 7 MAKE YOUR OWN COMPA7

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 =

54;

LESSON 7 MAKE YOUR OWN COMPATIBL

11.435+568=

12. 295 + 706 =

13. 455 + 456

14.263+738-

15.375+526=

16.276+727=

17. 459+544--

18. 2500 + 501 =

19. 425 + 176 =

20. 725 + 277 =

If 867 + 133 = 1000, what is 867 + 135?868 + 132? 8.67 + 1.33?

If 655 + 1345 = 2000, what is 655 + 1355?645 + 1355? 6.55 + 13.45?

Copyright Co 1988 by Dale Seymour Publications

AACAMTAE SAAT1J IlU 11 IAlifld UIf±U

In mental math, when a subtraction problemneeds regrouping. . .

DON'T DO THIS ...

2^.6-8=13.6

75-36 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

10. $10.00 -- $3.50

DO THISI

35=(a0= -75

MENTAL MATH IN JUNIOR HIGHCopyright 0 1988 by pale Seymour Publications 49


POWER BUILDER A1. 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 =

8. 8.2 - 1.9 -

9. 5.4 - 2.6 =

10. 9 - 7.8 =

THINK ITTHROUGH


POWER BUILDER B

e

Copyright ( 1988 by Dale Seymour Publications

LESSON 5 SUBTRACTING IN P

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 ITTHROUGH

50

20. $10.00 - $8.63 =

17.6.35--2.55=

18. 8.37 - 4.38 =

19. $20 . 00 - $3.75 =

LESSON 5 SUBTRACTING IN PART

The difference between two numbers is 25.If the numbers are tripled , what is the differencebetween the numbers?

The difference between two numbers is 19.If the numbers are doubled, what is the differencebetween the numbers?

Copyright O 1988 by Dale Seymour Publications

LESSON 15 SEARCHING FOR COMPATIBLES

Two numbers that totala nice "tidy" sum (like 10,or 100, or 1000) are calledcompatible numbers.

45 and 55 are compatible.So are 360 and 640.

Compatible numbers make mental math easy!Learn to recognize them.

Find compatible pairs.

4 60 40 71

56 75 29 30

44 33 12 67

96 70 88 25

E


400 300 550 600

510 620 250 100

630 900 700 380

450 750 490 370

TRY THESE. USE YOUR HEAD.Think about compatible numbers .J

number pairs that pairs of numberstotal 1 00. Write as total 1000?many as you can inone minute. GO!

ENTAL MATH IN THE MIDDLE GRADESCopyright C 1987 by Date Seymour Pubtications

1. On scrap paper, list 2. How many different

61

MENTAL MATH IN THE MIDDLE GRADES

POWER BUILDER A1. 35 + --100

2. 94 + = 100

3. 31 + = 100

4. 46 + = 100

5.25+ =100

6.100--17=

7. 100-53=

8.100-62=

9. 100-95=

10. 100 - 39 =

THINK ITTHROUGH

o0

Copyright © 1987 By Dale Seymour Publications


POWER BUILDER B1. 50+ = 100

2. 93 + = 100

3. 49 + = 100

4.15+ =100

5. 33+ = 100

6.100-75-

7. 100-8=

8. 100 - 29 =

9.100-80=

10. 100 - 42 =

THINK ITTHROUGH

62


11. 700+ = 1000

12. 975+ = 1000

13. 499 + = 1000

14. 450 + = 1000

15.95+_ =1000


11. 400+ = 1000

12. 250 + = 1000

13. 950 + = 1000

14. 899 + = 1000

15. 375+ = 10()0

16. 1000 - 501 =

20.1000-645=

17.1000-695=

18. 1000 -- 99 =

19.1000-725=

How many different pairs of whole numbersadd to 100?

16. 1000 - 125 =

17. 1000 - 901 =

18. 1000 - 255 =

19. 1000 - 650 =

20. 1000 - 575 =

How many different pairs of even wholenumbers add to 100?

Copyright V 1987 By Dale Seymour Publications

2. 54 4. 31

Remember : Adding thesame amount to bothnumbers leaves thedifference unchanged!

-39 -54 7. 90 --- 35 10. 93 -- 39

MENTAL MATH IN THE MIDDLE GRADESCOsyright © 1987 By Dale Seymour Publications

Which problem in each pair is easier? Why?

`Making tens" can help yousubtract in your head.


POWER BUILDER ALESSON 14 BALANCING IN SUBTRACTION

53-28=1 11.83-25=.

44-192 12.46-29=.

3 71 - 35 = 13. 71 -38=.

85 - 294 14.82-26=.

50-28=5 15. 66- 18=.

45 - 17 =6 16.80-29=.

81 -39=7 17. 46- 18=.

8. 56 - 37 = 18. 94 - 49 =

37 - 16 =9 19. 90-65=.

10. 42 - 28 = 20. 73 - 56 =

THINK ITTHROUGH


POWER BU I LDER B

Copyright 1987 By Dale Seymour Publications

LESSON 14 BALANCING IN SUBTRACTION

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 ITTHROUGH

60

Subtract the largest two-digit even number'from the largest three-digit even number.

Subtract the smallest three-digit odd numberfrom the smallest four-digit odd number.

When you add the same amount to each numberjai a subtraction problem, the answer does not change.

Adding }o both numbers balances the problem.

1Q 1417

_ --7 + 3 -iO7

"'-,SAM 6 F\"" SAMC

p rF F E T^ENc€ t i______ :

Balancing can sometimes makesubtraction easier to do in your head.

MENTAL MATH TIPAdd whatever you need to changethe subtrahend (second number)into an easily subtracted number.

ETRY 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 -- 3999MENTAL MATH IN JUNIOR HIGHCopyright © 1988 by pale Seymour Publications 67


POWER BUILDER ALESSON 12 SUBTRACTING BY BALANCING

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 numberfrom the largest 4-digit even number.

POWER BUILDER B1.76-39= 11.457-199-

2.84--48= 12. 845 - 188 =

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 = 20. 9635 - 8988 =

Subtract the largest 4-digit odd numberfrom the smallest 5-digit even number.

Mental Math - 2 Digit MultiplyingTeacher'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 Mathematicshttp:/f'ww,h--,math.nvu.edu/-braarns/links/em-arith.htmiCheck out the site below to see a neat way the Egyptians actually did multiply by usingdoubling: http://members.cox.net'e-lessons2.IE t!E3 I 2MathMulti l yin Measurin theE v tianWa df

For your reference , here are one properties of math that may help the students'understanding with these menEal math strategies:Commutative Property of Addition:5+6=6+5

Associative Property of Addition5 + 6 + 4 can be thought of as either(5+6)+4 or 5+(6+4)

Commutative property of Multiplication5x6=6x5

Associative Property of Multiplication5 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 ourEgyptian box method on the student worksheet:

5x(3+7)=5x 10=50

or we may do 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 alsosimplifies.

Ex: (9x6)+(9x4)=9x(6+4)=9x 10=90

The above examples were taken from Stein's Refresher Mathematics 7`'' Ed., Allyn & Bacon , Inc. 1980

2 Digit by 2 Digit Multiplication Name:Explaining in different ways Activity

Rather than try to multiply numbers by using a memorized method below:

Ex. 52x 16312520832

Let's try to explain how we can do multiplying by breaking up the numbers into tens andones.

52 = 50 + 2 and 16 = 10 + 6 a)Then let's multiply each of the 4 numbers and write theanswers inside the spaces in the box. b) Lastly, let's add up the four numbers from theboxes.

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:

(50 + 2)x(10+6) = (50 x 10) + (50 x 6) + (2 x 10) + (2 x 6

Let's practice!

1) 24 x 22

Add the four boxes totals up

2) 25 x 34

3) 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:

LESSON 15 FRONT-END MULTIPLICATION

524x 3

I

Multiplying in your head is easier ifyou break a number into parts andmultiply the front-end numbers first.

• Break up 524.

• Multiply from the frontto the back ...

• Add as you go along.

500+20+4

Focus on the left (front-end) .5digits by covering the others . X 3

TRY THESE IN VOLMultiply from the front.

4x55

2. 4 x 76

3.45x6

4.8x25 7.2x545

5. 4 x 625 8. 8 x 625

6.405x3 9.450x5

10. 3 x 235

MENTAL MATH IN JUNIOR HIGHCopyright 0 1988 by Dale Seymou r Publications 73

MENTAL MA'T'H IN JUNIOR HIGH

POWER BUILDER A1. 6x28= 11. 5x218=

2. 5x82= 12.2-x849=

3.7x36= 13.6x55=

4. 5x66= 14. 3x428=

5. 4x84= 15.7x450=

6.6x45= 16. 4x825=

7. 8x53= 17. 5x315=

8.9x72= 18.3x675=

9. 4x126- 19. 4 x 925 =

10.4x325= 20. 6x215=

Look at the number sentences in the box.Find a pattern and use it to mentally calculate15 x 37 and 21 x 37.

LESSON 15 FRONT-END MULTIPLICATION

3x37=1116 x 37 = 2229 x 37 = 333

12 x 37 = 444

Copyright © 1988 by Dale Seymour Publications


POWER BUILDER B

LESSON 15 FRONT-END MUL71PLIC

1. 7x27= 11.5x219=

2. 5x62= 12.2x849=

3. 8x46= 13. 4x65=

4. 5x66= 14. 3x428=

5. 4x84= 15.7x450=

6. 6x45= 16.4x825=

7. 8x53= 17. 5x315=

8. 9x72= 18.3x675=

9.3x126= 19.8x525-

10. 4x625= 20. 5x319=

Look at the number sentences in the box.Find a pattern and use it to mentally calculate28 x 15,873 and 42 x 15,873.

7x15,873 =111,11114 x 15,873 = 222,22221 x 15,873 = 333,333

Copyright C© 1988 by Dale Seymour Publications

DOUBLING

How muchfor two o fthose?

Doubling numbers is something we doevery day. Here's an easy way to do it

LESSON 28

in your head:Double a number by doubl i ngeach of its parts . Then add .

PouLE 126

Double 34 3. Double 912 7. Double 64

2. Double 81 4. Double 47 8. Double 75

5. Double 29 9 . Double 54

6. Double 430 10. Double 720

ENTAL MATH IN THE MIDDLE GRADESCopyright © 1987 by Dale Seymour Publications 95


POWER BUILDER A1. Double 23 =

2. Double 62 =

3. Double 210 =

4. Double 207 =

5. Double 45 =

6. Double 508

7. Double 57

8. Double 98 =

9. Double 250 _..

10. Double 900 =

THINK ITTHROUGH

LESSON 28 DOUBLING

11. Double 42

12. Double 91

13. Double 325 =

14. Double 36 =

15. Double 55

16. Double 86

17. Double 64

18. Double 128

19. Double 256

20. Double 512 =

Think of a number. Double it. Add 6. Divide by 2.Subtract the number you thought of first. Now dothe same thing with a new starting number. Canyou explain why your answer is always 3?

Copyright 0 1987 By Dale Seymour Publication


POWER BU ILDER B1. Double 43

2. Double 74

3. Double 113

4. Double 16 =

5. Double 85 =

6. Double 700

7. Double 87 =

8. Double 97 =

9. Double 65 =

10. Double 840

THINK ITTHROUGH

0

LESSON 28 DOUBLING

11. Double 34

12. Double 83 =

13. Double 424 =

14. Double 409 =

15. Double 75

16. Double 27

17. Double 54 =

18. Double 108

19. Double 216

20. Double 432

Think of a number. Multiply it by 4. Subtract 8.Divide by 4. Add 2. Now do the same thing witha new starting number . Can you explain your answer?

96 Copyright © 1987 By Dale Seymour Publications

LESSON 29 HALVING AND DOUBLING

Here's a trick to makemental multiplicationeasier.

If one number is even,you can cut it in half anddouble the other number

This sometimes gives youan easier problem.

0 2 X 30THAT'S EASY!

60

You can even keep on 18x8halving and doubling,if it helps ... 18 X 8

36X44 4

72X2

144TRY THESE IN YOUR HEAD.

Halve one , double the other.

1.4X17

2. 6 X 45

0 <j 0

3.5X68 7.8X13

4. 35 X 4 8. 12 X 150

5.25X16 9.8X45

6. 125 X 12 10. 55 X 6

4x 5

HALF OF 4 . . . DOUBLE 15

MENTAL MATH IN THE MIDDLE GRADESCopyright ©1987 by Dale Seymour Publications 7


POWER BUILDER A1. 4 x 13 = 11. 14 x 15

2. 6x 15= 12. 15x32=

3. 8x35= 13. 14x25=

4. 23 x 4 = 14. 18x25=

8.35x6= 15. 250 x 16 =

6.4x55 16. 150x6=

7. 6x65= 17. 150 x 14 =

8. 8x 15= 18.125x8=

9. 37x4= 19. 14x35=

10.25x6= 20. 12 x 150

THINK ITTHROUGH


Use mental math to decide which of the followingequals 64 x 32:

64x16 128x1632 x 128 128x64

Copyright 0 1987 By Dale S yr o ' Publications


POWER BUILDER B1. 4x 14=

2. 6 x 25 =

3. 8x45=

4. 24x4=

5.45x6=

6.4x65=

7. 6x55=

8.8x55=

9. 47x4=

10. 75 x 6 =

THINK ITTHROUGH

98

11. 18 x 15 =

12. 16 x 25 =

13. 15 x 64

14. 24 x 15

15.225x8=

16. 150x8=

17. 16 x 12 =

18. 125x6=

19. 18x35=

20. 15 x 120 =


Use mental math to decide which of the followingequals 48 x 144:

24x96 24x7296 x 288 96 x 72

Copyright, 1987 By Da'i Seymour Publications

LESSON 36 COMPATIBLE FACTORS

How would you do thisin your head? Multiplyingthe numbers in order, stepby step, is NOT the answer.

To make multiplication easier,search for compatible factors.

Then rearrange the factors tosimplify your figuring.

25x5x9x2x4

TRY THESE IN YOUR HEAD.Search for compatible factors .

MENTAL MATH IN JUNIOR HIGHCopyright 0 1988 by Dale Seymour Publications 1 23


POWER BUILDER A1. 5x7x2=

2.2x13x5=

3. 2x6x15=

4. 15x4x5=

5.20x7x5=

6. 2x7x5x6=

7. 15x7x2x3=

8. 6x4x5x25=

9. 11 x4x2x25=

1©.25x5x4x8


POWER BUILDER B

Copyright © 1988 by Dale Seymour Puhticati

LESSON 36 COMPA-n L-rA

1. 4x6x25= 11.2x3x5x13=

2. 2x29x5= 12. 9x8x5x2=

3. 7x15x2= 13. 5x3x9x2=

4. 4x 15x5= 14. 7x5x3x4=

5. 5x3x 12= 15.15x4x5x5=

6. 6x4x5x2x5= 16. 11 x5x5x8=

7. 11 x5x2x6= 17.7x5x20x8=

8.3x4x25x13= 18. 25x9x5x4=

9. 12x3x4x25= 19. 50x3x8x3=

10. 4x 13x25x2= 20. 125x-11 x2x4=

LESSON 36 COMPATIBLE Ei

11.15x3x4x2=

12. 4x4x 15x5=

13. 5x5x6x2x2=

14. 5x7x5x4=

15. 9x3x4x5=

16. 13x2x3x5=

17.5x7x7x2=

18. 5x5x8x2x4=

19. 11 x2x6x25=

20.9x8x50x2=

The dimensions of a large tank are25 m by 25 m by 8 m. What is thevolume of water it can hold?

Fifteen workers each worked 40 hours a weekfor 5 weeks at a rate of $8.00 an hour.Calculate the cost of the payroll.

Copyright C© 1988 by Dale Seymour Publiatio

LESSON 36 MAKE-YOUR-OWN COMPATIBLE FACTORS

Here's a trick that can simplifymental multiplication .. .

Rearrange one or bothof the numbers.

Your aim is to findcompatible pairs.

a

24 X 25

A6X4X25

COY'

Can you find a different wayto rearrange 24 X 25?

6 X 100 = 600

TRY THESE IN YOUR HEAD.

Rearrange to find compatible pairs.

I

5. 48X 15

6. 24 X 500

7. 12X15

8. 18X500

9. 12X35

10. 15 X 26

MENTAL MATH IN THE MEDDLE GRADESCopyright 9 1987 by Dale Seymour Publications 1 1 1

8X15 3. 15X16

2. 15X24 4. 35X50

2qx25


POWER BUILDER A1.4x35=

2. 4x45=

3. 15 x 14 =

4.24x15=

5. 15x 18=

6.12x25=

7. 5x24=

8.8x25=

9. 5x32=

10. 25x 16=

00

THINK ITTHROUGH


11. 22 x 15 =

12. 25 x 18

13.45x16=

14. 15x36=

15. 35 x 12

16. 60 x 25 =

17.55x40=

18.45x80=

19. 25 x 180

20. 450 x 8 =

Knowing that 25 x 25 = 625, mentally calculate24 x 25, 25 x 26, 25 x 27, and 25 x 23.

Copyright 0 1987 By Dale Seymour Publications


POWER BU I LDER B1.6x25=

2. 35x6=

3.55x4=

4.6x45=

5. 45 x 8

6.6x55=

7. 45x8=

8. 8x25=

9. 15x22=

10. 25 x 14 =

00

THINK ITTHROUGH

112


11. 25 x 18 =

12. 25 x 28 =

13.45x12=

14. 15x26=

15. 35x 14=

16. 40 x 35 =

17. 50 x 24 =

18. 250x 16=

19. 40 x 450 =

20. 15 x 180 =

Knowing that 50 x 50 = 2500, mentally calculate49 x 50, 50 x 51, 48 x 50, and 50 x 52.

Copyright 0 1987 By Dale Seymour Publications

LESSON 37 MAKING COMPATIBLE FACTORS

". -) simplify this multiplication,rearrange one or both ofthe numbers.

The trick is to lookfor pairs of factorsthat are compatible.

Then complete themultiplication in steps.

Can you find anotherpair of compatible factorsto check your calculation?

TRY THESE IN YOUR HEAD.b wnk o cor ma e you patible facto

r

28 x 25

7x4x25

7 x 100

700

4. 12x25 7. 15x36

5.18x15 8.36x25

5. 28 x 50 9. 32 x 500

10. 12 x 150MENTAL MATH IN JUNIOR HIGHCopyright 0 1988 by Date Seymour Publications 125

rs.


POWER BUILDER A

Rearrange the factors in these problems andcalculate the products mentally:1.5x8 25x1.2 2.5x48

1. 35x4= 11. 22x15=

2. 4x45= 12. 25 x 18

3. 15x14= 13. 45x16=

4.24x15= 14. 15 x 36 =

5. 15 x 18 15. 35x12=

6. 12x25= 16. 60x25=

7.5x24= 17. 55x40=

8. 18x50= 18. 45x80=

9. 25x16= 19. 25x 180=

10. 5x32= 20. 450x8-

THINK ITTHROUGH


POWER BUILDER B1. 6x25= 11. 25x18=

2. 35x6= 12.25x28=

3. 55x4= 13. 45x12

4. 6x45= 14. 15x26=

5. 45x8= 15. 35x14=

6. 6x55= 16. 40x35=

7. 35x8= 17.50x24=

8. 8x25= 18.250x16=

9. 15x22= 19.40x450=

10. 25 x 14 20. 15 x 180

THINK ITTHROUGH

LESSON 37 MAKING COMPATIBL

Copyright © 1988 by Dale Seymour Publ

LESSON 37 MAKING COMPAn. - t

The dimensions of a box are 15 by 10 by 24. If onedimension is doubled, what is the new volume? Whathappens to the volume if one dimension is halved andanother doubled? What if all the dimensions are doubled?

126Copyright C 1988 by Date Seymour Pubficai

is Iotice what happens when one-factor is multiplied by 10 ...The product is also multiplied by 10.

5x315

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 5X3 x3 x3 10 x3

-`emember these steps:

• Remove the trailing zeros.

• Multiply the remaining numbers.

• Tack on ALL the zeros.

60 x 300

OX 3 l e

18

TRY THESE IN YOUR HEAD.Tack on trailing zeros.

4 x 20 4. 5

2. 4 x 50 5. 3

3. 50 x 20 6. 7

000

r

0 x 50 7. 90 x 30

00 x 9 8. 5 x 8000

x 800 9. 30 x 500

10. 200 x 300MENTAL MATH IN JUNIOR HIGHCopyright 0 1988 by Dale Seymour Publications

71


POWER BUILDER A1. 7x30=

2.8x60=

3. 9x20=

4. 5x40=

5.500x9=

6.300x8=

7.5x800=

8. 30 x 200 =

9. 400 x 60

10.70x500=

THINK ITTHROUGH

LESSON 14 TACK ON TRA1UNG ZI

11. 50x600=

12. 300 x 50

13. 90x200=

14. 7 x 8000 =

15. 50x6000=

16. 800 x 700

17. 900 x 500 =

18. 7000 x 60 =

19. 300 x 700 =

20. 50 x 8000 =

List all the different products that can be formed bymultiplying any two numbers on this card.

MENTAL MATH IN JLNJIOf HIGH

POWER BUILDER B1. 6x40=

2. 7x50=

3. 8x30=

4. 5x60=

5. 500 x 7 =

6.300x6=

7.5x400=

8. 20 x 400

9. 600 x 40 =

10. 500 x 90

LESSON 14 TACK ON TAAILtNL

11.50x200=

12.500x50=

13.70x400=

14. 3x600=

15. 50 x 8000

16. 600 x 300

17. 800 x 500 =

18. 8000 x 60 =

19. 200 x 600 =

20. 50 x 4000

List all the different products that can be formedby multiplying any two numbers on this card.

I- DIVIDE IN YOUR HEAD l Numbers with trading zeros

1200: 4

4x3

are easy to divide in your head.

• Remove the trailing zeros.

• Divide the remaining numbers.

• Tack the trailing zeros ontoyour answer.

• Check by multiplying.

TRY THESE IN YOUR HEAD.Cut off and tack on the trailing zeros.

1 . 1200 - 2 4. 7

2. 2400:8 5. 4

3. 1000-.- 5 6. 12

2800 7. 9)27,000

860 8. 3600 ^ 6

2400 9. 3500 --35

10. 15 3000MENTAL MATH IN JUNIOR HIGHCopyright 0 1988 by Date Seymour Publications 77


POWER BUILDER A

LESSON 17 TACK ON TRAILING ZEI

1. 2400 -1- 6 = 11. 1800-2=

2. 320 + 8 = 12. 918,000 =

3. 7 420 = 13.42 ,000-+-6=

4. 7 3500 = 14.6 120,000

5. 4 2 00 15. 2500 + 25 =

6. 7 4900 = 16. 48 , 000 + 6 =

7. 540+6 17.72,000+8=

8. 5600 + 8 = 18. 7 21,000 =

9. 2410 + 3 _ 19. 5 250,000 =

10.5 3500 = 20.3 21,000 =

THINK ITTHROUGH

How many 5-cent stamps can you buy for $25?


POWER BUILDER B

Copyright0 1988 by Date Seymour Public" •s

LESSON 17 TACK ON TRAILING I J

1.3200+4= 11.1600-2=

2. 40008 12.8 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+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?

Copyright © 1988 by Dale Seymour Publications

LESSON 18 CANCEL COM MON TRAILING ZEROS

You can divide both numbersin a division problem by thesame amount without changingthe answer.

Using this idea, it's easy tosimplify a problem when bothnumbers have trailing zeros.

Iov350: 2 4

SHORTCUT:

Cancel the commontrailing zeros.

Check by multiplying.

8000 : 400

80QSo .4 =20

2 ^o x4TRY THESE IN YOUR HEAD.Cancel the common trailing zeros.

8

1. 900030 4. 800_ 2 7. 500050

2. 900 -- 300 5. 1000 ^ 50 8. 3600 -;- 900

3. 9000 = 3000 6. 2000 -- 50 9. 10,000 = 100

10. 1,000,000 ;- 2000

MENTAL MAN IN JUNIOR HIGHCopyright 0 1988 by pale Seymour Pub ications 79


POWER BUILDER A1. 800 + 40 =

2. 12,000 + 600 =

3. 15,000 + 30 =

4. 2400 + 80 =

5. 60 3600

6. 90 72,000

7. 400 32,006 =

8. 50)35O=

9. 800 4800 =

10. 4900 + 70

LESSON 18 CANCEL COMMON TAAILING ZER

11. 600 1200 =12. 50 40,000

13. 72,000 --^- 900 =

14. 800 3200

15. 30,000 + 60 =

16.45 ,000-90=

17. 500 20,000

18. 70 4200

19. 81 ,000 + 900 =

20. 45 ,000 + 50 =

The state gets a tax of 100 for every dollar ofgasoline sold. How many dollars does the stateget for gasoline sales of $400,000?



POWER BUILDER B

LESSON 18 CANCEL COMMON TRAILING 4

1. 600 + 30 - 1 1. 300 1 200

2. 16,000 + 400 12. 50 30,000 =

3. 18,000 + 60 = 13. 56,000 + 700

4. 3200 + 80 = 14. 406 2800

5. 50 2500 = 15. 40,000 + 80 =

6. 80 6400 = 16. 54,000 - 90 =

7. 300 27,000 17. 500 30,000

8. 50 4 00 = 18. 80 7200.

9. 600 4800 19. 63,000 - 900 =

10. 8100 -_.- 90 = 20. 35,000 + 50 --

THINK ITTHROUGH

80

The state gets a tax of 150 for every dollarof gasoline sold. How much money doesthe state get on gasoline sales of $600,000?

Copynght 1988 by Dale Seymour Publications

LESSON 13 BALANCING WITH DECIMALS

Which problem wouldyou rather do in your head?

They look different,but they are reallythe same problem.

That's how balancing canmake a problem easier.

How could you makethese problems easierby balancing?

Balance by adding 0.04.. .

3.42 + 0.04 -► 3.46- 6 + 0.04 -+- 2.

TRY TH ESE IN YOUR HEAD.Use balancing to make them easier.

E

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 HIGHCopyright 0 1989 by Dale Seymour Publications 69


POWER BUILDER ALESSON 13 BALANCING WITH DEC

1. 4.7 - 2.9 = 11. 6.24 - 3.86 =

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. 9.32-2.94- 19. 23 .543 - 13.985 =

10. 8.15 - 5.79 = 20. 45 .007 - 19.998 =

THINK ITTHROUGH

Take the largest 3-digit decimal lessthan one and double it. What do youneed to add to get a sum of 4?

MENTAL MATH IN JUMOR HIGH

Copyright C 1988 by Dale Seymour Pub!

LESSON 13 BALANCING WITH Or-

'ions

POWER BUILDER B1. 5.6-3.9- 11. 5.21 -1.893

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 - 0.99 = 15. 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 lessthan one and triple it. What do you needto subtract to have a difference of 1 ?

When pairs of decimal numbersadd to a whole number, we cansay they are compatible.

It works withmoney amounts,and it works withplain decimals.


$0.52 $6.90 $5.40

$9.60 $0.67 $0.48

$9.33 $2.50 $3.10

TRY THESE IN YOUR HEAD .

$0.85 $0.29 $0.35 0.85 6.20 5.10

$0.71 $0.15 $0.41 2.75 4.55 9.15

$0.34 $0.65 $0.66 3.80 4.90 7.25

(opic'Ai3LE Pn*(Rs

$ I.io+0.901.74^ 0.26

$8'45+ S1'55

3,7 + 1.3


12.8 4.15 0.85

5.37 2.4 94.63

9.15 3.85 2.6

Find compatible pairs 2. Find compatible pairsthat add to $1.00. that add to 10.

MENTAL MATH IN JUNIOR HIGHCopyright 0 1988 by Date Seymour Publications 55


POWER BUILDER ALESSON 8 SEARCHING FOR COMPA71 OECIM

1. $0-52+ $1.00 11. $4.95 + = $5.002. $0.69 + = $1.00 12. $3.69 + --$5.003. +0.36=1 13. + 1.63 = 5

4. +0.88-1 14. 1.7+ = 5

5. 0.41 + 1 15. 8.2 + = 10

6. $2.45 + -$10.00 16. +4.4=10

7. $4.51 + = $10.00 17. + 17.64 = 20

8. 9.38+ =10 18. 0.74+ = 10

9. +3.69=10 19. 9.345+ --10

10. + 5.74 = 10 20. + 4.745 = 5

Megan has only dimes and quarters. She hasthe same number of quarters as dimes. If shehas $3.85, how many quarters does she have?


POWER BUILDER B

LESSON 8 SEARCHING FOR COMPATIBLE DEC].

1. $0.64 + = $1.00 11. $3.72 + = $5.002. $0.73+ = $1.00 12. $3.57 + = $5.003. +$0.44=$1.00 13. +1.59=54. +0.77=1 14. 2.6+ = 55.0.39+ =1 15. 7.3+ = 106. $3.35+ = $ 10.00 16. + 5.5 = 107. $6.52+ = $10.00 17. + 18.38 = 208. 8.28+ == 10 18. 0.74+ = 109. + 4.59 = 10 19. 9.125 + = 10

10. +6.68= 10 20. + 4.085 = 5

Josh has only dimes and quarters . He has the samenumber of quarters as dimes . The total value of thequarters is 750 more than the total value of the dimes.How much money does he have?


number sense - wikispacessense+&+mental+math.pdf · number sense mental mathematics. mental...

Documents