Download Mental Mathematics

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
+ 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 -4
- 9 4 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 : 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 52
Half 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 416
Now we have 2 x 416 (instead of 16 x 52) = 832!
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)x9=270.
5. Commutative Property
This is multiplication with only two numbers. It is much dike 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) and 3 x 5 (= 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 20x40= 800,20x7 = 140,3x40 = 120, and 3 x 7 = 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.
1.7+3.6
1 + 3 = 4
7 tenths + 6 tenths = 1 and 3 tenths
4 + 1 and 3 tenths = 5.3
) ooa
TRY THESE IN YOUR HEAD.
Add from the left.
1. 22+39
4. 526+48
7. 4.5+2.5
2.46+38
5. 329+36
8. 6.4+1.8
3. 56 + 37
6. 236 + 120
9. 26.5+2.7
10. 43.8 + 10.8
MENTAL MATH IN JUNIOR HIGH
Copyright 0 1988 by Bale Seymour Pubfications
41
LESSON 1
MENTAL MATH IN JUNIOR HIGH
ADDING FROM THE U
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 =
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
December 22 is the first day of winter. March 21
is the fast day of winter. How many days does
winter offcally have?
Copyright ® 1988 by D_ w Seymour Publics
LESSON 1
MENTAL MATH IN .Jr1IOR HIGH
Af)DfNG FROM THE ...r"
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=
2©.65.8+12.2=
June 22 is the first day of summer. September 21
is the last day of summer . How many days does
summer officially have?
Copyright 4 1988 by Dale Seymour Publications
Mental Math Addition Practice (Carrying from ones into th e tens)
243
637
627
+649
+ 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.
TRY THESE IN YOUR HEAD.
Make compatibles and adjust.
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
MENTAL MATH IN JUNIOR HIGH
LESSON 7
MAKE YOUR OWN COMPATIBL
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 =
THINK IT
THROUGH
1'
;
)
20. 725 + 277 =
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 COMPA7
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 Co 1988 by Dale Seymour Publications
54;
AACAMTAE SAAT1J IlU
11 IAlifld UIf±U
In mental math, when a subtraction problem
needs regrouping. . .
DON'T DO THIS ...
DO THISI
35=(a0= -75
2^.6-8=13.6
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 JUNIOR HIGH
Copyright 0 1988 by pale Seymour Publications
49
MENTAL MATH IN JUNIOR HIGH
LESSON 5
SUBTRACTING IN PART
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 =
THINK IT
THROUGH
The difference between two numbers is 25.
If the numbers are tripled , what is the difference
between the numbers?
e
Copyright ( 1988 by Dale Seymour Publications
MENTAL MATH IN JUNIOR HIGH
LESSON 5
SUBTRACTING IN P
POWER 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 O 1988 by Dale Seymour Publications
LESSON 15
SEARCHING FOR COMPATIBLES
E
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 recognize them.
Find compatible pairs.
Find compatible pairs.
4
60
40
71
400
300
550
600
56
75
29
30
510
620
250
100
44
33
12
67
630
900
700
380
96
70
88
25
450
750
490
370
TRY THESE. USE YOUR HEAD.
Think about compatible numbers .J
1. On scrap paper, list
number pairs that
total 1 00. Write as
many as you can in
2. How many different
pairs of numbers
total 1000?
one minute. GO!
ENTAL MATH IN THE MIDDLE GRADES
Copyright C 1987 by Date Seymour Pubtications
61
MENTAL MATH IN THE MIDDLE GRADES
LESSON 15
SEARCHING FOR COMPATIBLES
POWER BUILDER A
1. 35 +
--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+
= 10()0
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=
o0
How many different pairs of whole numbers
add to 100?
THINK IT
THROUGH
Copyright © 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
How many different pairs of even whole
numbers add to 100?
Copyright V 1987 By Dale Seymour Publications
Which problem in each pair is easier ?
Why?
`Making tens" can help you
subtract in your head.
Remember: Adding the
same amount to both
numbers leaves the
difference unchanged!
2. 54
-39
MENTAL MATH IN THE MIDDLE GRADES
COsyright © 1987 By Dale Seymour Publications
4. 31
-54
7. 90 --- 35
10. 93 -- 39
MENTAL MATH IN THE MIDDLE GRADES
LESSON 14
BALANCING IN SUBTRACTION
POWER 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=
7 . 81 -39=
17. 46- 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 BU I LDER 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
THINK IT
THROUGH
60
20. 83 -- 49 =
Subtract the smallest three-digit odd number
from the smallest four-digit odd number.
When you add the same amount to each number
jai a subtraction problem, the answer does not change.
Adding }o both numbers balances the problem.
1Q _--7
14 + 3
17
-iO
7
"'-,SAM 6
p rF F E T^ENc€
F\"" SAMC
t i______ :
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.
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 -- 3999
MENTAL MATH IN JUNIOR HIGH
Copyright © 1988 by pale Seymour Publications
67
LESSON 12
MENTAL MATH IN JUNIOR HIGH
POWER BUILDER A
85 - 4 9=
11. 469 -198 _
2.73-59=
12. 753 - 187 =
3. 84-37=
13. 641 - 285 =
4. 62-285. 126-89 -
14. 704 - 475 =
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 =
1.
20. 9544 - 7985 =
10. 632 -88
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 - 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 number
from the smallest 5-digit even number.
SUBTRACTING BY BALANCING
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:/f'ww,h--,math.nvu.edu/-braarns/links/em-arith.htmi
Check out the site below to see a neat way the Egyptians actually did multiply by using
doubling: http://members.cox.net'elessons2.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 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:
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 also
simplifies.
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
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:
(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:
FRONT-END MULTIPLICATION
LESSON 15
Multiplying in your head is easier if
you break a number into parts and
multiply the front-end numbers first.
524
x 3
I
• Break up 524.
500+20+4
• Multiply from the front
to the back ...
• Add as you go along.
Focus on the left (front-end)
digits by covering the others .
.5
X
3
TRY THESE IN VOL
Multiply from the front.
4x55
4.8x25
7.2x545
2. 4 x 76
5. 4 x 625
8. 8 x 625
3.45x6
6.405x3
9.450x5
10. 3 x 235
MENTAL MATH IN JUNIOR HIGH
Copyright 0 1988 by Dale Seymour Publications
73
LESSON 15
MENTAL MA'T'H IN JUNIOR HIGH
FRONT-END MULTIPLICATION
POWER BUILDER A
1. 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 calculate
15 x 37 and 21 x 37.
3x37=111
6 x 37 = 222
9 x 37 = 333
12 x 37 = 444
Copyright © 1988 by Dale Seymour Publications
LESSON 15
MENTAL MATH IN JUNIOR HIGH
FRONT-END MUL71PLIC
POWER BUILDER B
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 calculate
28 x 15,873 and 42 x 15,873.
7x15,873 =111,111
14 x 15,873 = 222,222
21 x 15,873 = 333,333
Copyright C© 1988 by Dale Seymour Publications
LESSON 28
DOUBLING
How much
for two of
those?
Doubling numbers is something we do
every day. Here's an easy way to do it
in your head:
Double a number by doubl i ng
each of its parts . Then add .
PouLE 126
Double 34
2. Double 81
3. Double 912
7. Double 64
4. Double 47
5. Double 29
8. Double 75
9. Double 54
6. Double 430 10. Double 720
ENTAL MATH IN THE MIDDLE GRADES
Copyright © 1987 by Dale Seymour Publications
95
LESSON 28
MENTAL MATH IN THE MIDDLE GRADES
DOUBLING
POWER BUILDER 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. Double 86
7. Double 57
17. Double 64
8. Double 98 =
18. Double 128
9. Double 250 _..
19. Double 256
20. 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
Copyright 0 1987 By Dale Seymour Publication
MENTAL MATH IN THE MIDDLE GRADES
LESSON 28
DOUBLING
POWER BUI LDER B
1. Double 43
11. Double 34
2. Double 74
12. Double 83 =
3. Double 113
13. Double 424 =
4. Double 16 =
14. Double 409 =
5. Double 85 =
15. Double 75
6. Double 700
16. Double 27
7. Double 87 =
17. Double 54 =
8. Double 97 =
18. Double 108
9. Double 65 =
19. Double 216
10. Double 840
20. Double 432
0
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 © 1987 By Dale Seymour Publications
LESSON 29
HALVING AND DOUBLING
Here's a trick to make
mental multiplication
easier.
If one number is even,
you can cut it in half and
double the other number
4x 5
HALF OF 4 . . . DOUBLE 15
0
2 X 30
This sometimes gives you
an easier problem.
THAT'S EASY!
60
1 8x8
You can even keep on
halving and doubling,
if it helps ...
18 X 8
0 <j 0
36X4
4
4
72X2
144
TRY THESE IN YOUR HEAD.
Halve one , double the other.
1.4X17
3.5X68
7.8X13
2. 6 X 45
4. 35 X 4
8. 12 X 150
5.25X16
9.8X45
6. 125 X 12
10. 55 X 6
MENTAL MATH IN THE MIDDLE GRADES
Copyright ©1987 by Dale Seymour Publications
7
LESSON 28
MENTAL MATH IN THE MIDDLE GRADES
HALVING AND DOUBLING
POWER BUILDER A
1. 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 IT
THROUGH
Use mental math to decide which of the following
equals 64 x 32:
64x16
128x16
32 x 128
128x64
Copyright 0 1987 By Dale S yr o ' Publications
MENTAL MATH IN THE MIDDLE GRADES
LESSON 29
HALVING AND DOUBLING
POWER BUILDER B
1. 4x 14=
11. 18 x 15 =
2. 6 x 25 =
12. 16 x 25 =
3. 8x45=
13. 15 x 64
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. 75 x 6 =
THINK IT
THROUGH
98
20. 15 x 120 =
Use mental math to decide which of the following
equals 48 x 144:
24x96
24x72
96 x 288
96 x 72
Copyright,
1987 By Da'i 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.
25x5x9x2x4
To make multiplication easier,
search for compatible factors.
Then rearrange the factors to
simplify your figuring.
TRY THESE IN YOUR HEAD.
Search for compatible factors .
MENTAL MATH IN JUNIOR HIGH
Copyright 0 1988 by Dale Seymour Publications
1 23
LESSON 36
MENTAL MATH IN JUNIOR HIGH
COMPATIBLE Ei
POWER BUILDER A
1. 5x7x2=
11.15x3x4x2=
2.2x13x5=
12. 4x4x 15x5=
3. 2x6x15=
13. 5x5x6x2x2=
4. 15x4x5=
14. 5x7x5x4=
5.20x7x5=
15. 9x3x4x5=
6. 2x7x5x6=
16. 13x2x3x5=
7. 15x7x2x3=
17.5x7x7x2=
8. 6x4x5x25=
18. 5x5x8x2x4=
9. 11 x4x2x25=
19. 11 x2x6x25=
1©.25x5x4x8
20.9x8x50x2=
The dimensions of a large tank are
25 m by 25 m by 8 m. What is the
volume of water it can hold?
Copyright © 1988 by Dale Seymour Puhticati
MENTAL MATH IN JUNIOR HIGH
LESSON 36
COMPA-n L-rA
POWER BUILDER B
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=
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.
Copyright C© 1988 by Dale Seymour Publiatio
LESSON 36
MAKE-YOUR-OWN COMPATIBLE FACTORS
2qx25
Here's a trick that can simplify
mental multiplication .. .
a
Rearrange one or both
of the numbers.
24 X 25
Your aim is to find
compatible pairs.
A
6X4X25
COY'
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. 15X24
I
MENTAL MATH IN THE MEDDLE GRADES
Copyright 9 1987 by Dale Seymour Publications
3. 15X16
7. 12X15
4. 35X50
8. 18X500
5. 48X 15
9. 12X35
6. 24 X 500
10. 15 X 26
111
MENTAL MATH IN THE MIDDLE GRADES
LESSON 36
MAKE-YOUR-OWN COMPATIBLE FACTORS
POWER BUILDER A
1.4x35=
11. 22 x 15 =
2. 4x45=
12. 25 x 18
3. 15 x 14 =
13.45x16=
4.24x15=
14. 15x36=
5. 15x 18=
15. 35 x 12
6.12x25=
16. 60 x 25 =
7. 5x24=
17.55x40=
8.8x25=
18.45x80=
9. 5x32=
19. 25 x 180
10. 25x 16=
20. 450 x 8 =
00
Knowing that 25 x 25 = 625, mentally calculate
24 x 25, 25 x 26, 25 x 27, and 25 x 23.
THINK IT
THROUGH
Copyright 0 1987 By Dale Seymour Publications
LESSON 36
MENTAL MATH IN THE MIDDLE GRADES
MAKE-YOUR-OWN COMPATIBLE FACTORS
POWER BUI LDER B
1.6x25=
11. 25 x 18 =
2. 35x6=
12. 25 x 28 =
3.55x4=
13.45x12=
4.6x45=
14. 15x26=
5. 45 x 8
15. 35x 14=
6.6x55=
16. 40 x 35 =
7. 45x8=
17. 50 x 24 =
8. 8x25=
18. 250x 16=
9. 15x22=
19. 40 x 450 =
10. 25 x 14 =
20. 15 x 180 =
00
THINK IT
THROUGH
112
Knowing that 50 x 50 = 2500, mentally calculate
49 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 of
the numbers.
The trick is to look
for pairs of factors
that are compatible.
Then complete the
multiplication in steps.
28 x 25
7x4x25
7 x 100
700
Can you find another
pair of compatible factors
to check your calculation?
TRY THESE IN YOUR HEAD.
b a k e you r o wn co m patible facto rs.
r
4. 12x25
7. 15x36
5.18x15
8.36x25
5. 28 x 50
9. 32 x 500
10. 12 x 150
MENTAL MATH IN JUNIO R HIGH
Copyright 0 1988 by Date Seymour Publications
125
MENTAL MATH IN JUNIOR HIGH
LESSON 37
MAKING COMPATIBL
POWER BUILDER A
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=
20. 450x8-
10. 5x32=
THINK IT
THROUGH
Rearrange the factors in these problems and
calculate the products mentally:
25x1.2
1.5x8
2.5x48
Copyright © 1988 by Dale Seymour Publ
MENTAL MATH IN JUNIOR HIGH
LESSON 37
MAKING COMPAn.
-t
POWER BUILDER B
1. 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 IT
THROUGH
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?
126
Copyright C 1988 by Date Seymour Pubficai
is Iotice what happens when one
-factor is multiplied by 10 ...
The product is also multiplied by 10.
5
x3
15
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
x3 10
x3
r
-`emember these steps:
60 x 300
• Remove the trailing zeros.
• Multiply the remaining numbers.
• Tack on ALL the zeros.
OX 3 l e
18 000
TRY THESE IN YOUR HEAD.
Tack on trailing zeros.
4 x 20
4. 5 0 x 50
7. 90 x 30
2. 4 x 50
5. 3 00 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 TRA1UNG ZI
POWER BUILDER A
1. 7x30=
11. 50x600=
2.8x60=
12. 300 x 50
3. 9x20=
13. 90x200=
4. 5x40=
14. 7 x 8000 =
5.500x9=
15. 50x6000=
6.300x8=
16. 800 x 700
7.5x800=
17. 900 x 500 =
8. 30 x 200 =
18. 7000 x 60 =
9. 400 x 60
19. 300 x 700 =
10.70x500=
20. 50 x 8000 =
THINK IT
THROUGH
List all the different products that can be formed by
multiplying any two numbers on this card.
LESSON 14
MENTAL MATH IN JLNJIOf HIGH
POWER BUILDER B
1. 6x40=
11.50x200=
2. 7x50=
12.500x50=
3. 8x30=
13.70x400=
4. 5x60=
14. 3x600=
5. 500 x 7 =
15. 50 x 8000
6.300x6=
16. 600 x 300
7.5x400=
17. 800 x 500 =
8. 20 x 400
18. 8000 x 60 =
9. 600 x 40 =
19. 200 x 600 =
10. 500 x 90
20. 50 x 4000
List all the different products that can be formed
by multiplying any two numbers on this card.
TACK ON TAAILtNL
I- DIVIDE IN YOUR HEAD
l
1200: 4
with trading zeros
are easy to divide in your head.
Numbers
• Remove the trailing zeros.
• Divide the remaining numbers.
• Tack the trailing zeros onto
your answer.
4x3
• Check by multiplying.
TRY THESE IN YOUR HEAD.
Cut off and tack on the trailing zeros.
1 . 1200 - 2
4. 7 2800
7. 9)27,000
2. 2400:8
5. 4 860
8. 3600 ^ 6
3. 1000-.- 5
6. 12 2400
9. 3500 - 35
10. 15 3000
MENTAL MATH IN JUNIOR HIGH
Copyright 0 1988 by Date Seymour Publications
77
LESSON 17
MENTAL MATH IN JUNIOR HIGH
TACK ON TRAILING ZEI
POWER BUILDER A
1. 2400 -1- 6 =
11. 1800-2=
2. 320 + 8 =
12. 918,000 =
3. 7
420 =
13.42 , 000-+-6=
4. 7
350 0 =
14.6 120,000
5. 4 2 00
15. 2500 + 25 =
6. 7
16. 48 , 000 + 6 =
4900 =
7. 540+6
17.72,000+8=
8. 5600 + 8 =
18. 7 21,000 =
9. 2410
19. 5 250,000 =
+
3_
10.5 3500 =
20.3 21,000 =
THINK IT
THROUGH
How many 5-cent stamps can you buy for $25?
Copyright0 1988 by Date Seymour Public" •s
MENTAL MATH IN JUNIOR HIGH
LESSON 17
TACK ON TRAILING I
POWER BUILDER B
1.3200+4=
11.1600-2=
2. 40008
12.8 16,000 =
3. 4 280 =
13.42,000+6=
4. 7
14.4 160,000 =
4200 -
5. 9 270 =
15. 1500 + 15 =
6.
16. 36,000 + 6 =
8
4800 =
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
J
CANCEL COM M ON TRAILING ZEROS
LESSON 18
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.
Iov
350: 2
4
8000 : 400
SHORTCUT:
80Q
Cancel the common
trailing zeros.
So .4 =20
2 ^o x4
8
Check by multiplying.
TRY THESE IN YOUR HEAD.
Cancel the common trailing zeros.
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 HIGH
Copyright 0 1988 by pale Seymour Pub ications
79
MENTAL MATH IN JUNIOR HIGH
LESSON 18
CANCEL COMMON TAAILING ZER
POWER BUILDER A
600
=
1200
1. 800 + 40 =
11.
2. 12,000 + 600 =
12. 50 40,000
3. 15,000 + 30 =
13. 72,000 --^- 900 =
4. 2400 + 80 =
14. 800
5. 60
3600
15. 30,000 + 60 =
6. 90
72,000
16.45 ,000-90=
7. 400
3200
17. 500 20,000
32,006 =
8. 50)35O=
18. 70 4200
9. 800
19. 81 ,000 + 900 =
4800 =
10. 4900 + 70
20. 45 ,000 + 50 =
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?
Copyright ® 1988 by Dale Seymour Publications
LESSON 18
MENTAL MATH IN JUNIOR HIGH
CANCEL COMMON TRAILING 4
POWER BUILDER B
1. 600 + 30 -
1 1. 30 0
2. 16,000 + 400
12. 50
3. 18,000 + 60 =
13. 56,000 + 700
4. 3200 + 80 =
14. 406
5. 50 2500 =
15. 40,000 + 80 =
6. 80 6400 =
16. 54,000 - 90 =
7. 300 27,000
17. 500 30,000
8. 50
18. 80
9. 600
4 00 =
4800
10. 8100 -_.- 90 =
1 200
30,000 =
2800
7200.
19. 63,000 - 900 =
20. 35,000 + 50 --
THINK IT
THROUGH
The state gets a tax of 150 for every dollar
of gasoline sold. How much money does
the state get on gasoline sales of $600,000?
80
Copynght
1988 by Dale Seymour Publications
LESSON 13
BALANCING WITH DECIMALS
E
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
6 + 0.04 -+- 2.
That's how balancing can
make a problem easier.
How could you make
these problems easier
by balancing?
TRY TH ESE 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 1989 by Dale Seymour Publications
69
MENTAL MATH IN JUNIOR HIGH
LESSON 13
BALANCING WITH DEC
POWER BUILDER A
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. 6-3-2-85. 5.14 -0.98 =
14. 22.62-15,89=
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 =
15. 36.03 - 25.95 =
THINK IT
THROUGH
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 C 1988 by Dale Seymour Pub!
LESSON 13
MENTAL MATH IN JUMOR HIGH
POWER BUILDER B
1. 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 less
than one and triple it. What do you need
to subtract to have a difference of 1 ?
BALANCING WITH Or-
'ions
When pairs of decimal numbers
add to a whole number, we can
say they are compatible.
(opic'Ai3 L E Pn*(Rs
It works with
money amounts,
and it works with
plain decimals.
$ I.io+0. 90
1.74^ 0.26
$8'45+ S 1'55
3,7 + 1.3
Find compatible pairs.
$0.52
$9.33
$5.40
$6.90
$9.60
Find compatible pairs.
$0.67
$2.50
12.8
$0.48
4.15
2.4
5.37
$3.10
9.15
0.85
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 0 1988 by Date 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 COMPA71
OECIM
POWER BUILDER A
1. $0-52+
2. $0.69 +
3.
$1.00
= $1.00
+0.36=1
+0.88-1
1
4.
5. 0.41 +
6. $2.45 +
-$10.00
= $10.00
=10
+3.69=10
+ 5.74 = 10
7. $4.51 +
8. 9.38+
9.
10.
11. $4.95 +
12. $3.69 +
13.
14.
15.
16.
17.
18.
19.
20.
= $5.00
--$5.00
+ 1.63 = 5
=5
= 10
+4.4=10
1.7+
8.2 +
+ 17.64 = 20
= 10
0.74+
9.345+
--10
+ 4.745 = 5
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?
LESSON 8
MENTAL MATH IN JUNIOR HIGH
SEARCHING FOR COMPATIBLE DEC].
POWER BUILDER B
1. $0.64 +
2. $0.73+
3.
4.
5.0.39+
6. $3.35+
7. $6.52+
8. 8.28+
9.
10.
= $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 750 more than the total value of the dimes.
How much money does he have?
Copyright ® 1988 by Dale Seymour Publications