Download SCO A6

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GRADE 7
MENTAL MATH STRATEGIES
SCO A6
Develop and apply divisibility rules for 3,4,6, and 9.
Knowledge of the divisibility rules will provide a valuable tool for
mental arithmetic and general development of operation sense.
The divisibility rules are as follows:
A number is divisible by
 2 if it is even
 3 if the sum of the digits is divisible by 3
 4 if the number formed by the last two digits is divisible by 4
 5 if it ends in a 5 or a 0
 6 if the number is divisible by 3 and even
 9 if the sum of the digits is divisible by 9
 10 if it ends in a 0
N.B. Divisibility tests for 7 and 8 are not as simple as the tests for
the other numbers from 1 through 10. Just go ahead and do the
division.
Example: Try 360 - The number 360 ends in 0, the sum of its digits
is 9, and the number formed by its last two digits is 60. The
divisibility rules tell you that 2,3,4,5,6,9, and 10 are some of the
factors of 360.
Try 175: The number 175 ends in 5, the sum of its digits is 13
and the number formed by its last two digits is 75. The divisibility
rules tell you that 5 is a factor of 175 and 2,3,4,6,9, and 10 are not.
Try this one: There will be 138 people at a party. Can the
host fill tables of 5? No, 138 is not divisible by 5. Can the host fill
tables of 6? Yes, 138 is divisible by 6.
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Divisibility tests were much more useful before calculators
were readily available. Today they are mainly studied as an
opportunity to provide additional number sense, and because
they provide a tool that is useful in mental computation
activities. It is also important to learn how to test for
divisibility on a calculator.
SCO B1
Use estimation strategies to assess and justify the
reasonableness (accuracy) of calculation results for integers
and decimal numbers.
Facility with basic facts and mental computation skills are required
for estimation. Estimation should not be considered an exercise
one does only when called upon to do so, but an integral part of
doing any computation, whether it is with a calculator or with
pencil and paper.
Strategies that may be used for estimation include:
1. Rounding: Change the problem to one that is easier to work
with mentally by substituting ‘nicer’ numbers
(multiples of 10) of similar magnitude;
For example, 213 x 48  210 x 50 = 10 500
or 213 x 48  200 x 50 = 10 000
2. Front-end strategy: Perform operations from left to right:
For example: $2.39 + $4.56 + $2.97 + $2.28 + $5.78 =?
Front 2+4+2+2+5 = $15
End 39 + 56  $1 97  $1 28 + 78  $1
1+1+1 = $3
Total = $15 + $3 = $18.00
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*Also see Lessons 1, 3,14, and 15 from Mental Math in the
Junior High.
3. Special number strategy: turn one of the numbers into 1, 10,
100, etc., for easy multiplication or division
For example: 324.4  0.97 = ?
Since 0.97 is almost 1 then the estimate would be 324  1= 324.
Another strategy is to double both the dividend and divisor if
dividing by 5
For example: 342.5  5, think 342.5  5  680  10 = 68
*Also see Lesson 44 from Mental Math in the Junior High.
4. Clustering strategy: round a quantity of numbers to the same
number and multiply the quantity by the rounded number
For example:
$389.22 + $420.27 + $396.45 + $403.67 + $395.50, think: all of
these numbers are about $400.00 so 5 x $400.00 = $2000.00
5. Compatible Numbers: Look for number combinations that
result in 10, 100, 1000, etc.
For example,467+281+241+325, think: 467 + 241 is close to
700 and 281 + 325 is close to 600, 700 + 600 = 1300.
Note: These pairs of numbers are deemed compatible because
the two ten’s digits together make another hundred in each case.
*Also see lessons 6, 7, and 8 from Mental Math in the Junior
High.
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SCO B2
Use mental math strategies for calculations involving
integers and decimal numbers.
Many of the strategies for estimation can be used in mentally
calculating exact answers.
1. Front-end Strategy: For example, –46 + (-38):
Add the tens and ones separately and combine.
-40 + (-30) = -70 -6 + (- 8) = -14 -70 +(-14) = -84
Add tens of one number to the other complete number, and then
add the ones. -46 + (-30) = -76
-76 + (-8) = -84
Round one number, add the other, and adjust the rounding.
[-46 + (-4) = -50] -50 +(-38) = -88
-88 – (-4) = -84
Round one number, add the other, and adjust the rounding.
[-38 + (-2) = -40] -46 + (-40) = -86
-86 – (-2) = -84
Also see Lessons 2, 4, and 5 from Mental Math in the Junior
High.
2. Compatible Numbers: Analyze the numbers to see if
compatible sums resulting in 10, 100, 1000, etc. are present.
For example,
-28 + 63 + 37 + 33 +(-72) = [-28 + (-72)] + [63 + 37] + 33
= -100 + 100 +33
= 33
*Also see Lessons 6, 7, and 8
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3. Compatible Factors: Analyze the numbers to see if
compatible products resulting in 10, 100, 1000, etc. are
present.
For example,
-8 x 137 x 125 = (-8 x 125) x 137 = -1000 x 137 = -137 000.
Another variation of the same strategy is to break one or more
of the numbers into pairs of factors that may be compatible.
For example: 75 x 28
Think: 75 = 25 x 3 and 28 = 4 x 7, regroup the factors using
the associative property: 25 x 4 = 100 and 3 x 7 = 21, therefore,
21 x 100 = 2100.
*Also see Lessons 36 and 37 from Mental Math in the Junior
High.
4. Working by Parts: Break a number into two parts and find
the missing factor; one (or both) of the parts would be a
multiple of 10, 100, 1000, etc.
For example, 5472  9,
Think: 5472 is 5400 + 72, 5400 = 600 x 9 and 72 = 9 x 8;
so, 600 +8 = 608.
*Also see Lesson 42 and 43 from mental Math in the Junior
High.
5. Double and Halve: Double one factor and halve the other.
(Half as many groups, which are twice as large, result in the
same product.)
For example: 486 x 500 is the same as 243 x 1000 = 243 000.
*Also see Lessons 33 and 34 from Mental Math in the Junior
High.
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For divisors or multiples of more than two digits, the use of
technology is expected.
SCO B3:
Demonstrate an understanding of the properties of operations with
decimal numbers and integers.
1. Distributive Property: this can be useful in mental
computations,
For example: in multiplying 6 x (-84), think: -84 = -80 + (-4)
so 6 (-80 + (-4)) = 6 x (-80) + 6 x (-4)
= -480 + (-24)
= -504
Or (-7) x 68 + (-7) x 32 might be thought of as follows:
68 + 32 = 100 and -7 x 100 = -700.
Also see Lesson 16 from Mental math in the Junior High.
2. Associative Property: Changing the grouping of factors
does not change the product.
For example, 25 x 82 x 4 = 4 x 25 x 82 = 100 x 82 = 8200
3. Commutative Property: Changing the order for addition
and multiplication questions does not change the answer.
Subtraction and division are not Commutative.
*Also see Lesson 37 from Mental Math in the Junior High.
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SCO B7
Multiply mentally a fraction by a whole number and vice
versa.
The ability to multiply a fraction by a whole number is necessary
for working mentally with calculation of percents. It is therefore
necessary to develop multiplication of fractions to the point where
students can apply mental strategies when working with simple
cases of multiplying a whole number by a fraction.
Full development of fractions will occur in grade 8.
It is important to use concrete models and pictorial
representations to aid in the visualization process. The following
materials will help the students picture the fractions they are
multiplying: fraction factory pieces, circle models, fraction tiles,
fraction bars, pattern blocks, tangrams, geoboards.
1. Use manips:
Start with an example like:
1
4
x 12.
This is viewed as dividing 12 objects into 4 sets; how many are
in each set?
1
Then use 8 x ,
4
This can be viewed as 8 sets of
1
4
and students should be able to
form the 8 quarters into 2 wholes.
1
Now use 3 x , This can be viewed as 3 sets of
12
1
’s
12
which is
3
’s.
12
Students should be able to make a link between finding
dividing by 4.
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and
4
8
The formal algorithm is not the goal in grade
7, but will be developed in grade 8.
2. Strategy
: When
multiplying a whole number by a fraction, it’s okay to move
the denominator of the fraction underneath the whole
number, and it’s desirable if it will simplify the calculation!
For example:
3
5
45
x 45 is the same as 3 x 5 which is 3 x 9 = 27
This strategy works because of the Associative Principal.
be broken into
1
5
x 3, so
3
5
x 45 is the same as 3 x
can multiply in whatever order is easiest. (3 x
2
3
x 36 is the same as 2 x
52 x
3
4
=
52
4
x3
=
36
3
1
5
3
’s
5
can
x 45. They
45
)
5
which is 2 x 12 = 24
13 x 3 = 39
*Also see Lesson 29 from Mental Math in the Junior High.
SCO B8 and B9
Estimate and determine percent when given the part and the
whole.
Estimate and determine the percent of a number.
Students should be able to compute 10%, 50%, and 1%
mentally. This skill can be used to help with other mental
computations.
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1. Strategies
 Recognize that 50% is half of a number so 21 out of 40
represents slightly more than 50%.
1
 Recognize 25% as and then divide by 4 as a means of
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finding 25% of a number.
3
 Recognize 75% as .
4
 Compute 1% and then multiply.
For example, to find 6% of 400, find 1% of 400 and then
multiply by 6. (4 x 6 = 24)
*Also see Lessons 45, 46, 47, 48, 49, and 50 from Mental math
in the Junior High.
Revised April 2004