change plus
Algebraic
Notation
Method
collection
situations
change minus
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change minus
A change situation that can be
represented by subtraction. In a
change minus situation, the starting
number, the change, or the result
will be unknown.
Example:
Unknown
Unknown Unknown
Start
Change
Result
n-2=3
5-n=3 5-2=n
Situations that involve putting
together (joining) or taking apart
(separating) groups.
collection situations
Algebraic Notation Method
A strategy in which a factor is
decomposed to create simpler algebraic
expressions, and the distributive
property is applied. The multiplication
shown here uses this method.
Example:
9 • 28
= 9 • (20 + 8)
= (9 • 20) + (9 • 8)
= 180 + 72
= 252
change plus
A change situation that can be
represented by addition. In a
change plus situation, the starting
number, the change, or the result
will be unknown.
Example:
Unknown Unknown Unknown
Start
Change
Result
n+2=5 3+n=5 3+2=n
comparison
situation
combination
situation
Equal-Shares
Drawing
Digit-by-Digit
Method
Math Expressions
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8
8
8
5
7 3,822
-35
32
54
7 3,822
-35
32
- 28
42
one digit at
a time.
546
7 3,822
-35
32
- 28
42
- 42
Digit-by-Digit Method
A method used to solve a division
problem.
Example:
Put in only
8
4×
32
Equal-Shares Drawing
A diagram that shows a number
separated into equal parts.
Number of combinations = 3 × 2 = 6
peanut butter
cheese on
turkey on
on
white bread white bread
white bread
white
bread
turkey
peanut butter
cheese on
turkey on
on
wheat bread wheat bread
wheat bread
cheese
wheat
bread
peanut
butter
Different Sandwich Combinations
A table can sometime be used to show all
possible combinations; multiplcation can be used
to calculate the number of combinations.
combination situation
A situation in which two amounts are compared by
addition or by multiplication. An additive comparison
situation compares by asking or telling how much
more (how much less) one amount is than another.
A multiplicative comparison situation compares
by asking or telling how many times as many one
amount is as another. The multiplicative comparison
may also be made using fraction language. For
example, you can say, “Sally has one fourth as much
as Tom has,” instead of saying “Tom has 4 times as
much as Sally has.”
comparison situation
Factor
Fireworks
Expanded
Notation
Method
Fast Array
Factor
Triangle
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24
3
24
8
÷
4 × 8
÷
32
Factor Triangle
A diagram that shows a factor pair
and the product.
Example:
6
4
Fast Array
A quick way of drawing an array
to solve a multiplication or division
problem. It shows the number of
items in each row or column but
does not show every single item.
×
3 × 2
6
40 × 60
40 × 7
3 × 60
3 ×7
= 2400
= 280
= 180
= + 21
2,881
67 = 60 + 7
× 43 = 40 + 3
43 × 67
Expanded Notation Method
A method used to solve
multiplication and division problems.
2
12
Factor Fireworks
Shows how a whole number can
be broken down into a product of
prime factors.
New Groups
Below
Method
New Groups
Above
Method
Rectangle
Sections
Method
PartialQuotients
Method
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- 160 20
18
- 16 2
2 22
22 R2
30
100
+
15
50
5
100
50
30
+ 15
195
Partial-Quotients Method
A method used to solve division
problems where the partial
quotients are written next to the
division problem instead of above it.
Example: 8 178
3
+
10
10
Rectangle Sections Method
A method using rectangle drawings
to solve multiplication or division
problems.
13 × 15
Example:
++197
443
New Groups Above Method
A strategy for multi-digit addition.
The new groups are placed above
the existing groups. This is the
common method of addition.
11
Example:
246
++197
11
443
New Groups Below Method
A strategy for multi-digit addition.
The new groups are placed below
the existing groups on the line
waiting to be added.
Example:
246
Repeated
Groups
situation
Shortcut
Method of
Multiplication
solution
equation
situation
equation
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situation equation
An equation that shows the action
or the relationship in a problem.
Example: 35 + n = 40
solution equation
An equation that shows the
operation to perform in order to
solve the problem.
Examples: n = 40 - 35
Step 2
7
28
×9
252
Step 1
7
28
×9
2
Shortcut Method of Multiplication
A strategy for multiplying. It is the
current common method in the
United States.
Repeated Groups situation
A multiplication situation in which
all groups have the same number of
objects.