TITLE: Math Shortcuts
AUTHOR: Randy Bartholomew, Barnett Elementary, Payson, UT
GRADE LEVEL/SUBJECT: (4)
OVERVIEW:
The following math shortcuts will help students master some
of the more difficult concepts by presenting a simpler
method or helpful way of understanding the processes.
The methods have been used successfully in the fourth grade
math curriculum. Students have shown a substantial gain in
understanding and retaining the learning.
OBJECTIVE(s): The philosophy behind these shortcuts is to
relate the new learning more closely to previously learned
materials with the idea that elementary students are like
computers and must be reprogrammed each time new learning is
attempted unless a way can be found to tie the new learning
to the old in a quick and easy way.
ADD THE DIGITS
PURPOSE: To check multiplication problems of 2 or more
digits.
Students should have mastered multiplication and addition
facts.
Example 58
x37
-----
406
+1740
-----
2146
Now "ADD THE DIGITS" of each part to check for
correctness.
58 (5+8=13, 1+3=4) 4
x37 (3+7=10, 1+0=1) x1
----- --
406 (4+0+6=10, 1+0=1) 1
+1740 (1+7+4+0=12, 1+2=3) +3
----- --
2146 4
If any part of the check problem is incorrect, then the
student can see where the mistake was made. The concept
of casting out nines is the same, but students may not
understand it as well as they do addition.
SHORT DIVISION
PURPOSE: To teach students the initial concept of division
by one digit without the confusion of the long division
form.
ACTIVITIES AND PROCEDURES:
This method has been used very successfully to introduce the
concept of division in relation to multiplication. Students
who have mastered the multiplication facts should have no
difficulty with one-digit division. The long-division form
is taught after the students understand the short-division
form and can divide any number by one digit.
Rationale: From first grade, students have learned to add
and subtract problems from right to left starting with the
ones place. The long-division form attempts to teach
students to work from left to right, which goes counter to
all previous learning. Also students must master a series of
steps (divide, multiply, subtract, bring-down, remainder)
which uses several difficult math concepts and the concept
of "bring down" which can be very confusing. With short-
division the student uses the multiplication facts to break
the number and find how many are left over.
Example: ____
3) 7 (Have students use this form)
Student Asks: __2_
How many 3's are in 7? (Answer: 2) 3) 7
Student Asks: __2_r1
How many are "left over"? (Answer: 1) 3) 7
To Check: 3x2+1=7
Example: ______
3) 7 2 (Have students space the numbers))
Student Asks: __2___
How many 3's are in 7? (Answer 2) 3) 7 2
__2___
How many are "left over"? (Answer 1) 3) 7 2
(Place the 1 between, near the 2) 1
Explain: The 2 is now 12.
(Relates in form to carrying.)
Student Asks: __2_4_
How many 3's are in 12? (Answer:4) 3) 7 2
How may are left over? (Answer:0)
To Check: 3x24=72
____________
Difficult example: 4) 6 3 7 5
Student Asks: __1__________
How many 4's in 6? (1) 4) 6 3 7 5
__1__________
How many left over (2) 4) 6 3 7 5
2
__1_5________
How many 4's in 23? (5) 4) 6 3 7 5
How many left over? (3) 2 3
__1_5_9______
How many 4's in 37? (9) 4) 6 3 7 5
How many left over? (1) 2 3 1
__1_5_9_3_r3__
How many 4's in 15? (3) 4) 6 3 7 5
How many left over? (2) 2 3 1
The problem is now complete.
To check: 4x1593+3=6375
All major calculations were done by the student as
"headwork".
This procedure works with any number divided by one digit.
The long division form should be introduced after students
have mastered short division.
These shortcuts have helped my class tremendously. My post-
test average for division was 94%.
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John Kurilecjmk@ofcn.org