Date: 1994

Grade Level(s): 4

Subject(s):

• Mathematics/Arithmetic

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.

Objectives:

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:  58x37= 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.

Procedure:
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: How many 3's are in 7? (Answer: 2)

  __2_
3)   7

Student Asks: How many are left over? (Answer: 1)

  __2_r1
3)   7

To Check: (3x2) + 1 = 7

Example:

 _______
3) 7 2                 (Have students space the numbers)

Student Asks: How many 3's are in 7? (Answer 2)

  _2___
3) 7 2    

How many are "left over"? (Answer 1)

 __2___
3)  7  2
       1

(Place the 1 between, near the 2).  Explain: The 2 is now 12.  (Relates in form to carrying.)

Student Asks: How many 3's are in 12? (Answer: 4).  How may are left over? (Answer: 0) 

  _2_4_
3) 7  2

To Check:  3 x 24 = 72

Difficult example:

 _________
4) 6 3 7 5

Student Asks: How many 4's in 6? (1)

  _1______
4) 6 3 7 5

How many left over (2)

  _1______
4) 6 3 7 5
      2

How many 4's in 23 (5) How many left over? (3)

  _1_5____
4) 6  3  7 5
      2  3

How many 4's in 37? (9) How many left over? (1)

  _1_5_9___
4) 6  3  7 5
      2  3 1

How many 4's in 15? (3) How many left over? (3)

  _1_5_9_3_r3
4) 6  3  7  5
      2  3  1

The problem is now complete.
To check: (4 x 1593) + 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%.

Useful Internet Resource:
* Ask Dr. Math: Casting Out Nines to Check Arithmetic
http://mathforum.org/library/drmath/view/55926.html

May 1994

These lesson plans are the result of the work of the teachers who have attended the Columbia Education Center's Summer Workshop. CEC is a consortium of teacher from 14 western states dedicated to improving the quality of education in the rural, western, United States, and particularly the quality of math and science Education. CEC uses Big Sky Telegraph as the hub of their telecommunications network that allows the participating teachers to stay in contact with their trainers and peers that they have met at the Workshops.