Subtraction Table: Best Tool for Beginners

Super Easy Subtraction with the Table

Children from very early ages learn counting. Counting evolves into addition. The logical and easiest step after addition for children is subtraction. Why learn it before multiplication or division? The reasons could be because addition is the sum of two or more counted objects and subtraction is simply the difference of two or more counted objects. Children more naturally grasp this simple relationship between addition and subtraction than multiplication or division. Moreover, multiplication and division have a relationship. Division is the reciprocal of multiplication. The ideal approach is to learn multiplication and division sequentially.

To guide our students do subtraction for the first time, a subtraction table may be helpful. A Subtraction Table is a visual aid that is designed to help children learn how to subtract numbers and ease the learning process.

       Let us review a subtraction example. Using the Subtraction Table, subtract seven minus two and get an answer five: 9 - 7  = 2. The intersection of row nine and column seven is two; five is the answer. For children starting to learn subtraction, this is easy. They can use their fingers to solve the problem. It is also an excellent introduction to the Subtraction Table. In this example, we looked at the positive answer case. If the row number is greater in value than the column number, the answer to the problem is always positive i.e. greater than or equal to one.

        Now let us examine the case where the answer is zero. Using the table, subtract seven minus seven and get an answer zero: 7 - 7 = 0. The intersection of row seven and column seven of the table is zero; zero is the answer. In this example, we looked at the zero answer case. If the row number is equal in value to the column number, the answer to the subtraction problem is always zero.

        The last case is the most difficult for children to grasp, the case where the subtraction answer is a negative number. Using the table, subtract seven minus nine and get an answer minus two: 7 - 9 = -2. The intersection of row seven and column nine of a Subtraction Table is minus two; minus two is the answer. For children starting to learn subtraction, this can be a difficult concept. How can there be a negative of something? The best way to address this by a rule: subtract the row number from the column number and put a minus sign in front of the answer. In this example, we looked at the negative answer subtraction case. If the row number is less in value than the column number, the answer to the problem is always negative i.e. less than or equal to minus one.

How do we get most out of the Subtraction Table?

1. First, get familiar with the table.
2. Start at row number one. Move to column number one. The intersection of row one and column one is the answer: zero.
3. Subtract columns one through three from row seven. The answers are 6, 5, and 4 respectively.
4. Subtract columns one through eight from row two. The answers are 1, 0, -1, -2, -3, -4, -5, and -6 respectively.
5. Let us jump ahead. Now let us increase the level of difficulty. Subtract columns one through twelve from row five. The answers are 4, 3, 2, 1, 0, -1, -2, -3, -4, -5, -6, and -7 respectively.
6. If you are doing well so far, try a test. Solve the following problems in your head and then compare your answers to the table: two minus six, nine minus nine, four minus three, ten minus five, and seven minus twelve. The problem answers are 4, 0, -1, -5, and 5 respectively.

If you got four out of five problems correct, create your own tests. Calculate the answer in your head, and check your answer using the Subtraction Table.

Super Fast and Easy: Two-Digits Multiplication

Multiplying Two-Digit Numbers: The Other Way

While some of our kids or students easily give the answer in the above multiplication, others still  find it difficult to multiply such in a small amount of time. This post may help them to do such. With proper guidance of the teacher or the parent, they can do it.

Let us first recall one topic in Algebra which is the Product of Two Binomials

(a +b)² = a² + b² + 2ab

(a-b) ²  = a² + b² - 2ab

(a + b)  (c + d) = ac + ad + bc + bd.

(a + b)  ( c - d) = ac - ad + bc - bd.

(a-b) (c+ d) = ac + ad -bc -bd.

(a-b) (c-d) = ac - ad - bc + bd.

Let us rewrite a number k as a sum or difference of two numbers in the form

(a +b) or (a-b)

where a is equal to the nearest multiple of 10 less than or just greater than the number K.

For examples:

63 would be rewritten as (60 + 3) and 68 would be rewritten as (70 - 2).

73 would be rewritten as 70 + 3 and 79 would be rewritten as 80 - 1.

Let us first calculate 47 * 47

Let us use the formula

(a-b) * (a-b) = a * a - 2 * a * b + b *b where a = 50 and b = 3.

So 47 * 47 = 2500 - 300 + 9 = 2209.

Let us calculate 73 * 73

Let us use these formula (a+b) * (a + b) = a*a + 2 * a * b + b*b

73 * 73 = (70 + 3) ( 70 + 3 ) = 4900 + 420 + 9 = 5329.

26 * 26 = ( 30 - 4)(30 - 4) = 900 - 240 + 16 = 676.

Use (a +b)(c +d) to calculate 93 * 57 as

(90 + 3) * ( 50 + 7 )

= 4500 + 630 + 150 + 21,                                           Simplify this further by writing this as,

4500 + 600 + 30+100 + 50 + 20 + 1, 4500 + 700 + 101        which can be easily calculated as

5301

Let us similarly calculate 47 * 69 using the expression for (a-b) (c-d).

Let us rewrite the product as

=(50-3) (70-1).                        Expanding the individual terms as

=3500 - 50 - 210 + 3,              rewriting this as

=3500 - 200 - 60 + 3;              rewriting this as

=3500 - 200 - 60 + 3;              So

=3243.

Using the expression (a-b)(c+d) let us calculate 43 * 88,

Rewrite the original expression as

(40 +3)*(90 - 2). Compute individual sums as

= 3600 - 80 + 270 - 6

= 3870 - 86

= 3870 - 70 - 16

= 3800 - 16

= 3784.

Let us use the expression (a-b) (c-d) = ac - ad -bc + bd to calculate 57 * 79 quickly.

Rewriting this as

(60 - 3) * (80 -1)

= 4800 - 60 - 240 + 3

= 4800 - 300 + 3

= 4503.

This can be extended to 3 digit numbers, where the sums of products can be easily formulated using distributive laws of addition and multiplication respectively.

Let us try to calculate 153 * 94 quickly, by rewriting 153 as 100 + 50 +3 and 94 as 90 + 4.

Let us write the original product as

(100+ 50 + 3) (90 + 4);                                   the sum of individual terms is

9000 + 400 + 4500 + 200 + 270 + 12.            Sort is by thousands, hundredths and Units place.

13000 + 1000 + 300 + 82;                              quickly calculate the sum as

14382.