Subtraction Games for 2nd Graders

Conceptual Understanding of Subtraction

Following are some conceptual structures to introduce subtraction to second graders

PARTITIONING

This process involves splitting one group or quantity into two or more smaller sub-groups or sub-quantities. This is exactly the inverse of the combining model of addition where children combine two or more quantities to form a larger quantity.

Some examples of this partitioning model are:

A typical word problem to illustrate this kind of subtraction is:

“In a box, there are 5 toys, 3 are Jack’s and the rest are Jill’s, how many are Jill’s?”

REDUCTION

This type of subtraction involves reducing or diminishing the value of a quantity. It is also described as “take away” method. This should be clear that take away and subtraction is not interchangeable terms. There are other types of subtraction that do not follow the takeaway approach. The reduction model for subtraction is the inverse of the augmentation model for addition which is adding to a quantity.

Some examples of the reduction model are:

The number line method is one where you start at 5 and take away 2. Hence you left with 3

A typical word problem to illustrate this kind of subtraction is:

“Jack had 5 candies. He ate 2, how many are left now?”

COMPARATIVE DISTANCE

This type of subtraction involves comparing two quantities and expressing the difference of the two as a numerical value. It is important to note here that ‘find the difference between 5 and 2’ is equivalent to ‘find the difference between 2 and 5’. This is not true for the other types of subtraction like reduction. (i.e. 5 take away 2 is not equivalent to 2 take away 5).

Some comparative difference examples are:

There are 3 more red cubes than green

The difference is 3 cubes

A word problem to illustrates this kind of subtraction will be ‘How many more candies than Jill does Jack have?

COMPLEMENTARY ADDITION

This type of subtraction that is ‘Complementary Addition’ is also called ‘Additive Difference’ i.e. finding the difference or complement by adding instead of subtracting.

Examples of this type are as follows:

A typical word problem to represent this type of subtraction is:

“Jack has 2 candies, how many more does he need to have 5 candies?”

Methods for Subtraction

COUNTING UP FROM THE SMALLER TO THE LARGER NUMBER

This method combines the use of complementary addition with the use of number line.

We have seen how to subtract simple one-digit numbers using the number line format.

This method is for second graders to understand binary subtraction. Let us look at an example of subtracting two-digit numbers.

For example, Subtract 50-36

As discussed in complementary addition we try to understand subtraction as an addition problem. Hence 50-36 =?  is same as solving the equation 36 +? = 50. So, we start from 36, add 4 to get 40 and then add 10 to get to 50. Total we added 4 + 10 = 14. Hence, 50-36 = 14

              Similarly, for 84-56, we have:

COMPENSATION (TAKE TOO MUCH, ADD BACK)

The heading is self-explanatory. This method can be used to solve two-digit addition and subtraction sums. For subtraction problems first subtract too much and then add back whatever extra was subtracted from the minuend.

For example, 84-56

In this case look at the next closest tens to 56 (i.e. 60) now, subtract from 84, which is, 84—60 = 24. Now add back whatever extra was subtracted. 60 was 4 more than 56. Therefore add 4 more to 24. 24+4 =28

EXPANDED FORM OF STANDARD ALGORITHM

This stage is the beginning of the standard algorithm used for subtraction but in a much-expanded format. The purpose of this expansion is to allow the child to understand the algorithm, rather than merely rote learning the method.

Initially, the expanded format is introduced for subtractions in which the digits of the first number are all larger than the corresponding digits of the second number thus allowing the child to become familiar with the new algorithm without the complications of decomposition Notice that this algorithm relies heavily on the child’s ability to partition numbers into their constituent parts e.g. 88 = 80 + 8, 63 = 60 +3.

Let us look at an example for subtraction of two-digit numbers: 88 – 57

This same algorithm can be modelled using Dienes’ base 10 apparatus. Dienes’ base 10 is one of the best math aids that helps children to visualize subtraction. The base 10 blocks make understanding of subtraction with regrouping easy. It involves trading some of the bigger blocks with blocks of smaller value.

First let us look at examples where the digits of the first number are all larger than the corresponding digits of the second number. In all these cases, subtraction without regrouping model works.

We first partition the first number into tens and ones: 47 = 40 + 7. Place the corresponding Dienes' blocks in the relevant columns. Notice that for addition with Dienes' apparatus both numbers are represented but for subtraction only those blocks representing the first number are needed. Then, partition the second number into tens and ones: 32 = 30 + 2.

Remove Dienes' blocks equivalent to the second number from the first, one column at a time.

Combine the values of the blocks which remain after the subtraction has been performed to produce the required answer: 10 + 5 = 15.

If the algorithm is to be modelled using Dienes’ base 10 apparatus, it is recommended that the algorithm is practiced initially using just the Dienes’ apparatus, then using the apparatus alongside written recording and, finally, using written recording without the apparatus. The aim of using Dienes’ apparatus should be to ensure understanding of the algorithm and not as a prop for calculation.

EXPANDED FORM OF STANDARD ALGORITHM (WITH DECOMPOSITION)

This is the same method as above but with the complication that some of the digits in the first number are smaller than the corresponding digits in the second number. This makes the use of decomposition necessary. Decomposition is a different way of partitioning numbers.

Let us consider this example of regrouping subtraction of 2-digit numbers using decomposition of tens into ones (using the exchange 1 ten = 10 ones).

In the example, number 81 has been decomposed as 81 = 70 + 11. Note, however, that it has been partitioned in the standard way (81 = 80 + 1) before decomposing the 80 + 1 into 70 + 11. It is important that the child understands that the overall value of the first number in the subtraction has not changed; it is still 81.

This algorithm can be modelled using Dienes’ base 10 apparatus as follows:

Let us look at an example of subtraction 34-9

Another example of subtraction with regrouping is: Subtract 29 from 56.

First, partition the first number into tens and ones: 56 = 50 + 6. Place the corresponding Dienes' blocks in the relevant columns. Then, partition the second number into tens and ones: 29 = 20 + 9.

A decision needs to be made at this point as to whether the subtraction of individual columns is immediately feasible or not. In this example, the ones columns are not. One of the tens is decomposed into 10 ones. This finally makes the subtraction column by column feasible. Column by column, blocks equivalent to the values of the digits in the second number are removed from the first number. The combined value of the blocks remaining from the first number after the subtraction has been performed, produce the answer: 20 + 7 = 27.

Subtraction with regrouping worksheets should be practiced along with this so that the base 10 block model is not used as a calculative tool but just as a conceptual clarity tool.

This same model of 2 digit subtraction with regrouping can be extended to 3 digit subtraction with regrouping in which one of hundreds is decomposed into 10 tens and one of tens is decomposed to 10 ones.

Kids enjoy fun math games and activities rather than solving sums on subtraction worksheets or math educational books. Math worksheets seem as a burden with too many sums whereas the same worksheet converted into learning games becomes fun. Math activities to teach addition and subtraction with regrouping can be best modelled through the block 10 place value system. Using money for teaching the main order to understand the same as solving problems of daily life we can use currency instead. Counting money and using currency helps students to relate more and visualize better. This method of adding and subtracting positive and negative numbers with regrouping is also called subtraction with renaming.

Let one of the subtraction word problems be as follows: 52 take away 28 equals what?

Draw the problem or use objects, for example, count out/up the number 52 in tens and units using money:

Circle the bigger number in the units’ column:

At this point ask children, can you take 8 cents away from 2 cents? No? So, what can you do?

Your child could say this little rhyme “More on top? No need to stop! More on the floor? Go next door. Get one ten. That’s ten units more.”

Notice that one 10c coin is equal to (the same as) ten 1c coins:

Help your child to exchange (swop) one 10c coin for ten 1c coins. Show your child four 10c coins and twelve 1c coins.

Now ask: How many 10c coins do you have now? How many 1c coins?

Record the data proceedings so far also

Try working out the problem. Begin with the units’ column and then proceed with the tens column.

12 cent, take away 8 cent, leaves 4 cent.

Four 10c coins, take away two 10c coins, leaves two 10c coins.

And the answer is: Fifty-two take away twenty-eight equals twenty-four. 52 – 28 = 24

Hence, we see how simple adding and subtracting can be provided effective addition and subtraction games and strategies are used. This similar approach of subtracting numbers via the context of money can be extended to subtracting negative numbers, subtracting integers, subtracting fractions, subtracting mixed numbers, subtracting decimals, subtracting polynomials and even subtracting vectors in the later grades. To know how to add and subtract is the most basic operational skill that is used in daily life to solve math problems. A strong conceptual clarity will thereby help kids to grasp other mathematics topics well in successive grades too.