Subtraction Games for 3rd Graders

Addition and Subtraction as Inverse Operations

Addition and subtraction are called inverse operations as one is the mathematical ‘opposite’ of the other. Hence, they should be taught parallelly, alongside each other rather than as two separate entities. It is essential that children are taught to appreciate and make use of this mathematical relationship when developing and using mental calculation strategies. By comparing the different conceptual structures for addition and subtraction learnt so far, we can clearly see that each addition model has a corresponding subtraction model as its inverse:

To solve add and subtract math word problems care must also be taken in knowing well not just the addition and subtraction definition but also the terminology. Here is a list of terms curated to help solve math problems related to addition and subtraction.

For addition: plus, add, sum, total, together, addition, increase by, more, additional

For subtraction: minus, subtract from, subtraction, Less, difference, decrease by, take away, deduct

Requirements for Mental Addition and Subtraction Strategies

Place Value of Digits in any Number

Sound understanding of the aspects of place value system is utmost important for children to achieve success in mental addition and subtraction tasks. It is therefore important that children can partition numbers or decompose numbers into their constituents. This is possible when children can recognize the value of the digits within the numbers, thus 53 = 50 + 3; 527 = 500 + 20 + 7 and, at later stages, they should be able to understand the same for decimals. For example, 9·23 = 9 + 0·2 + 0·03.

Comparison of Numbers

Children will also need familiarity with the relative sizes of numbers. They should recall this from all the ordering activities in place value system

Relationship between Addition and Subtraction

The inverse relationship between addition and subtraction should also be clear to kids as knowing strategies for addition will help them unlock the subtraction strategies automatically.

Some Mental Strategies for Addition and Subtraction

1. Rearranging Numbers

Rearrange the numbers such that the larger number is put first when adding using the commutative law. For example, 6 + 14, is more easily tackled as 14 + 6.

2. Using Repeated Operations

Repeated subtraction is easier to do when subtracting a large number.

For example, subtracting 300 by subtracting 100 three times.

3. Compensation or Adjustment

At times, it is easier to calculate by adjusting the numbers a little. For example, calculating with 10 is easier than calculating with 9. Calculating 43 + 19 by working out 43 + 20 and then subtracting 1 is a much more efficient way of solving the problem.

4. Using Near Doubles or Halves

When the numbers involved in the math problem are consecutive numbers.

For example, calculating 18 + 17 as ‘double 17 and add 1’ or ‘double 18 and subtract 1’

Or calculating 24 − 13 as (24 − 12) – 1

5. Partitioning

Decomposing or partition a number is sometimes more efficient.

For example, 27 + 34 = (20 + 30) + (7 + 4) = 50 + 11 = 61

or 27 + 34 = (20 + 30) + 7 + 4 = 50 + 7 + 4 = 57 + 4 = 61

6. Bridging to the next Multiple of 10 or 100

For example, calculating 238 + 17 by splitting 17 into 2 + 15.

So, 238 + 17 = (238 + 2) + 15 = 240 + 15 = 255

7. Using Inverse Relationships

We know addition and subtraction are inversely related so subtractive problems can be made additive. For example, calculating 100 − 70 by knowing that 70 + 30 = 100 and that addition and subtraction are inverse operations.

8. Rounding (to approximating answers)

For example, to solve 24 + 67, round of 24 to 20, 67 to 70 and then add 20 + 70 = 90

This approach is mostly used in multiple choice answers where knowing the approximate answer also works.

9. Using Equivalences

Equivalence relation may be used in some cases.

For example, calculating 101 − 35 by noting that this is equivalent to calculating 100 – 34 or calculating 5 + 8 by noting that this is equivalent to 5 + 10 - 2 = 15 - 3 = 13

Progression in Written Methods for Subtraction

By this grade students are already well versed with double digit subtraction with regrouping and without regrouping. In 3rd grade children should be equipped with four digits and three digit subtraction with regrouping. Following methods can be used to define subtraction and extend their knowledge of two digit subtraction with regrouping to subtract integers, decimals, mixed numbers and fractions. Time subtraction, vector subtraction and matrix subtraction will also be done in a similar way in later stages.

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.

Compensation ( Take too much, add back)

To solve subtraction problems using this method first subtract too much and then add back whatever extra was subtracted from the minuend.

For example, 783-356

 

Expanded form of Standard Algorithm

This method is used when 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.

1. by decomposition of tens into ones (using the exchange 1 ten = 10 ones)

 

       

2. decomposition of hundreds into tens (using the exchange 1 hundred = 10 tens)

3. decomposition of hundreds into tens and then tens into ones

         

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

Let us look at an example of subtraction 243-179

Step 1: Partition the first number into hundreds, tens and ones: 243 = 200 + 40 + 3. Place the corresponding Dienes' blocks in the relevant columns.

Step 2: Partition the second number into hundreds, tens and ones:179 = 100 + 70 + 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 tens and ones’ columns are not.

Step 3: One of the hundreds blocks is      decomposed into 10 tens in order to make subtraction of the tens digits feasible.

However, the ones column is still infeasible so a further decomposition is necessary.

Step 4: This time, one of the tens is decomposed into 10 ones. This finally makes the subtraction column by column feasible.

Step 5: Column by column, blocks equivalent to the values of the digits in the second number are removed from the first number.

Step 6: The blocks remaining from the first number after the subtraction has been performed are now placed in the answer box and their values combined to produce the answer: 60 + 4 = 64.

Contraction of Expanded Algorithm to Standard Algorithm

This method is a contracted form of the algorithm detailed above. Following are some examples for three digit subtraction with regrouping.

For example,

1. with decomposition of tens into ones (using the exchange 1 ten = 10 ones)

2. with decomposition of hundreds into tens (using the exchange 1 hundred = 10 tens)

3. with decomposition of hundreds into tens and then tens into ones

Extend to Larger Numbers

Use the same algorithms for long subtraction. For larger numbers, introduce the expanded formats first, then contract these to the standard algorithm. Once comfortable with the standard algorithm, use the same to practice adding and subtracting negative numbers worksheets then proceed to adding and subtracting integers worksheet, then practice adding and subtracting decimals worksheets and finally practice adding and subtracting mixed numbers worksheet.

Extend to Negative Numbers

Children should be taught how to add and subtract integers through learning games and activities first and then through standard algorithms. How to subtract integers or how to subtract negative numbers is easy provided students are comfortable with standard subtraction algorithm. However, the rules for adding and subtracting negative numbers must be laid down for children to be able to correctly represent all the problems of mathematics.

Extend to Decimal Numbers

Children may continue to align the right-most digits of each number vertically even when this is not appropriate for decimal numbers. It is important that children are taught, even for whole number subtractions, to align the digits of the second number underneath those of the first number according to the place value of each of the digits, that is, tens under tens, ones under ones, tenths under tenths etc. Once this concept is clear, emphasize on rules of adding and subtracting decimals. Provide kids with subtracting decimals worksheet to practice the standard algorithm well.