Pariksha SaathiLesson 2 of 8 · Mathematics, Classes I-V (Paper I)
Addition and Subtraction: Regrouping, Properties, Estimation and Word Problems
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What the operations mean
Children meet addition first as putting together (3 red and 4 blue balloons) and as adding on (5 birds, then 2 more arrive). Subtraction appears in three different situations: taking away (8 sweets, 5 eaten), comparing (Rani has 8 and Sonu has 5; how many more does Rani have?) and finding a missing part (I have 5; how many more to make 8?). All three are 8 - 5 = 3, but comparing and missing-part problems are harder because nothing is visibly removed. Math-Magic starts with objects, pictures and stories, uses the number line for jumps forward and back, and builds number facts such as all the ways to make 10 (1 + 9, 2 + 8 and so on) before any written method.
Carrying in addition
Write ones under ones and tens under tens, and add from the right. When a column adds up to 10 or more, ten of that place are exchanged for 1 of the next place: this is the carry (हासिल in Hindi-medium books). In 368 + 475, the ones give 8 + 5 = 13, so write 3 and carry 1 ten; the tens give 6 + 7 + 1 = 14, so write 4 and carry 1 hundred; the hundreds give 3 + 4 + 1 = 8, and the sum is 843. When several numbers are added the carry can be more than 1: in 79 + 86 + 98 the ones add up to 23, so 2 tens are carried and the sum is 263. A child who writes 13 under the ones, or forgets the carry, has learnt the steps without the idea of exchange.
Borrowing in subtraction
When the top digit is smaller, take 1 from the next place and exchange it for 10 of this place. In 802 - 357 the ones cannot do 2 - 7 and the tens place is empty, so 8 hundreds become 7 hundreds and 10 tens, and then 10 tens become 9 tens and 10 ones: 12 - 7 = 5, 9 - 5 = 4 and 7 - 3 = 4, giving 445. The best-known error is subtracting the smaller digit from the larger in every column, which turns 802 - 357 into 555. The child is following a rule, not guessing, so the teacher asks her to check by adding (357 + 555 is 912, not 802) and rebuilds the exchange with base-ten blocks or money. Subtracting from 1,000 has a shortcut children can discover: take each digit from 9 and the last one from 10, so 1,000 - 643 = 357.
Properties and the inverse link
Addition is commutative (36 + 47 = 47 + 36) and associative ((18 + 25) + 75 = 18 + (25 + 75) = 118), and adding 0 leaves a number unchanged, so 0 is the additive identity. Subtraction has none of these properties: 9 - 4 is not 4 - 9, and (10 - 4) - 3 = 3 while 10 - (4 - 3) = 9. Because subtraction undoes addition, one fact gives three more: if 845 + 238 = 1,083, then 238 + 845 = 1,083, 1,083 - 238 = 845 and 1,083 - 845 = 238. The same link solves missing numbers and missing digits. In 3?7 + 25? = 623, the ones need 7 + 6 = 13, and the tens then need 6 + 5 + 1 = 12, so both missing digits are 6 (367 + 256 = 623).
Mental methods and estimation
Children who understand place value invent their own methods: 498 + 267 is 500 + 267 - 2 = 765 (compensation), and 1,000 - 643 can be found by counting up: 643 to 650 is 7, to 700 is 50 more and to 1,000 is 300 more, 357 in all. Estimation rounds the numbers before calculating, to judge whether an answer is reasonable: 487 + 312 is about 500 + 300 = 800 (exactly 799), and 6,812 - 2,195 is about 6,800 - 2,200 = 4,600 to the nearest hundred (exactly 4,617). Rounding to the nearest thousand (7,000 - 2,000 = 5,000) is quicker but rougher, so the situation decides how close an estimate must be.
Word problems
Key words mislead. In 'Rani has 45 marbles, which is 18 more than Sonu has; how many does Sonu have?' the word more hides a subtraction: 45 - 18 = 27. Children should picture the situation with a drawing or a bar model and ask what is known and what is missing, instead of hunting for clue words. Two-step problems join the operations: a library has 1,245 books, buys 368 and gives away 179, so it now has 1,245 + 368 - 179 = 1,434. A train with 1,250 seats carrying 987 passengers has 263 empty seats. Asking children to make up their own story for a sum such as 50 - 18 shows whether they understand what subtraction means.
Key facts
- Three meanings of subtraction: take away (8 sweets, 5 eaten), compare (8 and 5, how many more?) and missing part (5, how many more to make 8?); all give 3.
- Carrying: 368 + 475 = 843 (8 + 5 = 13, carry 1; 6 + 7 + 1 = 14, carry 1; 3 + 4 + 1 = 8).
- A carry can be more than 1: 79 + 86 + 98 = 263, because the ones add up to 23.
- Borrowing across a zero: 802 - 357 = 445; the error 'smaller digit from larger in every column' gives 555.
- Subtracting from 1,000: take each digit from 9 and the last from 10, so 1,000 - 643 = 357.
- Addition is commutative and associative with identity 0; subtraction is neither: (10 - 4) - 3 = 3 but 10 - (4 - 3) = 9.
- One fact gives three: 845 + 238 = 1,083, so 1,083 - 238 = 845 and 1,083 - 845 = 238.
- Missing digits: 3?7 + 25? = 623 is 367 + 256 = 623, so both missing digits are 6.
- Compensation: 498 + 267 = 500 + 267 - 2 = 765; estimation: 6,812 - 2,195 is about 6,800 - 2,200 = 4,600 (exactly 4,617).
- 'Rani's 45 marbles are 18 more than Sonu's' means Sonu has 45 - 18 = 27: the word more does not always mean add.
Quick revision
Addition: putting together or adding on; subtraction: take away, compare or missing part; carry when a column reaches 10 (368 + 475 = 843); borrow across zeros (802 - 357 = 445, not 555); addition is commutative and associative with identity 0, subtraction is not; 845 + 238 = 1,083 gives 1,083 - 238 = 845; estimate by rounding (6,800 - 2,200 = 4,600); picture the problem, do not trust key words.