Pariksha SaathiLesson 1 of 8 · Mathematics, Classes I-V (Paper I)
Numbers and Place Value: Lakhs, Expanded Form, Indian and International Systems, Roman Numerals
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From counting to numbers
Before a child can use numbers she needs pre-number ideas: matching one object with one object (one-to-one correspondence), sorting and classifying by a property, and arranging things in order of size (seriation). Counting rests on five principles described by Gelman and Gallistel: one number name for each object (one-one), names always in the same order (stable order), the last name tells how many (cardinality), anything can be counted (abstraction), and the order of counting does not change the total (order irrelevance). A child who counts 7 beads correctly but, asked how many, starts again from 1 has not yet grasped cardinality. Math-Magic builds numbers in stages in Class 1 (1 to 9, 10 to 20, 21 to 50 and beyond), and in Class 2 children bundle sticks into tens, the bridge to place value.
Place value, face value and expanded form
Our numerals form a base-ten place-value system with ten digits, 0 to 9, in which each place is worth 10 times the place to its right: 10 ones make 1 ten, 10 tens make 1 hundred and 10 ten thousands make 1 lakh. The face value of a digit is the digit itself; its place value is the digit multiplied by the value of its place. In 5,08,374 the 8 has face value 8 and place value 8,000, a difference of 7,992. The place value of 0 is always 0. When a digit repeats, each copy has its own place value: in 63,536 the two 3s are worth 3,000 and 30, with sum 3,030 and difference 2,970. The expanded form writes a number as the sum of its place values: 5,08,374 = 5,00,000 + 8,000 + 300 + 70 + 4. Zeros drop out of the expanded form, and putting them back is where children slip: 4,00,000 + 6,000 + 9 = 4,06,009.
Forming and naming numbers
The greatest 5-digit number is 99,999; adding 1 gives 1,00,000, the smallest 6-digit number. There are 900 three-digit numbers (100 to 999). For the greatest number from given digits, write them in descending order; for the smallest, in ascending order, but never begin with 0. With 0, 3, 5 and 7 used once each, the greatest 4-digit number is 7,530 and the smallest 3,057, a difference of 4,473. The predecessor of 1,00,000 is 99,999. Number names follow the periods: 3,40,040 is three lakh forty thousand forty. A child who writes two thousand three as 20003 is writing the number as it sounds; place-value cards laid over each other (2000, then 3 on top) show why it is 2,003.
Indian and international systems
The Indian system has the ones period (ones, tens, hundreds) and then periods of two places each: thousands (thousands, ten thousands), lakhs (lakhs, ten lakhs) and crores (crores, ten crores). The international system uses periods of three places: ones, thousands, millions and billions. So after the first three digits from the right, Indian commas come after every two digits and international commas after every three. 10 lakh = 1 million, 1 crore = 10 million and 100 crore = 1 billion. 7,25,40,816 is read seven crore twenty-five lakh forty thousand eight hundred sixteen; in the international system it is written 72,540,816 and read seventy-two million five hundred forty thousand eight hundred sixteen. A crore has 7 zeros.
Comparing and ordering
The number with more digits is greater (10,000 > 9,999). With the same number of digits, compare from the leftmost place and stop at the first place where they differ: 4,05,999 < 4,50,000 because in the ten-thousands place 0 is less than 5. Ascending order runs from the smallest to the greatest, descending the other way, and on a number line every number is greater than those to its left. The symbols > and < open towards the greater number; children who mix them up can say the sentence aloud (45 is greater than 38) before writing it.
Roman numerals
The Roman system has seven symbols, I = 1, V = 5, X = 10, L = 50, C = 100, D = 500 and M = 1,000, and no zero. A symbol after a greater or equal one is added (VI = 6, LX = 60); a smaller symbol before a greater one is subtracted (IV = 4, XC = 90). Only I, X and C are subtracted, each only from the next two greater symbols: I from V and X, X from L and C, C from D and M. I, X, C and M may be repeated up to three times; V, L and D are never repeated or subtracted. So 99 is XCIX, not IC; 49 is XLIX, not IL; 444 is CDXLIV; and 1947 is MCMXLVII.
In the classroom
Place value is understood, not memorised, when children bundle sticks, exchange ten 1-rupee coins for a 10-rupee note, use an abacus or base-ten blocks and read numbers from a place-value chart. Common errors are skipping zeros when reading (4,05,030 is four lakh five thousand thirty) and confusing place value with face value. Asking children to show 146 in two ways, as 1 hundred 4 tens 6 ones and as 14 tens 6 ones, builds the flexible regrouping that addition and subtraction need.
Key facts
- Face value and place value: in 5,08,374 the 8 has face value 8 and place value 8,000; the difference is 7,992.
- Repeated digit: in 63,536 the two 3s have place values 3,000 and 30; their sum is 3,030 and difference 2,970.
- Expanded form: 4,00,000 + 6,000 + 9 = 4,06,009; the empty places are held by zeros.
- Greatest and smallest 4-digit numbers from 0, 3, 5, 7 (each once): 7,530 and 3,057; difference 4,473.
- 99,999 + 1 = 1,00,000, the smallest 6-digit number; there are 900 three-digit numbers and 9,000 four-digit numbers.
- Systems: 10 lakh = 1 million, 1 crore = 10 million, 100 crore = 1 billion; 7,25,40,816 (Indian) = 72,540,816 (international).
- Comparing: more digits means greater; with equal digits compare from the left, so 4,05,999 < 4,50,000.
- Roman symbols: I 1, V 5, X 10, L 50, C 100, D 500, M 1,000; no zero; V, L and D are never repeated or subtracted.
- Roman numbers: 49 = XLIX, 99 = XCIX (not IC), 444 = CDXLIV, 1947 = MCMXLVII.
- Counting principles (Gelman and Gallistel): one-one, stable order, cardinality, abstraction and order irrelevance.
Quick revision
Pre-number ideas: matching, sorting, seriation; the last number said gives how many (cardinality); place value = digit × value of its place, face value = the digit, place value of 0 is 0; 99,999 + 1 = 1,00,000; Indian periods of 3, 2, 2 digits, international 3, 3, 3; 10 lakh = 1 million, 1 crore = 10 million; compare from the left; Roman I, V, X, L, C, D, M, only I, X, C subtract, 99 = XCIX.