[{"data":1,"prerenderedAt":1267},["ShallowReactive",2],{"layer:number-system:investigate":3},{"layer":4,"contentHash":1247,"dependencyHashes":1248,"approval":1261,"releaseId":1266},{"schemaVersion":5,"conceptId":6,"locale":7,"depth":8,"revision":5,"title":9,"subtitle":10,"summary":11,"objectives":12,"estimatedMinutes":18,"plate":19,"blocks":40,"sourceIds":1242,"reviewStatus":1243,"authoring":1244},1,"number-system","en","investigate","Testing big-number ideas","Predict first, then try it: shifting digits, rollovers, rounding traps and estimation errors","Make predictions about place value and then test them: what moving a digit does, how many numbers of each size exist, when a successor gains a digit, which numbers round to the same value, how far off an estimate can be, and why 6174 keeps appearing.",[13,14,15,16,17],"Predict and test how moving a digit one place changes its value, and explain it with place value.","Investigate patterns in counting numbers: how many n-digit numbers there are and how many digits it takes to write them.","Find every number that rounds to a given value, and show why rounding twice can give a different answer.","Compare estimates with exact answers and decide which rounding gives a good enough estimate.","Test claims about forming numbers and Roman numerals, deciding whether they are always, sometimes or never true.",45,{"title":20,"rows":21},"Lesson plate",[22,25,28,31,34,37],{"label":23,"value":24},"Depth","Investigate",{"label":26,"value":27},"Reading time","≈ 45 minutes",{"label":29,"value":30},"Prior knowledge","Place value, rounding (Understand)",{"label":32,"value":33},"Chapters","11",{"label":35,"value":36},"Labs","6 labs and games",{"label":38,"value":39},"Habit","Predict → test → explain",[41,45,51,57,75,104,123,136,153,167,172,183,218,233,254,259,271,288,298,309,314,325,351,360,365,385,390,401,406,416,429,432,453,458,469,474,487,511,565,578,589,593,604,618,623,636,669,672,688,692,720,738,743,756,783,787,792,803,814,860,863,892,901,921,932,937,940,949,958,963,968,971,983,992,1001,1005,1017,1037,1041,1075,1200,1206,1210,1215,1232],{"id":42,"type":43,"markdown":44},"intro-i","prose","Mathematicians do not just follow rules: they poke at them. *What happens if…? Is it always true? Can I find an example where it fails?* This layer is built around that habit. Each chapter starts with a **prediction**: commit to an answer before you read on. Then test it with a lab, a table or a calculation, and finally explain **why** it happens using place value.\n\nWrong predictions are not failures. They are the most useful kind, because they show you exactly where your picture of numbers needs fixing.",{"id":46,"type":47,"variant":48,"title":49,"markdown":50},"try-notebook","callout","try_it","Keep an investigation notebook","For each chapter write three lines: **I predict…**, **I found…**, **Because…**. By the end you will have your own record of ten small discoveries about numbers.",{"id":52,"type":53,"title":54,"eyebrow":55,"navLabel":56},"ch01","chapter","What happens when a digit moves?","Chapter 01","1 Moving digits",{"id":58,"type":59,"prompt":60,"options":61,"explanation":74},"pred-shift","prediction","Write a 0 on the end of 4,375 to get 43,750. What happened to the value of the digit 7?",[62,65,68,71],{"id":63,"label":64},"a","It stayed 70",{"id":66,"label":67},"b","It became 700",{"id":69,"label":70},"c","It became 7,000",{"id":72,"label":73},"d","It went up by 10","**b: it became 700.** In 4,375 the 7 is in the tens place (70). Putting a 0 on the end pushes every digit one place to the left, and each place is worth ten times the one to its right, so the 7 is now in the hundreds place: 700. Every digit became ten times as big, so the whole number did too: 4,375 × 10 = 43,750.",{"id":76,"type":77,"caption":78,"columns":79,"rows":83},"table-shift","table","Shifting 2,508 one place at a time",[80,81,82],"Operation","Result","Place value of the 5",[84,88,92,96,100],[85,86,87],"Start","2,508","500",[89,90,91],"× 10","25,080","5,000",[93,94,95],"× 100","2,50,800","50,000",[97,98,99],"× 1,000","25,08,000","5,00,000",[101,102,103],"× 10,000","2,50,80,000","50,00,000",{"id":105,"type":106,"component":107,"componentVersion":5,"config":108,"objective":121,"textAlternative":122},"lab-pv-shift","interactive","place-value",{"places":109,"system":110,"initial":111,"show":112,"challenges":115},8,"indian",2508,{"names":113,"expanded":113,"neighbours":114,"bothSystems":113},true,false,[116,117,118,119,120],25080,250800,2508000,25080000,8052,"Build 2,508 and then build it ten, hundred, thousand and ten thousand times bigger; watch every digit shift left.","The lab starts at 2,508 in an eight-column Indian chart (up to crores), and shows the name in both systems. The challenges ask for 25,080, 2,50,800, 25,08,000 and 2,50,80,000. Each time, the same pattern of counters (2, 5, 0, 8) moves one column further left, and a new 0 fills the ones column.\n\nIn words: two thousand five hundred eight; twenty-five thousand eighty; two lakh fifty thousand eight hundred; twenty-five lakh eight thousand; two crore fifty lakh eighty thousand. Internationally, 25,080,000 is twenty-five million eighty thousand.\n\nThe last challenge, 8,052, uses the same digits in reverse order and is a completely different number: digits alone mean nothing without their places.",{"id":124,"type":59,"prompt":125,"options":126,"explanation":135},"pred-swap","Swap the two digits of a 2-digit number, for example 72 → 27, and subtract the smaller from the bigger (72 − 27 = 45). Try 83 → 38, and 91 → 19. What do all the differences have in common?",[127,129,131,133],{"id":63,"label":128},"They are all even",{"id":66,"label":130},"They are all multiples of 9",{"id":69,"label":132},"They all end in 5",{"id":72,"label":134},"Nothing: they are random","**b: always a multiple of 9.** 72 − 27 = 45, 83 − 38 = 45, 91 − 19 = 72: 45 = 9 × 5 and 72 = 9 × 8. Why? A 2-digit number with digits a and b is worth 10a + b. Swapped, it is 10b + a. The difference is 9a − 9b = 9 × (a − b). The digit difference shows up inside it too: for 91 it is 9 − 1 = 8 and 9 × 8 = 72. Place value turns a puzzle into a certainty.",{"id":137,"type":138,"title":139,"items":140},"steps-why-zero","steps","Why “multiply by 10” means “put a 0 on the end”",[141,144,147,150],{"title":142,"text":143},"Every digit moves up one place","Ten ones make a ten, ten tens make a hundred: so ten copies of any place fill exactly the next place up.",{"title":145,"text":146},"The ones place is left empty","After the shift nothing is in the ones place, so we write 0 there as a placeholder.",{"title":148,"text":149},"It works for 100 and 1,000 too","Multiplying by 100 is multiplying by 10 twice: two shifts, two zeros. By 1,000: three shifts, three zeros.",{"title":151,"text":152},"Dividing undoes it","43,750 ÷ 10 = 4,375: every digit moves one place right and the 0 disappears. (If the ones digit is not 0, the answer is not a whole number.)",{"id":154,"type":155,"itemId":156,"prompt":157,"check":158,"hints":162,"feedback":164},"pr-shift","practice","number-system.investigate-shift","In 56,789 the digit 6 is worth 6,000. Multiply the number by 10. By how much does the value of the 6 increase?",{"kind":159,"answer":160,"tolerance":161},"number",54000,0,[163],"What is the 6 worth in 5,67,890?",{"correct":165,"incorrect":166},"Right: in 5,67,890 the 6 is in the ten thousands place, worth 60,000. It increased by 60,000 − 6,000 = 54,000.","After × 10 the 6 moves to the ten thousands place: 60,000. The increase is 60,000 − 6,000 = 54,000, which is 9 times its old value.",{"id":168,"type":53,"title":169,"eyebrow":170,"navLabel":171},"ch02","Is more digits always bigger?","Chapter 02","2 More digits?",{"id":173,"type":59,"prompt":174,"options":175,"explanation":182},"pred-digits","Claim: \"A whole number with more digits is always bigger than one with fewer digits.\" Is it…",[176,178,180],{"id":63,"label":177},"Always true",{"id":66,"label":179},"Sometimes true",{"id":69,"label":181},"Never true","**a: always true, for whole numbers written the normal way.** The smallest number with n + 1 digits is 1 followed by n zeros, and it is exactly 1 more than the largest number with n digits (n nines). So even the smallest longer number beats the largest shorter one. The claim needs the condition *written the normal way*: 000,045 has six characters but is just 45, and decimals like 0.123 are smaller than 5 despite having more digits. Mathematicians love finding the exact conditions under which a claim holds.",{"id":184,"type":77,"caption":185,"columns":186,"rows":190},"table-gap","The biggest n-digit number and its successor",[187,188,189],"Largest n-digit number","Successor","Digits in successor",[191,195,199,203,207,211,215],[192,193,194],"9","10","2",[196,197,198],"99","100","3",[200,201,202],"999","1,000","4",[204,205,206],"99,999","1,00,000","6",[208,209,210],"9,99,999","10,00,000","7",[212,213,214],"99,99,999","1,00,00,000","8",[216,217,193],"99,99,99,999","1,00,00,00,000",{"id":219,"type":106,"component":107,"componentVersion":5,"config":220,"objective":231,"textAlternative":232},"lab-pv-rollover",{"places":221,"system":110,"initial":222,"show":223,"challenges":224},9,99999,{"names":113,"expanded":114,"neighbours":113,"bothSystems":113},[225,226,227,228,229,230],100000,999999,1000000,9999999,10000000,99999999,"Explore rollovers: build the largest numbers of each size, look at their successors, and see a new place appear.","The lab starts at 99,999 with the predecessor (99,998) and successor (1,00,000) shown. Add one more counter to the ones column and every column rolls over: ten ones become a ten, ten tens a hundred, and so on until a single counter lands in the lakhs column.\n\nThe challenges pair each largest number with its successor: 9,99,999 and 10,00,000 (nine lakh ninety-nine thousand nine hundred ninety-nine, then ten lakh, which is one million); 99,99,999 and 1,00,00,000 (one crore, ten million); and finally 9,99,99,999 (nine crore ninety-nine lakh ninety-nine thousand nine hundred ninety-nine, or 99,999,999 internationally). Each time the successor has one more digit, and it is always 1 followed by zeros.",{"id":234,"type":155,"itemId":235,"prompt":236,"check":237,"hints":249,"feedback":251},"pr-digits-true","number-system.investigate-always-true","Which statement is **always** true for whole numbers?",{"kind":238,"options":239,"correct":248},"choice",[240,242,244,246],{"id":63,"label":241},"The successor of a number has the same number of digits",{"id":66,"label":243},"The successor of an n-digit number has n or n + 1 digits",{"id":69,"label":245},"The predecessor of a number always has fewer digits",{"id":72,"label":247},"A number ending in 0 has a predecessor ending in 1",[66],[250],"Try some examples with lots of 9s and 0s.",{"correct":252,"incorrect":253},"Right: the successor gains a digit only when the number is all 9s; otherwise it keeps the same number of digits.","Test each one: a fails for 999 → 1,000. c fails for 58 → 57. d fails for 40 → 39. Only b survives every test.",{"id":255,"type":53,"title":256,"eyebrow":257,"navLabel":258},"ch03","How many numbers? How many digits?","Chapter 03","3 Counting digits",{"id":260,"type":59,"prompt":261,"options":262,"explanation":270},"pred-digits-100","You write every number from 1 to 100 on the board. How many digits do you write altogether?",[263,264,266,268],{"id":63,"label":197},{"id":66,"label":265},"189",{"id":69,"label":267},"192",{"id":72,"label":269},"200","**c: 192.** Split by size: 1 to 9 are 9 one-digit numbers (9 digits); 10 to 99 are 90 two-digit numbers (180 digits); 100 is one three-digit number (3 digits). Total 9 + 180 + 3 = 192.",{"id":272,"type":77,"caption":273,"columns":274,"rows":278},"table-digit-count","Digits needed to write every number from 1 up to…",[275,276,277],"Up to","Count of numbers","Digits used",[279,280,281,283,286],[193,193,33],[197,197,267],[201,201,282],"2,893",[284,284,285],"10,000","38,894",[205,205,287],"4,88,895",{"id":289,"type":290,"title":291,"problem":292,"steps":293},"we-pages","worked_example","How many pages?","A printer used 1,002 digits to number the pages of a book, starting from page 1. How many pages does the book have?",[294,295,296,297],"Pages 1–9 use 9 digits. Pages 10–99 use 90 × 2 = 180 digits. Running total: 189.","Digits left: 1,002 − 189 = 813. From page 100 on, each page uses 3 digits.","813 ÷ 3 = 271 three-digit pages: pages 100 to 370.","So the book has **370 pages**. Check: 1002 digits. ✓",{"id":299,"type":155,"itemId":300,"prompt":301,"check":302,"hints":304,"feedback":306},"pr-sevens","number-system.investigate-count-sevens","How many times do you write the digit **7** when writing all the numbers from 1 to 100?",{"kind":159,"answer":303,"tolerance":161},20,[305],"Count the 7s in the ones place and the tens place separately.",{"correct":307,"incorrect":308},"Right: 10 times in the ones place (7, 17, …, 97) and 10 times in the tens place (70 to 79). Note 77 has two 7s and is counted in both lists.","Count by place: ones place gives 7, 17, 27, …, 97 (10 times); tens place gives 70, 71, …, 79 (10 times). Total 20.",{"id":310,"type":53,"title":311,"eyebrow":312,"navLabel":313},"ch04","Where the two systems agree and differ","Chapter 04","4 Two systems",{"id":315,"type":59,"prompt":316,"options":317,"explanation":324},"pred-first-diff","Start counting from 1 and write each number in both systems. What is the **first** number whose commas look different in the two systems?",[318,319,320,322],{"id":63,"label":201},{"id":66,"label":284},{"id":69,"label":321},"1,00,000 = 100,000",{"id":72,"label":323},"10,00,000 = 1,000,000","**c: one lakh.** Up to 99,999 both systems put exactly one comma after the thousands. At six digits the Indian system adds a comma after two more digits (1,00,000) while the International system waits for three (100,000). Numbers from 1,000 to 99,999 look identical in both systems.",{"id":326,"type":77,"caption":327,"columns":328,"rows":334},"table-commas-count","How many commas does an n-digit number need?",[329,330,331,332,333],"Digits","Indian example","Indian commas","International example","International commas",[335,337,340,342,344,346,349],[198,200,336,200,336],"0",[202,338,339,338,339],"9,999","1",[341,204,339,204,339],"5",[206,208,194,343,339],"999,999",[210,212,194,345,194],"9,999,999",[214,347,198,348,194],"9,99,99,999","99,999,999",[192,216,198,350,194],"999,999,999",{"id":352,"type":290,"title":353,"problem":354,"steps":355},"we-comma-rule","A rule for the number of commas","Find a rule for how many commas an n-digit number needs in each system (for n of 4 or more), and test it on a 12-digit number.",[356,357,358,359],"International: after the first 3 digits, a comma comes every 3 digits. Commas = (n − 1) ÷ 3, rounded down.","Indian: the first comma comes after 3 digits, then every 2. Commas = (n − 2) ÷ 2, rounded down, for n of 4 or more.","Test with 12 digits, 123456789012: International 123,456,789,012 has 3 commas, and (12 − 1) ÷ 3 = 3.67 → 3. ✓","Indian 1,23,45,67,89,012 has 5 commas, and (12 − 2) ÷ 2 = 5. ✓",{"id":361,"type":47,"variant":362,"title":363,"markdown":364},"obs-commas","observation","A pattern in the table","From 6 digits on, the Indian system needs one more comma than the International system, and the gap keeps growing slowly: each Indian period holds 2 digits and each International period holds 3. For a 9-digit number it is 4 commas against 2.",{"id":366,"type":155,"itemId":367,"prompt":368,"check":369,"hints":380,"feedback":382},"pr-same-commas","number-system.investigate-same-commas","Which of these numbers is written with the **same commas** in both systems?",{"kind":238,"options":370,"correct":379},[371,373,375,377],{"id":63,"label":372},"1,23,456",{"id":66,"label":374},"98,765",{"id":69,"label":376},"12,34,567",{"id":72,"label":378},"1,000,000",[66],[381],"How many digits does each number have?",{"correct":383,"incorrect":384},"Right: 98,765 has only five digits, and up to 99,999 both systems place a single comma after the thousands.","Numbers with 4 or 5 digits look the same in both systems. From 6 digits on, the Indian and International commas differ, so only 98,765 works.",{"id":386,"type":47,"variant":387,"title":388,"markdown":389},"mis-read-lakh","misconception","Reading an Indian comma as a thousands comma","Shown 12,50,000, some students who are used to International commas read the first group as *twelve thousand*… and get stuck. In the Indian system the comma after 12 marks **lakhs**, so it is *twelve lakh fifty thousand*. Internationally the same number is 1,250,000, *one million two hundred fifty thousand*. Before reading any big number, decide which system its commas follow: if any group in the middle has **two** digits, it is Indian.",{"id":391,"type":59,"prompt":392,"options":393,"explanation":400},"pred-names","Which system needs **more different period names** to read every number up to 99,99,99,999 (99 crore)?",[394,396,398],{"id":63,"label":395},"Indian",{"id":66,"label":397},"International",{"id":69,"label":399},"They need the same number","**a: Indian.** Up to that number, the Indian system uses three names beyond the ones period: *thousand, lakh, crore*. The International system uses only *thousand, million*, because 999,999,999 is still under a billion. Having more names means the Indian system reads big numbers with smaller chunks (two digits each), which many people find easier to say aloud; the International system reuses *hundred*, as in *nine hundred ninety-nine million*.",{"id":402,"type":53,"title":403,"eyebrow":404,"navLabel":405},"ch05","Playing with digits","Chapter 05","5 Playing with digits",{"id":407,"type":59,"prompt":408,"options":409,"explanation":415},"pred-perm","How many different 3-digit numbers can you make from the digits 2, 5 and 8, using each once?",[410,411,412,413],{"id":63,"label":198},{"id":66,"label":206},{"id":69,"label":192},{"id":72,"label":414},"27","**b: 6.** Choose the first digit (3 ways), then the second (2 ways left), then the last (1 way): 3 × 2 × 1 = 6. They are 258, 285, 528, 582, 825, 852. If one of the digits is 0, for example 0, 5, 8, only 4 work, because 058 and 085 are not 3-digit numbers.",{"id":417,"type":59,"prompt":418,"options":419,"explanation":428},"pred-nine","Take any digits, say 5, 2, 9, 1. Subtract the smallest number you can make from them (1,259) from the greatest (9,521). Is the answer always a multiple of 9, whatever digits you start with?",[420,422,424,426],{"id":63,"label":421},"Yes, always",{"id":66,"label":423},"Only for 4-digit numbers",{"id":69,"label":425},"Only when there is no 0",{"id":72,"label":427},"No, it depends","**a: always.** 9,521 − 1,259 = 8,262 = 9 × 918. The reason: every power of ten is one more than a multiple of 9 (10 = 9 + 1, 100 = 99 + 1, 1,000 = 999 + 1…), so any number leaves the same remainder on division by 9 as the **sum of its digits**. Two arrangements of the same digits have the same digit sum, so they leave the same remainder, and their difference leaves none. This is the idea behind the old *divisibility test for 9*.",{"id":430,"type":43,"markdown":431},"kaprekar","Here is a famous investigation described in the 1950s by the Indian mathematician **D. R. Kaprekar**, a school teacher in Devlali, Maharashtra (it is often dated to a talk he gave in 1949 and was published in 1955). Take any 4-digit number whose digits are not all the same. Make the greatest and the smallest numbers from its digits (keep any zeros, so the smallest may start with 0), and subtract. Repeat with the answer.",{"id":433,"type":77,"caption":434,"columns":435,"rows":440},"table-kaprekar","Kaprekar’s routine starting from 3,524",[436,437,438,439],"Step","Greatest","Smallest","Difference",[441,445,449],[339,442,443,444],"5,432","2345","3,087",[194,446,447,448],"8,730","0378","8,352",[198,450,451,452],"8,532","2358","6,174",{"id":454,"type":47,"variant":455,"title":456,"markdown":457},"aha-6174","aha","6174 is a trap you cannot escape","Every 4-digit number whose digits are not all equal reaches **6,174** in at most **7** steps, and then stays there, because 7,641 − 1,467 = 6,174. This number is called **Kaprekar’s constant**. Nobody made it up: it falls out of place value and subtraction. Try your year of birth or your house number.",{"id":459,"type":155,"itemId":460,"prompt":461,"check":462,"hints":464,"feedback":466},"pr-kap","number-system.investigate-kaprekar","Start Kaprekar’s routine with 2,111. What is the first difference?",{"kind":159,"answer":463,"tolerance":161},999,[465],"Arrange the digits 2, 1, 1, 1 in descending, then ascending order.",{"correct":467,"incorrect":468},"Right: greatest 2,111, smallest 1,112, difference 999. Keep going and you will reach 6,174.","The greatest arrangement of 2, 1, 1, 1 is 2,111 and the smallest is 1,112. 2,111 − 1,112 = 999.",{"id":470,"type":53,"title":471,"eyebrow":472,"navLabel":473},"ch06","Which numbers round to the same value?","Chapter 06","6 Rounding ranges",{"id":475,"type":59,"prompt":476,"options":477,"explanation":486},"pred-range","Rounded to the nearest hundred, a number is **500**. Which of these could it **not** be?",[478,480,482,484],{"id":63,"label":479},"450",{"id":66,"label":481},"549",{"id":69,"label":483},"449",{"id":72,"label":485},"501","**c: 449.** Numbers from 450 up to 549 round to 500 (450 rounds up by the halfway rule; 549 rounds down). 449 has a tens digit of 4, so it rounds down to 400. The whole numbers that round to 500 form a band of exactly 100 numbers: 450, 451, …, 549.",{"id":488,"type":77,"caption":489,"columns":490,"rows":494},"table-bands","Every whole number that rounds to a given value",[491,492,438,493],"Rounded value","Rounded to nearest","Largest",[495,499,500,503,507],[496,193,497,498],"70","65","74",[87,197,479,481],[91,201,501,502],"4,500","5,499",[504,284,505,506],"40,000","35,000","44,999",[508,205,509,510],"3,00,000","2,50,000","3,49,999",{"id":512,"type":106,"component":513,"componentVersion":5,"config":514,"objective":563,"textAlternative":564},"lab-sort-round","sort-game",{"prompt":515,"bins":516,"items":523,"seconds":562},"Rounded to the nearest thousand, does each number become 5,000?",[517,520],{"id":518,"label":519},"yes","Rounds to 5,000",{"id":521,"label":522},"no","Does not",[524,527,531,534,538,542,546,550,554,558],{"id":525,"label":501,"bin":518,"why":526},"n0","The hundreds digit is 5, so 4,500 rounds to 5,000.",{"id":528,"label":529,"bin":521,"why":530},"n1","4,499","The hundreds digit is 4, so 4,499 rounds to 4,000.",{"id":532,"label":502,"bin":518,"why":533},"n2","The hundreds digit is 4, so 5,499 rounds to 5,000.",{"id":535,"label":536,"bin":521,"why":537},"n3","5,500","The hundreds digit is 5, so 5,500 rounds to 6,000.",{"id":539,"label":540,"bin":518,"why":541},"n4","4,950","The hundreds digit is 9, so 4,950 rounds to 5,000.",{"id":543,"label":544,"bin":518,"why":545},"n5","5,050","The hundreds digit is 0, so 5,050 rounds to 5,000.",{"id":547,"label":548,"bin":521,"why":549},"n6","4,099","The hundreds digit is 0, so 4,099 rounds to 4,000.",{"id":551,"label":552,"bin":518,"why":553},"n7","5,001","The hundreds digit is 0, so 5,001 rounds to 5,000.",{"id":555,"label":556,"bin":521,"why":557},"n8","6,000","The hundreds digit is 0, so 6,000 rounds to 6,000.",{"id":559,"label":560,"bin":518,"why":561},"n9","4,501","The hundreds digit is 5, so 4,501 rounds to 5,000.",60,"Sort numbers by whether they round to 5,000, to discover the band of numbers from 4,500 to 5,499.","Ten number cards, 60 seconds. Rounds to 5,000: 4,500, 5,499, 4,950, 5,050, 5,001 and 4,501. Does not: 4,499 (rounds to 4,000), 5,500 (rounds to 6,000), 4,099 (4,000) and 6,000 (already 6,000).\n\nThe deciding digit when rounding to thousands is the hundreds digit. The band that rounds to 5,000 runs from 4,500 (the lowest halfway point, which rounds up) to 5,499 (the highest number still below the next halfway point, 5,500).",{"id":566,"type":59,"prompt":567,"options":568,"explanation":577},"pred-double","Aman rounds **2,449** to the nearest ten (2,450), then rounds that to the nearest hundred. Rani rounds 2,449 straight to the nearest hundred. Do they get the same answer?",[569,571,573,575],{"id":63,"label":570},"Yes, both get 2,400",{"id":66,"label":572},"Yes, both get 2,500",{"id":69,"label":574},"No: Aman gets 2,500 and Rani gets 2,400",{"id":72,"label":576},"No: Aman gets 2,400 and Rani gets 2,500","**c.** Aman: 2,449 → 2,450 → 2,500 (the tens digit of 2,450 is 5). Rani: the tens digit of 2,449 is 4 → 2,400. Rani is right: 2,449 is 49 away from 2,400 but 51 away from 2,500. Rounding twice lets a small push upward at the first step become a big jump at the second. This is called **double rounding**, and it is why you always round from the original number.",{"id":579,"type":155,"itemId":580,"prompt":581,"check":582,"hints":584,"feedback":586},"pr-double-count","number-system.investigate-double-count","How many whole numbers from 1 to 1,000 give a **different** answer when rounded to the nearest ten and then the nearest hundred, compared with rounding straight to the nearest hundred?",{"kind":159,"answer":583,"tolerance":161},50,[585],"Test 44, 45, 49 and 50. Which ones go wrong?",{"correct":587,"incorrect":588},"Right: the trouble-makers end in 45, 46, 47, 48 or 49. There are 10 such blocks (45–49, 145–149, …, 945–949), so 10 × 5 = 50.","Try a few: 45, 145, 2,449 all go wrong. Which last two digits cause it? 45 to 49. Count them in each hundred: 5, and there are 10 hundreds up to 1,000: 50.",{"id":590,"type":47,"variant":362,"title":591,"markdown":592},"obs-double","How often does double rounding go wrong?","A quick computer check of every number from 0 to 999 finds exactly **50** that give a different answer when rounded to tens first and then hundreds: those whose last two digits are 45, 46, 47, 48 or 49 (such as 45, 146, 2,449). That is 5 out of every 100 numbers, 5%. Rare enough that people do not notice, common enough to matter.",{"id":594,"type":155,"itemId":595,"prompt":596,"check":597,"hints":599,"feedback":601},"pr-band-lakh","number-system.investigate-band-lakh","What is the **largest** whole number that rounds to 3,00,000 when rounded to the nearest lakh?",{"kind":159,"answer":598,"tolerance":161},349999,[600],"What is halfway between 3,00,000 and 4,00,000?",{"correct":602,"incorrect":603},"Right: 3,50,000 is the halfway point and rounds up to 4,00,000, so the largest is 3,49,999.","The band for 3,00,000 runs from 2,50,000 to 3,49,999. At 3,50,000 the ten thousands digit is 5, which rounds up.",{"id":605,"type":106,"component":606,"componentVersion":5,"config":607,"objective":616,"textAlternative":617},"lab-round-timed","rounding-race",{"roundTo":608,"range":613,"rounds":614,"secondsPerRound":615},[609,610,611,612],10,100,1000,10000,{"min":610,"max":222},12,15,"Race the clock: round numbers up to 99,999 to the nearest 10, 100, 1,000 or 10,000 in 15 seconds each.","Twelve rounds with 15 seconds each. A number between 100 and 99,999 appears on a number line with the two nearest multiples of 10, 100, 1,000 or 10,000 marked. Tap the one it rounds to. Correct answers build a streak; the clock rewards quick, confident rounding.\n\nStrategy: find the deciding digit (one place to the right of the rounding place) and ignore everything after it. For 67,452 to the nearest 10,000 the deciding digit is 7, so the answer is 70,000; to the nearest 1,000 the deciding digit is 4, so 67,000; to the nearest 100 it is 5, so 67,500; to the nearest 10 it is 2, so 67,450.",{"id":619,"type":53,"title":620,"eyebrow":621,"navLabel":622},"ch07","How good is an estimate?","Chapter 07","7 Estimate errors",{"id":624,"type":59,"prompt":625,"options":626,"explanation":635},"pred-est-prod","Estimate 149 × 149 by rounding each number to its greatest place (100 × 100 = 10,000). The exact answer is…",[627,629,631,633],{"id":63,"label":628},"about 10,000",{"id":66,"label":630},"about 15,000",{"id":69,"label":632},"more than 20,000",{"id":72,"label":634},"less than 10,000","**c: 22,201.** Rounding 149 down to 100 throws away almost a third of each number, and in a product those losses multiply. The estimate is less than half the true answer. Rounding to the nearest ten instead gives 150 × 150 = 22,500, very close. For products, rounding more gently (to two non-zero digits) often gives a much better estimate.",{"id":637,"type":77,"caption":638,"columns":639,"rows":644},"table-est-err","Estimating products by rounding each number to its greatest place",[640,641,642,643],"Product","Estimate","Exact","Error (as % of exact)",[645,650,655,660,665],[646,647,648,649],"438 × 267","400 × 300 = 1,20,000","1,16,946","3%",[651,652,653,654],"4,812 × 3,276","5,000 × 3,000 = 1,50,00,000","1,57,64,112","5%",[656,657,658,659],"149 × 149","100 × 100 = 10,000","22,201","55%",[661,662,663,664],"68 × 72","70 × 70 = 4,900","4,896","0.1%",[666,667,668,649],"912 × 48","900 × 50 = 45,000","43,776",{"id":670,"type":43,"markdown":671},"est-err-why","Look at the error column. The estimate is excellent when the rounding errors cancel (one number rounded up, the other down, like 68 × 72 → 70 × 70) and poor when both are pushed the same way by a lot (149 × 149 → 100 × 100). A good estimator watches **which way** each number was rounded, and adjusts: *both rounded down, so the real answer is bigger than my estimate*.",{"id":673,"type":77,"caption":674,"columns":675,"rows":679},"table-sum-est","Estimating the sum 2,349 + 1,872 + 4,450 + 3,017 + 968 + 5,321 = 17,977 by rounding to different places",[676,677,678],"Round each number to the nearest","Estimated sum","Error",[680,682,685],[193,681,198],"17,980",[197,683,684],"18,000","23",[201,686,687],"17,000","977",{"id":689,"type":47,"variant":362,"title":690,"markdown":691},"obs-sum-est","Coarser rounding, bigger error (usually)","Rounding to tens gives an almost perfect estimate but is hardly quicker than adding exactly. Rounding to thousands is fast but can be off by hundreds. Choosing the rounding place is a trade-off between speed and accuracy, and the right choice depends on why you need the answer.",{"id":693,"type":106,"component":694,"componentVersion":5,"config":695,"objective":718,"textAlternative":719},"lab-sprint-est-i","arith-sprint",{"operations":696,"ranges":700,"rounds":614,"secondsTotal":704,"estimateFirst":113,"wordProblems":705},[697,698,699],"+","-","×",{"a":701,"b":702},{"min":611,"max":222},{"min":610,"max":703},9999,180,[706,710,714],{"prompt":707,"answer":708,"operation":697,"unit":709},"A stadium sold 48,750 tickets on Saturday and 52,380 on Sunday. How many tickets were sold in total?",101130,"tickets",{"prompt":711,"answer":712,"operation":698,"unit":713},"A village needs 1,25,000 litres of water a week and its tank holds 87,600 litres. How many more litres are needed?",37400,"litres",{"prompt":715,"answer":716,"operation":699,"unit":717},"A school orders 365 notebooks at ₹48 each. What is the cost in rupees?",17520,"₹","Estimate, then calculate, with bigger numbers and a three-minute clock; see how close your estimates get.","Twelve questions in three minutes. Each shows a sum, difference or product with a first number up to 99,999 and a second up to 9,999. You give an estimate first and then the exact answer; the game shows the gap.\n\nSome rounds are word problems that ask only for the exact answer; estimate them in your head first: 48,750 + 52,380 tickets (estimate 49,000 + 52,000 = 1,01,000; exact 1,01,130); 1,25,000 − 87,600 litres (estimate 1,25,000 − 88,000 = 37,000; exact 37,400); 365 × ₹48 notebooks (estimate 400 × 50 = ₹20,000, which is too high because both numbers were rounded up; exact ₹17,520).\n\nTry to notice when both numbers were rounded in the same direction and say whether your estimate is too high or too low.",{"id":721,"type":155,"itemId":722,"prompt":723,"check":724,"hints":733,"feedback":735},"pr-est-dir","number-system.investigate-estimate-direction","To estimate 612 × 387, Mira uses 600 × 400 = 2,40,000. Without calculating exactly, is the true answer bigger or smaller than her estimate?",{"kind":238,"options":725,"correct":732},[726,728,730],{"id":63,"label":727},"Definitely bigger",{"id":66,"label":729},"Definitely smaller",{"id":69,"label":731},"Cannot tell without calculating",[69],[734],"Which way was each number rounded?",{"correct":736,"incorrect":737},"Right: 612 was rounded down but 387 was rounded up, so the errors pull in opposite directions. You cannot tell which wins without more work. (Exact: 2,36,844, just smaller.)","One number was rounded down (612 → 600) and one up (387 → 400). The effects pull in opposite directions, so a quick glance cannot tell. In fact 612 × 387 = 2,36,844.",{"id":739,"type":53,"title":740,"eyebrow":741,"navLabel":742},"ch08","How big is a crore, really?","Chapter 08","8 Sizing a crore",{"id":744,"type":59,"prompt":745,"options":746,"explanation":755},"pred-crore-sec","If you counted one number every second, day and night without stopping, how long would it take to count to **one crore**?",[747,749,751,753],{"id":63,"label":748},"About a week",{"id":66,"label":750},"About four months",{"id":69,"label":752},"About three years",{"id":72,"label":754},"About thirty years","**b: about four months.** One day has 86,400 seconds, and 1,00,00,000 ÷ 86,400 ≈ 115.7 days, just under 116 days. Counting to a **billion** (100 crore) would take about 31.7 years of non-stop counting. Our sense of big numbers is poor: a crore and a billion *sound* similar, but one is a hundred times the other.",{"id":757,"type":758,"title":759,"note":760,"scale":761,"rungs":762},"ladder-seconds","ladder","How long is a big number of seconds?","Log scale. Each step is ten times the one below.","log",[763,766,769,772,775,779],{"label":764,"value":611,"display":765},"1 thousand seconds","≈ 17 minutes",{"label":767,"value":225,"display":768},"1 lakh seconds","≈ 28 hours",{"label":770,"value":227,"display":771},"10 lakh = 1 million seconds","≈ 11.6 days",{"label":773,"value":229,"display":774},"1 crore seconds","≈ 116 days",{"label":776,"value":777,"display":778},"10 crore seconds",100000000,"≈ 3.2 years",{"label":780,"value":781,"display":782},"100 crore = 1 billion seconds",1000000000,"≈ 31.7 years",{"id":784,"type":47,"variant":48,"title":785,"markdown":786},"try-thickness","How thick is a lakh of paper?","Measure the thickness of a notebook with 100 sheets (about 1 cm is typical). Then a lakh of sheets is 1,000 such notebooks: about 1,000 cm = 10 m, as tall as a three-storey building. A crore of sheets would be 100 times taller: about 1 km. Check your own notebook and redo the sums.",{"id":788,"type":53,"title":789,"eyebrow":790,"navLabel":791},"ch09","Investigating Roman numerals","Chapter 09","9 Roman puzzles",{"id":793,"type":59,"prompt":794,"options":795,"explanation":802},"pred-roman-long","Which number from 1 to 100 has the **longest** Roman numeral?",[796,797,799,801],{"id":63,"label":196},{"id":66,"label":798},"88",{"id":69,"label":800},"49",{"id":72,"label":197},"**b: 88 = LXXXVIII**, eight symbols. 99 is only XCIX (4 symbols) thanks to subtraction, and 100 is just C. The longest numerals come from digits that need many symbols: 8 needs VIII (four) in every place, so 88 = LXXX + VIII. The longest numeral below 4,000 is MMMDCCCLXXXVIII = 3,888, with 15 symbols.",{"id":804,"type":59,"prompt":805,"options":806,"explanation":813},"pred-roman-short","Is a Roman numeral ever **shorter** than the same number written in our digits?",[807,809,811],{"id":63,"label":808},"Never: Roman numerals are always longer",{"id":66,"label":810},"Yes, for a few numbers such as 10, 50, 100 and 1,000",{"id":69,"label":812},"Yes, for most numbers above 1,000","**b.** X (1 symbol) is shorter than 10 (2 digits), and M is shorter than 1,000 (4 digits). A computer check of 1 to 3,999 finds exactly **55** such numbers, including 10, 50, 100, 101, 105, 110, 150, 200, 400, 500 and 1,000. They are round numbers that happen to match single Roman symbols. For the other 3,944 numbers, the Roman numeral is as long or longer. The claim \"Roman numerals are always longer\" is **not always true**, and one counterexample (X) is enough to show it.",{"id":815,"type":77,"caption":816,"columns":817,"rows":823},"table-roman-len","Symbols needed for each digit in any place",[818,819,820,821,822],"Digit","Ones","Tens","Hundreds","Symbols",[824,828,832,836,840,844,848,852,856],[339,825,826,827,339],"I","X","C",[194,829,830,831,194],"II","XX","CC",[198,833,834,835,198],"III","XXX","CCC",[202,837,838,839,194],"IV","XL","CD",[341,841,842,843,339],"V","L","D",[206,845,846,847,194],"VI","LX","DC",[210,849,850,851,198],"VII","LXX","DCC",[214,853,854,855,202],"VIII","LXXX","DCCC",[192,857,858,859,194],"IX","XC","CM",{"id":861,"type":43,"markdown":862},"roman-place","The table reveals a secret: Roman numerals quietly use place value after all. Every digit from 1 to 9 has a fixed *pattern* (I, II, III, IV, V, VI, VII, VIII, IX), and the tens and hundreds just repeat that pattern with different letters (X-L-C and C-D-M instead of I-V-X). What Romans lacked was a single set of symbols reused in every place, and a zero to hold an empty place.",{"id":864,"type":106,"component":865,"componentVersion":5,"config":866,"objective":890,"textAlternative":891},"lab-match-roman","match-pairs",{"prompt":867,"mode":868,"pairs":869},"Match each Roman numeral with its value.","connect",[870,873,875,878,881,884,887],{"a":871,"b":872},"XIV","14",{"a":874,"b":800},"XLIX",{"a":876,"b":877},"XCIV","94",{"a":879,"b":880},"CDXLIV","444",{"a":882,"b":883},"MDCLXVI","1,666",{"a":885,"b":886},"MMXXVI","2,026",{"a":888,"b":889},"MMMCMXCIX","3,999","Connect Roman numerals with the numbers they stand for, splitting each numeral into place-value chunks.","Seven Roman numerals and seven numbers to connect. The pairs are: XIV = 14; XLIX = 49; XCIV = 94; CDXLIV = 444; MDCLXVI = 1,666; MMXXVI = 2,026; MMMCMXCIX = 3,999. Strategy: split each numeral into thousands, hundreds, tens and ones chunks. For example CDXLIV splits as CD | XL | IV = 400 + 40 + 4 = 444, and MMMCMXCIX splits as MMM | CM | XC | IX = 3,000 + 900 + 90 + 9 = 3,999, the largest number you can write with the standard symbols.",{"id":893,"type":290,"title":894,"problem":895,"steps":896},"we-roman-order","Ordering Roman numerals","Arrange XCIX, CI, LXXXIX, XC and CX from smallest to largest.",[897,898,899,900],"Length is no guide: LXXXIX has six symbols but XC has only two. Convert each first.","XCIX = 90 + 9 = 99; CI = 101; LXXXIX = 80 + 9 = 89; XC = 90; CX = 110.","Order the values: 89, 90, 99, 101, 110.","**LXXXIX, XC, XCIX, CI, CX.** Unlike our digits, a longer Roman numeral is not necessarily bigger: there is no \"count the digits\" shortcut.",{"id":902,"type":155,"itemId":903,"prompt":904,"check":905,"hints":916,"feedback":918},"pr-roman-longer","number-system.investigate-roman-longer","Test the claim: *“A Roman numeral with more symbols is always bigger.”* Which pair is a counterexample?",{"kind":238,"options":906,"correct":915},[907,909,911,913],{"id":63,"label":908},"XX (20) and X (10)",{"id":66,"label":910},"VIII (8) and X (10)",{"id":69,"label":912},"CC (200) and C (100)",{"id":72,"label":914},"III (3) and II (2)",[66],[917],"Find a longer numeral with a smaller value.",{"correct":919,"incorrect":920},"Right: VIII has four symbols but is smaller than X, which has one. One counterexample shows the claim is false.","Look for a pair where the longer numeral is the smaller number: VIII = 8 is longer than X = 10 but smaller.",{"id":922,"type":155,"itemId":923,"prompt":924,"check":925,"hints":927,"feedback":929},"pr-roman-sum","number-system.investigate-roman-sum","Add in Roman numerals: **XLVII + LXXVIII**. Give the answer as an ordinary number.",{"kind":159,"answer":926,"tolerance":161},125,[928],"Convert both to ordinary numbers first.",{"correct":930,"incorrect":931},"Right: 47 + 78 = 125, which is CXXV in Roman numerals.","Convert first: XLVII = 47 and LXXVIII = 78. Then 47 + 78 = 125 (CXXV). Doing it without converting is very hard, which is the point!",{"id":933,"type":53,"title":934,"eyebrow":935,"navLabel":936},"ch10","Investigating real big numbers","Chapter 10","10 Real data",{"id":938,"type":43,"markdown":939},"real-data","Real large numbers are messy: they come rounded, in different systems, and sometimes with mistakes. A good investigator checks them. Here is one way to test a claim you see in the news or in a quiz: convert it to plain digits, check the size with a rough estimate, and ask whether the answer is sensible.",{"id":941,"type":290,"title":942,"problem":943,"steps":944},"we-claim","Checking a claim","A social media post says: \"India has 140 crore people, that is 14 billion!\" Is it right?",[945,946,947,948],"Convert crore to plain digits: 140 crore = 140 × 1,00,00,000 = 1,40,00,00,000.","Regroup in threes: 1,400,000,000, which is 1.4 billion.","Use the key fact as a check: 100 crore = 1 billion, so 140 crore = 1.4 billion.","The post is wrong by a factor of 10: it should say **1.4 billion**, not 14 billion. (The whole world has only about 8 billion people.)",{"id":950,"type":290,"title":951,"problem":952,"steps":953},"we-rate","A crore of steps?","A child walks about 8,000 steps a day. Roughly how many days would it take to walk one crore steps?",[954,955,956,957],"Estimate: 1,00,00,000 ÷ 8,000.","Cancel three zeros from each: 10,000 ÷ 8 = 1,250 days.","That is about 1,250 ÷ 365 ≈ 3.4 years.","So a crore of steps is roughly three and a half years of normal walking.",{"id":959,"type":47,"variant":960,"title":961,"markdown":962},"careful-sources","careful","Check where a big number came from","Two websites can give different values for \"the population of Mumbai\" because they count different areas (the city or the whole metropolitan region), different years, or use estimates instead of a census. When numbers disagree, first check that they are measuring the same thing at the same time.",{"id":964,"type":53,"title":965,"eyebrow":966,"navLabel":967},"ch11","Estimation detectives: real Indian contexts","Chapter 11","11 Estimation detectives",{"id":969,"type":43,"markdown":970},"detective-intro","In real life nobody hands you a neat sum. You meet a situation, decide which numbers matter, round them sensibly, and check whether the answer is believable. This chapter works through four such situations. In each, **predict first**, then estimate, then compare with the exact answer.",{"id":972,"type":59,"prompt":973,"options":974,"explanation":982},"pred-crowds","A cricket ground hosts five matches with crowds of 48,215, 51,870, 39,940, 62,105, 55,330. Without adding exactly, the total crowd is closest to…",[975,977,978,980],{"id":63,"label":976},"25,000",{"id":66,"label":509},{"id":69,"label":979},"25,00,000",{"id":72,"label":981},"2,50,00,000","**b: about 2,50,000 (two and a half lakh).** Each crowd is roughly 50,000, and 5 × 50,000 = 2,50,000. Rounding each to the nearest ten thousand gives 50,000 + 50,000 + 40,000 + 60,000 + 60,000 = 2,60,000; the exact total is 2,57,460. The options differ by factors of ten, so even a very rough estimate picks the right one; that is what estimation is best at.",{"id":984,"type":290,"title":985,"problem":986,"steps":987},"we-rain-roof","How much rain falls on a roof?","Mumbai receives very roughly 2,200 mm of rain in a year, most of it in the monsoon. How many litres fall on a flat roof of 100 square metres?",[988,989,990,991],"Convert the rainfall to metres: 2,200 mm = 2.2 m (1,000 mm = 1 m).","Volume of water = area × depth = 100 × 2.2 = 220 cubic metres.","1 cubic metre holds 1,000 litres, so 220 × 1,000 = **2,20,000 litres** (two lakh twenty thousand).","A family using about 500 litres a day would need 2,20,000 ÷ 500 = 440 days to use it: more than a year of water from one roof, if it could all be stored.",{"id":993,"type":290,"title":994,"problem":995,"steps":996},"we-train-seats","Seats on a long-distance train","A train has 22 coaches. Most are sleeper coaches with 72 berths; suppose all 22 are. It runs every day of the year. Estimate the number of berths it offers in a year, then calculate exactly.",[997,998,999,1000],"Estimate: 22 ≈ 20, 72 ≈ 70, 365 ≈ 400. 20 × 70 × 400 = 5,60,000.","Exact: 22 × 72 = 1584 berths a day; 1584 × 365 = **5,78,160** berths a year.","The estimate is about 3% low: 22 and 72 were rounded down, 365 was rounded up by more, and the effects nearly cancel.","Said aloud: about five and three-quarter lakh berths, or about 0.58 million.",{"id":1002,"type":47,"variant":387,"title":1003,"markdown":1004},"mis-est-exact","“My estimate is wrong because it does not match the exact answer”","An estimate is **supposed** to differ from the exact answer. It is good if it has the right size (the right number of digits and a similar first digit) and is quick to find. It is bad only when it misleads, for example if it is ten times too big because a zero was dropped. Judge an estimate by whether it answers the question you had, not by whether it is exact.",{"id":1006,"type":155,"itemId":1007,"prompt":1008,"check":1009,"hints":1011,"feedback":1014},"pr-est-crowd","number-system.investigate-estimate-crowd","Estimate the total crowd from the five matches above by rounding each crowd to the nearest **thousand** instead. What is your estimate?",{"kind":159,"answer":1010,"tolerance":161},257000,[1012,1013],"Look at the hundreds digit of each crowd.","Add the rounded numbers in thousands: 48 + 52 + 40 + 62 + 55.",{"correct":1015,"incorrect":1016},"Right: 48,000 + 52,000 + 40,000 + 62,000 + 55,000 = 2,57,000, only 460 from the exact total.","Round each: 48,000, 52,000, 40,000, 62,000, 55,000. Add them: 2,57,000.",{"id":1018,"type":155,"itemId":1019,"prompt":1020,"check":1021,"hints":1032,"feedback":1034},"pr-est-choose","number-system.investigate-estimate-choose","A family buys a fridge for ₹38,990 and a washing machine for ₹27,490. They have ₹70,000. Which quick estimate best tells them whether they can afford both?",{"kind":238,"options":1022,"correct":1031},[1023,1025,1027,1029],{"id":63,"label":1024},"₹30,000 + ₹20,000 = ₹50,000, so yes easily",{"id":66,"label":1026},"₹39,000 + ₹27,500 = ₹66,500, so yes, with about ₹3,500 to spare",{"id":69,"label":1028},"₹40,000 + ₹30,000 = ₹70,000, so exactly enough",{"id":72,"label":1030},"You cannot estimate with money",[66],[1033],"How close is the total to ₹70,000?",{"correct":1035,"incorrect":1036},"Right: rounding to the nearest ₹500 keeps enough accuracy for a close decision. Exact total: ₹66,480, leaving ₹3,520.","When the answer is close to the limit, round gently. Option a rounds far too much; option c rounds both up and suggests there is no money left. b is closest: the exact total is ₹66,480.",{"id":1038,"type":1039,"prompt":1040},"reflect-i","reflection","Pick one prediction in this layer that you got wrong. What did you believe before, what did the test show, and how does place value explain the real answer?",{"id":1042,"type":1043,"title":1044,"terms":1045},"gloss-i","glossary","Investigation words",[1046,1049,1052,1056,1060,1064,1068,1071],{"term":1047,"meaning":1048},"Prediction","A definite guess made before testing, so the test can show whether you were right.",{"term":1050,"meaning":1051},"Always \u002F sometimes \u002F never true","Ways to classify a claim. One counterexample is enough to show a claim is not always true.",{"term":1053,"meaning":1054,"example":1055},"Counterexample","An example that shows a claim is false.","999 shows that \"the successor has the same number of digits\" is not always true.",{"term":1057,"meaning":1058,"example":1059},"Kaprekar’s constant","6,174: the number every 4-digit number (digits not all equal) reaches by repeatedly subtracting its smallest arrangement from its largest.","7,641 − 1,467 = 6,174",{"term":1061,"meaning":1062,"example":1063},"Double rounding","Rounding a number in two stages (for example to tens, then to hundreds), which can give a different answer from rounding once.","2,449 → 2,450 → 2,500, but 2,449 → 2,400 directly",{"term":1065,"meaning":1066,"example":1067},"Rounding band","The set of all numbers that round to the same value.","450 to 549 round to 500",{"term":1069,"meaning":1070},"Estimation error","The difference between an estimate and the exact value, sometimes given as a percentage of the exact value.",{"term":1072,"meaning":1073,"example":1074},"Arrangement (permutation)","One way of ordering a set of digits or objects.","258 and 852 are two arrangements of 2, 5, 8",{"id":1076,"type":1077,"title":1078,"questions":1079},"quiz-i","quiz","Investigate check",[1080,1093,1104,1115,1127,1139,1148,1161,1174,1187],{"itemId":1081,"prompt":1082,"options":1083,"correct":69,"why":1092},"number-system.inv-q-shift","The digit 3 in 4,381 is moved one place to the left by multiplying by 10. What is its new place value?",[1084,1086,1088,1090],{"id":63,"label":1085},"30",{"id":66,"label":1087},"300",{"id":69,"label":1089},"3,000",{"id":72,"label":1091},"30,000","4,381 × 10 = 43,810. The 3 moves from hundreds (300) to thousands (3,000).",{"itemId":1094,"prompt":1095,"options":1096,"correct":63,"why":1103},"number-system.inv-q-swap","63 − 36 equals…",[1097,1098,1100,1102],{"id":63,"label":414},{"id":66,"label":1099},"33",{"id":69,"label":1101},"37",{"id":72,"label":684},"63 − 36 = 27 = 9 × 3, and 6 − 3 = 3. The difference of a 2-digit number and its reverse is always 9 × (difference of the digits).",{"itemId":1105,"prompt":1106,"options":1107,"correct":66,"why":1114},"number-system.inv-q-count","How many 4-digit numbers are there?",[1108,1109,1111,1113],{"id":63,"label":338},{"id":66,"label":1110},"9,000",{"id":69,"label":1112},"8,999",{"id":72,"label":284},"From 1,000 to 9,999: 9,999 − 1,000 + 1 = 9,000.",{"itemId":1116,"prompt":1117,"options":1118,"correct":69,"why":1126},"number-system.inv-q-band","Which is the largest whole number that rounds to 3,000 when rounded to the nearest thousand?",[1119,1120,1122,1124],{"id":63,"label":889},{"id":66,"label":1121},"3,500",{"id":69,"label":1123},"3,499",{"id":72,"label":1125},"3,449","3,500 rounds up to 4,000, so the largest is 3,499.",{"itemId":1128,"prompt":1129,"options":1130,"correct":69,"why":1138},"number-system.inv-q-double","Rounding 1,346 to the nearest ten and then the nearest hundred gives…",[1131,1133,1135,1137],{"id":63,"label":1132},"1,300",{"id":66,"label":1134},"1,350",{"id":69,"label":1136},"1,400",{"id":72,"label":201},"1,346 → 1,350 → 1,400. Rounding directly gives 1,300, the correct nearest hundred.",{"itemId":1140,"prompt":1141,"options":1142,"correct":66,"why":1147},"number-system.inv-q-perm","How many 3-digit numbers can be made from 0, 4, 7 using each digit once?",[1143,1144,1145,1146],{"id":63,"label":198},{"id":66,"label":202},{"id":69,"label":206},{"id":72,"label":192},"6 arrangements, minus the 2 starting with 0 (047, 074): 407, 470, 704, 740.",{"itemId":1149,"prompt":1150,"options":1151,"correct":66,"why":1160},"number-system.inv-q-crore","Counting one number per second non-stop, one crore takes about…",[1152,1154,1156,1158],{"id":63,"label":1153},"12 days",{"id":66,"label":1155},"116 days",{"id":69,"label":1157},"3 years",{"id":72,"label":1159},"32 years","1,00,00,000 ÷ 86,400 ≈ 115.7 days.",{"itemId":1162,"prompt":1163,"options":1164,"correct":66,"why":1173},"number-system.inv-q-roman","Which has the most symbols?",[1165,1167,1169,1171],{"id":63,"label":1166},"XCIX (99)",{"id":66,"label":1168},"LXXXVIII (88)",{"id":69,"label":1170},"XLIX (49)",{"id":72,"label":1172},"C (100)","LXXXVIII has 8 symbols; XCIX and XLIX have 4; C has 1.",{"itemId":1175,"prompt":1176,"options":1177,"correct":66,"why":1186},"number-system.inv-q-est","Which estimate of 68 × 72 is best?",[1178,1180,1182,1184],{"id":63,"label":1179},"60 × 70",{"id":66,"label":1181},"70 × 70",{"id":69,"label":1183},"70 × 80",{"id":72,"label":1185},"60 × 80","Rounding each to the nearest ten gives 70 × 70 = 4,900. One was rounded up and one down, so the errors nearly cancel: exact 4,896.",{"itemId":1188,"prompt":1189,"options":1190,"correct":69,"why":1199},"number-system.inv-q-claim","140 crore is the same as…",[1191,1193,1195,1197],{"id":63,"label":1192},"14 million",{"id":66,"label":1194},"140 million",{"id":69,"label":1196},"1.4 billion",{"id":72,"label":1198},"14 billion","100 crore = 1 billion, so 140 crore = 1.4 billion.",{"id":1201,"type":1202,"conceptId":1203,"relation":1204,"explanation":1205},"conn-i-patterns","connection","patterns","related_to","Counting digits, Kaprekar’s routine and the multiples of 9 from reversed numbers are number patterns you can predict and explain.",{"id":1207,"type":1202,"conceptId":1208,"relation":1204,"explanation":1209},"conn-i-prime","prime-and-composite","Differences like 9 × (a − b) are always multiples of 9, a first taste of reasoning about factors.",{"id":1211,"type":1202,"conceptId":1212,"relation":1213,"explanation":1214},"conn-i-data","data-handling","applied_in","Checking whether real data makes sense, and rounding it sensibly, is the first step of any data investigation.",{"id":1216,"type":1217,"title":1218,"points":1219},"cheat-i","summary","What we found",[1220,1221,1222,1223,1224,1225,1226,1227,1228,1229,1230,1231],"**Multiplying by 10** shifts every digit one place left, so each place value becomes ten times as big.","**Reversing a 2-digit number** changes it by 9 × (difference of the digits).","**More digits means bigger** for whole numbers written normally, because 10ⁿ is one more than the largest n-digit number.","**The successor gains a digit** only when the number is all 9s.","**Digits to write 1 to 100:** 192. There are 9 × 10ⁿ⁻¹ numbers with n digits.","**The systems first differ at one lakh:** from 1,000 to 99,999 the commas are identical.","**Kaprekar’s constant:** 4-digit numbers (digits not all equal) reach 6,174 in at most 7 steps.","**Rounding bands:** 450–549 round to 500 (nearest 100). **Round once**, from the original number: double rounding changes 5% of results.","**Estimates of products** can be far off when both numbers are rounded the same way; note the direction of rounding.","**A crore seconds ≈ 116 days; a billion seconds ≈ 32 years.**","**Real estimates:** round gently when the answer is close to a limit, coarsely when only the size matters; judge an estimate by whether it answers your question.","**Roman numerals** reuse a digit pattern in each place but have no zero; 88 = LXXXVIII is the longest up to 100.",{"id":1233,"type":1234,"sourceIds":1235},"sources-i","sources",[1236,1237,1238,1239,1240,1241],"number-system-ncert-class6-knowing-numbers","number-system-ncert-class7-large-numbers","number-system-khan-place-value","number-system-mathsisfun-roman","number-system-wiki-indian-numbering","number-system-mathsisfun-place-value",[1236,1237,1238,1239,1240,1241],"needs_review",{"generatedBy":1245,"notes":1246},"claude-code","Draft generated with Python-checked numbers; pending owner review.","a1bf7c33c6d4bddff9ab8d71469c5f667454f0bb9f7591a89e9a7ecf6891fe59",{"component:place-value@1":1249,"logic:practice":1250,"component:sort-game@1":1251,"component:rounding-race@1":1252,"component:arith-sprint@1":1253,"component:match-pairs@1":1254,"source:number-system-khan-place-value":1255,"source:number-system-mathsisfun-place-value":1256,"source:number-system-mathsisfun-roman":1257,"source:number-system-ncert-class6-knowing-numbers":1258,"source:number-system-ncert-class7-large-numbers":1259,"source:number-system-wiki-indian-numbering":1260},"cae1e81c06fcfcc0152eb325ac8fb01f568e21a19ad76a75d79194e17231ae72","3d6b0fe1b15255975a32b0fcd94e8019bc959ad45cbf12e136e86149549c6878","b164f45a2c8ca08f26c450768ff0231e113e9fe45381eddb34dc6d0548596c38","6c2f2d540b01b2d7cd170d87a127ebd380947b02b030b91412433582d09e54f1","c0c63ed40e1bca5ba6d446d43886b887a3a7709e5cd6679419f64fd3ba62afe6","2a8ee4ac87460b4e1175a4bb13c96b03d577db06dde95670eb7fcfe4ad787899","8d110718a4c12de7f84b3e0653db1fd8b1147917a7dc31c48861b97c25731e0d","6a40a6a059b316a1b3d0621b353f1e4e944ed797310ad87c02f106fc0469f729","ef0f2abd2b3d7e393f1878aa51359c7a18194adf57596671bc0dff39cd1aa2f5","4e646d96c29f6b98469c9d11ee30da4e543f849b673fd067975aa685d8fef8a3","9dc5f1741ed9fad68db69882c1cc09c8d4b3cf5902d5d0273891bad1da1d10c0","045032c28b7bfe33f8bc73c5c09ee8dd8076293c90dbe922d8a461e059edd6e3",{"state":1262,"reviewer":1263,"selfReview":113,"reviewedAt":1264,"method":1265},"approved","The library owner","2026-09-20T10:18:37.581Z","owner_bulk","preview-7e1cbbcc4f",1789899597427]