[{"data":1,"prerenderedAt":1130},["ShallowReactive",2],{"layer:properties-of-numbers:understand":3},{"layer":4,"contentHash":1107,"dependencyHashes":1108,"approval":1124,"releaseId":1129},{"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":1102,"reviewStatus":1103,"authoring":1104},1,"properties-of-numbers","en","understand","The properties, precisely","Closure, commutative, associative and distributive laws, and the special numbers 0 and 1","State each property of whole numbers exactly, in words and with letters; see why it holds for + and × but fails for − and ÷; learn why division by zero is undefined; and use the properties for fast, reliable mental maths.",[13,14,15,16,17],"State closure, commutative, associative and distributive properties with letters, and say for which operations each holds.","Give a counterexample to show a property fails for subtraction or division.","Use the distributive property and area model for calculations like 98 × 25, 12 × 105 and 23 × 14.","Explain why 0 and 1 are identities, and why a ÷ 0 and 0 ÷ 0 are undefined.","Apply and justify the even\u002Fodd rules for sums, differences and products.",40,{"title":20,"rows":21},"Lesson plate",[22,25,28,31,34,37],{"label":23,"value":24},"Depth","Understand",{"label":26,"value":27},"Reading time","≈ 40 minutes",{"label":29,"value":30},"Prior knowledge","Turn-around facts, grouping (Discover)",{"label":32,"value":33},"Chapters","11",{"label":35,"value":36},"Labs","Brackets lab, sprint, property match, always-or-not sort",{"label":38,"value":39},"Notation","a, b, c for any whole numbers",[41,45,70,76,79,104,110,125,130,135,163,166,171,195,200,216,219,247,251,256,271,274,287,296,301,320,325,340,343,367,370,380,389,398,402,406,410,453,458,462,465,491,496,517,520,547,552,556,579,584,602,605,609,614,618,641,644,654,666,671,702,706,711,741,745,782,844,905,1050,1054,1069,1075,1080,1084,1088],{"id":42,"type":43,"markdown":44},"intro-understand","prose","In Discover you met the properties through chairs, laddoos and kirana bills. Now we make them **precise**. That means three things:\n\n1. **Say exactly which numbers** a rule is about (natural numbers? whole numbers?).\n2. **Say exactly which operation** it is about. A rule that is true for addition may be false for subtraction.\n3. **Write it with letters**, so that one short line covers every number at once.\n\nWriting a + b = b + a is not showing off. It is a promise: \"whatever whole numbers you put in place of a and b, the two sides will be equal.\" That promise, made once, saves us from checking 3 + 5, 17 + 42, 1,000 + 1 and every other pair separately. By the end of this layer you should be able to state each property in words and with letters, give an example and a counterexample, and use each one to calculate faster.",{"id":46,"type":47,"tone":48,"items":49},"spec-understand","spec","blue",[50,54,58,62,66],{"label":51,"big":52,"value":53},"Numbers","W = {0, 1, 2, …}","In this layer \"number\" means a whole number unless we say otherwise.",{"label":55,"big":56,"value":57},"Letters","a, b, c","Stand for any whole numbers. The same letter means the same number each time it appears.",{"label":59,"big":60,"value":61},"Brackets","( )","Mean \"do this part first\". (8 − 3) − 2 and 8 − (3 − 2) are different calculations.",{"label":63,"big":64,"value":65},"Holds for","+ and ×","Closure, commutativity and associativity hold for + and × of whole numbers.",{"label":67,"big":68,"value":69},"Fails for","− and ÷","None of those three hold for − and ÷ of whole numbers; one counterexample shows it.",{"id":71,"type":72,"title":73,"eyebrow":74,"navLabel":75},"ch1","chapter","Natural numbers, whole numbers and the number line","Chapter 01","1 Number families",{"id":77,"type":43,"markdown":78},"families-prose","The **natural numbers** are 1, 2, 3, 4, … The **whole numbers** are 0, 1, 2, 3, … The only difference is the number 0. (A few books outside India include 0 among the natural numbers. NCERT does not, and neither do we.)\n\nSome facts that follow directly:\n\n- **There is no largest natural or whole number.** If someone claims N is the largest, then N + 1 is larger. This is a tiny proof by contradiction, and it is airtight.\n- **The smallest natural number is 1; the smallest whole number is 0.**\n- **Successor:** the successor of n is **n + 1**. Every whole number has exactly one successor.\n- **Predecessor:** the predecessor of n is **n − 1**. Every whole number except 0 has one. 0 has no whole-number predecessor; 1 has predecessor 0, which is whole but not natural.\n- **Between two whole numbers** there is a definite count of whole numbers. Between 10 and 20 (not counting 10 and 20) there are 20 − 10 − 1 = 9 of them: 11 to 19.",{"id":80,"type":81,"caption":82,"columns":83,"rows":87},"ops-number-line","table","The four operations on the whole-number line",[84,85,86],"Operation","Number-line picture","Example",[88,92,96,100],[89,90,91],"Addition a + b","Start at a, jump b steps right","3 + 4: from 3, jump 4 right, land on 7",[93,94,95],"Subtraction a − b","Start at a, jump b steps left","7 − 4: from 7, jump 4 left, land on 3",[97,98,99],"Multiplication a × b","From 0, make a jumps of size b","3 × 4: jumps land on 4, 8, 12",[101,102,103],"Division a ÷ b","How many jumps of size b take you from 0 to a?","12 ÷ 4: jumps 4, 8, 12, so 3 jumps",{"id":105,"type":106,"variant":107,"title":108,"markdown":109},"nuance-subtract-left","callout","nuance","Where the road runs out","On the whole-number line, 3 − 5 asks you to jump 5 steps left from 3, which would take you past 0 and off the road. That is the number-line view of why whole numbers are not closed under subtraction. The road does continue to the left, through −1, −2, −3, … (the **integers**), but those are not whole numbers. The Extend layer walks along that part of the road.",{"id":111,"type":112,"itemId":113,"prompt":114,"check":115,"hints":119,"feedback":122},"pr-between","practice","properties-of-numbers.understand-between","How many whole numbers lie strictly **between** 89 and 150?",{"kind":116,"answer":117,"tolerance":118},"number",60,0,[120,121],"Count 90, 91, …, 149.","Use 150 − 89 − 1.",{"correct":123,"incorrect":124},"Right: 150 − 89 − 1 = 60. The numbers are 90 to 149.","The numbers are 90, 91, …, 149. That is 149 − 90 + 1 = 60 numbers, or 150 − 89 − 1 = 60.",{"id":126,"type":72,"title":127,"eyebrow":128,"navLabel":129},"ch2","Closure: does the answer stay in the family?","Chapter 02","2 Closure",{"id":131,"type":106,"variant":132,"title":133,"markdown":134},"def-closure","definition","Closure","A set of numbers is **closed under an operation** if, whenever you take **any** two numbers from the set and do the operation, the answer is **always** in the set too.\n\n- Whole numbers are closed under **addition**: if a and b are whole, a + b is whole.\n- Whole numbers are closed under **multiplication**: if a and b are whole, a × b is whole.\n- Whole numbers are **not** closed under **subtraction**: 3 − 5 is not whole.\n- Whole numbers are **not** closed under **division**: 5 ÷ 2 is not whole, and 5 ÷ 0 is not defined at all.",{"id":136,"type":81,"caption":137,"columns":138,"rows":144},"closure-sets-table","Closure for natural and whole numbers, with a counterexample wherever it fails",[139,140,141,142,143],"Set","Addition","Subtraction","Multiplication","Division",[145,150,154,158],[146,147,148,147,149],"Natural numbers","Closed","Not closed: 4 − 4 = 0 is not natural","Not closed: 3 ÷ 4 is not natural",[151,147,152,147,153],"Whole numbers","Not closed: 4 − 7 is not whole","Not closed: 7 ÷ 2 is not whole",[155,147,156,147,157],"Even numbers","Not closed: 4 − 8 is not whole","Not closed: 6 ÷ 4 is not whole",[159,160,161,147,162],"Odd numbers","Not closed: 3 + 5 = 8 is even","Not closed: 5 − 3 = 2 is even","Not closed: 3 ÷ 5 is not whole",{"id":164,"type":43,"markdown":165},"closure-why-prose","Notice the difference between the two halves of this table. A \"Not closed\" entry needs **one counterexample** and it is done. A \"Closed\" entry is a claim about **infinitely many** pairs, so no list of examples can prove it. Why, then, are we sure the whole numbers are closed under addition?\n\nBecause of what addition *means*. Adding b to a means taking b more steps to the right on the number line starting from a. Every step lands on the next whole number, and the road never runs out to the right. So you always land on a whole number. Multiplication is repeated addition (a jumps of size b), so it inherits the same guarantee. For odd numbers under multiplication, the reason is the pairing argument you will see in chapter 9.",{"id":167,"type":106,"variant":168,"title":169,"markdown":170},"misc-closure-examples","misconception","\"It worked for ten examples, so it is closed\"","Testing examples is a great way to **suspect** a property, but not to **prove** it. Consider the claim \"odd numbers are closed under subtraction\". 9 − 4 is no use (4 is even). Try 9 − 5 = 4: even already. The claim fails at the very first honest test. Other claims survive many tests and then fail on a big number. A property is only established when you have a reason that covers every case.",{"id":172,"type":112,"itemId":173,"prompt":174,"check":175,"hints":189,"feedback":192},"pr-closure","properties-of-numbers.understand-closure","Which set is closed under **addition**?",{"kind":176,"options":177,"correct":188},"choice",[178,180,182,185],{"id":179,"label":159},"a",{"id":181,"label":155},"b",{"id":183,"label":184},"c","Numbers from 1 to 100",{"id":186,"label":187},"d","Numbers ending in 5",[181],[190,191],"Try adding two members and see if you escape.","For \"1 to 100\", try 60 + 70.",{"correct":193,"incorrect":194},"Yes. Even + even = even, always: two sets of pairs make a set of pairs.","Odd: 3 + 5 = 8 escapes. 1 to 100: 60 + 70 = 130 escapes. Ending in 5: 5 + 15 = 20 escapes. Only even numbers are closed.",{"id":196,"type":72,"title":197,"eyebrow":198,"navLabel":199},"ch3","The commutative property","Chapter 03","3 Commutative",{"id":201,"type":202,"items":203},"formulas-commutative","formulas",[204,207,210,213],{"expression":205,"caption":206},"a + b = b + a","Commutative property of addition. For all whole numbers a and b.",{"expression":208,"caption":209},"a × b = b × a","Commutative property of multiplication. For all whole numbers a and b.",{"expression":211,"caption":212},"a − b ≠ b − a (in general)","Subtraction is not commutative. Equal only when a = b.",{"expression":214,"caption":215},"a ÷ b ≠ b ÷ a (in general)","Division is not commutative. Equal only when a = b (and a ≠ 0).",{"id":217,"type":43,"markdown":218},"comm-why-prose","**Why is addition commutative?** Put a counters in a row and b counters after them. Read the row from left to right and you count a + b. Read it from right to left and you count b + a. It is the same row of counters, so the totals are equal.\n\n**Why is multiplication commutative?** Arrange counters in a rectangle with a rows and b columns. Counting row by row gives a × b; counting column by column gives b × a. It is the same rectangle. This works for every rectangle, however big.\n\n**Why do subtraction and division fail?** One counterexample each is enough. 9 − 4 = 5, but 4 − 9 is not even a whole number. 20 ÷ 5 = 4, but 5 ÷ 20 = ¼. In fact 8 − 3 and 3 − 8 are *opposites* (5 and −5), and 20 ÷ 5 and 5 ÷ 20 are *reciprocals* (4 and ¼). Swapping turns the answer inside out.",{"id":220,"type":81,"caption":221,"columns":222,"rows":226},"comm-uses","Where the commutative property quietly helps",[223,224,225],"Situation","Hard way","Easy way (swapped)",[227,231,235,239,243],[228,229,230],"Counting on","4 + 87: count 87 steps from 4","87 + 4: count 4 steps from 87 → 91",[232,233,234],"Times tables","Learn 9 × 3 separately","Already know 3 × 9 = 27",[236,237,238],"Written multiplication","6 × 4,378 with 4,378 on the bottom line","4,378 × 6 with the single digit below",[240,241,242],"Checking a total","Add the column top to bottom","Add again bottom to top: the total must match",[244,245,246],"Seating","8 rows of 12 chairs","12 rows of 8: still 96 chairs",{"id":248,"type":106,"variant":107,"title":249,"markdown":250},"nuance-order-actions","Commutative is special, not normal","It is easy to think \"of course order does not matter\". But in everyday life order usually **does** matter. Put on socks, then shoes: fine. Shoes, then socks: silly. Add rice to boiling water, or boiling water to rice: different results. Walk 3 km north then turn right, or turn right then walk north: you end up in different places. Addition and multiplication are unusual, and lucky for us, in being commutative.",{"id":252,"type":72,"title":253,"eyebrow":254,"navLabel":255},"ch4","The associative property","Chapter 04","4 Associative",{"id":257,"type":202,"items":258},"formulas-associative",[259,262,265,268],{"expression":260,"caption":261},"(a + b) + c = a + (b + c)","Associative property of addition: the pair you add first does not matter.",{"expression":263,"caption":264},"(a × b) × c = a × (b × c)","Associative property of multiplication: the pair you multiply first does not matter.",{"expression":266,"caption":267},"(a − b) − c ≠ a − (b − c)","Subtraction is not associative (equal only when c = 0).",{"expression":269,"caption":270},"(a ÷ b) ÷ c ≠ a ÷ (b ÷ c)","Division is not associative. With a ≠ 0, the two sides are equal only when c = 1.",{"id":272,"type":43,"markdown":273},"assoc-vs-comm","Students often mix up associative and commutative. Here is the clean difference:\n\n- **Commutative** is about **order**: which number comes first. 5 + 8 versus 8 + 5.\n- **Associative** is about **grouping**: which pair you do first, *with the order unchanged*. (5 + 8) + 2 versus 5 + (8 + 2).\n\nIn real calculations we usually use **both** together. To do 25 × 17 × 4, we swap 17 and 4 (commutative) to get 25 × 4 × 17, then group (25 × 4) × 17 (associative) = 100 × 17 = **1700**. Because + and × have both properties, you can **rearrange a long sum or product in any order and any grouping** you like. That is the real superpower.",{"id":275,"type":276,"title":277,"problem":278,"steps":279},"we-assoc-counterexample","worked_example","Showing subtraction is not associative","Is (50 − 20) − 10 equal to 50 − (20 − 10)?",[280,281,282,283,284,285,286],"Left side: brackets first. 50 − 20 = 30.","Then 30 − 10 = **20**.","Right side: brackets first. 20 − 10 = 10.","Then 50 − 10 = **40**.","20 ≠ 40, so the two groupings give different answers.","One counterexample settles it: **subtraction is not associative**.","Why the difference? In 50 − (20 − 10), the 10 is subtracted from the 20 first, which *reduces* how much is taken from 50. So the 10 ends up effectively **added back**: 50 − 20 + 10 = 40.",{"id":288,"type":276,"title":289,"problem":290,"steps":291},"we-assoc-div","Showing division is not associative","Compare (64 ÷ 8) ÷ 2 with 64 ÷ (8 ÷ 2).",[292,293,294,295],"Left: 64 ÷ 8 = 8, then 8 ÷ 2 = **4**.","Right: 8 ÷ 2 = 4, then 64 ÷ 4 = **16**.","4 ≠ 16. Division is not associative.","Look closer: the right side is 4 times the left side, and 4 = 2 × 2. Dividing by (8 ÷ 2) is the same as dividing by 8 and then *multiplying* by 2, while the left side divides by 2 again.",{"id":297,"type":106,"variant":298,"title":299,"markdown":300},"careful-left-right","careful","With − and ÷, work from left to right","Because subtraction and division are not associative, an expression like 50 − 20 − 10 would be ambiguous unless we agree on a rule. The agreed rule is: **work from left to right**. So 50 − 20 − 10 = 30 − 10 = 20, and 64 ÷ 8 ÷ 2 = 8 ÷ 2 = 4. The full set of rules for mixed expressions is in the [Order of operations](\u002Ftopics\u002Forder-of-operations) topic.",{"id":302,"type":303,"component":304,"componentVersion":5,"config":305,"objective":314,"textAlternative":315,"help":316},"lab-order-ops-understand","interactive","order-ops",{"expressions":306,"showRuleCard":313},[307,308,309,310,311,312],"(50 - 20) - 10","50 - (20 - 10)","(64 ÷ 8) ÷ 2","64 ÷ (8 ÷ 2)","6 × (10 + 4)","6 × 10 + 6 × 4",true,"Work out pairs of expressions step by step and see when brackets change the answer and when they do not.","A step-by-step expression lab. Each expression is reduced one operation at a time; you choose which operation to do next.\n\nPair 1: (50 − 20) − 10 = 30 − 10 = 20, but 50 − (20 − 10) = 50 − 10 = 40. The brackets change the answer: subtraction is not associative.\n\nPair 2: (64 ÷ 8) ÷ 2 = 8 ÷ 2 = 4, but 64 ÷ (8 ÷ 2) = 64 ÷ 4 = 16. Division is not associative.\n\nPair 3: 6 × (10 + 4) = 6 × 14 = 84, and 6 × 10 + 6 × 4 = 60 + 24 = 84. Different routes, same answer: this is the distributive property.",{"hints":317},[318,319],"Always do the bracket first.","Compare the final answers of each pair.",{"id":321,"type":72,"title":322,"eyebrow":323,"navLabel":324},"ch5","The distributive property","Chapter 05","5 Distributive",{"id":326,"type":202,"items":327},"formulas-distributive",[328,331,334,337],{"expression":329,"caption":330},"a × (b + c) = a × b + a × c","Multiplication distributes over addition.",{"expression":332,"caption":333},"a × (b − c) = a × b − a × c","Multiplication distributes over subtraction (for whole numbers, when b ≥ c).",{"expression":335,"caption":336},"(b + c) × a = b × a + c × a","The same from the right, because × is commutative.",{"expression":338,"caption":339},"(b + c) ÷ a = b ÷ a + c ÷ a","Division distributes from the right only (a ≠ 0).",{"id":341,"type":43,"markdown":342},"dist-meaning","The distributive property is the only one of the big three that links **two different operations**. It says a multiplier can be \"handed out\" to every part of a sum or difference.\n\nThe picture is a rectangle. A rectangle 7 wide and 23 long has area 7 × 23. Cut it into a 7 × 20 piece and a 7 × 3 piece. The area has not changed, so 7 × 23 = 7 × 20 + 7 × 3 = 140 + 21 = 161. Every use of the distributive property is a rectangle being cut into pieces.\n\nTwo-digit by two-digit works the same way, cutting both sides. For **23 × 14**, split 23 = 20 + 3 and 14 = 10 + 4. The rectangle falls into four pieces:",{"id":344,"type":81,"caption":345,"columns":346,"rows":351},"area-23-14","Area model for 23 × 14: split both numbers, multiply each pair, add the four pieces",[347,348,349,350],"×","20","3","Row total",[352,357,362],[353,354,355,356],"10","10 × 20 = 200","10 × 3 = 30","230",[358,359,360,361],"4","4 × 20 = 80","4 × 3 = 12","92",[363,364,365,366],"Total","280","42","230 + 92 = 322",{"id":368,"type":43,"markdown":369},"dist-longmult","Check: 23 × 14 = 322. ✓ The column method of long multiplication you learned in school is exactly this table, written compactly: first 23 × 4 = 92, then 23 × 10 = 230, then add. **Long multiplication is the distributive property on paper.**\n\nThe property also works **backwards**, which is called taking out a **common factor**. In 37 × 68 + 37 × 32, both products share the 37. So 37 × 68 + 37 × 32 = 37 × (68 + 32) = 37 × 100 = 3,700. What looked like two hard multiplications became one easy one.",{"id":371,"type":276,"title":372,"problem":373,"steps":374},"we-98x25","Mental maths: 98 × 25","Work out **98 × 25** without paper.",[375,376,377,378,379],"98 is close to 100: write 98 = 100 − 2.","Distribute: 98 × 25 = 100 × 25 − 2 × 25.","100 × 25 = 2,500 and 2 × 25 = 50.","2,500 − 50 = 2,450.","So 98 × 25 = 2,450.",{"id":381,"type":276,"title":382,"problem":383,"steps":384},"we-12x105","Mental maths: 12 × 105","Work out **12 × 105**.",[385,386,387,388],"Split 105 = 100 + 5.","12 × 105 = 12 × 100 + 12 × 5.","= 1,200 + 60.","= **1,260**. Check: 12 × 105 = 1,260. ✓",{"id":390,"type":276,"title":391,"problem":392,"steps":393},"we-999","Mental maths: 47 × 999","Work out **47 × 999**.",[394,395,396,397],"999 = 1,000 − 1.","47 × 999 = 47 × 1,000 − 47 × 1.","= 47,000 − 47.","= **46,953**. Check: 47 × 999 = 46,953. ✓",{"id":399,"type":106,"variant":168,"title":400,"markdown":401},"misc-dist-half","Only multiplying the first part","The most common error is to write 5 × (20 + 3) as 5 × 20 + 3 = 103. The 5 must go to **both** parts: 5 × 20 + 5 × 3 = 100 + 15 = 115. Check: 5 × 23 = 115. If you picture the rectangle, you cannot leave one piece out.",{"id":403,"type":106,"variant":168,"title":404,"markdown":405},"misc-dist-times","Distributing over multiplication","The distributive property is about × **over + or −**. It does **not** say 2 × (3 × 4) = (2 × 3) × (2 × 4). Check: 2 × (3 × 4) = 2 × 12 = **24**, but (2 × 3) × (2 × 4) = 6 × 8 = **48**. Doubling a product means doubling **one** factor, not both.",{"id":407,"type":106,"variant":107,"title":408,"markdown":409},"nuance-div-right","Division distributes only from the right","Sharing works piece by piece: (40 + 8) ÷ 4 = 40 ÷ 4 + 8 ÷ 4 = 12. That is how short division works: 48 ÷ 4 is 4 tens ÷ 4 and 8 ones ÷ 4.\n\nBut the other way round fails: 24 ÷ (2 + 4) = 24 ÷ 6 = **4**, while 24 ÷ 2 + 24 ÷ 4 = 12 + 6 = **18**. Splitting the *divisor* is not allowed.",{"id":411,"type":303,"component":412,"componentVersion":5,"config":413,"objective":447,"textAlternative":448,"help":449},"lab-sprint-understand","arith-sprint",{"operations":414,"ranges":415,"rounds":422,"secondsTotal":423,"estimateFirst":313,"wordProblems":424},[347],{"a":416,"b":419},{"min":417,"max":418},11,99,{"min":420,"max":421},2,9,12,120,[425,429,433,436,440,443],{"prompt":426,"answer":427,"operation":347,"unit":428},"A shop sells 98 notebooks at ₹25 each. Use 98 = 100 − 2. How many rupees is that?",2450,"₹",{"prompt":430,"answer":431,"operation":347,"unit":432},"A school buys 12 cartons with 105 pencils each. Use 105 = 100 + 5. How many pencils?",1260,"pencils",{"prompt":434,"answer":435,"operation":347,"unit":428},"A stall sells chai for ₹37: 68 cups in the morning and 32 in the evening. Total takings in rupees?",3700,{"prompt":437,"answer":438,"operation":347,"unit":439},"A hall has 23 rows of 14 seats. Split into 20 + 3 rows. How many seats?",322,"seats",{"prompt":441,"answer":442,"operation":347,"unit":428},"A pack costs ₹999. What do 47 packs cost? Use 1,000 − 1.",46953,{"prompt":444,"answer":445,"operation":347,"unit":446},"Cricket: a team scores 6 runs off each of 19 balls. Use 6 × 20 − 6. How many runs?",114,"runs","Multiply two-digit numbers by one-digit numbers with estimate-first, then use the distributive property on shop and school word problems.","A timed sprint (120 seconds, 12 rounds; up to half are word problems from the list below). Each plain question multiplies a two-digit number (11 to 99) by a one-digit number (2 to 9). You first give a rounded estimate, then the exact answer. A good method: split the two-digit number into tens and ones, e.g. 7 × 46 = 7 × 40 + 7 × 6 = 280 + 42 = 322.\n\nWord problems (no estimate step) and answers:\n1. 98 notebooks at ₹25: 100 × 25 − 2 × 25 = 2,500 − 50 = ₹2,450.\n2. 12 cartons of 105 pencils: 1,200 + 60 = 1,260 pencils.\n3. Chai at ₹37 for 68 + 32 cups: 37 × 100 = ₹3,700.\n4. 23 rows of 14 seats: 20 × 14 + 3 × 14 = 280 + 42 = 322 seats.\n5. 47 packs at ₹999: 47,000 − 47 = ₹46,953.\n6. 6 runs off each of 19 balls: 120 − 6 = 114 runs.",{"hints":450},[451,452],"Estimate first by rounding the two-digit number to the nearest ten.","Near 100 or 1,000? Multiply by the round number and adjust.",{"id":454,"type":72,"title":455,"eyebrow":456,"navLabel":457},"ch6","Identity elements: 0 for addition, 1 for multiplication","Chapter 06","6 Identities",{"id":459,"type":106,"variant":132,"title":460,"markdown":461},"def-identity","Identity element","An **identity element** for an operation is a number that leaves every number unchanged under that operation, **whichever side it is on**.\n\n- **0 is the additive identity:** a + 0 = 0 + a = a for every whole number a.\n- **1 is the multiplicative identity:** a × 1 = 1 × a = a for every whole number a.",{"id":463,"type":43,"markdown":464},"identity-prose","Why \"whichever side\"? Look at subtraction. It is true that a − 0 = a for every a. But 0 − a is **not** a (for example 0 − 5 is not 5; it is not even a whole number). So 0 only works on the right. For division, a ÷ 1 = a but 1 ÷ a is usually not a. So subtraction and division have no true identity; 0 and 1 are only \"right-hand identities\" for them.\n\nIdentities matter more than they look. The whole idea of **undoing** an operation is built on them. Subtracting 7 undoes adding 7 because a + 7 − 7 = a + 0 = a. Dividing by 7 undoes multiplying by 7 because a × 7 ÷ 7 = a × 1 = a. When you learn about negative numbers and fractions you will meet **inverses**: numbers that combine with a to give the identity. The inverse of 7 for addition is −7; for multiplication it is 1\u002F7.",{"id":466,"type":81,"caption":467,"columns":468,"rows":473},"identity-table","Does the operation have an identity in the whole numbers?",[84,469,470,471,472],"Candidate","Right side","Left side","Identity?",[474,479,484,488],[140,475,476,477,478],"0","9 + 0 = 9 ✓","0 + 9 = 9 ✓","Yes: 0",[142,480,481,482,483],"1","9 × 1 = 9 ✓","1 × 9 = 9 ✓","Yes: 1",[141,475,485,486,487],"9 − 0 = 9 ✓","0 − 9 is not 9 ✗","No",[143,480,489,490,487],"9 ÷ 1 = 9 ✓","1 ÷ 9 is not 9 ✗",{"id":492,"type":72,"title":493,"eyebrow":494,"navLabel":495},"ch7","The properties of zero, and why you cannot divide by it","Chapter 07","7 Properties of zero",{"id":497,"type":202,"items":498},"formulas-zero",[499,502,505,508,511,514],{"expression":500,"caption":501},"a + 0 = a","Adding zero changes nothing (additive identity).",{"expression":503,"caption":504},"a − 0 = a","Subtracting zero changes nothing. a − a = 0.",{"expression":506,"caption":507},"a × 0 = 0 × a = 0","Zero property of multiplication.",{"expression":509,"caption":510},"0 ÷ a = 0 (a ≠ 0)","Zero shared among any number of people is zero each.",{"expression":512,"caption":513},"a ÷ 0 is undefined","No number times 0 gives a (when a ≠ 0).",{"expression":515,"caption":516},"0 ÷ 0 is undefined","Every number times 0 gives 0, so no single answer.",{"id":518,"type":43,"markdown":519},"divzero-prose","**Why is a × 0 = 0?** a × 0 means a groups of 0, or 0 added to itself a times: 0 + 0 + … + 0 = 0. And by the commutative property, 0 × a is the same.\n\n**Why is 0 ÷ a = 0?** Division asks \"what times a gives 0?\" The answer 0 works, because 0 × a = 0. And nothing else works: if q is not 0, then q × a is not 0 either. So 0 ÷ a = 0, exactly one answer.\n\n**Why is a ÷ 0 undefined?** Every division can be checked by multiplication: 20 ÷ 4 = 5 because 5 × 4 = 20. So 20 ÷ 0 = q would need **q × 0 = 20**. But q × 0 = 0 for every q. No number works. The question has **no answer**, so we call it **undefined**. It is not \"infinity\" and it is not \"0\"; it simply is not a number.\n\n**What about 0 ÷ 0?** Now we need q × 0 = 0. That is true for q = 0, q = 1, q = 7, q = 1,000 … **every** number works! A calculation that could equal anything is useless, so 0 ÷ 0 is also left undefined.",{"id":521,"type":81,"caption":522,"columns":523,"rows":527},"divzero-table","Checking divisions by multiplication",[143,524,525,526],"Needs a number q with","How many q work?","Verdict",[528,533,538,543],[529,530,531,532],"20 ÷ 4","q × 4 = 20","Exactly one: 5","20 ÷ 4 = 5",[534,535,536,537],"0 ÷ 4","q × 4 = 0","Exactly one: 0","0 ÷ 4 = 0",[539,540,541,542],"20 ÷ 0","q × 0 = 20","None","Undefined",[544,545,546,542],"0 ÷ 0","q × 0 = 0","Every number",{"id":548,"type":106,"variant":549,"title":550,"markdown":551},"aha-sharing-zero","aha","The sharing picture","Think of division as **repeated subtraction**: 20 ÷ 4 asks \"how many times can I take 4 away from 20?\" Take away 4 five times and you reach 0, so the answer is 5.\n\nNow 20 ÷ 0: \"how many times can I take 0 away from 20?\" You take away 0, still 20. Again, still 20. You could go on for ever and never reach 0. There is no count that finishes the job. That is division by zero.",{"id":553,"type":106,"variant":107,"title":554,"markdown":555},"nuance-zero-product","The zero-product rule","If a product is zero, at least one factor must be zero. If a × b = 0, then a = 0 or b = 0 (or both). For instance, if 17 × □ = 0, the box must be 0. This simple fact becomes the main tool for solving equations in algebra.",{"id":557,"type":112,"itemId":558,"prompt":559,"check":560,"hints":574,"feedback":576},"pr-zero-true","properties-of-numbers.understand-zero-true","Which statements are **true**? (Choose all that apply.)",{"kind":176,"options":561,"correct":573},[562,564,566,568,570],{"id":179,"label":563},"0 ÷ 25 = 0",{"id":181,"label":565},"25 ÷ 0 = 0",{"id":183,"label":567},"25 × 0 = 0",{"id":186,"label":569},"0 ÷ 0 = 1",{"id":571,"label":572},"e","25 − 25 = 0",[179,183,571],[575],"Check each division with multiplication.",{"correct":577,"incorrect":578},"Correct: 0 ÷ 25 = 0, 25 × 0 = 0 and 25 − 25 = 0. Division by 0 is undefined, including 0 ÷ 0.","25 ÷ 0 would need q × 0 = 25: impossible. 0 ÷ 0 would need q × 0 = 0: every q works, so it is undefined, not 1.",{"id":580,"type":72,"title":581,"eyebrow":582,"navLabel":583},"ch8","The properties of one","Chapter 08","8 Properties of one",{"id":585,"type":202,"items":586},"formulas-one",[587,590,593,596,599],{"expression":588,"caption":589},"a × 1 = 1 × a = a","Multiplicative identity.",{"expression":591,"caption":592},"a ÷ 1 = a","Dividing by 1 changes nothing.",{"expression":594,"caption":595},"a ÷ a = 1 (a ≠ 0)","Any non-zero number divided by itself is 1.",{"expression":597,"caption":598},"1 × 1 × 1 × … = 1","Multiplying 1 by itself any number of times is still 1.",{"expression":600,"caption":601},"a + 1 = successor of a","Adding 1 moves one step right on the number line.",{"id":603,"type":43,"markdown":604},"one-prose","The number 1 has a few more jobs:\n\n- **Building numbers.** Every natural number can be made by adding 1 again and again, starting from 1. That is what \"counting\" is.\n- **Neither prime nor composite.** A prime has exactly two factors; 1 has only one factor (itself). This is why 1 is left out of both lists in the [Prime and composite numbers](\u002Ftopics\u002Fprime-and-composite) topic.\n- **A factor of everything.** 1 divides every whole number exactly.\n- **Making equivalent forms.** Because multiplying by 1 changes nothing, and 1 can be written as 4 ÷ 4 or 100 ÷ 100, you can rewrite a number without changing its value. That is the secret behind equivalent fractions (½ = 2\u002F4) and changing units (1 m = 100 cm).",{"id":606,"type":106,"variant":168,"title":607,"markdown":608},"misc-one-zero","Mixing up 0 and 1","The additive identity is **0**, the multiplicative identity is **1**. Watch for these slips:\n\n- 8 × 1 = 9 ✗ (that is 8 + 1). Correct: 8 × 1 = 8.\n- 8 × 0 = 8 ✗ (that is 8 + 0). Correct: 8 × 0 = 0.\n- 8 ÷ 8 = 0 ✗ (that is 8 − 8). Correct: 8 ÷ 8 = 1.",{"id":610,"type":72,"title":611,"eyebrow":612,"navLabel":613},"ch9","Even and odd: rules and reasons","Chapter 09","9 Even and odd",{"id":615,"type":106,"variant":132,"title":616,"markdown":617},"def-parity","Even and odd","A whole number is **even** if it can be split into pairs with nothing left over; equivalently, it is 2 times some whole number (2 × k). A whole number is **odd** if one is left over after pairing; it is 2 × k + 1 for some whole number k.\n\nThe last digit tells you: even numbers end in 0, 2, 4, 6, 8; odd numbers end in 1, 3, 5, 7, 9. That works because 10 is even, so every ten pairs up perfectly and only the ones digit can leave a single over. **Zero is even** (0 = 2 × 0).",{"id":619,"type":81,"caption":620,"columns":621,"rows":626},"parity-full","All the even\u002Fodd rules for the four operations (for subtraction, the larger number first)",[622,623,624,625],"First","Second","Sum \u002F difference","Product",[627,631,635,638],[628,628,629,630],"even","even (8 + 4 = 12, 8 − 4 = 4)","even (8 × 4 = 32)",[628,632,633,634],"odd","odd (8 + 3 = 11, 8 − 3 = 5)","even (8 × 3 = 24)",[632,628,636,637],"odd (9 + 4 = 13, 9 − 4 = 5)","even (9 × 4 = 36)",[632,632,639,640],"even (9 + 3 = 12, 9 − 3 = 6)","odd (9 × 3 = 27)",{"id":642,"type":43,"markdown":643},"parity-why","**Why?** Think of each odd number as \"some pairs plus one spare\".\n\n- **odd + odd:** the two spares join to make a new pair, so nothing is left over: even.\n- **even + odd:** only one spare, so odd.\n- **even × anything:** a × b with b even means a groups, each made of pairs. All pairs: even.\n- **odd × odd:** 3 × 5 means 3 groups of (2 pairs + 1 spare). The pairs stay paired. The 3 spares are an odd number of spares: two of them pair up, one is left over. Odd.\n\n**Several numbers at once:** a sum of whole numbers is odd exactly when it contains an **odd number of odd numbers**. So 3 + 5 + 7 (three odds) is odd: 3 + 5 + 7 = 15. And 1 + 3 + 5 + 7 (four odds) is even: 1 + 3 + 5 + 7 = 16. A product is odd only if **every** factor is odd.",{"id":645,"type":276,"title":646,"problem":647,"steps":648},"we-parity-puzzle","Can it be done?","Can you choose **five odd numbers** that add up to **50**?",[649,650,651,652,653],"Pair up the odd numbers: odd + odd = even, and odd + odd = even again. That uses four of them, giving an even total.","The fifth odd number is left: even + odd = odd.","So the sum of five odd numbers is always **odd**.","50 is even. So it is **impossible**, whatever odd numbers you try.","This is a proof: it covers every possible choice without testing any of them.",{"id":655,"type":112,"itemId":656,"prompt":657,"check":658,"hints":660,"feedback":663},"pr-parity-count","properties-of-numbers.understand-parity-count","How many of these are odd? 13 × 7, 13 + 7, 28 × 5, 15 + 16, 99 × 99, 44 − 21",{"kind":116,"answer":659,"tolerance":118},4,[661,662],"Use the rules, not the calculations.","odd × odd is odd; a sum or difference of one odd and one even is odd.",{"correct":664,"incorrect":665},"Yes, four: 13 × 7 = 91, 15 + 16 = 31, 99 × 99 = 9,801 and 44 − 21 = 23. The even ones are 13 + 7 = 20 (odd + odd) and 28 × 5 = 140 (even factor).","Check each with the rules: 13 × 7 odd × odd → odd; 13 + 7 odd + odd → even; 28 × 5 has an even factor → even; 15 + 16 → odd; 99 × 99 → odd; 44 − 21 → odd. Four are odd.",{"id":667,"type":72,"title":668,"eyebrow":669,"navLabel":670},"ch10","Mental maths methods and the property behind each","Chapter 10","10 Mental maths",{"id":672,"type":81,"caption":673,"columns":674,"rows":677},"methods-table","Six mental maths methods, each one a property in disguise",[675,86,676],"Method","Property used",[678,682,686,690,694,698],[679,680,681],"Compensation (add)","198 + 57 = 200 + 55 = 255: move 2 from 57 to 198","Associative: 198 + 57 = 198 + (2 + 55) = (198 + 2) + 55",[683,684,685],"Compensation (multiply)","39 × 6 = 240 − 6 = 234","Distributive: (40 − 1) × 6",[687,688,689],"Doubling and halving","35 × 18 = 70 × 9 = 630","Associative: 35 × (2 × 9) = (35 × 2) × 9",[691,692,693],"Times 5","86 × 5 = 860 ÷ 2 = 430","5 = 10 ÷ 2",[695,696,697],"Times 25","36 × 25 = 3,600 ÷ 4 = 900","25 = 100 ÷ 4",[699,700,701],"Times 101","64 × 101 = 6,400 + 64 = 6,464","Distributive: 64 × (100 + 1)",{"id":703,"type":106,"variant":298,"title":704,"markdown":705},"careful-comp-sub","Compensation in subtraction goes the same way","For subtraction, change **both** numbers by the **same** amount in the **same** direction. 83 − 38: add 2 to both to get 85 − 40 = **45**. (On the number line, sliding both numbers keeps the gap the same.) Check: 83 − 38 = 45.\n\nIf instead you change only the second number, you must adjust the other way: 83 − 38 = 83 − 40 + 2 = 45. Subtracting 2 too many means you must **add** 2 back. That \"add back\" is where most mistakes happen.",{"id":707,"type":72,"title":708,"eyebrow":709,"navLabel":710},"ch11","Everything in one table","Chapter 11","11 Summary",{"id":712,"type":81,"caption":713,"columns":714,"rows":716},"master-table","Which properties hold for whole numbers? (✓ = always true, ✗ = fails for some numbers)",[715,140,141,142,143],"Property",[717,721,725,729,735],[133,718,719,718,720],"✓","✗ (3 − 5)","✗ (5 ÷ 2)",[722,718,723,718,724],"Commutative","✗ (5 − 3 vs 3 − 5)","✗ (6 ÷ 3 vs 3 ÷ 6)",[726,718,727,718,728],"Associative","✗ (8 − 3) − 2 vs 8 − (3 − 2)","✗ (8 ÷ 4) ÷ 2 vs 8 ÷ (4 ÷ 2)",[730,731,732,733,734],"Identity","✓ 0","✗ (only a − 0 = a)","✓ 1","✗ (only a ÷ 1 = a)",[736,737,738,739,740],"Distributive","× distributes over +","× distributes over −","links × with + and −","only from the right: (a + b) ÷ c",{"id":742,"type":106,"variant":107,"title":743,"markdown":744},"nuance-dmas","A note on \"DMAS\"","Some syllabus lists expand DMAS as \"Distributive, Multiplicative, Additive, Subtractive\". That is not right. **DMAS** (or BODMAS, BIDMAS, PEMDAS) names the **order of operations**: Division and Multiplication before Addition and Subtraction. The **distributive property** is a property of numbers, covered here; the order of operations is a convention for reading expressions, covered in [Order of operations](\u002Ftopics\u002Forder-of-operations). They are linked (the distributive property is why 6 × (10 + 4) = 6 × 10 + 6 × 4), but they are different ideas.",{"id":746,"type":303,"component":747,"componentVersion":5,"config":748,"objective":776,"textAlternative":777,"help":778},"lab-match-understand","match-pairs",{"prompt":749,"mode":750,"pairs":751},"Match each statement to the property it shows.","connect",[752,755,758,761,764,767,770,773],{"a":753,"b":754},"17 + 29 = 29 + 17","Commutative property of addition",{"a":756,"b":757},"(6 × 5) × 2 = 6 × (5 × 2)","Associative property of multiplication",{"a":759,"b":760},"9 × (10 − 1) = 90 − 9","Distributive property over subtraction",{"a":762,"b":763},"4,321 × 1 = 4,321","Multiplicative identity",{"a":765,"b":766},"0 + 58 = 58","Additive identity",{"a":768,"b":769},"999 × 0 = 0","Zero property of multiplication",{"a":771,"b":772},"12 + 19 is a whole number","Closure under addition",{"a":774,"b":775},"(40 + 8) ÷ 4 = 10 + 2","Division distributes from the right","Connect each number statement with the name of the property it demonstrates.","A matching game with eight statements and eight property names.\n\n17 + 29 = 29 + 17 → commutative property of addition (order swapped).\n(6 × 5) × 2 = 6 × (5 × 2) → associative property of multiplication (grouping changed).\n9 × (10 − 1) = 90 − 9 → distributive property over subtraction.\n4,321 × 1 = 4,321 → multiplicative identity.\n0 + 58 = 58 → additive identity.\n999 × 0 = 0 → zero property of multiplication.\n12 + 19 is a whole number → closure under addition.\n(40 + 8) ÷ 4 = 10 + 2 → division distributes from the right (both parts are divided by 4).",{"hints":779},[780,781],"Swapped order → commutative. Moved brackets → associative.","Two operations in one statement → distributive.",{"id":783,"type":303,"component":784,"componentVersion":5,"config":785,"objective":838,"textAlternative":839,"help":840},"lab-sort-understand","sort-game",{"prompt":786,"bins":787,"items":794,"seconds":118},"Is each statement true for ALL whole numbers, or false for at least one?",[788,791],{"id":789,"label":790},"always","Always true",{"id":792,"label":793},"not","Not always true",[795,798,802,805,809,811,815,818,822,826,830,834],{"id":796,"label":205,"bin":789,"why":797},"s1","Commutative property of addition.",{"id":799,"label":800,"bin":792,"why":801},"s2","a − b = b − a","Counterexample: 5 − 3 = 2 but 3 − 5 is not 2.",{"id":803,"label":329,"bin":789,"why":804},"s3","Distributive property.",{"id":806,"label":807,"bin":792,"why":808},"s4","a × (b × c) = (a × b) × (a × c)","Counterexample: 2 × (3 × 4) = 24 but 6 × 8 = 48.",{"id":810,"label":591,"bin":789,"why":592},"s5",{"id":812,"label":813,"bin":792,"why":814},"s6","1 ÷ a = a","Counterexample: 1 ÷ 2 = ½, not 2.",{"id":816,"label":817,"bin":789,"why":507},"s7","a × 0 = 0",{"id":819,"label":820,"bin":792,"why":821},"s8","a ÷ a = 1","Fails for a = 0: 0 ÷ 0 is undefined.",{"id":823,"label":824,"bin":792,"why":825},"s9","(a − b) − c = a − (b − c)","Counterexample: (8 − 3) − 2 = 3 but 8 − (3 − 2) = 7.",{"id":827,"label":828,"bin":789,"why":829},"s10","a + b is a whole number","Whole numbers are closed under addition.",{"id":831,"label":832,"bin":792,"why":833},"s11","a ÷ b is a whole number","Counterexample: 7 ÷ 2 = 3½.",{"id":835,"label":836,"bin":789,"why":837},"s12","(a + b) ÷ c = a ÷ c + b ÷ c (c ≠ 0)","Division distributes from the right (answers may be fractions, but the two sides are equal).","Decide whether each general statement with letters holds for every whole number, and find a counterexample for those that do not.","A sorting game with two bins: \"Always true\" and \"Not always true\".\n\nAlways true: a + b = b + a; a × (b + c) = a × b + a × c; a ÷ 1 = a; a × 0 = 0; a + b is a whole number; (a + b) ÷ c = a ÷ c + b ÷ c when c is not 0.\n\nNot always true, each with a counterexample: a − b = b − a (5 − 3 vs 3 − 5); a × (b × c) = (a × b) × (a × c) (24 vs 48 for 2, 3, 4); 1 ÷ a = a (1 ÷ 2 = ½); a ÷ a = 1 (fails for a = 0); (a − b) − c = a − (b − c) (3 vs 7 for 8, 3, 2); a ÷ b is whole (7 ÷ 2 = 3½).",{"hints":841},[842,843],"Try small numbers such as 2, 3 and 4, and do not forget 0.","One failure is enough for \"Not always true\".",{"id":845,"type":846,"title":847,"terms":848},"glossary-understand","glossary","Precise vocabulary",[849,851,853,857,860,863,866,869,871,873,876,879,882,886,890,894,898,902],{"term":146,"meaning":850},"The counting numbers 1, 2, 3, … (NCERT does not include 0).",{"term":151,"meaning":852},"The numbers 0, 1, 2, 3, … : the natural numbers together with 0.",{"term":854,"meaning":855,"example":856},"Closed (under an operation)","A set is closed under an operation if combining any two of its members always gives a member of the same set.","Whole numbers are closed under +; not under −.",{"term":722,"meaning":858,"example":859},"An operation is commutative if the order of the two numbers does not change the result: a ∗ b = b ∗ a for all a, b.","6 × 7 = 7 × 6",{"term":726,"meaning":861,"example":862},"An operation is associative if the grouping does not change the result: (a ∗ b) ∗ c = a ∗ (b ∗ c).","(2 + 3) + 4 = 2 + (3 + 4)",{"term":736,"meaning":864,"example":865},"Multiplication distributes over addition: a × (b + c) = a × b + a × c. It also distributes over subtraction.","4 × 27 = 4 × 20 + 4 × 7",{"term":460,"meaning":867,"example":868},"A number that leaves every number unchanged under an operation, on either side.","0 for +, 1 for ×",{"term":766,"meaning":870},"Zero: a + 0 = 0 + a = a.",{"term":763,"meaning":872},"One: a × 1 = 1 × a = a.",{"term":769,"meaning":874,"example":875},"Any number multiplied by 0 gives 0.","4,321 × 0 = 0",{"term":542,"meaning":877,"example":878},"An expression with no meaningful value. Division by zero is undefined.","7 ÷ 0",{"term":880,"meaning":881},"Indeterminate","Used for 0 ÷ 0: many values would fit, so none can be chosen.",{"term":883,"meaning":884,"example":885},"Counterexample","A single case showing a general statement is false.","(8 − 3) − 2 ≠ 8 − (3 − 2)",{"term":887,"meaning":888,"example":889},"Common factor","A number that multiplies every term in a sum, which can be taken outside a bracket.","37 × 68 + 37 × 32 = 37 × 100",{"term":891,"meaning":892,"example":893},"Compensation","Adjusting a number to make it friendly, then correcting the answer.","39 × 6 = 40 × 6 − 6",{"term":895,"meaning":896,"example":897},"Parity","Whether a whole number is even or odd.","17 has odd parity.",{"term":899,"meaning":900,"example":901},"Area model","Picturing a product as a rectangle and cutting it into smaller rectangles.","23 × 14 = 200 + 30 + 80 + 12",{"term":903,"meaning":904},"Inverse operation","An operation that undoes another: subtraction undoes addition, division undoes multiplication.",{"id":906,"type":907,"title":908,"questions":909},"quiz-understand","quiz","Check your understanding",[910,923,936,949,959,972,985,998,1011,1024,1037],{"itemId":911,"prompt":912,"options":913,"correct":181,"why":922},"properties-of-numbers.understand-q-comm","Which equation shows the commutative property of multiplication?",[914,916,918,920],{"id":179,"label":915},"3 × (4 × 5) = (3 × 4) × 5",{"id":181,"label":917},"3 × 4 = 4 × 3",{"id":183,"label":919},"3 × (4 + 5) = 12 + 15",{"id":186,"label":921},"3 × 1 = 3","Commutative means the order of the two numbers is swapped.",{"itemId":924,"prompt":925,"options":926,"correct":179,"why":935},"properties-of-numbers.understand-q-assoc","Which equation shows the associative property of addition?",[927,929,931,933],{"id":179,"label":928},"(7 + 8) + 2 = 7 + (8 + 2)",{"id":181,"label":930},"7 + 8 = 8 + 7",{"id":183,"label":932},"7 × (8 + 2) = 56 + 14",{"id":186,"label":934},"7 + 0 = 7","Associative: same order, different grouping.",{"itemId":937,"prompt":938,"options":939,"correct":181,"why":948},"properties-of-numbers.understand-q-dist","15 × 102 equals:",[940,942,944,946],{"id":179,"label":941},"1,500 + 2",{"id":181,"label":943},"1,500 + 30",{"id":183,"label":945},"1,530 + 102",{"id":186,"label":947},"150 + 30","15 × 102 = 15 × 100 + 15 × 2 = 1,500 + 30 = 1,530.",{"itemId":950,"prompt":951,"options":952,"correct":183,"why":958},"properties-of-numbers.understand-q-closure","Which operation are whole numbers closed under?",[953,954,955,956],{"id":179,"label":141},{"id":181,"label":143},{"id":183,"label":142},{"id":186,"label":957},"None of them","Product of two whole numbers is always whole. Subtraction (3 − 5) and division (5 ÷ 2) can leave the set.",{"itemId":960,"prompt":961,"options":962,"correct":181,"why":971},"properties-of-numbers.understand-q-divzero","Why is 9 ÷ 0 undefined?",[963,965,967,969],{"id":179,"label":964},"Because 9 ÷ 0 = 0",{"id":181,"label":966},"Because no number q gives q × 0 = 9",{"id":183,"label":968},"Because 9 is odd",{"id":186,"label":970},"Because it equals 9","A division is correct when the answer times the divisor gives the dividend. Every q × 0 = 0, never 9.",{"itemId":973,"prompt":974,"options":975,"correct":183,"why":984},"properties-of-numbers.understand-q-00","What is the problem with 0 ÷ 0?",[976,978,980,982],{"id":179,"label":977},"It equals 0",{"id":181,"label":979},"It equals 1",{"id":183,"label":981},"Every number q satisfies q × 0 = 0, so there is no single answer",{"id":186,"label":983},"There is no problem","A calculation with every number as a possible answer has no definite value, so it is left undefined.",{"itemId":986,"prompt":987,"options":988,"correct":181,"why":997},"properties-of-numbers.understand-q-identity","Why is 0 not a true identity for subtraction?",[989,991,993,995],{"id":179,"label":990},"Because 7 − 0 is not 7",{"id":181,"label":992},"Because 0 − 7 is not 7",{"id":183,"label":994},"Because 0 is even",{"id":186,"label":996},"It is a true identity","An identity must work on both sides. 7 − 0 = 7, but 0 − 7 is not 7.",{"itemId":999,"prompt":1000,"options":1001,"correct":181,"why":1010},"properties-of-numbers.understand-q-parity","The sum of 7 odd numbers is:",[1002,1004,1006,1008],{"id":179,"label":1003},"always even",{"id":181,"label":1005},"always odd",{"id":183,"label":1007},"sometimes even",{"id":186,"label":1009},"always 7","Six odds pair up into an even total, and the seventh odd makes it odd. An odd count of odd numbers gives an odd sum.",{"itemId":1012,"prompt":1013,"options":1014,"correct":179,"why":1023},"properties-of-numbers.understand-q-common","46 × 57 + 46 × 43 equals:",[1015,1017,1019,1021],{"id":179,"label":1016},"4,600",{"id":181,"label":1018},"2,622",{"id":183,"label":1020},"46 × 57 × 43",{"id":186,"label":1022},"460","Take out the common factor 46: 46 × (57 + 43) = 46 × 100 = 4,600.",{"itemId":1025,"prompt":1026,"options":1027,"correct":181,"why":1036},"properties-of-numbers.understand-q-assocdiv","What is (48 ÷ 6) ÷ 2 compared with 48 ÷ (6 ÷ 2)?",[1028,1030,1032,1034],{"id":179,"label":1029},"Both 4",{"id":181,"label":1031},"4 and 16",{"id":183,"label":1033},"16 and 4",{"id":186,"label":1035},"Both 16","(48 ÷ 6) ÷ 2 = 8 ÷ 2 = 4; 48 ÷ (6 ÷ 2) = 48 ÷ 3 = 16. Division is not associative.",{"itemId":1038,"prompt":1039,"options":1040,"correct":183,"why":1049},"properties-of-numbers.understand-q-mistake","Kabir wrote 7 × (30 + 4) = 210 + 4 = 214. What went wrong?",[1041,1043,1045,1047],{"id":179,"label":1042},"Nothing",{"id":181,"label":1044},"He should have added first",{"id":183,"label":1046},"He did not multiply the 4 by 7",{"id":186,"label":1048},"He should have multiplied 30 by 4","The 7 multiplies both parts: 210 + 28 = 238.",{"id":1051,"type":1052,"prompt":1053},"reflect-understand","reflection","In your own words, explain the difference between the commutative and associative properties to a younger child, using one example with chairs or sweets for each. Then explain why neither works for subtraction.",{"id":1055,"type":1056,"title":1057,"points":1058},"cheat-understand","summary","Cheat sheet",[1059,1060,1061,1062,1063,1064,1065,1066,1067,1068],"**Whole numbers** W = 0, 1, 2, … ; natural numbers start at 1. No largest number; successor n + 1; 0 has no whole predecessor.","**Closure:** W is closed under + and ×, not under − (3 − 5) or ÷ (5 ÷ 2). One counterexample shows \"not closed\".","**Commutative:** a + b = b + a, a × b = b × a. Fails for − and ÷.","**Associative:** (a + b) + c = a + (b + c), (a × b) × c = a × (b × c). Fails for − and ÷, so those go left to right.","**Distributive:** a × (b + c) = ab + ac and a × (b − c) = ab − ac. It is why long multiplication works. Division distributes only from the right.","**Identities:** 0 for addition, 1 for multiplication, on both sides. Subtraction and division have none.","**Zero:** a × 0 = 0; 0 ÷ a = 0; a ÷ 0 and 0 ÷ 0 are undefined. If a × b = 0 then a or b is 0.","**One:** a × 1 = a; a ÷ 1 = a; a ÷ a = 1 (a ≠ 0); 1 is neither prime nor composite.","**Parity:** same + same = even; mixed = odd; product odd only if all factors odd. Sum of n odd numbers is odd when n is odd.","**Mental maths:** 98 × 25 = 2,500 − 50; 12 × 105 = 1,200 + 60; 64 × 101 = 6,400 + 64; ×25 = ×100 ÷ 4.",{"id":1070,"type":1071,"conceptId":1072,"relation":1073,"explanation":1074},"conn-order","connection","order-of-operations","related_to","Order of operations is the agreed reading of expressions like 50 − 20 − 10. It is needed precisely because − and ÷ are not associative.",{"id":1076,"type":1071,"conceptId":1077,"relation":1078,"explanation":1079},"conn-four","four-operations","helps_understand","Long multiplication, short division and checking answers all rely on the distributive, commutative and associative properties.",{"id":1081,"type":1071,"conceptId":1082,"relation":1073,"explanation":1083},"conn-prime","prime-and-composite","The number 1 is neither prime nor composite, and even\u002Fodd rules tell you that 2 is the only even prime.",{"id":1085,"type":1071,"conceptId":1086,"relation":1073,"explanation":1087},"conn-hcf","hcf-and-lcm","Taking out a common factor, as in 37 × 68 + 37 × 32 = 37 × 100, is the distributive property; HCF finds the biggest such factor.",{"id":1089,"type":1090,"sourceIds":1091},"sources-understand","sources",[1092,1093,1094,1095,1096,1097,1098,1099,1100,1101],"properties-of-numbers-ncert-class6-whole-numbers","properties-of-numbers-ncert-class7-integers","properties-of-numbers-ncert-class8-rational-numbers","properties-of-numbers-mathsisfun-properties","properties-of-numbers-mathsisfun-divide-by-zero","properties-of-numbers-wiki-commutative","properties-of-numbers-wiki-distributive","properties-of-numbers-wiki-division-by-zero","properties-of-numbers-wiki-parity","properties-of-numbers-wiki-brahmagupta",[1092,1093,1094,1095,1096,1097,1098,1099,1100,1101],"needs_review",{"generatedBy":1105,"notes":1106},"claude-code","Draft generated with Python generators; every stated number computed and asserted. Pending owner review.","c22246dc8f54c45b312f563d4202157bd3ad99fbba4efe7f202f9ab402ecdf0b",{"logic:practice":1109,"component:order-ops@1":1110,"component:arith-sprint@1":1111,"component:match-pairs@1":1112,"component:sort-game@1":1113,"source:properties-of-numbers-mathsisfun-divide-by-zero":1114,"source:properties-of-numbers-mathsisfun-properties":1115,"source:properties-of-numbers-ncert-class6-whole-numbers":1116,"source:properties-of-numbers-ncert-class7-integers":1117,"source:properties-of-numbers-ncert-class8-rational-numbers":1118,"source:properties-of-numbers-wiki-brahmagupta":1119,"source:properties-of-numbers-wiki-commutative":1120,"source:properties-of-numbers-wiki-distributive":1121,"source:properties-of-numbers-wiki-division-by-zero":1122,"source:properties-of-numbers-wiki-parity":1123},"3d6b0fe1b15255975a32b0fcd94e8019bc959ad45cbf12e136e86149549c6878","3d630ebc066352df791b020d545e1876ce4620d55c8c1c082414d6b4717255c5","c0c63ed40e1bca5ba6d446d43886b887a3a7709e5cd6679419f64fd3ba62afe6","2a8ee4ac87460b4e1175a4bb13c96b03d577db06dde95670eb7fcfe4ad787899","b164f45a2c8ca08f26c450768ff0231e113e9fe45381eddb34dc6d0548596c38","f8a42fd8c82c266f4310450d74f1ae0449b3dba177afcf000b24aafa2839b629","a37b85a36742e06f34cdc083710eacdd662e2f7493395766e4b8c5632bfade3a","56b97dbf2988a58e1aa10f693ec9dd1039e1de14e795b1bdfa19980b6b18c1fa","aab8fe53a32660ef6307b4dbd6f5af02319957d9aecf3c0396895bc01ab46e2e","acb8b92cad8387724d72528e70ef68195351e5b2f03b580c5da191fde82dc67a","32e7279091a5877afe54d6da75cdd06a04f2ff8677338777578592d6dd98a338","a3e0fb47a2115f5962c9795b138bf38cd9c2c4874bc66e1b834717b9a01908c1","befe26bd7ec4e1c3911549d33a91844ae3eb5bd63388711f61822d398d3d576f","046256cc76cdb1b7cca72a1b391bef686855afdfb6a68d18b45a38723d1fe93d","01c5bcf03886aec6f9c2c18c9ec5333b19ce1d2ca6bfb4f8544c4ca9f1578e27",{"state":1125,"reviewer":1126,"selfReview":313,"reviewedAt":1127,"method":1128},"approved","The library owner","2026-09-20T10:18:37.581Z","owner_bulk","preview-7e1cbbcc4f",1789899597123]