[{"data":1,"prerenderedAt":1068},["ShallowReactive",2],{"layer:properties-of-numbers:discover":3},{"layer":4,"contentHash":1045,"dependencyHashes":1046,"approval":1061,"releaseId":1067},{"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":1040,"reviewStatus":1041,"authoring":1042},1,"properties-of-numbers","en","discover","Rules that numbers always follow","Turn-around facts, friendly groups, breaking apart and the magic of 0 and 1","Meet the properties of numbers through chairs, laddoos, kirana bills and socks: why 4 × 6 = 6 × 4, why you can add in any order, how breaking numbers apart makes sums easy, and what 0 and 1 do.",[13,14,15,16,17],"Tell natural numbers from whole numbers and place them on a number line.","Use turn-around facts and friendly grouping to add and multiply faster, and explain why they fail for − and ÷.","Break a number apart to multiply in your head, like 6 × 14 = 60 + 24.","Describe what 0 and 1 do in each operation, and why you cannot divide by zero.","Predict whether a sum or product is even or odd without calculating it.",30,{"title":20,"rows":21},"Lesson plate",[22,25,28,31,34,37],{"label":23,"value":24},"Depth","Discover",{"label":26,"value":27},"Reading time","≈ 30 minutes",{"label":29,"value":30},"Prior knowledge","Adding, subtracting and times tables",{"label":32,"value":33},"Chapters","10",{"label":35,"value":36},"Labs","Turn-around match, rule sort, even–odd sort, sprint",{"label":38,"value":39},"Big idea","Numbers obey rules you can use",[41,45,51,72,78,81,105,108,123,128,131,153,156,161,176,202,207,240,245,248,269,272,277,289,294,297,314,317,330,334,375,380,383,414,417,422,436,441,444,467,471,492,497,500,525,528,542,599,604,607,622,626,631,634,656,667,679,684,689,725,796,865,989,993,1007,1013,1018,1022,1026],{"id":42,"type":43,"markdown":44},"intro-chairs","prose","Your school is holding a function in the hall, and your class has to set out **24 chairs**. Riya puts them in **4 rows of 6**. Arjun says, \"It would look better as **6 rows of 4**.\" They argue for five minutes about which way needs more chairs.\n\nYou already know the answer, don't you? It is **24 chairs either way**. Turning the whole arrangement sideways does not add or remove a single chair: 4 × 6 = 24 and 6 × 4 = 24.\n\nThat tiny fact is an example of a **property of numbers**: a rule that is true not just for 4 and 6, but for *every* pair of numbers you could ever pick. This lesson is about those rules. They are not extra things to memorise. They are the reasons your mental maths already works, and once you can see them, hard sums start to look easy.",{"id":46,"type":47,"variant":48,"title":49,"markdown":50},"intro-how","callout","observation","How to use this lesson","Each chapter meets one rule through everyday things: chairs, sweets, shopping at a kirana shop, cricket scores. Whenever you see a **prediction**, choose an answer *before* reading on. Whenever you see a **lab**, play a few rounds: the games are where the rules turn into speed.",{"id":52,"type":53,"tone":54,"items":55},"intro-spec","spec","amber",[56,60,64,68],{"label":57,"big":58,"value":59},"Big question","Does order matter?","When you add or multiply, you can swap numbers around. When you subtract or divide, you usually cannot.",{"label":61,"big":62,"value":63},"Two special numbers","0 and 1","Adding 0 or multiplying by 1 changes nothing. Multiplying by 0 wipes everything out.",{"label":65,"big":66,"value":67},"Breaking apart","6 × 14","Split 14 into 10 and 4: 60 + 24 = 84. Big sums become small ones.",{"label":69,"big":70,"value":71},"Even and odd","pairs","Whether a number splits into pairs decides what happens when you add or multiply.",{"id":73,"type":74,"title":75,"eyebrow":76,"navLabel":77},"ch1","chapter","Counting numbers and whole numbers","Chapter 01","1 Number families",{"id":79,"type":43,"markdown":80},"natural-prose","Long before anyone wrote numbers down, people counted: one goat, two goats, three goats. The numbers we count with, **1, 2, 3, 4, 5, …**, are called **natural numbers** (or counting numbers). The dots mean *they never stop*. However big a number you think of, you can always add 1 and get a bigger one.\n\nNow think about an empty biscuit tin. How many biscuits are in it? None. We need a number for \"nothing\", and that number is **0**. When we put 0 together with all the counting numbers, we get the **whole numbers**: **0, 1, 2, 3, 4, …**\n\nSo every natural number is a whole number, but one whole number, 0, is not a natural number.",{"id":82,"type":83,"caption":84,"columns":85,"rows":92},"number-line-table","table","The start of the number line: each whole number sits one step to the right of the one before it",[86,87,88,89,90,91],"Number","0","1","2","3","4",[93,96,99,102],[94,95,95,95,95,95],"Is it a whole number?","Yes",[97,98,95,95,95,95],"Is it a natural number?","No",[100,101,87,88,89,90],"Number just before it","none (among whole numbers)",[103,88,89,90,91,104],"Number just after it","5",{"id":106,"type":43,"markdown":107},"number-line-prose","Picture the whole numbers as marks on a long straight road, equally spaced, starting at 0 and running off to the right for ever. This is the **number line**.\n\n- Moving **right** means getting bigger. Adding 3 means *jump 3 steps right*.\n- Moving **left** means getting smaller. Subtracting 3 means *jump 3 steps left*.\n- Multiplying 4 × 3 means *make 4 jumps of 3 steps each*, starting at 0: you land on 12.\n\nThe number just after a number is its **successor** (the successor of 99 is 100). The number just before is its **predecessor** (the predecessor of 100 is 99). Every whole number has a successor, but 0 has no whole-number predecessor: there is nothing to its left on our road. (In the Extend layer you will see what lives to the left of 0.)",{"id":109,"type":110,"itemId":111,"prompt":112,"check":113,"hints":117,"feedback":120},"pr-successor","practice","properties-of-numbers.discover-successor","What is the successor of 9,999?",{"kind":114,"answer":115,"tolerance":116},"number",10000,0,[118,119],"The successor is the number that comes just after.","Add 1.",{"correct":121,"incorrect":122},"Yes: 9,999 + 1 = 10,000.","The successor is one more: 9,999 + 1 = 10,000.",{"id":124,"type":74,"title":125,"eyebrow":126,"navLabel":127},"ch2","Turn-around facts: order does not matter for + and ×","Chapter 02","2 Turn-around facts",{"id":129,"type":43,"markdown":130},"laddoo-prose","Amma puts **5 laddoos** on one plate and **3 laddoos** on another. How many laddoos altogether? 5 + 3 = 8. Now start with the second plate: 3 + 5 = 8. Of course. The laddoos are the same laddoos; you only counted them in a different order.\n\nMathematicians call this the **commutative property of addition**. \"Commute\" means to travel back and forth, like a daily commute: the numbers can swap places and the answer stays the same. Teachers often call these **turn-around facts**.\n\nIt works for big numbers too: 348 + 57 = 405 and 57 + 348 = 405. And it gives you a free trick. To add 2 + 69, turn it around and count on from the bigger number: 69, then 70, 71. Much faster than counting on 69 steps from 2!",{"id":132,"type":83,"caption":133,"columns":134,"rows":138},"chairs-grid","Riya's 4 rows of 6 chairs, drawn with ● for each chair. Turn the page sideways and you see 6 rows of 4",[135,136,137],"Row","Chairs","Count",[139,143,145,147,149],[140,141,142],"Row 1","● ● ● ● ● ●","6",[144,141,142],"Row 2",[146,141,142],"Row 3",[148,141,142],"Row 4",[150,151,152],"Total","4 rows of 6","4 × 6 = 24",{"id":154,"type":43,"markdown":155},"array-prose","Look at the chairs again, but this time read *down* the columns instead of along the rows. There are **6 columns**, and each column has **4 chairs**. So the same picture shows 6 × 4 as well as 4 × 6. One picture, two multiplications, the same 24 chairs.\n\nAn arrangement of objects in neat rows and columns is called an **array**. Arrays are everywhere: eggs in a tray (5 rows of 6 = 30), windows on a building, seats in a cinema, stamps on a sheet. Every array is a picture of the **commutative property of multiplication**.\n\nThis is why you only need to learn *half* of the times tables. If you know 7 × 8 = 56, you already know 8 × 7 = 56.",{"id":157,"type":47,"variant":158,"title":159,"markdown":160},"aha-half-tables","aha","Half the tables for free","A 10 × 10 multiplication grid has 100 facts. But thanks to turn-around facts, 3 × 9 and 9 × 3 are the same fact. Only the 10 \"square\" facts (1 × 1, 2 × 2, … 10 × 10) have no partner. So instead of 100 facts you really need 10 + 90 ÷ 2 = **55** facts. Turn-around thinking cuts your homework in half.",{"id":162,"type":163,"prompt":164,"options":165,"explanation":175},"predict-subtract-order","prediction","Meena has ₹10 and spends ₹4. Can you swap the numbers in **10 − 4** and get the same answer?",[166,169,172],{"id":167,"label":168},"a","Yes, 10 − 4 and 4 − 10 give the same answer",{"id":170,"label":171},"b","No, 4 − 10 is a different problem, and with whole numbers you cannot even do it",{"id":173,"label":174},"c","Only if the numbers are both even","**No.** 10 − 4 = 6: Meena has ₹6 left. But 4 − 10 means \"have ₹4, spend ₹10\", which you cannot do with whole numbers; you would be ₹6 short. Order matters for subtraction.\n\nDivision is the same. 12 ÷ 3 = 4 (12 sweets shared by 3 children, 4 each). But 3 ÷ 12 means 3 sweets shared by 12 children: nobody gets a whole sweet. So **turn-around facts only work for adding and multiplying.**",{"id":177,"type":83,"caption":178,"columns":179,"rows":184},"order-table","Does swapping the two numbers keep the answer the same?",[180,181,182,183],"Operation","Try it","Swapped","Same answer?",[185,190,194,198],[186,187,188,189],"Addition","7 + 2 = 9","2 + 7 = 9","Yes, always",[191,192,193,189],"Multiplication","7 × 2 = 14","2 × 7 = 14",[195,196,197,98],"Subtraction","7 − 2 = 5","2 − 7: not a whole number",[199,200,201,98],"Division","8 ÷ 2 = 4","2 ÷ 8: not a whole number",{"id":203,"type":47,"variant":204,"title":205,"markdown":206},"misc-subtract-swap","misconception","\"Take the smaller from the bigger\"","Many learners working on 52 − 38 look at the ones column, see 2 − 8, and just do 8 − 2 = 6 instead, getting 26. That is *swapping*, and subtraction does not allow it. The correct way is to regroup (borrow): 12 − 8 = 4 in the ones, 4 − 3 = 1 in the tens, so 52 − 38 = **14**. Checking by adding back is a great habit: 14 + 38 = 52. ✓",{"id":208,"type":209,"component":210,"componentVersion":5,"config":211,"objective":233,"textAlternative":234,"help":235},"lab-turnaround-match","interactive","match-pairs",{"prompt":212,"mode":213,"pairs":214},"Match each fact to its turn-around partner, or to the reason it has no partner.","connect",[215,218,221,224,227,230],{"a":216,"b":217},"6 + 9 = 15","9 + 6 = 15",{"a":219,"b":220},"7 × 8 = 56","8 × 7 = 56",{"a":222,"b":223},"25 + 75 = 100","75 + 25 = 100",{"a":225,"b":226},"4 rows of 6 chairs","6 rows of 4 chairs",{"a":228,"b":229},"12 × 5 = 60","5 × 12 = 60",{"a":231,"b":232},"9 − 3 = 6","No partner: 3 − 9 is different","Connect each addition or multiplication fact with its turn-around fact, and spot the subtraction that cannot be turned around.","A matching game with six cards on the left and six on the right. Each left card is a fact; the right cards are the facts with the numbers swapped.\n\n6 + 9 = 15 matches 9 + 6 = 15. 7 × 8 = 56 matches 8 × 7 = 56. 25 + 75 = 100 matches 75 + 25 = 100. \"4 rows of 6 chairs\" matches \"6 rows of 4 chairs\": both are 24 chairs. 12 × 5 = 60 matches 5 × 12 = 60.\n\nThe odd one out is 9 − 3 = 6. Its right-hand card says \"No partner: 3 − 9 is different\", because subtraction cannot be turned around. The game shows that swapping works for + and × only.",{"simplerExplanation":236,"hints":237},"Look for the card with the same two numbers in the opposite order and the same answer.",[238,239],"Adding and multiplying can be turned around.","One card is a subtraction. Can that be turned around?",{"id":241,"type":74,"title":242,"eyebrow":243,"navLabel":244},"ch3","Grouping: which pair should you do first?","Chapter 03","3 Grouping",{"id":246,"type":43,"markdown":247},"kirana-prose","At the kirana shop you buy rice for **₹38**, dal for **₹45** and oil for **₹55**. The shopkeeper adds them up in his head in a flash. How?\n\nDoing it in the written order, you would do 38 + 45 = 83, then 83 + 55 = 138. That works, but the numbers are awkward. The shopkeeper spots that **45 and 55 make 100**, so he does that pair first: 45 + 55 = 100, then 100 + 38 = 138. Same answer, far less effort.\n\nWhen you add three numbers, you can choose which two to add first. We show the pair we do first with brackets: (38 + 45) + 55 and 38 + (45 + 55) both give 138. This is the **associative property of addition**. To \"associate\" means to keep company with: you choose which numbers keep company first.",{"id":249,"type":250,"title":251,"items":252},"steps-friendly","steps","The friendly-pairs method for adding a long list",[253,257,261,265],{"title":254,"tag":255,"text":256},"Scan the list","12 + 7 + 8 + 13 + 20","Do not start adding yet. Look for pairs that make 10, 20, 50 or 100.",{"title":258,"tag":259,"text":260},"Pair up","12 + 8, 7 + 13","Here 12 + 8 = 20 and 7 + 13 = 20. Turn-around and grouping let you pair any two numbers in the list.",{"title":262,"tag":263,"text":264},"Add the friendly totals","20 + 20 + 20","The two pairs and the leftover 20 give three 20s.",{"title":266,"tag":267,"text":268},"Finish","= 60","12 + 7 + 8 + 13 + 20 = 60. Check by adding in the written order: 19, 27, 40, 60. ✓",{"id":270,"type":43,"markdown":271},"assoc-mult-prose","The same trick works for multiplication. Try **2 × 7 × 5**. In the written order, 2 × 7 = 14, then 14 × 5 = 70. But if you first multiply 2 × 5 = 10, then 10 × 7 = 70, it is almost instant.\n\nA cricket example: at a practice net there are **3 bowlers**, each bowls **5 overs**, and each over is **6 balls**. Total balls = 3 × 5 × 6. Group (3 × 5) × 6 = 15 × 6 = 90, or 3 × (5 × 6) = 3 × 30 = 90. Both give **90 balls**.",{"id":273,"type":47,"variant":274,"title":275,"markdown":276},"careful-assoc-sub","careful","Grouping does not work for subtraction","Try (20 − 8) − 2. First 20 − 8 = 12, then 12 − 2 = **10**.\n\nNow group differently: 20 − (8 − 2). First 8 − 2 = 6, then 20 − 6 = **14**.\n\nDifferent answers! Subtraction is **not** associative. Division is not either: (24 ÷ 4) ÷ 2 = 6 ÷ 2 = **3**, but 24 ÷ (4 ÷ 2) = 24 ÷ 2 = **12**. That is why, with subtraction and division, you must work **left to right** unless brackets tell you otherwise.",{"id":278,"type":110,"itemId":279,"prompt":280,"check":281,"hints":283,"feedback":286},"pr-friendly-add","properties-of-numbers.discover-friendly-add","Use friendly pairs to add: **27 + 36 + 73 + 64**.",{"kind":114,"answer":282,"tolerance":116},200,[284,285],"Which number goes with 27 to make 100?","27 + 73 = 100 and 36 + 64 = 100.",{"correct":287,"incorrect":288},"Yes: (27 + 73) + (36 + 64) = 100 + 100 = 200.","Pair 27 with 73 (that is 100) and 36 with 64 (another 100). Total 200.",{"id":290,"type":74,"title":291,"eyebrow":292,"navLabel":293},"ch4","Breaking apart: the distributive property","Chapter 04","4 Breaking apart",{"id":295,"type":43,"markdown":296},"garden-prose","A school garden has **6 rows** of plants with **14 plants** in each row. How many plants?\n\n14 is a bit big to multiply in your head, so **break it apart** into 10 and 4. Think of the garden as two smaller gardens side by side: a part with 6 rows of 10, and a part with 6 rows of 4.\n\n- 6 rows of 10 = 60 plants.\n- 6 rows of 4 = 24 plants.\n- Together: 60 + 24 = 84 plants.\n\nSo 6 × 14 = 6 × 10 + 6 × 4 = 84. This is the **distributive property**. \"Distribute\" means to hand out, like distributing sweets: the 6 is handed out to *both* parts, the 10 and the 4.",{"id":298,"type":83,"caption":299,"columns":300,"rows":305},"garden-table","The garden of 6 rows × 14 plants split into two beds: 10 plants wide and 4 plants wide",[301,302,303,304],"Part","10 plants per row","4 plants per row","Row total",[306,309],[307,33,91,308],"Each of the 6 rows","14",[310,311,312,313],"All 6 rows","6 × 10 = 60","6 × 4 = 24","60 + 24 = 84",{"id":315,"type":43,"markdown":316},"break-more-prose","You can break a number apart in any way that helps. For **7 × 12**, split 12 into 10 + 2: 7 × 10 + 7 × 2 = 70 + 14 = 84. For **8 × 25**, split 25 into 20 + 5: 8 × 20 + 8 × 5 = 160 + 40 = 200.\n\nIt even works with subtraction. **9 × 19** is awkward, but 19 is just 20 − 1. Nine twenties are 180, and then take away one 9: 180 − 9 = 171. So 9 × 19 = 9 × 20 − 9 × 1 = 171. Shopkeepers use this all the time with prices like ₹99 or ₹199.",{"id":318,"type":319,"title":320,"problem":321,"steps":322,"help":328},"we-pens","worked_example","Buying pens at ₹99","A teacher buys **5 pens** that cost **₹99 each**. How much does she pay?",[323,324,325,326,327],"₹99 is just ₹1 less than ₹100.","5 pens at ₹100 would cost 5 × 100 = ₹500.","But each pen is ₹1 cheaper, so take away 5 × 1 = ₹5.","500 − 5 = 495. She pays **₹495**.","Check the long way: 5 × 99 = 495. ✓",{"simplerExplanation":329},"Pretend each pen costs ₹100, then give back ₹1 for every pen.",{"id":331,"type":47,"variant":204,"title":332,"markdown":333},"misc-distribute","Hand the multiplier to everyone","When you break apart 6 × 14 into 10 and 4, the 6 must multiply **both** parts. A common slip is to write 6 × 10 + 4 = 64, forgetting to multiply the 4. Think of sweets: if 6 children each get 10 toffees **and** 4 chocolates, you need 6 × 10 toffees **and** 6 × 4 chocolates, not just 4 chocolates in total. The right answer is 60 + 24 = **84**.",{"id":335,"type":209,"component":336,"componentVersion":5,"config":337,"objective":369,"textAlternative":370,"help":371},"lab-sprint-discover","arith-sprint",{"operations":338,"ranges":341,"rounds":347,"secondsTotal":116,"estimateFirst":348,"wordProblems":349},[339,340],"+","×",{"a":342,"b":345},{"min":343,"max":344},2,12,{"min":343,"max":346},20,10,false,[350,354,358,362,365],{"prompt":351,"answer":352,"operation":340,"unit":353},"Chairs are set out in 6 rows of 9. How many chairs? (Turn it around to 9 rows of 6 if that is easier.)",54,"chairs",{"prompt":355,"answer":356,"operation":339,"unit":357},"Rice ₹38, dal ₹45, oil ₹55. Add the pair that makes 100 first. What is the total in rupees?",138,"₹",{"prompt":359,"answer":360,"operation":340,"unit":361},"A garden has 6 rows of 14 plants. Break 14 into 10 + 4. How many plants?",84,"plants",{"prompt":363,"answer":364,"operation":340,"unit":357},"Five pens cost ₹99 each. Think 5 × 100 − 5. How many rupees altogether?",495,{"prompt":366,"answer":367,"operation":340,"unit":368},"A tray holds 5 rows of 6 eggs. How many eggs in 2 trays? Try 2 × 5 first.",60,"eggs","Answer quick additions and multiplications, then use turn-around, grouping and breaking-apart tricks on everyday word problems.","A sprint game. Plain questions add or multiply two numbers (the first from 2 to 12, the second from 2 to 20). Ten rounds, no timer.\n\nUp to half of the rounds are word problems drawn from these five, each with a hint about which property to use:\n1. 6 rows of 9 chairs: 6 × 9 = 54 chairs.\n2. ₹38 + ₹45 + ₹55: add 45 + 55 = 100 first, then 100 + 38 = ₹138.\n3. 6 rows of 14 plants: 6 × 10 + 6 × 4 = 60 + 24 = 84 plants.\n4. 5 pens at ₹99: 5 × 100 − 5 = 500 − 5 = ₹495.\n5. 2 trays of 5 rows of 6 eggs: 2 × 5 = 10, then 10 × 6 = 60 eggs.\n\nThe lesson of the game: when numbers look hard, swap, regroup or break apart until they look easy.",{"hints":372},[373,374],"For × questions, put the easier number second and count in steps.","Look for pairs that make 10 or 100.",{"id":376,"type":74,"title":377,"eyebrow":378,"navLabel":379},"ch5","Zero: the number that changes everything, or nothing","Chapter 05","5 Magic of zero",{"id":381,"type":43,"markdown":382},"zero-add-prose","Zero behaves in two very different ways, depending on what you do with it.\n\n**Adding or subtracting 0 changes nothing.** If you have 7 marbles and your friend gives you 0 more, you still have 7: 7 + 0 = 7. If you give away 0 marbles, you still have 7: 7 − 0 = 7. Because adding 0 leaves every number exactly as it was, 0 is called the **additive identity**. \"Identity\" means the number keeps its identity: it stays itself.\n\n**Multiplying by 0 wipes everything out.** Think of 5 boxes with 0 pencils in each. How many pencils? None: 5 × 0 = 0. Turn it around: 0 boxes of 5 pencils is also no pencils: 0 × 5 = 0. Whatever the number, times zero is zero. Even 1,000,000 × 0 = 0.",{"id":384,"type":83,"caption":385,"columns":386,"rows":391},"zero-table","What happens when zero meets each operation",[387,388,389,390],"What you do","Example","Answer","Everyday picture",[392,397,401,405,409],[393,394,395,396],"Add 0","8 + 0","8","Nobody gives you any more sweets",[398,399,395,400],"Subtract 0","8 − 0","You give away no sweets",[402,403,87,404],"Multiply by 0","8 × 0","8 empty plates hold no sweets",[406,407,87,408],"0 divided by a number","0 ÷ 8","Share 0 sweets among 8 friends: each gets 0",[410,411,412,413],"A number divided by 0","8 ÷ 0","Not possible","Share 8 sweets among 0 friends? Makes no sense",{"id":415,"type":43,"markdown":416},"zero-divide-prose","That last row is the famous one. **You cannot divide by zero.**\n\nHere is a way to see it. Division undoes multiplication. 12 ÷ 3 = 4 because 4 × 3 = 12. So for 8 ÷ 0 we need a number that, multiplied by 0, gives 8. But *anything* times 0 is 0, never 8. No such number exists. So 8 ÷ 0 has no answer, and mathematicians say it is **undefined**.\n\nTry it on a calculator or phone: type 8 ÷ 0 and it will show *Error* or *Cannot divide by zero*. The calculator is not broken. It is telling you the question has no answer.",{"id":418,"type":47,"variant":419,"title":420,"markdown":421},"try-zero-calc","try_it","Ask a calculator","With a grown-up's phone or a calculator:\n\n1. Type 0 ÷ 5. You get 0 (sharing nothing among 5 people).\n2. Type 5 ÷ 0. You get an error message.\n3. Type 5 ÷ 1, then 5 ÷ 0.1, then 5 ÷ 0.01. Watch the answers get **bigger and bigger** (5, 50, 500) as the divisor gets closer to 0. There is no final answer waiting at 0; the numbers just run away.",{"id":423,"type":163,"prompt":424,"options":425,"explanation":435},"predict-zero","Which of these is equal to **0**?",[426,428,430,432],{"id":167,"label":427},"0 + 9",{"id":170,"label":429},"9 × 0",{"id":173,"label":431},"9 − 0",{"id":433,"label":434},"d","9 ÷ 0","**9 × 0 = 0.** Nine groups of nothing is nothing.\n\n0 + 9 = 9 and 9 − 0 = 9, because adding or subtracting 0 changes nothing. And 9 ÷ 0 is not 0: it has **no answer at all**, because no number times 0 can make 9.",{"id":437,"type":74,"title":438,"eyebrow":439,"navLabel":440},"ch6","One: the number that keeps things the same","Chapter 06","6 Magic of one",{"id":442,"type":43,"markdown":443},"one-prose","The number **1** is to multiplication what 0 is to addition.\n\n- **Multiplying by 1 changes nothing.** One box with 9 mangoes holds 9 mangoes: 9 × 1 = 9, and 1 × 9 = 9. So 1 is the **multiplicative identity**.\n- **Dividing by 1 changes nothing.** Share 9 mangoes with 1 person (just yourself!) and you get all 9: 9 ÷ 1 = 9.\n- **A number divided by itself is 1** (as long as it is not 0). Share 9 mangoes among 9 people and each gets 1: 9 ÷ 9 = 1.\n\nBut be careful: **adding 1 does change a number.** 9 + 1 = 10. Adding 1 gives the next number on the number line, the successor. So 0 is special for adding, and 1 is special for multiplying. Mixing them up is one of the most common slips in maths.",{"id":445,"type":83,"caption":446,"columns":447,"rows":450},"zero-one-table","Zero and one side by side",[180,448,449],"With 0","With 1",[451,455,459,463],[452,453,454],"Adding","5 + 0 = 5 (no change)","5 + 1 = 6 (next number)",[456,457,458],"Multiplying","5 × 0 = 0 (wiped out)","5 × 1 = 5 (no change)",[460,461,462],"Dividing","5 ÷ 0: not possible","5 ÷ 1 = 5 (no change)",[464,465,466],"Special name","Additive identity","Multiplicative identity",{"id":468,"type":47,"variant":204,"title":469,"markdown":470},"misc-times-zero","\"Multiplying by zero leaves the number alone\"","Some learners write 6 × 0 = 6, mixing it up with 6 + 0 = 6. Say the multiplication in words to catch yourself: \"six groups of zero\". Six empty baskets contain **no** fruit. 6 × 0 = **0**.",{"id":472,"type":110,"itemId":473,"prompt":474,"check":475,"hints":487,"feedback":489},"pr-zero-one","properties-of-numbers.discover-zero-one","Which statement is **true**?",{"kind":476,"options":477,"correct":486},"choice",[478,480,482,484],{"id":167,"label":479},"15 × 1 = 16",{"id":170,"label":481},"15 + 0 = 0",{"id":173,"label":483},"15 × 0 = 0",{"id":433,"label":485},"15 ÷ 0 = 15",[173],[488],"Say each one in words: \"15 groups of 0\" is how many?",{"correct":490,"incorrect":491},"Right. 15 groups of nothing is nothing.","15 × 1 = 15, 15 + 0 = 15, and 15 ÷ 0 has no answer. Only 15 × 0 = 0 is true.",{"id":493,"type":74,"title":494,"eyebrow":495,"navLabel":496},"ch7","Even and odd: the pairing rules","Chapter 07","7 Even and odd",{"id":498,"type":43,"markdown":499},"even-odd-prose","Take some socks and try to put them into pairs. With **8 socks** you make 4 pairs and nothing is left over, so 8 is **even**. With **7 socks** you make 3 pairs and **one sock is left alone**, so 7 is **odd**.\n\n- Even numbers: 0, 2, 4, 6, 8, 10, 12, … (they end in 0, 2, 4, 6 or 8).\n- Odd numbers: 1, 3, 5, 7, 9, 11, 13, … (they end in 1, 3, 5, 7 or 9).\n\nZero is even: zero socks make zero pairs with nothing left over.\n\nNow the magic: you can predict whether an answer will be even or odd **without working it out**. Just think about the lonely leftover socks.",{"id":501,"type":83,"caption":502,"columns":503,"rows":507},"even-odd-add","Adding even and odd numbers: think about leftover socks",[504,388,505,506],"Add","Leftover socks","Answer is",[508,513,517,522],[509,510,511,512],"even + even","6 + 4 = 10","none + none = none","even",[514,515,516,512],"odd + odd","5 + 3 = 8","one + one = a new pair!",[518,519,520,521],"even + odd","6 + 3 = 9","none + one = one","odd",[523,187,524,521],"odd + even","one + none = one",{"id":526,"type":43,"markdown":527},"odd-odd-prose","The surprising row is **odd + odd = even**. Each odd number has one lonely sock. Put two odd numbers together and the two lonely socks become a pair! So 7 + 9 = 16 is even, and so is 99 + 101 = 200.\n\nMultiplying has its own rules: a number times an even number is always even, because you get groups of pairs. Only **odd × odd** is odd: 3 × 5 = 15, 7 × 7 = 49.",{"id":529,"type":83,"caption":530,"columns":531,"rows":533},"even-odd-mult","Multiplying even and odd numbers",[532,388,506],"Multiply",[534,536,539],[535,152,512],"even × even",[537,538,512],"even × odd","4 × 3 = 12",[540,541,521],"odd × odd","3 × 5 = 15",{"id":543,"type":209,"component":544,"componentVersion":5,"config":545,"objective":593,"textAlternative":594,"help":595},"lab-sort-evenodd","sort-game",{"prompt":546,"bins":547,"items":552,"seconds":116},"Without working out the answer, decide: will it be even or odd?",[548,550],{"id":512,"label":549},"Even answer",{"id":521,"label":551},"Odd answer",[553,557,561,565,569,573,577,581,585,589],{"id":554,"label":555,"bin":512,"why":556},"i1","23 + 45","odd + odd = even. (23 + 45 = 68.)",{"id":558,"label":559,"bin":521,"why":560},"i2","36 + 51","even + odd = odd. (36 + 51 = 87.)",{"id":562,"label":563,"bin":521,"why":564},"i3","17 × 3","odd × odd = odd. (17 × 3 = 51.)",{"id":566,"label":567,"bin":512,"why":568},"i4","15 × 4","Anything times an even number is even. (15 × 4 = 60.)",{"id":570,"label":571,"bin":512,"why":572},"i5","100 + 250","even + even = even. (100 + 250 = 350.)",{"id":574,"label":575,"bin":521,"why":576},"i6","49 − 20","odd − even = odd. (49 − 20 = 29.)",{"id":578,"label":579,"bin":512,"why":580},"i7","31 − 11","odd − odd = even. (31 − 11 = 20.)",{"id":582,"label":583,"bin":521,"why":584},"i8","7 × 9","odd × odd = odd. (7 × 9 = 63.)",{"id":586,"label":587,"bin":521,"why":588},"i9","1 + 3 + 5","1 + 3 is even, and even + 5 (odd) is odd. (1 + 3 + 5 = 9.)",{"id":590,"label":591,"bin":512,"why":592},"i10","0 × 13","The answer is 0, and 0 is even.","Predict whether each sum, difference or product is even or odd using the pairing rules, without calculating.","A sorting game with two bins, \"Even answer\" and \"Odd answer\", and ten cards.\n\nEven answer: 23 + 45 (odd + odd; it is 68), 15 × 4 (anything times even; 60), 100 + 250 (even + even; 350), 31 − 11 (odd − odd; 20) and 0 × 13 (the answer 0 is even).\n\nOdd answer: 36 + 51 (even + odd; 87), 17 × 3 (odd × odd; 51), 49 − 20 (odd − even; 29), 7 × 9 (odd × odd; 63) and 1 + 3 + 5 (three odd numbers; 9).\n\nThe rules: adding or subtracting two numbers of the same kind gives even; mixed kinds give odd. A product is odd only when every number multiplied is odd.",{"hints":596},[597,598],"Look only at the last digit of each number.","Two lonely socks make a pair.",{"id":600,"type":74,"title":601,"eyebrow":602,"navLabel":603},"ch8","Staying in the family: closure","Chapter 08","8 Staying in the family",{"id":605,"type":43,"markdown":606},"closure-prose","Here is a question that sounds odd at first: if you add two whole numbers, is the answer **always** a whole number?\n\nTry some: 3 + 5 = 8, 0 + 12 = 12, 4,567 + 8,999 = 13,566. Yes, every time. Adding whole numbers never takes you outside the family of whole numbers. We say the whole numbers are **closed under addition**: the family is like a closed room, and adding never lets you escape.\n\nMultiplication is the same: 7 × 6 = 42, 0 × 9 = 0. Always a whole number. So whole numbers are **closed under multiplication** too.\n\nBut subtraction can let you escape! 5 − 8 is not a whole number. (On a thermometer you would say −3, a number *less than zero*, which is not in our family.) And division can escape too: 7 ÷ 2 = 3½, a fraction. So whole numbers are **not closed** under subtraction or division.",{"id":608,"type":83,"caption":609,"columns":610,"rows":613},"closure-table","Do whole numbers stay whole?",[180,611,612],"Always gives a whole number?","An example that escapes",[614,617,618,620],[186,615,616],"Yes, closed","none can",[191,615,616],[195,98,619],"5 − 8 is not a whole number",[199,98,621],"7 ÷ 2 = 3½, not a whole number",{"id":623,"type":47,"variant":158,"title":624,"markdown":625},"aha-one-example","One example is enough to say \"no\"","To show a rule is **not** always true, you need just **one** example where it fails. 5 − 8 alone proves subtraction is not closed. Such an example is called a **counterexample**, and hunting for counterexamples is one of the most powerful tools in maths. But to show a rule **is** always true, examples are not enough: you need a reason that works for every number. You will see such reasons in later layers.",{"id":627,"type":74,"title":628,"eyebrow":629,"navLabel":630},"ch9","Tricks that are really properties","Chapter 09","9 Mental maths tricks",{"id":632,"type":43,"markdown":633},"tricks-intro","Every clever shortcut you have ever learned is one of these properties in disguise. Here are five favourites. Try each one on paper before looking at the answers.",{"id":635,"type":250,"title":636,"items":637},"steps-tricks","Five mental maths tricks and the property behind each",[638,642,646,649,653],{"title":639,"tag":640,"text":641},"Count on from the bigger","turn-around","3 + 68: start at 68 and count 3 more → 71. Swapping is allowed because addition is commutative.",{"title":643,"tag":644,"text":645},"Make a hundred","grouping","25 + 69 + 75 = 169: do 25 + 75 = 100 first, then add 69.",{"title":647,"tag":644,"text":648},"Times 5 = half of times 10","16 × 5: 16 × 10 = 160, half is 80. So 16 × 5 = 80.",{"title":650,"tag":651,"text":652},"Break apart","distributive","4 × 23 = 4 × 20 + 4 × 3 = 92: 80 + 12 = 92.",{"title":654,"tag":651,"text":655},"One less than a round number","3 × 49 = 3 × 50 − 3 = 147: 150 − 3 = 147.",{"id":657,"type":319,"title":658,"problem":659,"steps":660,"help":665},"we-doubling","Doubling and halving","Work out **14 × 5** in your head.",[661,662,663,664],"Half of 14 is 7, and double 5 is 10.","Halving one number and doubling the other keeps the product the same.","So 14 × 5 = 7 × 10 = **70**.","Check: 14 × 5 = 70. ✓ It works because 14 × 5 = (7 × 2) × 5 = 7 × (2 × 5) = 7 × 10.",{"simplerExplanation":666},"Imagine 14 rows of 5 chairs. Push pairs of rows together: you get 7 rows of 10 chairs, still 70 chairs.",{"id":668,"type":110,"itemId":669,"prompt":670,"check":671,"hints":673,"feedback":676},"pr-trick-25","properties-of-numbers.discover-times-25","Use grouping to find **4 × 37 × 25** quickly.",{"kind":114,"answer":672,"tolerance":116},3700,[674,675],"Which two numbers multiply to make 100?","4 × 25 = 100. Now multiply 100 by 37.",{"correct":677,"incorrect":678},"Yes: 4 × 25 = 100, and 100 × 37 = 3,700.","Group 4 × 25 = 100 first, then 100 × 37 = 3,700.",{"id":680,"type":74,"title":681,"eyebrow":682,"navLabel":683},"ch10","Putting it together","Chapter 10","10 Wrap up",{"id":685,"type":47,"variant":686,"title":687,"markdown":688},"nuance-dmas","nuance","Distributive is not DMAS","Some lists explain the letters DMAS as \"Distributive, Multiplicative, Additive, Subtractive\". That is a mix-up. **DMAS** (like BODMAS or PEMDAS) is the **order of operations**: Division, Multiplication, Addition, Subtraction, the agreed order for working out a sum such as 3 + 4 × 2. It is not a list of properties.\n\nThe **distributive property** (breaking apart, like 6 × 14 = 6 × 10 + 6 × 4) lives here, in this topic. The order of operations has its own topic: [Order of operations](\u002Ftopics\u002Forder-of-operations).",{"id":690,"type":83,"caption":691,"columns":692,"rows":696},"big-table","The rules you have met, all together",[693,694,388,695],"Rule","Everyday name","Works for",[697,702,706,710,713,716,720],[698,699,700,701],"Commutative","Turn-around","3 + 9 = 9 + 3 = 12","+ and ×, not − or ÷",[703,704,705,701],"Associative","Grouping","(2 × 7) × 5 = 2 × (7 × 5) = 70",[707,65,708,709],"Distributive","6 × 14 = 60 + 24 = 84","× over + and over −",[465,711,712,186],"Adding 0","8 + 0 = 8",[466,714,715,191],"Times 1","8 × 1 = 8",[717,718,719,191],"Zero property","Times 0","8 × 0 = 0",[721,722,723,724],"Closure","Staying in the family","3 + 5 = 8 is whole","+ and × for whole numbers",{"id":726,"type":209,"component":544,"componentVersion":5,"config":727,"objective":790,"textAlternative":791,"help":792},"lab-sort-rules",{"prompt":728,"bins":729,"items":742,"seconds":116},"Which rule is each example showing?",[730,733,736,739],{"id":731,"label":732},"turn","Turn-around (commutative)",{"id":734,"label":735},"group","Grouping (associative)",{"id":737,"label":738},"break","Breaking apart (distributive)",{"id":740,"label":741},"zeroone","Zero or one",[743,747,751,755,759,763,767,771,775,779,783,787],{"id":744,"label":745,"bin":731,"why":746},"r1","8 + 5 = 5 + 8","The two numbers swap places.",{"id":748,"label":749,"bin":731,"why":750},"r2","4 rows of 6 chairs = 6 rows of 4 chairs","4 × 6 = 6 × 4: the array is turned sideways.",{"id":752,"label":753,"bin":734,"why":754},"r3","(38 + 45) + 55 = 38 + (45 + 55)","Same order, different pair done first.",{"id":756,"label":757,"bin":734,"why":758},"r4","2 × 7 × 5: do 2 × 5 first","Choosing which pair to multiply first. (Strictly, moving the 5 next to the 2 also uses turn-around.)",{"id":760,"label":761,"bin":737,"why":762},"r5","6 × 14 = 6 × 10 + 6 × 4","14 is broken into 10 + 4 and the 6 multiplies both parts.",{"id":764,"label":765,"bin":737,"why":766},"r6","5 × 99 = 500 − 5","99 = 100 − 1, so 5 × 99 = 5 × 100 − 5 × 1.",{"id":768,"label":769,"bin":740,"why":770},"r7","37 + 0 = 37","Adding 0 changes nothing.",{"id":772,"label":773,"bin":740,"why":774},"r8","37 × 1 = 37","Multiplying by 1 changes nothing.",{"id":776,"label":777,"bin":740,"why":778},"r9","37 × 0 = 0","Anything times 0 is 0.",{"id":780,"label":781,"bin":737,"why":782},"r10","7 × 12 = 70 + 14","12 = 10 + 2, so 7 × 12 = 7 × 10 + 7 × 2.",{"id":784,"label":785,"bin":734,"why":786},"r11","(3 × 5) × 6 = 3 × (5 × 6)","Brackets moved, order kept.",{"id":788,"label":789,"bin":731,"why":746},"r12","25 × 4 = 4 × 25","Sort everyday examples into the rule they show: turn-around, grouping, breaking apart, or the special numbers 0 and 1.","A sorting game with four bins and twelve cards.\n\nTurn-around (commutative): 8 + 5 = 5 + 8; 4 rows of 6 chairs = 6 rows of 4 chairs; 25 × 4 = 4 × 25. The numbers swap places.\n\nGrouping (associative): (38 + 45) + 55 = 38 + (45 + 55); 2 × 7 × 5 done as 2 × 5 first; (3 × 5) × 6 = 3 × (5 × 6). The order stays; only the pair you do first changes.\n\nBreaking apart (distributive): 6 × 14 = 6 × 10 + 6 × 4; 5 × 99 = 500 − 5; 7 × 12 = 70 + 14. One number is split and the multiplier goes to each part.\n\nZero or one: 37 + 0 = 37; 37 × 1 = 37; 37 × 0 = 0.",{"hints":793},[794,795],"Swapped numbers → turn-around. Moved brackets → grouping.","A number split into two parts → breaking apart.",{"id":797,"type":798,"title":799,"terms":800},"glossary-discover","glossary","Words to know",[801,805,809,813,817,821,825,829,833,836,839,842,845,849,853,857,861],{"term":802,"meaning":803,"example":804},"Natural numbers","The counting numbers 1, 2, 3, 4, … They go on for ever.","You count 1, 2, 3 goats; you never count \"zero goats\".",{"term":806,"meaning":807,"example":808},"Whole numbers","Zero together with all the natural numbers: 0, 1, 2, 3, …","0 is a whole number but not a natural number.",{"term":810,"meaning":811,"example":812},"Number line","A straight line with numbers marked at equal steps; bigger numbers lie further right.","Adding 3 is a jump of 3 steps to the right.",{"term":814,"meaning":815,"example":816},"Successor","The number that comes just after a given number: add 1.","The successor of 99 is 100.",{"term":818,"meaning":819,"example":820},"Predecessor","The number that comes just before a given number: subtract 1.","The predecessor of 100 is 99.",{"term":822,"meaning":823,"example":824},"Property","A rule that is true for every number of a certain kind, not just for one example.","For any two numbers, a + b = b + a.",{"term":826,"meaning":827,"example":828},"Commutative property","Swapping the order of two numbers does not change the answer. True for addition and multiplication.","4 × 6 = 6 × 4",{"term":830,"meaning":831,"example":832},"Associative property","When combining three numbers, it does not matter which pair you do first. True for addition and multiplication.","(2 × 7) × 5 = 2 × (7 × 5)",{"term":834,"meaning":835,"example":761},"Distributive property","Multiplying a sum (or difference) is the same as multiplying each part and then adding (or subtracting).",{"term":465,"meaning":837,"example":838},"The number 0, because adding 0 leaves any number unchanged.","25 + 0 = 25",{"term":466,"meaning":840,"example":841},"The number 1, because multiplying by 1 leaves any number unchanged.","25 × 1 = 25",{"term":721,"meaning":843,"example":844},"A family of numbers is closed under an operation if the answer is always in the same family.","Whole number + whole number = whole number.",{"term":846,"meaning":847,"example":848},"Even number","A whole number that splits exactly into pairs; it ends in 0, 2, 4, 6 or 8.","0, 2, 14, 100",{"term":850,"meaning":851,"example":852},"Odd number","A whole number that leaves one over when split into pairs; it ends in 1, 3, 5, 7 or 9.","1, 7, 15, 99",{"term":854,"meaning":855,"example":856},"Counterexample","One example that shows a rule is not always true.","5 − 8 is not a whole number, so subtraction is not closed.",{"term":858,"meaning":859,"example":860},"Array","Objects arranged in equal rows and columns.","Eggs in a tray: 5 rows of 6.",{"term":862,"meaning":863,"example":864},"Undefined","Has no answer that makes sense. Dividing by zero is undefined.","8 ÷ 0 is undefined.",{"id":866,"type":867,"title":868,"questions":869},"quiz-discover","quiz","Check what you found",[870,883,896,909,922,932,945,954,967,979],{"itemId":871,"prompt":872,"options":873,"correct":173,"why":882},"properties-of-numbers.discover-q-turnaround","Riya sets out 7 rows of 9 chairs. Arjun sets out 9 rows of 7. Who has more chairs?",[874,876,878,880],{"id":167,"label":875},"Riya",{"id":170,"label":877},"Arjun",{"id":173,"label":879},"Both have 63 chairs",{"id":433,"label":881},"You cannot tell","Turning an array sideways does not change the number of chairs: 7 × 9 = 9 × 7 = 63.",{"itemId":884,"prompt":885,"options":886,"correct":173,"why":895},"properties-of-numbers.discover-q-subtract","Which of these CANNOT be turned around without changing the answer?",[887,889,891,893],{"id":167,"label":888},"12 + 30",{"id":170,"label":890},"12 × 30",{"id":173,"label":892},"30 − 12",{"id":433,"label":894},"30 + 12","30 − 12 = 18, but 12 − 30 is not even a whole number. Subtraction is not commutative.",{"itemId":897,"prompt":898,"options":899,"correct":173,"why":908},"properties-of-numbers.discover-q-group","What is the quickest first step for 5 × 17 × 2?",[900,902,904,906],{"id":167,"label":901},"5 × 17",{"id":170,"label":903},"17 × 2",{"id":173,"label":905},"5 × 2",{"id":433,"label":907},"Add 5 + 17","5 × 2 = 10, and 10 × 17 = 170. Grouping the friendly pair first makes it easy. 5 × 17 × 2 = 170.",{"itemId":910,"prompt":911,"options":912,"correct":170,"why":921},"properties-of-numbers.discover-q-break","Which is the same as 8 × 13?",[913,915,917,919],{"id":167,"label":914},"8 × 10 + 3",{"id":170,"label":916},"8 × 10 + 8 × 3",{"id":173,"label":918},"8 + 10 × 3",{"id":433,"label":920},"80 + 3","The 8 must multiply both parts of 13 = 10 + 3: 8 × 10 + 8 × 3 = 80 + 24 = 104.",{"itemId":923,"prompt":924,"options":925,"correct":170,"why":931},"properties-of-numbers.discover-q-zero","Share 0 laddoos among 4 friends. How many does each get?",[926,927,928,929],{"id":167,"label":91},{"id":170,"label":87},{"id":173,"label":88},{"id":433,"label":930},"It is impossible","0 ÷ 4 = 0. There is nothing to share, so each friend gets 0. (It is dividing BY zero that is impossible.)",{"itemId":933,"prompt":934,"options":935,"correct":173,"why":944},"properties-of-numbers.discover-q-divzero","Why does 6 ÷ 0 have no answer?",[936,938,940,942],{"id":167,"label":937},"Because the answer is 0",{"id":170,"label":939},"Because the answer is 6",{"id":173,"label":941},"Because no number times 0 gives 6",{"id":433,"label":943},"Because 6 is even","Division undoes multiplication. We would need a number that, times 0, gives 6. Every number times 0 is 0, so there is no answer.",{"itemId":946,"prompt":947,"options":948,"correct":170,"why":953},"properties-of-numbers.discover-q-one","Which number can you multiply by without changing anything?",[949,950,951,952],{"id":167,"label":87},{"id":170,"label":88},{"id":173,"label":89},{"id":433,"label":33},"1 is the multiplicative identity: 45 × 1 = 45.",{"itemId":955,"prompt":956,"options":957,"correct":167,"why":966},"properties-of-numbers.discover-q-oddodd","47 + 35: even or odd?",[958,960,962,964],{"id":167,"label":959},"Even",{"id":170,"label":961},"Odd",{"id":173,"label":963},"It depends",{"id":433,"label":965},"Neither","odd + odd = even: the two leftover ones make a pair. 47 + 35 = 82, which is even.",{"itemId":968,"prompt":969,"options":970,"correct":170,"why":978},"properties-of-numbers.discover-q-closure","Which of these shows that whole numbers are NOT closed under division?",[971,972,974,976],{"id":167,"label":200},{"id":170,"label":973},"9 ÷ 2 = 4½",{"id":173,"label":975},"0 ÷ 5 = 0",{"id":433,"label":977},"10 ÷ 10 = 1","9 ÷ 2 is not a whole number, so dividing two whole numbers can take you outside the family. One counterexample is enough.",{"itemId":980,"prompt":981,"options":982,"correct":167,"why":988},"properties-of-numbers.discover-q-natural","Which whole number is NOT a natural number?",[983,984,985,986],{"id":167,"label":87},{"id":170,"label":88},{"id":173,"label":33},{"id":433,"label":987},"100","Natural numbers start at 1. Whole numbers add 0 to the family.",{"id":990,"type":991,"prompt":992},"reflect-discover","reflection","Think of one time this week when you added or multiplied in your head: a bill, a score, a count of things. Which property (turn-around, grouping or breaking apart) could have made it easier? Write the sum both ways.",{"id":994,"type":995,"title":996,"points":997},"cheat-discover","summary","Cheat sheet",[998,999,1000,1001,1002,1003,1004,1005,1006],"**Natural numbers:** 1, 2, 3, … **Whole numbers:** 0, 1, 2, 3, … Both go on for ever along the number line.","**Turn-around (commutative):** 4 × 6 = 6 × 4 and 5 + 3 = 3 + 5. Works for + and ×, never for − or ÷.","**Grouping (associative):** (45 + 55) + 38 = 45 + (55 + 38). Do the friendly pair first. Works for + and ×.","**Breaking apart (distributive):** 6 × 14 = 6 × 10 + 6 × 4 = 84; 5 × 99 = 500 − 5 = 495. Multiply *every* part.","**Zero:** a + 0 = a; a × 0 = 0; 0 ÷ a = 0; but a ÷ 0 has no answer.","**One:** a × 1 = a; a ÷ 1 = a; a ÷ a = 1 (a not 0). Adding 1 gives the next number.","**Even and odd:** odd + odd = even; even + odd = odd; odd × odd = odd; anything × even = even.","**Closure:** whole numbers stay whole when you add or multiply, but not always when you subtract or divide.","**DMAS is order of operations**, not a list of properties. See the Order of operations topic.",{"id":1008,"type":1009,"conceptId":1010,"relation":1011,"explanation":1012},"conn-four-ops","connection","four-operations","helps_understand","The four operations topic uses these properties inside every method: column addition, long multiplication and checking by reverse operations.",{"id":1014,"type":1009,"conceptId":1015,"relation":1016,"explanation":1017},"conn-order-ops","order-of-operations","related_to","The distributive property explains why 6 × (10 + 4) and 6 × 10 + 6 × 4 give the same answer, while order of operations tells you which to do first.",{"id":1019,"type":1009,"conceptId":1020,"relation":1016,"explanation":1021},"conn-number-system","number-system","Natural and whole numbers, the number line, successors and predecessors all come from the number system.",{"id":1023,"type":1009,"conceptId":1024,"relation":1016,"explanation":1025},"conn-patterns","patterns","Even and odd numbers alternate on the number line: one of the simplest number patterns, and the reason the pairing rules work.",{"id":1027,"type":1028,"sourceIds":1029},"sources-discover","sources",[1030,1031,1032,1033,1034,1035,1036,1037,1038,1039],"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",[1030,1031,1032,1033,1034,1035,1036,1037,1038,1039],"needs_review",{"generatedBy":1043,"notes":1044},"claude-code","Draft generated with Python generators; every stated number computed and asserted. Pending owner review.","cc000602f35992202233892683772ff9d263b73df082be7897f53d3cd05be5e9",{"logic:practice":1047,"component:match-pairs@1":1048,"component:arith-sprint@1":1049,"component:sort-game@1":1050,"source:properties-of-numbers-mathsisfun-divide-by-zero":1051,"source:properties-of-numbers-mathsisfun-properties":1052,"source:properties-of-numbers-ncert-class6-whole-numbers":1053,"source:properties-of-numbers-ncert-class7-integers":1054,"source:properties-of-numbers-ncert-class8-rational-numbers":1055,"source:properties-of-numbers-wiki-brahmagupta":1056,"source:properties-of-numbers-wiki-commutative":1057,"source:properties-of-numbers-wiki-distributive":1058,"source:properties-of-numbers-wiki-division-by-zero":1059,"source:properties-of-numbers-wiki-parity":1060},"3d6b0fe1b15255975a32b0fcd94e8019bc959ad45cbf12e136e86149549c6878","2a8ee4ac87460b4e1175a4bb13c96b03d577db06dde95670eb7fcfe4ad787899","c0c63ed40e1bca5ba6d446d43886b887a3a7709e5cd6679419f64fd3ba62afe6","b164f45a2c8ca08f26c450768ff0231e113e9fe45381eddb34dc6d0548596c38","f8a42fd8c82c266f4310450d74f1ae0449b3dba177afcf000b24aafa2839b629","a37b85a36742e06f34cdc083710eacdd662e2f7493395766e4b8c5632bfade3a","56b97dbf2988a58e1aa10f693ec9dd1039e1de14e795b1bdfa19980b6b18c1fa","aab8fe53a32660ef6307b4dbd6f5af02319957d9aecf3c0396895bc01ab46e2e","acb8b92cad8387724d72528e70ef68195351e5b2f03b580c5da191fde82dc67a","32e7279091a5877afe54d6da75cdd06a04f2ff8677338777578592d6dd98a338","a3e0fb47a2115f5962c9795b138bf38cd9c2c4874bc66e1b834717b9a01908c1","befe26bd7ec4e1c3911549d33a91844ae3eb5bd63388711f61822d398d3d576f","046256cc76cdb1b7cca72a1b391bef686855afdfb6a68d18b45a38723d1fe93d","01c5bcf03886aec6f9c2c18c9ec5333b19ce1d2ca6bfb4f8544c4ca9f1578e27",{"state":1062,"reviewer":1063,"selfReview":1064,"reviewedAt":1065,"method":1066},"approved","The library owner",true,"2026-09-20T10:18:37.581Z","owner_bulk","preview-7e1cbbcc4f",1789899597107]