[{"data":1,"prerenderedAt":1191},["ShallowReactive",2],{"layer:four-operations:discover":3},{"layer":4,"contentHash":1167,"dependencyHashes":1168,"approval":1184,"releaseId":1190},{"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":1162,"reviewStatus":1163,"authoring":1164},1,"four-operations","en","discover","Four ways to change a number","Adding, subtracting, multiplying and dividing: what each one means and when to use it","Meet the four operations through a kirana-shop trip, cricket scores, egg trays and shared laddoos. Learn what each operation means, how they undo each other, how to pick the right one from a story, and how to check that an answer is sensible.",[13,14,15,16,17],"Explain addition as joining and subtraction as both taking away and comparing.","Describe multiplication as equal groups, arrays and 'times as many'.","Tell sharing from grouping in division, and decide what a remainder means in a story.","Use inverse operations and fact families to check answers.","Choose an operation by reading the whole story, and estimate to see if an answer makes sense.",35,{"title":20,"rows":21},"Lesson plate",[22,25,28,31,34,37],{"label":23,"value":24},"Depth","Discover",{"label":26,"value":27},"Reading time","≈ 35 minutes",{"label":29,"value":30},"Prior knowledge","Counting, and times tables up to 10",{"label":32,"value":33},"Chapters","10",{"label":35,"value":36},"Labs","6: sprints, sorting, matching, rounding",{"label":38,"value":39},"Signs used","+  −  ×  ÷  =",[41,45,51,57,60,78,108,129,134,139,142,163,173,178,194,199,202,207,219,235,246,259,278,283,286,289,292,323,327,336,349,369,374,377,394,418,421,425,437,450,463,481,496,501,504,517,521,530,560,572,577,580,584,626,703,723,728,731,741,746,755,786,797,802,837,840,869,873,884,888,893,898,982,988,993,997,1002,1135,1148],{"id":42,"type":43,"markdown":44},"intro-question","prose","Every day you change numbers without noticing. You buy things, share things, count things in rows and work out how much is left. Almost all of it uses just **four operations**: **adding**, **subtracting**, **multiplying** and **dividing**.\n\nThis layer is about what each one *means*, not about long methods. By the end you should be able to read a little story, see the numbers in it, and say: \"Ah, that is a sharing problem, so it is division\", and then check that your answer makes sense.",{"id":46,"type":47,"variant":48,"title":49,"markdown":50},"how-to-use","callout","observation","How to use this layer","Read the chapters in order the first time. Whenever you meet a **prediction**, choose an answer *before* reading on; guessing first and then checking is how ideas stick. The games (labs) are for playing more than once: try to beat your own score.",{"id":52,"type":53,"title":54,"eyebrow":55,"navLabel":56},"ch1","chapter","A trip to the kirana shop","Chapter 01","1 The kirana shop",{"id":58,"type":43,"markdown":59},"kirana-story","Ammu's mother sends her to the kirana shop at the end of the lane with a **₹200 note**. Here is what happens:\n\n1. She buys a bag of rice for **₹64** and a packet of dal for **₹38**. The shopkeeper, Ramesh uncle, works out the bill.\n2. He gives Ammu her **change** from the ₹200 note.\n3. Ammu also wants biscuits for her little brothers. Each packet costs **₹10** and she wants **4** packets, so she asks what that would cost.\n4. At home, she has a jar of **24 toffees** to share equally among **4** cousins.\n\nFour little problems. Four different ways of changing numbers. Can you tell which is which before you read on?",{"id":61,"type":62,"prompt":63,"options":64,"explanation":77},"predict-kirana","prediction","Which operation does Ramesh uncle use to find the **change** from Ammu's ₹200 note (step 2)?",[65,68,71,74],{"id":66,"label":67},"a","Adding: put the rice and dal together",{"id":69,"label":70},"b","Subtracting: take the bill away from ₹200",{"id":72,"label":73},"c","Multiplying: equal groups",{"id":75,"label":76},"d","Dividing: sharing equally","**Subtracting.** First the bill is found by **adding**: ₹64 + ₹38 = **₹102**. Then the change is what is left when the bill is taken away from the note: ₹200 − ₹102 = **₹98**.\n\nThe biscuits are **multiplying**: 4 equal packets of ₹10 cost 4 × ₹10 = ₹40. The toffees are **dividing**: 24 shared equally among 4 gives 6 each. One shop visit, all four operations.",{"id":79,"type":80,"caption":81,"columns":82,"rows":87},"table-kirana","table","Ammu's shop trip: four questions, four operations",[83,84,85,86],"What Ammu wants to know","Operation","Number sentence","Answer",[88,93,98,103],[89,90,91,92],"How much is the bill?","Addition (putting together)","64 + 38","₹102",[94,95,96,97],"How much change?","Subtraction (what is left)","200 − 102","₹98",[99,100,101,102],"What do 4 biscuit packets cost?","Multiplication (equal groups)","4 × 10","₹40",[104,105,106,107],"How many toffees each?","Division (fair sharing)","24 ÷ 4","6 toffees",{"id":109,"type":110,"tone":111,"items":112},"spec-four","spec","amber",[113,117,121,125],{"label":114,"big":115,"value":116},"Addition","+","Putting amounts together. The answer is the **sum**.",{"label":118,"big":119,"value":120},"Subtraction","−","Taking away, or finding how far apart two numbers are. The answer is the **difference**.",{"label":122,"big":123,"value":124},"Multiplication","×","Adding equal groups quickly. The answer is the **product**.",{"label":126,"big":127,"value":128},"Division","÷","Sharing equally, or making equal groups. The answer is the **quotient**.",{"id":130,"type":47,"variant":131,"title":132,"markdown":133},"aha-story-first","aha","Read the story, not just the numbers","The numbers alone never tell you what to do: 24 and 4 could be added (28), subtracted (20), multiplied (96) or divided (6). It is the **story** that decides. Are things being joined? Taken away? Compared? Repeated in equal groups? Shared out? Ask that question first and the operation usually picks itself.",{"id":135,"type":53,"title":136,"eyebrow":137,"navLabel":138},"ch2","Adding: putting together","Chapter 02","2 Adding",{"id":140,"type":43,"markdown":141},"add-meaning","**Adding** means **putting amounts together** to find how many there are altogether.\n\nA city bus at the Charminar stop has **27** passengers. At the next stop **15** more climb on and nobody gets off. How many are on the bus now? We are joining a new group to the old one, so we add: 27 + 15 = **42**.\n\nThe numbers being added are called **addends**, and the answer is the **sum**. In 27 + 15 = 42, the addends are 27 and 15, and the sum is 42.",{"id":143,"type":144,"title":145,"items":146},"steps-count-on","steps","Counting on: the oldest adding trick",[147,151,155,159],{"title":148,"tag":149,"text":150},"Start with the bigger number","8","To find 8 + 5, don't count from 1. Hold 8 in your head.",{"title":152,"tag":153,"text":154},"Count on the smaller one","9, 10, 11, 12, 13","Say the next five numbers, one for each finger.",{"title":156,"tag":157,"text":158},"The last number is the sum","13","8 + 5 = 13. Starting from the bigger number means fewer counts.",{"title":160,"tag":161,"text":162},"Make a ten","8 + 2 + 3","Or split 5 into 2 and 3: 8 + 2 = 10, then 10 + 3 = 13. Tens are easy to add.",{"id":164,"type":165,"title":166,"problem":167,"steps":168},"we-cricket","worked_example","Runs from three batters","In a school cricket match, Kabir scores **34**, Sana scores **27** and Dev scores **18**. How many runs did the three make together?",[169,170,171,172],"The runs are being **put together** into one team total, so this is addition.","Add two at a time: 34 + 27 = 61.","Then add the third: 61 + 18 = **79 runs**.","Order does not matter for adding: 18 + 27 + 34 is also 79. You can start with whichever pair is easiest.",{"id":174,"type":47,"variant":175,"title":176,"markdown":177},"nuance-add-order","nuance","Adding in any order","7 + 3 and 3 + 7 are both 10. Swapping the numbers in an addition never changes the sum, and when you add three numbers you can group them however you like. This is why a shopkeeper can add a long bill in whatever order is easiest, for example pairing ₹25 with ₹75 to make ₹100 first. (The *properties of numbers* topic gives these rules their proper names.)",{"id":179,"type":180,"itemId":181,"prompt":182,"check":183,"hints":188,"feedback":191},"pr-add-bus","practice","four-operations.d-add-passengers","A train compartment has 45 passengers. At Vijayawada, 38 more get in and nobody gets off. How many passengers are there now?",{"kind":184,"answer":185,"tolerance":186,"unit":187},"number",83,0,"passengers",[189,190],"More people are joining, so put the two groups together.","Try 45 + 40 = 85, then take away the extra 2.",{"correct":192,"incorrect":193},"Right: 45 + 38 = 83. Joining groups means adding.","Joining means adding: 45 + 38. One way: 45 + 40 = 85, and since we added 2 too many, 85 − 2 = 83.",{"id":195,"type":53,"title":196,"eyebrow":197,"navLabel":198},"ch3","Subtracting: taking away and comparing","Chapter 03","3 Subtracting",{"id":200,"type":43,"markdown":201},"sub-meaning","Subtraction has **two** everyday meanings, and it helps to know both.\n\n**1. Taking away.** You had **35** sweets and ate **12**. How many are left? 35 − 12 = **23**. Something is removed, and we find what remains.\n\n**2. Comparing (finding the difference).** Riya is **138 cm** tall and Arjun is **129 cm**. How much taller is Riya? Nothing is taken away from anybody! We are asking how far apart two numbers are: 138 − 129 = **9 cm**.\n\nEither way, the answer to a subtraction is called the **difference**.",{"id":203,"type":47,"variant":204,"title":205,"markdown":206},"mis-sub-takeaway","misconception","“Subtraction only means taking away”","Many people think subtraction is only for \"left over\" problems. But \"How many **more**?\", \"How much **taller**?\", \"How many **fewer**?\" and \"How many **still needed**?\" are all subtraction too, even though nothing disappears.\n\nA team needs **186** runs to win and has **142**. The runs still needed are 186 − 142 = **44**. No runs are taken away; we are finding the gap between where they are and where they want to be.",{"id":208,"type":165,"title":209,"problem":210,"steps":211,"help":217},"we-change","Change from ₹100 by counting up","A toy car costs **₹67**. You pay with a **₹100** note. How much change should you get?",[212,213,214,215,216],"This is subtraction: ₹100 − ₹67.","Many shopkeepers **count up** instead of taking away: from ₹67 to ₹70 is **₹3**.","From ₹70 to ₹100 is **₹30**.","Add the jumps: ₹3 + ₹30 = **₹33**.","Check by adding back: ₹67 + ₹33 = ₹100. ✓",{"simplerExplanation":218},"Change is the gap between the price and the note. Jump from 67 to the next ten, then to 100, and add the jumps.",{"id":220,"type":144,"title":221,"items":222},"steps-sub-kinds","Three questions that all mean subtract",[223,227,231],{"title":224,"tag":225,"text":226},"How many are left?","take away","35 sweets, 12 eaten: 35 − 12 = 23 left.",{"title":228,"tag":229,"text":230},"How many more (or fewer)?","compare","Riya 138 cm, Arjun 129 cm: Riya is 9 cm taller.",{"title":232,"tag":233,"text":234},"How many still needed?","missing part","Target 186, scored 142: 44 runs still needed.",{"id":236,"type":62,"prompt":237,"options":238,"explanation":245},"predict-order-sub","Is 9 − 4 the same as 4 − 9?",[239,241,243],{"id":66,"label":240},"Yes, order never matters",{"id":69,"label":242},"No: swapping the numbers changes a subtraction",{"id":72,"label":244},"Only when the numbers are small","**No.** 9 − 4 = 5, but you cannot take 9 away from 4 with ordinary counting numbers (you would need negative numbers, which you will meet later). Unlike adding, **subtraction is not the same when you swap the numbers**, so always start from the whole amount and take the part away from it.",{"id":247,"type":180,"itemId":248,"prompt":249,"check":250,"hints":253,"feedback":256},"pr-sub-change","four-operations.d-sub-change","Kavya buys a notebook for ₹58 and pays with a ₹100 note. How much change does she get?",{"kind":184,"answer":251,"tolerance":186,"unit":252},42,"₹",[254,255],"Count up from 58 to 60, then from 60 to 100.","Then add your jumps together.",{"correct":257,"incorrect":258},"Yes: 100 − 58 = 42. Counting up: 2 + 40 = 42.","Change is 100 − 58. Count up: 58 → 60 is 2, 60 → 100 is 40, so the change is ₹42.",{"id":260,"type":180,"itemId":261,"prompt":262,"check":263,"hints":273,"feedback":275},"pr-sub-compare","four-operations.d-sub-compare","Which question is a **comparing** subtraction (nothing is taken away)?",{"kind":264,"options":265,"correct":272},"choice",[266,268,270],{"id":66,"label":267},"Meena had 20 marbles and lost 6. How many are left?",{"id":69,"label":269},"The Rajdhani has 18 coaches and the local train has 12. How many more coaches does the Rajdhani have?",{"id":72,"label":271},"A crow ate 5 of the 30 grains. How many grains remain?",[69],[274],"In which story does nothing get lost, eaten or removed?",{"correct":276,"incorrect":277},"Right. Nobody takes coaches off either train; we compare 18 and 12: the difference is 6.","In (a) and (c) something is lost or eaten. In (b) we only compare two trains: 18 − 12 = 6 more coaches.",{"id":279,"type":53,"title":280,"eyebrow":281,"navLabel":282},"ch4","Multiplying: equal groups, fast","Chapter 04","4 Multiplying",{"id":284,"type":43,"markdown":285},"mul-meaning","An egg tray at the market holds **30** eggs. How many eggs are in **4** trays?\n\nYou could add: 30 + 30 + 30 + 30 = 120. But because every tray holds the **same** number, there is a shortcut: **multiply**. 4 × 30 = **120**.\n\nThat is the first big idea of multiplication: **repeated addition of equal groups**. We read 4 × 30 as \"4 groups of 30\" or \"4 times 30\". The answer is called the **product**.",{"id":287,"type":43,"markdown":288},"mul-array","The second picture is an **array**: things in neat rows and columns. For the school assembly, **6** rows of **8** chairs are set out.\n\n```\n□ □ □ □ □ □ □ □\n□ □ □ □ □ □ □ □\n□ □ □ □ □ □ □ □\n□ □ □ □ □ □ □ □\n□ □ □ □ □ □ □ □\n□ □ □ □ □ □ □ □\n```\n\n6 rows of 8 is 6 × 8 = **48** chairs. Now turn your head sideways: you see 8 columns of 6, which is 8 × 6. It is the same chairs, so it must be the same number, 48. An array shows at a glance why **the order of multiplying does not matter**.",{"id":290,"type":43,"markdown":291},"mul-scale","The third picture is **scaling**, or \"times as many\". Meera has **12** stamps. Her brother has **3 times as many**. That means three lots of Meera's collection: 3 × 12 = **36** stamps.\n\nWords like **double** (× 2), **triple** (× 3) and \"**five times as much**\" are all multiplication.",{"id":293,"type":80,"caption":294,"columns":295,"rows":299},"table-mul-pictures","Three pictures of multiplication",[296,297,85,298],"Picture","Story","Product",[300,305,310,315,319],[301,302,303,304],"Equal groups","4 trays of 30 eggs","4 × 30","120",[306,307,308,309],"Array","6 rows of 8 chairs","6 × 8","48",[311,312,313,314],"Scaling","3 times as many as 12 stamps","3 × 12","36",[301,316,317,318],"7 autos, 3 wheels each","7 × 3","21",[301,320,321,322],"9 bicycles, 2 wheels each","9 × 2","18",{"id":324,"type":47,"variant":204,"title":325,"markdown":326},"mis-mul-bigger","“Multiplying always makes a number bigger”","With the counting numbers you use most, multiplying by 2, 3, 10 and so on does make things bigger. But look at these:\n\n- **1 × 25 = 25.** One group of 25 is just 25. Multiplying by 1 changes nothing.\n- **0 × 25 = 0.** Zero groups of 25 is nothing at all. If you buy 0 packets of chips, you spend ₹0 whatever the price.\n\nLater you will multiply by numbers smaller than 1, such as a half, and the answer gets **smaller**. So a safer rule is: multiplying by a number **bigger than 1** makes things bigger.",{"id":328,"type":165,"title":329,"problem":330,"steps":331},"we-auto","Wheels in the auto stand","An auto stand has 7 autos and 9 bicycles. How many wheels are there altogether?",[332,333,334,335],"Autos: 7 groups of 3 wheels, so 7 × 3 = 21.","Bicycles: 9 groups of 2 wheels, so 9 × 2 = 18.","The two lots are **put together**, so add: 21 + 18 = **39 wheels**.","Notice that one question used two operations: multiply to count each kind of vehicle, then add. Many real problems work like this.",{"id":337,"type":180,"itemId":338,"prompt":339,"check":340,"hints":343,"feedback":346},"pr-mul-laddoo","four-operations.d-mul-boxes","A sweet shop packs laddoos in boxes of 12. How many laddoos are in 5 boxes?",{"kind":184,"answer":341,"tolerance":186,"unit":342},60,"laddoos",[344,345],"Every box holds the same number, so you have equal groups.","5 × 12 is 5 × 10 plus 5 × 2.",{"correct":347,"incorrect":348},"Yes: 5 × 12 = 60. Five equal groups of 12.","5 boxes of 12 is 5 × 12. Split 12 into 10 and 2: 5 × 10 = 50 and 5 × 2 = 10, so 60 laddoos.",{"id":350,"type":351,"component":352,"componentVersion":5,"config":353,"objective":363,"textAlternative":364,"help":365},"lab-tables","interactive","arith-sprint",{"operations":354,"ranges":355,"rounds":360,"secondsTotal":186,"estimateFirst":361,"wordProblems":362},[123],{"a":356,"b":359},{"min":357,"max":358},2,10,{"min":357,"max":358},12,false,[],"Practise times-table facts up to 10 × 10 until they come quickly.","This game asks twelve quick questions from the times tables up to 10 × 10. There is no timer, so you can think.\n\nA question looks like **6 × 7 = ?** (answer 42: six groups of seven). The division facts that undo these come in the next chapter.\n\nTips that make the tables easier:\n- **× 2** is doubling; **× 4** is doubling twice; **× 8** is doubling three times.\n- **× 5** is half of **× 10**: 5 × 8 = 40 because 10 × 8 = 80.\n- **× 9** is **× 10** take away one group: 9 × 7 = 70 − 7 = 63.\n- Turn-arounds help: if you know 3 × 8 = 24, you also know 8 × 3 = 24.\n\nYour streak grows with each correct answer in a row. Try it again later and see if your best score improves.",{"hints":366},[367,368],"Use a fact you know: 6 × 7 is one more 7 than 5 × 7 = 35.","Stuck on 7 × 8? Remember 5, 6, 7, 8: 56 = 7 × 8.",{"id":370,"type":53,"title":371,"eyebrow":372,"navLabel":373},"ch5","Dividing: sharing and grouping","Chapter 05","5 Dividing",{"id":375,"type":43,"markdown":376},"div-meaning","Division also has two everyday meanings.\n\n**1. Sharing (fair shares).** 18 chocolates are shared equally among 3 children. How many does each child get? Hand them out one at a time, round and round, until they are gone: each child gets **6**. We know the number of groups and want **the size of each share**.\n\n**2. Grouping (how many groups?).** A teacher has **36** pencils and puts **6** in each box. How many boxes can she fill? Keep taking out 6 at a time: she fills **6** boxes. We know the size of each group and want **how many groups**.\n\nBoth are written the same way, **36 ÷ 6 = 6**. The number being divided (36) is the **dividend**, the number we divide by (6) is the **divisor**, and the answer (6) is the **quotient**.",{"id":378,"type":80,"caption":379,"columns":380,"rows":385},"table-share-group","Sharing or grouping? Same sign, different question",[381,382,383,384],"Kind","You know…","You want…","Example",[386,391],[387,388,389,390],"Sharing","How many groups","How many in each group","18 chocolates among 3 children → 6 each",[392,389,388,393],"Grouping","40 bangles in sets of 8 → 5 sets",{"id":395,"type":144,"title":396,"items":397},"steps-repeated-sub","Grouping as taking away again and again: 20 ÷ 5",[398,402,406,410,414],{"title":399,"tag":400,"text":401},"Start","20","20 marbles, put in bags of 5.",{"title":403,"tag":404,"text":405},"Fill bag 1","20 − 5 = 15","15 marbles left.",{"title":407,"tag":408,"text":409},"Fill bag 2","15 − 5 = 10","10 left.",{"title":411,"tag":412,"text":413},"Fill bag 3","10 − 5 = 5","5 left.",{"title":415,"tag":416,"text":417},"Fill bag 4","5 − 5 = 0","Nothing left. We took 5 away **4** times, so 20 ÷ 5 = 4.",{"id":419,"type":43,"markdown":420},"div-remainder","Sometimes things do not share out perfectly. Nani has made **23** laddoos for **5** grandchildren. Each child gets **4** laddoos, which uses 4 × 5 = 20, and **3** are left over.\n\nThe amount left over is called the **remainder**. We write 23 ÷ 5 = 4 remainder 3, or **4 R 3**.\n\nA remainder is always **smaller than the number you are dividing by**. If 5 or more were left, you could give everyone one more!",{"id":422,"type":47,"variant":204,"title":423,"markdown":424},"mis-cant-share","“If it doesn't divide exactly, you can't share it”","You *can* divide 23 laddoos among 5 children: each gets 4 and 3 are left. What you do with the 3 depends on the story. Nani might keep them, break them into pieces, or give them to the neighbours. A remainder is not a mistake; it is part of the answer.",{"id":426,"type":165,"title":427,"problem":428,"steps":429,"help":435},"we-autos","How many autos?","**14** people need to get to the station. An auto can carry **3** passengers. How many autos are needed?",[430,431,432,433,434],"This is grouping: how many groups of 3 fit into 14?","14 ÷ 3 = 4 remainder 2, because 4 × 3 = 12 and 14 − 12 = 2.","4 autos carry 12 people, but 2 people are still standing on the road!","They need one more auto, so the answer is **5 autos**.","The remainder changed the answer. Always go back to the story to decide what the remainder means.",{"simplerExplanation":436},"Nobody can be left behind, so even a part-full auto counts. That is why we round up here.",{"id":438,"type":62,"prompt":439,"options":440,"explanation":449},"predict-remainder","30 children go on a picnic. Each mat seats 4 children. How many mats do they need?",[441,443,445,447],{"id":66,"label":442},"7 mats",{"id":69,"label":444},"7 remainder 2 mats",{"id":72,"label":446},"8 mats",{"id":75,"label":448},"120 mats","**8 mats.** 30 ÷ 4 = 7 remainder 2: seven mats seat 28 children, and the other 2 still need somewhere to sit, so an eighth mat is needed. \"7 remainder 2 mats\" is not something you can buy! And 120 would be 30 × 4, which is multiplying, not grouping.",{"id":451,"type":180,"itemId":452,"prompt":453,"check":454,"hints":457,"feedback":460},"pr-div-rem","four-operations.d-div-remainder","29 bananas are shared equally among 4 monkeys, and the leftover bananas are kept aside. How many bananas does each monkey get?",{"kind":184,"answer":455,"tolerance":186,"unit":456},7,"bananas",[458,459],"Which times-table fact of 4 is closest to 29 without going over?","4 × 7 = 28.",{"correct":461,"incorrect":462},"Right: 29 ÷ 4 = 7 remainder 1. Each monkey gets 7, and 1 banana is kept aside.","4 × 7 = 28 and 4 × 8 = 32, which is too many. So each monkey gets 7 and 1 is left over.",{"id":464,"type":180,"itemId":465,"prompt":466,"check":467,"hints":476,"feedback":478},"pr-div-kind","four-operations.d-div-kind","\"40 bangles are packed in sets of 8. How many sets?\" Is this sharing or grouping?",{"kind":264,"options":468,"correct":475},[469,472],{"id":470,"label":471},"share","Sharing: we know the number of groups",{"id":473,"label":474},"group","Grouping: we know the size of each group",[473],[477],"What does the number 8 tell you: how many sets, or how many bangles in each set?",{"correct":479,"incorrect":480},"Yes, grouping: each set has 8 bangles, and we want the number of sets: 40 ÷ 8 = 5.","We already know each set has 8 bangles. We want how many sets, so it is grouping: 40 ÷ 8 = 5.",{"id":482,"type":351,"component":352,"componentVersion":5,"config":483,"objective":491,"textAlternative":492,"help":493},"lab-div-facts",{"operations":484,"ranges":485,"rounds":360,"secondsTotal":186,"estimateFirst":361,"wordProblems":490},[127],{"a":486,"b":489},{"min":487,"max":488},4,100,{"min":357,"max":358},[],"Practise division facts by dividing numbers up to 100 by 2 to 10, using the times tables backwards.","This game asks twelve division questions. The number you divide by (the divisor) is from 2 to 10, and the number being divided is at most 100. Every division comes out exactly, with nothing left over. There is no timer.\n\nA question looks like **42 ÷ 7 = ?** It asks: how many sevens make 42? Or: if 42 is shared among 7, how many each? Either way the answer is **6**, because 7 × 6 = 42.\n\nTips:\n- Turn every division into a missing-number multiplication: 56 ÷ 8 asks \"8 × what = 56?\" The answer is 7.\n- Dividing by 2 is halving: 90 ÷ 2 = 45. Dividing by 10: 70 ÷ 10 = 7.\n- Some answers are bigger than 10: 96 ÷ 4 = 24, because 4 × 24 = 96. Split it: 80 ÷ 4 = 20 and 16 ÷ 4 = 4.\n\nCheck each answer by multiplying back.",{"hints":494},[495],"Which times-table fact has this number as its answer?",{"id":497,"type":53,"title":498,"eyebrow":499,"navLabel":500},"ch6","Fact families: operations that undo","Chapter 06","6 Fact families",{"id":502,"type":43,"markdown":503},"inverse-meaning","Adding and subtracting are a pair: **one undoes the other**. If you add 6 to 9 you get 15; subtract 6 from 15 and you are back at 9. Multiplying and dividing are a pair in the same way.\n\nOperations that undo each other are called **inverse operations**. Three numbers that belong together make a **fact family**:",{"id":505,"type":80,"caption":506,"columns":507,"rows":510},"table-fact-families","Two fact families",[508,509],"Numbers","Facts in the family",[511,514],[512,513],"9, 6, 15","9 + 6 = 15 · 6 + 9 = 15 · 15 − 6 = 9 · 15 − 9 = 6",[515,516],"7, 8, 56","7 × 8 = 56 · 8 × 7 = 56 · 56 ÷ 8 = 7 · 56 ÷ 7 = 8",{"id":518,"type":47,"variant":131,"title":519,"markdown":520},"aha-check-inverse","The undo button checks your answer","Because inverse operations undo each other, you always have a way to **check**:\n\n- You worked out 50 − 17 = 33? Add back: 33 + 17 = 50. ✓\n- You worked out 56 ÷ 8 = 7? Multiply back: 7 × 8 = 56. ✓\n- You found 23 ÷ 5 = 4 remainder 3? Multiply and add the remainder: 4 × 5 + 3 = 23. ✓\n\nIf the undo does not bring you back to where you started, something went wrong.",{"id":522,"type":165,"title":523,"problem":524,"steps":525},"we-missing","Find the missing number","Ravi thinks of a number, adds 17, and gets 50. What was his number?",[526,527,528,529],"Write it as a puzzle: ? + 17 = 50.","To undo adding 17, **subtract** 17.","50 − 17 = **33**.","Check: 33 + 17 = 50. ✓",{"id":531,"type":351,"component":532,"componentVersion":5,"config":533,"objective":558,"textAlternative":559},"lab-fact-families","match-pairs",{"prompt":534,"mode":535,"pairs":536},"Match each fact with the fact from the same family that undoes it.","connect",[537,540,543,546,549,552,555],{"a":538,"b":539},"7 × 8 = 56","56 ÷ 8 = 7",{"a":541,"b":542},"9 + 6 = 15","15 − 6 = 9",{"a":544,"b":545},"6 × 9 = 54","54 ÷ 9 = 6",{"a":547,"b":548},"48 + 25 = 73","73 − 25 = 48",{"a":550,"b":551},"4 × 12 = 48","48 ÷ 12 = 4",{"a":553,"b":554},"100 − 36 = 64","64 + 36 = 100",{"a":556,"b":557},"81 ÷ 9 = 9","9 × 9 = 81","Pair up facts from the same fact family, seeing how adding undoes subtracting and multiplying undoes dividing.","This game shows two columns of number facts. Connect each fact on the left with the fact on the right that **undoes** it.\n\nThe seven pairs are:\n- 7 × 8 = 56 goes with 56 ÷ 8 = 7 (dividing undoes multiplying).\n- 9 + 6 = 15 goes with 15 − 6 = 9 (subtracting undoes adding).\n- 6 × 9 = 54 goes with 54 ÷ 9 = 6.\n- 48 + 25 = 73 goes with 73 − 25 = 48.\n- 4 × 12 = 48 goes with 48 ÷ 12 = 4.\n- 100 − 36 = 64 goes with 64 + 36 = 100 (adding undoes subtracting).\n- 81 ÷ 9 = 9 goes with 9 × 9 = 81.\n\nNotice that the same three numbers appear in both facts of each pair. That is what makes them a fact family. Whenever you finish a sum, you can use its partner fact to check it.",{"id":561,"type":180,"itemId":562,"prompt":563,"check":564,"hints":566,"feedback":569},"pr-inverse","four-operations.d-inverse-divide","Complete the fact family: 9 × ? = 72. What is the missing number?",{"kind":184,"answer":565,"tolerance":186},8,[567,568],"Undo the multiplying by dividing: 72 ÷ 9.","Count up in nines: 9, 18, 27, …",{"correct":570,"incorrect":571},"Yes: 72 ÷ 9 = 8, and 9 × 8 = 72.","Divide to undo: 72 ÷ 9 = 8. Check: 9 × 8 = 72.",{"id":573,"type":53,"title":574,"eyebrow":575,"navLabel":576},"ch7","Which operation? Reading the story","Chapter 07","7 Which operation?",{"id":578,"type":43,"markdown":579},"choose-op","Here is a question to ask of every word problem: **what is happening in the story?**\n\n- Groups are being **joined**, or you want a **total** of different amounts → **add**.\n- Something is **taken away**, or you are **comparing** two amounts, or finding **how many more are needed** → **subtract**.\n- **Equal groups** are repeated, or something is \"**times as many**\" → **multiply**.\n- Something is **shared equally**, or split into **equal groups** → **divide**.\n\nWords such as \"altogether\", \"left\", \"each\" and \"share\" can be clues, but read carefully: the whole story matters more than any single word.",{"id":581,"type":47,"variant":204,"title":582,"markdown":583},"mis-keywords","“The word ‘more’ always means add”","Look at this: *\"Anu has 25 stickers. That is 8 **more** than Bala has. How many does Bala have?\"*\n\nThe word \"more\" appears, but Bala has **fewer** stickers than Anu. We need 25 − 8 = **17**. Keyword spotting would have given 33, which is wrong.\n\nPicture the story instead: Anu's pile is the bigger one, so Bala's must be smaller.",{"id":585,"type":586,"title":587,"prompt":588,"options":589},"explorer-ops","explorer","Pick an operation to see what kind of story it tells","Choose one of the four operations.",[590,600,609,617],{"id":591,"label":114,"chain":592,"badge":596,"note":599},"ex-add",[593,594,595],"Two or more amounts","Put them together","Find the total (sum)",{"text":597,"tone":598},"Joining","yes","Stories about a **total**: runs in an innings, the cost of a kirana bill, children in two classes together, rainfall over two days. Example: 34 + 27 + 18 = 79 runs.",{"id":601,"label":118,"chain":602,"badge":606,"note":608},"ex-sub",[603,604,605],"A whole or two amounts","Take away or compare","Find what is left (difference)",{"text":607,"tone":598},"Taking away · comparing","Stories about what is **left**, **how many more or fewer**, **how much taller or cheaper**, or **how many still needed**. Example: 186 − 142 = 44 runs still needed to win.",{"id":610,"label":122,"chain":611,"badge":614,"note":616},"ex-mul",[301,612,613],"Number of groups × size of each","Find the total (product)",{"text":615,"tone":598},"Equal groups · times as many","Stories with **equal** groups: trays of eggs, rows of chairs, the price of several identical items, or \"3 times as many\". Example: 4 × 30 = 120 eggs.",{"id":618,"label":126,"chain":619,"badge":623,"note":625},"ex-div",[620,621,622],"A total","Share equally or make equal groups","Each share or no. of groups (quotient)",{"text":624,"tone":598},"Sharing · grouping","Stories about **fair shares** or **how many groups**. Watch out for remainders. Example: 23 laddoos among 5 children gives 4 each with 3 left over.",{"id":627,"type":351,"component":628,"componentVersion":5,"config":629,"objective":701,"textAlternative":702},"lab-which-op","sort-game",{"prompt":630,"bins":631,"items":644,"seconds":186},"Read each little story. Which operation would you use to answer it?",[632,635,638,641],{"id":633,"label":634},"add","Add (+)",{"id":636,"label":637},"sub","Subtract (−)",{"id":639,"label":640},"mul","Multiply (×)",{"id":642,"label":643},"div","Divide (÷)",[645,649,653,657,661,665,669,673,677,681,685,689,693,697],{"id":646,"label":647,"bin":633,"why":648},"s1","Sita has ₹35 and her aunt gives her ₹20. How much does she have now?","Money is being put together: 35 + 20 = 55.",{"id":650,"label":651,"bin":633,"why":652},"s2","A class has 16 girls and 19 boys. How many children altogether?","Two groups joined into one total: 16 + 19 = 35.",{"id":654,"label":655,"bin":633,"why":656},"s3","It rained 23 mm on Monday and 41 mm on Tuesday. What was the total rainfall?","A total of two amounts: 23 + 41 = 64 mm.",{"id":658,"label":659,"bin":636,"why":660},"s4","A kite seller had 50 kites and sold 32. How many are left?","Kites are taken away: 50 − 32 = 18.",{"id":662,"label":663,"bin":636,"why":664},"s5","The Qutub Minar is about 73 m tall and a 10-storey building roughly 30 m. How much taller is the Minar?","Comparing two heights: 73 − 30 = 43 m.",{"id":666,"label":667,"bin":636,"why":668},"s6","Anu has 25 stickers, 8 more than Bala. How many does Bala have?","Bala has fewer, even though the word is 'more': 25 − 8 = 17.",{"id":670,"label":671,"bin":636,"why":672},"s7","A team needs 120 runs and has 97. How many more runs do they need?","Finding the gap to a target: 120 − 97 = 23.",{"id":674,"label":675,"bin":639,"why":676},"s8","One mango costs ₹15. What do 6 mangoes cost?","Six equal prices: 6 × 15 = ₹90.",{"id":678,"label":679,"bin":639,"why":680},"s9","A cinema hall has 12 rows of 20 seats. How many seats?","An array of rows and columns: 12 × 20 = 240.",{"id":682,"label":683,"bin":639,"why":684},"s10","Priya read 9 pages. Her brother read 4 times as many. How many did he read?","'Times as many' is scaling: 4 × 9 = 36.",{"id":686,"label":687,"bin":642,"why":688},"s11","32 children split into 4 equal teams. How many in each team?","Equal sharing into 4 teams: 32 ÷ 4 = 8.",{"id":690,"label":691,"bin":642,"why":692},"s12","60 eggs are packed in trays of 30. How many trays?","Grouping by 30: 60 ÷ 30 = 2.",{"id":694,"label":695,"bin":642,"why":696},"s13","₹90 is shared equally by 3 friends. How much does each get?","Fair sharing: 90 ÷ 3 = ₹30.",{"id":698,"label":699,"bin":639,"why":700},"s14","A hen lays 1 egg a day. How many eggs in 3 weeks?","3 weeks is 3 groups of 7 days: 3 × 7 = 21 eggs.","Decide from the story, not from single keywords, which of the four operations answers each question.","This game shows 14 story cards and four bins: add, subtract, multiply and divide. Sort each card by asking what is happening in the story.\n\n**Add:** ₹35 plus ₹20 from an aunt (55); 16 girls and 19 boys altogether (35); 23 mm and 41 mm of rain in total (64 mm).\n\n**Subtract:** 50 kites, 32 sold, how many left (18); Qutub Minar about 73 m against a roughly 30 m ten-storey building, how much taller (43 m); Anu's 25 stickers are 8 more than Bala's, so Bala has 25 − 8 = 17; need 120 runs, have 97, so 23 more.\n\n**Multiply:** mangoes at ₹15 each, 6 bought (6 × 15 = 90); 12 rows of 20 seats (240); 4 times as many pages as 9 (36); one egg a day for 3 weeks (21).\n\n**Divide:** 32 children in 4 equal teams (8 each); 60 eggs in trays of 30 (2 trays); ₹90 among 3 friends (₹30 each).\n\nWatch the sticker card: it says 'more' but needs subtracting, because Bala has fewer.",{"id":704,"type":180,"itemId":705,"prompt":706,"check":707,"hints":718,"feedback":720},"pr-choose","four-operations.d-choose-op","\"A school bus makes 4 trips and carries 45 children each trip. How many children does it carry in all?\" Which operation?",{"kind":264,"options":708,"correct":717},[709,711,713,715],{"id":66,"label":710},"Addition only: 4 + 45",{"id":69,"label":712},"Subtraction: 45 − 4",{"id":72,"label":714},"Multiplication: 4 × 45",{"id":75,"label":716},"Division: 45 ÷ 4",[72],[719],"Is the same number of children carried every trip?",{"correct":721,"incorrect":722},"Yes: 4 equal trips of 45 is 4 × 45 = 180 children.","Each trip carries the same number, so it is equal groups: 4 × 45 = 180. (Adding 45 four times gives the same answer, more slowly.)",{"id":724,"type":53,"title":725,"eyebrow":726,"navLabel":727},"ch8","Does my answer make sense?","Chapter 08","8 Sensible answers",{"id":729,"type":43,"markdown":730},"est-meaning","Before or after you calculate, ask: **roughly** how big should the answer be? A quick rough answer is called an **estimate**.\n\nThe easiest way to estimate is to **round** each number to the nearest ten (or hundred) first. To round to the nearest ten, look at the ones digit: **0–4, round down**; **5–9, round up**. So 48 → 50 and 31 → 30.\n\nEstimate 48 + 31: 50 + 30 = **80**. The exact answer is 79, very close. If you had written 709 or 17, the estimate would warn you straight away.",{"id":732,"type":165,"title":733,"problem":734,"steps":735},"we-estimate-catch","Catching a mistake with an estimate","Chintu adds 38 + 45 and writes **713**. Is that sensible?",[736,737,738,739,740],"Round: 38 → 40 and 45 → 50.","Estimate: 40 + 50 = **90**.","713 is far bigger than about 90, so something went wrong.","Where did 713 come from? Ones: 8 + 5 = 13. Tens: 3 + 4 = 7. Chintu wrote the whole 13 next to the 7, making 7|13. But 13 ones is **1 ten and 3 ones**, so the extra ten belongs with the tens.","Correct answer: 38 + 45 = **83**, which is close to the estimate of 90. ✓",{"id":742,"type":47,"variant":743,"title":744,"markdown":745},"careful-estimate","careful","An estimate is a check, not the answer","If a question asks for the exact cost, \"about ₹100\" is not enough: you still need the exact answer. Use the estimate to check it. For example, 5 packets at ₹19 is **about** 5 × ₹20 = ₹100; the exact cost is 5 × ₹19 = **₹95**. Both are useful: the estimate tells you ₹100 in your pocket is enough, and the exact answer tells you the change.",{"id":747,"type":351,"component":748,"componentVersion":5,"config":749,"objective":753,"textAlternative":754},"lab-round-tens","rounding-race",{"roundTo":750,"range":751,"rounds":565,"secondsPerRound":186},[358],{"min":358,"max":752},99,"Round two-digit numbers to the nearest ten on a number line, the first step in making quick estimates.","In this game a two-digit number appears on a number line between two tens, for example 47 between 40 and 50. Pick the ten it is closer to.\n\n- 47 is 7 steps past 40 but only 3 steps before 50, so it rounds to **50**.\n- 42 is only 2 past 40, so it rounds to **40**.\n- A number ending in **5**, such as 65, is exactly halfway. The usual rule is to round **up**, so 65 → **70**.\n\nThe quick rule: look at the ones digit. 0, 1, 2, 3 or 4: round down. 5, 6, 7, 8 or 9: round up.\n\nThere are 8 rounds and no timer. Once rounding feels easy, use it to estimate sums: 47 + 42 is about 50 + 40 = 90 (the exact answer is 89).",{"id":756,"type":351,"component":352,"componentVersion":5,"config":757,"objective":784,"textAlternative":785},"lab-add-sub",{"operations":758,"ranges":760,"rounds":360,"secondsTotal":186,"estimateFirst":361,"wordProblems":763},[115,759],"-",{"a":761,"b":762},{"min":358,"max":752},{"min":358,"max":752},[764,767,770,773,777,781],{"prompt":765,"answer":766,"unit":252,"operation":115},"At the kirana shop, rice costs ₹64 and dal costs ₹38. What is the total bill?",102,{"prompt":768,"answer":769,"unit":252,"operation":759},"Ammu pays for a ₹102 bill with a ₹200 note. How much change does she get?",98,{"prompt":771,"answer":251,"unit":772,"operation":115},"A bus has 27 passengers and 15 more get on. How many passengers now?","people",{"prompt":774,"answer":775,"unit":776,"operation":759},"A team needs 186 runs to win and has scored 142. How many more runs do they need?",44,"runs",{"prompt":778,"answer":779,"unit":780,"operation":759},"Riya is 138 cm tall and Arjun is 129 cm. How much taller is Riya?",9,"cm",{"prompt":782,"answer":783,"unit":776,"operation":115},"Kabir scored 34, Sana 27 and Dev 18. How many runs did they make together?",79,"Add and subtract two-digit numbers, including kirana-shop, bus and cricket stories, and check each answer is sensible.","This game gives twelve adding and subtracting questions with two-digit numbers. Up to half of them are short stories, picked at random from the six below. There is no timer. When you subtract, the bigger number always comes first.\n\nThe stories are:\n- Rice ₹64 and dal ₹38: total bill 64 + 38 = **₹102**.\n- Paying ₹102 with a ₹200 note: change 200 − 102 = **₹98**.\n- 27 passengers and 15 get on: 27 + 15 = **42**.\n- Need 186 runs, have 142: 186 − 142 = **44** more.\n- Riya 138 cm, Arjun 129 cm: Riya is **9 cm** taller.\n- Runs 34, 27 and 18: total **79**.\n\nFor the plain sums, use friendly tens: 57 + 28 = 57 + 30 − 2 = 85. For subtraction, count up: 72 − 45: 45 → 50 is 5, 50 → 72 is 22, so the answer is 27. After each answer, ask whether it is sensible: a difference must be smaller than the bigger number.",{"id":787,"type":180,"itemId":788,"prompt":789,"check":790,"hints":792,"feedback":794},"pr-estimate","four-operations.d-estimate-sum","Estimate 62 + 29 by rounding each number to the nearest ten. What is your estimate?",{"kind":184,"answer":791,"tolerance":186},90,[793],"62 rounds to 60. What does 29 round to?",{"correct":795,"incorrect":796},"Yes: 60 + 30 = 90. (The exact answer is 91, very close.)","Round 62 down to 60 and 29 up to 30. The estimate is 60 + 30 = 90.",{"id":798,"type":53,"title":799,"eyebrow":800,"navLabel":801},"ch9","Operations all around you","Chapter 09","9 All around you",{"id":803,"type":80,"caption":804,"columns":805,"rows":808},"table-around","The four operations in an ordinary Indian day",[806,807,84,86],"Where","Question",[809,813,817,821,825,829,833],[810,811,123,812],"Train journey","A train covers 80 km every hour. How far in 3 hours?","80 × 3 = 240 km",[814,815,123,816],"Tiffin box","Amma packs 4 rotis a day for 6 school days. How many rotis?","4 × 6 = 24",[818,819,123,820],"Rangoli","A square rangoli has 7 rows of 7 dots. How many dots?","7 × 7 = 49",[822,823,127,824],"PE lesson","32 children make 4 equal teams. How many in each?","32 ÷ 4 = 8",[826,827,123,828],"School trip","Bus fare is ₹15 per child for 30 children. Total fare?","30 × 15 = ₹450",[830,831,119,832],"Cricket","Target 150, scored 128. Runs still needed?","150 − 128 = 22",[834,835,115,836],"Rain gauge","12 mm rain in the morning and 17 mm at night. Total?","12 + 17 = 29 mm",{"id":838,"type":43,"markdown":839},"around-prose","Notice how often **multiplication** turns up: whenever the same amount repeats, such as the same fare for every child or the same distance every hour, multiplying is the fast way to the total. **Division** shows up whenever people want things to be fair. Adding and subtracting keep track of totals and gaps.\n\nBig real problems often need **more than one** operation. For a school trip, the teacher might multiply to find the total bus fare (₹450), add the cost of snacks, then subtract from the money collected to see what is left. You will meet many of these multi-step problems in the deeper layers.",{"id":841,"type":842,"title":843,"items":844},"timeline-signs","timeline","Where the signs came from",[845,849,853,857,861,865],{"time":846,"title":847,"text":848},"Long ago","Words, not signs","For thousands of years people did arithmetic with counting boards, pebbles and words. Traders in India, Egypt and China all added, multiplied and shared long before there were signs for it.",{"time":850,"title":851,"text":852},"628 CE","Brahmagupta's rules","The Indian mathematician Brahmagupta wrote down rules for calculating with zero, such as 'a number plus zero is the number itself'.",{"time":854,"title":855,"text":856},"1489","+ and − in print","Johannes Widman's arithmetic book, printed in Leipzig, is the first book to use + and − in print — at first to mark surpluses and deficits in trade problems, not as signs for the operations.",{"time":858,"title":859,"text":860},"1557","The = sign","Robert Recorde, in *The Whetstone of Witte*, chose a pair of parallel lines for 'is equal to', in his words 'bicause noe 2 thynges can be moare equalle'.",{"time":862,"title":863,"text":864},"1631","The × sign","William Oughtred used the cross × for multiplication in *Clavis Mathematicae*, written about 1628 and printed in London in 1631.",{"time":866,"title":867,"text":868},"1659","The ÷ sign","Johann Rahn used ÷ for division in *Teutsche Algebra*. Many countries write division with a colon or a slash instead, like 12 : 3 or 12 \u002F 3.",{"id":870,"type":47,"variant":175,"title":871,"markdown":872},"nuance-zero-share","Sharing with nobody","You can share 12 sweets among 3 children (4 each), among 1 child (all 12) and even share 0 sweets among 3 children (0 each). But what would it mean to share 12 sweets among **0** children? There is nobody to give them to, so the question has no sensible answer. That is why mathematicians say **you cannot divide by zero**. The deeper layers explain why more carefully.",{"id":874,"type":180,"itemId":875,"prompt":876,"check":877,"hints":879,"feedback":881},"pr-cricket","four-operations.d-sub-target","Target 150, and the team has scored 128. How many more runs are needed to reach 150?",{"kind":184,"answer":878,"tolerance":186,"unit":776},22,[880],"Count up from 128 to 130, then to 150.",{"correct":882,"incorrect":883},"Correct: 150 − 128 = 22.","Count up: 128 → 130 is 2, 130 → 150 is 20. So 22 runs are needed.",{"id":885,"type":886,"prompt":887},"reflect-day","reflection","Think about yesterday. Find one moment when you (or someone at home) added, one when you subtracted, one when you multiplied and one when you divided, even if nobody wrote anything down. Which operation did you use most?",{"id":889,"type":47,"variant":890,"title":891,"markdown":892},"try-shop","try_it","Play shop at home","Set up a pretend kirana shop with 6–8 things from the kitchen. Give each a price tag between ₹5 and ₹50.\n\n1. Buy three things and add up the bill.\n2. Pay with a ₹100 \"note\" and count up to find the change.\n3. Buy 4 of the same thing: multiply.\n4. Share a packet of biscuits fairly among your family: divide, and decide what to do with any remainder.\n\nEstimate every answer first, then work it out exactly and see how close you were.",{"id":894,"type":53,"title":895,"eyebrow":896,"navLabel":897},"ch10","Words, links and a quick check","Chapter 10","10 Words and quiz",{"id":899,"type":900,"title":901,"terms":902},"glossary-discover","glossary","Words for the four operations",[903,907,911,915,919,923,927,931,935,938,942,946,950,954,958,962,966,970,974,978],{"term":904,"meaning":905,"example":906},"addition","Putting amounts together to find a total. Sign: +.","15 + 9 = 24",{"term":908,"meaning":909,"example":910},"addend","Each number that is being added.","In 15 + 9 = 24, the addends are 15 and 9.",{"term":912,"meaning":913,"example":914},"sum","The answer to an addition; the total.","The sum of 15 and 9 is 24.",{"term":916,"meaning":917,"example":918},"subtraction","Taking one amount away from another, or finding how far apart two amounts are. Sign: −.","24 − 9 = 15",{"term":920,"meaning":921,"example":922},"difference","The answer to a subtraction: what is left, or the gap between two numbers.","The difference between 138 and 129 is 9.",{"term":924,"meaning":925,"example":926},"multiplication","Adding equal groups quickly; also 'times as many'. Sign: ×.","4 × 30 = 120 eggs in 4 trays",{"term":928,"meaning":929,"example":930},"product","The answer to a multiplication.","The product of 6 and 8 is 48.",{"term":932,"meaning":933,"example":934},"equal groups","Groups that all have the same number of things. Multiplication counts them quickly.","5 boxes of 12 laddoos",{"term":936,"meaning":937,"example":307},"array","Objects arranged in rows and columns, each row the same length.",{"term":939,"meaning":940,"example":941},"division","Sharing a total equally, or finding how many equal groups fit in it. Sign: ÷.","36 ÷ 6 = 6",{"term":943,"meaning":944,"example":945},"sharing","Division where you know the number of groups and want the size of each share.","18 chocolates among 3 children",{"term":947,"meaning":948,"example":949},"grouping","Division where you know the size of each group and want how many groups.","40 bangles in sets of 8",{"term":951,"meaning":952,"example":953},"dividend","The number being divided.","In 36 ÷ 6, the dividend is 36.",{"term":955,"meaning":956,"example":957},"divisor","The number you divide by.","In 36 ÷ 6, the divisor is 6.",{"term":959,"meaning":960,"example":961},"quotient","The answer to a division (how many in each share, or how many groups).","In 36 ÷ 6 = 6, the quotient is 6.",{"term":963,"meaning":964,"example":965},"remainder","What is left over when a number cannot be divided exactly. It is always smaller than the divisor.","23 ÷ 5 = 4 remainder 3",{"term":967,"meaning":968,"example":969},"inverse operations","Operations that undo each other: + and −, × and ÷.","15 − 6 = 9 undoes 9 + 6 = 15.",{"term":971,"meaning":972,"example":973},"fact family","A set of related facts made from the same three numbers.","7 × 8 = 56, 8 × 7 = 56, 56 ÷ 8 = 7, 56 ÷ 7 = 8",{"term":975,"meaning":976,"example":977},"estimate","A quick, rough answer, often made by rounding the numbers first.","48 + 31 is about 50 + 30 = 80.",{"term":979,"meaning":980,"example":981},"round","Replace a number by a nearby friendly number, such as the nearest ten.","47 rounds to 50; 42 rounds to 40.",{"id":983,"type":984,"conceptId":985,"relation":986,"explanation":987},"conn-number-system","connection","number-system","helps_understand","Knowing tens, hundreds and thousands (place value) is what lets you add, subtract, multiply and divide bigger numbers correctly.",{"id":989,"type":984,"conceptId":990,"relation":991,"explanation":992},"conn-properties","properties-of-numbers","related_to","Why 3 + 7 = 7 + 3 and 6 × 8 = 8 × 6, and other rules that make calculating easier, are the properties of numbers.",{"id":994,"type":984,"conceptId":995,"relation":986,"explanation":996},"conn-order","order-of-operations","When one sum has several operations, such as 3 + 4 × 5, you need rules for which to do first.",{"id":998,"type":984,"conceptId":999,"relation":1000,"explanation":1001},"conn-data","data-handling","applied_in","Finding an average (mean) of scores or rainfall means adding them all and then dividing.",{"id":1003,"type":1004,"title":1005,"questions":1006},"quiz-discover","quiz","Quick check: the four operations",[1007,1020,1033,1046,1059,1072,1084,1097,1110,1123],{"itemId":1008,"prompt":1009,"options":1010,"correct":69,"why":1019},"four-operations.d-quiz-change","Ammu buys things costing ₹45 and ₹30 and pays with ₹100. What two steps find her change?",[1011,1013,1015,1017],{"id":66,"label":1012},"Multiply 45 by 30, then divide by 100",{"id":69,"label":1014},"Add 45 + 30, then subtract the total from 100",{"id":72,"label":1016},"Subtract 30 from 45, then add 100",{"id":75,"label":1018},"Divide 100 by 2","First add to find the bill: 45 + 30 = 75. Then take it away from ₹100: 100 − 75 = ₹25.",{"itemId":1021,"prompt":1022,"options":1023,"correct":66,"why":1032},"four-operations.d-quiz-compare","Which of these is a subtraction question?",[1024,1026,1028,1030],{"id":66,"label":1025},"How many more runs did Rohit score than Virat?",{"id":69,"label":1027},"How many runs did they score together?",{"id":72,"label":1029},"How many runs in 6 overs at 8 runs an over?",{"id":75,"label":1031},"How many balls in 5 overs?","'How many more' compares two amounts, which is subtraction. The others are adding (together) and multiplying (equal groups).",{"itemId":1034,"prompt":1035,"options":1036,"correct":72,"why":1045},"four-operations.d-quiz-array","A mango orchard has 9 rows of 7 trees. How many trees?",[1037,1039,1041,1043],{"id":66,"label":1038},"16",{"id":69,"label":1040},"56",{"id":72,"label":1042},"63",{"id":75,"label":1044},"72","Rows of equal length form an array: 9 × 7 = 63. 16 is what you get by adding 9 + 7, which ignores the rows.",{"itemId":1047,"prompt":1048,"options":1049,"correct":66,"why":1058},"four-operations.d-quiz-share","36 pencils are shared equally among 4 children. How many does each get?",[1050,1052,1054,1056],{"id":66,"label":1051},"9",{"id":69,"label":1053},"32",{"id":72,"label":1055},"40",{"id":75,"label":1057},"144","Fair sharing is division: 36 ÷ 4 = 9, because 4 × 9 = 36.",{"itemId":1060,"prompt":1061,"options":1062,"correct":69,"why":1071},"four-operations.d-quiz-remainder","What is the remainder when 17 sweets are shared among 5 children?",[1063,1065,1067,1069],{"id":66,"label":1064},"0",{"id":69,"label":1066},"2",{"id":72,"label":1068},"3",{"id":75,"label":1070},"5","Each child gets 3 (that uses 15), and 17 − 15 = 2 are left. The remainder must be smaller than 5.",{"itemId":1073,"prompt":1074,"options":1075,"correct":72,"why":1083},"four-operations.d-quiz-boats","25 people cross a river in boats that each carry 6. How many boat trips are needed so everyone crosses?",[1076,1078,1080,1081],{"id":66,"label":1077},"4",{"id":69,"label":1079},"4 remainder 1",{"id":72,"label":1070},{"id":75,"label":1082},"150","25 ÷ 6 = 4 remainder 1. Four trips carry 24 people; the last person still needs a trip, so the answer is 5.",{"itemId":1085,"prompt":1086,"options":1087,"correct":69,"why":1096},"four-operations.d-quiz-inverse","Which fact can you use to check that 63 ÷ 7 = 9?",[1088,1090,1092,1094],{"id":66,"label":1089},"63 + 7 = 70",{"id":69,"label":1091},"9 × 7 = 63",{"id":72,"label":1093},"63 − 9 = 54",{"id":75,"label":1095},"9 + 7 = 16","Multiplication undoes division. If 9 × 7 = 63, then 63 ÷ 7 really is 9.",{"itemId":1098,"prompt":1099,"options":1100,"correct":69,"why":1109},"four-operations.d-quiz-more-trap","Tina has 40 beads. That is 12 more than Uma. How many beads does Uma have?",[1101,1103,1105,1107],{"id":66,"label":1102},"52",{"id":69,"label":1104},"28",{"id":72,"label":1106},"480",{"id":75,"label":1108},"12","Tina has more, so Uma has fewer: 40 − 12 = 28. Adding just because you see 'more' gives 52, which is wrong.",{"itemId":1111,"prompt":1112,"options":1113,"correct":69,"why":1122},"four-operations.d-quiz-estimate","Which is the best estimate for 49 + 22?",[1114,1116,1118,1120],{"id":66,"label":1115},"about 50",{"id":69,"label":1117},"about 70",{"id":72,"label":1119},"about 100",{"id":75,"label":1121},"about 700","Round 49 to 50 and 22 to 20: 50 + 20 = 70. The exact sum is 71.",{"itemId":1124,"prompt":1125,"options":1126,"correct":72,"why":1134},"four-operations.d-quiz-times","Meena has 8 bangles. Her sister has 3 times as many. How many does her sister have?",[1127,1129,1130,1132],{"id":66,"label":1128},"11",{"id":69,"label":1070},{"id":72,"label":1131},"24",{"id":75,"label":1133},"83","'3 times as many' means 3 lots of 8: 3 × 8 = 24.",{"id":1136,"type":1137,"title":1138,"points":1139},"cheat-sheet","summary","Cheat sheet",[1140,1141,1142,1143,1144,1145,1146,1147],"**Add (+):** put amounts together. The numbers are **addends**; the answer is the **sum**. Order does not matter: 7 + 3 = 3 + 7.","**Subtract (−):** take away *or* compare. The answer is the **difference**. 'How many left?', 'How many more?' and 'How many still needed?' are all subtraction. Order matters: 9 − 4 is not 4 − 9.","**Multiply (×):** equal groups, arrays, 'times as many'. The answer is the **product**. 4 trays of 30 eggs = 4 × 30 = 120.","**Divide (÷):** sharing (how many each?) or grouping (how many groups?). dividend ÷ divisor = **quotient**.","**Remainder:** what is left over; always smaller than the divisor. 23 ÷ 5 = 4 remainder 3. The story decides what to do with it (round up for autos and boats!).","**Inverse operations:** + and − undo each other; × and ÷ undo each other. Use the undo to check: 56 ÷ 8 = 7 because 7 × 8 = 56.","**Choosing:** read the whole story and ask what is happening: joining, taking away, comparing, equal groups or sharing. Single keywords like 'more' can trick you.","**Estimate:** round to the nearest ten first. 48 + 31 ≈ 50 + 30 = 80. If your exact answer is far from the estimate, check again.",{"id":1149,"type":1150,"sourceIds":1151},"sources-discover","sources",[1152,1153,1154,1155,1156,1157,1158,1159,1160,1161],"four-operations-ncert-math-magic-5","four-operations-khan-arithmetic","four-operations-bbc-bitesize-ks2","four-operations-britannica-arithmetic","four-operations-ncert-math-6","four-operations-mactutor-symbols-operation","four-operations-mactutor-symbols-relation","four-operations-mactutor-indian-numerals","four-operations-wiki-arithmetic","four-operations-mathsisfun-estimation",[1152,1153,1154,1155,1156,1157,1158,1159,1160,1161],"needs_review",{"generatedBy":1165,"notes":1166},"claude-code","Draft generated locally with Python-checked numbers; pending owner review.","1d0ea202189c12bc3c4c0b15058dee3bd3683ba43ba2cbef1dc0a73c46d29603",{"logic:practice":1169,"component:arith-sprint@1":1170,"component:match-pairs@1":1171,"component:sort-game@1":1172,"component:rounding-race@1":1173,"source:four-operations-bbc-bitesize-ks2":1174,"source:four-operations-britannica-arithmetic":1175,"source:four-operations-khan-arithmetic":1176,"source:four-operations-mactutor-indian-numerals":1177,"source:four-operations-mactutor-symbols-operation":1178,"source:four-operations-mactutor-symbols-relation":1179,"source:four-operations-mathsisfun-estimation":1180,"source:four-operations-ncert-math-6":1181,"source:four-operations-ncert-math-magic-5":1182,"source:four-operations-wiki-arithmetic":1183},"3d6b0fe1b15255975a32b0fcd94e8019bc959ad45cbf12e136e86149549c6878","c0c63ed40e1bca5ba6d446d43886b887a3a7709e5cd6679419f64fd3ba62afe6","2a8ee4ac87460b4e1175a4bb13c96b03d577db06dde95670eb7fcfe4ad787899","b164f45a2c8ca08f26c450768ff0231e113e9fe45381eddb34dc6d0548596c38","6c2f2d540b01b2d7cd170d87a127ebd380947b02b030b91412433582d09e54f1","07eb90693160dd6149271155357cff693c59a2006a36d3cc67c133d9f2900790","66e407fe40575b09f96c71af650fbd183b777a5b7047ed8fcc6b693b637804f6","ef6b590392c4299a9ce1be700f25412c2a52dea7c431f6dd28b6d7c1ce15374d","e5cf10b627ae5e1c1d0cb78cbdabf4a54d8dba727ef143b6e7088d5ca2bb4555","66320b0f77aa291c64a3a1723ffbc6328deb70617e06c528f28f7639f21db386","a754bcfcbcd2bf977cb2025f85d7dc4dece098b942626139cf73b2aa90595a9d","4fa46f082bd73cea931140c432a48f7ee35a7f7d6bbecb43d270505bf6c94d00","c66e4c408e97e6c5758c1b1160148cfabaf1f5e9ca4caba87383f8ca4e389c95","1856ee000472d9371b3855e38b691fbd35b221cc502b33bc887ffd8f161b636a","3fd5b7fcbfec31c612f8356e67126230dae21df1a69f1c1a06216d510015a231",{"state":1185,"reviewer":1186,"selfReview":1187,"reviewedAt":1188,"method":1189},"approved","The library owner",true,"2026-09-20T10:18:37.581Z","owner_bulk","preview-7e1cbbcc4f",1789899597064]