[{"data":1,"prerenderedAt":1234},["ShallowReactive",2],{"layer:order-of-operations:discover":3},{"layer":4,"contentHash":1216,"dependencyHashes":1217,"approval":1228,"releaseId":1233},{"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":1211,"reviewStatus":1212,"authoring":1213},1,"order-of-operations","en","discover","One line of maths, one answer","Why 2 + 3 × 4 is 14 everywhere in the world, and the simple rules that make it so","Meet the puzzle 2 + 3 × 4 through a shopping bill, learn why everyone needs one agreed order, and practise the three rules: brackets first, then × and ÷, then + and −, with partners going left to right.",[13,14,15,16,17],"Explain, using a shopping bill, why 2 + 3 × 4 is 14 and not 20.","Say why an agreed order of operations is needed, like traffic rules.","Work out expressions with brackets, × ÷ and + − in the right order.","Use left to right for partner operations and avoid the \"+ before −\" trap.","Turn short shopping and cricket stories into one line of maths.",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","Times tables and simple division",{"label":32,"value":33},"Chapters","13",{"label":35,"value":36},"Labs","Order-ops game, sort game, memory match, sprint",{"label":38,"value":39},"Key rule","Brackets → × ÷ → + −",[41,47,53,59,65,86,91,109,114,117,145,150,154,159,164,185,198,213,225,246,251,256,287,291,304,315,327,332,338,343,355,367,440,445,448,468,488,498,519,547,552,555,586,596,601,652,664,669,674,677,681,693,698,701,728,748,753,766,771,776,805,817,829,841,845,849,854,857,908,918,928,941,952,964,968,973,976,1000,1012,1024,1028,1033,1072,1179,1192,1198,1202],{"id":42,"type":43,"markdown":44,"help":45},"intro-puzzle","prose","Here is a tiny puzzle. Work it out in your head before you read on:\n\n**2 + 3 × 4 = ?**\n\nMany people say **20**. They read from left to right like a sentence: 2 + 3 is 5, and 5 × 4 is 20.\n\nOther people say **14**. They do the multiplying first: 3 × 4 is 12, and 2 + 12 is 14.\n\nSame numbers. Same signs. Two different answers. That cannot be right! If a shopkeeper and a customer read the same bill and got different totals, there would be an argument. If two engineers read the same instruction and got different numbers, a bridge might not fit together.\n\nMathematicians all over the world have agreed on **one** way to read a line like this, so that everyone gets the same answer. The agreed answer is **14**. This lesson is about that agreement: what it says, why it makes sense and how to use it without getting tricked.",{"simplerExplanation":46},"When a sum has more than one kind of sign, people could do the parts in different orders and get different answers. So there is a rule that everyone follows. For 2 + 3 × 4, the rule says multiply first: 3 × 4 = 12, then 2 + 12 = 14.",{"id":48,"type":49,"variant":50,"title":51,"markdown":52},"how-to-use","callout","observation","How to use this lesson","Read the chapters in order the first time. Whenever you see a **prediction**, choose an answer *before* you read the explanation: guessing first and then checking is one of the best ways to remember. The games in this lesson have no time pressure unless you switch it on, so play them as many times as you like.",{"id":54,"type":55,"title":56,"eyebrow":57,"navLabel":58},"ch1","chapter","A shopping bill that settles the argument","Chapter 01","1 The shopping bill",{"id":60,"type":43,"markdown":61,"help":62},"bill-story","Riya goes to the kirana shop near her house. She buys **one pencil for ₹2** and **three erasers at ₹4 each**.\n\nHow much does she pay? Think about it the way the shopkeeper does:\n\n- The pencil costs ₹2.\n- Three erasers at ₹4 each cost 3 × 4 = ₹12.\n- Altogether: 2 + 12 = **₹14**.\n\nNow write the bill as one line of maths: **2 + 3 × 4**. The real answer, the money that actually changes hands, is ₹14. Nobody would pay ₹20! The ₹20 answer would mean Riya bought *three* lots of (a pencil and an eraser), which is not what happened.\n\nSo the rule \"multiply before you add\" is not a random trick. It matches what the numbers **mean**. The 3 × 4 is one chunk (the cost of the erasers). It has to be worked out before it can be added to anything else.",{"simplerExplanation":63,"anotherExample":64},"3 × 4 is the price of the erasers, one chunk of money. Work out each chunk first, then add the chunks: 2 + 12 = 14.","A cricket batter hits 1 single and 2 fours: 1 + 2 × 4. The fours are worth 2 × 4 = 8 runs, so the score is 1 + 8 = 9 runs, not (1 + 2) × 4 = 12.",{"id":66,"type":67,"caption":68,"columns":69,"rows":73},"tbl-bill","table","Riya's bill, line by line",[70,71,72],"Item","How many × price","Cost",[74,78,82],[75,76,77],"Pencil","1 × ₹2","₹2",[79,80,81],"Eraser","3 × ₹4","₹12",[83,84,85],"**Total**","2 + 3 × 4","**₹14**",{"id":87,"type":49,"variant":88,"title":89,"markdown":90},"aha-chunks","aha","Multiplying makes chunks","Think of every × as glue. In **2 + 3 × 4**, the glue sticks the 3 and the 4 together into one chunk worth 12. The + sign sits between two separate things: the ₹2 pencil and the ₹12 chunk of erasers.\n\nThat is why × is done before +: you must finish building each chunk before you can add chunks together.",{"id":92,"type":93,"prompt":94,"options":95,"explanation":108},"predict-cricket","prediction","In a cricket over, a batter scores **2 runs** and then hits **3 sixes**. Which line of maths gives the batter's runs, using the agreed rule?",[96,99,102,105],{"id":97,"label":98},"a","2 + 3 × 6 = 20",{"id":100,"label":101},"b","2 + 3 × 6 = 30",{"id":103,"label":104},"c","2 × 3 + 6 = 12",{"id":106,"label":107},"d","(2 + 3) × 6 = 30","**2 + 3 × 6 = 20.** The three sixes are one chunk: 3 × 6 = 18 runs. Add the 2 runs: 2 + 18 = 20. Writing (2 + 3) × 6 = 30 would mean 5 sixes, which the batter did not hit. Check: 2 + 3 × 6 = 2 + 18 = 20.",{"id":110,"type":55,"title":111,"eyebrow":112,"navLabel":113},"ch2","Why the world needs one agreed order","Chapter 02","2 Why a rule?",{"id":115,"type":43,"markdown":116},"traffic","Imagine a city where some drivers keep to the left and others keep to the right. Every junction would be a crash waiting to happen. In India we all agree to drive on the **left**. There is nothing magic about the left side (many countries drive on the right), but *everyone agreeing* is what keeps us safe.\n\nThe order of operations is the same kind of agreement. There is nothing magic about it, but because every textbook, every scientist, every exam and every well-made calculator follows the **same** order, a line like 2 + 3 × 4 means exactly one thing everywhere in the world: in Chennai, in Chandigarh, in Japan and in Brazil.\n\nMaths is a language. A language only works if the speaker and the listener share the same rules.",{"id":118,"type":67,"caption":119,"columns":120,"rows":124},"tbl-agreements","Agreements that make life work",[121,122,123],"Where","The agreement","What goes wrong without it",[125,129,133,137,141],[126,127,128],"Roads in India","Keep to the left","Cars meet head-on",[130,131,132],"Cricket","The umpire's signals mean the same to everyone","Nobody knows if it was a four or a six",[134,135,136],"Reading English or Hindi","Read left to right, top to bottom","Words come out jumbled",[138,139,140],"Calendar","1 January starts the year everywhere the Gregorian calendar is used","Birthdays and exams on different days for different people",[142,143,144],"Maths","Brackets first, then × and ÷, then + and −","The same sum gives different answers",{"id":146,"type":49,"variant":147,"title":148,"markdown":149},"nuance-convention","nuance","A rule people chose, not a law of nature","Nobody *discovered* that multiplication comes before addition the way someone discovered that the Earth goes round the Sun. People **chose** this order because it is the most useful one (you saw why with Riya's bill). Mathematicians could have picked a different rule, but then everybody would have to write many more brackets. The rule we use keeps writing short and clear.",{"id":151,"type":152,"prompt":153},"reflect-agree","reflection","Think of one more rule from your everyday life that only works because *everyone* follows it (at school, at home, in a game or on the road). What would happen if half the people followed a different rule?",{"id":155,"type":55,"title":156,"eyebrow":157,"navLabel":158},"ch3","Rule one: × and ÷ before + and −","Chapter 03","3 × and ÷ first",{"id":160,"type":43,"markdown":161,"help":162},"rule1","Here is the first part of the agreement:\n\n> When there are no brackets, do all the **multiplying (×) and dividing (÷)** first. Then do the **adding (+) and subtracting (−)**.\n\nLet us try it on a few lines. Watch how each line shrinks, one step at a time, until only one number is left. That last number is the **value** of the line.\n\n- 10 − 2 × 3 = 10 − 6 = 4\n- 6 + 8 ÷ 2 = 6 + 4 = 10\n- 5 × 2 + 3 × 4 = 10 + 3 × 4 = 10 + 12 = 22\n- 20 − 12 ÷ 4 = 20 − 3 = 17\n\nIn the third line there are *two* chunks (5 × 2 and 3 × 4). Work out both chunks, then add them.",{"simplerExplanation":163},"Look for × and ÷ signs first. Work them out. Then do the + and − that are left.",{"id":165,"type":166,"title":167,"items":168},"steps-rule1","steps","How to work out 7 + 4 × 5 − 6",[169,173,177,181],{"title":170,"tag":171,"text":172},"Look along the line","spot the signs","The signs are +, × and −. Only one of them is a × or ÷ sign.",{"title":174,"tag":175,"text":176},"Do the × first","4 × 5 = 20","Replace 4 × 5 with 20. The line becomes 7 + 20 − 6.",{"title":178,"tag":179,"text":180},"Now + and −","7 + 20 = 27","Only + and − are left, so work from left to right.",{"title":182,"tag":183,"text":184},"Finish","27 − 6 = 21","One number is left: the value is **21**.",{"id":186,"type":187,"title":188,"problem":189,"steps":190,"help":196},"we-rule1","worked_example","Mangoes and bananas","Amma buys 4 kg of mangoes at ₹60 a kg and a dozen bananas for ₹50. Write one line of maths for the total, and work it out.",[191,192,193,194,195],"The mangoes cost 4 × 60 rupees. The bananas cost 50 rupees.","One line: **4 × 60 + 50**.","× first: 4 × 60 = 240. The line becomes 240 + 50.","Then +: 240 + 50 = **₹290**.","Check that it makes sense: the mangoes alone cost ₹240, so the total must be a bit more than ₹240. ✓",{"simplerExplanation":197},"Price of the mangoes (4 × 60) plus the price of the bananas (50).",{"id":199,"type":200,"itemId":201,"prompt":202,"check":203,"hints":207,"feedback":210},"pr-rule1","practice","order-of-operations.discover-rule1","Work out **9 + 6 × 3**.",{"kind":204,"answer":205,"tolerance":206},"number",27,0,[208,209],"Find the × sign first.","6 × 3 = 18. Now add the 9.",{"correct":211,"incorrect":212},"Yes: 6 × 3 = 18, then 9 + 18 = 27.","Do the × first: 6 × 3 = 18. Then 9 + 18 = 27. (If you got 45, you added first: that is the trap.)",{"id":214,"type":200,"itemId":215,"prompt":216,"check":217,"hints":219,"feedback":222},"pr-rule1b","order-of-operations.discover-rule1b","Work out **30 − 18 ÷ 3**.",{"kind":204,"answer":218,"tolerance":206},24,[220,221],"Find the ÷ sign first.","18 ÷ 3 = 6. Now take it away from 30.",{"correct":223,"incorrect":224},"Right: 18 ÷ 3 = 6, then 30 − 6 = 24.","Divide first: 18 ÷ 3 = 6. Then 30 − 6 = 24. (Subtracting first, 30 − 18 = 12, then 12 ÷ 3 = 4, is the trap.)",{"id":226,"type":227,"component":228,"componentVersion":5,"config":229,"objective":239,"textAlternative":240,"help":241},"lab-orderops-easy","interactive","order-ops",{"expressions":230,"showRuleCard":238},[84,231,232,233,234,235,236,237],"10 - 2 × 3","6 + 8 ÷ 2","20 - 12 ÷ 4","5 × 2 + 3 × 4","4 × 60 + 50","9 + 6 × 3","30 - 18 ÷ 3",true,"Choose which operation to do next, and watch each line shrink step by step until only its value is left.","This game shows one line of maths at a time. You tap the operation you think should be done next. If it is allowed, that part is worked out and the line gets shorter. A rule card reminds you: brackets first, then × and ÷, then + and −.\n\nThe lines, with the correct steps:\n\n- 2 + 3 × 4 = 2 + 12 = 14\n- 10 − 2 × 3 = 10 − 6 = 4\n- 6 + 8 ÷ 2 = 6 + 4 = 10\n- 20 − 12 ÷ 4 = 20 − 3 = 17\n- 5 × 2 + 3 × 4 = 10 + 3 × 4 = 10 + 12 = 22\n- 4 × 60 + 50 = 240 + 50 = 290\n- 9 + 6 × 3 = 9 + 18 = 27\n- 30 − 18 ÷ 3 = 30 − 6 = 24\n\nIn every line, the × or ÷ is done before the + or −. In 5 × 2 + 3 × 4 there are two × signs. Doing either one first gives 22 in the end, but the game follows the tidy habit of taking the leftmost × or ÷ first, so start with 5 × 2.",{"simplerExplanation":242,"hints":243},"Tap the × or ÷ first. When there are none left, tap the + or −.",[244,245],"If you see a × or ÷, that is your first move.","When two × or ÷ signs are both there, the one on the left is always safe to do first.",{"id":247,"type":55,"title":248,"eyebrow":249,"navLabel":250},"ch4","Rule two: brackets say “do me first”","Chapter 04","4 Brackets first",{"id":252,"type":43,"markdown":253,"help":254},"brackets","Sometimes we really *do* want to add first. For example, Riya buys **3 packets**, and each packet has **2 pencils and 4 erasers**. How many things did she buy?\n\nEach packet has 2 + 4 = 6 things, and there are 3 packets. We need to add *first* and then multiply. To show that, we put the adding inside **brackets**:\n\n**3 × (2 + 4)**\n\nBrackets are like a box with a label saying *\"work me out first\"*. Anything inside the brackets is finished before it is used for anything else.\n\n- 3 × (2 + 4) = 3 × 6 = 18\n- (2 + 3) × 4 = 5 × 4 = 20\n- (10 − 2) × 3 = 8 × 3 = 24\n- 20 ÷ (2 + 3) = 20 ÷ 5 = 4\n\nCompare the first line with **3 × 2 + 4**, which is 6 + 4 = 10. Same numbers, same signs, different answer: the brackets changed the meaning.",{"simplerExplanation":255},"Brackets are a 'do me first' box. Work out the inside of the box before anything else.",{"id":257,"type":67,"caption":258,"columns":259,"rows":263},"tbl-brackets-change","Same numbers, different brackets, different answers",[260,261,262],"Written as","Steps","Value",[264,267,271,275,279,283],[84,265,266],"2 + 3 × 4 = 2 + 12 = 14","14",[268,269,270],"(2 + 3) × 4","(2 + 3) × 4 = 5 × 4 = 20","20",[272,273,274],"10 − 2 × 3","10 − 2 × 3 = 10 − 6 = 4","4",[276,277,278],"(10 − 2) × 3","(10 − 2) × 3 = 8 × 3 = 24","24",[280,281,282],"12 ÷ 2 + 4","12 ÷ 2 + 4 = 6 + 4 = 10","10",[284,285,286],"12 ÷ (2 + 4)","12 ÷ (2 + 4) = 12 ÷ 6 = 2","2",{"id":288,"type":49,"variant":88,"title":289,"markdown":290},"aha-brackets","Brackets let you say what you mean","The rule \"× before +\" is the *default*. Brackets are how you **overrule** it when you need to. So the number sentence 2 + 3 × 4 = 14 and the number sentence (2 + 3) × 4 = 20 are both correct. They just describe different situations.\n\nYou can also add brackets that change nothing, just to make things extra clear: 2 + (3 × 4) is still 14.",{"id":292,"type":93,"prompt":293,"options":294,"explanation":303},"predict-brackets","Which of these has the **biggest** value?",[295,297,299,301],{"id":97,"label":296},"5 + 5 × 5",{"id":100,"label":298},"(5 + 5) × 5",{"id":103,"label":300},"5 × 5 + 5",{"id":106,"label":302},"5 + (5 × 5)","**(5 + 5) × 5 = 50** is the biggest. The others are all 30: 5 + 5 × 5 = 5 + 25 = 30, 5 × 5 + 5 = 25 + 5 = 30 and 5 + (5 × 5) = 5 + 25 = 30. Only the brackets in (b) force the adding to happen first, making a bigger number to multiply.",{"id":305,"type":187,"title":306,"problem":307,"steps":308},"we-sweets","Sweets for the class","A teacher has 5 boxes of laddoos. Each box has 12 laddoos. 4 laddoos are broken and thrown away. The rest are shared equally among 8 children. How many does each child get? Write one line of maths.",[309,310,311,312,313,314],"Total laddoos: 5 × 12.","Take away the broken ones: 5 × 12 − 4. This whole amount must be shared, so it goes in brackets: **(5 × 12 − 4) ÷ 8**.","Inside the brackets, × first: 5 × 12 = 60. The brackets now hold 60 − 4.","Finish the brackets: 60 − 4 = 56. The line is 56 ÷ 8.","Divide: 56 ÷ 8 = **7 laddoos each**.","Without the brackets, 5 × 12 − 4 ÷ 8 would mean dividing only the 4 broken laddoos among the children, which makes no sense (and does not even give a whole number).",{"id":316,"type":200,"itemId":317,"prompt":318,"check":319,"hints":321,"feedback":324},"pr-brackets","order-of-operations.discover-brackets","Work out **4 × (6 + 3)**.",{"kind":204,"answer":320,"tolerance":206},36,[322,323],"Brackets first: 6 + 3.","6 + 3 = 9. Now 4 × 9.",{"correct":325,"incorrect":326},"Yes: brackets first, 6 + 3 = 9, then 4 × 9 = 36.","Work out the brackets first: 6 + 3 = 9. Then 4 × 9 = 36.",{"id":328,"type":55,"title":329,"eyebrow":330,"navLabel":331},"ch5","Rule three: partners share, left goes first","Chapter 05","5 Left to right",{"id":333,"type":43,"markdown":334,"help":335},"partners","× and ÷ are **partners**: neither one is more important than the other. + and − are partners too.\n\nWhen only partners are left, just go from **left to right**, the way you read.\n\n- 10 − 3 + 2 = 7 + 2 = 9\n- 8 ÷ 4 × 2 = 2 × 2 = 4\n- 20 − 5 − 5 = 15 − 5 = 10\n- 24 ÷ 6 × 2 = 4 × 2 = 8\n\nIt is tempting to think \"+ before −\" or \"÷ before ×\", but that is **not** the rule. Try doing 10 − 3 + 2 with the + first: 3 + 2 = 5, then 10 − 5 = 5. That is wrong! If you have ₹10, spend ₹3 and then someone gives you ₹2, you have ₹9, not ₹5.",{"simplerExplanation":336,"anotherExample":337},"Two partners side by side (× with ÷, or + with −): do whichever comes first when you read from the left.","You have 20 marbles. You lose 5, then lose 5 more. You have 20 − 5 − 5 = 10 left. Doing 5 − 5 first would give 20 − 0 = 20, as if you had lost nothing!",{"id":339,"type":49,"variant":340,"title":341,"markdown":342},"misc-plus-first","misconception","“Addition always comes before subtraction”","Some people remember a word like DMAS or BODMAS and think the letters give a strict order: D, then M, then A, then S. But **A (add) and S (subtract) are partners** with equal rank, and so are **D (divide) and M (multiply)**. The letters have to go in *some* order in the word, that is all.\n\nThe real rule: do × and ÷ together from left to right, then + and − together from left to right.\n\n- 10 − 3 + 2 = **9** (not 5)\n- 8 ÷ 4 × 2 = **4** (not 1)",{"id":344,"type":93,"prompt":345,"options":346,"explanation":354},"predict-lr","What is **16 ÷ 4 × 2**?",[347,348,350,352],{"id":97,"label":286},{"id":100,"label":349},"8",{"id":103,"label":351},"32",{"id":106,"label":353},"6","**8.** ÷ and × are partners, so go left to right: 16 ÷ 4 × 2 = 4 × 2 = 8. If you did the × first (4 × 2 = 8, then 16 ÷ 8 = 2) you fell into the trap.",{"id":356,"type":200,"itemId":357,"prompt":358,"check":359,"hints":361,"feedback":364},"pr-lr","order-of-operations.discover-left-right","Work out **15 − 6 + 4**.",{"kind":204,"answer":360,"tolerance":206},13,[362,363],"+ and − are partners: go left to right.","15 − 6 = 9 first.",{"correct":365,"incorrect":366},"Yes: 15 − 6 = 9, then 9 + 4 = 13.","Left to right: 15 − 6 = 9, then 9 + 4 = 13. (Adding first gives 15 − 10 = 5, which is the trap.)",{"id":368,"type":227,"component":369,"componentVersion":5,"config":370,"objective":434,"textAlternative":435,"help":436},"lab-sort-first","sort-game",{"prompt":371,"bins":372,"items":385,"seconds":206},"What should you do FIRST in each line? Sort every card into the right bin.",[373,376,379,382],{"id":374,"label":375},"mul","Do a × first",{"id":377,"label":378},"div","Do a ÷ first",{"id":380,"label":381},"br","Do the brackets first",{"id":383,"label":384},"left","Do the + or − on the left",[386,390,394,398,402,406,410,414,418,422,426,430],{"id":387,"label":388,"bin":374,"why":389},"i1","7 + 3 × 2","× comes before +, so start with 3 × 2 = 6. Then 7 + 6 = 13.",{"id":391,"label":392,"bin":377,"why":393},"i2","20 − 12 ÷ 3","÷ comes before −, so start with 12 ÷ 3 = 4. Then 20 − 4 = 16.",{"id":395,"label":396,"bin":380,"why":397},"i3","(7 + 3) × 2","Brackets first: 7 + 3 = 10. Then 10 × 2 = 20.",{"id":399,"label":400,"bin":383,"why":401},"i4","10 − 4 + 3","Only + and − here, so go left to right: 10 − 4 = 6 first, then 6 + 3 = 9.",{"id":403,"label":404,"bin":377,"why":405},"i5","18 ÷ 3 × 2","÷ and × are partners: the ÷ is on the left, so 18 ÷ 3 = 6 first. Then 6 × 2 = 12.",{"id":407,"label":408,"bin":374,"why":409},"i6","5 × 4 ÷ 2","× and ÷ are partners: the × is on the left, so 5 × 4 = 20 first. Then 20 ÷ 2 = 10.",{"id":411,"label":412,"bin":380,"why":413},"i7","9 − (2 + 3)","Brackets first: 2 + 3 = 5. Then 9 − 5 = 4.",{"id":415,"label":416,"bin":383,"why":417},"i8","6 + 9 − 5","Only + and −: left to right. 6 + 9 = 15, then 15 − 5 = 10.",{"id":419,"label":420,"bin":380,"why":421},"i9","40 ÷ (4 + 4)","Brackets first: 4 + 4 = 8. Then 40 ÷ 8 = 5.",{"id":423,"label":424,"bin":374,"why":425},"i10","1 + 2 × 3 × 4","× before +. Work out 2 × 3 × 4 = 24 first, then 1 + 24 = 25.",{"id":427,"label":428,"bin":377,"why":429},"i11","100 − 50 ÷ 5","÷ before −: 50 ÷ 5 = 10. Then 100 − 10 = 90.",{"id":431,"label":432,"bin":383,"why":433},"i12","12 − 7 + 1","Left to right: 12 − 7 = 5, then 5 + 1 = 6.","Decide what the first step is in each line: a ×, a ÷, a bracket, or the leftmost + or −.","This is a sorting game with 12 cards and 4 bins: \"Do a × first\", \"Do a ÷ first\", \"Do the brackets first\" and \"Do the + or − on the left\".\n\n- Bracket cards: (7 + 3) × 2 = 20, 9 − (2 + 3) = 4, 40 ÷ (4 + 4) = 5.\n- × first: 7 + 3 × 2 = 13, 5 × 4 ÷ 2 = 10 (× is on the left), 1 + 2 × 3 × 4 = 25.\n- ÷ first: 20 − 12 ÷ 3 = 16, 18 ÷ 3 × 2 = 12 (÷ is on the left), 100 − 50 ÷ 5 = 90.\n- Leftmost + or −: 10 − 4 + 3 = 9, 6 + 9 − 5 = 10, 12 − 7 + 1 = 6.\n\nThe pattern: brackets beat everything; × and ÷ beat + and −; partners go left to right.",{"hints":437},[438,439],"Look for brackets first.","No brackets? Look for × or ÷. If there are both, the left one goes first.",{"id":441,"type":55,"title":442,"eyebrow":443,"navLabel":444},"ch6","The whole rule on one card","Chapter 06","6 The rule card",{"id":446,"type":43,"markdown":447},"rule-card","Put the three rules together and you have the whole order of operations. In India it is often remembered with the word **DMAS** (or its longer cousin **BODMAS**):\n\n- **D** is for **Division**\n- **M** is for **Multiplication**\n- **A** is for **Addition**\n- **S** is for **Subtraction**\n\nBODMAS adds **B** for **Brackets** at the start and **O**, which Indian and British books usually read as **Of** (as in \"half *of* 10\") and other books read as **Orders**, meaning powers. You will meet \"of\" properly in the next layer.\n\nThe word is a memory helper, and it hides one trap you already know about: D and M are partners, and A and S are partners.",{"id":449,"type":166,"title":450,"items":451},"steps-rulecard","The order of operations",[452,456,460,464],{"title":453,"tag":454,"text":455},"Brackets","first","Work out anything inside brackets ( ) before anything else.",{"title":457,"tag":458,"text":459},"× and ÷","second, left to right","Multiply and divide, whichever comes first as you read from the left.",{"title":461,"tag":462,"text":463},"+ and −","last, left to right","Add and subtract, whichever comes first as you read from the left.",{"title":465,"tag":466,"text":467},"One number left","done","When only one number is left, that is the value of the whole line.",{"id":469,"type":470,"tone":471,"items":472},"spec-rulecard","spec","blue",[473,476,480,484],{"label":453,"big":474,"value":475},"( )","Always first. They are the \"do me first\" box.",{"label":477,"big":478,"value":479},"Divide \u002F multiply","÷ ×","Partners. Left to right.",{"label":481,"big":482,"value":483},"Add \u002F subtract","+ −","Partners. Left to right, after all × and ÷ are done.",{"label":485,"big":486,"value":487},"Memory word","DMAS","Division, Multiplication, Addition, Subtraction. Longer versions: BODMAS, BIDMAS, PEMDAS.",{"id":489,"type":187,"title":490,"problem":491,"steps":492},"we-full","Using all three rules","Work out **3 + (8 − 2) × 5 − 12 ÷ 4**.",[493,494,495,496,497],"Brackets first: 8 − 2 = 6. The line is 3 + 6 × 5 − 12 ÷ 4.","× and ÷ next, left to right. 6 × 5 = 30. The line is 3 + 30 − 12 ÷ 4.","12 ÷ 4 = 3. The line is 3 + 30 − 3.","+ and − last, left to right. 3 + 30 = 33. Then 33 − 3 = 30.","The value is **30**.",{"id":499,"type":200,"itemId":500,"prompt":501,"check":502,"hints":514,"feedback":516},"pr-which-first","order-of-operations.discover-which-first","In **18 − 2 × (3 + 4)**, what do you do first?",{"kind":503,"options":504,"correct":513},"choice",[505,507,509,511],{"id":97,"label":506},"18 − 2",{"id":100,"label":508},"2 × 3",{"id":103,"label":510},"3 + 4",{"id":106,"label":512},"It does not matter",[103],[515],"Look for brackets.",{"correct":517,"incorrect":518},"Yes: brackets first. 3 + 4 = 7, then 2 × 7 = 14, then 18 − 14 = 4.","Brackets always come first: 3 + 4 = 7. Then 2 × 7 = 14, and 18 − 14 = 4.",{"id":520,"type":227,"component":521,"componentVersion":5,"config":522,"objective":542,"textAlternative":543,"help":544},"lab-match-values","match-pairs",{"prompt":523,"mode":524,"pairs":525},"Flip two cards at a time. Find each line of maths and its value.","memory",[526,527,528,531,533,536,539,540],{"a":84,"b":266},{"a":268,"b":270},{"a":529,"b":530},"10 − 3 + 2","9",{"a":532,"b":274},"8 ÷ 4 × 2",{"a":534,"b":535},"20 − 12 ÷ 4","17",{"a":537,"b":538},"3 × (2 + 4)","18",{"a":232,"b":282},{"a":234,"b":541},"22","Match each expression to its value using the agreed order: brackets, then × and ÷, then + and −.","A memory game with 16 face-down cards: 8 lines of maths and their 8 values. Turn over two at a time; if they match, they stay face up.\n\nThe pairs are: 2 + 3 × 4 = 14; (2 + 3) × 4 = 20; 10 − 3 + 2 = 9; 8 ÷ 4 × 2 = 4; 20 − 12 ÷ 4 = 17; 3 × (2 + 4) = 18; 6 + 8 ÷ 2 = 10; 5 × 2 + 3 × 4 = 22.\n\nNotice the two cards that use the same numbers, 2 + 3 × 4 and (2 + 3) × 4. They have different values, 14 and 20, because the brackets change which step comes first.",{"hints":545},[546],"Work out the value in your head before you flip the second card.",{"id":548,"type":55,"title":549,"eyebrow":550,"navLabel":551},"ch7","Maths lines from real life","Chapter 07","7 Real-life lines",{"id":553,"type":43,"markdown":554},"reallife","Every time you work out a bill, a score or a share, you are secretly using the order of operations. Turning a story into one line of maths is a skill you will use for the rest of your life, so let us practise it.\n\nThe trick is to ask: **what are the chunks?** Each \"how many × how much\" is a chunk. Chunks get added (or taken away) at the end. If something must be worked out *before* it is multiplied or shared, put it in brackets.",{"id":556,"type":67,"caption":557,"columns":558,"rows":561},"tbl-stories","From story to one line of maths",[559,560,262],"Story","One line",[562,566,570,574,578,582],[563,564,565],"3 pens at ₹12 and 2 notebooks at ₹40","3 × 12 + 2 × 40","₹116",[567,568,569],"A batter hits 4 fours, 3 sixes and 5 singles","4 × 4 + 3 × 6 + 5","39 runs",[571,572,573],"₹100 note, buy 2 samosas at ₹15: change?","100 − 2 × 15","₹70",[575,576,577],"4 friends share 2 pizzas cut into 8 slices each","2 × 8 ÷ 4","4 slices each",[579,580,581],"3 bags, each with 5 red and 7 blue marbles","3 × (5 + 7)","36 marbles",[583,584,585],"An auto fare: ₹30 to start plus ₹15 for each of 6 km","30 + 15 × 6","₹120",{"id":587,"type":187,"title":588,"problem":589,"steps":590},"we-pens","Pens and notebooks","Kabir buys **3 pens at ₹12 each** and **2 notebooks at ₹40 each**. He pays with a ₹200 note. How much change does he get?",[591,592,593,594,595],"Pens: 3 × 12. Notebooks: 2 × 40. Total bill: 3 × 12 + 2 × 40.","× first: 3 × 12 = 36 and 2 × 40 = 80. Bill = 36 + 80 = ₹116.","Change: 200 minus the **whole** bill. The bill must be worked out first, so it goes in brackets: **200 − (3 × 12 + 2 × 40)**.","200 − 116 = **₹84**.","Careful: without brackets, 200 − 3 × 12 + 2 × 40 = 200 − 36 + 80 = 244, which *adds* the notebook money back instead of taking it away. The brackets are needed.",{"id":597,"type":49,"variant":598,"title":599,"markdown":600},"careful-brackets-change","careful","Taking away a whole bill needs brackets","When you subtract a total that has more than one part, put the total in brackets.\n\n- 200 − (36 + 80) = 200 − 116 = 84 ✓\n- 200 − 36 + 80 = 164 + 80 = 244 ✗\n\nThe second line takes away the pens but then **adds** the notebooks. A shopkeeper who made that mistake would be giving away money!",{"id":602,"type":227,"component":603,"componentVersion":5,"config":604,"objective":646,"textAlternative":647,"help":648},"lab-sprint-shop","arith-sprint",{"operations":605,"ranges":608,"rounds":613,"secondsTotal":206,"estimateFirst":614,"wordProblems":615},[606,607],"×","÷",{"a":609,"b":612},{"min":610,"max":611},2,10,{"min":610,"max":611},16,false,[616,621,625,629,633,636,639,643],{"prompt":617,"answer":618,"unit":619,"operation":620},"3 pens at ₹12 each and 2 notebooks at ₹40 each. What is the total bill?",116,"₹","+",{"prompt":622,"answer":623,"unit":624,"operation":620},"A batter hits 4 fours and 3 sixes. How many runs is that altogether?",34,"runs",{"prompt":626,"answer":627,"unit":619,"operation":628},"You pay for 2 samosas at ₹15 each with a ₹100 note. How much change do you get?",70,"-",{"prompt":630,"answer":631,"unit":632,"operation":607},"2 pizzas are each cut into 8 slices and shared equally by 4 friends. How many slices each?",4,"slices",{"prompt":634,"answer":320,"unit":635,"operation":606},"3 bags each hold 5 red and 7 blue marbles. How many marbles in all?","marbles",{"prompt":637,"answer":638,"unit":619,"operation":620},"An auto charges ₹30 to start plus ₹15 for each km. What is the fare for 6 km?",120,{"prompt":640,"answer":641,"unit":642,"operation":607},"5 boxes of 12 laddoos; 4 break. The rest are shared by 8 children. How many each?",7,"laddoos",{"prompt":644,"answer":645,"unit":619,"operation":620},"Riya buys 1 pencil at ₹2 and 3 erasers at ₹4 each. What does she pay?",14,"Warm up your times tables, then turn short shopping and cricket stories into one line of maths and solve them.","A quick practice game with 16 rounds. It asks times-table and division questions with numbers from 2 to 10 (every division has a whole-number answer). Mixed in with them are story problems, picked at random from this list:\n\n- 3 pens at ₹12 and 2 notebooks at ₹40: 3 × 12 + 2 × 40 = ₹116.\n- 4 fours and 3 sixes: 4 × 4 + 3 × 6 = 34 runs.\n- 2 samosas at ₹15 paid with ₹100: 100 − 2 × 15 = ₹70 change.\n- 2 pizzas of 8 slices shared by 4: 2 × 8 ÷ 4 = 4 slices each.\n- 3 bags of 5 red and 7 blue marbles: 3 × (5 + 7) = 36.\n- Auto fare, ₹30 plus ₹15 a km for 6 km: 30 + 15 × 6 = ₹120.\n- 5 boxes of 12 laddoos, 4 broken, shared by 8: (5 × 12 − 4) ÷ 8 = 7 each.\n- A ₹2 pencil and 3 erasers at ₹4: 2 + 3 × 4 = ₹14.\n\nThere is no timer, so take your time and write the line of maths before you work it out.",{"hints":649},[650,651],"Find the chunks: each \"how many × how much\" is one chunk.","Work out each chunk, then add or take away.",{"id":653,"type":200,"itemId":654,"prompt":655,"check":656,"hints":658,"feedback":661},"pr-auto","order-of-operations.discover-auto-fare","A shared auto charges **₹10 to get in** plus **₹8 for every kilometre**. Meena rides **7 km**. Write one line of maths and find her fare in rupees.",{"kind":204,"answer":657,"tolerance":206,"unit":619},66,[659,660],"The kilometres make one chunk: 8 × 7.","Then add the ₹10.",{"correct":662,"incorrect":663},"Yes: 10 + 8 × 7 = 10 + 56 = ₹66.","The line is 10 + 8 × 7. Multiply first: 8 × 7 = 56. Then 10 + 56 = ₹66.",{"id":665,"type":49,"variant":666,"title":667,"markdown":668},"tryit-bill","try_it","Be the shopkeeper","Next time someone at home comes back from the market, ask to see the bill (or ask what they bought).\n\n1. Write the whole bill as **one** line of maths, like 2 × 45 + 3 × 20 + 60.\n2. Work it out using the order of operations.\n3. Check your answer against the bill.\n4. Now ask: how much change from ₹500? Remember the brackets!",{"id":670,"type":55,"title":671,"eyebrow":672,"navLabel":673},"ch8","Calculators do not always agree","Chapter 08","8 Calculator clash",{"id":675,"type":43,"markdown":676},"calc","Here is a surprise. Type **2 + 3 × 4 =** into two different calculators.\n\n- A **simple** calculator (the cheap kind with big buttons, often used in shops) usually shows **20**. It works out each step the moment you press the next sign: 2 + 3 = 5, then 5 × 4 = 20.\n- A **scientific** calculator (the kind used in secondary school) or the calculator app on most phones shows **14**, because it waits until you press = and then follows the order of operations.\n\nSo a simple calculator does *not* follow the agreed rule. Is it broken? No: it was designed for adding up a list of prices one after another. To get 14 on a simple calculator, you must press the keys in a helpful order: **3 × 4 = 12**, then **+ 2 = 14**.",{"id":678,"type":49,"variant":598,"title":679,"markdown":680},"careful-calc","Know your calculator","Before you trust a calculator with a long line of maths, test it with **2 + 3 × 4**. If it says 14, it follows the order of operations. If it says 20, it works step by step as you type, so you must do the × and ÷ parts first yourself (or use the memory keys M+ and MR).",{"id":682,"type":93,"prompt":683,"options":684,"explanation":692},"predict-calc","A shopkeeper types **5 + 2 × 10 =** on a simple step-by-step calculator. What will it show?",[685,687,689,691],{"id":97,"label":686},"25",{"id":100,"label":688},"70",{"id":103,"label":690},"52",{"id":106,"label":535},"**70.** A step-by-step calculator does 5 + 2 = 7 first, then 7 × 10 = 70. The agreed answer is 5 + 20 = 25, which a scientific calculator would show.",{"id":694,"type":55,"title":695,"eyebrow":696,"navLabel":697},"ch9","Be a maths detective","Chapter 09","9 Spot the mistake",{"id":699,"type":43,"markdown":700},"detective","Now that you know the rules, you can catch mistakes, even ones made by grown-ups. A mistake in the order of operations usually looks like one of these:\n\n1. **Adding before multiplying:** reading 2 + 3 × 4 like a sentence and getting 20.\n2. **Ignoring the brackets:** working out 3 × (2 + 4) as 3 × 2 + 4 = 10 instead of 18.\n3. **The letter trap:** doing + before − (or ÷ before ×) just because of the order of letters in DMAS.\n\nA good detective does not just say \"wrong!\". They find the **exact step** where the mistake happened and fix it from there.",{"id":702,"type":67,"caption":703,"columns":704,"rows":708},"tbl-detective","Three pieces of working. Where is the mistake?",[705,706,707],"Working","Mistake?","Fixed working",[709,713,717,721,725],[710,711,712],"6 + 4 × 5 = 10 × 5 = 50","Added before multiplying","6 + 4 × 5 = 6 + 20 = 26",[714,715,716],"20 − 8 + 2 = 20 − 10 = 10","Did + before − (the letter trap)","20 − 8 + 2 = 12 + 2 = 14",[718,719,720],"2 × (5 + 3) = 10 + 3 = 13","Ignored the brackets","2 × (5 + 3) = 2 × 8 = 16",[722,723,724],"18 ÷ 3 × 2 = 18 ÷ 6 = 3","Did × before ÷ (the letter trap)","18 ÷ 3 × 2 = 6 × 2 = 12",[726,727,726],"7 + 3 × 2 = 7 + 6 = 13","No mistake!",{"id":729,"type":200,"itemId":730,"prompt":731,"check":732,"hints":743,"feedback":745},"pr-detective","order-of-operations.discover-detective","Arjun wrote: **40 − 10 + 5 = 40 − 15 = 25**. Is he right?",{"kind":503,"options":733,"correct":742},[734,736,738,740],{"id":97,"label":735},"Yes, 25 is right",{"id":100,"label":737},"No, the answer is 35",{"id":103,"label":739},"No, the answer is 45",{"id":106,"label":741},"No, the answer is 15",[100],[744],"+ and − are partners. Which one is on the left?",{"correct":746,"incorrect":747},"Correct: go left to right. 40 − 10 = 30, then 30 + 5 = 35. Arjun added first.","Arjun added 10 + 5 first. Left to right: 40 − 10 = 30, then 30 + 5 = 35.",{"id":749,"type":49,"variant":750,"title":751,"markdown":752},"example-dice","example","A dice game for two players","You need 3 dice (or 3 slips of paper with numbers 1 to 6).\n\n1. Roll the 3 dice. Say you get 2, 5 and 4.\n2. Each player secretly makes the **biggest** number they can, using each number once, with + and × and brackets if they like.\n3. Show your lines and work them out *using the order of operations*. Biggest correct value wins the round.\n\nWith 2, 5 and 4: 2 + 5 × 4 = 22, but (2 + 5) × 4 = 28, and 2 × 5 × 4 = 40. Can you beat 40? (Hint: with a 1, adding can beat multiplying: for 1, 5 and 4, (1 + 5) × 4 = 24 beats 1 × 5 × 4 = 20.)",{"id":754,"type":93,"prompt":755,"options":756,"explanation":765},"predict-dice","You roll **1, 1 and 6**. Which line gives the biggest value?",[757,759,761,763],{"id":97,"label":758},"1 × 1 × 6",{"id":100,"label":760},"1 + 1 + 6",{"id":103,"label":762},"(1 + 1) × 6",{"id":106,"label":764},"1 + 1 × 6","**(1 + 1) × 6 = 12.** The others: 1 × 1 × 6 = 6, 1 + 1 + 6 = 8, 1 + 1 × 6 = 7. Multiplying by 1 does not make anything bigger, so it is better to add the 1s first (using brackets) and then multiply.",{"id":767,"type":55,"title":768,"eyebrow":769,"navLabel":770},"ch-kitchen","Maths in the kitchen","Chapter 10","10 In the kitchen",{"id":772,"type":43,"markdown":773,"help":774},"kitchen","Kitchens are full of order-of-operations maths. Think about what Amma works out when guests are coming:\n\n- **Rotis:** each of the 3 guests eats 4 rotis, and she makes 5 extra for tomorrow's tiffin. That is **4 × 3 + 5** rotis: 4 × 3 + 5 = 12 + 5 = 17.\n- **Rice:** the recipe says 2 cups of rice feed 4 people. For 12 people she needs **12 ÷ 4 × 2** cups: 12 ÷ 4 × 2 = 3 × 2 = 6. First find how many \"lots of 4 people\" there are, then use 2 cups for each lot.\n- **Laddoos:** she makes 3 trays of 8 laddoos and 6 get eaten while cooling (it happens!). Left over: **3 × 8 − 6** = 18.\n\nIn every case the multiplying makes a chunk first, and the adding or taking away happens at the end.",{"simplerExplanation":775},"Rotis for guests: 4 each for 3 guests is one chunk (12). Then add the 5 extra: 17.",{"id":777,"type":67,"caption":778,"columns":779,"rows":780},"tbl-kitchen","Kitchen stories as one line of maths",[559,560,262],[781,785,789,793,797,801],[782,783,784],"4 rotis each for 3 guests, plus 5 for tiffin","4 × 3 + 5","17 rotis",[786,787,788],"2 cups of rice feed 4 people; how much for 12?","12 ÷ 4 × 2","6 cups",[790,791,792],"3 trays of 8 laddoos, 6 eaten","3 × 8 − 6","18 laddoos",[794,795,796],"2 packets of 6 eggs, 3 eggs used in a cake","2 × 6 − 3","9 eggs",[798,799,800],"5 glasses of lassi; each needs 2 spoons of sugar and 1 of cardamom mix","5 × (2 + 1)","15 spoons",[802,803,804],"1 litre of milk (1,000 ml) minus 4 cups of 150 ml for tea","1,000 − 4 × 150","400 ml left",{"id":806,"type":187,"title":807,"problem":808,"steps":809,"help":815},"we-lassi","Sweet lassi for the family","Grandma makes lassi for **6** people. Each glass needs **200 ml of curd** and **50 ml of water**. How many millilitres of liquid does she need altogether?",[810,811,812,813,814],"One glass needs 200 + 50 ml. This must be worked out before multiplying by 6 glasses, so it goes in brackets.","One line: **6 × (200 + 50)**.","Brackets first: 200 + 50 = 250 ml per glass.","Then 6 × 250 = **1500 ml**, which is 1.5 litres.","Another way that gives the same answer: 6 × 200 + 6 × 50 = 1,200 + 300 = 1500 ml. (Curd for everyone plus water for everyone.)",{"simplerExplanation":816},"Work out one glass first (250 ml), then six glasses.",{"id":818,"type":200,"itemId":819,"prompt":820,"check":821,"hints":823,"feedback":826},"pr-rotis","order-of-operations.discover-rotis","For a party, Amma makes **3 rotis each for 8 children** and **4 rotis each for 5 adults**. How many rotis is that?",{"kind":204,"answer":822,"tolerance":206},44,[824,825],"Children: 3 × 8. Adults: 4 × 5.","24 + 20.",{"correct":827,"incorrect":828},"Yes: 3 × 8 + 4 × 5 = 24 + 20 = 44 rotis.","Two chunks: 3 × 8 = 24 and 4 × 5 = 20. Then 24 + 20 = 44.",{"id":830,"type":200,"itemId":831,"prompt":832,"check":833,"hints":835,"feedback":838},"pr-rice","order-of-operations.discover-rice","A recipe uses **3 cups of flour for every 4 people**. How many cups for **16 people**? Work out **16 ÷ 4 × 3**.",{"kind":204,"answer":834,"tolerance":206},12,[836,837],"÷ and × are partners: left to right.","16 ÷ 4 = 4 lots of people.",{"correct":839,"incorrect":840},"Yes: 16 ÷ 4 = 4 lots, and 4 × 3 = 12 cups.","Left to right: 16 ÷ 4 = 4, then 4 × 3 = 12 cups. (Doing 4 × 3 first gives 16 ÷ 12, which is not even a whole number: a clue it is wrong.)",{"id":842,"type":49,"variant":666,"title":843,"markdown":844},"tryit-recipe","Turn a family recipe into maths","Ask someone at home for a recipe you all like (poha, dal, kheer…).\n\n1. Write down how much of each thing it needs and how many people it feeds.\n2. Your family has guests: work out the amounts for **twice** as many people. Write each one as a line of maths, like 2 × 3 cups.\n3. Now work out the total cost of the ingredients, as a sum of chunks: kg × price + packets × price + …\n4. Check your lines with the order of operations before you go shopping!",{"id":846,"type":49,"variant":598,"title":847,"markdown":848},"careful-reading","Reading maths is not like reading a story","When we read a story, we go word by word from left to right. It is natural to do the same with maths, and that is exactly how the \"2 + 3 × 4 = 20\" mistake happens.\n\nA good habit: before you start, **scan the whole line** and put a finger on every × and ÷. Those are your first jobs. Only then go back to the start.",{"id":850,"type":55,"title":851,"eyebrow":852,"navLabel":853},"ch-cricket","The cricket scoreboard","Chapter 11","11 Cricket scores",{"id":855,"type":43,"markdown":856},"cricket","A cricket scorer is always doing order-of-operations maths. A batter's runs are made of **chunks**: fours, sixes, and runs taken by running. The team total adds the batters' runs and the **extras** (wides, no-balls and byes).\n\nSuppose Smriti hits **7 fours** and **2 sixes** and runs **19** more. Her score is\n\n**7 × 4 + 2 × 6 + 19** = 7 × 4 + 2 × 6 + 19 = 28 + 2 × 6 + 19 = 28 + 12 + 19 = 40 + 19 = 59 runs.\n\nWithout the order of operations, the scorer would get nonsense: reading left to right, 7 × 4 = 28, 28 + 2 = 30, 30 × 6 = 180… a score nobody made!",{"id":858,"type":859,"title":860,"prompt":861,"options":862},"explorer-scoring","explorer","How runs are counted","Pick a part of the scoreboard to see how its runs are worked out.",[863,876,887,898],{"id":864,"label":865,"chain":866,"badge":872,"note":875},"boundaries","Boundaries",[867,868,869,870,871],"Count the fours","Count the sixes","4 × fours","6 × sixes","Add the two chunks",{"text":873,"tone":874},"Multiply first, then add","yes","A batter with 5 fours and 3 sixes scored 5 × 4 + 3 × 6 = 20 + 18 = 38 runs from boundaries. The scorer works out each product before adding. Writing (5 + 3) × 4 would pretend all 8 boundaries were fours.",{"id":877,"label":878,"chain":879,"badge":884,"note":886},"running","Running between wickets",[880,881,882,883],"Count singles","Count twos","Count threes","1 × singles + 2 × twos + 3 × threes",{"text":885,"tone":874},"A sum of products","12 singles, 4 twos and 1 three: 1 × 12 + 2 × 4 + 3 × 1 = 12 + 8 + 3 = 23 runs. Every kind of run is a chunk.",{"id":888,"label":889,"chain":890,"badge":895,"note":897},"extras","Extras",[891,892,893,894],"Wides","No-balls","Byes and leg-byes","Add them all",{"text":896,"tone":874},"Only addition","Extras are just added. If a team gets 6 wides, 2 no-balls and 4 byes, that is 6 + 2 + 4 = 12 extras. With only + signs, order does not matter at all.",{"id":899,"label":900,"chain":901,"badge":905,"note":907},"total","Team total",[902,903,904,900],"Each batter's runs","Add them up","Add the extras",{"text":906,"tone":874},"Brackets optional","Team total = (sum of batters' runs) + extras. Because it is all addition, brackets change nothing here. The order of operations only matters inside each batter's chunk of fours and sixes.",{"id":909,"type":187,"title":910,"problem":911,"steps":912},"we-team","Adding up an innings","Three batters score: Riya **4 fours and 11 other runs**; Anu **2 sixes, 3 fours and 8 other runs**; Meera **27 runs**. The team also gets **9 extras**. What is the team total?",[913,914,915,916,917],"Riya: 4 × 4 + 11 = 16 + 11 = 27.","Anu: 2 × 6 + 3 × 4 + 8 = 12 + 12 + 8 = 32.","Meera: 27.","Total: 27 + 32 + 27 + 9 = **95 runs**.","As one line: 4 × 4 + 11 + 2 × 6 + 3 × 4 + 8 + 27 + 9. The order of operations works out all the products first, exactly like the scorer.",{"id":919,"type":187,"title":920,"problem":921,"steps":922},"we-extras","Wides, no-balls and a boundary","In one over, the bowler bowls **2 wides** (1 run each), **1 no-ball** (1 run, and the batter also hits a four off it), and the other balls give **3 singles**. How many runs came from the over?",[923,924,925,926,927],"Wides: 2 × 1 = 2 runs.","No-ball: 1 run for the no-ball plus the 4 the batter hit: 1 + 4 = 5 runs.","Singles: 3 × 1 = 3 runs.","One line: **2 × 1 + (1 + 4) + 3 × 1**. The brackets around (1 + 4) are optional here (they only group an addition), but they show that both runs came from the same ball.","Total: 2 + 5 + 3 = **10 runs**.",{"id":929,"type":93,"prompt":930,"options":931,"explanation":940},"predict-over","In one over a bowler gives away: **a four, a six, two singles and a wide** (1 run). Which line gives the runs in the over?",[932,934,936,938],{"id":97,"label":933},"4 + 6 + 2 × 1 + 1",{"id":100,"label":935},"(4 + 6 + 2) × 1 + 1",{"id":103,"label":937},"4 + 6 + 2 + 1",{"id":106,"label":939},"4 × 6 + 2 + 1","**(a) 4 + 6 + 2 × 1 + 1 = 13.** Two singles are 2 × 1 = 2 runs. Options (b) and (c) happen to give 13 too, because multiplying by 1 changes nothing, but (a) is the line that matches the story. (d) would mean 4 lots of 6.",{"id":942,"type":200,"itemId":943,"prompt":944,"check":945,"hints":946,"feedback":949},"pr-smriti","order-of-operations.discover-cricket-score","A batter hits **6 fours**, **1 six** and runs **14** more. How many runs did she score?",{"kind":204,"answer":822,"tolerance":206},[947,948],"Fours: 6 × 4 = 24.","24 + 6 + 14.",{"correct":950,"incorrect":951},"Yes: 6 × 4 + 1 × 6 + 14 = 24 + 6 + 14 = 44.","Multiply first: 6 × 4 = 24 and 1 × 6 = 6. Then 24 + 6 + 14 = 44.",{"id":953,"type":200,"itemId":954,"prompt":955,"check":956,"hints":958,"feedback":961},"pr-team-total","order-of-operations.discover-team-total","A team's batters score **34, 27 and 45**, and the team gets **3 × 4** runs from byes that went for four, plus **5** wides. What is the team total? Work out **34 + 27 + 45 + 3 × 4 + 5**.",{"kind":204,"answer":957,"tolerance":206},123,[959,960],"× first: 3 × 4 = 12.","34 + 27 + 45 + 12 + 5.",{"correct":962,"incorrect":963},"Yes: 34 + 27 + 45 + 12 + 5 = 123.","Multiply first: 3 × 4 = 12. Then add left to right: 34 + 27 = 61, + 45 = 106, + 12 = 118, + 5 = 123.",{"id":965,"type":49,"variant":340,"title":966,"markdown":967},"misc-bigger-first","“Do the biggest numbers first”","Some children think you should start with the biggest numbers, or with whatever looks easiest. But the order depends on the **signs**, not the size of the numbers.\n\nIn 100 − 10 × 5, you do 10 × 5 first (= 50), even though 100 is the biggest number. The answer is 100 − 50 = **50**, not 90 × 5 = 450.",{"id":969,"type":55,"title":970,"eyebrow":971,"navLabel":972},"ch-train","Buying train tickets","Chapter 12","12 Train tickets",{"id":974,"type":43,"markdown":975},"train","The Sharma family is going by train from Nagpur to their grandparents' town. An adult ticket costs **₹120**, a child's ticket costs **₹60**, and there is a **₹20** booking charge for the whole booking.\n\nThere are 2 adults and 3 children. The fare is\n\n**2 × 120 + 3 × 60 + 20** = 2 × 120 + 3 × 60 + 20 = 240 + 3 × 60 + 20 = 240 + 180 + 20 = 420 + 20 = 440.\n\nThen they buy **5 cups of chai at ₹10** on the platform. The whole trip costs **2 × 120 + 3 × 60 + 20 + 5 × 10** = ₹490.",{"id":977,"type":166,"title":978,"items":979},"steps-train","Working out the Sharma family fare",[980,984,988,992,996],{"title":981,"tag":982,"text":983},"Find the chunks","how many × price","Adults 2 × 120, children 3 × 60, booking charge 20.",{"title":985,"tag":986,"text":987},"Multiply","240 and 180","Work out each product: 2 × 120 = 240 and 3 × 60 = 180.",{"title":989,"tag":990,"text":991},"Add","240 + 180 + 20","Now add the chunks from left to right: 420 + 20 = 440.",{"title":993,"tag":994,"text":995},"Check","about right?","Roughly 250 + 200 = 450. ₹440 is close. ✓",{"title":997,"tag":998,"text":999},"Change","from ₹500","Change = 500 − (2 × 120 + 3 × 60 + 20) = 500 − 440 = ₹60. The bracket keeps the whole fare together.",{"id":1001,"type":200,"itemId":1002,"prompt":1003,"check":1004,"hints":1006,"feedback":1009},"pr-train","order-of-operations.discover-train-fare","A metro ticket costs **₹30 for adults** and **₹15 for children**. What do **3 adults and 4 children** pay?",{"kind":204,"answer":1005,"tolerance":206,"unit":619},150,[1007,1008],"Adults: 3 × 30. Children: 4 × 15.","90 + 60.",{"correct":1010,"incorrect":1011},"Yes: 3 × 30 + 4 × 15 = 90 + 60 = ₹150.","Multiply first: 3 × 30 = 90 and 4 × 15 = 60. Then 90 + 60 = ₹150.",{"id":1013,"type":200,"itemId":1014,"prompt":1015,"check":1016,"hints":1018,"feedback":1021},"pr-train-change","order-of-operations.discover-train-change","They pay for those metro tickets (₹150) with a **₹200 note**. Write the change as one line with brackets, **200 − (3 × 30 + 4 × 15)**, and work it out.",{"kind":204,"answer":1017,"tolerance":206,"unit":619},50,[1019,1020],"Brackets first: the whole fare.","200 − 150.",{"correct":1022,"incorrect":1023},"Yes: 200 − 150 = ₹50.","Brackets first: 3 × 30 + 4 × 15 = 150. Then 200 − 150 = ₹50.",{"id":1025,"type":49,"variant":340,"title":1026,"markdown":1027},"misc-no-brackets-change","Real mistake: “200 − 3 × 30 + 4 × 15”","Many children write the change as 200 − 3 × 30 + 4 × 15. Work it out: 200 − 90 + 60 = **170**. That is more than the ₹150 fare, so something is wrong! The line takes away the adults' tickets but **adds** the children's.\n\nWhenever you take away a whole bill, wrap the bill in brackets.",{"id":1029,"type":55,"title":1030,"eyebrow":1031,"navLabel":1032},"ch10","Words to know and a check-up","Chapter 13","13 Words and check-up",{"id":1034,"type":1035,"title":1036,"terms":1037},"glossary-discover","glossary","Words for this topic",[1038,1042,1046,1049,1053,1056,1059,1061,1064,1068],{"term":1039,"meaning":1040,"example":1041},"Operation","Something you do to numbers to get a new number. The four basic operations are addition, subtraction, multiplication and division.","+, −, × and ÷ are operation signs.",{"term":1043,"meaning":1044,"example":1045},"Expression","A line of numbers joined by operation signs (and maybe brackets) that has a value. It has no = sign.","2 + 3 × 4 is an expression.",{"term":262,"meaning":1047,"example":1048},"The single number an expression works out to.","The value of 2 + 3 × 4 is 14.",{"term":1050,"meaning":1051,"example":1052},"Simplify","Work an expression out step by step until only one number is left.","6 + 8 ÷ 2 simplifies to 10.",{"term":1054,"meaning":1055},"Order of operations","The agreed order for working out an expression: brackets, then × and ÷ from left to right, then + and − from left to right.",{"term":453,"meaning":1057,"example":1058},"Pairs of symbols such as ( ) that group part of an expression. Whatever is inside is worked out first.","(2 + 3) × 4 = 20",{"term":486,"meaning":1060},"A memory word for the order of operations: Division, Multiplication, Addition, Subtraction. D and M share a rank; A and S share a rank.",{"term":1062,"meaning":1063},"BODMAS","A longer memory word: Brackets, Of, Division, Multiplication, Addition, Subtraction.",{"term":1065,"meaning":1066,"example":1067},"Left to right","The rule for partners: when two operations of the same rank are next to each other, do the one on the left first.","10 − 3 + 2 = 7 + 2 = 9",{"term":1069,"meaning":1070,"example":1071},"Convention","A rule people agree to follow so that everyone understands things the same way.","Driving on the left in India is a convention.",{"id":1073,"type":1074,"title":1075,"questions":1076},"quiz-discover","quiz","Quick check-up",[1077,1087,1096,1108,1117,1130,1143,1156,1169],{"itemId":1078,"prompt":1079,"options":1080,"correct":100,"why":1086},"order-of-operations.dq-1","What is 4 + 5 × 2?",[1081,1082,1083,1084],{"id":97,"label":538},{"id":100,"label":266},{"id":103,"label":33},{"id":106,"label":1085},"11","× first: 5 × 2 = 10. Then 4 + 10 = 14.",{"itemId":1088,"prompt":1089,"options":1090,"correct":97,"why":1095},"order-of-operations.dq-2","What is (4 + 5) × 2?",[1091,1092,1093,1094],{"id":97,"label":538},{"id":100,"label":266},{"id":103,"label":33},{"id":106,"label":1085},"Brackets first: 4 + 5 = 9. Then 9 × 2 = 18.",{"itemId":1097,"prompt":1098,"options":1099,"correct":100,"why":1107},"order-of-operations.dq-3","What is 12 − 4 + 3?",[1100,1102,1103,1105],{"id":97,"label":1101},"5",{"id":100,"label":1085},{"id":103,"label":1104},"19",{"id":106,"label":1106},"1","+ and − are partners: left to right. 12 − 4 = 8, then 8 + 3 = 11.",{"itemId":1109,"prompt":1110,"options":1111,"correct":100,"why":1116},"order-of-operations.dq-4","What is 12 ÷ 2 × 3?",[1112,1113,1114,1115],{"id":97,"label":286},{"id":100,"label":538},{"id":103,"label":530},{"id":106,"label":353},"÷ and × are partners: left to right. 12 ÷ 2 = 6, then 6 × 3 = 18.",{"itemId":1118,"prompt":1119,"options":1120,"correct":100,"why":1129},"order-of-operations.dq-5","Why do we need an order of operations?",[1121,1123,1125,1127],{"id":97,"label":1122},"To make sums harder",{"id":100,"label":1124},"So the same expression means the same thing to everyone",{"id":103,"label":1126},"Because calculators need it",{"id":106,"label":1128},"Because addition is the most important","Without an agreed order, one line of maths could have several answers. The rule makes every expression mean exactly one thing.",{"itemId":1131,"prompt":1132,"options":1133,"correct":103,"why":1142},"order-of-operations.dq-6","In DMAS, what does the D stand for?",[1134,1136,1138,1140],{"id":97,"label":1135},"Distributive",{"id":100,"label":1137},"Double",{"id":103,"label":1139},"Division",{"id":106,"label":1141},"Difference","D is Division, M is Multiplication, A is Addition, S is Subtraction.",{"itemId":1144,"prompt":1145,"options":1146,"correct":97,"why":1155},"order-of-operations.dq-7","3 pens cost ₹12 each and a ruler costs ₹10. Which line gives the total?",[1147,1149,1151,1153],{"id":97,"label":1148},"3 × 12 + 10",{"id":100,"label":1150},"3 × (12 + 10)",{"id":103,"label":1152},"3 + 12 × 10",{"id":106,"label":1154},"(3 + 12) × 10","Pens: 3 × 12 = 36, plus the ruler 10: 3 × 12 + 10 = 46. The brackets in (b) would mean 3 rulers.",{"itemId":1157,"prompt":1158,"options":1159,"correct":100,"why":1168},"order-of-operations.dq-8","A simple shop calculator shows 20 for 2 + 3 × 4. Why?",[1160,1162,1164,1166],{"id":97,"label":1161},"It is broken",{"id":100,"label":1163},"It does each step as you type, left to right",{"id":103,"label":1165},"It always multiplies first",{"id":106,"label":1167},"It rounds the answer","Simple calculators work step by step: 2 + 3 = 5, then 5 × 4 = 20. A scientific calculator follows the order of operations and shows 14.",{"itemId":1170,"prompt":1171,"options":1172,"correct":100,"why":1178},"order-of-operations.dq-9","What is 30 − 10 ÷ 5?",[1173,1174,1176,1177],{"id":97,"label":274},{"id":100,"label":1175},"28",{"id":103,"label":686},{"id":106,"label":286},"÷ first: 10 ÷ 5 = 2. Then 30 − 2 = 28.",{"id":1180,"type":1181,"title":1182,"points":1183},"cheat-discover","summary","Cheat sheet",[1184,1185,1186,1187,1188,1189,1190,1191],"**Why a rule?** One line of maths must mean one thing to everyone, like everyone driving on the same side of the road.","**Brackets first.** ( ) is a \"do me first\" box: (2 + 3) × 4 = 5 × 4 = 20.","**Then × and ÷.** 2 + 3 × 4 = 2 + 12 = 14. Multiplying makes chunks; finish the chunks before adding them.","**Then + and −.** Only after every × and ÷ is done.","**Partners go left to right.** 10 − 3 + 2 = 9 and 8 ÷ 4 × 2 = 4. + does not beat −; ÷ does not beat ×.","**DMAS** = Division, Multiplication, Addition, Subtraction. BODMAS adds Brackets and Of.","**Stories to maths:** find the chunks (\"how many × how much\"), add them, and use brackets when a whole total is taken away or shared.","**Calculators:** simple ones work step by step (2 + 3 × 4 → 20); scientific ones follow the rule (→ 14).",{"id":1193,"type":1194,"conceptId":1195,"relation":1196,"explanation":1197},"conn-four-ops","connection","four-operations","helps_understand","You need to add, subtract, multiply and divide confidently before deciding which to do first.",{"id":1199,"type":1194,"conceptId":1200,"relation":1196,"explanation":1201},"conn-properties","properties-of-numbers","The distributive property explains why multiplying before adding gives the right answer for bills like 3 × 12 + 2 × 40.",{"id":1203,"type":1204,"sourceIds":1205},"sources-discover","sources",[1206,1207,1208,1209,1210],"order-of-operations-ncert-class7","order-of-operations-mathsisfun-bodmas","order-of-operations-khan-intro","order-of-operations-wikipedia","order-of-operations-openstax-prealgebra",[1206,1207,1208,1209,1210],"needs_review",{"generatedBy":1214,"notes":1215},"claude-code","Draft generated with Python-checked arithmetic (× ÷ left to right, then + − left to right); pending owner review.","314881a4df86180155c762d6b9df2254fe60744f94b52437e3173879d6f120b0",{"logic:practice":1218,"component:order-ops@1":1219,"component:sort-game@1":1220,"component:match-pairs@1":1221,"component:arith-sprint@1":1222,"source:order-of-operations-khan-intro":1223,"source:order-of-operations-mathsisfun-bodmas":1224,"source:order-of-operations-ncert-class7":1225,"source:order-of-operations-openstax-prealgebra":1226,"source:order-of-operations-wikipedia":1227},"3d6b0fe1b15255975a32b0fcd94e8019bc959ad45cbf12e136e86149549c6878","3d630ebc066352df791b020d545e1876ce4620d55c8c1c082414d6b4717255c5","b164f45a2c8ca08f26c450768ff0231e113e9fe45381eddb34dc6d0548596c38","2a8ee4ac87460b4e1175a4bb13c96b03d577db06dde95670eb7fcfe4ad787899","c0c63ed40e1bca5ba6d446d43886b887a3a7709e5cd6679419f64fd3ba62afe6","44599222ec19c395d411822012f005a3d35a5fcd89e9d519498d4099163b78b4","92959283b293565e5dbcac7d99680a63df55eef50520c997ab808634afaa0062","e7cadfedc70b54014195d6d8bc2740fb10a7e8aed3509c11f1efbde6e7698ade","a23188943c5655abd93ffecc7ea2a47277d0ffeae0cf3637ec684845500e891a","88ccf3653f7d9723bbb33a67a6548ccff64980f752cba0cd94a32bc60f72bd44",{"state":1229,"reviewer":1230,"selfReview":238,"reviewedAt":1231,"method":1232},"approved","The library owner","2026-09-20T10:18:37.581Z","owner_bulk","preview-7e1cbbcc4f",1789899598347]