[{"data":1,"prerenderedAt":1168},["ShallowReactive",2],{"layer:order-of-operations:deepen":3},{"layer":4,"contentHash":1150,"dependencyHashes":1151,"approval":1162,"releaseId":1167},{"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":1145,"reviewStatus":1146,"authoring":1147},1,"order-of-operations","en","deepen","Why the rule is the rule","Repeated addition, the distributive property, powers, the vinculum, history and how machines read maths","Justify the order of operations: why × comes before + (repeated addition, the distributive property), why partners go left to right (negatives and reciprocals), where powers fit, the vinculum and history of brackets, expression trees, RPN and edge cases.",[13,14,15,16,17],"Explain why × is done before + using repeated addition and the distributive property.","Explain the left-to-right rule by rewriting − as adding a negative and ÷ as multiplying by a reciprocal.","Place powers and the vinculum correctly in the order and simplify long multi-bracket expressions.","Describe how expression trees and Reverse Polish Notation represent the order of operations.","Reason about when rewriting steps are valid, and handle edge cases such as division by zero.",45,{"title":20,"rows":21},"Lesson plate",[22,25,28,31,34,37],{"label":23,"value":24},"Depth","Go deeper",{"label":26,"value":27},"Reading time","≈ 45 minutes",{"label":29,"value":30},"Prior knowledge","Understand and Investigate layers",{"label":32,"value":33},"Chapters","13",{"label":35,"value":36},"Labs","Long order-ops, 2 match games, sort",{"label":38,"value":39},"Big idea","A convention built on real properties",[41,45,51,57,62,89,94,99,110,148,153,158,186,191,209,214,217,251,256,271,276,281,291,332,336,341,344,354,364,374,394,406,411,414,439,443,446,474,486,491,494,560,581,586,589,620,633,645,650,653,663,672,676,696,708,713,716,759,764,774,787,799,856,861,864,868,877,889,902,907,910,914,918,923,970,1100,1115,1121,1126,1131,1135],{"id":42,"type":43,"markdown":44},"intro","prose","You can now use the order of operations confidently. This layer asks harder questions. **Why** does multiplication come before addition, and not the other way round? **Why** left to right for − and ÷? Where did brackets come from, and why do some old books use a bar instead? How do calculators and computers actually read an expression? And what happens at the edges: powers, fractions, the word \"of\", dividing by zero?\n\nBy the end you should be able to *justify* the rule, not just use it, and to simplify long expressions of the kind found in olympiad papers and Class 8 textbooks.",{"id":46,"type":47,"variant":48,"title":49,"markdown":50},"how-to","callout","observation","How to read this layer","This layer is about reasons. When you meet a claim, pause and ask \"why?\" before reading the explanation. Try the arguments with your own numbers. A reason you can explain to a friend is worth more than ten rules you have memorised.",{"id":52,"type":53,"title":54,"eyebrow":55,"navLabel":56},"ch1","chapter","Why multiplication comes before addition","Chapter 01","1 Why × before +",{"id":58,"type":43,"markdown":59,"help":60},"why-mult","Multiplication is **repeated addition**: 3 × 4 means 4 + 4 + 4. So the expression 2 + 3 × 4 is short for\n\n**2 + 4 + 4 + 4 = 14**.\n\nWriting 3 × 4 is a *compact way of writing a sum*. When you see 2 + 3 × 4, the \"3 × 4\" is really a bundle of three 4s. It would be strange to break the bundle open and add the 2 to just the 3 (the *count* of the bundle), which is what (2 + 3) × 4 does: it means five 4s.\n\nThe same goes for money: 3 × ₹12 + 2 × ₹40 is ₹12 + ₹12 + ₹12 + ₹40 + ₹40. A bill is a **sum of products**, and sums of products are by far the most common shape of real-world calculations. The order of operations is designed so that this common shape needs **no brackets at all**.",{"simplerExplanation":61},"3 × 4 is short for 4 + 4 + 4. So 2 + 3 × 4 is 2 + 4 + 4 + 4 = 14. Multiplying first is just unpacking the shorthand.",{"id":63,"type":64,"caption":65,"columns":66,"rows":70},"tbl-shapes","table","Which shape needs brackets?",[67,68,69],"Situation","Natural expression","Brackets needed?",[71,75,79,82,86],[72,73,74],"Bill: 3 items at ₹12, 2 at ₹40","3 × 12 + 2 × 40","No",[76,77,78],"Perimeter of a 7 by 5 rectangle","2 × 7 + 2 × 5 (or 2 × (7 + 5))","No (or yes, if written the second way)",[80,81,74],"Cricket: 4 fours, 3 sixes, 5 singles","4 × 4 + 3 × 6 + 5",[83,84,85],"Sharing a total among 4","(360 + 40) ÷ 4","Yes",[87,88,74],"Algebra: 3 apples and 2 bananas","3a + 2b (meaning 3 × a + 2 × b)",{"id":90,"type":47,"variant":91,"title":92,"markdown":93},"aha-algebra","aha","Algebra would be unreadable the other way round","In algebra you will write things like **3x + 2y** or **5a − 4b + 7**. Each term is a product: 3 × x, 2 × y. Because × binds more tightly than +, these need no brackets.\n\nIf the rule were \"+ before ×\", you would have to write (3 × x) + (2 × y) every single time, and every textbook would be covered in brackets. Sums of products are everywhere in mathematics, so making products bind tightly saves the most ink.",{"id":95,"type":43,"markdown":96,"help":97},"distributive","There is a deeper link with the **distributive property**:\n\n**a × (b + c) = a × b + a × c**\n\nFor example, 3 × (10 + 2) = 3 × 10 + 3 × 2 = 30 + 6 = 36. The right-hand side has no brackets, and it only means the right thing *because* × is done before +. The order of operations and the distributive property fit together: the distributive property tells you how to **remove** brackets, and the order of operations tells you how to **read** the result.\n\nThat is also why the syllabus pairs this topic with Properties of numbers. But note: the distributive property is a *property* (something always true about numbers); DMAS is a *convention* (an agreed way of reading). DMAS stands for Division, Multiplication, Addition, Subtraction, not \"distributive\".",{"anotherExample":98},"Mental maths: 7 × 98 = 7 × (100 − 2) = 700 − 14 = 686. The distributive property turns one hard product into two easy ones.",{"id":100,"type":101,"title":102,"problem":103,"steps":104},"we-dist-area","worked_example","The area picture of a × (b + c)","Show with an area argument that 6 × (5 + 3) = 6 × 5 + 6 × 3.",[105,106,107,108,109],"Draw a rectangle 6 units tall and 5 + 3 = 8 units wide. Its area is 6 × (5 + 3) = 6 × 8 = 48 squares.","Now draw a vertical line splitting the width into 5 and 3. You get two rectangles: 6 by 5 and 6 by 3.","Their areas are 6 × 5 = 30 and 6 × 3 = 18. Together: 30 + 18 = 48.","Same rectangle, same area: **6 × (5 + 3) = 6 × 5 + 6 × 3**.","Notice that the right-hand side, 6 × 5 + 6 × 3, only means \"30 + 18\" because × is done before +. The convention makes the property short to write.",{"id":111,"type":112,"component":113,"componentVersion":5,"config":114,"objective":142,"textAlternative":143,"help":144},"lab-match-dist","interactive","match-pairs",{"prompt":115,"mode":116,"pairs":117},"Connect each expression to its distributive rewrite (both sides have the same value).","connect",[118,121,124,127,130,133,136,139],{"a":119,"b":120},"7 × (100 − 2)","7 × 100 − 7 × 2 = 686",{"a":122,"b":123},"6 × (5 + 3)","6 × 5 + 6 × 3 = 48",{"a":125,"b":126},"12 × 15","12 × 10 + 12 × 5 = 180",{"a":128,"b":129},"25 × 44","25 × 40 + 25 × 4 = 1,100",{"a":131,"b":132},"9 × 99","9 × 100 − 9 × 1 = 891",{"a":134,"b":135},"(20 + 3) × 4","20 × 4 + 3 × 4 = 92",{"a":137,"b":138},"15 × 7 + 15 × 3","15 × (7 + 3) = 150",{"a":140,"b":141},"8 × 26 − 8 × 6","8 × (26 − 6) = 160","Match each product or sum to its distributive-property rewrite, seeing how the order of operations lets the rewrite drop brackets.","A connect-the-pairs game with 8 pairs. Each left card is an expression; each right card is the same value written with the distributive property.\n\n- 7 × (100 − 2) ↔ 7 × 100 − 7 × 2 = 686\n- 6 × (5 + 3) ↔ 6 × 5 + 6 × 3 = 48\n- 12 × 15 ↔ 12 × 10 + 12 × 5 = 180\n- 25 × 44 ↔ 25 × 40 + 25 × 4 = 1,100\n- 9 × 99 ↔ 9 × 100 − 9 × 1 = 891\n- (20 + 3) × 4 ↔ 20 × 4 + 3 × 4 = 92\n- 15 × 7 + 15 × 3 ↔ 15 × (7 + 3) = 150\n- 8 × 26 − 8 × 6 ↔ 8 × (26 − 6) = 160\n\nThe last two go the other way: a common factor is taken out and put in front of a bracket. The rewrites without brackets work only because × is done before + and −.",{"hints":145},[146,147],"Split the bigger number into a friendly sum or difference, like 98 = 100 − 2.","For the last two, look for a number that appears in both products.",{"id":149,"type":53,"title":150,"eyebrow":151,"navLabel":152},"ch2","Why partners go left to right","Chapter 02","2 Why left to right",{"id":154,"type":43,"markdown":155,"help":156},"why-lr","In Class 7 you meet **integers** and **fractions**, and they give a beautiful explanation of the left-to-right rule.\n\n**Subtracting is adding the opposite.** 10 − 3 is the same as 10 + (−3). So\n\n10 − 3 + 2 = 10 + (−3) + 2.\n\nNow everything is addition, and addition can be done in *any* order: 10 + (−3) + 2 = 10 + 2 + (−3) = 9. The only danger is forgetting that the − belongs to the 3. Going left to right is simply a safe way to keep each sign attached to its number.\n\n**Dividing is multiplying by the reciprocal.** 8 ÷ 4 is the same as 8 × ¼. So\n\n8 ÷ 4 × 2 = 8 × ¼ × 2 = 4,\n\nin any order. The trap 8 ÷ (4 × 2) = 1 would be 8 × ¼ × ½: it has quietly turned the × 2 into × ½.",{"simplerExplanation":157},"Think of every − as '+ a negative number' and every ÷ as '× a fraction'. Then order stops mattering, as long as each sign stays with its own number.",{"id":159,"type":64,"caption":160,"columns":161,"rows":165},"tbl-signed","Rewriting − as + and ÷ as ×",[162,163,164],"Expression","Rewritten","Value",[166,170,174,178,182],[167,168,169],"10 − 3 + 2","10 + (−3) + 2","9",[171,172,173],"50 − 20 + 10 − 5","50 + (−20) + 10 + (−5)","35",[175,176,177],"8 ÷ 4 × 2","8 × ¼ × 2","4",[179,180,181],"36 ÷ 6 ÷ 2","36 × ⅙ × ½","3",[183,184,185],"1 ÷ 2 × 3","1 × ½ × 3","3\u002F2 (= 1½)",{"id":187,"type":47,"variant":188,"title":189,"markdown":190},"nuance-fraction-bar","nuance","Why grown-up maths rarely uses ÷","After primary school, mathematicians hardly ever write ÷. They use a **fraction bar** instead: 8 ÷ 4 × 2 is written as (8\u002F4) × 2 or as a stacked fraction with 8 on top and 4 underneath, followed by × 2. The fraction bar shows exactly what is divided by what, so the left-to-right question never comes up.\n\nThe ÷ sign, called the **obelus**, is mostly used in school arithmetic and on calculator keys. Many countries (Germany, for example) use a colon, 8 : 4, for division instead.",{"id":192,"type":193,"prompt":194,"options":195,"explanation":208},"predict-half","prediction","What is **1 ÷ 2 × 4** using the standard convention?",[196,199,202,205],{"id":197,"label":198},"a","⅛",{"id":200,"label":201},"b","2",{"id":203,"label":204},"c","½",{"id":206,"label":207},"d","8","**2.** Left to right: 1 ÷ 2 = ½, then ½ × 4 = 2. Rewritten, 1 × ½ × 4 = 2. If you did 2 × 4 first you would get ⅛, which is what 1 ÷ (2 × 4) means.",{"id":210,"type":53,"title":211,"eyebrow":212,"navLabel":213},"ch3","Powers: the next level up","Chapter 03","3 Powers",{"id":215,"type":43,"markdown":216},"powers","Just as multiplication is repeated addition, a **power** (also called an index or exponent) is repeated multiplication: 3² = 3 × 3 = 9 and 2³ = 2 × 2 × 2 = 8. You will study powers properly in Class 7 and 8, but they fit neatly into the order of operations.\n\nBy the same logic as before, a power is a compact bundle of multiplications, so it **binds even more tightly than ×**. The full order is:\n\n1. Brackets\n2. Powers (the I in BIDMAS, the E in PEMDAS; some Indian books fold this into the O of BODMAS as \"orders\")\n3. × and ÷, left to right\n4. + and −, left to right\n\n- 2 + 3² = 2 + 9 = **11**, but (2 + 3)² = 5² = **25**.\n- 2 × 3² = 2 × 9 = **18**, but (2 × 3)² = 6² = **36**.\n- 10 − 2³ = 10 − 8 = **2**.",{"id":218,"type":64,"caption":219,"columns":220,"rows":222},"tbl-powers","Powers inside expressions",[162,221,164],"Power first",[223,227,231,235,239,243,247],[224,225,226],"2 + 3²","2 + 9","11",[228,229,230],"(2 + 3)²","5²","25",[232,233,234],"2 × 3²","2 × 9","18",[236,237,238],"(2 × 3)²","6²","36",[240,241,242],"4² − 3²","16 − 9","7",[244,245,246],"(4 − 3)²","1²","1",[248,249,250],"5 × 2³ ÷ 4","5 × 8 ÷ 4","10",{"id":252,"type":47,"variant":253,"title":254,"markdown":255},"misc-square-sum","misconception","“(a + b)² is a² + b²”","It is tempting to \"share\" a square over a bracket: (2 + 3)² = 2² + 3² = 4 + 9 = 13? No! (2 + 3)² = 5² = **25**.\n\nThe square applies to the **whole** bracket: (2 + 3)² = (2 + 3) × (2 + 3). Using the distributive property twice, that is 2 × 2 + 2 × 3 + 3 × 2 + 3 × 3 = 4 + 6 + 6 + 9 = 25. The two middle terms (6 + 6) are exactly what the mistake forgets.",{"id":257,"type":258,"itemId":259,"prompt":260,"check":261,"hints":265,"feedback":268},"pr-powers","practice","order-of-operations.deepen-powers","Evaluate **3 × 4² − (10 − 2 × 3)²**.",{"kind":262,"answer":263,"tolerance":264},"number",32,0,[266,267],"Bracket first: 10 − 2 × 3 = 10 − 6 = 4.","Then powers: 4² = 16 and 4² = 16.",{"correct":269,"incorrect":270},"Yes: 3 × 16 − 16 = 48 − 16 = 32.","Bracket: 10 − 6 = 4. Powers: 4² = 16 twice. Then 3 × 16 = 48, and 48 − 16 = 32.",{"id":272,"type":53,"title":273,"eyebrow":274,"navLabel":275},"ch4","The vinculum and a short history of brackets","Chapter 04","4 Vinculum & history",{"id":277,"type":43,"markdown":278,"help":279},"vinculum","Before round brackets became common, mathematicians grouped things with a horizontal line drawn **over** them, called a **vinculum** (Latin for \"bond\" or \"chain\"). A line over 5 − 2 meant exactly what (5 − 2) means today.\n\nSome Indian textbooks still include the vinculum in \"simplify\" questions, and call it the **bar**. The full BODMAS-style order in those books is: **bar first, then ( ), then { }, then [ ]**, then of, ÷, ×, +, −. Since the bar is always the innermost grouping, this is just \"innermost first\" again.\n\nThe vinculum is not extinct. You use it every day in two disguises:\n\n- The **fraction bar**: in a stacked fraction with 6 + 4 on top and 2 below, the bar groups the whole top, so it means (6 + 4) ÷ 2 = 5, not 6 + 4 ÷ 2 = 8.\n- The **square-root sign**: the line that stretches over the numbers in √(16 + 9) is a vinculum, so it means √25 = 5, not √16 + 9 = 13.",{"simplerExplanation":280},"A vinculum is a line drawn over part of an expression. It works exactly like brackets. The line in a fraction and the top of a square-root sign are both vincula.",{"id":282,"type":101,"title":283,"problem":284,"steps":285},"we-vinculum","A textbook-style \"bar\" question","Simplify: 40 − [10 + {12 − (8 − *bar over 5 − 2*)}] where the bar sits over 5 − 2.",[286,287,288,289,290],"The bar is the innermost grouping: 5 − 2 = 3. The expression is 40 − [10 + {12 − (8 − 3)}].","Round brackets: 8 − 3 = 5. The expression is 40 − [10 + {12 − 5}].","Curly brackets: 12 − 5 = 7. The expression is 40 − [10 + 7].","Square brackets: 10 + 7 = 17. The expression is 40 − 17.","Finally 40 − 17 = **23**.",{"id":292,"type":293,"title":294,"items":295},"timeline-brackets","timeline","How grouping symbols came to be",[296,300,304,308,312,316,320,324,328],{"time":297,"title":298,"text":299},"1484","A line under the group","Nicolas Chuquet, in *Le Triparty en la Science des Nombres*, draws a bar **under** the parts that belong together. Historians of notation record this as the earliest grouping symbol of any kind.",{"time":301,"title":302,"text":303},"c. 1550","Square brackets","Square brackets [ ] appear in the manuscript of Rafael Bombelli’s *Algebra* (the book itself was printed in 1572).",{"time":305,"title":306,"text":307},"1556","Round brackets appear","Round parentheses are found in Niccolò Tartaglia’s printed arithmetic *General trattato di numeri e misure* in Italy — the earliest printed work in which Cajori names them.",{"time":309,"title":310,"text":311},"1593","Braces join in","Braces { } and square brackets are both used in the 1593 edition of François Viète’s *Zetetica*. (Books often credit Viète with the overbar too, but that attribution is doubted.)",{"time":313,"title":314,"text":315},"1646","The bar moves on top","Frans van Schooten, editing Viète’s collected works, throws the parentheses away and puts a horizontal bar **above** the grouped parts. This overbar is the vinculum you still meet in Indian textbooks.",{"time":317,"title":318,"text":319},"1700s","Brackets win","Parentheses are only rarely found in printed mathematics before the eighteenth century; the vinculum was used far more. Through the 1700s brackets steadily take over, partly because a bar over the text is awkward to typeset.",{"time":321,"title":322,"text":323},"1800s–1900s","Rules written down","School textbooks spell out the order of operations explicitly as arithmetic is taught to millions of children.",{"time":325,"title":326,"text":327},"1900s","Memory words","Mnemonics like PEMDAS (USA) and BODMAS (UK, India) become common in school teaching.",{"time":329,"title":330,"text":331},"1970s on","Machines join in","Scientific calculators and programming languages build the order of operations into their design.",{"id":333,"type":47,"variant":188,"title":334,"markdown":335},"nuance-history","Nobody “invented” the order of operations","There is no single inventor and no date when the rule was decided. \"× before +\" grew naturally with algebra, where terms like 3x + 2y needed it, and was used long before anyone wrote it down as a rule. Left to right for − and ÷ was never quite as universal. Even today there are small disagreements at the edges, which you will meet in the Extend layer. The dates in the timeline are approximate: historians find earlier and later examples.",{"id":337,"type":53,"title":338,"eyebrow":339,"navLabel":340},"ch5","Long expressions, handled calmly","Chapter 05","5 Long expressions",{"id":342,"type":43,"markdown":343},"long","Olympiad and Class 8 questions often give long expressions with every feature at once. The method never changes:\n\n1. **Scan** the whole expression. Mark every bracket pair and match them up.\n2. **Innermost first.** Inside each bracket, use the normal order.\n3. **One step per line**, rewriting the whole expression each time (you can copy the parts that did not change).\n4. **Look for shortcuts** before grinding: the distributive property, cancelling, or noticing a factor of 0.\n5. **Check** with an estimate or a second method.",{"id":345,"type":101,"title":346,"problem":347,"steps":348},"we-long1","Everything at once","Simplify **[84 ÷ {3 × (9 − 2)} + 5] × (18 − 4 × 4) − 6 ÷ 3**.",[349,350,351,352,353],"Innermost brackets: (9 − 2) = 7, and (18 − 4 × 4) = 18 − 16 = 2. The expression is [84 ÷ {3 × 7} + 5] × 2 − 6 ÷ 3.","Curly bracket: 3 × 7 = 21. The expression is [84 ÷ 21 + 5] × 2 − 6 ÷ 3.","Square bracket: ÷ first, 84 ÷ 21 = 4, then 4 + 5 = 9. The expression is 9 × 2 − 6 ÷ 3.","× and ÷ left to right: 9 × 2 = 18 and 6 ÷ 3 = 2. The expression is 18 − 2.","Finally **16**.",{"id":355,"type":101,"title":356,"problem":357,"steps":358},"we-long2","Spot the shortcut","Simplify **37 × 64 + 37 × 36 − (125 − 5 × 25) × 999**.",[359,360,361,362,363],"Do not multiply 37 × 64 straight away! Look first.","The bracket: 125 − 5 × 25 = 125 − 125 = **0**. So (0) × 999 = 0. That whole term vanishes.","The first two terms share a factor of 37: 37 × 64 + 37 × 36 = 37 × (64 + 36) by the distributive property.","64 + 36 = 100, so this is 37 × 100 = 3,700.","The answer is 3,700 − 0 = **3,700**.",{"id":365,"type":101,"title":366,"problem":367,"steps":368},"we-long3","A BODMAS question with \"of\"","Simplify **3 of 12 ÷ 4 + [60 − {6 × (7 − 3)}] ÷ 9**, doing \"of\" before ÷ and ×.",[369,370,371,372,373],"\"3 of 12\" means 3 × 12 = 36, done first as a chunk. It is outside the brackets, so write it as (36): the expression is 36 ÷ 4 + [60 − {6 × (7 − 3)}] ÷ 9.","Round bracket: 7 − 3 = 4. Curly: 6 × 4 = 24. Square: 60 − 24 = 36. The expression is 36 ÷ 4 + 36 ÷ 9.","÷ first: 36 ÷ 4 = 9 and 36 ÷ 9 = 4.","Then 9 + 4 = **13**.","Here \"of\" made no difference (3 × 12 ÷ 4 = 9 either way), but in something like 36 ÷ 3 of 4 it matters: 36 ÷ 12 = 3 versus 36 ÷ 3 × 4 = 48.",{"id":375,"type":112,"component":376,"componentVersion":5,"config":377,"objective":388,"textAlternative":389,"help":390},"lab-orderops-long","order-ops",{"expressions":378,"showRuleCard":387},[379,380,381,382,383,384,385,386],"[84 ÷ [3 × (9 - 2)] + 5] × (18 - 4 × 4) - 6 ÷ 3","37 × 64 + 37 × 36 - (125 - 5 × 25) × 999","[60 - [6 × (7 - 3)]] ÷ 9 + 36 ÷ 4","(144 ÷ 12 - 2 × 3) × [5 + 45 ÷ (4 + 5)]","[(17 - 5) × (13 - 8) - 10] ÷ [2 × (9 - 4)]","1000 - [250 ÷ 5 × 4 + 3 × (60 - 5 × 11)]","48 ÷ [2 × (3 + 5)] × [7 × 8 - 6 × 9]","99 - 9 × [9 - (9 - 9 ÷ 9)]",true,"Reduce long, olympiad-style expressions with several layers of brackets, one step at a time.","Eight long expressions (curly brackets are written as square brackets, since the game uses ( ) and [ ] only). Their values:\n\n- [84 ÷ [3 × (9 − 2)] + 5] × (18 − 4 × 4) − 6 ÷ 3 = 16\n- 37 × 64 + 37 × 36 − (125 − 5 × 25) × 999 = 3,700\n- [60 − [6 × (7 − 3)]] ÷ 9 + 36 ÷ 4 = 13\n- (144 ÷ 12 − 2 × 3) × [5 + 45 ÷ (4 + 5)] = 60\n- [(17 − 5) × (13 − 8) − 10] ÷ [2 × (9 − 4)] = 5\n- 1,000 − [250 ÷ 5 × 4 + 3 × (60 − 5 × 11)] = 785\n- 48 ÷ [2 × (3 + 5)] × [7 × 8 − 6 × 9] = 6\n- 99 − 9 × [9 − (9 − 9 ÷ 9)] = 90\n\nOne in full: 99 − 9 × [9 − (9 − 9 ÷ 9)] = 99 − 9 × [9 − (9 − 1)] = 99 − 9 × [9 − 8] = 99 − 9 × 1 = 99 − 9 = 90. In the second, the bracket (125 − 5 × 25) is 0, so the 999 term vanishes.",{"hints":391},[392,393],"Find every innermost bracket first.","In the second one, look at the last bracket before you multiply big numbers.",{"id":395,"type":258,"itemId":396,"prompt":397,"check":398,"hints":400,"feedback":403},"pr-long","order-of-operations.deepen-long","Simplify **[(17 − 5) × (13 − 8) − 10] ÷ [2 × (9 − 4)]**.",{"kind":262,"answer":399,"tolerance":264},5,[401,402],"Round brackets: 12, 5 and 5.","Top square bracket: 12 × 5 − 10 = 50. Bottom: 2 × 5 = 10.",{"correct":404,"incorrect":405},"Yes: 50 ÷ 10 = 5.","Round brackets give 12, 5, 5. Then [12 × 5 − 10] = 50 and [2 × 5] = 10. 50 ÷ 10 = 5.",{"id":407,"type":53,"title":408,"eyebrow":409,"navLabel":410},"ch6","How machines read expressions","Chapter 06","6 Trees and stacks",{"id":412,"type":43,"markdown":413},"trees","A computer cannot \"see\" an expression the way you do. It turns it into a structure called an **expression tree**. The operation done **last** sits at the top; each operation has two branches for the two things it combines.\n\nFor 2 + 3 × 4:\n\n- The top of the tree is **+** (it is done last).\n- Its left branch is **2**.\n- Its right branch is **×**, which has branches **3** and **4**.\n\nFor (2 + 3) × 4 the tree is different: × at the top, with branches (+ with 2 and 3) and 4.\n\nWorking out the tree from the bottom up is exactly the order of operations. Brackets and the precedence rules are just a way of writing a tree in a single line. That is why the same expression must always give the same tree: otherwise two computers could get different answers.",{"id":415,"type":64,"caption":416,"columns":417,"rows":421},"tbl-tree","Reading an expression as a tree",[162,418,419,420],"Done last (top of tree)","Left branch","Right branch",[422,426,430,433,436],[423,424,201,425],"2 + 3 × 4","+","3 × 4",[427,428,429,177],"(2 + 3) × 4","×","2 + 3",[167,431,432,201],"+ (the rightmost)","10 − 3",[175,434,435,201],"× (the rightmost)","8 ÷ 4",[437,424,438,425],"5 × 2 + 3 × 4","5 × 2",{"id":440,"type":47,"variant":91,"title":441,"markdown":442},"aha-lr-tree","Left to right means the rightmost partner is on top","Look at 10 − 3 + 2 in the table. Because partners go left to right, the **rightmost** + is done last, so it sits at the top of the tree. The tree is ((10 − 3) + 2). This is called being **left-associative**: repeated partners group from the left.",{"id":444,"type":43,"markdown":445},"rpn","In the 1920s the Polish logician **Jan Łukasiewicz** showed that if you write each operation *before* its two numbers (now called **Polish notation**), you need **no brackets and no precedence rules at all**. In the 1950s computer scientists turned it around, putting the operation *after* its two numbers. This is **Reverse Polish Notation (RPN)**, or postfix:\n\n- 2 + 3 × 4 becomes **2 3 4 × +**\n- (2 + 3) × 4 becomes **2 3 + 4 ×**\n\nTo evaluate RPN, read left to right with a stack (a pile of numbers): push numbers onto the pile; when you meet an operation, take the top two numbers, combine them, and push the result back.\n\nFor 2 3 4 × +: pile 2; pile 2, 3; pile 2, 3, 4; × takes 3 and 4 → pile 2, 12; + takes 2 and 12 → **14**.\n\nFamous scientific calculators made by Hewlett-Packard used RPN, and computers still use stacks like this inside them to evaluate expressions.",{"id":447,"type":112,"component":113,"componentVersion":5,"config":448,"objective":469,"textAlternative":470,"help":471},"lab-match-rpn",{"prompt":449,"mode":116,"pairs":450},"Connect each Reverse Polish (postfix) expression to the ordinary expression it means.",[451,454,457,460,463,466],{"a":452,"b":453},"2 3 4 × +","2 + 3 × 4 = 14",{"a":455,"b":456},"2 3 + 4 ×","(2 + 3) × 4 = 20",{"a":458,"b":459},"10 3 − 2 +","10 − 3 + 2 = 9",{"a":461,"b":462},"8 4 ÷ 2 ×","8 ÷ 4 × 2 = 4",{"a":464,"b":465},"7 2 − 1 3 + ×","(7 − 2) × (1 + 3) = 20",{"a":467,"b":468},"20 12 4 ÷ −","20 − 12 ÷ 4 = 17","Translate between bracket-free Reverse Polish Notation and ordinary expressions, to see that precedence is really about tree shape.","A connect-the-pairs game with 6 pairs:\n\n- 2 3 4 × + ↔ 2 + 3 × 4 = 14\n- 2 3 + 4 × ↔ (2 + 3) × 4 = 20\n- 10 3 − 2 + ↔ 10 − 3 + 2 = 9\n- 8 4 ÷ 2 × ↔ 8 ÷ 4 × 2 = 4\n- 7 2 − 1 3 + × ↔ (7 − 2) × (1 + 3) = 20\n- 20 12 4 ÷ − ↔ 20 − 12 ÷ 4 = 17\n\nIn postfix, each operation acts on the two most recent results, so no brackets or precedence rules are needed. The ordinary forms need brackets exactly when the tree has a + or − below a × or ÷.",{"hints":472},[473],"Use a stack: push numbers; an operation combines the top two.",{"id":475,"type":258,"itemId":476,"prompt":477,"check":478,"hints":480,"feedback":483},"pr-rpn","order-of-operations.deepen-rpn","Evaluate the RPN expression **6 2 + 5 × 4 −**.",{"kind":262,"answer":479,"tolerance":264},36,[481,482],"6 2 + gives 8. Then 8 5 × gives 40.","Then 40 4 − gives 36.",{"correct":484,"incorrect":485},"Yes: it means (6 + 2) × 5 − 4 = 36.","Stack: 6, 2 → + → 8; 8, 5 → × → 40; 40, 4 → − → 36. In ordinary form, (6 + 2) × 5 − 4 = 36.",{"id":487,"type":53,"title":488,"eyebrow":489,"navLabel":490},"ch7","Reasoning with expressions","Chapter 07","7 Reasoning",{"id":492,"type":43,"markdown":493},"reasoning","Knowing the rule lets you **prove** things about numbers, not just compute. A few examples:\n\n**Claim:** for any numbers, a − (b − c) = a − b + c.\n*Reason:* taking away (b − c) means taking away b, which is c too much (you only needed to take away b − c), so give c back: a − b + c.\n\n**Claim:** 9 × n + n = 10 × n for every n.\n*Reason:* 9 × n + n = 9 × n + 1 × n = (9 + 1) × n = 10 × n, by the distributive property. (Try it: 9 × 7 + 7 = 63 + 7 = 70.)\n\n**Claim:** the expressions (a + b) × c and a + b × c are equal only if a = 0 or c = 1.\n*Reason:* (a + b) × c = a × c + b × c. Setting it equal to a + b × c gives a × c = a, which forces a = 0 or c = 1.\n\nIn each case, the order of operations tells us exactly what the expression means, and properties of numbers do the rest.",{"id":495,"type":112,"component":496,"componentVersion":5,"config":497,"objective":555,"textAlternative":556,"help":557},"lab-sort-valid","sort-game",{"prompt":498,"bins":499,"items":506,"seconds":264},"Is each rewriting step valid for every number, or is it a mistake? Sort the cards.",[500,503],{"id":501,"label":502},"valid","Valid step",{"id":504,"label":505},"invalid","Invalid step",[507,511,515,519,523,527,531,535,539,543,547,551],{"id":508,"label":509,"bin":501,"why":510},"v1","7 × (a + 3) → 7 × a + 21","Distributive property: 7 × a + 7 × 3 = 7 × a + 21.",{"id":512,"label":513,"bin":504,"why":514},"v2","7 × (a + 3) → 7 × a + 3","The 7 must multiply both parts: 7 × a + 21.",{"id":516,"label":517,"bin":501,"why":518},"v3","50 − (a + 8) → 50 − a − 8","Taking away a total means taking away each part.",{"id":520,"label":521,"bin":504,"why":522},"v4","50 − (a − 8) → 50 − a − 8","The − inside flips: 50 − a + 8.",{"id":524,"label":525,"bin":501,"why":526},"v5","a × b + a × c → a × (b + c)","Distributive property in reverse (taking out a common factor).",{"id":528,"label":529,"bin":504,"why":530},"v6","(a + b)² → a² + b²","(a + b)² = a² + 2 × a × b + b². Test a = b = 1: 4 versus 2.",{"id":532,"label":533,"bin":501,"why":534},"v7","a ÷ (b × c) → a ÷ b ÷ c","Dividing by a product is dividing by each factor in turn (b, c not 0).",{"id":536,"label":537,"bin":504,"why":538},"v8","a ÷ b × c → a ÷ (b × c)","Left to right gives (a ÷ b) × c. Test 8 ÷ 4 × 2: 4, not 1.",{"id":540,"label":541,"bin":501,"why":542},"v9","a − b + c → a + c − b","Each number keeps its own sign; order of + and − terms can then change.",{"id":544,"label":545,"bin":504,"why":546},"v10","a − b + c → a − (b + c)","That would take c away instead of adding it. a − (b − c) is the correct bracketed form.",{"id":548,"label":549,"bin":501,"why":550},"v11","2 × a × 3 → 6 × a","Multiplication is commutative and associative: 2 × 3 × a = 6 × a.",{"id":552,"label":553,"bin":504,"why":554},"v12","2 + a × 3 → (2 + a) × 3","× binds first: 2 + 3 × a. Test a = 1: 5 versus 9.","Judge whether each algebra-style rewriting step is always valid, using the order of operations and number properties.","A sorting game with 12 rewriting steps and two bins.\n\nValid: 7 × (a + 3) → 7 × a + 21; 50 − (a + 8) → 50 − a − 8; a × b + a × c → a × (b + c); a ÷ (b × c) → a ÷ b ÷ c; a − b + c → a + c − b; 2 × a × 3 → 6 × a.\n\nInvalid: 7 × (a + 3) → 7 × a + 3 (the 7 must multiply the 3 too); 50 − (a − 8) → 50 − a − 8 (should be + 8); (a + b)² → a² + b² (missing 2 × a × b); a ÷ b × c → a ÷ (b × c) (left to right gives (a ÷ b) × c); a − b + c → a − (b + c); 2 + a × 3 → (2 + a) × 3.\n\nA quick test with small numbers catches every invalid step.",{"hints":558},[559],"Test each step with a = 1, b = 2, c = 3 (or 8, 4, 2).",{"id":561,"type":258,"itemId":562,"prompt":563,"check":564,"hints":576,"feedback":578},"pr-proof","order-of-operations.deepen-proof","Which argument correctly shows that **99 × n + n = 100 × n** for every n?",{"kind":565,"options":566,"correct":575},"choice",[567,569,571,573],{"id":197,"label":568},"It works for n = 1 and n = 2, so it is always true",{"id":200,"label":570},"99 × n + n = 99 × n + 1 × n = (99 + 1) × n = 100 × n",{"id":203,"label":572},"99 × n + n = 99 × (n + n) = 100 × n",{"id":206,"label":574},"Add first: 99 × (n + n)",[200],[577],"Write n as 1 × n.",{"correct":579,"incorrect":580},"Yes: writing n as 1 × n and using the distributive property proves it for every n.","Examples cannot prove \"always\". (c) and (d) wrongly add n + n first. The proof: n = 1 × n, so 99 × n + 1 × n = (99 + 1) × n = 100 × n.",{"id":582,"type":53,"title":583,"eyebrow":584,"navLabel":585},"ch-minimal","How few brackets do you need?","Chapter 08","8 Fewest brackets",{"id":587,"type":43,"markdown":588},"minimal","The order of operations exists to **save brackets**. So here is a good test of understanding: given a *fully bracketed* expression, remove every bracket you can **without changing the value for any numbers**.\n\nRules of thumb that follow from the convention:\n\n- Brackets around a **× or ÷ chunk** that is being added or subtracted can go: (a × b) + c → a × b + c.\n- Brackets on the **left** of a partner chain can go, because left to right does them first anyway: (a − b) + c → a − b + c, and (a ÷ b) × c → a ÷ b × c.\n- Brackets on the **right** of a − or ÷ must stay (unless you flip signs): a − (b + c) and a ÷ (b × c) need them.\n- Brackets around a + or − that is then multiplied or divided must stay: (a + b) × c.\n- Brackets around a whole sum that is added can go: a + (b + c) → a + b + c, because addition is associative.",{"id":590,"type":64,"caption":591,"columns":592,"rows":595},"tbl-minimal","Removing unnecessary brackets",[593,594,164],"Fully bracketed","Fewest brackets",[596,600,603,606,609,612,616],[597,598,599],"((2 + 3) × 4) − 1","(2 + 3) × 4 − 1","19",[601,602,33],"(2 + (3 × 4)) − 1","2 + 3 × 4 − 1",[604,605,169],"(12 ÷ 4) × 3","12 ÷ 4 × 3",[607,608,246],"12 ÷ (4 × 3)","12 ÷ (4 × 3) (must stay)",[610,611,250],"(20 − 5) − (3 + 2)","20 − 5 − (3 + 2)",[613,614,615],"(6 × 5) + (8 ÷ 2)","6 × 5 + 8 ÷ 2","34",[617,618,619],"9 + (4 + 7)","9 + 4 + 7","20",{"id":621,"type":193,"prompt":622,"options":623,"explanation":632},"predict-minimal","Which brackets in **(a − b) − (c − d)** can be removed without changing the value for any numbers?",[624,626,628,630],{"id":197,"label":625},"Both pairs",{"id":200,"label":627},"Only the first pair",{"id":203,"label":629},"Only the second pair",{"id":206,"label":631},"Neither","**Only the first pair.** (a − b) is on the left of a − chain, so left to right does it first anyway: a − b − (c − d). The second pair sits to the right of a −, so it must stay (or you must flip its sign: a − b − c + d). Test with 10, 4, 3, 1: (10 − 4) − (3 − 1) = 6 − 2 = 4, and 10 − 4 − 3 + 1 = 4 ✓, but 10 − 4 − 3 − 1 = 2 ✗.",{"id":634,"type":258,"itemId":635,"prompt":636,"check":637,"hints":639,"feedback":642},"pr-minimal","order-of-operations.deepen-minimal","How many of the three bracket pairs in **((8 × 3) − (10 ÷ 2)) + (4 × 5)** can be removed without changing the value?",{"kind":262,"answer":638,"tolerance":264},3,[640,641],"Each pair surrounds a × or ÷ chunk, or sits on the left of a + − chain.","Rewrite: 8 × 3 − 10 ÷ 2 + 4 × 5.",{"correct":643,"incorrect":644},"Yes: all three. 8 × 3 − 10 ÷ 2 + 4 × 5 = 24 − 5 + 20 = 39, the same value.","All three can go: 8 × 3 − 10 ÷ 2 + 4 × 5 = 24 − 5 + 20 = 39, exactly what the bracketed version gives.",{"id":646,"type":53,"title":647,"eyebrow":648,"navLabel":649},"ch-fracdec","The same rule with fractions and decimals","Chapter 09","9 Fractions, decimals",{"id":651,"type":43,"markdown":652},"fracdec","The order of operations does not care what kind of numbers you use. Fractions, decimals and (later) negative numbers obey exactly the same levels. What changes is that the arithmetic inside each step is harder, so mistakes in the *order* are easier to miss.\n\n- **½ + ⅓ × 6** = ½ + 2 = **2½**. (× first: ⅓ × 6 = 2.) Reading left to right would give (½ + ⅓) × 6 = ⅚ × 6 = 5.\n- **0.5 × 4 + 1.5** = 2 + 1.5 = **3.5**.\n- **2.4 ÷ 0.6 × 2** = 4 × 2 = **8** (left to right), not 2.4 ÷ 1.2 = 2.\n- **¾ of (20 − 4)** = ¾ × 16 = **12**.\n\nWith fractions, the \"of\" and the fraction bar often do the bracketing for you: in a stacked fraction such as (3 + 5) over (6 − 2), the bar says: work out the whole top, work out the whole bottom, then divide: 8 ÷ 4 = 2.",{"id":654,"type":101,"title":655,"problem":656,"steps":657},"we-frac","A fraction expression, step by step","Simplify **⅔ × 9 − 1½ ÷ ¾ + ⅚**.",[658,659,660,661,662],"Terms: ⅔ × 9, then 1½ ÷ ¾, then ⅚.","First term: ⅔ × 9 = 18\u002F3 = 6.","Second term: 1½ ÷ ¾ = 3\u002F2 × 4\u002F3 = 12\u002F6 = 2. (Dividing by ¾ is multiplying by its reciprocal 4\u002F3.)","Now + and − left to right: 6 − 2 + ⅚ = 4 + ⅚ = **4⅚**.","Check by estimating: about 6 − 2 + 1 = 5, and 4⅚ is just under 5. ✓",{"id":664,"type":101,"title":665,"problem":666,"steps":667},"we-dec","A shopping bill with decimals","Petrol costs ₹104.50 a litre. A scooter rider buys **4 litres** and pays with **₹500**, then buys a **₹12.50** snack with the change. How much money is left?",[668,669,670,671],"Change from petrol: 500 − 4 × 104.50. The × is done first, so no brackets are needed here.","4 × 104.50 = 418. Change: 500 − 418 = 82.","After the snack: 500 − 4 × 104.50 − 12.50 = 82 − 12.50 = **₹69.50**.","As one expression: 500 − (4 × 104.50 + 12.50) = 500 − 430.50 = 69.50, the same. Taking away each item in turn is the same as taking away the total.",{"id":673,"type":47,"variant":253,"title":674,"markdown":675},"misc-decimal-lr","“Decimals change the rules”","Faced with 1.2 + 0.3 × 4, many students add first because the decimal numbers \"look like they belong together\". The rule does not change: 0.3 × 4 = 1.2, then 1.2 + 1.2 = **2.4** (not 1.5 × 4 = 6).\n\nSimilarly ½ + ½ × ½ is ½ + ¼ = **¾**, not 1 × ½ = ½.",{"id":677,"type":258,"itemId":678,"prompt":679,"check":680,"hints":690,"feedback":693},"pr-frac","order-of-operations.deepen-fractions","What is **½ + ¼ × 8 − ⅓ × 3**?",{"kind":565,"options":681,"correct":689},[682,684,686,688],{"id":197,"label":683},"1½",{"id":200,"label":685},"5",{"id":203,"label":687},"6",{"id":206,"label":204},[197],[691,692],"× first in each term: ¼ × 8 and ⅓ × 3.","½ + 2 − 1.",{"correct":694,"incorrect":695},"Yes: ¼ × 8 = 2 and ⅓ × 3 = 1, so ½ + 2 − 1 = 1½.","× first: ¼ × 8 = 2 and ⅓ × 3 = 1. Then ½ + 2 − 1 = 1½.",{"id":697,"type":258,"itemId":698,"prompt":699,"check":700,"hints":702,"feedback":705},"pr-dec","order-of-operations.deepen-decimals","Evaluate **2.5 × 4 − 1.2 ÷ 0.4**.",{"kind":262,"answer":701,"tolerance":264},7,[703,704],"2.5 × 4 = 10.","1.2 ÷ 0.4 = 3.",{"correct":706,"incorrect":707},"Yes: 10 − 3 = 7.","× and ÷ first: 2.5 × 4 = 10 and 1.2 ÷ 0.4 = 3. Then 10 − 3 = 7.",{"id":709,"type":53,"title":710,"eyebrow":711,"navLabel":712},"ch-formulas","Formulas: expressions with letters","Chapter 10","10 Formulas",{"id":714,"type":43,"markdown":715},"formulas-prose","In Class 6 and 7 you start writing rules with letters, called **formulas**. A formula is an expression waiting for numbers. The order of operations is what lets a formula be written without dozens of brackets.\n\nAn app-cab company charges **₹50 to start, ₹12 per km after the first 2 km, and ₹2 per minute**:\n\n**fare = 50 + 12 × (d − 2) + 2 × t**, where d is the distance in km and t the time in minutes.\n\nFor a 10 km, 25-minute ride: substitute d = 10 and t = 25, *putting each number where its letter was*:\n\n50 + 12 × (10 − 2) + 2 × 25 = 50 + 12 × 8 + 50 = 50 + 96 + 50 = **₹196**.\n\nNotice the bracket (d − 2): without it, 12 × d − 2 would charge for every km and then take off just ₹2.",{"id":717,"type":64,"caption":718,"columns":719,"rows":723},"tbl-formulas","Everyday formulas and the order they hide",[720,721,722,164],"Formula","Where it is used","Example",[724,729,734,739,744,749,754],[725,726,727,728],"P = 2 × (l + b)","Perimeter of a rectangle","l = 12 m, b = 7 m","38 m",[730,731,732,733],"A = l × b − s × s","Garden area minus a square pond","l = 20, b = 15, s = 4","284 m²",[735,736,737,738],"C = 50 + 12 × (d − 2) + 2 × t","Cab fare","d = 10, t = 25","₹196",[740,741,742,743],"S = runs × 100 ÷ balls","Strike rate","63 runs, 42 balls","150",[745,746,747,748],"M = (a + b + c) ÷ 3","Mean of three marks","68, 75, 82","75",[750,751,752,753],"F = 9 × C ÷ 5 + 32","Celsius to Fahrenheit","C = 35 (a hot day)","95 °F",[755,756,757,758],"B = u × r + f","Electricity bill","250 units at ₹6, ₹120 fixed charge","₹1,620",{"id":760,"type":47,"variant":761,"title":762,"markdown":763},"careful-substitute","careful","Substituting: keep the structure, add brackets when needed","When you put a number in place of a letter, the formula's structure must not change. Two traps:\n\n- **Letters side by side mean ×.** If P = 2l + 2b and l = 12, write 2 × 12, not 212.\n- **Substituting an expression needs brackets.** If a formula says 3 × n and n = 4 + 1, you must write 3 × (4 + 1) = 15, not 3 × 4 + 1 = 13.\n\nWhen in doubt, put every substituted value in brackets first, and remove the brackets you do not need afterwards.",{"id":765,"type":101,"title":766,"problem":767,"steps":768},"we-celsius","Why the temperature formula works in any order","Convert **35 °C** to Fahrenheit using F = 9 × C ÷ 5 + 32, and check that 9 ÷ 5 × C + 32 gives the same.",[769,770,771,772,773],"Substitute C = 35: 9 × 35 ÷ 5 + 32.","× and ÷ left to right: 9 × 35 = 315, then 315 ÷ 5 = 63.","Then + : 63 + 32 = **95 °F**.","The other order: 9 ÷ 5 = 1.8, then 1.8 × 35 = 63, then 63 + 32 = 95. Same, because the ÷ 5 stays attached to the 5.","What would be wrong: 9 × 35 ÷ (5 + 32), which divides by 37. The + 32 is the last step; it must not be pulled into the divisor.",{"id":775,"type":258,"itemId":776,"prompt":777,"check":778,"hints":781,"feedback":784},"pr-formula","order-of-operations.deepen-formula","An electricity bill is **units × rate + fixed charge**. Find the bill for **180 units at ₹7 per unit** with a **₹150** fixed charge.",{"kind":262,"answer":779,"tolerance":264,"unit":780},1410,"₹",[782,783],"180 × 7 first.","1,260 + 150.",{"correct":785,"incorrect":786},"Yes: 180 × 7 + 150 = 1,260 + 150 = ₹1,410.","Substitute: 180 × 7 + 150. Multiply first: 1,260. Then add 150: ₹1,410.",{"id":788,"type":258,"itemId":789,"prompt":790,"check":791,"hints":793,"feedback":796},"pr-substitute","order-of-operations.deepen-substitute","The cab fare is **50 + 12 × (d − 2) + 2 × t**. What is the fare for **d = 7 km** and **t = 15 minutes**?",{"kind":262,"answer":792,"tolerance":264,"unit":780},140,[794,795],"Bracket: 7 − 2 = 5.","50 + 60 + 30.",{"correct":797,"incorrect":798},"Yes: 50 + 12 × 5 + 2 × 15 = 50 + 60 + 30 = ₹140.","Substitute: 50 + 12 × (7 − 2) + 2 × 15. Bracket 5; products 60 and 30; total 50 + 60 + 30 = ₹140.",{"id":800,"type":801,"title":802,"prompt":803,"options":804},"explorer-levels","explorer","The levels of the order of operations","Choose a level to see what belongs there and how it is handled.",[805,818,827,837,847],{"id":806,"label":807,"chain":808,"badge":814,"note":817},"grouping","Level 1: grouping",[809,810,811,812,813],"Bar (vinculum)","( )","{ }","[ ]","Fraction bar, √ bar",{"text":815,"tone":816},"Innermost first","yes","Anything that groups: brackets of every shape, the vinculum, the bar of a fraction and the bar of a square root. Work from the innermost group outwards. Inside each group, all the other levels apply in order.",{"id":215,"label":819,"chain":820,"badge":824,"note":826},"Level 2: powers",[821,822,823],"Squares and cubes","Other indices","Roots",{"text":825,"tone":816},"Before × and ÷","2 × 3² = 2 × 9 = 18. A power applies only to the number (or bracket) directly below it: 2 × 3² squares the 3, not the 2 × 3. Stacked powers are done from the top down: 2^(3^2) = 2⁹ = 512.",{"id":828,"label":829,"chain":830,"badge":834,"note":836},"of","Level 2½: “of”",[831,832,833],"Fraction of","Percent of","Treated as a chunk",{"text":835,"tone":816},"BODMAS O","In BODMAS-style textbooks, “of” is done before ordinary × and ÷: 36 ÷ ½ of 12 = 36 ÷ 6 = 6. Many mathematicians avoid the question by writing brackets: 36 ÷ (½ × 12).",{"id":838,"label":839,"chain":840,"badge":844,"note":846},"muldiv","Level 3: × and ÷",[841,842,843],"Leftmost × or ÷","Next × or ÷","Until none left",{"text":845,"tone":816},"Left to right","Equal rank. 24 ÷ 4 × 3 = 18. Because ÷ is × a reciprocal, you may regroup freely as long as each ÷ stays attached to its own number: 24 × 3 ÷ 4 = 18 too.",{"id":848,"label":849,"chain":850,"badge":854,"note":855},"addsub","Level 4: + and −",[851,852,853],"Leftmost + or −","Next + or −","One number left",{"text":845,"tone":816},"Equal rank. 30 − 8 + 5 = 27. Because − is + a negative, you may reorder terms as long as each keeps its sign: 30 + 5 − 8 = 27.",{"id":857,"type":53,"title":858,"eyebrow":859,"navLabel":860},"ch-negatives","Negative numbers and the minus sign","Chapter 11","11 Negative numbers",{"id":862,"type":43,"markdown":863},"negatives","In Class 7 you meet **integers**, and the − sign starts doing two jobs: *subtract* and *negative*. The order of operations still works, but you must read each − carefully.\n\n- **5 − 8 + 4**: left to right, 5 − 8 = −3, then −3 + 4 = **1**. (Or reorder: 5 + 4 − 8 = 1.)\n- **−2 × 3 + 10**: the −2 is a negative number. −2 × 3 = −6, then −6 + 10 = **4**.\n- **10 − (−3)**: subtracting a negative is adding: 10 + 3 = **13**.\n- **−3²**: by the usual convention the power is done first, then the negative: −(3²) = **−9**. If you mean negative three, squared, write (−3)² = 9.\n- **20 ÷ (−4) × 2**: left to right, −5 × 2 = **−10**.",{"id":865,"type":47,"variant":253,"title":866,"markdown":867},"misc-neg-bracket","Real mistake: −(a + b) = −a + b","When a minus sign sits in front of a bracket, it applies to **everything** inside. A very common error is 15 − (6 + 4) = 15 − 6 + 4 = 13. Correct: 15 − 10 = **5**, or 15 − 6 − 4 = 5.\n\nThink of −(6 + 4) as −1 × (6 + 4) = −6 − 4. It is the distributive property again, with −1 as the multiplier.",{"id":869,"type":101,"title":870,"problem":871,"steps":872},"we-temps","Temperatures in Leh","In winter, Leh records these night temperatures over 4 nights: **−12 °C, −8 °C, −15 °C, −5 °C**. Find the mean, using (sum) ÷ (count).",[873,874,875,876],"Sum, left to right: −12 + (−8) = −20; −20 + (−15) = −35; −35 + (−5) = −40.","Count: 4 nights.","Mean: (−12 − 8 − 15 − 5) ÷ 4 = −40 ÷ 4 = **−10 °C**.","Without the bracket, −12 − 8 − 15 − 5 ÷ 4 would divide only the last temperature: −12 − 8 − 15 − 1¼ = −36¼, which is colder than every night. The size check (a mean lies between the smallest and largest values) catches it.",{"id":878,"type":258,"itemId":879,"prompt":880,"check":881,"hints":883,"feedback":886},"pr-neg","order-of-operations.deepen-negatives","Evaluate **4 − 3 × 5 + 20**. (A negative number appears along the way.)",{"kind":262,"answer":882,"tolerance":264},9,[884,885],"3 × 5 = 15 first.","4 − 15 = −11, then −11 + 20.",{"correct":887,"incorrect":888},"Yes: 4 − 15 + 20 = −11 + 20 = 9.","× first: 3 × 5 = 15. Then left to right: 4 − 15 = −11, and −11 + 20 = 9. (Or reorder: 4 + 20 − 15 = 9.)",{"id":890,"type":193,"prompt":891,"options":892,"explanation":901},"predict-neg-square","A student types **-4^2** into a spreadsheet and gets **16**, but her textbook says −4² = −16. Who is right?",[893,895,897,899],{"id":197,"label":894},"The spreadsheet: −4² is always 16",{"id":200,"label":896},"The textbook: by the usual maths convention the square is done before the negative",{"id":203,"label":898},"Both, because they follow different conventions",{"id":206,"label":900},"Neither: the answer is −8","**(c), with (b) as the school convention.** In school maths and most programming languages, −4² means −(4²) = −16. Microsoft Excel publishes a precedence table that puts negation *above* powers, so =-4^2 is (−4)² = 16; spreadsheets that copy Excel's rules do the same. Neither is broken; they follow different precedence rules. The safe habit: always write (−4)² or −(4²), so nobody has to guess.",{"id":903,"type":53,"title":904,"eyebrow":905,"navLabel":906},"ch8","Edge cases","Chapter 12","12 Edge cases",{"id":908,"type":43,"markdown":909},"edges","A good rule should say what happens in awkward corners.\n\n- **Dividing by zero.** 12 ÷ (5 − 5) = 12 ÷ 0 has **no value**: no number times 0 gives 12. An expression that needs division by 0 is *undefined*, however it is bracketed.\n- **Zero divided by something.** 0 ÷ 7 = 0 is perfectly fine.\n- **Non-whole results.** In whole-number arithmetic, 7 + 5 ÷ 2 gives 7 + 2½ = 9½. The order of operations still works; the answer just is not whole.\n- **Negative results along the way.** 3 − 5 + 4 = −2 + 4 = 2. With integers (Class 7) nothing breaks; if you only know whole numbers, you can reorder as 3 + 4 − 5 = 2.\n- **Unary minus.** In −3 + 5, the − is not \"subtract\": it means *negative three*. The value is 2.\n- **Very long chains of the same operation.** 2 − 2 − 2 − 2 = −4 by left to right; 64 ÷ 4 ÷ 4 ÷ 4 = 1.",{"id":911,"type":47,"variant":761,"title":912,"markdown":913},"careful-zero","Undefined is not the same as zero","12 ÷ 0 is **not** 0 and not 12. It has no answer at all. Check: if 12 ÷ 0 were some number k, then k × 0 would have to be 12. But anything × 0 is 0. So no k works. That is why calculators show \"Error\" or \"Math ERROR\" and why every division in this site's games is chosen so it comes out exact and never divides by 0.",{"id":915,"type":916,"prompt":917},"reflect-convention","reflection","The order of operations is a *convention*, but the distributive property is a *fact*. In your own words, explain the difference. Could a different country choose a different distributive property? Could it choose a different order of operations? What would be the cost?",{"id":919,"type":53,"title":920,"eyebrow":921,"navLabel":922},"ch9","Words, check-up and summary","Chapter 13","13 Wrap-up",{"id":924,"type":925,"title":926,"terms":927},"glossary-deepen","glossary","Deeper vocabulary",[928,931,935,939,943,946,949,952,955,959,962,966],{"term":929,"meaning":930},"Repeated addition","Multiplication as adding equal groups: 3 × 4 = 4 + 4 + 4.",{"term":932,"meaning":933,"example":934},"Distributive property","a × (b + c) = a × b + a × c. Multiplication distributes over addition (and subtraction).","7 × 98 = 7 × 100 − 7 × 2",{"term":936,"meaning":937,"example":938},"Power \u002F index \u002F exponent","Repeated multiplication. In 2³, 2 is the base and 3 is the index.","2³ = 2 × 2 × 2 = 8",{"term":940,"meaning":941,"example":942},"Reciprocal","The number that multiplies to give 1. Dividing by a number is multiplying by its reciprocal.","The reciprocal of 4 is ¼.",{"term":944,"meaning":945},"Vinculum","A horizontal bar used to group terms. Survives as the fraction bar and in the square-root sign.",{"term":947,"meaning":948},"Obelus","The ÷ sign.",{"term":950,"meaning":951},"Expression tree","A diagram of an expression with the last operation at the top and each operation branching to what it combines.",{"term":953,"meaning":954},"Left-associative","Repeated operations of the same level group from the left: a − b − c = (a − b) − c.",{"term":956,"meaning":957,"example":958},"Reverse Polish Notation (RPN)","Postfix notation: operations come after their two numbers. Needs no brackets.","2 3 4 × + means 2 + 3 × 4",{"term":960,"meaning":961},"Stack","A pile where you can only add or take from the top. Used to evaluate RPN.",{"term":963,"meaning":964,"example":965},"Undefined","Having no value, like any division by 0.","12 ÷ (5 − 5)",{"term":967,"meaning":968,"example":969},"Unary minus","A minus sign that makes a number negative rather than subtracting.","−3 + 5 = 2",{"id":971,"type":972,"title":973,"questions":974},"quiz-deepen","quiz","Reasoning check-up",[975,988,1001,1014,1025,1038,1051,1064,1074,1087],{"itemId":976,"prompt":977,"options":978,"correct":197,"why":987},"order-of-operations.dpq-1","Why is 2 + 3 × 4 read as 2 + (3 × 4)?",[979,981,983,985],{"id":197,"label":980},"Because 3 × 4 is shorthand for 4 + 4 + 4, a single bundle",{"id":200,"label":982},"Because 4 is the biggest number",{"id":203,"label":984},"Because addition is harder",{"id":206,"label":986},"Because calculators say so","Multiplication packs a repeated addition into one symbol. 2 + 3 × 4 = 2 + 4 + 4 + 4 = 14.",{"itemId":989,"prompt":990,"options":991,"correct":200,"why":1000},"order-of-operations.dpq-2","Which uses the distributive property correctly?",[992,994,996,998],{"id":197,"label":993},"5 × (8 + 2) = 5 × 8 + 2",{"id":200,"label":995},"5 × (8 + 2) = 5 × 8 + 5 × 2",{"id":203,"label":997},"5 + (8 × 2) = 5 + 8 × 5 + 2",{"id":206,"label":999},"5 × (8 + 2) = (5 × 8) × 2","a × (b + c) = a × b + a × c. Both 50.",{"itemId":1002,"prompt":1003,"options":1004,"correct":200,"why":1013},"order-of-operations.dpq-3","Rewrite 20 − 6 + 4 using only addition.",[1005,1007,1009,1011],{"id":197,"label":1006},"20 + 6 + 4",{"id":200,"label":1008},"20 + (−6) + 4",{"id":203,"label":1010},"20 + (−6) + (−4)",{"id":206,"label":1012},"20 + (−10)","Each − belongs to the number after it: 20 + (−6) + 4 = 18.",{"itemId":1015,"prompt":1016,"options":1017,"correct":200,"why":1024},"order-of-operations.dpq-4","What is 2 × 3²?",[1018,1019,1020,1022],{"id":197,"label":238},{"id":200,"label":234},{"id":203,"label":1021},"12",{"id":206,"label":1023},"64","Powers before ×: 3² = 9, then 2 × 9 = 18. (2 × 3)² would be 36.",{"itemId":1026,"prompt":1027,"options":1028,"correct":197,"why":1037},"order-of-operations.dpq-5","A vinculum over 7 − 4 means…",[1029,1031,1033,1035],{"id":197,"label":1030},"7 − 4 is done first, like brackets",{"id":200,"label":1032},"7 − 4 is ignored",{"id":203,"label":1034},"7 − 4 is squared",{"id":206,"label":1036},"Divide by 7 − 4","The vinculum is a bar used as a grouping symbol, exactly like ( ).",{"itemId":1039,"prompt":1040,"options":1041,"correct":200,"why":1050},"order-of-operations.dpq-6","In Reverse Polish Notation, what does 5 1 + 2 × mean?",[1042,1044,1046,1048],{"id":197,"label":1043},"5 + 1 × 2 = 7",{"id":200,"label":1045},"(5 + 1) × 2 = 12",{"id":203,"label":1047},"5 × 1 + 2 = 7",{"id":206,"label":1049},"5 + 12 = 17","Push 5 and 1, + gives 6; push 2, × gives 12. That is (5 + 1) × 2.",{"itemId":1052,"prompt":1053,"options":1054,"correct":197,"why":1063},"order-of-operations.dpq-7","What is 37 × 64 + 37 × 36?",[1055,1057,1059,1061],{"id":197,"label":1056},"3,700",{"id":200,"label":1058},"3,600",{"id":203,"label":1060},"370",{"id":206,"label":1062},"37,000","Take out the common factor: 37 × (64 + 36) = 37 × 100 = 3,700.",{"itemId":1065,"prompt":1066,"options":1067,"correct":206,"why":1073},"order-of-operations.dpq-8","What is 18 ÷ (6 − 6)?",[1068,1070,1071,1072],{"id":197,"label":1069},"0",{"id":200,"label":234},{"id":203,"label":181},{"id":206,"label":963},"6 − 6 = 0, and division by 0 has no value.",{"itemId":1075,"prompt":1076,"options":1077,"correct":200,"why":1086},"order-of-operations.dpq-9","Which statement about (a + b)² is always true?",[1078,1080,1082,1084],{"id":197,"label":1079},"It equals a² + b²",{"id":200,"label":1081},"It equals a² + 2 × a × b + b²",{"id":203,"label":1083},"It equals 2 × (a + b)",{"id":206,"label":1085},"It equals a × b × 2","(a + b) × (a + b) = a × a + a × b + b × a + b × b.",{"itemId":1088,"prompt":1089,"options":1090,"correct":197,"why":1099},"order-of-operations.dpq-10","Where is the + in the expression tree of 10 − 3 + 2?",[1091,1093,1095,1097],{"id":197,"label":1092},"At the top",{"id":200,"label":1094},"At the bottom-left",{"id":203,"label":1096},"Inside the − branch",{"id":206,"label":1098},"There is no +","Left to right means the rightmost partner, the +, is done last, so it is at the top: (10 − 3) + 2.",{"id":1101,"type":1102,"title":1103,"points":1104},"cheat-deepen","summary","Cheat sheet",[1105,1106,1107,1108,1109,1110,1111,1112,1113,1114],"**× before +** because multiplication is repeated addition: 2 + 3 × 4 = 2 + 4 + 4 + 4. Sums of products (bills, 3x + 2y) then need no brackets.","**Distributive property:** a × (b + c) = a × b + a × c. It removes brackets; the order of operations reads the result. It is a property, not a DMAS letter.","**Left to right** because − is + a negative and ÷ is × a reciprocal: 10 − 3 + 2 = 10 + (−3) + 2; 8 ÷ 4 × 2 = 8 × ¼ × 2.","**Powers** bind tighter than ×: 2 × 3² = 18, (2 × 3)² = 36. (a + b)² ≠ a² + b².","**Full order:** brackets (innermost; the bar first) → powers → of → × ÷ left to right → + − left to right.","**Vinculum:** an overbar used as a bracket; lives on as the fraction bar and the square-root sign.","**History:** brackets spread from the 1500s–1700s; the rule grew with algebra; mnemonics are 20th-century.","**Machines:** expressions become trees (last operation on top). RPN (2 3 4 × +) needs no brackets; stacks evaluate it.","**Edge cases:** division by 0 is undefined; 0 ÷ 7 = 0; non-whole and negative results are fine.","**Long expressions:** scan, innermost first, one step per line, look for shortcuts (a zero factor, a common factor).",{"id":1116,"type":1117,"conceptId":1118,"relation":1119,"explanation":1120},"conn-properties","connection","properties-of-numbers","helps_understand","The distributive, associative and commutative properties explain why the order of operations is the way it is.",{"id":1122,"type":1117,"conceptId":1123,"relation":1124,"explanation":1125},"conn-patterns","patterns","related_to","Expressions like 9 × n + n = 10 × n describe number patterns; the order of operations makes each rule unambiguous.",{"id":1127,"type":1117,"conceptId":1128,"relation":1129,"explanation":1130},"conn-prime","prime-and-composite","applied_in","Prime factorisations like 2³ × 3 × 5 = 120 are expressions whose value depends on doing powers before ×.",{"id":1132,"type":1117,"conceptId":1133,"relation":1129,"explanation":1134},"conn-electricity","electricity","Formulas such as power = current² × resistance, or costs = units × tariff + fixed charge, rely on the same order.",{"id":1136,"type":1137,"sourceIds":1138},"sources-deepen","sources",[1139,1140,1141,1142,1143,1144],"order-of-operations-wikipedia","order-of-operations-miller-grouping","order-of-operations-wikipedia-vinculum","order-of-operations-ncert-class7","order-of-operations-mathsisfun-bodmas","order-of-operations-excel-precedence",[1139,1140,1141,1142,1143,1144],"needs_review",{"generatedBy":1148,"notes":1149},"claude-code","Draft generated with Python-checked arithmetic (× ÷ left to right, then + − left to right); pending owner review.","5c9c39106a1272340c866da34f326969beca40950fc93f04590ab2dff91a450e",{"component:match-pairs@1":1152,"logic:practice":1153,"component:order-ops@1":1154,"component:sort-game@1":1155,"source:order-of-operations-excel-precedence":1156,"source:order-of-operations-mathsisfun-bodmas":1157,"source:order-of-operations-miller-grouping":1158,"source:order-of-operations-ncert-class7":1159,"source:order-of-operations-wikipedia":1160,"source:order-of-operations-wikipedia-vinculum":1161},"2a8ee4ac87460b4e1175a4bb13c96b03d577db06dde95670eb7fcfe4ad787899","3d6b0fe1b15255975a32b0fcd94e8019bc959ad45cbf12e136e86149549c6878","3d630ebc066352df791b020d545e1876ce4620d55c8c1c082414d6b4717255c5","b164f45a2c8ca08f26c450768ff0231e113e9fe45381eddb34dc6d0548596c38","7c20b35268a810137167a94207d30541be7d945d944d4fd6ac2383362ba2f153","92959283b293565e5dbcac7d99680a63df55eef50520c997ab808634afaa0062","b8e2b4b4403541ce11f0ad7556454fb58e597e57642a953ab207231ad44c83c1","e7cadfedc70b54014195d6d8bc2740fb10a7e8aed3509c11f1efbde6e7698ade","88ccf3653f7d9723bbb33a67a6548ccff64980f752cba0cd94a32bc60f72bd44","cd5db84b556a6a4123ba798fa0e81eb83042de63e5a7b5b3db0091913e668d92",{"state":1163,"reviewer":1164,"selfReview":387,"reviewedAt":1165,"method":1166},"approved","The library owner","2026-09-20T10:18:37.581Z","owner_bulk","preview-7e1cbbcc4f",1789899598952]