[{"data":1,"prerenderedAt":1262},["ShallowReactive",2],{"layer:order-of-operations:investigate":3},{"layer":4,"contentHash":1246,"dependencyHashes":1247,"approval":1256,"releaseId":1261},{"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":1241,"reviewStatus":1242,"authoring":1243},1,"order-of-operations","en","investigate","Brackets under the microscope","Predict, test and explain: moving brackets, missing signs, calculators and targets","Experiment with the order of operations: count how many values brackets can make, find when brackets change nothing, test always\u002Fsometimes\u002Fnever statements, fill in missing signs, compare calculators and hit targets.",[13,14,15,16,17],"Find all the values an expression can take when brackets are moved, and explain why four numbers have five bracketings.","Decide when brackets change a value, using associativity and the order of operations.","Test conjectures about removing brackets after − and ÷, and use counterexamples.","Predict what basic and scientific calculators show for a key sequence.","Solve missing-sign and target puzzles by reasoning and working backwards.",40,{"title":20,"rows":21},"Lesson plate",[22,25,28,31,34,37],{"label":23,"value":24},"Depth","Investigate",{"label":26,"value":27},"Reading time","≈ 40 minutes",{"label":29,"value":30},"Prior knowledge","Understand: terms, nested brackets",{"label":32,"value":33},"Chapters","13",{"label":35,"value":36},"Labs","3 order-ops, 2 sort games, 1 match",{"label":38,"value":39},"Big idea","One counterexample breaks \"always\"",[41,47,53,59,77,103,108,131,152,167,172,175,203,208,274,295,300,303,316,346,351,363,368,371,440,452,457,460,526,530,543,558,581,586,589,617,628,639,644,647,677,681,694,698,703,706,721,733,738,741,773,783,788,793,796,824,837,841,853,865,870,873,922,935,948,958,969,974,977,982,985,1022,1026,1036,1045,1065,1070,1099,1203,1217,1223,1228,1233],{"id":42,"type":43,"markdown":44,"help":45},"intro","prose","So far you have *followed* the order of operations. In this layer you will **experiment** with it. What happens if you move a bracket? How many different answers can the same four numbers make? When do brackets change nothing at all? Is a rule you notice *always* true, or only sometimes?\n\nThat is how mathematicians work: **predict, try, compare, and test**. You will make guesses (called **conjectures**), try to break them with a single example that does not fit (a **counterexample**), and keep only the ideas that survive.",{"simplerExplanation":46},"In this layer you play with brackets and signs, guess what will happen, and check your guesses.",{"id":48,"type":49,"variant":50,"title":51,"markdown":52},"how-to","callout","observation","Your investigation kit","For every investigation, keep a simple record:\n\n- **Predict**: write what you think will happen and why.\n- **Test**: try at least three examples, including an odd one (with 0, 1, or a big number).\n- **Record**: a small table of results.\n- **Conclude**: always true, sometimes true, or never true? Give a reason or a counterexample.",{"id":54,"type":55,"title":56,"eyebrow":57,"navLabel":58},"ch1","chapter","How many answers can brackets make?","Chapter 01","1 Moving brackets",{"id":60,"type":61,"prompt":62,"options":63,"explanation":76},"predict-count","prediction","Take **2 + 3 × 4 − 1** (value 13 with no brackets). You may add brackets anywhere, as many as you like, but you must keep the numbers and signs in the same order. How many **different values** can you make?",[64,67,70,73],{"id":65,"label":66},"a","Just 1: brackets never change anything",{"id":68,"label":69},"b","2",{"id":71,"label":72},"c","4",{"id":74,"label":75},"d","Hundreds","**4 different values: 11, 13, 15 and 19.** There are exactly five ways to fully bracket four numbers, and two of them happen to give the same value (13). See the table below.",{"id":78,"type":79,"caption":80,"columns":81,"rows":85},"tbl-five","table","All five ways to bracket 2 + 3 × 4 − 1",[82,83,84],"Fully bracketed","Shortest way to write it","Value",[86,90,93,96,99],[87,88,89],"((2 + 3) × 4) − 1","(2 + 3) × 4 − 1","19",[91,92,33],"(2 + (3 × 4)) − 1","2 + 3 × 4 − 1 (no brackets needed)",[94,94,95],"(2 + 3) × (4 − 1)","15",[97,98,33],"2 + ((3 × 4) − 1)","2 + (3 × 4 − 1)",[100,101,102],"2 + (3 × (4 − 1))","2 + 3 × (4 − 1)","11",{"id":104,"type":43,"markdown":105,"help":106},"five-why","Why exactly five? Every fully bracketed expression has a **last operation**, the one done at the very end. Here there are three signs, so there are three choices for the last one:\n\n- Last is **+**: the left side is just 2, and the right side is 3 × 4 − 1, which can be bracketed in 2 ways.\n- Last is **×**: left is 2 + 3 (1 way), right is 4 − 1 (1 way): 1 × 1 = 1 way.\n- Last is **−**: left is 2 + 3 × 4 (2 ways), right is 1: 2 ways.\n\nTotal: 2 + 1 + 2 = **5**. With five numbers the same thinking gives 14 ways, with six numbers 42 ways, then 132, 429… These are called the **Catalan numbers**, and they turn up all over mathematics (you will meet them again in the Extend layer).",{"simplerExplanation":107},"Pick which sign is done last. Then count the ways to bracket each side. Add up.",{"id":109,"type":110,"component":111,"componentVersion":5,"config":112,"objective":125,"textAlternative":126,"help":127},"lab-orderops-brackets","interactive","order-ops",{"expressions":113,"showRuleCard":124},[114,115,116,117,118,119,120,121,122,123],"(2 + 3) × 4 - 1","2 + 3 × 4 - 1","(2 + 3) × (4 - 1)","2 + (3 × 4 - 1)","2 + 3 × (4 - 1)","(3 + 4) × 5 - 2","3 + 4 × (5 - 2)","2 × (3 + 4 × 5)","(2 × 3 + 4) × 5","2 × (3 + 4) × 5",true,"Step through the same numbers with brackets in different places and see how the value changes.","Ten expressions built from the same numbers with brackets moved around.\n\nFrom 2, 3, 4, 1 with +, ×, −:\n- (2 + 3) × 4 − 1 = 5 × 4 − 1 = 20 − 1 = 19\n- 2 + 3 × 4 − 1 = 2 + 12 − 1 = 14 − 1 = 13\n- (2 + 3) × (4 − 1) = 5 × (4 − 1) = 5 × 3 = 15\n- 2 + (3 × 4 − 1) = 2 + (12 − 1) = 2 + 11 = 13\n- 2 + 3 × (4 − 1) = 2 + 3 × 3 = 2 + 9 = 11\n\nFrom 3, 4, 5, 2: (3 + 4) × 5 − 2 = 7 × 5 − 2 = 35 − 2 = 33, 3 + 4 × (5 − 2) = 3 + 4 × 3 = 3 + 12 = 15\n\nFrom 2, 3, 4, 5 with ×, +, ×: 2 × (3 + 4 × 5) = 2 × (3 + 20) = 2 × 23 = 46, (2 × 3 + 4) × 5 = (6 + 4) × 5 = 10 × 5 = 50, 2 × (3 + 4) × 5 = 2 × 7 × 5 = 14 × 5 = 70",{"hints":128},[129,130],"Always start with the innermost bracket.","Compare the values: which bracket position gives the biggest?",{"id":132,"type":110,"component":133,"componentVersion":5,"config":134,"objective":150,"textAlternative":151},"lab-match-brackets","match-pairs",{"prompt":135,"mode":136,"pairs":137},"Connect each bracketing of 2 × 3 + 4 × 5 to its value.","connect",[138,141,143,145,147],{"a":139,"b":140},"2 × 3 + 4 × 5","26",{"a":121,"b":142},"46",{"a":122,"b":144},"50",{"a":123,"b":146},"70",{"a":148,"b":149},"(2 × 3) + (4 × 5)","26 as well (same as no brackets)","Match each way of bracketing 2 × 3 + 4 × 5 with its value, and notice which bracketings change nothing.","A connect-the-pairs game about the numbers 2, 3, 4, 5 with ×, + and ×.\n\n- 2 × 3 + 4 × 5 = 6 + 4 × 5 = 6 + 20 = 26\n- 2 × (3 + 4 × 5) = 2 × (3 + 20) = 2 × 23 = 46\n- (2 × 3 + 4) × 5 = (6 + 4) × 5 = 10 × 5 = 50\n- 2 × (3 + 4) × 5 = 2 × 7 × 5 = 14 × 5 = 70\n- (2 × 3) + (4 × 5) = 26: these brackets only surround what would be done first anyway, so they change nothing.\n\nThe five bracketings give four different values: 26, 46, 50 and 70.",{"id":153,"type":154,"itemId":155,"prompt":156,"check":157,"hints":161,"feedback":164},"pr-biggest","practice","order-of-operations.investigate-biggest","Using **3 + 4 × 5 − 2** with the numbers and signs in this order, add brackets to make the **biggest** value you can. What is it?",{"kind":158,"answer":159,"tolerance":160},"number",33,0,[162,163],"Try making the + happen first.","(3 + 4) × 5 − 2 = 7 × 5 − 2.",{"correct":165,"incorrect":166},"Yes: (3 + 4) × 5 − 2 = 35 − 2 = 33. The other values are 21 and 15.","The biggest is (3 + 4) × 5 − 2 = 35 − 2 = 33. Without brackets you get 21; with 3 + 4 × (5 − 2) you get 15.",{"id":168,"type":55,"title":169,"eyebrow":170,"navLabel":171},"ch2","When brackets change nothing","Chapter 02","2 No difference",{"id":173,"type":43,"markdown":174},"assoc","Brackets do **not** always change the value. Test these:\n\n- (2 + 3) + 4 = 9 and 2 + (3 + 4) = 9.\n- (2 × 3) × 4 = 24 and 2 × (3 × 4) = 24.\n- 2 + (3 × 4) = 14, the same as 2 + 3 × 4.\n\nIn the first two, the expression uses **only +** or **only ×**. Adding and multiplying are **associative**: how you group them does not matter. In the third, the brackets just surround a part that would have been done first anyway.\n\nNow test subtraction and division:\n\n- (20 − 8) − 2 = 12 − 2 = 10, but 20 − (8 − 2) = 20 − 6 = 14.\n- (48 ÷ 4) ÷ 2 = 12 ÷ 2 = 6, but 48 ÷ (4 ÷ 2) = 48 ÷ 2 = 24.\n\nSubtraction and division are **not** associative. That is precisely why the order of operations needs the left-to-right rule for them.",{"id":176,"type":79,"caption":177,"columns":178,"rows":183},"tbl-assoc","Does grouping matter? A test table",[179,180,181,182],"Operation","Test 1","Test 2","Always the same?",[184,189,194,199],[185,186,187,188],"+","(5 + 7) + 9 = 21; 5 + (7 + 9) = 21","(100 + 1) + 0 = 101; 100 + (1 + 0) = 101","Yes: addition is associative",[190,191,192,193],"×","(5 × 7) × 2 = 70; 5 × (7 × 2) = 70","(9 × 1) × 0 = 0; 9 × (1 × 0) = 0","Yes: multiplication is associative",[195,196,197,198],"−","(20 − 8) − 2 = 10; 20 − (8 − 2) = 14","(9 − 5) − 0 = 4; 9 − (5 − 0) = 4","No: test 1 is a counterexample",[200,201,202,198],"÷","(48 ÷ 4) ÷ 2 = 6; 48 ÷ (4 ÷ 2) = 24","(8 ÷ 4) ÷ 1 = 2; 8 ÷ (4 ÷ 1) = 2",{"id":204,"type":49,"variant":205,"title":206,"markdown":207},"aha-counter","aha","One counterexample is enough","Look at the − row: test 2 *happened* to give the same answer both ways. If you had only tried that one, you might have concluded \"subtraction is associative\". But test 1 breaks it.\n\nIn mathematics, **one** example that fails is enough to show a rule is not always true. No number of examples that work can prove a rule is always true; for that you need a reason. This is the difference between checking and proving.",{"id":209,"type":110,"component":210,"componentVersion":5,"config":211,"objective":269,"textAlternative":270,"help":271},"lab-sort-diff","sort-game",{"prompt":212,"bins":213,"items":220,"seconds":160},"Do the brackets change the value? Sort each pair.",[214,217],{"id":215,"label":216},"same","Same value",{"id":218,"label":219},"diff","Different value",[221,225,229,233,237,241,245,249,253,257,261,265],{"id":222,"label":223,"bin":215,"why":224},"p1","(4 + 5) + 6 and 4 + (5 + 6)","Only additions: grouping never matters. Both are 15.",{"id":226,"label":227,"bin":215,"why":228},"p2","(4 × 5) × 6 and 4 × (5 × 6)","Only multiplications: both are 120.",{"id":230,"label":231,"bin":218,"why":232},"p3","(30 − 10) − 5 and 30 − (10 − 5)","15 versus 25. Subtraction is not associative.",{"id":234,"label":235,"bin":218,"why":236},"p4","(64 ÷ 8) ÷ 2 and 64 ÷ (8 ÷ 2)","4 versus 16. Division is not associative.",{"id":238,"label":239,"bin":215,"why":240},"p5","7 + (3 × 2) and 7 + 3 × 2","The brackets surround the × that is done first anyway. Both are 13.",{"id":242,"label":243,"bin":218,"why":244},"p6","(7 + 3) × 2 and 7 + 3 × 2","20 versus 13. The brackets force the + first.",{"id":246,"label":247,"bin":215,"why":248},"p7","(18 ÷ 3) × 2 and 18 ÷ 3 × 2","Left to right already does 18 ÷ 3 first. Both are 12.",{"id":250,"label":251,"bin":218,"why":252},"p8","18 ÷ (3 × 2) and 18 ÷ 3 × 2","3 versus 12. The brackets make the ÷ act on 3 × 2.",{"id":254,"label":255,"bin":215,"why":256},"p9","(12 − 5) + 3 and 12 − 5 + 3","Left to right does 12 − 5 first anyway. Both are 10.",{"id":258,"label":259,"bin":218,"why":260},"p10","12 − (5 + 3) and 12 − 5 + 3","4 versus 10. The bracket takes away both 5 and 3.",{"id":262,"label":263,"bin":218,"why":264},"p11","(9 − 9) × 5 and 9 − 9 × 5","0 versus 9 − 45, which is negative (−36). Very different.",{"id":266,"label":267,"bin":215,"why":268},"p12","(6 × 2) + (8 ÷ 4) and 6 × 2 + 8 ÷ 4","Both brackets hold × or ÷ chunks that are done first anyway. Both are 14.","Decide whether adding brackets to an expression changes its value, and spot which kinds of brackets are \"free\".","A sorting game with 12 pairs of expressions and two bins: same value or different value.\n\nSame: (4 + 5) + 6 and 4 + (5 + 6) (15); (4 × 5) × 6 and 4 × (5 × 6) (120); 7 + (3 × 2) and 7 + 3 × 2 (13); (18 ÷ 3) × 2 and 18 ÷ 3 × 2 (12); (12 − 5) + 3 and 12 − 5 + 3 (10); (6 × 2) + (8 ÷ 4) and 6 × 2 + 8 ÷ 4 (14).\n\nDifferent: (30 − 10) − 5 = 15 vs 30 − (10 − 5) = 25; (64 ÷ 8) ÷ 2 = 4 vs 64 ÷ (8 ÷ 2) = 16; (7 + 3) × 2 = 20 vs 13; 18 ÷ (3 × 2) = 3 vs 12; 12 − (5 + 3) = 4 vs 10; (9 − 9) × 5 = 0 vs 9 − 45 = −36.\n\nBrackets change nothing when they only group what the rule would do first anyway, or when only + or only × is involved.",{"hints":272},[273],"Work out both sides. Or ask: do the brackets force a different first step?",{"id":275,"type":154,"itemId":276,"prompt":277,"check":278,"hints":290,"feedback":292},"pr-free","order-of-operations.investigate-free-brackets","In which expression can the brackets be removed **without** changing the value?",{"kind":279,"options":280,"correct":289},"choice",[281,283,285,287],{"id":65,"label":282},"40 − (10 + 5)",{"id":68,"label":284},"40 ÷ (10 ÷ 5)",{"id":71,"label":286},"40 − (10 × 2)",{"id":74,"label":288},"(40 − 10) × 2",[71],[291],"Would the rule do the bracketed part first anyway?",{"correct":293,"incorrect":294},"Yes: 40 − 10 × 2 already does 10 × 2 first, so both are 20.","Only (c) has free brackets: 40 − 10 × 2 does the × first anyway, so both give 20. The others change: 40 − (10 + 5) = 25 but 40 − 10 + 5 = 35; 40 ÷ (10 ÷ 5) = 20 but 40 ÷ 10 ÷ 5 is 4 ÷ 5, not even whole; (40 − 10) × 2 = 60 but 40 − 10 × 2 = 20.",{"id":296,"type":55,"title":297,"eyebrow":298,"navLabel":299},"ch3","Opening a bracket after − or ÷","Chapter 03","3 Opening brackets",{"id":301,"type":43,"markdown":302},"open-minus","Here is a real investigation. Compare 20 − (8 + 2) with 20 − 8 + 2.\n\n- 20 − (8 + 2) = 20 − 10 = **10**\n- 20 − 8 + 2 = 12 + 2 = **14**\n\nThey differ. Now try 20 − 8 − 2 = 12 − 2 = **10**. That matches!\n\n**Conjecture:** *when you remove a bracket that has a − in front of it, the signs inside flip: + becomes − and − becomes +.*\n\n- 20 − (8 + 2) = 20 − 8 − 2 ✓ (both 10)\n- 20 − (8 − 2) = 20 − 8 + 2? Left: 20 − 6 = 14. Right: 12 + 2 = 14 ✓\n\nIt makes sense: taking away (8 − 2) means taking away 8 but then giving back the 2, because you took away 2 too many.",{"id":304,"type":61,"prompt":305,"options":306,"explanation":315},"predict-divide","By the same thinking, which of these equals **48 ÷ (4 × 2)**?",[307,309,311,313],{"id":65,"label":308},"48 ÷ 4 × 2",{"id":68,"label":310},"48 ÷ 4 ÷ 2",{"id":71,"label":312},"48 × 4 ÷ 2",{"id":74,"label":314},"None of them","**48 ÷ 4 ÷ 2.** Dividing by 4 × 2 means dividing by 4 and then by 2. Check: 48 ÷ (4 × 2) = 48 ÷ 8 = 6, and 48 ÷ 4 ÷ 2 = 12 ÷ 2 = 6. But 48 ÷ 4 × 2 = 12 × 2 = 24, which is different. Removing a bracket after ÷ flips × to ÷ (and ÷ to ×), just as removing one after − flips + and −.",{"id":317,"type":79,"caption":318,"columns":319,"rows":323},"tbl-open","Testing the sign-flip conjecture",[320,321,322],"With brackets","Brackets removed, signs flipped","Both values",[324,328,332,336,339,342],[325,326,327],"20 − (8 + 2)","20 − 8 − 2","10 and 10",[329,330,331],"20 − (8 − 2)","20 − 8 + 2","14 and 14",[333,334,335],"100 − (30 − 10 + 5)","100 − 30 + 10 − 5","75 and 75",[337,310,338],"48 ÷ (4 × 2)","6 and 6",[340,308,341],"48 ÷ (4 ÷ 2)","24 and 24",[343,344,345],"50 + (8 − 3)","50 + 8 − 3","55 and 55",{"id":347,"type":49,"variant":348,"title":349,"markdown":350},"nuance-plus","nuance","After a + or ×, nothing flips","Look at the last row: 50 + (8 − 3) = 50 + 8 − 3. When the bracket has a **+** in front, you can simply remove it; nothing flips. The same goes for a bracket after × if it holds only × and ÷: 2 × (6 ÷ 3) = 2 × 6 ÷ 3 = 4.\n\nBut careful: 2 × (6 + 3) is **not** 2 × 6 + 3. A bracket holding + or − after a × needs the distributive property: 2 × 6 + 2 × 3. That is the subject of the Deepen layer.",{"id":352,"type":154,"itemId":353,"prompt":354,"check":355,"hints":357,"feedback":360},"pr-open","order-of-operations.investigate-open-bracket","Without working out the bracket first, rewrite **75 − (25 − 10)** with no brackets and evaluate it.",{"kind":158,"answer":356,"tolerance":160},60,[358,359],"Removing a bracket after −: the − inside becomes +.","75 − 25 + 10.",{"correct":361,"incorrect":362},"Yes: 75 − 25 + 10 = 50 + 10 = 60. Check: 75 − 15 = 60.","75 − (25 − 10) = 75 − 25 + 10 = 60. Check with the bracket: 25 − 10 = 15 and 75 − 15 = 60.",{"id":364,"type":55,"title":365,"eyebrow":366,"navLabel":367},"ch4","Always, sometimes or never?","Chapter 04","4 Always or never?",{"id":369,"type":43,"markdown":370},"asn","Mathematicians love statements that are **always** true. But many statements are only **sometimes** true, and some are **never** true. Deciding which, and backing it up, is a key skill.\n\n- To show **sometimes**, give one example where it works and one where it fails.\n- To show **never** or **always**, you need a reason that covers every possible number, not just a few tests.\n\nTry the sort game below. For each statement, test it with a few numbers (include 0 and 1) before you decide.",{"id":372,"type":110,"component":210,"componentVersion":5,"config":373,"objective":434,"textAlternative":435,"help":436},"lab-sort-asn",{"prompt":374,"bins":375,"items":385,"seconds":160},"Is each statement always, sometimes or never true for whole numbers? Test with a few numbers first.",[376,379,382],{"id":377,"label":378},"always","Always true",{"id":380,"label":381},"sometimes","Sometimes true",{"id":383,"label":384},"never","Never true",[386,390,394,398,402,406,410,414,418,422,426,430],{"id":387,"label":388,"bin":380,"why":389},"s1","a + b × c = (a + b) × c","True when a = 0 (0 + b × c = b × c) or c = 1, e.g. 2 + 3 × 1 = 5 = (2 + 3) × 1. False for 2 + 3 × 4 = 14 ≠ 20.",{"id":391,"label":392,"bin":380,"why":393},"s2","a × b + c = a × (b + c)","True when a = 1 or c = 0. False for 2 × 3 + 4 = 10 ≠ 14.",{"id":395,"label":396,"bin":377,"why":397},"s3","(a + b) + c = a + (b + c)","Addition is associative: grouping never changes a sum.",{"id":399,"label":400,"bin":377,"why":401},"s4","a − (b + c) = a − b − c","Taking away a total is the same as taking away each part.",{"id":403,"label":404,"bin":380,"why":405},"s5","a − (b − c) = a − b − c","Only when c = 0. Usually a − (b − c) = a − b + c instead. 10 − (5 − 2) = 7 but 10 − 5 − 2 = 3.",{"id":407,"label":408,"bin":377,"why":409},"s6","a ÷ (b × c) = a ÷ b ÷ c","Dividing by a product is dividing by each factor in turn (b and c not 0).",{"id":411,"label":412,"bin":380,"why":413},"s7","(a + b) × c is bigger than a + b × c","(a + b) × c − (a + b × c) = a × (c − 1). Bigger when a > 0 and c > 1; equal when a = 0 or c = 1; smaller when c = 0 and a > 0.",{"id":415,"label":416,"bin":377,"why":417},"s8","a × (b + c) = a × b + a × c","This is the distributive property.",{"id":419,"label":420,"bin":380,"why":421},"s9","a + b × c equals (a + b) × c + 1","With a = 1, b = 0, c = 0: 1 + 0 = 1 and (1 + 0) × 0 + 1 = 1, true. With 2, 3, 4: 14 versus 21, false.",{"id":423,"label":424,"bin":380,"why":425},"s10","a − a × 0 = 0","a − a × 0 = a − 0 = a, so it equals 0 only when a = 0. (People who subtract first get (a − a) × 0 = 0 and think it is always true.)",{"id":427,"label":428,"bin":377,"why":429},"s11","a + 1 × b = a + b","1 × b = b, done first, so a + 1 × b is always a + b.",{"id":431,"label":432,"bin":383,"why":433},"s12","(a + 1) × 0 = a + 1 × 0 when a is a counting number","The left side is always 0; the right side is a + 0 = a, which is at least 1 for counting numbers.","Test algebra-style statements about brackets and order with real numbers and decide if each is always, sometimes or never true.","A sorting game with 12 statements and three bins.\n\nAlways true: (a + b) + c = a + (b + c); a − (b + c) = a − b − c; a ÷ (b × c) = a ÷ b ÷ c; a × (b + c) = a × b + a × c; a + 1 × b = a + b.\n\nSometimes true: a + b × c = (a + b) × c (only if a = 0 or c = 1); a × b + c = a × (b + c) (only if a = 1 or c = 0); a − (b − c) = a − b − c (only if c = 0); (a + b) × c bigger than a + b × c (when a > 0 and c > 1); a + b × c = (a + b) × c + 1 (for some numbers only); a − a × 0 = 0 (only when a = 0, since a − a × 0 = a).\n\nNever true: (a + 1) × 0 = a + 1 × 0 for counting numbers a (0 on the left, a on the right).",{"hints":437},[438,439],"Try a = 2, b = 3, c = 4 first, then try 0 and 1.","One failing example rules out \"always\". One working example rules out \"never\".",{"id":441,"type":442,"title":443,"problem":444,"steps":445},"we-asn","worked_example","Proving a \"sometimes\" statement","When is **a + b × c** equal to **(a + b) × c**? Find all the cases.",[446,447,448,449,450,451],"Try numbers: a = 2, b = 3, c = 4: 14 versus 20. Not equal.","a = 0, b = 3, c = 4: 12 versus 12. Equal! a = 2, b = 3, c = 1: 5 versus 5. Equal!","Find the reason. (a + b) × c = a × c + b × c (distributive property).","So the two are equal when a + b × c = a × c + b × c, that is, when **a = a × c**.","a = a × c happens exactly when **a = 0** or **c = 1**. Every other choice gives different values.","Conclusion: **sometimes true**, precisely when a = 0 or c = 1. Tests suggested it; the reason proved it.",{"id":453,"type":55,"title":454,"eyebrow":455,"navLabel":456},"ch5","Fill in the missing signs","Chapter 05","5 Missing signs",{"id":458,"type":43,"markdown":459},"missing","Here is a different kind of puzzle. The numbers are fixed, but the signs are hidden: **6 ▢ 3 ▢ 2**. Each box can be +, −, × or ÷. How many different values can you get?\n\nThere are 4 × 4 = **16** ways to fill the boxes. Before you look at the table, predict: which filling gives the biggest value? Which gives 0? Are any two fillings equal?",{"id":461,"type":79,"caption":462,"columns":463,"rows":466},"tbl-missing","All 16 ways to fill 6 ▢ 3 ▢ 2 (using the order of operations)",[464,465,84],"Expression","After the first step",[467,470,474,478,482,486,490,494,498,502,506,510,514,517,520,523],[468,469,102],"6 + 3 + 2","9 + 2",[471,472,473],"6 + 3 − 2","9 − 2","7",[475,476,477],"6 + 3 × 2","6 + 6","12",[479,480,481],"6 + 3 ÷ 2","6 + 3\u002F2","15\u002F2 (not whole)",[483,484,485],"6 − 3 + 2","3 + 2","5",[487,488,489],"6 − 3 − 2","3 − 2","1",[491,492,493],"6 − 3 × 2","6 − 6","0",[495,496,497],"6 − 3 ÷ 2","6 − 3\u002F2","9\u002F2 (not whole)",[499,500,501],"6 × 3 + 2","18 + 2","20",[503,504,505],"6 × 3 − 2","18 − 2","16",[507,508,509],"6 × 3 × 2","18 × 2","36",[511,512,513],"6 × 3 ÷ 2","18 ÷ 2","9",[515,516,72],"6 ÷ 3 + 2","2 + 2",[518,519,493],"6 ÷ 3 − 2","2 − 2",[521,522,72],"6 ÷ 3 × 2","2 × 2",[524,525,489],"6 ÷ 3 ÷ 2","2 ÷ 2",{"id":527,"type":49,"variant":50,"title":528,"markdown":529},"obs-missing","What the table shows","- The biggest value is 6 × 3 × 2 = **36**.\n- Two fillings give **0**: 6 − 3 × 2 and 6 ÷ 3 − 2.\n- Two fillings give **4**: 6 ÷ 3 + 2 and 6 ÷ 3 × 2.\n- Two fillings (6 + 3 ÷ 2 and 6 − 3 ÷ 2) do not give whole numbers, because 3 ÷ 2 is done first.\n- 6 + 3 × 2 = 12 but, if you forgot the rule, you would say 18. Many of these puzzles are designed to catch that.",{"id":531,"type":61,"prompt":532,"options":533,"explanation":542},"predict-target","Fill the boxes in **8 ▢ 4 ▢ 2** to make **16**. Which works?",[534,536,538,540],{"id":65,"label":535},"8 + 4 × 2",{"id":68,"label":537},"8 × 4 ÷ 2",{"id":71,"label":539},"8 ÷ 4 × 2",{"id":74,"label":541},"8 − 4 × 2","**8 × 4 ÷ 2 = 16** and also **8 + 4 × 2 = 16**! Both (a) and (b) work: 8 × 4 ÷ 2 = 32 ÷ 2 = 16 and 8 + 4 × 2 = 8 + 8 = 16. The others: 8 ÷ 4 × 2 = 2 × 2 = 4 and 8 − 4 × 2 = 8 − 8 = 0. Puzzles can have more than one solution, so always check them all.",{"id":544,"type":110,"component":111,"componentVersion":5,"config":545,"objective":553,"textAlternative":554,"help":555},"lab-orderops-signs",{"expressions":546,"showRuleCard":124},[475,547,548,537,535,549,550,551,552],"6 × 3 - 2","6 - 3 × 2 + 9","12 ÷ 4 + 2 × 3","12 - 4 ÷ 2 × 3","9 × 2 - 12 ÷ 3","5 + 10 ÷ 5 × 3 - 1","Work through expressions made by filling in missing signs, and check which ones hit the target you predicted.","Nine expressions, each one a possible answer to a missing-signs puzzle. Step through each and compare the value with what you predicted.\n\n- 6 + 3 × 2 = 6 + 6 = 12\n- 6 × 3 − 2 = 18 − 2 = 16\n- 6 − 3 × 2 + 9 = 6 − 6 + 9 = 0 + 9 = 9\n- 8 × 4 ÷ 2 = 32 ÷ 2 = 16\n- 8 + 4 × 2 = 8 + 8 = 16\n- 12 ÷ 4 + 2 × 3 = 3 + 2 × 3 = 3 + 6 = 9\n- 12 − 4 ÷ 2 × 3 = 12 − 2 × 3 = 12 − 6 = 6\n- 9 × 2 − 12 ÷ 3 = 18 − 12 ÷ 3 = 18 − 4 = 14\n- 5 + 10 ÷ 5 × 3 − 1 = 5 + 2 × 3 − 1 = 5 + 6 − 1 = 11 − 1 = 10\n\nTwo different fillings of 8 ▢ 4 ▢ 2 both make 16.",{"hints":556},[557],"Predict the value before you tap anything.",{"id":559,"type":154,"itemId":560,"prompt":561,"check":562,"hints":576,"feedback":578},"pr-missing","order-of-operations.investigate-missing-signs","Which filling of **10 ▢ 5 ▢ 5** gives **3**?",{"kind":279,"options":563,"correct":575},[564,566,568,570,572],{"id":65,"label":565},"10 − 5 − 5",{"id":68,"label":567},"10 ÷ 5 + 5",{"id":71,"label":569},"10 − 5 ÷ 5",{"id":74,"label":571},"10 ÷ 5 × 5",{"id":573,"label":574},"e","None of these",[573],[577],"Work each one out with the order of operations.",{"correct":579,"incorrect":580},"Right: the values are 0, 7, 9 and 10, so none gives 3. (With brackets, (10 + 5) ÷ 5 = 3 would work.)","Check each: 10 − 5 − 5 = 0, 10 ÷ 5 + 5 = 7, 10 − 5 ÷ 5 = 9, 10 ÷ 5 × 5 = 10. None is 3. Brackets would help: (10 + 5) ÷ 5 = 3.",{"id":582,"type":55,"title":583,"eyebrow":584,"navLabel":585},"ch6","Biggest and smallest","Chapter 06","6 Biggest & smallest",{"id":587,"type":43,"markdown":588},"bigsmall","Take the digits **1, 2, 3, 4**. Use each once, with any of + and ×, and any brackets. What is the biggest value?\n\nYou might guess 1 × 2 × 3 × 4 = 24. But try (1 + 2) × 3 × 4 = **36**. Adding the 1 to something first is better than multiplying by 1, because multiplying by 1 changes nothing.\n\nThis leads to a useful observation: **for numbers bigger than 2, multiplying beats adding; but 1s should be added, not multiplied.** (2 and 2 are a tie: 2 + 2 = 2 × 2 = 4.)",{"id":590,"type":79,"caption":591,"columns":592,"rows":597},"tbl-add-mult","When does multiplying beat adding?",[593,594,595,596],"Pair","a + b","a × b","Winner",[598,602,604,607,610,612,615],[599,600,485,601],"1 and 5","6","add",[603,69,489,601],"1 and 1",[605,72,72,606],"2 and 2","tie",[608,485,600,609],"2 and 3","multiply",[611,600,513,609],"3 and 3",[613,477,614,609],"5 and 7","35",[616,513,493,601],"0 and 9",{"id":618,"type":61,"prompt":619,"options":620,"explanation":627},"predict-small","Using **2, 3, 4, 5** once each with + and × and any brackets, what is the **smallest** possible value?",[621,623,624,625],{"id":65,"label":622},"14",{"id":68,"label":140},{"id":71,"label":89},{"id":74,"label":626},"120","**14**: just add them all, 2 + 3 + 4 + 5 = 14. Every number is at least 2, so replacing any + by × can never make things smaller (for numbers 2 or bigger, a × b ≥ a + b). The biggest is 2 × 3 × 4 × 5 = 120.",{"id":629,"type":154,"itemId":630,"prompt":631,"check":632,"hints":633,"feedback":636},"pr-biggest-digits","order-of-operations.investigate-biggest-digits","Using **1, 1, 5, 6** once each, with + and × and brackets, what is the **biggest** value you can make?",{"kind":158,"answer":356,"tolerance":160},[634,635],"Do not multiply by 1s: add them to something first.","Try (1 + 1) × 5 × 6 or (1 + 5) × (1 + 6).",{"correct":637,"incorrect":638},"Yes: (1 + 1) × 5 × 6 = 2 × 30 = 60. It beats (1 + 5) × (1 + 6) = 42 and 1 × 1 × 5 × 6 = 30.","Compare: 1 × 1 × 5 × 6 = 30, (1 + 5) × (1 + 6) = 42, (1 + 1) × 5 × 6 = 60. The biggest is 60.",{"id":640,"type":55,"title":641,"eyebrow":642,"navLabel":643},"ch7","Investigating calculators","Chapter 07","7 Calculator tests",{"id":645,"type":43,"markdown":646},"calc-inv","In Understand you met two kinds of calculator. Now test them like a scientist. For each key sequence below, **predict** what a basic (step-by-step) calculator shows and what a scientific calculator shows, then check if you can.\n\nThe basic calculator works like reading a sentence: it finishes each operation before starting the next, and it has **no brackets**. The scientific calculator waits for = and follows the order of operations.",{"id":648,"type":79,"caption":649,"columns":650,"rows":655},"tbl-calc","Basic versus scientific: key sequences tested",[651,652,653,654],"Keys pressed (then =)","Basic shows","Scientific shows","Agree?",[656,659,663,666,667,670,673],[657,501,622,658],"2 + 3 × 4","no",[660,661,662,658],"20 − 4 × 3","48","8",[664,513,513,665],"10 − 3 + 2","yes",[539,72,72,665],[668,600,669,658],"6 + 12 ÷ 3","10",[671,672,622,658],"5 × 4 − 2 × 3","54",[674,675,676,658],"100 − 10 × 5 + 5","455","55",{"id":678,"type":49,"variant":50,"title":679,"markdown":680},"obs-calc","When do they agree?","The two calculators agree whenever the sequence **never puts + or − before a × or ÷**: 10 − 3 + 2 and 8 ÷ 4 × 2 are fine, because for partners the basic calculator's left-to-right habit is exactly the rule.\n\nThey disagree when a + or − comes before a × or ÷, because the basic calculator does the + or − too early. Notice 5 × 4 − 2 × 3: the basic calculator does 20 − 2 = 18, then 18 × 3 = 54, instead of 20 − 6 = 14.",{"id":682,"type":61,"prompt":683,"options":684,"explanation":693},"predict-reorder","You have only a basic calculator and want 100 − 10 × 5 + 5. Which key sequence gives the correct value (55)?",[685,687,689,691],{"id":65,"label":686},"100 − 10 × 5 + 5 =",{"id":68,"label":688},"10 × 5 = then 100 − that + 5, typed as: 100 − 50 + 5 =",{"id":71,"label":690},"5 + 100 − 10 × 5 =",{"id":74,"label":692},"It is impossible on a basic calculator","**(b).** Work out the chunk 10 × 5 = 50 first, then type 100 − 50 + 5 = 55. Option (a) gives (100 − 10) × 5 + 5 = 455, and (c) gives (105 − 10) × 5 = 475.",{"id":695,"type":696,"prompt":697},"reflect-calc","reflection","A shopkeeper uses a basic calculator all day and never gets bills wrong. How can that be, if the calculator ignores the order of operations? Think about how bills are usually typed in, item by item.",{"id":699,"type":55,"title":700,"eyebrow":701,"navLabel":702},"ch8","Hitting a target","Chapter 08","8 Hit the target",{"id":704,"type":43,"markdown":705},"target","Now combine everything: signs, brackets and the order of operations. A **target puzzle** gives you numbers and a target; you must hit the target exactly.\n\nExample: use **3, 5, 7, 2** in this order with any signs and brackets to make **30**.\n\n- 3 × 5 + 7 × 2 = 15 + 14 = 29. Close!\n- (3 + 5 + 7) × 2 = 15 × 2 = **30**. ✓\n- 3 × (5 + 7) − 2 × 3? Not allowed: that uses 3 twice.\n\nA good strategy is to work **backwards**: 30 = 15 × 2, so can the first three numbers make 15? 3 + 5 + 7 = 15. Done.",{"id":707,"type":110,"component":111,"componentVersion":5,"config":708,"objective":716,"textAlternative":717,"help":718},"lab-orderops-targets",{"expressions":709,"showRuleCard":124},[710,711,712,713,714,715],"(3 + 5 + 7) × 2","(9 - 3) × (2 + 2)","[6 × (4 - 1)] ÷ 2 + 1","(7 + 5) × (8 - 6)","[20 - (6 + 4)] × 5","(10 + 2) ÷ 3 × 6","Step through target-puzzle solutions and confirm each one really hits its target.","Six solutions to target puzzles. Step through each to check it:\n\n- (3 + 5 + 7) × 2 = (8 + 7) × 2 = 15 × 2 = 30\n- (9 − 3) × (2 + 2) = 6 × (2 + 2) = 6 × 4 = 24\n- [6 × (4 − 1)] ÷ 2 + 1 = [6 × 3] ÷ 2 + 1 = 18 ÷ 2 + 1 = 9 + 1 = 10\n- (7 + 5) × (8 − 6) = 12 × (8 − 6) = 12 × 2 = 24\n- [20 − (6 + 4)] × 5 = [20 − 10] × 5 = 10 × 5 = 50\n- (10 + 2) ÷ 3 × 6 = 12 ÷ 3 × 6 = 4 × 6 = 24\n\nTargets: 30, 24, 10, 24, 50 and 24. Several different starting numbers can reach 24: it is a favourite target because it has so many factors.",{"hints":719},[720],"Work backwards: which two numbers multiply to the target?",{"id":722,"type":154,"itemId":723,"prompt":724,"check":725,"hints":727,"feedback":730},"pr-target","order-of-operations.investigate-target","Put brackets into **2 + 4 × 5 − 1** (keep the order) to make the value as big as possible. What is the biggest value?",{"kind":158,"answer":726,"tolerance":160},29,[728,729],"Try the + first: (2 + 4) × 5 − 1.","Or both brackets: (2 + 4) × (5 − 1).",{"correct":731,"incorrect":732},"Yes: (2 + 4) × 5 − 1 = 30 − 1 = 29 is the biggest.","The options are 2 + 4 × 5 − 1 = 21, (2 + 4) × 5 − 1 = 29, (2 + 4) × (5 − 1) = 24, 2 + 4 × (5 − 1) = 18. The biggest is 29.",{"id":734,"type":55,"title":735,"eyebrow":736,"navLabel":737},"ch-zero-one","Zero and one: the troublemakers","Chapter 09","9 Zero and one",{"id":739,"type":43,"markdown":740},"zero-one","When you test a rule, always include **0** and **1**. They behave in special ways, and they are where wrong rules most often *seem* to work.\n\n- Multiplying by 1 changes nothing: 7 + 1 × 5 = 7 + 5 = 12, the same as if the × were not there.\n- Multiplying by 0 wipes everything out: 7 + 0 × 5 = 7 + 0 = 7, but (7 + 0) × 5 = 35.\n- Adding or subtracting 0 changes nothing: 9 × 4 − 0 = 36.\n- Dividing by 1 changes nothing: 20 ÷ 1 × 3 = 60.\n- Dividing **by** 0 is not allowed at all: 5 ÷ (3 − 3) has no value.\n\nA classic trick question is **9 − 9 × 0 + 1**. People who go left to right say (0) × 0 + 1 = 1. The rule says 9 − 9 × 0 + 1 = 9 − 0 + 1 = 9 + 1 = 10.",{"id":742,"type":79,"caption":743,"columns":744,"rows":747},"tbl-zero","Zero and one inside expressions",[464,745,84,746],"Steps","Watch out",[748,752,756,760,764,768],[749,750,477,751],"7 + 1 × 5","7 + 1 × 5 = 7 + 5 = 12","× 1 does nothing",[753,754,473,755],"7 + 0 × 5","7 + 0 × 5 = 7 + 0 = 7","0 × anything = 0",[757,758,614,759],"(7 + 0) × 5","(7 + 0) × 5 = 7 × 5 = 35","brackets change it",[761,762,72,763],"0 ÷ 4 + 4","0 ÷ 4 + 4 = 0 + 4 = 4","0 ÷ 4 = 0 is fine",[765,766,669,767],"9 − 9 × 0 + 1","9 − 9 × 0 + 1 = 9 − 0 + 1 = 9 + 1 = 10","not 1",[769,770,771,772],"5 ÷ (3 − 3)","5 ÷ 0","no value","cannot divide by 0",{"id":774,"type":61,"prompt":775,"options":776,"explanation":782},"predict-zero","What is **1 + 1 × 0 + 1**?",[777,778,779,780],{"id":65,"label":493},{"id":68,"label":489},{"id":71,"label":69},{"id":74,"label":781},"3","**2.** × first: 1 × 0 = 0. Then 1 + 0 + 1 = 2. Full chain: 1 + 1 × 0 + 1 = 1 + 0 + 1 = 1 + 1 = 2. Reading like a sentence gives (2) × 0 + 1 = 1, which is wrong.",{"id":784,"type":49,"variant":785,"title":786,"markdown":787},"careful-zero","careful","Zero can hide a mistake","Suppose you test the wrong rule \"a + b × c = (a + b) × c\" with a = 0. You get b × c = b × c: it *works*! If that was your only test, you would believe a false rule. This is why a good investigation always tries several kinds of numbers, not only the easiest one.",{"id":789,"type":55,"title":790,"eyebrow":791,"navLabel":792},"ch-strike","Cricket statistics: does the order of × and ÷ matter?","Chapter 10","10 Strike rates",{"id":794,"type":43,"markdown":795},"strike","A batter's **strike rate** is the runs scored per 100 balls: **runs × 100 ÷ balls**. If Shafali scores 45 runs off 30 balls, her strike rate is 45 × 100 ÷ 30 = 4,500 ÷ 30 = 150.\n\nHere is an investigation. The formula could also be written **runs ÷ balls × 100** or **100 ÷ balls × runs**. Do these always give the same value?\n\n- 45 ÷ 30 × 100: 45 ÷ 30 = 1½, then 1½ × 100 = 150.\n- 100 ÷ 30 × 45: 100 ÷ 30 = 3⅓, then 3⅓ × 45 = 150.\n\nSame value! Each ÷ stays attached to its own number (30), so moving the × and ÷ around does not change the answer. But look at the *path*: only the first order kept every step a whole number. When you work by hand, **multiply first and divide last** to avoid fractions along the way.",{"id":797,"type":79,"caption":798,"columns":799,"rows":804},"tbl-strike","Strike rate three ways (runs × 100 ÷ balls)",[800,801,802,803],"Runs, balls","runs × 100 ÷ balls","runs ÷ balls × 100","All steps whole?",[805,810,813,816,820],[806,807,808,809],"45, 30","4,500 ÷ 30 = 150","3\u002F2 × 100 = 150","only the first way",[811,812,808,809],"60, 40","6,000 ÷ 40 = 150",[814,815,808,809],"36, 24","3,600 ÷ 24 = 150",[817,818,819,809],"80, 50","8,000 ÷ 50 = 160","8\u002F5 × 100 = 160",[821,822,823,809],"100, 80","10,000 ÷ 80 = 125","5\u002F4 × 100 = 125",{"id":825,"type":61,"prompt":826,"options":827,"explanation":836},"predict-economy","A bowler's **economy rate** is runs given ÷ overs. Which expression gives the economy for **42 runs in 6 overs and 30 runs in 4 overs** combined?",[828,830,832,834],{"id":65,"label":829},"42 ÷ 6 + 30 ÷ 4",{"id":68,"label":831},"(42 + 30) ÷ (6 + 4)",{"id":71,"label":833},"42 + 30 ÷ 6 + 4",{"id":74,"label":835},"(42 ÷ 6 + 30 ÷ 4) ÷ 2","**(b) (42 + 30) ÷ (6 + 4) = 72 ÷ 10 = 7.2 runs per over.** Total runs over total overs. Option (d), averaging the two economies (7 and 7.5), gives 7.25, which is close but wrong, because the bowler bowled more overs in the first spell. Option (c) is 42 + 5 + 4 = 51, nonsense without brackets. Both sets of brackets in (b) are needed.",{"id":838,"type":49,"variant":205,"title":839,"markdown":840},"aha-average-of-averages","An average of averages is not always the average","The economy puzzle shows a surprising fact: (a ÷ b + c ÷ d) ÷ 2 is usually **not** equal to (a + c) ÷ (b + d). They are equal when the two spells have the same number of overs (b = d), but not otherwise. Brackets are not just about the order of steps; they decide **which quantity you are actually calculating**.",{"id":842,"type":154,"itemId":843,"prompt":844,"check":845,"hints":847,"feedback":850},"pr-strike","order-of-operations.investigate-strike-rate","Rohit scores **72 runs off 48 balls**. What is his strike rate (**72 × 100 ÷ 48**)?",{"kind":158,"answer":846,"tolerance":160},150,[848,849],"Multiply first: 72 × 100 = 7,200.","7,200 ÷ 48.",{"correct":851,"incorrect":852},"Yes: 7,200 ÷ 48 = 150.","Multiply first, then divide: 72 × 100 = 7,200 and 7,200 ÷ 48 = 150.",{"id":854,"type":154,"itemId":855,"prompt":856,"check":857,"hints":859,"feedback":862},"pr-economy","order-of-operations.investigate-economy","A bowler gives **24 runs in 4 overs**, then **36 runs in 2 overs**. Using **(24 + 36) ÷ (4 + 2)**, what is the overall economy rate?",{"kind":158,"answer":858,"tolerance":160},10,[860,861],"Total runs ÷ total overs.","60 ÷ 6.",{"correct":863,"incorrect":864},"Yes: 60 ÷ 6 = 10 runs per over. (Averaging the two spells, (6 + 18) ÷ 2 = 12, would be wrong.)","Total runs 24 + 36 = 60, total overs 4 + 2 = 6, so 60 ÷ 6 = 10. The average of the spell economies, (6 + 18) ÷ 2 = 12, gives the wrong answer because the spells had different lengths.",{"id":866,"type":55,"title":867,"eyebrow":868,"navLabel":869},"ch-split","Splitting a bill: where do the brackets go?","Chapter 11","11 Splitting bills",{"id":871,"type":43,"markdown":872},"split","Four friends eat at a dhaba. The bill is **₹640** for food, plus **₹80** for drinks that only **two** of them had. How much should each person pay?\n\nThis is a real investigation, because there are several *fair* answers, and each is a different expression:\n\n- **Everyone shares everything:** (640 + 80) ÷ 4 = (640 + 80) ÷ 4 = 720 ÷ 4 = 180 each.\n- **Food shared by all, drinks by the two drinkers:** non-drinkers pay 640 ÷ 4 = 160; drinkers pay 640 ÷ 4 + 80 ÷ 2 = 640 ÷ 4 + 80 ÷ 2 = 160 + 80 ÷ 2 = 160 + 40 = 200.\n- **Check the second plan adds up:** 2 × 160 + 2 × 200 = 2 × 160 + 2 × 200 = 320 + 2 × 200 = 320 + 400 = 720 = 640 + 80 ✓.\n\nThe common *wrong* expression is 640 + 80 ÷ 4 = 640 + 20 = 660 per person, which charges everyone almost the whole bill! The ÷ grabbed only the 80.",{"id":874,"type":875,"title":876,"prompt":877,"options":878},"explorer-split","explorer","Four ways to split a ₹720 dhaba bill","Choose a rule for sharing and see its expression.",[879,890,901,911],{"id":880,"label":881,"chain":882,"badge":887,"note":889},"equal","Equal shares",[883,884,885,886],"Add food and drinks","(640 + 80)","Share by 4","₹180 each",{"text":888,"tone":665},"Brackets needed","(640 + 80) ÷ 4 = 720 ÷ 4 = 180. Simple and quick; the two who did not have drinks pay ₹20 more than their share of what they ate and drank.",{"id":891,"label":892,"chain":893,"badge":898,"note":900},"fair","Pay for what you had",[894,895,896,897],"Food ÷ 4","Drinks ÷ 2","Drinkers add both","₹160 or ₹200",{"text":899,"tone":665},"Two terms","Non-drinkers: 640 ÷ 4 = 160. Drinkers: 640 ÷ 4 + 80 ÷ 2 = 160 + 40 = 200. Check: 160 + 160 + 200 + 200 = 720.",{"id":902,"label":903,"chain":904,"badge":909,"note":910},"tip","Add a 10% tip",[905,906,907,908],"Total 720","Tip 720 ÷ 10 = 72","(720 + 72) ÷ 4","₹198 each",{"text":888,"tone":665},"(720 + 720 ÷ 10) ÷ 4 = (720 + 72) ÷ 4 = 792 ÷ 4 = 198. The tip is worked out inside the bracket: ÷ before +.",{"id":912,"label":913,"chain":914,"badge":919,"note":921},"wrong","The slip",[915,916,917,918],"640 + 80 ÷ 4","÷ grabs only 80","640 + 20","₹660 each!",{"text":920,"tone":658},"Wrong: missing brackets","640 + 80 ÷ 4 = 660. Four people paying ₹660 would pay ₹2,640 for a ₹720 bill. A quick size check (each share must be less than the whole bill) exposes the missing brackets.",{"id":923,"type":61,"prompt":924,"options":925,"explanation":934},"predict-split","Six friends share a **₹900** bill equally, but one friend has a **₹150** discount coupon that applies to the whole bill. Which expression gives each person's share?",[926,928,930,932],{"id":65,"label":927},"(900 − 150) ÷ 6",{"id":68,"label":929},"900 − 150 ÷ 6",{"id":71,"label":931},"900 ÷ 6 − 150",{"id":74,"label":933},"(900 ÷ 6) − 150 ÷ 6 × 6","**(a) (900 − 150) ÷ 6 = 750 ÷ 6 = ₹125.** The coupon reduces the whole bill, so the subtraction must happen before sharing. (b) = 900 − 25 = 875 per person; (c) = 150 − 150 = 0 per person; (d) = 150 − 150 = 0. Checking that six shares add up to the bill after the coupon (6 × 125 = 750) confirms (a).",{"id":936,"type":154,"itemId":937,"prompt":938,"check":939,"hints":942,"feedback":945},"pr-split","order-of-operations.investigate-split-bill","Five friends share a **₹450** pizza bill equally, and three of them also share a **₹90** dessert. How much does a dessert-eater pay? Use **450 ÷ 5 + 90 ÷ 3**.",{"kind":158,"answer":940,"tolerance":160,"unit":941},120,"₹",[943,944],"Two terms: 450 ÷ 5 and 90 ÷ 3.","90 + 30.",{"correct":946,"incorrect":947},"Yes: 90 + 30 = ₹120.","÷ first in each term: 450 ÷ 5 = 90 and 90 ÷ 3 = 30. Then 90 + 30 = ₹120.",{"id":949,"type":442,"title":950,"problem":951,"steps":952},"we-auto-investigate","When does a missing bracket not matter?","An auto fare is **₹30 to start plus ₹15 per km**: 30 + 15 × km. A careless app programmer writes (30 + 15) × km instead. For which trip lengths does the app still charge the right fare?",[953,954,955,956,957],"Test some trips. 1 km: 30 + 15 × 1 = 45 and (30 + 15) × 1 = 45. Same!","2 km: 30 + 15 × 2 = 60 but (30 + 15) × 2 = 90. Different.","5 km: 30 + 15 × 5 = 105 but (30 + 15) × 5 = 225. Different, and the gap is growing.","Find the reason: (30 + 15) × km = 30 × km + 15 × km. This equals 30 + 15 × km only when 30 × km = 30, that is, when **km = 1**.","Conclusion: the bug is invisible on a 1 km test ride, but overcharges every longer trip by 30 × (km − 1) rupees. This is why testers try several inputs, not just the easiest one.",{"id":959,"type":154,"itemId":960,"prompt":961,"check":962,"hints":963,"feedback":966},"pr-auto-gap","order-of-operations.investigate-auto-gap","With the buggy app, a **6 km** auto ride is charged (30 + 15) × 6 instead of 30 + 15 × 6. How many rupees too much is charged?",{"kind":158,"answer":846,"tolerance":160,"unit":941},[964,965],"Work out both fares.","(30 + 15) × 6 = 270 and 30 + 15 × 6 = 120.",{"correct":967,"incorrect":968},"Yes: 270 − 120 = ₹150, which is 30 × (6 − 1).","Correct fare 30 + 90 = 120; buggy fare 45 × 6 = 270. The overcharge is 270 − 120 = ₹150.",{"id":970,"type":49,"variant":971,"title":972,"markdown":973},"misc-one-test","misconception","“It worked on my example, so it is right”","Students (and programmers) often test a formula with one easy number, see the right answer, and stop. The auto-fare bug passes the 1 km test perfectly. The halving slip (a + b ÷ 2 instead of (a + b) ÷ 2) passes whenever b = 0. And 2 + 2 × 2 happens to equal 2 × 2 + 2, hiding which step was done first.\n\nA good test uses **several** inputs, including awkward ones, and checks that the answer is sensible in size.",{"id":975,"type":696,"prompt":976},"reflect-fair","Which way of splitting the dhaba bill do you think is fairest, and why? Write the expression for your choice. Does it change if one friend had a birthday and the others want to pay for her?",{"id":978,"type":55,"title":979,"eyebrow":980,"navLabel":981},"ch-sheet-exp","Spreadsheet experiments","Chapter 12","12 Spreadsheet tests",{"id":983,"type":43,"markdown":984},"sheet-exp","A spreadsheet is a perfect laboratory for the order of operations: type a formula, press Enter, and the machine shows exactly how it reads it. Try this experiment (or predict the results if you have no computer to hand). Put **10** in A1, **4** in A2 and **2** in A3. Then type each formula into a new cell and **predict before you press Enter**.",{"id":986,"type":79,"caption":987,"columns":988,"rows":992},"tbl-sheet-exp","Predict, then test: A1 = 10, A2 = 4, A3 = 2",[989,990,991],"Formula","Means","Result",[993,996,999,1002,1006,1010,1014,1018],[994,995,662],"=A1-A2+A3","10 − 4 + 2",[997,998,72],"=A1-(A2+A3)","10 − (4 + 2)",[1000,1001,501],"=A1\u002FA3*A2","10 ÷ 2 × 4",[1003,1004,1005],"=A1\u002F(A3*A2)","10 ÷ (2 × 4)","5\u002F4 (shown as 1.25)",[1007,1008,1009],"=A1+A2*A3","10 + 4 × 2","18",[1011,1012,1013],"=(A1+A2)*A3","(10 + 4) × 2","28",[1015,1016,1017],"=(A1+A2+A3)\u002F3","(10 + 4 + 2) ÷ 3","16\u002F3 (shown as 5.33)",[1019,1020,1021],"=A1+A2+A3\u002F3","10 + 4 + 2 ÷ 3","44\u002F3 (shown as 14.67)",{"id":1023,"type":49,"variant":50,"title":1024,"markdown":1025},"obs-sheet","What the experiment shows","Every pair of formulas differs only by brackets, and every pair gives different results: 8 and 4, 20 and 1.25, 18 and 28, 5.33 and 14.67. The spreadsheet never guesses what you *meant*; it follows the order of operations exactly. That is why a formula that looks right but gives a strange number almost always has a missing bracket.",{"id":1027,"type":61,"prompt":1028,"options":1029,"explanation":1035},"predict-sheet","With A1 = 10, A2 = 4 and A3 = 2, what does **=A1*A2\u002FA3-A2** give?",[1030,1031,1032,1034],{"id":65,"label":505},{"id":68,"label":485},{"id":71,"label":1033},"−20",{"id":74,"label":493},"**16.** × and \u002F left to right: 10 × 4 = 40, 40 ÷ 2 = 20; then 20 − 4 = 16. As a chain: 10 × 4 ÷ 2 − 4 = 40 ÷ 2 − 4 = 20 − 4 = 16.",{"id":1037,"type":442,"title":1038,"problem":1039,"steps":1040},"we-sheet-gst","Testing a GST column","A shop's spreadsheet has prices in column B and needs the price with 18% GST in column C. Someone types **=B2+B2*18\u002F100** and someone else **=(B2+B2)*18\u002F100**. Test both with B2 = 500.",[1041,1042,1043,1044],"First formula: 500 + 500 × 18 ÷ 100. × and ÷ first, left to right: 500 × 18 = 9,000, 9,000 ÷ 100 = 90. Then 500 + 90 = **590**. Correct.","Second formula: (500 + 500) × 18 ÷ 100 = 1,000 × 18 ÷ 100 = **180**. That is only double the tax, not the price with tax.","A third correct version is =B2*118\u002F100: 500 × 118 ÷ 100 = 590. It is shorter because B2 + 18% of B2 = B2 × 118%.","Test with a second value too, say B2 = 200: the first formula gives 200 + 36 = 236, and 200 × 118 ÷ 100 = 236. ✓",{"id":1046,"type":154,"itemId":1047,"prompt":1048,"check":1049,"hints":1060,"feedback":1062},"pr-sheet-fix","order-of-operations.investigate-sheet-fix","A shop spreadsheet should give the **average of three prices** in C2, C3, C4, but shows a number bigger than all three. The formula is **=C2+C3+C4\u002F3**. Which is the fix?",{"kind":279,"options":1050,"correct":1059},[1051,1053,1055,1057],{"id":65,"label":1052},"=C2+C3+(C4\u002F3)",{"id":68,"label":1054},"=(C2+C3+C4)\u002F3",{"id":71,"label":1056},"=C2\u002F3+C3+C4",{"id":74,"label":1058},"=3\u002F(C2+C3+C4)",[68],[1061],"Which part should be divided by 3?",{"correct":1063,"incorrect":1064},"Yes: the whole total must be divided, so it goes in brackets.","The whole total must be divided by 3: =(C2+C3+C4)\u002F3. Option (a) is the same as the broken formula; (d) divides 3 by the total.",{"id":1066,"type":55,"title":1067,"eyebrow":1068,"navLabel":1069},"ch10","What you found out","Chapter 13","13 Wrap-up",{"id":1071,"type":1072,"title":1073,"terms":1074},"glossary-investigate","glossary","Words for investigating",[1075,1079,1083,1087,1090,1092,1095],{"term":1076,"meaning":1077,"example":1078},"Conjecture","A statement you think is true, based on examples, but have not yet proved.","Removing a bracket after − flips the signs inside.",{"term":1080,"meaning":1081,"example":1082},"Counterexample","One example that shows a statement is not always true.","(20 − 8) − 2 = 10 but 20 − (8 − 2) = 14.",{"term":1084,"meaning":1085,"example":1086},"Associative","An operation is associative if grouping does not matter: (a ∘ b) ∘ c = a ∘ (b ∘ c). + and × are; − and ÷ are not.","(2 × 3) × 4 = 2 × (3 × 4)",{"term":1088,"meaning":1089,"example":94},"Bracketing","A way of placing brackets in an expression to fix the order.",{"term":82,"meaning":1091,"example":87},"Every operation has its own pair of brackets, so no rule is needed to read it.",{"term":1093,"meaning":1094},"Catalan numbers","1, 2, 5, 14, 42, 132, … The number of ways to fully bracket 2, 3, 4, 5, 6, 7, … numbers.",{"term":1096,"meaning":1097,"example":1098},"Target puzzle","A puzzle where numbers must be combined with operations to reach an exact value.","Make 24 from 7, 5, 8, 6.",{"id":1100,"type":1101,"title":1102,"questions":1103},"quiz-investigate","quiz","Investigation check-up",[1104,1113,1126,1139,1150,1162,1172,1181,1190],{"itemId":1105,"prompt":1106,"options":1107,"correct":71,"why":1112},"order-of-operations.iq-1","How many different values can brackets give 2 + 3 × 4 − 1 (order kept)?",[1108,1109,1110,1111],{"id":65,"label":489},{"id":68,"label":781},{"id":71,"label":72},{"id":74,"label":485},"Five bracketings give 19, 13, 15, 13 and 11: four different values.",{"itemId":1114,"prompt":1115,"options":1116,"correct":71,"why":1125},"order-of-operations.iq-2","Which operation is associative?",[1117,1119,1121,1123],{"id":65,"label":1118},"Subtraction",{"id":68,"label":1120},"Division",{"id":71,"label":1122},"Multiplication",{"id":74,"label":1124},"None","(a × b) × c = a × (b × c) always. Subtraction and division fail, e.g. (20 − 8) − 2 ≠ 20 − (8 − 2).",{"itemId":1127,"prompt":1128,"options":1129,"correct":68,"why":1138},"order-of-operations.iq-3","100 − (40 − 15) equals…",[1130,1132,1134,1136],{"id":65,"label":1131},"100 − 40 − 15",{"id":68,"label":1133},"100 − 40 + 15",{"id":71,"label":1135},"100 + 40 − 15",{"id":74,"label":1137},"100 − 55","Removing a bracket after − flips the − inside to +: 100 − 40 + 15 = 75, the same as 100 − 25.",{"itemId":1140,"prompt":1141,"options":1142,"correct":68,"why":1149},"order-of-operations.iq-4","Is \"a + b × c = (a + b) × c\" always, sometimes or never true?",[1143,1145,1147],{"id":65,"label":1144},"Always",{"id":68,"label":1146},"Sometimes",{"id":71,"label":1148},"Never","It is true only when a = 0 or c = 1. For 2, 3, 4 it gives 14 versus 20.",{"itemId":1151,"prompt":1152,"options":1153,"correct":68,"why":1161},"order-of-operations.iq-5","A basic calculator and a scientific one will always agree on…",[1154,1155,1157,1159],{"id":65,"label":657},{"id":68,"label":1156},"12 − 3 + 5",{"id":71,"label":1158},"10 − 2 × 3",{"id":74,"label":1160},"1 + 8 ÷ 2","12 − 3 + 5 has only + and −, and the basic calculator's left-to-right habit matches the rule. Both show 14.",{"itemId":1163,"prompt":1164,"options":1165,"correct":68,"why":1171},"order-of-operations.iq-6","Using 1, 2, 3, 4 once each with + and × and brackets, the biggest value is…",[1166,1168,1169,1170],{"id":65,"label":1167},"24",{"id":68,"label":509},{"id":71,"label":669},{"id":74,"label":661},"(1 + 2) × 3 × 4 = 36. Adding the 1 beats multiplying by it.",{"itemId":1173,"prompt":1174,"options":1175,"correct":68,"why":1180},"order-of-operations.iq-7","Which filling of 6 ▢ 3 ▢ 2 gives 0?",[1176,1177,1178,1179],{"id":65,"label":487},{"id":68,"label":491},{"id":71,"label":524},{"id":74,"label":503},"6 − 3 × 2 = 6 − 6 = 0. (6 ÷ 3 − 2 = 0 also works.)",{"itemId":1182,"prompt":1183,"options":1184,"correct":71,"why":1189},"order-of-operations.iq-8","How many ways are there to fully bracket 5 numbers?",[1185,1186,1187,1188],{"id":65,"label":485},{"id":68,"label":669},{"id":71,"label":622},{"id":74,"label":1167},"The Catalan numbers 1, 2, 5, 14, 42… give the count: 14 ways for five numbers.",{"itemId":1191,"prompt":1192,"options":1193,"correct":68,"why":1202},"order-of-operations.iq-9","What is enough to show a statement is NOT always true?",[1194,1196,1198,1200],{"id":65,"label":1195},"Three examples that work",{"id":68,"label":1197},"One counterexample",{"id":71,"label":1199},"A calculator",{"id":74,"label":1201},"A teacher saying so","A single example where it fails is enough.",{"id":1204,"type":1205,"title":1206,"points":1207},"cheat-investigate","summary","Cheat sheet",[1208,1209,1210,1211,1212,1213,1214,1215,1216],"**Moving brackets** can change the value: 2 + 3 × 4 − 1 can be 11, 13, 15 or 19. Four numbers can be fully bracketed in 5 ways; five numbers in 14 (Catalan numbers).","**Free brackets:** brackets change nothing when they surround what the rule does first anyway, or when only + or only × is involved (associative).","**− and ÷ are not associative:** (20 − 8) − 2 = 10 but 20 − (8 − 2) = 14. That is why partners go left to right.","**Opening brackets:** after −, flip + and − inside; after ÷, flip × and ÷ inside. After + (or × with only × ÷ inside) nothing flips.","**Always \u002F sometimes \u002F never:** one counterexample kills \"always\"; a reason is needed to prove \"always\".","**Missing signs:** 6 ▢ 3 ▢ 2 has 16 fillings; the order of operations decides each value.","**Biggest values:** multiply numbers above 2, add 1s. (1 + 2) × 3 × 4 = 36.","**Calculators:** basic and scientific agree unless a + or − comes before a × or ÷.","**Targets:** work backwards from the target: 30 = 15 × 2, so make 15 from the rest.",{"id":1218,"type":1219,"conceptId":1220,"relation":1221,"explanation":1222},"conn-properties","connection","properties-of-numbers","helps_understand","Associativity and the distributive property, tested here with brackets, are studied in Properties of numbers.",{"id":1224,"type":1219,"conceptId":1225,"relation":1226,"explanation":1227},"conn-patterns","patterns","related_to","Counting bracketings gives the Catalan numbers 1, 2, 5, 14, 42, a number pattern with its own rule.",{"id":1229,"type":1219,"conceptId":1230,"relation":1231,"explanation":1232},"conn-data","data-handling","applied_in","The mean is (sum of values) ÷ (number of values). Forgetting the brackets and dividing only the last value is a classic slip.",{"id":1234,"type":1235,"sourceIds":1236},"sources-investigate","sources",[1237,1238,1239,1240],"order-of-operations-wikipedia","order-of-operations-mathsisfun-bodmas","order-of-operations-ncert-class7","order-of-operations-wikipedia-24-game",[1237,1238,1239,1240],"needs_review",{"generatedBy":1244,"notes":1245},"claude-code","Draft generated with Python-checked arithmetic (× ÷ left to right, then + − left to right); pending owner review.","2970b5b9c53307644c494d779ed3a727d2d2d3cb1d07e0b45f3248114a4ad87f",{"component:order-ops@1":1248,"component:match-pairs@1":1249,"logic:practice":1250,"component:sort-game@1":1251,"source:order-of-operations-mathsisfun-bodmas":1252,"source:order-of-operations-ncert-class7":1253,"source:order-of-operations-wikipedia":1254,"source:order-of-operations-wikipedia-24-game":1255},"3d630ebc066352df791b020d545e1876ce4620d55c8c1c082414d6b4717255c5","2a8ee4ac87460b4e1175a4bb13c96b03d577db06dde95670eb7fcfe4ad787899","3d6b0fe1b15255975a32b0fcd94e8019bc959ad45cbf12e136e86149549c6878","b164f45a2c8ca08f26c450768ff0231e113e9fe45381eddb34dc6d0548596c38","92959283b293565e5dbcac7d99680a63df55eef50520c997ab808634afaa0062","e7cadfedc70b54014195d6d8bc2740fb10a7e8aed3509c11f1efbde6e7698ade","88ccf3653f7d9723bbb33a67a6548ccff64980f752cba0cd94a32bc60f72bd44","1022e875bb64d7d92d0b0eb62e2dfa81ff1c82aef5e67c620cafa77e2d0e8e1e",{"state":1257,"reviewer":1258,"selfReview":124,"reviewedAt":1259,"method":1260},"approved","The library owner","2026-09-20T10:18:37.581Z","owner_bulk","preview-7e1cbbcc4f",1789899597928]