[{"data":1,"prerenderedAt":1255},["ShallowReactive",2],{"layer:properties-of-numbers:investigate":3},{"layer":4,"contentHash":1232,"dependencyHashes":1233,"approval":1249,"releaseId":1254},{"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":1227,"reviewStatus":1228,"authoring":1229},1,"properties-of-numbers","en","investigate","Always, sometimes or never?","Predict, test and explain: counterexamples, grouping gaps, parity patterns and shortcut showdowns","Test claims about whole numbers the way mathematicians do: predict, hunt for counterexamples, measure how badly subtraction and division fail to swap or regroup, discover patterns and shortcuts, and explain why the true ones must be true.",[13,14,15,16,17],"Use a testing routine and counterexamples to classify claims as always, sometimes or never true.","Measure and explain the gap between (a − b) − c and a − (b − c).","Discover and justify shortcuts for multiplying by 9, 11, 99 and 101.","Investigate parity patterns for consecutive numbers and sums of odd numbers.","Decide whether families such as multiples, squares and odd numbers are closed under + and ×.",45,{"title":20,"rows":21},"Lesson plate",[22,25,28,31,34,37],{"label":23,"value":24},"Depth","Investigate",{"label":26,"value":27},"Reading time","≈ 45 minutes",{"label":29,"value":30},"Prior knowledge","The properties stated precisely (Understand)",{"label":32,"value":33},"Chapters","10",{"label":35,"value":36},"Labs","5 (two sorts, sprint, memory match, brackets lab)",{"label":38,"value":39},"Key skill","Finding counterexamples",[41,45,66,72,105,120,128,134,139,142,174,188,193,196,228,241,265,270,289,294,297,331,344,347,391,396,399,427,440,478,483,488,491,517,520,533,567,580,660,665,668,704,770,775,778,803,807,829,832,843,848,851,868,903,908,942,947,950,977,980,994,1006,1051,1176,1180,1194,1200,1204,1209,1213],{"id":42,"type":43,"markdown":44},"intro-investigate","prose","So far you have been *told* the properties. In this layer you become the mathematician. You will meet claims, some true and some false, and your job is to **predict, test and decide**:\n\n- **Always true**: true for every number you could try. To be sure, you need a *reason*, not just examples.\n- **Sometimes true**: true for some numbers, false for others. Find one of each.\n- **Never true**: false for every number. Again, you need a reason.\n\nKeep a notebook (or the back of an old calendar) beside you. Every chapter has a **prediction** to commit to before you read on, and most have a lab where you can test your ideas quickly. Being wrong in a prediction is not failure; it is the moment the idea gets stored in your memory.",{"id":46,"type":47,"tone":48,"items":49},"spec-investigate","spec","copper",[50,54,58,62],{"label":51,"big":52,"value":53},"Your tools","test · compare · explain","Try small numbers, try 0 and 1, try big numbers, then look for the reason.",{"label":55,"big":56,"value":57},"Killer move","one counterexample","A single failure proves a claim is not always true.",{"label":59,"big":60,"value":61},"Weak move","many examples","A hundred successes still do not prove \"always\". They only make you suspect it.",{"label":63,"big":64,"value":65},"Labs here","5","Always–sometimes–never sort, closure sort, shortcut sprint, shortcut match, brackets lab.",{"id":67,"type":68,"title":69,"eyebrow":70,"navLabel":71},"ch1","chapter","How to test a claim about numbers","Chapter 01","1 Testing claims",{"id":73,"type":74,"title":75,"items":76},"steps-testing","steps","A testing routine that catches most false claims",[77,81,85,89,93,97,101],{"title":78,"tag":79,"text":80},"Try an easy example","e.g. 2 and 3","Small numbers are fast and show how the claim behaves.",{"title":82,"tag":83,"text":84},"Try 0","the troublemaker","Zero breaks many claims: a × 0, a ÷ 0, a − 0.",{"title":86,"tag":87,"text":88},"Try 1","the quiet one","One often makes \"bigger\" claims fail: a × 1 is not bigger than a.",{"title":90,"tag":91,"text":92},"Try equal numbers","a = b","Some claims fail only when the two numbers are the same, or only when they differ.",{"title":94,"tag":95,"text":96},"Try odd and even","mix them","Parity claims often fail for one mix but not another.",{"title":98,"tag":99,"text":100},"Try something big","1,000 or more","A few claims work for small numbers and fail later.",{"title":102,"tag":103,"text":104},"Look for the reason","why?","If nothing fails, find an argument (pairs, arrays, number line) that covers every case.",{"id":106,"type":107,"prompt":108,"options":109,"explanation":119},"predict-bigger","prediction","Claim: \"When you multiply two whole numbers, the answer is bigger than both of them.\" Always, sometimes or never true?",[110,113,116],{"id":111,"label":112},"a","Always true",{"id":114,"label":115},"b","Sometimes true",{"id":117,"label":118},"c","Never true","**Sometimes true.** 4 × 5 = 20 is bigger than both. But the testing routine finds failures fast:\n\n- Try 0: 7 × 0 = 0, which is smaller than 7.\n- Try 1: 7 × 1 = 7, which is not bigger than 7.\n\nFor whole numbers, the claim is true exactly when **both numbers are at least 2**. \"Multiplication makes bigger\" is a belief many people carry into secondary school, where it fails again with fractions: ½ × 8 = 4.",{"id":121,"type":107,"prompt":122,"options":123,"explanation":127},"predict-sum-bigger","Claim: \"a + b is bigger than a.\" Always, sometimes or never true, for whole numbers?",[124,125,126],{"id":111,"label":112},{"id":114,"label":115},{"id":117,"label":118},"**Sometimes true.** It is true whenever b is at least 1. It fails for b = 0: a + 0 = a, which is equal, not bigger. The additive identity is exactly the number that makes this claim fail. A careful mathematician would say \"a + b ≥ a, always\" (bigger than **or equal to**).",{"id":129,"type":130,"variant":131,"title":132,"markdown":133},"aha-reasons","callout","aha","Examples suggest; reasons decide","Try \"the sum of two odd numbers is even\" with 1 + 3, 5 + 7, 11 + 13, 99 + 101. All even. Convincing, but still not a proof. The **reason**: each odd number is pairs plus one spare; two spares make a pair; so the total is all pairs. That argument works for every odd number that exists, including ones nobody has ever written down. That is the difference between checking and proving.",{"id":135,"type":68,"title":136,"eyebrow":137,"navLabel":138},"ch2","Investigation: can subtraction or division ever be swapped?","Chapter 02","2 Swapping − and ÷",{"id":140,"type":43,"markdown":141},"swap-prose","You know that subtraction is not commutative in general. But is a − b = b − a **never** true, or only **sometimes** false? Test pairs. With whole numbers, a − b only makes sense when a ≥ b, and b − a only when b ≥ a. Both make sense together only if a = b, and then both are 0. So:\n\n**a − b = b − a is sometimes true: exactly when a = b.**\n\nDivision is similar. 6 ÷ 6 = 1 either way. But 6 ÷ 3 = 2 while 3 ÷ 6 = ½. And 0 ÷ 0 is undefined, so the equal pair must not be 0. **a ÷ b = b ÷ a is sometimes true: exactly when a = b and neither is 0.**",{"id":143,"type":144,"caption":145,"columns":146,"rows":152},"swap-table","table","Swapping in subtraction and division: only equal pairs survive",[147,148,149,150,151],"Pair a and b","a − b","b − a","a ÷ b","b ÷ a",[153,158,163,167,172],[154,64,155,156,157],"9 and 4","not a whole number","9\u002F4 = 2¼","4\u002F9",[159,160,155,161,162],"20 and 5","15","4","5\u002F20 = ¼",[164,165,165,166,166],"12 and 12","0","1",[168,169,155,170,171],"100 and 1","99","100","1\u002F100",[173,165,165,166,166],"7 and 7",{"id":175,"type":107,"prompt":176,"options":177,"explanation":187},"predict-swap-gap","With integers (allowing negatives), 10 − 3 = 7 and 3 − 10 = −7. For **any** a and b, how are a − b and b − a related?",[178,180,182,184],{"id":111,"label":179},"They are always equal",{"id":114,"label":181},"They are opposites: same size, opposite signs",{"id":117,"label":183},"b − a is always bigger",{"id":185,"label":186},"d","There is no pattern","**They are opposites.** Swapping in subtraction does not scramble the answer randomly; it flips the direction on the number line. From 3 to 10 is 7 steps right; from 10 to 3 is 7 steps left. The two answers add to zero: (a − b) + (b − a) = 0. Division has a matching pattern: a ÷ b and b ÷ a are **reciprocals**, and they multiply to 1 (for example 4 × ¼ = 1).",{"id":189,"type":68,"title":190,"eyebrow":191,"navLabel":192},"ch3","Investigation: how wrong does a bad grouping go?","Chapter 03","3 Grouping gaps",{"id":194,"type":43,"markdown":195},"gap-prose","Subtraction is not associative: (a − b) − c and a − (b − c) can differ. But by **how much**? Rather than just saying \"different\", let's measure the gap. Work out both groupings for several triples and subtract.",{"id":197,"type":144,"caption":198,"columns":199,"rows":204},"gap-table","The gap between the two groupings of a − b − c",[200,201,202,203],"a, b, c","Left grouping","Right grouping","Gap",[205,209,214,219,224],[206,207,208,161],"20, 8, 2","(20 − 8) − 2 = 10","20 − (8 − 2) = 14",[210,211,212,213],"50, 20, 10","(50 − 20) − 10 = 20","50 − (20 − 10) = 40","20",[215,216,217,218],"100, 30, 7","(100 − 30) − 7 = 63","100 − (30 − 7) = 77","14",[220,221,222,223],"60, 15, 15","(60 − 15) − 15 = 30","60 − (15 − 15) = 60","30",[225,226,227,165],"35, 12, 0","(35 − 12) − 0 = 23","35 − (12 − 0) = 23",{"id":229,"type":107,"prompt":230,"options":231,"explanation":240},"predict-gap","Look at the gap column and the value of c in each row. What is the gap, always?",[232,234,236,238],{"id":111,"label":233},"Always 0",{"id":114,"label":235},"Always c",{"id":117,"label":237},"Always 2 × c",{"id":185,"label":239},"Always a − b","**Always 2 × c.** The rows give gaps 4, 20, 14, 30 and 0 for c = 2, 10, 7, 15 and 0.\n\nThe reason: in (a − b) − c you take away c. In a − (b − c), the c is taken off the b first, so you take away **less** from a, by exactly c. That turns \"subtract c\" into \"add c\". From −c to +c is a jump of 2c. And that also tells you when the two groupings agree: only when 2 × c = 0, that is, **c = 0**.",{"id":242,"type":144,"caption":243,"columns":244,"rows":246},"div-gap-table","Division: how many times bigger is the right grouping?",[200,201,202,245],"Right ÷ left",[247,251,256,261],[248,249,250,161],"64, 8, 2","(64 ÷ 8) ÷ 2 = 4","64 ÷ (8 ÷ 2) = 16",[252,253,254,255],"300, 10, 5","(300 ÷ 10) ÷ 5 = 6","300 ÷ (10 ÷ 5) = 150","25",[257,258,259,260],"72, 6, 3","(72 ÷ 6) ÷ 3 = 4","72 ÷ (6 ÷ 3) = 36","9",[262,263,264,166],"90, 9, 1","(90 ÷ 9) ÷ 1 = 10","90 ÷ (9 ÷ 1) = 10",{"id":266,"type":130,"variant":267,"title":268,"markdown":269},"obs-div-gap","observation","The division pattern","For division, the right grouping is always **c × c** times the left one (4 = 2 × 2, 25 = 5 × 5, 9 = 3 × 3, 1 = 1 × 1). The reason is the same as for subtraction: dividing by (b ÷ c) divides by b but *multiplies* by c, while the left grouping divides by c. The two groupings agree only when c × c = 1, that is c = 1 (or when a = 0, since then both sides are 0).",{"id":271,"type":272,"component":273,"componentVersion":5,"config":274,"objective":283,"textAlternative":284,"help":285},"lab-order-ops-investigate","interactive","order-ops",{"expressions":275,"showRuleCard":282},[276,277,278,279,280,281],"(100 - 30) - 7","100 - (30 - 7)","(72 ÷ 6) ÷ 3","72 ÷ (6 ÷ 3)","(25 + 17) + 83","25 + (17 + 83)",true,"Reduce pairs of expressions step by step and measure how much the grouping changes the answer.","A step-by-step expression lab. For each expression you choose the next operation and see it reduce.\n\n(100 − 30) − 7 = 70 − 7 = 63, while 100 − (30 − 7) = 100 − 23 = 77. The gap is 14, which is 2 × 7.\n\n(72 ÷ 6) ÷ 3 = 12 ÷ 3 = 4, while 72 ÷ (6 ÷ 3) = 72 ÷ 2 = 36. The right grouping is 9 = 3 × 3 times bigger.\n\n(25 + 17) + 83 = 42 + 83 = 125, and 25 + (17 + 83) = 25 + 100 = 125. For addition the grouping makes no difference, but the second is easier to do in your head.",{"hints":286},[287,288],"Do the brackets first, then compare the pair.","For subtraction, is the gap twice the last number?",{"id":290,"type":68,"title":291,"eyebrow":292,"navLabel":293},"ch4","Investigation: multiplying by 9, 11, 99 and 101","Chapter 04","4 Special multipliers",{"id":295,"type":43,"markdown":296},"special-prose","Some multipliers are just one away from a round number: 9 = 10 − 1, 11 = 10 + 1, 99 = 100 − 1, 101 = 100 + 1, 999 = 1,000 − 1. The distributive property turns each into a shortcut. Before reading the table, try to predict each answer using the round number.",{"id":298,"type":144,"caption":299,"columns":300,"rows":306},"special-table","Near-round multipliers using the distributive property",[301,302,303,304,305],"n","n × 9","n × 11","n × 99","n × 101",[307,313,319,325],[308,309,310,311,312],"23","230 − 23 = 207","230 + 23 = 253","2,300 − 23 = 2,277","2,300 + 23 = 2,323",[314,315,316,317,318],"47","470 − 47 = 423","470 + 47 = 517","4,700 − 47 = 4,653","4,700 + 47 = 4,747",[320,321,322,323,324],"58","580 − 58 = 522","580 + 58 = 638","5,800 − 58 = 5,742","5,800 + 58 = 5,858",[326,327,328,329,330],"86","860 − 86 = 774","860 + 86 = 946","8,600 − 86 = 8,514","8,600 + 86 = 8,686",{"id":332,"type":107,"prompt":333,"options":334,"explanation":343},"predict-101","Look at the n × 101 column: 23 × 101 = 2,323, 47 × 101 = 4,747. Without calculating, what is **68 × 101**?",[335,337,339,341],{"id":111,"label":336},"6,868",{"id":114,"label":338},"6,808",{"id":117,"label":340},"68,068",{"id":185,"label":342},"7,068","**6,868.** Multiplying a two-digit number by 101 writes it twice: 68 × 101 = 6,800 + 68 = 6,868. Because 6,800 ends in two zeros, the 68 slots neatly into the gap. Try a three-digit number: 368 × 101 = 36,800 + 368 = 37,168. Now the parts overlap, so the \"write it twice\" pattern breaks. The distributive property still works; the pattern was a lucky special case.",{"id":345,"type":43,"markdown":346},"eleven-prose","The **× 11** column hides a famous trick. For a two-digit number, write the two digits apart and put their **sum** in the middle:\n\n- 23 × 11: 2 _ 3, middle 2 + 3 = 5 → **253**.\n- 54 × 11: 5 _ 4, middle 9 → **594**.\n- 86 × 11: 8 _ 6, middle 14 → carry the 1 → **946**.\n\nWhy? 86 × 11 = 860 + 86. Line them up and the tens digit of 86 lands in the same column as the ones digit of 860's tens place, so the middle column adds the two digits. It is the distributive property plus place value.",{"id":348,"type":272,"component":349,"componentVersion":5,"config":350,"objective":385,"textAlternative":386,"help":387},"lab-sprint-investigate","arith-sprint",{"operations":351,"ranges":353,"rounds":355,"secondsTotal":359,"estimateFirst":360,"wordProblems":361},[352],"×",{"a":354,"b":357},{"min":355,"max":356},12,99,{"min":358,"max":358},11,90,false,[362,366,370,374,378,382],{"prompt":363,"answer":364,"operation":352,"unit":365},"A bus has 47 seats. How many seats in 99 buses? (Use 100 buses, then take away one busload.)",4653,"seats",{"prompt":367,"answer":368,"operation":352,"unit":369},"A railway ticket costs ₹68. What do 101 tickets cost?",6868,"₹",{"prompt":371,"answer":372,"operation":352,"unit":373},"A dabbawala delivers 58 tiffins a day. How many in 9 days? (58 × 10 − 58)",522,"tiffins",{"prompt":375,"answer":376,"operation":352,"unit":377},"A cricket team scores 86 runs in each of 11 matches. Total runs?",946,"runs",{"prompt":379,"answer":380,"operation":352,"unit":381},"For a science project, a class counts 999 rice grains into each of 23 packets. How many grains altogether?",22977,"grains",{"prompt":383,"answer":384,"operation":352,"unit":365},"A theatre has 36 rows of 25 seats. Use 36 × 100 ÷ 4. How many seats?",900,"Test the × 11 shortcut on random two-digit numbers, then use near-round multipliers (9, 99, 101, 999, 25) on word problems.","A 90-second sprint of 12 rounds; up to half of them are word problems drawn from the list below. Every plain question is a two-digit number (12 to 99) times 11. Use \"split the digits, put their sum in the middle, carry if the sum is 10 or more\". Example: 75 × 11 → 7, 7 + 5 = 12, 5 → carry → 825.\n\nWord problems and answers:\n1. 47 seats × 99 buses: 4,700 − 47 = 4,653 seats.\n2. ₹68 × 101 tickets: 6,800 + 68 = ₹6,868.\n3. 58 tiffins × 9 days: 580 − 58 = 522 tiffins.\n4. 86 runs × 11 matches: 860 + 86 = 946 runs.\n5. 999 grains × 23 packets: 23,000 − 23 = 22,977 grains.\n6. 36 rows × 25 seats: 3,600 ÷ 4 = 900 seats.",{"hints":388},[389,390],"× 11: add the two digits and place the sum in the middle.","× 99: multiply by 100 and subtract one copy.",{"id":392,"type":68,"title":393,"eyebrow":394,"navLabel":395},"ch5","Experiment: sneaking up on division by zero","Chapter 05","5 Dividing by zero",{"id":397,"type":43,"markdown":398},"sneak-prose","We cannot divide by 0, but we **can** divide by numbers closer and closer to 0 and watch what happens. This is exactly what mathematicians did to understand the problem. Predict the pattern before reading the table.",{"id":400,"type":144,"caption":401,"columns":402,"rows":406},"sneak-table","Dividing 12 by smaller and smaller numbers",[403,404,405],"Division","Answer","Check by multiplying",[407,411,415,419,423],[408,409,410],"12 ÷ 1","12","12 × 1 = 12",[412,413,414],"12 ÷ 0.1","120","120 × 0.1 = 12",[416,417,418],"12 ÷ 0.01","1,200","1,200 × 0.01 = 12",[420,421,422],"12 ÷ 0.001","12,000","12,000 × 0.001 = 12",[424,425,426],"12 ÷ 0.0001","120,000","120,000 × 0.0001 = 12",{"id":428,"type":107,"prompt":429,"options":430,"explanation":439},"predict-sneak","As the divisor shrinks towards 0, what happens to 12 ÷ (divisor)?",[431,433,435,437],{"id":111,"label":432},"The answer shrinks towards 0",{"id":114,"label":434},"The answer settles at 12",{"id":117,"label":436},"The answer grows without limit",{"id":185,"label":438},"The answer becomes 1","**It grows without limit.** Each time the divisor becomes 10 times smaller, the answer becomes 10 times bigger: 12, 120, 1,200, 12,000, 120,000, … There is no \"last\" answer waiting at 0; the numbers run off for ever. (With negative divisors, like −0.001, the answers run off in the **negative** direction instead.) That is another reason 12 ÷ 0 is left **undefined** rather than given a value.",{"id":441,"type":144,"caption":442,"columns":443,"rows":447},"zero-one-exp","Zero and one in all four operations: fill in your predictions first, then check",[444,445,446],"Expression","Value","Rule",[448,452,455,458,461,465,468,472,475],[449,450,451],"36 + 0","36","Additive identity",[453,155,454],"0 − 36","Subtraction is not commutative",[456,165,457],"36 × 0","Zero property",[459,165,460],"0 ÷ 36","Zero divided by a non-zero number",[462,463,464],"36 ÷ 0","undefined","Division by zero",[466,450,467],"36 × 1","Multiplicative identity",[469,470,471],"1 ÷ 36","1\u002F36","Not an identity from the left",[473,166,474],"36 ÷ 36","A number divided by itself",[476,463,477],"0 ÷ 0","Every number would fit",{"id":479,"type":130,"variant":480,"title":481,"markdown":482},"careful-calculator","careful","Calculators say different things","A basic calculator shows \"E\" or \"Error\" for 12 ÷ 0. A phone may show \"Can't divide by 0\". Some computer programs show \"Infinity\" or \"NaN\" (Not a Number). None of these is a number you can calculate with. If you see one, it is a signal that a division by zero sneaked into your work.",{"id":484,"type":68,"title":485,"eyebrow":486,"navLabel":487},"ch6","Investigation: sums of consecutive numbers and odd numbers","Chapter 06","6 Parity patterns",{"id":489,"type":43,"markdown":490},"consec-prose","**Consecutive** numbers follow one another: 7, 8, 9. Here are three claims to investigate. Test each with a few examples, then look for a reason.\n\n1. The sum of **two** consecutive numbers is always odd.\n2. The sum of **three** consecutive numbers is always a multiple of 3.\n3. The product of **two** consecutive numbers is always even.",{"id":492,"type":144,"caption":493,"columns":494,"rows":499},"consec-table","Testing the three claims (the middle column shows the sum is 3 × the middle number)",[495,496,497,498],"Start","Two consecutive: sum","Three consecutive: sum","Two consecutive: product",[500,504,509,513],[161,501,502,503],"4 + 5 = 9","4 + 5 + 6 = 15 = 3 × 5","4 × 5 = 20",[505,506,507,508],"7","7 + 8 = 15","7 + 8 + 9 = 24 = 3 × 8","7 × 8 = 56",[409,510,511,512],"12 + 13 = 25","12 + 13 + 14 = 39 = 3 × 13","12 × 13 = 156",[255,514,515,516],"25 + 26 = 51","25 + 26 + 27 = 78 = 3 × 26","25 × 26 = 650",{"id":518,"type":43,"markdown":519},"consec-why","All three claims survive, and each has a neat reason:\n\n1. Of two consecutive numbers, **one is even and one is odd**. Even + odd = odd. Always.\n2. Three consecutive numbers are (middle − 1), middle, (middle + 1). The −1 and +1 cancel, so the sum is **3 × middle**. Always a multiple of 3. (Look at the table: 12 + 13 + 14 = 39 = 3 × 13.)\n3. One of the two is even, and anything × even is even. Always.",{"id":521,"type":522,"title":523,"problem":524,"steps":525},"we-four-consec","worked_example","A full investigation: four consecutive numbers","Is the sum of **four** consecutive numbers ever a multiple of 4?",[526,527,528,529,530,531,532],"Test: 1 + 2 + 3 + 4 = 10, 5 + 6 + 7 + 8 = 26, 10 + 11 + 12 + 13 = 46. None is a multiple of 4 (10, 26 and 46 all leave remainder 2 when divided by 4).","Conjecture: the sum is **never** a multiple of 4.","Look for a reason. Call the first number n. The four numbers are n, n + 1, n + 2 and n + 3.","Add them: four n's and 1 + 2 + 3 = 6 extra. So the sum is 4 × n + 6.","4 × n is a multiple of 4. Adding 6 = 4 + 2 leaves a remainder of **2**, whatever n is.","So the sum is never a multiple of 4. It is, however, always even (4 × n + 6 = 2 × (2 × n + 3)).","Compare with three consecutive numbers, whose sum is always a multiple of 3. A pattern that works for 3 does not automatically work for 4: that is why we test.",{"id":534,"type":144,"caption":535,"columns":536,"rows":541},"odd-sum-table","Adding the first few odd numbers",[537,538,539,540],"How many odd numbers","Sum","Total","Pattern",[542,544,548,552,556,559,563],[166,166,166,543],"1 × 1",[545,546,161,547],"2","1 + 3","2 × 2",[549,550,260,551],"3","1 + 3 + 5","3 × 3",[161,553,554,555],"1 + 3 + 5 + 7","16","4 × 4",[64,557,255,558],"1 + 3 + 5 + 7 + 9","5 × 5",[560,561,450,562],"6","1 + 3 + 5 + 7 + 9 + 11","6 × 6",[505,564,565,566],"1 + 3 + 5 + 7 + 9 + 11 + 13","49","7 × 7",{"id":568,"type":107,"prompt":569,"options":570,"explanation":579},"predict-oddsum","Using the table, what is the sum of the first **20** odd numbers (1 + 3 + 5 + … + 39)?",[571,573,575,577],{"id":111,"label":572},"200",{"id":114,"label":574},"400",{"id":117,"label":576},"390",{"id":185,"label":578},"420","**400 = 20 × 20.** The sum of the first n odd numbers is always n × n, a **square number**. Picture it: a 1 × 1 square, then add an L-shape of 3 to make 2 × 2, then an L of 5 to make 3 × 3, and so on. Each odd number is exactly the L-shaped border needed to grow the square by one. Check: 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 + 21 + 23 + 25 + 27 + 29 + 31 + 33 + 35 + 37 + 39 = 400.",{"id":581,"type":272,"component":582,"componentVersion":5,"config":583,"objective":654,"textAlternative":655,"help":656},"lab-sort-asn","sort-game",{"prompt":584,"bins":585,"items":592,"seconds":653},"For whole numbers: is each claim always, sometimes or never true?",[586,588,590],{"id":587,"label":112},"always",{"id":589,"label":115},"sometimes",{"id":591,"label":118},"never",[593,597,601,605,609,613,617,621,625,629,633,637,641,645,649],{"id":594,"label":595,"bin":587,"why":596},"c1","The sum of two odd numbers is even.","The two spare ones make a pair.",{"id":598,"label":599,"bin":591,"why":600},"c2","The sum of three odd numbers is even.","Two odds make even, and even + odd is odd. Always odd, so never even.",{"id":602,"label":603,"bin":589,"why":604},"c3","a × b is bigger than a.","True for 3 × 4; false for 3 × 1 or 3 × 0.",{"id":606,"label":607,"bin":589,"why":608},"c4","a − b = b − a","Only when a = b, e.g. 5 − 5 = 0 both ways.",{"id":610,"label":611,"bin":587,"why":612},"c5","The product of two consecutive numbers is even.","One of the two is even.",{"id":614,"label":615,"bin":587,"why":616},"c6","The sum of three consecutive numbers is a multiple of 3.","It equals 3 × the middle number.",{"id":618,"label":619,"bin":591,"why":620},"c7","An odd number times an even number is odd.","Any product with an even factor is even.",{"id":622,"label":623,"bin":589,"why":624},"c8","(a − b) − c = a − (b − c)","Only when c = 0; otherwise the gap is 2 × c.",{"id":626,"label":627,"bin":589,"why":628},"c9","a × 0 = a","Only when a = 0. For any other a, a × 0 = 0.",{"id":630,"label":631,"bin":587,"why":632},"c10","a + 0 = a","0 is the additive identity.",{"id":634,"label":635,"bin":591,"why":636},"c11","The sum of two consecutive numbers is even.","One is even, one is odd, so the sum is always odd.",{"id":638,"label":639,"bin":589,"why":640},"c12","a ÷ b = b ÷ a","Only when a = b and they are not 0.",{"id":642,"label":643,"bin":587,"why":644},"c13","1 + 3 + 5 + … (first n odd numbers) is a square number.","The sum of the first n odd numbers is n × n.",{"id":646,"label":647,"bin":587,"why":648},"c14","A number ending in 5 times a number ending in 5 ends in 5.","Only the ones digits decide the ones digit: 5 × 5 = 25 ends in 5.",{"id":650,"label":651,"bin":589,"why":652},"c15","a + b is odd.","Odd when one is even and one odd, e.g. 2 + 3; even for 2 + 4.",0,"Classify claims about whole numbers as always, sometimes or never true, using counterexamples and reasons.","A sorting game with three bins and fifteen claims.\n\nAlways true: sum of two odd numbers is even; product of two consecutive numbers is even; sum of three consecutive numbers is a multiple of 3; a + 0 = a; the first n odd numbers add to a square; (ends in 5) × (ends in 5) ends in 5.\n\nSometimes true: a × b bigger than a (fails for b = 0 or 1); a − b = b − a (only a = b); (a − b) − c = a − (b − c) (only c = 0); a × 0 = a (only a = 0); a ÷ b = b ÷ a (only a = b, not 0); a + b is odd (only for one odd, one even).\n\nNever true: the sum of three odd numbers is even; odd × even is odd; the sum of two consecutive numbers is even.",{"hints":657},[658,659],"For \"sometimes\", find one example that works and one that fails.","For \"always\" or \"never\", look for a reason about pairs.",{"id":661,"type":68,"title":662,"eyebrow":663,"navLabel":664},"ch7","Investigation: which families are closed?","Chapter 07","7 Closed families",{"id":666,"type":43,"markdown":667},"closure-inv-prose","Closure is not only about whole numbers. You can ask it about **any** family of numbers. Take the **multiples of 3**: 0, 3, 6, 9, 12, … Add two of them: 12 + 21 = 33, a multiple of 3. The reason: 3 × a + 3 × b = 3 × (a + b), by the distributive property. So multiples of 3 are closed under addition. The same argument works for multiples of **any** number.\n\nNow try **square numbers** (1, 4, 9, 16, 25, …). Multiply two: 4 × 9 = 36 = 6 × 6, a square. Add two: 1 + 4 = 5, not a square. So squares are closed under multiplication but **not** under addition. Before the lab, make your own predictions about the families below.",{"id":669,"type":144,"caption":670,"columns":671,"rows":675},"closure-inv-table","Predict first, then check: closed or not? (A counterexample is given where it fails)",[672,673,674],"Family","Under +","Under ×",[676,680,684,688,692,696,700],[677,678,679],"Multiples of 3","Closed: 3a + 3b = 3(a + b)","Closed",[681,682,683],"Odd numbers","Not: 1 + 3 = 4","Closed: odd × odd = odd",[685,686,687],"Square numbers","Not: 1 + 4 = 5","Closed: (a × a) × (b × b) = (a × b) × (a × b)",[689,690,691],"Numbers ending in 1","Not: 1 + 11 = 12","Closed: ones digits 1 × 1 = 1",[693,694,695],"Numbers ending in 5","Not: 5 + 15 = 20","Closed: ones digits 5 × 5 = 25",[697,698,699],"{0, 1}","Not: 1 + 1 = 2","Closed: 0 × 0, 0 × 1, 1 × 1 are all 0 or 1",[701,702,703],"Numbers 1 to 10","Not: 6 + 7 = 13","Not: 4 × 5 = 20",{"id":705,"type":272,"component":582,"componentVersion":5,"config":706,"objective":764,"textAlternative":765,"help":766},"lab-sort-closure",{"prompt":707,"bins":708,"items":714,"seconds":763},"Is the family closed under the operation shown?",[709,711],{"id":710,"label":679},"closed",{"id":712,"label":713},"open","Not closed",[715,719,723,727,731,735,739,743,747,751,755,759],{"id":716,"label":717,"bin":710,"why":718},"f1","Whole numbers under addition","Adding whole numbers always lands on a whole number.",{"id":720,"label":721,"bin":712,"why":722},"f2","Whole numbers under subtraction","3 − 5 is not a whole number.",{"id":724,"label":725,"bin":712,"why":726},"f3","Natural numbers under subtraction","4 − 4 = 0, and 0 is not a natural number.",{"id":728,"label":729,"bin":710,"why":730},"f4","Even numbers under multiplication","Anything × even is even.",{"id":732,"label":733,"bin":712,"why":734},"f5","Odd numbers under addition","3 + 5 = 8 is even.",{"id":736,"label":737,"bin":710,"why":738},"f6","Odd numbers under multiplication","odd × odd = odd.",{"id":740,"label":741,"bin":710,"why":742},"f7","Multiples of 5 under addition","5a + 5b = 5(a + b).",{"id":744,"label":745,"bin":712,"why":746},"f8","Square numbers under addition","1 + 4 = 5 is not square.",{"id":748,"label":749,"bin":710,"why":750},"f9","Square numbers under multiplication","(a × a)(b × b) = (ab)(ab).",{"id":752,"label":753,"bin":712,"why":754},"f10","Whole numbers under division (non-zero divisor)","7 ÷ 2 = 3½.",{"id":756,"label":757,"bin":710,"why":758},"f11","Numbers ending in 6 under multiplication","Ones digits: 6 × 6 = 36 ends in 6.",{"id":760,"label":761,"bin":712,"why":762},"f12","Numbers ending in 6 under addition","6 + 6 = 12 ends in 2.",120,"Decide whether each family of numbers is closed under the given operation, finding a counterexample when it is not.","A timed sorting game (120 seconds) with two bins: Closed and Not closed.\n\nClosed: whole numbers under addition; even numbers under multiplication; odd numbers under multiplication; multiples of 5 under addition; square numbers under multiplication; numbers ending in 6 under multiplication.\n\nNot closed, with a counterexample: whole numbers under subtraction (3 − 5); natural numbers under subtraction (4 − 4 = 0); odd numbers under addition (3 + 5 = 8); square numbers under addition (1 + 4 = 5); whole numbers under division (7 ÷ 2); numbers ending in 6 under addition (6 + 6 = 12).",{"hints":767},[768,769],"Try to escape: pick two members and see whether the answer is still in the family.","For ones-digit families, only the ones digits matter when multiplying.",{"id":771,"type":68,"title":772,"eyebrow":773,"navLabel":774},"ch8","Patterns that reveal properties","Chapter 08","8 Patterns",{"id":776,"type":43,"markdown":777},"patterns-intro","Some number patterns look like magic. Almost always, the magic is a property of numbers in disguise. Work out a few rows yourself, predict the next row, then read the explanation.",{"id":779,"type":144,"caption":780,"columns":781,"rows":784},"ones-pattern","The ones pattern",[782,783],"Calculation","Result",[785,788,791,794,797,800],[786,787],"1 × 9 + 2","11",[789,790],"12 × 9 + 3","111",[792,793],"123 × 9 + 4","1,111",[795,796],"1234 × 9 + 5","11,111",[798,799],"12345 × 9 + 6","111,111",[801,802],"123456 × 9 + 7","1,111,111",{"id":804,"type":130,"variant":267,"title":805,"markdown":806},"obs-ones","Why the ones appear","Multiplying by 9 is multiplying by 10 and taking one away (distributive). So 123 × 9 = 1,230 − 123 = 1,107, and adding 4 gives 1,111. Each row's shift-and-subtract leaves exactly the right digits for the added number to fill in with 1s. Check the next row yourself: 1234567 × 9 + 8 = 11,111,111.",{"id":808,"type":144,"caption":809,"columns":810,"rows":811},"pattern-37","The 37 pattern",[782,783],[812,814,817,820,823,826],[813,790],"37 × 3",[815,816],"37 × 6","222",[818,819],"37 × 9","333",[821,822],"37 × 12","444",[824,825],"37 × 15","555",[827,828],"37 × 27","999",{"id":830,"type":43,"markdown":831},"pattern-37-why","Why do 111, 222, 333 … appear? Because 37 × 3 = 111. Then 37 × 6 = 37 × (3 × 2) = (37 × 3) × 2 = 111 × 2 = 222, by the **associative** property. In the same way 37 × 27 = 111 × 9 = 999.\n\nAnd the square pattern of ones: 11 × 11 = 121, 111 × 111 = 12,321, 1,111 × 1,111 = 1,234,321. Split 111 × 111 by the distributive property into 111 × 100 + 111 × 10 + 111 × 1 = 11,100 + 1,110 + 111. Stack them and the columns add up to 1, 2, 3, 2, 1. The pattern survives until **111,111,111 × 111,111,111 = 12,345,678,987,654,321**, then carrying spoils it.",{"id":833,"type":107,"prompt":834,"options":835,"explanation":842},"predict-37","Using the pattern, what is **37 × 18**?",[836,837,838,840],{"id":111,"label":822},{"id":114,"label":825},{"id":117,"label":839},"666",{"id":185,"label":841},"777","**666.** 18 = 3 × 6, so 37 × 18 = (37 × 3) × 6 = 111 × 6 = 666. Check: 37 × 18 = 666.",{"id":844,"type":68,"title":845,"eyebrow":846,"navLabel":847},"ch9","Experiment: which shortcut is fastest?","Chapter 09","9 Shortcut showdown",{"id":849,"type":43,"markdown":850},"showdown-prose","Here is an experiment you can do with a friend and a stopwatch. Each of you does the same calculation, one the \"straight\" way and one with a property shortcut. Swap roles and repeat. Most people find the shortcuts win by a lot, but only once they have practised spotting them.",{"id":852,"type":144,"caption":853,"columns":854,"rows":857},"halving-table","Doubling and halving 16 × 25: the product never changes",[855,856],"Step","Product",[858,860,862,864,866],[859,574],"16 × 25",[861,574],"8 × 50",[863,574],"4 × 100",[865,574],"2 × 200",[867,574],"1 × 400",{"id":869,"type":144,"caption":870,"columns":871,"rows":875},"showdown-table","Straight way versus shortcut",[782,872,873,874],"Straight way","Shortcut","Property",[876,880,885,889,893,898],[859,877,878,879],"Long multiplication","4 × 100 = 400 (halve twice, double twice)","Associative",[881,882,883,884],"198 + 47","Column addition","200 + 45 = 245","Associative (compensation)",[886,877,887,888],"63 × 99","6,300 − 63 = 6,237","Distributive",[890,891,892,879],"125 × 8 × 7","Left to right","1,000 × 7 = 7,000",[894,895,896,897],"7 × 38 + 3 × 38","Two multiplications","10 × 38 = 380","Distributive (backwards)",[899,900,901,902],"1 + 2 + 3 + … + 10","Add one by one","5 pairs of 11 = 55","Commutative and associative",{"id":904,"type":130,"variant":905,"title":906,"markdown":907},"misc-halving","misconception","Halving both numbers","Doubling-and-halving only works when you halve **one** number and double the **other**. If you halve both, the product shrinks to a quarter: 16 × 25 = 400, but 8 × 12½ = 100. For addition the rule is different: you can move an amount from one number to the other (add to one, subtract from the other), as in 198 + 47 = 200 + 45.",{"id":909,"type":272,"component":910,"componentVersion":5,"config":911,"objective":936,"textAlternative":937,"help":938},"lab-match-shortcuts","match-pairs",{"prompt":912,"mode":913,"pairs":914},"Match each calculation to its shortcut form. The answers are included so you can check.","memory",[915,918,921,923,924,927,930,933],{"a":916,"b":917},"98 × 25","2,500 − 50 = 2,450",{"a":919,"b":920},"12 × 105","1,200 + 60 = 1,260",{"a":859,"b":922},"4 × 100 = 400",{"a":886,"b":887},{"a":925,"b":926},"45 × 101","4,500 + 45 = 4,545",{"a":928,"b":929},"35 × 18","70 × 9 = 630",{"a":931,"b":932},"37 × 68 + 37 × 32","37 × 100 = 3,700",{"a":934,"b":935},"86 × 5","860 ÷ 2 = 430","Find pairs in a memory game: each calculation matches the property shortcut that makes it easy.","A memory game: sixteen face-down cards, eight calculations and eight shortcut forms. Turn two at a time to find matching pairs.\n\n98 × 25 ↔ 2,500 − 50 = 2,450 (98 = 100 − 2).\n12 × 105 ↔ 1,200 + 60 = 1,260 (105 = 100 + 5).\n16 × 25 ↔ 4 × 100 = 400 (halve 16 twice, double 25 twice).\n63 × 99 ↔ 6,300 − 63 = 6,237.\n45 × 101 ↔ 4,500 + 45 = 4,545.\n35 × 18 ↔ 70 × 9 = 630 (double 35, halve 18).\n37 × 68 + 37 × 32 ↔ 37 × 100 = 3,700 (common factor).\n86 × 5 ↔ 860 ÷ 2 = 430 (× 5 = × 10 ÷ 2).",{"hints":939},[940,941],"Look for 99, 101, 105 or 98: numbers near 100.","Look for 25 and 5, which pair with 4 and 2.",{"id":943,"type":68,"title":944,"eyebrow":945,"navLabel":946},"ch10","Properties in real life","Chapter 10","10 Real life",{"id":948,"type":43,"markdown":949},"chairs36","**Arranging chairs.** The school hall needs **36 chairs** in a rectangle. How many different arrangements are there? The rows × columns pairs are: 1 × 36, 2 × 18, 3 × 12, 4 × 9, 6 × 6, 9 × 4, 12 × 3, 18 × 2, 36 × 1. That is **9** arrangements if you count 4 rows of 9 and 9 rows of 4 as different (the hall is longer one way than the other). If you only care about the *shape*, turn-arounds are the same, leaving **5** shapes: 1 × 36, 2 × 18, 3 × 12, 4 × 9 and 6 × 6. The square 6 × 6 is its own turn-around, which is why the count is odd. (Only square numbers have an odd number of arrangements, because only a square number has an arrangement that is its own turn-around.)",{"id":951,"type":144,"caption":952,"columns":953,"rows":956},"bill-table","A kirana bill: the total is ₹700 whichever order you add in",[954,955],"Item","Price (₹)",[957,960,963,966,969,972,975],[958,959],"Atta 5 kg","245",[961,962],"Sugar 2 kg","90",[964,965],"Toor dal 1 kg","155",[967,968],"Tea 250 g","110",[970,971],"Soap","45",[973,974],"Biscuits","55",[539,976],"700",{"id":978,"type":43,"markdown":979},"bill-prose","**Shopping totals.** The shopkeeper does not add the bill from top to bottom. He spots partners: atta + dal = 245 + 155 = 400, sugar + tea = 90 + 110 = 200, soap + biscuits = 45 + 55 = 100. Total **₹700**. Commutative and associative properties together promise that the total is the same in any order.\n\n**Cricket.** A batter's score is the same however the runs arrive: 4 + 6 + 1 or 1 + 6 + 4. But the *order* of overs matters for a run chase: needing 20 off the last over is different from needing 20 off the first. Real life often mixes things where order matters with things where it doesn't. Spotting which is which is the skill.",{"id":981,"type":982,"itemId":983,"prompt":984,"check":985,"hints":988,"feedback":991},"pr-chairs","practice","properties-of-numbers.investigate-chairs-48","How many rectangular rows × columns arrangements are there for **48 chairs**, counting 6 × 8 and 8 × 6 as different?",{"kind":986,"answer":987,"tolerance":653},"number",10,[989,990],"List factor pairs of 48: 1 × 48, 2 × 24, …","Each factor pair gives two arrangements (turned around).",{"correct":992,"incorrect":993},"Yes: 48 has 5 factor pairs (1 × 48, 2 × 24, 3 × 16, 4 × 12, 6 × 8), and each can be turned around, giving 10.","Factor pairs: 1 × 48, 2 × 24, 3 × 16, 4 × 12, 6 × 8. Each can be turned around, so 5 × 2 = 10 arrangements.",{"id":995,"type":982,"itemId":996,"prompt":997,"check":998,"hints":1000,"feedback":1003},"pr-gap","properties-of-numbers.investigate-gap","Without calculating both sides fully: how much bigger is 500 − (120 − 35) than (500 − 120) − 35?",{"kind":986,"answer":999,"tolerance":653},70,[1001,1002],"The gap is always 2 × c.","c = 35.",{"correct":1004,"incorrect":1005},"Right: the gap is 2 × 35 = 70. (500 − (120 − 35) = 415 and (500 − 120) − 35 = 345.)","The gap between the groupings is 2 × c = 2 × 35 = 70: 415 versus 345.",{"id":1007,"type":1008,"title":1009,"terms":1010},"glossary-investigate","glossary","Words for investigating",[1011,1015,1017,1019,1021,1024,1028,1032,1036,1039,1043,1047],{"term":1012,"meaning":1013,"example":1014},"Conjecture","A statement you think is true because of evidence, but have not yet proved.","\"The sum of three consecutive numbers is a multiple of 3.\"",{"term":112,"meaning":1016},"True for every number the statement talks about. Needs a reason that covers every case.",{"term":115,"meaning":1018},"True for some numbers and false for others. Show one example of each.",{"term":118,"meaning":1020},"False for every number. Needs a reason that covers every case.",{"term":1022,"meaning":1023},"Counterexample","One case where a statement fails, which proves it is not always true.",{"term":1025,"meaning":1026,"example":1027},"Consecutive numbers","Whole numbers that follow one another without gaps.","7, 8, 9",{"term":1029,"meaning":1030,"example":1031},"Square number","A number made by multiplying a whole number by itself.","1, 4, 9, 16, 25",{"term":1033,"meaning":1034,"example":1035},"Multiple","The result of multiplying a number by a whole number.","Multiples of 3: 0, 3, 6, 9, …",{"term":1037,"meaning":1038},"Reciprocal","The number you multiply by to get 1: the reciprocal of 4 is ¼.",{"term":1040,"meaning":1041,"example":1042},"Compensation","Adjusting numbers to make them friendly, then correcting.","198 + 47 = 200 + 45",{"term":1044,"meaning":1045,"example":1046},"Doubling and halving","Halving one factor and doubling the other leaves a product unchanged.","16 × 25 = 8 × 50 = 4 × 100",{"term":1048,"meaning":1049,"example":1050},"Factor pair","Two whole numbers that multiply to give a number.","4 and 9 are a factor pair of 36.",{"id":1052,"type":1053,"title":1054,"questions":1055},"quiz-investigate","quiz","What did your investigations show?",[1056,1069,1082,1092,1105,1118,1131,1142,1152,1163],{"itemId":1057,"prompt":1058,"options":1059,"correct":114,"why":1068},"properties-of-numbers.investigate-q-asn1","The product of two consecutive whole numbers is…",[1060,1062,1064,1066],{"id":111,"label":1061},"always odd",{"id":114,"label":1063},"always even",{"id":117,"label":1065},"sometimes even",{"id":185,"label":1067},"never even","One of any two consecutive numbers is even, and anything × even is even.",{"itemId":1070,"prompt":1071,"options":1072,"correct":117,"why":1081},"properties-of-numbers.investigate-q-asn2","\"a − b = b − a\" for whole numbers is…",[1073,1075,1077,1079],{"id":111,"label":1074},"always true",{"id":114,"label":1076},"never true",{"id":117,"label":1078},"true only when a = b",{"id":185,"label":1080},"true only when b = 0","Both sides exist only when a = b, and then both are 0.",{"itemId":1083,"prompt":1084,"options":1085,"correct":114,"why":1091},"properties-of-numbers.investigate-q-gap","(90 − 40) − 15 and 90 − (40 − 15) differ by:",[1086,1087,1088,1090],{"id":111,"label":160},{"id":114,"label":223},{"id":117,"label":1089},"40",{"id":185,"label":165},"The gap is 2 × c = 30: (90 − 40) − 15 = 35, 90 − (40 − 15) = 65.",{"itemId":1093,"prompt":1094,"options":1095,"correct":111,"why":1104},"properties-of-numbers.investigate-q-101","57 × 101 =",[1096,1098,1100,1102],{"id":111,"label":1097},"5,757",{"id":114,"label":1099},"5,707",{"id":117,"label":1101},"57,057",{"id":185,"label":1103},"5,777","5,700 + 57 = 5,757.",{"itemId":1106,"prompt":1107,"options":1108,"correct":114,"why":1117},"properties-of-numbers.investigate-q-11","72 × 11 =",[1109,1111,1113,1115],{"id":111,"label":1110},"7,272",{"id":114,"label":1112},"792",{"id":117,"label":1114},"782",{"id":185,"label":1116},"729","Digits 7 and 2, middle 7 + 2 = 9: 792.",{"itemId":1119,"prompt":1120,"options":1121,"correct":111,"why":1130},"properties-of-numbers.investigate-q-oddsum","1 + 3 + 5 + … + 29 (the first 15 odd numbers) =",[1122,1124,1126,1128],{"id":111,"label":1123},"225",{"id":114,"label":1125},"215",{"id":117,"label":1127},"450",{"id":185,"label":1129},"196","The first n odd numbers add to n × n: 15 × 15 = 225.",{"itemId":1132,"prompt":1133,"options":1134,"correct":117,"why":1141},"properties-of-numbers.investigate-q-closure","Which family is closed under addition?",[1135,1136,1137,1139],{"id":111,"label":685},{"id":114,"label":681},{"id":117,"label":1138},"Multiples of 7",{"id":185,"label":1140},"Numbers ending in 3","7a + 7b = 7(a + b). The others fail: 1 + 4 = 5, 1 + 3 = 4, 3 + 13 = 16.",{"itemId":1143,"prompt":1144,"options":1145,"correct":185,"why":1151},"properties-of-numbers.investigate-q-sneak","12 ÷ 0.001 =",[1146,1148,1149,1150],{"id":111,"label":1147},"0.012",{"id":114,"label":409},{"id":117,"label":417},{"id":185,"label":421},"12,000 × 0.001 = 12. Smaller divisors give bigger answers.",{"itemId":1153,"prompt":1154,"options":1155,"correct":111,"why":1162},"properties-of-numbers.investigate-q-37","37 × 24 =",[1156,1158,1159,1160],{"id":111,"label":1157},"888",{"id":114,"label":841},{"id":117,"label":828},{"id":185,"label":1161},"848","24 = 3 × 8, so 37 × 24 = 111 × 8 = 888.",{"itemId":1164,"prompt":1165,"options":1166,"correct":111,"why":1175},"properties-of-numbers.investigate-q-halve","Which is equal to 24 × 35?",[1167,1169,1171,1173],{"id":111,"label":1168},"12 × 70",{"id":114,"label":1170},"12 × 17½",{"id":117,"label":1172},"48 × 70",{"id":185,"label":1174},"12 × 35","Halve 24, double 35: 12 × 70 = 840 and 24 × 35 = 840.",{"id":1177,"type":1178,"prompt":1179},"reflect-investigate","reflection","Invent your own claim about even and odd numbers or about consecutive numbers. Test it with at least five examples, including 0 and 1. Is it always, sometimes or never true? Write the reason, or the counterexample, in one or two sentences.",{"id":1181,"type":1182,"title":1183,"points":1184},"cheat-investigate","summary","Cheat sheet",[1185,1186,1187,1188,1189,1190,1191,1192,1193],"**Testing routine:** easy numbers, 0, 1, equal numbers, odd\u002Feven mixes, big numbers, then look for the reason.","**One counterexample** kills an \"always\"; examples alone never prove one.","**a − b = b − a** only when a = b; **a ÷ b = b ÷ a** only when a = b ≠ 0. In general they are opposites and reciprocals.","**Grouping gaps:** a − (b − c) is 2 × c more than (a − b) − c; for division the ratio is c × c.","**Near-round multipliers:** × 9 = × 10 − 1; × 11 = × 10 + 1; × 99 = × 100 − 1; × 101 = × 100 + 1; × 999 = × 1,000 − 1.","**Dividing by numbers near 0** gives answers that grow without limit, which is why a ÷ 0 is undefined.","**Parity patterns:** two consecutive numbers sum to odd; three sum to 3 × middle; their product is even; first n odd numbers sum to n × n.","**Closure hunts:** multiples of any number are closed under +; squares and odd numbers are closed under × but not +.","**Real life:** chairs for 36 have 9 arrangements (5 shapes); a bill adds to the same total in any order.",{"id":1195,"type":1196,"conceptId":1197,"relation":1198,"explanation":1199},"conn-patterns","connection","patterns","related_to","Sums of odd numbers making squares, 37 × 3 = 111 and 11 × 11 = 121 are number patterns explained by properties of numbers.",{"id":1201,"type":1196,"conceptId":1202,"relation":1198,"explanation":1203},"conn-prime","prime-and-composite","Listing every rows × columns arrangement of chairs is listing factor pairs, the starting point for primes and composites.",{"id":1205,"type":1196,"conceptId":1206,"relation":1207,"explanation":1208},"conn-four","four-operations","applied_in","Compensation, doubling and halving and near-round multipliers make everyday calculations in the four operations faster.",{"id":1210,"type":1196,"conceptId":1211,"relation":1198,"explanation":1212},"conn-number-system","number-system","The × 11 trick and the patterns of ones rely on place value: every shift left is × 10.",{"id":1214,"type":1215,"sourceIds":1216},"sources-investigate","sources",[1217,1218,1219,1220,1221,1222,1223,1224,1225,1226],"properties-of-numbers-ncert-class6-whole-numbers","properties-of-numbers-ncert-class7-integers","properties-of-numbers-ncert-class8-rational-numbers","properties-of-numbers-mathsisfun-properties","properties-of-numbers-mathsisfun-divide-by-zero","properties-of-numbers-wiki-commutative","properties-of-numbers-wiki-distributive","properties-of-numbers-wiki-division-by-zero","properties-of-numbers-wiki-parity","properties-of-numbers-wiki-brahmagupta",[1217,1218,1219,1220,1221,1222,1223,1224,1225,1226],"needs_review",{"generatedBy":1230,"notes":1231},"claude-code","Draft generated with Python generators; every stated number computed and asserted. Pending owner review.","d64951d572aef13908b0fcab1e9b6e04e9b80f0c23e1524245ddb273a2bb7335",{"component:order-ops@1":1234,"component:arith-sprint@1":1235,"component:sort-game@1":1236,"component:match-pairs@1":1237,"logic:practice":1238,"source:properties-of-numbers-mathsisfun-divide-by-zero":1239,"source:properties-of-numbers-mathsisfun-properties":1240,"source:properties-of-numbers-ncert-class6-whole-numbers":1241,"source:properties-of-numbers-ncert-class7-integers":1242,"source:properties-of-numbers-ncert-class8-rational-numbers":1243,"source:properties-of-numbers-wiki-brahmagupta":1244,"source:properties-of-numbers-wiki-commutative":1245,"source:properties-of-numbers-wiki-distributive":1246,"source:properties-of-numbers-wiki-division-by-zero":1247,"source:properties-of-numbers-wiki-parity":1248},"3d630ebc066352df791b020d545e1876ce4620d55c8c1c082414d6b4717255c5","c0c63ed40e1bca5ba6d446d43886b887a3a7709e5cd6679419f64fd3ba62afe6","b164f45a2c8ca08f26c450768ff0231e113e9fe45381eddb34dc6d0548596c38","2a8ee4ac87460b4e1175a4bb13c96b03d577db06dde95670eb7fcfe4ad787899","3d6b0fe1b15255975a32b0fcd94e8019bc959ad45cbf12e136e86149549c6878","f8a42fd8c82c266f4310450d74f1ae0449b3dba177afcf000b24aafa2839b629","a37b85a36742e06f34cdc083710eacdd662e2f7493395766e4b8c5632bfade3a","56b97dbf2988a58e1aa10f693ec9dd1039e1de14e795b1bdfa19980b6b18c1fa","aab8fe53a32660ef6307b4dbd6f5af02319957d9aecf3c0396895bc01ab46e2e","acb8b92cad8387724d72528e70ef68195351e5b2f03b580c5da191fde82dc67a","32e7279091a5877afe54d6da75cdd06a04f2ff8677338777578592d6dd98a338","a3e0fb47a2115f5962c9795b138bf38cd9c2c4874bc66e1b834717b9a01908c1","befe26bd7ec4e1c3911549d33a91844ae3eb5bd63388711f61822d398d3d576f","046256cc76cdb1b7cca72a1b391bef686855afdfb6a68d18b45a38723d1fe93d","01c5bcf03886aec6f9c2c18c9ec5333b19ce1d2ca6bfb4f8544c4ca9f1578e27",{"state":1250,"reviewer":1251,"selfReview":282,"reviewedAt":1252,"method":1253},"approved","The library owner","2026-09-20T10:18:37.581Z","owner_bulk","preview-7e1cbbcc4f",1789899597456]