[{"data":1,"prerenderedAt":1240},["ShallowReactive",2],{"layer:four-operations:investigate":3},{"layer":4,"contentHash":1220,"dependencyHashes":1221,"approval":1234,"releaseId":1239},{"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":1215,"reviewStatus":1216,"authoring":1217},1,"four-operations","en","investigate","Predict, test, check","Estimating first, changing the numbers, making sense of remainders and catching keyword traps","Predict before you calculate and test with labs and tables: estimate sums and products, see what happens when numbers change, decide what a remainder means in a story, catch misleading keywords and check answers by undoing them.",[13,14,15,16,17],"Estimate sums, differences and products by rounding and use the estimate to judge an exact answer.","Predict how a sum, difference, product or quotient changes when one or both numbers change, and test the prediction.","Decide from the story whether a remainder should be rounded up, rounded down, reported or split.","Choose the operation from the structure of a word problem instead of trusting keywords.","Check answers with inverse operations and use counter-examples to test 'always true' claims.",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","Column methods and vocabulary (Understand)",{"label":32,"value":33},"Chapters","11",{"label":35,"value":36},"Labs","Rounding race, 2 sprints, 2 sort games, order-ops",{"label":38,"value":39},"Units used","₹, km, kg, L, quintals",[41,45,51,57,60,77,95,113,129,139,144,149,162,187,192,205,228,241,246,259,295,299,312,325,329,334,345,378,389,429,433,444,465,470,473,493,564,569,580,591,596,599,603,647,715,735,740,743,754,765,770,773,806,810,821,852,858,878,883,903,953,964,976,981,984,997,1001,1006,1048,1178,1182,1196,1200,1205],{"id":42,"type":43,"markdown":44},"intro","prose","A good mathematician is a little like a good detective. Before doing a long calculation, a detective asks: **roughly what should the answer be?** After the calculation, they ask: **how can I check it?** And when they notice a pattern, they ask: **is this always true, or did I just get lucky with my examples?**\n\nThis layer is about those three habits. You will **predict** before you calculate, **test** your prediction with a lab or a table of examples, and **explain** what you find. Some predictions will turn out right. Some will surprise you. The surprises are where the learning happens.\n\nYou already know what the four operations *mean* and how the column methods *work*. Now we poke at them: what happens if we change one number? What does a remainder really tell us? Why do the words in a story sometimes point the wrong way?",{"id":46,"type":47,"variant":48,"title":49,"markdown":50},"intro-how","callout","observation","How to use this layer","Every time you see a **prediction** block, commit to an answer *before* you open the explanation. Write it on paper if you like. Then use the lab or table to test it. Keep a small notebook page called **\"Things that are always true\"** and another called **\"Things that only look true\"**. By the end you will have a surprising list on both.",{"id":52,"type":53,"title":54,"eyebrow":55,"navLabel":56},"ch1","chapter","Estimate first, then calculate","Chapter 01","1 Estimate first",{"id":58,"type":43,"markdown":59},"est-why","Imagine a shopkeeper adds up a bill of ₹4,872 and ₹3,195 and gets ₹80,067. Something has gone badly wrong, and a quick **estimate** catches it at once: 4,872 is about 5,000, 3,195 is about 3,000, so the total must be about **₹8,000**. The exact answer is ₹8,067, which sits comfortably close to the estimate. The wrong answer, ₹80,067, is ten times too big: a digit slipped into the wrong column.\n\nAn estimate is not a lazy answer. It is a **prediction** that tells you the size of the real answer, so you can spot mistakes of place value, a forgotten carry, or a wrong operation.",{"id":61,"type":62,"title":63,"items":64},"est-methods","steps","Three ways to estimate",[65,69,73],{"title":66,"tag":67,"text":68},"Round to the leading digit","front-end","Keep only the first digit and replace the rest with zeros: 4,872 → 5,000 (rounded) or 4,000 (cut off). Quick, rough, good for spotting place-value slips.",{"title":70,"tag":71,"text":72},"Round to a chosen place","nearest 10 \u002F 100 \u002F 1,000","Round each number to the same place: 4,872 → 4,900 to the nearest hundred. Slower but closer to the exact answer.",{"title":74,"tag":75,"text":76},"Use friendly numbers","compatible numbers","Choose numbers that are easy together: 4,872 ÷ 6 ≈ 4,800 ÷ 6 = 800, because 48 is in the 6 times table.",{"id":78,"type":79,"prompt":80,"options":81,"explanation":94},"pred-est-product","prediction","Without calculating exactly, which is the best estimate of **68 × 42**?",[82,85,88,91],{"id":83,"label":84},"a","About 280",{"id":86,"label":87},"b","About 2,800",{"id":89,"label":90},"c","About 28,000",{"id":92,"label":93},"d","About 1,100 (because 68 + 42 = 110)","**About 2,800.** Round each factor: 68 → 70 and 42 → 40, and 70 × 40 = 7 × 4 × 100 = 2,800. The exact product is 2,856, very close. Option (d) is a classic slip: it *adds* the numbers instead of multiplying. Option (a) and (c) are place-value errors: count the zeros carefully (70 × 40 has two zeros).",{"id":96,"type":97,"component":98,"componentVersion":5,"config":99,"objective":107,"textAlternative":108,"help":109},"lab-rounding","interactive","rounding-race",{"roundTo":100,"range":103,"rounds":105,"secondsPerRound":106},[101,102],100,1000,{"min":101,"max":104},99999,10,0,"Round large numbers to the nearest hundred or thousand quickly, the first skill of estimating.","This game shows a random number between 100 and 99,999 on a number line, along with the two nearest multiples of 100 (or of 1,000) on either side. You choose which one it is closer to. There are 10 rounds, untimed, with points for each correct choice and a streak counter.\n\nThe rule you practise: look at the digit just to the **right** of the place you are rounding to. If it is 5 or more, round **up**; if it is 4 or less, round **down**.\n\nExamples you might meet:\n- 4,872 to the nearest hundred: the tens digit is 7, so round up to **4,900**. To the nearest thousand: the hundreds digit is 8, so **5,000**.\n- 63,149 to the nearest thousand: the hundreds digit is 1, so round down to **63,000**.\n- 850 to the nearest hundred: the tens digit is 5, so round up to **900** (halfway rounds up by the usual school rule).\n- 99,960 to the nearest hundred: round up to **1,00,000**, one lakh.\n\nTry saying each rounded number aloud in the Indian system before you pick.",{"simplerExplanation":110,"hints":111},"Find the two 'round' numbers on either side, then ask which one is nearer. If it's exactly in the middle, go up.",[112],"Only one digit decides: the one just after the place you are rounding to.",{"id":114,"type":97,"component":115,"componentVersion":5,"config":116,"objective":127,"textAlternative":128},"lab-sprint-est-add","arith-sprint",{"operations":117,"ranges":120,"rounds":105,"secondsTotal":106,"estimateFirst":125,"wordProblems":126},[118,119],"+","-",{"a":121,"b":123},{"min":102,"max":122},9999,{"min":101,"max":124},999,true,[],"Make a rounded estimate before each exact addition or subtraction, then compare the two.","This sprint gives 10 additions and subtractions of a four-digit number and a three-digit number, untimed. **Before** you type the exact answer, it asks you for an estimate. Then it shows how close your estimate was.\n\nA good way to estimate here is to round both numbers to the nearest hundred. For example, 6,428 + 781: 6,400 + 800 = **7,200**; the exact answer is 7,209. For 5,013 − 468: 5,000 − 500 = **4,500**; exact 4,545.\n\nNotice two things as you play. Your estimate is usually within about 100 of the exact answer, because each rounding moves a number by at most 50. And if your exact answer is far from your estimate, recheck the columns: a missed carry or borrow usually changes the answer by 10, 100 or 1,000.",{"id":130,"type":131,"title":132,"problem":133,"steps":134},"we-est-check","worked_example","Using an estimate to catch a mistake","Riya works out 389 × 21 and gets **1,167**. Use an estimate to decide whether to trust her answer.",[135,136,137,138],"Round each factor: 389 is close to 400 and 21 is close to 20.","400 × 20 = 8,000. So the answer should be somewhere near **8,000**.","Riya's answer, 1,167, is far too small. It is close to 389 × 3 = 1167, so she probably treated the 2 in 21 as 2 ones instead of 2 tens: 389 × 2 + 389 × 1 = 389 × 3.","Exact: 389 × 1 = 389; 389 × 20 = 7,780; total 7,780 + 389 = **8,169**. This is close to 8,000. ✓",{"id":140,"type":47,"variant":141,"title":142,"markdown":143},"est-nuance","nuance","When an estimate can fool you","Rounding both numbers **up** makes an estimate too big; rounding both **down** makes it too small. For a product this effect grows: 45 × 45 rounds to 50 × 50 = 2,500, but the exact answer is 2025. That is still the right *size*, which is the job of an estimate. For subtraction, rounding can even flip a close call: 5,049 − 4,951 is exactly 98, but rounding both to the nearest thousand gives 5,000 − 5,000 = 0. When two numbers are close, estimate with a finer place (nearest ten) or just calculate.",{"id":145,"type":53,"title":146,"eyebrow":147,"navLabel":148},"ch2","What happens if…? Changing the numbers in a sum or difference","Chapter 02","2 Shifting numbers",{"id":150,"type":79,"prompt":151,"options":152,"explanation":161},"pred-diff-shift","83 − 47 = 36. Now add **3** to *both* numbers: (83+3) − (47+3). What happens to the difference?",[153,155,157,159],{"id":83,"label":154},"It gets 3 bigger",{"id":86,"label":156},"It gets 6 bigger",{"id":89,"label":158},"It stays exactly the same",{"id":92,"label":160},"It gets 3 smaller","**It stays the same.** 86 − 50 = 36. Think of two children standing on a staircase: if both climb 3 steps, the *gap* between them does not change. This is the secret behind a great mental trick: change a subtraction into an easier one with the same difference. 83 − 47 becomes 86 − 50, which is easy: 36.",{"id":163,"type":164,"caption":165,"columns":166,"rows":170},"tab-diff-shift","table","Testing: add the same number to both parts of 83 − 47",[167,168,169],"Change to both","New subtraction","Difference",[171,175,178,181,184],[172,173,174],"+0","83 − 47","36",[176,177,174],"+3","86 − 50",[179,180,174],"+10","93 − 57",[182,183,174],"−7","76 − 40",[185,186,174],"+100","183 − 147",{"id":188,"type":47,"variant":189,"title":190,"markdown":191},"aha-staircase","aha","Same gap, easier numbers","**Subtraction is really about the gap.** Add the same amount to both numbers, or take the same amount from both, and the gap stays the same. So ₹1,000 − ₹368 is the same as ₹999 − ₹367 (take 1 from both), and 999 − 367 needs **no borrowing at all**: 632. Shopkeepers giving change from a ₹1,000 note use this without thinking.",{"id":193,"type":79,"prompt":194,"options":195,"explanation":204},"pred-sum-move","58 + 37 = 95. Suppose you move 2 from the second number to the first: 60 + 35. What happens to the sum?",[196,198,200,202],{"id":83,"label":197},"It stays the same",{"id":86,"label":199},"It goes up by 2",{"id":89,"label":201},"It goes down by 4",{"id":92,"label":203},"It goes up by 4","**It stays the same:** 60 + 35 = 95. You have just moved 2 sweets from one plate to another: the total on both plates is unchanged. This is called **compensation**, and it turns awkward sums into friendly ones: 58 + 37 = 60 + 35 = 95.",{"id":206,"type":164,"caption":207,"columns":208,"rows":212},"tab-compare-rules","Sum and difference behave differently when you shift numbers",[209,210,211],"What you do","Effect on a sum a + b","Effect on a difference a − b",[213,217,221,225],[214,215,216],"Add 5 to both numbers","Sum goes up by 10","Difference unchanged",[218,219,220],"Add 5 to the first, take 5 from the second","Sum unchanged","Difference goes up by 10",[222,223,224],"Add 5 to the first only","Sum goes up by 5","Difference goes up by 5",[226,223,227],"Add 5 to the second only","Difference goes **down** by 5",{"id":229,"type":230,"itemId":231,"prompt":232,"check":233,"hints":236,"feedback":238},"prac-shift","practice","four-operations.i-shift-difference","Use the same-gap trick: 7,002 − 3,998. (Hint: add 2 to both numbers first.) What is the difference?",{"kind":234,"answer":235,"tolerance":106},"number",3004,[237],"Adding 2 to both gives 7,004 − 4,000.",{"correct":239,"incorrect":240},"Yes: 7,004 − 4,000 = 3,004. No borrowing across all those zeros!","Add 2 to both numbers: 7,002 + 2 = 7,004 and 3,998 + 2 = 4,000. Then 7,004 − 4,000 = 3,004.",{"id":242,"type":53,"title":243,"eyebrow":244,"navLabel":245},"ch3","What happens if…? Changing the factors in a product","Chapter 03","3 Changing factors",{"id":247,"type":79,"prompt":248,"options":249,"explanation":258},"pred-double-one","24 × 15 = 360. What is **48 × 15**? Predict before you multiply.",[250,252,254,256],{"id":83,"label":251},"362",{"id":86,"label":253},"720",{"id":89,"label":255},"1440",{"id":92,"label":257},"375","**720.** Doubling one factor doubles the product. Picture an array of 24 rows of 15 chairs: doubling the rows to 48 gives twice as many chairs. Doubling *both* factors makes four copies of the original array: 48 × 30 = 1440, which is 4 × 360.",{"id":260,"type":164,"caption":261,"columns":262,"rows":267},"tab-products","Testing what happens to 24 × 15 = 360",[263,264,265,266],"Change","Product","Value","How many times the original",[268,273,277,282,285,290],[269,270,271,272],"Original","24 × 15","360","360 ÷ 360 = 1",[274,275,253,276],"Double the first","48 × 15","720 ÷ 360 = 2",[278,279,280,281],"Double both","48 × 30","1,440","1,440 ÷ 360 = 4",[283,284,271,272],"Halve the first, double the other","12 × 30",[286,287,288,289],"First × 10","240 × 15","3,600","3,600 ÷ 360 = 10",[291,292,293,294],"Both × 10","240 × 150","36,000","36,000 ÷ 360 = 100",{"id":296,"type":47,"variant":189,"title":297,"markdown":298},"aha-halve-double","Halve one, double the other","If you halve one factor and double the other, the product does not change. This makes hard products easy:\n\n16 × 25 = 8 × 50 = 4 × 100 = **400**.\n\n35 × 14 = 70 × 7 = **490**.\n\nA cricket scorer who needs 125 × 8 can think 250 × 4 = 500 × 2 = **1000**.",{"id":300,"type":79,"prompt":301,"options":302,"explanation":311},"pred-bigger","Is this statement true? **Multiplying a number always makes it bigger.**",[303,305,307,309],{"id":83,"label":304},"Always true",{"id":86,"label":306},"True for whole numbers except when you multiply by 0 or 1",{"id":89,"label":308},"Never true",{"id":92,"label":310},"Only true for even numbers","**It is not always true.** 9 × 1 = 9 (no change) and 9 × 0 = 0 (smaller!). With whole numbers bigger than 1 it does make things bigger. Later you will meet fractions: 9 × ½ = 4½, which is smaller again. One example that breaks a rule is called a **counter-example**, and a single counter-example is enough to show a rule is not *always* true.",{"id":313,"type":79,"prompt":314,"options":315,"explanation":324},"pred-zero","A shop sold 0 packets of biscuits on each of 7 days. How many packets did it sell in the week?",[316,318,320,322],{"id":83,"label":317},"7",{"id":86,"label":319},"0",{"id":89,"label":321},"70",{"id":92,"label":323},"You cannot multiply by zero","**0.** 7 × 0 means seven groups of nothing, which is nothing. Also 0 × 7 = 0: no groups of 7. Any number multiplied by zero is zero. (Dividing *by* zero is a different story: that is explored in the Go deeper layer.)",{"id":326,"type":47,"variant":48,"title":327,"markdown":328},"obs-est-errors","Why some estimates drift","Estimate 31 × 19 by rounding to tens: 30 × 20 = 600, and the exact product is 589, very close, because one factor went down and the other went up, so the errors partly cancel. Estimate 55 × 55: 60 × 60 = 3,600, but the exact product is 3,025, because both factors were pushed **up** by 5. When both factors round the same way, expect the estimate to drift further.",{"id":330,"type":53,"title":331,"eyebrow":332,"navLabel":333},"ch4","What happens if…? Dividing by bigger and bigger numbers","Chapter 04","4 Testing division",{"id":335,"type":79,"prompt":336,"options":337,"explanation":344},"pred-bigger-divisor","360 laddoos are packed into boxes. If each box holds **more** laddoos, what happens to the number of boxes?",[338,340,342],{"id":83,"label":339},"More boxes are needed",{"id":86,"label":341},"Fewer boxes are needed",{"id":89,"label":343},"The number of boxes stays the same","**Fewer boxes.** Dividing by a bigger number gives a smaller answer when the dividend stays the same: 360 ÷ 4 = 90, but 360 ÷ 12 = 30. The table below shows the full pattern. Notice that 360 divides exactly by every one of these numbers, which is why the ancient Babylonians (and our clocks and compasses) liked 60 and 360 so much.",{"id":346,"type":164,"caption":347,"columns":348,"rows":351},"tab-360","Sharing 360 into equal groups: bigger divisor, smaller quotient",[349,350],"Division","Quotient",[352,355,358,361,364,367,370,373,375],[353,354],"360 ÷ 2","180",[356,357],"360 ÷ 3","120",[359,360],"360 ÷ 4","90",[362,363],"360 ÷ 5","72",[365,366],"360 ÷ 6","60",[368,369],"360 ÷ 8","45",[371,372],"360 ÷ 9","40",[374,174],"360 ÷ 10",[376,377],"360 ÷ 12","30",{"id":379,"type":79,"prompt":380,"options":381,"explanation":388},"pred-rem-size","Divide 100 by every number from 3 to 12. Could a remainder ever be **equal to or bigger than** the divisor?",[382,384,386],{"id":83,"label":383},"Yes, sometimes",{"id":86,"label":385},"No, never",{"id":89,"label":387},"Only when the divisor is odd","**Never.** If the remainder were as big as the divisor, you could make one more whole group, so you had not finished dividing. The remainder is always **less than the divisor**: dividing by 7 can leave 0, 1, 2, 3, 4, 5 or 6, never 7 or more. Check the table.",{"id":390,"type":164,"caption":391,"columns":392,"rows":395},"tab-rem100","Testing: 100 divided by 3 to 12",[349,350,393,394],"Remainder","Remainder \u003C divisor?",[396,401,404,407,411,415,418,420,423,426],[397,398,399,400],"100 ÷ 3","33","1","yes",[402,403,319,400],"100 ÷ 4","25",[405,406,319,400],"100 ÷ 5","20",[408,409,410,400],"100 ÷ 6","16","4",[412,413,414,400],"100 ÷ 7","14","2",[416,417,410,400],"100 ÷ 8","12",[419,33,399,400],"100 ÷ 9",[421,422,319,400],"100 ÷ 10","10",[424,425,399,400],"100 ÷ 11","9",[427,428,410,400],"100 ÷ 12","8",{"id":430,"type":47,"variant":48,"title":431,"markdown":432},"obs-rem-pattern","Spot the pattern in the table","Look down the remainder column: 1, 0, 0, 4, 2, 4, 1, 0, 1, 4. Every one is smaller than its divisor. Notice too that **100 ÷ 4, 100 ÷ 5 and 100 ÷ 10 leave remainder 0**: 4, 5 and 10 are **factors** of 100. Finding numbers that divide exactly is the start of the topic on prime and composite numbers.",{"id":434,"type":230,"itemId":435,"prompt":436,"check":437,"hints":439,"feedback":441},"prac-rem-max","four-operations.i-largest-remainder","What is the **largest** remainder you can possibly get when you divide a whole number by 9?",{"kind":234,"answer":438,"tolerance":106},8,[440],"The remainder must be less than the divisor.",{"correct":442,"incorrect":443},"Right: 8. If you had 9 or more left over, you could make another group of 9.","The remainder must be smaller than 9, so the largest possible is 8. For example 17 ÷ 9 = 1 remainder 8.",{"id":445,"type":230,"itemId":446,"prompt":447,"check":448,"hints":460,"feedback":462},"prac-check-div","four-operations.i-check-division","Mohan says **250 ÷ 7 = 34 remainder 12**. What is wrong?",{"kind":449,"options":450,"correct":459},"choice",[451,453,455,457],{"id":83,"label":452},"Nothing, it is correct",{"id":86,"label":454},"The remainder 12 is bigger than 7, so another 7 fits: the answer is 35 remainder 5",{"id":89,"label":456},"The quotient should be 30",{"id":92,"label":458},"250 cannot be divided by 7",[86],[461],"Is the remainder smaller than the divisor?",{"correct":463,"incorrect":464},"Yes. 7 × 35 = 245 and 250 − 245 = 5, so 250 ÷ 7 = 35 r 5. Check: 7 × 35 + 5 = 250. ✓","A remainder of 12 is bigger than 7, so one more group of 7 fits: 12 − 7 = 5. The answer is 35 remainder 5, since 7 × 35 + 5 = 250.",{"id":466,"type":53,"title":467,"eyebrow":468,"navLabel":469},"ch5","What does the remainder mean? It depends on the story","Chapter 05","5 Remainders in stories",{"id":471,"type":43,"markdown":472},"rem-intro","Here are three stories. Each one is a division with a remainder, but each needs a **different** final answer.\n\n1. **250 students** are going on a school trip. Each bus holds **45**. 250 ÷ 45 = 5 remainder 25. How many buses? Not 5! That would leave 25 children at school. You need **6 buses**. The answer is **rounded up**.\n2. You have **₹500** and notebooks cost **₹45** each. 500 ÷ 45 = 11 remainder 5. You can buy **11 notebooks**, with ₹5 left over. You cannot buy part of a notebook, so the answer is **rounded down**.\n3. **50 laddoos** are shared among **8 cousins**. 50 ÷ 8 = 6 remainder 2. Each cousin gets 6, and the **2 left over** might go to Grandma, or be cut into pieces and shared. Here the remainder itself is part of the answer.\n\nSame kind of calculation, three different answers. The **story** decides, not the arithmetic.",{"id":474,"type":62,"title":475,"items":476},"rem-questions","Four questions to ask about a remainder",[477,481,485,489],{"title":478,"tag":479,"text":480},"Does everyone or everything need a place?","round up","Buses, boxes, tables, trips: a leftover still needs one more. 250 students, 45 per bus → 6 buses.",{"title":482,"tag":483,"text":484},"Can I only use complete groups?","round down","Buying items, filling full packets, cutting whole pieces of cloth: the leftover is wasted or kept. ₹500 at ₹45 → 11 notebooks.",{"title":486,"tag":487,"text":488},"Is the leftover what they are asking for?","remainder is the answer","\"How many laddoos are left over?\" → 2. \"How much change?\" → ₹5.",{"title":490,"tag":491,"text":492},"Can the leftover be split further?","share the remainder","Rupees can become paise, a roti can be torn in half: 7 rotis for 2 people → 3½ each.",{"id":494,"type":97,"component":495,"componentVersion":5,"config":496,"objective":560,"textAlternative":561,"help":562},"lab-sort-rem","sort-game",{"prompt":497,"bins":498,"items":511,"seconds":106},"Each story ends in a division with a remainder. What should you do with the remainder?",[499,502,505,508],{"id":500,"label":501},"up","Round up",{"id":503,"label":504},"down","Round down",{"id":506,"label":507},"rem","The remainder is the answer",{"id":509,"label":510},"split","Split the remainder",[512,516,520,524,528,532,536,540,544,548,552,556],{"id":513,"label":514,"bin":500,"why":515},"bus","250 students, 45 seats per bus. How many buses?","250 ÷ 45 = 5 r 25. The 25 students still need a bus: 6 buses.",{"id":517,"label":518,"bin":503,"why":519},"notebooks","₹500 to spend, notebooks cost ₹45. How many can you buy?","500 ÷ 45 = 11 r 5. You cannot buy part of a notebook: 11.",{"id":521,"label":522,"bin":506,"why":523},"laddoo-left","50 laddoos shared equally among 8 cousins. How many are left over?","50 ÷ 8 = 6 r 2. The question asks for the leftover: 2.",{"id":525,"label":526,"bin":500,"why":527},"boxes","100 mangoes packed 12 to a box. How many boxes to pack them all?","100 ÷ 12 = 8 r 4. The last 4 mangoes need a box too: 9 boxes.",{"id":529,"label":530,"bin":503,"why":531},"ribbon","A 100 cm ribbon is cut into 15 cm pieces. How many full pieces?","100 ÷ 15 = 6 r 10. Only 6 full pieces; 10 cm is too short.",{"id":533,"label":534,"bin":506,"why":535},"change","₹200 buys as many ₹30 pens as possible. How much change?","200 ÷ 30 = 6 r 20. The change is the remainder: ₹20.",{"id":537,"label":538,"bin":509,"why":539},"rotis","7 rotis shared equally between 2 friends. How much each?","7 ÷ 2 = 3 r 1. The last roti can be torn in half: 3½ rotis each.",{"id":541,"label":542,"bin":503,"why":543},"teams","40 players make teams of 11. How many full teams?","40 ÷ 11 = 3 r 7. Only 3 full teams; 7 players are left out (or become substitutes).",{"id":545,"label":546,"bin":500,"why":547},"trips","A lift carries 6 people. 27 people want to go up. How many trips?","27 ÷ 6 = 4 r 3. The last 3 people need a 5th trip.",{"id":549,"label":550,"bin":509,"why":551},"money-split","₹25 shared equally between 2 sisters. How much each?","25 ÷ 2 = 12 r 1. The last rupee becomes 100 paise: ₹12.50 each.",{"id":553,"label":554,"bin":506,"why":555},"weeks","Holidays last 45 days. How many days more than full weeks?","45 ÷ 7 = 6 r 3. Six full weeks and 3 extra days: the answer is 3.",{"id":557,"label":558,"bin":500,"why":559},"shelves","130 books, each shelf holds 25. How many shelves to hold every book?","130 ÷ 25 = 5 r 5. The last 5 books need a sixth shelf.","Decide what a remainder means in a real story: round up, round down, give the leftover, or split it.","This sorting game shows 12 word-problem cards. Each one is a division that does not come out exactly. Drag or tap each card into one of four bins: **Round up**, **Round down**, **The remainder is the answer**, or **Split the remainder**.\n\n- Round up: 250 students with 45 seats per bus (5 r 25 → 6 buses); 100 mangoes, 12 per box (8 r 4 → 9 boxes); 27 people, lift for 6 (4 r 3 → 5 trips); 130 books, 25 per shelf (5 r 5 → 6 shelves).\n- Round down: ₹500 for ₹45 notebooks (11 r 5 → 11); 100 cm ribbon into 15 cm pieces (6 r 10 → 6); 40 players into teams of 11 (3 r 7 → 3).\n- Remainder is the answer: 50 laddoos among 8, how many left (2); change from ₹200 buying ₹30 pens (₹20); 45 days, how many days beyond full weeks (3).\n- Split the remainder: 7 rotis between 2 (3½ each); ₹25 between 2 (₹12.50 each).\n\nThe lesson: always reread the question after dividing. Ask, \"What exactly are they asking for?\"",{"simplerExplanation":563},"After dividing, picture the leftover. Does it still need a bus? Is it wasted? Is it what they asked for? Can it be cut up?",{"id":565,"type":47,"variant":566,"title":567,"markdown":568},"mis-rem-decimal","misconception","“The remainder goes after a decimal point”","Some learners write 250 ÷ 45 = 5 r 25 as **5.25**. That is wrong. The remainder 25 means 25 students, not 25 hundredths. As a decimal, 250 ÷ 45 is about 5.56 (the leftover 25 out of 45 is a bit more than half a bus). Keep \"remainder\" and \"decimal\" separate until you learn to convert one into the other.",{"id":570,"type":230,"itemId":571,"prompt":572,"check":573,"hints":575,"feedback":577},"prac-rem-auto","four-operations.i-autos","An auto-rickshaw can carry 3 children. How many autos are needed to take 38 children to school?",{"kind":234,"answer":574,"tolerance":106},13,[576],"38 ÷ 3 = 12 remainder 2. Do the last 2 children need an auto?",{"correct":578,"incorrect":579},"Yes: 13 autos. 12 autos carry 36 children and the last 2 need one more.","38 ÷ 3 = 12 r 2. Twelve autos carry 36 children, but 2 are still waiting, so you need 13 autos.",{"id":581,"type":230,"itemId":582,"prompt":583,"check":584,"hints":586,"feedback":588},"prac-rem-down","four-operations.i-packets","A factory has 1,000 biscuits and packs 24 in each packet. How many **full** packets can it make?",{"kind":234,"answer":585,"tolerance":106},41,[587],"1,000 ÷ 24: try 24 × 40 = 960.",{"correct":589,"incorrect":590},"Right: 41 full packets (24 × 41 = 984), with 16 biscuits left over.","24 × 41 = 984 and 24 × 42 = 1,008, which is too many. So 41 full packets, with 1,000 − 984 = 16 biscuits left.",{"id":592,"type":53,"title":593,"eyebrow":594,"navLabel":595},"ch6","Can keywords be trusted? Testing the “clue word” rule","Chapter 06","6 Keyword traps",{"id":597,"type":43,"markdown":598},"kw-intro","Many of us were taught clue words: **\"altogether\" means add, \"left\" means subtract, \"each\" means multiply, \"share\" means divide.** Let us test that rule the way a scientist would: look for stories where it gives the wrong answer.\n\n- *\"Ravi has 12 marbles. That is 5 **more** than Sita. How many does Sita have?\"* The word \"more\" shouts *add*, but Sita has **fewer**: 12 − 5 = **7**.\n- *\"Meena had some stickers. She gave away 8 and has 15 **left**. How many did she start with?\"* \"Left\" shouts *subtract*, but the start was 15 + 8 = **23**.\n- *\"Each packet has 6 pens. I need 48 pens. How many packets?\"* \"Each\" shouts *multiply*, but the answer is 48 ÷ 6 = **8**.\n- *\"A train covered 245 km in the morning and 180 km in the afternoon. How much **farther** did it go in the morning?\"* No \"minus\" word anywhere, yet it is 245 − 180 = **65 km**.\n\nClue words are hints about the **situation**, not instructions. The reliable method is to picture what is happening.",{"id":600,"type":47,"variant":566,"title":601,"markdown":602},"mis-keywords","“Find the keyword and you know the operation”","Keywords work often enough to be tempting and fail often enough to be dangerous. Examiners know this, and many test questions deliberately use a word like \"more\" in a subtraction problem. Instead of hunting for words, ask:\n\n- **Am I joining** amounts, or **separating\u002Fcomparing** them?\n- Are the groups **equal**? If so, is the unknown the **total**, the **number of groups** or the **size of each group**?\n- What is **unknown**: the start, the change, or the result?",{"id":604,"type":164,"caption":605,"columns":606,"rows":611},"tab-structures","The real structures behind word problems",[607,608,609,610],"Structure","What is unknown","Example","Operation",[612,617,621,625,629,634,638,642],[613,614,615,616],"Join","Result","32 people on a bus, 9 get on. How many now?","32 + 9 = 41 (add)",[613,618,619,620],"Start","Some people on a bus, 9 get on, now 41. How many at first?","41 − 9 = 32 (subtract)",[622,614,623,624],"Separate","₹100, spend ₹37. How much left?","100 − 37 = 63 (subtract)",[626,169,627,628],"Compare","Mount Everest 8,849 m, K2 8,611 m. How much taller?","8,849 − 8,611 = 238 (subtract)",[630,631,632,633],"Equal groups","Total","8 boxes, 12 pencils each. How many pencils?","8 × 12 = 96 (multiply)",[630,635,636,637],"Number of groups","96 pencils, 12 per box. How many boxes?","96 ÷ 12 = 8 (divide)",[630,639,640,641],"Size of each group","96 pencils in 8 boxes equally. How many per box?","96 ÷ 8 = 12 (divide)",[643,644,645,646],"Scale","Bigger amount","A mango costs ₹15; a melon costs 4 times as much.","15 × 4 = 60 (multiply)",{"id":648,"type":97,"component":495,"componentVersion":5,"config":649,"objective":713,"textAlternative":714},"lab-sort-keywords",{"prompt":650,"bins":651,"items":664,"seconds":106},"Read each story carefully (don't trust the clue words!). Which operation solves it?",[652,655,658,661],{"id":653,"label":654},"add","Add (+)",{"id":656,"label":657},"sub","Subtract (−)",{"id":659,"label":660},"mul","Multiply (×)",{"id":662,"label":663},"div","Divide (÷)",[665,669,673,677,681,685,689,693,697,701,705,709],{"id":666,"label":667,"bin":656,"why":668},"marbles","Ravi has 12 marbles, 5 more than Sita. How many does Sita have?","Sita has fewer: 12 − 5 = 7. 'More' describes Ravi, not the operation.",{"id":670,"label":671,"bin":653,"why":672},"stickers","Meena gave away 8 stickers and has 15 left. How many did she start with?","The start is the part she kept plus the part she gave: 15 + 8 = 23.",{"id":674,"label":675,"bin":662,"why":676},"packets","Each packet has 6 pens. How many packets make 48 pens?","The unknown is the number of equal groups: 48 ÷ 6 = 8.",{"id":678,"label":679,"bin":656,"why":680},"farther","A train ran 245 km before lunch and 180 km after. How much farther before lunch?","Comparing two amounts: 245 − 180 = 65 km.",{"id":682,"label":683,"bin":659,"why":684},"times","A pen costs ₹12. A bag costs 25 times as much. Cost of the bag?","Scaling up: 12 × 25 = ₹300.",{"id":686,"label":687,"bin":653,"why":688},"fewer","Class A has 38 students, 6 fewer than Class B. How many in Class B?","B has more: 38 + 6 = 44. 'Fewer' describes A.",{"id":690,"label":691,"bin":659,"why":692},"rows","A hall has 18 rows with 24 chairs in each. How many chairs?","Equal rows, total unknown: 18 × 24 = 432.",{"id":694,"label":695,"bin":662,"why":696},"per-day","A farmer picks 1,260 mangoes in 7 days, the same each day. How many per day?","Size of each group: 1,260 ÷ 7 = 180.",{"id":698,"label":699,"bin":653,"why":700},"total-left","After spending ₹245 Kavya has ₹55 left. How much did she have?","Start = spent + left: 245 + 55 = ₹300.",{"id":702,"label":703,"bin":662,"why":704},"each-share","Four friends together paid ₹360 for a pizza, equally. How much each?","Share equally: 360 ÷ 4 = ₹90. 'Together' does not mean add here.",{"id":706,"label":707,"bin":656,"why":708},"runs-need","The target is 287 runs. The team has scored 198. How many more runs are needed?","The missing part: 287 − 198 = 89. 'More' again, but it is subtraction.",{"id":710,"label":711,"bin":659,"why":712},"altogether-mul","12 buses each carry 45 pilgrims. How many pilgrims altogether?","Equal groups: 12 × 45 = 540. 'Altogether' with equal groups means multiply.","Choose the operation from the situation, not from clue words, including stories built to trick keyword-hunters.","This game gives 12 story cards to sort into Add, Subtract, Multiply or Divide. Many cards contain a misleading clue word on purpose.\n\n- Subtract: \"Ravi has 12, 5 more than Sita\" (Sita has 7); \"245 km vs 180 km, how much farther\" (65 km); \"target 287, scored 198, how many more\" (89).\n- Add: \"gave away 8, has 15 left, how many at start\" (23); \"38 students, 6 fewer than Class B\" (B has 44); \"spent ₹245, ₹55 left, how much at first\" (₹300).\n- Multiply: \"₹12 pen, bag 25 times as much\" (₹300); \"18 rows of 24 chairs\" (432); \"12 buses of 45 pilgrims altogether\" (540).\n- Divide: \"6 pens per packet, how many packets for 48\" (8); \"1,260 mangoes in 7 equal days\" (180 a day); \"₹360 shared equally by 4\" (₹90).\n\nFor each card, picture the situation: joining, separating, comparing, equal groups or scaling. Then decide what is unknown.",{"id":716,"type":230,"itemId":717,"prompt":718,"check":719,"hints":730,"feedback":732},"prac-kw","four-operations.i-keyword-trap","*\"A shirt costs ₹450. That is ₹120 more than a T-shirt.\"* What does the T-shirt cost?",{"kind":449,"options":720,"correct":729},[721,723,725,727],{"id":83,"label":722},"₹570",{"id":86,"label":724},"₹330",{"id":89,"label":726},"₹3.75",{"id":92,"label":728},"₹54,000",[86],[731],"Which item is cheaper?",{"correct":733,"incorrect":734},"Right: the T-shirt is cheaper, so 450 − 120 = ₹330. The word 'more' was describing the shirt.","The shirt costs more, so the T-shirt costs less: 450 − 120 = ₹330. Adding (₹570) is the keyword trap.",{"id":736,"type":53,"title":737,"eyebrow":738,"navLabel":739},"ch7","Checking by undoing: inverse operations","Chapter 07","7 Undo to check",{"id":741,"type":43,"markdown":742},"inv-intro","Every operation has a partner that **undoes** it. Addition and subtraction undo each other; multiplication and division undo each other. So every answer can be tested by running the calculation backwards.\n\n- If 36,48,275 − 9,87,654 = 26,60,621, then 26,60,621 + 9,87,654 must give back 36,48,275. It does.\n- If 7,392 ÷ 16 = 462, then 462 × 16 must give back 7,392. It does.\n- If 1,000 ÷ 24 = 41 remainder 16, then 24 × 41 + 16 must give back 1,000: 984 + 16 = 1,000. ✓",{"id":744,"type":79,"prompt":745,"options":746,"explanation":753},"pred-inverse","Pooja worked out 5,006 − 2,789 = 3,327. She checks by adding 3,327 + 2,789. What will she get, and what does it tell her?",[747,749,751],{"id":83,"label":748},"5,006, so she is right",{"id":86,"label":750},"6,116, so her subtraction is wrong",{"id":89,"label":752},"5,006, but that proves nothing","**6,116.** 3,327 + 2,789 = 6116, not 5,006, so her subtraction must be wrong. The correct difference is 5,006 − 2,789 = **2217**, and 2217 + 2,789 = 5,006. ✓ A common slip when borrowing across the zeros in 5,006 is to forget that the borrowed hundred and ten have already been used, which gives an answer 1,000 or 100 too big.",{"id":755,"type":230,"itemId":756,"prompt":757,"check":758,"hints":760,"feedback":762},"prac-inverse-div","four-operations.i-inverse-find-dividend","A number divided by 23 gives quotient 17 and remainder 9. What is the number?",{"kind":234,"answer":759,"tolerance":106},400,[761],"Undo the division: dividend = divisor × quotient + remainder.",{"correct":763,"incorrect":764},"Yes: 23 × 17 + 9 = 391 + 9 = 400.","Multiply back and add the remainder: 23 × 17 = 391, then 391 + 9 = 400.",{"id":766,"type":53,"title":767,"eyebrow":768,"navLabel":769},"ch8","Is it always true? Testing claims about the operations","Chapter 08","8 Always true?",{"id":771,"type":43,"markdown":772},"always-intro","A claim like \"you can add in any order\" is a statement about **every** pair of numbers, not just the ones you tried. Testing ten examples cannot *prove* it is always true (there are infinitely many numbers), but it can build confidence. And **one** failed example proves it is **not** always true. Let us test some claims.",{"id":774,"type":164,"caption":775,"columns":776,"rows":782},"tab-order","Does order matter? a ○ b compared with b ○ a",[777,778,779,780,781],"Numbers a, b","a + b \u002F b + a","a − b \u002F b − a","a × b \u002F b × a","a ÷ b \u002F b ÷ a",[783,789,795,801],[784,785,786,787,788],"15 and 6","21 \u002F 21","9 \u002F below 0","90 \u002F 90","2.5 \u002F 0.40",[790,791,792,793,794],"100 and 4","104 \u002F 104","96 \u002F below 0","400 \u002F 400","25 \u002F 0.04",[796,797,798,799,800],"48 and 12","60 \u002F 60","36 \u002F below 0","576 \u002F 576","4 \u002F 0.25",[802,803,804,785,805],"7 and 3","10 \u002F 10","4 \u002F below 0","2.33 \u002F 0.43",{"id":807,"type":47,"variant":48,"title":808,"markdown":809},"obs-order","What the table shows","Addition and multiplication gave the same answer both ways in every test: they are **commutative**. Subtraction and division did not: 15 − 6 = 9 but 6 − 15 goes below zero, and 48 ÷ 12 = 4 but 12 ÷ 48 = 0.25. So \"order does not matter\" is true for + and ×, and **false** for − and ÷. This is why, in a subtraction or division word problem, you must decide carefully which number comes first.",{"id":811,"type":79,"prompt":812,"options":813,"explanation":820},"pred-odd","Add two odd numbers: 7 + 9, 13 + 21, 35 + 101… Is the sum **always** even?",[814,816,818],{"id":83,"label":815},"Yes, always even",{"id":86,"label":817},"Always odd",{"id":89,"label":819},"Sometimes even, sometimes odd","**Always even.** Each odd number is a pair-able number with one left over. Put two odd numbers together and the two leftovers make another pair, so nothing is left: the sum is even. Try a few big ones to feel confident: 999 + 1,001 = 2,000. This is an argument, not just examples, which is why we can say *always*.",{"id":822,"type":164,"caption":823,"columns":824,"rows":828},"tab-claims","Claims tested with hundreds of examples by computer, then explained",[825,826,827],"Claim (whole numbers)","Verdict","Why",[829,833,836,839,842,846,849],[830,831,832],"odd + odd is even","Always","Two leftovers make a pair.",[834,831,835],"odd × odd is odd","No factor of 2 appears anywhere.",[837,831,838],"even × anything is even","A factor of 2 is already there.",[840,831,841],"the sum of three numbers in a row divides by 3","k + (k+1) + (k+2) = 3 × (k+1): three times the middle number.",[843,844,845],"a − b is less than a","Not always","Counter-example: 9 − 0 = 9, which is not less than 9.",[847,844,848],"a ÷ b is less than a","Counter-examples: 9 ÷ 1 = 9, and 0 ÷ 5 = 0.",[850,831,851],"the product of two numbers in a row is even","One of any two neighbours is even.",{"id":853,"type":854,"conceptId":855,"relation":856,"explanation":857},"conn-properties","connection","properties-of-numbers","helps_understand","Commutative, associative and distributive properties are the 'always true' rules you have been testing here, stated and named carefully.",{"id":859,"type":230,"itemId":860,"prompt":861,"check":862,"hints":873,"feedback":875},"prac-counter","four-operations.i-counterexample","Which example is a **counter-example** to the claim \"the difference of two numbers is always smaller than both of them\"?",{"kind":449,"options":863,"correct":872},[864,866,868,870],{"id":83,"label":865},"20 − 5 = 15",{"id":86,"label":867},"10 − 3 = 7",{"id":89,"label":869},"9 − 8 = 1",{"id":92,"label":871},"100 − 99 = 1",[83],[874],"Is the difference smaller than both numbers?",{"correct":876,"incorrect":877},"Yes: 15 is **not** smaller than 5, so the claim fails. One counter-example is enough.","Check each: in 20 − 5 = 15, the answer 15 is bigger than 5, so the claim fails there. That is the counter-example.",{"id":879,"type":53,"title":880,"eyebrow":881,"navLabel":882},"ch9","Real problems: choose, estimate, calculate, check","Chapter 09","9 Real problems",{"id":884,"type":62,"title":885,"items":886},"four-steps","A routine for any word problem",[887,891,895,899],{"title":888,"tag":889,"text":890},"Picture it","understand","Draw a bar, a box or a quick sketch. What is happening: joining, separating, comparing, equal groups, scaling?",{"title":892,"tag":893,"text":894},"Predict","estimate","Round the numbers and guess the size of the answer before calculating.",{"title":896,"tag":897,"text":898},"Calculate","method","Use the column method, a mental strategy or a calculator, whichever is most reliable here.",{"title":900,"tag":901,"text":902},"Check and answer","sense","Undo the calculation, compare with your estimate, and write the answer with its unit (₹, km, kg).",{"id":904,"type":97,"component":115,"componentVersion":5,"config":905,"objective":951,"textAlternative":952},"lab-sprint-words",{"operations":906,"ranges":909,"rounds":914,"secondsTotal":106,"estimateFirst":125,"wordProblems":915},[118,119,907,908],"×","÷",{"a":910,"b":911},{"min":101,"max":124},{"min":912,"max":913},2,9,20,[916,920,924,928,931,934,938,941,944,948],{"prompt":917,"answer":918,"unit":919,"operation":907},"A train from Chennai to Delhi travels 2,182 km. The return trip is the same. How many km for the round trip?",4364,"km",{"prompt":921,"answer":922,"unit":923,"operation":118},"A kirana shop sold rice worth ₹18,450 on Monday and ₹22,785 on Tuesday. What was the total?",41235,"₹",{"prompt":925,"answer":926,"unit":927,"operation":119},"A farmer harvested 3,250 kg of wheat and sold 1,875 kg. How many kg are left?",1375,"kg",{"prompt":929,"answer":930,"unit":923,"operation":907},"A school buys 36 chairs at ₹845 each. What is the total cost?",30420,{"prompt":932,"answer":933,"unit":923,"operation":908},"₹7,560 is shared equally among 12 families. How much does each family get?",630,{"prompt":935,"answer":936,"unit":937,"operation":119},"A team needs 287 runs to win and has made 198. How many more runs are needed?",89,"runs",{"prompt":939,"answer":914,"unit":940,"operation":908},"A water tank holds 5,000 L. A family uses 250 L a day. How many days will a full tank last?","days",{"prompt":942,"answer":943,"unit":919,"operation":907},"Mumbai to Pune is 148 km by road. A taxi makes the one-way trip 25 times in a month. Total km?",3700,{"prompt":945,"answer":946,"unit":947,"operation":118},"A village has 4,68,352 people. The next village has 97,648 more. How many in the next village?",566000,"people",{"prompt":949,"answer":950,"unit":525,"operation":908},"A bag of 1,440 laddoos is packed in boxes of 24. How many boxes?",60,"Solve real Indian word problems: choose the operation, estimate, calculate exactly and check by undoing.","This untimed sprint has 20 rounds. Up to half of them are word problems, picked at random from the ten below; the rest are generated sums with all four operations, where you estimate first. For each word problem, make your own quick estimate on paper, then give the exact answer.\n\n1. Chennai–Delhi round trip, 2,182 km each way: 2,182 × 2 = **4,364 km**.\n2. Kirana sales ₹18,450 + ₹22,785 = **₹41,235**.\n3. Wheat 3,250 kg − 1,875 kg sold = **1,375 kg** left.\n4. 36 chairs at ₹845: 36 × 845 = **₹30,420**.\n5. ₹7,560 shared by 12 families: **₹630** each.\n6. Runs needed: 287 − 198 = **89**.\n7. A 5,000 L tank at 250 L a day lasts **20 days**.\n8. Mumbai–Pune 148 km × 25 trips = **3,700 km**.\n9. Village of 4,68,352 plus 97,648 more = **5,66,000** people.\n10. 1,440 laddoos in boxes of 24 = **60 boxes**.\n\nEach time, decide the structure first (join, separate, compare, equal groups, scale), then check by undoing.",{"id":954,"type":131,"title":955,"problem":956,"steps":957},"we-two-step","A two-step problem: the school trip","Class 6 has 42 students and 3 teachers going to a science museum. A bus costs ₹4,500 for the day and each ticket costs ₹60. What is the total cost?",[958,959,960,961,962,963],"**Picture it.** Two separate costs: the bus (one fixed amount) and tickets (equal groups: one per person).","**People:** 42 + 3 = 45 people need tickets.","**Estimate:** about 50 × 60 = 3,000 for tickets, plus 4,500 for the bus: roughly ₹7,500.","**Tickets:** 45 × 60 = ₹2,700.","**Total:** ₹4,500 + ₹2,700 = **₹7,200**. This is close to the ₹7,500 estimate. ✓","**Bonus:** shared equally among the 42 students, that is ₹7,200 ÷ 42 = 171 remainder 18. The class would probably charge ₹172 each (round **up**, so the costs are covered).",{"id":965,"type":230,"itemId":966,"prompt":967,"check":968,"hints":970,"feedback":973},"prac-crop","four-operations.i-crop-yield","A farmer's field gives 48 quintals of paddy. He keeps 6 quintals for his family and sells the rest at ₹2,300 per quintal. How many rupees does he receive?",{"kind":234,"answer":969,"tolerance":106,"unit":923},96600,[971,972],"First find how many quintals he sells.","Then multiply by the price.",{"correct":974,"incorrect":975},"Correct: (48 − 6) × 2,300 = 42 × 2,300 = ₹96,600.","He sells 48 − 6 = 42 quintals. 42 × 2,300 = ₹96,600. Estimate check: 40 × 2,000 = 80,000, so ₹96,600 is the right size.",{"id":977,"type":53,"title":978,"eyebrow":979,"navLabel":980},"ch10","Two steps at once: writing a problem as one expression","Chapter 10","10 One expression",{"id":982,"type":43,"markdown":983},"expr-intro","When a problem has two steps, you can write it as a single **expression**. The school trip above becomes **4500 + 45 × 60**. But careful: which operation happens first? If you add first you get 4,545 × 60 = 272,700, which is absurd for a class trip. The rule is that **multiplication and division are done before addition and subtraction**, unless brackets say otherwise. So 4500 + 45 × 60 = 4,500 + 2,700 = 7,200.\n\nThe order-of-operations topic studies this properly. Here is a quick lab to test it on problems you have already met.",{"id":985,"type":97,"component":986,"componentVersion":5,"config":987,"objective":995,"textAlternative":996},"lab-order-ops","order-ops",{"expressions":988,"showRuleCard":125},[989,990,991,992,993,994],"4500 + 45 × 60","500 - 11 × 45","(250 + 175) ÷ 5","287 - (98 + 100)","7560 ÷ 12 - 30","(48 - 6) × 2300","Pick which operation to do first in two-step word-problem expressions and watch each reduce to its answer.","This lab shows six expressions that come from word problems. You choose which operation to do next; the expression shrinks one step at a time. A rule card reminds you: brackets first, then × and ÷, then + and −.\n\n- 4500 + 45 × 60 (school trip): 45 × 60 = 2,700 first, then 4,500 + 2,700 = **7,200**.\n- 500 - 11 × 45 (₹500, eleven ₹45 notebooks): 11 × 45 = 495, then 500 − 495 = **5** rupees change.\n- (250 + 175) ÷ 5 (two classes sharing 5 buses equally): 425 ÷ 5 = **85** per bus.\n- 287 - (98 + 100) (runs still needed after two partnerships): 198, then 287 − 198 = **89**.\n- 7560 ÷ 12 - 30 (each family's share minus a ₹30 fee): 630 − 30 = **600**.\n- (48 - 6) × 2300 (quintals of paddy sold at ₹2,300 each): 42 × 2,300 = **96,600** rupees.\n\nTry doing the wrong step first in your head and see how far off the answer goes.",{"id":998,"type":854,"conceptId":999,"relation":856,"explanation":1000},"conn-order","order-of-operations","Once each operation is reliable, the next question is which one to do first when several appear together in one expression.",{"id":1002,"type":53,"title":1003,"eyebrow":1004,"navLabel":1005},"ch11","What you found out","Chapter 11","11 Wrap-up",{"id":1007,"type":1008,"title":1009,"terms":1010},"glossary-investigate","glossary","Words for investigating",[1011,1014,1018,1022,1025,1029,1033,1037,1041,1045],{"term":893,"meaning":1012,"example":1013},"A sensible approximate answer found by rounding or using friendly numbers, used to predict and to check.","68 × 42 ≈ 70 × 40 = 2,800",{"term":1015,"meaning":1016,"example":1017},"leading digit","The first (left-most) non-zero digit of a number; it tells you most about the number's size.","In 4,872 the leading digit is 4 (thousands).",{"term":1019,"meaning":1020,"example":1021},"front-end estimation","Estimating using only the leading digits of each number.","4,872 + 3,195 ≈ 4,000 + 3,000",{"term":75,"meaning":1023,"example":1024},"Numbers chosen because they work neatly together, making an estimate easy.","4,872 ÷ 6 ≈ 4,800 ÷ 6 = 800",{"term":1026,"meaning":1027,"example":1028},"compensation","Changing numbers to make a calculation easier, then adjusting (or choosing a change that needs no adjustment).","58 + 37 = 60 + 35",{"term":1030,"meaning":1031,"example":1032},"inverse operation","The operation that undoes another: subtraction undoes addition, division undoes multiplication.","462 × 16 = 7,392 checks 7,392 ÷ 16 = 462",{"term":1034,"meaning":1035,"example":1036},"counter-example","A single example showing that a claim is not always true.","9 × 1 = 9 disproves 'multiplying always makes bigger'.",{"term":1038,"meaning":1039,"example":1040},"commutative","An operation where order does not change the answer: true for + and ×, false for − and ÷.","6 × 15 = 15 × 6",{"term":1042,"meaning":1043,"example":1044},"interpreting a remainder","Deciding from the story whether to round up, round down, give the leftover, or split it.","250 students, 45 per bus → 6 buses",{"term":1046,"meaning":1047,"example":989},"expression","A calculation written with numbers and operation signs, without an equals sign.",{"id":1049,"type":1049,"title":1050,"questions":1051},"quiz","Check yourself",[1052,1065,1078,1090,1103,1115,1126,1139,1152,1165],{"itemId":1053,"prompt":1054,"options":1055,"correct":86,"why":1064},"four-operations.i-quiz-estimate","Which is the best estimate of 612 × 49?",[1056,1058,1060,1062],{"id":83,"label":1057},"About 3,000",{"id":86,"label":1059},"About 30,000",{"id":89,"label":1061},"About 300,000",{"id":92,"label":1063},"About 660","612 ≈ 600 and 49 ≈ 50, so 600 × 50 = 30,000. The exact product is 29,988.",{"itemId":1066,"prompt":1067,"options":1068,"correct":86,"why":1077},"four-operations.i-quiz-gap","Which subtraction has the same answer as 1,000 − 457?",[1069,1071,1073,1075],{"id":83,"label":1070},"1,001 − 456",{"id":86,"label":1072},"999 − 456",{"id":89,"label":1074},"999 − 458",{"id":92,"label":1076},"1,000 − 500","Take 1 from both numbers: 999 − 456 = 543, same as 1,000 − 457 = 543. No borrowing needed.",{"itemId":1079,"prompt":1080,"options":1081,"correct":86,"why":1089},"four-operations.i-quiz-double","36 × 25 = 900. What is 72 × 50?",[1082,1084,1085,1087],{"id":83,"label":1083},"1,800",{"id":86,"label":288},{"id":89,"label":1086},"900",{"id":92,"label":1088},"1,850","Both factors doubled, so the product is 4 times as big: 4 × 900 = 3,600.",{"itemId":1091,"prompt":1092,"options":1093,"correct":83,"why":1102},"four-operations.i-quiz-halve","Which product is equal to 18 × 35?",[1094,1096,1098,1100],{"id":83,"label":1095},"9 × 70",{"id":86,"label":1097},"36 × 70",{"id":89,"label":1099},"9 × 17",{"id":92,"label":1101},"20 × 33","Halve 18 and double 35: 9 × 70 = 630, and 18 × 35 = 630.",{"itemId":1104,"prompt":1105,"options":1106,"correct":92,"why":1114},"four-operations.i-quiz-rem","Which of these remainders is impossible when dividing by 6?",[1107,1108,1110,1112],{"id":83,"label":319},{"id":86,"label":1109},"3",{"id":89,"label":1111},"5",{"id":92,"label":1113},"6","The remainder must be less than the divisor, so dividing by 6 can only leave 0 to 5.",{"itemId":1116,"prompt":1117,"options":1118,"correct":86,"why":1125},"four-operations.i-quiz-bus","314 pilgrims travel in buses of 50 seats. How many buses are needed?",[1119,1120,1121,1123],{"id":83,"label":1113},{"id":86,"label":317},{"id":89,"label":1122},"6 remainder 14",{"id":92,"label":1124},"6.14","314 ÷ 50 = 6 r 14. The 14 pilgrims still need a bus, so round up: 7 buses.",{"itemId":1127,"prompt":1128,"options":1129,"correct":86,"why":1138},"four-operations.i-quiz-keyword","\"Anu has ₹85. That is ₹20 less than Babu.\" How much does Babu have?",[1130,1132,1134,1136],{"id":83,"label":1131},"₹65",{"id":86,"label":1133},"₹105",{"id":89,"label":1135},"₹1,700",{"id":92,"label":1137},"₹4.25","Anu has less, so Babu has more: 85 + 20 = ₹105. 'Less' did not mean subtract.",{"itemId":1140,"prompt":1141,"options":1142,"correct":83,"why":1151},"four-operations.i-quiz-check","Which calculation checks that 2,145 ÷ 15 = 143?",[1143,1145,1147,1149],{"id":83,"label":1144},"143 × 15",{"id":86,"label":1146},"2,145 − 15",{"id":89,"label":1148},"143 + 15",{"id":92,"label":1150},"2,145 × 15","Undo division by multiplying: 143 × 15 = 2,145. ✓",{"itemId":1153,"prompt":1154,"options":1155,"correct":86,"why":1164},"four-operations.i-quiz-counter","Which is a counter-example to 'dividing always makes a number smaller'?",[1156,1158,1160,1162],{"id":83,"label":1157},"20 ÷ 4 = 5",{"id":86,"label":1159},"9 ÷ 1 = 9",{"id":89,"label":1161},"100 ÷ 10 = 10",{"id":92,"label":1163},"8 ÷ 2 = 4","9 ÷ 1 = 9 is not smaller than 9, so the claim fails.",{"itemId":1166,"prompt":1167,"options":1168,"correct":89,"why":1177},"four-operations.i-quiz-order","Which pair of operations are both commutative (order does not matter)?",[1169,1171,1173,1175],{"id":83,"label":1170},"+ and −",{"id":86,"label":1172},"× and ÷",{"id":89,"label":1174},"+ and ×",{"id":92,"label":1176},"− and ÷","a + b = b + a and a × b = b × a always. Subtraction and division change when you swap the numbers.",{"id":1179,"type":1180,"prompt":1181},"reflect","reflection","Think of a real situation from your own week (a shop, a game, a journey, a festival) that involved a division with a remainder. What did the people involved actually do with the remainder, and was it rounding up, rounding down, or something else?",{"id":1183,"type":1184,"title":1185,"points":1186},"cheat-sheet","summary","Cheat sheet",[1187,1188,1189,1190,1191,1192,1193,1194,1195],"**Estimate first:** round to the leading digit, to a chosen place, or to compatible numbers. The estimate predicts the size of the answer and catches place-value slips.","**Same gap:** adding or taking the same number from both parts of a subtraction keeps the difference. 1,000 − 368 = 999 − 367 = 632.","**Compensation in sums:** move an amount from one addend to the other and the sum stays the same. 58 + 37 = 60 + 35.","**Products:** double one factor → product doubles; double both → ×4; halve one and double the other → no change. × 0 gives 0; × 1 changes nothing.","**Division:** bigger divisor → smaller quotient. The remainder is always less than the divisor.","**Remainders in stories:** round up (buses, boxes), round down (items you can buy, full pieces), give the remainder (leftover, change) or split it (rupees into paise).","**Keywords mislead:** 'more', 'left', 'each' and 'altogether' describe situations, not operations. Picture the structure: join, separate, compare, equal groups, scale.","**Check by undoing:** subtraction checks addition; multiplication checks division; divisor × quotient + remainder = dividend.","**Always true?** + and × are commutative; − and ÷ are not. One counter-example is enough to disprove a claim.",{"id":1197,"type":854,"conceptId":1198,"relation":856,"explanation":1199},"conn-number","number-system","Rounding to the nearest ten, hundred or thousand, which drives every estimate here, is built on place value.",{"id":1201,"type":854,"conceptId":1202,"relation":1203,"explanation":1204},"conn-data","data-handling","applied_in","Checking whether a mean or total is sensible uses exactly the estimate-first habit from this layer.",{"id":1206,"type":1206,"sourceIds":1207},"sources",[1208,1209,1210,1211,1212,1213,1214],"four-operations-ncert-math-6","four-operations-khan-arithmetic","four-operations-bbc-bitesize-ks2","four-operations-mathsisfun-long-division","four-operations-britannica-arithmetic","four-operations-wiki-arithmetic","four-operations-mathsisfun-estimation",[1208,1209,1210,1211,1212,1213,1214],"needs_review",{"generatedBy":1218,"notes":1219},"claude-code","Draft generated locally with Python-checked numbers; pending owner review.","1da7697c8a34db5c423a2b1309619553763e50786c283eaf75b0b8be89ee0751",{"component:rounding-race@1":1222,"component:arith-sprint@1":1223,"logic:practice":1224,"component:sort-game@1":1225,"component:order-ops@1":1226,"source:four-operations-bbc-bitesize-ks2":1227,"source:four-operations-britannica-arithmetic":1228,"source:four-operations-khan-arithmetic":1229,"source:four-operations-mathsisfun-estimation":1230,"source:four-operations-mathsisfun-long-division":1231,"source:four-operations-ncert-math-6":1232,"source:four-operations-wiki-arithmetic":1233},"6c2f2d540b01b2d7cd170d87a127ebd380947b02b030b91412433582d09e54f1","c0c63ed40e1bca5ba6d446d43886b887a3a7709e5cd6679419f64fd3ba62afe6","3d6b0fe1b15255975a32b0fcd94e8019bc959ad45cbf12e136e86149549c6878","b164f45a2c8ca08f26c450768ff0231e113e9fe45381eddb34dc6d0548596c38","3d630ebc066352df791b020d545e1876ce4620d55c8c1c082414d6b4717255c5","07eb90693160dd6149271155357cff693c59a2006a36d3cc67c133d9f2900790","66e407fe40575b09f96c71af650fbd183b777a5b7047ed8fcc6b693b637804f6","ef6b590392c4299a9ce1be700f25412c2a52dea7c431f6dd28b6d7c1ce15374d","4fa46f082bd73cea931140c432a48f7ee35a7f7d6bbecb43d270505bf6c94d00","794b6ef61d62320200593d8724a8525d6a81168da99708651826563b823043a5","c66e4c408e97e6c5758c1b1160148cfabaf1f5e9ca4caba87383f8ca4e389c95","3fd5b7fcbfec31c612f8356e67126230dae21df1a69f1c1a06216d510015a231",{"state":1235,"reviewer":1236,"selfReview":125,"reviewedAt":1237,"method":1238},"approved","The library owner","2026-09-20T10:18:37.581Z","owner_bulk","preview-7e1cbbcc4f",1789899598648]